doctor of philosophy harris (coach) and olav solgaard for reading this dissertation, especially in...

AN OPTICALLY CONTROLLED OPTOELECTRONIC SWITCH:

FROM THEORY

TO

50 GIGAHERTZ BURST-LOGIC DEMONSTRATION

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF APPLIED PHYSICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Micah B. Yairi

November 2001

ii

© Copyright by Micah Yairi 2002

All Rights Reserved

micah

micah

micah

iv

AbstractFor high-speed communication, it is essential to multiplex, demultiplex, and

switch individual data bits at very rapid rates. Similarly, in wavelength division

multiplexed (WDM) systems the ability to change wavelengths dramatically increases the

potential connectivity of such transmission systems. This dissertation presents work on a

unique optically controlled optical gate that is capable of both high speed optical gating

and wavelength conversion.

The optically controlled optical gates (OCOG) described herein alter the

reflection of a surface-normal pulse of light in response to the presence or absence of a

control light pulse. Low required switching energy is possible for two reasons: (1)

separation of photogenerated electrons and holes creates large changes in the electric

field and (2) the absorption of the multiple quantum wells in a p-i-n diode is strongly

field-dependent due to the quantum confined Stark effect. The recovery mechanism used

in these devices is based on diffusive conduction, a novel optoelectronic behavior that

enables fast gating. In essence, the localized voltage change that builds up in the vicinity

of the incident light pulse relaxes in an analogous manner to a voltage pulse in a two-

dimensional dissipative transmission line. This recovery is a local effect and can,

therefore, be made fast -- on the order of picoseconds; it is not constrained by the overall

RC time constant of the device. With proper design, multiple insulating and conducting

layers within a device may be used to modify the voltage relaxation process, further

enhancing OCOG switching speed.

Three generations of optically controlled quantum well optical gates were

investigated. For each generation, both the theory of operation and experimental results

are presented. Our multi-layered dual-diode device exhibits a 7 ps FWHM switching

time that requires a switching energy of only 40 fJ/�m2. This device has also

demonstrated burst-logic operation at 50 GHz. These optically controlled optical gates

are not only low power, but they are scalable in 2D arrays and integrable with silicon

circuitry, offering intriguing possibilities for applications.

v

Acknowledgements

There are many colleagues whose assistance and support made this work possible.

Ellen Judd, Chris Coldren, Petar Atanackovic, Pauline Prather, and Tom Carver helped

from growing the wafers to simulating quantum well absorption spectra. Most important,

however, has been Volkan Demir whose hard work, unending energy, enthusiasm, and

optimism has helped sustain my research effort. Working alongside him day in and day

out into the wee hours of the night has been a true pleasure. I am indebted to him.

I would also like to take this opportunity to express my deep appreciation to all

the faculty members at Stanford University who have so graciously shared their

knowledge and wisdom with me during my graduate studies at this wonderful institute of

higher education. In regard to the work reported herein, special thanks go to Professors

Jim Harris (Coach) and Olav Solgaard for reading this dissertation, especially in such a

timely fashion and for their thoughtful comments and suggestions.

A very special place in my thanks is reserved for Professor David Miller, my

advisor and mentor. I count it a small miracle that I was able to join the Miller group and

work so closely with him. It has been an incredible experience, being the first student

actually in the group, seeing it grow and flourish from just his office and a couple of

rooms full of discarded equipment into a dynamic world-class laboratory with nearly

twenty students. His light-handed approach has allowed me the freedom to commit a few

mistakes and helped me learn how to conduct careful and useful experiments. His

advice, particularly that there IS a logical reason for whatever might be occurring in the

laboratory, and that it CAN be understood, has helped me become a better physicist.

And, of equal or greater worth, Professor Miller provided the necessary support and

encouragement; he always did right by me. I cannot thank him enough.

I have had a truly wonderful time at Stanford. And this is largely due to all my

friends – The Miller Group – with whom I’ve been able to work every day!: Helen,

Volkan, Vijit, Gordon, Bianca, Diwakar, Ryohei, Henry, Sameer, Noah, Aparna, Martina,

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Christof, Petar, and Hatije, Ray, and Yang. Sometimes it was so much fun I didn’t get

any work done all day long. It has been great to wake up in the morning and know you

get to spend the whole day working with friends; I can’t imagine life here without you

guys!

I would also like to thank my long-time friends and roomates: Gaeron McClure,

John Hauser, Chris Nicholas, Ken Brownfield, Gil Winograd, Jitendra Mohan, Joost

Bakker, Mitchell Golner, John Fay, and Diwakar Agarwal. They have stood by me

through thick and thin for all the sorrows and joys I have faced, managing to tolerate my

faults, and helped me to reach this point.

Finally, there is my family. Mom, Father, Dani, and Keren. Words can never

express my love.

vii

Table of Contents Page #

Chapter 1: Introduction 1

1.1 Establishing General Switching Principles 2

1.1.1 What aspects of an optical signal can be switched? 2

1.1.2 What methods are available to enable optical switching? 2

1.1.3 What properties are desired in an optical switch? 3

1.2 Comparing Optical Switches 5

1.2.1 Comparisons based on nonlinearity 5

1.2.2 Switching speed and switching energy comparisons 7

1.3 Summary 9

References 11

Chapter 2: Methods of Optical Switching 14

2.1 Optical-Electrical-Optical Devices (OEOs) 15

2.2 Electronically-Controlled Optical Switches 16

2.2.1 Electrooptic Modulators 17

2.2.2 Electro-absorption modulators 20

2.2.3 Other voltage-controlled devices 21

2.3 Optically-controlled Switches 22

2.3.1 Coherent switches 23

2.3.2 Incoherent switches 30

2.4 Self-electrooptic-effect devices (SEEDS) 37

References 42

Chapter 3: OCOG Device Concept 51

3.1 First generation OCOG-1 51

3.1.1 Principles of OCOG-1 operation 51

3.1.2 Advantages and Disadvantages of OCOG-1 57

3.1.3 Uses of OCOG-1 59

viii

3.2 Second generation OCOG-2 60

3.2.1 Principles of OCOG-2 operation 60

3.2.2 OCOG-2 Advantage and Disadvantages 63

3.3 Third generation OCOG-3 63

References 65

Chapter 4: Theory of Optically Controlled Optical Gates 66

4.1 Theory of Diffusive Conduction 66

4.1.1 Qualitative Descriptions of Enhanced Diffusion 66

4.1.2 General Modeling Approaches 71

4.1.3 Giant Ambipolar Diffusion 74

4.1.4 Diffusive Conduction 78

4.1.5 Comparison Between Approaches 81

4.1.6 Uses, Limits, and Limitations of Enhanced Diffusion 83

4.2 Theory of Multilayer Diffusive Conduction 85

4.2.1 2-layer case 85

4.2.2 N-layer case 89

4.2.3 Discussion 90

References 95

Chapter 5: Experimental Methodology 98

5.1 Device Growth 98

5.2 Device Processing 98

5.3 Device Characterization 100

5.3.1 Electrical characterization 100

5.3.2 Optical characterization 101

5.4 Pump-probe Set-up 103

5.4.1 General description 103

5.4.2 Second-harmonic generation 106

5.4.3 Creation of 4-pulse burst data stream 107

References 108

ix

Chapter 6: Results and Analysis 109

6.1 First Generation OCOG-1 109

6.1.1 Small signal experimental results and simulation 111

6.1.2 Large signal experimental results and simulation 112

6.1.3 Signal gain (large signal) experimental results and simulation 114

6.2 Second Generation OCOG-2 115


6.2.2 Further test of Diffusive Conduction 117


6.3 Third Generation OCOG-3 126



6.3.3 Multiple pulse (small signal) experimental results and simulation 132

References 133

Chapter 7: Simulation Methods 134

7.1 Introduction 134

7.2 OCOG-1: Charge transport modeling in p-i(MQW)-n Devices 135

7.2.1 Governing equations 135

7.2.2 Simulation implementation 140

7.3 Large-Signal model: OCOG-2 146

7.4 Small signal FFT Models: OCOG-3 152

References 158

Chapter 8: Discussion and Future Directions 161

8.1 Ultimate Limits of OCOG devices 161

8.2 How do OCOGs compare to other optical switches? 166

8.3 Future research directions 161

8.4 Conclusions (Summary) 173

References 174

x

Appendix A: Examples of Multilayer Diffusive Conduction Calculation 176

Appendix B: Analysis of Photocurrent Spectra 180

Appendix C: Optically-Controlled Optical Gate Device Designs 194

xi

List of Tables

Page #Chapter 1

Table 1.1 Comparison of Optical Switch Energies and Repetition Rates 8

Chapter 7

Table 7.1 OCOG-1 Key Simulation Parameters 143

Chapter 8

Table 8.1 Comparison Between Selected All-Optical Switches 170

Appendix A

Table A.1 Multiple-Layer Variables 176

Appendix B

Table B.1 Legend Notation 185

Table B.2 Comparison Between Simulated and Graphically-Determined

Exciton Resonances 190

Appendix C

Table C.1 OCOG-1 Structure Design 192



xii

List of Illustrations Page #

Chapter 1

Figure 1.1 The nonlinear coefficient of the refractive index, n2, (top) and afigure-of-merit for switching (bottom) as a function ofrelaxation time for a variety of materials. 6

Chapter 2

Figure 2.1 Schematic diagram of an OEO switch 15

Figure 2.2 Schematic diagram of an electrically-controlled waveguideinterferometric switch 18

Figure 2.3 Schematic illustration of optical parametric generation 24

Figure 2.4 Illustration of switching behavior of a soliton dragging gate 27

Figure 2.5 Schematic diagram of a terahertz optical asymmetricdemultiplexer 31

Figure 2.6 A schematic illustration of cross-grain modulation in asemiconductor optical amplifier 33

Figure 2.7 Hypothetical response of a bistable device 36

Figure 2.8 Schematic illustration of an R-SEED 38

Chapter 3

Figure 3.1 Schematic diagram of OCOG-1 52

Figure 3.2 Schematic illustrations of OCOG-1 device dynamics 52

Figure 3.3 Description of the Quantum Confined Stark Effect (QCSE) 53

Figure 3.4 Illustration of a mesh of resistors and capacitors 54

Figure 3.5 Conceptual illustrations of the resistance per square andcapacitance per unit area of a p-i-n diode with an incidentcontrol pulse with a relative small spot size and the resultingvoltage due to the separation of the photogenerated carriers 55

Figure 3.6 Illustration of relaxation of the initial Gaussian voltagedistribution due to diffusive electrical conduction 56

xiii

Figure 3.7 Illustration of the different electric field magnitudes due tocarrier separation depending on whether of not carriers are ableto escape from quantum wells in the intrinsic region 58

Figure 3.8 Device schematic of OCOG-2 61

Figure 3.9 Schematic illustrations of OCOG-2 device dynamics 62

Figure 3.10 Illustration of induced voltage in a dual-layer OCOG device 63

Figure 3.11 Schematic of the p-i-n-p-i(MQW)-n structure of OCOG-3 64

Chapter 4

Figure 4.1 Schematic diagrams showing how an incident light pulse maycreate effective lateral electric fields in a p-i-n structure 68

Figure 4.2 Schematic p-i-n structure showing distributed resistance andcapacitance 70

Figure 4.3 Schematic of 1D RC transmission line with a single or withtwo resistive planes 80

Figure 4.4 Schematic of two-layer 1D RC structure 85

Figure 4.5 Bottom diode voltage behavior (-VCB) at pulse center (r=0) 89

Figure 4.6 Changes in the voltage relaxation of the bottom most layer in amulti-layered OCOG devices as the number of layers increases 92

Figure 4.7 Small signal response of OCOG-3 compared to simulations ofboth 3 and 2-layer structures. 93

Chapter 5

Figure 5.1 Image of a wire-bonded OCOG-2 device 100

Figure 5.2 Contour graph of photocurrent spectra of OCOG-1 102

Figure 5.3 OCOG-1 reflectivity as a function of reverse bias for variouswavelengths of picosecond pulses 102

Figure 5.4 Schematic of optical pump-probe set-up 104

Figure 5.5 Photographs of pump-probe set-up 105

Figure 5.6 Optical sub-system for generating a 4-pulse bit stream 106

xiv

Chapter 6


Figure 6.2 Small signal response of OCOG-1 111

Figure 6.3 OCOG-1 Reflectivity modulation of probe (signal) pulse as afunction of time across at various pump (control) pulse powers 113

Figure 6.4 Change in reflected probe (signal) power normalized againstpump (control) pulse power – equivalent to signal gain forOCOG-1 114


Figure 6.6 Small-signal response of OCOG-2 117

Figure 6.7 Comparison between data and simulation of OCOG-2dynamics for a different values of the radius of the incidentlight pulse 118

Figure 6.8 Figure 6.8: Illustration of on-center and off-center behavior inan OCOG-2 device at four instances in time 120

Figure 6.9 The measured and simulated device response for variousseparation distances between the probe and control pulses 121

Figure 6.10 Figure 6.10: Simulation of the spatial and temporal responseof the bottom layer in an OCOG-2 device 122

Figure 6.11 OCOG-2 response to large signal control inputs 123

Figure 6.12 Normalized change in OCOG-2 reflectivity as a function oftime for various control pulse powers 124

Figure 6.13 Schematic of OCOG-3 127

Figure 6.14 Small signal response of OCOG-3 128

Figure 6.15 Large signal response of OCOG-3 129

Figure 6.16 Overall and initial OCOG-3 device response at a various inputpowers 130

Figure 6.17 “Turn-on” of OCOG-2 device response for various inputpowers 131

Figure 6.18 Multiple-pulse, small-signal response of OCOG-3 with 20 pspulse separation 132

xv

Chapter 7

Figure 7.1 Schematic flow-chart of time-iterative large-signal computersimulation 141

Figure 7.2: Conceptual illustration of the behavior of a p-i-n device at aparticular lateral (x,y) point in which voltage diffusion occursresults in the overall voltage between the doped layersrecovering from the change in voltage due to the separation ofphotogenerated carriers before those carriers have been sweptout of the intrinsic region 149

Figure 7.3: G(t), the normalized incident light pulse that createsphotogenerated carriers 153

Figure 7.4 R(t), the change in voltage shielding across the top intrinsiclayer due to the absorption of a instantaneous light pulse 154

Figure 7.5 D(t), the voltage decay in the bottom layer of a two-layerdevice due to diffusive conduction of a “time slice” of voltageshielding 155

Figure 7.6 Convolution of G(t), R(t), D(t), and S(t) from Figs. 7.3, 7.4, and7.5 above, providing the small-signal voltage response of thebottom layer of a two-layer device 156

Figure 7.7 Small signal simulation of multiple pulses with variousrepetition periods 157

Chapter 8

Figure 8.1 Small-signal simulation of next-generation OCOG device,OCOG-4 164

Figure 8.2 Reported demonstrations of high bit rate data transmission in afiber optic cable 167

Figure 8.3 Schematic illustration of a dual-diode optically controlledwaveguide switch (OCWS) 171

Figure 8.4 Conceptual view of a reconfigurable NxN wavelengthconverter based on OCWS devices 172

Appendix A

Figure A.1 Illustration depicting Ii and Ji 176

xvi

Appendix B

Figure B.1 OCOG-1 photocurrent. 181

Figure B.1.b OCOG-1 photocurrent, expanded view 182

Figure B.2 Differential photocurrent (second derivative) of OCOG-1 183

Figure B.3 Simulation of heavy hole exciton and 1st excited electron-stateexciton resonance splitting 185

Figure B.4 Simulation of Stark ladders for adjacent wells, ground stateelectron and hole and 1st excited state electron and hole 186

Figure B.5 Simulation of next-to-adjacent well Stark ladder splittings withheavy hole exciton resonance 187

Figure B.6 Simulation of various Stark ladders and the heavy hole excitonresonances 187

Figure B.7 OCOG-1 photocurrent data and simulation 188

Figure B.8 Theoretical Stark ladders overlapped against photocurrentspectra 189

1

Chapter I: Introduction

A new world that fundamentally changes the way people communicate, learn, and

process information is emerging. Advances such as virtual multiuser white and

video/computer phones that can connect hundreds of millions of people together across

the planet are not in the distant future but merely a few years away. The current

backbone of the telecommunications network has evolved to the point where most data is

sent optically through fiber optic cables. Realizing this future vision, however, requires

an optical communications network able to route orders of magnitude more information

than it can today, transporting data over long distances at fast bit rates between large

numbers of people. Optical switches are crucial for such data manipulation. Current

optical switches, however, are insufficient for this task; they are too slow, expensive, and

do not scale well with either the rapidly increasing number of users or bitstream data

rates. The focus of this dissertation is a new type of optical switch -- an optically

controlled optical gate (OCOG) -- that has demonstrated its potential to help overcome

these hurdles. An OCOG changes the reflectivity or transmission of an optical signal

beam by virtue of a separate optical control pulse at speeds approaching a hundred

gigabits per seconds (Gbps). It provides switching capability that is low powered, fast,

and scalable while maintaining integrability with electronics and flexibility of use.

This introduction places OCOGs in the context of other optical switches. This is

done by first establishing general optical switching principles through answering the

following questions:

∙ What aspects of an optical bitstream can be switched?

∙ What methods are available to enable optical switching?

∙ What properties are desired in an optical switch?

∙ What constraints are placed on optical switches from other components of a

communication network?

Next, the optical switches are evaluated by comparing data on switching energy and

switching time. The more common assessment method, that of comparing the

magnitudes of the nonlinear coefficients of switches, is deemed to be of limited viability.

2

1.1 ESTABLISHING GENERAL SWITCHING PRINCIPLES

1.1.1 What aspects of an optical signal can be switched?The essence of a communication switch is the ability to change the channel along

which data flows. These channels are the switch’s degrees of freedom. For optical data

switches these degrees of freedom include intensity, spatial location, frequency, phase,

polarization, pulse time location, and spatial mode. Optical data are usually encoded by

intensity modulation.i It is also possible to encode data (e.g., soliton pulses) in such a

way that the temporal location of an optical pulse within a given time window of a bit

determines whether the pulse represents a 1 or a 0. Data channels themselves are

differentiated by the remaining degrees or freedom of optical signals, i.e., spatial location

(which fiber optic cable it is in) or wavelength. In some situations these channels are

capable of propagating simultaneously along a single fiber without interfering with each

other. For example, using wavelength division multiplexing (WDM) allows several

optical data streams, channels, to propagate independently along a fiber, each at a

different wavelength. Alternatively, in time division multiplexing (TDM), different

channels use the same wavelength but are differentiated by their time slots within the data

stream. Unless a special, more expensive, fiber is used, the polarization of a beam of

light is quickly randomized as it propagates, severely hindering the use of polarization as

a means for differentiating channels. The focus in this discussion will be primarily on

optical switches that change the wavelength or redirect an input optical stream or

multiplex/demultiplex a TDM bitstream.

1.1.2 What methods are available to enable optical switching?

Three general methods are used to control switching. One, the optical signal may

be detected, converted into an electrical signal and processed, and subsequently

re-emitted as a new optical signal. This is referred to as OEO. Two, the optical signal

maintains its optical form, but its channel may be controlled and redirected using an

3

electrical signal. Three, the optical signal maintains its optical form, but its channel may

be controlled and redirected using an optical signal.

1.1.3 What properties are desired in an optical switch?The primary function of switches in a communications network is to either

redirect or multiplex/demultiplex input bitstreams into the proper output channels.

Without these essential features, a network will not function. Providing this functionality

requires or is improved by:

∙ high speed, both for switching speed and switch repetition rate∙ large magnitude signal change, both in signal energy and in relative change (contrast ratio, CR)∙ low latency (delay)∙ low energy consumption and energy dissipation∙ scalability (expandable in 1-D or 2-D arrays)∙ small physical size∙ cascadablity (provide logic level restoration)∙ history independence∙ low cost∙ non-critical, non-sensitive set-up and operation∙ integrability with signal input, output, and control∙ good input/output isolation

Switches must not only incorporate many of the above attributes, but they are also

constrained by other components of the network, including the transmitters, receivers,

types of data, the medium through which the data travel between transmitters and

receivers, and the physical elements of network control. It is assumed here that the data

are binary bitstreams and that the medium for data transmission is fiberoptic cable as

opposed to electrical cable. (As discussed in Appendix A, fiberoptic cable provides

many advantages compared to electrical cable for high bit rate, long-distance

communication). Because the data travels in an optical form, optical switches are

required.

Transmitters are capable of providing 1-10 GHz data using active components,

such as edge-emitting lasers (e.g., distributed feedback lasers, DFBs) or vertical cavity

surface emitting lasers (VCSELs),[1-4] and up to 40 GHz data using modulators.[5-8]

i Frequency and phase modulation are also possible, though uncommon. These are referred to asfrequency- and phase-shift keying, FSK and PSK.

4

Depending on the application, switches must be able to handle such bit rates, or even

faster rates if the data is to be muliplexed in time.

System-level operation and control of the network also exert a strong influence on

what is required of switches. At present, most intelligence or logic capability is handled

by electronic circuitry. Optical logic is not currently competitive and will not be

competitive for the foreseeable future, at least for logically complex operations.

Therefore, any complex intelligence aimed at controlling switches, such as routing and

higher-level operations, will reach the switches only as electrical signals. This suggests

that electrically-controlled optical switches will be needed. On the other hand,

optically-controlled optical switches in some situations can be more efficient and

significantly faster than their electrically-controlled counterparts. When little intelligence

is required, they can provide functionality that is difficult to do electronically, such as

high speed multiplexing or demultiplexing of bitstreams. Similarly, wavelength

conversion is particularly efficient using optically controlled devices. Hybrid

architectures may be imagined which combine the best of both types of control signals.

For example, it is possible to design a routing system that switches large blocks of data

electronically but requires high speed bit-level switching – such as wavelength

conversion – that may be done much more efficiently with optically controlled devices.

Consequently, optically-controlled optical switches may be needed in future

communication networks.

Another system-level constraint on switches comes from the nature of the data to

be routed. The length of data packets and individual bit duration may strongly impact the

required speed of the switch as suggested in the previous paragraph. Meeting the

potential requirement of very short bit duration in future TDM bitstreams may be

possible only by using optically controlled switches. The higher level structure of future

data networks is an active area of research, and no definite conclusions may be drawn

about the required switching speeds at this time.

5

1.2 COMPARING OPTICAL SWITCHES

1.2.1 Comparisons based on nonlinearityIs there a metric, such as a particular physical attribute, that measures how well a

switch functions compared to others? All switches are inherently nonlinear devices and,

therefore, the nonlinearity of a switch is one measure that may provide a good

figure-of-merit. A linear device treats signal and control inputs independently; no effect

on the signal input is possible due to the presence or absence of a control input.

Consequently there is no switching. A nonlinear interaction, however, does not treat the

inputs independently, hence one input effects the other. In an optical switch, nonlinearity

is generally due to coupling between the propagating electric field of light and the

charged particles, principally electrons, of the particular material of the switch itself.

Many switches have a well-defined, measurable material nonlinearity. For

example, optical parametric switches are rigorously described by a power series-based

nonlinearity in terms of nonlinear susceptibilities (i.e. �(2), �(3), etc.) In semiconductors,

changes in absorption and the index of refraction due to band-gap filling and related

effects, are similarly well-described by a nonlinear complex dielectric function.[9] For a

given type of switch, a larger nonlinear coefficient generally implies a lower required

input control energy and hence a more efficient switch.

Some authors have compared a measure of nonlinearity of materials, such as the

index of refraction’s dependence on the intensity of the incident light, n2, for certain types

of switches (Fig. 1.1 (top)). Figures-of-merit which take into account other factors, such

as the effective absorption length of a switch, or the degree to which diffraction limited

light can be focused, have also been suggested. One such figure-of-merit, n2/��3, is

plotted in Fig. 1.1 (bottom).[10] n2 is a measure of the second-order nonlinearity of the

index of refraction; one expects that switch efficiency will improve with increasing

nonlinearity. On the other hand, the output of the switch degrades as the signal is

absorbed, proportional to the absorption, ��multiplied by interaction length, �,

suggesting a reduction in the figure-of-merit as � and � increase. Finally, the ability to

focus the light to a small spot in order to increase its intensity is limited by diffraction

`which is proportional to �2. This type of figure-of-merit works well for comparing

6

certain switches whose basis for switching is similar (e.g. waveguide switches).

Stegeman and Wright also use a similar figure-of-merit: n2/��.[11]

Figure 1.1: (top) The nonlinear coefficient of the refractive index, n2, as a function of relaxation time (thetime the material needs to recover from a pulse of light), for a variety of materials. (bottom) Afigure-of-merit for switching based on n2 modified for the power used (absorption, ��multiplied byinteraction length, �), and the ability to focus the light to a small spot in order to increase its intensity (�2).Note that the relative strength of certain materials, such as glass, is more prominent for the figure-of-meritthan with n2 alone. (From Ref. [10])

7

Making direct comparisons between wider varieties of switches, however, is

difficult using nonlinear coefficients as a figure-of-merit. Some switches have different

functional dependencies on the same nonlinear susceptibilities. For example, switches

based on sum-frequency mixing are dependent on the square of the second-order

nonlinearity of susceptibility of the material, (�(2))2. Switches based on cascaded second

harmonic generation, however, depend on the fourth power of the second-order

nonlinearity, (�(2))4, and Kerr-like switches are linearly proportional to the third-order

nonlinearity, �(3). Simply knowing the magnitude of �(2) or �(3) is insufficient to

determine which switch is more efficient. For other switches, such as those that are

bi-stable or nonlocal, (e.g., self-electrooptic effect devices (SEEDs)), nonlinear

susceptibilities do not properly describe their switching behavior.ii Finally, for certain

switches (e.g., micro-electro-mechanical systems (MEMS) mirrors) switching is

independent of the material nonlinearity itself. Therefore, another means for making

comparisons is needed.

1.2.2 Switching speed and switching energy comparisonsA simple method for comparing switches is to focus on the desired optical

switching attributes listed earlier. There is an interplay between some of these attributes

(improving one may worsen another) and the specific uses of a switch determine which

characteristics are most important. A switched signal should have a good contrast ratio

(CR) and a large change in the energy in the output beam; many figures-of-merit look at a

combination of these two factors. A high contrast ratio may be obtained by removing the

effective DC floor of the signal, and that may be accomplished by running two signal

streams in a differential configuration. Though slightly more complicated and twice as

energy intensive, this ability suggests that the change in energy of the signal is of more

fundamental importance. If it can be assumed that a certain change of the signal must be

obtained for a particular application, reasonable questions would be: “how much control

signal energy is required to accomplish such switching” and, “how fast would such a

switch operate?” The broad functionality of these two characteristics, switching time and

ii Under certain circumstances, “effective” nonlinear susceptibilities may be determined.[12]

8

switching energy, make them a useful comparison tool for switches. Table 1.1 presents

just such a list where switching energy is defined as the energy required by the device to

enable strong switching operation.

