# Global simulation of a gallium arsenide metal -semiconductor -metal photodetector for the conversion of optical signals into microwaves

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GLOBAL SIMULATION OF A GALLIUM ARSENIDE METALSEMICONDUCTOR-METAL PHOTODETECTOR FOR THE CONVERSION OF OPTICAL SIGNALS INTO MICROWAVES A dissertation submitted in partial fulfillment o f the requirements for the degree o f Doctor of Philosophy in Chemical Physics at Virginia Commonwealth University By David B. Ameen M.S., Physics, Virginia Commonwealth University, 1996 M.S., Biophysics, University o f Virginia, 1987 B.S., Science Education, University o f Virginia, 1976 Director: Dr. Gregory B. Tait, Associate Professor o f Electrical Engineering Virginia Commonwealth University Richmond, Virginia May, 2000 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number. 9963200 UMI* UMI Microform9963200 Copyright 2000 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. College o f Humanities and Sciences Virginia Commonwealth University This is to certify that the dissertation prepared by David B. Ameen entitled Global Simulation o f a Gallium Arsenide M etal-Semiconductor-M etal P hotodetectorfor the Conversion o f O ptical Signals into M icrowaves has been approved by his committee as satisfactory completion o f the dissertation requirement for the degree o f Doctor o f Philosophy. Dr. Gregory S? Tajtf School o f Engineering, Research Director Dr. Donald D. Shillady, College ooff unities and Sciences, Committee Chairman Dr. M. Sarny El-Shall, College o f Humanities and Sciences <c . Dr. B ijanjL Rao, College o f Humanities and Sciences ivtvq. Dr. Shiv N. Khanna, College o f Humanities and Sciences Dr. Fr8dM . Hawkridge, Chemistryfihairman D. Gottfir can, College o f Humanities and Sciences Dr. Jack L. Haar, Dean, School o f Graduate Studies Vil as, D ate1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For my wife Betsy and my Mom and Dad ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements A special thanks goes to Dr. Gregory Tait who sponsored this work and taught me all that I know about the physics o f semiconductors. A special thanks also goes to Dr. Marilyn Bishop and Dr. Tom McMullen who taught me physics and who have worked with me for the past six years of graduate study, especially on the research for the M.S. degree. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS List of F igu res.......................................................................................................................viii List of T a b les..................................................................................................................... xiii A bstract................................................................................................................................ xv 1 Introduction..................................................................................................................... 1 1.1 Using Light to Transmit Radar M icrowave......................................................... 1 1.2 Overview o f Semiconductor Device Simulation................................................. 8 1.2.1 Behavioral M odels.................................................................................... 8 1.2.2 Physics-Based Numerical M odels............................................................10 1.2.3 Global M odels............................................................................................ 14 1.2.4 Most Commonly Used Photodetector...................................................... 16 1.3 Goals o f Dissertation.............................................................................................. 19 1.3.1 Synopsis o f O bjectives............................................................................. 19 1.3.2 First Objective............................................................................................ 19 1.3.3 Second Objective....................................................................................... 22 1.3.4 Third O bjective......................................................................................... 23 2 Characteristics o f Bulk Semiconductors.................................................................. 24 2.1 Introduction.............................................................................................................24 2.2 Physical P roperties................................................................................................ 27 2.2.1 Crystal Structure........................................................................................ 27 2.2.2 Energy-Momentum Relationship............................................................. 27 2.2.3 Free Electrons and H oles....................................................................... 31 2.2.4 Density o f States.................................................................................... 37 2.3 Carrier D ensity....................................................................................................... 40 2.3.1 Electron Distribution Function............................................................. 40 2.3.2 Intrinsic Semiconductors....................................................................... 44 2.3.3 Doping the Semiconductor................................................................... 47 2.3.4 Debye Screening L ength........................................ 50 2.4 Generation o f Carriers by Illumination............................................................. 51 2.4.1 Light Absorption.................................................................................... 51 2.4.2 Carrier Generation Rate Due to Constant Illumination....................... 53 2.4.3 Converting Light into Microwaves....................................................... 57 2.4.4 Generation Rate with Oscillating Light Intensity............................... 62 2.4.5 Responsivity and Bandwidth................................................................. 64 2.5 Recombination..................................................................................................... 70 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5.1 2.5.2 Scattering and Radiative D ecay............................................................. 70 Recombination-Generation Statistics....................................................... 76 3 T ransport P ro p e rtie s.................................................................................................... 3.1 Introduction........................................................................................................... 3.2 Semiclassical M odel............................................................................................ 3.2.1 R u les........................................................................................................ 3.2.2 B asis........................................................................................................ 3.2.3 L im its...................................................................................................... 3.3 Drifi-Diffusion Transport M odel...................................................................... 3.3.1 Boltzmann Transport Equation............................................................. 3.3.2 Drifi-Diffusion Approximation............................................................. 3.3.3 Drift-diffusion E quations................................................................... 3.3.4 Current Continuity Equations............................................................... 3.3.5 Poisson’s E quation................................................................................ 3.3.6 One-Dimensional Transport Equationsin n andp .............................. 3.4 Equilibrium Solution........................................................................................... 3.4.1 Physics o f the Schottky B arrier........................................................... 3.4.2 Schottky Barrier in Equilibrium......................................................... 3.4.3 Boltzmann Distribution Derived from Transport Equations............ 3.4.4 Numerically Generated Equilibrium Solution................................... 3.5 DC Steady State Equations for Electrons......................................................... 3.5.1 Physical Description Using E-x D iagram s........................................ 3.5.2 Mathematical Characterization........................................................... 3.6 Inclusion o f Light and Time in the Transport Equations................................ 3.6.1 Inclusion o f H o le s................................................................................ 3.6.2 Effects o f L ig h t..................................................................................... 3.6.3 Final Form o f the Time-Dependent Transport Equations................ 3.7 Carrier M obility................................................................................................ 3.7.1 Complex Behavior o f M obility........................................................... 3.7.2 Mobility M odels.................................................................................. 3.7.3 Empirical-Fit Mobility Relationships................................................. 81 81 85 85 88 89 91 91 93 96 98 99 100 102 102 109 110 117 130 130 133 140 140 142 143 146 146 150 154 4 Boundary Conditions on the C u rre n t Density....................................................... 4.1 Introduction.......................................................................................................... 4.2 Review of Current Density Boundary Condition M odels............................... 4.2.1 Combined Drift-Diffusion/ThermionicEmission M odel................. 4.2.2 Revisions to the Combined M odel..................................................... 4.3 Derivation o f New Current Density Boundary C ondition........................... 4.3.1 Electron Current D ensity................................................................... 4.3.2 Hole Current D ensity ......................................................................... 4.4 The Schottky and Ohmic L im its....................................................................... 4.4.1 The Schottky L im it.............................................................................. 4.4.2 The Ohmic L im it.................................................................................. 4.4.3 Final Form o f Boundary Conditions................................................... 157 157 160 160 162 167 167 174 181 181 195 199 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5 DC Steady-State Solution in the Absence of L ight........................................... 201 5 Sim ulation o f the Isolated D evice....................................................................... 5.1 Introduction......................................................................................................... 5.2 Numerical Techniques....................................................................................... 5.2.1 Poisson Solver........................................................................................ 5.2.2 Transport Solver..................................................................................... 5.2.3 Norm alization......................................................................................... 5.2.4 Fixed Point Iteration for Current Density at B oundary...................... 5.3 Method o f Illumination under B ia s.................................................................. 5.4 Test o f New Current Density Boundary Condition........................................ 5.4.1 Comparison to Analytical I-V Characteristics....................................... 5.4.2 Comparison of Simulation with Experiment........................................ 5.4.3 Schottky Diode Simulations Compare Form ulations......................... 5.4.4 MSM Photodetector Simulations Compare Formulations................. 5.5 Limitations o f Simulator.................................................................................... 5.5.1 Preliminary Study................................................................................... 5.5.2 Limits Imposed by Voltage, Intensity, L ength.................................... 5.6 Device Perfomance Study.................................................................................. 5.6.1 Effects o f Barrier H eight....................................................................... 5.6.2 Substrate Growth Temperature Effects................................................. 5.6.3 Effects o f Device L ength....................................................................... 211 211 214 214 219 225 226 230 235 235 238 240 245 250 250 252 260 260 263 273 6 G lobal Sim ulation....................................................................................................... 6.1 Introduction........................................................................................................ 6.2 Physical Model o f the Embedding C ircuit....................................................... 6.2.1 Photomixer Circuit................................................................................. 6.2.2 Characterizing the Embedding Circuit................................................. 6.3 The Convolution................................................................................................. 6.3.1 Impulse Response D efined................................................................... 6.3.2 Embedding Circuit Voltage Given by a Convolution........................ 6.3.3 Impedance Function.............................................................................. 6.3.4 Calculate Impulse Response................................................................. 6.3.5 Discrete Convolution............................................................................ 6.4 Testing the Global Simulator............................................................................ 6.5 Effects o f Global Simulator on Device Performance..................................... 6.5.1 Interaction o f Time Constants Determines Bandwidth...................... 6.5.2 Comparison of Device and Global Simulations.................................. 277 277 280 280 283 288 288 289 293 297 300 306 315 315 326 7 Conclusion.................................................................................................................... 7.1 N ew Accomplishments...................................................................................... 7.2 Global Sim ulations............................................................................................. 7.3 Current Density Boundary Condition............................................................... vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 331 331 333 341 Appendices......................................................................................................................... A Sym bols................................................................................................................ B Model Param eters.............................................................................................. C Telephone Communication................................................................................ D Mixing W aves..................................................................................................... E Electrical Response to Illum ination................................................................. F Derivation o f Drift-Diflusion E quations........................................................ 344 344 352 355 357 359 361 Bibliography....................................................................................................................... 371 V ita ................................................................................................................................... vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 377 LIST OF FIGURES 1.1.1 Phased-array antennae on the U.S. Navy’s Aegis (actual number o f radiating elements is not shown) [after Fisher and Bahl PI].................................................... 3 1.1.2 Radar microwaves generated by photomixing (heterodyning) light..................... 7 1.2.1 MSM photodetector with interdigitated electrodes that are placed on top o f the substrate (top formation) [after Buchal and Loken [291].................................. 17 Short section o f a GaAs MSM photodetector with electrodes in trench formation in (a) 3D view and (b) cross-section [after Buchal and Loken P91].. 21 2.2.1 Cubic unit cell for GaAs with lattice constant a [after Blakemore p6]]............. 28 2.2.2 E-k plot of GaAs [from Blakemore psl]................................................................ 30 2.2.3 Generalized E -k diagrams showing carriers and bands for two types o f semiconductors regarding bandgaps: (a) direct gap and (b) indirect g a p . . . . 33 Partial GaAs E -k diagram showing various energy levels for an unintentional donor Si doping density o f N D = 3.5 x 10 ‘2 cm -3, and with a deep level donor trap [after Blakemore psi]............................................. 46 Generalized E -k diagrams showing light absorption for two types of semiconductors regarding band gaps: (a) direct gap and (b) indirect g a p . . . . 52 1.3.1 2.3.1 2.4.1 2.4.2 2.4.3 2.5.1 3.2.1 A single MSM unit showing the substrate and contacts, the dimension symbols, the cross-sectional areas, a differential depth element, and a monochromatic light wave oriented as described in the t e x t .................. 54 Optical responsivity and bandwidth are determined from the (a) timedependent photocurTent and graphed (b) on frequency response curve 66 Generalized E -k diagrams showing decay for two types o f semiconductors regarding bandgaps: (a) direct gap and (b) indirect gap 72 Schematic view o f the semiclassical model [after Ashcroft and Mermin [38]]........................................................................................................... 86 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4.1 E-x diagrams illustrating the formation o f the Schottky barrier: (a) isolated substances just prior to physical contact; and (b) after band bending is completed................................................................................................................ 3.4.2 Generalized E-x diagram and physical model o f a Schottky diode at equilibrium..................................................... 3.4.3 Generalized E-x diagram and physical model o f an MSM at equilibrium ... 103 Ill 113 3.4.4 Equilibrium: numerically generated (a) E-x diagram (using both the Poisson and transport solvers) and (b) device profile of Schottky diode CTH 562....................................................................................................... 120 3.4.5 Equilibrium: numerically generated (a) £-x diagram (using both the Poisson and transport solvers) and (b) profile o f MSM with 0.6 eV Schottky barriers................................................................................................... 121 3.4.6 Equilibrium: numerically generated (a) E-x diagram (using both the Poisson and transport solvers) and (b) profile o f MSM with 1.0 eV Schottky barriers................................................................................................... 122 3.5.1 Generalized E-x diagram o f a Schottky diode under conditions o f (a) forward bias and (b) reverse bias.............................................................. 132 3.5.2 Generalized E-x diagram o f an MSM under conditions o f bias......................... 134 Drift velocity and mobility // as a function o f electric field S in GaAs for (a) electrons and (b) holes.................................................................... 148 3.7.2 Monte Carlo simulation o f a 2.5 pm long n-GaAs sample under a constant and uniform electric field £ = 7 kV/cm, showing the average total electron kinetic energy [taken from Tait and Krowne I201]............................................. 152 4.4.1 Effective barrier lowering due to the image force effect for a high constant field ( = l x l 0 7 V/cm).................................................................. 183 3.7.1 4.4.2 Critical width x c for Schottky barrier (a) occurs at the tunnel probability peak and (b) is used as limiting value x“ to determine the barrier height lowering ...................................................................................................... 188 4.4.3 Schottky barrier on Ohmic limit at x = 0, generated by simulator..................... 197 4.5.1 Steady state: numerically generated (a) £-x diagram (full transport ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5.2 4.5.3 solver) and (b) profile o f Schottky diode CTH 562 at -0 .7 V forward bias.. . 202 Steady state: numerically generated (a) E-x diagram (full transport solver) and (b) profile o f MSM at 1.0 V bias with 0.6 eV Schottky barriers................................................................................................................... 203 Comparison o f full transport and electron-only results for steady state: (a) E -x diagram and (b) profile o f MSM at 1.0 V bias with 0.6 eV Schottky barriers.................................................................................................. 208 4.5.4 Steady state: numerically generated (a) E-x diagram (full transport solver) and (b) profile o f MSM at 1.0 V bias with 1.0 eV Schottky barriers................................................................................................................. 5.3.1 209 Time evolution to AC steady state for 1.0 eV barrier MSM under 1 V bias that is exposed to light o f 1 KW/cm2 intensity and modulation frequency 30 GHz............................................................................................... 231 Simulated I-V characteristics for Schottky diodes with 0.6 eV barrier [(a) and (b)] and 1.0 eV barrier [(c) and (d)]..................................................... 237 5.4.2 Comparison o f experimental and simulated I-V characteristics for Schottky diodes................................................................................................... 239 5.4.1 5.4.3 Comparison o f Schottky diode I- V characteristics for current density boundary condition formulations for (a) 0.6 eV and (b) 1.0 eV barriers 241 5.4.4 Photocurrent generated by illumination of a 1.0 eV barrier Schottky diode with 1 KW/cm2 constant light at -10 V bias............................................. 244 5.4.5 Frequency response curves for four formulations o f current density boundary condition for 1 KW/cm2 light intensity with the (a) FD and (b) FI mobility models, and for 25 KW/cm2 light intensity with the (c) FD and (d) FI mobility models....................................................................247 5.5.1 Convergence as a function o f voltage for 10 GHz light at 1 KW/cm2 intensity for 1.1 pm device length with FD mobility model............................ 254 5.5.2 Convergence as a function o f light intensity for 10 GHz light at 1 V bias for 1.1 pm device with FD mobility model............................................... 255 5.5.3 Convergence as a function o f light intensity for 10 GHz light at 10 V bias for 1.1 pm device length with FD mobility model.................................... 256 5.5.4 Convergence as a function o f light intensity and voltage for 10 GHz light x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for 0.7 (Am device length with FD mobility model (NOTE: Scale of (a) is adjusted)......................................................................................................... 257 Comparison o f E-x diagrams for (a) low (0.6 eV) and (b) high (1.0 eV) Schottky barriers with (c) associated profiles for MSM illuminated by 1 KW/cm2............................................................................................................... 261 Comparison o f frequency response curves for different recombination times....................................................................................................................... 264 Electric fields for FI mobility model, used to compare the driving force that pushes the carriers to the contacts of the MSM.................................. 267 Comparison o f frequency response curves for different device lengths, for (a) FI mobility model and (b) FD mobility model....................................... 274 Photomixer circuits, including (a) single element (Rl), (b) bias tee, and (c) Thevenin equivalent bias tee.................................................................. 282 Physical picture o f a convolution, showing the (a) current function, (a) the impulse response function, (c) the discrete evolution o f the current, and (d) the discrete evolution o f the first four impulse-responsecurrent products.................................................................................................... 290 Original impedance functions for device capacitances o f (a) 20 fF and ( b ) 1 0 0 f F.............................................................................................................. 295 Discretized impulse responses for device capacitances o f (a) 20 fF and (b) 100 fF............................................................................................................ 298 Original (before) and reconstructed (after) impedance functions for device capacitances o f (a) 20 fF and (b) 100 fF for 50 fs time step............... 301 Original (before) and reconstructed (after) impedance functions for device capacitances o f (a) 20 fF and (b) 100 fF for 500 fs tim e step.............. 302 6.4.1 Comparison o f device and global simulations at equilibrium........................... 310 6.4.2 Comparison o f device and global simulations for 1 V DC bias........................ 311 6.4.3 Comparison o f device and global simulations for 1 V DC bias under 1 KW/cm2 constant illumination....................................... 312 5.6.1 5.6.2 5.6.3 5.6.4 6.2.1 6.3.1 6.3.2 6.3.3 6.3.4 6.3.5 6.4.4 Comparison o f band diagrams for device and global simulations for 500 fs time step, 20 fF capacitance, and FD mobility....................................... xi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 314 6.5.1 Equivalent photomixer circuit approximations, including (a) output branch and (b) parasitic branch........................................................................... 321 6.5.2 Current fractions for (a) 20 fF and (b) 100 fF parasitic capacitances................ 322 6.5.3 Comparison o f frequency response curves for device and global sim ulators.. 328 Appendix C Transmission o f signals in telephone communication........................... 356 Appendix D Mixing waves............................................................................................. 358 Appendix E Electrical response to illumination........................................................... 360 xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES 3.3.1 Transport equations in n andp ................................................................................ 101 3.4.1 Current densities for the Schottky diode CTH 562 and the MSMs at equilibrium............................................................................................................. 124 3.6.1 Final state variables................................................................................................. 144 3.6.2 Final transport equations......................................................................................... 145 4.2.1 Four formulations o f the combined DD/TE current density boundary condition................................................................................................................. 166 4.3.1 Electron current density statistics for limiting cases o f electron drift velocity................................................................................................................... 174 4.3.2 Hole current density statistics for limiting cases o f hole drift velocity 180 4.4.1 Effective lowering o f the Schottky barrier due to image force effects and tunneling........................................................................................................... 186 4.4.2 Final boundary conditions....................................................................................... 200 4.5.1 Current densities for the Schottky diode CTH 562 and the 0.60 eV barrier MSM at steady state................................................................................. 204 4.5.2 Current densities for 1.00 eV barrier MSM at steady state.................................. 210 5.2.1 Transport solver equations for boundaries and bulk in residual function form and with the RHS in discretized form (LHS discretized form is analogous to Equation 5.2.10) for time step th................................................... 223 5.2.2 Normalization factors............................................................................................. 226 5.4.1 Experimental Schottky diode material parameters.............................................. 240 5.4.2 Electron DD velocities and densities at Schottky boundary for diodes under 0.5 V forward bias in the absence o f lig h t............................................. 242 xiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.4.3 Electron and hole DD velocities and densities at Schottky boundary o f 1.0 eV barrier diode under unmodulated 1KW/cm2 illumination for FD mobility model at —10 V bias.................................................................. 245 5.4.4 Frequency response curves at high light intensity (25 KW/cm2) and 1 V bias................................................................................................................... 248 5.4.5 Peak current electron and hole DD velocities and densities at left (electron reverse biased) MSM contact at 10 GHz and 1 V bias..................... 249 5.4.6 Peak current electron and hole DD velocities and densities at right (electron forward biased) MSM contact at 10 GHz and 1 V bias..................... 249 5.5.1 Preliminary convergence efficiency study performed by changing five device parameters one at a time and determining the ANIPT................. 252 5.6.1 Responsivity and bandwidth for long and short recombination lifetimes 265 5.6.2 Interaction of transit time and effective recombination lifetime for long and short recombination lifetimes............................................................... 269 5.6.3 Detailed analysis o f simulation results associated with short recombination lifetimes....................................................................................... 273 5.6.4 Comparison of responsivity and bandwidth for three MSM lengths................. 275 5.6.5 Comparison of transit times r, for three MSM lengths..................................... 276 6.2.1 Frequency domain equivalent voltages and impedances for the passive elements in the equivalent circuit model o f the distributed microwave circuit.................................................................................................................... 285 6.3.1 Parameters for discretized impulse response functions..................................... 297 6.4.1 Testing for the vanishing o f the sums o f the discrete impulse response series (with for unfiltered samples, and hn for smoothed samples) 308 6.5.1 Optical responsivity and bandwidth for device, 20 fF global, and 100 fF global simulations.............................................................................. 327 xiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT GLOBAL SIMULATION OF A GALLIUM ARSENIDE METALSEMICONDUCTOR-METAL PHOTODETECTOR FOR THE CONVERSION OF OPTICAL SIGNALS INTO MICROWAVES By David Bruce Am een, Ph.D. A dissertation submitted in partial fulfillment of the requirements for the degree o f Doctor of Philosophy in Chemical Physics at Virginia Commonwealth University. Virginia Commonwealth University, 2000. Director: Dr. Gregory B. Tait, Associate Professor o f Electrical Engineering The conversion o f light into microwaves by a semiconductor photodetector, a process called photomixing, is studied using simulations. The photomixing process is presently used in the fiber-optic transmission of telephone signals. This study anticipates the use of photomixing in phased-array antennas to generate radar microwaves due to the transmission advantages gained through fiber-optics. Device and global simulators are developed for use as tools in the design o f photomixer circuits, and to explore the internal mechanisms o f photodetector operation. The photomixer circuit that is modeled consists of two parts: (1) a gallium arsenide metal-semiconductor-metal photodetector with trench electrodes; and (2) an embedding circuit that has a bias tee, a voltage source, and a device parasitic capacitance. xv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A device simulator is constructed to model electron and hole transport in the photodetector under conditions o f illumination with modulated light. The device simulator is based on the physics o f the drift-diffusion approximation o f the Boltzmann transport equation, and includes six nonlinear first-order partial differential equations. The equations are discretized and solved numerically through a Newton-Raphson technique that calculates six state variables as a function o f position and time. The simulation uses a current density boundary condition that is derived in this study from first principles regarding the semiclassical model. The device simulator characterizes photodetector performance, as measured by the optical responsivity and bandwidth. Results indicate that photodetector performance is affected by the mobility model, recombination time constant, voltage, light intensity, and device length. A global simulator is developed to model the photomixer circuit. The global simulator integrates the device simulator with an efficient convolution that models the embedding circuit. The embedding circuit produces an impulse response that is characterized in the frequency domain as the impedance function. The impedance function is solved, discretized, and inverse fast Fourier transformed into the time domain to generate the discretized impulse response. Global simulations determine the effect of the embedding circuit on device performance, and the results indicate that the parasitic capacitance is significant The global simulation achieves accuracy through the convolution and the new current density boundary condition. Efficiency is achieved through truncation o f the impulse response sequence and through the extrapolation of the current in conjunction with a fixed-point iteration scheme for the discretized convolution. xvi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 CHAPTER 1 INTRODUCTION 1.1 USING LIGHT TO TRANSMIT RADAR MICROWAVES We live in an age that demands high speed communication. Even though electronic technology has developed extensively over much o f the past century to meet this demand, there is a continual push for improvement. An important example of improvement in high speed communication has already occurred in the telephone system (see Appendix C). In the past, telephone systems transmitted information exclusively as electrical signals along copper wires. Presently, the copper wires are being replaced by fiber-optic cables, which transmit information as light signals. As pointed out by Agrawal, fiber-optic cables provide a six-order magnitude increase in signal transmission capability as compared to signal transmission over copper cables [1]. This improvement results because light affords a higher frequency signal and a lower signal power attenuation. Since the fiber-optic system offers advantages over more traditional methods o f signal transmission, there is great interest in replacing the existing transmission media with fiber-optic technology for other communication systems as well. One such system for which fiber-optic signal transmission shows promise is the radar microwave system utilized by the military. An essential component o f the fiber-optic system is the semiconductor photodetector. The photodetector converts the light signal transmitted Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 along the optical fiber into signals that more closely resemble the final output o f the communication process. There is a high degree of interest in converting radar systems to fiber-optic-based transmission systems due to the recent development of phased-array antennas [2]-[7]. When radar microwaves were first used by ships in the navy to detect other ships and aircraft, the antenna that projected the radar beam had to be mechanically rotated in order to change the direction o f the beam. Mechanical rotation slowed the antenna response. This problem has since been overcome through the use o f phased-array antennas to send out the radar signal. Figure 1.1.1 illustrates the phased-array antenna used by the U.S. Navy Aegis [8],[9]. In a phased-array system, the base station produces microwave signals, and coaxial cables pipe the signals to the antenna. The diameter of the coaxial cables is approximately the wavelength o f the microwaves being transmitted, which varies between 1 mm and 1 cm. Each o f the four antennae on the Aegis (only two antennae are shown) contains 4000 thousand tiny emitters arranged as an array on a flat surface that is 12 ft across. The emitters are supplied by separate microwave feeds from a branched distribution network. The relative phases of the separate microwaves are varied so that when the microwaves are emitted from the antenna they interfere with each other in a controlled way. This interference results in a narrow beam that can be directed at any angle between 0° - 180°, depending on the particular phase configuration that is chosen. Phased-array antennas are therefore able to project radar without physically rotating the disc. Since there is no need for mechanical rotation, the antenna is able to respond much faster (within microseconds) to changes in the beam direction coordinates, allowing for quicker detection of objects that move within the vicinity o f the ship. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 Base station Array antenna Radar beam Radiating element Branching feed network Front view Side view Figure 1.1.1 Phased-airay antennae on the U.S. Navy's Aegis (actual number of radiating elements is not shown) [after Fisher and Bahl (*]]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 The use o f microwave cables to carry the signal from the base station to the antenna has disadvantages, however, as pointed out by Hickey [7]. The antenna should be placed in a location on the ship high up in the air where its view is unobstructed, and as far away from the base station as possible to minimize the electromagnetic interference originating from the base station. A high, remote placement o f the antenna requires long microwave cables to link the antenna to the base station. The problem with long cables is that microwave signals undergo substantial attenuation in the cables. For example, a typical microwave signal has its power reduced by 90% for every ten feet o f cable. This high signal power loss prevents the antenna from being placed in as remote a location as desired, so electromagnetic interference is still a problem. Note for example the close proximity o f the antenna to the base station on the Aegis in Figure 1.1.1. Furthermore, the cables are large, heavy, and expensive, and these traits pose a significant disadvantage due to the high cable density in phased-array systems. The disadvantages o f microwave coaxial cables can be overcome i f the coaxial cables are replaced by optical fibers. Since optical fibers use infrared light to carry signals, the range o f operation frequencies changes. For example, while microwaves have millimeter wavelengths and frequencies in the 30 - 300 GHz range, infrared light has micrometer wavelengths and frequencies in the 30 —300 THz range. The shorter wavelength enables waveguides for light to have a much smaller diameter than waveguides for microwaves, allowing optical fibers to be compact and low in weight Since optical fibers are thin glass fibers, they are flexible and inexpensive. They are immune to interference because their covering is opaque. Most importantly, the light signal that propagates along the optical fiber undergoes dramatically less attenuation than Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 microwaves along coaxial cables. For example, a typical light signal is reduced by 50 % for every kilometer o f optical fiber. Optical fibers can transmit a signal greater than onethousand times the distance that is possible for microwave coaxial cables for the same reduction in signal power. If optical fibers are to replace the existing microwave coaxial cables in the phased-array antenna system, then two additional signal processing steps must be added to the existing system. First, since light carries the signal, the light must be prepared so that the higher frequency light (—100 THz) carries the lower frequency microwave signal (-100 GHz). This is accomplished by transmitting two laser beams of slightly off-set frequencies along the fiber-optic cable from the source to the destination, as shown in Figure 1.1.2. The difference in frequencies between the two laser beams is equal to the frequency of the microwave signal that is to be generated at the destination end o f the transmission. The second additional signal processing step is the conversion o f the two light beams into the desired microwave signal at the destination end. This conversion step can be accomplished effectively by using a semiconductor photodetector, also shown in the figure. W hen the photodetector absorbs the two laser beams, it mixes the light intensity together in a process called photomixing or heterodyning (see Appendix D). In photomixing, the intensity is modulated so that it consists o f an underlying carrier signal whose frequency is in the optical range and an envelope signal whose frequency is in the microwave range. The final result o f this photomixing process is the generation o f a microwave whose frequency is the difference frequency o f the two laser beams. The photomixing process offers a better method for transmitting radar microwaves than is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 provided by coaxial cables. The motivation o f the present work is to use a simulation to study the operation o f the photodetector during the photomixing process. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Signal Source Laser 1 ( k,) m il Laser 2 (v2) radar microwave (vr v2) Fiber-optic cable Photodetector **rv' * = 1 » ~ ... j microwave envelope Modulated intensity optical carrier Figure 1.1.2 Radar microwaves generated by photomixing (heterodyning) light. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 1.2 OVERVIEW OF SEMICONDUCTOR DEVICE SIMULATION Semiconductor devices and the circuits that contain the semiconductor devices are simulated in order to understand their behavior, both from a scientific viewpoint and from the viewpoint o f design and manufacture. The two major approaches toward modeling a device or a circuit are through behavioral models and through physics-based numerical models. In the earlier years o f modeling, the emphasis was on the separate development o f the simulation for the device and the simulation for the circuit in which the device was placed. Since the circuit in which a device is embedded affects the performance of the device, the more recent aim is to develop global models. In global models, the device and its embedding circuit are simulated together in an integrated approach. This section addresses the historical development o f the models, with an emphasis on the advantages and disadvantages o f each. The section concludes with a description o f the most commonly used photodectector for photomixer circuits, which has its electrodes on top of the semiconductor substrate. Emphasis is placed on the disadvantages o f this photodetector to set the stage for the choice o f photodetector to be modeled in the present study, which has its electrodes embedded within the semiconductor substrate. 1.2.1 Behavioral Models Behavioral models for devices consist o f equivalent circuits with analytical expressions that predict the current at the terminals o f the device. As described by Clarke, a behavioral device model consists o f the configuration o f lumped circuit elements that best represents the function o f the device under a unique set o f operating conditions [10]. Analytical expressions specific to each configuration are used to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 characterize the terminal current. For example, there are several different configurations for the pn junction diode, depending on whether the operating conditions are for DC behavior, transient analysis, or small signal AC behavior. Each of these circuit configurations is simulated using the analytical expressions specific to that particular array of circuit elements. One o f the most widely used o f the equivalent circuit simulators is SPICE, developed in 1975 by Nagel. The major advantage of the behavioral models based on equivalent circuit simulations is that much less computational time and space is required compared to the physics-based numerical simulations. However, as pointed out by Tait, these behavioral models have limitations [11]. They operate in a quasi-static mode that assumes that the carriers adjust relatively rapidly to successive changes in local environmental conditions such as electric field. This is only true at low frequency operation, so that the carriers have time to complete their response to an incremental change in local conditions. For high frequency operation, the time between successive local environmental changes becomes too short for the carriers to complete their response before the next change occurs. In this high frequency case the carriers are actually in a dynamic or full timedependent mode, so that the quasi-static models reduce accuracy. Another problem is that equivalent circuits can be difficult to construct since they require substantial insight into device operation. Finally, if the behavioral model consists o f analytical expressions, these expressions must be curve fitted to the results obtained from full time-dependent carrier transport simulations or from experiments. Such experiments and/or simulations may not exist, and even if they do, the model can only be applied to restricted operating conditions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 1.2.2 Physics-Based Numerical Models In device simulations that use physics-based numerical models, the current in the semiconductor device is determined by solving a system o f transport equations. These equations are derived predominantly from the Boltzmann transport equation (BTE), which is a differential equation whose solution provides the time- and space-dependent momentum distribution function for carriers in phase space [12]. Due to the complexity o f the BTE, only the Monte Carlo simulation method is able to numerically solve this equation without significant approximations. However, the Monte Carlo method is too computationally intensive for widespread use in design optimization. Therefore, the BTE is used to derive a more solvable set o f transport equations by forming successively higher order velocity moments o f the fundamental equation. Varying degrees of approximations are then made in the resulting transport equation set to decouple the equations from each other. The more approximations that are made, the less computationally intensive the solution o f the resulting transport equations becomes, but this is accompanied by a loss in accuracy. In the first level o f approximation o f the BTE, the resulting transport equation set forms the hydrodynamic model, and consists o f the first three velocity moments o f the BTE: charge continuity, momentum conservation; and energy conservation [13]. The full hydrodynamic set consists of at least eight (and often nine) equations. An additional set o f approximations, notably the decoupling o f the momentum and energy conservation equations, is valid when the electron carrier temperature is in equilibrium with the rest of the semiconductor. This decoupling leads to the drift-diffusion model, which includes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 only the first two moments o f the Boltzmann transport equation. The full drift-diffusion set consists o f six equations. Due to its lower computational demand, the drift-diffusion model is more often used than the hydrodynamic model. The six equations in the driftdiffusion set include the charge balance or Poisson equation (derived from Maxwell’s equations), the definition o f the electric field as the gradient o f the electrostatic potential, the drift-diffusion equations for electrons and holes (from momentum conservation), and the current continuity equations for electrons and holes. Since the equations in these models are coupled nonlinear partial differential equations, analytical solutions are not possible. Many numerical approaches have been developed to determine the current and other quantities in specific devices. These numerical methods require that the differential equations be discretized and solved self-consistently in an iterative process, usually through a Newton iteration technique. Some o f the more important physics-based numerical methods are now discussed from an historical basis. One o f the earliest attempts to solve the drift-diffusion transport equations throughout a semiconductor device was made in the mid-1960’s by Gummel in his work with transistor simulations [14]. He devised what became known as the Gummel solution method, in which the equations in the set were separately solved one at a time by using the results from the solution o f the previous equation. Each cycle through the equation set comprised one iteration. The iterations continued until there was insignificant change in the subsequent solutions. This circular iteration process resulted in convergence, but the convergence rate was linear and therefore slow. In the late 1960’s, Gummel teamed with Scharfetter to devise a landmark numerical procedure, again with the drift-diffusion model [15]. In order to increase the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 convergence rate and to deal effectively with strongly coupled equations, as occurs when current injection into the device is high, it was necessary to solve the transport equations simultaneously. This required more stringent criteria to keep the potentially unstable process from diverging. One o f the biggest problems with numerical solutions o f discretized second order differential equations involving flow variables such as current density is that spatial oscillations arise, as discussed by Roache [16]. This is especially prevalent in the current continuity equations when they are solved by substituting the current density from the drift-diffusion equations. The problem arises because the carrier concentrations and current densities are exponential in nature, which can lead to sharp spatial changes. Not only are the oscillations unphysical, they can cause dramatic divergence in the numerical solution. Scharfetter and Gummel inserted exponential terms into the drift-diffusion equations to prevent the oscillations from occurring, thus greatly extending the range over which the simultaneous-equation Newton method could be used. It was shown later by Kreskovsky, however, that the insertion o f these exponential terms introduced some error by causing artificial contributions to the current [17]. The drift-diffusion model demonstrated satisfactory accuracy until devices were designed that produced unusually large electric fields, carrier gradients, or current densities, as discussed by Snowden [13]. In such devices, electrons were heated to the point that their temperature could no longer be considered in equilibrium with the rest o f the semiconductor. This hot electron effect was especially prevalent in the newer submicron devices, where the electric field and subsequent current densities could get very high due to the shorter lengths. To maintain accuracy in the solution, devices that exhibited substantial electron heating had to be simulated using the hydrodynamic model, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 which allowed for a variable temperature. This was made apparent through simulation studies such as one conducted by Hjelmgren, in which the device was 0.12 pm in length [18]. The simulations were conducted both aLlowing for and not allowing for hot electron effects, and the results were compared with experiment. The inclusion o f hot electron effects in the simulation model significantly enhanced the agreement between the simulated current and the experimental current. Snowden points out that when devices become so small that carrier transport is determined completely by non-stationary processes, Monte Carlo simulations can be used to ensure the accurate characterization o f transport [19]. In the Monte Carlo method, the motion of each carrier in the system is tracked by analyzing the effect o f local electric fields and individual scattering events [20]. Random numbers with specified probability distributions are generated to govern the time o f flight between scattering events, the type of scattering mechanism, and the final momentum state after scanering. Moglestue was successful in using the Monte Carlo method to simulate small signal AC behavior in diodes [21]. Due to the excessive amount o f computer time required, the simulations were restricted to transient cases as opposed to a true AC analysis over several cycles. The Monte Carlo approach has been especially useful in characterizing phenomenological parameters to be used during separate hydrodynamic simulations. For example, Tait and Krowne used Monte Carlo simulations to determine such fielddependent transport parameters as the electron temperature, energy and momentum relaxation times, and the electron effective mass [20],[22],[23]. These quantities were then used to calculate the field-dependent mobility and diffusivity for subsequent use in hydrodynamic device simulations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 1.2.3 Global Models Much o f the work in the 1990's in semiconductor modeling has been aimed at developing global simulations for semiconductor devices. The need for a global approach arises because device performance is affected by the circuit, especially through impedance at the contact points between the device and the rest o f the circuit. The global models fall into three main categories: (1) behavioral device/dynamic circuit; (2) physics-based device/static circuit; and (3) physics-based device/dynamic circuit. The first category uses a behavioral model to represent the semiconductor device. In general, devices are not incorporated into the global circuit simulator in their physicsbased numerical form as this would be too computationally demanding. Instead, the device is integrated as a less complex behavioral model such as an equivalent circuit with an analytical expression. As explained earlier, this tends to reduce the accuracy o f the final current-voltage relationship generated by the simulation. Furthermore, only quantities at the device terminals can be calculated. Physical quantities within the device such as local electric fields, which are o f significant interest from a scientific viewpoint, cannot be determined with behavioral models o f the device. The second category o f global modeling, physics-based device/static circuit, uses a full time-dependent physics-based numerical model of the semiconductor device, but does not model the embedding circuit in the full time-dependent dynamic mode. In one example of this approach, Tait used the harmonic balance technique to simulate the embedding circuit in the frequency domain [11]. Although AC steady state solutions were obtained, since the current-voltage relationship was solved in the frequency domain, these solutions did not fully characterize the time-dependent response o f the circuit to the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 device. In a second example, Ashour et al used an equivalent embedding circuit model so that the embedding circuit could be simulated directly in the time-domain [24]. However, a major simplification was made. The voltage response o f the circuit to the device current was treated as being instantaneous. In actual circuits, the voltage response of the circuit has a finite time width, which has a significant effect on the current-voltage relationship as a function o f time. The simplification therefore reduced the accuracy of the solution compared to a full time-dependent approach, which would require that the finite time width o f the circuit response be characterized. The third category o f global modeling uses the full time-dependent physics-based numerical approach with no major simplifications in either the device model or the embedding circuit model, so that accuracy can be maximized. One o f the newest examples o f this approach is the combined electromagnetic and solid state simulator developed by Imtiaz and El-Ghazaly [25],[26]. The electromagnetic simulator is combined with a hydrodynamic-based device simulator due to the monolithic nature of the solid state circuits. Monolithic means that the various components are integrated on a common chip. The close proximity o f these circuit elements causes the electromagnetic waves that are generated by a given component to interfere with the operation of the companion components on the chip. This interference is significant when the wavelengths o f the signals approach the size of the components on the chip, which is more likely to occur at higher frequencies. Electromagnetic wave effects in circuit elements are characterized by the electromagnetic simulator, which solves Maxwell’s equations in the time domain. The resulting electric and magnetic fields are then used by the device simulator to calculate the current densities. The process repeats when the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 current densities are used by the electromagnetic simulator to update the fields. These advanced global circuit simulators provide the most accurate results to date for high frequency circuits. An important disadvantage to this electromagnetic approach is that Maxwell’s equations must be solved for the entire circuit at each time step, which makes this technique so computationally intensive as to require massively parallel machines. A second approach in the physics-based device/dynamic circuit category uses the convolution of the circuit response to accurately characterize the time-dependent behavior of the embedding circuit. The convolution allows for the finite time-width o f the circuit response in the calculation o f the current-voltage relationship. Its use in device analysis was first suggested by Evans and Scharfetter thirty years ago [27]. However, the earlier techniques for the implementation o f the convolution were too computationally demanding to be useful in global simulations. Very recently, Tait and Jones developed a technique that speeds the convolution process by making use o f a Kaiser filter to dramatically reduce the number o f sample points needed in the convolution [28]. Using this modification of the convolution, they produced a fully timedependent physics-based global simulation for transferred-electron oscillators. 1.2.4 Most Commonly Used Photodetector In a recent study conducted by Buchal and Loken, the performance o f the most commonly used MSM photodetector was analyzed experimentally, in which the electrodes (contacts) are placed on top o f the semiconductor substrate [29]. The authors pointed out two important advantages of this photodetector, which is pictured in Figure 1.2.1. First, this planar MSM with interdigitated electrodes provided a large light Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Light Interdigitated electrodes Si substrate KEY: £ electric field • electron o hole Figure 1.2.1 MSM photodetector with interdigitated electrodes that are placed top o f the substrate (top formation) [after Buchal and Loken I29!]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 absorption area to maximize power absorption capability. Secondly, interdigitation of the electrodes reduced the time it took for photogenerated charges to travel to the electrodes, so that the device could respond faster. The authors also concluded that there were weaknesses inherent in an arrangement in which the electrodes were placed on top of the substrate. From the cross section of this MSM, it can be seen that the large substrate depth causes the electric field strength to diminish substantially with increasing distance from the electrodes. It was found that lower frequency light was not substantially absorbed until it had penetrated far into the silicon substrate, where the photogenerated carriers had a large distance to travel to get to the contacts and a weak electric field to propel them. Both o f these effects lengthened the transit time and reduced the response speed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 13 13.1 GOALS OF DISSERTATION Synopsis o f Objectives Based on the motivation presented in section 1.1 and on the present state of the science o f semiconductor device simulation as presented in section 1.2, this study chose and accomplished the following three objectives: (1) Develop an efficient and accurate convolution-based, fully time-dependent global simulation for a photomixer circuit that utilizes a gallium arsenide (GaAs) metalsemiconductor-metal (MSM) photodetector with trench electrodes that is modeled in its physcis-based form; (2) Use the simulation to characterize the performance o f the GaAs MSM photodetector with trench electrodes; and (3) Derive a new current density boundary condition that is based on first principles regarding the semiclassical model for solid state systems, and therefore gives a more accurate characterization o f the physics of the device. 