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Microwave nonlinearities in photodiodes

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O rder N um ber 9508080
Microwave nonlinearities in photodiodes
Williams, Keith Jake, Ph.D.
University of Maryland College Park, 1994
UMI
300 N. Zeeb Rd.
Ann Arbor, MI 48106
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ABSTRACT
Title o f D issertation:
M icrowave N onlin earities in Photodiodes
K eith Jake "Williams, Doctor o f Philosophy, 1994
D issertation directed by:
Dr. M ario D agenais
Professor of Electrical E ngineering
T he n o n lin e a r itie s in p-i-n photodiodes h a v e b een m ea su red and
n u m e r ic a lly m odeled.
H arm onic d istortion , r esp o n se red u ction , and
sin u so id a l output distortion m ea su rem en ts w ere m ade w ith two sin g le ­
frequency offset-phased-locked Nd:YAG la sers, w h ich provided a source
dynam ic range greater th an 130 dB, a 1 M Hz to 50 GHz frequency ra n g e,
and optical powers up to 10 mW.
A sem i-classical approach w as used to solve th e carrier transport in
a one-dim ensional p-i-n photodiode structure. T his required the sim u lta ­
neous solution of three coupled non linear differential equations: P o isso n ’s
equation and th e hole and electron continuity equations.
Space-charge
electric field s, loading in th e extern al circuit, and absorption in u n d e­
pleted region s n ext to th e in trin sic region all contributed to th e n o n lin e a r
behavior described by th ese equations.
N u m erical sim u la tio n s w ere perform ed to in v estig a te and isolate
th e various non linear m ech a n ism s.
It w as found th a t for in trin sic region
electric fields below 50 kV /cm , the n o n lin earities w ere in fluenced p rim a r­
ily by th e space-charge electric-field-induced change in hole and electron
velocities. B etw een 50 and 100 kV/cm, the non lin earities were found to be
in flu en ced p rim arily by ch a n g es in electron velocity for frequencies
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above 5 GHz and byp-region absorption below 1 GHz. Above 100 kV/cm,
only p-region absorption could explain the observed nonlinear behavior,
where on1'.” 8 to 14 nm of undepleted absorbing material next to the intrin­
sic region was necessary to model the observed second harmonic distor­
tions of -60 dBc at 1 mA.
Simulations were performed at high power densities to explain the
observed response reductions and time distortions.
A radially inward
component of electron velocity was discovered, and under certain condi­
tions, was estimated to have the same magnitude as the axial velocity. The
model was extended to predict that maximum photodiode currents of 50
mA should be possible before a sharp increase in nonlinear output occurs.
For capacitively-limited devices, the space-charge-induced nonlinearities
were found to be independent of the intrinsic region length, while external
circuit loading was determined to cause higher nonlinearities in shorter
devices. Simulations indicate that second harmonic improvements of 40 to
60 dB may be possible if the photodiode can be fabricated without unde­
pleted absorbing regions next to the intrinsic region.
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Microwave Nonlinearities in Photodiodes
by
Keith Jake W iliam s
Dissertation submitted to the Faculty of the Graduate School
of The University of Maryland in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
1994
Advisory Committee:
Professor Mario Dagenais, Advisor
Professor Christopher Davis
Professor Marty Peckerar
Professor Aristos Christou
Dr. Ronald Esman
Dr. Lew Goldberg
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ACKNOWLEDGEMENTS
I would like to thank the members on my dissertation committee for
giving their precious time: my advisor, Dr. Mario D agenais, Dr.
Christopher Davis, Dr. Marty Peckerar, Dr. Aristos Christou, Dr. Ronald
Esman, and Dr. Lew Goldberg. I would also like to thank NRL Code 5750
for the use of their Convex computer, the NRL Center for Computational
Science for the use of their Cray Computer, Dr. John Bowers of the
University of California Santa Barbara for useful discussions, Dr. Greg
Olsen and Epitaxx for detectors, and Dr. Joseph E Weller for his support.
Special appreciation is extended to Dr. Ronald Esman for enlightening dis­
cussions and proofreading of the manuscript, and to my wife, Vicki, for
her support, especially during the preparation of the manuscript.
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TABLE OF CONTENTS
Section
Page
List of Figures.....................................................................................................
v
Chapter I. Introduction..................................................................................
1
Chapter II. Measurement System.................................................................
8
Chapter III. Photodiode Device P hysics.....................................................
15
Generalized Transport Equations............................................................ 15
p-i-n Photodiode Structure......................................................................
16
Simplifying Assumptions..........................................................................
17
Carrier Transport Properties of InG aAs................................................. 21
Diffusion Current Lim itations................................................................... 25
The Nonlinear Transport Equations..........................................................29
Chapter IV. Numerical Techniques............................................................ 35
Introduction................................................................................................ 35
Solving Poisson’s Equation....................................................................... 37
The Continuity Equations and Diffusion Saturation.........................
39
Calculations of Output Current..............................................................
44
A Linear Approximation for the Gaussian...........................................
45
Carrier Spreading Approximation for the p-InGaAs Contact . . . .
47
Numerical T ests........................................................................................ 50
.Chapter V. Determination of Dominant Nonlinear Mechanisms . . . .
62
Introduction................................................................................................ 62
Five Volt Measurements and Sim ulations..........................................
66
Ten Volt Measurements and Simulations............................................
76
Fifteen Volt Measurements and Sim ulations........................................100
Sum m ary...................................................................................................... 101
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Chapter VI. Low Power Density Nonlinearities in Different
p-i-n Structures...................................................................................... 104
Introduction...................................................................................................104
0.95-pm Device Measurements and Sim ulations................................. 104
0.5-pm Device Measurements and Sim ulations....................................110
0.2-pm Device Measurements and Sim ulations....................................117
Additional Measurements.......................................................................... 126
Summa r y ...................................................................................................... 128
Chapter VII. High Power Density Nonlinearities: A 0.95 pm Device . 130
Introduction..................................................................................................130
Measurement Data...................................................................................... 131
Simulation Results.......................................................................................139
Two-Dimensional Carrier Flow................................................................. 152
Sum m ary...................................................................................................... 159
Chapter VIII. Reduction in Nonlinear Output and Extrapolation
to Higher P ow ers.......................................................................................161
Introduction................................................................................................. 161
0.95-pm Long Intrinsic Region Devices................................................... 161
Extrapolation to Higher Powers................................................................165
Sum m ary...................................................................................................... 174
Chapter IX. Conclusion.....................................................................................176
References.............................................................................................................182
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LIST OF FIGURES
Number
Page
1.1 Measured fundamental and harmonic power of a p-i-n
PD at a fundamental frequency of 1 G H z...........................................
3
2.1 Heterodyne laser sys. for PD NL meas. The individual laser
powers can be adj. to yield mod. depths from 0 to 100%.....................10
2.2 Lens coupling system with a variable optical spot size for
use with laboratory mounted devices................................................... 12
3.1 Simple model of a p-i-n photodiode structure......................................... 17
3.2 A depiction of the photodiode band diagram ...................................... 20
3.3 Electron velocity vs E-field for electron mobilities of 8,000 and
10,000 cm2/Vs. Experimental data from Ref. [8]............................... 22
3.4 Hole velocity vs E-field for hole mobilities of 150 and 250 cm2/Vs. . 23
3.5 Hole and electron diffusion constants versus electric field
according to equations 3.22 and 3.23.......................................................28
4.1 Algorithm flow ch art...................................................................................36
4.2 Diode partitions and carrier approximations......................................... 37
4.3 Bin (i)...............................................................................................................38
4.4 Bins (i-1), (i), and (i+1).................................................................................39
4.5 Illus. for the derivation of diffusion. From Ref. [11]............................41
4.6 Electron diffusion between bins (i) and (i+1).......................................... 42
4.7 Intensity vs norm, radius for a Gaussian and the approx. 1-D
functions. Both functions yield the same total pow er........................46
4.8 Intensity3/2 versus radial position for a Gaussian and the
e*2 and e-1 approximating functions......................................................... 46
4.9 Beam diameter versus position for estimating the hole
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spreading in the p-region....................................................................... 48
4.10 Beam diameter versus position for estimating the hole
spreading in the p-region....................................................................... 49
4.11 Carrier densities and E-field under dark conditions............................51
4.12 Carrier densities and transit time currents for cons, and
exp. ilium. In all cases, no carrier diffusion is assumed................. 53
4.13 Simulated and analytical impulse response of the test diode.
Analytical results based on equations 4.20 and 4.21........................... 55
4.14 Simulated impulse response of the model PD. Modeled
with and without restrictions on the carrier diffusion vel.................56
4.15 Carrier densities and electric field under SS conditions....................57
5.1
Measured fund, and harm, power vs. PD appl. reverse bias
volt, at 100 MHz. Fiber pigtailed. Ave. PD current = 1 mA
5.2
63
Measured fund, and harm, power vs. PD appl. reverse bias
volt, at 1 GHz. Fiber pigtailed. Ave. PD current = 1 mA..................63
5.3
Measured fund, and harm, power vs. PD appl. reverse bias
volt, at 5 GHz. Fiber pigtailed. Ave. PD current = 1 mA..................64
5.4
Measured fund, and harm, power vs. PD appl. reverse bias
volt, at 10 GHz. Fiber pigtailed. Ave. PD current = 1 mA................ 64
5.5
Meas. fund, and harm, power vs. current at 1 GHz. Appl.
V = -5 V. Fiber pigtailed. 40 and 60 dB per dec. tend, incl................67
5.6 Meas. fund, and harm, power vs. current at 5 GHz. Appl.
V = -5 V. Fiber pigtailed. 40 and 60 dB per dec. tend, incl...............68
5.7
Meas. and sim. harm, power at 5 GHz.
Sim. ss = 5 pm w/hole
mob. of 200, 230, and 260 cm2/Vs. Data from fig. 5.6....................... 69
5.8
Meas. and sim. harm, power at 5 GHz.
Sim. ss = 6 pm w/hole
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mob. of 200 ,2 3 0 , and 260 cm2/Vs. D ata from fig. 5 . 6 ........................... 70
5.9
M eas. and sim. harm, power a t 5 GHz. S S = 7 p m ...............................70
5.10 Harm, power at 5 GHz. Sim . mod. w ith pe = 8,000 cm2/V s, incr.
the rec. tim e, om itting scat., and adding a 50 Ohm load.....................72
5.11
The s-chg E-field in the i-region. SS = 7 pm. V = -5 V . ......................74
5.12
The change in carrier velocities in th e i-region......................................74
5.13 M eas. and sim . harm, power vs current a t 1 GHz. Appl.
V = -5 V. Fiber pigtailed. pp = 150 and 175 cm2/s. S S = 7 pm . . . . 76
5.14
I-region E-field for a doping density o f 5.0 x 1015 cm-3 ........................ 77
5.15
M eas. fund, and harm, power vs PD current at 1 G H z ......................78
5.16
M eas. fund, and harm, power vs PD current at 5 G H z ......................78
5.17 M eas. and sim . harm, power vs current a t 5 GHz for hole
mob. of 150 and 200 cm2/Vs. Applied V = -10 V. SS = 7pm
5.18
79
Sim. fund, and harm, power for an "ideal" PD at 5 GHz................... 81
5.19 Sim ulated harm, power vs PD current for an "ideal" PD at 5 GHz
incl. only diffusion. Appl. V = -10 V. pp = 200 cm2/ V s ....................... 82
5.20 Sim. harm, for an "ideal" PD incl. the field-dependent e-velocity
w ith spc-chg effects. Applied V = - 1 0 V .................................................... 83
5.21 Sim. harm, for an "ideal" PD incl. the field-dependent e-velocity
w ith spc-chg effects, scattering, and a 50 Ohm lo a d ............................ 84
5.22
The s-chg E-field in the i-region. SS = 7.0 p m ...................................... 85
5.23
The change in carrier velocities in the intrinsic region......................86
5.24
Sim. harm, power for an "ideal" PD incl. only p-abs............................87
5.25
M eas. and sim. harm, power at 5 GHz w/o p-region ab s......................88
5.26
M eas. and sim. harm, power at 5 GHz w/o the e-vel N L ..................
5.27
M eas. and sim. harm, power at 1 GHz. V = -10 V . ..............................90
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88
5.28 M eas. and sim . harm, power at 1 GHz w/o p-absorption.
|ip = 200 cm2/Vs. V = -10V. Exp. data from figure 5 .1 5 ....................... 91
5.29
M eas. and sim. harm, power at 1 GHz w/o p -a b s .............................
5.30
M eas. and sim. harm, power at 1 GHz. SS = 6.0 pm .......................... 93
92
5.31 M easured and sim ulated harm onic power vs current at 1 GHz.
V = -10 V. \h ria b le positions for hole spreading.................................
95
5.32 M eas. and sim. harm, power a t 1 GHz. SS = 6.0 p m ........................ 96
5.33 M eas. and sim . harm, power for var. electron sca tt............................... 96
5.34
M eas. and sim. harm, power at 5 GHz. SS = 5 and 7 pm ..............
98
5.35
M eas. and sim. harm, power at 1 GHz. V = -10 V ..............................98
5.36
M eas. and sim . harm, a t 5 GHz w / and w/o p-abs................................ 99
5.37
M eas. and sim. harm, power at 1 GHz. V = -5 V ............................. 99
5.38
M eas. and sim. harm, at 5 GHz w / and w/o p-abs...........................
101
5.39 Regions of applied bias w here different nonlinear mech.
dom inate the second harm. M eas. data from fig. 5.3......................... 102
6.1
Doping profile for the 0.95-pm long i-region d e v ic e .......................... 105
6.2
Diode bin w idth vs position, X, utilized in device sim s........................ 106
6.3
M eas. and sim . harm, power vs ave. PD current at 10 GHz.
V = -5 V. Pp = 150 and 175 cm2/Vs. SS = 7.0 pm ...................................107
6.4
M eas. and sim. harm, power vs PD current at 10 G H z ................... 107
6.5
M eas. and sim. harm, power a t 10 GHz. SS = 7.0 p m ..................... 108
6.6
Carrier densities and E-field at 1 mA. 0.95-pm PD .......................... 109
6.7
The space-charge E-field in the i-region. SS = 7.0 pm ..................... 109
6.8
0.95 pm device characteristics and sim. param eters............................110
6.9
Doping profile vs pos. for the 0.5 pm long i-region P D ..................... I ll
6.10 M eas. fund, and harm, power vs PD reverse bias voltage at
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ix
100 MHz. Fiber pigtailed. A verage PD current = 1 m A .................... 112
6.11
M eas. fund, and harm, power v s reverse bias v o lta g e ...................... 112
6.12 M eas. fund, and harm, power v s PD reverse bias voltage a t
5 GHz. Fiber pigtailed. Average PD current = 1 m A ..........................113
6.13
M eas. fund, and harm, power v s reverse bias v o lta g e ...................... 113
6.14
M eas. and sim . harm, power a t 5 GHz. 0.5-|im P D ............................115
6.15
Carrier den sities and E-field a t 1 mA. 0.50-|im P D ........................
116
6.16 Space-charge E-field in the i-region due to the photogen. carrier
densities. Average PD currents of 100 pA and 1 m A .......................... 116
6.17
0.5 pm device characteristics and sim. p a r a m e te r s.......................
6.18
Doping d en sity vs pos. for the 0.2-um p h otodiod e................................118
117
6.19 M eas fund, and harm, power v s reverse bias voltage a t 100 MHz.
Incident e*2 S S = 10 pm. M odulation depth = 100% ........................
119
6.20
M eas fund, and harm, power vs reverse bias voltage.........................120
6.21
M eas fund, and harm, power vs reverse bias voltage a t 5 GHz.
Incident e'2 SS = 10 pm. M odulation depth = 100% ........................
120
6.22
M eas fund, and harm, power vs reverse bias voltage.........................121
6.23
M eas. and sim . harm, power a t 5 GHz negl. p-abs..............................122
6.24 Sim . harm , power at 5 GHz. One sim. exc. gen. near th e p-i
interface, and th e second sim . negl. e-flow into th e i-region . . . . 123
6.25
M eas. and sim. harm, power at 5 GHz. SS = 7 pm .............................124
6.26
Carrier d ensities and E-field w ith a current of 1 m A ......................... 125
6.27 Space-charge E-field in the i-region due to the photogen. carrier
densities. 0.2-pm long intrinsic region. SS = 7 pm .............................125
6.28
0.2 pm device characteristics and sim. param eters............................ 126
6.29
M easurem ent data for eight P D s................................................................127
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X
7.1
Large-signal relative FR o f a 0.95-jim P D ................................................. 132
7.2
N orm lg-sig output of a 0.95-pm PD a t 150 M Hz....................................133
7.3
Norm lg-sig output of a 0.95-pm PD a t 500 MHz.Ave. currents
o f 100 and 1400 pA w ith an e*2 SS o f 5.75 ± 0.25 pm .............................. 134
7.4
Norm lg-sig output of a 0.95-pm PD at 500 M H z .................................. 134
7.5
Fund, and harm, power v s current for a 0.95-pm P D .......................... 135
7.6
Fund, and harm onic pow er v s PD current for a 0.95-pm PD.
Incident e-2 SS of 5.75 ± 0.25 pm. 500 MHz fund, frequency. . . . .
136
7.7
Fund, and harm, power v s current for a 0.95-pm P D ......................... 136
7.8
SS relative FR of a -5 V-biased 0.95-pm P D .............................................138
7.9
M eas. and sim . SS FR of a 0.95-pm PD. Ave. currents of 800
and 1000 pA. Pp = 230 cm2/Vs. D ata from Fig 7 . 8 ............................... 139
7.10
M eas. and sim. SS FR of a 0.95-pm PD ................................................
7.11
Carrier dens, and E-field in th e i-region at 100 pA............................. 141
7.12
Carrier dens, and E-field in the i-region at 800 pA............................. 142
7.13
Carrier dens, and E-field in th e i-region at 1000 pA...........................142
7.14
Lg-sig FR o f a -5 V-biased 0.95-pm P D ..................................................... 143
7.15
Lg-sig FR of a -5 V-biased 0.95-pm PD. Ave currents o f 800 and
1000 pA. V = -5 V. pp = 150 cm2/Vs. D ata from fig 7 . 1 ...................
7.16
140
144
Sim lg-sig output of a PD at 150 M H z .................................................. 145
7.17 Sim lg-sig output of a PD a t 500 MHz. Ave currents of 100
and 1400 pA. SS = 3 pm. pp = 150 cm2/Vs. V = -5 V ........................... 146
7.18
DC-coupled sim lg-sig output of a PD a t 500 M H z ..............................147
7.19
DC-coupled sim lg-sig output of a PD at 500 M H z ..............................147
7.20 Sim harm onic power versu s PD current at 5 GHz. Sim . w ith
param eters leading to b est fits in fig 7.10. D ata from fig 7.7. . . . 148
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7.21 Sim harm, power vs PD current at 5 GHz. Sim. with SS of 3.6
and 4.0 pm. V = -5 V. pp = 200 cm W s. Data from fig 7.7.............. 149
7.22 Sim harm, power vs PD current at 5 GHz........................................ 150
7.23 Sim harm, power vs PD current at 5 GHz........................................ 151
7.24 Sim harm, power vs PD current at 500 MHz......................................151
7.25 Rep. of the potential in the i-region vs radial position...................... 153
7.26 Rep. of 2-D flow due to the radial potential from fig 7 .2 5 ................155
7.27 Linear approx. function for the Gaussian int. profile...................... 156
7.28 Procedure for obtaining a radial electric field estim ate...................156
7.29
Ratio of the est. electron radial vel. to the ele. axial v e l ..................157
7.30
Ratio of the est. hole radial vel. to the hole axial vel.........................159
8.1
I-region E-field for various i-region doping d en sities...................... 162
8.2
Sim. harm, power negl. p-abs for var. i-region doping.................... 163
8.3
The space-charge E-field in the i-region............................................. 164
8.4
Diff. in electron velocity from 10 pA to 1000 pA.................................164
8.5
Space-charge E-field in the i-region due to the photogen. carrier
densities. 0.95-pm long intrinsic region. SS = 7 p m ....................... 166
8.6 Sim. harm, power at 5 GHz with and w/o p-abs................................. 167
8.7
Sim. harm, power at 5 GHz with and w/o p-abs................................. 168
8.8
Sim. harm, power at 5 GHz with and w/oa 50 D lo a d ..................... 169
8.9 Sim. harm, power at 5 GHz with and w/o p-abs................................. 171
8.10 Sim. harm, power at 5 GHz w/ and w/o a 50 Q. load...................... 172
8.11
Sim. harm, power at 5 GHz with and w/o p-abs................................ 173
8.12
Sim. harm, power with and w/o a 50 Q load...................................... 175
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1
L INTRODUCTION
N on lin ea r distortion in optical system s h a s b een stu d ied by m a n y
r e se a r c h e r s.1-10 A lth ough m ost of the research in th e non lin ear behavior
of th ese system s h a s b een concentrated on so u r ce s,1'4 for exam ple la ser
diodes and extern al m odulators, there h as been a lim ited am ount o f work
done con cern in g th e n o n lin ea rities in receivers.5'11
One im portant
receiver com ponent, th e p-i-n photodiode (PD), h a s received very little
attention. The reason for th is lack of work on the p-i-n PD h as been two­
fold, 1) th e devices w ere a ssu m ed to be very lin ear devices i f the electric
field is kept h igh enough in the in trin sic (depletion) region to cause the
carrier velocities to saturate, and 2) source n o n lin ea rities lim ited m ea ­
surem en t dynam ic range.
N everth eless, nonlinear behavior in PDs can be an im portant lim it­
in g factor in h igh -fid elity an alog and digital com m u nication sy stem s.
CATV applications requiring studio quality video tra n sm issio n strive for
h a rm on ic d istortion (H D ) and in ter-m o d u la tio n d isto rtio n products
(IMD) betw een -60 and -80 dBc. Also, m any radar and electronic w arfare
system s require HD or IMD levels of -50 dBc or better.
M any fiber-optic sy stem s are now ta k in g ad van tage o f recently
available h ig h optical pow er sources for in crea sed perform ance.
For
in sta n ce, heterodyne detection tak es advantage o f h igh local oscillator
power to obtain shot-noise-lim ited perform ance in PD s. Optical p rea m ­
p lifiers, both Erbium doped fiber am plifiers and sem iconductor optical
am p lifiers, are used to reduce system link loss. U sin g h igh power optical
sources w ith external m odulators also low ers system link loss by in c r ea s­
in g the m odulated optical power w ithout in creasin g the drive m icrow ave
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
2
power. Of course, harmonics traditionally grow with a power-law behav­
ior, for example second harmonics will grow proportional to the square of
optical power, third harmonics will grow proportional to the cube of opti­
cal power, etc. Thus nonlinearities become more important in systems
with high incident optical powers. All of these basic optical systems will
be affected, if not limited, by the nonlinearities in the common component
—the PD. Hence, it is extremely important to understand the nonlinear
behavior in p-i-n PDs so that steps can be taken to minimize it, and so
that the fundamental limits of devices can be explored.
Nonlinearities will exist at some level in PDs for many reasons. It
is usually a poor assumption that an electric field which is high enough
to saturate the carrier velocities w ill yield, by itself, high linearity.
Saturated carrier velocities are only one of the many assumptions that
are required to linearize the equations that describe the transport of car­
riers in PDs. The exclusion of electric field dependence of the diffusion
constants, electric field screening due to high space-charge densities,
highly-doped (undepleted) absorbing regions where electric fields are
low, trap sites, recombination, heterojunctions, and non-zero load resis­
tances all add nonlinear terms to the transport equations. This study
concentrates on five of these effects:
1) Generation in highly-doped
absorbing regions where the electric field is low, 2) diffusion, 3) recombi­
nation, 4) the effects of space-charge induced electric field screening, and
5) load resistance. These effects will be analyzed for their relative contri­
bution to the nonlinear behavior in PDs.
A sample of the harmonic data obtained from a commercial p-i-n
PD is shown in figure 1.1. The device in figure 1.1 has a 3-dB bandwidth
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
3
-10
9
£ -30
T3
w -50
u
a>
I -70
••
PQ
n
CL.
■
S
>? -90
03
I -n o
u
i
r"
■ t i* *
♦
♦ " ! ........
♦
▲............
•
■
♦
▲
-!3°
-150
1
0.1
■
■
■
■
■
■
Fundam ental
2nd Harmonic
3rd Harmonic
4th Harmonic
40dB/dec
■
1
A v e ra g e P h o to d io d e C u r r en t (m A )
10
Figure 1.1 Measured fundamental and harmonic power of a p-i-n
photodiode at a fundamental frequency of 1 GHz.
of 22 GHz with a 0.95-pm long intrinsic region. The applied bias was -2 V
and the device was pigtailed with a single mode optical fiber. From the
figure, the device displays two distinct regions of nonlinear behavior. For
average PD currents below 300 to 500 pA, the growth in harmonic power
approximately follows power-laws; however, for average PD currents
above 500 pA, the growth in harmonic power deviates significantly from
a power-law behavior.
The overall scope of this dissertation is three-fold: demonstration of
a technique for wide dynamic range (>130 dB) m easurement of device
nonlinearities, the modeling (physics) of photodetector carrier transport,
and the obtaining of new insight into PD operation by relating nonlineari­
ties from the simulation output to measured output. The second part of
this work, device modeling, will help to predict and determine the lim its
of the nonlinear behavior. To carry out the numerical simulation portion
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
4
of this dissertation, a program was developed to solve the three coupled
nonlinear differential equations required for the description of carrier
transport in semiconductor materials. The simulations include all three
regions of the semiconductor material, instead of just the intrinsic region
as found in previous work,12-14 since nonlinearities may originate from
undepleted regions of the device where there is absorption. This compli­
cates the num erical solution significantly since diffusion cannot be
ignored. The associated numerical instabilities involved with including
diffusion have been well documented in the literature.15 Various tech­
niques15'16 have been used to deal with diffusion instabilities — here a
new technique is introduced which is based on basic solid state physics.
The simulations are first used to accurately model and understand
device nonlinear behavior. After adequate agreement has been obtained
between experiments and sim ulations, the various properties of the
device, such as recombination, carrier mobilities, absorption in unde­
pleted semiconductor regions, the intrinsic properties of InGaAs, and
the intrinsic region length are modified to determine their relative con­
tributions to the nonlinear behavior for various PD operating conditions.
The work presented in the following eight chapters provides the
first comprehensive study of the nonlinear behavior in p-i-n PDs and
reveals that nonlinearities in certain devices can be quite high even when
illuminated with low to medium power levels.
Also, operation under
high power densities is found to cause not only harmonic distortion, but
also response reduction and phase distortions.
Chapter II outlines the various methods for measurement of har­
monic levels in PDs or other photosensitive devices.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
5
Chapter III reviews the physics governing PD carrier transport
and the associated nonlinear transport equations. Specific properties of
InGaAs and InP materials, including diffusion constants and the elec­
tric field dependence of the carrier drift velocities, are covered along with
the structure of the p-i-n PDs under investigation here.
Chapter IV discusses the numerical methods employed to solve the
set of nonlinear equations governing carrier transport (Poisson's equa­
tion and the carrier continuity equations) in one spatial dimension.
Special attention is paid to specific limitations on the diffusion current
derived from first principles rather than using numerical treatments.
Chapter V presents measurement data with sim ulation results
and analysis to sort out the various nonlinear mechanism s and their
contribution to PD nonlinearity in various regions of PD applied voltage.
Chapter VI presents additional data with simulation results and
analysis pertaining to nonlinearities in several photodiodes under mod­
erate power densities where the generated carrier densities are low
enough not to cause the intrinsic region electric field to collapse.
Chapter VII presents high power density measurements and sim ­
ulations of a device where the generated carrier densities are high
enough to screen the intrinsic region electric field.
Chapter VIII investigates ways of reducing PD nonlinearity and
ways to increase the maximum PD current before the effects investigated
in Chapter VII occur.
Chapter IX gives conclusions. It summarizes the main results of
this research and discusses suggestions for further work.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
6
1.
W. S u sa k i,
"Recent P ro g ress in
su p e r lin e a r In G aA sP
L aser
D iodes,” OFC 91, Paper WG5, p. 92.
2.
A.
H . G n au ck ,
e t a l.,
M o d u la tio n for
CATV
"C om parison o f D ir ec t a n d E xternal
L ig h tw a v e
T r a n s m is s io n
at
1.5p.m
Wavelength," E lectron. L ett., 28, p. 1875,1992.
3.
G. E. Bodeep and T. E. D arcie, "Comparison o f Second- and T hirdOrder D istortion in In te n sity M odulated InG aA sP L asers and a
LiNbOg External Modulator," OFC 89, Paper WK2.
4.
R. B. C hilds and V. A. O’B y m e , "Predistortion L in earization of
D irectly M odulated DFB Lasers and E xternal M odulators for AM
Video Transmission," OFC 90, Paper WH6.
5.
R .D . E sm a n a n d K .J. W illiam s, " M ea su rem en t o f H arm onic
D istortion in M icrowave Photodetectors," IE E E Photon. Tech. L e tt.,
PTL-2, p. 502,1990.
6.
K. J. W illiam s an d R. D. E sm an, "Observation o f Photodetector
N onlinearities," E lectron. L ett., 28, p. 731,1992.
7.
M. D en ta n and B. de C rem oux, "N um erical S im u la tio n o f the
N o n lin e a r
R esp on se
Illum ination,"
8.
R.
R.
H ayes
of
a
p -i-n
P h o to d io d e U n d e r
H igh
J. o f L igh tw ave Tech., JLT8, p. 1137,1990.
and
D .L .
P e r se c h in i,
“N o n lin e a r ity o f
p-i-n
Photodetectors,” IE E E Photonics Tech. L ett., PTL-5, p. 70,1993.
9.
T. Ozeki and E. H. Hara, "M easurem ents of N o n lin ea r D istortion in
Photodiodes," Electron. L ett., 12, p. 80,1976.
10. D. K u h l, e t a l., "Influence o f Sp ace C h a rg es on th e Im p u lse
R esp on se o f InG aA s M etal-Sem iconductor-M etal Photodetectors,"
J. o f L igh tw ave Tech., JLT10, p. 753,1992.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
7
11. A. R. Williams, et al., "High Frequency Saturation Measurements of
an InGaAs/InP Waveguide Photodetector," Electron. Lett., 29, p.
1298,1993.
12. G. Lucovsky, et al., "Transit-Time Considerations in p-i-n Diodes," J.
o f Appl. Physics, 35, p. 622,1964.
13. R. Sabella and S. Merli, "Analysis of InGaAs p-i-n Photodiode
Frequency Response," IEEE J. o f Quantum Elec., JQE-29, p. 906,
1993.
14. J.M. Zhang and D.R. Conn, "State-Space Modeling of the PIN
Photodetector," J. o f Lightwave Tech., JLT10, p. 603,1992.
15. S.J. Polak, et al., "Semiconductor Device Modeling from the
Numerical Point of View," Intl. J. for N um erical Methods in Eng.,
24, p. 763,1987.
16. H. Yi, et al., "Novel Method to Control Num erical Solution
Oscillation of Diffusion-Drift Equation," Electron. Lett., 2S, p. 1487,
1990.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
8
H. MEASUREMENT SYSTEM
To fully characterize and analyze the nonlinear behavior in PDs,
an amplitude modulated source free from harmonic content is essential.
A measure of the source quality is its dynamic range or the range of opti­
cal power for which optical nonlinearities in the source can be neglected.
In order to simulate a wide variety of current optical systems, the source
also m ust have a modulation depth that is variable from very low values
for m ultichannel systems to values at least greater than 50% for single
channel systems.
The direct modulation of laser diodes is a poor candidate as a
source for testing nonlinearities in PDs. This source presents some very
difficult problems for manufacturers wishing to construct devices w ith
high dynamic range.
Several researchers1'5 have measured and ana­
lyzed the dynamic range of laser diodes and their results have shown
that a dynamic range of 60 dB or better for both HD and IMD is difficult
when the optical modulation depth (OMD) is greater than 4%. For 50%
modulation depths, values of HD and IMD are a disappointing 20 dB.6
Light emitting diodes (LEDs) have been used7 to study the nonlinearities
in avalanche photodiodes.
However, this m easurem ent technique7
requires several assumptions about the relationship between actual har­
monic levels and the intermodulation or second order products m ea­
sured. Additionally it provides only about 60 dB dynamic range.
External modulation of a CW optical source with a Mach-Zehnder
(MZ) electro-optic modulator is another possible source for testing the
nonlinear behavior in PDs. The modulator, biased at quadrature tom ax-
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
9
imize the linearity of the sinusoidal transfer characteristic, produces an
amplitude modulated optical output according to the applied electrical
signal input. Since the light output from the modulator is proportional to
the sine of the input voltage, a large modulation signal causes harmon­
ics of the modulation frequency f to appear in the optical output. The
amplitude of these harmonics is described by a Bessel function expansion
of sin(msin(cot)-H}>), where co = 2nf. The power in the harmonics of f
depend quite heavily on the modulation depth m and bias point 0; how­
ever, these nonlinearities have been measured by several researchers2'6-8
(for m = 0.02) to be as low as -55 dBc for IMD and -85 dBc for HD. By uti­
lizing modulator linearizing circuits,4 slightly better results of -65 dBc
have been achieved for IMD. As the MZ modulation depth approaches
50%, values of HD and IMD approach 20 dB.6-9 Therefore the use of exter­
nally modulated sources for reliable measurements of PD nonlinear
properties is very limited.
Recently, an attractive new source10-11 has been developed for the
testing of nonlinearities in photodiodes. This new source meets every
requirement of the ideal source for testing PDs due to its unique way of
generating the frequency f.