Table 1.1: Comparison of Optical Switch Energies and Repetition Rates *

Type of Switch Repetition Rate (GHz) Enengy (pJ) Reference (~1/switching time)

Electrically-ControlledElectrooptic modulators **

Polymer 40 10 [13]Lithium Niobate 40 28 [6]

40 9 [14]Semiconductor 40 35 [15]

Electroabsorption modulatorsSemiconductor 40 0.9 [16]

50 0.140 [17]Tunable lasers 0.1# 10 # [18]MEMS 2 MHz 0.5 [19]

Optically-ControlledCoherent

OPG 100 5-10 [20]NOLM 640 3.5 [21]Soliton Gate 200 5.8 [22]NLDC 1000 180 [23]Virtual Switches 500 40 [24]

IncoherentFiber-SOA devices 80 0.2 (1.1)*** [25]Waveguide

XGM 100 0.1 (0.9)*** [26]XPM 160 0.05 (1.1)*** [27]XAM 20 6 [28]

Surface-Normal 100 6 [29]

SEED 25 4 [30]

OCOG 50 1.5 [31]

* Table 1.1 presents a representative listing of optical switches only.

** The electrooptic modulators cited here are all travelling-wave modulators.

*** These devices also require significant external power (e.g., SOAs are current-driven forwardbiased diodes). The required optical control power is listed first and the total power is listed inparentheses.

# The values listed here for tunable lasers often are for tuning between two closely spacedwavelengths. Typically, tunable lasers are made to tune across many wavelengths andconsequently tune more slowly and require greater energy.

9

Comparing the total switching energies and speeds (Table 1.1),

electronically-controlled switches are generally slower than optically-controlled devices

as they are limited to the control circuitry speeds. Fiber-based switches, such as soliton

gates and nonlinear optical loop mirrors (NOLMs) can switch very fast but they are bulky

and are difficult to use in large numbers. Recently, optical parametric generation (OPG)

has undergone significant advances particularly with regard to making phase-matching

easier (e.g. using periodically-poled materials). Not only is it fast but is also becoming

low-powered. Switches based on this principle hold great potential. Incoherent

optically-controlled waveguide devices such as semiconductor optical amplifier (SOA)

switches based on cross phase modulation (XPM) are perhaps the best general-purpose

optical switches: they have demonstrated fast, low-powered switching and eve provide

gain; they are, however, generally limited to one-dimensional arrays. Two-dimensional

scaling is readily achieved using surface-normal switches, but most are based on energy-

intensive saturable absorption. Optically controlled optical gates offer both high speed

and low energy switching and are scalable in two dimensions. Moreover, their

optoelectronic nature allows them to be simultaneously integrated with circuitry, a unique

and useful combination.

1.3 SUMMARYThe preceding introduction has attempted to set optically controlled optical gates

(OCOGs) within the larger context of optical switches. The role optical switches play in

enabling the next revolution in communications was highlighted, general switching

principles were established, and key characteristics of optical switches were compared.

Against this background the present thesis addresses the following general

question: What are the limits of optically-controlled optical gates (OCOGs)? Answers to

the follow specific questions are pursued:

∙ What are the underlying physical principles that govern the operation of

OCOG devices and how do they constrain its performance?

∙ What switching energies are required?

∙ How fast can OCOGs operate and over what wavelength range?

10

The remainder of this dissertation is presented in seven additional sections:

(Chapter 2) literature review of optical switches; (Chapter 3) description of the concept of

the optically-controlled optical gate; (Chapter 4) the theory behind OCOG operation;

(Chapter 5) description of the experimental set-up used to test the behavior of OCOGs;

(Chapter 6) results; (Chapter7) simulation details; (Chapter 8) conclusions and thoughts

on directions for future research.

11

REFERENCES

1. Okayasu, M., N. Ishihara, and S. Tohono, "10 Gbit/s 1.3 �m DFB-LD module in0.5cc ceramic package for LAN/MAN applications," Elec. Lett., vol. 37, pp. 303-304 (2001).

2. Peters, F.H. and M.H. MacDougal, "Hihg-Speed High-Temperature Operation ofVerical-Cavity Surface-Emitting Lasers," IEEE Phot. Tech. Lett., vol. 13, pp.645-647 (2001).

3. Steinle, G., et al., "Data transmission up to 10Gbit/s with 1.3 �m wavelengthInGaAsN VCSELs," Elec. Lett., vol. 37, pp. 632-634 (2001).

4. Tomkos, I., et al., "10-Gb/s Transmission of 1.55-�m Directly Modulated Signalover 100 km of Negative Dispersion Fiber," IEEE Phot. Tech. Lett., vol. 13, pp.735-737 (2001).

5. Dagli, N., "Wide-Bandwidth Lasers and Modulators for RF Photonics," IEEETrans. Microwave Theory and Tech., vol. 47, pp. 1151-1171 (1999).

6. Burns, W.K., et al., Low Drive Voltage, 40GHz LiNb03 Modulators in OpticalFiber Communication Conference 1999 & International Conference on IntegratedOptics and Optical Fiber Communication 1999 (OFC/IOOC, 1999).

7. Leclerc, O., et al., "Simultaneously regenerated 4x40Gbit/s dense WDMtransmission over 10,000km using single 40 Ghz InP Mach-Zehnder mondulator,"Elec. Lett., vol. 36, pp. 1574-1575 (2000).

8. Leclerc, O., et al., "40 Gbit/s polarisation-insensitive and wavelength-independentInP Mach-Zehnder modulator for all-optical regeneration," Elec. Lett., vol. 35, pp.730-731 (1999).

9. Haug, H. and S. Schmitt-Rink, "Basic mechanisms of the optical nonlinearities ofsemiconductors near the band edge," J. Opt. Soc. Am. B, vol. 2, pp. 1135-1142(1985).

10. Walker, A.C., "A comparison of optically nonlinear phenomena in the context ofoptical information processing," Optical Computing and Processing, vol. 1, pp.91-106 (1991).

11. Stegeman, G.I. and E.M. Wright, "All-optical waveguide switching," Optical andQuantum Electronics, vol. 22, pp. 95-122 (1990).

12. Khurgin, J.B., et al., "Cascaded optical nonlinearities: Microscopic understandingas a collective effect," J. Opt. Soc. Am. B, vol. 14, pp. 1977-1983 (1997).

13. Lee, S.-S., et al., "Optical Intensity Modulator Based on a Novel ElectroopticPolymer Incorporating a High �B Chromophore," IEEE J. Quant. Elec., vol. 36,pp. 527-532 (2000).

12

14. Hopfer, S., Y. Shani, and D. Nir, "A Novel, Wideband, Lithium NiobateElectrooptic Modulator," J. Lightwave Tech., vol. 16, pp. 73-77 (1998).

15. Sakamoto, S.R., A. Jackson, and N. Dagli, "Substrate Removed GaAs-AlGaAsElectrooptic Modulators," IEEE Phot. Tech. Lett., vol. 11, pp. 1244-1246 (1999).

16. Satzke, K., et al., "Ulrahigh-bandwidth (42 GHz) polarisation-independent ridgewaveguide electroabsorption modulator based on tensile strained InGaAsPMQW," Elec. Lett., vol. 31, pp. 2030-2032 (1995).

17. Kawano, K., et al., "Polarisation-insensitive travelling-wave electrodeelectroabsorption (TW-EA) modulator with bandwidth over 50 GHz and drivingvoltage less than 2V," Elec. Lett., vol. 33, pp. 1580-1581 (1997).

18. Coldren, C., Personal Communication (2001).

19. Solgaard, O., F.S.A. Sandejas, and D.M. Bloom, "Deformable grating opticalmodulator," Opt. Lett., vol. 17, pp. 688-690 (1992).

20. Parameswaran, K., Personal Communication (2001).

21. Yamamoto, T., E. Yoshida, and M. Nakazawa, "Ultrafast nonlinear optical loopmirror for demultiplexing 640 Gbit/s TDM signals," Elec. Lett., vol. 34, pp. 1013-1015 (1998).

22. Islam, M.N., C.E. Soccolich, and D.A.B. Miller, "Low-energy ultrafast fibersoliton logic gate," Opt. Lett., vol. 15, pp. 909-911 (1990).

23. Kang, J.U., G.I. Stegeman, and J.S. Aitchison, "All-optical multiplexing offemtosecond signals using an AlGaAs nonlinear directional coupler," Elec. Lett.,vol. 31, pp. 118-119 (1995).

24. Kan'an, A.M., et al., "1.7-ps Consecutive Switching in an Integrated Multiple-Quantum-Well Y-Junction Optical Switch," IEEE Phot. Tech. Lett., vol. 8, pp.1641-1643 (1996).

25. Kelly, A.E., et al., "80 Gbit/s all-optical regenerative wavelength coversion usingsemicoductor optical amplifier based interferometer," Elec. Lett., vol. 35, pp.1477-1478 (1999).

26. Ellis, A.D., et al., "Error free 100 Gbit/s wavelength conversion using gratingassisted cross-gain modulation in 2mm long semiconductor amplifier," Elec. Lett.,vol. 34, pp. 1958-1959 (1998).

27. Tajima, K., S. Nakamura, and Y. Ueno, "Ultrafast all-optical signal processingwith Symmetric Mach-Zhender type all-optical switches," Optical and QuantumElectronics, vol. 33, pp. 875-897 (2001).

28. Cho, P.S., D. Mahgerefteh, and J. Goldhar, "All-Optical 2R Regeneration andWavelength Conversion at 20 Gb/s Using an Electroabsorption Modulator," IEEEPhot. Tech. Lett., vol. 11, pp. 1662-1664 (1999).

13

29. Loka, H.S. and P.W.E. Smith, "Ultrafast All-Optical Switching with anSymmetric Faby-Perot Device Using Low-Temperature-Grown GaAs: Materialand Device Issues," IEEE J. Quant. Elec., vol. 36, pp. 100-111 (2000).

30. Serkland, D.K., et al., Fast-switching symmetric self-electrooptic-effect device at865 nm in CLEO 2000 (Opt. Soc. Am., 2000).

31. Yairi, M.B., H.V. Demir, and D.A.B. Miller, "Optically controlled optical gatewith an optoelectronic dual diode structure - theory and experiment," Optical andQuant. Elec., vol. 33, pp. 1035-1054 (2001).

14

Chapter II: Methods of Optical Switching

Extraordinarily wide varieties of optical switches have been developed: the

mechanisms alone upon which switching is based range from electrical flipping of

mirrors to creation of virtual particles, while their potential uses vary from network

restoration (e.g, if a fiber optic cable is broken) to all-optical logic. Light interacts

with matter usually via the coupling of its electromagnetic wave with electrons. Thus,

it is not so surprising that at the heart of every optical switch lies some means of

(nonlinear) electronic change in the interacting material. Although this dissertation

focuses on one particular type of switch, it is important to know about the other types

of optical switches that have or are currently being investigated and used. Brief

descriptions of a number of representative optical switches are provided below. These

descriptions are organized primarily by switching method: optical-electrical-optical

conversion (OEO), electronic control, optical control, and self-electrooptic effect.

Within these categories, both the principles of operation and characteristics (including

advantages and disadvantages) are provided for each type of switch.i

In Chapter 1 it was suggested that useful parameters for comparing optical

switches are the switching time and energy. These values provide a sense of the

efficiency of the optical switch, although they do not incorporate numerous other

parameters that might be important, such as latency, size, complexity, reliability,

dynamic ranges, sensitivity, or cost. Nevertheless, they do provide a first-order

classification scheme applicable across a wide variety of different types of switches.

Table 1.1 provided this information for a number of different switches, many of which

are described in this chapter.

i The subject matter of the literature review is broad. An attempt has been made to specifically referencecharacteristic early work and occasionally general reviews published for the types of switches described andstate-of-the-art references, particularly when energy and switching times are presented.

15

2.1 OPTICAL-ELECTRICAL-OPTICAL DEVICES (OEOS)Principles of operation

The conversion of optical data to electronic bits and back to optical data is the

primary means by which optical bitstreams are switched and routed today. OEO

switches consist of three components: (1) a receiver for the incoming optical data,

transforming them into an electrical signal, usually a p-i-n photodiode or avalanche

photodiode; (2) electronic circuitry that provides data processing and routing

capability; and (3) a transmitter or transmitters that transform the final electronic

signal into optical data, as is illustrated in Fig. 2.1. These devices are typically edge-

emitting solid state lasers that are either directly modulated or are integrated with

external modulators. [1, 2]

Electronics: 3-R Logic Routing

Photodetector Laser

Figure 2.1: Schematic diagram of an OEO switch. Incoming light is absorbed by a photodetector. Theresulting electrical signal is electronically cleaned-up and processed. A laser or external modulator usesthe final electronic signal to retransmit the incoming data.

Characteristics

One reason OEOs are in wide-spread use today is that they provide a direct

means for inserting intelligence into the switch via electronic logic. This is an easy

task to accomplish in these devices because the signal becomes entirely electronic

during the conversion process. Pulse reshaping, regeneration (in amplitude), and

retiming (the “3 R’s” of signal regenerators) of the electronic signal using electronic

logic are transferable to the re-emitted optical data. Moreover, the data can be time-

division (de)multiplexed while the data are in their electronic form and then be re-

emitted at any wavelength or spatial location as long as the appropriate laser is

electronically accessible.

16

These advantages, however, come at a price. OEOs are limited by the speed of

the electronic circuitry and the rate at which the outgoing data can be modulated.

Typical switching rates are between 1-10 GHz and the circuits are designed to operate

only at a single bit rate.[2-4] The retransmitted bit rate may also be constrained by the

chirp (optical frequency sweeping) placed on the output data, particularly if the lasers

are directly modulated. Additionally, OEOs only switch data that are of a digital,

intensity-modulated format, and will only work with a given data format – they do not

have data format transparency.

One of the most prohibitive disadvantages of OEOs is their cost. The price of

each output laser is of the order of ten thousand dollars, an extremely expensive price

for a single element of a switch if such switches need to be used in large quantities.[5]

New technology under development may eventually lower the price. These include

vertical-cavity surface-emitting lasers (VCSELs),[6] new generations of distributed

feedback (DFB) edge-emitting lasers, and cheaper cooling systems. However, the

limitations discussed here will become more constraining as the bit rate and number of

channels increase, and may result in OEOs giving way to other types of optical

switches. Such OEOs are also in practice difficult to integrate, limiting cost reduction,

because the technologies of the different components are not only disparate but each is

used near its current operating limit.

2.2 ELECTRONICALLY-CONTROLLED OPTICAL SWITCHES

The switches at the moment that offer the stiffest competition to OEOs are

electronically-controlled optical modulators.[7-9] Electronic control provides the

means to interface with electronic logic. This category of devices does not use active

photoemissive elements such as lasers; instead the switch alters the pre-existing bit

stream directly either at a packet or bit level. By leaving the optical data relatively

undisturbed (compared to OEOs), latency and power consumption can be reduced.

Greater data format transparency is also possible. On the other hand, these types of

modulators do not provide 3R ability or the option of wavelength conversion; they are

17

purely space and/or time switches. This type of functionality may be obtained if they

are able to be integrated as part of a more elaborate OEO switch.

2.2.1 Electrooptic modulatorsPrinciples of operation

One of the most common types of electrically-controlled switches are

electrooptic modulators. They come in a variety of waveguide configurations, usually

Mach-Zehnder interferometers (symmetric or asymmetric, with single-arm or two-arm

control[10]) but also coupled waveguides and intersecting X switches. By inducing a

phase shift, usually � radians, in one arm of the switch versus the other, the intensity

of the recombined light at the output port(s) may be controlled.

One basis for changing the phase is the linear electrooptic (Pockels) (2)� effect,

namely, applying an electric field changes the index of refraction of the material as

expressed by (2) 3

0

0 1 2E n rEn � �

� �� where r is the appropriate electrooptic tensor

coefficient. In some switches other methods for electrically controlling the relative

phase are also possible. If the (2)� -dependent term is small compared to the (3)

� term,

(3) 20

0 1

En � �

� �� ; this may occur, for example, in centro-symmetric crystals or in the

presence of large electric fields. Consequently, change in the index of refraction is

proportional to E2, the Kerr effect. Another related phase-shifting mechanism is the

quantum-confined Stark effect (QCSE) – based on (3)� behavior -- in which an

applied electric field strongly shifts the bandedge in a multiple quantum well structure

and hence the index (due to the Kramers-Kronig relationship). Switches may also be

designed based on the index change due to changes in carrier density.[11, 12] This

particular effect is actually often a detrimental side-effect in optical switches because it

adds chirp to the output data bits.

18

V+

_

Figure 2.2: Schematic diagram of an electrically-controlled waveguide interferometric switch. Theincoming light pulse is split at the left-most Y-branch. Changing the voltage, for example, may alter thephase of the bottom-propagating pulse so that it is either in-phase or out-of-phase when it recombineswith the top-propagating pulse at the right-most Y-branch.

There are three principle types of materials used for electrooptic switches:

dielectric, semiconductor, and polymer. For switches based on the Pockels effect, the

electric field needed to induce a � phase shift depends on the nonlinear coefficient r of

the material. Dielectric materials, such as lithium niobate (LiNbO3) typically have

relatively large r, approximately 30 pm/V. This is offset, however, by their smaller

index of refraction n and their larger (low frequency) dielectric constant �, which also

reduces device switching speed.[13, 14] Some organic polymers have similarly large

electrooptic coefficients due to large effective molecular dipoles. Electrooptic

polymer materials are an active area of research; they are potentially capable of very

high-speed modulation at reasonable voltages, and have the added advantage of

fabrication integration with standard electronic circuitry. High power operation,

however, can lead to polymer degradation.[15, 16] The semiconductor material

systems of GaAs/AlGaAs, InGaAsP/InP, and InGaAs/InAlAs have smaller r (e.g., 1.4

pm/V for GaAs) although this is somewhat offset by a larger index of refraction.[17,

18] The QCSE is often a much stronger effect, able to shift the index of refraction one

or two orders of magnitude more than the Pockels effect can given the same electric

field. This electrorefraction is enhanced due to its proximity to the bandgap where it is

resonant. However, this proximity also limits the device length due to band-tail

absorption. In all waveguide devices, residual loss becomes important to consider in

longer waveguides.

19

Electrically-controlled thermooptical devices also fall into this category of

modulators.[19-22] Here, an applied current is passed through a resistor that is

adjacent to part of the waveguide. As the current increases, the resistor heats up. The

phase of the light passing through that arm of the waveguide is altered due to an

expansion of the material and a shift in its temperature-dependent refractive index.

The principle drawback of this type of switch is the relatively slow heating and cooling

periods required – on the order of milliseconds. On the other hand, thermooptic

switches use low voltages and can be used with silica-on-silicon waveguides, making

integration with silicon circuitry potentially significantly easier than optical switches

made of other material.

Characteristics

The energy required to induce sufficient change in phase is determined by the

necessary electric field, the capacitance and resistance of the device, as well as the

electrical driving components. Switching speed is generally limited by the overall RC

time constant of the device as well as the relaxation time of the material.ii Travelling-

wave modulators reduce the switching time by effectively turning on only small

sections of the device at a given time, shrinking the RC time constant. These benefits

come at the cost of greater device complexity .

Electrically-controlled interferometric switches are typically waveguide

devices, making it simple to scale the number of devices made in the lateral direction

(dimension) perpendicular to the waveguides. Such devices can both modulate a

beam and switch it physically from one path to another; when the two arms recombine

out-of-phase, eliminating output in the forward direction, the resulting light instead is

reflected and may be, with proper design, redirected. Another notable characteristic is

low chirp. Finally, coupling between edge-emitting lasers and waveguides in general

is particularly lossy due to the elliptical beam-profile of the laser, although mode

matching can help eliminate some of these effects. The most common type of 10 GHz

ii The relaxation time is the time for the induced dipoles that arise in the presence of the applied field torelax once the field has been removed. In most nonlinear optical materials this is very rapid.

20

external modulator currently commercially available is an electooptic lithium niobate

Mach-Zehnder modulator.

2.2.2 Electro-absorption modulatorsPrinciples of operation

Electro-absorption modulators are another type of electrically-controlled

optical switch.[23, 24] An applied field electric field changes the absorption of the

device, directly modulating the transmission of the optical datastream. Consequently,

the beam path does not need to be split and recombined (as it does for an

interferometer). These devices are typically reverse-biased p-i-n diodes whose

intrinsic region is either bulk semiconductor, in which case the absorption shift is due

to the Franz-Keldysh effect, or is made of multiple quantum wells, in which case the

quantum confined Stark effect (QCSE) provides the switching mechanism. The

physical design of this type of switch is more flexible than that of waveguide

interferometers, since both waveguide and surface-normal devices (including those

that use Fabry-Perot resonances to enhance their switching behavior[25]) are possible.

Characteristics

The limitations of electroabsorption devices are similar to those of

interferometers. The RC time constant of the switch and driving circuitry limit the speed

and increase switching energy. The primary difference between the two categories lies in

the magnitude of the electric field that needs to be applied. Generally speaking, the

energy required to obtain complete switching (e.g., 10 dB contrast ratio (CR)) is

significantly lower than for electrooptic devices, although heat dissipation may be a

concern in this type of switch. One advantage compared to interferometers is that the

device design is simpler. Additionally, there is no oscillatory behavior of the switch with

increasing switching energy, allowing a larger dynamic range for the control energy (in

electrooptic devices, applying too large a field results in a greater than � phase shift,

reducing the modulation depth). Electroabsorption devices cannot, however, easily

switch a beam from one path to another, though there are some clever exceptions.[26, 27]

21

2.2.3 Other voltage-controlled devicesOther electrically-controlled switches include MEMS mirrors and liquid crystal

modulators.

2.2.3.1 MEMS mirrors

Principles of operation

Micro-electronic-mechanical systems (MEMS) mirrors are lithographically-

defined micro-mirrors. Depending on the specific design, an applied voltage typically

translates or rotates mirrors. As a consequence, an incoming beam of light may be

directed to a new spatial location. MEMS mirrors are usually broadband, capable of

reflecting all of the wavelength channels that may be present in the incident light.

Characteristics

The dimensions of MEMS mirrors vary a great deal in size, from around 15x15

�m for display devices to 500x500 �m for cross-bar switches.[28, 29] The

corresponding capacitance combined with large switches voltages (e.g., 100V) results

in large switching energies (~4000 pJ). Switching time is limited by the mechanical

resonance of the switch and is generally on the order of milliseconds. Some MEMS

mirrors, based on modifying a grating or resonance structure instead of moving the

entire mirror in order to modify the beam, can have MHz switching rates.[30-32] On

the other hand, although they are far too slow to switch individual bits, MEMS mirrors

may switch long packets of data or be used for restoration in a network. Switching is

format independent, low loss, scalable in two dimensions, and practically cross-talk

free.

2.2.3.2 Liquid Crystals and Photorefractive Materials


Another method of electrically controlling light is by applying a voltage to a liquid

crystal. The molecules in a liquid crystal each have a dipole and thus rotate when an

electric field is applied across them, changing the index of refraction of the crystal and its

22

polarizing effects. These changes may be used in a variety of ways for optical

switching.[33, 34] For example, without an electric field the molecules become

randomly oriented, providing an index of refraction that is an average of the ordinary and

extraordinary indicies of the material (n0 and ne). When an electric field is applied, the

molecules orient themselves and thus the index of refraction for normally incident light

becomes, say, n0. This change in index with field can be used, for example, to make an

index-grating-based mirror become strongly reflecting in one state or transparent in

another. Alternatively, by pre-biasing the crystal with one electric field and then

controlling a second field perpendicular to the first, the orientation of the molecules shift,

providing an electrically-controlled polarizer. When combined with passive external

polarizers, this can be used as an efficient means for directing the outgoing light.

Characteristics

Generally, liquid crystals require switching times between micro- to milliseconds,

limited by the time it takes for the molecules to reorient themselves when the field

changes. The voltages required vary, too, but can range up to 100s of volts. On the

positive side, liquid crystals tend to be cheap to manufacture and scale well in one and

two-dimensional arrays.[33]

There are also electrically-controlled diffraction gratings that can be created in

photorefractive crystals, such as in para-electric crystals (e.g. KLTN) -- such devices

have an index of refraction proportional to the square of the electric field. These switches

have significantly faster switching times, on the order of nanoseconds, and are used in

commercial systems (Trellis Photonics).[35]. Also, acousto-optic and magneto-optic

effects in crystals can similarly be used to create, say, a diffraction grating that may be

used for optical switching.[36]

2.3 OPTICALLY-CONTROLLED SWITCHES

Optically-controlled switches change their state due to either an optical control

signal or the intensity of the optical data signal itself; no electronic control is required.

Two broad categories of all-optical switches are “coherent” and “incoherent” switches.

23

The former uses only virtual (transient) electronic properties whereas the latter

involves absorption and the creation of real charge carriers.

2.3.1 Coherent switchesCoherent switches are all-optical devices whose switching function is primarily

governed by phase-dependent interactions between electric fields of the optical beams.

Several types fall under this category, including optical parametric generation,

nonlinear optical loop mirrors, soliton gates, nonlinear coupled mode devices, and

virtual gates.

2.3.1.1 Optical Parametric Generation (OPG)


An archetype of coherent switches is the general category of those based on

optical parametric generation, a physical process where photons at certain wavelengths

are coherently absorbed in a material and re-emitted at a different wavelength. The

switch produces an output (either generating or depleting a signal) in a

communication channel at a wavelength different from the control beam. Such a

process is possible due to the nonlinear response of certain materials to an electric

field of the incident light. This nonlinearity is well stated by a complex dielectric

coefficient that depends on a power-series expansion:� � � � � �� 1 2 3 2

0( ) 1 ( ) ( ) ( ) ...E E� � � � � � � � ��

Because parametric processes typically operate far from the resonant frequency

of the device’s material, the dielectric coefficient is largely real. There is little

incoherent (loss of phase information) absorption of light. The incident wavelengths

of light are virtually absorbed as the light interacts with the material, creating a driving

field at new frequencies. Under the proper circumstances, the excited material relaxes

by coherently emitting photons at a new frequency or frequencies instead of at the

original wavelength.

24

�input

�pump�new

Nonlinear Material

Figure 2.3: Schematic illustration of optical parametric generation using, say, PPLN. The input datasignal at �input propagates with a strong c.w. beam at �pump. As they propagate in the nonlinear material,an identical data stream is generated at a new wavelength, �new= 2��pump��input� The input and newwavelength data streams are phase-matched to each other so that as they propagate the power of the newsignal continues to grow. In this figure, the different colored sections in the material are differently-poled regions of lithium niobate that enable quasi-phase matching to be maintained.

Several types of potential nonlinear interactions are possible, though usually

only one is dominant. Depending on the specific materials and wavelengths selected,

different susceptibilities dominate the dielectric coefficient, determining the behavior

of the switch. When the second-order nonlinearity �(2) is the leading term, several

processes, including second harmonic generation (�’=�1+�1), difference frequency

mixing (�’=�1-�2), and sum frequency mixing (�’=�1+�2), may be used to encode

data into a new channel. Third-order nonlinearity �(3) provides additional wavelength

combinations, such as (�’=�1+�1+�2) or (�’=�1+�1-�1) through 4-wave mixing. [37-

39] Second-order nonlinearities may also be cascaded to create effective third-order

processes (e.g., �’=�1+�1, followed by �’’=��’-�1). [40, 41] This is useful when a

third-order type of functionality is desired but otherwise difficult to achieve because

the material has predominately a second order nonlinearity (as in LiNbO3).