13.2 First Objective The purpose for developing a photomixer circuit simulator is to provide a method for studying the operation of the photodetector and its embedding circuit. Due to the ease with which parameters can be varied, simulations provide information relative to both the scientist and the design engineer much more readily and cost-effectively than experimentation. The photomixer circuit is chosen in anticipation o f its future use by the military to convert light into radar microwaves. Using a physics-based device model in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 the simulator increases the accuracy and enables the researcher to see what happens inside of the device. The simulation is global because the circuit in which the device is embedded affects the performance o f the device, and vice-versa. By making the simulation convolution-based, the circuit can be characterized in the full time-dependent mode, which further increases accuracy. Using a Kaiser filter to modify the convolution reduces the computational demand to such an extent that the convolution-based approach is preferred over the other major global simulation method with the electromagnetic solver that requires massively parallel processing. The GaAs MSM is chosen as the photodetector simulation device due to its high speed optical response, its large signal-to-noise ratio, and its high-power capability [6]. Figure 1.3.1 illustrates the photodetector that is modeled in the present study, which has its electrodes embedded in trenches within the GaAs substrate. The trench formation for the electrode arrangement is chosen because recent work by Buchal and Loken and by Laih et al. indicates that the trench formation increases the optical response speed as compared to the more conventional arrangement in which the contacts are placed on top of the substrate [29],[30]. There are two reasons for the increase in optical response speed. Note in Figure 1.3.1 for the trench formation that the electric field strength does not diminish with depth the way that it does when the contacts are arranged on top as shown in Figure 1.2.1. Also note that for the trench formation, the distance that charges must travel to reach the contacts is relatively short throughout the depth o f the substrate. The actual MSM photodetector modeled in this study contains many repeating MSM units, as indicated in Figure 1.3.1. The semiconductor substrate represented by the clear portion of the figure is GaAs. A single MSM unit consists o f two oppositely- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 (a) GaAs substrate (b) Interdigitated electrodes Light KEY: Single MSM unit ~Selectric field • electron o hole Figure 1.3.1 Short section o f a GaAs MSM photodetector with electrodes in trench formation in (a) 3D view and (b) cross-section [after Buchal and Loken t29!]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 charged contacts and the semiconductor substrate between the contacts, as shown in the cross sectional view. Two light beams with slightly offset frequencies enter the device substrate and are absorbed. Each absorbed photon creates an electron and a hole that serve as the charge carriers. The voltage applied across the photodetector causes the charge carriers to move to the electrodes, generating an electric current called photocurrent. This photocurrent has an AC component whose frequency matches the difference frequency o f the original two light beams. Therefore, the AC component generates an electromagnetic wave with this difference frequency. Since the difference frequency is in the microwave range, the photodetector effectively converts light into microwaves. 1.3.3 Second Objective The second objective o f this study is to characterize the performance o f the photodetector through simulations. The performance is measured in terms o f the following two figures o f merit: (1) optical responsivity, which is the ratio o f AC photocurrent to the light power; and (2) bandwidth, which is the upper limit to the microwave frequencies that the photodetector can generate. The parameters that determine how a device performs interact in a very complicated way. The relationships between these parameters and the two figures o f merit for device performance do not lend themselves to analytical expressions. Often, even qualitative relationships are difficult to predict with an acceptable degree of certainty. The only effective and accurate way to characterize the performance o f this complex device is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 through simulations. One motivation is to gain a deeper insight into how MSM photodetectors function from a strictly scientific point o f view. A second motivation is to provide a more cost-effective alternative than manufacturing the device and characterizing its performance through experimentation. Special emphasis is placed on the effects of variations in the applied voltage, time constants, carrier mobility models, light intensity, device length, and embedding circuit capacitance. 1.3.4 Third Objective The third objective arose because the models that are now used to calculate the current density at the boundary o f the device are based in part on conflicting assumptions. The original model for the current density at the boundary has undergone three major revisions since it was first proposed over thirty years ago [31]. In any one o f the newer versions o f the model, some o f the revisions are incorporated while others or not, so that there is no consistency between the newer versions [32]-[34]. For example, there is disagreement on how to normalize the carrier velocity, so that the carrier velocity generated by alternate versions can differ by a factor of two. In order to resolve the conflicts, the present work derives the current density boundary condition from first principles regarding the semiclassical model of solid state systems. In so doing, it is shown that all three revisions as well as a new revision discovered in this derivation process should be included in the current density boundary condition. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 CHAPTER 2 CHARACTERISTICS OF BULK SEMICONDUCTORS 2.1 INTRODUCTION The absorption of light and the subsequent generation o f microwaves is accomplished by the semiconductor portion of the photodetector. Semiconductor material is unique in that it can be made to act as an electronic switch. Under a particular set o f conditions, the semiconductor can be relatively non-conducting. Then a change in one o f its conditions electronically alters the semiconductor, allowing it to conduct electricity. In the case of photodetectors, light activates the electronic switch by increasing the density o f mobile charge. This chapter focuses on the bulk properties o f semiconductors that enable them to behave as electronic switches under illumination. Section 2.2 presents the physical properties of semiconductors in general and those o f GaAs in particular. Semiconductors form crystalline structures with periodic potentials created by the regular spacing o f the core atoms. Schrodinger’s equation can be solved for such a structure, establishing the relationship between the energy E and wavevector k o f the electronic states o f the crystal as energy bands. The unique properties o f semiconductors result from the energy gap between the valence and conduction bands. Thermal energy promotes electrons from the valence to the conduction band, generating free electrons and holes, which become the charge carriers Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 of the semiconductor. The curvature in the bands determines the effective mass of the charge carriers. The periodic quality o f the crystal also allows the density o f energy states to be determined. In section 2.3, the density o f states and the effective mass are used together with the Fermi-Dirac distribution function to derive the formula for carrier density, which is required for the characterization o f current. The concept of intrinsic carrier density is introduced in reference to semiconductors that have no impurities. Impurities called dopants are added to semiconductors to increase the carrier density, thereby enhancing conductivity. The final form o f the carrier density relationships depend on the dopant density, and are expressed in terms o f the intrinsic carrier density. The Debye screening length is introduced as a predictor o f the internal electric field, which is required to move charge through the semiconductor. Section 2.4 explains how light generates the excess electron-hole pairs required for the photocurrent. The generation rate for the electron-hole pairs is linked directly to the light intensity. When two monochromatic light waves o f differing frequencies are photomixed they create an intensity component that oscillates at the difference frequency. By choosing the initial light frequencies properly, the difference frequency is in the microwave range, to which the photodetector can respond electrically. The result is the generation o f microwaves. The two figures o f merit for measuring the performance of the photodetector in the photomixing process are optical responsivity and bandwidth. Section 2.5 discusses recombination, the process whereby free electrons lose energy and return to the valence band to occupy empty states. Recombination resupplies the valence band with electrons so that the processes o f photo- and thermal-generation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 can continue indefinitely. The loss o f energy during recombination can be mediated by the thermal process o f scattering or by the radiative process o f optical photon emission. Direct gap semiconductors are more efficient at photon emission than are indirect gap semiconductors, which indirectly leads to the choice o f GaAs over Si as the semiconductor o f choice for photodetectors. Scattering is the dominant mechanism for recombination due to the existence o f traps in the bandgap that facilitate phonon exchange with the lattice. This dominance requires that the thermal recombination- generation rate be characterized for use in the carrier transport equations that calculate photocurrent. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 2.2 2.2.1 PHYSICAL PROPERTIES Crystal Structure The active substrate in the MSM photodetector modeled in the present study is gallium arsenide (GaAs). This semiconductor is a III-V compound because gallium and arsenic are members o f groups IH and V, respectively, o f the periodic table [35]. When these two elements combine, Ga contributes three valence electrons and As contributes five valence electrons. The eight shared electrons are divided among four bonds, with two electrons per bond. The bonding arrangement is such that a single atom of one element is surrounded by four equidistant neighbors of the other element at the comers of a tetrahedron, so that GaAs is crystalline. Fig. 2.2.1 illustrates the cubic unit cell for GaAs. The crystalline structure is zincblende, which in general refers to crystals consisting o f two interpenetrating face-centered cubic lattices. For GaAs, one sublattice is gallium and the other sublattice is arsenic. Each cubic unit cell contains four galliumcentered tetrahedrons, and the length o f a side o f the unit cell is a = 5.65 A [36]. 2.2.2 Energy-Momentum Relationship The energy-momentum relationship for the electrons in a semiconductor is determined by solving the Schrodinger equation for an approximate one-electron problem [35] v y f)t^ [£ -(f(r)l(i(? )= 0 , n Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.2.1) 28 [0 1 0 ] Ga As [100] [001] ► Figure 2.2.1 Cubic unit cell for GaAs with lattice constant a [after Blakemore^36! ] . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 where E is the total energy of the electron, (p(r) is the one-electron wave function, U (r ) is the crystal potential energy, m is the mass o f the electron, and h is Planck’s constant divided by 2 Tt. The periodic potential energy U (r) arises due to the atomic cores at the lattice points, so U(r) has the periodicity o f the lattice. According to Bloch’s theorem, the solution <p^( r) o f the Schrodinger equation in a periodic potential can be written as the product o f a plane wave characterized by the wave vector k and a function un(k, r ) that has the periodicity o f the crystal lattice (with j = V—T ): (p.(F) = expO'k r ) u a( k , r ) , (2.2.2) where n is the band index. The wave function given by Equation 2.2.2 is substituted into the Schrodinger equation (2.2.1), and this is solved numerically to establish the relationshipbetween electron energy Ea(k) and electronwave vector k . Fig. 2.2.2 shows the E-k plot for GaAs. The numerical method used togenerate this E-k plot is called the nonlocal empirical pseudopotential model (EPM) as developed by Chelikowsky and Cohen [37]. The plot in Fig. 2.2.2 is actually a more detailed version o f their original E-k plot for GaAs, with this enlarged version having been constructed by Blakemore [36]. E-k plots are said to provide the relationship between electron energy and momentum because the electron crystal momentum p(jc) is related to its wave vector k through [38] p ( k ) = hk . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.2.3) 30 Energy: E - E v (eV) 3 T* 3 0 0 K 0.40 eV 2 1.71 eV 0 (VI) Heavy 1.90 eV 1.42 eV 0 .3 4 eV holes Light holes" ( V2) > Sp li t- of f band (V3) A X(IOO) Wave vector k Figure 2.2.2 E-k plot o f GaAs [from BlakemoreP6!]. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 31 The basis for the unique properties of semiconductors, as revealed by the E-k plot in Fig. 2.2.2, is that there are two sets o f bands with an energy gap between them. The energy gap, called the bandgap, represents a region in which there are no allowed electronic states. The set o f bands below the bandgap are called valence bands, with the following bands depicted in the figure: heavy holes, light holes, and the split-off band. Note that each band has a single peak occurring at k = 0. The set of bands above the bandgap are called conduction bands, with two such bands depicted in the figure. Note that the lowest conduction band has three valleys, one at k = 0 and two at each end o f the graph. The minimum energy in the conduction bands, termed Ec, is the energy corresponding to the point T6. Similarly, the maximum energy in the valence bands, termed Ev, is the energy associated with the point rg. The bandgap energy Eg represents the difference in energy between the conduction band minimum E c and the valence band maximum Er , and is given by Eg = E C- E , , (2.2.4) which for GaAs is 1.42 eV. 2.2.3 Free Electrons and Holes A given electronic state En (&) within a particular band n is considered filled if it is occupied by an electron and empty if it is not occupied by an electron. At absolute zero, the valence bands are completely filled with electrons and the conduction bands are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 completely empty o f electrons. When the temperature is above absolute zero, thermal energy causes the promotion o f some electrons from the valence bands into the conduction bands. Thermal promotion is enabled through scattering events, in which electrons gain energy from the lattice through collisions with the core atoms. Since the promoted electrons have an abundance o f empty states to move into within the conduction band, they are considered to be free electrons. Every promoted electron leaves behind an empty electronic state in the valence band that is called a hole. Relatively few valence band electrons are promoted to the conduction bands, so that even at elevated temperatures the conduction bands contain mostly empty states while the valence bands contain mostly filled states. This can be seen in Figure 2.2.3, which is a generalized E-k diagram showing the two types o f semiconductors regarding bandgap. Direct gap semiconductors, as shown in part (a), have Ec centered directly above £„. Indirect gap semiconductors, as shown in part (b), have Ec and E r displaced from each other relative to the k -axis. For example, GaAs is a direct gap semiconductor while Si is an indirect gap semiconductor. Since electrons seek the minimum energy in the band, the electrons that are promoted to a conduction band tend to be located in the region o f the energy minimum or valley o f the band, which can be seen in Figure 2.2.3. The holes that remain behind in a valence band tend to be located in the region o f the energy maximum or peak of the band because the electrons in that band move to the lower energy region. To help understand this behavior of holes, they are ofren described as behaving like air bubbles in water, which tend to float to the top. By convention, the energy o f electrons is positive when Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 (a) Direct gap (b) Indirect gap Lowest conduction band valley valley peak peak ► k =0 k =0 Highest valence band KEY: • electron o hole Figure 2.2.3 Generalized E-k diagrams showing carriers and bands for two types of semiconductors regarding band gaps: (a) direct gap and (b) indirect gap. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 measured upward while the energy o f holes is positive when measured downward on the E-k plots. Finally, as will be explained in Section 2.3, only the lowest conduction band has significant electron density, and only the heavy hole and light hole valence bands have significant hole density. Therefore, unless otherwise noted, the remainder o f this paper will refer only to the lowest conduction band and only to the light hole and heavy hole valence bands. The motion o f free electrons and holes can be treated semi-classically by assigning them effective masses, and the procedure for making this assignment is now explained. Fig. 2.2.2 reveals that the shape o f the energy valley about k = 0 in the conduction band as well as the shape o f the peaks about k = 0 in the valence bands is very nearly parabolic at energies that are not far removed from the respective minimum and maxima. Since most o f the conduction band electrons are located in the valley region and most o f the valence band holes are found in the peak regions, this motivates the representation o f the band energy near k = 0 by a Taylor series expansion about k - 0 [38]. For both sets o f bands, the Taylor series expansion gives d 2E. * .4 £ £ k,kJt (2.2.5) where the first order term vanishes as the first derivative o f an extremum and the second order term is rewritten as 5 2£ . k,k, = ——— k 2, - £ £ 2m. ♦ 2 i-TZ.rjZ'y.r dktdkj 0 i j Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.2.6) 35 in which 1/ m n * is defined as the inverse effective mass tensor for the electron in band n as given by _ 1 d 2E„ 1 m» * J ij h 2 dk'dk, ' j i,J = x,y,z (2.2.7) _ o By orienting the coordinate axes along the principal axes o f the constant energy surfaces in k -space, the o ff diagonal terms o f the tensor vanish [39]. For GaAs, the constant energy surfaces are spherical for energies not far removed from Ec for the conduction band and from Er for the valence band. This further simplifies the tensor so that it becomes a scalar, which means that the effective mass is completely independent o f orientation. By defining the effective mass in this manner, the effect o f the crystalline field is incorporated into the effective mass, so that the dynamics o f the electrons and holes can be treated as if the carriers are semi-classical wave packets [40]. For the electrons in the conduction band valley about k = 0, the energy is expressed as (2 .2 .8 ) where mn * is the effective mass o f the electrons in the lowest conduction band valley. From this formulation, it is apparent that the conduction band edge Ec can be identified with the electron potential energy since the electron kinetic energy is identified as the term containing the wave vector [40]. The top two valence bands are nearly degenerate in energy, especially in the region o f the peak about k = 0 . The curvature o f the peaks o f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 each o f these bands differs, so the effective masses differ as well. For the holes in the peaks o f the two highest valence bands, the energy is expressed as (2.2.9) where mpl * and * refer to light and heavy holes, respectively. The uppermost o f these two bands has the least curvature, so its holes are the heavy holes designated by mass . The hole potential energy is identified as the valence band edge E , . The electron and hole effective masses are determined by fitting experimental results to parabolic bands [35]. One o f the best experimental methods for determining the effective mass o f carriers is the cyclotron resonance effect. The effective mass o f electrons in the conduction band valley about k = 0 for GaAs is determined by taking a consensus o f the results o f various experiments used to measure this quantity, including interband magnetoabsorption, Faraday rotation, magnetophonon resonance, cyclotron resonance, and Zeeman spectroscopy [36]. The actual value is presented as the ratio o f the effective mass to the electron rest mass m0. The electron effective mass ratio at T = 300° K , which is the temperature at which the present study models the photodetector, is given by mn * Im 0 = 0.063. The effective masses o f the holes in the light and heavy hole valence bands is determined by taking a consensus o f the results o f experiments that included cyclotron resonance and hall effect measurements [41]. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 values o f the hole effective mass ratios at T = 300° K. are mpl * / mQ= 0.087 and m ph * t m o = 0*480 . 2.2.4 Density of States With the relationships between E(k) and k now established for both electrons and holes, the density of single particle states can be determined for the conduction and valence bands to be used later for calculating the carrier densities. The first step is to determine the density of states in k -space, and then convert this so that the density of states becomes a function o f energy. The process begins with a rectangular crystal o f dimensions Lx , Ly , and L. and volume V [39],[42]. The assumption is made that the probability wave solutions {p*(r)j be periodic at the boundaries o f the crystal. This requires that the plane wave term exp(ik •r ) of the Bloch solution (2.2.2) be periodic in the crystal since the term that has the lattice periodicity uB(ic,r) is automatically periodic in the crystal. The periodic boundary conditions lead to the following expressions for the x ,y , and z components o f the allowed wave vectors: kx =^y~nx, ky = ^ - n y , k z = y ^ n .; x y (nx,ny,n z = 0 , ± 1 , ± 2 , * ) . (2.2.10) L. If all o f the allowed k -states are plotted in k -space, a rectangular lattice is obtained, such that the volume Vk in k -space for each k -point is given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 (2.2.11) where V is the crystal volume. The density o f states g a (Jc) for band o is found by taking the reciprocal o f Vk and allowing for both spin possibilities in each k -state, which yields the following expression: *■ (*)= - A , (2.2.12) where the crystal volume has been divided out because the density of states is the number of k -states per unit volume o f k -space dk per unit volume o f r -space dr . Note that the density o f states in k -space is constant and therefore not actually a function o f k . The next step is to make the density o f states a function o f energy. This requires finding the number o f quantum states g„(E)dE between the constant energy surfaces in k -space corresponding to energies E and E + dE. The process is made easier when the constant energy surfaces in k -space are spherical, as is the case for GaAs for spheres of small k -radius. The quantity g B(E)dE is found by integrating the density o f states g„(k) over the volume dVk in k -space representing the region between the two constant energy surfaces E and E + dE. Equations 2.2.8 and 2.2.9 are used to change variables from k to E for electrons and holes, respectively. The result leads to the density o f states g„(E) for the conduction band electrons, as given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 (2.2.13) - 1'2 j E - E c , and to the density o f g pi(E ) for the light(/) and heavy(/i) valence band holes, as given by ft J E ,-E i = l.h, (2.2.14) with the assumption here being that the energy E for heavy and light holes is degenerate in that part o f the two peaks that is significantly populated by holes. The density o f states is used to determine the carrier density. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 23 2 3 .1 CARRIER DENSITY Electron Distribution Function The derivation o f the formula to calculate charge carrier density begins with the number o f electrons N(E)dE as a function o f the electron energy E whose energy is in a range dE about E, with N(E)dE given by [42] N{E)dE = f ( E ) g niE )d E , in which (2.3.1) is the distribution function o f the system. The distribution function fiJE) is defined as the average number of electrons that occupy a single-particle electronic state o f energy E. Due to the Pauli exclusion principle, the Fermi-Dirac distribution function is used for electrons, giving / ( £ ) = -------1+ exp 1------------- (2.3.2) k .T where k B is Boltzmann's constant, T is the Kelvin temperature, and E F is the chemical potential energy although it is called the Fermi energy in the context o f semiconductors. In order to determine the conduction band electron density n, the Fermi-Dirac distribution function (Equation 2.3.2) and the conduction band density o f states (Equation 2.2.13) are substituted into Equation 2.3.1, which is then integrated over all energies in the conduction band, to give [39] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 « - / / ( £ ) « , (E)dE = 8 JI t c :— m, hJ ' J— £c 1+ exp * 3/2 V ^ eT < /£ , (2.3.3) LV where the upper limit on the energy Elop is replaced by oo. This is justified by the rapid drop in the integrand toward zero for increasing energy above Ec, so that a negligible amount is added to n as a result o f integrating beyond the value o f E,op. The probability that a given state o f energy E is occupied by an electron is called the occupancy, and is given by the distribution function f[E). Therefore, the probability that a given state o f energy E is empty is 1—/ (E ) . This is identical to the probability that a given state o f energy E is occupied by holes. The valence band density p for light (/) and heavy (h) holes is determined then by replacing J[E) by 1 - / ( £ ) in Equation 2.3.1. Then as was done with electrons, Equations 2.3.2 (Fermi-Dirac distribution) and 2.2.14 (the valence band density of states) are substituted into the adjusted form o f Equation 2.3.1. The resulting equation is integrated over all energies in valence band i to give P. = j[l-/(£ )k „ (£ )d E = X/£“ T £ d E , el + exp J .T Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.3.4) 42 where the lower limit on the energy E ^ , ^ is replaced by - oo for the same reason as was used above for electron density. The two integrals for n (Equation 2.3.3) and p (Equation 2.3.4) can be rearranged into a more convenient form to obtain [39] and (2.3.5) where N c and N v are the effective density o f states for the conduction band and valence band, respectively, as given by and (2.3.6) in which Fu2 is the Fermi integral o f order X A (a tabulated function) as given by (2.3.7) and where tjc = ( £ f —Ec)/(k BT) and 7 , = (£ , - EF)f(kBT) . Using Equation 2.3.6 and the previously quoted values for the constants in the equation, the effective density of states (at 300° K) is 4.0x10” cm ' 3 for electrons and 9 .0 x l0 18 cm -3 for holes. Since the Fermi integral as given by Equation 2.3.7 must be solved numerically, it would be useful if there were a way to approximate the Fermi integral so that the carrier density equation (2.3.5) could be solved in closed form. For most practical situations in which a semiconductor is used, including the photodetector of the present study, there is a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. way to approximate the Fermi integral so that an analytical solution can be achieved. From the denominator o f the Fermi integral in its original form, it is apparent that if the argument o f the exponential becomes large enough, the exponential term will dominate the denominator so that the number one present in the denominator can be ignored. When the Fermi energy E f is relatively far away from either edge o f the band gap, this domination by the exponential term is achieved. Specifically, if |£ - EF\ > 3kBT , where E represents an energy in either the conduction or valence bands, then the Fermi integral can be approximated so that the carrier densities follow a Boltzmann distribution, with the density o f conduction band electrons given by (2.3.8) and the density o f valence band holes given by p = AT, exp ~^— {Ev - E f ) . _kBT (2.3.9) Semiconductors in which the carrier density can be characterized by Equations 2.3.8 and 2.3.9 are said to be nondegenerate. It was stated earlier that only the lowest conduction band and the highest two valence bands have significant charge carrier densities. The reason for this is now seen to be due to the exponential drop off in carrier density as the carrier energy moves farther from the Fermi energy (see Equations 2.3.8 and 2.3.9). For example, the ratio o f split-off Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 band hole density to the combined light and heavy hole density can be estimated using Equation 2.3.9, which leads to N. = 2 .0 x 10-6 exp where the energy difference between the light (and heavy) hole band and the split-off band at k = 0 is 0.34 eV (see Figure 2.2.2), and the split-off hole effective mass ratio mso * / mo =0.154 is approximately the same order as the average o f the light and heavy hole effective mass ratio [39]. This proves that the holes in the split-off band make a negligible contribution to the total hole density in the valence bands and need not be considered in the temperature range of the present study. 2.3.2 Intrinsic Semiconductors It is customary to express carrier densities in terms of the intrinsic carrier density and the intrinsic Fermi energy, which are now defined. Pure semiconductors, ones that have no impurities, are called intrinsic semiconductors. Intrinsic semiconductors have equal electron and hole densities, so that n = p = n,, where ni represents either the electron or the hole density and is called the intrinsic carrier density. For an intrinsic semiconductor, the electron density is given by n, = jV. exp (2.3.10) Lk*T Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 where £, is the Fermi energy in an intrinsic semiconductor, and is called the intrinsic Fermi energy. Similarly, the hole density for an intrinsic semiconductor is given by n, = N r exp — (Er - £ , ) . k BT (2.3.11) These last two equations can be used to calculate the intrinsic carrier density nk. Multiplying Equations 2.3.10 and 2.3.1 1 together and using the definition o f the band gap energy E g given by Equation 2.2.4, the intrinsic carrier density ni is given by (2.3.12) which when evaluated gives n, = 2.2x10 6 cm -3 for GaAs. [NOTE: In calculating quantities that involve k BT , it is customary to convert k aT to eV, which is accomplished by using the charge on the electron q, giving k BT = 0.02586 e V .] These same three equations can be used to determine the intrinsic Fermi energy E, relative to the valence band edge Er . By setting Equations 2.3.10 and 2.3.11 equal to each other and using the definition o f the bandgap (Equation 2.2.4), the intrinsic Fermi energy E, is given by (2.3.13) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 Lowest conduction band 0.0058 eV -E , &E- c = 0.30 eV AEn , = 0.67 eV E = 1.42 eV 1.12 eV 0.67 eV 0.75 eV Highest valence band ► /c Figure 2.3.1 Partial GaAs E-k diagram showing various energy levels for an unintentional donor Si doping density o f ND = 3.5 x 1012 cm'3, and with a deep level donor trap [after Blakemore I36!]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 in which E t - Er = 0 .7 5 eV . Figure 2.3.1 is a partial E-k plot for GaAs showing the location o f the intrinsic Fermi energy E , . 2.3.3 Doping the Semiconductor The carrier density is largely determined by impurities or dopant species that are added to semiconductors to increase the electrical conductivity by providing additional mobile charge carriers. If the dopant species donates electrons to the conduction band, the dopant is called a donor and the semiconductor is considered n-doped. If the dopant species accepts electrons from the valence band, thereby increasing the hole density in the valence band, the dopant is called an acceptor and the semiconductor is considered pdoped. In the case o f GaAs, adding group VI elements such as S, Se, and Te provides electron donors [43]. The group VI atom replaces the group V atom As so that there is now an extra valence electron that has no empty orbital to form a bond with Ga. This extra electron is only loosely bound to the lattice, and so is easily promoted to the conduction band. Adding group II elements such as Be, Mg, and Zn provides electron acceptors. The group II atom replaces the group III atom Ga so that there is now an empty valence orbital in one of the four bonds between the group II atom and As. A valence electron is easily promoted into this empty orbital, leaving an extra hole in the valence band. Donor and acceptor densities are given by N D and N A, respectively. In the present study, GaAs is modeled as being unintentionally n-doped with donor Si at a density o f N D = 3.5xl0 12 atoms/cm3. Unintentional doping indicates that there are impurities even in supposedly pure semiconductors. Silicon, a Group IV atom, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 is a popular «-dopant for GaAs. The Si atom replaces a Ga atom, thereby offering an extra valence electron for promotion to the conduction band. The Si donor electronic state E d is located in the bandgap very close to the conduction band edge Ec at an energy that is 0.0058 eV below Ec, as can be seen in Figure 2.3.1 [39], Since the average kinetic energy o f a particle is on the order o f kBT , donor electrons have more than enough energy to be promoted into the conduction band at room temperature. It is therefore assumed here that the dopant species is completely ionized, so that the resulting ion density is equal to the original dopant density N D. The carrier density for semiconductors that are doped with impurities can be conveniently expressed in terms o f the intrinsic Fermi energy Et and the intrinsic carrier density «,. By solving Equation 2.3.10 for the effective density o f conduction band states N c, and substituting this into Equation 2.3.8, the electron density for doped semiconductors can be expressed as n = n, exp (2.3.14) L*bT providing the semiconductor is nondegenerate. Likewise, by solving Equation 2.3.11 for the effective density o f valence band states N r , and substituting this into Equation 2.3.9, the hole density for doped nondegenerate semiconductors can be expressed as — l ( E , - E F) . P = ”' eX\ k , T Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.3.15) 49 The intrinsic carrier density ni and the intrinsic Fermi energy E, are constants, so all that remains in order to calculate the electron density n and the hole density p is to determine the Fermi energy E F as a function o f dopant density N D. For unintentionally donor-doped GaAs, the Fermi energy is determined by assuming that the donor species is completely ionized. Since the doping density N D is six orders o f magnitude greater than the intrinsic carrier density ni , the electron density n is assumed to be due completely to the ionized donors, so that n = N D. Substituting N D for n in Equation 2.3.14 and solving for the Fermi energy EF gives E f = £, + k BT In (2.3.16) in which &EFj = E f —E, = 0.37 eV (see Figure 2.3.1). calculated that E , - E r = 0.75 eV , adding these last Since it was previously two expressions gives EF - Ev = 1.12 e V , which places the Fermi energy in the band gap at an energy 0.30 eV below the conduction band edge Ec as is shown in Figure 2.3.1. This is well within the range of nondegeneracy, since the semiconductor does not become significantly degenerate until the Fermi energy EF is closer than about 3kBT = 0.08 eV to the conduction band edge Ec. Other energy separations shown in the figure include AE cF —Ec —E F —0.30 eV and AEcj —Ec —E, —0.67 e V . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 2 3 .4 Debye Screening Length When a charge is placed near a semiconductor, the mobile charge carriers within the semiconductor rearrange themselves in response to the electric field from the perturbing charge [43]. The rearrangement o f the semiconductor charges creates a net local electric field that opposes the electric field of the perturbing charge. This tends to balance out the perturbing field, or screen the rest o f the semiconductor from the perturbing field. The screening effect causes an exponential drop o ff in the perturbing field as a function o f distance from the charge. The Debye length LD is a measure o f the shielding distance, or the distance at which the field falls off by a factor o f 1/e. For ndoped semiconductors, the Debye length LD is given by (2.3.17) where K s is the dielectric constant o f the semiconductor and s 0 is the permittivity of free space. Substituting the GaAs dielectric constant K s =13.18 [44] along with the unintentional donor doping density used in the present study gives a Debye screening length (at 300°K) L D —2.32 pm . This means that for the 1.1 pm length MSM used in the present study, the electric field produced when the metal electrode is brought into physical contact with the semiconductor (to form the MSM) extends throughout the length of the device without substantial screening at this low doping. This is advantageous for the photodetector since it is this electric field that pushes the excess charges created by light through the substrate, creating the desired photocurrent. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 2.4 2.4.1 GENERATION OF CARRIERS BY ILLUMINATION Light Absorption As pointed out earlier, the addition o f thermal energy to a semiconductor generates electron-hole pairs, thereby increasing the density o f both types o f charge carriers. The second way in which electron-hole pairs can be generated is through the application o f light. Illuminating a semiconductor with light o f energy greater than the bandgap energy leads to the absorption o f photons by valence band electrons. The excited electrons are promoted to the conduction band, leaving behind holes in the valence band, so that the density o f both types o f charge carriers is increased. This section presents the formulation necessary to calculate the carrier generation rate due to light absorption. Figure 2.4.1 presents a generalized E-k plot that shows the photogeneration o f an electron-hole pair. Light absorption occurs in the valence band in regions to each side of the valence band peak, because these side regions are more likely to contain electrons. However, the electrons in the side regions require more energy for promotion across the band gap since there is a much greater energy separation between the valence and conduction bands in the side regions. Therefore, if the transitions are still to be in the optical range, they must not occur too far from the valence band peak. In the present study, 800 nm light is used, which corresponds to an energy o f 1.55 eV. Therefore, the photon absorption events must occur close to the valence band peak (see the E-k plot of GaAs given in Figure 2.2.2). The low hole density in the split-off band indicates that the region close to the peak o f the split-off band serves as a potential source o f valence band Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 (a) Direct gap (b) Indirect gap EG) EG) valley valley peak peak Photon absorbed Photon absorbed ► /c = 0* Figure 2.4.1 Generalized E-k diagrams showing light absorption for two types of semiconductors regarding band gaps: (a) direct gap and (b) indirect gap. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ► 53 electrons for light absorption. However, the use o f 1.55 eV light precludes photon absorption by the electrons in the split-off band. Even the highest energy electrons in this band would require photons with a minimum energy o f 1.76 eV to promote the electrons across the bandgap. 2.4.2 Carrier Generation Rate Due to Constant Illumination The carrier generation rate GL due to constant illumination is given by * dt dp dt (2.4.1) where the L (for light) designation serves to separate photo- from thermal-generation (thermal generation is characterized in the next section with thermal recombination because both rates are incorporated into a single equation). GL is actually the average rate o f change in carrier density. Light is attenuated as it penetrates deeper into the semiconductor substrate, so that the instantaneous carrier generation rate is a function o f depth. To calculate the average carrier generation rate GL, the semiconductor substrate is modeled as a three-dimensional rectangular solid of length Ls =1.1 pm along the xaxis, depth d = 1.0 pm along the y-axis, and width W = 35 pm along the z-axis, as shown in Figure 2.3.2. The cross-sectional area exposed to light AB = 38.5 pm 2 is in the x-z plane. Light penetrates into the substrate normal to and along the y-axis. The substrate is divided into infinitesimal differential volume elements dV, with dV = A^dy Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 Plane polarized light ik [xz d GaAs substrate Contact Figure 2.4.2 A single MSM unit showing the substrate and contacts, the dimension symbols, the cross-sectional areas, a differential depth element, and a monochromatic light wave oriented as described in the text. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 as shown in the figure. Within a given volume element dV, the instantaneous generation rate is assumed constant The number of electron-hole pairs dNL that are photogenerated in volume d V per time dt is equal to the number o f photons -dy that are absorbed in d V per dt due to the one-to-one correspondence between photon absorption and electron-hole pair generation; i.e. (2.4.2) The number o f photons d y m an increment o f light energy dE is given by (2.4.3) where vis the frequency o f the light, so that h v is the energy per photon. Equation 2.4.3 can be used to rewrite Equation 2.4.2 as (2.4.4) Since the instantaneous light intensity d^jf is defined by (2.4.5) Equation 2.4.4 can be rewritten as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The average generation rate can be extracted by integrating both sides o f Equation 2.4.6 over the volume o f the substrate: — fd N L = - — f^ - A ^ d y d tJ L h v * dy dt \ Gl = - - L - [ j { y = d ) - j ( y = 0)]. hvd (2.4.7) The light intensity <J(y) as a function o f depth y for light that penetrates a medium is given by [24],[43] s7(y) = J o O " r )exP(~ ay) . where (2-4.8) is the initial light intensity (at y = 0 , the position where the light first enters the substrate), a (800 nm) = 1.7 x 10 4 cm "1 is the absorption coefficient, and r is the reflectance as given by [45] r = ' P. ~ n 2 (2.4.9) Vn i + n 2 in which n, and n 2 are the indices o f refraction for media 1 and 2, respectively. Based on n, =1.0 (air) and n 2 =3.6 (GaAs), the reflectance is r = 0.32. An adjustment to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 Equation 2.4.8 must be made to allow for the fraction o f the light that actually enters the substrate. The portion o f the initial beam intensity ^J0 that impinges on the contacts does not enter the substrate and therefore makes no contribution to the generation o f carriers. The fraction o f the initial intensity that enters the substrate is given by Ls/ (Ls + Lc) , where Lc =0.66 pm is the length o f the contact, so that the fraction o f photons that enter the substrate is 0.6. (see Figure 2.4.2). Making this adjustment and substituting for <J{y) from Equation 2.4.8 into Equation 2.4.7 gives the baseline generation rate GL0 G lo = - L* (l - r )[l - exp(—a d )], h v d Ls + Le (2.4.10) where baseline refers to light that is not modulated. The term in brackets gives the fraction o f those photons that enter the substrate that are absorbed by the substrate, which is 0.82. The total fraction o f the initial intensity that leads to the generation o f excess electron-hole pairs is the product o f the three fractions given in Equation 2.4.10; i.e., the fractions due to the contacts, the reflection at the interface between the air and substrate, and the incomplete photon absorption in the substrate; so that 0 . 1 6 ^ actually generates electron-hole pairs. 2.4.3 Converting Light into Microwaves Microwaves can be created by a semiconductor through light absorption. If voltage is applied across the semiconductor while it is being illuminated, then electric Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ! 58 current called photocurrent is generated in the semiconductor. When the light intensity is made to oscillate at frequencies to which the semiconductor can respond electrically, the photocurrent in the semiconductor oscillates at the frequency of the oscillating light intensity. Since oscillating electric charge generates electromagnetic radiation, the AC signal generated in the semiconductor produces an electromagnetic wave with the oscillation frequency. Light intensity is made to oscillate by mixing together two monochromatic light beams that have different frequencies in a process called photomixing or heterodyning. The frequency o f the electromagnetic wave produced by the oscillating semiconductor charge is equal to the difference in frequencies between the two monochromatic light waves. If the frequencies o f the two light waves are chosen properly, then the electromagnetic radiation produced by photomixing has a frequency in the microwave range. As demonstrated in the previous subsection, the generation rate for the electronhole pairs responsible for the photocurrent is directly related to the absorbed light intensity. The time-dependent variation in intensity that results from photomixing is based on the time-dependence o f the amplitude o f the photomixed light. Therefore, the relationship between the intensity and the amplitude o f the photomixed light must be derived in order to characterize the time-dependence o f the photocurrent and the resulting microwave. The first step in the derivation is to determine the resultant amplitude for two light waves that are superimposed, as occurs during photomixing. The photomixing model developed in this study assumes that the two light waves are harmonic plane waves, are polarized in the same direction, and are in phase. The equation for the timedependent amplitude 5 , (/) o f the electric field o f the first light wave is given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 (0=5,.o sin(<o,/), where g lQ is the maximum amplitude and (2.4.11) is the angular frequency of the light, with the relationship between frequency v and angular frequency being qj = 2i t v . Likewise, the time-dependent amplitude g 2(t) of the second light wave is given by ^ ( 0 = ^ 2.0 sin(o>2/ ) , (2.4.12) The time-dependent resultant amplitude g R(t) of the superposed light waves is given by S R(/) = 5, (0 + S 2(0 = g u0 sin(<0,O + g 20 sin(o>2t ) . (2.4.13) The second step in deriving the relationship between the intensity and amplitude for the photomixed light is to derive the general relationship between light intensity and amplitude. The light intensity is equivalent to the time average o f the Poynting vector S , as given by [46] where fj. is the relative permeability o f the medium and g ± B . The magnitude B o f the magnetic field can be expressed in terms o f the magnitude g o f the electric field if the electromagnetic wave that generates these fields is harmonic, i.e. o f the form Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 S z O ,0 k = 5-o sin(fcy - a t ) k , (2.4.15) in which the wave propagates along the ^-axis (in keeping with the model) with wave number k, and is arbitrarily polarized along the z-axis so that the magnetic field Bx( y , t )i is along the x-axis (see Figure 2.4.2 for this light wave). When the Maxwell equation that expresses Faraday’s law V x5= -— dt (2.4.16) is applied to the electric field given by Equation 2.4.15, the result is d S .(y ,t)t dBx(y,t)~ — =------- 1= ------ay at ? i = k£ .0c o s ( k y - a t ) i , (2.4.17) where d £ .(y ,t) /d x = 0 for plane waves with the phase plane parallel to the xz-plane. Integrating the magnitude o f the magnetic field over time and evaluating at point (y,t) yields yj yj j dBx (y , r) = - kE:0 jcos(fcy-cot)dr yjo y/o k Bx(y,t) - Bx(y,t) = —5 .0 sin(*y- a t ) 6) k £ :0 sin(ky - a t 0) 6) Jq Bx(y ,t) = —5-o sin(Ay - a t ) a Bx(y ,t) = J w S x (y , t ) , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.4.18) 61 where the wave velocity in the medium is given by v = a)/k, e is the relative permittivity, and the velocity is given by v = \ f . Substituting the last part of Equation 2.4.18 into Equation 2.4.14 (with S . = S ) gives the intensity in terms of the magnitude o f the electric field: Finally, the relationship between intensity and amplitude for the photomixed light is constructed by substituting the time-dependent resultant amplitude from Equation 2.4.13 into Equation 2.4.19 to give + S l0S 20 {cos[(<w, - a ) 2>]-cos[(fi>, +<y2>]}),, (2.4.20) where the following two trig identities were used to generate the final step in the equation: sin 2 a = 1/2 - (l/ 2 )cos 2 a ; and sin a sin b = (l/ 2 )[cos(a - b) - cos(a + A)]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 2.4.4 Generation Rate with Oscillating Light Intensity Before performing the time average in the intensity equation (2.4.20), note that o f the four terms that depend on time, three have frequencies in the range of light, namely 2o)\, 2qj2, and cox + a>2. The fourth time-dependent term has a much lower, difference frequency, cox —co2, which can be in the microwave range if the two initial light frequencies are properly chosen. The photodetector absorbs the two constant components o f the intensity as well as the four time-dependent components. However, the photodetector is only able to respond electrically to the time variation in intensity for the low frequency component. Therefore, to accurately characterize the photodetector electrical response to the intensity o f photomixed light, the time average is taken over an interval that retains the time dependence o f the low frequency component [47], Mathematically, this time interval is the product o f the three high frequency periods, or T = (Tt/o), ){jt/to2)\2x /{ g) x + a)2)]. This particular value for T is chosen so that the high frequency terms o f the intensity average to zero because there are a whole number of cycles in T for each o f the high frequency components. Physically, the high frequency components contribute no additional charge to the excess charge density because successive intensity peaks and troughs impinge upon the semiconductor before the excess charges that they create can be swept out. and so average to the baseline intensity created by the DC baseline. The time averaging o f Equation 2.4.20 over an interval that is on the order of a few optical periods makes intensity ^f(t) a function o f time, as given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 *?(') = ) dr - + fJ ^ c o s ( 2 a ,t) d t - cos(2*m)<* T ^^7 &10&20 COS[(^1 —^ 2 )* ] ' f j ^ SwS20 COS^ ' + C>*)/1£* ’ (2.4.21) in which the time-dependent portion o f the integrand in the fourth term is so slowly varying relative to the integral period that it essentially remains constant over T and can be factored out of the integrand. The final result of the integration yields *2(0 = + S i )+ «*[(<», ' a>2 )']• (2.4.22) +ei)*Ms„e,,angm - 0>M (2.4.23) which can be equivalently expressed as by shifting the initial time by nJ2. The final form of ^7(/) is given by O W = Oro 6 + ^ sin(2^v„ t )], (2.4.24) where the initial intensity <J0 , the modulation index M, and the modulation frequency vm are respectively given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 = y j f ( s i +*»). Z Vr1 M = 2 f '° £ “ , and ®I0 + ^20 k „ = |k ,-^ |. (2.4.25) [M varies between 0 and 1, depending on the relative optical intensity o f the two light beams that are photomixed, with 0 associated with a single beam and 1 associated with two beams of equal intensity.] The value for the time-dependent intensity ^ { t ) given by Equation 2.4.25 is substituted for ^J0 in the equation (2.4.10) for the generation o f carriers to give the time-dependent generation rate GL(t) as GL(t) = G lo [l + M sin(2;rvm/)], (2.4.26) with the baseline generation rate G LQ given by Equation 2.4.10. Both y je and M are simulation input parameters. 2.4.5 Optical Responsivity and Bandwidth As pointed out in Section 1.3.3, the two figures o f merit by which the photodetector is measured are the optical responsivity and the bandwidth.Since these two figures of merit depend on the photogeneration o f current, they characterized. are now The photocurrent i{t,vm) is a function o f time and the modulation frequency vm through »(*. vm) = ' dc + i'xc(O sin(2;zvIB0 , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.4.27) 65 where i°AC is the constant or direct current (DC) portion of the photocurrent, ) sin(2;rvm/) is the alternating current (AC) portion, and iAC(vm) is the peak o f the AC signal for a given modulation frequency v m. The DC portion ix o f the photocurrent is associated with the constant intensity term in Equation 2.4.23, while the AC portion iQ Ac ( v m)s™{2xvmt) o f the photocurrent is associated with the time-dependent intensity term in Equation 2.4.23. The AC peak i°AC(v m) is the output that is sought through the simulation o f the MSM photodetector. Part (a) o f Figure 2.4.3 plots the photocurrent as a function of time for two modulation frequencies. From these plots, it is apparent that the AC peak i°AC(y m) can be calculated by i°AC(v m) = where /max , (2.4.28) and imin (t, v m) are the maximum and minimum AC current values, respectively, associated with the modulation frequency vm. [NOTE: Equation 2.4.27 implies that the dark current is negligible since it is not included in the equation. The maximum dark current iAjiax in the present study occurs at a bias o f 10 V, which produces iAjaax = 4 . 0 x l 0 -7 A . Even for the lowest light intensity used in this study (1 KW/cm2), the ratio o f the maximum dark current to the total photocurrent (8.6x1 O'5 A) is about 1/200, which is insignificant.] For relatively low modulation frequencies, the peak o f the AC signal has the same maximum value i°ACmm, because the photodetector is fast enough to produce a complete Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 (a) (b) ' ac ( ^ ) 'AC,max 1AC, max opt ~ U Bandwidth m m.low m,thr 3-db Figure 2.4.3 Optical responsivity and bandwidth are determined from the (a) time-dependent photocurrent and graphed (b) on frequency response curve. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 current response to the oscillating portion o f the light intensity (see Appendix E). The photocurrent i(t,v mJaw) plotted in part (a) o f Figure 2.4.3 is an example o f the photocurrent associated with a modulation frequency vmJaw that is within the low frequency range. Part (b) o f the figure, called the frequency response curve, plots the AC peak i°c (vm) as a function of modulation frequency vm. Note that the value o f the AC peak is constant for modulation frequencies within the low frequency range, where i°AC( O = i°ACjnxt - As the frequency is increased, a threshold modulation frequency v mJhr is reached for which the photodetector is unable to form a complete current response, as shown in part (b). Subsequent increase in vm results in progressively smaller current responses to the oscillations in light intensity, so that the AC peak i°AC(vm) decreases with increasing frequency. The photocurrent i(t, ) plotted in part (a) is an example o f the photocurrent associated with a modulation frequency that is above the threshold frequency v mJhr. Eventually the modulation frequency becomes so high that the photodetector is unable to make any electrical response to the oscillating portion of the light intensity, and only the constant DC portion o f the photocurrent remains. This is indicated on the frequency response curve in part (b) where the AC peak i°AC(ym) goes to zero. The optical responsivity Ropt is the ratio o f the low frequency constant value o f the frequency response i°Cjnix to the peak light power P°A associated with the timedependent portion o f the light that strikes the substrate, and is given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 (2.4.29) in which P°a is given by (2.4.30) P°u where the product M^J0 returns the peak value o f the time-dependent portion of the light intensity given in Equation 2.4.23, as can be seen by using the definitions of M and found in Equation 2.4.25. The bandwidth is defined as that frequency at which the AC power peak P°c (ym) associated with the AC peak i°AC(ym) drops to one-half of the AC power peak P°Cmax associated with the AC peak i°Cmax at the constant low frequency value. The bandwidth is also called the 3 - db point because the base-ten log o f the ratio P°c ( v ^ ) / PACjnMX =1/2 is —3.0 decibels. Since power is proportional to the square o f the current, the 3 - db point is likewise defined by 10-log Du ‘ = 1 0 -log -0 I .<• m«v ~ = -3 .0 , (2.4.31) which leads to P a C (y -j-d b nO ) *AC ( v 3- d b ) A 2 f°-4C.m»x J_ ^ so that at the 3 - db point Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.4.32) From Equation 2.4.33, the bandwidth is formally defined as that frequency at which the AC peak i°AC(y 3_<a.) *s equal to l/V 2 o f its constant low frequency value /°Cjnlx, as can be seen in both parts of Figure 2.4.3. Part (b) illustrates that the bandwidth is the width o f the modulation frequencies from vm = 0 to vm = . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 2.5 2.5.1 RECOMBINATION Scattering and Radiative Decay Up to this point, two processes have been presented that generate carriers, thermal generation and photogeneration. Along with this set o f generation processes is the simultaneous occurrence o f the set o f reverse processes called recombination. In recombination, a conduction band electron loses enough energy that it enters an empty valence band state and effectively recombines with a hole. The two major mechanisms by which electrons lose their energy in semiconductors are scattering and radiative decay. The loss o f energy through scattering is called thermal recombination, and is the reverse process o f the scattering-mediated thermal generation o f electron-hole pairs. Radiative recombination, the reverse process o f light absorption, occurs when the recombining conduction band electron emits light o f energy greater than or equal to the bandgap energy. The relative importance o f either type of energy loss mechanism depends on the characteristic time between successive occurrences of that mechanism, such that the faster a mechanism occurs the more it tends to dominate. As will be explained in this subsection, scattering plays a more dominant role in recombination than does radiative decay in GaAs. A third energy loss mechanism, Auger recombination, occurs when two conduction band electrons collide so that one electron gives up enough energy to recombine, while the other electron thermalizes the excess energy. This process is important only at high doping, and is not considered here [39]. Scattering involves an interaction between a carrier and another particle or the electrostatic field associated with that particle. Pierret states that the two most important Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 scattering events for device quality semiconductors are phonon scattering and ionized impurity scattering [39]. Phonon scattering refers to collisions between the carriers and the lattice atoms that affect the lattice vibrational states. Phonon scattering is subdivided into lower energy interactions involving acoustic phonons and higher energy events involving optical phonons. In GaAs, optical phonon scattering is a dominant lattice scattering mechanism. The second type o f scattering event, impurity scattering, refers to the Coulombic interaction between the carriers and ionized impurities such as dopants. These interactions can be attractive or repulsive and result in the deflection o f the carrier. Chen and Sher calculate a combined phonon and impurity scattering rate for GaAs that is on the order o f lxlO 13 s_l [48]. Radiative decay is the emission o f an optical photon when the electron drops across the bandgap from the conduction band into the valence band. There are two types o f radiative decay processes, depending on whether the semiconductor is a direct gap or indirect gap material. In a direct gap semiconductor, the minimum energy Ec o f the conduction band occurs at the same wave vector £ as the maximum energy E r o f the valence band. GaAs is an example o f a direct gap semiconductor. Note that in the GaAs E-k diagram o f Fig. 2.2.2, both E c and Er occur at k = 5. In an indirect gap semiconductor, E c and E v do not occur at the same k value. Figure 2.5.1 shows how radiative decay occurs in a direct gap in [part (a)] and in an indirect gap [part (b)] semiconductor. Recall that most o f the electrons are in the conduction band valley about E c, and most of the holes are in the valence band peak about E r. Therefore, the most Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 (a) Direct gap (b) indirect gap E(V) E(/c) valley K - J i —H valley Photon emitted W/W-P- Photon emitted Phonon emitted peak peak ic =o' Figure 2.5.1 Generalized £-£ diagrams showing radiative decay for two types o f semiconductors regarding band gaps: (a) direct gap and (b) indirect gap. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 likely occurrence o f recombination through radiative decay is from the k -state associated with Ec to the k -state associated with E r . If Ee and £ , have the same k , as occurs with direct gap semiconductors, then the emission o f a photon automatically conserves momentum since k does not change. Such transitions are vertical on the E-k diagram and involve only the electron and emitted photon. If Ec and E, have different k values, as occurs with indirect gap semiconductors, then the loss o f a photon does not by itself conserve momentum. In order for radiative decay to occur in indirect gap semiconductors, the photon loss must be accompanied by a phonon interaction to conserve momentum. For example, in part (b) o f the figure, Ec has a higher k value than Er . As the electron radiates a photon it must simultaneously lose a phonon to the lattice so that the final momentum o f the electron is consistent with the ic value o f its destination in the valence band. Therefore, radiative decay in an indirect gap semiconductor involves the interaction o f an electron, a photon (vertical transition), and a phonon (horizontal transition). The simultaneous vertical and horizontal transitions associated with radiative decay in indirect gap semiconductors makes this event much less likely to occur than the vertical transition exhibited by direct gap semiconductors. Saleh and Teich quote radiative decay lifetimes o f lxlO -7 s and lx lO -2 s for GaAs and Si, respectively, which confirms the much greater likelihood for radiative decay in GaAs than in Si [45]. This is the major reason that GaAs has taken the forefront over Si as the semiconductor o f choice in photodetectors. Even though photodetectors base their action on light absorption Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 rather than emission, the optical use o f semiconductors began with light-emitting devices, notably lasers. Since GaAs is five orders of magnitude more efficient at light emission than Si, GaAs was extensively developed in optical systems before the growth of the light absorption technology. Therefore, even though the two semiconductors are comparable regarding light absorption, the technology was already in place to use GaAs. [A review o f Figure 2.4.1 demonstrates that both direct gap and indirect gap semiconductors absorb light as a vertical transition, so that there is no inherent advantage for one type o f semiconductor over the other regarding photogeneration.] In GaAs, the two mechanisms for recombination, radiative and nonradiative decay, are comparable. Electrons that are excited to higher energy states within the conduction band at first lose energy almost exclusively through successive scattering events, since the scattering rate is about six orders o f magnitude greater than the rate of radiative decay, based on the previously quoted scatter rate and radiative lifetime (the rate o f radiative decay is the reciprocal o f the lifetime, or 107 s '1). The maximum scatter rate o f 1013 s -1 is in effect because the full range o f phonon energy exchanges is available to the excited electron, including the lower-energy phonon exchanges. The step-wise descent of the electron from the upper energy region o f the conduction band, called thermalizatioii, causes the excited electron to fall quickly into the valley of the conduction band [45]. In order for the electron to drop across the gap, however, it must release a much larger quantum o f energy than was required during each step of the thermalization process. The high energy requirement lowers the probability that a phonon interaction will drop the electron across the band gap. The lower probability for such a high energy scattering event when the electron is in the conduction band valley Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 causes the lifetime for nonradiative transitions across the bandgap to lengthen to a value on the order o f 10-7 s , which is the same order as the lifetime for radiative decay, making these two recombination mechanisms comparable in importance. The nonradiative decay rate, 107 s*1, is this high due to the presence o f impurities and defects located near the middle o f the band gap, called traps or recombinationgeneration (R-G) centers. These R-G centers act as stepping stones for the electron as it falls across the gap, reducing the magnitude o f the energy quanta that must be removed from the electron by about one-half, thus raising the probability for a phonon interaction. The centers are denoted as R-G centers because they facilitate the generation o f electron hole pairs as well as recombination. Due to the presence o f these R-G centers, a Shockley-Read-Hall mechanism for modeling recombination and generation via the R-G centers is the central process for recombination [24]. Before discussing the R-G mechanism, one final note is made regarding nonradiative decay in GaAs. When the electric field due the applied voltage reaches about 3.3xlO 3 V/cm for GaAs at 300° K, intervalley transfer becomes important [43]. Electrons in the T valley begin to attain enough energy to be promoted to the L valley. The E-k plot of GaAs in Figure 2.2.2 indicates that the energy necessary to lift an electron from the T valley to the L valley is 0.29 eV. The major consequence o f intervalley transfer is that carrier mobility is reduced significantly, and this will be addressed later. Regarding recombination, the same basic processes occur as for recombination from the T valley. However, energy decay across the bandgap takes longer due to the greater Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 separation between the conduction and valence bands in the region of the L valley. Therefore, intervalley transfer reduces the recombination rate. 2.S.2 Recombination-Generation Statistics Recombination-generation (R-G) statistics characterize the thermal generation and recombination o f carriers via traps in the mid-region o f the band gap, and this statistical approach is also used to characterize radiative recombination in the present study as well. Thermal generation occurs when a valence band electron is promoted first to a mid-gap trap, and then is promoted a second time to the conduction band. Each promotion step is enabled by a scattering interaction in which the electron absorbs a phonon from the lattice. Thermal recombination is the reverse process. A conduction band electron drops first into a mid-gap trap, and then drops a second time into a valence band vacancy, with the electron releasing a phonon to the lattice for each step in its descent. Based on the similarity in the form of the R-G mechanism and the form o f the radiative decay mechanism presented by Selberherr, the R-G mechanism is used to characterize radiative as well as nonradiative decay [49]. The derivation of the R-G rate equation is now summarized from the formulation presented by Pierret [39]. The R-G rate relationships for electrons and holes are given in generalized form as dn ~dt _^ R -G ~~dt e m it dn ~~dt and c a p tu re dp dt R -G dt em it dp dt Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 where the positive term on the RHS o f each equation represents emission by the mid-gap trap and the negative term represents capture by the mid-gap trap. For nondegenerate semiconductors, in which almost all o f the conduction band states are empty, the rate equations can be rewritten as dn ~dt = e n n T - Cn ( N T ~ n T ) n ’ “ d R -G 5p dt = ep{NT - n T) - c pnTp , (2.5.2) R -G where c„ and cp are the capture coefficients for electrons and holes, respectively; en and ep are the emission coefficients for electrons and holes, respectively; N T is the tGtal trap density; and nT is the density of traps occupied by an electron. By assuming equilibrium, in which the rate o f change o f carrier density is zero for both carriers, the emission coefficients can be eliminated from Equation 2.5.2, giving dn It = cnnTnx —cn(N t - n T)n, R -G and ^ at = (N t “ nT )Pi ~ cpnr p , (2.5.3) R -G where the trap constants are given by ( N t - nT)n n \ = n, and NT-n T (2.5.4) The trap constants n} and p, are calculated as functions o f the trap energy Er and N t through the same statistical approach that was previously used to determine the electron and hole density, resulting in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 «, = «, exp E t - e A V k .T and e, p x = n, exp I - et k ,T ) (2.5.5) in which the equation for a deep-level donor trap such as the one used in the present study is given by Ylj = N t 1 + exp f ET - E F^ -i (2.5.6) For the GaAs model, a deep-level donor trap o f energy ET = 0.67 eV above the valence band edge E y is used, as shown in Figure 2.3.1 [39]. Then using Equation 2.5.5, the trap constants are calculated, with nx = 1 .0 x l0 5 cm ”3 and p x — 4.9xlO 7 cm ”3. Next, steady state conditions are assumed, in which the rates of change in electron and hole densities are equal. This allows N T and nT to be eliminated between Equations 2.5.3 and 2.5.4, which leads to the R-G rate equation with the thermal recombination-generation rate RG given by RG= ^ I dt R -G cjp dt R - G n, - n p Tp{n + nx)+T„{p + pxy (2.5.7) in which the electron and hole lifetime constants r„ and r , respectively are given by " cmN T ’ and r„ = cpN t Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.5.8) 79 and nlp l = nf is used. A positive RG value indicates net generation, while a negative RG value indicates net recombination. Even though the R-G rate equation is derived for steady state conditions, it is typically used to approximate non-steady state conditions as well, and this is done in the present study. The R-G rate equation is also used to characterize radiative recombination due to the similarity o f the R-G rate equation to the radiative recombination rate equation, as given by [49] R-raj = copl(nf - np), (2.5.9) where R ^ is the radiative recombination rate and copt is the optical capture rate. Since the nonradiative and radiative lifetimes are comparable, copf is comparable to the reciprocal of the denominator o f the RG rate given by Equation 2.5.8, and justifies the use o f the R-G rate equation for the radiative recombination mechanism. The final step in this formalism is the choosing o f the recombination lifetimes that will be used in the RG rate. According to Mathiessen’s rule, the effective recombination lifetime r nceff due to both the radiative and nonradiative events is given by [45] 1 where and 1 1 (2.5.10) are the nonradiative and radiative time constants respectively. Using the previously quoted values for these constants leads to Tnc tff =5x10"* s . Based Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 on this result as well as the values used by Ashour [24], the maximum recombination lifetime constants chosen for this study for both electrons and holes are r„ = r = 10”®s . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 CHAPTER 3 TRANSPORT PROPERTIES 3.1 INTRODUCTION The two models used to characterize transport in semiconductors are the semiclassical model and the Boltzmann transport equation. The semiclassical model establishes the distribution o f electrons in phase space as the Fermi-Dirac distribution, and provides the equations o f motion that govern the evolution o f the electron’s position and momentum in response to electric and magnetic fields. Section 3.2 presents the three rules o f the semiclassical model, provides the basis for the equations o f motion, and discusses the limitations to the semiclassical model. The Boltzmann transport equation applies the semiclassical equations o f motion to a system of electrons to characterize the evolution o f their distribution over time, from which the electric current is ultimately determined. Section 3.3 explains how the solution o f the Boltzmann transport equation is facilitated by the integration o f velocity moments o f the equation. This results in sets of transport equations that effectively remove the distribution function by expressing the carrier density as a total and the carrier velocity as an average. O ne such set, based on the drift-diffusion approximation, includes the original form o f the six equations that model transport in the photodetector. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 The equilibrium solution is presented in section 3.4 as a guide to understanding device operation as well as a test o f simulation accuracy, with both the Poisson solver and the full transport solver being tested. Semiconductors act as electronic switches because the energy bands are bent at the boundary between the substrate and the metal electrode. The bending bands form a Schottky barrier that limits electron flux into and out o f the semiconductor. The two devices that are simulated are Schottky diodes, which have a single Schottky barrier, and MSMs, which have Schottky barriers at each electrode. The Schottky diode is studied both as an instructional tool and in preparation for the test of the new boundary condition in Chapter 5, where simulated current-voltage characteristic curves will be compared to characteristics obtained from experiment. A t equilibrium, the electron fluxes across the Schottky barrier are equal in magnitude and cancel each other so that there is no current. The test for zero current is one o f the measures o f simulation accuracy. The solution o f the drift-diffusion transport equations under equilibrium conditions requires solving only the Poisson equation in second order form, and enables the statistically-derived carrier density presented in Chapter 2 to be linked through the electrostatic potential to the carrier density in the transport model. Parameter profiles are generated to further test simulation accuracy as well as to provide insight into device operation. The method for simulating the electron-only DC steady state is discussed in section 3.5, while the actual simulation appears in Chapter 4 with the introduction o f the new boundary condition. Current is generated in semiconductors when the opposing electron fluxes across the Schottky barrier become unequal in magnitude. This occurs because the barrier height is fixed relative to electrons that originate on the metal side, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 but can be varied relative to electrons that originate on the semiconductor side by applying bias. The bias lifts or lowers the semiconductor electrostatic potential energy band. Since the carrier density is an exponential function o f electrostatic potential energy, small changes in the electrostatic potential energy lead to large changes in the current magnitude. When bias is applied, there is no longer a common Fermi energy for the metal-semiconductor system. This requires the introduction o f the quasi-Fermi potential energy in order to retain the same equations for the characterization of carrier density as are used under equilibrium conditions. The solution o f the transport equations for the electron-only DC steady state requires solving four o f the six equations in the drift-diffusion set. Since the goal o f the study is to simulate photomixing, the simulation is designed to operate under AC steady state conditions, with the AC signal frequency equal to the light modulation frequency. Therefore, section 3.6 presents the final form of the driftdiffusion transport model, which includes six state variables and six first order partial differential equations. Light produces equal densities of excess holes and electrons, so holes are included in the transport equations, as is both the light generation rate and the thermal recombination-generation rate. The driving term for the time-dependence of the transport model is the light generation rate, with the time-dependence propagating into each state variable through the coupling o f the differential equations. Section 3.7 discusses the complex behavior o f carrier mobility and presents the two models o f carrier mobility used in the present study. The field-dependent mobility model is applied for devices that are longer than 1.0 pm. The field-independent mobility model is more accurate when devices become significantly shorter than 1.0 mm, and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 when the modulation frequency is in the upper microwave range. Since the MSM model is 1.1 pm and the modulation frequencies that are simulated extend into the higher regions of the microwave band, both mobility models are used. The mobilities are calculated using empirical-fit relationships. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 3.2 3.2.1 SEMICLASSICAL MODEL Rules The semiclassical model, which is applied to the electrons in a crystal, is described by Ashcroft and Mermin [38]. This model characterizes the evolution o f the electron’s position r and wave vector A: in the presence o f external electric and magnetic fields, and in the absence o f collisions. Although the semiclassical model makes use o f quantum mechanics, the semiclassical model cannot be considered quantum theory. The semiclassical model uses quantum mechanics in two ways: (1) the Schrodinger equation for a single electron is numerically solved to generate the E-k band structure for a given crystalline material; and (2 ) the velocity o f the electron is the group velocity, the velocity o f the wave packet that characterizes the electron’s position. This requires that the wave packet be significantly wider than the lattice constant a, as shown in Figure 3.2.1. The equations o f motion that describe the behavior o f the electron in applied electric and magnetic fields, however, are classical equations. In order for the fields to be treated as ordinary classical forces, they must vary little over the dimensions o f the wave packet, as shown in the same figure. The model is considered semiclassical rather than classical regarding the equations o f motion because even though the external fields are treated classically, the periodic field from the lattice ions is not treated classically. The periodic potential of the core ions is included in the quantum mechanical derivation o f the E-k band structure through the use o f Bloch’s theorem. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 Lattice constant Spread of wave packet <4 Wavelength of applied field Figure 3.2.1 Schematic view o f the semiclassical model [after Ashcroft and Mermin t38!]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 Based on En{k) from the E-k band structure, in which n is the band index, the semiclassical model characterizes the evolution o f r , k , and n through the following three rules: (1) The band index n remains constant, so that interband transitions are ignored. (2) Two equations o f motion describe the time evolution o f r and k in the presence o f external electric S ( r ,t) and magnetic , . (a) _ fr\ ~ 1 dEa(k) < U * )= r= n ok fields: hk m, (3.2.1) where o { k \ is the (group) velocity o f the electron in band n for a parabolic dispersion relation; and (b) PB( r ,k ,t)= h k = - q \s { r ,t)+ v n[k)x /f(r,f)], (3.2.2) where Fn(r,k,r) is the Lorentz force on the electron, q is the electron charge, and c is the speed o f light. (3) The electron distribution in band n in the infinitesimal volume element die of k - space is given by the Fermi distribution / [ £ . ( * ) ] ^ r = — 1— 4it i— . ' ' '■ exp* Y j [ E n( k ) - E F }^ + 1 where the Fermi energy EF is considered the chemical potential for semiconductors. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.2.3) 88 3.2.2 Basis The basis for the first equation o f motion (3.2.1) is that the velocity o f the electron is the group velocity of the underlying wave packet. This is valid providing there is little variation in energy over all levels appearing in the wave packet, a requirement that is met when the wave packet spread is small compared to the dimensions o f the Brillouin zone (the Brillouin zone contains all o f the ^-states associated with the crystal volume V). The basis for the second equation o f motion (3.2.2) is that for a static electric field this equation guarantees conservation o f energy. If the field S is given by the negative gradient o f a scalar potential W, then the motion o f the electron wave packet should be such that the energy E.\t(o\-<i'rH')] 0-2.4) remains constant, where k refers to the electron wave packet. Testing this by taking the time derivative gives Dn( k \ { h i - q § } , (3.2.5) which vanishes if Hk = q g ; i.e. if the force given by the electric field is as given by the Lorentz force o f Equation 3.2.2. The fact that the Lorentz force contains the magnetic field component does not prevent Equation 3.2.5 from vanishing because the force exerted by the magnetic field is perpendicular to the velocity and so would vanish with Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 the dot product o f the velocity if it were included in Equation 3.2.5. This proves that the Lorentz equation guarantees conservation o f energy. 3.2.3 Limits Ashcroft and Mermin recommend that two criteria be met if the use o f the semiclassical model is to remain valid in a given solid-state system [38]. The first criterion is that interband transitions are forbidden, since the semiclassical model ignores them. Interband transitions can be caused by the electric field that results from the applied voltage, or from illumination. Results reported by Pierret indicate that electric breakdown, interband transitions caused by high applied voltage, do not occur in GaAs unless the applied voltage is above 50 V for a moderately high doping o f ~ 10 16 cm '3(an even higher voltage is required for doping that is lower than this, as occurs in the present study) [43]. In our simulation, interband transitions do not occur due to the electric field because the applied voltage is not more than 10 V. However, since the purpose o f this study is to use light to promote valence band electrons into the conduction band, the interband transition criterion is not met in the presence o f light. Even in the absence o f light, the thermal recombination term also causes interband transitions. A second criterion is that the wavelength X o f the electromagnetic fields be much longer than the lattice constant a; otherwise, the wave packet assumption has no meaning. Since the minimum wavelength of the light used in the present study is 8000 A, this criterion is met. [NOTE: A third, magnetic field criterion is not discussed here since no significant magnetic fields exist in the present study.] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 Because interband transitions do occur in our simulation, the use o f the semiclassical model is potentially limited in its ability to produce accurate results. This study proposes, however, that such a limitation only occurs if one band is modeled. When electron-hole pairs are created or annihilated, if both bands are modeled, then the system can be characterized accurately with the semiclassical model. Since the charges in both the conduction and valence bands are tracked in the simulations o f the present study, the semiclassical model is still expected to be acceptably accurate in characterizing the system. The creation and recombination o f electron-hole pairs updates the charge densities in the conduction and valence bands; otherwise, the mathematical machinery is still a drift-diffusion process. Therefore, in terms o f physical mechanisms, the electron and hole carrier and current densities are expected to evolve in the same manner as would occur in the absence o f interband transitions. Evidence for this last statement can be found by looking ahead to Figure 4.5.3, in which two DC steady state simulations were conducted in the absence o f light. One simulation uses an electron-only solver and does not have a thermal recombination term, so that there are no interband transitions. The second simulation includes electrons and holes and uses the thermal recombination term, so there are interband transitions. In part (a), it is evident that the band diagrams are almost identical, which confirms that the physical mechanisms are essentially the same whether or not interband transitions are modeled. Also note in part (b) that there is a 50 % correction in electron density by including the holes along with the electrons. This confirms what was stated earlier, that by including both the conduction and valence bands, a more accurate solution is obtained. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 33 33.1 DRIFT-DIFFUSION TRANSPORT MODEL Boltzmann Transport Equation The semiclassical model provides the velocity and force equations that characterize the evolution of the position and wave vector o f the electron in a given energy band. Since the physical quantity that is ultimately sought in device simulations is the electric current, a formulation is required that is consistent with the semiclassical model and yet provides current. The Boltzmann transport equation (BTE) fits this requirement because it utilizes the velocity and force as given by the semiclassical model. Furthermore, the statistics used to characterize the carrier densities are ultimately derived from Fermi-Dirac statistics as required in the semiclassical model. The final transport formulation is actually a set o f equations derived from the BTE (see Appendix F) [13],[49]-[52]. The set o f equations used by the present study is derived in part by using the nondegenerate limit of the Fermi distribution function, i.e. a Boltzmann distribution. The BTE characterizes electron dynamics as particles o f a fluid by calculating the distribution function f( r ,D ,t) in a six-dimensional phase space (here the wave vector k is replaced with velocity u through v = h k / m * , and the band index is understood). According to the Liouville theorem, in the absence o f collisions, the distribution o f carriers f ( r ,D ,t ) at point r,u will follow its trajectory in phase space unchanged, so that the total time derivative of the distribution function is zero: — = — + V - / - F + V - / - D = 0. dt dt rJ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.3.1) 92 By using the definition o f velocity v - r and the definition o f force F = m * u , Equation 3.3.1 can be rewritten as df df = + dt dt F = 0. m* + (3.3.2) Collisions have the effect o f scattering carriers into and out of the phase space point r, D . To account for the deviation from zero in the total time derivative d f I dt due to the / dt\ , the final form o f the change in the distribution function caused by collisions BTE becomes df F f + v , / . u- + v „ . / . - = n dt (3.3.3) where the collision term is characterized by an integral that gives the probability that electrons will scatter into a differential phase space volume element centered about point r, v and out o f a volume element centered about point r, O'. A full solution to the BTE with the scattering term is extremely difficult as this form o f the BTE is an integro-differential equation with seven independent variables. Therefore, in practice, a more solvable set of equations is derived from the BTE. This is done by forming progressively higher order velocity moments with the BTE, in which the BTE is multiplied by velocity u and then integrated over 5 . A set o f assumptions is made in order to simplify the resulting moment equations. Various models are derived in this way, with each model characterized by the moments of the BTE that are used as well Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 as by the simplifying assumptions that transform these moments into more easily solved transport equations. 3 3 .2 Drift-Diffusion Approximation In the present study, the drift-diffusion approximation is the basis for the set of transport equations that are used. The drift-diffusion model uses both the zeroth and the first moments o f the BTE, as well as Poisson’s equation. The derivation o f the moments is beyond the scope o f the present work, so only the moment results are presented here. The zeroth velocity moment for electrons is given by [50]-[52] which according to Blotekjaer means that the increase in electron density equals the increase in electron density due to collisions minus the electron outflow [52]. The first velocity moment for electrons is given by [50]-[52] (3.3.5) where the average momentum density (p) and the pressure tensor P are respectively given by (p) = m * n(p) and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.3.6) 94 Blotekjaer physically interprets the first moment by stating that the increase in momentum density equals the increase in momentum density due to the combination of collisions and force density (from electric and magnetic fields), minus the decrease in momentum density due to the combination o f momentum density outflow and electron thermal pressure [52]. After forming the velocity moments with the BTE, a set of simplifying assumptions is made so that the equations can be solved numerically with an acceptably low computational demand [13],[49]-[52]. [NOTE: To keep the explanation generalized so that it applies to both electrons and holes, no subscripts differentiating between electrons and holes are used at this point The carriers are denoted by n for electrons, but can be replaced by p for holes by making the sign positive for the charge q in the Lorentz force.] These assumptions and their results include the following: (1) The distribution function is assumed to be a displaced Maxwellian as given by (3-3.7) This assumption causes the pressure tensor to become diagonal: P:j - nkBT8t]. (2) The collision terms are approximated by relaxation time expressions, with the original collision term in the BTE given by (3.3.8) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 where / 0 is the equilibrium distribution and r is the relaxation time. Applying this approximation to the two moment equations gives J[(3fjd t^ d O = 0 for the zeroth moment and ^ v \ d f l d t \ d v = - n (p )/r- for the first moment, in which ris the momentum relaxation time. (3) The magnetic field strength is assumed to be negligible, so that the Lorentz force is given by F = - q S . Applying these first three assumptions leads to the zeroth moment being given by | 1 = - V ,.« ( 0 ) , (3.3.9) and the first moment being given by dt (4) = _ ^ p ) - q n S - n { v ) V f - ( p ) - V f {nkBT ), (3.3.10) The electron temperature is assumed to be equal to the lattice temperature T so that the last term in Equation 3.3.10 becomes k aT V f n . (5) The average velocity (u) is assumed to change so slowly that the term n(u )V ? -(p) is much smaller than the other terms in Equation 3.3.10 and therefore vanishes. (6) In device modeling, there is at least an order o f magnitude difference between the device and circuit responses, so that the device can be considered quasi-static, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 which allows nd(j>)fdt to vanish. Applying assumptions (4) - (6 ) enables the first moment to be given by o= -^> ■qnS - k BT V f n . (3.3.11) rp 3.3.3 Drift-Diffusion Equations The drift-diffusion equations that result from the above simplifying assumptions constitute two o f the six equations in the drift-diffusion model, and are the basis for the name o f this model. These two equations characterize the electron current density J „ and the hole current density J p . The drift-diffusion equation for electrons is formed from Equation 3.3.11 by multiplying the equation by the electron charge - q and the momentum relaxation time z pJt, dividing through by the electron effective mass mn *, and rearranging terms to give r \ p.« - q n ( v H) = qn \ m n * S +q J r k BT zP^>- \ B V «, (3.3.12) V m n* J where now the terms are specific to electrons as denoted by the subscripts (the gradient operator is understood to be for position r ). Current density is defined as the flow o f positive charge, even when the charge carriers are electrons, so J„ = -qn(p„ ). Using this definition, the final form o f the electron drift-diffusion equation is given by J n —Qn Mi, & + q D„ V n , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.3.13) 97 where the electron mobility //„ and the electron diffusion constant Dn are respectively given by and (3.3.14) W ith the definition for hole current density given by J p —qp(up) (where combined velocity for light and heavy holes), the hole drift-diffusion equation is given by (where an equation analogous to Equation 3.3.11 is multiplied by + q ) J P = q p ttPs - q D p v p , (3.3.15) in which the combined hole mobility fi p and the combined electron diffusion constant D p are respectively given by and (3.3.16) with m p * representing the average effective mass for light and heavy holes. The first term on the RHS o f Equations 3.3.13 and 3.3.15 is called the drift term because it represents the component o f current density that is due to carrier drift, which is caused by the electric field S . The second term on the RHS o f these two equations is called the diffusion term because it represents the component o f current density that is due to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 diffusion, which is caused by the concentration gradient V/i (V p). Note that the carriers diffuse in the direction that is opposite to the concentration gradient. 3 3 .4 Current Continuity Equations The current continuity equations constitute two more of the six equations in the drift-diffusion transport model. Current continuity means that the change in charge density per unit time within a differential volume element must be accounted for by the net charge flux through the volume element in combination with charge density changes per unit time due to sources or sinks within that volume element. The simplified form o f the zeroth velocity moment given by Equation 3.3.9 is called the particle continuity equation. To convert this into the Boltzmann electron current continuity equation, the RHS o f Equation 3.3.9 is multiplied and divided by the electron charge - q , giving dt q (3.3.17) [A positive divergence corresponds to a net positive charge moving out o f the volume. This occurs when there is a net flow o f electrons into the volume, which accounts for the positive change in electron density within that volume.] The companion Boltzmann hole current continuity equation is given by (3.3.18) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 These last two equations do not account explicitly for charge sources as occurs with generation nor for charge sinks as occurs with recombination. Therefore, the rates of change in carrier density due to illumination by light GL(t) and thermal recombinationgeneration RG are added phenomenologically [13],[50] to the Boltzmann current continuity equations, giving at (3.3.19) q for electrons [see Equation 2.4.26 for GL(t) rate], and &- = ~ V - J , + G L(t) + KG at (3.3.20) q for holes (see Equation 2.5.7 for RG rate). 3 3 .5 Poisson’s Equation The two drift-diffusion equations and the two current continuity equations represent four equations with five unknown state variables: electron n and hole p densities, electron J„ and hole J p current densities, and electric field S . Therefore, a fifth equation is required. The fifth equation is derived from Maxwell’s equations, specifically through Gauss’s law and Faraday’s law, as given respectively by [13],[49],[50] V -D = p and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.3.21) 100 where p is the total charge density, B is the magnetic induction, and D is the electric displacement vector. The electric displacement D is given by D=eS, (3.3.22) in which the permittivity e , which is formally a tensor, is treated as a scalar in the present study because the substrate is assumed to be homogeneous regarding its dielectric properties [49]. Substituting for the electric displacement vector D from Equation 3.3.22 into Gauss’ law and rearranging gives the fifth equation as V - 5 = —. e (3.3.23) Since the boundary value o f the electrostatic potential *F is readily derived from the applied voltage, is designated as the sixth state variable. To derive the necessary sixth equation, the quasi-static assumption that was applied earlier is invoked again, so that Faraday’s law can be given by V x | = 0 . This allowsthe electric field to be derived from a scalar potential alone to provide the sixth equation, or [13],[49] g = - V 'F . 3 3.6 (33.24) One-Dimensional Transport Equations in n and p The six transport equations derived above are adjusted to reflect one-dimensional transport (along the x-axis). The trench electrode formation o f the MSM used in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 present study causes the electric field that propels the carriers to act in one-dimension. The transport equations are still in terms o f the carrier densities n and p, although this will later be changed for numerical reasons. The six equations, without the explicit xdependence notation put in yet, are listed in Table 3.3.1 along with the name o f each equation. T ab le 3.1.1 Transport equations in n and p. Equation Name and Number Equation Definition o f electric field 3.3.25 Poisson’s equation 3.3.26 Electron drift-diffusion equation 3.3.27 Hole drift-diffusion equation 3.3.28 Electron current continuity equation 3.3.29 Hole current continuity equation 3.3.30 dn _ 1 a /, + G l + RG dt q dx dp _ dt 1&p + G l + RG q dx Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 3.4 3.4.1 EQUILIBRIUM SOLUTION Physics o f the Schottky Barrier The carrier density formulas as given by Equations 2.3.14 and 2.3.15 indicate that the carrier density is a function o f the difference in energy between the Fermi energy E F and the intrinsic Fermi energy £ ,. According to these two equations, there is no explicit dependence o f carrier density on position in the semiconductor. With these equations in their present form, the only way to change the separation between E F and £, is to alter the doping density. However, the independence o f carrier density from position remains true only for semiconductor substrates that are kept separated from metals and away from external electric fields (or the applied potentials that cause electric fields). Electric fields cause the separation between E F and E, to vary with position, so carrier density becomes dependent on position. The right side o f part (a) o f Figure 3.4.1 shows the energy configuration for an isolated semiconductor (GaAs) prior to making physical contact with the metal to the left Note that throughout the semiconductor length, the various energy levels or bands remain constant as a function o f position x. To be useful in electrical circuits, however, the semiconductor must be joined to metal electrodes. After the semiconductor is brought into physical contact with the metal, as shown in part (b) of the figure, the energy bands (except for the Fermi energy E F) become functions of position x. This phenomenon is called band bending, and it creates the Schottky barrier on which the unique abilities o f semiconductor photodetectors depend. Band bending is shown graphically by diagrams such as Figure 3.4.1 that depict energy £ versus position x, and these diagrams called E-x plots. This E-x diagram illustrates the Schottky barrier Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 (a) Semiconductor E0 = 5.07 Ee = 1.00 E, = 0.33 -Fm boundary Ev s • 0.42 (b) Neutral bulk region Schottky barrier i-T ------1 En * 4.37 \z i 1 1 ^ F z ,i I Ec = 0.30 c,Fs t 1f £ ,*-0 .3 7 Charge-depletion region Figure 3.4.1 £-x diagrams illustrating the formation o f the Schottky barrier: (a) isolated substances just prior to physical contact; and (b) after band bending is completed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 for the semiconductor material and doping used in the present study, and assumes a device length that is greater than the Debye screening length. In order to adapt the carrier transport equations to the metal-semiconductor system, it is necessary to develop a physical model o f the Schottky barrier. The first step in constructing a model o f the Schottky barrier is to choose a physically meaningful reference point. Fortunately, there is a point for which the metal and semiconductor have the same energy, and this point occurs at the boundary between them, which is at x = 0 in Figure 3.4.1. When the metal and semiconductor are placed into physical contact, the vacuum energy levels £ 0 o f each substance become continuous at the boundary, and the common value o f £ 0 at the boundary becomes the reference point [54]. The vacuum energy level m ust be common to the metal and semiconductor at their boundary or a violation of conservation of energy would occur. For example, if the semiconductor vacuum level were at a higher energy than the metal vacuum level (at the boundary), then an electron moving freely from the semiconductor into the metal would effectively gain energy even though no work had been done on it while crossing the boundary. Subsequent ejection of the electron on the metal side by absorption of energy A£ would leave the electron with more kinetic energy than ejection on the semiconductor side by absorption o f the same energy A£ . Since this hypothesized example violates energy conservation, the vacuum energy levels must be continuous at the boundary. In the second step o f constructing the model o f the Schottky barrier, the separation between the energy levels prior to band bending is established, as pictured in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 part (a) o f Figure 3.4.1. The metal work function determines the separation between the vacuum energy E0 and the metal Fermi energy EFm . The metal work function 4>m used in the present work is a generalized value that is chosen to reflect experimental Schottky barrier heights, which leads to choosing <Pm = 5.07 eV [18]. For convenience, the zero for energy is chosen as the metal Fermi energy E Fm, so the vacuum energy is E0 = 5.07 e V . For GaAs, the semiconductor electron affinity x —4-07 eV determines the location o f the semiconductor conduction band edge Ec relative to the vacuum level E0 [35]. The conduction band edge is below the vacuum energy by 0 m- z * 80 Ec = 1.00 e V . After Ec is located, the valence band edge E r is positioned below Ec by the band gap energy Eg = 1.42 eV , giving E r = -0.42 e V . The semiconductor intrinsic Fermi energy E, is positioned below E c by the difference in these two energies AEcj = 0.67 e V , giving E, = 0.33 e V . Likewise, the semiconductor Fermi energy E Fi is located below Ec by the difference between these two energies AEcFs = 0.30 eV , so that EFs = 0.70 e V . After the metal is placed in contact with the semiconductor, electrons begin to exchange between the metal and semiconductor. The alignment o f the vacuum energy levels reveals that when the metal is first placed in contact with the semiconductor, the Fermi energies o f the metal and the semiconductor are not equal. As shown in part (a) o f Figure 3.4.1, the semiconductor Fermi energy is greater than the metal Fermi energy E Fm. Therefore, there is a net flow o f electrons from the semiconductor into the metal. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 This lowers the Fermi energy o f the semiconductor and raises the Fermi energy o f the metal. The uneven electron exchange continues until the new Fermi energies in the metal and semiconductor are equal. At this point, electron exchange proceeds at equal and constant rates. The entire metal-semiconductor system now has a common Fermi energy E f that is constant as a function o f position x, as can be seen in part (b) o f the figure. Even though the Fermi energy o f the metal was raised in this process, the metal work function is unchanged since the crystalline structure o f the metal is essentially unchanged. Therefore, the vacuum energy on the metal side is still 5.07 eV above the common Fermi energy. By choosing the common Fermi energy E F as energy zero, the metal energy level diagram remains identical to its previous configuration prior to band bending, as can be seen in the figure. Effectively, the entire set of bands has been shifted downward to realign the common Fermi energy EF with the former metal Fermi energy ^Fm • A direct result o f the establishment o f the common Fermi energy E F is the creation of an internal electric field S mt, as shown in part (b) o f Figure 3.4.1. Due to the net gain in electrons by the metal from the semiconductor region adjacent to the metal, negative charge appears on the surface o f the metal and positive charge appears in the semiconductor region next to the metal. The excess electrons remain on the surface o f the metal because metals cannot support the internal electric field that would result if the excess charge resided in the interior o f the metal. In the semiconductor, the positive charge is spread throughout what is now called a charge-depletion region, because the positive donor species are immobile core ions that have become uncovered by the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 electrons that formerly balanced the charge o f those core ions. The charge-depletion region extends as far into the semiconductor as necessary to transfer enough electrons to equalize the two Fermi energies. An internal electric field S mt is generated by the charge separation between the negative metal surface and the positive charge-depletion region o f the semiconductor. The combination o f the two charged regions is much like a parallel plate capacitor, except that the thin positive plate o f the capacitor has been replaced by a thick or extended region o f charge in the semiconductor. As with a capacitor, the internal electric field exists only between the two charged regions. The remainder o f the semiconductor bulk has zero internal electric field because the positive charge-depletion region screens the negative metal surface. By tracing the path o f an electron through the charge-depletion region from the metal-semiconductor boundary to the neutral semiconductor bulk, the manner in which the conduction band edge E c bends can be understood, as shown in part (b) o f Figure 3.4.1. As the electron is moved away from the metal, its electrostatic potential energy drops because the electron is moving in the direction that the field would push it, and the conduction band edge Ee that represents the electron potential energy is lowered. Since the field strength is diminishing due to screening, the curvature o f the conduction band becomes less steep with increasing distance from the boundary. This is what causes the conduction band to have a concave upward curvature with the steepest portion at the boundary. When the electron reaches the end o f the charge-depletion region, the field is completely screened. There is no further lowering o f the electron potential energy as the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 electron continues into the neutral bulk region, and the conduction band remains flat throughout the neutral bulk. In the third step o f constructing the model o f the Schottky barrier, the positioning o f the energy levels in the neutral bulk region is determined. The position of the conduction band edge Ec in the neutral bulk region is determined by the energy difference AEc Fs relative to the common Fermi energy EF. The neutral bulk region o f the semiconductor behaves as if the semiconductor were still isolated from the metal, so the separation between the conduction band edge Ec and the common Fermi energy EF must equal the separation between Ec and the Fermi energy E Fs for the isolated semiconductor, i.e. to AEcjrs. Therefore, Ee —0.30 eV in the neutral bulk region. The positioning o f the remaining two energy levels can be determined from the location of the conduction band energy Ec. Since the bandgap energy depends on the crystal structure, which is not changed significantly anywhere in the semiconductor, the valence band edge £ , is everywhere parallel to the conduction band edge £ e. Also, since the intrinsic Fermi energy E, is defined as being at a constant energy separation AEcj from the conduction band edge Ec, the intrinsic Fermi energy is likewise parallel to the conduction band edge throughout the semiconductor. Therefore, in the neutral bulk region, Er = —1.12 eV and £, = -0.37 e V . The bending o f the conduction band creates a barrier to electron flow from the semiconductor into the metal, called the Schottky barrier, which is pictured in part (b) o f Figure 3.4.1. The Schottky barrier also blocks the flow of electrons from the metal into Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 the semiconductor. Physically, the Schottky barrier is the electrostatic repulsion exerted by the negatively charged metal surface on conduction band electrons that approach the surface from either side o f the boundary. The cross-hatched regions in each material that are shown in the figure represent essentially filled electronic states. The mobile electrons on the metal side o f the boundary are found in the energy levels above the Fermi energy E F. The height o f the Schottky barrier <Pbm relative to electrons that originate in the metal is in theory equal to the difference between the metal work function and the semiconductor electron affinity, or <Pbm = 0 bs relative to the electrons - ^ = 1.00 e V . The height o f the barrier in the semiconductor is given by <&bs = &bm - AEe Fs = 0.70 e V . As will be seen in the next section, the height o f the Schottky barrier <Pbs on the semiconductor side can be changed by applying bias, which is the basis for the operation o f semiconductor devices. Finally, to complete the band bending configuration, the energy values of the three parallel bands £ c, £ ,, and £„ in the charge-depletion region are sketched approximately in Figure 3.4.1 so that these bands reflect the decreasing concave curvature explained previously. The calculation of the exact energy values for these three bands in the charge-depletion region awaits the use o f the transport solver. 3.4.2 Schottky Barrier in Equilibrium In order to understand current in a biased device that has a Schottky barrier, it is first necessary to understand carrier fluxes at the Schottky barrier under equilibrium conditions. At equilibrium, no current exists because the carrier fluxes in the - x - and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 + x -directions are equal in magnitude and cancel each other. Under bias, current results because the carrier fluxes are changed from their equilibrium values such there is a net carrier flux through the device. Devices that have a single Schottky barrier are called Schottky diodes, and devices with back-to-back Schottky barriers are called MSMs. Even though the photodetector modeled in the present study is an MSM, Schottky diodes are also characterized because the test o f the new boundary condition developed in this study is performed with Schottky diodes. The new boundary condition is tested by comparison to experimental current-voltage (I-V) curves, and these are typically obtained from experiments with Schottky diodes since Schottky diodes can be forward biased, while MSMs effectively operate only in reverse bias. A Schottky diode has a Schottky contact at one end and an Ohmic contact at the other end. Ohmic contacts are created by special doping o f the semiconductor in the region adjacent to the metal, such that there is effectively no barrier to electron flux between the metal and the semiconductor (see section 4.4 for details). Figure 3.4.2 presents an E-x diagram that illustrates a Schottky diode under equilibrium conditions, with the Schottky barrier at x = 0 and the Ohmic contact at x = LS. The figure includes a physical model o f the Schottky diode beneath the E-x diagram. The electrons in the E-x diagram are approximated as a Boltzmann distribution such that the electron density decreases exponentially with increasing energy (plotted vertically), and are not plotted as a function of position. Although electrons move in both the + x - and -x-directions in the semiconductor and in the metal, only the electrons that move toward the Schottky barrier are involved in the boundary flux, so only these are shown in the figure. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I ll Boundary (Schottky barrier) Boundary (Ohmic) Metal Semiconductor Metal 0 net e 'flu x £• • 2 *2 * 2 *2 *» •2*2#2 * 2 » 2 ^ i t& m ’/nt C harge depletion region % x =0 x = Ls Figure 3.4.2 Generalized E-x diagram and physical model of a Schottky diode at equilibrium. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 electrons on the semiconductor side whose energy is below &hs are blocked from entering the metal, limiting the electron flux out o f the semiconductor. The electrons on the metal side whose energy is below <Pbm are blocked from entering the semiconductor, limiting the electron flux into the semiconductor. Since the electron flux into and out o f the diode at the Schottky boundary cancel each other for equilibrium, the net flux at the boundary is zero and there is no current in the device. MSMs have a Schottky contact at each end. Figure 3.4.3 illustrates the E-x diagram and physical model o f a symmetric MSM in equilibrium. Again the net flux at each Schottky boundary is zero, so there in no current in the device. 3 .43 Boltzmann Distribution Derived from Transport Equations In the formulation o f the drift-diffusion transport model from the BTE, the distribution function was approximated as a shifted Boltzmann distribution. Therefore, the solution of the drift-diffusion transport model at equilibrium should lead to a distribution that is Boltzmann in form, if not shifted. The determination o f the carrier distribution from the transport equations is important for four reasons: (1) It confirms that at equilibrium the drift-diffusion model is consistent with the semiclassical model regarding carrier density. (2) It provides a mathematical link between the electrostatic potential inherent in the transport equations and the intrinsic Fermi energy £, inherent in the statistical derivation o f carrier density from the Feimi-Dirac distribution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 113 Boundary (Schottky barrier) Boundary (Schottky barrier) Metal Semiconductor »4 »4 » « 4 « kVel J |! j v Metal m« 0 net e ' flux f :: <!• • ' ( i«Se •i* zv T J \ 0 net e'flux t i i t fce>i^^S>S»S«8 S 2 > ! > N i.^ Ei — + — + - - i a . - - Y - 1 "»**- ■■ -V - -B 'z s m — + + t * Charge depletion region I x =0 — * Charge depletion region x = L« Figure 3.43 Generalized £-x diagram and physical model of an MSM at equilibrium. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 114 (3) An added benefit to this mathematical link is that the carrier density can be replaced as a state variable by the potential term. This greatly aids the numerical treatment because the potential energy varies much more slowly than the carrier density, which varies exponentially with position. (4) It proves that the energy bands are a function o f position. To derive the carrier density distribution at equilibrium, two assumptions are made regarding equilibrium: (1) the current densities J n and J p are zero, and (2) the change in carrier density n (and p) as a function o f time is zero. From the first assumption, the divergence o f the current densities in the two current continuity equations (3.3.29 and 3.3.30) can be set to zero. Since there is no illumination, GL(t) is zero, which along with the second assumption makes RG zero, so that all the terms in the current continuity equations vanish at equilibrium. From the first assumption, the driftdiffusion equations 3.3.27 and 3.3.28 respectively become and (3.4.1) The solutions to these equations are given by /i(x) = /i(0)exp (3.4.2) P(x) = p(0)cxp (3.4.3) for electrons and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 for holes, in which x = 0 is the position o f the Schottky boundary. Note that the carrier densities given by Equations 3.4.2 and 3.4.3 are functions o f position through the electrostatic potential 5F(x) and have the familiar Boltzmann form of nondegenerate semiconductors that were previously derived statistically. The solutions given by Equations 3.4.2 and 3.4.3 are used to link the electrostatic potential 5PXx) to the statistically-derived carrier density formulas given by Equations 2.3.14 and 2.3.15. It was previously demonstrated through physical arguments that the three parallel energy bands £ c, £,., and £„ bend in accordance with the electrostatic potential energy; i.e., that these three bands are parallel to the electrostatic potential energy. Since the transport equations define - q V { x ) as the electrostatic potential energy, -q*F (x) can be expressed in terms o f any one o f the parallel bands to within an additive constant representing the initial value -q'F iO ) . It is convenient to choose £, due to the use o f £, in the Equations 2.3.14 and 2.3.15 for carrier density. Therefore, the electron electrostatic potential energy - g!£(x) is expressed as -q*F(x) = £ , ( x ) - £ f , (3.4.4) where the Fermi energy E F is constant at equilibrium as shown earlier. The electrostatic potential 5£(x) is derived from the electrostatic potential energy -q 'F {x) by dividing —q*F{x) by the electron charge - q . The reference energy for the metal-semiconductor system, as provided through Equation 3.4.4, is the constant Fermi energy E F. Since the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 E-x diagrams are plotted with the system Fermi energy E F set to zero, Equation 3.4.4 can be simplified to —q'Pi.x) = E ,(x ), so that V{x) = —E ,( x ) /q . N ow the carrier densities can be expressed in terms o f the electrostatic potential E(x) o f the system and still make use o f the statistically-derived semiconductor constants such as the effective density o f states N c and the Schottky barrier height on the metal side (iV ,), the intrinsic carrier density ni , . This is done by first substituting the definition for electrostatic potential given by Equation 3.4.4 into the carrier density Equations 2.3.14 and 2.3.15 to express the carrier densities as (3.4.5) for electrons and (3.4.6) for holes. To complete the link between the statistical and transport formulations, the electrostatic potential at the Schottky boundary SPfO) must be determined in terms o f the statistically derived parameters. The statistically-derived equation (2.3.8) for electron density at x = 0 gives n(0 ) = N e exp< kkb1T / ’ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.4.7) 117 which is then set equal to the new equation (3.4.5) for electron density and solved for the boundary value o f the electron electrostatic potential energy —qf'iO) to give - q f'iO ) = 0 bm - k gT\n N. or (nA 5P(0) = v (3.4.8) j Using the previously quoted values for the statistically-derived parameters for the device used in the present study, Equation 3.4.8 is used to calculate the electrostatic potential energy and potential at the Schottky boundary, giving -q f'iO ) = 0.33 eV or f'iO) = -0.33 V as expected according to Equation 3.4.4. 3.4.4 Numerically Generated Equilibrium Solution At equilibrium, transport is completely characterized by solving Poisson’s equation in its second order differential equation form. As explained in the last subsection, all the terms in the current continuity equations (3.3.29 and 3.3.30) vanish, which reduces the original drift-diffusion equation set from six to four. Since the current densities J„(x) = 0 and J p(x) = 0 for all values o f x at equilibrium, the number o f state variables that remain unknown is also reduced from six to four, leaving n(x), p(x), fix ), and Six) yet to be determined. From the last subsection, it was found that n{x) and p(x) could be expressed in terms o f the electrostatic potential fix ), which further reduces the number o f state variables needed in the transport solver to two, i.e. ¥(x) and Six). It was also explained that the transport solver behaves better when the electrostatic potential is used in place o f carrier density, so this substitution should be performed anyway. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 Therefore, the electron and hole densities given by Equations 3.4.5 and 3.4.6 are substituted into Poisson’s equation (3.3.26). The number o f equations is also reduced to two since the drift-diffusion equations were solved to derive npF(x)] and p[!F(x)]. Finally, the definition o f electric field g(x) given by Equation 3.3.25 is substituted into Poisson’s equation, which changes Poisson’s equation from a first order to a second order differential equation with the single state variable f f x) as the unknown: a r 2(x) _ q \ dxz e\ r^ (x ) ~ n, exp I k aT = f { " ' ex p p qH x) - N kBr _ d +Na . (3.4.9) Equation 3.4.9 is solved using the numerical method employed in the present study, which is discussed in detail in Chapter 5. Poisson’s equation (3.4.9) in second order form can be solved since there are two known boundary conditions. The boundary condition for ¥f0) is given by Equation 3.4.8, because x = 0 always represents a Schottky contact. The boundary condition for ¥ { L S) is given by Equation 3.4.8 when V{LS) represents a Schottky contact, as occurs when the device being simulated is an MSM. However, an Ohmic boundary condition is required for *F(LS) when a Schottky diode is the device that is being simulated. The Ohmic boundary condition is determined by assuming that the semiconductor material at x = Ls behaves as i f it were in the neutral bulk region so that the electron density n(Ls) is equal to the donor doping density N D (again assuming complete ionization o f the donor species, and that the intrinsic density offers negligible contribution to the electron Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 density relative to the doping density). Substituting N D for n(Ls) in Equation 3.4.5 and solving for the electrostatic potential energy - q'F{Ls) gives (3.4.10) In this subsection, the simulation results for a Schottky diode and two MSM’s are presented to help explain device operation at equilibrium and to test whether the simulator produces the expected results. The Schottky diode represents one o f the four experimental diodes that is used to test the current density boundary condition developed in this study, with this test being performed in Chapter 5. The Schottky diode, called CTH 562, has an n-doped GaAs substrate with N D = 2 .5 x l0 16 cm -3, a barrier height @bm= 0.879 e V , and a current cross-sectional area A}_ = 8.0 pm2. Figure 3.4.4 presents the simulation results associated with this Schottky diode. The MSMs have an unintentionally n-doped GaAs substrate with N D = 3.5xlO 12 cm -3, a current crosssectional area Ay.= 3 5 pm 2, and a barrier height of either <Pim= 0.60eV or < P bm= 1.00 eV . These two barrier heights are studied because GaAs Schottky barriers vary in height over a wide range, from a low o f 0.66 eV reported by Adachi [53] to a high o f 1.027 eV reported by Hjelmgren [18]. Figures 3.4.5 and 3.4.6 present the simulation results for the two MSM’s. By studying the three figures, it is evident that the high doping level o f the Schottky diode relative to the MSM provides an additional benefit regarding the testing o f the simulation. The unintentional doping density o f the MSM’s is too low to produce any noticeable curvature in the bending bands because the depletion Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 Transport Soivsr (a) 1 Poisson Solver Ec Ec Ev Ev > 0 LU 1 2 0.0 0.2 0.4 0.6 1.0 0.8 X (nm ) 15 Electron density (/cm3) Electric field (V/cm) Electron drift (A/cm2) Electron diffusion (A/cm *) Hole density (/cm3) JD (0 O 0) O) O -10 -15 0.0 0.2 0.4 0.6 0.8 1.0 X (nm ) Figure 3.4.4 Equilibrium: numerically generated (a) E-x diagram (using both the Poisson and transport solvers) and (b) device profile o f Schottky diode CTH 562. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 (a) 1 0 i Ul -1 Transport Solvsr • Poisson Solvsr Ec Ec Ev Ev 2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 (b) 10 5 _Q) (Q O CO O O 0 5 Electron density (/cm ^ Electric field (V/cm) Electron drift (A/cm ^ Electron diffusion (A/cm 2) Hole density (/cm3) -10 -15 -20 0.0 0.2 0.4 0.6 0.8 1.0 1.2 x(nm) F igure 3.4.5 Equilibrium: numerically generated (a) E-x diagram (using both the Poisson and transport solvers) and (b) profile o f MSM with 0.6 eV Schottky barriers. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 1.0 0.5 > 3 •s ■m 0.0 LU -0.5 Transport Solvsr - Poisson Solvsr Ec 1.0 ▲ 0.0 0.2 0 .4 0.6 Ev 0.8 1.0 1.2 x(nm) (b) 15 10 a> re o CO o> o 5 0 -5 Hole density (/cm 3) Electric field (V/cm) Hole drift (A/cm2) Hole diffusion (A/cm2) Electron density (/cm3) -10 -15 0.0 0.2 0.4 0.6 0.8 1.0 1.2 x(nm) Figure 3.4.6 Equilibrium: numerically generated (a) E-x diagram (using both the Poisson and transport solvers) and (b) profile o f MSM with 1.0 eV Schottky barriers. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 region extends throughout the semiconductor, which makes the electric field change slowly, as seen in part (a) o f Figures 3.4.5 and 3.4.6. This prevents comparing the E-x diagram results for the low-doped MSM’s to the typical E-x diagrams for Schottky barriers. The numerical solution for the high-doped diode, however, produces the typical band bending results, as can be seen in part (a) of Figure 3.4.4. Both o f the numerical device solvers used in this study are tested under equilibrium conditions. The Poisson solver determines the electrostatic potential F(x) at equilibrium by solving Poisson’s equation (3.4.9). The other five state variables are then calculated in post-processing steps and are used along with 'Fix) as the initial values for the transport solver, which solves all six transport equations for any condition o f bias. The transport solver provides current checks that the Poisson solver is unable to provide. As illustrated in part (a) o f Figures 3.4.4 - 3.4.6, both solvers generate identical equilibrium solutions for the Schottky diode as well as for the two MSMs. Since the total current density at equilibrium is expected to be zero, this serves as the second test and the primary check on the equilibrium solution. The total current density J T is given by [13] (3.4.11) where e d S jd t is the displacement current density and J is the combined particle current density as given by (3.4.12) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 with Equations 3.4.11 and 3.4.12 applying to one-dimension. For all situations o f bias, even time-dependent bias, the total current density J T is constant across the device. This is proven by taking the divergence o f the Maxwell equation that expresses Ampere’s law: V - ( V x / / ) = 0 = V- J + £ ---dt (3.4.13) where the divergence o f the curl on the LHS is identically zero. The constancy o f the total current density J T is used as a simulation check for all situations o f bias, including equilibrium, for which the constant value for J T happens to be zero. At equilibrium, the displacement current density e d S /d t must vanish to satisfy Equation 3.4.13, since the combined particle current density J is zero. [In steady state, the displacement current also vanishes because there are no time-dependent electric fields.] Table 3.4.1 shows that the combined particle current density J, the displacement current density e d S /d t, and the total current density J T for the three devices at equilibrium are all essentially zero as expected (current density values are approximate averages across the device). Table 3.4.1 equilibrium. Current densities for the Schottky diode CTH 562 and the MSMs at Diode CTH 562 Electron (A/cm2) ~ 10' “' Hole (A/cm2) - 10*" Displacement (A/cm2) ~ I 0 *,u Total (A/cm2) - 10*,u MSM (0.60 eV barrier) - 10~7 ~io-* - - 10’7 - 1 0 14 MSM (1.00 eV barrier) - 1 0 18 - 10*3 - - 10'5 - 10*12 Device Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 One o f the major benefits of the physics-based device simulator employed in the present study is that it provides information regarding important physical quantities in the device interior that can lead to a greater understanding o f device operation. Such quantities include the carrier densities, the electric field, and the drift and diffusion components o f the current densities. Part (b) o f Figures 3.4.4 - 3.4.6 presents the magnitudes o f these quantities on a log scale called a device profile (only magnitudes can be given due to the use o f a log plot). The behavior o f these quantities is tracked to learn why the device behaves as it does under specific operating conditions. The analysis o f these quantities also serves as a third check on the simulation, since there are certain expected interactions based on the physics of device operation. [NOTE: The use o f log plots causes quantities that change sign to form a dip in the profile, as seen in Figures 3.4.5 and 3.4.6 for the MSMs. The change in sign results due to symmetry; i.e., MSMs are back-to-back Schottky diodes.] The E-x diagrams and the corresponding device profiles behave as expected for equilibrium, which is now discussed through an examination o f Figures 3.4.4 - 3.4.6. Emphasis is placed on the Schottky diode since this shows the most variation in state variables as a function of position. The electron density and electric field pictured in Figure 3.4.4 for the Schottky diode are examined first. The parallel bands in the region o f the Schottky contact at x = 0 have the concave curvature characteristic o f the charge-depletion region o f Schottky barriers. These bands continue to bend throughout the charge-depletion region, which extends to about 0.3 pm. The electron density rises from a minimum at the Schottky contact to a maximum value at the end o f the charge-depletion region, and then remains constant throughout the neutral bulk region. In the charge-depletion region, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 downward sloping bands are consistent with the increasing electron density, because the difference E F - Et becomes less negative and then more positive, which exponentially increases electron density (see Equation 2.3.14). Since the magnitude o f the electric field is the slope o f the bands, the electric field magnitude should attain its highest value where the bands are steepest, i.e. adjacent to the Schottky boundary. The electric field profile is consistent with this expectation, as the electric field peaks at the Schottky boundary. Only the electron current density o f the Schottky diode in Figure 3.4.4 is examined since the hole current density is negligible, due to the relatively low hole density. Recall that current density consists of two components, a drift component driven by the electric field and a diffusion component driven by the carrier concentration gradient, with each o f these components being the product of the charge density and the driving factor. The drift and diffusion components are. calculated by the transport solver in post-processing steps using the final state variables, with the electron drift current density given by J n.dr= t (3.4.14) and the electron diffusion current density given by J n j f — J n ~ J n .d r (3.4.15) (see Equations 3.3.25 and 3.3.27), with analogous relationships for hole current density. In the Schottky diode, the profile indicates that the magnitudes o f both the drift and diffusion electron current densities rise from a minimum at the Schottky contact to a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 maximum at the end o f the charge-depletion region, and then decrease more gradually throughout the remainder o f the device. In the charge-depletion region, the magnitudes of both driving terms are high, but the electron density is low, so the drift and diffusion electron current densities start out low and increase with increasing electron density. In the neutral region, where the electron density is a constant maximum, the magnitudes o f the two driving terms decrease to a minimum, which brings the drift and diffusion electron current densities back down. The simulation results for the MSMs under equilibrium conditions, pictured in Figures 3.4.5 and 3.4.6, are also as expected. A changing separation between the Fermi energy E F and the intrinsic Fermi energy E, is required to produce a changing charge density. The imperceptible bending o f the bands in the E-x diagrams shown in part (a) of the figures indicates that the charge-depletion region has a fairly constant charge density, which is confirmed by the flat profiles. In conclusion, since the E-x diagrams for both the Schottky diode and the MSMs behave as expected, and since the corresponding device profiles are consistent with the E-x diagrams, the simulation appears at least from this standpoint to produce accurate results. A fourth test o f simulation accuracy at equilibrium is whether the two current density components are equal and opposite in magnitude so that the net current has the expected value o f zero. In the diode and the 0.60 eV barrier MSM, the electron current density dominates the hole current density due to the higher electron density provided by n-doping. Part (b) o f Figures 3.4.4 and 3.4.5 indicates that the equilibrium criterion of equal electron current density components is met. For the diode (Figure 3.4.4), the electron drift component moves in the —x -direction, consistent with the electric field Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 pointing in the - x direction (the negative sign does not appear on the plot). The electron diffusion component moves in the + x -direction, consistent with the electron concentration gradient that also points in the + x -direction (recall that electron current density is in the opposite direction to the electron velocity). Therefore, the two electron current density components are equal and opposite, canceling each other so that no electron current results. In the left half o f the 0.60 eV barrier MSM (Figure 3.4.5), the electric field points in the - x -direction and the concentration gradient points in the + x direction, so the electron current density components are in the same direction as for the diode. Due to symmetry in the MSM, however, all directions are reversed for the right half of the MSM. For the 1.00 eV barrier MSM (Figure 3.4.6), the hole density and resultant hole current density components are dominant over the electron quantities in spite o f the ndoping. The high Schottky barrier creates a much greater energy separation between the metal Fermi energy EFm and the semiconductor Fermi energy E Fs prior to bringing the metal into physical contact with the semiconductor than occurs with the 0.60 eV MSM. This requires more electrons to be drawn from the semiconductor into the metal to establish the common Fermi energy EF subsequent to physical contact. As can be seen in the figure, the semiconductor is almost completely depleted of electrons, and the hole density is elevated over five orders o f magnitude above the intrinsic hole density. This occurs due to the activation o f the thermal recombination-generation term in the current continuity equations, which acts to balance changes in the carrier densities. Recall that the denominator o f the RG rate term of Equation 2.5.7 expresses the difference between Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 the np product and the square o f the intrinsic carrier density rtf. At equilibrium, this difference m ust equal zero, which requires the elevation o f p to balance the decrease in n brought on by the high barrier. In the left half o f the 1.00 eV barrier MSM, the hole drift component is in the -x-direction in response to the electric field, while the hole diffusion component is in the + x -direction in response to the hole density gradient. As with the 0.60 eV barrier MSM, the two components of the dominant current density are equal and opposite, leading to no current. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 3.5 3.5.1 DC STEADY STATE EQUATIONS FOR ELECTRONS Physical Description Using E-x Diagrams When a constant voltage is applied across a semiconductor device, the device operates in a DC steady state characterized by constant current. Current is generated because the applied voltage changes the Schottky barrier height <Pbs on the semiconductor side without changing the barrier height 0 bm on the metal side. This creates a net electron flux across the Schottky barrier, resulting in current through the device. The presence o f the charge-depletion region causes essentially all o f the applied potential difference to drop across the semiconductor portion o f the device, because the charge-depletion region has a higher electrical resistance. Since there is negligible drop in electrostatic potential across the metal portion of the contact, the band structure o f the metal does not change, which essentially leaves the barrier height on the metal side of the Schottky barrier at its equilibrium value. The magnitude o f the current across the Schottky barrier depends on whether the barrier operates in forward or reverse bias, with forward bias characterized by large current and reverse bias characterized by small current. Schottky diodes operate in either complete forward bias or complete reverse bias because they contain only one Schottky contact. MSMs operate with one Schottky contact in forward bias and the other in reverse bias. This section focuses on the electron-only device, so that the effect o f holes can be determined when they are included in the formulation in the next section. To create a forward bias current in the Schottky diode, the voltage at the Schottky barrier at x - 0 is designated the reference voltage and is grounded to give V(Q) = 0 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131 The voltage at the Ohmic contact at x = Ls is m ade negative by setting V(LS) = - V A, which lifts the parallel semiconductor bands Ec, E f , and E v, as shown in part (a) o f Figure 3.5.1. Since these parallel bands change according to the electron electrostatic potential energy, each band is raised by (-q)( -VA) = q V A at the Ohmic contact relative to equilibrium. The peak o f the Schottky barrier, however, remains fixed since the metal at the Schottky contact is held to ground and <Pim does not change. The net result is that the barrier height &bs for electrons in the semiconductor is reduced relative to the equilibrium height (compare part (a) with Figure 3.4.2). This leads to an exponential increase in the electron density within the semiconductor that is able to surmount the barrier and exit the semiconductor. Since the electron flux from the metal into the semiconductor at the Schottky contact at x = 0 remains at its equilibrium value, there is a net electron flux out o f the semiconductor, resulting in. current into the semiconductor. The Schottky diode can also be reverse biased, so that the current is out o f the semiconductor. If the applied voltage at the Ohmic contact is made positive by setting V(LS) = +VA, then the parallel energy bands o f the semiconductor are lowered relative to the peak of the Schottky barrier, as can be seen in part (b) o f Figure 3.5.1. Each o f the three bands drops by (—q){+VA) = -qVA at the Ohmic contact relative to equilibrium. This increases the barrier height 0 Aj for electrons in the semiconductor relative to the equilibrium height, and causes an exponential decrease in the semiconductor electron density that is able to surmount the barrier and exit the semiconductor. Since the electron flux from the metal into the semiconductor at the Schottky contact at x = 0 remains at its equilibrium value, there is a net electron flux into the semiconductor, resulting in current Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ^ ib m #2 °Z°Z°Z*» • 2 *2 * 2 » 2 « Z ^ I Fn t +<?Va Ei x = L, x =0 (b) net e~ flux lbs Z°2*2°2°2*k. Z*Z*Z*Z*2*Z^w 2t2*2e2«2>2>i>, ■Fn Figure 3.S.1 Generalized E-x diagram o f a Schottky diode under conditions o f (a) forward bias and (b) reverse bias. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 out o f the device. The current is low because the Schottky barrier to electrons originating in the metal 0 bm blocks most of these electrons from entering the semiconductor. The MSM is biased by grounding the Schottky contact at x = 0 and applying a voltage at the Schottky contact located at x = Ls . Figure 3.5.2 shows the MSM with a positive voltage V(LS) = +VA applied at the Schottky contact at x = Ls . This lowers the three parallel energy bands at x - L s by (~q)(+VA) = - q V A relative to equilibrium (compare to Figure 3.4.3). This band lowering extends all the way back to x = 0 by an amount that continually decreases, so that at x = 0 the bands are located at their respective equilibrium energies. The result o f this band lowering is that the Schottky contact at x = 0 is placed under reverse bias, while the Schottky contact at x = Ls is placed under forward bias for electrons. However, since current is continuous, the current through the device is limited by the reverse bias contact, causing the MSM to effectively operate under reverse bias. If the contact at x —Ls has a negative voltage applied to it instead o f a positive voltage, then the contact at x - L s becomes reverse biased while the contact at x = 0 is forward biased, and the MSM still operates under an effective reverse bias. 3.5.2 Mathematical Characterization When a semiconductor has a nonzero voltage applied across it, it is no longer in equilibrium. Recall that the common Fermi energy E F is established at equilibrium when the electron flux from the semiconductor into the metal equals the electron flux Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 134 i net e ' flux net e" flux »i ►« » » » A <Pb, m ••••••••• I «*«*»•«•■•«•»*>. y - V .•2*2*2 2*2*7“ ^^SSK «S«Sn<- «^2£2s2s2is2sSs2iSj -Fn X =0 Figure 3.5.2 Generalized £-x diagram o f an MSM under conditions o f bias. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 from the metal into the semiconductor. These two fluxes are o f necessity unequal when there is current in the device, so in steady state the common Fermi energy E F no longer exists for the entire metal-semiconductor system. The Fermi energy only has meaning for the metal, as seen in Figures 3.5.1 and 3.5.2, which show that the Fermi energy o f the metal contact at x = 0 does retain the common Fermi energy E F. The retention o f EF by the metal results because the metal essentially has no voltage difference across it, and behaves as if it were in equilibrium regarding the charge density. Since the Fermi energy E F no longer exists in the semiconductor under bias, the electron carrier density «(x) in the semiconductor can no longer be characterized by Equation 2.3.14 in its present form. It is assumed, however, that if the proper adjustment is made regarding E F , then the equation will retain its general equilibrium form and still accurately characterize electron density. The adjustment is to replace E F in the carrier density formula with a new potential energy term E ^ (x) called the electron quasi-Fermi potential energy. The electron quasi-Fermi potential energy is defined as that potential energy that when used in place o f the Fermi energy E F gives the actual carrier density that exists in the device as a function of position x. The electron quasi-Fermi potential energy can be thought o f as a correction factor, so that the general form o f the statistical representation o f carrier density for the equilibrium case can be retained for the non equilibrium case. For the equilibrium case, the electron quasi-Fermi potential energy is necessarily zero. For the non-equilibrium case, the electron carrier density n(x) is given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 n(x) = n, exp (3.5.1) The electron quasi-Fermi potential energy E ^ i x ) is shown in Figures 3.5.1 and 3.5.2. In order to reestablish the link between this statistically-derived carrier density equation (3.5.1) and the transport equations that use the electrostatic potential JP(x), Equation 3.5.1 is altered by adding and subtracting the Fermi energy E F in the exponent, leading to (3.5.2) where 0 „ (x) is the electron quasi-Fermi potential and is defined through (3.5.3) Now that bias has been applied to the device, the electron current density J n(x) as given by the drift-diffusion equation (3.3.27) is no longer zero. For the electron-only device, there are four unknowns or state variables, including the electrostatic potential 'F(x), the electric field S(x) , the electron quasi-Fermi potential 0„ (x), and the electron current density J n( x ) . Four equations are required in order to determine the state variables, and include the following: (1) the definition of the electric field S(x) as the gradient o f the electrostatic potential, given by Equation 3.3.25; (2) Poisson’s equation (3.3.26) for the electron-only device in its first-order form, but Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 with the electron density n(pc) replaced by the electrostatic potential *F(x) as the state variable, or (3) the electron drift-diffusion equation for the electron-only device in DC steady state, as derived from the equilibrium form given by Equation 3.3.27, into which is substituted the electron density for the non-equilibrium case as given by Equation 3.5.2, to give (3-5.5) [this equation is solved for the derivative o f the electron quasi-Fermi potential energy in the transport solver]; and (4) the electron current continuity equation for the electron-only device in DC steady state (with no light, so that the light generation rate and the thermal RG rate are zero), as derived from the equilibrium form given by Equation 3.3.29, into which is substituted the electron density for the non-equilibrium case as given by Equation 3.5.2, and solved for the derivative o f the electron current density to give Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 a /„ 0 ) (3.5.6) 8x The electrostatic potential ^ (x ) at each o f the two semiconductor boundaries provide two o f the four boundary conditions necessary to solve the four transport equations. As demonstrated in Figures 3.5.1 and 3.5.2, the intrinsic Fermi energy E,(x) that represents the electrostatic potential energy —q ¥ ( x ) is shifted by the applied voltage by an amount equal to +qVA. Since the applied voltage at x = 0 is VA = 0 , the boundary value for the electrostatic potential !F(0) is the equilibrium value as given by Equation 3.4.8. For the MSM, the electrostatic potential for the contact at x = Ls is determined by adding +qVA to the equilibrium value o f - q ¥ { L s) defined by Equation 3.4.8 (for a symmetric MSM), giving - q ¥ { L , ) = 0 im- k BT In or (3.5.7) where the sign in the first equation is —for a positive applied voltage and ■+• for a negative applied voltage (reversed for second equation). For Schottky diodes, the electrostatic potential ¥ ( L S) for the Ohmic contact is determined by adding +qVA to the equilibrium value of - q ¥ ( L s) defined by Equation 3.4.10, giving ¥ ( L S) = k T (N \ In —2- ± V it <f I ”, R e p r o d u c e d with permission of the copyright owner. Further reproduction prohibited without permission. (3.5.8) 139 where the sign in the equation to the left is —for a positive applied voltage and + for a negative applied voltage (reversed for equation to the right). For the MSM, the third and fourth boundary conditions are the electron current densities at each Schottky contact, J„(0) and J n (Ls) , with the equation for electron current density at the boundary yet to be derived as the new boundary condition. For the Schottky diode, the third boundary condition is the electron current density J n(0) at the Schottky contact at x = 0 , while the fourth boundary condition is the electron quasi-Fermi potential <Pn(Ls) at the Ohmic contact. The current density boundary condition is the topic of much o f Chapter 4, and the quasi-Fermi potential boundary condition is presented in Section 4.4 where the Ohmic limit is discussed. The numerical solution to the electron-only device in DC steady state awaits the development of both o f these boundary conditions, and will be presented in Section 4.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140 3.6 INCLUSION OF LIGHT AND TIME IN THE TRANSPORT EQUATIONS 3.6.1 Inclusion o f Holes The GaAs modeled in this study is n-doped, so the density o f holes is negligible for the semiconductor in its isolated state. However, in high barrier MSMs as well as in all devices under illumination, holes become significant. Under illumination, the photodetector generates just as many excess holes as electrons, both o f which have a potentially higher density than the electrons that exist prior to light exposure. Therefore, if photocurrent is to be accurately characterized, holes must be included in the transport equations. This requires the insertion o f a hole quasi-Fermi potential energy Efp as the correction factor for the hole density given by Equation 2.3.15, analogous to what was done for electron density. For the non-equilibrium case, the hole carrier density p(x) is given by (3.6.1) In order to reestablish the link between this statistically-derived carrier density equation (3.6.1) and the transport equations that use the electrostatic potential *F(x) , Equation 3.6.1 is reworked by adding and subtracting the Fermi energy E F in the exponent, leading to (3.6.2) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 where 4>p(x) is the hole quasi-Fermi potential and is defined through the hole quasiFermi potential energy —q<Pp{x) by - q4>p(x) = EJp( x ) - E F . (3.6.3) Since the hole photocurrent is significant under conditions o f illumination, the hole current density J p(x) must be characterized, which requires the inclusion o f both the hole drift-diffusion equation and the hole current continuity equation in the final transport set. The hole drift-difiusion equation is derived from the equilibrium form given by Equation 3.3.28, into which is substituted the hole density p(x) for the non equilibrium case as given by Equation 3.6.2. Then Equation 3.3.28 is reworked to give = - qn, e x p j ^ - [ - 5P(x) + 0 p(x )] |^ p ^ x) - Mpk BT j - n , e x p j - ^ [ - ¥{x) + <t>p (x)]J = - W pn, exp | - ^ r [ - ^ (x ) + 0 p(x)]| • (3-6.4) The complete form o f the hole current continuity equation is derived from Equation 3.3.30, and is presented in the last subsection along with the complete form of the companion electron current continuity equation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 142 3.6.2 Effects o f Light Illumination by light affects the entire transport process through the light generation rate GL(t) as given by Equation 2.4.26. The light generation rate is a sinusoidal function o f time that is modeled to have the same value at all positions x in the substrate. This rate is independent o f the six state variables, and drives the changes in these variables as well as the changes in the thermal recombination-generation (RG) rate RG(x,t) given by Equation 2.5.7. The effects of the light generation rate GL(t ) begin in the current continuity equations (3.3.29 and 3.3.30), where GL(t) appears. Increases in GL(t) elevate the carrier densities n(x) andp(x), which in turn increase the np product in the denominator o f the RG rate. As a result, the np product becomes greater than the square o f the intrinsic carrier density nf, and the RG rate becomes negative. This favors the recombination path of the RG process whereby excess carriers created by generation are removed. Decreases in the light generation rate GL(j) reduce the carrier densities n(x) and p(x), which in turn decreases the np product in the denominator o f the RG rate, slowing down the removal of excess carriers. Since the continuity equations are coupled to the drift-diffusion equations (3.3.27 and 3.3.28) through the carrier densities n(x) andp(x), a change in light generation causes changes in the current densities J„(x) and J p(x) in the drift-diffusion equations. The drift-diffusion equations are coupled back to the current continuity equations through the current densities J„(x) and J p( x ) . Therefore, changes in J„(x) and J p(x) originally initiated by a change in light generation ultimately feed back to the current continuity equations from the drift-diffusion equations, acting as a second alteration o f the carrier Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 143 densities n(pc) and p(x) in the current continuity equations. Since the current continuity equations are also coupled to Poisson’s equation (3.3.26) through the carrier densities n(x) and p(x), a change in light generation causes a change in the electric field as well. Finally, the coupling o f Poisson’s equation to the definition o f electric field equation (3.3.25) produces a change in the electrostatic potential 'F(x). The change in electrostatic potential *F(x) feeds back to the drift-diffusion equations due to coupling through !P(x), and the current densities J„{x) and J p(x) are affected a second time. This complicated, dynamic interconnection between the six state variables is the reason that a numerical solver is required to solve the transport equations. 3 .6 3 Final Form o f the Time-Dependent Transport Equations To accurately characterize the photocurrent generated under conditions o f illumination by photomixed light, the time-dependence inherent in the transport model is included in the final equation set. The time dependence appears explicitly in the current continuity equations through the light generation term GL(t) as defined by Equation 2.4.26. Since the oscillating light intensity implicitly effects all o f the state variables, this implicit time-dependence is incorporated by making all o f the state variables a function o f time t as well as o f position x. The final list o f six state variables is presented in Table 3.6.1. The carrier densities n(x,t) and p(x,/) are now formally replaced by the quasiFermi potentials <Pn(x,t) and <Pp(x,t) as state variables. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 144 Table 3.6.1 Final state variables. Sym bol N am e Derivative Definition n x ,o Electrostatic potential Definition o f electric field S(x,t) Electric Field Poisson’s equation &n(X,t) Electron quasi-Fermi potential Electron drift-diffusion equation ®P(XJ) Hole quasi-Fermi potential Hole drift-diffusion equation J n(x,t) Electron current density Electron current continuity equation J p (x ,t) Hole current density Hole current continuity equation The final set o f six transport equations is presented in Table 3.6.2. Each transport equation is solved for the first order partial derivative present in that equation because this is the form required by the numerical solver used to characterize the state variables. In this derivative form, the RHS o f each equation can be treated as the derivative definition o f the LHS, as reflected in Table 3.6.1. The source o f each derivative equation is now summarized. The definition o f electric field (Equation 3.6.5) is as presented previously by Equation 3.3.25. Poisson’s equation (Equation 3.6.6) is in the first order form presented by Equation 3.5.4 with the necessary terms for holes added. The electron and hole drift-diffusion equations (3.6.7 and 3.6.8) come from Equations 3.5.5 and 3.6.4, respectively. 3.6.10) Finally, the electron and hole current continuity equations (3.6.9 and come from Equations 3.3.29 and 3.3.30, respectively. The thermal recombination-generation rate RG(x,t) used in the current continuity equations is given Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 145 by Equation 2.5.7, and is here made a function o f position and time since this equation contains the carrier densities. Table 3.6.2 Final transport equations. Name Definition o f electric field 3.6.5 Poisson’s Equation 3.6.6 Equation d*F(x,t) _ ^ ^ ~ S(x,t) dx dS(x,t) dx qf exp]7^ r [ ¥ r( x , / ) - 0 n(x,/)] k j \ A driSSSL equation 3.6.7 J H °)? . dnft-diffusion equation 3.6.8 & J W , J p(.x,l) f q r ,1 = ------------- exp*! ——VF(x,t)-& (x,t)\\ «V>< W J ---& Electron current continuity equation 3.6.9 ^ ( — ^ H o le. . currentit conti continuity quatior equation 3.6.10 dJD(x,t) f d f q — p-' — = - q — n, exp< [- F{x,t) + <fip (x,/)]J - G l (t) - RG(x,t) Kdt r [kBT } =q rd ~dtn‘ eXPi [ “ ° L(r)" RG(X' 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 146 3.7 3.7.1 CARRIER MOBILITY Complex Behavior o f Mobility Carrier mobility is an important transport parameter that measures the ease with which mobile charges move through a semiconductor. The greater the mobility, the greater the current. In Section 3.3, carrier mobility fj. was introduced through the driftdiffusion equations, specifically as a parameter in the drift term. The drift term J ^ in the current density equations was expressed in that section by Jjr^q n u S , where n representselectrons or (3.7.1) holes, and thesubscriptsforterms specific to the electrons or holes are understood. The drift current density can also be expressed as Jdr (3.7.2) where the drift velocity 0dr is understood to be an average [43]. Equating these two expressions for J dr gives Ddr= f i S , (3.7.3) for which the mobility n can be expressed as the ratio of the magnitudes o f the drift velocity to the electric field, or = Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.7.4) 147 Carrier mobility is difficult to characterize because it becomes dependent on electric field at higher field strengths. For low electric fields, the mobility fj is constant, and Equation 3.7.3 predicts that the drift velocity v Jr increases linearly with increasing field strength S. The linear behavior o f the drift velocity for low electric fields is shown in Figure 3.7.1, which plots the mobility and the drift velocity for both electrons and holes in GaAs. Note the linear behavior in the drift velocity for the electrons in part (a) and the holes in part (b) for the electric field region below - 103 V /cm . However, as the field strength continues to increase, the drift velocity no longer increases linearly. For the GaAs holes pictured in part (b), the drift velocity increases monotonically while approaching a maximum value called the saturation velocity . This indicates that beyond the low-field regime, mobility becomes a function o f electric field. Furthermore, for the GaAs electrons pictured in part (a), there is a threshold electric field at about 3.3 xlO 3 V/cm beyond which the electron drift velocity actually decreases before leveling off, indicating that electron mobility has an even more complicated dependence on electric field than simply causing the drift velocity to approach an asymptote. By examining the defining expression for mobility fi given by Equations 3.3.14 and 3.3.16, the reason for the asymptotic behavior o f the carrier drift velocity v dr can be explained [43]. These two equations can be generalized to both electrons and holes by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Electrons (a) cn E fjn (cm2/Vs) 10® 106 c ^ 105 drift velocity mobility 10° 10® 10® 105 10® S (V/cm) (k) Holes 10® (cm2/Vs) 10® E o Q. ■Q drift velocity mobility 10° 102 103 104 5 (V/cm) Figure 3.7.1 Drift velocity vdr and mobility //as a function o f electric field S in for (a) electrons and (b) holes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 149 in which r is the mean free time between collisions of the carrier (also called the momentum relaxation time). According to this relationship, as the time between scattering events decreases, the mobility decreases. The mean free time between scattering events depends significantly on the number of phonon scattering modes that are available to the carrier. At low electric fields, the average carrier energy is low, so only the acoustic phonon modes are available for energy exchange between the carriers and the lattice. Increasing the electric field strength within this low-field regime does not significantly change the probability o f scattering events. The mean free time between collisions remains essentially constant, which keeps mobility constant, so that the increase in drift velocity is linear with increasing field. Since the carrier energy does increase with increasing electric field, a point is reached where a new set o f higher energy optical modes becomes available to the carriers [54]. The probability o f a scattering event increases significantly, which decreases the mean free time between collisions, thereby reducing the mobility. Furthermore, since more energy is transferred from the carriers to the lattice during optical phonon exchange, the drift velocity approaches an upper limit. The expression for mobility given by Equation 3.7.5 can also be used to explain the decrease in electron drift velocity for GaAs after the electric field reaches a threshold value of approximately 3.3 xlO 3 V/cm [39]. The elevated electric field is able to transfer enough energy to T valley electrons that some are promoted into the higher energy L valley, a process called intervalley transfer. The electron effective mass in the L valley is mnL* = 0.55m0 [39], which is almost an order o f magnitude higher than the electron Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 effective mass (0.063 m0) in the I~ valley. Due to the inverse relationship between fi and m *, the electron mobility in the L valley is an order of magnitude lower than for electrons in the r valley. Since the net electron mobility is the weighted average of the mobilities o f the electrons in each valley, the mobility begins to decrease as more electrons are transferred from the T to the L valley. The sudden drop in the electron mobility and drift velocity seen in part (a) o f Figure 3.7.1 is due to the onset o f intervalley electron transfer. As electric the field continues to increase, the transfer of electrons from the L valley back into the T valley balances the forward electron transfer, and the electron drift velocity approaches an asymptote as occurs with holes. 3.7.2 Mobility Models The two mobility models include the field-independent (FI) model, which is used in the case o f low electric field, and the field-dependent (FD) model. Under ordinary operating conditions, the electric field in devices is elevated substantially above the lowfield regime. Since mobility depends on the electric field when the field is elevated, the FD mobility model is used when the device operates under ordinary conditions. Besides a high electric field, ordinary operating conditions include a relatively long device and a relatively low modulation frequency. As will be explained shortly, the FI mobility model is more accurate than the FD mobility model when the device is short or the modulation frequency is high, even when the electric field is elevated. Both mobility models assume that the carriers are able to undergo enough collisions to reach a steady state regarding energy exchange with the lattice. For example, if a low energy carrier is injected into a semiconductor that has a relatively high Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 electric field, then the field will begin to accelerate the carrier. This process is illustrated in Figure 3.7.2 through a Monte Carlo study performed by Tait and Krowne [20]. As the carrier accelerates, it undergoes collisions with the lattice. With each collision, some of the energy that the carrier has gained from the electric field is transferred to the lattice. Between each collision the carrier gains more energy from the electric field, so that on average, the carrier has a higher energy after a given collision than it had after the previous collision. The higher the carrier energy, however, the more energy it can transfer to the lattice during a given collision. Therefore, after some minimum number of collisions, the carrier energy has increased to a maximum as it reaches a steady state regarding energy exchange. As shown in the figure, steady state has been reached by the time the carrier has traveled 1.0 jam into the substrate. This steady state length is consistent with a study reported by Snowden [13], who concluded that non-equilibrium transport effects become significant in GaAs devices that are shorter than 1.0 pm [13]. There are two sets o f operating conditions that do not permit the carrier to reach the steady state required for the accurate application o f either mobility model, although one model is considered more accurate than the other depending on the condition as well as the carrier injection energy. These two operating conditions occur either when the device length is relatively short, or when the modulation frequency of the electric field is relatively high. For short devices, there are not enough mean free paths for a steady state to be established. If the injected carriers begin with low energy, then they are unable to attain enough energy to engage the optical phonon modes. The carriers behave as if the electric field were low, and are more accurately characterized by the FI mobility model, even in situations in which the electric field is elevated. If the injected carriers begin Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 152 0.15 0.12 > ~ 0.09 * > CD tr lu 0.06 z INDICATES ENSEMBLE AVERAGE < w > IN CELL OF LENGTH AX * 0.05 Jim LU 0.03 0.5 10 1.5 2.0 2.5 DISTANCE, x (/im) Figure 3.7.2 Monte Carlo simulation o f a 2.5 pm long n-GaAs sample under a constant and uniform electric field S —l kV/cm, showing the average total electron kinetic energy [taken from Tait and Krowne P°l]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 153 with high energy, then they can engage the optical modes upon injection and are therefore more accurately characterized by the FD mobility model. For fields that reverse themselves at high frequency, there are not enough mean free times for the carriers to reach steady state before the field has changed direction again. I f the injected carriers have low energy, then the FI mobility model is more accurate even in cases o f elevated electric field. If the injected carriers begin with high energy, then they can engage the optical phonon modes at first, but eventually lose the high energy through collisions and are unable to regain the high energy due to the rapidly changing electric fields. Therefore, at high modulation frequencies, the FI mobility model is more accurate for both high and low carrier injection energies. In the present study, both mobility models are used. The device length is 1.1 pm, which is longer than the 1.0 pm threshold for steady state regarding energy exchange between the injected carriers and the lattice. Therefore, the FD mobility model is considered more reliable based on device length. Furthermore, most of the injected electrons result by promotion from the valence band due to absorption of 1.55 eV light. Since the bandgap energy is 1.42 eV, the electrons injected into the conduction band have approximately 1 eV o f kinetic energy, which is equivalent to the kinetic energy attainable by applying a voltage o f 1 V across the device. Since a 1 V applied voltage is associated with elevated electric fields, the FD mobility model appears to be the more accurate choice based on the injection energy as well. However, as the modulation frequency is increased, the injected electrons ultimately lose the high energy and tend to behave more as if they were in the low-field regime. This prompts the use o f the FI mobility model in conjunction with the FD mobility model. The final bandwidths are then reported as being Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ! 154 located somewhere between the bandwidth limits determined by using both the FD and FI mobility models. 3.7.3 Empirical-Fit Mobility Relationships Due to the complicated behavior o f carrier mobility, empirical-fit relationships are widely used to characterize this transport parameter [39]. The empirical-fit relationships used in the present study are based on a model presented by Lundstrom et al [55]. According to this model, the low-field or field-independent electron mobility H Ljt (T, N d , N A) is a function o f the temperature T, and o f the donor N D and acceptor N a doping concentrations, and is given by 7200 M i „ ( r , N 0 , N t )= [1 + 5 51x l 0 -u (Afo 0.233 f S O O K ^ cm 2 V -s (3.7.6) The field-dependent electron mobility n„{&) is given by u (5 ) - ^ L-n !&cs, cm' i + te * jr v-s* where the nominal saturation velocity (3 7 7) ( 3 '7 7 ) and the critical field S CJ, for electrons is respectively given by »'*** = (1.28-0.0015 K * T ) x l 0 7 cm /sand S CJ, = (5 .4 -T /2 1 5 K ) x l 03 V /c m . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 155 Also according to this model, the low-field or field-independent hole mobility V l.p n d>N a ) is given by '300 K ^2-7 cm 2 ll + 3.17*10-, , (A r„+ JV jJ 0266 { T J V - s (3.7.8) The field-dependent hole mobility fi p{0) is given by (3.7.9) where the critical field for holes is S cp = l.9 5 x l0 4 V/cm. For the present study, the mobility parameters required by Equations 3.7.6 - 3.7.9 include the following: T = 300° K , N d = 3.5x10 !2 cm -3, and N A = 0 . Equations 3.7.7 and 3.7.9 are used to generate the plots given in Figure 3.7.1. Several transport parameters are now calculated for the system modeled in the present study. The low-field mobilities for electrons and holes are determined from Equations 3.7.6 and 3.7.8, respectively, which give n Ljl - 7200 cm2/V - s and fxL p =380 cm2/V —s . The low-field diffusion constants are determined by deriving the Einstein relationships for electrons and holes from Equations 3.3.14 and 3.3.16, respectively: and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.7.10) 156 Substituting the low-field mobilities into Equation 3.7.10 generates the diffusion constants as DLjt = 186 cm 2/s and DLp =9.83 cm 2/s. The saturation velocities for electrons and holes are determined from the product o f the respective field-dependent mobilities (Equations 3.7.7 and 3.7.9) and the electric field, by allowing the electric field to increase until the velocities approach asymptotes, which gives and = 7 .3 x 1 0 6 cm/s. = 8 .3 x l0 6 cm/s The low-field mean free time between collisions for electrons and holes is calculated from Equation 3.7.5 using the respective low-field mobilities and effective masses, which gives r^ , = 2 .6 xL 0-13 s and r^ , = 6 . 1x l 0 ~u s. The upper limits to the mean free paths o f electrons (x„) and holes (xp) are calculated using * (3.7.11) which gives x„ = 216 A and x p = 45 A. These values for the mean free paths represent the upper limits because they are based on carriers that have both the maximum possible mobility and the maximum possible drift velocity, a combination that cannot occur since mobility drops to a minimum at the saturation velocity. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 157 CHAPTER 4 BOUNDARY CONDITIONS ON THE CURRENT DENSITY 4.1 INTRODUCTION One o f the m ajor objectives o f this study is to develop a more physically reasonable model o f current density at the Schottky boundary than the existing models. The three major existing models are based on the combined drift-diffusion/thermionic emission formulation. Drift-diffusion accurately characterizes the current density in the bulk region since drift-diffusion includes scattering, which is an essential process in the bulk. Thermionic emission accurately characterizes the current density at the boundary because thermionic emission is not constrained by a potentially undefined quantity that appears in the drift-diffusion formulation, and scattering is not meaningful at the boundary. The combined formulation enables accuracy to be achieved throughout the model of the device, and allows the current density to be numerically determined in a self-consistent manner regarding the boundary and the bulk. Section 4.2 reviews the historical development o f three important existing versions of the combined drift-diffusion/thermionic emission formulation for modeling current density at the boundary. With the original model, four physical mechanisms can be identified as being important in the characterization of current density at the boundary: carrier normalization, velocity o f carriers out o f the semiconductor, velocity o f carriers Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 158 into the semiconductor, and carrier fraction. The second version o f the combined formulation revised the physical mechanisms concerning normalization and the velocity of the carriers out o f the semiconductor. The third model revised the physical mechanism regarding velocity o f the carriers into the semiconductor, but returned to the original normalization. As a result, the original model and the two later models have significant differences regarding the physics o f current density at the boundary. The present study resolves to settle these differences by deriving the current density boundary condition from first principles regarding the semiclassical model for solid state systems, which is the topic o f section 4.3. This approach leads automatically to the three revised physical mechanisms adopted in part by the later models, and results in a fourth revised mechanism concerning carrier fraction. Section 4.4 discusses the Schottky and Ohmic limits as they apply to the current density boundary condition. A Schottky barrier always forms at the boundary between a metal and a semiconductor. The Schottky limit refers to a barrier that blocks the flow o f electrons across the boundary. The Ohmic limit occurs when the semiconductor is highly doped in a narrow region adjacent to the metal, which effectively removes the barrier to the flow o f electrons. In the Scottky limit, there are two processes that effectively lower the barrier without completely removing it. The image force effect is the effective lowering o f the barrier by an electron that approaches the negatively-charged metal surface at the boundary and repels away some o f the negative charge that creates the barrier. Electrons can tunnel through the barrier in the region near its narrow tip, which also effectively lowers the barrier. Both the image force effect and tunneling are built into the model o f the current density at the boundary. The section ends by presenting the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 159 final form o f the complete set o f boundary conditions for both Schottky diodes and MSMs. The DC steady-state solution in the absence o f light is studied in section 4.5, both to test the simulator and to gain insight into device operation under these conditions. An effective electron-only solution is generated by the full transport solver using all six partial differential equations, which is made possible due to the w-doping and low barrier height. A set o f E-x diagrams and device profiles are generated for the CTH 562 Schottky diode and the low-barrier MSM. A purely electron-only solution is also generated using a reduced form o f the solver that does not include holes or thermal recombination-generation. The results o f the full transport solver and the purely electrononly solver are compared to assess the effects o f holes and thermal recombinationgeneration in n-doped GaAs. Finally, an effective hole-only solution is generated by the full transport solver through simulation o f the high barrier MSM. The high barrier leads to almost complete electron depletion of the substrate, causing the thermal recombination-generation rate to compensate by making holes the primary carrier. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 160 4.2 REVIEW OF CURRENT DENSITY BOUNDARY CONDITION MODELS 4.2.1 Combined Drift-Diffusion/Thermionic Emission Model In the earliest stages o f modeling transport in Schottky diodes, two theories emerged regarding the current density. As described by Sze [35], these theories included the drift-diffusion theory developed by Schottky in 1938 and the thermionic emission theory formulated by Bethe in 1942. The drift-diffusion theory assumed that scattering must be included, and used the drift-difiusion Equations 3.5.5 and 3.6.4 for the electron and hole current densities, respectively. A problem with this theory was that the spatial derivative o f the quasi-Fermi potential d<P/dx in the drift-diffusion equations could become undefined at the Schottky boundary under conditions o f bias. On the metal side, the quasi-Fermi potential energy is equal to the metal Fermi energy E F since there is no change in the metal from equilibrium. However, there is no physical reason for the quasi-Fermi potential energy to be continuous at the boundary. If a discontinuity did exist, then the current densities given by the drift-diffusion equations would be undefined. To prevent this, the drift-diffusion theory assumed that the quasi-Fermi potential energy on the semiconductor side joined smoothly to the metal Fermi energy E f at the Schottky boundary. At the other extreme, the thermionic emission theory assumed that the quasiFermi potential energy was constant and equal to (~q)(±VA) all the way from the Ohmic contact to the Schottky boundary. The characterization of current density by this theory did not require the quasi-Fermi potential energy to be continuous across the Schottky boundary since the current density (using electrons as the example) is given by [35] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 161 J n =<lv0 JJ, n - q u ° J,n0, where (4 .2 . 1) is the magnitude o f the equilibrium electron surface velocity (originally termed recombination velocity), nQ is the equilibrium electron density, and the metal is on the left side o f the boundary (with + i to the right). The first term on the RHS o f Equation 4.2.1 represents the current density due to electrons moving from the semiconductor into the metal, while the second term represents the current density due to electrons moving from the metal into the semiconductor. A problem with the thermionic emission theory was that scattering was not included its formulation. The combined drift-diffusion/thermionic emission (DD/TE) model o f carrier transport is presently used in simulations to characterize current density for devices that have a Schottky contact [31]-[35]. The DD/TE model was developed to allow for the more physically reasonable possibility that the quasi-Fermi potential energy is discontinuous at the boundary. The drift-diffusion equation is used to characterize the current density in the bulk region right up to the boundary between the metal and the semiconductor. To allow for discontinuities, the thermionic emission equation is used to characterize the current density at the boundary. This equation is not used in the bulk because the effects o f scattering are critical for the accurate characterization of transport in the bulk. With these two equations, the current density at the boundary and in the bulk can be determined self-consistently through a numerical solver. This allows the quasiFermi potential energy to seek its most accurate value rather than being pre-determined as it was in the two previous theories. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 162 The combined DD/TE formulation was first developed by Crowell and Sze about thirty-five years ago [31]. Focusing on electrons as the carrier, at equilibrium n = n0,so J n = 0 . Under bias, there is net current density across the Schottky boundary. Since the current density component originating in the metal remains constant even under bias, the current density component originating in the semiconductor must vary. In this original formulation o f the combined DD/TE model, the only variation that was allowed was in the electron current density n. The variation in n depended on the deviation of the electron quasi-Fermi potential from its equilibrium value. This variation in n allowed the current density component originating in the semiconductor to vary, as required for net current. The surface velocity 0 °^, however, was not allowed to vary with bias. Also, for both components o f the current density, the full density n and n0 were assumed to move across the boundary in this original version of the combined DD/TE formulation. Finally, the electron surface velocity for the electrons originating in the semiconductor was not treated separately from the electron surface velocity for the electrons originating in the semiconductor. This original formulation o f the current density boundary condition is termed formulation-1. 4.2.2 Revisions to the Combined Model Over the years since the combined DD/TE formulation was first proposed, three physical mechanisms within the original formulation have been revised: 1) defining how the surface velocity is normalized; 2) allowing the surface velocity to vary; and 3) using separate surface velocities for electrons that move into and out of the semiconductor. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 163 The equilibrium electron surface velocity v°SJt is the average velocity for those semiconductor electrons that move toward the metal due to their thermal energy. In the original version o f the combined DD/TE model, u SaJt was calculated by Sze as [35] 0 v SQJ, = exp[(- m„ * )/(2 kBT ^ d v nJC ----------------------------------------Jexp[(-m„ \_ k j_ 2 ran. ' (4.2.2) — «3 where unj is the component o f the electron velocity in the direction o f the current density (the x-direction). The numerator is integrated over those electrons that move in the -x -d ire c tio n . Since the normalization chosen by Sze was over all velocities in the x-direction, both positive and negative, the final result for u°J r, in the original version was one-half of the actual average velocity for those electrons that move toward the metal. Therefore, the first suggested revision to the original formulation was to use only those velocities associated with electrons that moved in one direction as the normalization for the surface velocity. This unidirectional normalization was suggested by several groups, including Baccarani and Mazzone [56], Berz [57], Adams and Tang [32], and Nylander et al [33]. At equilibrium, the distribution o f electron velocities on the semiconductor side o f the Schottky boundary is balanced, so that half the electrons move in the -x -d ire c tio n while the other half move in the + x-direction. (The same is true for the electrons on the metal side o f the boundary). Under bias, the velocity distribution on the semiconductor side o f the boundary is shifted away from equilibrium, favoring velocities in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 164 -x -d ire c tio n for forward bias while favoring velocities in + x-direction for reverse bias. This causes the electron surface velocity v SJt to vary with the applied voltage, rather than remain constant as was proposed in the original model. Therefore, the second revision to the original formulation was to allow the electron surface velocity to vary under bias. This was first proposed by Adams and Tang [32], and subsequently by Nylander et al [33]. Their formulation o f the combined DD/TE current density boundary condition, termed formulation-2 , contains two revised mechanisms: surface velocity normalization and a variable surface velocity. a unidirectional Formulation-2 is still commonly used [58],[59]. In the original formulation, the same surface velocity was applied to both the electrons that originated in the metal and those that originated in the semiconductor. Although this is true for equilibrium, under bias the surface velocities are no longer equal. The surface velocity for the electrons originating in the metal remains at the equilibrium value. Therefore, the third revision, as proposed by Darling [34], was to separately determine the two surface velocities, and apply the variable surface velocity only to the electrons that originated in the semiconductor. Darling, however, used the original bidirectional normalization for the surface velocity so that the surface velocity was half its actual value. His formulation, termed formulation-3, contains two revised mechanisms: a variable surface velocity and the separation of the metal and semiconductor surface velocities. The present study proposes a fourth revision to the combined DD/TE model for current density. Since only half o f the electrons on either side o f the boundary move Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ! 