Such a source is used for this work and is
based on heterodyning two single frequency lasers. This source creates a
microwave signal at the frequency f by mixing two optical frequencies
separated by f and has no inherent mixing components to create frequen­
cies at integer multiples of f.
Consider the case of two single frequency lasers with electric fields,
E^t) = Eolcos(co1t+<()1) and E2(t) = Eo2cos(eo2t+<{>2), incident on and hetero­
dyned in a photodetector as shown in figure 2.1. The resulting time-
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
10
PLL
Nd:YAG
Laser
Microwave
Source
PD
To Photodiode
Under Test
Nd:YAG
Laser
Polarization Maintaining
Fiber-Optic Coupler
Figure 2.1 Heterodyne laser system for photodiode nonlinearity
measurements. The individual laser powers can be adjusted to yield
modulation depths from 0 to 100%.
average current generated by an ideal detector is proportional to the
square of the total electric field and is given by:
-
(E ^ + E y
I(t)oei
+ E o lE o2 COs(tO]t - C0 2t + <!>!- $ 2 ) •
(2.1)
The first term in equation 2.1 is the average PD current while the
second term is a signal at the frequency 0 ) 2-0 0 2 . If the frequencies are
adjusted such that 0 ) 2 -002 = 2rcf is a microwave or RF frequency, the het­
erodyned signal can be detected by a PD. Notice that since there is no
term in equation 2.1 that contains an integer multiple of f, the source is
completely free of nonlinearities.
However, without a phase-lock-loop
(PLL), the beatnote will exhibit a linewidth approximately twice that of
the laser linewidth.
Commercial Lightwave Electronics model 120
single-frequency lasers are ideal candidates for heterodyning since they
are available with very narrow linewidths (<100 kHz). Additional essen­
tial features are output powers up to 150 mW at 1319 nm and tunability
over 50 GHz.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
II
With heterodyning only (no phase-lock) and temperature stabiliza­
tion of the laser cavities as provided by the manufacturer, the frequency f
is stable enough to measure the signals generated by the photodiode at f,
2 f,... , and nf directly with a microwave spectrum analyzer. With an HP
model 8566B spectrum analyzer operating at a resolution bandwidth
(RBW) of 1 MHz and video averaging or with a 100 kHz resolution band­
width without averaging, the m inimum detectable electrical power is
approximately -90 dBm. The RBW m ust be set sufficiently higher than
its m inim um of 10 Hz because of the frequency drift (1 MHz/min) and
linew idth (1 to 50 kHz) of the lasers and associated beatnote.
Additionally, at an upper PD current lim it of 2.5 mA, the microwave
power generated at the fundamental frequency f is approximately -10
dBm. This results in an 80 dB dynamic range, which represents a 15 to
60 dB improvement (depending on the modulation depth) over the direct
modulation of laser diodes or the external modulation of lasers with a MZ
modulator approaches.
A higher dynamic range for the heterodyne system (figure 2.1) can
be obtained by offset-phase-locking the two lasers. Phase-locking12-13 the
two lasers at the difference frequency f produces a spectrally pure beat
note with negligible linewidth (< 1 Hz) and drift. The RBW can then be
reduced to less than 1 Hz, allowing the detection of electrical signals with
signal powers o f -140 dBm (10 Hz RBW, HP8566B)and below, depending
on the spectrum analyzer. This increases the dynamic range of the m ea­
surement system to 130 dB, a value unsurpassed by any other PD nonlin­
earity measurement technique.
To test a PD, the single mode fiber-optic output of the source in fig­
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
12
ure 2.1 can be used in two ways. It can be fused directly to the fiber-optic
pigtail of a fiber-coupled PD or it can be used with laboratory-mounted
unpigtailed PDs as shown in figure 2.2. The output Gaussian beam from
the fiber is collimated with a spherical lens and is focused with a second
spherical lens onto the PD under test. With identical focusing and colli­
m ating lenses, a spot size equal to the spot size of the light in the fiber
(approximately 10 pm) will be incident on a PD placed in the focal plane
of the focusing lens. With another combination of focusing len ses10 the
m inimum spot size can be reduced allowing higher power densities
inside the photodiode. Larger spot sizes can also be achieved by translat­
ing the PD in the Z direction (defocusing) as shown in figure 2.2.
Output
From Laser
Heterodyne
Figure 2.1
Vbias
A
A
fcyUA O u tp U t TO
>F S
V
\J
Collimating
Lens
Focusing
Lens
-II— — J Spectrum
Analyzer
Bias Tee
(Cut Away View)
Figure 2.2 Lens coupling system with a variable optical spot size for use
with laboratory mounted devices.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
13
1.
W. Susaki,
"Recent Progress in superlinear InGaAsP Laser
Diodes," OFC 91, Paper WG5, p. 92.
2.
A. H. Gnauck, et al., "Comparison of Direct and External
M odulation for
CATV
L ightw ave T ransm ission a t
1.5|im
Wavelength," Electron. Lett., 28, p. 1875,1992.
3.
G. E. Bodeep and T. E. Darcie, "Comparison of Second- and ThirdOrder Distortion in Intensity Modulated InGaAsP Lasers and a
LiNbOg External Modulator," OFC 89, Paper WK2.
4.
R. B. Childs and V. A. O’Bym e, "Predistortion Linearization of
Directly Modulated DFB Lasers and External Modulators for AM
Video Transmission," OFC 90, Paper WH6.
5.
C. H. Cox, et al,. "An Analytic and Experimental Comparison of
Direct and External Modulation in Analog Fiber-Optic Links," IEEE
Trans, on Microwave Theory and Tech., MTT-38, p. 501,1990.
6.
W. E. Stephens and T. R. Joseph, "System Characteristics of Direct
Modulated and Externally Modulated RF Fiber-Optic Links," IEEE
J. of Lightwave Tech., LT-5, p. 380, 1987.
7.
T. Ozeki and E. H. Hara, "Measurements of Nonlinear Distortion in
Photodiodes," Electron. Lett., 12, p. 80, 1976.
8.
C. H. Bulmer, "Sensitive, Highly Linear Lithium Niobate interfer­
ometers for Electromagnetic Field Sensing," Appl. Phys. Lett., 53, p.
2368,1988.
9.
B. H. Kolner and D. W. Dolfi, "Intermodulation Distortion and
Compression in an Integrated Electrooptic Modulator,"
A pplied
Optics, 26, p. 3676,1987.
10. R.D. Esm an and K.J. W illiams, "Measurement of Harmonic
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
14
D istortion in M icrow ave Photodetectors," IE E E Photon. Tech. L e tt.,
PTL-2, p. 502,1990.
11.
K. J. W illiam s and R. D . E sm an , "Observation o f Photodetector
Nonlinearities," E lectron. L ett., 28, p. 731,1992.
12.
K. J. W illiam s, "Offset P h ase Locking o f Nd:YAG N onp lanar R in g
Lasers," MS T hesis, U niv. o f M aryland, 1989.
13.
K.J. W illiam s, e t a l., "6-34 GHz O ffset P h ase Locking o f Nd:Y A G
1319 nm N onplanar R ing Lasers," Electron. L ett., 25, p. 1242,1989.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
15
m . PHOTODIODE DEVICE PHYSICS
Generalized Transport Equations
The basic equations governing carrier transport in any semicon­
ductor region are Poisson’s equation and the carrier continuity equa­
tions. Poisson's equation relates the potential 'F, to the charge distribu­
tion inside any region and is given by:
-V-V'F = -|(p - n + N d-N a) ,
(3.1)
where 4*= 'F(x,y,z,t) is the potential at the position (x,y,z) at some time t, p
= p(x,y,z,t) is the hole density, n = n(x,y,z,t) is the electron density, Na is
the density of ionized acceptor dopants present in the crystal, N d is the
density of ionized donor dopants present in the crystal, q is the unit of
charge (here positive), and £ is the permitivity of the semiconductor mate­
rial. The electric field, E = E(x,y,z,t), in the semiconductor region is given
by the negative gradient of the potential *F and when substituted into
Poisson's equation (3.1) yields Gauss's Law which is given by:
y .E = —(p -n + N d -N a) .
(3.2)
£
The continuity equations for holes and electrons control the conser­
vation of carriers in any volume. With a mid-bandgap approximation for
the recombination centers,1 the continuity equations are given by:
|p =
at
3t
~
p n -n f
_1
(p + n + 2nj)xp q
r
p n ~--^ -
(p + n + 2nj)xn
(33)
F
+ -V -J n,
q
(3.4)
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
16
where G = G(x,y,z,t) is the generation of carriers in the volume, t p and xn
are the hole and electron recombination tim es, respectively, and n; is the
intrinsic carrier density. Jp and Jn are the hole and electron currents,
respectively, which are expressed as:
Jp —Jpdrift
Jn
Jpdiff »
Jndrift JndifF
(3.5)
(3.6)
where Jpdrift and Jndrift are the hole and electron drift currents, respec­
tively, and Jpdiff and Jn(jifr 21-6 the hole and electron diffusion currents,
respectively. The total current in the semiconductor region is the sum of
the hole and electron currents (equations 3.5 and 3.6) with the addition of
the displacement current and is given by:
(3.7)
Equations 3.2 to 3.4 describe the carrier flow within a semiconduc­
tor region and solutions to a particular problem require the sim ultaneous
solution of these coupled equations with the appropriate boundary condi­
tions and the particular expressions for the currents (equations 3.5 and
3.6). In particular, photodiodes made with InGaAs and InP material
will be considered here.
p-i-n Photodiode Structure
The basic photodiode structure under investigation here (figure 3.1)
is either a single or double heterojunction device made from InP and
InGaAs. InGaAs lattice-matched to InP has a bandgap2 of 0.75 eV at 300
K and is sensitive to optical radiation if the wavelength is 1650 nm or
shorter. InP, however, has a bandgap2 of 1.35 eV at 300 K and is sensitive
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
17
only to radiation having a wavelength of 920 nm or shorter. Therefore
these are an ideal pair of semiconductor materials for detection of radia­
tion in the 1300-nm to 1550-nm wavelength range, a region of importance
for low-loss optical fiber communication systems.
The generic device (figure 3.1) under investigation is composed of a
highly-doped n-InP substrate, an intrinsic layer of unintentionally-doped
n-InGaAs, and a degenerately-doped p-InP or p-InGaAs cap layer. The
X=Wp
x=0
v
X =W p+W i
V
P
V
p-InGaAs
or
p-InP
-
X =W p +W i+ W n= W
n-InGaAs
n
n-InP
Substrate
n-Side
Illumination
p-Side
Illumination
Applied Reverse Bias, Va
Load Resistor
Figure 3.1 Simple model of a p-i-n photodiode structure.
incident light is assumed to pass through an aperture in either the n- or
p-side ohmic contact of the device. Actual devices may take on many
forms such as planar or m esa3>4 structures.
However, the simplified
model will allow many of the physical mechanism s responsible for PD
nonlinearities (NL) to be studied.
Simplifying Assumptions
Throughout this work, several simplifying assumptions on the
behavior and physical properties of the device in figure 3.1 will be made
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
18
by eliminating unknown or little known aspects of the device. The valid­
ity of any such assumptions is deemed appropriate if sufficient agree­
ment is obtained between experimental measurements and the sim u­
lated results. A one-dimensional (one spatial dimension) analysis will be
considered here. This will allow some aspects of the device NL to be studied, however; as will be explained later, two-dimensional simulations
will be required to further study device nonlinear behavior.
The second simplifying assumption is that the p- and n-contacts
are ohmic and, as such, offer no barrier to carrier flow. This permits the
majority carrier density near the contact/semiconductor interface to be
approximated5 by the density in the bulk region. This can be stated m ath­
ematically by the following boundary conditions at x=0 and x=W:
at x = 0,
(3.8)
at x = W
(3.9)
An additional boundary condition is needed for the solutions of
equations 3.2 to 3.4. This boundary condition relates the given applied
voltage V a and the built-in diffusion potential Vbi to the electric field in
the semiconductor region and is given by:
(3.10)
with the built-in potential given by the usual expression:
kT . IN a(0)Nd(W)
(3.11)
where k is Boltzmann’s constant and T is the absolute temperature. Note
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
19
that in equations 3.10 and 3.11, the heterojunction potential steps have
been omitted in the formulation of the potential and electric field. This is
equivalent to treating the device (from the point of view of the electric
field) as a homojunction p-i-n diode. The transport of carriers across the
heterojunction(s) will be discussed in the next section.
The light incident on the detector (figure 3.1) enters either through
an opening in the p-side or the n-side contact depending on device con­
struction.
For the purposes here, the incident contact surface is
assumed to be anti-reflection coated and thus 100% of the incident light
enters the semiconductor material. The generation rate G (eqns 3.3 and
3.4) can be expressed as the absorption of photons at a position x m ulti­
plied by the time dependence of the generated light. The absorption is
exponential in the regions where a single photon has enough energy to
generate a hole-electron pair, which for 1300 to 1550-nm light is only in
the InGaAs regions. For single-pass p-side illumination, the generation
rate is expressed as:
G(x,t) = G0(t)e'<xx ,
(3.12)
where G0(t) is the time-dependent generation rate per volume and a is
the absorption coefficient for InGaAs.
For single-pass n-side illum ina­
tion, the generation rate is expressed as:
G (x,t) = G0Ct)e'a(wp+Wi'x ) ,
(3.13)
where x = (wp+w;) is the InP/InGaAs interface (figure 3.1).
The band diagram of the device is shown in figure 3.2, where a pInGaAs cap layer is assumed with a reverse bias voltage of a few volts.
The InGaAs/InP heterojunction depicted in figure 3.2 has a valence band
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
20
n-InGaAs
p-InGaAs
H'JhP
0.37 eV
0.23 eV
0.1 eV
0.1 eV net
Conduction
Band Barrier
0.5 eV net Valence
Band Barrier
Figure 3.2 A depiction of the photodiode band diagram.
discontinuity6 of 0.37 eV and a conduction band discontinuity6 of 0.23 eV.
The reduction in the conduction band discontinuity and the increase in
the valence bands discontinuity of approximately 0.1 eV is the result of
the difference in the doping of the intrinsic n-InGaAs, N<j = 1015 cm'3,
and the n-InP, N d = 1017 cm-3. The band discontinuities are barriers to
current flow and their effect is to trap carriers.7 Here the effects of the
heterojunctions will be reduced because of several approximations.
Electrons will be allowed to flow without restriction across the con­
duction band barrier since, near the 0.1 eV barrier, the electron can have
significant velocity (energy) towards the barrier due to the high electric
field and, with a low effective mass, may travel across the barrier without
restriction. Holes on the other hand will not tend to flow across the 0.5 eV
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
21
valence band barrier since they inherently tend to flow away from the
barrier and, with their higher effective mass, will be less likely to pene­
trate into the InP. Therefore, for modeling purposes, holes will not be
allowed to move past the heterojunction.
Carrier Transport Properties of InGaAs
The solutions of equations 3.2 to 3.4 with the associated current
equations 3.5 and 3.6 require expressions for the drift and diffusion cur­
rents. This requires values for carrier mobilities, diffusion constants,
and carrier velocities with their respective relationships to the doping
concentrations and the electric field. In general the equations for the
drift currents Jpdrift and Jndriftare:
Jpdnn = -<3PVp(E) ,
(3.14)
Jndrift = qnvn(E) ^
(3.15)
where vp(E) and vn(E) are the electric-field-dependent hole and electron
drift velocities, respectively. For electric fields below 4 kV/cm for elec­
trons and below 20 kV/cm for holes, the drift velocities are the usual
expressions of mobility times the electric field. However, for high speed
operation, it is desirable to operate the device with electric fields high
enough to saturate both carrier velocities to minimize the carrier transit
times. To model a particular device, it is necessary to utilize the entire
velocity-versus-field relationships for the drift velocities because carriers
will be in regions of the device where the electric field may be high
enough to saturate carrier velocities (intrinsic region) and in regions
where the fields are low (contacts and interface regions).
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
22
The electron velocity versus electric field has been measured8 for
various samples of InGaAs at electric fields from 10 to 100 kV/cm. A n
empirical expression is used to describe vn(E) for electrons in InGaAs:
( E )= e (i v ^ | | | )
1+PE
(3.16)
where |in is the electron low-field mobility in InGaAs, vnhf is the high
field electron velocity, and P is a fitting parameter.
Equation 3.16 is plotted in figure 3.3 for electron mobilities of 10,000
and 8,000 cm 2/Vs with the experimental data from reference [8]. The cor­
responding fitted parameters are P = 1.0 x 10‘7 and 0.8 x 10'7, respectively,
and in both cases vnhf = 5.4x 106 cm/s. Equation 3.16 differs slightly from
the empirical formula given byDentan and de Cremoux.9 Equation 3.16
was chosen because it provides a better fit to the experimental data8 from
20 to 100 kV/cm, even though it fails to give a high enough peak electron
2.0
so
u = 8,000 & beta = 0.8 x 10
u = 10,000 & beta = 1.0 x 10
Experimental Data [8]
o
1-0
13
>
g 0.5
+3
o
,-2
s
0.0
0
20
40
60
E lectric F ie ld (kV /cm )
80
100
Figure 3.3 Electron velocity versus electric field for electron mobilities
of 8,000 and 10,000 cm2/Vs. Experimental data from Reference [8].
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
23
velocity. For simulation purposes, the low-field electron mobility used
will be between 6,000 and 10,000 cm 2/Vs since the measured values in d if­
ferent semiconductor samples vary between 6,000 and 10,500 cm2/V s.2
The electric field dependence of the hole velocity in InGaAs has not
received as much attention as the electron velocity; however, m easure­
ments have been made for the saturated hole velocity.4 A high-field hole
velocity of vphf = 4.8 x 106 cm/s may be used with the following empirical
formula:
U. v . ,E
v E )>
w + lt ;Ef
« - 17>
to analytically describe the hole velocity versus electric field (figure 3.4)
for hole mobilities of 150 and 250 cm2/V s.
The value for the hole mobility used in our studies is between 50 to
0.6
CO
g 0.5
i
u
X
* 0.3
o
ji =250 cm /V s
>
Experim ental Data [4]
• i 0-1
0.0
0
20
40
60
E le ctr ic F ield (kV /cm )
80
100
Figure 3.4 Hole velocity versus electric field for hole mobilities of
2
150 and 250 cm /Vs. Experimental data from Reference [4],
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
24
80 percent of the only reported10 (300 cm 2/Vs) value in the literature. This
is reasonable considering the fact that the electron mobility varies appre­
ciably from sample to sample. The exact electric field dependence of the
hole mobility in InGaAs below 50 kV/cm has not yet been measured. For
example, the hole mobility in equation 3.17 needs to be at least 10% lower
for a best fit to the empirical formula of Dentan and de Cremoux9 below
50 kV/cm.
Therefore, the hole mobility will be left as an adjustable
parameter for this study so long as the values used are within the 150-300
cm2/Vs range.
It has also been reported2 that the carrier mobilities decrease as
the doping densities exceed 1016 /cm 3 due to ionized impurity or free car­
rier scattering. To incorporate this information into the transport equa­
tions, the hole and electron mobilities will be reduced by the following
empirical relationships:
Pp
RP =
lV2
P + n
Ph
m’
n
,
( 3 .1 8 )
,
(3.19)
.
------- ----------r
iV 2
l+ £ ± ^
nh .
where ph and n h are the carrier densities where the respective mobilities
have decreased by 1/V2. These empirical relationships are approxima­
tions to data2 and are not meant to distinguish between the different scat­
tering mechanisms such as hole-hole, electron-hole, electron-electron, or
carrier-ionized-dopant scattering.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
25
Diffusion Current Limitations
Limitations on the diffusion current has not received much atten­
tion in the literature with the exception of several11’12 numerical treat­
ments. The diffusion current for electrons is typically expressed as:
Jndifr = q D n || •
(3.20)
This equation implies that, as the carrier density gradient increases, the
diffusion current will increase without bound. This w ill certainly not
occur in any semiconductors since carrier scattering and thermal veloc­
ity limitations will limit the carriers effective velocity whether the carrier
moves by drift or diffusive mechanisms.
When equations 3.5 and 3.6, containing the diffusion terms, are
substituted into the continuity equations 3.3 and 3.4, the three coupled dif­
ferential equations become second order in x.
The resulting drift-
diffusion problem becomes analytically untractable and leads to instabili­
ties11,12 in num erical solutions.
To avoid these complexities, many
researchers9,13'15 disregard the diffusion terms in the continuity equa­
tions and solve drift-only problems in regions where the diffusion terms
are believed to be less important. This is rationalized13 with the argu­
ment that once the voltage exceeds kT/q, the contribution of the diffusion
current to the total current is negligible. While this may be true, carrier
diffusion is crucial for an accurate determination of the internal PD elec­
tric field which depends on the charge location, itself a strong function of
the diffusion terms. Additionally, reference [13] assum es that the gener­
ated carrier densities are small such that the electric field is not per­
turbed, which may not be true for the high incident powers under consid­
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
26
eration here. In fact, the basic solution to a p-n junction under dark con­
ditions (zero total current) requires the delicate balance between the drift
and diffusion currents.
To circum vent the added complexity o f the diffusion term s,
researchers13'15 have limited their modeling to the intrinsic region only
or divided the problem up into distinct regions. In this study, the diffu­
sion terms will not be neglected. Although this adds considerable com­
putation time to the simulations, a complete treatment of diffusion is nec­
essary since an accurate description of the electric field is important for
high power effects as will be demonstrated in Chapter VII.
Equation 3.20 therefore needs to be modified for the effects of veloc­
ity saturation.
The expression for the diffusion constant for non­
degenerate semiconductors is given by the Einstein relationship:
kT
D =— p .
(3.21)
q
The dependence of the diffusion constant with electric field is linked to
the mobility since kT/q is constant at a given carrier temperature. A n
expression for the mobility as a function of electric field for electrons in
m aterials with intravalley scattering (GaAs, InGaAs, InP, and etc.) is
given by Boer17 (eqn 33.23). Although the exact characteristic of electrons
in InGaAs is not known, the dependence from equation (33.23)17 suggests
that at high fields, the mobility has a 1/E dependence. Experimental veri­
fication of the diffusion constant versus electric field is difficult to m ea­
sure since the diffusion current is masked by the drift current at high
fields. However, theoretical calculations18'21 in InP and GaAs result in
diffusion constants with a 1/E^ dependence where £, is between 0.5 to 1,
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
27
depending on carrier temperature and model assumptions. Additionally
these calculations predict a diffusion constant peak near an electric field
where the drift velocity peaks. From the results in GaAs and InP, an
estimation of the diffusion constant will be made for electrons in InGaAs
and will be expressed as:
kT
— n(E = 0)
Dn = '
q_________
V4
/ _ A2
r_
4 E
E
>
1 -2
H--d
3 VEd
VJ
^E P
(3.22)
where Ep is the electric field where the diffusion constant peaks and
p(E=0)is given by equation 3.19. The polynomial and its coefficients in
the denominator of equation 3.22 were chosen to portray the behavior of
the diffusion constant18-21 in GaAs and InP in three respects: 1) the slope
of the diffusion constant at E = 0 is zero, 2) the peak in the diffusion con­
stant is approximately 130% of its value at E = 0 at a field near the drift
velocity peak, and 3) the characteristic roll off at high electric fields is \ =
0.75. Equation 3.22 is plotted in figure 3.5 with Ep = 4 kV/cm.
The diffusion constant for holes has not received much attention in
the literature. Since the hole does not experience a velocity peak at a par­
ticular electric field (scattering into a higher-effective-mass valley), the
hole diffusion constant can be approximated by the following expression:
D p ='
kT v p(E)
q
E
(3.23)
where vp(E) is given by equation 3.17. Equation 3.23 is also plotted in fig­
ure 3.Z. Note that the diffusion constant for electrons is 20 to 50 times
greater than that of the hole for E < 100 kV/cm. Equations 3.22 and 3.23
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
28
«£o
'Z
P
co
to
P
o
0
350
300
-- Holes
Electrons
250
5.0
200
150
&
Q 100
p
2
50
+2
O
1
0 0
1
co
4.0 ©
P
O
3.0 oP
co
r+2.0 S»
P
P
*1
7.0 ffl
0
6.0 a
a
1.0 O
20
40
80
60
E lectric F ield (kV /cm )
0.0
100
CO
Figure 3.5 Hole and electron diffusion constants versus electric
field according to equations 3.22 and 3.23.
are only estimates for the diffusion constants and, as will be discussed
shortly, further reductions in the diffusion current will be required.
Even with the diffusion constant limitations implied by equations
3.22 and 3.23, the diffusion current and effective diffusion carrier "veloc­
ity" can quickly exceed the thermal velocity or the saturated carrier veloc­
ities. Boer has analyzed diffusion5"17-22-23 and applied his results to sem i­
conductors.
His work leads to the hypothesis that, since diffusion is
derived from the difference between two random walk currents22 (a for­
ward and reverse current), as the carrier density gradient becomes
increasingly high, the contribution of the reverse current is negligible
compared to the forward current.
The maximum diffusion current
therefore, is the maximum forward current which is expressed as the
Richardson-Dushman current.23
For carriers following Boltzmann
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
29
statistics the maximum current through a planar surface17 is given by:
qn
"ndiff.max —
v rms >
( 3 .2 4 )
where vrms is the thermal velocity for electrons. A corresponding equa­
tion exists for holes. Boer goes on to suggest22 that this current is further
limited for high built-in electric fields (high enough to saturate the drift
velocity) to a current equal to the saturated drift current near the deple­
tion region edges where drift and diffusion currents must be equal. Boer
therefore suggests that the maximum diffusion current in high carrier
density gradients and high built-in fields should be limited by the m axi­
mum drift currents, with the maximum diffusion currents given by:22
where
v ndiff_sa t
and
^ndiff.max — 9 n v ndiff-sat ,
( 3 .2 5 )
c^pdifr,max — QPv pdiff-sat >
( 3 .2 6 )
v pdiff.sat
are the saturated diffusion "velocities" and in
some sense are equal or nearly equal22 to the saturated drift velocities.
The Nonlinear Transport Equations
Gauss's law (eqn. 3.2) combined with the continuity equations (3.3
and 3.4) w ith the appropriate substitutions for the currents J n and Jp,
(eqns 3.5 and 3.6), and the expressions for the drift currents, (eqns 3.14
and 3.15), yield the three simplified transport equations:
V-E = —(p-n + N d - N a) ,
e
^£ = 0
3t
p n ~ n(p + n + 2ni)xp
v ^ P -p ^ P -i^ L
p 3x
3x q 3x
(3.27)
,
(3.28)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30
<*L= G
3t
Pn ~ n'
+v ^ + n^ L +l^ i! L
(p + n + 2nj)xn
" 3x
3x q 3x
,
(3.29)
where the diffusion current has not been simplified but may be reduced
with the help of equations 3.25 and 3.26. Equations 3.27 to 3.29 are linear
only if the following three conditions are satisfied:
1) The carrier velocities, vn and v , are independent of the carrier densi­
ties, n and p.
2) The diffusion current terms in equations 3.28 and 3.29 are linear with
carrier density.
3) The recombination terms are simplified or neglected.
Since the carrier velocities are related to the electric field via equa­
tions 3.16 and 3.17 and are further related to the carrier densities via
equation 3.27, the first condition is satisfied only when the electric field is
not effected in the PD due to the generated carrier densities; however,
this does not restrict the velocities from being functions of position or
time. This approximation is the low-level-injection condition.
To define the low-level-injection condition, let the total hole and
electron concentrations be defined as follows:
P = P0 + P'
>
(3.30)
n = n0 + n'
,
(3.31)
where p0 and n0 are the dark-condition carrierconcentrations and p' and
n' are the excess carrier concentrations due to the incident light.
Gauss's law (eqn. 3.27) can then be expressed as:
V 'E =
^ ( p o + p ’ -Ho-H + N d - N j .
(3.32)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
31
The electric field can then be expressed as the sum of a dark electric field
( E d a r k
) and a space charge electric field (Esc) given by:
(3.33)
E = Edark + Esc
which allows Gauss's law to be divided into two equations; one equation
describing Edark and the other describing Esc, given by:
^ ' ®dark — ^ (Po "
~
)
>
(3.34)
(3.35)
Low-level-injection is defined when p’ «
p0J n' «
n 0, and p' = n'.
Operating in the low-level-injection condition therefore allows Esc = 0, or
equivalently, it assum es that the electric field is unchanged from its
value under otherwise dark conditions. To a first approximation, this
breaks the connection between the electric field and the carrier velocities.
The second condition is satisfied only if the diffusion constants are
independent of the carrier densities. The carrier velocities and diffusion
constants may be functions of the generated carrier densities by the fol­
lowing mechanisms:
1)
Space-charge fields, which violate the low-level-injection conditions,
result in carrier-density-dependent electric fields. This can result in
changes in the electron velocity via equation 3.16, changes in the
hole velocity via equation 3.17, and changes in the diffusion con­
stants via equations 3.22 and 3.23.
2)
The flow of current in the p-region, being directly proportional to the
electric field, modifies the carrier velocities of the generated carriers
in this region.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
32
3)
Lower carrier mobilities via equations 3.18 and 3.19 from scattering
at high carrier densities can modify the carrier velocities and the
diffusion constants with increasing carrier densities.
4)
Photodiode potential drops from current flow in the external load
resistance lowers the internal electric field and results in carrierdensity-dependent carrier velocities.
5)
Saturation of trap sites.
6)
Carrier flow near or across the heterojunctions.
7)
The generated carriers not reaching their steady-state velocities
instantly due to their finite acceleration and scattering times.
The relative contribution of the first four of these conditions to PD nonlin­
ear behavior will be the focus Chapters V through VIII.
The solution to the transport equations (3.27 to 3.29) thus requires
the substitutions of the appropriate equations given throughout this chap­
ter and results in a system of three coupled nonlinear differential equa­
tions for the currents in a p-i-n PD. In principle, using these equations
with the boundary conditions (3.8 to 3.11), the resulting system of equa­
tions can be solved. Analytical solutions for equations 3.27 to 3.29 are
extrem ely
d ifficu lt
and
have
led
to
nu m erou s
num erical
solutions.9’11-12-14-15-24>25 Chapter IV will discuss the approach taken here
to solve this system of equations as well as presenting the resultant physi­
cal approach for the diffusion current limitations, which was proposed in
equations 3.25 and 3.26.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
33
1.
S.M. Sze, "Physics of Semiconductor Devices," 2nd Edition, John
Wiley and Sons, pp. 35-51, 1981.
2.
T.P. Pearsall, Editor, "GalnAsP Alloy Semiconductors", John Wiley
and Sons, p.456, 1982.
3.
Y.G. Wey, etaL , "Ultrafast Graded Double-Heterostructure G aln A s/
InP Photodiode," Appl. Physics Lett., 58, p. 2156,1991.
4.
P. Hill, et al., "Measurement of Hole Velocity in n-iype InGaAs,"
Appl. Physics Lett., 50, p. 1260,1987.
5.
K.W. Boer, "Survey of Semiconductor Physics," Van Nostrand
Reinhold, New York, Volume II, p. , 1992.
6.
K.W. Boer, "Survey of Semiconductor Physics," Van Nostrand
Reinhold, New York, Volume II, p. 316, 1992.
7.
J.E. Bowers and C.A. Burrus,"Ultrawide-Band Long-Wavelength pi-n Photodetectors," J. of Lightwave Tech., JLT5, p. 1339,1987.
8.
T.H. Windhom, et al., "The Electron Velocity-Field Characteristic for
n-InGaAs at 300K," IEEE Electron Device Lett., EDL4J, p. 18, 1982.
9.
M. Dentan and B. de Cremoux, "Numerical Sim ulation of the
N on lin ear R esponse
of
a
p-i-n
Photodiode U nder
High
Illumination," J. o f Lightwave Tech., JLT8, p. 1137,1990.
10. T.P. Pearsall, et al., "Electron and Hole Mobilities in GalnAs,"
Gallium Arsenide and Related Compounds 1980, p. 639,1981.
11. H. Yi, et al., "Novel Method to Control N um erical Solution
Oscillation of Diffusion-Drift Equation," Electron. Lett., 26, p. 1487,
1990.
12. R.E. Bank, et al., "Numerical Methods for Semiconductor Device
Simulation," IEEE Trans, on Electron Devices, ED-30, p. 1031,1983.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
34
13. G. Lucovsky, et al., "Transit-Time Considerations in p-i-n Diodes," J.
o f Appl. Physics, 35, p. 622, 1964.
14. R. Sabella and S. Merli, "Analysis of InGaAs p-i-n Photodiode
Frequency Response," IEEE J. o f Quantum Elec., JQE-29, p. 906,
1993.
15. J.M. Zhang and D.R. Conn, "State-Space Modeling of the PIN
Photodetector," J. of Lightwave Tech., JLT10, p. 603,1992.
17. K.W. Boer, "Survey of Semiconductor Physics," Van Nostrand
Reinhold, New York, Volume I, 1990.
18. P.E. Bauhahn, et al., "Comparison of the Hot Electron Diffusion
Rates for GaAs and InP,” Electron. Lett., 9, p. 460,1973.
19. C. Hammar and B. Vinter, "Diffusion of Hot Electrons in n-Indium
Phosphide," Electron. Lett., 9, p. 9, 1973.
20. W. Fawcett and G. Hill, "Temperature Dependence of the VelocityField Characteristic of Electrons in InP," Electron. Lett., 11, p. 80,
1975.
21. P S . Cheung and C.J. Hearn, "The Diffusion of Electrons in
Semiconductors in High Electric Fields," J. o f Physics C: Solid State
Physics, 5, p. 1563, 1972.
22. K.W. Boer, "High-Field Carrier Transport in Inhomogeneous
Semiconductors," Ann. der Physik, p. 371, 1985.