Characteristics

In some respects parametric-based optical switches are ideal devices. Their

switching time is nearly instantaneous and very short-lived, so that switching rates are

only limited by the pulse length. High switching (conversion) efficiencies are

possible, particularly with waveguide configurations.iii [42] In passive devices, almost

no excess noise is added to the signal. The primary limitation in OPG devices is the

need for phase-matching between the incident and generated light. This may be

25

compensated by using such quasi-phase-matching techniques as periodically-polling

ferroelectric materials like LiNbO3 (PPLN),[43] or using bonding or regrowth

techniques for semiconductor material systems such as GaAs.[44, 45] Frequency

mixing in active devices, such as semiconductor optical amplifiers, have the

disadvantage of introducing amplified spontaneous emission; on the other hand, phase-

matching requirements are significantly relaxed due to their relatively high conversion

efficiency and resulting short required interaction lengths.[38, 39, 46] Other

limitations include frequency-matching requirements: the generated/modified

wavelength has a one-to-one relationship given particular input wavelengths.

Switching one wavelength arbitrarily to another is not easily accomplished. Finally,

group velocity walk-off restricts the pulse-length (and its associated pulse bandwidth)

and the length of the device that may be used.

2.3.1.2 Nonlinear Optical Loop Mirrors (NOLM)


Another switch based on coherent interaction between light pulses is the

nonlinear optical loop mirror.[47] For these and other interferometic all optical

devices, switching is often accomplished by making use of (3)� response (the Kerr

effect) of a material. Because (3) 2

0

0 1

En � �

� �� and the intensity of light is

proportional to E2, the change in index is proportional to the intensity of the incident

light. NOLMs themselves are optically-switched Sagnac interferometers. Incoming

light is split into two pulses via a coupler, and each of the resulting beams propagates

in opposite directions, clockwise (cw) and counter-clockwise (ccw) around a loop of

fiber (typical (3)� in glass ~ 10-15 cm2/W), recombining back at the coupler. If they

are in-phase when they intersect, the recombined light exits primarily down the output

port. If they are out-of-phase, they recombine and exit back along the input port. A

iii As with many optically-controlled switches, switching energy is not well-defined since the efficiency ofthe switch, and hence the energy required, changes with pulse length. We assume here that pulse length,when appropriate, will be on the order of picoseconds.

26

relative � phase shift may be obtained via self-phase modulation if an unbalanced

coupler is used.[47] Alternatively, the phase of one of the propagating signal pulses in

the fiber loop could be changed due to the presence of a copropagating control pulse

via cross-phase modulation. Control pulses of either different wavelengths[48, 49] or

different polarizations[50] allow smooth decoupling from signal pulses. Moreover,

using a control pulse allows a balanced (50/50) coupler to be used, maximizing the

potential contrast ratio of the device. Care must be taken so that the difference in

signal and control propagation velocities does not result in too short a walk-off

distance, limiting the duration of cross-phase modulation. Soliton pulses may also be

used in NOLMs.[51]

Characteristics

Switch latency is determined by the length of the fiber loop, typically on the

order of several hundred meters. Shorter lengths are possible but generally come at the

expense of higher switching energies. Repetition rate is limited by the pulse length

and, if there is walk-off between signal and control pulses, the separation time that

grows between them.iv

2.3.1.3 Soliton Gates


Soliton gates are nearly perfect all-optical switches, exhibiting most of the

desired characteristics for optical switching and logic. The propagation of the electric

field of light through nonlinear material may be described by a nonlinear Schroedinger

wave equation. In glass fiber, the Kerr-nonlinearity (intensity-dependent index of

refraction) is accounted for by adding a third-order term with a coupling constant

proportional to n2 (a (3)� term). The steady-state solution in which the pulse shape

iv Repetition rate may also be limited by different signal pulses affecting each other. Normally the controland the counter-propagating signal pulses’ interaction is ignored due to the short overlap time; the same isassumed for a signal pulse and any other counter-propagating signal pulses in the loop (particularly sincethe signal pulses are typically much weaker than the control pulse, and hence have an even smaller effect).Nevertheless, if the loop is long, it may be filled with sufficient pulses that such effects significantlydegrade switching. By limiting the repetition rate this degradation may be avoided.

27

remains constant in time as it propagatesv is known as a temporal, or Kerr, soliton. vi

[56, 57] Soliton pulses do not disperse as they propagate and hence will not spread out

and run into each other. Consequently, very high-bit rate pulse streams may be

envisioned.vii Soliton velocity itself is dependent on amplitude, wavelength, and, if the

medium is birefringent, polarization.

time

without control pulse

t=t0+�t“0”

Arrival time at photodetector

t=t0time

with control pulse

“1”

Figure 2.4: Switching behavior of a soliton dragging gate. As a soliton signal pulse travels down afiber, if a faster control soliton pulse is also present for part of the time, the control pulse will “walkthrough” the signal soliton, temporarily speeding up the signal while slowing down itself while they areoverlapped. As a result, the arrival time of the signal pulse at a photodetector at the end of the fiber willbe sooner (e.g., time t=t0) with a control pulse than without (t=t0+�t).

Soliton gates function by switching the temporal location of a soliton within a

given clock cycle. Solitons interact with each other in an intriguing manner, coupled

together via the nonlinear term of the Schroedinger equation. In birefringent fiber,

given two solitons of different polarization (and hence velocity), one will pass through

the other if timed properly. When traveling in the same direction and near to each

other, they “attract” each other, temporarily slowing the faster soliton and speeding up

v The index of refraction increases with intensity due to Kerr nonlinearity, and so the higher intensity partsof a pulse (e.g. the middle) travel slower than the lower intensity parts (front and back). The time derivativeof phase equals frequency. Consequently, the slowing phase velocity in front half of pulse results in aneffective negative frequency shift, while there is a positive frequency shift in the back At the same time,negative group velocity dispersion (GVD) makes lower frequencies propagate more slowly than higherfrequencies. When both effects are present, they may balance for certain pulse shapes, and the pulse shapeis maintained as the pulse propagates.vi By balancing the Kerr nonlinearity with the electrooptic effect (change of the index of refraction linearlydependent on the electric field), “spatial solitons” may be created.[52-54] Such solitons are confined, forexample, in one dimension, creating waveguides. Spatial solitons interact in a similar manner to temporalsolitons, attracting or repelling each other, and these effects may be used for switching, too. [55] Theseeffects are due to real absorption of light to create free charge carriers able to move and thus locally changethe electric field (and thus it is not a coherent effect).vii Research into long-haul high-bit rate soliton transmission has been a vibrant field, though not directlyrelevant to this thesis. References, [56],[54] are a good starting point. At present, the ability to transmit80 GBps over trans-oceanic distances (10,000 km) has been demonstrated.[58]

28

the slower soliton. This is referred to as soliton dragging.viii A special case of soliton

dragging, known as soliton trapping, occurs when such mutual attraction is sufficient

to keep two solitons trapped together as they propagate. “Billiard-ball”-like

interactions (elastic collisionsix) are also possible given two solitons traveling in

opposite directions with different group velocities. [59, 60] In this case, the solitons

are displaced away from each other in time. Soliton self-switching based on the

soliton self-frequency shift (dependent on the soliton pulse’s intensity) has also been

demonstrated.[61]

Characteristics

All-optical switches based on soliton interactions have been demonstrated.[59]

Making use of soliton dragging, a switching energy of 5.8 pJ operating at bit rates of

0.2 THz was demonstrated.[62] This switch has also demonstrated the feasibility of

using soliton gates for optical logic systems. It was cascadable, provided logic level

restoration and timing, and performed logic functions, and allowed good input/output

isolation. One drawback, as with most soliton switches, was the over 400 m of fiber

required. Consequently, those switches not only are physically large but also incur

significant latency. With specially made fiber, a low-latency switch reduced this

length to 50 m though the switching energy was larger (40 pJ).[63]

2.3.1.4 Nonlinear coupled mode devices


“Nonlinear coupled mode devices” capture a broad category of coherent all-

optical switches.[64] A (non-resonant) control pulse changes the index of refraction

of part of the switch. This index change switches the signal pulse between two ports

or modulates the signal pulse at a single port. Mach-Zehnder interferometers, coupled

waveguides, and X-switches all may fall under this heading. The nonlinear directional

coupler (NLDC) is a classic example.[65, 66] In this device, two waveguides are viii Similar behavior occurs given two solitons of the same polarization but different wavelengths.Depending on the relative phase, the solitons slow or speed each other up.

29

brought sufficiently close together so that the evanescent electric fields of light in one

waveguide overlap and couple with the other waveguide. The coupling results in the

optical intensity oscillating back and forth between the two waveguides as it

propagates. The coupling length is defined to be the distance light needs to propagate

in the coupled system to fully shift into the other waveguide; typical values are of the

order of millimeters in semiconductors. If the initial intensity of the light is

sufficiently high (or a control beam is present to create a comparable effect), the shift

in the index of refraction due to the waveguide material’s nonlinearity, 0 2n n I n� � ,

results in “walkoff”, preventing the evanescent field in the other waveguide from

building up. Hence, at low powers an NLDC acts as a switch that shifts the beam to

the other waveguide, while at high powers little switching occurs. Because the

coupling and nonlinearities involved are all due solely to the electric fields of light,

switching occurs on the order of femtoseconds, though there is latency. Other coupled

mode devices use the same nonlinear dependence on the intensity as the basis for their

switches as well.[67]

Characteristics

Some NLDC devices used wavelengths below half the bandgap of AlGaAs to

enable switching at 1.55 �m to avoid parasitic two-photon absorption creating long-

lived charge carriers.[68] This came at a cost of a weaker nonlinearity that required a

greater switching energy, and generally speaking it has been difficult to obtain large

contrast ratios. Fiber-based NLDCs using dual-core fiber with soliton pulses offered

reduced parasitic absoprtion while avoiding pulse break-up and distortion.[69]

ix Soliton collisions along the same (polarizing) axis may be described by nonlinear Schroedinger equationswhich are fully integrable, and hence are “elastic,” unlike soliton dragging gates whose coupled equationsare not integrable.

30

2.3.1.5 Virtual switches


Virtual switches are another form of coherent optical gates, making explicit use

of virtual carriersx as well as the energy band structure of semiconductors. An incident

control light detunedxi from the absorption resonance couples with the material only

during the duration of the pulse, providing a short-lived nonlinear behavior. The

coupling between photons and electrons may result in new eigenstates of the system,

shifting the resonant energy levels – an optical Stark effect (also referred to as the AC

Stark effect or dynamic Stark effect).[70-72] This shift changes the absorption and

index of refraction near the frequency of a signal beam, hence providing a switching

mechanism. At sufficiently high intensities, virtual excitons may also switch the

signal by bleaching the absorption.

Characteristics

The fast speed and potentially strong switching behavior of virtual switches are

offset by their high switching energies because the non-resonant interaction is a weak

effect. Devices with 1-2 picosecond gating times with limited contrast ratios (between

2-5:1) have been demonstrated with switching energies between 1-10 pJ/�m2.[73, 74]

Parasitic absorption below the band-gap (e.g. Urbach tail or two-photon absorption)

created long-lived carriers that also reduced the switching efficiency.xii

2.3.2. Incoherent switchesIncoherent all-optical switches are the complement to the coherent modulators

described above. Broadly speaking, incoherent switches function by absorbing an

incoming optical control pulse, creating electrons and holes. These charge carriers, in

x Virtual carriers are photo-excited carriers that remain phase coherent with the incident light pulse. Theyare able to relax, re-emitting light in phase with the original pulse. Because the carriers are usually notpresent in their excited state after the light pulse passes by, they are referred to as “virtual” particles.xi In semiconductor material systems, light is detuned to be below the absorption band so that it is notabsorbed.xii A related device has been suggested based on virtual charge screening. [75] With a DC applied biasacross a MQW stack, a slightly detuned below band-gap incident light beam could create virtual electronsand holes that would separate slightly within a quantum well, screening the field and hence changing therefractive index properties of the system (e.g. in essence a virtual quantum-confined Stark effect).

31

turn, alter the optical properties of the device and are thus responsible for the

switching behavior. The switching response can be quite strong, but the response time

of the devices is now limited by charge carrier response times and lifetimes.

2.3.2.1 SOAs in Fiber Loop Mirrors (SLALOM, TOAD)


TOADs and SLALOMs are implementations of incoherent switches based on

the incorporation of semiconductor optical amplifiers (SOAs) into fiber loop mirrors.

A semiconductor optical amplifier is placed off-center in an optical fiber loop mirror.

In the presence of a control pulse or intense signal pulse, light is absorbed in sufficient

amounts to create enough photogenerated carriers to significantly alter the index of

refraction of the SOA via cross-phase modulation (e.g., by saturation) – this is the

carrier density induced index change mentioned before. Referred to as SLALOMs

(semiconductor laser amplifier in a loop mirror), or in some circumstances TOADs

(terahertz optical asymmetric demultiplexer), it was demonstrated that a � relative

phase change between two counterpropagating pulses could occur in a few hundred

microns instead of many meters of fiber optic cable, as is needed by NOLMs.[76-79]

SOA

4-port device(50/50 coupler)

Signal Control

“0”“1”

Figure 2.5: Schematic diagram of a terahertz optical asymmetric demultiplexer (TOAD). The incomingsignal pulse is split at the 50/50 coupler into two counterpropagating pulses. Depending on the presenceor absence of a properly timed control pulse’s effect on the off-center SOA, imposing different phaseshifts to the signal pulses, the pulses will recombine either in-phase or out-of-phase. As a result, therecombined signal pulse will either continue to propagate along the fiber in its original direction or itreturn back along the fiber, respectively.

32

Characteristics

The principle advantage of SLALOMs and TOADs over NOLMs is a

dramatically reduced size and hence latency. Also, they may be integrated with

electronic circuitry. Extremely fast demultiplexing rates are possible (e.g.

250 GHz [80]). The repetition rate is constrained, however, by the recovery time of

the SOA, typically on the order of 100s of picoseconds to nanoseconds.[79] A few

methods have been developed to overcome this limitation, reducing the recovery time

to the order of a few to tens of picoseconds (these methods include operating at

transparency current, and hence minimal absorption; injecting a cw “holding” beam to

enhance recombination[81]; or designing a symmetric system to equalize the arms

using an additional pulse but also reducing the operating nonlinearity). Another useful

feature is that the control pulse may be of a different polarization or wavelength,

making it easy to separate from the signal and also enabling polarization or wavelength

switching.

2.3.2.2 Semiconductor Waveguides

A wide array of semiconductor waveguide devices may be used as incoherent

optically-controlled switches. Because the gain or absorption in a semiconductor can

be quite sensitive to the carrier density, changes in intensity from a control bitstream

can be used to modulate the characteristics of the semiconductor so that the bitstream

can be imprinted on another optical channel. Cross-gain, cross-phase, and cross-

absorption modulation (XGM, XPM, and XAM) are the common types of switches

and are described below.


If a semiconductor is forward biased it can potentially be used as a

semiconductor optical amplifier (SOAs).[82, 83] SOA all-optical switches may

function by interacting with a control beam of light (e.g. providing amplification)

which changes the remaining electron and hole density of the material, altering its

complex index of refraction.[84] If it is primarily the gain which is affected (reduced)

33

by the signal pulse, the switch is referred to as a cross-gain modulator. On the other

hand, cross-phase modulators (XPMs) rely on a change in the index of refraction.

SOA

Control beam blockCW signal beam

Control beam

Output signal data

Figure 2.6: A schematic illustration of cross-grain modulation in an SOA. The incoming c.w. signalbeam is amplified as it passes through the SOA. However, when the control beam is present, it too isamplfied, reducing the gain available for the CW beam. As a result, the inverse data stream of thecontrol beam becomes imprinted onto the signal beam. In this illustration the control beam is at adifferent wavelength than the signal and is therefore may be blocked with an appropriate filter.

If the semiconductor is reverse biased, changes in the gain are no longer possible.

However, the control pulses can instead be absorbed. This, too, changes the carrier

concentration. This can be used, for example, to separate the photogenerated carriers,

changing the voltage, and thus the absorption.[85]

It is important to mention that optically-induced changes due to carrier absorption

may also be induced in waveguides (particularly MQW devices) by other nonlinear

processes, such as band-filling and other absorption-saturation phenomena, altering the

dielectric coefficient. Such effects can be used to alter the phase of the signal beam (such

as two-photon absorption-induced changes in the index of refraction) or its intensity.[86]

Characteristics

Fully switching cross-gain modulators typically require carrier densities near

the saturation level of the device material and devices that are several hundred microns

long. Switching energies of 0.8 pJ per bit at 40 Gbits/s have been demonstrated.[87]

Cross-phase modulators may require significantly less switching energy than XGMs

because only a � phase shift is needed. Additionally, the smaller optical control signal

results in a correspondingly smaller chirp that is added to the signal. For example,

100 GHz XPM wavelength conversion has been demonstrated using only 125 fJ per

pulse.[88] On the down side, XPMs are more sensitive to the control power (they

have a smaller input power dynamic range) since excess energy adds too much phase

shift and require more complicated waveguide configurations such as interferometers.

34

XAMs also require less power than XGMs. However, cross-absorption modulation

can have significant parasitic absorption, resulting in much smaller output powers than

XGMs or XPMs.

2.3.2.3 Surface-Normal, Optically-controlled Switches


Surface-normal, optically-controlled switches differ from the previously

described devices in that the optical control and signal beams are incident from the top

of the device (perpendicular to the surface) instead of from the side. The interaction

lengths of such devices are limited to a few microns due to the surface-normal

geometry. Hence, strong intensity-dependent nonlinearity is needed in order to switch

the device. One class of devices uses nonlinear absorption and, possibly, nonlinear

refraction associated with absorption saturation effects. This is accomplished by

operating near the band-edge of a semiconductor where the complex index of

refraction’s nonlinear dependence on detuning, gapE �� , becomes large although

predominantly imaginary and hence absorbing. A strong control pulse can be

absorbed, creating carriers that saturate the absorption, reducing the absorption until

the carriers recombined. Hence, during that time the signal is more strongly

transmitted; the switch is “on”. As the saturation relaxes, the switch turns off. The

original research on this type of all-optical switch was first done by Takahashi, et. al.

[89]. Work progressed with MQW InGaAs/InAlAs devices[90, 91] and bulk GaAs,

too.[92, 93]

Characteristics

The standard characteristics of this type of device are strongly influenced by

the surface-normal geometry. The thin intrinsic region seen by the light pulses makes

it difficult to achieve a large contrast ratio (CR). This can be overcome by the use of

an internal DBR mirror to make the device reflection-based, both doubling the

effective active region’s thickness and allowing the overall reflectivity of the device to

be tailored so that it becomes a low-finesse asymmetric Fabry-Perot cavity. This can

provide nearly zero reflectivity in its “off” state, allowing large CR while maintaining

35

a bandwidth of about 40 nm.[93] Saturable absorption has a lifetime on the order of

nanoseconds. However, the use of low temperature-grown and Be-implanted GaAs

(LT-GaAs) can reduce the lifetime of the photoexcited carriers to picoseconds,

significantly enhancing the switch recovery time. Surface-normal devices tend to be

smaller than waveguide devices and easily scale in two dimensions to NxN formats.

They also avoid the tight alignment requirements of waveguide devices. Moreover,

there is no polarization dependence due to the planar symmetry with respect to the

light.xiii

2.3.2.4 Bistable Devices

Another type of all-optical switch is a bistable device – a device in which for

some (optical) control inputs there are two possible (optical) outputs. Which of the

two output states is produced depends on the history of the device. Moreover, small

changes in input power can result in changing from one state of the device to another

and thereby creating a large change in, say, the output power. This is a type of signal

gain.

xiii Actually, there was polarization dependence. The switches were sufficiently fast that the spin-state of theexcited electrons (which decays on the order of a few picoseconds) was still correlated with polarization ofthe pump light pulse. [89]) Such an effect becomes much less pronounced with a slower surface-normalswitch.

36

Out

put p

ower

Input power

1

2

3

4

StableUnstable

Voltage

Cur

rent

V0

Figure 2.7: Hypothetical response of a bistable device – a reverse biased p-i(MQW)-n photodiode inseries with a resistor. (Fig. 2.7b, LEFT) Load-line analysis of the coupled equations of the system. Thestraight line is related to the current/voltage relationship of the resistor while the curving line describesthe absorption (and hence photocurrent) of the MQW diode. Note that increased current implies greaterabsorption and so lower output power. As the input power rises, the diagonal load line increases itsangle with respect to the voltage axis where its intercept is pinned by the overall bias across the system,V0. By tracing out the stable points, a curve similar to Fig. 2.7b can be created. (Fig 2.7a, RIGHT)Output light power as a function of input light power. For certain input powers the output power can below (1) or high (3) depending on if the input power was previously low (and rising) or high (but falling),respectively. The sharp nonlinear responses at (2) and (4) enable large changes in output power withrelatively small changes in input power.

Typically, bistable devices rely on some form of feedback to achieve their

strong nonlinear and dual-state behavior. Such a system may be one that obeys two

separate but coupled differential equations whose solutions are the bistable points.

One example of this could be a device based on nonlinear refraction (the index of

refraction, and hence the path length, is a function of intensity) placed inside a

resonant cavity (whose transmission, and hence intensity, is dependent on the path

length between its ends). Here path length in the cavity depends on intensity in the

nonlinear device, but the intensity in the nonlinear device depends on the transmission

of the cavity that in turn depends on the path length – a feedback loop. “The

bistability then arises from the simultaneous conditions that the optical length of the

nonlinear material depends on the intensity of the lights in the resonator, and that the

intensity of light in a resonator depends on its optical length. This bistability results in

a hysteresis and switching in the behavior of the transmitted light intensity.” [94]

Positive feedback can drive this system to either a high or a low output state (the

bistable points) and dynamically what results is a hysteresis loop. With c.w. incident

37

light, a reverse-biased p-i(MQW)-n photodiode (with transparent substrate) in series

with a resistor is another example (see Fig. 2.7).

A large number of bistable passive devices based on a wide variety of

microscopic mechanisms have been studied.[95] The principle driving force behind

this work was the desire for a useful optical logic gate. Generally speaking, bistable

devices were limited in this regard due to their lack of a cheap and compact manner to

cascade them.

2.4 SELF-ELECTROOPTIC-EFFECT DEVICES (SEEDS)Self-electrooptic-effect devices (SEEDs) are another class of semiconductor

optically controlled optical switches whose switching behavior is due to

photogenerated charged carriers. The nonlinear switching of SEEDs is distinct from

the previously discussed optical switches, however, in that it is a non-local

phenomenon. That is, the device area for the signal datastream affected by the

photogenerated carriers may be physically distinct from the area where the control

light interacts with the device.xiv Furthermore, SEEDs may operate as bistable

switches, an intrinsically multi-valued nonlinear behavior that is not possible to

express by a power-series expansion (power series are inherently single-valued) of the

xiv The nonlinearity of SEEDs is different from the local material nonlinearity of the dielectric coefficient.Having said this, some aspects of SEED behavior have been described using an effective 3rd-ordernonlinearity. [96-98] The photogenerated carriers of the control beam create an electric field as theyseparate under bias. This may be viewed as a 2nd-order difference-frequency mixing effect, creating a DC(�’=�control-�control=0) field. This field then interacts with the signal beam (�’’=�signal+�’=��signal),modifying the dielectric coefficient to change, say, the absorption. This cascaded �(2) effect, or equivalent�

(3) functionality, may be quite large since the photogenerated carriers are able to move, creating very largeeffective dipole moments by adding their fields in concert. The effective �sc

(2) due to separated chargecompared to the local material �(2) may be expressed as:

� �

( 2)

1( 2)

sc r scd

d

� �

� ��

�

where dsc is the dipole due to separated charge, d is the dipole of the medium (e.g. semiconductor material)

due to the electric field of light, �r is the recombination time, and � �1

�� is the inverse linewidth of the

light.

38

dielectric coefficient. SEEDs generally have been designed to interact with surface-

normal light beams (though they also work in waveguides [99]). As a result, devices

on the scale of a few hundred square microns may be used, and large two-dimensional

arrays have been demonstrated.[100] Finally, SEEDs can be integrated with standard

silicon electronics (e.g. CMOS) using processes such as flip-chip bonding.[101] This

allows SEEDs to take advantage of the complex logic that silicon circuitry can

provide.

Vpi (MQW)n

-

+

incident light

output light

R

Figure 2.8: Schematic illustration of an R-SEED.

2.4.1 Principles of operationSEEDs were first invented in the early 1980s.[102] Since that time, there have

been many scientific papers written on the subject as it evolved from a curiosity to a

well-established concept and technology.[98, 103] Most SEEDs are optically sensitive

p-i(multiple quantum well)-n diodes connected with other electronic or optoelectronic

components, including other MQW photodiodes. An illustration of a SEED is

presented in Fig. 2.8. A control beam of light that interacts with the primary p-

i(MQW)-n diode is absorbed, creating photogenerated carriers. These photogenerated

carriers are able to move to other diodes or optoelectronic devices, altering their

absorption properties. If desired, the signal beam may, therefore, be designed either to

interact with the other optoelectronic components of the device than does the control

pulse. The control and signal beams do not need to be near one another.

The key to SEED operation is voltage-dependent absorption. The reverse-

biased MQW region’s heavy-hole exciton absorption peak near the band edge

39

red-shifts with increasing bias due to the quantum confined Stark effect (QCSE).[104]

Because absorption creates charged carriers that can then modify absorption, it is

possible to create circuits with feedback. With proper design, positive feedback can be

used to create bistable systems. Negative feedback, on the other hand, provides the

means to force the inherently nonlinear system to respond linearly to an input signal.

2.4.2 Types of SEEDsR-SEED

The first SEED, demonstrated by Miller et. al., was a p-i(MQW)-n diode

placed in series with a voltage supply and a resistor (see Fig. 2.8).[102] By changing

the input power, the device could be switched from one of the two stable points to the

other, thus changing the reflectivity of the diode. Changing the applied bias similarly

switched the device. Switching speed was RC-limited, and critical parameter settings

(either power or voltage) were needed for relatively low-power switching.

D-SEED

Replacing the resistor with a photodiode offered a wider variety of device

functionality.[105] Compared to the resistor-SEED, this diode-biased SEED

(D-SEED) required a smaller switching energy and was not limited by large RC time

constants. The switching time, however, was still constrained by the charging rate of

the MQW diode. Bistability was more readily achieved and with greater differences

between the two stable states. Moreover, it was possible to make use of negative

differential resistivity to achieve self-linearized modulation (the absorbed power

varied linearly with the imposed current with appropriate initial conditions).