165 across it at equilibrium, both n and n0 should be divided by two in the boundary current density equation (4.2.1). Even though this was not done explicitly in the original version, both n and nQ were effectively divided by two because o f the normalization. Berz [57] was the first to suggest the explicit division o f the electron densities by two. Although the division o f the electron densities by two correctly characterizes the electron fractions under equilibrium, under conditions o f bias the fraction o f electrons that move from the semiconductor into the metal is no longer one-half. The fraction is greater than one-half for forward bias and less than one-half for reverse bias. This variation in electron fraction results because the same velocity distribution that determines the surface velocity must also determine the fraction o f electrons that move into the metal. Therefore, the present study proposes a fourth revision, which is to allow the fraction o f electrons that move from the semiconductor into the metal to vary with the applied voltage. The proposed mechanism o f a variable carrier density is incorporated into a new version o f the combined DD/TE formulation for the current density boundary condition. The formulation is new both because o f the revised carrier density mechanism and because all three of the previously suggested revisions are included (the existing formulations each use no more than two revised mechanisms). Furthermore, the new formulation is derived mathematically from first principles regarding the semiclassical model. This derivation leads to the appearance o f an additional term in each current density component of Equation 4.2.1, with this additional term being the fraction of carriers that moves across the boundary. The derivation automatically retains the three previously suggested revised mechanisms as well. The new formulation developed in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 166 this study is termed ,formulation-4. Table 4.2.1 summarizes the four formulations by presenting the four physical mechanisms (original or revised) used by each. T able 4.2.1 Four formulations o f the combined DD/TE current density boundary condition. Physical Mechanisms Surface Semiconductor Velocity Surface Normalization Velocity 1 (original) bidirectional constant 2 •unidirectional •variable 3 bidirectional •variable 4 (new) *unidirectional •variable *indicates a revision o f the original model ( 1) Formulation Metal vs Semiconductor Surface Velocity same same •separate •separate Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fraction of Carriers into Metal constant constant constant •variable 167 4.3 DERIVATION OF NEW CURRENT DENSITY BOUNDARY CONDITION 4.3.1 Electron Current Density The electron current density boundary condition can be derived from first principles by beginning with the electron velocity distribution according to the semiclassical model. The electron current density J njc in the x-direction is given by J njc = Jg n f v njcdun , where un is the electron velocity and v njc is its x-component, f (4.3.1) is the v„ -dependent distribution function for electrons, and g n is the electron density o f states in 0„ -space. The electron density o f states g n is given by gH m. * 3 4it h (4.3.2) The integral in Equation 4.3.1 can be split into two halves, one that integrates o v er-u „ components and one that integrates over +- v HJ -components: Iy * j g n f V n s d V n s ~ 4 1 y= \ g n f \ 0 in which Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.3.3) 168 « ao l r. = f \d o nydu„^ . — « -« Each term in the sum o f Equation 4.3.3 can be expressed in terms o f an average velocity if use is made o f the electron density n as given by U as « = \ Z n f d v m = I,. \ g nf d o nj Jr l yz \ g j dvn j , (4.3.4) in which the two terms on the RHS represent the densities o f electrons that move in the —x - and + x -directions, respectively. If the first term in Equation 4.3.3 is multiplied and divided by the density o f electrons that move in the -x-direction, and the second term is multiplied and divided by the density o f electrons that move in the + x -direction, then the electron current density can be expressed in terms o f the average electron velocity in the - x - and + x -directions, o . and U . , respectively: ly. \ s n f d v n^ - q o ] I y . \ g mf d v u (4.3.5) I>. \ g » f ° n s d v n U . = — 2--------------I* \ g „ f d v n (4.3.6) where u I>, f g „ f v n Jd o , and Vttyr. = ' I>, Jg „ f d o n Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 169 Finally, if each term in the sum o f Equation 4.3.5 is multiplied and divided by the total electron density n as given by Equation 4.3.4, then the electron current density can be expressed in terms o f the fraction o f electrons that move in the - x - and + x directions, F and Fn j. , respectively: (4-3-7) where o F . = 00 and F . = (4.3.8) Note that the normalization for each velocity includes only those electrons that move in the direction indicated by the velocity, and not all of the electrons (revision 1). Also, the fraction of carriers that move in a given direction appears automatically in the equation for current density when current density is derived from first principles (revision 4). Furthermore, the use o f different integration limits in the numerators of the velocities separates them from each other, so that this procedure will automatically allow for the separate expression o f the surface velocity o f electrons originating in the semiconductor and metal (revision 3). Assuming that the doping concentration is low enough that the semiconductor material remains nondegenerate, the electron distribution function can be approximated Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. by a displaced Maxwell-Boltzmann distribution that depends on electron energy E [35],[32]-[34]: (4.3.9) The electron energy is given by (4.3.10) where udn is the electron drift-diffusion velocity in the x-direction, which is identical to the displacement in the electron distribution along the unj -axis. When the energy E given by Equation 4.3.10 is substituted into Equation 4.3.9, the form of the distribution function becomes a displaced Maxwellian. The use of the displaced-Maxwellian in the electron velocity distribution function introduces a carrier surface velocity that naturally varies with the carrier drift-diffusion velocity (revision 2). Now all four revisions have been introduced into the current density boundary condition by first principles. [This last revision can be considered to be from first principles because the Maxwell-Boltzmann distribution is the limiting case of the Fermi-Dirac distribution under nondegenerate conditions.] To evaluate the average electron velocities, the distribution function / given by Equation 4.3.9 (with the electron energy E given by Equation 4.3.10) is substituted into Equation 4.3.6. The average electron velocities in the - x - and +x-directions are respectively given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in which the error function sign for un^. is + for v djl < 0 and - for udjl > 0 ; the error function sign for Z7 is —for v dj% < 0 and + for v djt > 0 ; is given by 1 and the one-dimensional (and unidirectional) electron thermal velocity v llM is given by (4.3.13) which is the value returned by Equation 4.3.6 (using v with odjt = 0 , and evaluates to unjt = 2 .1 4 x l0 7 cm /s. ) for the equilibrium case, i.e. To evaluate the fraction of electrons that move in the - x - and + x-directions, the distribution function / given by Equation 4.3.9 (with the electron energy E given by Equation 4.3.10) is substituted into Equation 4.3.8. The fraction o f electrons that move in the - x - and + x-directions are respectively given by = i [ l ± e r f ( 0 ], = | [ l ±erf(A .)], Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.3.14) 172 in which the error function sign for F error function sign for F is + for udjt < 0 and - for udJ) > 0 , and the is + for vnd > 0 and - for v dj, < 0 . After calculating the average electron velocities and the electron fractions, the electron current density can be determined with Equation 4.3.7. Consider first the boundary o f a Schottky contact in which the metal is on the left and the semiconductor is on the right. The first term on the RHS o f Equation 4.3.7 represents the current density due to electrons that move from the semiconductor into the metal, while the second term represents the constant current density due to electrons that move from the metal into the semiconductor, and is equal to the first term at equilibrium. Therefore, adopting the usual notation for the current density boundary condition, the electron current density J n is given by sjt* n” (4.3.15) where the electron surface velocity uSJI = U , the fraction o f electrons that move from Jn = 9 \ the semiconductor into the metal Fn = , Fnjc. = 1 /2 , and n0 is the equilibrium electron density on the semiconductor side o f the contact. At equilibrium, uSJt evaluates to v lXjl, Fn evaluates to 1/2, and J n goes to zero as expected. The Schottky contact is placed in forward bias for electrons by making odM negative. As udM -> -<», v SJt approaches the electron drift-diffusion velocity v djt, and F„ approaches one. This latter result indicates that all o f the electrons on the semiconductor side o f the boundary move Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 173 into the metal, which is consistent with high forward bias. The Schottky contact is placed in reverse bias for electrons by making udn positive. Mathematically, as v djt -*■ -k», both uSJJ and Fn as well as n approach zero, and the electron current density is comprised solely o f the contribution due to electrons flowing from the metal into the semiconductor as expected for high reverse bias. For a Schottky contact in which the semiconductor is on the left and the metal is on the right, the first term on the RHS of Equation 4.3.7 represents the constant current density due to electrons that move from the metal into the semiconductor, and the electron current density is given by (4.3.16) where Fnjc. = 1I/2 , u ijr i = \|l>n a . ’, and F„» = Fn_r .. Table 4.3.1 summarizes the surface velocity, electron fraction, and current density associated with electrons for the limiting cases o f drift velocity for a Schottky barrier at both contacts. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 174 Table 4.3.1 velocity. Electron current density statistics for limiting cases o f electron drift or vd jt Vd j t —> -C O q n v d jt x=0 0 Vd j t Schottky B arrier at 0 +CO - ( l / 2k * o 0 /Ijf 0 Vd j t —>—CO {\/2)qnQuitijt x-L . v d jt Vd j t +00 - q n vd jt 4 3 .2 Hole C u rren t Density The hole current density J pjc in the x-direction is the sum o f the light- and heavy- hole current densities in the x-direction. As was done with electrons, J pjc is split into two halves, one that integrates over - -components and one that integrates over + upi^ -components (where / refers to /ight or /reavy), so that P -* .+ J p a ., p .1 in which the hole current densities in the - x - and + x -directions are given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.3.17) v ' 175 J Px =(f L l <.y-. \ s P, ( 1 - / )v p,ad u p ^ ’ (4.3.18) im ljl where g pi, 1 - / , and u piJC are the density o f states (Equation 4.3.2 with the appropriate hole effective mass), the distribution function, and the component o f the velocity in the xdirection, respectively, for holes in the /th (/ight or heavy) valence band; a = -oo and b = 0 for J P-x. while a = 0 and b = « for J P-x. ; and = J fap'.ydVp,.; ■ This is followed by the multiplication and division o f each unidirectional hole current density first by the density of holes that move in that direction and then by the total hole density, giving J p, =<luPs - FpJ -P + < l up S Fp s P ' (.4.3.19) where the average hole velocities in the —x - and + x -directions are given respectively by J . — ' and qp~ «j p J . = -£ d -; (4.3.20) <u> the fraction o f holes moving in the - x - and + x -directions are given respectively by F . =E" P and F . =— ; " P Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.3.21) 176 the density of holes moving in the —x - and -+-x -directions are given respectively by b (4.3.22) with a = -co and 6 = 0 for p~ while a = 0 and 6 = <» for p* ; and the total hole density is given by P= Z j gp' (l -f r)dVplJ . (4.3.23) Again assuming nondegeneracy, the hole distribution function can be approximated by a displaced Maxwell-Boltzmann distribution that depends on hole energy Eht: (4.3.24) The hole energy is given by (4.3.25) where m pi * and v d pi are the effective mass and the drift-diffusion velocity o f the holes in the rth valence band, respectively. To evaluate the average hole velocities, first the hole distribution functions given by Equation 4.3.24 (with the hole energies given by Equation 4.3.25) are substituted into Equations 4.3.18 and 4.3.22. Then the resulting hole current densities and associated Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 177 hole densities are substituted into Equation 4.3.20 to give the average hole velocities UpX and U , respectively: Z m p>*3'2 Upy °«.p> exp(~ )+ ud.P. [ 1± erf m Pi *m h ± erfU p/)]+ m p* *m [l ± erf in which the error function sign for & )]} )] (4.3.26) is + for udpi < 0 and - for oJp, > 0 ; the error function sign for Uppc. is + for udpi > 0 and - for ud pi < 0 ; Ap, is given by = 1 Vj.p, i = I, h; (4.3.27) Vt\.pi and the one-dimensional (and unidirectional) hole thermal velocity vtXp, is given by ( « .2 8 ) which is the value returned by Equation 4.3.26 for the equilibrium case, i.e. with v dp, = 0 . To evaluate the fraction o f holes that move in the - x - and + x -directions, the hole densities given by Equations 4.3.22 and 4.3.23, and the distribution function given by Equation 4.3.24 (with the hole energies given by Equation 4.3.25) are substituted into Equations 4.3.21. The fraction of holes that move in - x - and + x-directions are respectively given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 178 1 nip, *5/- [ l± e r f f c „ )]+mptl *3/2 [l± e rf(/lM )] FP -* x «„ ^ 2 +mplt (4.3.29) in which the error function sign for F n r. is + for udp/ < 0 and - for udB, ' d.p' > 0; the error function sign for F . is + for v dp, > 0 and - for udpi < 0 . After calculating the average hole velocities and the hole fractions, the hole current density can be determined with Equation 4.3.19. At the Schottky contact in which the metal is on the left and the semiconductor is on the right, the hole current density J is given by Po J , = q v « . p - - v s.pFp P \, where the hole surface velocity ui p = v (4.3.30) , the fraction o f holes that move from the semiconductor into the metal is Fp = F ^ . , Fpj. = 1/2, p Q is the equilibrium hole density on the semiconductor side o f the contact, and the combined one-dimensional hole thermal velocity l>„ is given by l2kBT m pl * * *3/2 \ m pf + m ph *2/2 (4.3.31) J which can be determined from Equation 4.3.26 for the equilibrium case {o d p, = 0 ), and evaluates to ulipt = 8 .5 2 x l0 6 cm /s. At equilibrium usp evaluates to ull p, Fp evaluates Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to 1/2, and J p goes to zero as expected. The Schottky contact is placed in forward bias for holes by making udpi negative. As v d p, —►-<», the hole surface usp velocity approaches the combined hole drift-diffusion velocity - v d pi. Since Fp 1, all o f the holes on the semiconductor side o f the boundary move from the semiconductor into the metal, which is consistent with forward bias. The Schottky contact is placed in reverse bias for holes by making ud pi positive. Mathematically, as ud p, —►+00 , both v s and Fp as well as p approach zero, and the hole current density is comprised solely o f the contribution due to holes flowing from the metal into the semiconductor as expected for high reverse bias. For a Schottky contact in which the semiconductor is on the left and the metal is on the right, the hole current density is given by r - on p -Po £ -\, where FPJC . = 1I/2 ,’ v S..P. = \ Iv p j c . ’, and Fn = FPn -r*1*. . P P (4.3.32) Table 4.3.2 summarizes the surface velocity, electron fraction, and current density associated with holes for the limiting cases o f drift velocity for a Schottky barrier at both contacts. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 180 T able 4.3.2 Hole current density statistics for limiting cases of hole drift velocity. Vs .P v d .p Vd . p —00 - q p v. x=0 0 Vs . p Vd . P Schottky B arrier at 0 +00 V* .p Vd . p {\/2 )q p 0 v tt\ . p 0 0 —00 —0 /2 )^ Po v ,\. x = L. us .p Vd . p v.d . P +00 <1P V.d . p Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 181 THE SCHOTTKY AND OHMIC LIMITS 4.4 4.4.1 The Schottky Limit When a metal is placed into physical contact with a semiconductor, a Schottky barrier always forms at the boundary between the semiconductor and the metal. This is true even for Ohmic contacts, although the special doping near the contact effectively removes the barrier. The Schottky limit refers to those metal-semiconductor (MS) contacts in which the Schottky barrier is able to block electrons from crossing the boundary, while the Ohmic limit refers to those MS contacts in which the special doping renders the Schottky barrier ineffective in blocking electrons. This subsection discusses the Schottky limit, and the next subsection discusses the Ohmic limit. As explained previously, the true peak o f the Schottky barrier remains fixed, so that the actual height o f the barrier relative to the metal Fermi energy E Fm is invariant. However, in the Schottky limit, there are two processes that effectively lower the barrier height to electrons originating on either side o f the barrier. These two processes, the image force effect and tunneling, are able to effectively lower the barrier height because the peak region o f a Schottky barrier is relatively narrow. Each o f these processes is a function o f the electric field, such that the greater the electric field the lower the effective barrier height becomes. The remainder o f this subsection explains how these two processes affect the electron current density at the boundary, and what modifications are made to the simulation to account for these effects. As an electron attempts to cross the boundary between the metal contact and the semiconductor, the image force effect lowers the effective barrier height due to the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 182 Coulombic interaction between the approaching electron and the electrons o f the metal surface. As the electron approaches the boundary, it repels electrons away from the region of the metal surface closest to the approaching electron. If the metal surface were neutral, the repulsion o f its electrons would leave a positive image charge on the metal surface with a total magnitude o f q, and with an effective position as far behind the metal surface as the approaching electron was in front o f it. Since the metal surface is negatively-charged, the approaching electron is only able to reduce the magnitude o f the negative charge on the metal. This diminishes the repulsive force that the negativelycharged metal surface exerts on the approaching electron, and effectively lowers the Schottky barrier to that electron. Figure 4.4.1 illustrates the effective barrier lowering caused by the image force effect for the 0.60 eV barrier. The magnitude o f the effective barrier lowering due to the image force effect is calculated by using the principle of superposition. The region o f the metal surface that becomes less negative due to the approaching electron can be equivalently treated as containing all o f the original negative charge plus the induced positive charge q. Then the total electrostatic force exerted on the approaching electron results from the combination o f the original electric field of the metal surface £ m and the field o f the induced image charge £ q . The original electric field o f the metal surface S m is approximated as being constant over the short distance that the image force effect occurs, and is set equal to the maximum value of the electric field between the negative metal surface and the positive charge-depletion region o f the semiconductor, or S m = The electric field S q due to the induced image charge q is given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. |. 183 0.7 Ec,m Ec.q Ec.eff 0.5 bm 0.4 c.max 0.3 0.2 O) 0.1 0.0 - 0.1 - 0.2 - 0.3 m 0 1 2 3 4 5 6 7 8 9 x(A) Figure 4.4.1 Effective barrier lowering due to the image force effect for a high constant field (Smax = lxlO 7 V/cm). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 184 +q q q (4.4.1) \6 n e x 1 ’ 4;ze(2x)2 where e is the permittivity o f the semiconductor and the left boundary o f the semiconductor is located as usual at x = 0 . In calculating the barrier height lowering, the focus is on the effect that the two electric fields have on the actual conduction band energy £ c(x ), because the actual barrier height is given by £ c(0 ); i.e. <Pbm = £ c(0). As explained earlier, any changes that occur in the electron potential energy cause identical changes in the conduction band energy Ec (x ). The superposition of the two electric fields S m and S q produce an effective conduction band energy Ectff{x) that establishes the effective barrier peak. Consider first the application o f the original electric field S m arising from the negativelycharged metal surface. The electron potential energy ECJH(x) that results just from S m is determined by Ecj.1*) X (4.4.2) so that (4.4.3) E c A x ) = E(<y)-q\SmJ \x . Figure 4.4.1 illustrates Ecjn (x) as the straight dashed line. Note that Ecm (x) very closely approximates the actual conduction band energy £ c(x) close to x = 0 ( £ c(x) is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 185 not shown). This is required since both Ec(x) and Ecjn (x) represent the electrostatic potential energy due to the negatively-charged metal surface in the absence o f an approaching electron. The electron potential energy Ecq(x) due just to the image charge is given by ( 4 -4 -4) The figure illustrates the electron potential energy Ec^(x) due just to the image charge by the curved dashed line. When Ecq (x) is superimposed upon Ecjn (x ), the final value of the effective conduction band energy Ec eff (x) is given by 2 Ec.eff O ) = ECJ*0 0 + E (x) = Ec(0) - q |5miX| x - lo x e x . (4.4.5) Ec,*ff (x) is shown in Figure 4.4.1 as the solid line. As predicted by Equation 4.4.5 and illustrated in Figure 4.4.1, there is a maximum value for the effective conduction band energy Ec eff (x). Note in the figure that the effective peak o f the Schottky barrier is not only lowered by , but shifts its position as well to xmix . The coordinates (xmix, E c m2X) o f the effective barrier peak are found by setting the derivative o f E ctff (x) (Equation 4.4.5) with respect to x equal to zero and solving for xmax, to give Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 186 (4 -4 6 ) and then substituting into Equation 4.4.5 to give 4its where the position and energy o f the effective barrier peak are functions o f the maximum electric field • Table 4.4.1 presents the coordinates o f the effective barrier peaks for a typical range o f maximum electric fields for the 0.60 eV barrier. Since the displacement in position is small relative to the usual length o f the device (L s = 1.1 p m ), the position o f the effective barrier peak is still treated as being at x = 0 . The table also gives the barrier height lowering A 0 bj = <Pbm —ECJiax due to the image force effect. [NOTE: A high electric field is chosen to illustrate the image force effect because the upper limit on field strength in the MSMs is about lx l0 s V/cm, which does not show the image force effect as clearly as a higher-than-normal field.] T able 4.4.1 tunneling. |S - I (V/cm) 103 104 103 10* Effective lowering of the Schottky barrier due to image force effects and ^BUX (pm) 1.7xl(T 5.2xl0 '3 1.7xl0 *3 5.2x10"* £c.m« (eV) 0.60 0.59 0.57 0.50 (eV) 0.00 0.01 0.03 0.10 A#*, (eV) 0.00 0.00 0.03 0.30 A<Pb (eV) 0.00 0.01 0.06 0.40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.00 0.98 0.90 0.33 187 The tunneling o f electrons through the Schottky barrier also effectively lowers the barrier height. The probability PT that an electron will tunnel through the barrier is given by the product o f the tunneling transmission coefficient T(E) at a given energy and the occupancy / ( £ ) at that same energy [58]-[60]. Since the electron energy £ is a function o f position x, both T(E ) and / ( £ ) can be converted into functions o f position. This enables finding the most probable width at which electrons tunnel through the Schottky barrier. This most probable width, called the critical width x c, is then used to estimate the effective lowering o f the Schottky barrier A0 A; due to tunneling. To derive the expression for the tunneling probability PT(x) as a function of position x, the Schottky barrier is approximated as a triangle as was done in deriving the image force effect. The slanted portion o f the triangle ECJH(x) is again given by Equation 4.4.3, where x is the width of the triangular barrier. The width o f the triangular barrier depends on the energy o f the electron, such that the width lessens with increasing electron energy, as can be seen in part (b) o f Figure 4.4.2. The tunneling transmission coefficient £ (£ ) for an electron o f energy £ that tunnels through a triangular barrier is given by the WKB approximation as (4.4.8) 0 where x is the barrier width for the electron o f energy £ and U0 is the barrier height. Substituting for U0 = £ c(0) and for ECJK(x) as given by Equation 4.4.3, and then Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 188 (a) 1.0 h. Transmission coefficient Occupancy Tunneling x 5.0x109 0.8 0.7 0.5 Q. 0.3 0.2 0.0 100 200 300 400 X(A) <b) E(eV) bm X,=X, x(A) Figure 4.4.2 Critical width xc for Schottky barrier (a) occurs at the tunnel probability peak and (b) is used as limiting value xc° to determine the barrier height lowering r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 189 integrating Equation 4.4.8, the tunneling transmission coefficient becomes a function o f x as given by 7\x) = exp V 4 ^ 2 m n * q ^max| _3/2 3h (4.4.9) / The occupancy / ( £ ) can be expressed as a function o f position x through the MaxwellBoltzmann approximation (Equation 4.3.9) for equilibrium ( E F = 0 ), where now the conduction band energy E (x) is a function o f x: f i x ) = exp (4.4.10) k BT in which E in Equation 4.3.910 is equal to the conduction band energy £ CJB(x) as given by Equation 4.4.3. Therefore, the probability PT(x) that an electron will tunnel through the barrier becomes a function o f the width o f the barrier x at that point (in energy) at which the electron tunnels, and is given by PT(X) = f W / ( x ) = exp 3h x 3/2 - kBT (4.4.11) The tunneling probability given by Equation 4.4.11 is used to determine the critical width xc. Part (a) o f Figure 4.4.2 illustrates T (x), f i x ) , and their product Pr (x ). The tunneling transmission coefficient T{x) is a decreasing function ofx, which Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 190 is expected because the barrier widens with increasing x. On the other hand the occupancy / ( x ) is an increasing function o f x, because the difference in energy between the conduction band Ec m(x) and the Fermi energy EF —0 decreases with increasing x. Therefore, the tunneling probability PT(x) is a peaked function, indicating that there is a most probable value o f xc at which tunneling will occur. The value o f the critical width xc is determined by setting the derivative o f PT(x) with respect to x equal to zero and solving for the critical width xc, giving Xc S m ^ k lT 2 (4'4 1 2 ) The critical width associated with a maximum electric field of \$m3X| = 1.7 x 105 V/cm is shown in the figure, with xc = 37.5 A. The critical width locates the most probable point in terms o f electron energy at which tunneling will occur for the given maximum electric field \Sm2X| . For example, part (b) o f the figure indicates that an electron with energy E = Ec (0) - q\Smtx \xc is more likely to tunnel through the barrier than electrons with any other energy when the electric field is given by \£„,„ | . The model that is finally developed for estimating the effective barrier height lowering A 0 bJ due to tunneling must allow for experimental findings. According to Equation 4.4.12, the critical width x c for tunneling depends on the maximum electric field \Smtx I. If for example the maximum electric field increases, the equation predicts Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I 191 that the critical width will increase. However, experiments indicate that the critical width has a maximum or limiting value x° that is independent o f the maximum electric field \Sm3x| . Muller and Kamins [53] report that appreciable tunneling occurs only if the critical width is less than 40 A, while Rhoderick and Williams [60] report an upper limit of 35 A in barrier height lowering due to the electric field. A limiting critical value half way between these two limits or x° = 37.5 A is chosen for the graph in Figure 4.4.2. The limiting critical value chosen in this study is x° =30.75 A, which is the value chosen by Jones in a recent study [58]. It is considered a representative value because the maximum electric field | = 1.4 x10s V/cm associated with this critical width is itself representative o f the upper limit for the maximum electric field. The limiting critical width x° is then used to estimate the barrier lowering by assuming that the critical width is no longer a function o f maximum electric field, so that this limiting critical width x° is used for all values o f maximum electric field. It is then assumed that all electrons of energy E 2 . E ' ( 0 ) - q \ £ m \x°' (4.4.13) will tunnel through the barrier, while all electrons o f energy £ < £ « ( 0 ) - g | S _ |* ; (4.4.14) will not tunnel through the barrier [58],[62]-[64]. This effectively lowers the barrier height by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 192 (4.4.15) Equation 4.4.14 indicates that as the maximum electric field gets stronger the effective barrier height lowering A 0 bJ due to tunneling increases, which can be seen in part (b) of the figure. The maximum electric field associated with the dashed line is given by l^max.21> l^max | ’ which leads to greater barrier lowering than for \Smix | ; i.e. A 0 bj2 > A 0 bJ. Table 4.4.1 gives the effective barrier height lowering A 0 bJ due to tunneling. The effective barrier height lowering due to both the image force effect and tunneling are considered independent processes. Therefore, the combined effective lowering of the Schottky barrier A 0 b is calculated by adding the individual effective barrier reductions associated with each process: A 0 b = A 0 bj + A<Pb , . (4.4.16) Table 4.4.1 presents the combined effective barrier lowering A 0 h and the fraction o f the effective barrier height relative to the original height. Note that since the electric fields typically range between lxlO 3 V/cm to lxlO 5 V/cm, the barrier is usually not reduced by more than 10 %. The combined effective lowering A&b o f the Schottky barrier must be incorporated into the current density boundary condition given by Equation 4.3.7. This is done by adjusting the electron density n as given by Equation 2.3.8 for the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 193 nonequilibrium case, so that the Fermi energy E F is replaced by the electron quasi-Fermi energy (x) and the conduction band energy Ec{x) becomes a function o f x. The effective barrier lowering is incorporated by subtracting A 0 h from Ec (0) = &bm, so that the nonequilibrium electron density n(0) at the Schottky boundary at x = 0 is given by " ( 0 ) = K exp j - r = [ £ > ( 0 ) - K . - A 0 ,)]} i.T k ,T Xexpf t V k ,T ~ £ ( « )]+ [ f * (0) - E f | ) exp(A 0,) = ”, expj-“ ~ [ ^ ( 0 ) - <P.(O)]lexp(A0,) = n \ 0) exp(A0fc) (4.4.17) where n'(0) represents the electron density prior to adjustment for barrier lowering; and the definitions o f the intrinsic carrier density n ,, the electrostatic potential f '( x ) , and the electron quasi-Fermi potential 0„(x) as given by Equations 2.3.10, 3.4.4, and 3.5.3, respectively, are used. The effective Schottky barrier is also lowered for the electrons originating in the metal by the same amount as for the electrons on the semiconductor side. Therefore, incorporating the effective Schottky barrier lowering into the electron current density boundary condition leads to Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission. 194 J nj<= I" Fn j- n - q v ^ . F #i)exp(A04), (4.4.18) This is the final form o f the electron current density boundary condition for Schottky contacts, and is applied at both contacts for the MSMs with the appropriate maximum electric fields associated with each contact. For Schottky diodes, Equation 4.4.18 is applied only at the Schottky contact. The Ohmic boundary condition is derived in the next subsection. The current density boundary condition for holes is given by factoring the effective barrier lowering into Equation 4.3.19, which leads to J = I? V - FPJC- P + FpX pJexp(A 0 s). (4.4.19) The barrier lowering effect is only applicable to situations when the Schottky barrier acts to block the flow o f carriers out of the semiconductor. It is also possible that the slope o f the bands precludes any blocking o f the carriers originating in the semiconductor. This can be seen by looking ahead to the E-x diagram in part (a) of Figure 4.5.2. A t x = 0 , electrons are blocked from leaving the semiconductor while holes are not, while at x = Lt the opposite is true. For the holes at x = 0 and the electrons at x = Ls , the Schottky comer no longer forms a triangular, peaked shape. The image force effect does not occur because both sources o f the superimposed electric fields point in the same direction, which causes the respective electrostatic potential energies to change in the same direction and no maximum in potential energy is formed. Tunneling does not occur because the comer is no longer narrow. Therefore, the barrier Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 195 lowering A 0 b vanishes, and the barrier lowering factor exp(A 0A) becomes unity, which effectively removes the barrier lowering effect 4.4.2 The Ohmic L im it By heavily doping the region o f the semiconductor adjacent to the metal contact, the Schottky barrier becomes so narrow that electrons can tunnel through it readily, which effectively removes the barrier so that the contact becomes Ohmic. The Ohmic limit refers to the level o f doping that is necessary to effectively remove the entire Schottky barrier to tunneling. In order to determine a representative doping and the associated doping region width necessary to create an Ohmic contact, the numerical solver is used to simulate a Schottky contact in the Ohmic limit. A GaAs Schottky diode with a Schottky barrier height o f 0.6 eV located at x = 0 is simulated. The complete removal o f the tunneling barrier corresponds to A&bJ = 0 bm = 0.6 eV . The value o f the maximum electric field \Smtx\ necessary to effectively remove the tunneling barrier is given by = (4-4.20) <P, according to Equation 4.4.14. Evaluating Equation 4.4.19 with x° =37.5 A leads to a maximum electric field o f = 1.6xlO 6 V/cm. After performing several trial simulations, it is found that the doping level required to create this value of |£ max| at x=0 is N d = 1 .4 x 1 0 19 cm '3. The simulation results corresponding Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to 196 N d =1.4x10 19 cm -3 are presented in Figure 4.4.3, which shows the E-x diagram o f a Schottky barrier in the Ohmic limit. Only part o f the diode is shown so that the narrow charge-depletion region x d can be more easily seen, since the width of xd is only about 150 A as expected. Note that in this simulation the entire substrate has the same highdoping density. For an actual diode, a similar Ohmic limit would exist if the low-doping density normally modeled in the present study were used with the high-doping density located only in the region ( x < 150 A) adjacent to the metal contact. In the present study, however, Ohmic contacts are not modeled this way because the simulation is dedicated to homogeneous substrates. The method used for modeling Ohmic contacts is presented below. When Schottky diodes are modeled in the present study, the boundary at x = 0 is treated as the Schottky contact while the boundary at x = Ls is treated as the Ohmic contact. The Ohmic contact requires a different set o f boundary conditions than the Schottky contact. The Ohmic limit provides quasi-Fermi potential boundary conditions that are used instead o f the current density boundary conditions that are required at the Schottky contact. Since the barrier at the Ohmic boundary is effectively removed and the depletion region is very narrow compared to the length o f the semiconductor, it is assumed that the semiconductor behaves as if the bulk region extended right up to the metal contact [35]. The entire region is considered charge neutral, so there is no drop in potential from the metal to the bulk semiconductor adjacent to the Ohmic contact. Therefore, the bulk region adjacent to the Ohmic contact is treated as if it were in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 197 Schottky barrier 0.5 Semiconductor Energy (eV) Metal - 0.0 - 0.5 Occupied states - 1.0 - 1.5 Charge-depletion region 0 250 500 750 1000 Position x (A) Figure 4 .4 3 Schottky barrier in Ohmic limit at x = 0, generated by simulator. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 198 equilibrium, in which case the electron density is given both by Equations 2.3.14 and 3.5.1,or n(Ls) = n, expi—?—[£f. - £ ,(£,)]) k BT ' = n, e x p j - l - ^ d , ) - £ ,(!,)]} , k BT (4.4.21) in which Er is the Fermi energy of the metal at x = Ls. This requires the electron quasiFermi potential energy E ^ i L ,) to be equal to the Fermi energy E r o f the metal at the Ohmic boundary. Since E r =+qVA and EJk(Ls) = -q4>n(LI ) , there is a boundary condition for the electron quasi-Fermi potential (Ls) at the Ohmic boundary, as given by: -q<Pna , ) = * q V A, and 0 n{Ls) = ±qVA. (4.4.22) Similarly for holes at the Ohmic boundary, Equations 2.3.15 and 3.6.1 must be equal, or P i t , ) = n, expir——[X H s ) - £ r ]l k BT ' = n, e x p |j L ; [ £ , C 4 ) - £ # ( i , ) l |. (4.4.23) Therefore, the hole quasi-Fermi potential energy E ^ ( L J must be equal to the Fermi energy Er o f the metal at the Ohmic boundary. Since Er = ^qV A and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 199 E/p (Ls ) = -q<Pp (Ls) , there is a boundary condition for the hole quasi-Fermi potential 0 p(Ls) at the Ohmic boundary, as given by - q ^ P{LI) = ^q V A, 4 .43 and 0 p( L J = ±qVA. (4.4.24) Final Form o f Boundary Conditions The final form of the boundary conditions are presented in Table 4.4.2. For the Schottky contact at x = 0 , the boundary conditions include the electrostatic potential !F(0) (from Equation 3.4.8), the electron current density J„(0) (from Equations 4.3.15 and 4.4.18), and the hole current density J p(0) (from Equations 4.3.30 and 4.4.19). For the Schottky contact at x - L s (for MSMs), the boundary conditions include the electrostatic potential 5F(Z,X) (from Equation 3.5.7), the electron current density J„(LS) (from Equations 4.3.16 and 4.4.18), and the hole current density J P(LS) (from Equations 4.3.32 and 4.4.19). For an Ohmic contact at x = Ls (for diodes), the boundary conditions include the electrostatic potential V{LS) (from Equation 3.5.8), the electron quasi-Fermi potential &„(LS) (from Equation 4.4.22), and the hole quasi-Fermi potential &P(LS) (from Equation 4.4.24). For both the MSM and the Schottky diode, these six boundary conditions satisfy the requirement that the number o f boundary conditions equal the number of equations to be solved simultaneously. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 200 Table 4.4.2 Final boundary conditions. MSM \ n. , 4.4.25 r :xp(A0A) •/„«>)=<? 4.4.26 J , (0) = ql un p ^ - ospFpP jexp(A0*) 4.4.27 r a s) = f »r \ N. <*>>. +L ——In k BT ±V. q q K ni y 4.4.28 J nil's) = d *>„.» ^ - o SJ,F„nJexp(Atf>6) 4.4.29 J P(Ls) = ?f os pFpp - v n p v ]exp(A04) ^ y 4.4.30 Schottky Diode 4.4.31 0 H(Ls) = ±VA 4.4.32 0 p (L s )= ± V a 4.4.33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 201 4.5 DC STEADY-STATE SOLUTION IN THE ABSENCE OF LIGHT The DC steady-state solution for electrons is achieved as described in Section 3.5, along with the boundary condition given by Equation 4.4.18 for Schottky contacts and Equation 4.4.22 for Ohmic contacts. The transport solver with its full set o f six transport equations is used because the full transport solver effectively produces the electron-only steady state solution for both unintentionally n-doped low-barrier MSMs and moderatelyhigh n-doped Schottky diodes. The electron density in both cases is so much greater than the hole density that the hole terms effectively drop out o f Poisson’s equation. The hole drift-diffusion equation leads to a negligible value for hole current density. With the light off, the generation rate is zero, so there is nothing to drive the thermal recombinationgeneration rate in the current continuity equations to produce substantial holes since the barrier height is relatively low. Therefore, using the full set o f equations under n-doping with a low barrier and no light essentially reduces the equations and state variables to those enumerated in Section 3.5 for the electron-only solution. The full transport solver simulations are run using the same CTH 562 Schottky diode and 0.60 eV barrier MSM that were used previously for the equilibrium case presented in Section 3.4. For the Schottky diode, the voltage at the x = Ls contact is - 0 .7 V , so the diode is in forward bias. For the MSM, the voltage at x = Ls is 1.0 V , so the Schottky contact at x = 0 is in reverse bias for electrons while the Schottky contact at x = Ls is in forward bias for electrons. The simulation results are presented as previously for the equilibrium case, using E-x diagrams and device profiles, as can be seen in Figures 4.5.1 and 4.5.2. Table 4.5.1 also presents the numerical values o f the current Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 202 (a) 1 E f 0 Uj Ec 1 Ev Efn •2 0.0 0.2 0.4 0.6 0.8 1.0 0.6 0.8 1.0 x(nm ) (b) 20 V © CO o CO Electron density (/cm *) Electric field (V/cm) Electron drift (A/cm2) Electron diffusion (A/cm2) Total current density (A/cm2) Hole density (/cm3) -10 0.0 0.2 0.4 *(nm ) Figure 4.5.1 Steady state: numerically generated (a) E-x diagram (full transport solver) and (b) profile o f Schottky diode CTH 562 at -0.7 V forward bias. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 203 1 0 1 Ec 2 Ev Efh Efp 3 0.0 (b) 0.2 0.4 0.6 x (n m ) 0.8 1.0 1.2 10 5 0) 8 0 9 -5 Electron density (/an 3) Electric field (V/cm) Electron drift (A/cm2) -•••— CO Electron diffusion (A/cm*) Total current density (A/cm2) Hole density (/cm3) -10 -15 0.0 0.2 0.4 0.6 x (n m ) 0.8 1.0 Figure 4.5.2 Steady state: numerically generated (a) E-x diagram (full transport solver) and (b) profile o f MSM at 1.0 V bias with 0.6 eV Schottky barriers. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.2 i I I 204 densities (approximate averages across the device). As expected, even in the low-doped case o f the MSM, the hole current density and hole density are negligible compared to their electron counterparts, as evidenced by the electron density being approximately 250 times greater than the hole density, and by the electron current density being approximately three orders o f magnitude greater than the hole current density. Also as expected, note that for the Schottky diode in part (a) o f Figure 4.5.1, at the x = 0 boundary the value for the eiectron-quasi-Fermi potential energy shows a discontinuity o f 0.61 eV relative to the Fermi energy. This confirms that the combined DD/TE formulation allows the quasi-Fermi potential to float at the Schottky boundary to a selfconsistently determined value, as described in section 4.2. For steady state, the displacement current is still expected to be zero, and Table 4.5.1 indicates that this is the case for both devices. The total current is expected to be constant, and this is shown to be the case in the profiles o f each device (the plot o f the total current density is mostly hidden beneath the plot of the electron current density in the two figures). T able 4.5.1 Current densities for the Schottky diode CTH 562 and the 0.60 eV barrier MSM at steady state. Diode CTH 562 Electron (A/cm2) 484 Hole (A/cm2) - 10"* Displacement (A/cm2) ~ 10*J Total (A/cm2) 484 MSM (0.60 eV barrier) -4.7x10'3 -4.4x1 O'* ~ - 10'^ -4.7x10‘5 Device Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 205 The electron density profiles also behave as expected in two important ways for the Schottky diode and the 0.60 eV barrier MSM. First, in the E-x diagram for the Schottky diode shown in part (a) o f Figure 4.5.1, the electron quasi-Fermi potential energy has a large separation from the intrinsic Fermi energy £ ,, so that their difference is relatively large. This is expected to correspond to a high electron density, and is consistent with the electron density profile in part (b). Also note that in the charge depletion region, as the quasi-Fermi energy E ^ moves farther away from the intrinsic Fermi energy £, with increasing x, the electron density increases exponentially as expected. The other extreme occurs in the 0.60 eV MSM in Figure 4.5.2, where the small separation between £ > and E, seen in the E-x diagram is consistent with a relatively low electron density in the corresponding profile. Secondly, the electron current density (equivalent to the total current density in electron-only devices) goes as the product o f the electron density and the derivative o f the electron quasi-Fermi potential energy . Since the electron current density (essentially the total current density) is constant in steady state, the slope o f the electron quasi-Fermi potential energy is expected to vary inversely with the electron density, and this behavior is observed in the profiles. Using the diode in Figure 4.5.1 as an example, at x = 0 the slope o f EM in the E-x diagram is at a maximum while the electron density in the profile is at a minimum. With increasing x, the slope o f Efa decreases and the electron density increases. For the MSM in Figure 4.5.2, the slope of E ^ is fairly constant, which is consistent with the electron density being relatively constant Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 206 The current density components exhibit more complicated behavior in the Schottky diode at steady state (Figure 4.5.1) than occurred in equilibrium, but the behavior is still as expected. In the charge depletion region, which extends about 0.2 pm into the diode, the diffusion component in the + x -direction is only slightly greater than the drift component in the opposite direction. Due to the high electron density, the slight difference in the components is enough to produce a relatively high total current density in the + x -direction o f 484 A/cm2. The current is diffusion driven in this region because the injection of electrons at the Ohmic contact simultaneously reduces the electric field in the charge depletion region and increases the electron concentration gradient due to the shortening o f the depletion width. In the section where the charge depletion turns into the bulk, the electric field reverses itself and points in the + x -direction due to the negative charge on the Ohmic contact. This causes the electron drift to reverse its direction in the bulk (at the dip in the profile plot). Since the electron density gradient is negligible in the bulk, the drift component is several orders o f magnitude higher than the diffusion component and drives the current in this region. The total current maintains its constancy in direction as well as magnitude because the driving drift component is now in the + x direction. The 0.60 eV MSM exhibits less complicated current density behavior, in which the drift component is several orders o f magnitude higher than the diffusion component throughout the device and drives the current in the - x -direction. This is caused by the net extraction of electrons out o f the MSM at the x = Ls contact, which simultaneously increases the electric field and reduces the electron density gradient. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 207 The electron-only steady state solution is also generated for the 0.6 eV barrier MSM by a reduced version o f the transport solver in which all terms associated with holes are removed. This purely electron-only simulator was the original form of the transport solver before holes were added. The purpose for this purely electron-only simulation is to determine how much error is introduced by not allowing for holes. Part (a) o f Figure 4.5.3 indicates that the band diagram is not significantly affected when holes are left out. However, part (b) indicates that the full transport solver predicts a higher electron density than does the purely electron-only solver, such that the error that results from using the purely electron-only solver is approximately 50% regarding electron density. The source o f the error in the purely electron-only solver is attributed to the absence o f the thermal recombination-generation rate term in the current continuity equations. The full transport solver calls for a higher electron density than the purely electron-only solver can produce. Since there is no light, so that the light generation term is off in the full transport solver, the only source o f the extra electrons is the thermal RG rate term, in which the np product simultaneously increases the electron density n at the expense o f the hole density p. The hole-only steady-state solution is generated with the full transport solver by simulating the 1.00 eV barrier MSM, as can be seen in Figure 4.5.4 and Table 4.5.2, in which the electron density is negligible. The RG rate in the current continuity equations is now fully engaged in producing holes to compensate for the electron-depletion. The hole current density is driven by drift in the —x -direction, as can be seen in the profile part o f the figure, where the hole drift component is several orders o f magnitude greater than the hole diffusion component. When the purely electron-only solver is used to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 208 (a) 1 0 I LU 1 Ec Ev Efh Ec (etectron-only) Efh (etedron-oniy) 2 3 0.0 (b) 0.2 0.4 0.6 x (^ m ) 0.8 1.0 1.2 0.8 1.0 1.2 10 8 E o & (A C 6 Q c g o 4 © Fun transport code Eledrorvonty code © LU 8> 2 0 0.0 0.2 0.4 0.6 x (n m ) Figure 4.5.3 Comparison o f full transport and electron-only results for steady state: (a) E-x diagram and (b) profile o f MSM at 1.0 V bias with 0.6 eV Schottky barriers. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 209 1 0 1 Ec 2 Ev Efh Efp 3 0.0 (b) 0.2 0.4 0.6 x(*im) 0.8 1.0 1.2 15 r 10 © 75 8 Hole diffusion (A/cm2) Total current density (A/an2) Electron density (/cnO Hole density (/cm3) Electric field (V/cm) Hole drift (A/cm2) 5 CO -0 -5 -10 0.0 0.2 0.4 0.6 0.8 1.0 x(p m ) Figure 4.5.4 Steady state: numerically generated (a) E-x diagram (full transport solver) and (b) profile o f MSM at 1.0 V bias with 1.0 eV Schottky barriers. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.2 210 simulate the 1.00 eV barrier MSM, numerical convergence fails before a steady state solution can be achieved. Evidently, not only is the full transport solver more physically representative of the actual transport process, the additional equations and terms included in the full solver also adds more flexibility to the numerical process so that solutions can be achieved under more extreme conditions such as those caused by the presence o f a high barrier. Table 4.5.2 Current densities for 1.00 eV barrier MSM at steady state. Device MSM (1.00 eV barrier) Electron (A/cm2) -8.9xl0'u Hole (A/cm2) -2.5xl0*‘ Displacement (A/cm2) ~0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Total (A/cm2) -2.5x10' 1 211 CHAPTERS SIMULATION OF THE ISOLATED DEVICE 5.1 INTRODUCTION The operating behavior o f the MSM photodetector is studied through device-only simulation. There are two reasons for simulating the isolated device rather than simulating the photomixer circuit, which consists o f the device integrated into its embedding circuit F irst the measurement of those variables that originate in the device can be performed faster and without being influenced by the effects o f the additional variables that are introduced by the embedding circuit Secondly, the accurate assessment o f the effects o f the embedding circuit variables on device performance requires prior knowledge o f device performance in isolation. Device performance is gauged using the two figures o f m erit optical responsivity and bandwidth. Section 5.2 explains the numerical techniques used to simulate the isolated device. The output o f the device-only simulation consists of the values o f the six state variables as functions o f position and time. Since the solution o f the transport equations is a numerical rather than analytical process, the differential equations are discretized, and the state variables are associated with grid points along the x-axis that form a one dimensional mesh. In the first phase o f the simulation, a Poisson solver is used to calculate the equilibrium solution by solving the second order form o f Poisson's equation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 212 for the electrostatic potential. The Poisson solver uses a Newton-Raphson technique that determines the electrostatic potential separately at each grid point. The advantage o f this approach is that global convergence is achieved. The equilibrium values o f the other five state variables are generated from the electrostatic potential in post-processing steps. The resulting equilibrium values o f the six state variables are the initial values for the transport solver. In the second phase of the simulation, the transport solver allows the state variables to evolve through time under a specific set o f operating conditions regarding voltage and light. The transport solver uses a Newton-Raphson approach that solves the transport equations simultaneously, which is a more efficient approach than the Poisson solver but has less convergence capability. Section 5.3 explains how the voltage and light are ramped up to constant baseline values before modulating the light to generate AC signals. The new current density boundary condition is tested in section 5.4. The I-V characteristics for Schottky diodes are simulated and compared to both analytical and experimental diode curves to test the accuracy o f the new boundary condition. The simulated I-V characteristics are then compared to diode curves simulated by using the three existing formulations for the current density boundary condition, with simulations conducted both in the absence and presence o f constant light. The objective o f these comparisons is to determine how the formulations differ regarding the physics o f device operation at the Schottky boundary. Finally, frequency response curves for the MSM photodetector are generated using all four boundary condition formulations in a second comparison test. The objective of these comparisons is to determine how the formulations differ regarding bandwidth and responsivity. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 213 The limitations o f the device simulator are explored in section 5.5. The two objectives are to measure convergence efficiency as a function o f parameter configuration, and to determine if there are parameter configurations that cause the simulation to fail. Section 5.6 is dedicated to the study o f device performance by focusing on the effects o f three device parameters. Barrier height is studied by using the two extremes o f 0.6 eV and 1.0 eV for Schottky barrier heights, as based on values reported in the literature. Substrate growth temperature effects are studied by using recombination time constants associated with low-growth- and conventional-growthtemperature GaAs. Previous studies indicate that there is an optimum growth temperature regarding the trade-off between increasing bandwidth and decreasing responsivity. The objective is to determine the frequency responses for a short (1 ps ) and long (1 ps) recombination times, corresponding to the low-growth- and conventionalgrowth-temperature regimes, respectively. Finally the effects o f device length are studied by comparing the frequency responses o f a shorter (0.7 pm) and longer (1.5 pm) device to the frequency response o f the standard MSM (1.1 pm) modeled in this study. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 214 5.2 5.2.1 NUMERICAL TECHNIQUES Poisson Solver In the first phase o f the simulation, the Poisson solver is used to calculate the equilibrium solution by solving the second order form o f Poisson's equation (3.4.9) for the electrostatic potential !P(x). Since the Poisson solver discretizes Poisson’s equation, the x-axis is first divided into a series o f N grid points with variable spacing called a mesh. The electrostatic potential !P(x) is determined only at the grid points, with the value o f the electrostatic potential at grid point x, being given by IP,. Initial values o f electrostatic potential !P° are assigned to each grid point x, prior to calculating the equilibrium solution. Since the Poisson solver is globally convergent, the initial values are not so critical. The two boundary values for the electrostatic potential are given by Equation 3.4.8 for Schottky contacts and by Equation 3.4.10 for Ohmic contacts. Since Equation 3.4.10 also characterizes the electrostatic potential in the bulk, this equation assigns the initial values o f electrostatic potential 5P,° for the remainder o f the mesh. After constructing and initializing the mesh, the Poisson solver discretizes the second derivative o f the electrostatic potential IP, at each grid point x ,. The second derivative o f the electrostatic potential is approximated through a finite difference scheme for second derivatives which is now derived [65],[66]. expansion o f the electrostatic potential VM A Taylor’s series at grid point x,+l gives +A x,P,' + ^ L < r " + ~ , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.2.1) 215 where Ax, = x,+, - x , . Similarly, a Taylor’s series expansion o f 5P,_, at grid point x,_, is given by (5-2.2) where Ax,_, = x, - x ,_ ,. Multiplying Equation 5.2.1 by Ax,_, and Equation 5.2.2 by Ax,, adding the resultant equations together, and solving for the second derivative o f the n electrostatic potential 5P, at x, gives the LHS o f Poisson’s equation (3.4.9) as d2¥ dx2 2 Ax, + Ax,_, 1 A*, Ax,-, J (y ,). (5.2.3) where terms greater than second order have been dropped, and g {(5P,) is a continuous function of IP, that is unique for each grid point x, since IP,,, and depend on x ,. The discretized form o f the RHS o f Poisson’s equation is given by q r, k*T j (5.2.4) where vv(!P,) is a continuous function o f IP, , thus completing the discretization o f Poisson’s equation. A Newton-Raphson technique is then used by the numerical solver to find a solution to the discretized fonn o f Poisson’s equation. The numerical solver adjusts the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 216 values o f *¥, at each grid point until the LHS o f Poisson’sequation equals the RHS, within some tolerance, for all the grid points. Once such a self-consistentsolution has been achieved, «,(!?,)= «{!?,), or (5.2.5) at every grid point x ,. Prior to achieving the self consistent solution for the ¥ , , the residual function / ( ¥ , ) , as defined by /,(* ',) = *, i r h M r , ) , (5.2.6) represents the difference between the LHS and the RHS o f the Poisson equation. Therefore, finding the solution for the set of ¥ , is equivalent to finding the roots o f the set o f residual functions f ( !P, ), which is the method used by the numerical solver. The problem now consists o f N —2 algebraic equations o f the residual function form given by Equation 5.2.6. The two equations (3.4.8 or 3.4.10) for the boundary values !F(0) and ¥ ( L S) , which are automatically in discretized form, are also cast in the residual function form, so that there are N residual functions. The steps employed by the numerical solver to determine the roots of the N residual functions are based on an iterative process that is structured as two nested loops. The outer loop sequentially selects the grid point x,, and the inner loop applies the Newton-Raphson technique at x, to find the root o f the residual function / " ( ¥ ,) , in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 217 which m is the outer loop step. The root o f the residual function f " ( ¥ ,) is determined by first linearizing the residual function f ” (IF,) through f . ^ r ' ) = / ; { r ; ) + ( r r ' - r / )/,■'(!r ; ) , (5.2.7) where the value o f the electrostatic potential at x, is IP/ for step j o f the inner loop, f mt y l ) the derivative f " (?P/) are determined using the original residual function, and yr/ +l is theroot o f the linear approximation o f f " ('Fi) for step j + 1.Since the root of the linearizedform o f f " ) occurs when / j*(*pr/*‘)= 0 , Equation 5.2.7 can be solved for !P/ +1 in terms o f the known quantities at step j, v ; ' = V! - A - f c l . (5.2.8) /r (* v ) The process repeats with *¥J*X becoming the value o f the electrostatic potential for the original function / " (*P() during step 7 + 1 . The inner loop iterations continue (in principle) for K +1 steps until the root to the original function is found, which occurs when the term in the denominator of the fraction in Equation 5.2.8 vanishes, i.e. / - (iP*) = 0 , so that the difference V f" - iF/ vanishes. The outer loop updates the electrostatic potential at every grid point for each step m. At step m +1, the initial value of the residual function f" * x(iP,) associated with grid point Xj is based on the values o f !PM and that were set in the previous pass through Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 218 the entire mesh via the outer loop. Since !P(_, and are expected to change with each pass through the mesh, the next time that grid point x, is encountered during step m + 1 , the root to the updated version f " * 1( f , ) of the residual function at x, will have to be determined again. Each pass through the mesh brings the set o f !P, closer to their final self-consistent values. The passes terminate when the difference between successive values o f the electrostatic potential at x, are less than some tolerance for all o f the x ,. In the present study, the tolerance is set to 2.59x10'* V . After the Poisson solver determines 1ri through the Newton-Raphson iterative technique, the equilibrium values for the remaining five state variables are calculated for each grid point x, based on the final IP, and on the equilibrium assumption o f zero current. The equilibrium electric field S, at x, is determined by approximating the derivative o f the electrostatic potential through the finite difference approach for first derivatives, which is now derived. Multiplying Equation 5.2.1 by Ax,2., and Equation 5.2.2 by Ax,2, subtracting the resultant equations, and solving for the first derivative of t the electrostatic potential !P, at x, gives v ' = Ax,-_1 . irr f TM . 'r * I Ax, (Ax,_, + Ax,) I 1 m Ax A*,-. Ax Ax > -V i Ax, " Ax , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.2.9) 219 t so that the electric field 2, at x, is given by 2, * -*¥, . The other four state variables <PnJ, <PpJ, J n j, and J pj are all set to zero at equilibrium. These discretized equilibrium values o f the six state variables become the initial values for the transport solver. 5.2.2 Transport Solver In the second phase o f the simulation, the transport solver calculates the values of the six state variables as a function o f time by solving all six transport equations (3.6.5 3.6.10) for any condition of bias and illumination. Since there are now six differential equations at each grid point, discretization leads to 6 N nonlinear algebraic equations. In order to achieve a significant increase in the convergence rate, these discretized equations are solved simultaneously rather than by the method o f the Poisson solver in which the state variable at each grid point is solved separately. The convergence criteria become more stringent, however, requiring the simulation to begin at equilibrium since the initial values at equilibrium are known through the Poisson solver. The LHS o f the six transport equations (3.6.5 - 3.6.10) are discretized through a trapezoidal finite difference scheme for first derivatives, as given by [50],[66] Sf * ' 2 Ax, Axf_, (5.2.10) where S f is the state variable k at grid point x ,. The RHS o f each o f these equations in discretized form is given in Table 5.2.1 for the bulk region ( x ,, i * l , N ) and at the boundaries (x, = 0 and x s - Ls ). The equations are presented in the table in the residual Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 220 function format used by the transport solver, and retain the equation number o f the undiscretized form. The time step At = th —th_x is small enough that the time derivative in the current continuity equations is adequately characterized by the difference between the values o f the carrier density at the current time step th and the previous time step th_x, divided by A t . After initializing the six state variables to the equilibrium values, the transport solver begins an iterative process in which the Newton-Raphson technique is applied simultaneously to all 6N discretized equations for each iteration. The linearized residual functions are structured in a form analogous to the following rearrangement of Equation 5.2.8: /,"'(y /)a p ,r l = - / " ( * 7 ) . where the correction factor (5-2.il) is given by s r /* 1 = ? 7 +I - r / . (5.2.12) With this arrangement o f the linearized form, it is apparent that when the correction factor vanishes, the root o f the original residual functionf " (!P,) since the vanishing o f SPf*1 requires. Unlike at x, will be found, the Poisson solver, the simultaneous solution approach used by the transport solver requires that each residual function f tk for state variable k and grid point x, become a function of all six state Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 221 variables at each grid point xy [49],[50]. For example, the residual function f * for the electrostatic potential !F, at x { is given by f7 2 J..X , (5.2.13) so that f * is a function o f 6jV variables (with x, = 0 and x N = Ls ). This causes the derivative of the residual function o f the electrostatic potential f ? at grid point x, to be expressed as f7 I K 6 pu-r (5.2.14) j mX *-1 0 5 j where 5* represents state variable k with the partial derivative evaluated at grid point j . The entire system consists o f 6N expressions analogous toEquation 5.2.14 for six state variables at each gridpoint. The application each o f the o f theNewton-Raphson linearization technique to this system leads to F ’{U)Sa = -F {u ), (5.2.15) where the state variable vector u , the correction vector 5 u , and the residual function vector F{u) are respectively given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 222 ~ S * i~ ssx ' St ~ fL fx* f? fL //' F{u) = f Jp SSp, SF 2 Vz (5.2.16) f? /p * . r i . and the Jacobian m atrix o f partial derivatives F '(u) is given by 'M L dFx ML Qgy d&nl Of? d 0 p.i df? df? a /- , SJ p-i df x• dF, _ML a / p„v ML dFx M L d*Fx ML dFx * fi9 dFx M L. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. . (5.2.17) 223 Equation 5.2.15 is solved with a nonlinear algebraic equation solver developed by IMSL that iteratively reduces the magnitude o f the correction vector Su until its magnitude is less than some tolerance. In the present study, the state variables that are monitored regarding the tolerance are the three potentials, and the tolerance is again 2.59 xlO -8 V . T able 5.2.1 Transport solver equations for boundaries and bulk in residual function form and with the RHS in discretized form (LHS discretized form is analogous to Equation 5.2.10) for time step th. [NOTE: It is understood that the subscript for nt refers to intrinsic and not to the grid point x ,.] Bounday a t x = 0 4.4.25 4.4.26 4.4.27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 224 Table 5.2.1 (cont.) B ulk Region f? Ji ft -M i J‘ &1 3.6.5 & 1 -(-ft) 3.6.6 - i- « ,e x p LV - * J + n, exp r*. _ - ^ — (\ - F ,i + 0 pj p j)/ + v d - na 3.6.7 exp[ _ 2 _ ( ^ + 0 cbc 3.6.8 7' ar ^ n , |_**r 3.6.9 dx -G l -R G , 3.6.10 I*1 — i /i, exp —— (- !P, + 0 .) A/I ' |_*«r —n, exp kBT V f + 0 P. JJ/ -G l -R G , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 225 Table 5.2.1 (cont.) Boundary a t x = Ls MSM 0 bm f s ^ s fu k aT , ±V. 4.4.28 v ”, j = J n.N ~ y ~ v „ F n” j exp(A0A) ( D ^ = ^„jv -<1 v spFpp - u , Xp ^ ~ e x p (A 0 j 4.4.29 4.4.30 Diode / ; = n - '* £ i r { ^ . 9 5.2.3 1 4.4.31 ft- = f ..v - ( ± ^ ) 4.4.32 /* ' 4.4.33 (± ^ ) Normalization The differential equations in the Poisson and transport solvers are normalized. This reduces the number o f repetitive operations involving coefficients. Most importantly, normalization makes the derivatives comparable in magnitude, which increases the effectiveness o f the numerics, especially for the Jacobian matrix. Table 5.2.2 lists the normalization factors and the variables that are normalized. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 226 Table 5.2.2 Normalization factors. Variable Normalization Factor Magnitude Carrier densities n, p "o 1 x 10 *° cm *-1 Potentials IP, <Pn, 0 p Position (distance) x Vt = * sT /q 2.586 x 10*2 V Vt /L d Do 5.96 x 10" V/cm 1 ciriVs D ./fT <FoDoI l d 38.7 cnT/Vs 3.69 x 101 C/cm~s Electric field S Diffusion coefficients Dn, Dp Mobilities , pp Current densities J„, J p 5.2.4 4.339 x 10** cm I' d = Velocities v s , v d Time t Do/L d 2.30 x 103 cm/s I'D/Do 1.88 x 10*‘‘ s Frequency co D o/L l 5.31 x lO ,0 Hz Fixed Point Iteration for Current Density at Boundary In order to determine the electron and hole current densities at the Schottky contacts, the carrier drift-diffusion velocities are calculated using a fixed-point iteration scheme suggested by Adams and Tang [32]. The drift-diffusion velocity v djn for electrons that enter the Schottky contact region is determined from the value o f the electron current density J„Mlk at the semiconductor mesh point closest to the boundary, with v djt given by = qn Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.2.18) 227 where J njndk is the value self-consistently derived from the drift-diffusion equations. The dedicated use for the simulator in this study is to characterize transport in MSMs, for which carrier injection occurs primarily within the device via illumination as opposed to injection from the metal contacts. Interior carrier injection enables a scattering quasi equilibrium to be reached by the time the excess carriers reach the contacts. Therefore, the bulk region in the present study goes right up to the Schottky boundary. During a given time step, the value o f J njm0t for the previous time step is used in Equation 5.2.18 to generate the initial value o f odjl for the present time step. The initial value o f udJI is then used to calculate the electron surface velocity (Equation 4.3.11) and electron fraction (Equation 4.3.14), which in turn are used in the electron current density boundary condition (Equations 4.4.18) for the present time step. This represents a single iteration in a fixed-point iteration scheme, in which each variable is a function o f the other as indicated in Equation 5.2.18; i.e., the iterations seek that fixed value o f udJ, that returns udn through the function 5.2.18. Iterations continue for the present time step until the difference between successive values o f udJ, are less than some tolerance. By making the time steps small enough, one iteration is sufficient. This same fixed-point iteration scheme is used to determine the hole current density at the Schottky boundary. Unlike the case for electrons, there are two carriers for which the drift-diffusion velocities must be calculated, i.e. light and heavy holes. An equation analogous to Equation 5.2.18 would yield a combined hole drift-diffusion velocity, and still leave the individual hole velocities v d pi and v d ptl undetermined. In Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 228 order to calculate udpl and udpll, it is assumed that the same effective force acts on both light and heavy holes, so the change in wave vector magnitude for each type o f hole is equal, or kdpt = kdph = kd p . The wavevector magnitude kd p must be extracted from the hole current density J pJndk in the bulk region. In order to do this, J pjmlk is expressed in terms of the individual contributions from each type o f hole: (5.2.19) where the densities o f light and heavy holes are determined by Pi = 2 xk„T i = l ,h . (5.2.20) Since the total hole density p is the sum o f p, and p h,p is given by P= '2 x k . T ',n exp (5.2.21) i = I, h . (5.2.22) The hole drift-diffusion velocities od pi are given by hkd.P Vd.p, = ------- -I * The wavevector magnitude kJ p is then derived by multiplying and dividing each term on the RHS o f Equation 5.2.19 by p, and using Equations 5.2.20 - 5.2.22 to give Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 229 , J p jm lk dp qph f m *V2 , _ m pl * 3/2 p ft m* /*' *V2 +rnrt p" *,/2 j (5.2.23) In the fixed-point iteration scheme to determine the hole current density at the boundary, kd p is derived from Equation 5.2.23 and is used in Equation 5.2.22 to generate the hole drift-diffusion velocities ud p, . The values for v J pi are then used to update the hole surface velocities (Equation 4.3.26) and hole fractions (Equation 4.3.29) for the hole current density boundary condition (Equations 4.4.19) in the next iteration o f the numerical solver. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 230 S3 METHOD OF ILLUMINATION UNDER BIAS The physical quantities associated with normal device operation such as current and voltage are determined while the device is in an AC steady state. An AC steady state means that the current peaks do not vary in magnitude over time. The evolution to AC steady state depends critically on the time step and how the voltage and illumination are applied. Furthermore, in order to adequately resolve the time-dependent operation o f the device resulting from exposure to modulated light during AC steady state, the time step At should be at least two orders o f magnitude smaller than the modulation period. The time step should also be large enough to allow the numerics to behave favorably and to provide a simulation time that is reasonable. Based on these upper and lower limits, the time step At ranges between 0.04 ps and 1.0 ps, depending on the modulation frequency. After the choice of an appropriate time step, the simulation begins by establishing the conditions o f constant voltage and baseline light intensity under which the device will operate as it evolves into an AC steady state. In order to ensure convergence during each time step so that this AC steady state can be reached, the ramping o f the bias and baseline illumination are conducted separately and gradually. A typical example o f the simulation method for device illumination under bias is presented in Figure 5.3.1. The figure illustrates the time evolution o f the 1.0 eV barrier MSM to AC steady state under illumination with light o f modulation frequency 30 GHz and intensity 1 KW/cm2 under a bias o f 1 V. The time step is 0.04 ps during the ramping o f the voltage and light, and is 0.33 ps during the modulation o f the light. Time zero represents the system with its initial equilibrium values for the state variables as provided Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 231 3.0e+025 1.2 0.8 < E, c £ 2.0e+025 0.6 1.5e+025 O ■o c. m 0.4 1.0e+025 0.2 93 0 co 1 5.0e+024 - 0.0 - 0.2 0 .0e +000 Voltage Current Generation rate Recombination rate x 1000 -0.4 - 0.6 Generation and Recombination Rates (e'h*-pairs/cm3s) 2.5e+025 1.0 -5.0e+024 -1.0e+025 50 100 150 200 250 300 Time (ps) Figure 5.3.1 Time evolution to AC steady state for 1.0 eV barrier MSM under 1 V bias that is exposed to light o f 1 KW/cm 2 intensity and modulation frequency 30 GHz. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 232 by the Poisson solver. The transport solver is allowed to operate for 30 ps without any voltage being applied. This allows for the decay o f the transient that results due to the slightly different variable configurations o f the two solvers at equilibrium, as seen by the spike in the current at t = 0. After the 30 ps pause, the voltage is ramped up linearly from 0 V to 1 V between 30 ps and 40 ps. The solver is then allowed to operate at constant voltage with no light for 8 ps to allow the voltage transient to completely settle out, as seen by the decay o f the rectangular current spike that was generated during the voltage ramp. Between / = 48 ps and 52 ps, the light is turned on using the following exponential ramp equation: , GL(t) = GL0 n k,At 30 (5.3.1) \^A t/nmp J in which the baseline generation rate Gt 0 is given by Equation 2.4.10, k, represents the light ramp time step, and ^ t%ramp represents the total light ramp time. This exponential ramp has the effect o f turning on the light extremely slowly at first, which gradually reconfigures the state variables into the illumination mode. Once in the illumination r mode, the system can be exposed to exponentially increasing light intensity and still achieve convergence. Note in Figure 5.3.1 that the recombination rate becomes elevated (from ~ - 10 * e h * -p airs/cm 3 prior to the light being turned on) to a value o f 5.9xlO 24 e~h* —pairs/cm 3 after the generation rate reaches its full baseline value o f 1.4x10“ e~h* - pairs/cm3, where e‘h* - pairs refers to electron-hole pairs. The graph Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 233 gives the negative o f the RG rate so that the phase can be more easily compared to the generation rate. Initially, the recombination rate is positive, which indicates that thermal generation dominates the RG rate term prior to illumination. During illumination, the RG rate is negative, which indicates that recombination dominates during illumination, hence the use o f recombination to designate the RG process. The solver is allowed to operate at constant light intensity until the recombination rate reaches a constant value, which has occurred by the time 108 ps have elapsed after completion o f the light ramp (at t = 160 p s ). Beginning at t = 160 ps, the light is modulated according to Equation 2.4.26 for the remainder o f the simulation. The modulation index M is set to 0.9 to prevent the system from reentering the nonillumination mode when the phase of the sine reaches 270°, which can cause the simulation to fail to converge. As indicated in Figure 5.3.1, there is a slight time lag for the recombination peaks relative to the generation peaks, and a somewhat longer though still slight time lag for the current peaks relative to the generation peaks. Note that the peaks in the current magnitude are consistent with the peaks in the generation rate because the current is in the - x -direction. The decay o f the modulation transient is best seen by tracking the reduction in peak height for the recombination curve, which appears to reach a steady state height after the first peak. The simulation continues until the change in successive current extrema is less than 0.5%, at which point AC steady state is assumed to have been achieved. The final state variables are those that are associated with the last peak in the current, as can be seen in the figure just past 300 ps.. The primary output o f the device simulation is the AC peak Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 234 /°c (vm) as a function o f modulation frequency. Although the AC peak is composed of the fundamental frequency component as well as the higher order harmonic components, the higher order components are negligible, so the AC peak is identified with the fundamental frequency. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 235 5.4 5.4.1 TEST OF NEW CURRENT DENSITY BOUNDARY CONDITION Comparison to Analytical I-V Characteristics The new current density boundary condition is first tested by comparing current voltage (I-V) characteristic curves generated by the simulation to analytically generated (I-V) curves. Schottky diodes are chosen as the devices to be simulated because analytical I-V curves do not exist for MSMs. In an I-V curve, the current at a given voltage represents the steady state current for that voltage. The analytical expression that calculates diode current l(V A) as a function o f the applied voltage VA is given by [54] (5.4.1) where the saturation current I s is given by (5.4.2) in which the barrier height on the semiconductor side o f the Schottky boundary 0 ^ refers to the equilibrium value (see part (b) of Figure 3.4.1). The device parameters match the baseline MSMs used in the present study, including an unintentionally n-doped GaAs substrate with N D = 3 .5 x l0 12 cm -3, time constants r„ = 1 0 “* s and r p = 10’* s, length Ls = 1.1 p m , current cross-sectional area 0 bm =0.6eV and <Pbm = 1 .0 e V . = 3.5 xlO ' 7 cm 2, and barrier heights The applied voltage ranges from reverse bias with Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 236 VA = -2.5 V to forward bias with VA - 0.5 V for the 0.6 eV barrier diode and VA = 0.9 V for the 1.0 eV barrier diode. [NOTE: The sign o f VA is opposite to the expression o f VA in the previous chapters and in the simulator because by convention VA = V (0 )-V (L s) for diode curves.] The time step is long enough to simulate steady state at each voltage increment, with At = 1 m s. Due to the device length, the FD mobility model is used, and the simulations are conducted in the absence o f light. Figure 5.4.1 presents the I-V curves generated by the simulation and by Equation 5.4.1. The top two graphs compare the total currents for the 0.6 eV barrier diode in part (a) and for the 1.0 eV diode in part (c). The bottom two graphs give the electron and hole currents generated by simulation that correspond to the total currents given in parts (a) and (c), with part (b) corresponding to part (a) and part (d) corresponding to part (c). For the 0.6 eV barrier diode shown in part (a) there is good agreement between the analytical and simulated results. The corresponding electron and hole curves in part (b) indicate that the dominant current is the electron current For the 1.0 eV barrier diode shown in part (c), there is poor agreement between the analytical and simulated results for the reverse bias region and improved agreement in the forward bias region. The corresponding electron and hole curves in part (d) show that the agreement between the analytical and simulated results is good only when the electron current dominates. These results are expected because the analytical expression for diodes given by Equation 5.4.1 is based on Schottky barriers to electron flux that are shaped as given in part (b) o f Figure 3.4.1. The corresponding barrier to hole flux from the metal has a completely different shape, as can be seen in the valence band region o f part (b) o f Figure 3.4.1, for which Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 237 (a) o Total Current Total Current Low Barrier High Barrier -5 £ •4—” < c £ -10 c ® § -15 O o> „ O o s -20 • —— -25 Analytical Simulated -10 -15 -20 Analytical Simulated -25 -30 -30 -3 -2 -1 0 -2 Voltage (V) (b) o 0 High Barrier Low Barrier I5 o 05 o 0 Voltage (V) -5 £ -1 •5 < -io r -10 C £ 3 o -15 __ -20 15 -20 Electron Hole -25 Electron Hole -25 -30 •30 -3 -2 -1 0 Voltage (V) 3 ■2 1 0 Voltage (V) Figure 5.4.1 Simulated I-V characteristics for Schottky diodes with 0.6 eV barrier [(a) and (b)] and 1.0 eV barrier [(c) and (d)]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 238 Equation 5.4.1 does not apply. Therefore, when the hole current is comparable to the electron current, agreement between the analytical and simulated I-V curves is expected to be poor. The good agreement between the simulated and analytical characteristics for the diode in which the electron current is dominant indicates that the new formulation o f the current density boundary condition is accurate. 5.4.2 Comparison of Simulation with Experiment The new current density boundary condition is tested by comparing current voltage (I-V) characteristic curves generated by the simulation to diode (I-V) curves obtained from experiment. The experimental diodes that are chosen have lengths that are representative o f the different ranges of applicability of the two mobility models. These lengths are presented along with the other device parameters in Table 5.4.1. The device parameters and experimental I-V characteristics for the four diodes are obtained from Zirath [67] and Hjelmgren [18], and the simulations are conducted in the absence o f light. Figure 5.4.2 shows the simulated and experimental I-V characteristics for the diodes, with the simulated characteristics including both the FD and the FI mobility models. For the 1.0 |im diode in part (a) and the 2.5 pm diode in part (b), the simulations that use the FD mobility model show good agreement with experiment for the entire range tested. For the 0.35 mm diode in part (c), the simulation that uses the FD mobility model shows better agreement with experiment than does the simulation that uses the FI mobility model for most o f the range tested. However, for the 0.12 pm diode in pan (d), the simulation that uses the FI mobility model shows better agreement with experiment than Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 239 (a) 2 CTH562: 1.00 1 Experiment FD simulation Rsimulation < E c <o O) o (b) 0 <u 3 o S1 1 ■2 ■3 0.4 2 M45-116: 2.50 nm 1 Experiment FD simulation FI simulation 0 1 2 0.6 0.8 1.0 ■3 0.4 1.2 0.6 Voltage (V) (c) (d) Experiment FD simulation R simulation 1 1.0 1.2 Voltage (V) CTH188: 0.35 jim 2 0.8 2 2E1: 0.12 um 1 Experiment FD simulation FI simulation 0 0 <u O) o £ o 1 /• 1 • • 2 3 -2 ■4 3 5 0.4 0.6 0.8 1.0 Voltage (V) 1.2 0.4 0.6 0.8 1.0 1.2 Voltage (V) Figure 5.4.2 Comparison o f experimental and simulated I-V characteristics for Schottky diodes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 240 does the simulation that uses the FD mobility model for most o f the range tested. These results are consistent with the 1.0 pm device length limit on the FD mobility model, since as the device length becomes progressively shorter than 1.0 pm, the accuracy o f the FD mobility model at predicting the I-V characteristic decreases. The good agreement with experiment over such a wide range o f diode lengths indicates that the simulation with the new formulation o f the current density boundary condition is accurate. T able 5.4.1 Experimental Schottky diode material parameters. Diode CTH 562 lb*J M45-116lA/J CTH 188io,J 2E1 5.4.3 Barrier (eV) 0.879 1.020 1.020 1.027 Doping (cnf*) 2.5x10*° 6 . 1x l 0 ,b 2 .0 x l 0 1A 4.0x10*° Length (pm) 1.00 2.50 0.35 0.12 Area (pm2) 8.0 28.0 3.8 2.5 Schottky Diode Simulations Compare Formulations Two sets o f I-V characteristics for the Schottky diodes are generated by simulations using all four formulations o f the current density boundary condition in order to compare the results, with the first set done in the absence o f light. The diodes have the same parameters as were used previously in the subsection that compared simulation to analytical curves; i.e., those o f the baseline MSMs, and only the FD mobility model is applied. The forward- and reverse-bias I-V characteristics are plotted in Fig. 5.4.3 for the 0.6 eV barrier diodes in part (a) and the 1.0 eV barrier diodes in part (b). Although the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 241 Low Barrier < -10 c £ 3 -15 o O) o -20 Formulation-1 Formulation-2 Formulation-3 Formulation-4 -25 -30 ■3 •2 •1 0 1 0 1 Voltage (V) (b) < High Barrier -10 -15 -20 Formulation-1 Formulation-2 Formulation-3 Formulation-4 -25 -30 •3 -2 1 Voltage (V) Figure 5.43 Comparison o f Schottky diode I- V characteristics for current density boundary condition formulations for (a) 0.6 eV and (b) 1.0 eV barriers. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 242 four formulations produce similar I-V characteristics, the electron drift-diffusion (DD) velocities and electron densities are different for forward bias, indicating that the physics o f the internal operation o f the device at the Schottky contact is different for the formulations. This can be seen in Table 5.4.2, which lists the electron and DD velocities and carrier densities at the Schottky boundary for both barrier heights under 0.5 V forward bias. As indicated in the table, the DD velocities are higher for the models that use a variable surface velocity while the electron densities are lower. The variable nature o f the surface velocity in formulations 2, 3, and 4 enables the now substantial driftdiffusion velocity to contribute to the surface velocity. The electrons are then swept out o f the semiconductor at a higher rate, which also lowers the electron density. This increase in the surface velocity is greater for models 2 and 4 because in model 3 the surface velocity is divided by two due to the normalization. T able 5.4.2 Electron DD velocities and densities at Schottky boundary for diodes under 0.5 V forward bias in the absence o f light. Form ulation 1 2 3 4 0.6 eV Barrier DD Velocity Carrier Density (cm/s) (cm*3) - 1. 1x 10 ' 4.1xl0‘‘t - 1.0x 10* 4.4x10“ -1.4x10' 3.2x10*^ - 1. 1x 10* 4.1x10" 1.0 eV Barrier DD Velocity Carrier Density (cm/s) (cm*3) 6 .6 x 10° -1.3x10' 2 . 1x 10' -8.2 x 10* 5.5x10* - 1.8x 10' 2 . 1x 10' -8.2 x 10* Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 243 In the second set o f comparison simulations, unmodulated light o f wavelength 800 tun and intensity I KW/cm 2 is introduced to see how it effects the DD velocity and carrier density. The same two Schottky diodes that were used in the absence of light are simulated, again with the FD mobility model. Since diode photodetectors operate under effective reverse bias, only reverse bias is simulated. Under reverse bias and illumination, the 0.6 eV barrier diode and the 1.0 eV barrier diode produce the same results regarding current, DD velocity, and carrier density; therefore, only the results from the 1.0 eV barrier diode are presented. Fig. 5.4.4 illustrates the electron, hole, and total currents across the length o f the device. Since all four formulations give the same current o f 0.086 mA, only formulation-4 is graphed. The results for the semiconductor side o f the Schottky contact (x = 0*) are presented in Table 5.4.3 for an applied voltage o f - 1 0 V . They indicate that when the Schottky diode is illuminated by light, the net flow o f electrons and holes just inside the Schottky contact is out o f the semiconductor even under conditions o f -10 V bias. This occurs because the light generates a high concentration o f excess electrons that leads to a large concentration gradient near the Schottky barrier. Near the boundary, the diffusion current that tends to sweep the electrons out o f the semiconductor and into the metal dominates the drift current that tends to push the electrons to the opposite contact. There are significant differences between the four formulations in both the DD velocities and the densities at the Schottky boundary. As indicated in Table 5.4.3, again the electron DD velocities are higher for the models that use a variable surface velocity while the electron densities are lower. As shown in both the table and Fig. 5.4.4, the holes form the dominant portion o f the current Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 244 0 Electron current Hole current Total current Log Current (A) ........ -9 -12 “ -15 -i -18 u0.0 0.2 l_ 0.4 0.6 0.8 1.0 1.2 * (n m ) Figure 5.4.4 Photocurrent generated by illumination o f a 1.0 eV barrier Schottky diode with 1 KW/cm 2 constant light at -10 V bias. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 245 density in the Schottky boundary region during illumination, indicating that the contact is effectively in high forward bias for holes. The hole DD velocity is significantly greater and the hole density is significantly less for formulation-4 than for the other models. Table 5.4.3 Electron and hole DD velocities and densities at Schottky boundary o f 1.0 eV barrier diode under unmodulated lKW/cm 2 illumination for FD mobility model at —10 V bias. Form ulation 1 2 3 4 5.4.4 ElectTons DD Velocity Carrier Density (cm/s) (cm'3) - 1.2 x 10° 2.5xlOlu -2.7x10° 9.8x10’ - 1.6 x 10° 1.9xl0,u -2 .0 x 10* 1.5xl0,u Holes DD Velocity Carrier Density (cm/s) (cm'3) -4.2x10b 3.6x10“ -5.7x10° 2.7x10“ -5.7x10* 2.7x10“ 1.4x10“ - l.lx l0; MSM Photodetector Simulations Compare Formulations Frequency responses to modulated light intensity are generated by simulation using all four formulations o f the current density boundary condition for an MSM photodetector. Each MSM unit cell consists o f two 1.0 eV Schottky contacts on 3.5 xlO 12 cm -3 n-doped GaAs, and the light wavelength is again 800 nm [the use of only the 1.0 eV barrier height is explained in the next section]. The electrodes are placed in trenches 1.1 pm apart to enhance the optical response, and the same one-dimensional model is applied as was done for the diodes [30,68]. Due to the use o f high-frequency modulated light, simulations are now run with both the FI and the FD mobility models. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 246 Figure 5.4.5 illustrates the results at I V bias, in which the current peak at the fundamental frequency is plotted versus the fundamental frequency. Parts, (a) and (b) are the frequency response curves at low peak light intensity (1 KW/cm2) for the FD and FI mobility models, respectively. Parts (c) and (d) are the frequency response curves at high peak light intensity (25 KW/cm2) for the FD and FI mobility models, respectively. For all four frequency response curves, formulation-4 produces a lower responsivity than the other three models. At high light intensity, however, formulation-4 produces a greater 3-db bandwidth than the other three versions. The higher bandwidth results because the current density boundary condition o f formulation-4 enables a faster response. The higher surface velocity is able to sweep the excess electrons out o f the semiconductor at a higher rate. That the bandwidth increase appears in the high intensity case but not in the low intensity case can be attributed to the greater excess charge density in the high intensity case, for which the effects o f a higher response speed are more significant. Formulation-4 responds significantly faster than formulation-2 for electrons as well as for holes. Formulation-2 allows the current density component from the metal to increase above the constant value that is assumed by formulation-4. This tends to counteract the sweeping o f electrons out o f the semiconductor, effectively lowering the response speed. The greatest differences in the frequency responses occur in the FI mobility case at high light intensity, and Table 5.4.4 presents these differences. The largest difference in response across the entire bandwidth curves for the three existing formulations as compared to formulation-4 is approximately 8 - 9 %. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 247 0 ») (a) 0.08 0.08 0.07 0.07 m 0.06 <u 0.06 < E, CL c £ 3 0.05 o Formulatton-1 FormUation-2 Formulation-3 Formulation-4 O 0.04 Formulation-1 Formulation-2 Formulation-3 Formulation-4 0.05 0.04 10® ^ 3<S6^q9 2 3 456^q10 2 3 10® ^ 3 456 iq 9 2 3 456^q10 2 3 Frequency (Hz) (c) (d) 1.4 1.4 1.3 1.3 < E 1.2 to <o 1.1 CL 1.0 £ w3 0.9 o Frequency (Hz) 0.8 Formulation-1 Formulation-2 Formulation-3 Formulation-4 0.7 aj Q. c £ 3 o 1.0 0.9 0.8 0.7 ^q8 2 3 4S6^q9 2 3 456^q10 2 3 10® ^ 3 456iq 9 2 3 4 56^q10 2 3 Frequency (Hz) Frequency (Hz) Figure 5.4.5 Frequency response curves for four formulations of current density boundary condition for 1 KW/cm 2 light intensity with the (a) FD and (b) FI mobility models, and for 25 KW/cm 2 light intensity with the (c) FD and (d) FI mobility models. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 248 T able 5.4.4 Frequency response curves at high light intensity (25 KW/cm2) and 1 V bias and FI mobility model. Formulation Bandwidth (GHz) 1 2 3 4 9 9 10 13 Responsivity at 9 GHz (AAV) 0.112 0.112 0.114 0.123 Maximum % Difference Relative to Formulation-4 8.9 8.9 7.3 ------- The physics at the Schottky boundaries demonstrate significant differences for formulation-4 as compared to the other three formulations. Tables 5.4.5 and 5.4.6 present the electron and hole DD velocities and carrier densities at both the left and right contacts o f the MSM that correspond to the results o f Table 5.4.4 for the 10 GHz modulation frequency. In each table the carrier density for both electrons and holes is significantly lower for formulation-4, and the DD velocity for both electrons and holes is significantly higher for formulation-4. Taken together, these effects lead to the lower responsivity and the higher bandwidth seen in the response curves for formulation-4. Table 5.4.6 shows that the greatest differences in the new and previous formulations are in the electron DD velocity and carrier density at the right (electron forward biased) contact, with the DD velocity being about 200 times higher and the carrier density being about 200 times lower for formulation-4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 249 Table 5.4.5 Peak current electron and hole DD velocities and densities at left (electron reverse biased) MSM contact at 10 GHz and 1 V bias. Form ulation 1 2 3 4 Electtrons DD Velocity Carrier Density (cm/s) (cm*3) - 1.2 x 10 ' 2 . 1x 10 “ -2 .8x 10 y 1.7x10" - 1.0 x 10* 1.6 x 10 “ -5.6x10* 8 .8x 10 ’° Holes Carrier Density DD Velocity (cm/s) (cm*3) -2 .6 x 10° 1.8x l 0 ‘° -3.2x10° 1.5xl0‘° -3.5x10* 1.4x10’° -8.9x10* 5.3x10" Table 5.4.6 Peak current electron and hole DD velocities and densities at right (electron forward biased) MSM contact at 10 GHz and 1 V bias. Form ulation 1 2 3 4 Electtrons DD Velocity Carrier Density (cm/s) (cm'3) 6 .6 x 10° 8 .8x 10 " 8.3x10* 7.0x10" 8.7x10* 6.7x10" 1.4x10* 4.5x10" Holes DD Velocity Carrier Density (cm/s) (cm*3) 2.7x10° 3.9x10" 3.3x10* 3.3x10" 3.6x10* 3.1x10" 7.4x10* 1.9x10" Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 250 5.5 5.5.1 LIMITS OF SIMULATOR Preliminary Study The performance o f the device simulator is tested over a wide range of selected parameters in an attempt to determine its application limits. The five selected parameters include mobility model, recombination time constant, voltage, light intensity, and device length. A n attempt is made to find parameter configurations that cause the simulator to fail to converge. In the process, it is also constructive to measure simulator performance for parameter configurations that do yield convergence. The criteria for measuring the simulator performance are (1) convergence efficiency, and (2) solution accuracy. The second criteria is a qualitative measure based on observing the AC signal output by the simulator, to determine whether the plot o f photocurrent as a function o f time has the same sinusoidal behavior as the modulated light The first criteria offers a quantitative measure o f the performance of the simulator. The relative performance o f the simulator is gauged by the average number o f iterations per time step (ANIPT) necessary to yield convergence, such that fewer iterations indicate better performance. To guide the process for varying specific parameters, a preliminary study is conducted in which the five device parameters are changed one at a time. Table 5.5.1 presents the ANIPT for a baseline parameter configuration as well as for five other configurations in which one of the five parameters is varied relative to the baseline. Each of the six simulations use light with a modulation frequency o f 10 GHz. The following trends are revealed by the table: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I [ 251 (1) the FI mobility model has a significantly lower ANIPT than the FD mobility model; (2) the shorter the recombination lifetime constant the lower the ANIPT; (3) the higher the voltage the lower the ANIPT; (4) the lower the light intensity the lower the ANIPT; and (5) the shorter the device length the lower the ANIPT. Previous simulation results presented in conjunction with the Schottky diodes in the absence o f light support finding (1) but apparently contradict findings (3) and (5), as seen in parts (c) and (d) o f Figure 5.4.2. Notice in the I-V characteristics presented there that the FD mobility model reaches an upper limit in voltage beyond which the simulations fail to generate a solution. Since these plots represent shorter devices, it appears that the combination o f higher voltage with a shorter device could present a limitation to the capability o f the simulator when modeling the MSMs. The figure reveals that the FI mobility model is able to converge throughout the voltage range tested for the shorter Schottky diode devices. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 252 Table 5.5.1 Preliminary convergence efficiency study performed by changing five device parameters one at a time and determining the ANIPT. Device Parameter Name Value Mobility FD FI Recombination 10-* s Time Constant 10*u s Voltage 1V 10 V Light 1 KW/cm2 Intensity 25 KW/cm2 Device 1.1 mm Length 0.7 mm Average Number of Iterations Per Time Step 5.5.2 Configuration vindicates Change Relative to Base) Base AMb AF A r rtc _ AI, X X X X X *X X X X X x *x X X X X X *x X X X X X *x X X X X X *x 3.6 4.0 9.2 5.2 4.0 3.6 Limits Imposed by Voltage, Intensity, and Length Based on the preliminary findings from Table 5.5.1 and on the Schottky diode I-V characteristics for shorter devices (Figure 5.4.2), the simulator limits are determined by using the FD mobility model with the longer recombination lifetime constant. Simulations are then conducted with increasing voltage, increasing light intensity, and decreasing device length, with a modulation frequency o f 10 GHz in each case. The results are presented in Figures 5.5.1 - 5.5.4 through a series o f plots that show both the photocurrent (left vertical axis) and the iterations to convergence (right vertical axis) as a function o f time. The photocurrent is shown for the modulation phase o f the simulation and does not include the ramp-up phase in which the device is prepared for modulation by gradually increasing the voltage and intensity to constant values. The vertical scales Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 253 o f the four plots in each figure are kept constant to aid in comparing the results (except for part (a) o f Figure 5.5.4). Figure 5.5.1 shows that increasing the voltage reduces the ANIPT as predicted by the preliminary findings, with a lower limit o f ANIPT = 3.6 being reached between 10 V [part (b)] and 50 V [part (c)]. Since efficient convergence occurs even for the 100 V simulation shown in part (d), the results indicate that there is no upper limit on the voltage that can be modeled by the device simulator. Figure 5.5.2 demonstrates that an increase in light intensity raises the ANIPT, which was also predicted by the preliminary findings. For example, at the first time step of the 1000 KW/cm 2 simulation pictured in part (d), the number o f iterations reaches a maximum value o f 49 and then drops to an average value o f 13.4. Even at such a high light intensity as 1000 KW/cm2, however, the simulator is able to produce physically reasonable results for low voltage, as indicated by the sinusoidal photocurrent curves. In conclusion, for the standard device length of 1.1 pm , increasing the voltage enhances the conversion efficiency, while increasing the light intensity reduces the conversion efficiency. Figure 5.5.3 illustrates the effect o f increasing both the voltage and the light intensity at the same time, in which the same four light intensities presented previously in Figure 5.5.2 are now simulated at 10 V. When the light intensity is increased beyond 100 KW/cm2, a point is reached at which the convergence efficiency becomes poor. Note in parts (c) and (d) o f the figure that the number o f iterations periodically reaches the maximum value o f 152 (off the graph) as allowed by the device code. At intensities o f 500 KW/cm 2 [part (c)] and 1000 KW/cm 2 [part (d)], ANIPT equals 88.1 and 42.7, respectively. The reliability of the device simulator is enhanced, however, by the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 254 1 Volt -0.00 (b ) 10 -0.05 6 -0-10 O i i i V V M 4 g o 2o = I 1 c £w 3 o -0.15 - 0.10 -0.15 - -0.20 • 100 300 0.20 100 500 (C) 50 Volt - •0.05 M C o 2 0.10 -0.15 - 0.00 c g3 o - Iterations - 500 100 Volt (d) 0.00 -0.05 C S 3 O 300 Time (ps) Time (ps) - 10 -0.05 < E | 10 Volt -0.00 Iterations (a) 0.10 -0.15 0 0.20 100 300 -0.20 100 500 Time (ps) 300 500 Time (ps) Current Iterations Figure 5.5.1 Convergence as a function o f voltage for 10 GHz light at 1 KW/cm 2 intensity for 1.1 fim device length with FD mobility model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 255 (a) 25 KW/cm2 (b ) -20 -20 < c & 2 & -40 Iterations in E c g 3 o 100 KW/cm2 0 -40 -60 -60 0 -80 100 300 0 -80 100 500 500 Time (ps) Time (ps) (C) 300 500 KW/cm2 1000 KW/cm2 (d) -20 < C cA O 2 E -40 -60 C Iterations -20 -40 ■ g 3 o -60 0 -80 100 300 0 -80 100 500 Time (ps) 300 500 Time (ps) Current Iterations Figure 5.5.2 Convergence as a function o f light intensity for 10 GHz light at 1 V bias for 1.1 fim device with FD mobility model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 256 25 KW/cm2 100 KW/cm2 (b) 100 \AAAA -25 80 -25 < E 60 -50 (0 co « v k_ 40 O 60 £ w a o -75 ‘ -50 40 -75 ' 20 f W 0 -100 100 300 W -100 100 500 Time (ps) 300 500 Time (ps) 500 KW/cm2 1000 KW/cm2 (d) 100 100 -25 -25 60 c £ 3 O -50 40 -75 20 0 -100 100 300 1c0 o 2 & 60 c -50 40 -75 20 -100 100 500 Time (ps) 300 500 Time (ps) Current Itsrations Figure 5.53 Convergence as a function o f light intensity for 10 GHz light at 10 V bias for 1.1 |im device length with FD mobility model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Iterations (C) 100 Iterations (a) 257 (a ) 1V, 500 KW/cm2 (b ) 1 V, 1 KW/cm2 30 30 “ ' nA A A A 20 < E Q) fc 3 o -0.4 »c o 2 10 c g3 o -0.6 Iterations -20 -0.2 -40 -60 -0.8 -80 100 300 100 500 500 Time (ps) Time (ps) (C) 300 1V. 1000 KW/cm2 10 V, 100 KW/cm2 (d) 30 J W -20 -20 M c o -40 2 20 1 Iterations 1 c g3 o \ A -40 10 -60 -60 i ii i i 0 -80 100 300 •80 500 100 300 500 Time (ps) Time (ps) Current Iterations Figure 5.5.4 Convergence as a function o f light intensity and voltage for 10 GHz light for 0.7 pm device length and with FD mobility model (NOTE: Scale o f (a) is adjusted). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 258 observation from the figure that even when the iterations do not reduce the state variable residuals below the tolerance, physically reasonable sinusoidal solutions are still obtained. Figure 5.5.4 illustrates the effects o f increasing the voltage and the light intensity for a shorter MSM, in which Ls = 0.7 p m . At low voltage (1 V), as the intensity increases to between 500 KW/cm 2 [part (b)] and 1000 KW/cm2 [part (c)], the initial time step o f the modulation phase takes between 20 and 30 iterations. The steps demonstrate efficient convergence, however, with remaining time ANIPT = 3.7 for both intensities. A t high voltage (10 V), the number o f iterations during the initial time step o f the modulation phase increases dramatically, often prohibitively. For example, with the 100 KW/cm 2 simulation shown in part (d), the number of iterations during the initial time step is 133 (off the graph). Furthermore, at light intensities of 1 KW/cm 2 and 500 KW/cm2, the simulation fails to generate a solution (these cases cannot be pictured). As with the low voltage cases for the short device, the high voltage simulations that are able to produce a solution demonstrate efficient convergence after the initial time step, with ANIPT = 4.1 for part (d). Apparently, it is only the first time step o f the modulation phase that poses a convergence problem in these cases of high voltage for the short device. These results indicate that once the state variables are properly configured, convergence is efficient for subsequent time steps even at high voltage and high light intensity. The problem is that at the end o f the ramping-up phase, the state variables are too far removed from a favorable configuration to allow the simulation to successfully enter the modulation phase. This finding reconciles the apparent inconsistency pointed out earlier, in which the simulations failed at higher voltages for the short Schottky Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 259 diodes while the preliminary studies indicate that shorter MSMs demonstrate more efficient convergence. To complete the study on the limits o f applicability for the device simulator, two additional simulations are run, one for a lower voltage than the baseline and one for a longer device length. The lower voltage simulation uses the baseline parameter configuration except that V = 0.5 V , and results in ANIPT = 5.1, which is not significantly different from the results for 1 V. The longer device length uses the baseline parameter configuration except that Ls =1.5 pm, and results in ANIPT = 3.6, which is significantly less than the AINPT for I , =1.1 pm . Based on these two simulations, it is concluded that there is no limit on the simulator for decreasing voltage or increasing length. This conclusion is confirmed by the simulation results for the long (2.50 pm) Schottky diode, as presented through the I-V characteristic in part (b) o f Figure 5.4.2, where it can be seen that the FD mobility model produces a solution across the entire voltage range that is tested. Therefore, the only limitations on the parameter range o f the device simulator is the combination o f a short device length (~ 0.7 pm and below) with a high voltage (~ 10 V and above), with light intensity having a mixed effect on this combination. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 260 5.6 5.6.1 DEVICE PERFORM ANCE STUDY Effects o f B arrier H eight Previous results with MSMs that are presented in section 4.5 show that in the absence o f light, barrier height does have a significant effect on the current. Comparing the total current densities in Figures 4.5.2 and 4.5.4 demonstrates that the total current density produced in the 1.0 eV barrier MSM ( - 2 .5 x 1 0 “* A/cm2) is over three orders o f magnitude higher than the total current density produced in the 0.6 eV barrier MSM ( - 4 . 7 x l 0 “5 A/cm2) [see also Tables 4.5.1 and 4.5.2]. This is reasonable based on the band diagrams in part (a) o f each figure. For the 0.6 eV barrier MSM, in which the electron current is dominant, the current is limited by the reverse biased contact for electrons at x = 0 , for which the barrier height to electron flux into the semiconductor is 0.6 eV. For the 1.0 eV barrier MSM, in which the hole current is dominant, the current is limited by the reverse biased contact for holes at x = Ls , for which the barrier height to hole flux into the semiconductor is 0.42 eV. The hole current therefore has a lower barrier in the 1.0 eV barrier MSM than the electron current has in the 0.6 eV barrier MSM. In the presence of light, however, the two MSMs produce the same current. This is evident in Figure 5.6.1, which presents the simulation results for both the 0.6 eV barrier MSM and the 1.0 eV barrier MSM exposed to a constant light intensity of 1 KW/cm 2 under a bias o f 1 V. Unlike the MSMs in the dark, for which the current source is injection from the metal, the current source for the MSMs exposed to the light is essentially the excess electron-hole pairs created by light within the substrate o f the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 261 (a) (b) Ec Ev Efn Efp 1 1 0 0 I I m LU 1 1 Et 2 Efn Efp 2 0.0 1.0 0.5 x(pm ) 0.0 0.5 x (p m ) 1.0 (c) © (0 o W o> o _l (Barrier High Barrier Electron density (/cm3) Hole density (/cm3) Electric field (V/cm) Total current (A/cm2) Electron density (/cm3) Hole density (/cm3) Electric field (V/cm) Total current (A/cm2) -10 -15 0.0 0.2 0.4 0.6 0.8 1.0 1.2 x (n m ) Figure 5.6.1 Comparison o f E-x diagrams for (a) low (0.6 eV) and (b) high (1.0 eV) Schottky barriers with (c) associated profiles for MSM illuminated by 1 KW/cm2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 262 MSM. This excess charge has no barrier to its exit from the MSM as it flows downhill in response to the electric field, with electrons flowing in the + x -direction and holes flowing in the - x -direction. The band diagrams indicate that the slope o f the electrostatic potential energy and therefore the electric field is the same for both MSMs, so the current must be the same for each. The profiles confirm the equivalence o f the electric fields and currents for the high- and low-barrier MSMs. Since all of the profile quantities are equivalent, it is apparent that there is essentially no difference in the operation of these two MSMs in the presence o f light. Since current is continuous, there must be compensation at the restriction points to allow for such a large current compared to the case in the absence of light. The restriction of electron flux into the semiconductor at x = 0 (equivalent to positive current out o f the semiconductor) is compensated by the high hole flux out o f the semiconductor at x = 0. Likewise, the restriction o f hole flux into the semiconductor at x = Ls is compensated by the high electron flux out o f the semiconductor at x = Ls (equivalent to positive current into the semiconductor). This compensation is evident in the profile o f part (b) of Figure 5.5.1, in which the drop in electron density at x = 0 is compensated by the rise in hole density at x = 0, and vice-versa at x = Ls . This finding, that barrier height has no effect on device operation for symmetric MSMs, is confirmed by similar results for all combinations o f voltage and light that are applied in the present study. The remainder o f the simulation results are reported for the 1.0 eV barrier MSM, since the most recent studies use this barrier height for GaAs Schottky barriers [58]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 263 5.6.2 Substrate Growth Temperature Effects The optical response speed of GaAs photodetectors can be increased by growing the GaAs substrate at lower temperatures than the conventional growth temperature of about 600° C [69]. At lower growth temperature, excess As deposits in the lattice and creates point defects. These point defects add electronic states within the bandgap that act as recombination centers, shortening the recombination lifetime and increasing the response speed and bandwidth. The shortening o f the recombination lifetime, however, causes the responsivity to decrease due to the reduction in the number o f carriers in the n o product Also, the increased scattering that results from the point defects lowers the mobility and therefore the carrier velocity, which further reduces the n o product and responsivity. The optimum combination o f speed and responsivity regarding the largesignal MSM photodetectors appears to occur when GaAs is grown in the intermediate temperature range o f about 350° C. One o f the objectives o f the parameter study portion o f the present work is to determine how the MSM modeled here responds across the range o f growth temperatures. Device simulations are conducted for low-growth- temperature (~ 200° C ) and conventional-growth-temperature GaAs. Low-growth- temperature GaAs is characterized by a short recombination lifetime constant t„ =10~1Z s, while conventional-growth-temperature GaAs is characterized by a long recombination lifetime constant = 10"* s , where r„ = xp for both time constants. The results o f the growth temperature effects are presented in Figure 5.6.2 through a comparison o f frequency response curves for eight different combinations of mobility model (FI and FD), voltage (IV and 10 V), and light intensity (1 KW/cm2 and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 264 :») (b) 1.0 oe c« t3 O 1.0 0.9 Ic 0.8 c S w 5 0.7 * O 10 ® 2 3 45 10 ® 2 34 5 ,10 2 3 4 5 0.9 FD Mobility 1.0 V 1.0 KW/cm2 0.8 0.7 ,11 2 2 3 45 10* Frequency (Hz) ,10 2 3 4 5 2 Frequency (Hz) (c) 1.0 g oc c s o3 1.0 0.9 Io FI Mobility 1.0 V 25.0 KW/cm2 0.8 O 109 2 345 FD Mobility 1.0 V 25.0 KW/cm2 0.8 0.7 10® 2 3 4 5 0.9 c 1010 2 3 45 10,11 0.7 2 Frequency (Hz) 0.9 1.0 0.9 FI Mobility 10.0 V 1.0 KW/cm2 0.8 2 3 45 ^q 9 FD Mobility 10.0 V 1.0 KW/cm2 C c g3 O 0.7 1(j8 2 3 4 5 1Q10 2 3 4 5 1011 0.8 0.7 108 2 2 3 45 Frequency (Hz) 1Q9 2 3 4 5 1Q10 2 3 4 5 ,11 2 Frequency (Hz) (g ) 00 1.0 c £ 3 o 2 (0 0-0 1.0 Ioc ,10 2 3 4 5 Frequency (Hz) (e) § o3 2 345 10 ® 2 345 0.9 0.8 0.7 x \ - FI Mobility 10.0 V 25.0 KW/cm2 \b A 10® 2 345 10® 2 345 1010 2 345 1011 2 Frequency (Hz) _ 1.0 | 1 0.9 r 1 0.8 - - 0.7 FD Mobifity 10.0 V 25.0 KW/cm2 _L. 10® 2 345 10® 2 3 4 5 1010 2 3 4 5 j q II 2 Frequency (Hz) Recombination Time Legend Figure 5 .6.2 Comparison o f frequency response curves for different recombination times. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 265 25 KW/cm ). Each graph compares the 3 -db bandwidths of the two curves associated with the short and long recombination time constants. The current axis represents the normalized AC peaks, which are normalized for ease o f comparison o f the bandwidths. Table 5.6.1 gives the numerical values o f the bandwidths as well as the responsivities for both recombination time constants o f each parameter set. The table shows that where there is a substantial increase in bandwidth, there is an accompanying substantial decrease in responsivity. This trend is consistent with the usual inverse relationship between these two figures o f merit. The table indicates that the increase in bandwidth and the decrease in responsivity when going from conventional-growth- to low-growthtemperature GaAs is dramatic for both mobility models at low voltage, but only for the FD mobility model at high voltage. The table also shows that in all but one case of increasing voltage the bandwidth increases substantially, while the responsivity increases substantially for the short recombination time curves but not so substantially for the long recombination time curves. T able 5.6.1 Responsivity and bandwidth for long and short recombination lifetimes. Parameter Combination FI, 1 V, 1 KW/cm" FD, 1 V, 1 KW/cm 2 FI, 1 V, 25 KW/cm 2 FD, 1 V, 25 KW/cm 2 FI, 10 V, 1 KW/cm 2 FD, 10 V, 1 KW/cm 2 FI, 10 V, 25 KW/cm 2 FD, 10 V, 25 KW/cm 2 W =10'* s (Long) Responsivity Bandwidth (A/W) (GHz) 0.21 19 14 0.21 0.16 13 8 0.15 188 0.22 0.22 33 188 0.22 0.22 33 Tn(P)s = IQ"12 s (Short) Responsivity Bandwidth (A/W) (GHz) 27 0.12 68 0.042 0.088 163 0.042 80 0.21 192 0.052 72 0.21 192 0.052 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 266 The quantities that determine bandwidth are calculated in post-processing steps in order to establish a quantitative description o f low-growth-temperature versus conventional-growth-temperature GaAs regarding these quantities. The two major determinants o f bandwidth for the device simulations are transit time and the effective recombination lifetime, with the shorter o f these two times being the major factor in limiting bandwidth. The transit time represents the average time for a given type o f carrier to travel from its generation point to the metal contact. It is assumed that carriers are generated equally at all grid points and travel to the right or left contact according to the sign o f the velocity, so that the average transit time for each carrier is given by P - ,. I V" V y _ y — ft!— _ * _ 4 - **l N - 1 ZrfZ-l j.\ ~ *I ° d j\ I* -I y N -l y X________ k *1 x k Zri Zrfl *♦! k *I (5.6.1) U ij ~ U, where i = n ,p (for electrons, holes), od is the drift-diffusion velocity, and P_,+ is the grid point at which the sign o f the carrier velocity switches from negative to positive. The transit time is affected by the mobility model, the applied voltage, and the screening o f the electric field by excess charge carriers. The FD mobility model reduces mobility, which increases transit time. Increasing the voltage reduces the transit time because a higher electric field is created and propels the charges faster. Figure 5.6.3 illustrates the electric fields associated with the different voltage and light intensity combinations. Note how the high light intensity (25 KW/cm2) coupled with low applied voltage ( I V ) significantly reduces the electric field through screening. The effective recombination Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 267 5 Log S (V/cm) 4 3 2 1V .1K W 1 V, 25 KW 10 V, 1 KW 10 V. 25 KW 1 0 0.0 0.2 0.4 0.6 0.8 1.0 X (n m ) Figure 5.6.3 Electric fields for FI mobility models, used to compare the driving force that pushes the carriers to the contacts of the MSM. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.2 268 lifetime is the reciprocal o f the average number o f recombination events that a single carrier undergoes per unit o f time, and is given by \ f R G ' w \ and n i.e ff I -I mud j -i rR G ■pirff IP (5.6.2) mud j in which i = s ,l (short, long) and mid refers to the middle o f the device. The effective recombination lifetime is comparable to the recombination time constant. Due to its flexibility, the effective recombination lifetime is considered more accurate than the recombination tim e constant for assessing the interaction between transit time and recombination lifetime that limits bandwidth. Table 5.6.2 provides electron and hole transit times r,M and t, p , respectively, and electron and hole effective recombination lifetimes r mtff and Tpi tjf ( / = s ,l), respectively, for both recombination time constants of each parameter set, as calculated by Equations 5.6.1 and 5.6.2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 269 Table 5.6.2 Interaction of transit time and effective recombination lifetime for long and short recombination lifetimes. Parameter Combination FI, IV , 1 KW/cm" FD, 1 V, 1 KW/cm 2 FI, 1 V, 25 KW/cm 2 FD, 1 V, 25 KW/cm" FI, 10 V, 1 KW/cm 2 FD, 10 V, 1 KW/cm 2 FI, 10 V, 25 KW/cm 2 FD, 10 V, 25 KW/cm 2 *■„(„), = 1 0 “* s (Long) Electrons Holes rM. T n lx ff (ps) (P S ) 0.72 4.1 4.9 10. 