23. S. Dushman, Rev. Modern Physics, 2, p. 381,1930.
24. O. Heinreichsberger, et al., "Fast Iterative Solution of Carrier
Continuity Equations for Three-Dimensional Device Simulation,"
SIAM J. of Sci. Stat. Comput., 13, p. 289,1992.
25. A. Yoshii, et al., "Investigation of Num erical Algorithms in
Semiconductor Device Simulation," Solid State Elec., 30, p. 813,1987.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
35
IV. NUMERICAL TECHNIQUES
Introduction
Solutions to the transport equations outlined in Chapter III have
been limited to numerical treatments with the exception of some simple
one-dimensional (1-D) problems. A sample of the overwhelming number
of papers on numerical techniques for semiconductor device sim ulation
are given in references [l]-[9]. The appropriate choice of numerical tech­
nique for the solution of the transport equations is not straight forward
and each method has its advantages and disadvantages.
Several numerical methods were considered including the finite
difference, finite element, and iterative methods. Iterative and finite ele­
m ent methods are usually considered two-dimensional (2-D) methods
and since the extension of this work to 2-D was not pursued, these m eth­
ods were not used.
Initial solutions with the finite difference method
using a two-point difference resulted in poor accuracy and num erical
instabilities. After some time modeling only the intrinsic region, an
open-loop time-step solution was chosen over the finite difference method.
Although this method may not be the most efficient (computationally), it
promised to give accurate and satisfactory results. A block diagram of
the algorithm is given in figure 4.1. The PD (figure 3.1) is divided into m
slices or bins, not necessarily of the same width. A piecewise-constant
approximation for the carrier densities is utilized in each slice as shown
in figure 4.2.
The algorithm begins by solving the steady-state transport equa­
tions with a given constant (possibly zero) generation function. The m ain
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36
Steady S tate Distributions
Tim e t = t + At
Generation and Recombination
Poisson's Equation & Electric Field
Field D ependant D rift Velocities and Diffusion C onstants
Carrier Transport
No
Output
Yes
O utput Current
No
Stop
Yes
End
Figure 4.1 Algorithm Flow Chart
loop (figure 4.1) begins by ad ju stin g th e carrier d en sities v ia gen eration
and recom bination during the tim e At and, w ith th e n ew carrier d en si­
ties, th en recalculates the electric field at the bin interfaces via a solution
o f P o isso n ’s equation. F ollow in g th e evaluation of a n ew electric field,
new valu es for the carrier velocities and diffusion con stan ts are ca lcu ­
lated a t the bin interfaces. The carriers are th en allow ed to m ove d u rin g
At from th eir respective bins according to their new velocities and diffu­
sion constants. If no output (current) is desired, the loop repeats its e lf for
a predeterm ined tim e. The algorithm is open-loop10 because th e solution
o f th e transport equations at the tim e t requires inform ation about the
carrier d en sities at the tim e t - At only once and does not iteratively solve
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37
•n-j
h
Diode Partition
Hole Concentration
nnr
H n rm n
E lectron Concentration
Figure 4.2 Diode Partitions and Carrier Approxim ations
for variables at tim e t. Specifics outlinin g each of the algorith m step s w ill
be presented in the n ext few sections.
S olving Poisson's E quation
The solutions to Poisson's equation (eqn. 3.1) or, equivalen tly to the
differential form o f G au ss's law (eqn. 3.2) u sin g a p iecew ise-co n sta n t
approxim ation for th e carrier d en sities is a sim ple calcu lation for a given
boundary condition (eqn. 3.9) as follow s.
R eferring to fig u re 4 .3 and
a ssu m in g a forw ard-difference approxim ation for th e derivative in equa­
tion 3.2, the electric field at the interface i+1,
E j+1,
is related to
Ej
by:
Ei+i - dE + Ej - -^-(pi -n ; +(N d -N g ^ d x j + E; _
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.1)
38
Interface (i)
_
Bin (1 )
Interface (i+1)
(N a -N d )i
P;
n;
Bin Width = dx.
E i+ l
E ?
Figure 4.3 Bin (i)
Beginning with E0 = k, the electric field inside the diode can be deter­
mined to within a constant, k, determined by the charges on the ohmic
contacts. After the calculation of E via equation 4.1, the electric field is
integrated by the trapezoidal method. The resulting potential, V E, is sub­
tracted from the applied and built-in voltages:
Verror = VE-(V a +Vbi) .
(4.2)
The resulting error potential, V error, is due to charge on the ohmic con­
tacts, since the field inside the contacts is uniquely determined (to within
a constant) by equation 4.1, and the potential between the contacts m ust
be equal to the applied and built-in voltages. Therefore an image charge
is placed on the contacts, which results in an additional constant electric
field component through the diode and is given by:
E
add
_
V
v error
W
(4.3)
where W is the separation between the ohmic contacts. The additional
electric field is added to the field at each interface, which is equivalent to
beginning the solution of equation 4.1 with E0 = k + Eadd. The integration
of E is repeated with the calculation of a new error potential, until the
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39
error falls below approximately 1 p V. A single iteration will not neces­
sarily give the correct answer for E since terms in the integration of E
can change signs, resulting in one or two additional iterations.
The Continuity Equations and Diffusion Saturation
The solutions to the continuity equations traditionally have m eant
the expansion of equations 3.3 and 3.4 directly into finite differences or
other numerical methods. Here the approach is to reanalyze the continu­
ity equations and to carry out their solutions by the way in which they are
created.
The continuity equations are a mathematical description of the
conservation of carriers within a given volume. Consider the bins (i-1),
(i), and (i+1) in figure 4.4. The change in carrier density within bin(i) is
equal to the number of carriers entering the bin minus the number of
carriers leaving the bin. The physical mechanisms for carriers entering
bin(i) are generation within bin(i), carrier drift from bin(i-l) and bin(i+l)
into bin(i), and carrier diffusion from bin(i-l) and bin(i+l) into bin(i). The
physical m echanism s for carriers leaving bin(i) are recombination
Interface (i)
Bin (i-1)
earner
flow
n i-l
Interface (i+1)
Bin (i)
n;
n i+1
Bin Width = dx ^
Bin Width = d x ;
Bin Width = dxi+1
Bin Area = area*.!
Bin Area = area;
Bin Area = area i+1
Figure 4.4 Bins (i-1), (i), and (i+1)
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40
w ith in bin(i), carrier drift from bin(i) into b in (i-l) and b in (i+ l), and car­
rier diffusion from bin(i) into b in (i-l) and b in (i+ l).
Therefore, d u rin g
every tim e step At, th e carrier d e n sitie s in each b in are reca lcu la ted
based on generation, recom bination, and carrier m ovem ent.
The carrier flow due to th e drift current a t interface (i) is evaluated
w ith th e follow ing sim ple exp ression s. Consider the case w hen th e drift
velocity at interface (i) is such th a t electrons w ill tend to flow out o f bin(i)
and into b in (i+ l), the num ber of carriers th a t flow during tim e At into
b in (i+ l) is just:
An = njVj+1areaiAt .
(4.4)
The drift term s in the continuity equations are thus treated by calculating
th e num ber of carriers th at m ove through interface (i), u sin g equation
4.4, and by adding and subtracting those carriers (divided by th e volum e)
from their respective bins. In the lim it of th is sim ple physical m odel, the
d istan ce that th e carriers travel in a sin g le tim e step m ay not be any
larger th an the m axim um of the velocity tim es the sm aller of the lea v in g
or entering bin w idths, in order to keep the carriers from m oving across
two bin boundaries. This resu lts in an upper lim it for the tim e step, At,
th a t th e algorithm m ay use.
Carrier diffusion is handled in a sim ilar w ay w ith a sim ple p h y si­
cal approach of lim itin g th e diffusion "velocity" (eqns 3.25 an d 3.26)
w hich is consisten t w ith the an alysis o f Boer11,12 and D u sh m a n .13 T his is
in contrast to the nu m erical tech n iq u es14,15 used to control the diffusion
current and prevent in stab ilities in drift-diflusion num erical solutions.
Follow ing the an alysis o f B oer,12 the diffusion current is derived by
th e difference in carrier flow from one position in space to an oth er as
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41
show n in figure 4.5. T h ese currents can be ex p ressed 12 a s forw ard and
reverse currents due to th eir Brow nian m otion as:
^ndifT
2 s xn
dn^ v rm
Q n„— —
2 J 3 dx
(4.5)
d n ^V“
- rm
2 s x*n
T
3 dx
(4.6)
Jndin- = q
+
w here xn is th e m ean tim e betw een collisions. The n e t diffusion cu rren t
is th e difference betw een equations 4.5 and 4.6 and is given by:
ndifT =
q
V rm ^ dn
3
(4.7)
dx
The diffusion constant is therefore defined as:
TQ
v rm
2 s x*n
(4.8)
The m axim u m diffusion current (eqn. 3.24) is derived from equations 4.5
and 4.6 a ssu m in g th at for high carrier density gradients th e reverse c u r ­
ren t (eqn. 4.6) v a n ish es. If th e distan ce dx is se t equal to th e m ea n free
path, dx = vrmsxn, equation 3.24 is obtained.
A direct analogy betw een th e discrete nature of a n u m erica l solun
n0
X
Figure 4.5 Illustration for the derivation of the diffusion current.
From Ref. [11]
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42
tion and the derivation for diffusion can be presented with a simple
model. Given the carrier densities between bins (i) and (i+1) in figure 4.6,
Interface (i)
Interface (i+1)
Bin (i)
n; =10
carrier
flow
16
Interface (i+2)
Bin (i+1)
~15
— -►
n i+l =10
Bin Width = dx j
Bin Width = dxj+1
Bin Area = area j
Bin Area = areai+1
Ei
E ; +1
E j+ 2
Figure 4.6 Electron diffusion between bins (i) and (i+1).
the diffusion current is expressed with equations 4.7 and 4.8 as:
(4.9)
Jn«T =!lD n| i
The derivative of the carrier density in finite arithmetic can be expressed
by the forward difference formula as:
dn _ (ni+1 - n ,)
dx ( dxi+1 + dxj ^ ,
I
2
where the denominator is the average of the bin widths.
(4.10)
Substituting
equation 4.10 into 4.9, the diffusion current can be rewritten as:
dndiff
9
2D r
2D r
n;
dxi+1 + dx; n i+i- q dxi+1 + dxi
(4.11)
By inspection of equation 4.11 and from the definition of diffusion current
(eqns. 4.5 and 4.6), the first term is the reverse diffusion current (eqn. 4.6)
and the second term is the forward diffusion current (eqn. 4.5). The
expressions in brackets are interpreted as the effective diffusion "veloci-
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43
ties" for discrete problems. For arbitrary carrier density gradients, the
term in the brackets must be limited by the maximum diffusion veloci­
ties, vndiff_max and vpdiff max , defined in equations 3.25 and 3.26. For dis­
crete problems the dXj's are fixed during any particular solution; there­
fore, a lim it is imposed on the diffusion constants for electrons and holes
given by:
_ (dxj+i + dxj)
•‘-'n-max
^
(4 12)
v n d iff-sa t »
(d*M + d * i)
■^p-max
2
(i±.X4)
(4 1 3 )
v pdifF-sat
If one were to mistakenly ignore the limits of equations 4.12 and
4.13 the calculation of the diffusion current between two adjacent bins
can be significantly overestimated. For example, let a 2 micron p-i-n PD
be equally divided into 200 bins, resulting in a bin width of 1.0 x 10'6 cm.
At a position in the PD where the electric field is 10 kV/cm, the electron
drift velocity is (eqn. 3.16) approximately 1.3 x 107 cm/s. The electron dif­
fusion constant at 10 kV/cm is estimated (eqn. 3.22) to be 150 c m2/Vs. The
maximum diffusion constant allowed (eqn. 4.12) is 13cm 2/Vs, assum ing
the maximum diffusion velocity is equal to the drift velocity in the bin.
Therefore if one were to ignore the limitations on the diffusion constant
imposed by equation 4.12, the diffusion current between bins would be
overestimated by a factor of 150/13 or greater than 11 times. Bin widths of
10'7 cm are desired to improve numerical accuracy and so could result in
an overestimation in the diffusion current by over 100 times.
For the purposes of this study, the maximum diffusion velocities
(eqns. 4.12 and 4.13) will be set equal to the saturated drift velocities.
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44
Carrier diffusion will be carried out through the calculation of diffusion
current (eqn. 4.11) with a corresponding expression for holes. An expres­
sion similar to one used for calculating the drift carrier movement (eqn.
4.4) is used to calculate the diffusion carrier movement during At. This
is believed to be the first time that the diffusion constants and hence the
diffusion currents have been limited by physical considerations when
finding the numerical solution of drift-diffusion problems.
Calculations of Output Current
The current, being the externally measured quantity, is calculated
from equation (3.7) as:
(4.14)
where x :=0 and x 2=W are points for the device in figure 3.1. The third
term in the integral is the displacement current. If the load resistance is
neglected for the device in figure 3.1, the displacement term is zero due to
the constant-voltage conditions between the ohmic contacts. For a real
device this is not the case since a 50 Q resistive load exists for the frequen­
cies of interest here. However, to distinguish between internal and exter­
nal PD nonlinear m echanism s, the load resistance will normally be
excluded. Any additional nonlinearities caused by the load resistance
will be discussed in Chapters V and VIII.
To achieve greater than 140 dB of dynamic range for computing PD
harmonics, the current needs to be calculated with seven significant dig­
its. The representation of a real number as an IEEE 32-bit (single preci­
sion) number is unsuitable since the number only has approximately
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45
seven significant digits and each numerical operation may reduce the
precision. This level of accuracy therefore requires 64-bit computation.
A Linear Approximation for the Gaussian
The solution to the transport equations (eans 3.2 to 3.4), in onespatial-dimension requires a 1-D approximation for the Gaussian distri­
bution of the generated carriers. If the problem was linear, the radius of
the corresponding approximating 1-D function would not affect the calcu­
lated current, so long as the total incident power remains the same. The
situation is more complicated for nonlinear problems. By approximating
the Gaussian radial intensity (I) with a constant value out to some radius
r0, the approximating intensity near the center of the Gaussian is lower
than the actual intensity while the approximating intensity at r0 is
greater than the actual intensity. If the approximating radius r0 is too
small, the device NL will be overestimated and vice versa.
Often the effective radius of the Gaussian is chosen16 to be the e"2
radius as shown in figure 4.7. As can see from the figure, about the
same amount of the Gaussian intensity is above the linear function as
below. The choice of the e'2 approximating radius is derived from the
effective area equation which is the square of the integral of the intensity
divided by the integral of the intensity squared, both integrated from r = 0
to r = infinity. For effects which are linearly related to the intensity, this
is a good approximation, however the NL here are proportional to higher
powers of the intensity, therefore requiring a smaller effective spot size.
For example, Raman processes, where the Raman gain is propor­
tional to the electric field cubed (I3/2), use the e'1 intensity spot size as the
effective spot size.17 This is shown in figure 4.8 where I3/2 is plotted with
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46
1.20
1.00
Gaussian Intensity
1-D Approximation
g
c
0.80
0.60
£
CO
C3 0.40
C
0.135 = e‘
0.20
0.00
0.0
0.5
1.0
1.5
2.0
R ad ia l P o sitio n r (L inear)
Figure 4.7 Intensity versus normalized radius for a Gaussian and the
approximating 1-D function. Both functions yield the same total power.
1.20
1.00
a
o
0.80
,3/2
0.60
CO
.£
CO
C 0.40
r = 0.707
03
a
0.20
0.0C
0.0
0.5
1.0
1.5
R ad ia l P o sitio n r (L in ear)
2.0
q/O
Figure 4.8 Intensity versus radial position for a Gaussian and the
2
1
e and e approximating functions.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
47
th e e_1 and e'2 in ten sity approxim ations. If the e*2 approxim ation is u sed
(figure 4.8) an u n d erestim a tio n o f th e R am an g a in r esu lts sin ce the
G au ssian function is high er th an the approxim ating function over m ost
o f the e'2 radius. Since the n o n lin ea r effects under investigation have not
b een stu d ied extensively, th e effective spot size w ill be an adjustable
p aram eter in the sim u lation s so long as reasonable resu lts are obtained
w ith values in the range 0.5 to 0.8 tim es the actual e*2 in ten sity spot size.
C arrier Spreading A pproxim ation for th e p-InGaAs C ontact
For diffraction lim ited focusing of the optical beam , th e generation
o f carriers w ith in a PD m ay take place w ithin dim ensions as sm all as the
w avelen gth .
For generation in th e in trin sic region, an approxim ation
w as m ade in the previous section for the radius o f the generated carrier
distribution based on the in cid en t spot size. However, for generation in
th e p-region, the app roxim ating radius m u st be m odified to account for
th e large num ber of majority carriers (holes) available for transport.
T he num ber of holes generated in the p-region is u su a lly sm a ll
(1016 to 1017 cm '3) com pared to the num ber of free holes available for
transport (1019 cm '3). Therefore, the p-region hole distribution is e ss e n ­
tia lly un ch anged w ith the presence o f light. The current in th e (unde­
pleted) p-region is proportional to the electric field and th e area corre­
sp on d in g to th e app roxim atin g fu nction.
T herefore, i f th e in trin sic
region approxim ating radius is used to calcu late the p-region current
in s te a d o f th e p h y sic a l d ia m eter , sig n ific a n t e lec tr ic fie ld over­
estim ation s result. For exam ple, let a 5-pm e'2 in ten sity optical beam be
in cid en t on a PD w ith a 1.0-|im long p-region havin g a 35 pm p h ysical
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
48
diameter. If an approximating radius of 3 pm (60%) is utilized through­
out the 1.0 pm cap layer instead of the 35 pm physical diameter, the elec­
tric field and hence the series resistance in this region would be overesti­
mated by 135 times (the ratio of the two areas). The over-estimated elec­
tric field will thus cause the p-region generated electrons (minority carri­
ers) to have a velocity 135 tim es greater than their actual velocity.
Therefore, to account for the large number of available holes in the pregion, an expansion function will be used to model the hole distribution
just inside the p-i interface.
One approximating function is plotted in figure 4.9. The approxi­
mating diameter (see pg. 45) for the incident Gaussian beam is utilized
for the beam diameter in the intrinsic region up to the edge of the deple­
tion region. From the edge of the depletion region to the p-contact, where
40.0
^
1 J 1
!
i
i
i
i
■ ■ ■ I
35-°
j l 30.0
o) 25.0
<x>
B 20.0 “
.2
Q 15.0 :
:
a
cs 10.0
a>
In trmsic
-
Region
j
> 1.0 pm
:
n .T r
Cap Layer
p-i interface
:
0.0 to 1.0 p. m
I
i , , , i__i__i__t_I
0.0 I_i__ i__ i__1__i__ i__i__
0.94
0.96
0.98
1.00
1.02
1.04
D iod e X P o sitio n (jpm)
Figure 4.9 Beam diameter versus position for estimating the hole
spreading in the p-region.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
49
the injected hole density into the p-region is negligible compared to the
majority carrier density, the approximating beam diameter is the physi­
cal diameter of the p-region.
This function best describes the actual
transport into the contact since the junction is abrupt (an assumption)
resulting in a sharp transition between the depleted i-region and the
undepleted p-region. However, the use of this beam diameter function
leads to a large discontinuity in the calculated current between these two
regions, even when bin widths of a few Angstroms are used.
When the beam diameter for holes was not allowed to change by
more than 20% from bin to bin to minimize discontinuities, the error
incurred in the calculated current (dc conditions) near the p-i interface
was observed to fall below 2% in any particular bin. Figure 4.10 shows an
example of this type of approximating function for a 35 pm diameter PD
consisting of a 1.0 pm p-type cap layer. The holes traveling from the
40.0
35.0
Intrinsic
30.0
o> 25.0
a»
B 20.0
CB
15.0
Region
> 1.0 pm
p-InGaAs
Cap Layer
p-i interface
0.0 to 1.0 pm
CD
10.0
5.0
0.0
0.94
0.96
0.98
1.00
D iod e X P o sitio n (pm )
1.02
1.04
Figure 4.10 Beam diameter versus position for estimating the hole
spreading in the p-region.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
50
intrinsic region to the p-region are allowed to expand from their gener­
ated diameter, 4.2 pm for this example, to the 35 pm physical diameter.
The electric field remains high ju st inside the p-i interface due to the
depletion depth into the p-region, so little spreading is allowed from 1.0 to
0.995 pm. After 0.995 pm, the electric field quickly decreases to below 10
V/cm, therefore the beam width is allowed to expand in this region. The
expansion takes place in about 5 nm, derived from an estimate of the
depletion depth (1-2 nm) plus a few nm to allow for hole diffusion into the
intrinsic region.
The electron beam diameter will be adjusted to have the same beam
diameter as the holes; however, the electron travels in the opposite direc­
tion and will be focusing into the intrinsic region. This is certainly not
valid for any real device since a radial electric field does not exist to focus
the carriers into the intrinsic region. However, the injected electron den­
sities will be very low, since most electrons recombine before reaching the
intrinsic region.
Furthermore, for electrons that reach the intrinsic
region, the increase in density caused by the artificial focusing will be
sm all compared to the generated electron densities in the intrinsic
region.
The electrons flowing out of the p-type cap layer cannot be
neglected since they may contribute to the PD nonlinearities because they
move in a region where the carrier velocities can change with increasing
carrier densities.
Num erical Tests
To test the validity of the numerical techniques utilized here, a few
test simulations will be carried out in this section which can be verified
by easily obtained analytical results. The following results are obtained
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
51
with the algorithm, written in FORTRAN, from a PD that will be mod­
eled and studied extensively in the next two chapters, a mesa type p-i-n
PD. The device consists of a 1.0-pm long p-GalnAs contact layer (Na 7xl018 cm-3), a 0.95-(im long n-GalnAs intrinsic layer (Nd ~ 5xl015 cm-3),
and an n-InP buffer layer (Nd~ 2xl017 cm-3). Additional properties of this
device can be found in Chapter VI.
For the first test, the PD will be reversed biased (all efficient p-i-n
PDs are reversed biased) with -5 V and operated under dark (Gen = 0 and
with no thermal generation) conditions. The total current should be very
small through the device, governed only by the injected minority carrier
density at the contacts which is quite small due to the high doping densi­
ties. Figure 4.11 shows the simulated PD carrier densities and electric
field under the conditions where the diffusion from each ohmic contact of
the device is observed along with a sudden change in the carrier densities
near the edges of the intrinsic region due to the sudden increase in car-
Electron Density
Hole Density
Electric Field
o
n-InP
£
w
C
<u
Q
p-InGaAs
-50
(''I
CD
u
ca
O
r
Applied Voltage = -5 V
Intrinsic InGaAs
0.00
0.25
0.50
0.75
1.00
o
*1
1.25
1.50
1.75
-100 o
3
-150
2.00
D iode X P o sitio n (pm )
Figure 4.11 Carrier densities and electric field under dark conditions.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
52
rier velocity from the high electric field.
The figure also shows the diffusion of carriers from their nom inal
values in the highly-doped layers into the depletion region as nearly
straight lines (on a log scale) representing an exponential behavior of the
carrier density near the edge of the depletion region. The figure also
shows that the electric field rapidly increases near the edges o f the intrin­
sic region and is approximately equal to zero in the contact layers. The
slope of the electric field in the intrinsic region agrees with Gauss's Law
where the change in the electric field is proportional to the ionized
dopants in the intrinsic region having a density of 5 x 1015 c m*3. The total
current out of the modeled PD in figure 4.11 was less than 10'10 amps
which demonstrates the degree of equality of the drift and diffusion cur­
rent components near the edges of the depletion region.
In a second test, the ideal transit time response is used to check the
algorithm output. The output can be compared to the analytical solution
of an ideal PD depicted in figure 4.12, where a 0.95-pm long intrinsic
region lies between metal plates and constant illumination is assum ed
instantaneously over the intrinsic region. The carriers drift across the
intrinsic region with their saturated velocities and into the plates, where
they instantly recombine. Therefore, the drift currents at any given time
are calculated from equation 4.14 and are given by:
vn
I l bsa
d l t(tn
V II
^
J =
—
v n sa t4 i
where
and
v p sa t(tp ~k )
t)f
J ^ c V n s a td x + v
q
1 .
v psatt p
j QPcVpsatdx ’
(4<15)
o
are the electron and hole transit times respectively and n c
and pc are the electron and hole instantaneous carrier densities. This
reduces to the triangular responses shown in the first column of figure
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
53
Constant Illumination
Exponential Illumination
n,p
n,p
n(0) = p(0)
at
t = 0
x
n
n
_N o Back
Diffusion
No Back
Diffusion
n(t0)
at
t = to
x
P
P
P(t0)
at
t = tn
No Back
Diffusion
No Back
Diffusion
x
J
total
J
Currents
Versus
Time
Figure 4.12 Carrier densities and transit time currents for constant and
exponential illumination. In all cases, no carrier diffusion is assumed.
4.12 and expressed as:
_ q n cv n s a t ( t n ~ t ) , clP c v p s a t ( t p
J=
t)
(4.16)
where the first term is the electron contribution and the second term is
the hole contribution to the total current.
Since real devices absorb light and generate carriers with an expo-
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
54
nential dependence, the triangular transit-time responses are replaced
by exponential responses as shown in the second column of figure 4.12.
Note that for both illumination conditions, the ratio of electron to hole
current at t = 0 is given by the ratio of saturated drift velocities and is
equal to 1.15. However, the electron current decreases more rapidly than
the hole current just after t = 0 since more electrons than holes exit the
intrinsic region sooner due to the n-side illumination conditions depicted
here. For holes, the transit-time response is obtained from equation 4.14
and is given as:
(4.17)
where pp is the peak hole density. Equation 4.17 simplifies to:
(4.18)
The electron transit-time response is sim ilarly obtained from equation
4.14 and simplifies to give:
J —^ E . (e avnsatt . g avnsattn )
n atn V
J >
(4.19)
where n p = pp is the peak electron density. Equations 4.18 and 4.19 are
plotted in the lower right hand curve of figure 4.12.
The simulated currents from the intrinsic region (that is the com­
parison of interest) of the PD under test are shown in figure 4.13 with the
calculated results from equations 4.18 and 4.19. The model PD was
pulsed from dark conditions at t = 0, with a flat pulse of light from t = 0 to
t = 10-15 s. It can clearly be seen that the hole and electron currents follow
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
55
Total Current
Simulated Device
Analytic Results
03
a>
S
•Electron Current
Hole Current
-w
s
otn
S3
o
0
5
10
15
T im e (p s)
20
25
Figure 4.13 Simulated and analytical impulse response of the test
diode. Analytical results based on equations 4.20 and 4.21.
the relationships of equation 4.18 and 4.19 except for the small tails near
the respective carrier transit times. The tails are a result of the diffusion
of carriers during transit in the opposite direction of the drift motion,
which was neglected in the analysis leading to equations 4.18 and 4.19.
The high density gradient causes a small portion of the carriers to
remain in the intrinsic region for many multiples of the transit tim e,
although the associated current is small compared to the total current.
An example of the transit time response when the diffusion velocity lim i­
tations formulated here and in Chapter III are not implemented is plot­
ted in figure 4.14. Although the overall shape of the transit time response
is accurate, the noise due to the oscillating hole densities in the p-contact
cause the current to oscillate.
To test the simulation under steady-state conditions, a constant
photocarrier generation rate equivalent to 50 pA of average PD current is
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
56
Total Current
w/ diffusion lim its
w/o diffusion limits
c
<
uu
u
3
O
0
5
10
15
T im e (p s)
20
25
Figure 4.14 Simulated impulse response of the model PD. Modeled
with and without restrictions on the carrier diffusion velocities.
used. The carrier densities and electric field are plotted in figure 4.15
and are computed until the total current reaches a steady-state (1 part in
106). Figure 4.15 shows the diffusion of electrons into the p-contact as
well as a slight dip in the electron density just before the electrons enter
the n-contact due to the increase in electron velocity at electric fields
below 20 kV/cm. The hole density is higher near the p-InGaAs contact
since holes flow to the left in the figure. The effect of the heterojunction is
clearly seen at X = 1.95 pm for holes since here it was assumed that the
hole could not travel over the 0.5 eV heterojunction barrier, hence a very
low hole density for x greater than 1.95 pm.
Although the previous tests are not absolute proof that the num eri­
cal simulations are working properly, the ultimate test of the utility of the
program will be the accuracy at which it provides results which agree
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
57
25
C
Electric Field
Electron Density.
Hole Density
-3-125 &
1012
i
0.0
i i
i I i i i i L
0.5
1.0
-150
2.0
D i o d e X P o s i t i o n (p .m )
Figure 4.15 Carrier densities and electric field under steady-state
conditions. Average PD current = 50 |iA. Applied voltage = -5 V.
w ith th e experim ents of the n ext three chapters. A lth ough some error is
expected from the assu m ptions m ade here such as the carrier sp read in g
in th e p-contact, i f resu lts are in close agreem ent w ith experim ent, alter­
ations to th e p h ysical ch aracteristics o f th e m odel PD w ill be m ade to
in vestigate the origin(s) of PD nonlinearities.
To sim u late PD non lin earities at all volta g es, th e m odel diode is
excited w ith a con stan t photocarrier gen eration fu nction u n til the c u r ­
ren t reaches a steady s t a t e d part in 106) at w hich tim e a sinusoidal s ig ­
nal stim u lates th e device for a num ber of w a v elen g th s. For H anning or
H am m in g w ind ow s, only a sin g le w a v elen g th is n ecessary; how ever,
th ese w indow s have very low off-peak rejection, resu ltin g in false c a lcu la ­
tions of actual signal NL caused by leakage into the m ain lobe of the FFT
filter. To en su re m in im al leakage, a B lackm an w in d o w 19 is used. T he
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
58
passband rejection of the Blackman window surpasses many other w in­
dow functions providing 60 to 80 dB of stop-band rejection and nulls that
are wider than other window functions. Also, the rejection increases if
one operates at integer multiples of the sam pling period.
The m ain
drawback of this style of window is that the -width of the main FFT lobe
requires at least three or four full wavelengths for greater than 100 dB
rejection of the fundamental frequency at the second harmonic, w hich
requires additional simulation time.
As discussed earlier, to achieve greater than 100 dB FFT dynamic
range, not only does the current need to be calculated in 64 bit math, but a
64-bit FFT is needed as well.
A single-precision FFT routine20 was
rewritten for double-precision use. Additionally, since no external capac­
itor circuit was included in the simulated diode, instantaneous changes
in current creates aliasing from high frequency components in the sim u ­
lated output, and so a digital pre-filter is required to ensure accurate FFT
results.
The FFT, windowing, and digital filter programs provide
approximately 115, 125, and 133 dBc of second, third, and fourth har­
monic dynamic range, respectively.
To accurately model these devices, as many diode bins as possible
would be desirable; however, computation time considerations will lim it
the number of bins simulated. 384 bins were chosen since it is a m ultiple
of 128 and 64 which allows efficient full-vector-length vectorization on the
three vector computers used throughout this study, while yielding suffi­
cient accuracy as will be evident from the results in Chapter V.
The
main computer utilized in this work, a Convex C-3820, has a vector
length of 128 and offers 125 MFlops peak per-processor double precision
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
59
(IEEE 64 bit) performance.
Highly vectorized and efficient algorithms
achieve alm ost 50% efficiency on vector machines; this algorithm
obtained an estimated 40% efficiency (50 MFlops). Some computing time
was also obtained on two other vector computers both of which have vec­
tor lengths of 64. For this algorithm, a Cray YMP-EL yielded 1 0 -1 5
MFlops of 64-bit performance, and a Cray YMP-C90 yielded 100 - 150
MFlops of 64-bit performance with this algorithm, both about 40-45% of
their peak per-processor performances. Several scaler personal comput­
ers were tested for their throughput with this algorithm and it was found
that the Macintosh Quadra 950(33 MHz Motorola 68040)and an IBM
clone (50 MHz Intel 486) both achieved about 1.5 MFlops of performance.
The program requires approximately 1 hour of CPU time per 4 ns (384
bins) of simulation time on a lOMFlop throughput machine. An estim a­
tion of the computer time required to calculate the harmonic content ver­
sus average PD current for five different currents at a fundamental fre­
quency of 1 GHz, requires about six to eight full wavelengths of time out­
put to achieve greater than 100 dB FFT dynamic range.
Eight wave­
lengths multiplied by five power levels equals 40 ns of simulation time or
about 10 hours of CPU time on a 10 MFlop machine or equivalently 67
hours on a 50 MHz Intel 486 machine. The total number of floating point
operations (MFlops) used throughout this study for program debugging
and production simulations equates to approximately 360,000,000 MFlops.
This is equivalent to 99 years of computer time on a Digital Equipment
VAX 11/780 (1980) and 569 years of computer time on a IBM 8 MHz 286
(1985).
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
60
1.
M. D en ta n and B. de C rem oux, "N um erical S im u la tio n o f the
N o n lin e a r
R e sp o n se
of
a
p -i-n
P h o to d io d e U n d e r
H ig h
Illum ination," J. o f L igh tw a ve Tech., JLT8, p. 1137,1990.
2.
R. S ab ella and S. M erli, "A nalysis o f InG aA s p-i-n Photodiode
Frequency R esponse," IE E E J. o f Q u a n tu m E lec., JQE-29, p. 906,
1993.
3.
J.M . Z hang and D .R . C onn, "State-Space M odeling o f th e P IN
Photodetector," J. o f L igh tw a ve Tech., JLT10, p. 603,1992.
4.
U. A sc h e r , et a l., "Conditioning of the Steady S tate Sem iconductor
D evice Problem," SIA M J. o f A p p lie d M ath., 49, p. 165,1989.