Integrated D-SEED

The SEEDs described so far have a limitation in that not only does the MQW

diode have to be (dis)charged, but that the external circuitry also adds load capacitance

and resistance. Monolithic integration of the photodetector of the D-SEED and the

MQW diode improved these limitations.[106] In one device, the second (control)

diode was grown on top of the first (signal) diode in a p-i-n-p-i-n structure. By

selecting proper layer compositions and optical wavelengths, the control light was

40

absorbed in the top diode while the longer-wavelength signal light passed through to

be absorbed by the lower diode. In the future, this type of integrated D-SEED may be

particularly useful for arrays of devices because it is compact. The OCOG devices

described in this dissertation can be viewed as next-generation integrated D-SEEDs.

S-SEED

Symmetric SEEDs (S-SEEDs) derive their name from the symmetry between

the control and signal diode: both are identical (though usually spatially separate)

p-i(MQW)-n diodes in this type of device.[107] The behavior of the device is

determined by the ratio of the two optical input intensities. Consequently, the state of

the device can be first set using low-power beams, then a large output change is

induced using high-power beams, providing a form of time-sequential gain. S-SEEDs

do not require critical biasing and have good input-output isolation. These qualities

are important characteristics for logic devices that other SEEDs lacked. Indeed, full

cascaded logic functionality was demonstrated using differential logic signals. The

fastest demonstrated SEED switching (turn-on time only) was 6 ps using an S-SEED

with extremely shallow quantum wells to overcome escape and transport across the

MQWs.[108]

T-SEED

It is also possible to incorporate active electronic components into SEEDs. A

beam of light incident on a phototransitor is used to control the current through a

p-i(MQW)-n diode onto which a second beam of light shone. As with other SEEDs,

altering the current changed the absorption of the diode and the transmission of the

light.[109] This was soon followed by a monolithic device using a p-n-p structure

replacing the separate phototransistor.[110] Such a device did not significantly lower

the required switching energy in spite of the gain the phototransistor was able to

provide, largely due to the Miller capacitance effect (where the effective capacitance of

the transistor is multiplied by its gain).[98]

FET-SEED

The FET-SEED bypassed the Miller capacitance limitation.[111] In this

device, the control light beam was absorbed in a vertical p-i(MQW)-n diode. The

41

charged carriers were used to generate a current that induced a change in voltage on

the gate of a field-effect transistor (FET). The resultant drain current modulated the

absorption of a second beam of light. The gain inherent in such a system was

evidenced by only requiring ~0.5 V change due to the input beam compared to ~5 V in

regular SEEDs. It is interesting to note that this work in FET-SEEDs led to integration

with CMOS electronics,[101] which in turn led to work on dense optical

interconnects.[112, 113]

2.4.3 CharacteristicsSEEDs have demonstrated a wide variety of device functionality, including

bistability, self-linearization, gain (using a small signal to effect the reflectivity of a

much larger signal), and logic capability[114], [107]. Most of the devices discussed

above were demonstrated at wavelengths near 850 nm, but SEEDs have also been

made that operate around 1500 nm.[115] Typical operating switching energies of

3-5 fJ/��m-2 were needed, not including the energy dissipation of other circuitry.

Switching time was generally limited by charging or discharging the depletion

capacitance of the MQW diode. There was an inverse relationship between the

switching power and the switching time. The faster the desired rate of switching, the

larger the number of carriers needed at a given moment, hence greater input power was

required. Compared to other all-optical switching technologies the main drawback of

SEEDs is their relatively slow speed, although this has improved signficantly. As

mentioned above, the fastest switching speed obtained was 6 ps.[108] An additional

limitation is that at high photogenerated carrier intensities, about 1012 cm-2 per

quantum well[116], saturation of the exciton absorption peaks becomes a competing

nonlinear process and also slows down device recovery speed.

42

REFERENCES

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51

Chapter III: OCOG Device Concept

In this chapter, we introduce the basic operating principles of the three

generations of devices studied. Detailed analysis and results will follow in subsequent

chapters.

3.1 FIRST GENERATION OCOG-13.1.1 Principles of OCOG-1 operation

The first generation optically-controlled optical gate produced a 50 ps window

that changed the absolute reflectivity of the signal pulse from approximately 30% to 60%,

a 2:1 contrast ratio, with a switching energy of 5 fJ/�m2. It was a simple reverse-biased

multiple quantum well p-i(MQW)-n diode (Fig. 3.1) with a fixed external voltage whose

reflectivity was able to be temporarily changed by a control pulse of light, as the

schematic in Fig. 3.2 shows. The device was initially made relatively opaque by setting

the bias voltage so that the heavy-hole exciton absorption peak was at the same

wavelength as the control pulse. The control light was thus absorbed in the MQW

intrinsic region, creating electrons and holes. These photogenerated carriers escaped

from the quantum wells and were then pulled toward the n and p regions, respectively.

As they separated, these carriers screened the applied electric field across the intrinsic

region of the device. As a result, the optical absorption of the quantum wells was

blue-shifted due to the quantum-confined Stark effect (QCSE), as suggested by Fig.

3.3.[1] While this screening lasted, the effective absorption of the device was reduced,

allowing a second, signal pulse to be strongly transmitted (or in the presence of a buried

mirror, as with our device, strongly reflected).

52

V

p

n

_

+

i (MQWs)

Figure 3.1: Schematic picture of OCOG-1, a reverse biased p-i(MQW)-n diode. Optical control and signalpulses are incident from the top of the device (surface-normal).

1st light pulse strongly absorbed in intrinsic region

As electrons and holes separate, absorption changes with effective voltage

2nd light pulse not strongly absorbed Absorption gate closes asvoltage diffuses away

V

V

V

V

p

n

_

+

i

1) t~0 2) t~10 ps

3) t~20 ps 4) t~50 ps

+ +

+

_

_ _

p

p p

n

n n

i

ii

(mirror)

Figure 3.2: Schematic illustration of OCOG-1 device behavior over time with incident pump (control) andprobe (signal) pulses. The control pulse is absorbed in the reverse-biased device, creating electrons andholes that subsequently moved to shield the voltage locally. Consequently, the absorption of the device isreduced such that the signal pulse can be strongly reflected. As the shielding voltage dissipates, the devicereturns to its opaque state. In this device, the wavelengths of the control and signal pulses may be at thesame.

53

The duration of the strongly reflecting “ON” state of the device lasted while the

bias voltage remained screened. Normally, a principle objection to such a device would

be that, although the gating could potentially be turned on rapidly, the turn-off time

would be long, perhaps corresponding to the external resistive-capacitive (RC) time

constant of the entire device of, say, several nanoseconds. However, another process,

diffusive conduction, was exploited to relax the voltage in a controllable manner on a

significantly shorter time scale.[2]

6V

10V

0V

Figure 3.3: Description of the Quantum Confined Stark Effect (QCSE). (Left) From [3]. The electron andhole wavefunctions and energies in the conduction and valence bands, respectively, for a quantum wellwithout an electric field (left side) and with an electric field (right side). In the presence of an electric fieldthe energy spacing between the ground electron and hole states shrinks, reducing the bandgap. Thequantum well barriers provide confinement of the wavefunctions under bias, resulting in a strong excitonicabsorption peak – though this weakens as the bias increases. (Right) An example of the absorption profilenear the bandgap in GaAs MQWs for a variety of biases. As the bias increases, the absorption peaksignificantly red-shifts. This functionality is used in OCOG devices to make their absorption sensitive tothe electric field.

54

x

y

V

Figure. 3.4. Conceptual mesh of resistors (the doped n and p layers) and capacitors (the intrinsic region)for analysis of the diffusive conduction. For simplicity, the resistance is shown only in the p layer.

The p and n regions of a diode can be considered as the resistive planes of a plate

capacitor (the intrinsic region) and can be modeled as a mesh of resistors with

capacitance between the top and bottom layers, as illustrated in Fig. 3.4. In such a mesh,

if the voltage is changed locally by injecting charge into some of the capacitors (e.g., via

photogeneration, see Fig. 3.5), the voltage can relax through the local electrical

conduction in the resistors. Analysis of this type of structure reveals that the voltage

obeys the following relationship:

� �� 2,

,xy

dV x yD V x y

dt� � (3.1)

which has the form of a diffusion equation. V is the voltage across the intrinsic layer at a

lateral position (x.y). It is important to understand that this does not represent the

diffusion of the generated carriers as in a bulk semiconductor; rather, it is a dissipative

electrical wave propagation that can actually travel much faster than the physical

movement of individual carriers. The effective diffusion constant, D, is given by

1

sq A

DR C

� (3.2)

where Rsq is the sum of the resistances per square of the top and bottom conducting layers

and CA is the capacitance per unit area. Diffusive conduction is described in detail in

Chapter 4.

55

pin

Control pulse

yx

Voltage

Figure 3.5: Conceptual illustrations of (left) the resistance per square and capacitance per unit area of ap-i-n diode with an incident control pulse with a relative small spot size and (right) the resulting voltagedue to the separation of the photogenerated carriers (left). Note that the voltage change induced by carriermovement inside the diode is always a reduction of the local voltage, though it is more convenient to treatthis change as a positive number, and hence as a “peak” that changes in time.

The optical beam used has a Gaussian lateral intensity profile that results in an

injected charge (and hence voltage) distribution that was initially Gaussian as well.

Given this initial condition, the solution to the diffusion equation has a simple analytic

expression for the local voltage change as a function of time t and radius r from the

center of the beam,

� ��

��

��

�

��

�

��

�

tDr

tVtrV M 4

exp,2

(3.3)

where

Dwo

8

2

�� (3.4)

and

Ao

totM Cw

QV 2

2�

� (3.5)

Here Qtot is the total charge created (on each capacitor plate) by the absorbed photons and

w0 is the 1/e2 intensity radius of the Gaussian beam.

56

10 5 0 5 100

0.2

0.4

0.6

0.8

1

Radial Distance, r

Volta

ge C

hang

e (A

rb. U

nits

) t=0

1

49

Figure 3.6: Illustration of relaxation of the initial Gaussian voltage distribution in diffusive electricalconduction from an initial Gaussian voltage distribution on the conducting capacitor plates at time t = 0.Curves are shown for times t = 0, 1, 4, and 9 units. Time is in units of the characteristic time,�. Distance isin units of the initial Gaussian spot radius, wo.

The absorption of photons in the quantum well layers and the subsequent

separation of the photogenerated electrons and holes due to the electric field leads to a

voltage reduction, or screening, in the p and n regions that laterally is essentially

Gaussian in shape. In time, the Gaussian distribution expands, reducing the local

magnitude of the voltage change as it spreads and returning the device to nearly its

original condition. Fig. 3.6 illustrates a Gaussian voltage change as a function of time

due to diffusive conduction. The generated charge does eventually move through the

external circuit, but the local voltage can be relaxed on a time scale much faster than this

external charge movement, and so it is not limited by the external circuitry’s relatively

long RC time constant. The characteristic time for this local voltage recovery, �, is

strongly dependent on controllable parameters: the doping levels of the p and n layers and

the spot size, w0. Hence, in this type of device, the turn-on time is controlled by the

emission and separation of charges from quantum wells, and the turn-off time is governed

by the diffusive electrical conduction. Both processes can operate in the picosecond

regime, allowing an optoelectronic gate to function on a short time scale.

Actually, the external circuitry and overall device RC response cannot be

dismissed quite so easily. This behavior effectively acts as a “global” device repetition

57

rate that is convolved with the “local” optical modulation recovery rate just described.

Through diffusive conduction, the localized photogenerated carrier density is quickly

smoothed out across the entire device. This process results in rapid local recovery but

creates a finite, albeit much smaller, global screening voltage with a relaxation rate

determined by the overall RC time constant. With periodic optical pulses consistently

generating carriers, a steady state global screening voltage may be reached. As the

repetition rate of the optical pulses increases, so does the effective bias point. This shift –

roughly proportional to the ratio of the RC value of the device to the time period between

pulses – may be easily compensated for by adjusting the original bias voltage and does

not of itself limit the repetition rate. Other mechanisms, such as restrictions on power

dissipation, might set the practical limit on the data rates the overall device can handle.

3.1.2 Advantages and disadvantages of OCOG-1An important feature of OCOG-1 is that it can be very sensitive (low required

switching power) compared to other optically-controlled switches that are based on

processes such as absorption saturation. The reason for this sensitivity is that the charge

generated in each well eventually contributes to screening in all the wells once the

charges have been transported to the doped layers (see Fig. 3.7). To take the simplest

case, if the carriers leave all the wells and gather at the electrodes, then, for N wells

altogether in the structure, the field screening will be correspondingly N times as large

compared to the case of in-well screening with the same number of total carriers.i

Suppose, for example, there are 50 quantum wells in the structure. The total number of

carriers per unit area required for a 5 x 104 V/cm field change (which is large enough to

cause significant electroabsorption change of transmission) is about 3 x 1011 cm-2. To

achieve this, an average of only about 1/50th of that carrier density is required to be

initially generated in each well, i.e., ~ 6 x 109 cm-2. That corresponds to an incident

optical energy of just ~2 fJ/�m2 instead of the greater than 100 fJ/�m2 needed for in-well

screening or a nonlinear optical process such as excitonic saturation. Hence, compared to

excitonic absorption saturation or in-well screening, we have a nonlinear effect that is at

58

least 50 times stronger in terms of the energy required to effect switching. The main cost

for this benefit of low-power operation is a longer turn-on time.

in p

i

electrons

holes

n p

Quantum Wells

Figure 3.7: Illustration of the different electric field magnitudes -- the thickness of the horizontal blackarrows -- across the intrinsic region of a p-i(MQW)-n diode depending on if (top) the electrons and holesstay in the quantum wells (in-well screening), or (bottom) there are four quantum wells and the electronsand holes fully separate. If there were 100 quantum wells and the carriers fully separate, the electric fieldwould be an 25 times larger than the illustrated 4 quantum well case. The increase in the potential electricfield due to carrier separation instead of in-well screening is clear.

There is a fundamental trade-off between the magnitude of the signal change due

to switching and its initial speed. A thicker MQW region enables a larger number of total

carriers to be absorbed, providing a potentially larger electric field and, hence, absorption

change. The average transport time for the photogenerated carriers increases at the same

time, however, and results in a longer turn-on period. In seeking to optimize these

effects, the reduction in CA due to the wider intrinsic region, and thus smaller �, must also

be taken into account.

It is worth noting that OCOG-1 can operate either when both control and signal

pulses have the same wavelength near the heavy-hole exciton resonance or when the

i Note that the field between two sheets of charge depends only on the total charge density in the sheets, noton the separation distance between the sheets (field dependence near spot-center is similar to a parallelplate capacitor).

59

control pulse is at a shorter wavelength. That pulse is still strongly absorbed in the MQW

region, allowing the capability for limited wavelength conversion.

Unlike most optically-controlled switches, OCOG devices may also be made in

2-D arrays due to their surface-normal configuration. Moreover, their optoelectronic

nature allows straight-forward integration with electronic circuitry, such as

complementary metal-oxide-semiconductors (CMOS), via well-established techniques

such as flip-chip bonding.[4] This surface-normal configuration limits the active region

where the absorption occurs to merely the thickness of the intrinsic region (typically

~1 �m thick). Consequently, the contrast-ratio (CR) of these types of devices is small,

e.g. 3 dB, compared to, say, waveguide devices which may have a CR of 10-20 dB. The

change in actual power, fortunately, is large, approximately 30%, so that a pair of

OCOGs could be effectively used in a differential pair configuration to significantly

improve the effective CR. Data from the different generations of OCOG devices is

presented in Chapter 6.

3.1.3 Other uses of OCOG-1If the recovery of the gate is fast, an OCOG can be used as a time-sensitive gate;

the second pulse is strongly transmitted only if it arrives within a short time window after

the first pulse. This type of switch may be useful for applications such as time division

demultiplexing when an interface is needed between the extremely fast bit rates possible

in optical fiber and the relatively slow speed of CMOS processing.

The device can also operate as a logic AND gate; only if both pulses arrive (in the

correct timing sequence) is the second pulse transmitted. The amount of photocurrent

generated by the absorption of the second pulse will be sensitive to the time of arrival of

the second pulse, so the device can operate as a gated photodetector. The device should

also permit a weak first pulse to gate a stronger second pulse, so the device may be able

to show signal gain. This gain is possible because a pulse, if its pulse width is shorter

than the carrier escape and transport time, has relatively little effect on itself; its electrical

effects grow after it has been absorbed.

60

3.2 SECOND GENERATION OCOG-23.2.1 Principles of OCOG-2 operation

The OCOG-2 was similar to OCOG-1 in that a control pulse of light was used to

temporarily change the absorption in a MQW diode structure, switching the transmission

of a second light pulse. The basic operating mechanisms were the same but used in a

slightly different manner in order to improve the speed and flexibility of the device.

OCOG-1 devices were limited by the escape time of the photogenerated carriers,

principally holes, from the quantum wells as well as by the drift time required to reach

the doped layers. Moreover, there was a trade-off in the design of the MQW region

between fast escape and strong absorption sensitivity to electric fields. To overcome

these limitations, the control and modulator functions were separated. OCOG-2 was a

dual diode-like structure, n-i-p-i(MQW)-n, stacked on top of a distributed Bragg reflector

(DBR) mirror, shown schematically in Fig. 3.8. The bottom (modulator) diode, which

contained multiple quantum wells in its intrinsic region, had its absorption switched when

an incident control pulse was absorbed in the top (control) diode. The results were

switching speeds with less than 10 ps FWHM switching times, significantly faster than

the OCOG-1 device.

By separating the control and modulation functions in OCOG-2s, it was possible

to optimize each section separately. A bulk intrinsic section in the top diode avoided

delay due to carrier escape from the QWs (since there were no QWs in the top diode!)

and also allowed the use of a potentially thinner intrinsic region and thus provide shorter

transit times. At the same time, the bottom MQW region could be thicker and the wells

optimized for absorption shift; both could improve the switching behavior without the

need to be overly concerned with sweep-out times.

61

_Va

Vb

+

+nip

i (MQWs)

n

Control Diode

Modulator Diode

Figure 3.8: Device schematic of OCOG-2 -- a double diode, n-i-p-i(MQW)-n structure with separate biascontrols for each layer.

Gate dynamics are illustrated in Fig. 3.9. The control diode was designed to be

transparent to the signal (probe) pulse but opaque to a control (pump) pulse at a shorter

wavelength. The modulator diode was reverse biased so that the MQWs were initially

substantially transparent to the signal as compared to when the gate was “on”; the system

was in its highly reflective state for the signal. Upon absorption of the control pulse in

the reverse-biased top diode, the photogenerated carriers locally vertically separated,

locally screening the bias in the vicinity of the control pulse spot. As a consequence of

the dual-diode structure, the reverse bias voltage on the bottom diode was locally

increased, raising the absorption of the MQWs and hence decreasing the reflectivity of

the device through the QCSE. The turn-on time of the device was determined by the

voltage build-up time due to carrier transport in the bulk intrinsic region of the top diode.

The turn-off time of the device was controlled by the local electrical relaxation of the

voltage across the diodes through diffusive conduction (described in detail in Chapter 4).

OCOG-2’s repetition rate was constrained, as with OCOG-1, by this fast on-off cycle

rather than the external RC time constants.

62

(1) t=0 (2) t=0-4ps

(3) t~4ps (4) t=4-20ps

The control pulse is absorbed in the top diode

The voltage build-up decays away and the device returns to its opaque state: OFF

Top Diode

Bottom Diode

Va

Vb

Control

Top Diode

Bottom Diode

Va

Vb

Signal

Top Diode

Bottom Diode

Va

Vb

Signal

Top Diode

Bottom Diode

Va

Vb

Voltage build-up changes absorption levelin bottom diode: ON

+ + + - - -_

+

+

_

+

+

_

+

+

_

+

+

Photogenerated carriers separate, shielding the voltage across the top diode

Figure 3.9: Schematic of the optically controlled optical gate (OCOG-2) device operation with picosecondpulses.

The dual diode structure can be modeled as a series of dielectric layers (the intrinsic

regions), each sandwiched between conducting layers (p and n regions). A key aspect of

OCOG-2s was that the top-most and bottom-most layers of the entire device were highly

conductive. Consequently, the voltage between these two layers was held effectively

constant. Therefore, any local voltage reduction (increase) in the top diode leads to a

corresponding local voltage increase (decrease) in the bottom diode so that the overall

voltage remains steady (Fig. 3.10).ii Because a voltage change could be induced in the

bottom diode due to changes in the top diode, the separation of the control and modulator

functions was possible in this device. Interestingly, for OCOG-2 devices, if the induced

voltage change is sufficiently large, the bottom p-i(MQW)-n diode may be temporarily

forward biased, opening an entirely new class of devices based on

optoelectronically-controlled optical gain.

ii Actually, the voltage was not held perfectly constant; the response time of the device was finite, albeitvery fast (on the order of 1 ps). Induced voltage was properly modeled by solving the coupled voltageequations of the device as described in Chapter IV.

63

Voltage

Time�VBOTTOM

�VTOP

Figure 3.10: Illustration of induced voltage in a dual-layer OCOG device. The overall voltage is heldconstant across the entire device due to highly conducting top-most and bottom-most layers. Consequently,a change in voltage in the top part of the structure (e.g., across the top intrinsic region) must becompensated for by an opposite change in voltage across the bottom part of the structure (e.g., the MQWintrinsic layer) so that the overall change in voltage is zero.

3.2.2 OCOG-2 advantages and disadvantagesAs indicated above, the primary advantage of the dual diode structure was that its

turn-on speed could be decoupled from QW escape times, a significant improvement over

OCOG-1. As a gated photodetector, the dual diodes make separating the control and

signal pulse photocurrents an inherent property of the device. To function properly, it

was important that the control pulse be fully absorbed in the top diode since leakage into

the bottom diode could have led to faulty signaling as well as excess carriers in the MQW

region. Unlike the OCOG-1, the wavelengths of the control and signal pulse in OCOG-2

must, therefore, be different.

3.3 THIRD GENERATION OCOG-3The third-generation optically-controlled optical gates were variations on the

basic configuration of OCOG-2. The control and MQW regions were separate, but the

number of layers and the conductivities of the p and n regions differed as the device was

altered to make it faster and more efficient. A p-i-n-p-i(MQW)-n structure was designed

as OCOG-3. Large-signal response times of 20 ps were demonstrated, again with about a

30% absolute reflectivity change; 12 ps FW10%M small-signal gating was also shown at

64

50 GHz burst-logic rates, about three times faster again than OCOG-2. These results are

presented in detail in Chapter 6.

As the top intrinsic region was screened due to the photogenerated carriers of the

control pulse, it induced the opposite change in voltage in the MQWs. Because the p and

n regions are reversed in this section compared to OCOG-2, the induced voltage added a

negative bias across the bottom intrinsic layer, red-shifting the exciton peak, increasing

the absorption. OCOG-3, consequently, was inverting.

Control Diode

Modulator Diode

Vt

+

Vb

+

_

_

ni

pin

p

Figure 3.11: Schematic of the p-i-n-p-i(MQW)-n structure of OCOG-3.

The top intrinsic layer was made thinner to increase the carrier sweep-out rate,

reducing the turn-on time. More critically, OCOG-3 was, in fact, a device with three

depletion regions; the depletion region between the middle n and p layers played a key

role in its operation. Adding this additional intervening layer in the device enabled a

faster turn-off time due to diffusive conduction recovery. The diffusive conduction

behavior due to the coupled layers of the dual diode’s layers changes the voltage

recovery’s functional dependence on time, resulting in faster turn-off times. (See details

in Chapter IV.) Although the gating time of the bottom (MQW) layer can shrink

significantly, the overall repetition rate of the device, however, is affected to a lesser

degree.

65

REFERENCES

1. Miller, D.A.B., et al., "Electronic Field Dependence of Optical Absorption nearthe Bandgap of Quantum Well Structures," Phys. Rev. B, vol. 32, pp. 1043-1060(1985).

2. Livescu, G., et al., "High-speed absorption recovery in quantum well diodes bydiffusive electrical conduction," Appl. Phys. Lett., vol. 54, pp. 748-750 (1989).

3. Miller, D.A.B., D.S. Chemla, and S. Schmitt-Rink, "Relation betweenelectroabsorption in bulk semiconductors and in quantum wells: The quantum-confined Franz-Keldysh effect," Phys. Rev. B, vol. 33, pp. 6976-6982 (1986).

4. Goossen, K.W., et al., "GaAs MQW modulators integrated with silicon CMOS,"IEEE Phot. Tech. Lett., vol. 7, pp. 360-362 (1995).

66

Chapter IV: Theory of Optically Controlled

Optical Gates

Diffusive conduction, as stated in previous chapters, was responsible for relaxing

the voltage changes due to photogenerated carrier separation in OCOGs. This chapter

provides a more in-depth discussion of diffusive conduction and related phenomena as

well as the means to accurately model OCOG device mechanisms.

4.1 THEORY OF DIFFUSIVE CONDUCTION

The response of semiconductor devices to photogenerated carriers has been a critical area

of research over the past several decades due to the wide variety of uses for

light-sensitive devices, such as photodetectors, optical switches, and lasers. Carrier

dynamics due to diffusion often play an important role in the behavior of these types of

devices and have been extensively investigated. For example, it has been shown that in

bulk material, ambipolar diffusion is the primary diffusion mechanism for

photogenerated carriers.[1-6] In the mid-to-late 1980s it was discovered, however, that

diffusion in semiconductor p-i-n diode and n-i-p-i structures could exhibit a response

several orders of magnitude faster than ambipolar diffusion. Two mechanisms were

separately proposed: diffusive conduction and giant ambipolar diffusion.[7, 8] Since that

time, work based on these phenomena has progressed.[9-14]

Unlike in a bulk semiconductor material, in reverse-biased diodes, n-i-p-i’s, and

biased n-i-n or p-i-p devices electrons and holes separate, building up a carrier

density-dependent screening potential between them. As will be shown, it is this

difference which accounts for the dramatically enhanced diffusion in diodes versus bulk

material.

4.1.1 Qualitative descriptions of enhanced diffusionEnhanced diffusion may be described from two different perspectives:

microscopic, focusing on charge motion, or macroscopic, observing the voltage dynamics

67

of the system. Diffusion in semiconductors is often approached microscopically. In bulk

semiconductor material in the absence of electric fields, photogenerated carrier dynamics

are well described by regular ambipolar diffusion: when a neutral distribution of excess

carriers is created in bulk semiconductor material, e.g., via photogeneration, the electrons

and holes predominantly move together. Local charge neutrality is approximately

maintained, in spite of the different mobilities of the charge carriers, because the

Coulomb attraction between an electron and hole is much stronger than the dispersive

effects of diffusion alone. As a result, electrons and holes diffuse together with a single

diffusion coefficient that equals a weighted average of the (isolated) electron and hole

diffusion coefficients.