0.084 6.4 0.085 6.5 1. 1x 10* 1.2 x l 04 1.5x101 .8 x 10l.lx lO 4 1.7xl0 4 l.lx lO 4 1.7xl0 4 *n(P)s = IQ"12 s (Short) Holes Electrons ns.tff T p t* t r rM (ps) (ps) (PS) (ps) (ps) (ps) 12 17 7.5 12 1.6 9.1 1.5 8.8 20 x l 04 5.6x104 4.9x102.8x 104 21x l 04 2.4x1020x 10* 2.4x10* 0.69 4.1 1.1 3.6 0.084 6.5 0.085 6.4 1.1 1.4 1.2 1.3 1.1 1.8 1.1 1.8 12 18 II 15 1.6 9.1 1.6 8.8 13 3.6 5.8 4.5 20 2.2 19 2.2 * p s . 'f f The analysis o f the bandwidth results begins with the FI mobility model at low voltage and low light intensity in part (a) o f Figure 5.6.2, where a slight separation between the responses associated with the long and short recombination time constants is apparent. As shown in Table 5.6.2, both the long and short recombination curves are limited by the transit times since these are shorter than the respective effective recombination lifetimes. The fact that there is significant separation between the two curves even though the transit times are almost identical is attributed to the effective recombination lifetimes for the short recombination curve. These are comparable to the transit times and therefore work synergistically with the transit times to increase the bandwidth o f the short recombination curve compared to the long recombination curve. For the corresponding low voltage, low intensity FD mobility case in part (b), the reduced Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 270 mobility lengthens the transit time for both carriers and reduces the effective recombination lifetime for holes, as shown by Table 5.6.2. For the long recombination curve, the bandwidth is reduced by 26 % compared to the FI case in part (a) because the transit time, which still determines bandwidth, is longer. However, for the short recombination time curve, the effective recombination lifetime for both electrons and holes is shorter than the corresponding transit times. The short recombination curve is now limited by the effective recombination lifetime, producing a dramatic increase o f 152 % in the bandwidth of the short recombination time curve compared to the FI case in part (a). For the FI mobility model at low voltage and high light intensity in part (c) o f Figure 5.6.2, there is a wide separation between the long and short recombination curves. For the long recombination curve, the bandwidth is reduced by 32 % compared to the low intensity case in part (a), largely because o f the seven-fold increase in the electron transit time. The transit time determines the bandwidth for the long recombination curve, and the increase in the electron transit time is due to the reduction in mobility produced by screening o f the electric field by the higher excess charge density caused by the increased light intensity (see Figure 5.6.3 for field reduction). For the short recombination curve, the effective recombination lifetime for both carriers is comparable to or shorter than the corresponding transit times (see Table 5.6.2). Again the effective recombination lifetime limits the bandwidth for the short recombination curve, leading to a dramatic increase o f 504 % in the bandwidth compared to the low intensity case in part (a). These same overall results are repeated in the corresponding FD mobility case of part (d), with both curves shifted to lower bandwidths relative to part (c) due to the reduction in mobility. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 271 For the high voltage cases pictured in parts (e) through (h) o f Figure 5.6.2, light intensity does not affect the outcome, so the results o f (e) also represent those o f (g), and the results of (f) also represent those o f (h). For the FI mobility model at high voltage in part (e), the average electric field is at its maximum (see Figure 5.6.3), so the transit time is reduced to a significantly shorter value than either the long or short effective recombination lifetimes (see Table 5.6.2). Therefore, the bandwidths for both the long and short recombination curves in part (e) are essentially determined by the transit time, resulting in little separation between the curves. For the corresponding FD mobility model in part (f), the reduced mobility causes the transit times to lengthen. Since the transit times for the long recombination curve still determine its bandwidth, the bandwidth o f the long recombination curve is shifted to a lower value by 82 % compared to the FI case in part (e). The bandwidth o f the short recombination curve is also shifted to a lower value (by 63 % ) compared to part (e). This downward shift is less than for the long recombination curve because the transit times for the short recombination curve become longer than the effective recombination lifetimes, so the effective recombination lifetimes limit the bandwidth for the short recombination curve, and these are shorter than the transit times o f the long recombination curve. A more detailed analysis o f the data associated with the short recombination curves is performed in an attempt to establish a relationship between the bandwidths and specific combinations o f transit time and effective recombination lifetime. Only the short recombination curves are included in this detailed analysis because the bandwidths for the long recombination curves are completely determined by the transit time as the transit time is four orders o f magnitude shorter than the effective recombination lifetime. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 272 short recombination curves are divided into three groups according to comparable bandwidth, as shown in Table 5.6.3. Then the fast and slow path to carrier removal is determined for each curve. The fast path is found by summing the shortest time for electron removal with the shortest time for hole removal between transit time and effective recombination lifetime. The slow path is found by summing the longest time for electron removal with the longest tim e for hole removal between transit time and effective recombination lifetime. The total time for carrier removal is the sum o f the transit times and effective recombination lifetimes for electrons and holes. Two conclusions can be drawn from the table: (1) the primary determinant o f bandwidth is the time for the fast path to carrier removal, as evidenced by the ordering of the groups, in which the bandwidth increases with decreasing fast path time (when comparing entire groups); and (2) for curves that have comparable fast path times, the bandwidth is secondarily determined by the time for the slow path to carrier removal, as evidenced within each group, in which the bandwidth increases with decreasing slow path time (this conclusion is violated once, for the curve with the 80 GHz bandwidth). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 273 T able 5.6*3 Detailed analysis of simulation results associated with short recombination lifetimes. Group A B C 5 .6 3 Bandwith (GHz) 27 68 72 72 80 163 192 192 Fast Path Slow Path (ps) (ps) 12.7 5.0 4.0 4.0 5.8 6.9 1.7 1.7 14.1 22.1 15.6 15.2 18.6 12.2 21.1 20.1 Total Time (PS) 26.8 27.1 19.6 19.2 24.4 19.1 22.8 21.8 Effects o f Device Length The effect o f device length is tested to determine whether the shorter devices have a higher bandwidth and lower responsivity as expected due to the shorter transit time. Besides the standard MSM length o f 1.1 pm, shorter (0.7 pm ) and longer (1.5 pm) MSMs are simulated for both the FI and FD mobility models. The normalized frequency response curves are presented in Figure 5.6.4, and the bandwidth and responsivities are listed in Table 5.6.4. The average transit times for electrons and holes as calculated by Equation 5.6.1 are listed in Table 5.6.5, and these correlate well overall with the bandwidths. For example, the shortest MSM (0.7 pm) has about half the transit time as the standard MSM (1.1 pm) and approximately twice the bandwidth for the FI mobility model. Similarly, the standard MSM has about half the transit time as the longest MSM Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 274 (a) 1.0 E W. o 0.9 c FI mobility c £ 3 O 0.8 0.7 microns 1.1 microns 1.5 microns 0.7 108 2 3 4 5 6 7 8 109 2 3 4 5 6 7 ^10 2 3 4 5 6 7 1Q11 2 3 4 5 6 7 1Qii Frequency (Hz) (b) 1.0 E oc 0.9 FD mobility c 3 O 0.8 0.7 microns 1.1 microns 1.5 microns 0 .7 108 2 3 4 5 6 7 8 109 2 3 4 5 67 ^10 Frequency (Hz) Figure 5.6.4 Comparision o f frequency response curves for different device lengths, for (a) FI mobility model and (b) FD mobility model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 275 (1.5 Jim) and about twice the bandwidth for the FI mobility model. The correlation with transit time and bandwidth is poor only for electrons in the FD mobility model. The results reveal two unexpected findings: (1) there is no significant reduction in responsivity even for large, two-fold increases in bandwidth; and (2) the bandwidths for the shortest and longest MSMs do not change for the two mobility models, while the bandwidth for the standard MSM increases by 36% for the FI mobility model as compared to the FD mobility model. The second unexpected result does not correlate well with the transit time data, in which Table 5.6.5 reveals that a marked increase in transit time for the shortest and longest MSMs does not reduce the bandwidth when comparing the two mobility models. These apparent anomalies reinforce the value of the simulation in determining relationships that do not lend themselves to analytical prediction. T able 5.6.4 Comparison o f responsivity and bandwidth for three MSM lengths. F I Mobility M odel FD M obility Model 0.7 pm 1.1 pm 1.5 pm 0.7 pm 1.1 pm 1.5 pm Responsivity (A/W) 0.21 0.21 0.20 0.21 0.21 0.20 Bandw idth (GHz) 46 19 10 46 14 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 276 Table 5.6.5 Comparison o f transit times r, for three MSM lengths. FI M obility M odel FD M obility M odel 0.7 pm 1.1 pm 1.5 pm 0.7 pm l .l pm 1.5 pm Electron rtjn (ps) 0.29 0.72 1.61 3.19 4.14 4.18 Hole t , p (ps) 5.22 11.7 18.8 9.01 17.2 24.9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 277 CHAPTER 6 GLOBAL SIMULATION 6.1 INTRODUCTION The global simulation is important because under actual conditions, the photodetector device is used as part o f a wider photomixer circuit that affects device performance. The global simulation calculates the AC signal that is output by the photomixer circuit in response to modulated light. This calculation requires the derivation o f the current-voltage relationship that characterizes the photomixer circuit response in the time domain. Section 6.2 begins the derivation by presenting the photomixer circuit as a combination of the photodetector device, the embedding circuit, and the DC voltage source that drives the photocurrent through the circuit. Since the embedding circuit in actual microwave photomixer circuits is constructed as a distributed network, the embedding circuit must be solved in the frequency domain in order to derive the photomixer current-voltage relationship. The solution o f the embedding circuit is begun by devising an equivalent circuit model, which consists o f a bias tee and the device parasitic capacitance. The function o f the bias tee is to separate the AC and DC signals, while the device parasitic capacitance is an inherent part o f the embedding circuit that shunts away a portion o f the usable AC signal. The bias tee is comprised of an inductance branch that carries the DC signal and a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 278 capacitive output branch that carries the AC signal to the load resistance. The presence o f the inductance and capacitances cause the embedding circuit to produce a finite rather than instantaneous response, called an impulse response, to the AC signal generated by the device. This leads to impedances at the ports o f the device that affect the photomixer current-voltage relationship. Section 6.3 explains how a convolution integral is used to characterize the impulse response in the photomixer current-voltage relationship. The impulse response is derivable because it is the Fourier transform o f the impedance function. The equivalent circuit model is used to characterize the impedance function in the frequency domain, which effectively solves the circuit. The derivation o f the photomixer current-voltage relationship is completed by returning to the time domain through an inverse fast Fourier transform o f the discretized impedance function. The resulting discretized impulse response is smoothed and truncated using a Kaiser filter to speed the time-domain simulation. The final photomixer current-voltage relationship is a fixed-point iteration scheme that utilizes a linear extrapolation for current, which further increases the simulation speed. The global simulator is tested in section 6.4 by using the principle that the integral o f the impulse response over time is equal to zero. A first test consists o f summing the discrete form of the four impulse responses that are used in the global simulations to determine whether the sums do in fact approach zero. A second test compares the device-only photocurrents to the global photocurrents for DC operating conditions, which include equilibrium, DC steady state in the absence o f light, and DC steady state in the presence o f light. Under DC operating conditions, the photocurrents are expected to be identical because the convolution vanishes for the global simulations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 279 The effects o f the global simulator on device performance are explored in section 6.5, with performance being gauged by optical responsivity and bandwidth. The bandwidth results can be explained through the interaction between three response times. These include the device response time, the output branch response time, and the response time of the branch that contains the parasitic capacitance. Each response time is derived by determining the frequency at which the response reduces the AC output by a factor o f l/V 2 . The output branch establishes a lower limit for the photomixer circuit bandwidth. The upper limit o f the bandwidth is determined by the longer of the two responses between the device response time and the capacitance branch response time. The comparison o f the device-only and the global simulation results are presented through frequency response curves for the same eight parameter combinations that were previously studied in Chapter 5 in conjunction with the isolated device. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 280 6.2 6.2.1 PHYSICAL MODEL OF THE EMBEDDING CIRCUIT Characterizing the Photomixer C ircuit The photomixer circuit consists o f the photodetector device, the DC voltage source v ^ ., and the linear embedding circuit. The photodetector device absorbs modulated light and converts it into photocurrent, and the DC voltage source drives the photocurrent through the embedding circuit. The embedding circuit isolates that portion o f the photocurrent that is used to generate microwaves, so that the final output of the global simulation is only that part o f the photocurrent that actually generates microwaves. In order to extract this portion o f the photocurrent, the current-voltage relationship that characterizes the entire photomixer circuit must be determined and built into the global simulator. The photomixer current-voltage relationship depends on the interaction between the three parts o f the photomixer circuit, and is largely a function of the architecture of the linear embedding circuit. It is necessary to characterize the operating mode o f the photomixer circuit so that the photomixer curTent-voltage relationship can be constructed. During the photomixing process, the operating mode o f the photomixer circuit is AC steady state. In AC steady state, the device has a time-dependent voltage vd (r) across it and outputs a timedependent current /(f), with the current being given by Equation 2.4.27, in which co = 27uvm (the m subscript is dropped for convenience). This equation indicates that the current output from the device has a sinusoidal AC component iAC(t) given by iAc (0 = i°Ac(*>) 0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (6-2.1) 281 and a constant DC component ix . [NOTE: The dependence o f voltage vd(t) and current i[t) on the frequency to is absorbed by the AC amplitude i°AC(at) , and is not expressed explicitly in vd (r) and i{t) to emphasize that the photomixer current-voltage relationship yet to be derived applies to the time domain.] The desired output o f the global simulation is the AC portion of the signal iAC(/) since this is what creates the microwaves. The fact that iAC(t) is a component o f the device current /(/) is the reason that the relationship between device voltage vd{t) and device current i(J) is derived. In the simplest possible embedding circuit, the only element is the load resistance RL that converts the AC portion o f the device current into microwaves, as pictured in part (a) of Figure 6.2.1. The current-voltage relationship o f the photomixer circuit associated with this simplest embedding circuit is given according to Kirchhoff s voltage law by vA t) = vDC- v daJlL(t), where with vdu.RL(0 = R JiO > (6.2.2) (t) represents the voltage drop across the embedding circuit, which is simply the load resistance in this case. Note that according to Equation 6.2.2, the voltage vd(t) across the device at time t is completely determined from the current /(/) at time t. As will be seen in section 6.3, this is not the case with the embedding circuit that is actually used. Also note that for the simplest embedding circuit, the current through the device is identical to the current through the entire circuit due to the series arrangement. This is true for any embedding circuit architecture since the source of both the DC and AC components is the photodetector device. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 282 (a) Voc -= r MSM *AAA/V— vcM,Rl (b) Embedding circuit i I = Iqc * *AC Figure 6.2.1 Photomixer circuits, inculding (a) single element (RL), (b) bias tee, and (c) Thevenin equivalent bias tee. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 283 6.2.2 Characterizing the Em bedding Circuit In actual embedding circuits, the AC component photocurrent /(/) is separated from the DC component iAC(t) o f the device so that the load resistance RL can perform an efficient microwave conversion. This requires introducing inductances and capacitances into the embedding circuit, because these elements present frequencydependent impedances that enable the AC and DC signals to be separated. The inductances and capacitances are passive elements that consist o f distributed networks of metal connectors whose dimensions are on the order o f the wavelength o f microwaves. Such distributed networks are more efficient regarding energy loss than lumped elements, and are able to transmit a wider range o f signal frequencies. Distributed circuits are solved in the frequency domain, and the resulting solution is transformed into the time domain to produce the required photomixer current-voltage relationship, such as Equation 6.2.2. The characterization o f the embedding circuit in the frequency domain is accomplished preferably through experimentation or by using an electromagnetic field simulator that solves Maxwell’s equations in three dimensions. Since both o f these methods are beyond the scope o f the present study, a third more accessible method for characterizing the circuit in the frequency domain is chosen here. The embedding circuit is characterized in the frequency domain by using an equivalent circuit model o f a typical microwave distributed circuit. Part (b) o f figure 6.2.1 depicts the equivalent circuit model o f the distributed microwave circuit used by the global simulation in the present study. The characterization o f the equivalent circuit model requires calculating the impedances of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 284 the passive elements that comprise this circuit The frequency-dependent impedances o f the passive elements are determined by phasor transforms o f the time domain voltages across each element [70]. The phasor transforms of the voltage vk(t) and current ik(t) associated with circuit element k are represented by Vk(Ja>) and I k(Ja>), respectively, as given by ^ [vi (')] = f', U “i) = ^ c j . 0») exp(/a) 4 „ ( ') i = !,< Jv)= i°cA a )< :xp (j0 ), and (62.3) where v°c* and /°Cjt are the AC voltage and current amplitudes associated with circuit element k, a and f$ are the respective phase angles, and co is the modulation frequency. Table 6.2.1 gives the time-domain and the frequency-domain equivalent voltages for each passive element of the equivalent circuit as these elements relate to the various branches o f the circuit (which are described below). It also presents the frequency-dependent impedances Z k(Jeo), which are calculated by dividing the frequency-domain voltage Vk (Jco) across the element by the frequency-domain current I k (jco) through the element. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 285 Table 6.2.1 Frequency domain equivalent voltages and impedances for the passive elements in the equivalent circuit model o f the distributed microwave circuit. Element and/or Branch Voltage Across Element Time Domain Frequency Domain Parasitic Capacitance c, Bias Tee Inductance Lb Bias Tee Capacitance CB Load Resistance *L Output Branch CB and R l Impedance h j°>Cp 1 j<oCp LgjOl I L jeaLg -p^\'A c.adT I ACjo j(o C B 1 jo )C B R - J a C j> R 1.1AC.a Rl ^ 8 Lb dt 7T" \*AC.od r + R - J aCjd B -J6)Cg T 7 7 -+ V .C , J6)Cg The embedding circuit portion o f the photomixer circuit as shown in part (b) of Figure 6.2.1 has three major parts: (1) a bias tee; (2) the load resistance RL; and (3) the parasitic capacitance o f the device Cp . The bias tee is used to separate the AC and DC signals, so that only AC signals generated by the photomixing are delivered to the load resistance RL. The separation of the HF and DC signals is enabled by the presence o f the inductance LB in the branch of the bias tee that contains the DC voltage source, called the inductance branch. Since the impedance o f the inductance is directly proportional to the signal frequency, the inductance acts to impede the AC signals so that the DC signal j'qc passes through the inductance branch (designated iL in Table 6.2.1 to allow for very low frequency time variant signals). The output branch o f the bias tee contains the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 286 capacitance C 8 , whose impedance is inversely proportional to the product o f the signal frequency and the capacitance. Therefore, the capacitor impedes the DC signal (whose frequency is zero) and admits the AC signal, so that the output branch passes the AC signal to the load resistance. A portion o f the AC signal iAC is carried by the branch of the embedding circuit with the parasitic device capacitance Cp , called the parasitic branch, while the remainder o f the AC signal iACo is carried by the output branch. Since the device parasitic capacitance is much smaller than the bias tee capacitance, lower frequency signals are carried primarily by the load resistance in the output branch. As the frequency is increased, more and more o f the signal is shunted through the parasitic capacitance. The output iACo from the photomixer circuit is that portion o f the signal that is carried by the load resistance and is the primary quantity sought through the global simulation. The photomixer current-voltage relationship associated with the actual embedding circuit has the same general form as occurred for the simplest embedding circuit given by Equation 6.2.2. According to Thevenin’s theorem, the embedding circuit can be replaced by a single equivalent element such that the voltage drop across this element is equivalent to the voltage drop v ^ ^ r ) across the actual embedding circuit, in which B T refers to bias tee [70]. The equivalent photomixer circuit then consists o f the device, the equivalent embedding circuit element, and the equivalent voltage vtq all in the same series arrangement as occurred with the simplest embedding circuit, and is illustrated in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 287 part (c) o f Figure 6.2.1 Again by KirchhofFs voltage law, the photomixer current- voltage relationship is given as Vd CO = % “ V d a J!T CO • (6 2.4) The equivalent voltage veq is determined according to Thevenin’s theorem as follows: the voltage across the ports a and b o f the device due to the rest o f the circuit is identical to the voltage at these same ports when the device is removed from the circuit leaving it open across a and b. When the photodetector device is removed form the circuit in part (b), the voltage across ports a and b equals the voltage across the inductance branch plus because this combination is in common with ports a and b. Since essentially DC signals are carried by the inductance branch, the time derivative o f the current diL / dt is zero, which causes the voltage across the inductance to approach zero according to Table 6.2.1. Therefore, the voltage across the inductance branch plus vx and therefore across the device is v ^ ., and the photomixer current-voltage relationship can be rewritten as Vd CO ~ V DC ~ The voltage drop across the embedding circuit V d u .B T CO • (6.2.5) (t) is characterized by a convolution in the time domain and a Thevenin equivalent impedance Z cil times the current phasor I { jo ) in the frequency domain, both o f which are explained in the next section. Part (b) o f the figure shows the impedance that the embedding circuit presents at the ports o f the device. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 288 63 6.3.1 TH E CONVOLUTION Im pulse Response Defined It was stated in section 6.2 that the construction o f the current-voltage relationship that characterizes the photomixer circuit used in this study would involve a convolution. The need for the convolution arises because the capacitances and inductance that appear in the embedding circuit cause the embedding circuit to create an impulse response to the AC signed. Capacitances and inductances are energy storage elements, so they take a finite time to complete their response to changes in voltage across them or current through them, rather than completing their response instantaneously. This is unlike resistances such as the load resistance depicted in Equation 6.2.1, which complete their response to voltage or current changes instantaneously. For example, from Table 6.2.1, the voltage v(/) at time t across a capacitance is given by the integral 1 (6.3.1) o which indicates that the value o f the current at every increment o f time between r = 0 and r = t is involved in determining the voltage across the capacitance at the single time t. The finite time width o f the response o f an energy storage element is characterized by the impulse response o f that element. The impulse response to current can be defined as the complete response over time made by an energy storage element that is exposed to an infinitely narrow (in time) pulse o f current. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 289 6 3 .2 Embedding C ircuit Voltage Given by a Convolution A convolution is the mathematical construct that characterizes the effect o f the impulse response on the time-domain current-voltage relationship of a circuit. For the photomixer circuit, the impulse response of the embedding circuit determines the voltage drop across the embedding circuit in response to the time-varying current /(/) through it. The physical description o f the convolution is pictured in Figure 6.3.1. The current function i(r) shown in part (a) is sinusoidal due to the form of the oscillating light intensity. The impulse response function h(t), depicted in part (b) as a decaying exponential, is the voltage per current per time, and represents the embedding circuit response that would result if the photodetector device produced a sharp spike o f current. This motivates characterizing the current function as a series o f current spikes depicted in the figure as current rectangles. A given current rectangle initiates a separate impulse response in the embedding circuit that superimposes on the previous impulse responses, all o f which are in different phases o f completing their decays. The magnitude o f the current rectangle determines the overall magnitude o f the impulse response initiated by that current rectangle. Note in part (d) that the impulse response is reversed in order to correctly pair its phase o f decay with the current rectangles as both the current and the impulse response evolve through time. The equation for the convolution in its discrete approximate form as well as in its exact integral form can be derived with the aid o f Figure 6.3.1 [70]. As seen in parts (c) and (d), at t0 the current /0 creates the first impulse response, which decays in time so that its value at time tk is hk . At each time step tk o f width A /, the contribution Av0> Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. «1) VoAt hjoAt Vo** — -ri V i Af \ h ^ t — ~A h^A t h^A t h ^ t h ^ t |j - A V s Af & Figure iJ .1 Physical picture o f a convolution, showing the (a) current function, (b) the impulse response function, (c) the discrete evolution o f the current, and (d) the discrete evolution of the first four impulse-response-current products. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 291 m a d e to th e v o l t a g e a c r o s s th e e m b e d d i n g c irc u it b y t h i s f i r s t im p u ls e r e s p o n s e is a p p r o x im a te d b y t h e p r o d u c t o f i0A / a n d t h e v a lu e o f t h e i m p u l s e r e s p o n s e p o i n t in tim e tk, o r A v0Jl » A 4/0A / . At c u r r e n t in c re m e n t. F o r e a c h t i m e s te p A vu « tk , a t th a t , th e c u r re n t /, c r e a t e s th e s e c o n d im p u ls e r e s p o n s e , so t h a t t h e c o n t r i b u t i o n m a d e to th e v o lta g e a t t i m e s t e p i m p u ls e r e s p o n s e is a p p r o x im a te d b y hk hkixA t , t h e v o lta g e tk b y th e s e c o n d a n d s o o n f o r e a c h s u c c e s s iv e vcklJl a c r o s s t h e e m b e d d in g c i r c u i t is t h e s u m o f t h e c o n tr ib u tio n s f r o m e a c h im p u ls e r e s p o n s e a s d e p e n d e n t u p o n t h e i r p h a s e s a t th a t p a r tic u la r t i m e s te p , s o t h a t is a p p r o x im a te d b y k vck,jc * (6.3.2) • j-0 F o r e x a m p le , a c c o r d in g to E q u a tio n 6 .3 .2 a n d a s s e e n in t h e f ig u r e , th e v o lta g e a t t3 is a p p r o x im a te d b y vc*,j * ( M o + M i + I n t h e lim it a s A / —» 0 , t h e v o lta g e (t) a t tim e k a n d s u m m a tio n i n d e x t is hoh W • j ( 6 .3 .3 ) g iv e n b y t h e c o n v o lu tio n in te g ra l t jh (t-T )i(r )d T , o v c4/( 0 = w h e r e th e tim e s t e p + ( 6 .3 .4 ) a r e re p la c e d b y t h e c o n tin u o u s v a r ia b le s a n d t, r e s p e c tiv e ly . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. t 292 The photomixer current-voltage relationship can now be completed by substituting the convolution integral o f Equation 6.3.4 for the embedding circuit voltage vcla(r) in Equation 6.2.5, giving (6.3.5) 0 In principle, knowledge o f the architecture of the embedding circuit is enough to determine the impulse response h it) , so that the photomixer current-voltage relationship becomes solvable during the simulation. However, since the distributed nature o f actual microwave circuits requires that the circuit be solved in the frequency domain, the objective o f the present work is to build a system that begins in the frequency domain. Keeping with this objective, the convolution integral is Fourier transformed to the frequency domain, where it becomes the complex, frequency-dependent impedance function (Jo>) for the embedding circuit. This is the starting point for the approach developed in this study. According to the principle that the Fourier transform o f a convolution yields the product o f the Fourier transform o f each original function in the convolution integral, the Fourier transform of v^ f/) given by Equation 6.3.4 leads to [72] — [v^.(t)cxp(j<ot)dt = — (1 fh (t)i(t-r )d T exp{jtot)dt /-r)/(r)d r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 293 t + J/j(/ - r )/(r) d z -f- jh (t - z ) i ( z ) d z \cw p{jot)dt « CO 90 VdaUa) = — J \ h ( t- z ) i( z ) e x p ( jo t) d T d t Vdu U & ) = 1 1 * \h{&) expO'cocr) d a -j= = J/(r)ex p O < a r)rfr (6.3.6) v ck. U * o ) = Z dk,(J o > )-1 ( / a > ) . where with a - t - r {jco) is the Fourier transform o f the voltage across the circuit and I (J o ) is the Fourier transform o f the current in the photomixer circuit The two integrals added to the RHS in the second step equal zero because the current does not exist for r < 0 and the impulse response does not exist for t > 0 . Once the embedding circuit impedance is determined in the frequency domain, the photomixer circuit is solved, because the impulse response can be constructed from the inverse fast Fourier transform (IFFT) of the impedance function: Jza ( j o ) exp( - j o 6 .3 3 t)d o (6.3.7) Impedance Function In the present study, the embedding circuit impedance function Z ^Jcoi) is directly calculable using laws analogous to those that determine resistance in the time domain. This can be done because the transforms o f the time-dependent voltages and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 294 currents preserves KirchhofFs voltage and current laws [70]. Thevenin’s theorem is again used, this tim e in the frequency domain, to determine the impedance o f the equivalent element that replaces the embedding circuit, as shown in part (c) o f Figure 6.2.1. According to Thevenin’s theorem, the impedance presented at ports a and b o f the device due to the rest o f the circuit is identical to the impedance at these same ports when the ports are short circuited, with all independent sources deactivated [70]. Using the photomixer circuit pictured in part (b) of Figure 6.2.1 with the voltage source v^deactivated and the impedance values listed in Table 6.2.1, the frequency-dependent impedance function (Jco) presented at the ports a and b o f the device is given by r 1 1 jcoCB The complex impedance function (6.3.8) (Jco) given by Equation 6.3.8 is evaluated using typical values for the passive elements in photomixer circuits containing an MSM photodetector and a bias tee. These values include a load resistance RL = 50 £2, a bias tee capacitance CB = 5 p F , a bias tee inductance Z.fl = 5 nH , and two device capacitances, Cp = 20 fF and C p = 100 fF . Figure 6.3.2 presents the two complex impedance functions used in the present study, one for each of the two device capacitances that are modeled. The frequencies have been converted from angular frequency ©to f —col2 k for the graphs. These impedance functions are used to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 295 (a) 60 20 fF Device Capacitance 40 a N © o c (Q ■a © a. E Real (Z) Imaginary (Z) 20 -0 -20 -40 10 20 30 Frequency (GHz) (b) 8 c (0 ■O © 50 100 fF Device Capacitance 60 § 40 40 20 Real (Z) Imaginary (Z) -0 E -20 -40 0 10 30 Frequency (GHz) 20 40 50 Figure 63.2 Original impedance functions for device capacitances of (a) 20 fF and(b) 100 fF. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 296 generate the impulse response functions through the IFFT, as presented in the next section. Before the impedance function undergoes the IFFT it is discretized, with the impedance samples given by Z ^ = Z ckl (m A f), m = 0 ..M , (6.3.9) where Af is the step in the frequency domain and the impedance values are calculated up to the Nyquist frequency M A f. The Nyquist frequency is one-half o f the reciprocal o f the time step At used in the device simulator, in which 1/ At represents the sampling rate in the time domain. The Nyquist sampling theorem states that the sampling rate must be at least twice the frequency o f the highest frequency to be Fourier transformed [71]. Otherwise, aliasing would occur, which is the representation o f higher frequency components by lower frequency components in the transformed function. For example, in runs for which At = 50 fs, the Nyquist frequency is 10 THz, so sinusoid components o f frequencies greater than 10 THz should n o t be used in the Fourier transform o f the time domain signal. If an attempt is made to use a IS THz sinusoid for instance, then the Fourier coefficient corresponding to the 15 T H z sinusoid would incorrectly increase the coefficient associated with a lower frequency sinusoid. The use o f the Nyquist theorem is relevant because the final form of the impulse response is Fourier transformed to test its agreement with the original impedance function. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 297 6.3.4 Calculate Impulse Response The original form o f the discrete impulse response function z cb(tk) is derived through the IFFT o f the discretized impedance function Z m previously characterized by Equation 6.3.9: Z da{m bf) => IFFT => z ^ n A l ) , (6.3.10) with the IFFT expression being given by [71] ( .2 m m '] 1 ^ d a jt To «-0 V 1 (6.3.11) To J where N To is the total number o f impedance samples. Figure 6.3.3 presents the four impulse responses in their original discretized form as points. Table 6.3.1 lists the parameters used to calculate each discretized impulse response. The total number of samples N To is given by twice the ratio o f the Nyquist frequency to the frequency step Af , with A/ = 0.1 G H z. Note from the figure that for a given device capacitance, the shape and magnitude o f the response depends on the width o f the time step A/ since this effects the scaling and the sinusoid frequency through N To (see Equation 6.3.11). T able 63.1 Parameters for discretized impulse response functions. Time Step (fs) Nyquist Frequency (Hz) 50 500 l x l 0 lj lx l0 u N To 2x103 2xl04 21901 3901 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. *2 22000 4000 298 (a) 20 fF Device 15 CO > d) CO 10 Original 50 fs step Smoothed 50 fs step Original 500 fs step Smoothed 500 fs step c o Q. CO 0) tr. 0) co 3 Q. E 10 15 Time (ps) 20 25 30 (b) 100 fF Device CO 4 © co e 3 Q. CO 2 o V cc a> CO 3 a. E Original 50 fs step Smoothed 50 fs step Original 500 fs step Smoothed 500 fs step 1 0 0 5 10 15 Time (ps) 20 25 Figure 6 .3 3 Discretized impulse responses for device capacitances o f (a) 20 fF and (b) 100 fF. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 299 Since the frequency step must be small enough to avoid aliasing effects in the impulse response, a large number o f impedance samples are required. Therefore, the corresponding number 2Af+l o f discrete-time impulse response samples is large, on the order of 104 — 10s. In order to achieve a fast global simulation, the impulse response sequence must be truncated, which could limit the accuracy o f the simulation. However, an asymmetric Kaiser filter is used to reshape the impulse response to preserve accuracy, with the Kaiser filter function given by [28] 0 <n<N, Cs ■/o / M - r n —N x ^ = N x < n < N 2’ (6.3.12) /„(/?) 0 N 2 <n , where c ,, /?, and TV, are shaping factors, N z is the truncation index for the sequence, I 0 is the modified Bessel function, and w„ is the filter weight for the nth impulse response sample. For all four discretized impulse response functions, cs = 1 and /? = 7 (see Table 6.3.1 for the values o f A'",). The smoothed value o f each impulse response sample hn is calculated by multiplying the unfiltered sample by its filter weight w„ , or K = *****• Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (6.3.13) 300 Figure 6.3.3 also presents the smoothed forms o f the four discretized impulse response functions as lines, each o f which shows the necessary agreement with the corresponding original forms. The smoothed forms are further tested for accuracy by performing fast Fourier transforms (FFT) to ensure that the smoothed impulse responses match the corresponding impedance functions from which they were originally derived, with the FFT expression being given by [71] 2 Tonn (6.3.14) Figures 6.3.4 and 6.3.5 present the results, which indicate that the original impedance response functional forms are preserved. The advantage gained by truncation is that N 2 is on the order o f 103 - 104, which is 5 - 10 times smaller than the original impulse response sequences and enhances the simulation speed. 6.3.5 Discrete Convolution The global simulation is prepared by ramping up the voltage to and the light intensity to its baseline value using just the device simulator. After the device current reaches DC steady state, the photomixed light intensity with a millimeter-wave beat frequency is applied at time zero, and the global simulator with the discrete convolution is activated. The IFFT o f the discretized complex impedance function (Equation 6.3.11) produces the discrete form o f the convolution, which when smoothed (Equation 6.3.13) and substituted into the photomixer current-voltage relationship given by Equation 6.3.5 produces the discretized form of the photomixer current-voltage relationship [28] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 301 (a) 60 20 fF Device Capacitance 40 a N < ou c (0 T3 a) O- Real (Z) before Real (Z) after Imaginary (Z) before Imaginary (Z) after 20 E -20 -40 10 20 . 30 Frequency (GHz) (b) 40 50 100 fF Device Capacitance 60 40 g, N 8 c (0 3a. Real (Z) before Real (Z) after Imaginary (Z) before Imaginary (Z) after -0 E -20 -40 20 40 Frequency (GHz) Figure 6 3 .4 Original (before) and reconstructed (after) impedance functions for device capacitances o f (a) 20 fF and (b) 100 fF for SO fs time step. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 302 (a) 60 20 fF Device Capacitance 40 a N a) o c Rl ■a 03 Q. E Real (Z) before Real (Z) after Imaginary (Z) before Imaginary (Z) after 20 -0 -20 -40 10 20 30 Frequency (GHz) (b) 40 50 100 fF Device Capacitance 60 40 a N 8 20 c RI *a . E Real (Z) before Real (Z) after Imaginary (Z) before Imaginary (Z) after 0 X -20 -40 0 30 Frequency (GHz) 20 40 Figure 6.3.5 Original (before) and reconstructed (after) impedance functions for device capacitances o f (a) 20 fF and (b) 100 fF for 500 fs time step. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 303 * -l v* = vdc ~ 2 bk-jtj + (v* ) ’ (6.3. 1 5) where k represents the time step level kA l, and v and / are the voltages and currents o f the device. Equation 6.3.15 is in fixed-point iteration form, and the Ath current sample is separated from the other terms because ik is the only unknown current sample. To speed the iterative process, the unknown ik can be approximated by the linear extrapolation ik * 2/*_, - ik_2, (6.3.16) providing the time step At is very small [28]. This approximation is valid because the single unknown current sample has small effect on a summation over so many samples. In other words, the unknown voltage vk at time step k is determined almost entirely by the current terms under the summation symbol, all of which are known. The global circuit algorithm is as follows: (1) predict the device current ik for the present time step based on the known previous currents; (2) use the discrete convolution to generate the device voltage vk for the present time step; (3) use the device simulator with the device voltage vk from step 2 to calculate the corrected current ik; (4) check the percent difference between the prediction and correction for ik ; and (5) iterate if necessary. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 304 Experience to date has shown that step (5) has not been necessary, which greatly speeds the global simulation. The final output is the current iMTo(t) that is passed to the output branch containing the load resistance. This output current is calculated by beginning with Kirchhoffs current law in the frequency domain (6.3.17) where I , I p, I L, and /„ are the currents through the device (and photomixer circuit), the parasitic branch, the inductance branch, and the output branch, respectively. Using the definition o f admittance Y - \ I Z and ohm’s law in the frequency domain V = Z I , Equation 6.3.17 can be rewritten as [70] (6.3.18) where the impedances are given by Table 6.2.1 (with Z„ = ZCj + RL). Then using Vd = l J Y 0, Equation 6.3.15 can be solved for the frequency domain output current to give Finally, the time domain current iACo( 0 at the fundamental frequency is calculated by performing the inverse phasor transform o f the complex output current given by Equation 6.3.19, leading to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. < = {flc* (a>)expOW[ ‘acA O “ Refc^(<»)expOW «P(<i>')} L c .o ( 0 = ' ° c . „ f c > ) s i n ( ® 0 , (6.3.20) where 0 = —n i l to produce the sine solution, and from which the AC peak i°M-0M extracted as explained in section 5.3 in order to construct the frequency response. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is 306 6.4 TESTING THE GLOBAL SIMULATOR The global simulator is tested by making use o f the principle that the integral o f the impulse response function over time equals zero. This can be proven by first assuming that only a DC signal passes through the photomixer circuit. Under this assumption, examination o f part (b) o f Figure 6.2.1, in conjunction with the formulas presented in Table 6.2.1, indicates that the parasitic and output branches behave as open circuits due to the infinite impedances o f the capacitances, so that these branches carry no current; while the inductance behaves as a short since it offers zero impedance. Therefore, under DC operating conditions, only the inductance branch carries current, and the embedding circuit presents zero impedance to the device. This can be verified by evaluating (J co) as given by Equation 6.3.8 in the limit as the frequency a - * 0 : -i lim Q>-* 0 lim 6)-* 0 j0 )C . + JQ)LB ja>CB lim A -i ~0 — +~0 j°>LB + - oj —► 0 lim jto L B a>-¥ 0 0. The vanishing o f (6.4.1) (jco) in Equation 6.4.1 causes V * (Jeo) in Equation 6.3.6 to vanish as well, which causes the IFFT o f V^ijcoi) to equal zero, or Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 307 = 0 = va ,(r)]DC, (6.4.2) DC so that the voltage drop across the embedding circuit vanishes. Since the current /(/) is constant under DC conditions, i^ . can be factored out o f the convolution given by Equation 6.3.4, leaving i t vcb(Oloc = 0 = j h ( t ~ T ) i ^ d x = ioc Jh i t - x ) d x , 0 0 (6.4.3) The only way for the far RHS o f Equation 6.4.3 to vanish is for the integral of the impulse response to vanish, since is nonzero. There are two ways in which the principle o f the vanishing sum o f the impulse response is used to test the global simulator. First, the discrete form o f the impulse response functions are summed to determine whether they actually approach zero. The results o f the summation test are presented in Table 6.4.1, which lists the raw and smoothed sums of the discrete impulse response samples for all four combinations of time step and device capacitance. A ratio is formed between each sum and the corresponding sum of the absolute values o f the samples, with the expectation that the ratio should be very small if the discrete sample sums are in fact approaching zero. Since the ratios are on the order o f 1 0 or smaller, the vanishing of the sum of the impulse responses is verified. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 308 Table 6.4.1 Testing for the vanishing o f the sums o f the discrete impulse response series (with z ^ j , for unfiltered samples, and hn for smoothed samples). Time Step/Device Capacitance 50 fs/20 <F 50 fs/100 fF 500 fs/20 fF 500 fs/100 fF Raw Smoothed Raw Smoothed Raw Smoothed Raw Smoothed -0.0382 -0.0409 -0.0231 -0.257 -4.14x10"* -1.47x10“* -7.08x10"* -5.66x10"* 102.47 102.47 97.32 97.31 102.60 102.60 97.44 97.44 ^dajt -3.7x10"* -4.0x10"* -2.4x10"* -2.6x10"* -4.0x10** -1.4x10** -7.3x10** -5.8x10** The second way in which the principle of the vanishing sum o f the impulse response can be used to test the global simulator is by direct comparison o f the global simulation results to the results o f the device-only simulator under DC operating conditions. Under DC operating conditions, both the device-only and the global simulations should produce identical photocurrents. The vanishing o f (r) in the photomixer current-voltage relationship under DC conditions shown by Equation 6.4.2 causes the voltage across the device to equal the DC voltage source, or vd(t) — Equation 6.2.5). (see This is identical to the voltage relationship for the device-only simulations. To perform the second test, the photocurrents o f the device-only simulations are compared to the results of the global simulations for three sets o f DC operating conditions for the FD mobility model: (1) equilibrium; (2) constant 1 V bias without illumination; and (3) constant 1 V bias with 1 KW/cm2 constant illumination. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 309 Four combinations o f global simulations are compared to the device-only simulations for each o f the above three cases o f DC operating conditions, and consist o f the four possible pairings o f the SO fs and 500 fs time steps with the 20 fF and 100 fF parasitic capacitances. The global simulations are run until more than one cycle o f each impulse response is completed to allow the full effect of the impulse response to manifest itself. For the 500 fs time step, 5000 time steps are completed, and for the 50 fs time step, 30000 time steps are completed. In each case, the photocurrent is sampled after the transient has decayed and the photocurrent becomes constant. The transient is a numerical effect that arises because the state variable configuration output from the device-only simulator and input into the global simulator is not in exactly the same configuration as the final configuration output by the global simulator. The results are presented by plotting the photocurrent generated by the deviceonly simulation on the same graph as the photocurrent generated by one of the four global parameter combinations, with both photocurrents corresponding to the same set o f DC operating conditions. Figure 6.4.1 compares the photocurrents under equilibrium conditions for all four global parameter combinations. The photocurrents are on the order o f 10"20 A , which are so small that they are within the range o f numerical noise, as evidenced by the fluctuations. Nonetheless, there is excellent agreement between the photocurrents generated by the device-only simulation and the global simulations, especially in consideration o f the very small scale. Under the DC operating conditions and the DC with constant illumination operating conditions illustrated in Figures 6.4.2 and 6.4.3, respectively, there is again excellent agreement between the device-only Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 310 (a) Time (ps) for Global Solver 2400 -15 2425 2450 2475 Time (ps) for Global Solver (b) 2500 1480 -15 1481 500 fs time step 20 fF capacitance -17 E <u c §3 o 3 o 03 O -21 g, • Device o Global -23 1484 1485 -19 <b -21 • Device o Global -23 -25 1483 50 Is time step 20 fF capacitance -17 -19 1482 -25 580 585 590 595 600 580 Time (ps) for Device Solver (c) 2425 2450 2475 590 595 600 Time (ps) for Device Solver Time (ps) for Global Solver 2400 -15 585 Time (ps) for Global Solver (d) 2500 1480 1481 1482 1483 1484 1485 -15 500 fs time step 100 fF capacitance -17 50 fs time step 100 fF capacitance -17 < c £W 3 o 03 o *19 c -19 3 o *21 • Device o Global -23 -21 • Device o Global -23 -25 -25 580 585 590 595 Time (ps) for Device Solver F i g u r e 6 .4 .1 600 580 585 590 595 Time (ps) for Device Solver C o m p a r i s i o n o f d e v ic e a n d g lo b a l s im u la tio n s a t e q u ilib riu m . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 600 311 (a) Time (ps) for Global Solver 2400 2425 2450 2475 Time (ps) for Global Solver (b) 2500 1480 1481 500 fs time step 20 fF capacitance 1482 1483 1484 1485 50 fs time step 20 fF capacitance o> Device Global • Device — Global -11 -11 580 585 590 595 600 580 Time (ps) for Device Solver 2400 -3 2425 2450 2475 590 595 600 Time (ps) for Device Solver Time (ps) for Global Solver (c) 585 Time (ps) for Global Solver (d) 2500 1480 1481 500 fs time step 100 fF capacitance 1482 1483 1484 1485 50 fs time step 100 fF capacitance •5 £ s O oo> -7 •9 • Device — Global Device Global -11 -11 580 585 590 595 Time (ps) for Device Solver 600 580 585 590 595 Time (ps) for Device Solver Figure 6.4.2 Comparision o f device and global simulations for 1 V DC bias. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 600 312 (a) Time (ps) for Global Solver 2400 2425 2450 2475 Time (ps) for Global Solver (b) 2500 1480 -2 -2 1481 500 fs time step 1483 1484 1485 50 fs time step 20 fF capacitance 20 fF capacitance -3 ■3 c g3 o 8* -4 O ol ■5 -4 -5 Device Global Device -6 -6 580 585 590 595 600 580 Time (ps) for Device Solver 2400 2425 2450 2475 585 595 600 Time (ps) for Global Solver (d) 2500 -2 1480 -2 1481 1482 1483 1484 1485 50 fs time step 100 fF capacitance 500 fs time step 100 fF capacitance ■3 -3 -4 -4 -5 -5 -6 580 590 Time (ps) for Device Solver Time (ps) for Global Solver (C) 3 o oo> 1482 Device •6 585 590 595 Time (ps) for Device Solver 600 580 585 590 595 Time (ps) for Device Solver Figure 6.4.3 Comparision of device and global simulations for 1 V DC bias under 1 KW/cm2 constant illumination. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 600 313 simulation and the global simulations regarding the photocurrents. This agreement is further confirmed by an E-x diagram comparison for the combination that uses a 500 fs time step and 20 fF parasitic capacitance. As shown in Figure 6.4.4, the device-only and global solvers produce identical E-x diagrams for all three operating conditions. Therefore, the second test also verifies the accuracy o f the global simulation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 314 a) Equilibrium % 05 u7 0.0 -0.5 0.4 0.0 (b) 0.6 x(nm) 0.8 1.0 1.2 0.8 1.0 1.2 1.0 1.2 1 V DC Bias 1.0 0.5 $• ^ 00 -0.5 - 1.0 -1.5 0.0 0.4 0.2 0.6 x(um) 1 V DC Bias, 1 KW/cm2 Constant Illumination (C) 1.0 0.5 I Uj 0.0 -0.5 - 1.0 -1.5 0.0 0.4 0.2 Device Solver Ec E Ev Efn Efp 0.6 0.8 Global Solver e Ec a E a Ev ♦ Efn O Efp Figure 6.4.4 Comparison o f band diagrams for device and global simulations for 500 fs time step, 20 fF capacitance, and FD mobility. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 315 6.5 6.5.1 EFFECTS OF GLOBAL SIMULATOR ON DEVICE PERFORMANCE Interaction of Time Constants Determines Bandwidth The presence o f impedance in the embedding circuit is expected to alter the performance o f the photomixer circuit relative to the device-only operation. The bandwidth should be reduced because the passive elements slow the response o f the photomixer circuit due to the finite time width o f the impulse response. The effect o f the embedding impedance on optical responsivity is more difficult to predict. Based on the usual inverse relationship between bandwidth and responsivity, an increase in responsivity is expected to accompany a decrease in bandwidth. However, the shunting o f some o f the AC signal into the parasitic branch o f the embedding circuit is expected to reduce the AC signal to the output branch, thereby lowering the responsivity. In order to determine the effect o f the embedding circuit on the two figures o f merit, the photocurrents that result from the global simulations are compared to those from the device-only simulations. To account for the results o f the comparison o f the device and global simulations in a quantitative way, attention is focused on the three time constants whose interaction determines the final bandwidth o f the global frequency responses. These time constants include the device response time Td, the output branch RC time constant parasitic branch RC time constant , and the p. These three time constants must be defined in a consistent manner if the comparison of their values is to be a valid tool for understanding how bandwidth is limited. Therefore, each time constant is defined to correspond to that frequency at which the given time constant reduces the output current to 1/V2 o f its Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 316 maximum constant value. The remainder o f this subsection explains how these time constants are derived and how they interact to determine the global bandwidth. The device response time r d is derived by treating the device as if it were under DC steady state conditions, so that the AC signal can be approximated as a small signal perturbation, and then assuming that a large signal produces approximately the same result regarding the definition o f the time constant. Under DC steady state conditions, the one-dimensional electron current continuity equation is given by ^ ) - i a / ' W + Gu - * W dt q dx 0= na(x) —n„(x) o — ——-----------------------------------------------(6.5.1) where the thermal recombination-generation rate RG0(x) is given by a relaxation time approximation for which neq(x) represents the equilibrium electron density, nQ(x) is the DC steady state electron density at position x, and rd represents an overall recombination time for the device. The overall recombination time xd is considered the device response time because it includes the two characteristic times that determine the device response time: the bulk recombination time and the transit time. The relaxation time approximation form o f the RG rate does not restrict itself to the recombination time constants r„ and r p the way that Equation 2.5.7 for the RG rate does. Therefore, the overall recombination time r . allows for recombination at the contacts as well as in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 317 bulk, which automatically incorporates the transit time. For a small sinusoidal perturbation, the electron current continuity equation is approximated as dn{x,t) ~ n ( x ,t) - n ( x ) - - * Gl o + G l exp(jto t)----------------— , ot X, (6.5.2) while the electron density is approximated by n(x,t) * rt0(x) + n(x)exp(j( 0 t) , (6.5.3) in which GL and n(x) are complex amplitudes. Substituting the electron density given by Equation 6.5.3 into the electron current continuity relation given by Equation 6.5.2 leads to y o n (x )e x p (/fi)/)sr Glo + Gl cxpijeot) n0(x) + n (x) expO'e o t) - n r (x) G l.o ■Gl exp(jcot)- j£on(x)exp(jct)t)*iG L c x p {jo )t)- n(x)exp(ja>t) n(x)exp{jeot) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (6.5.4) 318 where the quantity in brackets on the RHS o f the next to the last step represents the DC signal and therefore vanishes according to Equation 6.5.1. Solving for the electron density in Equation 6.5.4 gives w( * ) « — ■1 +J6)Td (6-5.5) Since n (x ) = nm„ ( x )e x p ( /» , the real part o f the electron density amplitude n„„ (x) is determined by taking the magnitude o f n(x) in Equation 6.5.5, giving (6.5.6) The electron density magnitude n ^ i x ) due to the AC signal is used to determine the small AC signal peak in the device, which will enable the device time constant rd to be defined in terms o f the device bandwidth vz A . The perturbed electron velocity is approximated as U (x,0 « v0(x) + u(x)exp(jcot), which leads to the approximation o f the electron current density as J „ (x,t)= q n (x,t)u (x,t) * qnQ(x)t>0(x) + qn0(x)u(x) cxp(jeo t) + qn(x) cxp(jcot)u0(x) + qn(x) exp(y cot)u(x) e x p (j cot) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (6.5.7) 319 * q n (x)v 0(x) exp (jto t) , (6.5.8) where n0(x) and the perturbed portion o f the electron velocity are assumed relatively small. [This is true in the large signal case that applies in the present study for n0(x) since the light causes n (x) » n0(x) , and is also true for u (x) because the electric field does not change appreciably over time.] Substituting for the magnitude o f the electron density given by Equation 6.5.6 into Equation 6.5.8 and multiplying by the current crosssectional area gives the small AC signal peak in the device as C c ) = \ J n A yz~ q , , l> 0a „ , Vl + (2 7CVm Td ) 2 (6.5.9) in which com = 2n v m and the x-dependence is understood. This produces the familiar bandwidth behavior described in section 2.4, in which the AC peak is constant for low modulation frequencies and eventually rolls off with increasing vm. The 3-db point occurs when the constant low frequency AC peak is reduced by 1/ -v/2, which occurs for the bandwidth frequency ^ 3 - ~ , 2Jtzd (6.5.10) and enables the device time constant r . to be defined in terms o f the device bandwidth as r rf= — . 