5.
0 . H ein reich sb erger, e t a l., "Fast Itera tiv e S o lu tio n o f C arrier
C ontinuity E quations for T h ree-D im en sion al D evice Sim ulation,"
SIA M J. o f Sci. S tat. C om put., 13, p. 289,1992.
6.
H .K . G u m m el,
"A S elf-C o n sisten t Ite ra tiv e S c h e m e for One-
D im en sion al Steady State Transistor C alculations," IEE E Trans, on
E lectron D e v ic e s ,, ED-31, p. 455,1964.
7.
A. Y oshii, e t a l.,
"In vestigation o f N u m erica l A lg o rith m s in
Sem iconductor D evice Sim ulation," S o lid S ta te Elec., 30, p. 813,1987.
8.
C.S. Rafferty, et a l., "Iterative M ethods in Sem iconductor D evice
Sim ulation," IEEE Trans, on E lectron D evices, E D 32, p. 2018,1985.
9.
R.E. B ank, et a l., "N um erical M ethods for Sem iconductor D evice
Sim ulation," IEEE Trans, on Electron D evices, ED-30, p. 1031,1983.
10. L .B .
R a il,
" C om p u tation al S o lu tio n o f
N o n lin e a r
Operator
Equations," Krieger Publishing, H untington, NY, 1979.
11. K.W.
B oer, " H igh -F ield C arrier T ransport in
In h o m o g en eo u s
Sem iconductors," A nn. d e r P h ysik, p. 371,1985.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
61
12. K.W. Boer, "Survey of Semiconductor Physics," Van Nostrand
Reinhold, New York, Volume I, 1990.
13. S. Dushman, Rev. Modern Physics, 2, p. 381,1930.
14. H. Yi, et al., "Novel Method to Control Num erical Solution
Oscillation of Diffusion-Drift Equation," Electron. Lett., 26, p. 1487,
1990.
15. S.J. Polak, et al., "Semiconductor Device M odeling from the
Numerical Point of View," Intl. J. for Num erical Methods in Eng.,
24, p. 763,1987.
16. A. Yariv, "Optical Electronics," Holt, Rinehart and Winston, 1985.
17. R.H. Stolen and E.P Ippen, "Raman Gain in Glass Optical
Waveguides," Appl. Physics Lett., 22, p. 276,1973.
18. R.B. Adler, et al., "Introduction to Semiconductor Physics," John
Wiley and Sons, pp. 173-180,1964.
19. H. Baher, "Analog and Digital Signal Processing," John Wiley and
Sons, 1990.
20. W.H. Press, et al., "Numerical Recipes The Art of Scientific
Computing," Cambridge University Press, pp. 397-407, 1989.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
62
V. DETERM INATION OF DOM INANT NONLINEAR M ECHANISMS
Introduction
In th is chapter, p-i-n photodiode (PD) non lin earities (NL) are in v e s­
tig a ted to d eterm in e th e d om in an t n o n lin ea r m e c h a n ism s for various
region s o f applied PD voltage and frequency. The basic device stru ctu re
u n d er in v estig a tio n in th is ch ap ter is a sin g le-h etero stru ctu re1 m e sa typ e device w ith a 0.95-(im long in trin sic region. T his chapter w ill only
an a ly ze a su b set of the m ea su rem en t and sim u la tio n data obtained on
th is device to study th e basic n on lin ear behavior of p-i-n PD s. A dd itional
inform ation for th is device can be found in C hapter VI.
D eterm in ation of the device param eters w hich dictate PD NL is a n
im m e n se problem due to th e n u m ero u s m e c h a n ism s (C hapter III)
w h ich resu lt in non linear behavior. H owever, sign ifican t in sig h t into PD
N L can be obtained from the v a st am ount of available m ea su rem en t data
o f PD h arm on ic distortion by d issection versu s applied voltage and fre­
quency. For exam ple, figures 5.1, 5.2, 5.3, and 5.4 show the NL of th e PD
operating a t 1 m A o f average PD current w ith a m odulation depth o f 100%
a t freq u en cies o f 100 M Hz, 1 GHz, 5 GHz, and 10 GHz, respectively,
w here several trends are observed:
1) Above a given applied voltage, w h ich depends on frequency, the NL
content is n early independent of voltage.
2) T he m in im u m valu e of th e second h arm on ic (-92, -88, -87 , and -92
dBm a t 100 M Hz, 1 GHz, 5 GHz, and 10 GHz, respectively) a t h ig h applied
voltages is n early independent of frequency.
3) T he tren d s o f th e third and fourth h arm on ics at h igh er applied volt-
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
63
■i r i i i i i
- 1 6 .4
i | i i i i i i i i i f""s~r
-16.6
-40
100 MHz
-60
U
♦
A
............
Modulation
j Depth = 100% “ -17
■ill1
A
^ A
=^A^....f^...A.^.
iA
A
A
A-A-....
AA j
-17.4
A
-17.6
15
-140
0
5
(d B m )
2 -120
-17.2
Pow er
AM I
A yO L
-80
-16.8
2f
3f
4f
F u n d a m e n ta l
6
t
10
D io d e A p p lie d R e v e r s e B i a s V o lt a g e (-V )
Figure 5.1 Measured fundamental and harmonic power versus
detector applied reverse bias voltage at 100 MHz. Fiber pigtailed.
Average detector current = 1 mA.
-20 i I i i i i i i i i i i i i i ■' ■t~ i' i"*i
-40
•
1 GHz
-80
Pow er
n,
F u n d a m e n ta l
S
1 i i i i i i i i -16
5
Ai
(d B m )
— Modulation
Depth = 100% i
■■ ■
10
15
D io d e A p p lie d R e v e r s e B ia s V o lt a g e (-V )
Figure 5.2 Measured fundamental and harmonic power versus
detector applied reverse bias voltage at 1 GHz. Fiber pigtailed.
Average detector current = 1 mA.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
64
5 GHz
Pow er
3
F u n d a m e n ta l
•
-so
(dB m )
Modulation
Depth = 100%
0
5
10
15
D io d e A p p lie d R e v e r s e B i a s V o lt a g e (-V )
Figure 5.3 Measured fundamental and harmonic power versus
detector applied reverse bias voltage at 5 GHz. Fiber pigtailed.
Average detector current = 1 mA.
■20 | i i i i i i i i i i i i i i i i i i i | i i i . i i i i i | -18
-20
+
-«
,1 *
•
2f
10 GHz
-22
Oi
-24
-26
£ -loo
e
-120
0
■■* ' ■ ■ ■' ■■ ■■ ■ ■■ '
5
10
(d B m )
Modulation
Depth = 100% !
Pow er
I -60
o
CU
3 -80
c
o
£
jr
F u n d a m en ta l
I
s
-28
15
D io d e A p p lie d R e v e r s e B ia s V o lt a g e (-V )
Figure 5.4 Measured fundamental and harmonic power versus
detector applied reverse bias voltage at 10 GHz. Fiber pigtailed.
Average detector current = 1 mA.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
65
ages are very similar for 100 MHz, 1 GHz, and 5 GHz (10 GHz third and
fourth harmonics are above the spectrum analyzer frequency range).
4) A peak in the NL output occurs at an applied voltage where the funda­
mental power is within 1 to 2 dB of its value at high applied voltages.
NL effects which have significant frequency dependence are due to
transit-time (dynamic) effects.2*3 The frequency dependence of transit­
time effects has been analyzed by Hayes and Persechini3 who find that
the second harmonic should decrease by -20 dB per-frequency-decade.
The frequency dependence in the NL can be related to the change in
transit-time delay imposed by different carrier velocities compared to the
period of the signal. The analysis presented in reference [3] accurately
describes the PD NL data in figures 5.1 to 5.4 only in the region of applied
voltages from -3 to -5 V, where significant frequency dependence in the
NL exists. At higher applied voltages, the NL is nearly independent of
frequency. Later in this chapter, simulations a t -5 V and 5 GHz will sug­
gest that the NL is a result of, not just the electric-field-dependent elec­
tron velocity as reference [3] assum es in their analysis, but it also
includes a significant contribution from the electric field dependence of
the holes and from the absorption of carriers in the p-region.
The results from reference [3] do not predict device NL behavior
above -10 V, since the NL is essentially independent of voltage and fre­
quency. Harmonic levels which are independent of frequency are usually
assumed to arise from static nonlinearities3 in contrast to transit time
nonlinearities.
An investigation into these NL at high electric fields,
which dominate practical systems will be analyzed in this chapter to
determine their true origin.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
66
T he “w indow ” provided by the voltage and frequency dependence of
th e n o n lin ea rities w ill be u sed to exam ine and separate th e various n o n ­
lin ear m e c h a n ism s.
For th e purposes here, th e n o n lin ea rities a t 1 and 5
GHz w ill be sim u lated at applied voltages o f -5, -10 and -15 V. A t -5 V, th e
second h a rm o n ic of both frequencies is decreasing w ith in crea sin g (n eg ­
ative) voltage. A t -10 V, th e second h arm on ic of 1 GHz h a s b een a t its
m in im u m for several V olts and the second harm onic o f 5 GHz h as ju s t
reached th e tra n sitio n betw een the d ecreasin g portion o f th e curve and
th e region w here th e second harm onic rem ain s u n ch anged.
A t -15 V ,
th e second h arm on ic of both frequencies h ave b een a t th eir respective
m in im u m s for several V olts. The follow ing study w ill begin by fittin g the
m odeled PD NL to th e m easu rem en t data a t -5 V. A fter good a g reem en t
is ob tain ed , th e v a rio u s n o n lin ea r m e c h a n ism s w ill be a lte r ed or
rem oved to d eterm ine w hich m ech a n ism (s) dictate PD n o n lin ea r behav­
ior.
T he stu d y w ill continue a t -10 and -15 V to describe th e behavior
observed in the data from figures 4.1 to 4.4.
F ive V olt M easurem ents an d Sim ulations
To stud y th e dependence o f the generated harm onics w ith in cid en t
optical power, th e phase-locked lasers in figure 2.1 are adjusted such th at
th ey h ave equal am p litud e and th at th eir polarization s are a lig n ed to
yield a 100% m odulation depth. Since th e la sers do not generate frequen­
cies a t m u ltip les (harm onics) o f the fu n d am en tal frequency f, th e m e a ­
su rem en t o f th e PD n o n lin ea rities is obtained by sim ply m e a su r in g the
harm onic pow er a t th e frequencies 2f, 3f, and etc. M easu rem en ts of the
second and third harm onics are u su ally enough for m ost sy stem s app li­
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
67
cations and the intermodulation products can, in some instances, be
derived from the third harmonic. Measuring and sim ulating terms up
to order four will increase our confidence in the third harmonic results.
The measurement data for the 0.95 (im device at frequencies of 1
and 5 GHz are plotted in figures 5.5 and 5.6, respectively. The modula­
tion depth is 100%, the applied voltage is -5 V, and the device is fiber pig­
tailed, so the estimated e-2 incident spot size is 10 pm. From figures 5.5
and 5.6, two main tendencies for this device are observed. For a given
average optical power (or, equivalently, average current), the harmonic
power increases as the frequency increases. For example, at 1 mA, the
second harmonic increases from -80 dBm to -65 dBm as the frequency
increases from 1 GHz to 5 GHz. The second tendency is the deviation
from power-law growth of the harmonic power as the average PD current
-10
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Depth = 100%
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-130
40 dB per Decade
1 GHz
60 dB per Decade
-150
0.1
1
10
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 5.5 Measured fundamental and harmonic power versus
average detector current at 1 GHz. Applied voltage = -5 V. Fiber
pigtailed. 40 dB and 60 dB per decade tendaneies included.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
68
-10
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PQ
"C
• ••
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D epth = 100% ■
-30
-50
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£
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40 dB per Decade
-70
5 GHz
♦ '
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-90
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-130
0.1
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1
A v e r a g e D e te c to r C u r r e n t (m A )
Figure 5.6 M easured fundam ental and harm onic power versus
average detector current a t 5 GHz. Applied voltage = - 5 V. Fiber
pigtailed. 40 and 60 dB per decade tendancies included.
in crea ses to 1 m A and beyond, b ein g slig h tly m ore noticeable in th e 1
GHz case.
F irst, a fit to th e 5 GHz data (figure 5.6) w as attem pted w ith the
sim ulation program , w here at this tim e, only th e hole m obility and s im u ­
lation spot size w ere adjustable p aram eters.
Since th e electric field is
below 20 kV /cm n ea r th e n-contact (figure 4.13), a ch an ge in th e hole
m obility or in cid en t spot size should have the greatest effect on th e sim u ­
lated n on lin earities because the hole velocity does not satu rate for electric
fields of 20 kV/cm u n less the hole m obility is greater th an 400 c m2/V s and
the carrier d en sities are directly proportional to the spot size. The sim u ­
lated resu lts for hole m obilities o f 200, 230, and 260 c m 2/V s w ith a spot
size o f 5.0 pm is show n in figure 5.7. The sim u la ted data show s that
w hile the second harm onic data is slightly u n derestim ated , the third and
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
69
Modulation
Depth = 100%
<u
-80
■■////II
A
yffiifx.....i .........
7
If ^
rf A
7/f a
;
A
B
2fD ata
♦
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▲
4fD ata
...
II
p.p-'^DU
............. OQA
10
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 5.7 Measured and simulated harmonic power at 5 GHz.
Simulated spot size = 5 |im with hole mobilities of 200, 230,
o
and 260 cm /Vs. Measurement data from figure 5.6.
fourth harmonics are overestimated. Figure 5.8 plots the sim ulation
results with a slightly larger spot size of 6.0 gm with the same three hole
mobilities. Note that now, the second and third harmonics are slightly
underestimated and the fourth harmonic is still overestimated. From
the trends of figures 5.7 and 5.8, increasing the spot size to 7.0 (im and
decreasing the hole mobility should yield better fits to the measured data.
The simulated data for hole mobilities of 100,150, and 200cm 2/Vs with a
spot size of 7.0 |im is shown in figure 5.9. Here the simulated data for the
second and third harmonics, with a hole mobility of 150 cm 2/Vs, fits quite
well to the experimental data, while the fourth harmonic is still slightly
overestimated. Although the best fit hole mobility is about 1/2 that of a
measured sam ple4 of InGaAs, it is not so unrealistic, since measured
values of electron mobility4 for InGaAs also vary by 50% depending on the
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
70
s
PQ
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u
CD
£
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Depth = 100%
03
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cO
£
ou
Pp=260
Hp=200
CD
Pp=230
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 5.8 Measured and simulated harmonic power at 5 GHz.
Simulated spot size = 6 pm with hole mobilities of 200, 230,
2
and 260 cm /Vs. Measurement data from figure 5.6.
Modulation
Depth = 100%
•
2fD ata
■
3fD ata
♦
4fD ata
•1 =200
Pp=150
Pp=100
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 5.9 Measured and simulated harmonic power at 5 GHz.
Simulated spot size= 7 pm with hole mobilities of 100, 150,
o
and 200 cm /Vs. Measurement data from figure 5.6.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
71
p articu lar m ea su rem en t sam p le. A dd itionally, th e electric field depen­
dence o f th e hole velocity for electric field s below 50 kV /cm is not w ell
know n and w as only estim ated in the form ulation o f equation 3.17.
In C hapter III, it w as show n th a t n o n lin ea r con tin u ity equations
r esu lt w h en th e carrier velocities are functions of the gen erated carrier
d e n sitie s. A space-ch arge electric field (equation 3.35) can ex p la in the
ex isten ce o f a non lin earity, i f th e space-charge field is h ig h en ou gh to
perturb th e dark electric field (equation 3.34). The resu ltin g change in
carrier velocities cau se non lin earities m easured as harm on ics in the PD
output. Since the electric field in the in trin sic region (see figu re 4.13) is
n ot h igh enough (10 to 20 kV/cm w ith -5 V applied voltage) to saturate th e
hole or electron velocities n ea r th e n-contact, a photogenerated spacecharge electric field w ill cause a change in th e carrier velocities. To ver­
ify th a t other n on lin ear m ech a n ism s are le ss im portant a t th is bias volt­
age, the sim u lation param eters yieldin g the best resu lts in figure 5.9 are
u sed in several additional sim u lation s w here th e other n on lin ear m e c h a ­
nism s are m odified or removed.
The ad d ition al sim u la tio n s consider: 1) th e electron m obility, 2)
recom bination tim es in the in trin sic region, 3) reduction in th e low -field
m obility due to scatterin g, 4) absorption in th e p-region, and 5) th e addi­
tion of a 50Q load resistor. A longer recom bination tim e in th e in tr in sic
region reduces the contribution from the recom bination term in th e conti­
n u ity eq u ation s, w h ile n e g le c tin g sca tter in g and p -region absorption
rem oves th ese n on lin ear term s com pletely. The addition o f a 50Q load
resistor w ill add to th e NL by low erin g th e applied poten tial due to c u r ­
ren t flow through th e load resistan ce. T his m odifies th e in tern a l electric
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
72
field and hence th e carrier velocities. A change in th e electron mobility
should not cause a change in the NL, even though th e electron m ay con­
tribute sign ifican tly to the overall NL, since th e velocity dependence ver­
su s electric field above 20 kV /cm is a w eak function o f th e electron m obil­
ity as can be seen in figure 3.3.
The sim u lation param eters for th e best-fit r esu lts o f figure 5.9 are
u tilized w ith th e above five m odifications. T he n o n lin ea r m e c h a n ism s
are changed one a t a tim e to com pare th eir relative contributions to the
total device NL, w ith the resu lts plotted in figure 5.10. T he sim u la tio n
resu lts in figure 5.10 were obtained by reducing the electron m obility 40%
to 6,000 c m2/V s, n e g le ctin g m obility reduction (sca tterin g ), n eg lectin g
absorption in th e p-region, in crea sin g th e recom b ination tim e in the
in trin sic region from 2 ns to 2 ps, and adding a 50£1 load resistan ce to the
i i i i i
-50
, M odulation
Depth = 100%
-60
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j-,
-70
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2fData
3fData
4f Data
--------- Best Fit
--------- pe = 6,000
----------- N o p-red
............No p-abs
----------- Tau = 2 p s
-120
— — - w /50 Ohm
-130
0.1
10
A v e r a g e D e te c to r C u r r e n t (m A )
Figure 5 10 Harmonic power a t 5 GHz. B est-fit sim ulation modified
2,
w ith 1| = 6,000
cm /Vs, increasing the i-region recom bination tim e,
om itting scattering and p-region absorption, and adding a 50 Ohm load.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
73
PD output circuit. As can be seen, little change in the simulated NL
results from varying these additional nonlinear mechanisms.
There­
fore, we conclude that the NL is most sensitive to the hole mobility and
the spot size at this particular applied bias.
In Chapter III the NL was assumed to arrive from the existence of
a space-charge electric field causing a carrier density dependence in the
carrier velocities. To obtain the relative size of the space-charge electric
field compared to the dark electric field, the simulated diode was illum i­
nated with a constant level of light equivalent to 0, 100, and 1000 pA of
current with a constant spot size of 7.0 pm.
The space-charge electric
fields at 100 and 1000 pA are simply the difference between the electric
field at 0 pA and their respective high-current electric fields.
These
resulting space-charge fields (figure 5.11) can be significant, as much as
10% of the highest electric field under dark conditions. The space-charge
electric field in figure 5.11 will therefore modify the carrier velocities
according to their electric field dependence (equations 3.16 and 3.17).
The change in hole and electron velocities at 100 and 1000 pA from
the space-charge electric field (figure 5.11) is plotted in figure 5.12. A sig­
nificant change in both the hole and electron velocities occurs at 1000 p A,
suggesting that the NL output has contributions from both carriers at
this applied voltage. However, as mentioned before, the sensitivity to the
exact electron mobility in the simulated results is weak, as demonstrated
by the simulation in figure 5.10, even though the electron contribution to
the total NL may be significant.
Previous work2 has speculated that the load resistance may be the
main contributor to the overall NL however, the results in figure 5.10
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
74
10
i
|
i
i
i ■ | —i" "i1
I I I
1— 1— r
6
0
1
5
2
'3
•^
fa
0)
0
he
03
-5
Oi
03 -10
O
o3
a
m
-15
•Es c @Id c = 100^
Es c @Id c = 1000^
■ 1 '
1
1.2
_L
_L
■ ■ ±
1.4
1.6
1.8
D iod e X P o s itio n (|im )
■J. — Li
Figure 5.11 The space-charge electric field in the intrinsic
region due to the photogenerated carriers. Spot size = 7.0 pm.
Applied voltage = -5 V. pp = 150 cm2/Vs. pn = 10,000 cm2/V s.
Electrons -1 0 0 pA
Holes -1 0 0 pA
Electrons -1000 pA
Holes -1000 pA
O 0.5
r—i
7 , -0.5
1
1.2
1.4
1.6
D iod e X P o sitio n (|im )
1.8
Figure 5.12 The change in carrier velocities in the intrinsic region
due to the space-charge fields in figure 5.11. Spot size = 7.0 pm.
Applied voltage = -5 V. pp = 150 cm2/Vs. pn = 10,000 cm2/V s.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
75
seem to contradict th is resu lt. T he underlying reason for th e in se n sitiv ­
ity in th e N L output w ith th e presence o f a load resista n ce is app arent
from the space-charge electric field data in figure 5.11. A load current o f
1 m A through 50Q resu lts in a 50 mV potential drop w h ich ca u ses a 0.5
kV /cm in tern a l electric field decrease throughout a 0.95-pm lon g in tr in ­
sic region. T h is electric field change is a t le a st an order o f m a g n itu d e
le ss than the space-charge electric field (figure 5.11) a t the sam e cu rren t.
T herefore, it appears th a t th e electric field change as a r esu lt o f an d
extern al 50Q resista n ce should n ot contribute sig n ifica n tly to th e overall
N L output. T h is effect how ever, m ay becom e m ore im p ortan t in devices
w ith 0.1 to 0.2 pm in trin sic region len g th s or larger in cid en t spot size s
because, for th e sam e load current, the in tern al electric field change is
inversely proportional to the length and independent o f spot size.
The h ig h est percentage change in th e carrier v elocities due to the
space-charge electric field (figure 5.11) occurs near th e n-contact, w h ere
th e electric field is the low est. A lower hole m obility in crea ses the s e n s i­
tivity to th e space-ch arge field w h en the total electric field is below 50
kV/cm . A t -5 V or low er applied voltages, the NL sen sitiv ity w ill th en be
additionally dependent on the doping level in th e in trin sic region as w ell
as th e in tr in sic region len g th .
T his prediction is confirm ed5 and is
understood because the electric field near th e n-contact w ould be low er i f
the in trin sic region len gth w ere slightly longer or i f th e doping den sity in
th e intrinsic region w as sligh tly higher.
S h iftin g to 1 GHz, sim u la tio n s w ith hole m obilities o f 150 and 175
c m W s and a sim u lation spot size of 7 pm, sim ilar to th e best-fit s im u la ­
tion param eters utilized in figure 5.9, are plotted in figure 5.13. W ith the
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
76
-50
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M odulation
D epth = 100%
PQ
T3
S-/
-70
I
0
a<
-90
g -110
cC
£
1 -130
• i-H
-150
0.1
1
10
A v e r a g e D e te c to r C u r r e n t (m A )
Figure 5.13 M easured and sim ulated harm onic power vs average PD
current at 1 GHz. Applied voltage = -5 V. Fiber pigtailed.
o
|i p = 150 and 175 cm /Vs. Sim ulated spot size = 7.0 pm.
exception of the second harm onic, the sim u lated data fits quite w ell to the
m easu red data. The discrepancy in the second harm onic w ill be reex a m ­
ined after sim ulations and fits to the -10 V NL data.
T en V olt M easurem ents an d S im u lation s
M easu rem en ts a t high er applied voltages an d hence high er elec­
tric fields should low er the NL dependence on th e exact hole m obility of
the device, sin ce the hole velocity nearly saturates for electric fields above
50 kV /cm .
H igh er electric fields th us resu lt in sm a ller ch an ges in the
hole velocity for a g iven change in the in tern a l electric field. A -10 V
reverse b ias w ill in crea se the electric field by ap p roxim ately 50 k V /cm
over th e entire in trin sic region w hich should be su fficien t to saturate the
hole velocity given the hole m obility is above 150 c m 2/V s. The electric field
at 100 pA w ith -10 V applied voltage is plotted in figure 5.14, w here the
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
77
so
-50
>
rd
%
-100
-o -150
QJ
H
-200
1.0
1.2
1.4
1.6
1.8
D io d e X P o s i t i o n (}im )
Figure 5.14 Intrinsic region electric field for an intrinsic region
doping density of 5.0 x 10 35 cm"3 . Diode applied voltage =-10 V.
electric field is seen to be above 70 kV/cm over the entire intrinsic region.
The measured PD NL at frequencies of 1 and 5 GHz are shown in
figures 5.15 and 5.16, respectively.
Several differences are observed
between the -10 V data (figures 5.15 and 5.16) and the -5 V data (figures
5.5 and 5.6). The individual NL components are 10 to 20 dB lower as
expected from the higher electric fields which result in a nearly satu­
rated hole velocity. The frequency dependence is also reduced substan­
tially, for example, the increase in second harmonic at 1 mA between the
two frequencies is 15 dB at -5 V applied bias voltage, but reduces to 3 dB at
-10 V. Note also that compared to the -5 V data, the -10 V nonlinearities
resemble traditional power-law behaviors.
Recall from figures 5.2 and 5.3 that at -10 V applied PD voltage the 1
GHz second harmonic no longer decreases with increasing voltage while
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
78
M odulation
Depth = 100%
-50 O.
-70
«J
-90
2
-110
40 dB per Decade
•
■
♦
▲
1 GHz
2f
3f
4f
10
A v e r a g e D e te c to r C u r r e n t (m A )
Figure 5.15 M easured fundam ental and harm onic power versus
in p u t optical power at 1 GHz. Applied voltage = -10 V. Fiber
pigtailed. 40 dB per decade tendancy included.
-10
S
CQ
"O
OQ
dU
I
-50
£
f£
a»
-70
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Depth = 100%
5 GHz
40 dB per Decade
s „ .
60 dB per Decade
-130
0.1
1
A v e r a g e D e te c to r C u r r e n t (m A )
Figure 5.16 M easured fundam ental and harm onic power versus
incident optical power at 5 GHz. Applied voltage = -10 V. Fiber
pigtailed. 40 and 60 dB per decade tendancies included.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
10
79
the 5 GHz second harmonic has just reached its minimum. Simulations
at -10 V applied PD voltage will be used to classify these two regions of
nonlinear behavior. Figure 5.17 shows the simulated nonlinearities with
a 7 pm simulation spot size and hole mobilities of 150 and 200 cm 2/V s.
Since the electric field in the entire intrinsic region is above 70 kV/cm
(figure 5.14), there should not be a significant dependence on the sim u­
lated data with the hole mobility. However, the sensitivity to hole mobility
in the simulated nonlinear data could be due to the NL generated in the
highly doped p-region where the the hole velocity is unsaturated.
The observation of residual nonlinearities at -10 V applied voltage,
where the electric field in the intrinsic region is sufficient to approxi­
mately saturate the hole velocity, is not unexpected due to the many non­
linear m echanism s in the continuity equations (3.3 and 3.4). Several of
-60
S
Modulation
Depth = 100%
-70
PQ
S
-80
2
-90
& -100
o
&
-110
CO
2 fD ata
3 fD ata
o -120
o
-130
p =150
-140
0.1
1
10
A v e r a g e D e te c to r C u r r e n t (m A )
Figure 5.17 Measured and simulated harmonic power vs current
at 5 GHz for hole mobilities of 150 and 200 cm2/Vs. Applied
V = -10 V. Spot size = 7 pm. Experimental Data from Figure 5.16.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
these mechanisms will be investigated to determine their contribution to
the total device NL at high applied voltages. The nonlinear m echanism s
under consideration here are:
1) Space charge fields which may modify the carrier velocities.
2) Free carrier scattering which can reduce the carrier mobilities and
possibly the carrier velocities.
3) Carriers absorbed in the p-type cap layer (where the electric field is
low) may contribute to the total device NL.
4) The diffusion term in equations 3.3 and 3.4 may contain a small non­
linear term due to the electric field dependence in the diffusion constant.
5) Loading in the external circuit which may modify the carrier veloci­
ties via a reduced internal electric field.
To determine the contribution from the above nonlinear terms, the
simulated diode is modified to produce a highly linear response.
To
accomplish this, the nonlinear m echanism s are removed or reduced
through the following techniques. The maximum diffusion “velocities”
(Chapter IV) may be lowered (a factor of 10 to 100 is sufficient) to reduce
its NL contribution by resulting in a relatively field-independent diffu­
sion. The fitting parameter, P (equation 3.16), in the electron velocity can
be reduced to cause the electron velocity to be nearly constant above 20
kV/cm. Absorption in the p-region and the mobility dependence on car­
rier density can both be removed. The diode can be simulated without a
load resistance. The simulated NL of such an “ideal” diode is shown in
figure 5.18 for simulation spot sizes of 5, 6, and 7 pm. As one can see the
device is very linear producing a -115 dBc second harmonic (dBc im plies
relative to the fundamental) at 115 pA limited by the FFT window and a -
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
81
-10
?
CQ
3
-30
-50
|
-70
Modulation
Depth = 100%
Fundamental
£
-90
©
£CO -110
£
o -130
5 pm ss
6 pm ss
7 pm ss
Second Harmonic
Third Harmonic
:
| *
-170
0.1
1
10
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 5.18 Simulated fundamental and harmonic power versus
detector current for an "ideal" detector at 5 GHz. Applied voltage
o
= -10 V. (ip = 200 cm /Vs. Simulated spot sizes of 5, 6, and 7 pm.
100 dBc second harmonic at 1 mA. The results above 1 mA are slightly
worse, possibly due to the rem aining small nonlinearities in carrier
velocities or diffusion. Nevertheless, the nonlinear terms may now be
added to quantify their relative contributions to the total device NL.
The above five nonlinear mechanisms can be reduced to three basic
NL sources, due to the inter-relationships between the nonlinear m echa­
nisms and the carrier velocities. Space-charge electric fields and loading
in the external circuit both modify the intrinsic region electric field, and
thus, the electron velocity. On the other hand, carrier scattering reduces
the carrier mobilities, thus changing the electric field dependence of the
carrier velocities. A space-charge induced change in the hole velocity
will be determined later to be negligible compared to the electron velocity
change. Therefore, if the electron velocity was not a function of electric
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
82
field above 50 kV/cm, there would be negligible NL contribution from
space-charge fields, scattering, or a load resistor.
The other two nonlin­
ear mechanisms, generation in the p-region and diffusion, are indepen­
dent of the other three terms. Therefore, simulations of the ideal diode to
determine the contribution from the five nonlinear m echanism s need
only to consider three dominating effects: 1) diffusion, 2) the electric field
dependent electron velocity, and 3) absorption in the p-type cap layer.
This is accomplished by taking the ideal diode and adding the NL term s,
one at a time to the model.
The first nonlinear term added to the ideal diode is diffusion. The
device NL of the simulated ideal diode with only diffusion added is shown
in figure 5.19. There is a slight increase from the ideal diode NL of figure
5.18, although the measured NL in the real device (see figure 5.16) are
-70
Modulation
Depth = 100%
?
PQ -90
. Second Harmonic
|.U 0
0
PU
j£ -130
CS
1
5 gm ss
6 gm ss
I -150
Third Harmonic
§
7 gm ss
-170
0.1
1
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 5.19 Simulated harmonic power versus detector current
for an "ideal" detector at 5 GHz including only diffusion.
Applied voltage = -10 V.
= 200 cm2/V s .
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
83
still 10 to 20 dB h igh er th an th e non lin earities in figure 5.19. T hus, diffu­
sion alone does not account for the observed NL.
T he addition o f th e field dependence of th e electron velocity adds NL
to th e id eal diode w henever there is a space-charge electric field, m obility
reduction due to scatterin g, or loading in th e external circuit. T he spacecharge field cannot be rem oved from th e sim u la tio n , bu t sca tterin g and
th e load resista n c e can.
Therefore th e id ea l diode w ill be m odified by
adding the electric-field-dependent electron velocity and w ill affect th e NL
via space-ch arge fields only. The sim u lated id eal diode N L under th ese
conditions is plotted in figure 5.20, w here th e NL h a s in crea sed by up to
15 dB over the diffusion-only resu lts o f figure 5.19. T he sim u la ted NL are
now w ith in a few dB of th e experim entally observed NL in figure 5.16.
T he add ition o f carrier sca tterin g (m obility reduction) or a load
-70
s
M odulation .
Depth = 100% -
Second Harm onic
m
-90
I
-110
0
cu
S2 -130
cS
1
& -150
Third H arm onic
§
-170
0.1
1
10
A v e r a g e D e te c to r C u r r e n t (m A )
Figure 5.20 Sim ulated harmonic power v s current for an "ideal"
detector at 5 GHz including the field dependent electron velocity
9
w ith space-charge effects. Applied V = -10 V.
= 200 cnT /V s
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
84
resistan ce w ill further modify the electric field at h ig h PD currents. T he
sim u lated diode lead in g to the resu lts in figure 5.20 is m odified to in clu d e
both o f th ese additional nonlinear m ech a n ism s and th e associated data is
plotted in figure 5.21. The resu lts show th a t th e device NL in clu d in g sca t­
terin g and a load resistor is about 1 dB less th a n th a t sim u lated diode in
figu re 5.20, w h ich excluded th ese two term s. T his m ay happen, since a
r e d u c tio n in
th e e le c tr o n m o b ility c a n
red u ce
th e electric-field-
dependence of th e electron velocity4 for electric fields above 20 kV/cm.