The material composition of p-i-n, n-i-p-i, and other similar semiconductor

structures is direction dependent; carrier motion in the direction perpendicular to the

layers, hereafter referred to as either z or “vertical”, may be quite different from that of

motion parallel to the planes, here defined as “lateral” or �. In these types of devices,

photogenerated electrons and holes in the intrinsic region separate in the vertical direction

due to the built-in and/or applied voltage across the layers of the device. Note that this

chapter only addresses lateral carrier dynamics, not vertical. Vertical carrier transport has

been a subject of extensive research; see, for example, Ref [15-19] and is included in the

simulations (Chapter 7). Understanding the effects that vertical charge separation has on

lateral carrier movement can be subtle. Briefly, the vertical separation of a localized

group of photogenerated carriers creates a lateral voltage gradient that pushes both

electrons and holes away much faster than ambipolar diffusion alone does.[20]

How and why does this happen? A schematic view of a p-i-n device is presented

in Fig. 4.1 (Top). The built-in/bias voltage across the intrinsic region is linearly related to

�np, the separation between the electron, �n, and hole, �p, quasi-Fermi levels. After a

pulse of light is absorbed, the photogenerated carriers vertically separate and the electric

field in the intrinsic region is screened, although only in the vicinity of the incident pulse

light beam. The results are illustrated in Fig. 4.1 (Middle). As more carriers are injected,

separate, and screen the field, the built-in and/or reverse bias voltage decreases in the

vicinity of the absorbed light beam pulse, and �np changes. The magnitude of this shift

in the quasi-Fermi levels is strongly dependent on the magnitude of the photogenerated

68

charge that has separated. This dependence is linear, at least for small voltage changes

and/or large reverse biases, because it results primarily from the reduction in voltage

from the vertical charge separation. An equivalent statement is that the derivative of �np

with respect to the density of the separated charge density is large or “giant”. This shift is

much larger than the typically logarithmic shift of quasi-Fermi levels found in a bulk

semiconductor that results from the shift of quasi-Fermi levels solely due to the statistical

mechanics of the change in carrier density. It is this fundamental difference – due to

charge separation - that is responsible for enhanced diffusion.

pin

bandgap

incident light pulse beam

separatedphotogeneratedcarriers

built-in/biasvoltage

z

ip n

valence band

quasi-Fermi level:hole and electron

conduction band

z

ip n

z

��

�

screenedvoltage

(“lateral” direction)

�p(��

�n(��

�p(��n(��

Voltage shielding

�

electron and holequasi-Fermi leveldifference, �np(�)

�n(��

�p(��

radial cross-sectionof incident lightpulse intensity

Figure 4.1: Schematic diagrams showing how an incident light pulse may create effective lateral electricfields in a reverse biased p-i-n structure. (TOP) A light pulse incident from the top on a p-i-n device isabsorbed in the intrinsic region, creating electrons (black circles) and holes (white circles) that quicklyvertically separate along z due to the built-in and/or reverse applied bias. (MIDDLE, LEFT) Where noincident light shines, the difference between the electron and hole quasi-Fermi levels, �np, is determined bythe built-in/reverse bias voltage. (MIDDLE, RIGHT) On the other hand, where the incident light isabsorbed, the vertically separated carriers shield the voltage. As a result, �np changes significantly because

69

it is an approximately linear – not logarithmic -- function of the separated photogenerated carrier density.(BOTTOM) Because of the vertical separation of the photogenerated carriers, �np has a lateral dependencethat mimics the lateral intensity variation of the incident light pulse. The resulting lateral gradients of boththe electron, �n, and hole, �p, quasi-Fermi levels produce electric fields in the n and p layers. These fieldshelp ‘push’ both electrons (in the n layer) and holes (in the p layer) laterally away and are what makesenhanced diffusion possible. The magnitude of these fields is proportional to the “giant” derivative of �npwith respect to the separated carrier density. Note also that these effective fields can act on the entirecarrier densities in the n and p regions, not merely the separated photogenerated carriers, further increasingthe effective diffusion.

Continuing with a microscopic perspective, if a pulse of light with a lateral

spatially varied profile, e.g., Gaussian, is absorbed in the intrinsic region, and the

photogenerated electrons and holes quickly separate vertically, �n and �p are forced

initially to have corresponding, though opposite, “lateral” Gaussian spatial dependence,

and consequently so does �np. Fig. 4.1 (Bottom) illustrates this situation. A gradient of a

quasi-Fermi level defines an effective electric field along that gradient (e.g. a gradient in

�n creates a field in the n layer). The lateral spatial variation of the input pulse – when

combined with vertical charge separation – thus creates a lateral electric field in both the

n and p layers. These fields help the carriers in both doped regions to disperse laterally.

The relatively large magnitudes of these extra fields are what account for the enhanced

diffusion effects of giant ambipolar diffusion.

Arguably, this process is not a diffusion process in the conventional sense. The

motion of the carriers can be viewed as a consequence of the electric fields,

corresponding to normal resistive transport. Note, too, that all of the carriers in the n and

p regions move in response to the lateral fields, not merely the additional photocarriers.

The mathematical equation describing the resulting movement of the carrier density of

the voltage pulse does have the form of a diffusion equation. The diffusion constant of

this equation, though, depends on the conductivity of the layers and the gradient of �np

with respect to separated charges – the capacitance between the doped layers. The

appearance of capacitance in the equations further clarifies that we are dealing with a

phenomenon different from conventional diffusion, in which capacitance would certainly

not appear. In this view of the process, it is known as diffusive conduction.[8]

From a macroscopic perspective, diffusive conduction is essentially an extension

of the voltage dynamics of a one-dimensional dissipative transmission line. A voltage

pulse in a transmission line can travel at a speed much faster than that of the individual

70

electrons as is well known in conventional inductive-capactive transmission lines (e.g., a

coaxial cable carrying signals at speeds near the velocity of light). The structures of

interest here are dissipative transmission lines, in which the series resistive impedance of

the p and n layers dominates over the inductive impedance leading to dissipative wave

propagation, but it is still true that the dissipative wave can move faster than the

individual electrons and holes. This is possible in part because the particles in the

medium exert strong forces on one another and because the medium of particles extends

throughout the length of the line. A p-i-n structure can be viewed as a two-dimensional

(“lateral”) version of a dissipative line, as illustrated in Fig. 4.2. The doped p and n

regions each have a resistance per square and there is also a capacitance per unit area

between them across the intrinsic region. When a spatially localized pulse of light is

absorbed (e.g., a light beam with a small spot size is absorbed in the center of a mesa

structure), the photogenerated electrons and holes in the intrinsic region will separate,

shielding the voltage. This results in a spatially localized voltage pulse. The behavior of

the pulse in this dissipative structure may be modeled by a diffusion equation. The result

is voltage diffusion that dissipates the voltage build-up across the entire device. As in a

dissipative transmission line, this response is not limited by individual carrier motion.

Instead, this diffusion depends only on the capacitance per unit area, the spot size, and the

resistance per square and, consequently, may be very fast.

pin

Figure 4.2: Schematic p-i-n structure showing distributed resistance and capacitance (for ease of viewing,resistance in the n layer has not been drawn). This type of structure is the 2D analog of a 1D dissipativetransmission line.

71

4.1.2 General modeling approachesWith an understanding of the qualitative behavior of enhanced diffusion, a

compelling question becomes: can this behavior be modeled from first principles? The

general response of p-i-n diodes and related structures to photogenerated carriers can be

determined from three relationships: (1) the forces present, including those due to the

photogenerated carriers; (2) the motion of all the carriers due to the forces present; and

(3) overall charge neutrality (an equal number of electrons and holes are created by

photogeneration). Combining the above relationships along with the initial and boundary

conditions allows a self-consistent description of the carrier dynamics to be found. Some

of the issues involving this process are discussed next.

The primary forces involved in semiconductor carrier dynamics are the Coulomb

attraction and/or repulsion due to the electric fields of space charges. These can be well

modeled by using, for example, Poisson’s equation.

Determining what should be the degree of accuracy of the equations governing

carrier motion is also an essential task in order to solve for the system dynamics. One of

the most fundamental approaches that may be considered is the use of the Boltzmann

Transport Equation (BTE) to express charge motion via the evolution of a charge

distribution function, ( , , )f tp r :[21]

coll

f f f ft t t t

� � � � � ��

� � � � � �

p rp r

(4.1)

where the last term is the change in f due to collisions and p and r are the momentum and

position vectors, respectively. Only two assumptions need to be made: that carriers may

be treated semi-classically (i.e., they have a well-defined position and momentum), and

that there is a sufficient number of carriers to meaningfully use a distribution function.

These are reasonable assumptions for many devices.[22] Unfortunately, making practical

use of this equation and solving for the unknown distribution function is difficult.

The BTE may be simplified, however, into a variety of more tractable expressions

by making appropriate simplifying assumptions.[22] One of the more dramatic

simplifications results in a drift-diffusion equation:

n n nq n qD n�� j E ∇ (4.2)

72

which expresses the current density of electrons, n, (or holes, p) as a function of ensemble

values (mobility, �n, and diffusion coefficient, Dn, each of which is dependent on the

distribution function, electric field, and temperature) combined with the electric field, E,

and the gradient of the charge density. q is the unit of charge. This equation is the basis

for describing carrier transport that will be used in this dissertation

In writing the drift-diffusion equation, several assumptions have been made:

magnetic fields have been assumed negligible; current due to carrier temperature

gradients (the thermoelectric effect) is small;i finally, mobility and diffusion coefficients

are not dependent on the detailed structure of the device (spatial variations are large

compared to scattering lengths). These last two assumptions are valid if the electric

fields are small or, if large, uniform. The enhanced diffuison transport discussed in this

paper only involves carrier transport of carriers in the lateral, doped planes. Thus, even

though the vertical dimensions of some layers may be small, since the transport is not in

that direction such variation is not critical. There are electric fields generated in the

lateral direction, as will be described below that play an important role in these transport

mechanisms. However, for spot sizes with radii of a few microns or larger, the variation

in electric field does occur over a distance large compared to the scattering length. As a

consequence, the validity of the assumptions remains uncompromised.

In order to use the drift-diffusion equation, the mobility and diffusion coefficient

must also be known a priori.ii Using these ensemble quantities implicitly removes

information regarding the statistical variances of these quantities. This is a safe

i Even if the theroelectric effect was not ignored, the magnitude of the resulting diffusion is within an orderof magnitude of the low-field regular carrier diffusion values and often significantly smaller.i As will beshown, the enhanced diffusion coefficients are two to three orders of magnitude larger, and thus ignoringthe temperature gradient seems reasonable. The actual diffusion coefficient will be slightly larger thanwhat is predicted.

ii The generalized Einstein relations provide the relationship[6]� �

� �

00

1/ 2

nnn

B n

eDk T

��

��

�

�

where � r is the Fermi-Dirac integral of order r, ( ) /n n c B nE k T� �� is the reduced chemical potential inwhich n� as the quasi-Fermi level, and D0

n is the Maxwell-Boltzmann diffusion coefficient. Here ndenotes electrons. A similar expression may be written for holes.

73

simplification if device behavior is not sensitive to such statistical fluctuations.iii These

quantities are also determined by using an expected value for (momentum) scattering

rate. Hence, only dynamics that occur on a time scale that is large compared to the

inverse of this scattering rate (typically on the order of a picosecond at room temperature)

are well defined.[22] It is worth noting that subsequent behavior based on these

assumptions describes the behavior of the ensemble of particles, not of individual

particles themselves.

Having discussed how to describe carrier motion, next we look more carefully at

the assumption of charge neutrality. Overall charge neutrality arises because

photogenerated carriers are always produced in electron and hole pairs. In bulk

semiconductors, as previously mentioned, local charge neutrality is also maintained in

the absence of external fields even if electrons and holes have different mobilities. The

Coulomb attraction between the carriers is significantly stronger than other prevailing

forces (e.g., such as diffusion, which could separate electrons and holes) and acts to keep

electrons and holes close together on the time-scales of interest. Local charge neutrality

is a key assumption of ambipolar diffusion and accounts for electrons and holes diffusing

together in spite of differing mobilities.[5, 23] In p-i-n’s and n-i-p-i’s, however, there are

built-in and/or applied fields along the z direction that separate the charge species.

Clearly, local charge neutrality no longer applies since the electrons and holes are

separated, typically on the order of one micron or less in many devices. However, the

separation is small enough (i.e. the Coulomb attraction is still sufficiently large) that an

effective local charge neutrality does continue to hold in the lateral directions. (Note: in

studying enhanced diffusion, we assume that no external fields are present in the lateral

directions.)

Even with a simplified expression for carrier transport, Eq. (4.2), and the

assumption of local charge neutrality in the lateral directions, solving for the carrier

dynamics is not easy. Each of the two approaches described below make additional

assumptions to make the math tractable and provide an analytic solution. Diffusive

conduction drops the explicit diffusion term of the transport equation. Giant ambipolar

iii Even at low energies, tens of thousands to hundreds of millions of electrons and holes may bephotogenerated in the intrinsic region. Therefore, if the device response is not sensitive to individualelectrons or holes, ignoring statistical fluctuations is often reasonable.

74

diffusion, on the other hand, simplifies the carrier density description by assuming a

Maxwell-Boltzmann (MB) distribution function in the doped regions.

4.1.3 Giant ambipolar diffusionModeling the effects of charge separation using the drift-diffusion equation is

clearly presented in the paper of Dohler and Gulden, et. al.[24] In the following

equations, subscripts n and p refer to electrons and holes, respectively; j is current

density, �� is mobility, n and p are the carrier densities, � is the quasi-Fermi level, and

( , , )r z � ��

� . Please note that in Sections 4.1.3 and 4.1.4 we temporarily reassign the

polar coordinate variable r to � in order to distinguish it from the general vector r� . A

Maxwell-Boltzmann distribution is assumed and, therefore, the Einstein relations relate

diffusion and mobility through a simple expression:

, pnn p

qDqDkT kT

� �� (4.3)

where Dn and Dp are the MB electron and hole diffusion constants. Consequently,

current density may be expressed as a function of the quasi-Fermi level gradient:

� �n n nj n r� ��

�

�

� �p p pj p r� ��

�

� (4.4)

If the electron and hole current densities could be expressed in terms of gradients of n and

p, respectively, we could write the relationships as diffusion equations:

n ndnj qD ndt �

� � ��

p pdpj qD pdt �

�

� � � �� (4.5)

And if, as will be shown, these current densities were equal, a single diffusion coefficient

would describe the dynamics. We next describe how such an equivalence arises and how

to express the current densities in terms of carrier gradients and in the process derive an

expression for the giant ambipolar diffuison constant, giantamb. diff.

D

In this dissertation we are concerned only with lateral carrier motion. Eq. (4.2) is

a separable equation and so treating the lateral components of the gradients alone, as in

75

e.g., Eq. (4.4), is a valid approach. In the equations below, 1�

� � �

� ��

�∇ .

Furthermore, in this derivation of giant ambipolar diffusion, it is assumed that the

photogenerated electrons and holes have already separated in the vertical direction across

the intrinsic region of the device. The assumption of vertical photogenerated carrier

separation is not required and does not affect the calculation of the diffusion coefficient;

it does, however, allow for a simplification of some of the equations. Therefore, if n0 and

p0 are the concentrations of electron and hole ionized dopants,

0n n n� � � 0p p p� � � (4.6)

where n, p, n0, p0, �n, and �p are no longer per unit volume but rather per unit area: n is

the electron density integrated across the thickness of the n-region; p is the hole density

integrated across the p-region. Similarly, �n and �p are the photogenerated carrier

densities integrated across the doped layers assuming they have vertically separated.

Thus, n, p, �n, and �p are functions of just the lateral dimensions (�,��).

It is assumed that at any given time there is “local” charge neutrality at each

lateral “point” in space. Consequently, there is no net lateral flow of charge. In other

words, because

p n� � � (4.7)

which, when combined with Eq. (4.6), leads directly to

p n� �

�� , (4.8)

we may write

n pj j� � (4.9)

in which jn and jp are the integrated lateral current densities in the n and p-layers.

The relationship between the quasi-Fermi levels and the carrier densities are

examined next. Changes in the number of available states result in the electron

quasi-Fermi level’s logarithmic dependence on the MB distribution of the carrier

density:[25].

. lnstatmechn i

i

nE kTn

��

� � � ��

(4.10)

76

where Ei is the Fermi level in the intrinsic region, T is the temperature, ni is the intrinsic

carrier concentration, and .

.statmechn� is the quasi-Fermi level due to this statistical

mechanics-based effect. Changes in the electron density therefore result in logarithmic

changes to the electron quasi-Fermi level. A similar relationship exists for changes in

hole carrier density and .

.statmechp� . Because the changes in electron and hole carrier densities

are equal (Eq. (4.7)), the change in both quasi-Fermi levels may be well described by

referring only to changes in electron or hole density. For example, the hole quasi-Fermi

level can be expressed as a function just of changes in electron density. Starting with.

. ln( )statmechp p� � (4.11a)

we can write.

.statmechp p�� (4.11b)

But because n p� � � , the hole quasi-Fermi level may be expressed as.

.statmechp n�� (4.11c)

We can also examine the difference between the quasi-Fermi levels:

np n p� � �� (4.12)

Changes in np� can thus be expressed either as a function of changes in n or as a function

of changes in p. It is also critical to account for the effect that carrier separation has on

the quasi-Fermi levels. The quasi-Fermi levels separation is directly proportional to the

voltage across the intrinsic region which itself is strongly determined by the electric field

due to carrier separation and thus depends (linearly) on the changes in separated

photogenerated carrier density. Hence,chargeseparationnp n� � � (4.13)

where �n is the separated carrier density and chargeseparationnp� is the quasi-Fermi level difference

due to the resulting voltage change. Combining both the electrostatic and statistical

mechanic dependencies of the photogenerated carrier densities,

77

. .. .( )

charge stat statseparation mech mech

np np n p� � � �� (4.14)

All the terms on the right hand side of Eq. (4.14) are one-to-one functions of the changes

in either carrier density, e.g., �n, as shown by Eqs. (4.10), (4.11c), and (4.13).

Because the relationship between carrier density and quasi-Fermi level is

monotonic (one-to-one), it may be inverted. Thus, the changes or the gradient in carrier

density are a function of np� .

� � np

np

nn

�

��

� �

��

� �

�

�� (4.15)

It is assumed that np� has no z (vertical) dependence; this is equivalent to assuming

well-defined, constant quasi-Fermi levels across the n and p regions.

Combining Eq. (4.15) with Eq. (4.8), we find that

� �� n p

np

nn p

� � � �

��

�

��

�

� � � �� (4.16)

Note the dependence on �. Finally, by combining Eq. (4.16) with Eq. (4.9) and Eq. (4.4)

and recalling that conductivity and mobility are related by the expression q n� �� ,

where q is the unit charge and n is the charge density,iv the relationship between the

gradients of quasi-Fermi level and current density may be found:

1

1

npn

n

p

nn� �

��

�

�

��

��

�

�

�� (4.17)

For electrons n n� �� ; a similar expression exists for holes. Using Eq. (4.9),

Eq. (4.6) may now be written as

p nj qD p qD n j� �

� � � �� (4.18)

with D an as-of-yet undefined proportionality constant – the diffusion coefficient.

Therefore, by substituting Eq. (4.17) into Eq. (4.18) the diffusion coefficient is found to

be

iv As the combination of Eqs. (4.10), (4.11c), and (4.13) in Eq. (4.14) suggests, np

n��

�

is a well-defined

quantity.

78

giant 2amb. diff.

1 n p np

n p

D Dq n

� � �

� �

��

� �(4.19)

As can be seen, this diffusion coefficient is directly related to the separation of the

quasi-Fermi levels and how quickly this energy difference changes with the carrier

density of separated photogenerated carriers, as had been suggested by the qualitative

description above. Investigations of giant ambipolar diffusion have measured diffusion

coefficients on the order of 104 V/cm2.[26]

The assumption of MB statistics is usually appropriate in silicon devices. In

AlGaAs material systems, however, degenerate electron populations are reached at lower

doping levels. This is not correctly modeled when a MB distribution is assumed.

Degeneracy increases the diffusion coefficient for ambipolar diffusion: in GaAs at room

temperature, the diffusion coefficient rises from about 40 to 130 cm2/sec as the electron

density grows from 1018 to 5x1019 cm-3 while in silicon it rises from approximately 20 to

60 cm2/sec as the carrier density grows from 1019 to 1021 cm-3.[6] This effect is likely

due to a higher effective temperature in degenerate systems, particularly of those carriers

that contribute to diffusion at the edge of the Fermi-sphere. This increase principally

affects the diffusion term, not the drift term, of a drift-diffusion equation. Since

enhanced diffusion is principally based on an effective induced drift,v a similar

proportional increase with carrier density and degeneracy is not expected. This suggests

that there will be only a small underestimation of the diffusion coefficient in degenerate

systems due to the assumption of a MB distribution in giant ambipolar diffusion. In fact,

the size of this error should be similar in magnitude to that due to the assumption of a

negligible diffusion term in diffusive conduction.

4.1.4 Diffusive conductionEquations governing the voltage behavior may be derived using a discrete

element approach to the problem. From the perspective of this paper, the simplification

v Recall, the gradient of the difference between the quasi-Fermi levels is similar to an induced electric fieldand, therefore, behaves as a “drift” term in a drift-diffusion equation. The magnitude of this term is ordersof magnitude larger than regular diffusion coefficients. Behavior similar to diffusive conduction, in whichthe regular diffusion terms are simply dropped, is therefore not surprising.

79

in the diffusive conduction approach (i.e., 0nqD n �∇ ) compared to the standard

drift-diffusion equation results in the following charge transport equation:

j q nE�� (4.20)

which is simply Ohm’s law. For a p-i-n like structure this expression may be integrated

to show

2sqRdV I

d� �� (4.21)

where Rsq is the sum of the resistance per square of the n and p layers. By using this

particular derivative in the expression another assumption has been implicitly made:

current does not flow in the “vertical” direction. This is a reasonable approximation in a

reverse-biased diode. From Q=CV, the relationship used to provide the field’s

dependence on carrier density is

2AdI V C ddt

�� (4.22)

where CA is the capacitance per unit area. Note that the local charge neutrality

relationship is implicitly assumed by using Q=CV.

A familiar manifestation of this phenomenon is signal propagation along a one

dimensional RC line (see Figure 4.3a). A voltage pulse travels along the line at a speed

determined by a diffusion coefficient proportional to its RC time constant, often much

more quickly than an individual electron may move. The equations that describe the

voltage are

dV IRdx

� � (4.23)

and

1dV dIdt C dx

� � (4.24)

Together, these equations provide a complete description of the voltage dynamics in the

form of a diffusion equation:2

x2

1dV V D Vdt RC x

��

�

�

� (4.25)

80

R

C

R1

R2

C

Figure 4.3: Schematic of 1D RC transmission line with a single (4.3a, left) or with two resistive planes(4.3b, right)

where D is the diffusion coefficient. For two resistors (Figure 4.3b), effective resistance

given by their sum:

1 2

1 1R ( )effective

DC R R C

� �

�

(4.26)

For the two-dimensional case, together equations (4.21) and (4.22) provide a

complete description of the voltage dynamics:

dV D Vdt �

�2

∇ (4.27)

The Laplacian is two-dimensional and the (voltage) diffusion coefficient is 1

sq A

DR C

�

where p layer n layersq sq sqR R R� �

� � and CA is the capacitance per unit area. This effective

diffusion coefficient depends on parameters (resistance and capacitance) that can be

controlled by proper device design. For the purpose of comparison, it is worth noting

that Eq. (4.27) may be equivalently written as

dn D ndt �

�2

∇ (4.28)

simply by using V=Q/C. Eq. (4.23) describes the diffusion of voltage, not individual

carrier motion. Similarly, Eq. (4.24) does not describe the diffusion of a particular

particle but rather an effective diffusion for the system of particles.

The resistances per square of the doped layers may be quite small (10’s-100’s �),

with a capacitance per �m2 on the order of 0.1 fF for a 1 �m thick intrinsic region.

81

Hence, the diffusion coefficient of diffusive conduction may be quite large, on the order

of 105 V/cm2 compared to a diffusion coefficient of only ~100 V/cm2 in bulk material.vi

If a pulse of light is incident on such a structure that is reverse biased and it is

absorbed in the intrinsic region, the photogenerated carriers vertically separate,vii

shielding the bias. This creates a lateral voltage distribution across the device face in the

shape of the incident pulse. If this pulse is Gaussian-shaped (as is typical), the analytical

solution to the differential diffusion equation is

� ��

2

, exp4MV r t V

t D t� �

� �

� �� (4.29)

where 20

2

TOTM

A

QVwC �

��

and 2

0 142

wD

�

� ��

, QTOT is the total charge of photogenerated

electrons, and w0 is the 1/e2 spot size radius.[8] In such cases, the voltage at the center of

the spot diffuses away with a predominantly hyperbolic time-dependence, �/(t+�). For a

properly designed device, the decay coefficient, �, may be on the order of picoseconds.

To reiterate, this fast diffusion is not a single carrier’s movement from the center

outwards. Rather it is the ensemble of electrons and holes across the whole extent of the

p and n regions each moving a small amount – just the same as when in a transmission

line the voltage signal is able to propagate much faster than an individual electron can.

Such voltage behavior has been verified.[10, 11, 27]

4.1.5 Comparison between approachesWhat is the relationship between the diffusion coefficients of diffusive

conduction, Eq. (4.27), and giant ambipolar diffusion, Eq. (4.19)? The answer is: they

vi The values for the diffusion coefficient of giant ambipolar diffusion and diffusive conduction presented inthis paper are measured results from test devices. Their values are different from each other simply due tothe differences in their material structure and doping compositions.vii In principle, the lateral voltage distribution could also cause the carriers in the intrinsic region to movehorizontally. We neglect such movement on the assumption that the spot radius, �0, is much larger than thethickness of the intrinsic region. Consequently, the lateral carrier gradient (and hence transport due toregular carrier diffusion) is small even when compared to the vertical carrier gradient.

82

are nearly the same. For these types of structures, the relationship between carrier

concentration and voltage is primarily linearviii

q n Q CV� � � (4.30)

Clearly, if voltage diffuses away at a given rate, the carrier concentration at that location

must change by a proportional amount. In this sense the similarity between the

microscopic and macroscopic approaches could be expected. This may be shown more

explicitly. Recalling thatix

1sqR l�

�� (4.31)

where l is the thickness of the doped layers, we can write

� �1 1

1 1n p

n pn p sq sq

n p

l R R� �

� �

� �

� �

� ��

(4.32)

Usingx

np

n nV �

� ��

� �(4.33)

and

� �q n AlQCV V

��

(4.34)

the capacitance per unit area may be related to the quasi-Fermi level difference:

� �2 2np

A

q AlV q lqn n C C��

� � ��

(4.35)

Combining this key relationship with Eq. (4.32) can then be used to show that, indeed,

� �giant diffusive2amb. diff. conduction

1 1 1n p npn p

n p Asq sq

D Dq n CR R

� � �

� �

��

� � �(4.36)

viii This ignores the logarithmic dependence of the quasi-Fermi levels on electron and hole density. This isa reasonable approximation as the magnitude of the linear effects in practical examples are larger by two ormore orders of magnitude.ix Although the conductivity is related to the resistance, Eq. (4.28) disregards the distinction between theassumed Maxwell-Boltzmann distribution of giant ambipolar diffusion and the unrestricted distribution ofdiffusive conduction.x The charge density-voltage relationship of Eq. (4.34) implicitly describes a functional relationship of achange in both the electron and hole density that are “vertically” separated.