2^3-(* Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (6.5.11) 320 This relationship is assumed to be approximately true for the large signals that are simulated in the present study. The output branch time constant r ^ a is derived by considering the photomixer circuit in the low frequency limit. At lower modulation frequencies, only the output branch passes significant AC signals since the bias tee capacitance C B is at least SO times larger than the parasitic capacitance Cp (see Table 6.2.1). In this low frequency range, the AC portion o f the circuit can be simplified to include just the output branch, as pictured in part (a) o f Figure 6.5.1, with the parasitic branch still open (and not shown). Using the Thevenin equivalent approach, the impedance Ztq o(J o ) for this output branch approximation is given by (6.5.12) + Rl , which leads to a frequency-dependent AC signal magnitude o f 1 (6.5.13) The term in brackets on the RHS o f Equation 6.5.13, called the output approximation fraction, represents the fraction of the maximum AC signal magnitude as a function o f frequency, in which the maximum AC signal magnitude is given by / RL, the value as o -> oo. Figure 6.5.2 graphs the output approximation fraction as a function o f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 321 (a) Output Branch Approximation + “T T T " Voc 54 MSM rl (b) = son Parasitic Branch Approximation 50 £2 MSM VDC Figure 6.5.1 Equivalent photomixer circuit approximations, inculding (a) output branch and (b) parasitic branch. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 322 (a) 20 fF 2.0 Actual Output Parasitic Approximation Output Approximation 1.5 o (0 c <u 1.0 0.5 0.0 2 3 4 56710 8 2 3 4 567^9 2 3 4 5 6 1Q10 2 3 4 5 6 1 0 11 2 3 456^12 Frequency (Hz) (b) 100fF 2.0 Actual Output Parasitic Approximation Output Approximation c o o CO Li. 1.5 1.0 7o»Jwr O 0.5 0.0 1Q7 2 3 4 5 6 7 1q 8 2 3 4 567^9 2 3 456^10 2 3 456^11 2 3 456^12 Frequency (Hz) Figure 6.S.2 Current fractions for (a) 20 fF and (b) 100 fF parasitic capacitances. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 323 frequency v for both parasitic capacitances used in the global simulations. The roll-off in frequency response occurs to the low frequency side, and there is no restriction on the AC signal amplitude to the high side o f the frequency response curve. This is the reverse o f the effect that occurs for the device bandwidth, where there is no restriction on the AC signal amplitude to the low frequency side o f the response curve, and the response rolls off to the high frequency side. The 3-db point for the output approximation fraction occurs when the constant high frequency AC peak is reduced by l/V2 , which occurs for the low roll-off frequency (6.5.14) roilJaw with (6.5.15) Based on Equation 6.5.15, the output branch time constant is which corresponds to a low roll-off frequency o f =250ps, = 0.6 GH z, as shown in Figure 6.5.2. If the device response time rd is longer than 250 ps, which corresponds to a device bandwidth t RCo prevents that is less than 0.6 GHz, then the output branch time constant any significant AC signal from being output except for cases in which the device bandwidth is comparable to 0.6 GHz. For device response times rd shorter than 250 ps, the output branch time constant „ has little effect on global bandwidth, except Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 324 again for cases in which the device bandwidth is comparable to 0.6 GHz. Otherwise, the bandwidth is determined by the interaction between the device response time Td and the parasitic branch time constant r * - , . [NOTE: As indicated in the figure, the low roll-off frequency is shifted toward higher frequencies in the actual output as compared to the output approximation due to the inductance branch, which shunts more current with decreasing frequency. To simplify the characterization o f the interaction o f the time constants, the inductance branch is not included in the formalism; however, its effect regarding interaction with the device time constant m ust be considered in those cases where the device time constant is comparable to the output branch time constant.] p is derived by considering the photomixer The parasitic branch time constant circuit in the high frequency lim it At higher modulation frequencies, the parasitic capacitance begins to shunt significant AC signal through its branch, as shown in Figure 6.5.2. At high frequency, the AC portion o f the photomixer circuit can be modeled by the parasitic branch approximation pictured in part (b) o f Figure 6.5.1, since the bias tee capacitance acts as a short. Again using the Thevenin equivalent approach, the impedance Z tq p(Joj) for this parasitic branch approximation is given by (6.5.16) V A which leads to a frequency-dependent AC signal magnitude o f (6.5.17) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 325 The square root term on the RHS o f Equation 6.5.17, called the parasitic approximation fraction, represents the fraction o f the constant limiting AC signal magnitude as a function o f frequency, in which the constant limiting AC signal magnitude is given by voc t R-l » t^ie value as <y —►0 . As shown in Figure 6.5.2, the parasitic approximation fraction is constant and equal to unity to the low frequency side o f the response, and increases beyond unity to the high frequency side o f the response, so there is no roll-off in the parasitic current. Unlike the output branch, which has a lower limit for its impedance due to the presence o f the load resistance, the parasitic branch is relatively limitless regarding the reduction in its impedance due to the low resistance o f the device. As current is shunted from the output branch, the actual AC signal output drops, as shown in the figure by the actual AC signal output curve. This motivates choosing the characteristic time constant for the parasitic branch approximation as that frequency at which the constant limiting current is increased by 4 l , so that the high frequency roll off is defined as VrollJligh —~Z ZTtTffCp (6.5.18) *kc.p = R lCp . (6.5.19) with This is a reasonable choice because the output and parasitic branch currents are inversely related, as shown in Figure 6.5.2. Due to this inverse relationship, an increase in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 326 parasitic branch current by -Jl causes a reduction in the output branch by l / V 2 , and thus preserves the required consistency in defining the time constants. Based on Equation 6.5.19, the parasitic branch tim e constants are —1 ps and = 5 p s , which correspond to high roll-off frequencies of v ^ , \ t =159 GHz and 1® , - 3 2 G H z, respectively, as shown in Figure 6.5.2. For the 20 fF parasitic capacitance case, if the device response time rd is longer than 1 p s, which corresponds to the device bandwidth being less than 159 G H z, then the global bandwidth is limited by the device response time r d . For a device response time rd shorter than 1 p s , so that the device bandwidth is greater than 159 GHz, the parasitic branch time constant rj£?p limits the global bandwidth. For the 100 fF parasitic capacitance case, if the device response time z d is longer than 5 p s , which corresponds to the device bandwidth w3-<n, being less than 32 G H z, then the global bandwidth is limited by the device response time r d . For a device response time zd shorter than 5 p s, so that the device bandwidth is greater than 32 G H z, the parasitic branch time constant r^ ° p limits the global bandwidth. 6.5.2 Comparison of Device and Global Simulations Three sets o f simulations are run, all with the long recombination lifetime constant (10~* s ) , and these include the device alone and the photomixer circuit with device parasitic capacitances o f 20 fF and 100 fF . The results are presented in Figure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 327 6.5.3 through a comparison o f the frequency response curves for eight different combinations o f mobility model (FI and FD), voltage ( I V and 10 V ), and light intensity (1 KW/cm2 and 25 KW/cm2). Each graph compares the frequency responses generated by the device simulation and the global simulations for the two parasitic capacitances. The bandwidth of the device-only response curve is used to determine the device response time rd according to Equation 6.5.11. responsivities R ^ and bandwidths Table 6.5.1 lists the optical for all three frequency responses for each o f the eight parameter combinations. The table also presents the device response times r d . The position o f the device response time zd relative to the 20 fF time constant z™ p = 1 ps or the lOOfF time constant = 5 p s determines which time constant limits the bandwidth. Table 6.5.1 Optical responsivity R ^ and bandwidth 100 fF global simulations. Device Parameters Rop, (A/W) FI, 1 V, 1 KW/cm* 0.21 FD, 1 V, 1 KW/cm2 0.21 0.16 FI, 1 V, 25 KW/cm2 0.15 FD, 1 V, 25 KW/cm2 FI, 10 V, 1 KW/cm2 0.22 FD, 10 V, 1 KW/cm2 0.22 FI, 10 V, 25 KW/cm2 0.22 FD, 10 V, 25 KW/cm2 0.22 v^-db (GHz) 19 14 13 8 188 33 188 33 for device, 20 fF global, and 20 fF Global (PS) 8.4 11.4 12.2 19.9 0.8 4.8 0.8 4.8 RofX (A/W) 0.21 0.21 0.15 0.14 0.22 0.22 0.22 0.22 Vy-db (GHz) 18 14 11 7 103 33 102 33 100 fF Global Ropt (A/W) 0.21 0.21 0.15 0.14 0.22 0.22 0.22 0.22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (GHz) 15 12 11 7 31 23 31 22 328 (■) 0.22 00 0.22 | 0.18 | 0.18 | 0.14 8 0.14 $ o.io 1 0.10 a. (c) 0.22 ? 0.18 7/ ~ \% "v 1 KW/cm2 tr 9 C t 10® 2 345 10® 2 345 101° 2 345 10” 2 Frequency (Hz) Frequency (Hz) R 1V 25 KW/cm2 0.10 0.18 0.14 (A O 0.10 a 0C 0.22 FD 1V 25 KW/cm2 0.18 8 0.14 « K 0.10 10® 2 345 109 2 345 1010 2 345 10 ” 2 Frequency (Hz) Frequency (Hz) (*) 0.22 8 c (d) | I 10a 2 3 45 ^q9 2 345 1Q10 2 345 1Q11 2 (e) f 0.22 0.18 S c I o: 10 V 1 KW/cm2 0.14 . FD 10 V 1 KW/cm2 0.10 2 345 109 2 345 ^glO 2 345 ^glt 2 10® 2 345 10® 2 345 ^qIO 2 345 Frequency (Hz) 0.18 0.18 8 c 0.14 e 0.10 tr Frequency (Hz) 0.22 0.22 a 10,11 2 (h) (8 ) 1 v 108 2 3 45 109 2 3 45 1Q10 2 345 1Q11 2 0.14 o o. 7 FD V KW/cm2 FI 10 V 25 KW/cm2 0.14 0.10 FD 10V 25 KW/cm2 10® 2 3 45 10 ® 2 345 ^glO 2 345 in 10’11 2 10® 2 345 10® 2 345 101° 2 345 10 ” 2 Frequency (Hz) Frequency (Hz) Solver Legend Device 20 fF Global 100 fF Global Figure 6.5.3 Comparison o f frequency response curves for device and global solvers. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 329 In the four cases o f low voltage, which include parts (a) - (d) o f Figure 6.5.3, the nearly equal bandwidths for all three curves on each graph is consistent with the device response time z d being longer than both the 20 fF time constant r 100 fF tim e constant = 1 ps and the = 5 p s . For these four cases, the device response time z d limits the global bandwidth. For the two FI mobility cases at high voltage given in parts (e) and (g), all three bandwidths separate on each graph. That the device bandwidth is greater than both global bandwidths is consistent with the device response time zd being shorter than the 20 fF as well as the 100 fF time constants. For both global simulations, the bandwidth is limited by the parasitic capacitance of the embedding circuit. For the two FD mobility cases at high voltage given in parts (f) and (h), only the 100 fF bandwidth separates from the other two curves. This result is consistent with the device response tim e z d being shorter than the 100 fF time constant than the 20 fF time constant = 5 ps and longer = 1 p s . For the 100 fF case, the global bandwidth is limited by the parasitic capacitance, while for the 20 fF case, the global bandwidth is limited by the device response time z d . Two findings are brought to light by the table and the graph regarding optical responsivity. First, the responsivity is essentially unaffected by the embedding circuit, even in those cases when the embedding circuit significantly reduces the bandwidth relative to the device-only simulations. This finding could not be predicted, and points to one o f the values o f using simulations. Secondly, for all cases o f increasing voltage, bandwidth increases substantially while responsivity increases only slightly, except for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 330 the cases o f high light intensity, in which the responsivity does increase substantially [compare (c) and (d) with (g) and (h)]. The field-screening effect due to the elevated charge density in the high light intensity cases is overcome by the electric field associated with the large potential applied across the device, so the response is fast. This elevates the velocity factor o f the nv product in the current, which raises the responsivity substantially as compared to the corresponding low voltage, high light intensity cases. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 331 CHAPTER 7 CONCLUSION 7.1 NEW ACCOMPLISHMENTS The broad purpose o f this work was to make a contribution to the science and engineering o f mixing light to produce radar microwaves, with the expectation that conventional radar microwave transmission will someday be replaced by a fiber-optic system. The method for this contribution was the development o f a global simulation for a particular photomixer circuit. The model o f the photomixer circuit has two parts: (a) a GaAs metal-semiconductor-metal (MSM) photodetector with trench electrodes; and (b) an embedding circuit with a bias tees, a driving voltage, and a parasitic capacitance. The specific objective o f this work was to construct a global simulation that is both accurate and efficient. In meeting this objective, two new accomplishments resulted: (1) the development o f a convolution-based global simulation for an MSM photodetector that accurately characterizes the embedding circuit, and achieves efficiency through (a) truncation o f the discrete impulse response sequence, and (b) a linear extrapolation to predict the current in conjunction with a fixed-point iteration scheme for the convolution; and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 332 (2) the development o f a physically reasonable current density boundary condition derived from first principles o f the semiclassical model, including the use o f (a) all three revisions that were previously made to the original model, and (b) a variable carrier density fraction as a new revision. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 333 7.2 GLOBAL SIMULATIONS The use and refinement o f global simulations, which integrate the device model into the embedding circuit model, has grown in the last ten years. An important subset o f globed simulations is concerned with modeling photomixer circuits. The two goals behind these simulations are (I) to provide a design tool for the efficient characterization o f photodetector performance, and (2) to enhance the understanding o f the internal mechanisms o f device operation from a scientific viewpoint. To meet the first goal, the simulation calculates the optical responsivity and the bandwidth, both o f which depend on the AC steady state solution. The second goal is accomplished through the study o f variables as functions o f position and time, where the emphasis is on solutions that reveal the progression toward steady state. One or both o f these goals are accomplished with varying degrees o f accuracy and efficiency, depending on which o f the four major simulation methods is used. These four methods include (1) behavioral models, (2) the physics-based device solver with a simplified circuit, (3) the harmonic balance technique, and (4) the physics-based, full time-domain solvers. In reviewing the major global simulation methods, it is instructive to center the discussion around the algorithm that summarizes all o f these simulations except for the harmonic balance technique. At each time step k, the global simulation does the following: (1) set the voltage vdJt applied to the device based on initial conditions or step 4; (2) set the light intensity ^Jk that illuminates the device based on the time step; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 334 (3) calculate the current ik based on vdJt and s j k, using the device solver; and (4) generate the device voltage vrfJk+I for the next time step k +1 based on ik, using the embedding circuit solver. The behavioral approach is the most efficient simulation method due to its simplification o f the device model through the use o f analytical expressions in step 3. Since only terminal currents are calculated, however, the scientific goal for simulations is not adequately met [3]. Furthermore, the quasi-static mode under which the device is assumed to operate limits behavioral model accuracy to low frequency conditions [11]. The approach that uses a physics-based device solver with a simplified circuit accomplishes both the design and scientific goals o f simulations, and is accurate and efficient regarding the modeling of the device [24]. The modeling of the embedding circuit, however, can be inaccurate since the convolution is not included in the photomixer current-voltage relationship for step 4, as given by VJ * = V D C ~ RJ ■ (7.2.1) According to Equation 7.2.1, the embedding circuit presents no impedance, so that the predicted bandwidth can only be limited by the device response time. As evidenced by Table 6.5.1, this leads to bandwidth errors in the FI mobility models that are as high as six-fold. The harmonic balance technique is accurate and efficient in modeling both the device and the photomixer circuit, but this technique does not permit the study o f transient behavior since the photomixer current-voltage relationship is solved in the frequency domain for AC steady state [11],[28]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 335 The most accurate time-domain approach for photodetector global simulations is the physics-based, full time-domain solvers, which model both the device (step 3) and the embedding circuit (step 4) in their time-dependent, physics-based forms. Furthermore, these full time domain solvers are very effective in accomplishing the design and scientific objectives o f simulations. The greater accuracy comes at a cost, however, in that the efficiency is reduced, sometimes drastically. In principle, the most accurate approach is Monte Carlo, and this is the best method for the purely scientific study of transient behavior [20]-[23]. For AC analysis and for design purposes such as the calculation of bandwidths and responsivities, Monte Carlo is too computationally demanding. The next level in accuracy is occupied by the combined electromagnetic and solid-state (CESS) simulator, which uses a hydrodynamic model to update the current density and a Maxwell equations solver to update the electric and magnetic fields [25],[26]. The CESS simulator achieves its high degree o f accuracy by accounting for the effects of electromagnetic wave propagation, which become more important with decreasing device dimensions and increasing modulation frequencies. However, solving Maxwell’s equations at each time step is so computationally intensive that the use of massively parallel machines is required. In order to achieve efficiency as well as accuracy in photomixer simulations, the first major objective o f this study was to develop a simulation that combines the driftdiffusion model o f carrier transport with an efficient convolution to characterize the embedding circuit. The drift-diffiision model consists o f six transport equations, which is few enough to enable an efficient device simulation and still achieve accuracy. The simulation solves the transport equations simultaneously using a numerical, Newton- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 336 Raphson technique. The use o f the convolution in step 4 accurately characterizes the photomixer current-voltage relationship, and allows the embedding circuit to limit the bandwidth as expected in actual photomixer circuits [68]. Interest in convolutions has grown in recent years, but they too can be computationally intensive. The development by Tait and Jones o f an asymmetric Kaiser filter enables the discrete impulse response sequence to be truncated so that the convolution is more efficient without losing accuracy, and the use of a linear extrapolation to predict the current in conjunction with a fixed-point iteration scheme further reduces the computational time [28]. In the present study, device and global simulations were conducted for several parameter combinations, both to test the accuracy o f the simulations and to achieve a greater understanding of device operation. The results o f the simulations were presented in two forms: (1) as frequency response curves to illustrate the bandwidth and responsivities and thereby meet the design goal of simulations; and (2) as a combination of E-x diagrams and the associated device profiles to meet the scientific goal. The frequency response curves were presented using both the FD and FI mobility models, which represent the lower and upper limits to the bandwidths, respectively, for a given parameter combination. Several studies were conducted to test the accuracy o f both the device and global simulations, and these were applied to models o f the Schottky diode and the MSM. The device simulator consists of the Poisson solver, which generates the initial equilibrium values for the state variables, and the transport solver that evolves the state variables over time under conditions o f bias and illumination. The Poisson and transport solvers generated identical E-x diagrams and associated device profiles under equilibrium Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 337 conditions. For the transport solver, the total current density was zero and the drift and diffusion components o f the current density canceled each other throughout the device. Additionally, the carrier density and electric field profiles were consistent with the E-x diagrams. Under conditions o f bias, the total current density remained constant across the device as expected, and the profiles were consistent with the corresponding E-x diagrams. The global simulation was tested in two ways through the principle that the integral o f the impulse response over time vanishes. First, the sums o f the discrete impulse response sequences were shown to approach zero. Secondly, also as expected, the comparison o f the global and device-only simulation results indicated that both simulations produced the same photocurrents under DC operating conditions, which included equilibrium and DC bias with and without constant light In conclusion, the tests indicate that the device and global simulations are accurate. The simulations are concluded to be efficient because bandwidth curves could be generated in computational times that ranged from approximately 2 hours for device simulations using the FI mobility model to approximately 14 hours for global simulations using the FD mobility model. The results o f the device and global simulations led to the following interesting and significant findings: (1) In the absence o f light, the high barrier (1.0 eV) MSM is almost completely depleted o f conduction band electrons, so that holes are the primary charge carriers, even though the GaAs substrate is n-doped. This is evidence o f the fundamental importance o f the thermal RG rate in the current continuity equations since this rate generates the excess holes that compensate for the loss o f electrons. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 338 (2) For the Schottky diode in DC steady state in the absence o f light, the current is driven by diffusion in the depletion region and by drift in the neutral region. Also, the drift component o f the current density reverses its direction in the neutral region relative to the charge-depletion region. (3) In the low-barrier (0.6 eV) MSM simulation that does not include holes in the transport equations, there is a 50% error in electron density even for n-doped substrates, for which the absence o f holes was not expected to have significant effect. The high-barrier (1.0 eV) MSM simulation that does not include holes fails to converge after just a few tim e steps. These findings reinforce the importance o f the RG rate term, and indicate that the presence o f the hole terms and the additional equations that holes require adds numerical resiliency as well as accuracy to the simulation. (4) In the MSM under illumination, both carriers have a net flux out o f the device at each boundary due to the high concentration gradients close to the boundaries that overcome the electric field. (5) The device simulator is robust, as evidenced by the following: (a) there are no limitations on applied voltage alone; (b) there is convergence inefficiency for high light intensity only when the voltage is also high; and (c) the simulation fails completely only for short device lengths (submicron) at high voltage under illumination. (6) In the presence o f light, the barrier height has no effect on the simulation results because the source of the excess carriers is essentially generation by light, and carrier Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 339 inflow restriction by the Schottky barrier is compensated by outflow o f the opposite carrier at that boundary. (7) For conventional-growth-temperature GaAs, the device response time and therefore the bandwidth is determined completely by the transit time. For low-growth- temperature GaAs, the device response tim e is determined by the interaction between the effective recombination lifetime and the transit time, with the shorter o f these two times having the greater effect in limiting the bandwidth. (8) As MSMs become shorter, the bandwidth increases significantly, while responsivity remains almost constant, even though these two figures o f merit typically have an inverse relationship. (9) The bandwidths for the photomixer circuits are determined by the interaction o f the device response time, the output branch response time, and the parasitic capacitance branch response time. The output branch response time determines the lower limit o f the bandwidth, while the upper limit is determined by the longer o f the two times between the device response time and the parasitic capacitance branch response time. The use of the drift-diffusion model in the present study has its limitations regarding accuracy. The derivation of the drift-diffusion model from the first two velocity moments o f the Boltzmann transport equation (BTE) depends on the assumption that the electron temperature is in equilibrium with the lattice temperature. For submicron devices, the electric fields can become so large that electrons have significantly higher temperatures than the surrounding lattice [13]. These hot electrons are more accurately modeled through the hydrodynamic equation set, which consists o f the first three moments o f the BTE [18]. However, since this raises the number o f partial Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 340 differential equations from six to at least eight and greatly increases the computational time, the present study focused on device lengths that are unlikely to exhibit hot electrons. A second limitation o f the Boltzmann transport formulation is the breakdown o f mobility models with shorter devices and higher frequencies. The FD mobility model requires a minimum number o f mean free paths and mean free times between collisions to be accurate. The FI mobility model is limited in its ability to accurately characterize shorter devices because the system is not in the quasi-equilibrium state regarding energy exchange between the carriers and the lattice that is assumed by the model. Based on these two limitations, the most important recommendation made in the present study for future work in the field o f photodetector simulations is to develop a more accurate bulk transport model for submicron devices. A second recommendation in the form o f an extension to the present work is to add the necessary transport equations to allow for hot electron effects. This will necessarily slow the simulation, but should still enable it to be more efficient than the CESS simulation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 341 73 CURRENT DENSITY BOUNDARY CONDITION If the model o f a boundary condition is to increase understanding o f the transport physics at the boundary, the boundary condition must be physically reasonable and not ju st accurate. The original combined drift-diffusion/thermionic emission model for calculating current density at the boundary was developed about thirty-five years ago [31]. Since that time, there have been three revisions to the physical mechanisms that comprise the model [32]-[34],[58]. These revisions include a unidirectional carrier normalization, a variable surface velocity, and the separation o f the semiconductor and metal surface velocities. However, in the major versions of the combined formulation in use today, there is no consensus on which physical revisions to use [31]-[34],[58]. The two points of contention in the existing models include the following: (1) when calculating the average surface velocity for carriers that cross the boundary, should the velocity distribution be divided by the number o f carriers that cross the boundary or by all of the carriers; and (2) should the surface velocity for the carriers that originate in the metal be held constant or set equal to the variable surface velocity o f the carriers that originate in the semiconductor. A third point o f contention grew out o f the present study: should the same carrier distribution that determines the average surface velocity be applied to determine the carrier density. Due to these three points o f contention, and to the motivation to understand the physics o f transport at the boundary, the second major objective o f this study was to develop the current density boundary condition from first principles regarding the semiclassical model o f solid state systems. According to the semiclassical model, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 342 current density can be determined by an integral that uses the nondegenerate MaxwellBoltzmann limit o f the Fermi-Dirac velocity distribution. The original integral is separated into two integrals, one over velocity components in the —x -direction and the other over velocity components in the + x -direction. Then each integral is multiplied and divided by the fraction o f carriers that move in the same direction as indicated by the integral, as well as by the total carrier density, which leads to the final form of the current density at the boundary. These steps automatically establish the carrier normalization as being unidirectional, allow for the separation o f the semiconductor and metal surface velocities, and introduce part o f the basis for a variable carrier fraction. By assuming that the velocity distribution in the presence o f an applied voltage is a drifted Maxwellian, the variable nature of the surface velocity and carrier fraction is automatically established. Since this new version is derived from first principles, it is presented here as a more physically reasonable formulation o f the current density boundary condition. The new current density boundary condition was tested by using the device simulator to generate I-V characteristics for Schottky diodes. The simulated curves agreed well with analytically generated curves and curves obtained from experiment, indicating that the new current density boundary condition is accurate. Simulation results for Schottky diodes and MSMs in the presence and absence o f light using all four formulations of the current density boundary condition were compared. The new formulation predicted the highest device response speed, for the following reasons: (1) the variable velocity allows drift to enhance the surface velocity so that excess carriers are swept out o f the semiconductor at a greater rate; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 343 (2) the unidirectional normalization prevents the surface velocity from being incorrectly divided by two, which would reduce the rate o f carrier removal significantly; and (3) the separation o f the metal and semiconductor surface velocities allows the metal surface velocity to keep its much lower constant value so that the metal carriers do not reenter the semiconductor at such a high rate that they lower the net rate o f carrier removal. The higher response speed increased the predicted bandwidth by an average o f 28 % for the FI mobility model with high light intensity and low voltage, and reduced the responsivity by an average o f 9 %. The greatest difference between the new and existing formulations was found in the physics o f transport at the boundary. At the electron forward biased contact o f the MSM under illumination, the electron drift-diffusion velocity was over two orders o f magnitude higher and the electron density two orders of magnitude lower than the corresponding values for the existing formulations. This finding has significant implications for shorter devices since the fraction o f the total current density belonging to the boundary regions increases with decreasing device length, so that the effects o f the physics o f transport at the boundary on overall device performance become more important. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX A SYMBOLS Lattice constant for GaAs Cross-sectional area o f substrate to light 4= Cross-sectional area of substrate to current a Light absorption coefficient B Magnetic induction vector fi Shaping factor in Kaiser filter c Speed o f light CB Bias tee capacitance Capture coefficient for electrons Parasitic capacitance Capture coefficient for holes Shaping factor in Kaiser filter X Electron affinity energy d Depth o f semiconductor substrate (along y-axis) Dn ( D l J Electron diffusion constant [low-field] D p L D l .p ] Hole diffusion constant [low-field] D Electric displacement vector E [ £ 0] Total electron energy [vacuum] Ec Conduction band edge energy Ed Si donor energy E f [ E Fm, E Fs ] Fermi energy [for metal, semiconductor] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 345 Electron quasi-Fermi potential energy E* Hole quasi-Fermi potential energy E* E g Bandgap energy E, Intrinsic Fermi energy Deep-level donor trap energy E j E. Valence band edge energy E c .rff Effective conduction band energy E Electron potential energy due to [metal contact, image charge] £[ S CJ,,ge.p] Electric field [critical field: electrons, holes] ^ t * .? j Electric field due to [metal contact, image charge] Emission coefficient for electrons Emission coefficient for holes eP [s0] £ A E) F IK ) Permittivity [of free space] Electron distribution function Force [Lorentz force on band n] F{u) Residual function vector E„ Fraction o f electrons that leave semiconductor Fraction o f electrons that move in the FS F + x-direction Fraction o f holes that leave semiconductor p Fraction o f holes that move in the + x-direction F p. * ' E u 2 Fermi integral o f order 14 fifo ) Residual function for Poisson’s equation at grid point x, f- Residual o f state variable k at grid point x, Metal work function Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 346 Height o f Schottky barrier to electrons from metal Height o f Schottky barrier to electrons from semiconductor *„S Electron quasi-Fermi potential Hole quasi-Fermi potential Barrier lowering [image force, tunneling] [G t .0] G l Generation rate due to light [baseline generation rate] Electron density o f states gn Hole density o f states g p LHS o f discretized Poisson equation g i P i ) r E-k diagram extremum energy at k = 0 r Number o f absorbed light photons h Planck’s constant divided by 2 n H Magnetic field intensity h it ) [h „ ] Impulse response function [smoothed, discrete samples] DC signal as for I-V curves [saturation current] I [ / ,] K t,v m) Photocurrent as a function o f time and modulation frequency i ac i t , co) AC component o f photocurrent hc.oit,CO) Output branch AC component o f photocurrent h e .p it,C O ) Parasitic branch AC component o f photocurrent AC peak as function of modulation frequency i °AC. Constant low frequency value for AC peak max DC component o f photocurrent l DC .max ] Modified Bessel function /o ^7I Dark current [maximum dark current] M Light intensity [initial light intensity] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 347 Total particle current density (ID ) J Diffusion current density Drift current density J* Electron current density j. Hole current density Jp Total current density (ID ; particle plus displacement) Jt Wave vector k Boltzmann’s constant 4 Wavevector magnitude for holes that causes net drift-diffusion k *.P K. Semiconductor dielectric constant Lb Inductance o f bias tee 4 Length o f contact (along x-axis) 4 Length o f semiconductor substrate (along x-axis) X Wavelength o f light K Adjusted drift-to-thermal velocity ratio for electrons Adjusted drift-to-thermal velocity ratio for holes 4 Electron rest mass m0 m n * {rnnX * ] Electron effective mass (T valley) [L valley] Average hole effective mass m P * m ph* Heavy hole effective mass m P, * Light hole effective mass m so * Split-off band hole effective mass M Modulation index M Permeability Mn [Ml*] Electron mobility [low-field or field-independent] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 348 Hole mobility [low-field or field-independent] n Electron density n Band index N{E) Number o f electrons o f energy E n a Acceptor dopant density Nc Effective conduction band density o f states ND Donor dopant density nl Number o f electron-hole pairs generated by light n, Intrinsic carrier density N To Total number o f impedance samples NT Total trap density nT Density o f traps occupied by electrons K [N ^N * ] Effective valence band density of states [light, heavy holes] Electron trap constant Shaping factor in Kaiser filter N2 Truncation index in Kaiser filter Light frequency Modulation frequency ^m jhr V y-d b Threshold modulation frequency Bandwidth frequency (3-db point) P (k) Crystal momentum P pO Momentum density L.i P°Ac(ym) PaC .^ Peak light power associated with time-dependent intensity AC power peak as a function of modulation frequency AC power peak associated with i% __ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 349 Pi Hole trap constant <P(r) One-electron wave function q Charge (magnitude) o f single carrier r Reflectance r Position vector K Load resistance R op, Optical responsivity RG Thermal recombination-generation rate P Charge density (total) Pm Density (mass density) Pm Particle (number) density S Poynting vector sf State variable k at grid point x, T Kelvin temperature T(E) Tunneling transmission coefficient T Period Momentum relaxation time (mean free time between collisions) T Device response time T n fT n s 'T n ,] Electron recombination lifetime constant [short, long] T p f r p s ’ T p tJ Hole recombination lifetime constant [short, long] Effective recombination lifetime (/ = n ,p ; j = long, short) rp .pJ Momentum relaxation time [electrons, holes] T RC. o Output branch (RC) time constant T R C ,p Parasitic branch (RC) time constant T'J Transit time (/ = n, p) u State variable vector Su Correction vector w„(k ,r ) Part o f Bloch solution that has periodicity o f lattice Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 350 V V Crystal volume [ V n ’ V p ] Group velocity [electron,hole] VA Applied voltage at metal contact vd Drift-diffusion velocity v* Diffusion velocity "dr Drift velocity vk Volume o f tjnjcr Average electron velocity in the + x-direction Average electron velocity in the + x-direction U pS Electron thermal velocity (unidirectional) Vas Hole thermal velocity (unidirectional) ° 'i.P V sa, k -point in k -space [V so, Saturation drift velocity [electrons,holes] Electron surface velocity [equilibrium] Vs.p [ » s .p J Hole surface velocity [equilibrium] vd Device voltage V DC DC voltage source Vckt W Voltage across embedding circuit W Width o f semiconductor substrate (along z-axis) Mr,) RHS of discretized Poisson equation w„ Filter weight for impulse samples CO Gram-molecular weight Angular frequency o f light Modulation angular frequency x [x „ ,x p] Mean free path [electrons,holes] x c [X°c ] Critical width for tunneling through barrier [limiting value] xd Width o f the charge-depletion region X, Grid point o f mesh Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Yk Admittance o f element k V Electrostatic potential Zk Impedance o f element k ^dajt Unfiltered impulse response samples Impedance function for embedding circuit Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 352 APPENDIX B M ODEL PARAMETERS GaAs Cryst:al Parameters Description Symbol Lattice constant a Band gap energy Value 5.65 A 1.42 eV Loc. 2.2 2.2 Src. [36] [36] Electron effective mass ratio mH*/m0 0.063 2.2 [36] Light hole effective mass ratio nip, */m0 0.087 2.2 [41] Heavy hole effective mass ratio mpH7 mo 0.48 2.2 [41] Split-off band hole effective m ass ratio 0.154 2.3 [39] Average hole effective mass ratio m«, */m0 mp */mo 0.28 3.3 L valley electron effective mass ratio ™n.L */mo 0.55 3.7 [39] Density Pm 5.32 g/cm3 3.2 [36] Particle density Gram-molecular weight p* w 2.21xl022cm'3 144.64 g/mol 3.2 3.2 GaAs Carrier Density Parameters Value Description Symbol SOO^K Temperature T Effective conduction band density o f states 4.0x10° cm'3 Nc Effective valence band density o f states 9.0x10** cm'3 Nr Intrinsic carrier density Loc. 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 Src. 2.3 [39] Intrinsic Fermi energy (relative to E r) E, 2.2x10* cm'3 0.02586 eV 3.5xl012 cm'3 0 0.75 eV Donor energy (relative to E c) Fermi energy (relative to £,.) Ed EF -0.0058 eV 1.12 eV 2.3 Difference E F- £, for doped GaAs NEFj 0.37 eV 2.3 Particle thermal energy Donor doping density Acceptor doping density ", k BT Nd n a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 353 GaAs Carrier Density Parameters (cont.) Description Symbol Value 0.30 eV Difference EC- E F for doped GaAs ^ c .F 0.67 eV Difference Ec- E, for doped GaAs * * eJ Loc. 2.3 Src. 2.3 GaAs dielectric constant Ks 13.18 2.3 [44] Debye screening length I'D ET 2.32 pm 0.67 eV 2.3 2.5 [39] Pi 1.0x10s cm'3 4.9x107 cm'3 2.5 2.5 Deep-level donor trap energy (relative to £ ,) Electron trap constant Hole trap constant MSM Photodetector System Parameters Description Symbol Value Cross-sectional area to light 38.5 pmz 4, Cross-sectional area to current 35.0 pm"4 ** Length o f substrate (along x-axis) 1.1 pm L, Length of contact 0.66 pm Lc Depth o f substrate (along y-axis) d 1.0 pm W Width o f substrate (along 2-axis) 35.0 pm Wavelength of light 800 nm X Refraction index 3.6 n2 1.7xl04 cm'1 a Absorption coefficient ( X = 800 nm) r Relfectance 0.320 Fraction o f photons that enter substrate 0.6 L j a , + l c) 0.817 Fraction o f photons that are absorbed 1 -ex p (-a d ) Fraction o f initial intensity absorbed Product (last 3) 0.157 M 0.9 Modulation index Maximum dark current (at 10 V) 4.0x10'7 A ^dkjaax Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Loc. 2.4 2.4 Src. 2.4 2.4 2.4 2.4 2.3 2.4 2.4 2.4 2.4 2.4 2.4 5.3 2.4 [45] [45] [45] 354 GaAs Dynamic and Transport Parameters Description Value Symbol Electron recombination lifetime (base) 10"* s 10** s Hole recombination lifetime (base) Metal work function Electron affinity Metal e* Schottky barrier (equil.) X Semiconductor e' Schottky barrier (equil.) Electron critical electric field Hole critical electric field S c* S ', Loc 2.5 Srce [24] 2.5 [24] 5.07 eV 3.4 [18] 4.07 eV 1.00 eV 3.4 3.4 [35] [18] 0.70 eV 3.4 4.00xl0J V/cm 3.7 1.95x10* V/cm 3.7 Field-independent electron mobility V l* 7200 cm2/V-s 3.7 [55] Field-independent hole mobility Ml.p 380 cm2/V-s 3.7 [55] Field-independent electron diffusion constant A. 186 cm2/s 3.7 Field-independent hole diffusion constant DP 9.83 cm2/s 3.7 8.3xl06 cm/s 3.7 [55] 7.3x10* cm/s 3.7 [55] 2.6x10‘13 s 3.7 6. lx l 0'14 s 3.7 216 A 45 A 3.7 3.7 2.14xlOTcm/s 4.3 Saturation drift velocity for electrons Saturation drift velocity for holes u~ . Electron momentum relaxation time Hole momentum relaxation time Tp, Electron mean free path Hole mean free path *p Electron thermal velocity (unidirectional) Hole thermal velocity (unidirectional) un.P 8.52x10* cm/s 4.3 Limiting value o f critical width x° 37.5 A 4.4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [58] 355 APPENDIX C TELEPHONE COMMUNICATION Telephone communication is the leader in the field o f signal transmission over lines. When telephones were first developed in the early 1900’s, signals were transmitted over wire pairs (see Figure C). In the 1940’s, coaxial cables replaced wire pairs, and improved transmission by two orders o f magnitude. In the mid 1970’s, fiber-optic systems replaced coaxial cables, and increased the transmission effectiveness by four orders o f magnitude. In the fiber-optic telephone system: (a) sound waves are converted into an electrical signal by the phone receiver; (b) the electrical signal modulates a carrier laser beam so that the beam carries the electrical signal as an envelope; (c) a fiber-optic cable transmits the modulated light; (d) the carrier laser and booster laser light intensities are mixed together by a semiconductor photodetector; (e) the photomixing process generates a microwave that now acts as the earner for the electrical signal; (f) a microwave detector picks the electrical signal o ff o f the microwave carrier; and (g) the electrical signal is converted back into the original sound by the telephone transmitter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 356 Sound Phone M l□ Carrier laser Wire pairs or Coaxial cable Amplitude modulator Fiber-optic cable Semiconductor Photomixing Microwave cable Sound Phone fWHI «/Vy/V> Figure C Transmission o f signals in telephone communication. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 357 APPENDIX D MIXING WAVES In the simplest case (Figure D), the waves have equal amplitudes, are in phase with each other, and have the same axis o f polarization. The frequencies o f the two waves differ slightly, with wave 1 having frequency 9 Hz and wave 2 having frequency 10 Hz, so that the difference between their frequencies is 1 Hz. When the two waves are superimposed, they interfere with each other such that the resultant wave has an envelope whose frequency is Vz o f the difference frequency, which in this case is Vz Hz. The important point is that the frequency o f the envelope is significantly lower than the frequency o f either primary wave. [When the intensities o f two waves are mixed, the envelope frequency is equal to the difference frequency due to the squaring o f the resultant electric field.] The resultant amplitude (derived by M. Bishop) is given by g l 0 sin(<y,/) + g 2Qsin(fi>2r) = [ j, o sin(<y,/) + i i 0 sin(®,/)]+ 0 sin(a>30 - S l 0 sin(<u2r)] + j [ s 2.o sin(<y2/) + S 20 sin(ry2r ) ] + j [ # 20 sin(<y,r)-£20 sm (ay)] (5,.o + £ 2.oJsinfai') + sin(a>2f ) ] - zr(suo ~ S 2.0Jsin(<a2f) - sin(*y,/)] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 358 Envelope: frequency = 1/2(v2-v1) Resultant wave Superimpose waves 1 and 2 Wave 1 v, = 9 Hz i----------- 1________i________i________i________i 0.0 0.2 0.4 0.6 0.8 1_______ i______ i _____ —i________i 1.0 1.2 1.4 1.6 1.8 2.0 Time (s) Figure D Mixing waves. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 359 APPENDIX E ELECTRICAL RESPONSE TO ILLUMINATION When a semiconductor photodetector is exposed to light, excess charges are created, and the voltage across the device sweeps these charges into the embedding circuit as photocurrent. If the light has a constant intensity, then it generates the same excess charge density at all times, and the photocurrent is DC. If the light has a low modulation frequency, then an incoming intensity peak generates the maximum charge density. The device responds fast enough to sweep this charge out o f the semiconductor before the intensity trough is encountered. The intensity trough generates the minimum charge density, which is swept out before the next intensity peak. As a result, AC current is generated, and electromagnetic (em) waves are produced. If the light has a high modulation frequency, then both the peak and trough impinge upon the semiconductor before the charges can be swept out, leading to an averaging out o f charge density so that DC is produced, and no em wave results. If the light has a medium modulation frequency, then there is time to sweep part o f the charge density out o f the semiconductor for a given intensity peak and trough, leading to AC current that has a smaller peak than for low frequencies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 360 Constant intensity Time 1 1 Photocurvent Time 2 DC IP Ife -S S a i - m m m m * Time Low frequency intensity Time 1 Time 2 i • • Photocurrent • High frequency intensity Time 1 Time 2 4 Photocurrent DC , K»X* S&SSSSR*?:-.-.-.- Time -----► Medium frequency intensity Time f Time 2 4 Photocurrent AC Time Figure E Electrical response to illumination. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 361 APPENDIX F DERIVATION OF DRIFT-DIFFUSION EQUATIONS The drift-diffusion equations are derived from the Boltzmann transport equation (BTE), which describes the motion o f carriers as the time evolution o f the distribution function f ( r , Q , t ) in a six-dimensional phase space. According to the Liouville theorem, in the absence o f collisions, the distribution o f carriers at point (D,k) will follow its trajectory in phase space unchanged, so that the total time derivative o f the distribution function is zero: HL= § L + ? L . * + V . ° e = o . dt dt dr dt du dt (F .i) Collisions have the effect o f scattering carriers into and out o f the point (r , 0 ) . Any deviation o f d f j d t from zero is due to the change in distribution caused by collisions , so that for actual semiconductor systems Equation F.I becomes where the velocity is defined by u - d rfdt, the force is defined by F - m * a = m *(<do/dt), and the gradients with respect to r and v appear in their usual form. This equation is then solved for the partial time derivative o f the distribution function, giving Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where the magnetic component o f the Lorentz force F = g ( |+ y x f l ) acting on the charge carriers is ignored based on the assumption that the magnetic induction B is negligible compared to the electrostatic Lorentzian component q £ . The more usable transport equations are a set o f equations derived from Equation F.3 by forming progressively higher order velocity moments o f that equation and integrating over v . This process transforms f ( r , D , t ) into the total carrier density n(r,t) (or p ( r , t ) ) and v ( k ) into the average carrier velocity (p (r,t) ) . For the present study, only the carrier transport equations derived from forming the zeroth and first moments of the velocity are needed. The carrier transport equation corresponding to the zeroth moment is found by integrating Equation F.3 over 0 , giving i f '-<& = -p ' f - M S - f V ' f - Z L a o + . (F.4) On the LHS o f Equation F.4, the time derivative is pulled out o f the integral, giving \^d o = J /d o = ^ , J dt dt J dt (F.4.a) where n represents the electron carrier density (and could be replaced by p for the hole carrier density). The first term on the RHS of Equation F.4 is integrated by parts, giving Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 363 - = —Jv? f u d v + JfV?-O dd . (F.4.b) The second term on the RHS of Equation F.4.b vanishes because the electronic velocity states un are independent o f position r (the same un-states exist at all positions). The first term is further evaluated by pulling the gradient operator out o f the integral, giving —V ? - \ f u d v = - V- • \ f d v - y — = -V - •n(v ) . Jf d v (F.4.c) The second term on the RHS of Equation F.4 is also integrated by parts, giving - W e f - ^ d o = —M K m m fZ d v + f / 7 - i d v ) . * (F.4.d) The first integral on the RHS of Equation F.4.d vanishes by the divergence theorem since the distribution f goes to zero as v approaches infinity. The second integral on the RHS also vanishes because the electric field £ is independent o f 0 . The integration o f the collision term on the RHS of Equation F.4 gives (F.4.e) The integrated collision term can be determined by realizing that although the u -state can change in the instantaneous time interval represented by d t , the position r cannot change. Since the v -dependence has been integrated out so that n depends only on r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 364 and /, and there is no change in n with respect to these two variables during dt, the collision term is zero. Substituting the four results obtained by evaluating each term in Equation F.4 back into that equation leads to the particle density continuity relationship, as given by ^ = - V , •»<£!). (F.5) The two current continuity relationships, one for electrons and the other for holes, are derived from Equation F.5 by introducing the electron charge q. These two relationships are expressed in one-dimension, in keeping with the one-dimensional current inherent in the model for the present study. Since the current associated with electrons is in the opposite direction o f the average electron velocity the electron current continuity equation is given by dn 1 5 / \ 1 dJ — = — — qn{ - u njl) = — — , dt q dx ' ' q dx (F.6) in which the one-dimensional electron current density is given by J nj = - ^ ( u n j ) . (F.7) The hole current density relationship is given by dp 1 5 / \ 1dJ = — -zr<lp{up^ ) ------------ > dt q dx ' ' q dx Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (F-8) 365 in which the one-dimensional electron current density as obtained from Equation 4.3.5 is given by (F.9) The carrier transport equation corresponding to the first moment is found by multiplying Equation F.3 by velocity and integrating over 5 , giving fD— dD = dt J = - J u V - f v d v - j u V - f - £ ^ d v + ju dt do (F.10) On the LHS of Equation F.10, the time derivative is pulled out o f the integral since 5 is independent of time, giving r-d f d r d \ v — d v = — \ f d v —.------ = — mu). 3 dt dt \fdv dt X 1 (F.lO.a) In anticipation o f simplifying Equation F.10 later on, the final result in Equation F.lO.a is expanded using the product rule to give — n(u) = n — (0) + ( u ) — n dt \ ! d t \ 7 \ / dt Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (F.lO.b) 366 Then the particle density continuity relationship given by EquationF.5 is substituted into the second term on the RHS o f Equation F.lO.b to give the simplifying form o f Equation F.lO.a: J U ( u ) = n ^ ( D ) - ( u ) V r •n(D) . (F.lO.c) The first term on the RHS of Equation F.10 is integrated by parts, giving - ju V ff u d u = -V . - [ ^ f v v d u — JyV- - u v d u ) . (F.lO.d) The second integral in Equation F.lO.d vanishes because v is independent o f the gradient variable r . Since the first integral contains the three-by-three tensor u u , Equation F.lO.d actually contains nine integrals o f the form ffU jV jd v . Although the distribution function under conditions o f bias is unknown, some function / must be assumed if this equation is to be evaluated. To make this equation solvable, the distribution function is chosen to be an even function so that only the diagonal terms are non-vanishing. This reduces the tensor to the three components o f a vector, and each component can be determined by integrating over its respective dimension. The assumed distribution function must also meet two further requirements that ensure that using the function leads to physically reasonable results. First, the function is required to be consistent with energy considerations. Specifically, the average electron kinetic energy density obtained with the distribution function for equilibrium conditions must equal (3/2)nkBT . Secondly, the distribution function must allow for bias, which causes a net displacement Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 367 in the average value o f v from (v) = 0 to a non-zero vector (p) . A distribution function / that satisfies the above three conditions is the displaced Maxwellian, as given by v3/2 I m — 1 expj"— —— |o - (y)|2 / =« 2n kbT .T J 1 2ksT l V /l _ . (F.lO.e) In anticipation o f using the above drifted Maxwellian distribution function, the drift by (v) is built into the first integral o f Equation F.l0.d by adding and subtracting the following terms within the integrand: f o ( y ) , f ( 0 ) u , and Equation F.lO.d then becomes - j u V Ff od d = —u(p) - (u)u + (p)(y))dv = —V- - -V ? = -V ? ~ m • J/(t»(o) + (u)u - (o)(u))dO j m * f ( v - (v)Xu ~ ( D ) ) d u - V . -n(u){0) = - V ? —l— P - n ( p ) V F tn -n(v), (F.lO.f) in which P is called the pressure tensor, and is defined by P,j = m* \ f { u i -[0 ) ^ 0 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (F.lO.g) 368 As stated previously, the distribution function/is chosen so that only the diagonal terms o f the pressure tensor are non-vanishing, leading to P0 = nkBTS,J (F.10.h) upon integration. Since the second term on the RHS o f Equation F.lO.c is equal to the third term on the RHS of Equation F.10.f, this term is common to both sides o f Equation F.10 and cancels out o f the first moment equation. The second term on the RHS o f Equation F.10 is integrated by parts twice, giving - f J = m* = f U ^ d O - I f v - V j d u - j / i V GDdv). (F-lO.i) The first integral on the RHS o f Equation F.lO.i vanishes by the divergence theorem, and the second integral vanishes because the electric field £ is independent o f the gradient variable u . The third integral is evaluated to give -a-f/f.V jO rfo-X f. m* J m* J m* (F.io.j) where 7 is the identity tensor. Prior to forming the velocity moment o f the third term on the RHS of Equation F.10, this collision term is approximated as the difference in the distribution / due to collisions and the equilibrium distribution f 0 (in the absence of collisions), divided by the average time between collisions r , giving Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 369 > 1 _dt _e /-/o (F.10.k) r When this term is multiplied by the velocity and integrated, it gives _ "<g > , " f a ) J r t t "fa , t (F.I 0.1) in which the second term vanished because the average velocity (y0} at equilibrium is zero. Equation F.10 is reconstructed from its evaluated terms, giving »±<0>— V, 5/ m* w ' + m* r (F.lO.m) which is then multiplied by r and solved for the fourth term on the RHS to give " (°> = ~ ~ ? ’^ ~ nTj S D^' (F-10-n) The third term on the RHS o f Equation F.lO.n is approximately zero since the divergence o f the average velocity is relatively small. The fourth term also approximately zero because the average collision time r is very short, making this term second-order small. Then the equation is multiplied by q to obtain the drift-difrusion form o f the current density J as J = qnp% —q D V f n , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (F.lO.o) 370 where the carrier mobility fj. is defined by (F.lO.p) and the diffusion coefficient D is defined by m* (F.lO.q) For electrons, the effective mass is mn *, and the electron mobility and electron diffusion coefficient are labeled fu„ and Dn, respectively. For holes, the effective mass is mp *, and the hole mobility and hole diffusion coefficient are labeled fi p and D p , respectively. Therefore, the drift-diffusion form o f the electron current density J nj for current in onedimension is given by (F.I 1) in which the sign for the diffusion term has been reversed since positive current moves in the direction opposite to the diffusion o f electrons. Likewise, the drift-diffusion form of the hole current density J pp[ for current in one-dimension is given by (F.12) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 371 BIBLIOGRAPHY [1] G. P. Agrawal, Fiber-Optic Communication Systems. Second Edition, 1997, New York: John Wiley & Sons, Inc., ch. 1. [2] A. Davidson and K. L. Dessau, “High Speed Detectors: Faster Measurements for Today’s Laboratory,” Photonics Spectra, pp. 110-114, Sept. 1998. [3] U. Gliese, T. N. Nielson, and S. Norskov et al., “Multifunctional Fiber-Optic Microwave Links Based on Remote Heterodyne Detection, ” IEEE Trans. Microwave Theory, Vol. 46, No. 5, pp. 458-468, May 1998. [4] R. P. Braun, G. 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Darling, “High-Field, Nonlinear Electron Transport in Lightly Doped Schottky-Barrier Diodes,” Solid-State Electronics, Vol. 31, No. 6, pp. 1031-1047, 1988. [35] S. M. Sze, Physics o f Semiconductor Devices. 2nd ed., John Wiley & Sons: New York, 1981. [36] J. S. Blakemore, “Semiconducting and Other M ajor Properties o f Gallium Arsenide,’V. Appl. Phys., Vol. 53, No. 10, pp. R123-R181, Oct. 1982. [37] J. R. Chelikowsky and M. L. Cohen, “Nonlocal Pseudopotential Calculations for the Electronic Structure o f Eleven Diamond and Zinc-Blende Semiconductors,” Phys. Rev. B, Vol. 14, No. 2, pp. 556, July 1976. [38] N. W. Ashcroft and N. D. Mermin, Solid State Physics. Jovanovich College Publishers: Fort Worth, 1976. Harcourt Brace [39] R. F. Pierret, Volume VI - M odular Series on Solid State Devices: Advanced Semiconductor Fundamentals. Addison-Wesley Publishing Co.: Reading, Mass., 1987. [40] R. F. Pierret, Volume I - M odular Series on Solid State Devices: Semiconductor Fundamentals. 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Prentice Hall PTR: [72] G. Arfken, M athematical M ethods fo r Physicists. Academic Press, Inc.: Boston, 1985. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 377 VITA David B. Ameen was bom on December 12,1952, in Munich, Germany, and is an American citizen. He graduated from Hopewell High School, Hopewell, Virginia, in 1971. He received his Bachelor o f Science in Science Education from the University o f Virginia, Charlottesville, Virginia in 1976 and subsequently taught in the public schools in Colonial Heights, Virginia for seven years. He received his Master o f Science in Biophysics from the University o f Virginia in 1987 and subsequently taught in private school in Petersburg, Virginia for four years. He was married on August 1, 1992 to Betsy A. Harrison. He received his Master o f Science in Physics from Virginia Commonwealth University, Richmond, Virginia in 1996. He became an instructor in the Department o f Physics at Virginia Commonwealth University in 1999. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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