A plot of the space-charge electric field (figure 5.22) is id en tical to
th e -5 V space-charge electric field. T his is expected since th e in tr in sic
reg io n carrier velocities are e sse n tia lly u n ch a n g ed , a lth o u g h th e NL
w ith th e diode biased a t -10 V is su b stan tially reduced from th e earlier -5
V case. Sin ce th e space-charge field h a s n ot changed, th e drop in the
-70
M odulation
Depth = 100%
Second Harm onic
S3
-o
-so
I
-110
Applied V = -10 V
♦u n = 2 0 0 cm2/V s
0
Ph
£
-130
V»J
5 pm ss
1
S
-150
Third H arm onic
7 pm ss
§
-170
0.1
1
10
A v e r a g e D e te c to r C u r r e n t (m A )
Figure 5.21 Sim ulated harmonic power v s current for an "ideal"
PD at 5 GHz including the field-dependent electron velocity w ith
space-charge effects, scattering, and a 50 Ohm load resistance.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
cs
A
Q
0
Es c @ Idc = 100^
0
0
Es c @ Id c = 1000^
a
15 L J
1.0
I
I
I I
1.2
I
I
I I
1.4
I
J
L
1.6
I
I
I I
1.8
L
D i o d e X P o s i t i o n ( j im )
Figure 5.22 The space-charge electric field in th e intrinsic
region due to th e photogenerated carriers. Spot size = 7.0 Jim.
Applied voltage = -10 V.
= 150 cm2/Vs. pn = 10,000 cm2/Vs.
in trin sic region electric field associated w ith a 50£2 load resistan ce is still
an order of m agn itud e less th an th e space-charge electric field. T h u s,
th e relative independence of the NL in figure 5.21 on a load resista n ce is
n ot un exp ected.
The NL output a t -10 V applied voltage red uces as a
resu lt of the reduction of the electric field dependence o f the carrier veloc­
itie s at h ig h electric fields. The r esu ltin g carrier velocity ch a n g es are
plotted in figure 5.23. Notice th at the hole velocity change is alm ost n e g li­
gible, as it should be since the hole velocity sa tu ra tes quickly above 50
kV/cm .
The electron velocity ch an ge is also reduced by a factor of ten
from th e r esu lts at -5 V applied voltage w hich r e su lts in th e overall
decrease in th e device NL.
T he third n on linear m ech a n ism under in v estig a tio n is th e addi­
tion of absorption in the p-region. T his adds to the NL o f the ideal diode by
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
86
^
s
w
0.6
0.4
£
£
0.2
I
0
Electrons -100 pA
Holes -1 0 0 pA
Electrons -1000 pA
Holes -1000 pA
e -0-2
03
g -0.4
•
as
A
^
-0.6
1
1.2
1.4
1.6
1.8
D io d e X P o s i t i o n ( |im )
Figure 5.23 The change in carrier velocities in the intrinsic
region due to the space-charge field. Spot size = 7.0 pm.
Applied voltage = -10 V. pp = 150 cm2/Vs. pn = 10,000 cm2/V s.
the generation and movement of carriers in a region where the carrier
velocities are rapidly changing, even though the carriers may quickly
recombine. The simulation results of the ideal diode, modified to include
p-region absorption, are plotted in figure 5.24. The nonlinearities in fig­
ure 5.24 are approximately 20 dB higher that those obtained when the
model diode includes all the nonlinear m echanism s.
This over-
estimation of the device NL is the result of neglecting mobility reduction
and diffusion, since the inclusion of these terms contribute to the carrier
movement in the p-region. For example, the electron velocity is propor­
tional to the electric field (the electric field is very low so the carrier veloc­
ities are equal to pE), which is related to the ratio of electron to hole veloc­
ity in the p-region. When mobility reduction is not included in the simu-
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
87
-60
Modulation .
Depth = 100% -
Second Harmonic
-80
0)
£ -100
o
O.
{£ -120
5 pm ss
05
Third Harmonic
|
§ -140
-160
0.1
6 pm ss
1
10
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 5.24 Simulated harmonic power versus detector current for
an "ideal" detector at 5 GHz including only the p-region absorption
nonlinearity. Applied V = -10 V. Spot sizes of 5, 6, and 7 pm.
lation, this ratio increases by a factor of 1.8, since the electron mobility (as
compared to the hole mobility) begins to decrease at lower carrier densi­
ties. Therefore, the net result is a higher change in electron velocity
when scattering is excluded. Enhancement or suppression of the other
nonlinear m echanism s may also occur when more than one NL is
included in the ideal diode simulations.
Therefore, the two dominant
terms, the electric field dependence of the electron velocity and the pregion absorption, will be further investigated by excluding each of these
two NL from the actual device.
The simulated results for the real diode excluding only p-region
absorption and excluding only the electric field dependence in the elec­
tron velocity above 20 kV/cm are plotted in figures 5.25 and 5.26, respec­
tively. Both simulations include the mobility as a function of carrier den-
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
88
-70
s
CQ
l-o
M odulation
D epth = 100%
Second Harm onic
-90
5h
% -110
S
2 fD a ta
£ -130
ca
|
b
3 fD a ta
5 pm ss
-150
6 pm ss
Third H arm onic
7 pm ss
-170
0.1
1
10
A v e r a g e D e te c to r C u r r e n t (m A )
Figure 5.25 M easured and sim ulated harm onic pow er vs detector
o
cu rren t at 5 GHz excluding p-region absorption. pp = 200 cm /Vs.
Applied V oltage = -10V.
Experim ental data from figure 5.16.
-60
a
PQ
M odulation
Depth = 100%
Second Harm onic
-80
% -100
o
8> -120
ca
|
§
— 5 pm ss
-140
Third H arm onic
--- 6 pm ss
— 7 pm ss
-160
0.1
1
10
A v e r a g e D e te c to r C u r r e n t (m A )
Figure 5.26 M easured and sim ulated harm onic power vs current
at 5 GHz excluding the electron velocity nonlinearity. Applied
V = -10V. (ip = 200 cm2/Vs.
Experim ental data from figure 5.16.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
89
sity and diffusion. From the figures it can be seen that either nonlinear­
ity can account for the observed device NL, although the p-region absorp­
tion NL results in a slight overestimation of the total device NL.
The simulation program may have difficulty accurately predicting
the NL contribution associated with the p-region absorption. The NL m ay
include the contribution from electrons reaching the intrinsic region
which, in their travel, encounter rapidly changing velocities. The den­
sity of these electrons will thus have a large impact on the amount of NL.
To account for carrier spreading (Chapter IV) in the p-region, a spread­
ing function was used near the p-i interface to sim ulate th is twodimensional effect for the one-dimensional model.
This assumption
however resulted in an artificial focusing of p-region-generated electrons
traveling into the intrinsic region.
The total number reaching the
intrinsic region was assumed to be small compared to the total number of
electrons generated there. Therefore this algorithm may be overestimat­
ing this NL slightly since the space-charge electric field, linked to the
change in electron velocity, can account for the observed device NL. I n
Chapter VI, additional simulations will determine what length of unde­
pleted p-region will result in the observed NL at high applied voltages.
Simulations at 1 GHz will be used to verify the dominant nonlinear
terms to the extent a 1-D model can accurately predict the p-region
absorption NL. The simulation results at 1 GHz and -10 V applied voltage
with hole mobilities of 175 and 200cm 2/Vs and a simulation spot size of 7
pm, parameters which produced good results at 5 GHz, are plotted in fig­
ure 5.27. Notice that at 1 GHz the simulated results do not agree well
with the experimental data. The contribution to the NL output at 5 GHz
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
90
-6 0
a
CQ
T3
V -/
u
a>
£
Modulation
Depth = 100%
-70
-80
-90
tS -loo
o
> -no
ca
0 -120
Sh
1
-130
-140
0.1
1
10
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 5.27 Measured and simulated harmonic power vs current
at 1 GHz. Applied V = -10 V. Fiber pigtailed. Spot size = 7.0 jim.
o
Pp = 175 and 200 cm /Vs. Measurement data from figure 5.15.
was seen to be influenced the most by two nonlinear mechanisms: a
space-charge-induced change in the electron velocity with increasing
carrier densities and the absorption in the p-region. It was determined
that the NL at 5 GHz could be explained by considering the change in the
electron velocity, without including the p-region absorption.
In fact,
including the p-region absorption, led to an overestimation of the second
harmonic by 10 dB at 5 GHz, similar to the results at 1 GHz (figure 5.27).
Therefore the PD is simulated again, this time neglecting p-region
absorption, to determine if the electron velocity NL can be used by itself to
model the device NL at 1 GHz. The simulated results with a hole mobility
of 200 c m2/Vs and simulation spot sizes of 5 and 7 pm are plotted in fig­
ure 5.28, where the NL is now 10 to 15 dB below the measurement data. A
lower hole mobility (100 cm 2/Vs) may increase the overall simulated
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
91
-70
?
CQ
S'
M odulation
Depth = 100%
-80
-90
03
£ -100
o
CL
0) -110
>
2 f D ata
3 fD a ta
oS
£ -120
4 f D ata
5 jam ss
7 Jim ss
-140
1
0.1
10
A v e r a g e D e te c to r C u r r e n t (m A )
Figure 5.28 M easured and sim ulated harm onic power v s PD
cu rren t at 1 GHz neglecting absorption in the p-region. Applied
o
V = -10 V. jip = 200 cm /Vs. M easurem ent data from figure 5.15.
device N L , so the PD is sim u lated a g a in (figure 5.29) w ith o u t p-region
absorption. Here again , th e sim u lated NL is 10 to 15 dB below the m e a ­
surem ent data. Therefore, the NL at 1 GHz w ith -10 V applied voltage is
dom inated by the p-region absorption and not th e space-ch arge-in duced
change in th e electron velocity.
One question still rem ains: w hy does the m odeled PD overestim ate
the NL at -10 V applied voltage at both frequencies of 1 and 5 GHz? From
the m od eling point o f view th is question m ay be altern ately phrased as:
W hat in flu en ces the NL in th e p-region? The an sw er to th is question m a y
not be a sim p le one, how ever, it is reasonable to a ssu m e that: 1) the
m inority carrier lifetim e in th e p-region w ill d eterm in e th e ex ten t of
w hich th e carriers generated in this region w ill contribute to the output,
2) th e tw o-d im en sion al focu sin g /d efo cu sin g ap p roxim ation a t th e p-i
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
92
-7 0
U
Modulation
Depth = 100%
-80
C O
■"O
s
t*
a>
£
o
CU
o>
>
<s
£
©
-90
-100
2fD ata
-110
3fD ata
4fD ata
-120
5 pm ss
Si
j* -130
-140
7 pm ss
0.1
1
10
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 5.29 Measured and simulated harmonic power vs PD
current at 1 GHz neglecting absorption in the p-region. Applied
V = -10 V.
= 100 cm /Vs. Measurement data from figure 5.15.
interface (Chapter IV) may influence the number of carriers which con­
tribute to the NL, 3) the ratio of carrier mobilities may effect the NL since
this ratio determines the electron velocity in the p-region, and 4) the elec­
tron mobility (and hence the electron velocity in the p-region) may effect
the NL, which is influenced by the functional dependence of the reduction
in electron mobility (equation 3.19) with increasing carrier density.
The p-region minority carrier lifetime was assumed to be 500 ps, or
about four times smaller than the intrinsic-region lifetime. The sim u ­
lated NL thus far were not affected by the exact value of the lifetime, how­
ever, at high electric fields, the nonlinearities are more subtle. M easure­
m ents6*8 of the minority carrier lifetime in InP, GaAs, and InGaAsP
indicate that the lifetimes in these materials are in the 10 to 500 ns range
(doping levels of 1016 cm*3) with a decrease to 0.1 to 5 ns at higher doping
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
98
levels (mid-1018 cm-3). The minority carrier lifetime in InGaAs has been
measured9 at low doping levels where values of 2 to 4 ns are obtained.
There is a lack of measurement data for p-InGaAs at high doping levels,
however, Landis et a l.10 have measured the minority carrier lifetime in
InP and have shown that the lifetime can be substantially different for pand n-type material. Minority carrier lifetimes o f400 ns for n-type m ate­
rial and 10 ns for p-type material are m easured10 with semiconductor
material doped at mid-1016 cm*3. Additionally, the lifetime in bothn- and
p-type materials decreases by up to two orders of magnitude at 1019 cm -3.
Therefore the model PD will be simulated again at 1 GHz to determine
the sensitivity of the device NL to the minority carrier lifetime.
The simulation results are plotted in figure 5.30 with minority car­
rier lifetimes of 100, 50, and 25 ps, a hole mobility of 200 cm2/Vs, and a
-70
s
CQ
Modulation
Depth = 100%
-80
w
-90
5-1
cu
£ -100
o
CU
2fData
3 f Data
4f Data
Tau=100ps
Tau=50ps
Tau=25ps
a> -110
>
ca
£ -120
oJ-H
-130
-140
0.1
1
10
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 5.30 Measured and simulated harmonic power vs PD current
at 1 GHz. Applied V = -10 V. Spot size = 6.0 pm. Measurement data
from figure 5.15. |ip = 200 cm2/Vs. pn = 10000 cm2/V s.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
94
sim u lation spot size of 6 |im . A m inority carrier lifetim e of 25 ps provides
th e best-fit to th e second harm onic data. A m inority carrier lifetim e o f 25
ps is about two orders of m agnitude le ss th an the lifetim e in the in trin sic
m a ter ia l.9 A lthough th is value m ay be sligh tly low, it is consisten t w ith
th e reductions (from th eir value a t low doping levels) observed in other
m a te r ia ls.6-8
T he m in ority carrier lifetim e is not th e only p ara m eter w h ich
in flu e n c e s th e p -region absorption NL
behavior.
A n o th er possible
p aram eter w hich m ay in flu en ce th e NL is the 2-D fo cu sin g /d efo cu sin g
approxim ation m ade for the p-region (Chapter IV).
T he location of the
focusing function m ay be moved closer to or further aw ay from th e in trin ­
sic region.
The electric field decreases very rapidly ju st after th e focal
position due to the h igh level o f p-region doping. Up u n til now , th is focal
position has been fixed at approxim ately 5 nm from the in trin sic region,
w here it w as placed based on an estim ate of the p-region depletion depth.
Figure 5.31 show s th e sim ulated resu lts w hen the focal position is moved
further aw ay (12 and 19 nm ) from th e in trin sic region and w hen the focal
position starts at th e in trin sic region edge (0 nm). M oving th e focal posi­
tion further aw ay cau ses more p-region-generated electrons to reach the
in trin sic region due to the increased electric field penetration into the pregion , resu ltin g in h igh er sim u la ted NL.
H ow ever, m ovin g th e focal
position to th e edge of th e in trin sic region seem s to have little additional
effect on th e sim ulated NL.
The m inority carrier lifetim e in the p-region d eterm ines how lon g
or eq u ivalen tly how m an y electron s contribute to th e NL.
A nother
param eter w hich in flu en ces th e p-region absorption non lin ear behavior
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
95
-60
a -70
PQ
S
-80
|
£
-90
c£ -loo
> -110
cS
I -120
I
-130
-140
0.1
1
A v e r a g e D e t e c t o r C u r r e n t (m A )
10
Figure 5.31 Measured and simulated harmonic power vs current
at 1 GHz. Applied V = -10 V. Variable positions for hole spreading
in the p-region. Spot size = 6.0 pm. Taup = 100 ps. pp = 200 cm /Vs.
is the electron velocity. The electron velocity is determined by the electron
mobility in the p-region because the electric field is below 1 kV/cm. The
electron mobility is influenced by two parameters: the low-doping-density
mobility pn and the scattering parameter nh (equation 3.19) which lowers
the electron mobility at high doping levels. To determine the sensitivity
in the sim ulated results to the electron mobility and the scattering
parameter, the PD is simulated with a fixed minority carrier lifetime of
100 ps while varying these two parameters.
The NL for electron mobilities of 6000,8000 and 10000 cm 2/Vs is
plotted in figure 5.32. The NL for electron scattering parameters of 0.5,1,
and 2 x 1017 cm-3 is plotted in figure 5.33 for an electron mobility of 8000
cm 2/Vs.
The scattering parameter (Chapter III) was estimated from
electrons in uncompensated n-InGaAs.4 The p-region is composed of
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
96
-60
s
— M odulation
? D epth = 100%
-70
P Q
3
u
03
£
-80
-90
2fD ata
(2 -100
3 fD ata
4 fD ata
> -110
CO
I
p
= 6000
-120
p
=8000
-130
p“ r> =10000
o
-140
0.1
10
1
A v e r a g e D e te c to r C u r r e n t (m A )
Figure 5.32 M easured and sim ulated harm onic pow er versus
current a t 1 GHz. Applied V = -10 V. Spot size = 6.0 pm.
p. = 6000,8000, and 10000 cm2/Vs. Tau = 100 ps. p = 2 0 0 c m 2/V s .
-60
M odulation
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-90
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n, =5 x 10
n u = l x 10
:
-130 -140
0.1
1
10
A v e r a g e D e te c to r C u r r e n t (m A )
Figure 5.33 M easured and sim ulated harm onic power v s current
a t 1 GHz for various electron scattering param eters. V = -10 V.
Spot size = 6.0 pm. pn = 8000 cm2/Vs. T aup = 100 ps. pp = 200 cm2/V s.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
97
compensated p-InGaAs. Therefore, some variation is expected due to the
different scattering m echanism s (electron-electron and electron-hole).
The results from figures 5.30 to 5.33 show how the various material
parameters affect the p-region absorption nonlinearities. Although the
exact m aterial parameters are not known, and the spreading function
was only an 1-D simplifying approximation, the simulation does provide
good results for reasonable material and simulation parameters.
Therefore the focal position will remain 5 nm away from the intrin­
sic region, the p-region minority carrier lifetime will be reduced to 100 ns,
the electron mobility will be reduced to 8000 c m W s, and the electron
scattering parameter will be reduced to 0.5 x 1017 cm-3. The sim ulations
at applied voltages of -5 and -10 V and frequencies of 1 and 5 GHz will be
recalculated to verify that the simulation contains a reasonable set of
assumptions to accurately predict device NL in regions where the spacecharge electric field and the p-region absorption lim it device nonlinear
performance.
Figures 5.34 and 5.35 show the measurement and simulated data
at 5 and 1 GHz, respectively, at an applied voltage of -10 V. The 5 GHz
simulation (figure 5.34) shows a slight improvement in the second har­
monic fit compared to the previous simulations (figure 5.17); however,
the simulated third harmonic now is underestimated by greater than 20
dB compared to the previous underestimation of 10 dB. The simulated
harmonics at 1 GHz (figure 5.35) have improved significantly over the
previous results (figure 5.27) for both the second and third harm onics,
with little observed change in the fourth harmonic.
Returning to -5 Volts applied bias, figures 5.36 and 5.37 show the
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
S6
-7 0
8
m
TS
w
Modulation :
Depth = 100% ~
-80
-90
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QJ
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2fD ata
3fD ata
5 pm ss
7 jim ss
0.1
1
10
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 5.34 Measured and simulated harmonic power vs current
at 5 GHz with a hole mobility of 150 cm'TVs. Applied V = -10 V.
Spot size = 5 and 7 pm. Experimental Data from figure 5.16.
-60
8
m
T3
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-70
-80
-90
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-130
5 um ss
7 pm ss
-140
0.1
1
10
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 5.35 Measured and simulated harmonic power versus
o
detector current at 1 GHz. Taup = 25 ps. pp = 150 cm /Vs. Applied
voltage = -10 V. Measurement data from figure 5.15.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
99
-5 0
Modulation
Depth = 100%
I
«
£§
-70
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£
£
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-90
2f Data
-100
3f Data
4 f Data
2o -no
w/pabs 7ss
-120
-130
no pabs 7ss
0.1
10
1
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 5.36 Measured and simulated harmonic power at 5 GHz.
9
Spot size = 7 pm with a hole mobility of 150 cm /Vs. With and
without p-region absorption. Measurement data from figure 5.6.
-50
P3
3
Modulation .
Depth = 100% -
-70
Sh
o -90
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Ph
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£
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2f Data
3f Data
4 f Data
w/pabs I no pabs I
s
-150
0.1
1
10
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 5.37 Measured and simulated harmonic power vs detector
current at 1 GHz. Applied voltage = -5 V. Fiber pigtailed. Spot size
o
= 7.0 pm. pp = 150 cm /Vs. Measurement data from figure 5.5.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
100
m ea su rem en t and sim u lated data a t 5 and 1 GHz, respectively, both w ith
and w ith o u t p-region absorption. T he 5 GHz sim u la tio n w ith p-region
absorption (figure 5.36) h a s not changed sig n ifica n tly com pared to the
previous sim u la tio n s (figure 5.10), and, exclu d in g p-region absorption
does not affect th e sim u lated NL. Since th e NL a t -5 V w a s determ ined by
a sp ace-ch arge-field -in d u ced ch a n g e in th e carrier v e lo citie s, th e pregion absorption n on lin ear m ech an ism does not contribute to th e device
NL a t th is applied bias and frequency. H ow ever, w hen th e frequency is
decreased to 1 GHz (figure 5.37), w here th e applied voltage is su ch that
th e NL is com prised of both m e c h a n ism s, som e subtle differences are
observed. N ote additionally th a t the 1 GHz resu lts at -5 V (figure 5.37)
now agree quite w ell in contrast to the previous sim ulations (figure 5.13).
F ifteen V olt M easurem ents an d Sim ulations
W hen the applied voltage is increased to -15 V, both th e 1 and 5 GHz
second h arm on ics have been at th eir m in im u m valu es for several V olts
(figures 5.2 and 5.3). If the NL in th is region of applied voltage is strictly
the r esu lt of the p-region absorption, as w as determ ined from the 1 GHz
sim u lation s a t -10 V, th e 5 GHz NL should now be due alm ost exclu sively
to th e p-region absorption. The m easu red and sim u lated NL a t -15 V
w ith and w ith ou t p-region absorption for spot sizes of 5 and 7 |im is plot­
ted in figu re 5.38. The NL is indeed dom inated by the p-region absorp­
tion, sin ce th e NL cannot be predicted (for a reasonable se t o f sim u la tio n
p aram eters) by in clu d in g only the space-charge-induced change in elec­
tron velocity.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
101
M odulation
Depth = 100%
PQ
s
Jh
03
£
PU
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2 f Data
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♦
>
cd
£
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3fD ata
no pabs 7ss
w/pabs 7ss
o
no pabs 5ss
w/pabs 5ss
A v e r a g e D e te c to r C u r r e n t (m A )
F igure 5.38 M easured and sim ulated harm onic power vs current
a t 5 GHz w ith and w ithout absorption in the p-region. Applied V =
-15 V. Spot sizes of 5 and 7 pm. Taup = 2 5 p s. pp = 150 cm 2 /V s .
S um m ary
In th is chapter, w e have in v estig a ted th e origins o f n o n lin ea rities
in p-i-n PD s and have, for th e first tim e, system atically identified w h ic h
n on lin ear m ech a n ism s are dom inant for different applied PD voltages.
The PD in trin sic region w idth, excitation frequency, and in cid en t spot
size all in flu en ce the exact bias voltage w here the tran sition betw een the
different reg im es occur.
T ypical r e su lts are plotted a g a in for conve­
nience in figure 5.39 (sam e as figure 5.3). Figure 5.39 show s th ese three
regim es:
1
) th e space-charge-induced change in carrier velocities dom i­
nate th e NL from the point w here the fu ndam en tal pow er sa tu ra tes (-3
V) to th e p oin t w here the NL stops d ecrea sin g (-10 V), 2) th e p-region
absorption n on lin ear m ech an ism dom inates for applied voltages greater
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
102
i i i i i
0
Other
Effects
S' -20
1 1
I
1 1 1 1 1 1 1 1
1
Space-Charge-Induced
Carrier Velocity Changes
■T 'T'V I I I
p-Region
Absorption
PQ
T -40
a>
(2
-60
1
-80
•
5 GHz t -
s
« -100
ffi
-120
0
5
10
D iod e A p p lied R ev erse B ia s V oltage (-V)
15
Figure 5.39 Regions of applied bias where different nonlinear
mechanisms dominate the second harmonic nonlinear output.
Measurement data from figure 5.3.
than -10 V, and 3) at low voltages (< 3 V) where significant twodimensional carrier flow and electric field redistribution (see Chapter
VII) dominate the device NL. The knowledge obtained from the results of
this chapter will allow device parameters such as the physical device
diameter, intrinsic region length, and intrinsic region doping densities
to be investigated to determine if improvements to the inherent device NL
can be made. These issues will be covered in Chapter VIII.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
103
1.
P. Hill, et al., "Measurement of Hole Velocity in n-Type InGaAs,"
Appl. Physics L e tt, 90, p. 1260,1987.
2. M. Dentan and B. de Cremoux, "Numerical Simulation of the Non­
linear Response of a p-i-n Photodiode Under High Illumination,"
J .o f Lightwave Tech., JLT-8 ,p. 1137,1990.
3.
R.R. Hayes and D.L. Persechini, “Nonlinearity of p-i-n Photodetec­
tors,” IEEE Photonics Tech. L e tt, PTL-5, p. 70,1993.
4.
T.P. Pearsall, Editor, "GalnAsP Alloy Semiconductors", John Wiley
and Sons, 1982.
5.
6
K.J. Williams, unpublished results.
. G.B. Lush, et al., “Determination of Minority Carrier Lifetimes in ntype GaAs and Their Implications for Solar Cells,” 22nd IEEE Photo­
voltaic Specialists Conference, p. 182, 1991.
7.
M. Kot and K. Zdansky, “Measurement of Radiative and Nonradiative Recombination Rate in InGaAsP-InP LED’s,” Quantum Elec­
tronics Lett., 28, p. 1746,1992.
8
. P. Jenkins, et al.,
“M inority Carrier Lifetim es in
Indium
Phosphide,” 22nd IEEE Photovoltaic Specialists Conference, p. 177,
1991.
9.
A.R. Adams, et al., "The temperature dependence of the Efficiency
and Threshold Current of In :_xGaxAsyP1_jr Lasers Related to Interva­
lence Band Absorption," Jpn. J. Appl. Phys., 19, p. L 621,1980.
10. G.A. Landis, et al., “Photoluminescence Lifetime Measurements in
InP Wafers,” 22nd IEEE Photovoltaic Specialists Conference, p. 636,
1991.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
104
VL LOW POWER DENSITY NONLINEARIT1ES IN DIFFERENT
p-i-n STRUCTURES
Introduction
This chapter will investigate p-i-n PD NL under low power density
conditions in three devices. Low power density conditions are defined as
power densities which generate low enough space-charge electric fields
(eqn. 3.35) such that no part of the intrinsic region electric field (eqn. 3.32)
collapses. Two basic device structures are under investigation here: one
double-heterostructure 1 mesa-type device with a
0 .2
-pm long intrinsic
region fabricated at the University of California Santa Barbara, and two
single-heterostructure 2 mesa-type devices w ith 0.5 and 0.95-pm long
intrinsic regions fabricated by GTE Laboratories.
These particular
devices were chosen because they all have bandwidths over 20 GHz and
they cover a range of intrinsic region thicknesses. Since the devices are
made by research groups, the layer structure and doping profile are
available -with reasonable accuracy. At the end of this chapter, data from
several additional PDs will be presented for completeness, although these
devices will not be modeled.
Q.95-jim D evice Measurements and Sim ulations
The 0.95-pm long intrinsic region device is pigtailed with a single
mode optical fiber and yields an incident intensity e -2 spot size of approxi­
mately 10 pm. The doping density versus position for the specific device
under investigation was not available; however, data for similar devices
was available. Although each device has slightly different characteris­
tics, the doping profile is estimated from a range of available measured
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
105
devices, w ith the specific profile u tilized for sim u la tio n purposes sh o w n
in figure 6.1. T he sim ulated device h a s a 0.95-pm un intentionally-doped
lattice-m atch ed n-InG aA s in trin sic region grow n on an InP substrate,
w ith a
1 .0
-pm h eavily doped p-InGaAs cap layer.
The bin sp acin g versus photodiode position is show n in figure 6.2
for th e 0.95 pm device. The bin w idths w ere shortened n ear the p-i an d
th e n-i interfaces to lim it the change in the electric field and carrier den­
sities betw een adjacent bins.
T he m easu red NL at 1 mA o f average PD current plotted v ersu s
detector applied voltage can be found in figu res 5.1 to 5.4 for 100 M Hz, 1
GHz, 5 GHz, and 10 GHz, respectively. T hese curves (figures 5.1 to 5.4)
and th e inform ation th ey contain for d eterm in in g th e dom inant n o n lin ­
ear m ech a n ism s w as the focus of C hapter V. T he sim u lation resu lts for
th is device can also be found in C hapter V (figures 5.34 to 5.36) for fre­
quencies of 1 and 5 GHz at applied voltages o f -5, -10, and -15 V olts and
10
n -T y p e In P
CO
> 1 .9 5 p m
p -T y p e In G a A s
0 - 1 . 0 j im
n -T y p e In G a A s
1 .0 - 1 . 9 5 p m
0.0
0.5
1.0
1.5
2.0
D io d e X P o s it io n (p m )
Figure 6.1 D oping profile for the 0.95-pm long intrinsic region device.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
0.0
0.5
1.0
1.5
2.0
D io d e X P o s i t i o n ((im )
Figure 6.2 Diode bin width versus position, X, utilized in device
sim ulations. 0.95-pm long intrinsic region.
will not be repeated here. The device NL at 10 GHz, with simulation
parameters which yielded good results at
1
and 5 GHz, are plotted in fig­
ures 6.3, 6.4 and 6.5 for applied voltages of -5, -10, and -15 Volts, respec­
tively. The NL in figures 6.3 to 6.5 are consistent with the results from
Chapter V. The -10 V plot (figure 6.4) demonstrates that the total NL is a
combination of a component from the space-charge-induced change in
electron velocity and a component from the p-region absorption. The nonlinearities from these mechanisms are approximately equal because the
simulated NL decreases by 3 dB when the p-region absorption nonlinear
mechanism is neglected. At an applied voltage of-15 V (figure 6.5), the
NL is almost entirely a function of the p-region absorption since the NL
decreases by 10 dB when this mechanism is excluded.
Various plots are generated to compare the properties of this device
to the shorter intrinsic region length devices. The carrier densities and
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
107
-10
Modulation ;
Depth = 100%—
-30
09
T3
03
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O
-50
-70
03
10 GHz
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cd -90
£
o
u
-110
-130
0.1
1
10
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 6.3 Measured and simulated harmonic power versus
average detector current at 10 GHz. Applied voltage = -5 V.
(ip = 150 and 175 cm2 /Vs. Spot size = 7.0 (im.
-10
|
Modulation
Depth = 100%
-30
3
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w/p abs
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no p abs | “
0.1
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10
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 6.4 Measured and simulated harmonic power versus
average detector current at 10 GHz. Applied voltage = -10 V.
2
|ip = 150 c m ffs . Spot size = 7.0 pm.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
108
-10
Modulation !
Depth = 100%—
-30
-50
-70
10 GHz
■■
-90
w/p abs
no p abs
-130
0.1
1
10
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 6.5 Measured and simulated harmonic power versus
current at 10 GHz. Applied V = -15 V. With and without
o
p-region absorption. pp = 150 cm /Vs. Spot size = 7.0 pm.
electric field are plotted in figure
6 .6
for an applied voltage of -15 V and 1
mA of average PD current. The space-charge electric field is plotted in
figure 6.7 for PD currents of 100 pA and 1 mA with a simulated spot size
of 7 pm. It will be shown that the space-charge electric field will have a
major impact on the maximum possible PD current before the onset of
high-power-density nonlinearities similar to those under investigation in
Chapter VII.
A summary of the simulation and device parameters are given in
figure 6 .8 . The left hand column in figure
6 .8
lists the physical and m ea­
sured characteristics for the device, while the right hand column lists
the simulation specific parameters which yielded the best-fits to the m ea­
surement data.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
109
-100
Field
-150
-200
10
0.0
0.50
1.0
( k V /c m )
03
Q
u
03
U
t-i
cc
-50
E lectric
03
+3
-250
2.0
1.5
D io d e X P o s i t i o n ( |im )
Figure
6 .6
Carrier densities and electric field with an average PD
current of 1 mA. 0.95-(im long intrinsic region. Applied V =-15 V.
a
o
>
fa
03
be
&
_
-5
O
g -10
sc
cS
&
CQ
-15
1.0
1.2
1.4
1 .6
1 .8
D io d e X P o s i t i o n (jim )
Figure 6.7 The space-charge electric field in the intrinsic region
due to the photogenerated carriers. 0.95-pm long intrinsic region.
Spot size = 7.0 pm. Applied voltage = -15 V.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
110
Physical Parameters
p-region Length
p-region Doping
WP
1.0 pm
Simulation Parameters
Hole Mobility
Up
150-175 cm2/Vs
Electron Mobility
Fn
6000-8000 cm2/Vs
Na
7 x 1018 cm*3
i-region Length
wi
0.95 pm
Hole Saturated
i-region Doping
Ndi 5 x 1015 cm-3
Velocity
n-region Length
wn
0.1 pm
Electron Saturated
n-region Doping
Velocity
Nd 2 x 1017 cm-3
Diameter
Electron Scattering
- 30 pm
Parameter
Device and Measurement
Characterisitics
Incident Spot Size
10 pm
DC Quantum
Efficiency
0.7 AAV
•
20 GKz
Figure
6 .8
Vnhf
5.4 x 10®cm/s
nh
1 x lO^cm*3
Ph
7 x lO^cm*3
P
0.6-0.8 x 10'7
Hole Scattering
Parameter
X
1319 nm
Fitting Parameter
Recombination
Laser Measurement
Wavelength
4.8 x 10®cm/s
Electron Velocity
T\
-3 dB Frequency
Response
Vphf
Time, i-region
Tn
Recombination
Tp
Time, p-region
Tn
2 ns
100 ps
0.95 pm device characteristics and simulation parameters.
