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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
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UMI Number. 9963200
UMI*
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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
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For my wife Betsy
and my Mom and Dad
ii
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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
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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
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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
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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...............................................................
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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 ...................................................................................................................................
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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
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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
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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
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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.......................................
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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
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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
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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
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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.
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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
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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
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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.
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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 (*]].
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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
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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
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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.
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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.
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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
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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.
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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
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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
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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,
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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.
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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
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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
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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
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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!].
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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.
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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
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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-
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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!].
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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
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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.
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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
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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
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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.
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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
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(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! ] .
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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 .
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(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!].
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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
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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
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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.
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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
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(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
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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
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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
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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
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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.
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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]
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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
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(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
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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
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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
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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)
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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!].
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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,
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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
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(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 .
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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.
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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
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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.
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►
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
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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.
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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
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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
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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
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!
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
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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
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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 ) ,
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(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)].
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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
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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
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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 ,
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(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
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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.
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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
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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
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(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 =
.
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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(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
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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 .
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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.
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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,
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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
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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.
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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.
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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!].
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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.
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(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
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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.]
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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.
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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
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(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
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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
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(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)
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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,
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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 ,
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(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
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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)
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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
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(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
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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
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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
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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.
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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
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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.
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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
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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
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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
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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
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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
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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.
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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.
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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.
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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
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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
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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 / ’
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(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.
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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
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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
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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.
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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.
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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.
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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)
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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
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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
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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
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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
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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
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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.
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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 .
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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
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^ 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.
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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
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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.
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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
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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
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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
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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 ”,
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(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.
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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)
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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.
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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
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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.
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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
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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
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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
=
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(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
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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.
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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
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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
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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
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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].
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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
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!
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 .
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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
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(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.
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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
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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
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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.
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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]
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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.
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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.
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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
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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
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!
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
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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
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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
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(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
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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
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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
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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 .)],
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(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
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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.
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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
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(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
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(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
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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
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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
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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.
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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
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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
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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
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|.
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).
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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.
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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
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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
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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.
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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.
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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
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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
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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.
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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
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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.
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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.
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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
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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
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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.
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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.
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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 " + ~ ,
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(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
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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
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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
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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 ,
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(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
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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
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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
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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.
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. (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
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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 ,
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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.
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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
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(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
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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
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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.
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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
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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.
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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
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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
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237
(a)
o
Total Current
Total Current
Low Barrier
High Barrier
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c
£
-10
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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
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-10
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£
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-20
15
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Electron
Hole
-25
Electron
Hole
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-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
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239
(a)
2
CTH562: 1.00
1
Experiment
FD simulation
Rsimulation
<
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c
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o
(b)
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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
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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
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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.
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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 »)
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0.08
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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)
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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"
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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:
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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.
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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
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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
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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.
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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.
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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).
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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
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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.
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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
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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.
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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].
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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
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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.
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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
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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.
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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
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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).
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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
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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.
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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
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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
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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
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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.
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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.
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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.
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(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.
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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.
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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
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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.
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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
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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
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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.
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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.
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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 .
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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
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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
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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.
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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.
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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
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*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.
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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 = *****•
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(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]
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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.
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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.
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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.
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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
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<
=
{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
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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.
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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.
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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
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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 .
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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.
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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.
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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.
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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.
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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
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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
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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)
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(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-(*
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(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
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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.
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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.
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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
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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)
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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
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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
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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
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(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.
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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
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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.
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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
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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.
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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;
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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].
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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-
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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
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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.
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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
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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
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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.
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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
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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;
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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.
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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]
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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
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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]
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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]
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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% __
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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
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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
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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
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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
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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
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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
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[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.
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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.
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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,/)]
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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.
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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.
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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.
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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
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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
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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
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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
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(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
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(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
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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
.
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(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
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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 ,
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(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)
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371
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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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