83

Diffusive conduction is anchored around V=IR, focusing on the voltage dynamics

of the system. If this is rewritten as I=V/R, the perspective changes, emphasizing carrier

dynamics. Giant ambipolar diffusion, on the other hand, starts from the expression for

current density, � �n n nj n r� �� . When this relationship is now compared to I=V/R, the

similarities become easy to see.

4.1.6 Uses, limits and limitations of enhanced diffusionThe different approaches have different advantages aside from whether one is

more interested in carrier or voltage dynamics. The diffusive conduction approach lends

itself well, for example, to modeling the coupled behavior in multiple-layer devices, such

as for a p-i-n-i-p or more general n-i-p-i structures.[27] An interesting feature that

diffusive conduction brings to light is that enhanced diffusion may occur before the

photogenerated carriers finish vertically separating. This is discussed also in Chapter 7

(see Fig. 7.2). As the photogenerated carriers begin to separate, they alter the voltage

across the intrinsic region. This voltage change immediately starts to diffuse away as the

free carriers in the doped regions respond to this localized voltage change. In fact, in

such a case the photogenerated carriers themselves are not directly involved in any lateral

diffusion in the system! This effect may be used, for example, in heterostructures where

the photogenerated carriers may find it difficult to cross over the barriers at the interfaces

between the intrinsic and doped regions. Nevertheless, the change in voltage their vertical

motion engenders may quickly diffuse away. This behavior is not clearly reflected in

giant ambipolar diffusion due to the assumption, in the derivation presented here (see Eq.

(4.19)), of the photogenerated carriers already being vertically separated and present in

the doped regions (recall we ignored the vertical transport responsible for that). If this

assumption was not made, the diffusion coefficient would not change, but the initial free

carriers in the doped regions would not drop out of the equations; it is those free carriers

that would then be responsible for the enhanced diffusion.

Having identified the assumptions made in deriving diffusive conduction and

giant ambipolar diffusion, we are now in a position to consider the effects on diffusive

conduction when some of the assumptions made in using the drift-diffusion equation fail.

When large (lateral) electric fields are present, the current density ceases to be a linear

84

function of field (e.g., mobility becomes field-dependent); as the saturated drift velocity

is reached carrier transport is no longer well described by Ohm’s law. Diffusion,

consequently, tends not be as large as expected.[22] This situation may arise when a

large number of photogenerated carriers are created within a small spot size and separate,

creating a large shielding voltage.

If the lateral device dimensions are small (e.g., on the order of 0.1 �m in GaAs),

scattering events may not occur as the carriers move from one side of the device to the

other. Instead, carrier transport begins to approach that ballistic transport. Similarly, if

the diffusive conduction coefficient is very fast – with a time constant � less than 1 ps –

so that it is smaller than the expected scattering time, it becomes important to account for

ballistic transport. Both of these effects tend to limit the carrier motion.

For both large electric fields and fast diffusion times, a magnetic field

proportional to dE/dt is created. The magnetic field also changes as a function in time,

creating an electric field opposite to that of the shielding field. The result is slowing of

the diffusion. In the diffusive conduction approach this would be modeled by including

the inductance per unit area; giant ambipolar diffusion would add a � �q v B� � term to the

drift-diffusion equation with careful attention now having to be paid to the different

directions of current flow. The ultimate limiting case is a device with negligible

resistance per unit area (e.g., conductor-like doped regions). This situation is analogous

to a two-dimensional lossless transmission line; the voltage decay is limited by the speed

of light in the medium.

85

4.2 THEORY OF MULTILAYER DIFFUSIVE CONDUCTION

4.2.1 2-layer case

R A

C ABlayer 1

layer 2

R B

R C

C CB

x

Figure 4.4: Schematic of two-layer 1D RC structure.

The voltage behavior of a multiple layer structure in more complicated than in

p-i-n devices, where “layer” refers to a plane of capacitors sandwiched by resistors. A

2-layer structure is shown in Fig. 4.4. The voltage across any particular layer is coupled

to all of the resistances and capacitances of the entire structure, hence the analysis is more

involved.

To tackle this problem, first the coupled partial differential equations governing

voltage for the system are found. The equations are re-expressed in an eigenvector basis

for which analytic solutions are known. Finally, the dynamics of the original voltages are

expressed as a weighted sum of the eigenvectors. This process is explored in more detail

below.

For the two-layer structure, we may write (using Ohm’s law)

'A

A A A AdV V V I Rdx

��

'C

C C C CdV V V I Rdx

�

� � � (4.37)

'B

B B B BdV V V I Rdx

��

Kirchoff’s equations provide the relationships

86

AAB

dI Idx

� �

CCB

dI Idx

� � (4.38)

BAB CB

dI I Idx

� �

and, using � �Q CVt�

��

,

A

AB AB

AB AB

dII dVdx

C C dt

��

(4.39)

C

CB CB

CB CB

dII dVdx

C C dt

��

In the above equations, subscripts A, B, and C refer to the top, middle, and bottom

resistive planes of Fig. 4.4, respectively, while subscripts AB and CB refer to currents and

voltages across layer 1 and layer 2, respectively. These equations may be combined (e.g.

taking the derivative with respect to ‘x’ of Eq. (4.37) and combining with Eq. (4.38) to

substitute for IA, IB, and IC in Eq. (4.39)) so that they reduce to two coupled equations in

terms of the voltages of interest:22

2 2CBAB AB

A CVV VD

t x x�

� ��

� � ��

2 2

2 2CB CB AB

C AV V VD

t x x�

� ��

� � ��

where (4.40)

BA

B A

RR R

� �

�

BC

B C

RR R

� �

�

� �

1 1A

AB A C C

DC R R �

�

� � �

1 1C

CB C A A

DC R R �

�

�

Although these expressions have been written in terms of one-dimensional equations, the

general form holds for two dimensions as well.

87

It is worth examining the coupled relationship suggested by Eq. (4.40). The

coupling strength of the second diode is determined by the relative magnitude of the

resistance of the middle resistive plane that acts as a voltage divider. As this resistance

drops, the middle plane becomes more conducting, isolating the voltage behavior of one

layer from the other. Conversely, as the resistance in the middle rises, the coupling

increases until ultimately the device functions as if it contained only a single intrinsic

region with an effective capacitance of CAB and CCB in series.

We can rewrite the coupled equations in a matrix form

r

d U D Udt

�

� �

� �� (4.41)

where r now refers to a polar coordinate variable (in place of �), r

1r r �

� ��

�∇ , and

AB

CB

VU

V� �

� � ��

� ;� � � �

� � � �

11 12

21 22

1 1 1

1 1 1

C

AB A C C AB A C C

A

CB C A A CB C A A

C R R C R RD DD

D DC R R C R R

�

� �

�

� �

�

��

(4.42)

To decouple these equations, the eigenvalues �� and eigenvectors �� of D�

are

determined, from which the transformation matrix S�

is found, relating the eigenvectors

to the original voltages, U� .

U S��

�� ;

1 1

1 1A A

A A A A

D DS

D D

� �

� �

�

� �

� ��

(4.43)

Next, the initial voltage conditions are transformed into the eigenvector basis:1

0 0S U��

�

� �� (4.44)

Hence, we may now write decoupled differential equations in which the diffusion

coefficients are simply the eigenvalues.

t� � �

��

��

�� (4.45)

88

where ��

is a diagonal matrix of the eigenvalues. It is, therefore, possible to solve for a

particular eigenvector (‘eigenvoltage’) ��(r,t) using the known solutions to the uncoupled

diffusion equation. Finally, these solutions may be transformed back into the original

basis set, providing the full dynamic response of the system:

� � � �, ,U r t S r t�

�

�� (4.46)

As an example, if our initial condition is a Gaussian voltage distribution across

the first layer with no voltage across the bottom layer (this happens if a short pulse of

light is absorbed in the top diode and the photogenerated carriers are whisked rapidly to

the p and n layers, as occurs in the OCOG-2), we have

00 0

VU

� ��

� (4.47)

0 00

0

111 1

A A

V

D D

��

�

� �

�

�

� �

� � � ��

��

� (4.48)

where

� � � �2

22 11 22 11 22 11 12 2142

D D D D D D D D��

� � � � �

� (4.49)

(see Eq. (4.42) for a description of Dij). Using the decoupled solution to the diffusion

equation given a Gaussian distribution (Eq. (4.29)),

� ��

20, exp

41 1A A

V rr tt t

D D

��

� � �

� �

�

�

� � �

� �

� ��

� �

� (4.50)

where 20 1

2 4�

��

�

�

� and Equation (4.46),

� ��

� �

� �

� �

1 1, ,

,, ,1 1

AB A A

CB

A A A A

V r t D D r tU r t

V r t r tD D

� � �

�

� �

� � �

�

� ��

� (4.51)

we find that at the center of the pulse (r=0) the voltages are given by

89

� �

� �

0 0

0

1 10,0, 1

1 1

A A

A AAB

CB

A A

A A

V Vt tD D

D DV r tV r t V

D t tD D

� �

� ��

� �

� �

� � �

� �

� �

� ��

� �

� �

� �

� �

� ��

� ��

� ��

(4.52)

Looking carefully at this result, we see that the voltage dynamics are governed by

two eigenvalue diffusive time constants, �+ and �-. If RA and RC << RB, �+ is very small

compared to �-. The voltage across the bottom diode, VCB, at first grows at a rate

primarily determined by �-, while VAB falls equally fast. After this increase, both VCB and

VAB decay at the (slower) rate determined by �-. This is shown in Fig. 4.5, validating our

more approximate original statement that highly conducting top-most and bottom-most

layers “hold the voltage constant across the device,” and the subsequent implication that

the voltage of the bottom diode mimics the opposite behavior of the top diode.

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

Vol

tage

(a.u

.)

Time (ps)

Figure 4.5: Bottom diode voltage behavior (-VCB) at pulse center (r=0). Note the two separate behaviors: avery fast rise time (hence voltage across entire device essentially kept constant) and a slower fall time dueto an impulse voltage on the top diode. The specific parameters used were the capacitance per unit area of

each of thee three depletion regions, C� =[0.67, 1.58, 0.10] fF/�m2, the resistance per square of each of thefour resistive planes, R� =[20, 300, 2000, 25] ohms per square, and the pot size radius, 3.5 �m.

4.2.2 N-layer caseThe analysis described above may be extended in a straightforward manner to

describe the voltage dynamics of a device with an arbitrary number of layers. This

90

general approach is composed of four steps: (1) determine the coupling diffusion matrix

D�

; (2) solve the eigenvector problem to describe a decoupled system; (3) apply the

solution to the regular (uncoupled) diffusion equation for the eigenvectors; (4) transform

the dynamic solution back to the original variables. Steps (2)-(4) are identical to those

just described. All that is left to do is determine the original coupling matrix, D�

.

Appendix B describes this process in more detail. The key result is that the voltage obeys

a diffusion equation

V D Vt

��

�

� �2∇ (4.53)

where D�

may be determined from the following relationship1 1

1[ ] 2[ ]

[ ]

1 1N N

N

D M MRC

� ��

�

� �(4.54)

The division of R� (a vector related to the resistance per square of each resistive plane)

and C� (the capacitance per unit area of each layer) mean element-by-element operations

instead of matrix operations, and 1

1[ ]NM�

�

and1

2[ ]NM�

�

are determined simply by the number

of layers. From this point forward the method described earlier in this section for

determining the voltage dynamics using eigenvalues and eigenvectors becomes

straightforward to apply.

4.2.3 DiscussionHaving the ability to accurately model the voltage behavior across multi-layer

resistive and capacitive stacks is essential for understanding a stacked diode OCOG. This

same analysis method may also be used to help study n-i-p-i devices. Multilayer

diffusive conduction analysis has also helped to reveal a useful insight. One might

suggest that adding extra layers simply presents extra resistance and capacitance,

reducing device speed. This is not the case. In fact, increasing the number of layers

significantly improves device speed.

The frequency response of an electrical RC filter chain circuit provides a good

analogy for multilayer OCOG behavior. Adding an extra RC filter changes the electrical

91

circuit’s filter response: each RC filter adds an extra 6 dB/octave decay, modified by the

extra load it adds.[28] In OCOGs, the presence of extra layers significantly speeds up the

voltage decay response, although at a cost of reduced voltage swing. This is particularly

relevant for diffusive conduction behavior since its single layer hyperbolic decay, �/(�+t),

although initially fast, slows down considerably at times large with respect to the

diffusion coefficient time constant, �. The time it takes a voltage pulse to fall by 90%, for

example, equals 9 �. The presence of multiple layers provides a method that enables the

voltage to drop more much more rapidly even at longer times. The result is not simply a

faster effective diffusion time constant; rather, the response function itself is changed,

more closely resembling the multiplication of the individual diffusive conduction decays.

92

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50 2 layers

3 layers

4 layers

5 layers

Rel

ativ

e C

hang

e in

Vol

tage

(a.

u.)

Time (ps)

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0 2 layers

3 layers

4 layers

5 layers

Nor

mal

ized

Cha

nge

in V

olta

ge

Time (ps)

Figure 4.6: (4.6a; top) Response of the bottom-most layer of a multilayer device when a unit voltagechange due to voltage shielding is placed across the top-most (control diode) layer for devices with anincreasing number of layers. Although the induced change in voltage is reduced, as the number of layersincreases the voltage decay function falls off faster than a hyberbolic decay. (4.6b; bottom) Thenormalized response of Fig. 4.6a. Simulation parameters were a spot size radius of 5 �m,R=[10; ..1000… ; 10] ��, and C=[0.3; ..0.3… ; 0.3] fF/�m2.

In Fig. 4.6a the voltage behavior across the bottom-most layer of a device is

graphed for a series of devices with progressively more layers assuming unit voltage

shielding is suddenly created across the top (control diode) layer. The magnitude of

induced voltage change diminishes as the number of layers increases. In essence, the

device has become a voltage divider. In addition, though, it is important to notice that the

decay of the induced voltage increases substantially as the number of layers increases,

falling faster than a hyperbolic decay. Comparing normalized data, Fig. 4.6b, makes this

93

point clearer. This may be of critical importance if fast turn-off times are required.

Without this change in the response function, faster turn-off times come at the large

expense of a dramatically reduced magnitude of voltage change (much more than the

reduction due to adding extra layers).xi This behavior is the 2-D analog of the sharper

filter response that arises in an RC filter chain as more RC filters are added.

-5 0 5 10 15 20 25 30 350.0

0.2

0.4

0.6

0.8

1.0 Data

3-Layer Simulation

2-Layer Simulation

Cha

nge

in R

efle

ctiv

ity (

a. u

.)

Time (ps)

Figure 4.7: Small signal response of the OCOG-3 compared to simulations of both 3 and 2-layer structures.Note the dramatic improvement in the device recovery as the number of layers increases.

The benefit of multiple layers can be clearly seen in Fig. 4.7. The dual diode

p-i-n-p-i-n OCOG-3 structure actually contains three depletion regions: two intrinsic

regions in the top and bottom diodes and one from the depletion region between the

middle n and p regions. Hence, a dual diode can in fact be a three layer structure. The

simulation of the OCOG-3 device was modeled with three layers of capacitance

(C� =[0.67; 1.58; 0.10] fF/�m2), each sandwiched by a layer of resistance

( R� =[20; 300; 2000; 25] ��) with these parameters based on measured or calculated

values. As Fig. 4.7 shows, when contrasted against a comparable two-layer structure

(e.g. ignoring the middle depletion region, equivalent to a n-i-p-i-n OCOG-2-like device

with C� =[0.67; 0.10] fF/�m2 and R� =[20; 300; 25] ��), the three-layer simulation

exhibits a much sharper response and is a significantly better fit to the data. The xi In point of fact, adding extra layers does result in a correspondingly smaller effective capacitance andresistance per unit area, which would itself result in a sharper voltage decay. However, the decay response

94

separation between when the rising and falling edges of the gate are at 10% of the peak

voltage change -- the full-width 10%-of-maxium (FW10%M) value -- of the simulated

two layer structure (22 ps) takes nearly twice as long as the three layer device (13 ps).

is actually faster than that due to the modified effective time constant, �effective; it falls off faster than�effective /��effective +t).

95

REFERENCES

1. Roosbroeck, W.v., "The Transport of Added Current Carriers in a Homogenoussemiconductor," Phys. Rev., vol. 91, pp. 282 (1953).

2. McKelvey, J.P., Solid State and Semiconductor Physics. Vol. Chapter 10, (RobertKreiger Publ. Co., Malabar, Florida, 1966).

3. Schetzina, J.F. and J.P. McKelvey, "Ambipolar Transport of Electrons and Holesin Anisotropic Crystals," Phys. Rev. B, vol. 2, pp. 1869-1874 (1970).

4. Shah, R.M. and J.F. Schetzina, "Excess-Carrier Transport in AnisotropicSemiconductors: The Photovoltaic effect," Phys. Rev. B, vol. 5, pp. 4014-4021(1972).

5. Ritter, D., E. Zeldov, and K. Weiser, "Ambipolar transport in amorphoussemiconductors in the lifetime and relaxation-time regimes investigated bysteady-state photocarrier grating technique," Phys. Rev. B, vol. 38, pp. 8296(1988).

6. Young, J.F. and H.M.v. Driel, "Ambipolar diffusion of high-density electrons andholes in Ge, Si, and GaAs: Many-body effects," Phys. Rev. B, vol. 26, pp. 2147-2158 (1982).

7. Lin, H., et al., "Anomalous In-Plane Drift and Diffusion Of Non-EquilibriumCharge Carriers In n-i-p-i Doping Superlattices," Surf. Sci., vol. 228, pp. 500-503(1990).


9. Schneider, H., et al., "Diffusive electrical conduction in high-speed p-i-nphotodetectors," Appl. Phys. Lett., vol. 60, pp. 2648-2650 (1992).

10. Yang, C.-M., et al., "Measurement of Effective Drift Velocities of Electrons andHoles in Shallow Multiple-Quantum-Well p-i-n Modulators," IEEE J. Quant.Elec., vol. 33, pp. 1498-1506 (1997).

11. Yairi, M.B., et al., "High-speed, otically controlled surface-normal optical switchbased on diffusive conduction," Appl. Phys. Lett., vol. 75, pp. 597-599 (1999).

12. Bradley, P.J., C. Rigo, and A. Stano, "Carrier Induced Transient Electric Field ina p-i-n InP-InGaAs Multiple-Quantum-Well Modulator," IEEE J. Quant. Elec.,vol. 32, pp. 43 (1996).

13. Ershov, M., "Lateral photocurrent spreading in single quantum well infraredphotodetectors," Appl. Phys. Lett., vol. 72, pp. 2865 (1998).

96

14. Streb, D., et al., "Extremely fast ambipolar diffusion in n-i-p-i dopingsuperlattices investigated by an all-optical pump-and-probe technique,"Superlattices and Microstructures, vol. 25, pp. 21-27 (1999).

15. Hutchings, D.C., C.B. Park, and A. Miller, "Modeling of cross-well carriertransport in a multiple quantum well modulator," Appl. Phys. Lett., vol. 59, pp.3009-3011 (1991).

16. Capasso, F., K. Mohammed, and A.Y. Cho, "Resonant Tunneling ThroughDouble Barriers, Perpendicular Quantum Transpot Phenomena in Superlattices,and Their Device Applications," IEEE J. Quant. Elec., vol. 22, pp. 1853 (1986).

17. Fraenkel, A., et al., "Vertical drift mobility of excited carriers in multi quantumwell structures," J. Appl. Phys., vol. 75, pp. 3536 (1994).

18. Dentan, M. and B.d. Cremoux, "Numerical Simulation of the Nonlinear responseof a p-i-n Photodiode under High Illumination," J. Lightwave Tech., vol. 8, pp.1137 (1990).

19. Rosencher, E., et al., "Emission and capture of electrons in Multiquantum-WellStructures," IEEE Trans. Quant. Elec., vol. 30, pp. 2875 (1994).

20. Poole, P.J., et al., "All-optical measurement of the giant ambipolar diffusionconstant in a hetero-nipi reflection modulator," Semicond. Sci. Tech., vol. 8, pp.1750-1754 (1993).

21. Harrison, W.A., Solid State Theory (General Publishing Company, Toronto,1979).

22. Lundstrom, M., Fundamentals of Carrier Transport. Modular Series on SolidState Devices, ed. G.W. Neudeck and R.F. Pierret. Vol. X, (Addison-Wesley,Reading, MA, 1990).

23. Herring, C., "Thoery of Transient Phenomena in the Transport of Holes in anExcess Semiconductor," Bell Sys. Tech. J., vol. 28, pp. 401 (1949).

24. Gulden, K.H., et al., "Giant Ambiploar Diffusion Constant of n-i-p-i DopingSuperlattices," Phys. Rev. Lett., vol. 66, pp. 373-376 (1991).

25. Pierret, R.F., Semiconductor Device Fundamentals (Addison-Wesley Publishing,Reading, Mass., 1996).

26. Streb, D., et al., "Carrier density dependence of the ambipolar diffusioncoefficient in GaAs n-i-p structures," Appl. Phys. Lett., vol. 71, pp. 1501-1503(1997).


97

28. Horowitz, P. and W. Hill, The Art of Electronics (Cambridge University Press,England, 1980).

98

Chapter V: Experimental Methodology

This chapter discusses the methodology used in the experiments. Descriptions of

the device growth, device processing, and device characterization are presented as are

details of the test-bed used for data collection.

5.1 DEVICE GROWTH

The OCOG devices were grown by solid source molecular beam epitaxy (MBE)

on either n+ or undoped GaAs (001) wafers. For each dopant, three test samples were

grown, each at a different growth temperature. Hall-measurements of these samples were

then taken and provided the basis to determine the activated doping value as a function of

temperature to within a factor of about two. In situ reflection high-energy electron

diffraction (RHEED) measurements were made just before device growth to determine

the growth rate of thin layers, such as quantum wells. For some devices, multiple

half-wavelength layers were grown and monitored using the reflection from the wafer of

a temperature-stabilized, 990 nm diode laser. The temperature dependence of the index

of refraction for GaAs and AlAs was calculated using the functional dependence

provided by [1]; the index for a particular AlxG1-xaAs material was approximated using a

linear combination of the GaAs and AlAs values. Growth rate calibrations for thick

layers (e.g. DBR layers) were thereby determined by using the measured period between

peaks in the reflection histogram and the appropriate index of refraction. For all the

devices, samples consisting of just the DBR mirror or the p-i (MQW)-n regions were

separately grown and tested (by taking photocurrent and reflectivity data as described

below) to make sure that the high-reflection bandwidth region of the DBR stack and the

voltage-dependent heavy-hole exciton peak were well overlapped.

5.2 DEVICE PROCESSING

All of the OCOG devices were processed in a similar manner. A set of masks

was designed and patterned that enabled up to four separate layers to be defined and

separately contacted. Standard lithography and wet etching were used to create devices

99

approximately 300x300 �m square. For accurate etching depths, thin 50 angstrom AlAs

stop-etch layers were included in the growth structure to stop citric acid etches through

AlGaAs layers; these stop-etch layers themselves were subsequently etched using a

hydrochloric acid mixture. Ohmic ring contacts (for an example, see Fig. 5.1) were made

to individual p and n layers by evaporating Ti/Au and Ge/Au/Ge/Au/Ni/Au compositions,

respectively. The contacts to n layers were annealed using a rapid thermal annealer

(440oC, 30 seconds) and were processed before any p layer contacts were deposited.

Silicon-nitride, SiN3, with a thickness ~104 nm, was typically deposited as an

antireflection coating centered at 850 nm; these AR coats were patterned by dry etching

using CF4 and O2 in a plasma asher (Phlegmatron). Finally, the processed wafers were

mounted on a ceramic 12-pin chip mount and wire-bonded. For the OCOG-3 devices, in

order to increase the conductivity of the top p-layer, 660 nm of indium-tin-oxide (ITO)

was deposited and annealed (500oC, 5 minutes); this layer also acted as an anti-reflection

coating in place of silicon-nitride. (The ITO used was 75% absorbing instead of an

expected 5-10%. We believe this was due to deposition difficulties and not inherent to

the device processing. Consequently, the ITO absorption has been discounted in

reflectivity and power calculations of Chapter 6.)

100

AR coating

Metalic rings

Bond pad

1st mesa 2nd mesa

Wire bonds

Figure 5.1: Image of a wire-bonded OCOG-2 device. The central square is the silicon-nitrideanti-reflection coating. The square concentric metallic rings and bond pads (100 �m square) contact the topn, p, and bottom n layers of the device.

5.3 DEVICE CHARACTERIZATION

5.3.1 Electrical characterizationTwo types of electrical characterization were conducted. A four-point probe

technique was used to determine the resistance per square of individual doped layers.

Four contacts, either soldered indium bumps or deposited ohmic contacts (the same as

described above), were approximately evenly placed in a row on a doped layer. A

ramped current was forced between two outer contacts while the induced voltage between

the two inner contacts was recorded by a parameter analyzer. The resistance equaled the

slope, �V/�I, multiplied by a correction factor of 4.53 for this geometry.[2]

The I-V curves of p-i-n stacks and p-n junctions were also measured. Of

particular interest was the reverse break-down voltage of the p-i-n and p-i(MQW)-n

layers; this revealed not only if a device might be functional, but also the maximum

voltage shift possible to induce in the device for appropriate layers. Moreover, it

provided a measurement of the dark current. For OCOG-2 and OCOG-3 devices, the

101

reverse-biased I-V dependence under optical illumination provided the means to

determine whether or not the top diode fully absorbed the control pulse and if it was

transparent to the signal pulse by measuring whether or not current was induced in the

bottom or top diode, respectively.

5.3.2 Optical characterizationComplete photocurrent spectra were obtained by measuring the current across a

p-i(MQW)-n diode layer in conjunction with a chopper and lock-in amplifier. This was

done while the applied reverse bias voltage as well as the wavelength of incident light,

were changed. Three light sources were used: a 0.25 meter SPEX spectrometer when

extended wavelength ranges (50-300 nm ranges) were desired; a Coherent continuous

wave tunable Ti-Sapphire laser for more accurate spectra; and, for femtosecond pulse

device response, the Tsunami Ti-Sapphire mode-locked tunable laser. Fig. 5.2 is an

example of typical spectra. Reflectivity as a function of voltage and wavelength spectra

were recorded in a similar method using a silicon photodetector, as exemplified by

Fig. 5.3.