0.5-pm Device M easurements and Simulations
The 0.5-pm long intrinsic region devices will be modeled with sim i­
lar material parameters that were assumed for the (earlier) 0.95-pm ver­
sions of the detector. The doping density versus position for the 0.5-pm
device is shown in figure 6.9. A 0.4-pm long p-InGaAs cap layer and a
Q.l-pm long n-InP substrate with the same doping concentrations as the
0.95-pm device w ill be assumed.
Since hole transport into the n-InP
region was forbidden (Chapter III), the physical length of the n-InP is
not required so long as enough region is modeled to prevent the depletion
region from reaching the n-contact. Any additional transparent InP acts
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
I ll
1
I
I
I
I
I
I
I
1
I
I
1
n -T y p e I n P
> 0 .9 p m
CO
I
-
j
n - T y p e In G a A s .
0 .4 - 0 .9 p m
o
Q
i i i i i i i I i i i i i i . L
1 0 15
0.0
0.2
0.4
0.6
0.8
1 .0
D io d e X P o s i t i o n (p m )
Figure 6.9 Doping profile versus position for the 0.5-pm long intrinsic
region device.
like a resistor since the electron current in this region is dominated by
the drift current. The simulation bin widths for the 0.5-pm device are
about half the size of the bin widths used for the 0.95-pm device.
Figures 6.10 to 6.13 plot the measured NL of this detector with an
average PD current of 1 mA and a 100% modulation depth for frequencies
of 100 MHz, 1 GHz, 5 GHz, and 10 GHz, respectively. The m easurem ent
data shows similar characteristics to the data from the 0.95-pm device
(figures 5.1 to 5.4). The NL decreases as the applied voltage increases up
to a given voltage where, for higher applied voltages, the NL is approxi­
mately unchanged. The transition between these regions are not as dis­
tinct compared to the 0.95-pm device; however, the same overall trends
are observed. For example, the transition from the two regions should be
frequency dependent in such a way that the transition should occur at
lower applied voltages for lower frequencies. The transitions occur (fig-
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
112
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a
-1 6 .5
-17.2
4
5
D io d e A p p lie d R e v e r s e B i a s V o lt a g e (-V )
Figure 6 .10 Measured fundamental and harmonic power versus
detector applied reverse bias voltage at 100 MHz. Fiber pigtailed.
Average detector current = 1 mA.
-16
-2 0
1 GHz
-60
-18
Modulation
Depth = 100%
-80
-19
-120
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1
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5
D io d e A p p lie d R e v e r s e B i a s V o lt a g e (-V )
Figure
6 .1 1
Measured fundamental and harmonic power versus
detector applied reverse bias voltage at 1 GHz. Fiber pigtailed.
Average detector current =
1
mA.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
113
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4
5
D io d e A p p lie d R e v e r s e B i a s V o lt a g e (-V )
Figure
6
.12 M easured fundam ental and harm onic power versus
detector applied reverse bias voltage a t 5 GHz. Fiber pigtailed.
A verage detector current = 1 mA.
-40
•
•
• i • • *
• ....... * ........* ........t ....
-5 0
• •
*
-18
.......
% -60
-70
2f
•
10 GHz
-19
B ■ B
M odulation
D epth = 100%
C3
-90
1
0
■ ■
1
1
1
■ ■ ■ ■ * =
2
i-20
1
■
1
1
3
1
■
1
■
-21
1
4
5
D io d e A p p lie d R e v e r s e B i a s V o lt a g e (-V )
Figure 6.13 M easured fundam ental and harm onic power versus
detector applied reverse bias voltage at 10 GHz. Fiber pigtailed.
A verage detector current = 1 mA.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(d B m )
I-
P ow er
/N
cS
.2
•-17
F u n d a m e n ta l
PQ
s
—i— -16
-j-- 1—r*
I I I I
1 1 1 1
114
ures 6.10 to 6.13) at applied voltages of approximately -3, -1, -1, and -2
Volts for frequencies of 100 MHz to 10 GHz, respectively, which contradict
the results from Chapter V. Simulations on this device will be limited to
5 GHz, since, due to the reduced bin width, the simulation time required
for a given time interval is about twice that of the 0.95-pm device. The
simulations will concentrate on the differences and possible advantages
over the 0.95-pm device nonlinearities.
If the dominant nonlinear mechanism is the p-region absorption
for high applied voltages, the NL should be independent of the intrinsic
region length. The data from figures 5.1 to 5.4 and figures 6.10 to 6.13
certainly support this hypothesis, with the NL approximately indepen­
dent of intrinsic region thickness (and frequency) for both devices. Later
in this chapter, the
0 .2
-p.m long intrinsic region device will also show
similar NL at high applied voltages. A question thus remains: do shorter
intrinsic region devices have any advantages in nonlinear performance if
the p-region absorption is excluded (which may require a p-type contact
in a high bandgap material such as InP or InGaAsP)?
The simulated nonlinearities of the 0.5-pm device with a spot size
of 7 pm, hole mobilities of 100 and 200 c m W s, and with p-region absorp­
tion are plotted in figure 6.14. Also plotted are the results with a hole
mobility of 100 cm 2/Vs without p-region absorption. The NL is indeed
dominated by the p-region absorption since the simulated second har­
monic decreases by 60 dB when p-region absorption is neglected. This is
a substantial decrease in NL compared to the results for the 0.95-pm
device operated a t -15 V (figure 5.36) where the decrease was only about 10
dB for the same simulation spot size. The difference in the second har-
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
115
-50
a -60
P Q
S
-70
|
-8 0
fS -90
<u
>
-100
G3
£
1-110
N
|- 1 2 0
5 GHz
no p abs
Second
Harmonic
-130
0.1
1
10
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 6.14 Measured and simulated harmonic power versus
average detector current at 5 GHz. 0.5-pm long intrinsic region.
Applied V = -4 V. Fiber pigtailed. Spot size = 7.0 pm.
monic between the two devices without p-region absorption may be the
result of several mechanisms. First, the electric field (figure 6.15) for the
0.5-pm device at -4 V is much flatter across the intrinsic region com­
pared to the electric field (figure
6 .6
) for the 0.95-pm device at -15 V even
though the electric field in the intrinsic region is higher for the 0.95-pm
device. Second, the space-charge electric field (figure 6.16) for the 0.5-pm
device is about half the space-charge electric field of the 0.95-pm device
(figure 6.7) operating with the same current density, since the electrons
and holes are separated, on average, by
1/2
the distance in the intrinsic
region of the 0.5-pm device compared to the 0.95-pm device. A summary
of the simulation and device parameters for the 0.5-pm device are given
in figure 6.17.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
116
E lectric
Field
ao
m
a
•H
+3
♦H
cn
125
0.0
0.20
0.40
0.60
0.80
(k V /c m )
•100
u
o3
-150
1.0
D io d e X P o s i t i o n (p.m )
Figure 6.15 Carrier densities and electric field with an average
detector current of 1 mA. 0.50-pm long intrinsic region.
4
o
2
0
fa
<u
bo
u -2
a
43
Oi
Q
OJ -4
a
Qh
CQ
sc
-6
0.40
0.50
0.60
0.70
0.80
0.90
D io d e X P o s i t i o n ( |im )
Figure 6.16 Space-charge electric field in the intrinsic region
due to the photogenerated carrier densities. Average detector
currents of 100 pA and 1 mA. 0.50-pm long intrinsic region.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
11 7
Physical Parameters
p-region Length
wp
p-region Doping
Na 7 x 1018 cm'3
i-region Length
wi
i-region Doping
Ndi 5 x 1015 cm-3
n-region Length
wn
n-region Doping
Nd 2 x 1017 cm"3
Diameter
0.4 pm
0.5 pm
0.1 pm
-
20 pm
Simulation Parameters
Hole Mobility
^P
150-175 cm/Vs
Electron Mobility
Fn
8000 cm 2/Vs
Vphf
4.8 x 106cm/s
Vnhf
5.4 x 106cm/s
nh
1 x lO ^ cm 3
Ph
7 x lO ^ cm 3
Hole Saturated
Velocity
Electron Saturated
Velocity
Electron Scattering
Device and Measurement
Parameter
Characterisitics
Hole Scattering
Incident Spot Size
-
10 pm
T\
0.3 AAV
-
> 24 GHz
DC Quantum
Efficiency
Wavelength
P
0 .8x 10*7
Recombination
Laser Measurement
K
Electron Velocity
Fitting Parameter
-3 dB Frequency
Response
Parameter
1319 nm
2 ns
Time, i-region
Recombination
XP
Time, p-region
^n
100 ps
Figure 6.17 0.5 pm device characteristics and simulation parameters.
0.2-fim Device Measurements and Sim ulations
The shortest of the tested device intrinsic lengths is a 0.2-pm
double-heterostructure device. The device is a 10 pm x 10 pm device hav­
ing a layer structure identical to 4 pm x 4 pm devices reported in refer­
ence [1], The layer structures are quite complicated from the modeling
point of view. Therefore, a three-part single-heterostructure model sim i­
lar to the previous devices will be utilized, with modifications to the
assumptions for the p-i heterojunction. The device size is approaching
the lim its where quantum effects and finite carrier acceleration times
may affect or even dominate carrier transport. Therefore the solutions to
the continuity and Poisson's equations with the assumptions made in
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
118
C hapters III and IV on su ch sm all geom etries m ay n o t yield accu rate
resu lts. H ow ever, th e m odel w ill he used to determ ine ju s t how m u c h
inform ation can be obtained w ith out in clu d in g m ore difficult (from the
m odeling point o f view ) transport m echanism s.
The doping d e n sitie s 1 for the 0.2-pm device are given in figure 6.18.
The heavily-doped p-InP layer, the n-InG aA s in trin sic region, and the nInP substrate extend from 0 to 0.19 pm , 0.19 to 0.39 pm, and 0.39 to 0.5
pm, respectively. T his device differs from the previous two devices by a n
addition o f a secon d heterostru ctu re c o n sistin g o f a la y e r o f p-InP
betw een th e p-InG aA s cap la y er and th e n -In G aA s in tr in sic region .
T his additional h eterostru ctu re keeps th e electron s gen erated in th e pInG aAs cap layer from trav elin g into the in trin sic r e g io n .3 The device
NL m ay be characterized by a sin gle heterostructure w ith ou t absorption
in the p-region, since the electrons generated in the p-InG aA s cap layer
do not m ake it to th e in trin sic region. The m easu red fu n d am en tal and
ao
CO
G
Q
bo
G
P<
o
Q
0
0.1
0.2
0.3
0.4
0.5
D io d e X P o s itio n (p m )
Figure 6.18 D oping density versus position for the 0.2-pm photodiode.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
11 9
harmonic powers are plotted in figures 6.19 to 6.22 versus applied voltage
for frequencies of 100 MHz, 1GHz, 5 GHz, and 10 GHz, respectively. The
nonlinearities are approsdmately frequency and applied voltage indepen­
dent, with a second harmonic of -78 dBm for applied voltages greater
than 1 V, and slightly better (-80 to -90 dBm) results at 100 MHz. These
results are characteristic of the p-region absorption nonlinear m echa­
nism discussed in Chapter V. In Chapter V, the nonlinearity from the pregion absorption was observed to be a function of how many electrons
made it to the intrinsic region by moving the focal position of the p-region
hole-expansion function and by varying the electron velocity in the pregion. However, the NL may contain a component from the generation
and movement of both carriers in the p-InGaAs cap layer, whether or not
electrons actually travel into the intrinsic region. Simulations at 5 GHz
-50
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£ -80
o
Oh
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2f
3f
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♦
9
4f
A
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•
-16.6 g
o*
SB
. -16.8 g
©
-
100 MHz j
B
■
♦
♦
-110
A
-120
t
0
♦
t
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........ ■
-17
-100
ffi
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O
-60
► ■
s
PQ
:
1
;
ft
B
■
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■
|
*
*
-17.2 J
7 .......... ♦ .... !.... * ..........* ..........1.....7 .........7
.............. .. ... i ........ . ♦. ...............i ........... ..........
♦
;
♦
A I A A
»
0.5
1
1
«
1
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1
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|
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1
-17.4 2.
K
-17.6 W
r
1.5
2
-17.8
D io d e A p p lie d R e v e r s e B i a s V o lt a g e (-V )
Figure 6.19 Measured fundamental and harmonic power vs applied
reverse bias voltage at 100 MHz. Average PD current = 1 mA.
o
Incident e spot size = 10 pm. Modulation depth = 100%.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
120
-20
pa
£
£
&
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-1 7
2f
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♦
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▲
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£
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-60
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■ gi ®
®
♦
I i .....
-1 0 0
g
▲
♦
;
A *
A
A
A
K
*2
1 GHz
3f
I
120
i
i l
-140
0
0.5
1
-19 |
<o
......
♦
<-i
N
A' ▲
.
aW
-20 3
.
1.5
2
D io d e A p p lie d R e v e r s e B i a s V o lt a g e (-V )
Figure 6.20 Measured fundamental and harmonic power vs applied
reverse bias voltage at 1 GHz. Average PD current = 1 mA.
o
Incident e spot size = 10 pm. Modulation depth = 100%.
-20
s
«
T3
I I
1
I | I I
1
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1
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1
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1
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£
3
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5 GHz S
-40
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£
-120
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i l l l
0
i ■l■
1 1 1 1
0.5
1
1 1.
♦
♦ '
i I i l l
1.5
2
>-S
-21 a*
Cd
-22
D io d e A p p lie d R e v e r s e B i a s V o lt a g e (-V )
Figure 6.21 Measured fundamental and harmonic power vs applied
reverse bias voltage at 5 GHz. Average PD current = 1 mA.
Incident e’ spot size = 10 pm. Modulation depth = 100%.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
0
0.5
1
1.5
2
D i o d e A p p l i e d R e v e r s e B i a s V o l t a g e ( -V )
Figure 6.22 M easured fundam ental and harm onic power vs applied
reverse bias voltage at 10 GHz. Average PD current = 1 mA.
o
Incident e' spot size = 10 pm. M odulation depth = 100%.
w ill help to ju stify or contradict th ese hypotheses.
The NL at 5 GHz neglectin g p-region absorption is plotted in figu re
6.23, w here the 0.2-pm device is sim ulated w ith and w ithout a 50
load
resistan ce. The m easu rem en t data w as tak en w ith an incid en t e - 2 in ten ­
sity spot size of 10 pm. The data show th a t th e device NL cannot be m od­
eled w ithout in clu d in g the p-region absorption, even though the ca rriers
are not perm itted to travel from th e p-InG aA s cap layer into th e in tr in sic
region.
T he device is sim u la ted w ith a 5 pm spot size, 50% o f th e e "2
in ten sity spot size, to m axim ize (w ithin reason) th e NL due to th e spacecharge electric field nonlinear m ech a n ism s.
Since th e device NL cannot
be m odeled w ith ou t absorption in the p-region (figure 6.23), additional
sim ulations are required to determ ine th e source of the NL behavior.
The non lin earities m ay be the resu lt o f carrier generation in the p-
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
122
-50
a
PQ
i iii
i i i i i i |'
-70
•
2fD ata
■
3fD ata
w/o 50 Ohm
T3
w/ 50 Ohm
o
> -110
o
2nd
H arm onic
3rd Harmonic
-150
0.1
1
10
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 6.23 Measured and simulated harmonic power vs current
at 5 GHz neglecting p-region absorption. Simulated with and
without a 50 Ohm load resistor. Spot size = 5 pm. Applied V= -2 V.
InGaAs cap layer even if the electrons do not enter the intrinsic region.
To test this hypothesis, two additional simulations are needed. One sim ­
ulation allows p-region absorption in the entire region except for
20
nm
which is nearest to the intrinsic region, allowing the electrons generated
in the p-region to recombine before they reach the intrinsic region. The
second simulation allows absorption in the entire p-region; however, the
electrons are prohibited from leaving the p-region, which is similar to the
effect of the second heterojunction. The simulated results (figure 6.24)
demonstrate that the NL still cannot be modeled simply by the generation
and movement of electrons or holes in the p-region, which implies that
the electrons must travel into the intrinsic region to effect the device NL.
A closer look at the device 1 reveals that between the intrinsic region
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
123
-5 0
2 f Data
-70
3f Data
No Gen
| -90
o
Ph
£ -1 1 0
CQ
£
o
v -130
No Move
2nd Harmonic
S
3rd Harmonic
-150
0.1
10
1
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 6.24 Simulated harmonic power at 5 GHz. One simulation
excludes genereration in the p-region near the p-i interface, and the
second simulation prohibits electron flow from the p- to the i-region.
(n-InGaAs) and the second heterostructure (p-InP), the device contains a
graded bandgap layer (GBL).
The GBL consists of alternating layers
totaling 3.6 nm oflnGaAs and InP which help to lower the band disconti­
nuities and prevent hole trapping at the heterojunction interface .4 The
depletion depth into the p-region is only about
1
nm due the high doping
level ( 8 x 1018 cm-3) of the InP, therefore much of the InGaAs here may be
undepleted. The device cannot be thought of as a simple p-n junction in
this region due to the small thicknesses (< Inm ) of the layers. However,
to determine how much undepleted InGaAs is necessary to yield the NL
in figure 6.23, the PD is simulated again, this time allowing p-region
absorption near the p-i interface. The simulated NL in figure 6.25 con­
sists of four different lengths (3.6, 8.2, 14, and 190 nm) of absorbing pInGaAs located between the p-InP and the intrinsic region. The results
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
124
-50
?
m
5
-70
|
-90
cS
2 f Data
3f Data
3.6 nm
8.2 nm
14 nm
190 nm
> -110
c3
£
o
b -130
-150
0.1
1
10
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 6.25 Measured and simulated harmonic power vs current
at 5 GHz. Simulations for various lengths of absorbing p-InGaAs
next to the i-region. 0.2-jim long intrinsic region. Spot size = 7 |im.
in figure 6.25 suggest that only about 8 to 14 nm of essentially undepleted
absorbing material next to the intrinsic region is required to model the
device nonlinearities.
Although the device only contains 3.6 nm of InGaAs near the
intrinsic region, the NL may still be solely from this region. The NL may
be enhanced by hole trapping at the heterojunction interface, which may
account for the decrease in the second harmonic at higher currents due
to filling of the trap sites. However, trapping mechanisms are not con­
sidered in the present model. Similar reductions in the second har­
monic at higher currents are observed at 100 MHz, 1 GHz, and 10 GHz.
The carrier densities and electric field for this device are plotted in
figure 6.26 for an applied voltage of -2 V and 1 mA of average current.
The space-charge electric field is plotted in figure 6.27 for PD currents of
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
1 25
-50
-75
-100
-125
-150
0.0
0.10
0.20
0.30
(k V /c m )
fi
O)
Q
Eh
a>
Si
ft
O
Field
CO
E lectric
-25
co
.2
*-P
—^ -175
0.50
0.40
D io d e X P o s i t i o n (p m )
Figure 6.26 Carrier densities and electric field with an average PD
current of
1
mA. 0.20-pm long intrinsic region. Simulated spot
size = 7 pm. Applied voltage = -2 V.
1.0
Eh
o
>: 0.5
© -0-5
bo
£
-l.o
oI
o
2ft -1-5
Cu
CO
2.0
sc
-
0.20
0.25
0.30
0.35
D io d e X P o s i t i o n (p m )
Figure 6.27 Space-charge electric field in the intrinsic region due
to the photogenerated carrier densities. Average PD currents of
100 pA and 1 mA. 0.2-pm long intrinsic region. Spot size = 7 pm.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
126
100 |iA and 1 mA with a sim ulated spot size of 7 pm.
Notice that the
space-charge electric field is about 1/5 (the ratio of the intrinsic region
lengths) the space-charge field of the 0.95-pm device (figure 6.7) for the
same current density. A summary of the simulation and device parame­
ters for the 0.2-pm PD are given in figure 6.28.
Physical Parameters
Simulation Parameters
Hole Mobility
PP
100-150 cm2/Vs
Electron Mobility
Fn
8000-10000 cm2/Vs
i-region Doping
Na 2 x 1019 cm'3
wi
0.20 pm
Ndi 5 x 1015 cm'3
Vphf
4.8 x 106cm/s
n-region Length
wn
Electron Saturated
v nhf
5.4 x 106cm/s
p-region Length
WP
p-region Doping
i-region Length
0.190 pm
0.115 pm
Hole Saturated
Velocity
n-region Doping
Velocity
Nd 2 x 1017 cm'3
Diameter
Electron Scattering
10 pm
nh
Device and Measurement
Parameter
Characterisitics
Incident Spot Size
10 pm
DC Quantum
Efficiency
n
Response
Parameter
Ph
7 x lO^cm"3
P
0.8-1.0 x 10'7
XP
2 ns
^P
Tn
100 ps
Electron Velocity
Fitting Parameter
Recombination
-
27 GHz
X
1319 nm
Time, i-region
Recombination
Laser Measurement
Wavelength
Hole Scattering
0.3 AAV
-3 dB Frequency
1 x 1017 cm"3
Time, p-region
Figure 6.28 0.2 pm device characteristics and simulation parameters.
Additional Measurements
Nonlinearity measurements on a wide range of devices from differ­
ent manufacturers were made to exam ine the differences, i f any,
between devices. Figure 6.29 is a listing of these measurements for eight
devices where the measurements have been reduced to seven character­
istic numbers. These characteristic numbers are the second and third
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
127
Detector
DC Q uantum
Efficiency (A/W)
50 MHz Q uantum
E fficiency (A/W)
B andw idth
-3 dB (GHz)
1
2
3
4
5
6
7
8
0.71
0.30
0.30
0.78
0.85
0 .2 1
0.70
0.84
0.71
0.30
0.30
0.78
0.70
0 .1 0
0.70
0.76
20
>24
27
18
14
22
4
12
>24
>24
>40
24
20
26
10
16
B andw idth
- 6 dB (GHz)
A pplied V oltage (-V)
(M in 2 f @ 5 GHz)
10
1
-6 8
5 GHz 2 f © 1 m A (dBc)
-76
-94
-72
-99
-71
5 GHz 3 f @ 1 m A (dBc)
-8 6
10 GHz 2f@ 1 m A (dBc)
-76
1 0 0 M H z 2 f@ lm A (d B c )
100 M Hz 3 f @ 1 m A (dBc)
1 GHz 2 f @ 1 m A (dBc)
1 GHz 3 f @ 1 m A (dBc)
-79
-55
-73
-55
-75
-57
0.5
-74
-8 6
-63
-83
-62
-82
-61
8
-54
-58
-45
-57
-6 8
-80
-55
15
-48
-78
-50
-87
-54
-89
-60
-77
-57
-73
-47
-6 8
N ot
C lear
-38
-42
-30
-38
-48
-63
-70
-1 0 0
-60
-92
-61
-90
-65
Figure 6.29 M easurem ent data for eight photodetectors. D etectors
three through eigh t are from six different m anufacturers.
harm onics of 100 M Hz, 1 GHz, and 5 GHz and the second harm onic of 10
GHz, all m easured at
1
mA, 100% m odulation depth, and a t the h ig h est
applied detector voltage allow ed by the m anufacturer. D etectors one, tw o,
and three are the 0.95-pm , 0.5-pm and 0.2-pm long devices d iscu ssed in
th is chapter. D etectors four through eigh t are from other m a n u fa ctu r­
ers. V ariation s in th e NL of up to a few dB can occur (for exam ple, see
figure 6.10 or 6.19) a t h igh bias voltages; therefore, the harm onic levels
stated are averages of the NL at high applied voltages and can be consid­
ered to have an error of up to ± 5 dB. W ith the exception of detector one,
no device appears to have low er NL at all four frequencies. The resu lts in
figure 6.29 help to ju stify th a t the NL are in trin sic properties o f PD s and
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
128
not isolated observations in m esa-type devices or devices from a p articu ­
lar m anufacturer.
S um m ary
In th is chapter, th e device ch aracteristics and best-fit sim u la tio n
param eters for th ree in trin sic region len g th devices w ere presented.
Sim ulations of th e 0.5-|im and 0.2-jim PDs at h ig h applied voltages and 5
GHz confirm th a t th e NL is dom inated by p-region absorption n ea r the
in trin sic region sim ila r to the resu lts obtained for th e 0.95-pm lon g
devices in C hapter V.
It w as found th a t only
8
to 14 nm o f undepleted
absorbing m aterial n ext to the in trin sic region is sufficient to resu lt in a
second harm onic of -60 dBc at 1 mA.
The photogenerated space-ch arge
electric field w as observed to scale in versely proportional to the in tr in sic
region th ick n ess for a given incident spot size.
T his im p lies th a t for a
given in cid en t optical spot size, shorter in trin sic region P D s w ill have
lower NL associated w ith th e change in carrier velocities. NL m e a su r e ­
m en ts from other devices at high applied voltages confirm th a t th ese
effects are in h eren t properties of currently available devices.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
129
1.
Y.G. Wey, et al., "Ultrafast Graded Double-Heterostructure G alnA s/
InP Photodiode,” Appl. Physics Lett., 58, p. 2156,1991.
2.
P. Hill et al., "Measurement of Hole Velocity in n-Type InGaAs,"
Appl.Physics Lett., 50, p. 1260,1987.
3.
Y.G. Wey, "High-Speed Double Heterostructure GalnAs/InP p-i-n
Photodiodes Theory, Fabrication and Measurement," University of
California Santa Barbara Ph.D. Dissertation, 1993.
4.
J.G. Wasserbauer, et al., “Specific Contact Resistivity of InGaAs/InP
p-Isotype Heterojunctions,” Electron. Lett., 28, p. 1568,1992.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
13 0
VH. HIGH POWER DENSITY NONLINEARITIES: A 0.95-pm
DEVICE
Introduction
This chapter will study p-i-n photodiode nonlinearities under high
power density conditions. High optical power densities obtained from the
focusing of light into a PD or from high incident optical powers result in
large space-charge densities (-
1 0 16
cm"3), which cause the electric field
to redistribute and collapse over part of the intrinsic region. These effects
can occur when a 0.95-pm long intrinsic region PD, biased at -5 V, is illu ­
minated with power densities greater than 3 kW/cm2. The nonlinear
effects associated with high power densities can result in additional NL
effects distinct from those of Chapter V. At high power densities the PD
can cause highly detrimental system characteristics, not only substan­
tially higher harmonic content, but also a reduction in diode highfrequency responsivity and (time domain) phase shifts. This decrease in
diode responsivity is shown (below) to be due to an increase in transit
across areas within the intrinsic region which have low electric fields
and unsaturated carrier velocities. The redistributed intrinsic region
electric field also produces a radial electric field component.
This is
shown to result primarily in electron movement towards the center of the
device, thereby increasing the electron density along the axis of the inci­
dent signal.
To avoid the high power density conditions, why not just increase
the incident optical spot size? To avoid reducing the net quantum effi­
ciency, an increase in spot size m ust be lim ited to the active area.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
131
Furtherm ore, for h igh frequency devices th e active area m u st be sm all to
ach ieve low capacitance.
For exam ple, 20-GHz devices are lim ited to
m axim u m device diam eters o f about 30 m icrons.
W ith the optical power
lev els considered here, a 30-|im device diam eter is large enough to avoid
th e h ig h
c a rr ie r
d e n s itie s u n d e r
in v e s tig a tio n in
th is
chapter.
N ev erth eless, m easu rem en ts and sim u la tio n s w ill be presen ted in this
chapter for sm aller incident spot sizes (<
10
Jim) to study and u n d ersta n d
th e high-pow er-density characteristics of PD s. The resu lts obtained from
m e a su r e m e n ts and sim u la tio n s a t sm all spot size s w ill th u s provide
in sig h t into th e non linear effects w hich m ay occur in the n ex t gen eration
o f h ig h current (10 to 100 mA) 20-GHz PDs; for u ltra-h igh speed (> 100
G Hz), sm all d iam eter (< 4 (im) PDs; and in current PD s w ith excessive
incid en t optical powers (> 100 mW).
M easurem ent D ata
M easu rem en ts of the frequency response w ith in crea sin g lev els of
optical pow er reveal th e on set of h ig h pow er n o n lin ea rities.
T he fre­
quency resp on se of a 0.95-|im PD plotted relative to th e resp o n se o f the
device at low average currents is show n in figure 7.1. The average cu r­
ren ts in th e device are 800 and 1000 jjA, the applied bias is -5 V, th e m od u­
lation depth is
100
%, and the m easured e ' 2 spot size of the in cid en t optical
b eam is 5.75 ± 0.25 |im . The response reductions observed in figu re 7.1
are very unexpected since the electric field (see figure 5.15 or 7.11) is h ig h
en ou gh in the depletion region to nearly satu rate th e carrier velocities
and th e load resista n ce (50 Cl) is too low to result in a sufficien t drop o f the
extern ally applied b ias voltage. M easu rem en ts o f th e low -pow er-density
resp on se confirm th e presence o f a fully-depleted in trin sic region sin ce
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
132
CQ
0
3
0
CD
800 pA
e -0.5
o
a
CO
Q -1
«
O
>
• »—( -1.5
C3
0)
■2
1000
0
5
10
pA
15
20
25
F r e q u e n c y (G H z)
Figure 7.1 Large-signal relative frequency response of a 0.95 pm PD.
2
Average currents of 800 and 1000 pA with an e' incident spot size of
5.75 ± 0.25 pm. 200 point resolution. 100% modulation depth.
the device response at 20 GHz does not improve by more than 0.5 dB for
applied bias voltages above -5 V. From figure 7.1 another anomaly m ust
be explained: the response at frequencies between 10 and 20 GHz are
more affected than for frequencies below 10 GHz and above 20 GHz.
To gain insight into this nonlinear behavior, the lasers in figure 2.1
were phased-locked to provide ±
1°
phase stability and the time domain
output of the PD was observed on an oscilloscope at low and high powers.
The frequency response in figure 7.1 shows the response ju st starting to
drop between 100 and 500 MHz. The sinusoidal outputs at 150 MHz with
100 and 1400 pA of average PD current with a 100% modulation depth are
plotted in figure 7.2. The amplitudes are normalized such that the sinu­
soids contained the same energy during a single cycle, since the average
PD current is approximately linear ( 1 %) with the average optical power.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
13 3
u
a
<a
ci
•i—
100 jiA
1400 (lA
C3
s
5h
o
£
0
2
4
6
8
10
T im e ( n s )
Figure 7.2 Normalized large-signal sinusoidal output of a 0.95-p.m
-5 V-biased PD at 150 MHz. Average currents of 100 and 1400 |iA with
o
an e’ incident spot size of 5.75 ± 0.25 |im. 100% modulation depth.
Figure 7.2 shows that the peak of the high-current sinusoid is delayed
from the low-current sinusoid by up to 400 ps, a factor of 20 greater than
the low-power transit time of the device. 150 MHz was specifically chosen
for this observation because the sinusoidal output was observed to
decrease very rapidly and recover just before the start of a new cycle.
When the frequency is increased to 500 MHz, the high-current
signal (figure 7.3) shows a large phase shift relative to the low-current
signal and some distortion. The distortion is more subtle in this case as
compared to the 150 MHz case since the PD response at 500 MHz is 0.5 dB
down from the low-power response and no longer recovers before the
start of a new cycle. However, it is apparent that the top half of the 500MHz high-current sinusoid is wider than the lower half. If the incident
power density is decreased by keeping the optical power the same and
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
134
100 pA
1400 pA
-u
3
a
3
O
<v
N
'cS
eSh
o
£
0.5
0
1.5
1
2
2.5
3
T im e ( n s )
Figure 7.3 Normalized large-signal sinusoidal output of a 0.95-pm
-5 V-biased PD at 500 MHz. Average currents of 100 and 1400 |iA
o
w ithae" incident spot size of 5.75 ± 0.25 pm. 100% modulation depth.
^--- V
u
a
<U
-- 100 pA S
- 1400 pA I
cc
su
o
Z
0
0.5
1
1.5
2
2.5
3
T im e ( n s )
Figure 7.4 Normalized large-signal sinusoidal output of a 0.95-p.m
-5 V-biased PD at 500 MHz. Average currents of 100 and 1400 pA
-2
with an e’ incident spot size of 7.5 ± 0.5 pm. 100% modulation depth.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
13 5
increasing the spot size to 7.5 pm, the PD response at 500 MHz recovers to
within 0.1 dB of its response at low powers. Figure 7.4 is a plot of the
sinusoidal time domain output at 500 MHz under these conditions and
shows that the output now is similar in shape to the output at 150 MHz in
figure 7.2. Therefore, as the frequency increases, the power density m ust
decrease to keep the PD response from decreasing.
The distortions of the sinusoids in figures 7.1 to 7.4 will result in
substantial harmonic distortion. The fundamental, second, third and
fourth harmonics of a 0.95-pm PD biased at -5 V are plotted in figures 7.5,
7.6 and 7.7 for frequencies of 150 MHz, 500 MHz and 5 GHz, respectively.
The growth rate in the harmonic content differs considerably from a
power-law dependence. The second harmonic increases from -110 dBm
at a current of 100 pA to nearly -40 dBm at 1 mA for all three frequen-10
-30
a -50
Z
o
Ph -70
a
S
150 MHz
-90
§
-130
0.1
1
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 7.5 Fundamental and harmonic power vs average PD current
2
for a -5 V-biased 0.95-pm PD. 100% modulation depth. Incident e"
spot size of 5.75 ± 0.25 pm. 150 MHz fundamental frequency.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
136
*
S? -90
m
•
■
♦
A
A
500 MHz
2f
3f
4f
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 7.6 Fundamental and harmonic power vs average PD current
for a -5 V-biased 0.95-pm PD. 100% modulation depth. Incident e'-2
spot size of 5.75 ± 0.25 pm. 500 MHz fundamental frequency.
-1 0
a
i
-30
i
i
i
i
••
» • • •*
* - • ..........
i i i i I
®©*'
i
rd
S -50
£
o
PU -70
a>
>
S -90
«■
. ■
-* -■
♦
L»
♦♦“ v
♦
♦
A
♦
A
^
»A
•
■
A
■
-130
5 GHz —
2f
_
3f
4f
■ ■
0.1
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 7.7 Fundamental and harmonic power vs average PD current
o
for a -5 V-biased 0.95-pm PD. 100% modulation depth. Incident e’
spot size of 5.75 ± 0.25 pm. 5 GHz fundamental frequency.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
137
cies—an increase by 70 dB in less than one decade of current.
At an applied voltage of -3 V, the PD response at 1 mA near 20 GHz
was observed to be more than 10 dB below the PD response at 100 (iA. The
low-power frequency response of the PD does not appreciably change for
applied voltages above -5 V and decreases only a couple of dB at 24 GHz
with applied voltages as low as -2 V. Therefore, external voltage drops of
50 mV (1mA across 50 Ci) are excluded from being the sole factor for the
observed 1 to 10 dB reductions in the PD response. With a 10 dB reduction
in the output power, the time variation of the output current has also
decreased substantially.
The frequency response of the device should
therefore be dependent, not on the time varying PD current, but on the
average PD current. To verify this hypothesis the frequency response is
measured at high power densities under small-signal conditions.
Sm all-signal m easurem ents are obtained with unequal laser
powers (figure 2.1). With proper adjustment, the same modulating cur­
rent (output microwave power) can be obtained while independently con­
trolling the average PD current. The relative frequency response (rela­
tive to the response at low powers) of a 0.95-pm PD, biased at -5 V, with
800 and 1000 pA of average current is plotted in figure 7.8. The frequency
response has a similar shape to the large-signal experimental results in
figure 7.1. The tendency of the response to decrease more at frequencies
near the middle of the frequency range remains.
From the high-power-density measurements of the large-signal
frequency response, the small-signal frequency response, and the sin u ­
soidal time domain distortions with the corresponding harmonics, sev­
eral conclusions can be formulated.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
138
0 .5
PQ
0
3
-0.5
m
S3
1
o
cu
cc
a) -1.5
«
CD
CD
>
800 pA
-2
J2 -2.5
3
1 0 0 0 ji A
-3
-3.5
0
10
5
15
20
25
F r e q u e n c y (G H z )
Figure 7.8 Sm all-signal relative frequency response o f a -5 V-biased
0.95-pm PD. Average PD currents o f 800 and 1000 pA w ith an e "2
incident spot size of 5.75 ± 0.25 pm. 1000 point resolution.
1
) Once the carriers are generated, m ost exit th e in trin sic region before
recom bining, since m easu rem en ts of the average current are lin ear w ith
incident power.
2) The effect m u st be in tern al, since the voltage drop in th e external cir­
cuit is insufficient to explain the observed effects.
3) If m ost of the generated carriers exit the in trin sic region, w ith m a n y
of th em delayed by several hu ndreds of picoseconds, th e in trin sic region
electric field m u st have redistributed sufficiently to ca u se th e observed
increase in carrier tra n sit tim es.
S im u la tio n s w ill provide sp ecific in fo rm a tio n abou t th e h ig h pow er-density carrier dynam ics and the electric field distributions in the
in trin sic region. T his inform ation can th en be usedTor predicting device
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
139
behavior when PD currents approach 10 to 100 mA with larger spot sizes.
Simulation. Results
To study these high power density effects, the model PD is sim u­
lated with the program discussed in Chapter IV. The simulated sm allsignal frequency response with spot sizes of 3.0, 3.1, and 3.2 pm is plotted
in figure 7.9. The sim ulations clearly show a PD response reduction,
where the best results are obtained using a spot size of between 3.0 and
3.1 pm for the 1 mA frequency response.
For the 800 pA frequency
response, the fit overestimates the response by up to 1 dB for frequencies
between
1
and 5 GHz. The best fit to this data was obtained with a sim u­
lated spot size of only 53% of the e ' 2 spot size, in contrast to the 65 to 75%
obtained with the low-power nonlinearities of Chapters V and VI. This
3.0 pm s s ----------- 3.2 pm ss
3.1 pm ss
CQ
0
s
o
co
G
o
1
a
CO
0)
PS
-2
03
>
800 pA
-4-3
cS
PS
■3
1000 pA
-4
0
5
10
15
20
25
F r e q u e n c y (G H z )
Figure 7.9 Measured and simulated small-signal relative frequency
response of a -5 V-biased 0.95-pm PD. Average currents of 800 and
1000 pA. pp = 230 cm2/V s. Measurement data from Fig 7.8.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
140
may be needed to account for two-dimensional effects discussed later.
The hole mobility used in the simulated results of figure 7.9 is also
about 2 0 %higher than the best-fit hole mobilities of the low-power-density
nonlinearities (Chapters V and VI). To compare the sensitivity in the
sim ulated results to hole mobility, figure 7.10 displays the simulated
small-signal frequency response for hole mobilities of 150, 200, and 230
cm 2/Vs with the corresponding best-fit spot sizes.
The results show
slightly better fits at higher frequencies with a higher hole mobility and
better fits at lower frequencies with a lower hole mobility. The sensitivity
in the simulated results to spot size was about the same as the results in
figure 7.9. The results in figure 7.10 show that the PD response reduction
is somewhat independent of the hole mobility, given the freedom to vary
the spot size. However, for a given spot size, higher hole mobilities (not
3.10 pm ss, Pp = 230
CQ
3.20 pm ss, Pp = 200
0
3
3.35 pm ss, p =150
03
02
G
O
a
m
03
K
03
>
-t-s
-1
800 pA
*2
CS
3
-
3
P h
11000 pA
0
5
10
15
20
25
F r e q u e n c y (G H z)
Figure 7.10 Measured and simulated small-signal relative frequency
response of a -5 V-biased 0.95-|a.m PD. Average currents of 800 and
1000 pA. Measurement data from Fig 7.8.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
141
higher saturated hole velocities) result in less response reduction.
Plots of the electric field and the carrier densities for a hole mobility
of 230 c m2/Vs and a spot size of 3.1pm are given in figures 7.11, 7.12, and
7.13 corresponding to DC photocurrents of 100 pA, 800 pA, and 1000 pA,
respectively. At 100 pA, the electric field (figure 7.11) is 15kV/cm at the
edge of the intrinsic region near the n-contact. As the average current
increases to 800 pA (figure 7.12) the electric field near the n-contact
decreases almost to zero, with the initial observation of an increase of the
carrier densities there. When the average current reaches 1 mA (figure
7.13) the electric field has collapsed over 10% (0.1 pm) of the intrinsic
region, in the region nearest to the n-contact. Here, the carrier densities
have increased to 6 . 0 to 8 . 0 x
1 0 16
cm -3 in a region where the electric field
is nearly zero. The electric fields in figures 7.11 to 7.13 establish that the
H
oct-
ST
-25
&
•
CD
-50
a
©
Q
O
*3
H- •
CD
-75
S-t
©
u
u
CO
o
.—
^
O
-100
1.0
1.2
1.4
1.6
1.8
-125
2.0
D io d e X P o s i t i o n (p m )
Figure 7.11 Carrier densities and electric field in the intrinsic
region at 100 pA. The intrinsic region extends from X = 1.0 pm
to X = 1.95 pm. Applied voltage = -5 V. Spot size = 3.1 pm.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
142
(c m
E lectric
D ensity
C arrier
-150
1.0
1.2
1.4
1.6
1.8
(k V /cm )
-100
Field
-50
-200
2.0
D io d e X P o s i t i o n (|j.m )
Figure 7.12 Carrier densities and electric field in the intrinsic
region at 800 pA. The intrinsic region extends from X = 1.0 pm
to X = 1.95 pm. Applied voltage = -5 V. Spot size = 3.1 pm.
(cm ’ )
E lectric
D en sity
C arrier
-150
1.0
1.2
1.4
1.6
1.8
( k V /c m )
-100
Field
-50
-200
2.0
D io d e X P o s i t i o n (p m )
Figure 7.13 Carrier densities and electric field in the intrinsic
region at 1000 pA. The intrinsic region extends from X = 1.0 pm
to X = 1.95 pm. Applied voltage = -5 V. Spot size = 3.1 pm.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
1 43
observed response reductions are indeed caused by the average PD cur­
rent, since it is the average current which determines the intrinsic
region electric field. Under large-signal conditions the intrinsic region
electric field can and does change during the sinusoidal cycle. However,
the electric field change due to the modulation results only in deviations
above and below the electric field determined by the average PD current.
Figure 7.14 shows the simulated large-signal results w ith a hole
mobility of 230 cm W s at average PD currents o f800 and 1000 pA. Figure
7.14 shows that the large-signal results do not fit quite as well as the
small-signal results (figures 7.9 and 7.10), although the large-signal sim ­
ulations seem to fit better at 800 pA than 1000 pA, in contrast to the
sm all-signal results which fit better at 1000 pA.
The results in figure
7.14 reflect the tendency for the response to remain high until 2 to 6 GHz,
0.5
3.07 pm ss
3.20 pm ss
0
3.3 pm ss
800 pA
1
-1.5
■2
1000 pA
-2.5
•3
5
0
10
15
20
25
F r e q u e n c y (G H z )
Figure 7.14 Large-signal relative frequency response of a -5 V-biased
0.95-pm PD. Average currents of 800 and 1000 pA. Applied voltage
2
= -5 V. pp = 230 cm /Vs. Measurement data from figure 7.1.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
144
where a sudden decrease (0.5- 1.0 dB) occurs. Simulation results with a
hole mobility of 150 cm 2/Vs (figure 7.15) show sim ilar characteristics,
although the best-fit spot size has increased from 3.2 to 3.5 pm. The spot
size for both figures 7.14 and 7.15, are consistent with the results for the
best-fit small-signal results, however, both large- and small-signal sim u­
lations require approximately
10
to
20
% smaller simulation spot sizes
than the best-fit results from Chapter V. The fits in figures 7.14 and
figure 7.15 suggest that the carrier dynamics under large-signal modula­
tion and collapsing electric fields are being modeled with sufficient accu­
racy to extend the simulations to the time domain responses.
The simulated sinusoidal output at 150 MHz with a hole mobility of
150cm 2/Vs and a spot size of 3.0 pm is plotted in figure 7.16, where the
output of the simulated PD is compared at 100 and 1400 pA. The data in
0.5
PQ
T3
3.40 pm ss
3.50 pm ss
0
— 3.70 pm ss
co -0.5
o
aCO 1
o
-1.5
0}
>
■2
T>
PS -2.5
1000 pA
■3
0
5
15
10
F r e q u e n c y (G H z )
20
25
Figure 7.15 Large-signal relative frequency response of a -5 V-biased
0.95-pm PD. Average currents of 800 and 1000 pA. Applied voltage
o
= -5 V. pp = 150 cm /Vs. Measurement data from figure 7.1.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
145
c3
05
fl
100 jiA
1400 pA
-^5
3
CX
-u
o
T3
<
Nu
•rH
su
o
£
o
2
4
6
8
10
T im e ( n s )
Figure 7.16 Simulated large-signal sinusoidal output of a 0.95-(im
PD at 150 MHz. Average currents of 100 and 1400 pA. Spot size =
3 pm. pp = 150 cm2/Vs. Applied V = -5 V. 100% modulation depth.
figure 7.16 agrees quite well with the measured results in figure 7.2. The
spot size required for this simulation is approximately
20
% less than the
required 3.5 pm to fit the frequency response at the same mobility; how­
ever, the average current has increased from 1.0 to 1.4 mA.
So despite
some small discrepancies, the algorithm does predict the overall shape of
the sinusoidal output at high power densities.
The simulation results at 500 MHz, with the same simulation
parameters used to obtain figure 7.16, are plotted in figure 7.17. The sim ­
ulated output shows sim ilar characteristics to the measured data in
figure 7.3, where there exists a 10 to 25° phase shift in the output, as well
as some observable distortion. Rather than normalizing the data, the
simulated data can be plotted in an absolute sense and compared to the
generation function. The measured data could not be plotted in this way
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
146
Gen
Shifted Jp
3
o
ro
o
N
•H
CO
au
o
£
0
0.5
1
1.5
2
2.5
3
T im e ( n s )
Figure 7.17 Simulated large-signal sinusoidal output of a 0.95-pm
PD at 500 MHz. Average currents of 100 and 1400 pA. Spot size =
3 pm. pp = 150 cm2 /Vs. Applied V = -5 V. 100% modulation depth.
due to the AC coupling (via the bias tee in figure
2 .2
) required to bias the
PD. The data in figure 7.17 can then be plotted again (figure 7.18)in an
absolute sense, where the input signal depicted in figure 7.18 is the
output of an ideal PD. Figure 7.18 contains more information about the
large-signal carrier dynamics since a direct comparison between the
generation function and the output can be made.
Figure 7.3 to 7.4 show that when the spot size is increased by 33%,
the PD response at 500 MHz recovers slightly and contains more observ­
able distortion. This is also the case in the modeled PD where the DCcoupled data in figure 7.19 shows a decrease in the phase shift to less
than a few degrees and similar distortion characterisitics. Figures 7.16
to 7.19 show that the dynamic nonlinear characteristics occurring in the
modeled diode agree with those of the actual device.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
147
3
-- Input
— Output
2
d
au>
1-4
d
1
-u
d
a-9
-<
d
O
0
0
0.5
1
1.5
2
2.5
3
T im e ( n s )
Figure 7.18 DC-coupled simulated large-signal output of a 0.95-pm
PD at 500 MHz. Average currents of 100 and 1400 pA. Spot size =
3 pm. pp = 150 cm2 /Vs. Applied V = -5 V. 100% modulation depth.
3
Filtered
2
-u
d
Q
U
U
d
O
1
-u
d
a
d
0
0
0.5
1
1.5
2
2.5
3
T im e ( n s )
Figure 7.19 DC-coupled simulated large-signal output of a 0.95-pm
PD at 500 MHz. Average currents of 100 and 1400 pA. Spot size =
4 pm. pp = 150 cm2 /Vs. Applied V = -5 V. 100% modulation depth.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
148
T he sim u la ted h arm on ic content o f th e device a t a fu n d a m en ta l fre­
quency of 5 GHz u sin g th e param eters w hich provided a best-fit to the fre­
quency respon se data in figure 7.10 and u s in g hole m obilities of 150 and
230 c m 2/Vs is plotted in figure 7.20. T he sim u lated harm onic data (figure
7.20) is su b stan tially overestim ated above 400 pA for both m ob ilities, and
n ot u n til th e current su rp a sses 900 pA do th e sim u la ted h a rm o n ics
approxim ate the m ea su rem en t data. T h is m ig h t be expected from the
sim u lation s in figure 7.10 w here th e sim u la ted response a t8 0 0 p A near 5
GHz h a s n ot dropped su fficien tly to resem b le th e m e a su r em e n t data.
Therefore th e m odeled dynam ic n on lin earity is sligh tly h igh er th a n that
w h ich is actu ally observed. T his m ay be m ore obvious if one con sid ers
th e sin u so id a l output. If th e sin u so id could be acquired on an oscillo­
scope w ith su fficien t bandw idth, the actual sin u soid at 800 pA would look
-30
!
I
1
I
I |
M odulation
■
2f Data
♦
3f Data
▲
4f Data
-130
0.1
1
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 7.20 Sim ulated harm onic power versus detector current
at 5 GHz. Sim ulated w ith param eters leading to b est frequency
response fits in figure 7.10. M easurem ent data from figure 7.7.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
149
sim ila r to the sinusoid in figure 7.17, w here the response h a s decreased
b y on ly
1
dB from the resp on se a t low pow ers.
On th e other hand, the
sim u la ted sin u so id w ould look sim ila r to th e sin u so id in figu re 7.19,
since th e sim ulated response h as decreased le ss th an 0.2 dB from the low
pow er respon se.
The sin u so id in figu re 7.19, how ever, h a s m ore d is­
cernible distortion th a n th e sin u soid in figu re 7.17 and th u s should also
have h igh er harm onic content. Therefore, to fit th e harm onic data, the
spot size w ill be reduced to 60 to 70% o f th e e *2 spot size w hich provided the
best-fits to harmonic data in C hapters V and VI.
The sim ulated resu lts u sin g a hole m obility o f 200 c m W s w ith spot
sizes o f 3.6 and 4.0 pm are plotted in figure 7.21. The data show s th a t the
sim u la ted th ird harm onic h a s a reasonab le fit to m ea su rem en t data,
w ith th e second harm onic slig h tly u n d erestim a ted and the fourth h a r ­
-30
M odulation
Depth = 100%
0}
%
0
-70
Q -i
£
03
2 f D ata
-90
3 fD a ta
4 f D ata
1 -no
~r=4
3.6 pm ss
4.0 pm ss
-130
0.1
1
A v e r a g e D e te c to r C u r r e n t (m A )
Figure 7.21 Sim ulated harmonic power versus PD current a t 5 GHz.
S im u lated w ith spot sizes of 3.6 and 4.0 pm. Applied voltage = -5 V.
o
p p = 200 cm /s . M easurem ent datafrom figure 7.7.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
150
m on ic slig h tly overestim ated, sim ila r to th e sim u la tio n ten d en cies in
C hap ter V (figure 5.13).
B etter fits in C hapter V w ere obtained by
d ecreasin g th e hole m obility and th e spot size.
S im u la tio n r esu lts for
hole m obilities of 175 and 150 cm 2/V s are plotted in figu res 7.22 and 7.23,
respectively. A best-fit is obtained w ith a hole m obility o f 150 c m W s and
a spot size of 4.5 pm (75% o f th e incident e -2 spot size) w h ich agrees w ith
the best-fit sim ulation param eters in Chapter V.
The frequency dependence in th e NL output (figures 7.5 to 7.7) h a s
two d istin ct regions of com parison. A t 200 pA, there ex ists a 22 dB differ­
ence in th e second harm on ic betw een the 150 M Hz and th e 5 GHz data,
w h ile at 1 mA the difference is less th an 3 dB. The sim u lated NL a t 500
M Hz is plotted in figure 7.24 u sin g a hole m obility o f 150 c m 2/Vs w ith spot
size o f 4.5 pm , values w hich produced the best-fits a t 5 GHz (figure 7.23).
-30
£
m
ro
M odulation
Depth = 100%
-50
o
P-I
>
as
£
2 f D ata
-90
3 f D ata
§ -110
4 f D ata
4.1 pm ss
4.5 pm ss
§
-130
0.1
1
A v e r a g e D e te c to r C u r r e n t (m A )
Figure 7.22 Sim ulated harmonic power versus PD current a t 5 GHz.
Sim u lated w ith spot sizes of 4.1 and 4.5 pm. Applied voltage = -5 V.
o
Pp = 175 cm /s . M easurem ent data from figure 7.7.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
15 1
Modulation
Depth = 100%
s
PQ
'O
u
<D
£
O
04
0)
>
aS
£
ou
u -110
/
.•
.......
/• *
/-A
,-k
k..........................
2fD ata
H
♦
3fD ata
▲ 4fD ata
---------- 4.5 pm ss
............. 4.9 pm ss
-130
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 7.23 Simulated harmonic power versus PD current at 5 GHz.
Simulated with spot sizes of 4.5 and 4.9 pm. Applied voltage = -5 V.
2
Pp = 150 cm /s. Measurement data from figure 7.7.
-30
s
CQ
T3
^—*
Modulation
Depth = 100%
-50
Jh
0)
£ -70
o
04
o -90
>
aS
£
oIn
o -110
2fD ata
3f Data
4 f Data
4.5 [im ss
-130
0.1
1
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 7.24 Simulated harmonic power vs PD current at 500 MHz.
Simulated with- a spot size of 4.5 pm. Applied voltage = -5 V.
2
Pp = 150 cm /s. Measurement data from figure 7.6.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
152
The overall NL is underestimated slightly; however, the results do pre­
dict the sudden increase in NL above 500 pA.
The sudden increase in NL above 500 pA can be linked to the
results of the electric field in the intrinsic region (figures 7.11 to 7.13). A s
the current increases to 800 pA, the electric field in the depletion region
decreases near the n-contact to zero. When the electric field decreases to
zero in a normally depleted absorbing region of a PD, the region becomes
quasi-neutral (undepleted) due to the photogenerated current.
In
Chapters V and VI, it was demonstrated that absorption in undepleted
regions next to the intrinsic region could result in nonlinearities.
One
factor controlling the amount of NL was the minority carrier lifetime,
where longer lifetimes resulted in higher nonlinearities. In contrast to
the highly-doped undepleted regions sim ulated in Chapter V, the
intrinsic region minority carrier lifetime is several nanoseconds due to
the low doping density. Additionally it is the holes which are originally
generated in the undepleted region that enter the depletion region.
Therefore the NL can be much higher since all the holes generated in the
undepleted region reach the depletion region before recombining. The
net result is the sudden increase in NL observed in figures 7.5 to 7.7.
TWo-Dimensiorial Carrier Flow
As the power density increases in the depletion region, the spacecharge electric field (equation 3.35) also increases. The result is a redis­
tribution of the electric field in the depletion region (figures 7.11 to 7.13).
The PD thus far has been considered to have exclusively one-dimensional
carrier movement in the intrinsic region. This was assumed since the
dark electric field in the intrinsic region is independent of the radial coor­
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
153
dinate, neglecting edge effects or other device asymmetries. This is a
good approximation because the intrinsic region, as far as the electric
field is concerned, is a parallel plate capacitor which has a diameter (35
pm) that is sufficiently larger than its width (1 pm). However, when the
electric field in the depletion region is perturbed according to the local­
ized carrier densities and the localized carrier densities are not uniform,
then the electric field may contain radial components. Carrier move­
ment, which was assumed to be exclusively axial, may now contain a
component in the radial direction.
A qualitative argument for the existence of a radial component of
electric field is undertaken here with the help of the low and high power
electric fields from figures 7.11 and 7.13, plotted on the same graph
(figure 7.25). Assum e that the device operating under dark conditions
—i—i— i—i—i—
50
From Figure 7.13
or High Powers
^
a
0
f -50
From Figure 7.11
or Low Powers
IB
Potential
at r = 6 pm
£ -loo
o
-M
J -150
•H
-200
0.75
Potential
at r = 0 pm
1.00
1.25
1.50
1.75
A xial P o sitio n X (m icron s)
2.00
Figure 7.25 Representation of the potential in the intrinsic region
versus radial position.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
154
has an internal electric field given in figure 7.11. Since there are no gen­
erated carriers anywhere in the depletion region, the potential, T(r,x), at
an arbitrary radial coordinate is just the integration of the electric field in
figure 7.11 from x = 0 to x = x. When light is incident on the PD equivalent
to 1 mA of average current, with a e-2 spot size of
generated carriers at r =
6
6
pm, the number of
pm is low enough that the electric field at r = 6
pm is unchanged from its value under dark conditions. The potential,
T(r = 6 pm,x), is given by the integration of the dark electric field in figure
7.25 from x = 0 to x. On the other hand, the intensity of the light at the
center of the Gaussian is enough to perturb the electric field, given by the
high-power curve in figure 7.25. The potential, 4/(r=0,x), is therefore
given by the integration of the redistributed electric field (figure 7.25)
from x = 0 to x. For any value of x, the potentials T(r=0,x) and
^ ^ = 6
pm,x) are not equal, given by the area difference in figure 7.25. In fact
the potential at the center of the Gaussian is always less than or equal to
the potential at the edge of the Gaussian. Electrons, which drift towards
a negative potential, will thus drift towards the center of the G aussian.
Holes on the other hand, which drift towards higher potentials, will tend
to drift away from the center of the Gaussian. This situation is depicted
in figure 7.26 where the carriers near the center and the edge of the
G aussian have only axial components of velocity and the carriers
between r = 0 and the edge of the Gaussian have radial velocity compo­
nents in addition to their axial velocity components.
Since there exist radial carrier velocity components from the above
arguments, the remaining problem is to determine how important the
radial velocity is compared to the axial velocity. Without a 2-D solution to
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
155
total
y
total
total
I n c id e n t
G a u s sia n
B eam
P h o t o d io d e
I n tr in s ic
R e g io n '
Figure 7.26 Representation of two-dimensional carrier flow due to
the radial potential from figure 7.25.
the transport equations (eqns 3.2 to 3.4), an estimation of the axial electric
field can be obtained from the solution of many 1-D problems with intensi­
ties equal to the intensities along the radial coordinate of the G aussian.
The 1-D solutions assume, by definition, that radial currents do not exist.
From the 1 -D results, this assumption can be proved or disproved by esti­
mating the magnitude of the expected radial carrier velocities.
The sim ulations begin with an approximation to the G aussian
radial intensity with j = 7 linear intensities (figure 7.27). A simulation
for each intensity, j, corresponding to a given radius rj, is performed to
obtain the electric field, ExCr^x). Once the electric field has been com-
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
156
•2
-1.5
-0.5
1
0
o
0.5
1
1.5
2
N o r m a li z e d e" R a d iu s
Figure 7.27 Linear approximation function for the Gaussian
intensity profile.
puted, the potential T(rj,x) is calculated by integration of the electric field
E^r^x). The potential T(r,x) is just the combination of all the individual
terms T(rj,x). The radial component of the electric field, Er(r,x), is then
calculated by taking the negative gradient of the potential T(r,x), w hich
can be used to estimate the radial carrier drift velocities from their rela­
tionships to the electric field (equations 3.16 and 3.17). This procedure is
outlined in figure 7.28.
=> Ex(rj,x)
Simulation:
Solutions
Integration:
JEx(rj,x) dx => '('(r^x)
X
0
Interpolation:
T (ri,x)
Gradient:
•V7(r,x)
=>
T(r,x)
=> E ^ x )
Figure 7.28 Procedure for obtaining a radial electric field estimate.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
157
T h is procedure w a s carried out for a spot size o f
6
jim and a n
average current of 1 m A for th e 0.95-jim long in trin sic region PD. T he
radial com ponents of th e velocity w ere com puted and com pared to the
axial com ponents of th e velocity. A three-dim ensional plot is required to
display the resu lts w hich are functions o f both x and r. The ratio o f elec­
tron a x ia l velocity (m ovem en t is a lw a y s tow ards la rg er x) to radial
velocity (towards the center of th e G aussian) is plotted in figure 7.29. T he
radial com ponent o f th e electron velocity near x = 1.4 jim h as peaked and
is over 1.5 tim es the value o f th e axial velocity. The radial com ponent of
the electron velocity n ear x = 1.95 jim (near the i-n ju nction) is also over
R a d ia l
V e lo c ity
A x ia l
V e lo c ity
1 2
Figure 7.29 Ratio of the estim ated electron radial velocity
to the electron axial velocity.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
158
1.5 times the size of the axial component. Thus an electron velocity in the
intrinsic region, near the steepest part of the Gaussian, has more radial
movement than axial movement in any given time interval. The average
ratio of the radial component of velocity relative to the axial component is
approximately one from figure 7.29.
The result is an electron movement in the radial direction which is
nearly equal to the electron movement in the axial direction. With an
intrinsic region width of 0.95 pm, the electrons can move (radially) up to
1 pm.
This results in a 10 to 20% overall decrease in the spot size,
accounting for the discrepancies in the simulation results earlier in the
chapter which required
10
to
response reductions at 1 mA.
20
% sm aller spot sizes to agree with
The reason that a small radial electric
field can cruse a high radial electron drift motion is related to the high
electron mobility. From figure 3.3, the radial component of the electric
field need only be a few kV/cm for the electron velocity to be equal to the
saturated electron velocity. Conversely for holes (figure 7.30), an electric
field of a few kV/cm does not result in a radial hole velocity which is com­
parable to the saturated hole velocity because of the low hole mobility (see
figure 3.4). Therefore the radial component of the hole drift velocity,
which is radially outward from r = 0, is small (figure 7.30) compared to
the axial component of the hole velocity. However, the radial hole velocity
in the region from X = 1.8 to X = 1.95 pm is not negligible. The diffusive
flow of both carriers was also calculated and was found to be small in
comparison to the results in figure 7.29. Although figures 7.29 and 7.30
were calculated for a PD where the response had already started to
decrease, the effects of a radial electric field will occur whenever there
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
159
,6
4
%
0
2
1
R a d ia l
V e lo c it y
A x ia l
V e lo c it y
0
1
Figure 7.30 Ratio of the estimated hole radial velocity
to the hole axial velocity.
are space-charge fields which are comparable to the dark electric field.
This is apparent from figure 7.29 where most of the radial electron move­
m ent occurs over the portion of the intrinsic region which is still
depleted.
Summary
The measurements and simulations in this chapter demonstrated
that several additional effects need to be included to fully understand the
nonlinear behavior of p-i-n PDs at power densities greater than 3
kW/cm2.
It was observed that when the hole and electron densities
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
160
exceed
1 0 16
cm-3, th e electric field in the in trin sic region m ay collapse
under certain b ias conditions.
A ccom panying th e collapse w ere addi­
tional NL effects su ch as a sudden in crease in NL, tim e dom ain distor­
tions, and response reductions. A radial com ponent o f electron drift c u r­
ren t (electron focusing) w as estim ated from a 1-D form u lation o f th e 2-D
in trin sic region potential. The estim ation predicts th a t the radial com po­
n e n t o f electron velocity can be as h ig h as th e a x ia l electron velocity
resu ltin g in sligh tly h ig h er carrier d en sities (up to 30%) along the a x is of
th e in cid en t sig n a l.
T h ese effects are p resen t w h en th e sp ace-ch arge
field is com parable in size to th e dark electric field , and n o t lim ited to
cases w here th e PD resp on se h as begun to decrease.
T w o-dim ensional
sim u la tio n s are therefore needed to further stu d y th e se effects on elec­
trons and holes.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
161
VHL REDUCTION IN NONLINEAR OUTPUT AND EXTRAPOLATION
TO HIGHER POWERS
Introduction
I f a PD can be m ade w ith a p-type cap layer from a sem icon ductor
m aterial w hich is transparent to the detection w a v elen g th , it h a s been
dem onstrated in C hapters V and VI th at sig n ifica n t im p rovem ent (> 60
dB) can be obtained in the second harm onic.
T his chapter w ill a s su m e
th a t it is possible to m ake a device w ith a transparent p-type cap layer and
further in vestigate w ays to decrease device NL. A dditionally, th is ch ap ­
ter w ill include sim u lation s to determ ine proper PD d esig n (from the NL
point o f view ) to achieve up to 50 mA of detector current w h ile m a in ta in ­
in g low NL and avoiding the nonlinear effects observed in Chapter VII.
0.95-jim L ong Intrin sic R egion D evices
For eq u ivalen t in trin sic-reg io n electric field s an d in c id en t spot
sizes, th e 0.5-pm device displayed low er n o n lin ea rities (see C hapter V I)
w hen th e p-region absorption w as neglected. One possible reason for the
n on lin ear su p p ression w as th at the in trin sic region electric field in the
0.5-pm device deviated from its average valu e le ss com pared to electric
field in the 0.95-pm device. To investigate w hether an approxim ately con­
s ta n t (u n d er dark conditions) in trin sic region electric field r e su lts in
low er NL, th e in trin sic region doping density can be reduced since the
slope of the electric field is proportional to the doping density. The in trin ­
sic region electric field under dark conditions for in trin sic region doping
densities of 1 x 1014, 1 x 1015, and 5 x 10 1 5 c nr 3 is plotted in figure 8.1.
The sim ulated device NL for the above three in trin sic region doping
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
162
-25
£ -100
-175
1.0
1.2
1.4
1.6
1.8
D io d e X P o s i t i o n (jim )
Figure 8.1 Intrinsic region electric field for intrinsic region
doping densities of 1 x 1014, 1 x 1015, and 5 x 1015 cm'3.
Applied voltage = -10 V.
densities is plotted in figure 8.2 for a simulated spot size of 7 jim where
the p-region absorption is neglected. A marginal improvement in the
second harmonic of 5 dB is achieved with lower intrinsic region doping
densities while the third harmonic actually increases by 5 dB for cur­
rents below 1 mA. A closer look at the resulting electron velocity change
versus doping density may help to clarify this result.
When the carrier velocities are approximately saturated in the
intrinsic region, the current (not current density) is proportional to the
carrier density, the carrier velocity, and the incident spot size.
This
implies that, for basically linear devices, the space-charge electric field is
independent of the intrinsic region doping level. The space-charge elec­
tric field for intrinsic region doping densities of 1 x 1014 and 5 x 1015 cm
3
with a PD current of 1 mA are plotted in figure 8.3, where little difference
is observed. Since the doping density determines the intrinsic region
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
163
B
m
S
u
o
£
o
M odulation
Depth = 100%
Second Harm onic
-100
-120
Ph
o -140
>
03
£
o
J-t
-160
•o
I—
I
Third H arm onic
N , . = 5 .0 x 1 0
§
0.1
1
10
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 8.2 Sim ulated harmonic pow er vs detector current a t 5 GHz
n eglectin g p-region absorption for various intrinsic region doping
densities. Spot size = 7 |im. Applied Voltage = -10 V.
electric field (figure 8.1), the resu ltin g difference in the device NL (figure
8 .2
) is due to th e change of th e electron velocity v ersu s position (recall
from figure 5.23 th a t the hole velocity change is negligible w hen th e elec­
tric field is greater than 50 kV/cm) from its electric field dependence.