102

0 2 4 6 8 10 12 14 16 18 20

1410

1415

1420

1425

1430

1435

1440

1445

1450

1455

1460

1465

1470

1475

Reverse Bias Voltage (V)

Pho

ton

Ene

rgy

(meV

)

Figure 5.2: Contour graph of photocurrent spectra of OCOG-1 in arbitrary units. Photocurrent increases from nearzero (blue) to larger values (red). The quadratic shift in energy with voltage of both the heavy and light hole excitonpeaks are clearly resolved. Strong inter-well coupling effects may also be seen. See Appendix B for a moredetailed analysis.

0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Wavelength (nm)

870

868

865

863

860

858

855

853

Abs

olut

e R

efle

ctiv

ity


Figure 5.3: OCOG-1 reflectivity as a function of reverse bias for various wavelengths of picosecond pulses.

103

5.4 PUMP-PROBE SET-UP

5.4.1 General descriptionExperimental investigations of the dynamic response of OCOG devices were

conducted using a standard pump-probe technique. Light pulses were generated by a

tunable Spectra Physics Tsunami Ti-Sapphire laser that was able to provide pulse widths

either 1-2 ps or about 80 fs long (depending on the internal setting of the laser) at a

repetition rate of 82 MHz. A given pulse was split to create a strong control (pump)

pulse and a weaker signal (probe) pulse. The dynamics of the OCOG devices were

studied by adjusting the path lengths of the control and signal beams so that the time

between the arrival of the two pulses could be finely controlled. Many reflectivity data

could be measured for any given time separation between control and signal. Hence, by

changing the relative separation in time between control and signal, the reflectivity of a

given device could be mapped out, e.g. at 0 ps separation, 0.1 ps, 1 ps, or 1000 ps. By

steadily adjusting the path length difference using a translation stage run by a

stepper-motor control, the time-dependent responses of OCOG devices were obtained

with femtosecond-scale resolution. A schematic of the optical bench set-up may be seen

in Fig. 5.4 and images of the actual optical bench equipment used in Fig.5.5.

104

Imaging optics

Lock-inDelay stage

TiSapphirefemptosecondlaser ~850nm

Device Voltage Control

PD

DPSSL532nm

2nd Harmonic Generation(BBO Crystal)

~425nm

Computer

Spectral and Polarization Filters

21

3

4

6

5

7

8

9

10

11

12

Figure 5.4: Schematic of optical pump-probe set-up. (1) tunable short pulse laser (Tsunami), (2) beamsplitter where pump and probe are separated, (3) BBO crystal used for 2nd harmonic generation,(4) retroflector on a variable delay stage, (5) beam splitter for recollimating pump and probe beams,(6) focusing lens, (7) OCOG device, (8) chopper, (9) photodetector, (10) lock-in amplifier, (11) imagingoptics, (12) voltage supplies.

105

1

2

3

4 5

6

10

12

5

10

6 7

8

9

11

4

Figure 5.5: Photographs of pump-probe set-up. The circled numbers correspond to the same numbereditems in Fig. 5. (1) tunable short pulse laser (Tsunami), (2) beam splitter where pump and probe areseparated, (3) BBO crystal used for 2nd harmonic generation, (4) retroflector on a variable delay stage,(5) beam splitter for recollimating pump and probe beams, (6) focusing lens, (7) OCOG device,(8) chopper, (9) photodetector, (10) lock-in amplifier, (11) imaging optics, (12) voltage supplies.

106

5.4.2 Second-harmonic generationIn the OCOG-2 and OCOG-3 experiments the frequency of the control pulse had

to be sufficiently larger than the signal pulse so that the top diode was strongly absorbing

of the control pulse yet transparent to the signal beam. To accomplish this, the frequency

of the control pulse was doubled (to approximately 425 nm) by using a pair of 10 cm

lenses to focus and subsequently recollimate the linearly polarized pump beam on a

properly oriented 1 mm thick BBO crystal. The polarization of the resulting

second-harmonic generated (SHG) light was perpendicular to the original pump light.

Following recollimation, the remainder of the original pump pulse was separated from

the SHG light using both polarizing and dichroic beam splitters as well as a low-pass

wavelength filter. Although the magnitude of the frequency difference between the

control and signal pulses was much larger than required by the OCOG devices, SHG was

used due to its relative ease of implementation.

1, 2, 3, 4

1, 3

2, 41

2

Figure 5.6: Optical sub-system for generating a 4-pulse bit stream. It consisted of a large beam splittersurrounded by four retroflectors, two of which (the left and right) were mounted on translation stages. Apulse of light entered from the left and split near the top-right of the beam splitter (beams 1 and 2); thosepulses were retroreflected and split again into a total of two groups of two pulses (1,3 and 2,4); finally, afteragain being retroflected, the pulses were overlapped and split once more. Half the light went to a beamdump (exiting the ‘top’ of the beam splitter in the figure) while the other half (pulses 1,2,3,4) exited on theright-hand side, continuing down the beam path.

107

5.4.3 Creation of 4-pulse burst data streamThe response of the device to multiple optical pulses was tested using a stream of

four control pulses that were focused onto the device. This type of pulse stream was

created using a large beam splitter (5 cm x 5 cm) and four corner cubes, one on each side

of the beam splitter, as illustrated in Fig 5.6. A single control pulse would pass through

the beam splitter and divide in half. Each of these two beams was reflected from a corner

cube back into the beam splitter and, consequently, split again. With careful alignment,

the initial two beams overlapped each other at that point. The result was four beams (two

pairs of two) that, in turn, were retroreflected by two more corner cubes and aligned to

overlap once more in the beam splitter before splitting a third time. The final result was

eight beams (two groups of four), half of which went to a beam dump and the other half

of which exited the beam splitter spatially overlapped to continue down-stream. Each of

these remaining beams was thus retroreflected from a unique combination of two of the

four corner cubes. As a result, by varying the individual corner cubes’ distances from the

beam splitter, it was possible to control each pulse’s relative time delay with respect to

the others. Because of the multiple splittings and reflections involved in creating the

pulse stream, the final power in each pulse was limited to the small-signal regime

(~1.3 pJ/pulse).

108

REFERENCES

1. Gehrsitz, S., et al., "The refractive index of AlxGa1-xAs below the band gap:Accurate determination and empirical modeling," J. Appl. Phys., vol. 87, pp.7825-7837 (2000).

2. Sze, S.M., Physics of Semiconductor Devices. 2nd Ed. ed (John Wiley and Sons,New York, 1981).

109

Chapter VI: Results and Analysis

In this chapter the experimental results of the switching behavior of the three

generations of OCOG devices are presented along with their simulated theoretical

responses. The close match between the simulations and data provide strong evidence

not only that our hypothesis that diffusive conduction is responsible for the switching

behavior but also that our models of the spatio-temporal response of multilayer structures

are reasonable.

6.1 FIRST GENERATION OCOG-1The first generation optically controlled optical gate, OCOG-1, was a simple

p-i(MQW)-n diode grown on top of a DBR mirror. The one micron thick Al10Ga90As p

and n layers were doped with beryllium and silicon, respectively, at a density of about

1018 atoms/cm3. The intrinsic region contained 94 GaAs quantum wells each 100 Å wide

and separated by extremely thin, 5 Å AlAs barriers. A Si3N4 antireflection coating was

deposited, centered at 850 nm. (Detailed descriptions of all generations of OCOG

structures can be found in Appendix C.) This device was grown to demonstrate that

diffusive conduction could be used as a switching mechanism for optical gates.

Quantum Well Stack

BraggMirror

p

n

300 �m

1 �m

i

Control

Signal

n

Figure 6.1: Schematic drawing of OCOG-1 p-i(MQW)-n structure.

110

The switching behavior of the OCOG-1 device is strongly influenced by carrier

motion. To allow the device to “turn-on”, the photogenerated electrons and holes must

separate, first escaping from the QWs and then moving toward the n and p layers,

respectively. In GaAs, at fields above roughly 1 V/�m charge carrier motion reaches

saturated drift velocities, moving at speeds near 0.1 �m/ps.[1] Hence, under such

conditions it takes approximately 10 picoseconds for an electron or hole to travel across a

one micron thick intrinsic region.

The other critical constraint on carrier motion is the escape time from the QWs.

There are a wide variety of avenues available for carriers to leave a quantum well. Two

principal methods are (1) to tunnel through the barrier or (2) to be thermally excited

above the barrier.[2-4] In the presence of an electric field, the wavefunction of the carrier

as well as the barrier shapes and wells are altered, modifying the escape mechanism.

This may have a significant effect on the escape time.

Numerous device structures for various applications have been designed to take

advantage of the different escape mechanisms. The strong coupling between QWs in

superlattices create mini-conduction bands that may be used to enhance carrier

sweep-out.[5] This is exemplified in quantum cascade lasers that rely on this behavior to

help empty the lowest energy state of their lasing structure and maintain a population

inversion.[6, 7] Extremely shallow quantum wells (ESQWs) have QW barrier heights

that are lower than the thermal energy (~25 meV), providing very rapid escape due to

thermal excitation.[8-10] Even modest electric fields (4 V/�m) reduce carrier

confinement sufficiently to eliminate exciton resonant absorption. ESQWs,

consequently, may be used in low-voltage switches.

One of the design goals of OCOG-1 was fast photogenerated carrier escape

combined with a structure that also allowed switching over a broad wavelength band.

ESQWs provide excellent escape times but have somewhat limited wavelength operation

due to the rapid extinction of the exciton peak. By using extremely thin barriers, it was

hoped that escape times could be enhanced while maintaining a wide operation

bandwidth. Transfer matrix method simulations [11, 12] of the ultra-thin barrier QWs

indicated that, as hoped, the exciton peak would significantly shift with voltage,

111

providing a wide wavelength operation region (see Appendix C). Escape times,

estimated from the resonance peak energy widths of the simulation (from 2 to about 20

ps), were slower than for ESQWs but sufficient for switching times faster than 100 ps.

6.1.1 Small signal experimental results and simulationAn example of the small-signal behavior of this device is shown in Fig. 6.2 using

a 60 fJ pump pulse with a 7 �m radius spot size at 860 nm with the device biased at

-8.3 V. Optically-controlled switching was clearly demonstrated, with a “turn-on” time

of about 17 ps and a “turn-off” time around 50 ps.

0 50 100 1500.45

0.46

0.47

0.48

0.49

0.50

0.51

0.52

Data

Simulation

Ref

lect

ivity

Time (ps)

Figure 6.2: OCOG-1 Small signal reflectivity data and simulation. The data was obtained using 860 nmlight with 20 �W control power and –8.36 V bias. Simulation parameters included D=0.4x105 cm2/s, 65 fJpulse energy, electron escape time = 3 ps, and hole escape time =20 ps.

Equally important, simulation results closely matched the data. The simulation

model included the following parameters:

� a Gaussian spatial distribution for the creation of electrons and holes based on the

spot size of the pulse

� effective escape times from the wells for both electrons and holes

112

� vertical field-dependent velocity – also for both carrier types

� a diffusion coefficient and voltage decay according to Eqs. (3.3)-(3.5)

� empirical results that provided absorption data as a function of bias voltage

The parameters used included a diffusion coefficient equal to 0.4x105 cm2/s, an electron

escape time of 3 ps, and a hole escape time of 20 ps. The close fit between data and the

simulations, particularly for the tails, provides strong support to the hypothesis that

diffusive conduction is, indeed, the mechanism primarily responsible for voltage decay.

Details of the simulation models used are described in Chapter 7.

The device recovery time was long, particularly in light of an expected voltage

diffusion coefficient on the order of a picosecond. We believe this may have been due to

a long hole escape time from the quantum wells. The long hole escape time would

provide a weak but extended “turn-on” time, preventing the device from quickly

recovering. Reducing the hole escape time, for instance by using lower QW barriers or a

tunneling-resonant device, should make this type of device much faster. Moreover, the

presence of these holes in the intrinsic region changes the spatial electric field screening

across the QWs which, when combined with the field-dependence of the absorption,

limits change in reflectivity.

6.1.2 Large signal experimental results and simulationUsing a 750 fJ pump pulse, OCOG-1 exhibited a large change in reflectivity such

as might be needed in a practical device. With a spot size of 7 �m radius (~5 fJ �m-2),

the control pulse was strongly absorbed by the heavy hole exciton with the laser tuned to

855 nm and the device biased at -6.3 V. As Fig. 6.3 shows, the probe pulse experienced

an absolute change in reflectivity of 0.3 with a contrast ratio of 1.8-to-1. The high

reflectivity state of the device induced by the pump pulse decayed within about 50 ps.

The device demonstrated this type of behavior when the laser was tuned between 855 and

865 nm, with the contrast ratio falling by about 30% at the high-wavelength end.

The energy needed for this switching was very low – less than one picojoule –

and compares favorably with the other switching mechanisms mentioned earlier. This

low level of required energy was possible due to the macroscopic motion of the

photogenerated carriers. The carriers from a single well, once reaching the n and p

113

regions, screened all of the QWs. Together, the carriers created in each of the 94 QWs

ultimately moved to create a much larger electric field change across any single QW than

would have been possible with carriers from that QW alone as described in Chapter 3.

Consequently, low switching energy arose naturally from the combination of strongly

field-dependent absorption of the QWs (the QCSE) and the macroscopic movement of

the photogenerated carriers across the entire width of intrinsic region.

0 20 40 60 80 100 120 140

0.3

0.4

0.5

0.6

0.7

0.8

1500 fJ

750 fJ

375 fJ

Ref

lect

ivity

Time (ps)

Figure 6.3: OCOG-1 Reflectivity modulation of probe (signal) pulse as a function of time with -6.3 V biasacross device and 855 nm light at various pump (control) pulse powers with a spot size of 50 �m2.Simulation [solid lines] reflectivity modulation at various input powers. The good fit of the tails to ahyperbolic decay is a good indication that diffusive conduction is responsible for the voltage dissipation.

The response of the OCOG-1 device changes dramatically with large control

pulse power, as Fig. 6.3, comparing medium (375 fJ, 750 fJ) and large-signal (1500 fJ)

data to simulations, illustrates. At these energies, compared to small-signal behavior,

sufficient carriers are created so that once a small fraction of them have escaped from

their QWs and separated, the voltage across the device is significantly shielded. When

this occurs, the escape time of the remaining carriers (particularly electrons) increases

while, simultaneously, the drift velocity of the carriers decreases. Diffusive conduction

eats away at the shielding voltage and ultimately is responsible for enabling the

114

remaining carriers to be swept out. Nevertheless, strong shielding remains prevalent for

an extended period of time and results in a slower gating response function. The change

in reflectivity also approaches a limiting value, in part due to the maximum shift in

absorption possible of the quantum wells as well as the finite voltage shift that may be

shielded. And, as already mentioned, the trapped carriers (in particular, holes) also

strongly modify the field across various regions of the intrinsic region,

Although details of the simulations are presented in Chapter 7, it is worth

mentioning here that the carrier escape time is field-dependent, particularly for the

electrons. Rapid field screening at larger optical powers may have increased the average

expected escape time for carriers still trapped. For the three curves shown, reasonable fits

were found with electron escape times of 3, 5, 7, and 10 ps for pump powers of 65

(Fig. 6.2), 375, 750, and 1500 fJ (Fig. 6.3), respectively; this was the only parameter

aside from pump power that was varied between the simulations.

6.1.3 Signal gain (large signal) experimental results and simulation

0 30 60 90 120 1500.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Change

in R

efle

cted

Sign

al E

nerg

y/C

ontr

ol E

nerg

y

Time (ps)

Figure 6.4: Change in reflected probe (signal) power normalized against pump (control) pulse power –equivalent to signal gain with the device biased at -6.0 V, pump power of 750 fJ and probe power of2.56 pJ at 855 nm.

115

OCOG-1 also demonstrated the ability to provide signal gain similar to that of a

three-terminal device. From this perspective, the control signal is the pump pulse; the

probe pulse is another input, and the output is its reflection. When the probe power is

small with respect to that of the control signal, the induced changes in the output

(reflected probe power) are also small compared to the pump’s power. If the incoming

probe power, however, were made large compared to the pump, the changes in reflected

probe power could potentially be larger than the control signal itself – in effect creating

gain in the reflected signal. Figure 6.4 demonstrates this effect, showing gain close to a

factor of 2. It was difficult to obtain larger gain with the OCOG-1 device by simply

turning up the probe power. The percentage change in reflectivity induced by the control

signal dropped at such high probe powers; this is likely due to self-screening effects that

are negligible at lower powers.

6.2 SECOND GENERATION OCOG-26.2.1 Small signal experimental results and simulation

The expectation of OCOG-2 devices was that the use of separate control and

modulator sections, accomplished by using a dual-diode structure, would result in both

faster turn-on and turn-off of the device. As will be shown, this was indeed the case.

Moreover, the OCOG-2 was used to further test our models of how diffusive conduction

behavior affects voltage decay. Small-signal data, data for different spot sizes, data that

examined the off-center response of the device, and large-signal behavior are presented

and discussed in this section.

116

Control

Distributed BraggMirror (DBR)

1 �mQuantum

WellStack

300 �mSignal

0.1 �m50 nm

Top “Control” Diode

Bottom “Modulator”

Diode

i

p

i

n

ni

Figure 6.5: Schematic diagram of OCOG-2, an n-i-p-i (MQW)-n device structure on top of a DBR mirror.

The dual-diode OCOG-2 we used was an n-i-p-i(MQW)-n structure atop a DBR

mirror. Details of the MBE-grown OCOG-2 device are as follows: a DBR mirror

centered at 855 nm at room temperature composed of 25 periods of alternating

Al0.08Ga0.92As and AlAs was grown on top of an n-doped GaAs substrate, followed by a

short “cleaning” superlattice (30 periods of alternating layers, each 20 Å thick, of GaAs

and AlAs) which provided a smooth surface following the DBR growth. The

bottom-most Al0.08Ga0.92As n layer was 5,000 Å thick and doped at about 1018 atoms/cm3,

followed by 69 MQWs with 50 Å AlAs barriers and 95 Å GaAs wells. On top of this a

1.2 �m 1018 cm-3 p, 0.3 �m i, and 500 Å 1018 cm-3 n region of Al0.08Ga0.92As each were

successively grown (including a final n-doped 50 Å GaAs cap layer). Finally, an

antireflection Si3N4 layer was sputtered on top. 4-point probe measurements of the

resistivity (resistance per square) indicated the bottom n layer to be ~10 �� while both

the p and top n layers were each ~300 ��, as had been designed. The contacts to the

doped layers were used to separately reverse bias the top and bottom diodes at -5.0 V

and –15 V, respectively.

The temporal device response to a control pulse was extracted by the previously

described pump-probe technique. A tunable short-pulse laser (Tsunami) was used to

provide ~80 fs pulses at 855 nm for the probe pulse while the control pulse was

frequency-doubled to 427 nm using a BBO crystal. This enabled the control beam to be

117

fully absorbed in the top diode while the wavelength of the signal pulse allowed it to

experience the voltage-sensitive absorption of the bottom diode. Recording the

reflectivity consequently provides an indirect measurement of the voltage dynamics of

the top diode.

0 5 10 15 20 25 30

0.0

0.2

0.4

0.6

0.8

1.0

Sim

Data

Nor

mal

ized

Cha

nge

in R

efle

ctiv

ity (

a.u.

)

Time (ps)

Figure 6.6: Small-signal OCOG-2 device behavior and simulation (0.66 pJ). The close fit between thesimulation and data is strong support for a simulation model based on diffusive conduction and inducedvoltage change across the bottom diode.

Small-signal response is presented in Figure 6.6. Simulations (detailed below)

match the data well, supporting both the premise of induced voltage change across the

bottom modulator diode and a turn-off response due to diffusive conduction.

6.2.2 Further tests of diffusive conductionWe conducted two additional experiments to test our hypothesis that the voltage

decay was indeed due to diffusive conduction. For the first test, the small-signal response

was taken for a variety of different spot sizes. Recalling that the value of the diffusive

time constant, �, depends on the square of the spot size radius, w0, for a Gaussian pulse,

as expressed in Eq. 6.1:

118

20 1

42 voltagediffusion

wD

�

� ��

(6.1)

it was expected that the recovery time of the OCOG device would diminish as the spot

size shrank. The data was compared to a series of small-signal simulations in which the

spot size was varied. The results are presented in Fig. 6.7. The close fit between the

simulation and data lent strong support to the diffusive conduction hypothesis. If the

recovery had been due to relaxation through the external circuitry, for example, there

should have been no dependence on the spot size of the incident light.

0 5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

Spot Size Radius

2.6 �m

5.0 �m

7.0 �m

13.0 �m

Nor

mal

ized

Cha

nge

in R

efle

ctiv

ity

Time (ps)

Figure 6.7: Comparison between data and simulation of OCOG-2 dynamics for a different values of theradius of the incident light pulse. The close fit between data and simulation is strong support for thehypothesis that recovery is based on diffusive conduction.

The second additional test of the diffusive conduction response examined the off-

axis response of the device. For almost all of the data presented in this dissertation, the

probe pulse was centered at the center of the incident pump pulse (r=0). Its reflectivity

was primarily determined by the voltage at spot center. To investigate off-center

response, the probe pulse was laterally displaced from the pump pulse. Equation (6.2)

expresses the radial dependence for a single-layer (OCOG-1) device.

119

� ��

2

, exp4M

rV r t Vt D t�

� �

� ��

(6.2)

The off-center, 0r � , response of diffusive conduction is distinct, as Eq. (6.2)

shows.i As the distance from the spot center increases, the initial rise-time of the voltage

shielding increases; it takes time for the initial shielding voltage to diffuse, increasing the

shielding at distant points. The turn-off time is also much longer; once the voltage

shielding does rise, its spatial gradient is much less (since it has already diffused from the

center of the spot), and so its decay is slow.

In a two-layer (OCOG-2) structure, off-center behavior is more complex. There

are two eigenmodes: eigenmode ( , )r t��

is the sum of the voltages across the top and the

bottom layers (the overall voltage across the device) while the other eigenmmode,

( , )r t��

, is the difference of the layers’ voltages. With highly conducting top-most and

bottom-most layers, the overall voltage ��

rapidly decays to near zero (the top and

bottom layers become oppositely biased while the longer decay time of ��

roughly

describes the voltage relaxation across a single layer. Each voltage eigenmode of the

system decays according to Eq. (6.2), dependent on its diffusion coefficient (eigenvalue)

– see Eq. (4.50). The voltage across a particular layer of the device decays as a weighted

sum of the eigenmodes.

As photogenerated carriers separate in the top layer, the voltage across that layer

is shielded. If such a voltage change were, instead, evenly distributed across both layers,

the reverse bias across the bottom diode would increase due to the reversed orientation of

its p and n layers compared to the top diode. On the other hand, if the overall voltage

change was zero, voltage shielding across the top layer would be mimicked by the

opposite voltage across the bottom layer, reducing the bottom layer’s reverse bias.

What happens in practice? (1) As photogenerated carriers separate in the top

layer, the overall voltage across the device does change, though it quickly relaxes due to

the highly conducting top-most and bottom-most layers – in essence, due to the fast

diffusion coefficient of ��

. At spot center, the overall voltage change is due entirely to

i OCOG-2 is a multilayer structure. Although Eq. (6.2) does not account for the multilayer behavior, itsradial dependence exhibits qualitatively the expected behavior.

120

the voltage across the top layer and so the voltage across the bottom layer never

increases. Off-center, however, that is not necessarily the case… (2) Because the

diffusion coefficient of the overall voltage (��

) is faster than that of the voltage across

the indivudual layers (��

), off center the overall voltage change rises faster than the

voltage in the top layer alone. The voltage across the bottom layer, consequently,

increases so that the sum of the top and bottom voltage changes equal the overall voltage

change. (3) As the overall voltage relaxes off-center (after it builds up), the top and

bottom layers become biased in opposite directions and grow as ��

continues to diffuse

into that region. (4) Eventually, ��

also decays and the change in voltages across both

the top and bottom layers drop to zero. These four steps have been illustrated in Fig. 6.8.

(1)

(2)

(3)

(4)

On spot center Off spot center

Overall voltage change

Voltage change in top layer

Voltage change in bottom layer

Figure 6.8: Illustration of on-center and off-center behavior in an OCOG-2 device at four instances in time.(1) voltage change due to photogenerated carrier separation, (2) shortly after carrier separation in which theoverall voltage has begun to significantly diffuse outwards (rising at off-center points), (3) overall voltagehas effectively fully diffused but voltage across individual layers is still present, decaying much more

121

slowly, (4) voltage across individual layers significantly decayed. Note that at time (2) for some off-centerpoints the induced voltage change is temporarily opposite to what might otherwise be expected.

The results of the collected data and simulation for OCOG-2 are presented in

Fig. 6.9. The simulations were made using the small-signal, multilayer simulation model

described in Chapter 7 that was modified to provide the response for off-center points.

Both the “turn-on” and “turn-off” times clearly increase as the probe pulse is further

separated from the pump. The “dip” in the induced voltage for off-center points is the

manifestation of the “reverse” shielding we expect to see in the bottom layer of the

OCOG-2 device as described above. The presence of this dip in both the simulation and

data is particularly strong evidence that the model for multi-layer diffusive conduction is

a good description of the physical behavior of the device. As Fig. 6.10 reveals, this “dip”

occurs at least to some degree for most off-center points outside of the incident spot.

0 20 40 60 80 100-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Curve

1

2

3

4

5

Control Spot

Probe Spot

5

4

3

2

1

Spot Separation

Data Simulation

5 �m 5 �m

10 �m 7.5 �m

15 �m 15 �m

17 �m 20 �m

20 �m 25 �m

Cha

nge

in V

olta

ge (

a.u.

)

Time (ps)

Figure 6.9: The measured (red) and simulated (black) device response for various separation distancesbetween the probe and control pulses. As the spot overlap diminishes, testing device response for off-axispoints, both the “turn-on” and “turn-off” slow.

122

-20 -10 0 10 200

5

10

15

20

25

Position (�m)

Tim

e (p

s)

0.19 -- 0.20 0.18 -- 0.19

0.17 -- 0.18 0.16 -- 0.17

0.15 -- 0.16 0.14 -- 0.15

0.13 -- 0.14 0.12 -- 0.13

0.11 -- 0.12 0.10 -- 0.11

0.09 -- 0.10 0.08 -- 0.09

0.07 -- 0.08 0.06 -- 0.07

0.05 -- 0.06 0.04 -- 0.05

0.03 -- 0.04 0.02 -- 0.03

0.01 -- 0.02 0 -- 0.01

-0.01 -- 0 -0.02 -- -0.01

-0.03 -- -0.02 -0.04 -- -0.03

-0.05 -- -0.04

Figure 6.10: Simulation of the spatial and temporal response of the bottom layer in an OCOG-2 device.The white central spot is the peak of the change in voltage greater than the normalized value of 0.2. Thepurple contours indicate where and when the reverse bias temporarily increases. Simulation parametersincluded a pulse length of 2 ps, a spot size of 5 �m, three resistive layers with R=[300; 300; 10] �� andtwo capacitive layers with C=[0.3; 0.1] fF/�m2.