T he change in electron velocity, due to the space-charge electric
field in figure 8.3, is plotted in figure 8.4. The resu lts in figure 8.4 are not
conclusive, since there does not seem to be a sign ifican tly low er c h a n g e
in the electron velocity for th e low er in trin sic region doping lev els. T he
h igh er in trin sic region doping level does how ever have a larger average
velo city ch a n g e com pared to th e low in tr in sic reg io n dop ing level,
alth ou gh th is m ay not be the u n d erly in g param eter w h ich d eterm in es
th e device NL. N everth eless, the resu lts in figure 8.2 do predict th a t a
slig h t decrease in th e second harm onic can be obtained w ith a slig h t
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
164
10
5
0
■5
sc
©
sc
-10
-15
1.0
1.2
1.4
1.6
1.8
D io d e X P o s i t i o n (fim )
Figure 8.3 The space-charge electric field in the intrinsic region.
Intrinsic region doping densities of
1
x 10 14 and 5 x 1015 cm"^.
Spot size = 7.0 pm. Applied voltage = -10 V.
CD
1
O
iC
©
T—I 0.5
©
©
a
0
Sh
s>>-0.5
•
o
o
'o
>
.1
1
1.2
1.4
1.6
1.8
D io d e X P o s i t i o n (jim )
Figure 8.4 Difference in electron velocities from 10 uA to 1000 pA
average currents. Intrinsic region doping densities of 1 x 1014
15
3
and 5 x 10 cm' . Spot size = 7.0 pm. Applied voltage = -10 V.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
16 5
increase in the third harmonic for lower intrinsic region doping levels.
Since the intrinsic region doping density has such a small effect on
the device NL, the majority of the NL remains to be the result of the
space-charge-induced electron velocity change. The electric field depen­
dence of the electron velocity is a fundamental property of InGaAs at a
given temperature; therefore, the only remaining parameter which can
reduce the device NL is the incident power density.
Decreasing the
power density (larger spot size) simply lowers the space-charge field.
Extrapolation to Higher Powers
When the carrier velocities are approximately saturated, the spacecharge electric field is simply proportional to the current in the intrinsic
region. This is shown in figure 8.5 where the space-charge electric field
is plotted at 10 mA of PD current with the space-charge field at
1
mA
scaled by a factor of ten. In Chapter VII it was observed that with large
generated carrier densities, the intrinsic region electric field could par­
tially collapse, causing the device NL to increase substantially. A n
approximation for the current at which this effect occurs, given the inci­
dent spot size, can be made with the space-charge field (figure 8.5) and
the dark electric field near the n-contact. In Chapter VII, the electric
field partially collapsed at approximately 0.8 to 1 mA of average current
(figures 7.12 and 7.13) with an incident e' 2 spot size of approximately 5.7
pm. An additional characteristic of the collapse was a sudden increase
(threshold effect) in the NL just before this collapse near 0.5 to 0.6 mA
(figures 7.6 and 7.7). With an e' 2 spot size of 5.7 pm, approximately onehalf the
10
pm e ' 2 spot size, the space-charge electric field is approxi-
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
16 6
100
50
0
-50
10 xE
&-100
-150
sc
sc
1.0
1.2
1.4
1.8
1.6
D io d e X P o s i t i o n ( |im )
Figure 8.5 The space-charge electric field in the intrinsic region
due to the photogenerated carriers. Spot size = 7.0 pm. 0.95-pm
long intrinsic region. Applied voltage = -10 V.
mately four tim es the space-charge field at 1 mA. An additional factor of
two arises if one considers the peak current under large signal condi­
tions. Eight tim es the space-charge electric field at
1
mA (figure 8.5)
yields an opposing space-charge electric field of 40 kV/cm atX = 1.7 pm ,
while the dark electric field atX = 1.7 pm (the position of the peak spacecharge field) is 35 to 40 kV/cm (figure 7.11). Therefore, the opposing
space-charge field is sufficient to collapse the intrinsic region electric
field in this region.
This electric field collapse is responsible for the
threshold in the device NL, noticeable at approximately 1/2 the fieldcollapsing current. We will define this current where the NL suddenly
increases as the threshold PD current, 0.5 mA for the above example.
With an incident e ' 2 spot size of 10 pm, the threshold PD current
can be approximated in a similar way. Given an applied voltage of -10 V ,
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
167
the electric field near X = 1.7 pm under dark conditions is approximately
95 kV/cm (figure 8.1). For a 100% modulation depth, the peak spacecharge electric field (figure 8.5) is 10 kV/cm at X = 1.7 pm for average
currents of 1 mA (peak currents of 2 mA). Therefore, the threshold PD
current is approximately 5 mA.
was simulated (figure
8 .6
To verify this approximation, the PD
) up to average PD currents of 20 mA with and
without p-region absorption. The device NL increases substantially near
5 mA in both cases, while a slight reduction in the device NL can be
achieved if the p-region absorption can be removed. In Chapter V it was
determined that at this applied voltage, frequency, and incident spot size,
the p-region absorption and the space-charge field induced change in
electron velocity resulted in approximately equal device NL. Therefore,
no significant improvement in NL was expected from excluding the p40
s
CQ
T3
I
w /pabs
w/o pabs
0
Fundamental
^ 0
O
P-c
SJ -80
CB
Second Harmonic
|
§ -120
5 mA Threshold
yw Fourth
U—Harmonic
Third
S
Harmonic
M odulation
Depth = 100%
-160
0.1
1
10
100
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure
8 .6
Simulated fundamental and harmonic power vs current
at 5 GHz with and without p-region absorption. 0.95-pm long intrinsic
o
region. Spot size = 7 pm. pp = 150 cmfVs. Applied V = -10 V.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
168
region absorption. The tendency for th e N L to decrease above 10 m A is a n
artifact of th e resp on se reduction.
For PD cu rren ts above th e cu rren t
w here the peak in th e NL occurs, the PD response decreases (see C hapter
V II).
W hen the PD response drops, th e dynam ic NL also d ecreases (for
exam ple, see figures 7.16 and 7.17) resu ltin g in less distortion.
To in crea se th e threshold PD current and reduce th e contribution
to th e device NL from the space-charge electric field, the in cid en t e *2 spot
size can be increased from 10 to 20 pm. T his should in crease the th r e sh ­
old PD current by a factor of four. T he sim u lated device NL under th ese
conditions w ith and w ithout p-region absorption is plotted in figure 8.7.
The figure show s th at the second harm onic in crea ses sharply a t approx­
im a tely 20 m A, th u s defin ing a th reshold PD current of 20 mA, or four
tim es th e previous threshold PD current o f 5 mA. Figure 8.7 also dem on-
Fundamental
6
PQ
0
'— ^
U
o>
£
o
CL
40
<x>
>
-80
£
O
Sh
o
120
Second Harmonic
w /pab s
w/o pabs
03
20 mA Threshold
• Fourth
Harmonic
Third ^
Harmonic
§
0.1
1
M od u lation
Depth = 100%
10
100
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 8.7 Sim ulated fundam ental and harm onic power v s current
at 5 GHz w ith and w ithout p-region absorption. Spot size = 14 pm.
0.95-pm long intrinsic region. pp = 150 cm /Vs. Applied V = -10 V.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
169
str a tes th a t th e p-region absorption n o n lin ea r m e c h a n ism now dom i­
n a tes th e device NL sin ce larger spot sizes resu lt in low er sp ace-ch arge
field s.
T he second, third, and fourth harm on ics decrease by approxi­
m a tely 3 0 ,1 0 , and 10 dB, respectively, for average PD currents below 20
mA. Above 20 mA, the device NL is dom inated by th e h igh pow er density
n o n lin ea rities stud ied in C hapter V II, w h ich are independent o f th e pregion absorption conditions.
S in ce the space-ch arge electric field h a s decreased by a factor of
four w ith th e increase in spot size from 7 to 14 (im, the device NL ca u sed
by a 50 Q output resistan ce m ay no longer be negligib le. A 50 £2 resistiv e
load w ill drop th e PD voltage by 1 V w hen the PD current reaches 20 m A .
Therefore, the device is sim u lated a g a in (figure
8 .8
) w ith th e load r e s is ­
tor. For PD currents below 10 mA, th e device NL is now dom inated by the
j~ Applied V oltage = -10 V
a
Fundamental
M odulation D epth = 100%
P Q
Threshold (w / 50 Ohm)
g
o
-40 i
r
pH
>
cd
w 1 50 Ohm
w/o 50 Ohm
-80 r
: Second Harmonic
T h resh o ld
(w/o 50 Ohm)
|
r
§ -120 jThird Harmonic
Fourth Harmonic
-160
0.1
1
10
100
A v e r a g e D e te c to r C u r r e n t (m A )
Figure
8 .8
Sim ulated fundam ental and harm onic power a t 5 GHz
w ith and w ithout a 50 Ohm load. No p-region absorption. 0.95-jim
9
long intrinsic region. Spot size = 14 pm. pp = 150 cm /V s.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
170
potential drop from the load resistor, which results in an 18 dB increase
of the second harmonic. The resistor also lowers the threshold PD cur­
rent from 20 mA to 14 mA, although this may be compensated for with a
higher applied voltage. At 14 mA of PD current, the intrinsic region elec­
tric field drops by 7 kV/cm due to the resistor, which is equivalent to a 7
mA space-charge field (spot size of 14 pm = 1/4 of figure 8.5), accounting
for the reduction in the threshold PD current. The tendency for the NL to
decrease above 20 mA (figure
8 .8
w/ 50 £1) is an artifact of the response
reduction as described for the results in figure
8 .6
. In this figure, the
presence of a resistor simply decreases the current where the NL peaks
as compared to the PD with a zero Ohm load.
Increasing the applied voltage from -10 to -15 V raises the electric
field by 50 kV/cm near the n-contact which should increase the threshold
PD current from 20-25 mA to 35-45 mA for a PD without a load resistor (0
£1), and from 14-18 mA to 20-28 mA with a 50 £2 load. The simulated
device NL with and without p-region absorption for a PD without a load
resistor is plotted in figure 8.9 where the threshold PD current has
increased to 40 or 50 mA. The second harmonic also decreases by 20 to 30
dB when the p-region absorption is neglected, similar to the results at -10
V (figure 8.7). Also note that the simulated third and fourth harm onics
in figure 8.9 (with p-region absorption) show behavior characteristic of
actual devices (see figures 7.7 or reference [1]). The tendency for the even
harmonics to peak when the odd harmonics dip and vice versa is proba­
bly due to the nonlinearities associated with the space-charge electric
field and the p-region absorption adding constructively and destructively.
This is the most likely explanation since the ripple is absent when the p-
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
171
40
S
PQ
|
Fundamental
0
Third
Harmonic
-40
w/pabs
w/o pabs
£
S>
03
•Threshold
-80
Second Harmonic
|
Fourth
Harmonic
§ -120
M odulation
Depth -100%
-160
0.1
1
10
100
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 8.9 Simulated fundamental and harmonic power vs current
at 5 GHz with and without p-region absorption. 0.95-pm long intrinsic
2
region. Spot size = 14 pm. pp = 150 cm /Vs. Applied V = -15 V.
region absorption is neglected. Similar characteristics are also observed
in figure 8.7.
The nonlinear output excluding p-region absorption for a PD with
and without an external resistor is plotted in figure
8 .1 0
where the
threshold PD current decreases to 30 mA when the external resistor is
included. The device NL is still higher with the presence of the resistor;
however, the increase in the second harmonic is only 7 dB, compared to
the 18 dB observed increase when the applied voltage was -10 V. This was
not unexpected because a higher average intrinsic region electric field
results in a smaller electron velocity change from a fixed electric field
decrease due to the load resistor. From the results in figure 8.10, a con­
ventional 0.95-pm long intrinsic region 20 GHz p-i-n PD could achieve a
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
172
40
M odulation
Depth = 100%
?
CQ
,-d
%
O
cu
8>
cd
Fundamental
0
T h resh o ld
^ 0
w / 50 Ohm
w/o 50 Ohm
-80
|
Second Harmonic
§ -120
T h r esh o ld
•^
Fourth
Harmonic
'.....Third...!
Harmonic
-160
0.1
1
10
100
A v e r a g e D e te c to r C u r r e n t (m A )
Figure 8.10 Sim ulated fundam ental and harm onic power a t 5 GHz
w ith and w ithout a 50 Ohm load. No p-region absorption. 0.95-jim
9
long intrinsic region. Spot size = 14 pm. fi = 1 5 0 c m /V s .
?
m axim u m th resh old PD current
o f 50 to 60 mA.
T h is is b ecau se the
device area cannot be in creased beyond 35 pm (figure
8 .8
a ssu m es a 20
pm or2 spot size) to achieve th e desired bandw idth. Therefore, th e in c i­
dent spot size cannot be increased beyond 30 pm resu ltin g in a factor of
two decrease in the power density. Furtherm ore, the applied voltage ca n ­
n ot be in crea sed sin ce th e electric field a t 60 mA is approaching 300
kV /cm n ear th e p-i in terface, w here th e device m a y b egin to display
avalanche g a in (resu ltin g in a low er frequency respon se) or break dow n
(device failure). A lso, fillin g the entire PD area w ith lig h t m ay introduce
additional NL from nonuniform electric field s p resen t n ea r th e edge of
the device.
T w o-dim en sion al carrier flow (C hapter V II) m ay further
reduce the threshold PD current from an in crease in the electron density
near the center of the incident signal.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
17 3
To determine if 0.5-jim PDs have any advantages over 0.95-|im PDs,
the 0.5-jim PD is simulated without a load resistor and with an applied
voltage (-5 V) which results in approximately the same peak intrinsic
region electric field as the 0.95-pm PD at -10 V. The simulated spot size is
10 pm (equivalent to a 14 pm e -2 incident spot size), which is near the
maximum allowed device diameter
(2 0
pm) for a capacitively-limited
20
GHz device. The results (figure 8.11) show that the 0.5-pm PD NL is
slightly betterthan the 0.95-pm PD NL. The 0.5-pm PD second harm onic
is -85 dBm at 10 mA compared to a second harmonic o f-77 dBm at 10 m A
(figure 8.9) for the 0.95-pm PD, both neglecting p-region absorption. Both
devices yield threshold PD currents of approximately 30 mA.
The insignificant improvement in nonlinear performance for the
0.5-pm device is the result of the smaller incident spot size resulting in a
Fundamental
w/pabs
w/o pabs
6
PQ
Threshold
Third
Harmonic
Second
Harmonic
Fourth
Harmonic
Modulation
Depth = 100%
• rH
-160
0.1
1
10
100
A v e r a g e D e t e c t o r C u r r e n t (m A )
Figure 8.11 Simulated fundamental and harmonic power vs current
at 5 GHz with and without p-region absorption. 0.5-pm long intrinsic
o
region. Spot size = 10 pm. pp = 150 cm /Vs. Applied V = -5 V.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
174
higher space-charge electric field. In general, for capacitively-limited
devices, the capacitance is proportional to the area divided by the intrin­
sic region thickness. Recall also that the space-charge electric field is
inversely proportional (Chapter VI) to the intrinsic region thickness.
Therefore, for capacitively-limited devices, the space-charge electric field
is independent of intrinsic region thickness. So although the 0.5-pm long
device exhibits 1/2 the space charge electric field (figure 6.16) as the 0.95pm long device (figure
6 .6
), the device area, and hence the incident spot
size, m ust also be twice as small for the 0.5-pm long device as the 0.95pm long device, accounting for the similarities in the device NL.
Actual microwave PDs experience an equivalent 50 Q. load resistor
(transm ission line) which results in lower threshold PD currents for
shorter intrinsic region devices. The same potential drop across a 50 D
load resistor results in double the intrinsic region electric field decrease
for the 0.5-pm long device compared to the 0.95 pm device, thereby lower­
ing the threshold PD current. Figure 8.12 displays this where the 0.5-pm
long device is simulated with and without a load resistor. The threshold
PD current has decreased from 30 mA to 12 mA.
These results imply
that thicker intrinsic region devices have both higher threshold PD cur­
rents and lower NL.
S um m ary
This chapter studied the device lim iting nonlinearities and the
implications that the intrinsic region length and device area have on the
device nonlinearity. A new concept of threshold PD current was intro­
duced and utilized to compare PD nonlinear behavior between devices
with different intrinsic region lengths at high PD currents.
It was
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
175
i
i rm
r r 't i ]
Fundamental-—|
T h resh o ld
Third
Harmonic
. Second .
Harmonic
T h resh o ld
/ - ri...' Fourth ...............
- j r . -
'
Applied V oltage = -5 V
1 ■■■■■■!
0.1
i Harmonic
M odulation D epth = 100%
■
■ ■ ■ ■ ■ ■ !
1
■
■
10
.
.
.
100
A v e r a g e D e te c to r C u r r e n t (m A )
Figure 8.12 Sim ulated fundam ental and harm onic power a t 5 GHz
w ith and w ithout a 50 Ohm load. No p-region absorption. 0.5-pm
n
long intrinsic region. Spot size = 10 pm. pp = 150 cm /V s.
show n th at the in trin sic region doping density has very little effect on the
total device N L, w hile longer in trin sic region len gth s low er device NL for
PD s w ith cap acitively-lim ited b an dw idths.
T herefore, a high -pow er
h igh -freq u en cy low -n on lin earity device should be d esign ed w ith three
m ain considerations. The in trin sic region should be a s thick as possible,
w ith in the allow ed tran sit tim e frequency response, to m inim ize the nonlin ea rities caused by a 50 Q. load resistor and to m axim ize the qu antu m
efficiency.
T he device area should be a s large as possible, w ith in the
allow ed capacitively-lim ited bandw idth, to m in im ize th e n o n lin ea rities
from the space-charge electric field. The device should also be fabricated
w ith out absorbing regions (undepleted) near the in trin sic region, to sub­
stan tially low er the nonlinearities generated in this region.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
176
DC CONCLUSION
T he n o n lin ea rities in p-i-n photodiodes h ave b een system a tica lly
m easu red and n u m erically m odeled. H arm onic distortion m e a su r e m e n ts
w ith g rea ter th a n 130 dB d yn am ic ran ge w ere m ade w ith tw o sin g le ­
frequency Nd:YAG la sers offset-phased-locked to a stable m icrow ave refer­
ence. The obtained dynam ic range is 50 to 70 dB high er w hen com pared to
an y other available source. The la ser sy stem w as also used to m easu re the
photodiode frequency response and the sin u soid al tim e output for low and
h ig h pow er d en sities. M easu rem en ts at pow er d en sities grea ter th a n 3
k W /cm 2 revealed reductions in the photodiode response o f a few dB. M ea­
su rem en ts of the sinusoid al tim e output at power d en sities o f 4 kW /cm 2 re­
v ealed ea sily discernible distortions such as second harm onics o f -20 dBc
and frequency dependent phase shifts o f 10 to 25°.
A sem i-cla ssica l approach to so lv in g photodiode carrier transport
w a s conducted.
T his approach required th e sim u lta n eo u s solu tion of
three coupled non linear differential equations: P oisson’s equation and the
h ole and electron continuity equations.
T h ese transport equations w ere
n u m erica lly solved in th e case o f a on e-d im en sion al In G aA s/In P p-i-n
structure, including th e undepleted p- and n-regions for com pleteness.
Several transport properties and ch aracteristics specific to In G a A s
w ere included in the m odel. The electric field dependence of both the hole
and electron velocity w ere included to account for velocity saturation (both
carriers) and velocity overshoot (electrons only). M obility reduction a t h ig h
carrier d en sities due to scatterin g m ech a n ism s w as in clu d ed to describe
th e carrier m ovem ent in the bulk region s w here th e carrier d en sities are
approaching 101 9 cm '3. D iffusion w as in clu d ed in the m odel. P articu lar
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
17 7
attention was given to the electric field dependence of the diffusion con­
stants and to lim iting the diffusion currents using a physical argument
rather than artificially limiting the numerical solution to avoid num erical
oscillations.
The effects of the heterojunction(s) were included by neglecting hole
transport from the intrinsic region into the n-type substrate where the het­
erojunction appears as a 0.5 eV barrier to hole flow. The 0.1 eV barrier to
electron flow from the intrinsic region into the n-type substrate was ne­
glected due to the high velocity of the electron towards the barrier com­
bined with its low effective mass. A second heterojunction, if present, was
included with similar assumptions.
After a discussion of the transport mechanisms, the transport equa­
tions were expanded to explain how nonlinear terms enter the equations.
Nonlinear terms arise when the carrier velocities are a function of the car­
rier densities. However, this does not restrict the carrier velocities from
being functions of position or time alone, as long as these functions do not
have an underlying carrier-density dependence.
Additional nonlinear
terms arise when the diffusion terms and the recombination terms have
underlying carrier-density dependencies.
Nonlinearities caused by carrier-dependent carrier velocities are
shown to be due to several intermediate mechanisms.
Space-charge elec­
tric fields, lower mobilities due to scattering, loading in the external cir­
cuit, and absorption in undepleted regions next to the intrinsic region all
contribute to the nonlinear output through the carrier-density-dependent
carrier velocity. As the generated carriers drift towards their respective
bulk regions, they induce a space-charge electric field.
As the space-
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
178
charge field in c r ea se s, a redistributed electric field r e su lts in c h a n g in g
h ole and electron velocities v ia P o isso n ’s equation and th e ir respective
electric-field d ep en d en cies.
S in ce th e electron v elo city does n ot fully-
satu rate for electric field s below 200 kV /cm , th is n o n lin ea r m e c h a n ism
can never be fu lly rem oved from the transport equations. P otential drops
in th e external circuit resu lt in lower in trin sic region electric fields c a u s ­
in g additional changes in th e hole and electron velocities. A nother ca rrier
velocity change occurs from th e decrease in th e carrier m obilities as a re­
su lt o f carrier-carrier scatterin g m ech an ism s.
A n em pirical relation sh ip
is used to incorporate th ese effects in the transport equations. A bsorption
in undepleted regions n ext to the in trin sic region w ere show n to introduce
n o n lin ea rities since th e electric field in th is region is proportional to the
total current.
Therefore, th e carriers generated in th ese region s travel
w ith velocities w hich are directly proportional to the current and hence the
carrier d en sities.
U sin g a very pow erful, high dynam ic ran ge, photodiode n o n lin ea r­
ity m ea su rem en t set-up and resorting to nu m erical a n a ly sis to sim u la te
carrier transport in a p-i-n diode structure, a system atic study into photo­
diode n on lin earities w as conducted. N um erical m od eling o f th e photodi­
ode response w as perform ed to in vestigate and isolate the various n o n lin ­
ear m ech a n ism s.
The sim u lation resu lts w ere first u sed to m ethodically
reduce the set of possible non lin ear m ech an ism s to sm aller, m an ageab le
subsets.
To reduce th e se t of possible non linear m ech a n ism s, three regions of
applied voltage and two frequencies w ere used. T he voltages and frequen­
cies w ere chosen to em ph asize the dom inant non lin ear m ech a n ism s
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
and
17 9
their region of importance.
It was determined that for intrinsic region
electric fields below 50 kV/cm (-5 V across a 0.95-pm long intrinsic region
photodiode), the nonlinearities were influenced primarily by the spacecharge electric-field-induced change in the hole and electron velocities.
Best-fit simulation results were obtained in this region of applied voltage
with a simulation spot size 70% of the e _2 incident intensity spot size and a
hole mobility of 150 cm 2 /Vs. For intrinsic region electric fields between 50
kV/cm and 100 kV/cm, the contribution to the overall nonlinearity was
shown to be influenced primarily by two mechanisms. The electron veloc­
ity change was shown to account for the nonlinear behavior at frequencies
above 5 GHz, while the p-region absorption was shown to account for the
nonlinear behavior at frequencies below
1
GHz. When the intrinsic region
electric field increases to greater than 100 kV/cm, only the p-region ab­
sorption could explain the observed nonlinear behavior for frequencies
below 10 GHz. It was determined that only
8
to 14 nm of undepleted ab­
sorbing material next to the intrinsic region was sufficient to cause second
harmonic distortion levels of -60 dBc at 1 mA of average photodiode cur­
rent. This is the first time that this region was shown to dominate the
nonlinearities at high electric fields.
After determining the origins of basic nonlinearities at low intensi­
ties, simulations were performed at high power densities. These sim ula­
tions provided valuable insight into the dynamics of carrier movement
under high power density operation. To predict the observed response re­
ductions and sinusoidal time-output distortions, 20 to 30% smaller spot
sizes were required. This was explained in terms of a two-dimensional
carrier motion in the intrinsic region. An approximation for the radial
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
180
electric field was implemented to determine the importance of the radial
flow. This approximation established the existence of a radial component
of electron velocity towards the center of the incident G aussian beam.
Under certain operating conditions, the magnitude of the average radial
electron velocity was estimated to be the same size as the axial electron ve­
locity. Two-dimensional movement (electron focusing) thus accounted for
the 20% decrease in spot size required to fit the measurement data. Very
good agreement at high power densities between measurement and sim u­
lation results were obtained.
The model was extended to predict the maximum current a photodi­
ode can handle before a sharp increase in nonlinear output occurs. For
capacitively-limited devices operating with the largest allowable incident
spot size, it was shown that the nonlinearities induced by the space-charge
electric field were independent of the intrinsic region length. However,
loading in the external circuit was determined to result in higher nonlin­
earities as the intrinsic region length decreases. It was thus concluded
that the photodiode with the lowest possible nonlinearities should be con­
structed without undepleted absorbing regions near the intrinsic region,
with the longest intrinsic region allowed from transit time considerations,
and with an incident spot size which fills the capacitively-limited device
area. With such a device, an improvement in the second harmonic of 40 to
60 dB can be obtained compared to currently available devices. It was also
determined that maximum photodiode currents of 30 to 50 mA should be
attainable in a p-i-n photodiode structure, with minimum nonlinearities.
Although this study significantly advances our understanding of
photodiode nonlinearities, more work remains. At high applied voltages,
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
181
th e n on lin earities are dom inated by th e p-region absorption. A bsorption in
u n depleted regions is com m on to m an y photodiode types. W aveguide de­
sig n s, p-i-n d esig n s, and MSM d esig n s a ll u tiliz e p-contacts to low er con­
tact r esista n ces. Double heterostructure d esig n s offer th e m o st p ro m isin g
hope for elim in a tin g th is source o f non lin earity, how ever, th e grading lay­
ers u tiliz ed n ear th e p-i interface and th e specifics con cern in g ca rrier
m o v e m e n t in th is region req u ire fu rth er in v e stig a tio n to d eterm ine
w h eth er th is n on lin ear m ech a n ism can be effectively reduced. Som e ad­
va n ta g e m ay be gained by u sin g a heterostru ctu re m a teria l w h ich h a s
only a slig h tly h igh er bandgap th a n th e absorbing m a teria l.
A lth o u g h
th is w ill lim it th e useful w avelen gth range w h ich attains h igh lin earity, it
m ay le sse n the requirem ents on th e grading layers.
Specific issu e s related to m aterial properties of InG aA s also needs to
be in vestigated . M aterial properties in highly-doped region s su ch as the
m in ority carrier lifetim e and carrier velocities versu s doping d en sity are
not w ell know n and require additional work. M ea su rem en ts o f the hole
velocity below 50 kV/cm are also needed. A dditional work is needed con­
cern in g carrier flow near th e p-i ju n ctio n .
T his w ill help to d eterm ine
w hich specific m ech an ism s control the n o n lin ea rities associated w ith the
p-region absorption. As devices are d esign ed to handle currents greater
th a n 10 m A , additional w ork is need ed to d eterm ine i f tw o-d im en sion al
carrier m ovem ent places any additional lim ita tio n s on th e m axim u m pos­
sible photo generated currents.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
18 2
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R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
CURRICULUM VITAE
Name: Keith Jake Williams
Permanent address: 827 Chatsworth Drive, Accokeek, MD 20607
Degree and date to be conferred: Ph.D., 1994
Date of birth: March 17,1964
Place of birth: Lincoln, Nebraska
Secondary education: Ashland-Greenwood High School, Ashland
Nebraska, 1982
Collegiate Institutions
Dates
Degree
Date Completed
University of Maryland
8/87-12/89
M.S.
December 1989
University of Nebraska
8/82-5/87
B.S.
May 1987
Major: Electrical Engineering
Publications
1.
K.J. Williams, et al., "Interferometric M easurement of LowFrequency Phase Noise Characteristics of Diode Laser-Pumped Nd:YAG
Ring Laser," Electron. Lett., 25, pp. 774-775,1989.
2.
K.J. Williams, et al., "6-34 GHz Offset Phase Locking of Nd:YAG
1319 nm Nonplanar Ring Lasers," Electron. Lett., 25, p. 1242,1989.
3.
R.D. Esman and K.J. W illiams, "Measurement of Harmonic
Distortion in Microwave Photodetectors," IEEE Photon. Tech. Lett., PTL2, p. 502,1990.
4.
R.D. Esman, K.J. W illiams, M.H. White, and V. Uzunoglu,
“Microwave Subcarrier and Clock Recovery by an Optically Injected
CPSO,” IEEE Photon. Tech. Lett., PTL-3, p. 179,1991.
5.
Z. Ma, M.H. White, K.J. Williams, R.D. Esman, and V. Uzunoglu,
“A High Performance Optically Injected Synchronous Oscillator,” IEEE
Photon. Tech. Lett., PTD4, p. 405,1992.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
.
K. J. W illiam s a n d R. D. E sm a n , "Observation o f Photodetector
N onlinearities," Electron. L ett., 28, p. 731,1992.
6
7.
L. Goldberg, R .D . E sm a n , and K.J. W illiam s, “G en eration and
Control o f M icrowave S ign a ls by O ptical T echniques,” IE E P ro ceed in g s- J,
139, p. 288,1992.
.
C. R a u sc h e r a n d K .J . W illia m s, “H eterod yn e R ecep tio n of
M illim eterw a v e-M o d u la ted O p tic a l S ig n a ls w ith a n
InP -B ased
T ran sistor,” To A ppear in M icrow ave T heory and T echniques D igest,
1994.
8
Patents
1.
E ffic ie n c y I m p r o v e m e n t fo r W ideban d F ib er-O p tic S ig n a l
Processing, Invention D isclosure 75,773, A ugust 25, 1993.
C onference P resentations
1.
A .D . Kersey. K .J. W illia m s, A. D an d rid ge, a n d J.F. W eller,
"C haracterization o f a D iode L aser-P um ped Nd:YAG R in g L aser for
Fiber S en sor A pplications," O ptical Fiber Sensors C onference, P a ris,
Paper T u-6-4,1989.
2.
K.J. W illiam s, e t a l ., “A ctive O ffset P h ase Locking o f Nd:YAG 1319nm
N o n p la n a r R in g L a s e r s ,” O p tic a l F ib e r C o m m u n ica tio n s
Conference, OFC ‘90, Paper ThC2, 1990.
3.
K.J. W illiam s, et a l., “H igh Frequency M icrowave P h ase R espon se
M easu rem en t Technique for Optical Input-O utput D evices,” C onference
on L asers and Electro-Optics, CLEO ‘90, Poster CTUH75, 1990.
4.
R.D. E sm an and K.J. W illiam s, “Subcarrier and Clock Recovery by
an O ptically Injected C oherent P h ase-L ocked Synchronous O scillator:
C oherent Subcarrier D em odulation,” C onference on L asers and ElectroOptics, CLEO ‘90, Paper CWD 2 , 1990.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
5.
A.D. Kersey, K.J. "Williams, and A. Dandridge, “Phase Noise
Reduction of a Diode Laser Pumped Nd:YAG Laser for Ultralow Noise
Fiber Interferometry,” Conference on Lasers and Electro-Optics, CLEO
‘90, Paper CTHP2,1990.
6 .
L. Goldberg, R.D. Esman, and K.J. Williams, “Optical Techniques
for Microwave Generation, Transm ission, and Control,” Microwave
Theory and Techniques Symposium, MTT-S 90, Invited Paper G -2,1990.
7.
K.J. Williams and A.D. Kersey, “Nd:YAG Nonplanar Ring Laser
Source for Fiber Sensors and Microwave Applications,” AFCEA ‘90
Conference, 1990.
.
K.J. W illiams, et al., “Photodetector Bandwidth Reduction and
Signal Distortion,” Optical Fiber Communications Conference, OFC ‘91,
Paper T h 0 5 ,1991.
8
9.
Y.G. Wey, M. Kamegawa, A. Mar, K.J. Williams, K. Giboney, D.L.
Crawford, J.E. Bowers, and M.J. Rodwell, “Hybrid Integration of an
InGaAs/InP PIN Photodiode w ith an U ltrafast Sam pling Circuit,”
Optical Fiber Communications Conference, OFC ‘91, Paper PD 1 2 - 1 , 1991.
10.
K.J. Williams and A.D. Kersey, “Nd:YAG Nonplanar Ring Laser
Source for Fiber Sensors and Microwave Applications,” OSA A nnual
Meeting, Invited Paper WF-1 , 1991.
11.
C. Rauscher and K.J. Williams, “A Heterodyne Receiver for 40GHz-Modulated 1.3-jim Optical Signals Using a Multi-Tasked InP-Based
HEMT,” Microwave Theory and Techniques Symposium, MTT-S 92, Paper
MM-2, 1992.
12.
K.J. W illiams and R.D. Esman, “Num erical Sim ulations of
Bandwidth Reduction in Microwave Photodetectors,” Optical Fiber
Communications Conference, OFC ‘93, Poster WH10, 1993.
Professional positions held:
Electrical Engineer, Naval Research Laboratory, Optical Sciences
Division, Washington, D.C., 5/87-present.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
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