123

-5 0 5 10 15 20 25 30 35-0.005

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

Pulse Power

6.0 pJ

5.3 pJ

4.6 pJ

4.0 pJ

3.3 pJ

2.6 pJ

2.0 pJ

1.3 pJ

0.66 pJ

Abs

olut

e C

hang

e in

Ref

lect

ivity

Time (ps)

Figure 6.11: OCOG-2 response to large signal control inputs. Device tested with probe (signal) pulsepower of 12 �W, 5 �m spot size radius, and –5 and –15 V bias across the top and bottom diode,respectively.

6.2.3 Large signal experimental results and simulationsThe response of the OCOG-2 device to larger control pulse signals is presented in

Fig. 6.11. At low energies (e.g. 0.66 pJ), as the pulse energy rises the magnitude of the

reflectivity change increases in a correspondingly linear fashion. As the pulse energy

continues to increase, however, the p-i-n recovery response begins to deviate from the

hyperbolic form, the �/(�+t) component of Eq. (6.2), of purely diffusive conduction-based

decay. The magnitude of the change in reflectivity increases at a slower and slower rate

with respect to increasing power, flattening out at the peak, and decays progressively

more slowly. At even higher powers (e.g. ~5.0 pJ) the magnitude of the change in

reflectivity clearly reaches a limit. Decay is quite slow and has a completely different

functional form than diffusive conduction. Turn-on time is nearly constant with

increasing control power. Actually, we believe the p-i-n response at large signals was

sensitive to the very short absorption length of the pump pulse (due to the short

wavelength we used). If instead, for example, photogeneration was approximately evenly

124

distributed across the intrinsic region (e.g., by using a wavelength much closer to the

bandedge), we expect that the initial voltage shielding (turn-on) time would shrink with

increasing incident power.

The fit between the data and simulation in Figure 6.12 provides support for the

following hypothesis describing the large signal response. When large control energies

are used, the magnitude of the overall reverse bias (applied bias plus the “built-in”

voltage) becomes a critical factor. If there are enough electrons and holes, as they

separate the shielding electric field they create rapidly reduces the local electric field to

near zero. As a result, carriers in that low-field region slow down and may stop drifting.

0 5 10 15 20 25 30

0.0

0.2

0.4

0.6

0.8

1.0

5.3, 5.5 pJ6.0, 5.8 pJ

4.0, 4.8 pJ

4.6, 5.2 pJ

1.3, 2.3 pJ0.66, 1.0 pJ

2.0, 3.1 pJ

2.6, 3.7 pJ

3.3, 4.4 pJ

Pulse Energy

Nor

mal

ized

Cha

nge

in R

efle

ctiv

ity (

a.u.

)

Time (ps)

Figure 6.12: Normalized change in reflectivity as a function of time for various control pulse powers for aspot size radius of 5 �m with –5 V and –15 V reverse bias applied across the top and bottom diodes,respectively. Values of the actual (left column) and simulated (right column) incident power are presentedin the figure’s legend. Simulation parameters included resistances per square of 300, 300, and 10 ��forthe top n, p, and bottom n layers with top and bottom layer intrinsic regions of 0.3 and 0.1 fF/�m2.

Carriers at the edge of the shielded region still see a largely unscreened field and

continue to drift. As they move, the shielded space expands in their wake. The result is

still a rapid “turn-on” but followed by a much slower decay as many of the carriers

become trapped in the intrinsic region. Two ongoing processes help alleviate this

125

situation. One is regular carrier diffusion in the z direction. The other process is

diffusive conduction. Diffusive conduction constantly eats away at the screening voltage.

As it does so, the electric field in the intrinsic region grows, allowing carriers to continue

to drift. As the carriers move, however, they once again shield the field and the process

repeats itself. The carriers move slowly but, eventually, they are completely extracted

from the intrinsic region. After that point, any remaining voltage build-up decays away,

limited only by diffusive conduction. At sufficiently large control energies, the overall

voltage (and hence reflectivity) becomes proportional to the percentage of the intrinsic

region that is still fully shielded. In this regime it is the voltage decay, and not just the

“turn on” that is limited by how quickly the carriers can be extracted from the intrinsic

region.

The above description and explanation make physical sense and the simulations

exhibit qualitative behavior similar to the device response. However, with the larger

intensities, the simulations do not match the data well: at longer times the modeled

voltage build-up decays away too quickly while at short times (roughly between 3 – 10

ps) the “dip” in the initial build-up response is not reproduced. The most likely reason

these effects occur is due to the simplification of only modeling the response of the

device at the center of the incident light. The off-center, 0r � , response of diffusive

conduction is distinct, as described in Section 6.2.2. The off-center small-signal response

is presented in Fig. 6.9. How off-center behavior becomes important with large signals is

addressed below.

With large incident light intensities, the voltage change in the center quickly

“saturates” (the device is fully shielded). The voltage shielding at off-center points,

however, continues to rise until they, too, saturate. Due to this non-linearity, the initial

Gaussian shape of the voltage gradient distorts, flattening out with increasing intensities

for large signals. This reduces the voltage diffusion speed. Additionally, the probe pulse

has a finite spot size (equal to the spot size of the pump pulse), and so it samples on-

center as well as off-center points. As expressed in Eq. (6.2), the response of off-center

voltage decay slows with increasing radial distance. Taken together, when pump

intensities are high, the resulting reflectivity measurement shows a slower decay due to

its off-center behavior. Modeling these effects would require full three-dimensional

126

analysis instead of the simplified two one-dimensional equations used here

As the incident power increases, the time needed for the change in reflectivity to

crest grows (and a ‘dip’ in reflectivity also develops at short times). This, too, is due to

the sampling of off-center points by the probe beam. As the distance from the spot

center increases, the initial rise-time of the voltage shielding lengthens; it takes time for

the initial shielding voltage to diffuse, increasing the shielding at distant points. Thus,

off-center points take longer to reach their maximum voltage change and, because the

probe beam samples them as well, the overall change in reflectivity peaks at a later time.

6.3 THIRD GENERATION OCOG-3The OCOG-3 consisted of a top p-i-n diode and a bottom p-i(MQW)-n diode on

the top of a DBR stack centered at 850 nm. The topmost p and intrinsic layers were only

50 and 100 nm thick, respectively, including a GaAs capping layer. Except for the

bottom diode’s intrinsic region (60 QWs, each consisting of 120 Å GaAs wells and 40 Å

AlAs barriers) and the DBR mirror, all other layers were Al0.3Ga0.7As. To increase the

conductivity of the top p-layer, 660 nm of indium-tin-oxide (ITO) was deposited. This

layer also acted as an anti-reflection coating.ii Device processing produced square mesas

300 �m wide on a side with wire-bonds to each of the four conducting layers of the

double-diode structure. This allowed independent biasing of each diode.

ii The ITO we used was 75% absorbing instead of an expected 5-10%. We believe this was due toprocessing difficulties and not inherent to the device operation. Consequently, in this paper we discountthe ITO absorption in reflectivity and power calculations.

127

BraggMirror

p

300�m

1�m

i

pi

Signal

Control

0.��m

Quantum WellStack

Top “Control” Diode

Bottom “Modulator”Diode

n

n

ITO

Figure 6.13: Schematic of OCOG-3, a p-i-n-p-i(MQW)-n device.

6.3.1 Small signal experimental results and simulationThe OCOG-3 devices performed as expected, providing faster switching times

while providing the potential for greater device functionality compared to OCOG-1 and

OCOG-2 devices. The close fit between simulation and data lends strong support to the

multi-layer theory of diffusive conduction as an appropriate model for OCOG behavior.

The significantly faster gate operation, in spite of thicker QW barriers and, hence, longer

escape times for electrons and holes, demonstrates a significant advantage of having

separate regions responsible for the gating and the electroabsorption shift functionality

that is provided by a dual diode structure. For small signal simulations, the diffusive

conduction response was calculated by the method described in Chapter 4. Resistance

per square and capacitance per unit area were the provided parameters. The result was

then convolved with the control pulse length and the carrier drift behavior in the top

diode (modeled separately) to determine the overall reflectivity response of the device.

Finally, that result was convolved with the signal (probe) pulse length to determine the

output as seen by the photodetector. This approach is similar to the Green’s function

methodology used elsewhere[13, 14], and is described more fully in Chapter 7.

128

-5 0 5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

Data

Simulation

Ref

lect

ivity

Cha

nge

(a.u

.)

Time (ps)

Figure 6.14: Small signal (70 fJ per pulse) response data and simulation of the optical gate with turn on-offtime with 2 ps control and signal pulses and spot size of 3.5 �m, demonstrating 13 ps full-width 10%maximum turn on-off time (horizontal, arrowed line).

Small signal (70 fJ switching energy) response of the device was tested using a

428 nm control pulse and an 857 nm signal pulse; the top diode was biased at –4.0 V and

the bottom diode biased at –2.7 V. The full-width 10% maximum turn on-off time was

13 ps for 2 ps pulses with a spot size of 3.5 �m. The theoretical simulations of the device

response are in good agreement with both sets of experimental data (i.e. both for

delta-function-like (very short) pulses and extended pulses in time), verifying the validity

of the multilayer diffusive conduction model. If the read-out pulse is deconvolved from

the results, the actual length of the time the gate is ‘on’ is just 10 ps.

6.3.2 Large signal experimental results and simulationLarge-signal device behavior is presented in Fig. 6.15. The spot size radius was

again 3.5 �m, while the switching energy was 1.5 pJ (39 fJ/�m2) using 2 ps pulses.

Excluding the parasitic absorption of the ITO (described above) and the top p-layer,

approximately a 2-to-1 contrast ratio was achieved with a change in absolute reflectivity

of about 30%. The optical gate opens and closes -- returns to 10% of maximum change --

FW10%M

129

within 20 ps; this time reduces to 17 ps when the read-out pulse is deconvolved from the

data.

-5 0 5 10 15 20 25 300.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70A

bsol

ute

Ref

lect

ivity

Time (ps)

Figure 6.15: Large signal response of the optical gate with 2 ps control and signal pulses and spot size of3.5 �m, demonstrating 20 ps full-width 10% maximum (horizontal, arrowed line) turn on-off time with a30% reflectivity change

The slower response of large signals likely stems from photogenerated carriers

fully screening the reverse bias across the top diode before they reach the doped layers,

slowing vertical carrier transport. This hypothesis is reinforced by the data shown in

Fig. 6.16a. As the power of the control pulse rose, the normalized device response stayed

constant until a critical value (e.g. one sufficient to fully screen to top diode’s bias) was

reached. Beyond that limit, the turn-off time significantly increases. As the voltage

applied across the top was diode decreased, making it easier to be fully shielded, the

magnitude of this critical power also decreased as expected. Moreover, as Fig. 6.16b

shows, as the power increased, the turn-on time decreased, indicating that when more

carriers were created the carriers did not need to move as far to produce a given voltage

shift (and that this shift was limited by the top diode reverse bias).

FW10%M

130

0 10 20 30

0.0

0.2

0.4

0.6

0.8

1.0

225�W

400�W

600�W

2000�W

Nor

mal

ized

Ref

lect

ivity

Time (ps)

0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

225�W

750�W

1350�W

2000�W

5000�W

Nor

mal

ized

Ref

lect

ivity

(a.u

.)

Time (ps)

Figure 6.16: Overall (6.16a, top) and initial (6.16b, bottom) OCOG-3 device response at a various input powersof a 100 fs, 5 �m radius pulse, with the top diode biased at –8.0V (top), and –1.0V (bottom).

131

1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

Increasing

Control Energy

Input Control Pulse Energy

'6.0pJ

'5.3pJ

'4.6pJ

'4.0pJ

'3.3pJ

'2.6pJ

'2.0pJ

'1.3pJ

'0.66pJ

Nor

mal

ized

Cha

nge

in R

efle

ctiv

ity

Time (ps)

Figure 6.17: “Turn-on” of OCOG-2 device response for various input powers of a 100 fs, 5 �m radiuspulse.

It is interesting to compare the qualitatively different turn-on responses of

OCOG-2 and OCOG-3 (Figs. 6.16b and 6.17). Although the turn-on time was reduced in

both devices as the input power increased, the improved faster response was much more

pronounced in the OCOG-3 device. This difference is likely due to the difference in the

thickness of the intrinsic region, 0.3 �m in OCOG-2 and 0.1 �m in OCOG-3, combined

with the very short absorption length, ~50 nm, of the control pulse light in both devices.

In the OCOG-2 device, most of the light is absorbed near one side of the device. Thus,

because the turn-on time will still be limited by the need for the carriers to travel across

the length of the intrinsic region, the turn-on time is only slightly dependent on the input

intensity. For OCOG-3 structures, however, the much thinner intrinsic region means that,

relatively, a significantly larger proportion of the intrinsic region directly absorbs

photogenerated carriers; to zeroeth order the photogenerated carriers are constant across

the device. In this case, a slight separation of the photogenerated holes and electrons will

fully shield the entire intrinsic region without needing to travel far at all.

132

6.3.3 Multiple pulse (small signal) experimental results and simulationFigure 6.18 shows the device’s response to four control pulses, each separated by

20 ps. Simulation results matched well, with the discrepancy likely due to imperfect

generation of matching pulse energies.iii This key result -- the ability of the device to

recover in a short time limited only by the fast diffusive conduction response, not the

external RC time constants -- is clearly evident. There was a slight increase in the base

reflectivity level for the later pulses due to the build-up of the decay of the previous

pulses. Simulations showed that this build-up of base reflectivity (critical if the device is

to be used as a modulator at these rates) leveled off to a manageable level not far from

what is already seen here.

0 20 40 60 80 100 120

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Data

Simulation

Ref

lect

ivity

Cha

nge

(a.u

.)

Time (ps)

Figure 6.18: Multiple-pulse, small-signal response of the optical gate with 20 ps pulse separation.

iii The beam splitter/retroreflector subsystem generated equal intensity pulses, but with varyingpolarizations. Although OCOG devices are largely polarization insensitive, the optical set-up usedpolarization-sensitive optical components. Hence, the energy of the pulses downstream from thesubsystem were attenuated to various degrees.

133

REFERENCES

1. Barry, D.M., C.M. Snowden, and M.J. Howes, A Numerical Simulation of HighSpeed GaAs Photodetectors in IEE Colloquium on Microwave Devices,Fundamentals and Applications (1988).

2. Schneider, H. and K.v. Klitzing, "Thermionic emission and gaussian transport ofholes in a GaAs/AlxGa1-xAs multiple-quantum-well structure," Phys. Rev. B, vol.38, pp. 6160-6165 (1988).

3. Larsson, A., et al., "Tunable Superlattice p-i-n Photodetectors: Characteristics,Theory, and Applications," IEEE J. Quant. Elec., vol. 24, pp. 787-801 (1988).

4. Fox, A.M., et al., "Quantum Well Carrier Sweep Out: Relation toElectroabsorption and Exciton Saturation," IEEE J. Quant. Elec., vol. 27, pp.2281-2295 (1991).

5. Capasso, F., K. Mohammed, and A.Y. Cho, "Resonant Tunneling ThroughDouble Barriers, Perpendicular Quantum Transport Phenomena in Superlattices,and Their Device Applications," IEEE J. Quant. Elec., vol. 22, pp. 1853-1868(1986).

6. Faist, J., et al., "Quantum Cascade Laser," Science, vol. 264, pp. 553-566 (1994).

7. Faist, J., et al., "Vertical transition quantum cascade laser with Bragg confinedexcited state," Appl. Phys. Lett., vol. 66, pp. 538-540 (1995).

8. Goossen, K.W., "Excitonic electroabsorption in extremely shallow quantumwells," Appl. Phys. Lett., vol. 57, pp. 2582-2584 (1990).

9. Feldmann, J., et al., "Fast escape of photocreated carriers out of shallow quantumwells," Appl. Phys. Lett., vol. 59, pp. 66-68 (1991).


11. Ricco, B. and M.Y. Azbel, "Physics of resonant tunneling. The one-dimensionaldouble-barrier case," Phys. Rev. B, vol. 29, pp. 1970-1981 (1984).

12. Macleod, H.A., Thin-Film Optical Filters. 2 ed (McGraw-Hill, New York, 1989).

13. Liu, H., High speed, low driving voltage vertical cavity MQW modulators foroptical interconnect and communication, in Electrical Engineering, (StanfordUniversity, Stanford, 2001).

14. Wang, H., et al., "Ultrafast Cross-Well carrier Transport in a Strained Multiple-Quantum-Well InGaAs-GaAs p-i-n Modulator," IEEE J. Quant. Elec., vol. 33, pp.192-197 (1997).

134

Chapter VII: Simulation Methods

The models used for the simulations that have been presented are described

below. Two general approaches were taken: a time-sequential finite difference model

was used when individual charge dynamics were of particular importance while a

Green’s function method was taken advantage of for small-signal simulations where the

different dynamic components of the switching behavior could be modeled independently

of each other. Both approaches allowed fast computation of simulation results.

7.1 INTRODUCTION

Carrier dynamics may be described by a combination of Poisson’s equation,

� �a dq n p N N�

��

E (7.1)

where n, p, Na, and Nd are the free electron, free hole, fixed negative charge, and fixed

positive charge densities, respectively where each is a function of ( , )r t� , and the

continuity equations,

n nn G Rt

��

�nj p p

p G Rt

��

�pj (7.2)

in which Gn is the generation/escape rate, Rn is the recombination/trapping rate, and nj is

the current density, for electrons; there are equivalent definitions for holes. In order to

describe the behavior of a particular device, a judicious choice in determining how to

express the functions in these equations and in how to solve them can provide a model

that is accurate yet requires only a short computation time. For OCOGs or similar p-i-n

devices, certain details, such as the use of small spot sizes compared to the area of the

device face and short pulses of light, enable many simplifications to be made. Three

different models of OCOG behavior were used to simulate device responses shown in

Chapter 6. The model used for OCOG-1 combined a time-iterative method to calculate

the vertical motion while using boundary conditions for the overall voltage between the

doped layers that were determined by diffusive conduction. From this format the two

other simulation methods branched, both of which made use of diffusive conduction

135

response functions for multi-layered devices. In one method, a detailed finite difference

approach was used to model the large-signal response of OCOG-2. The other method

was based on convolving the various impulse response functions of the device (a

small-signal model), particularly efficient for quickly modeling multi-pulse behavior, as

was done for OCOG-3s.

7.2 OCOG-1: CHARGE TRANSPORT MODELING IN

P-I(MQW)-N DEVICES

It turns out that for OCOG-1 devices we were able to make several simplifications

that allowed us to dramatically reduce what could be a very complicated, coupled

three-dimensional calculation. Carrier motion in the intrinsic region could be

approximated by looking at only vertical (z) motion while carrier behavior in the doped

layers was governed by (lateral) voltage diffusion that was also expressible in a

one-dimensional form. The reflectivity of the device as a function of time was modeled

by combining a time-iterative program, which determined at each step the electric field in

the intrinsic region along the z-axis (at the center of the incident control spot), with an

experimentally measured dependence of the absorption as a function of electric field. We

describe these models in detail below

7.2.1 Governing equationsIn a p-i-n diode without multiple quantum wells, all the carriers are free to move

and the electron and hole current densities of Eq. (7.2) can be expressed with (three

dimensional) drift-diffusion equations, Eq (4.2):i

n nq n qD n�� nj E ∇ p pq p qD p�� pj E ∇ (7.3)

where � and D are the appropriate mobility and carrier diffusion coefficients either for

electrons or holes.

When quantum wells are present, however, transport in the intrinsic region is

more complicated. Before a population of photogenerated carriers in a particular

i Refer to Chapter 4 for a description of when the drift-diffusion equation is an appropriate description ofcarrier transport.

136

quantum well can move, they must first escape from the quantum well by tunneling,

thermionic emission, or field-assisted escape. [1-4] Tunneling and thermionic emission

times for single quantum wells can be reasonably approximated based on Eq. (7-9) of

Ref. [4]. For the 100 Å GaAs wells and 5 Å AlAs barriers of OCOG-1, the calculations

indicate that tunneling (on the order of picoseconds) should be significantly faster for

both electrons and holes compared to thermionic emission (greater than a nanosecond).

However, the close spacing of the quantum wells suggests that tunneling actually

involves several quantum wells, and their combination might alter (reduce) the likelihood

of tunneling.[5] To account for such an effect we used a tunneling-resonance program

based on the specific QW structure of the device.[6] This simulation provided the energy

resonance widths (for a particular electric field) of the lower electron and hole levels.

These widths may be converted in lifetimes, or in this case escape times, using the

Heisenberg uncertainly principle: /tun FWHME� � �� (See Appendix C for more details).[7]

The results showed that though the effective tunneling times increase (~2 ps for electrons

and ~20 ps for holes with zero electric field), they are still much less than for thermionic

emission. Thus, the overall characteristic escape time for carriers (n for electrons, p for

holes) may be expressed as , ,n p n pescape tun� �� ; the trapped population of a QW decays as

,exp( / )n pescapet �� .

At the same time, free carriers have a chance of being recaptured by a quantum

well. The likelihood of this exponential process has a time constant of ,n ptrapping� . Carriers

trapped in this manner, however, may be more energetic than the initially photogenerated

carriers and can reside much closer to the top of the quantum well, particularly in deep

wells. Escape of these recaptured carriers, consequently, can be modeled by a separate

characteristic escape time, ,n pre escape��

. In our simulations, due to the ultra-thin but high

barriers we estimated that ,n ptrapping� =0.5 ps for both electrons and holes [3] but that

subsequent re-emission was even faster due to the higher energy levels into which the

carriers may be recaptured (as indirectly shown in Ref. [8]). Therefore, we did not model

the trapping and re-escape effects. The escape process -- and subsequent trapping and

re-emission processes had they been explicitly modeled -- were included as part of the

137

functions Gn and Rn of Eq. (7.2) for electrons (and in the similar functions for holes),

shown in Eqs. (7.14) and (7.15) below.

Carriers that have escaped are free to move due to drift and diffusion, Eq. (7.3).

This motion is primarily vertical since the voltage bias is along the z axis. We assumed

that, for the pulse energies we would use, the electric fields along the z axis in the

intrinsic region would remain large enough to ensure that motion due to drift was much

larger than that due to regular carrier diffusion. Regular carrier diffusion was therefore

not modeled.[9, 10]

Any voltage shielding which builds up due to this photogenerated carrier

separation smoothes itself out laterally across the device face as a result of diffusive

conduction described by Eqs. (7.4) and (7.5).[11] This voltage diffusion may be

equivalently described as “effective” lateral motion of the free carriers in the doped

layers. ii

2x yvoltage

diffusion

dV D Vdt

��

� (7.4)

V is the voltage due to shielding across the intrinsic region and

1voltagediffusion sq area

DR C

� (7.5)

where Rsq is the sum of the resistance per square of the p and n layers and the capacitance

per unit area is determined by the thickness of the intrinsic layer.

When the incident spot of light is small compared to the area of the device face,

voltage relaxation may occur on a picosecond time scale and is essentially independent of

the RC time constant of the external circuitry.iii If such an instantaneous surface-normal

light pulse is spatially Gaussian, V(r, t=0) is then given by:

� �2

20

, 0 exp/ 2M

rV r t Vw

� ��

� �(7.6)

where

ii Eqs. (3.1)-(3.5) have been reproduced here in Eqs. (7.4)-(7.7).iii The small overall voltage that builds-up across the entire device face does indeed relax through theexternal circuitry. For single pulse behavior this response may be ignored; when modeling manyconsecutive pulses or extended periods of time, this external relaxation should also be included.

138

20

2

TOTM

A

QVwC �

��

(7.7)

Here QTOT is the total photogenerated charge and w0 is the spot size radius. An analytic

solution to Eq. (7.4) is then given by:

� ��

2

, exp4impulse M

response

rV r t Vt D t�

� �

� ��

(7.8)

with2

0 142 voltage

diffusion

wD

�

� ��

(7.9)

The effective lateral carrier motion in the doped layers due to voltage diffusion is thus

several orders of magnitude faster than the lateral motion of carriers in the intrinsic

region which is primarily due to regular (ambipolar) carrier diffusion; voltagediffusion

D >> Dambipolar.

In many devices the thickness of the intrinsic layer is less than or equal to one or

two microns while the spot size of the incident light is several microns in diameter. As a

result, the relative time for photogenerated carriers of a pulse of light to escape and

vertically separate to the doped layers -- a few tens of picoseconds, and often much less --

before any significant (regular) lateral diffusion of the carriers in the intrinsic region

occurs. A simplifying assumption was therefore made: lateral carrier motion inside the

intrinsic region could be disregarded. Taking these ideas together, the device response

could be modeled by restricting carrier motion to be only along the z-axis except within

the doped regions where lateral voltage diffusion is also important.

To further simplify the calculations (particularly those related to diffusive

conduction), only the voltage at the center of the incident spot was modeled. The charge

density was determined by assuming that the incident power was evenly distributed

across an effective incident spot area of �w02/2. Also, the probe beam sampled

off-center voltages. This resulted in an indirect measurement of the voltage that was

smaller than the actual change at spot center. The correction factor was determined by

139

� �( , ) ( , ) 12( , )

pump probe

probe

V r V r r drd

V r r drd

� � �

� ��

��

(7.10)

As will be later described, this spot-center model may misjudge the device recovery time

of large signal inputs.

The thickness of the intrinsic region typically is significantly smaller than the spot

size width. Consequently, the charge density at the spot center may be approximated as

in an ideal parallel plate capacitor (no fringing field) so that the electric field due to the

charge density at a particular vertical location has constant magnitude between the doped

layers although its direction flips at the charge location. This is what allows the use of

the integral on the right-hand side of Eq. (7.11) below, significantly simplifying

Poisson’s equation.

A few other assumptions are also made:

∙ the incident pump and probe pulses have a Gaussian spot intensity distribution

∙ drift-diffusion equations are valid (e.g., we may ignore ballistic effects, temperature

gradients, etc.)

∙ recombination may be ignored (recombination times are typically on the order of

nanoseconds while the response of the devices considered here were 100 ps or less)

∙ the mobility of the carriers was kept independent of the electric field and carrier

density. We have adopted this commonly used assumption although it is not very

accurate [12-14]

Carrier motion within the device can thus be described using just the following few

equations. In the intrinsic region:

� �

( , )

( , ) ( , )

built in applied

bottom

top

E z t E E

q n z t p z t dz�

�

� �

��

�(7.11)

bottom and top refer to the bottom and top of the p-i-n deviceiv while n and p are the

non-steady state carrier densities. iv When used for multilayered devices such as OCOG-2, bottom and top refer to the bottom and top or thetopmost p-i-n layers

140

( , ) ( , )n n z nn z t G R j z t

t�

� � ��

( , ) ( , )p p z pp z t G R j z t

t�

� � ��

(7.12)

( , ) ( , ) ( , )n nj z t q n z t E z t�� ( , ) ( , ) ( , )p pj z t q p z t E z t�� (7.13)

1 1n initial recapturedinitial recaptured

n n

G g n n� �

� � � ��

� � � �(7.14)

1n trapping

n

R n�

� ��

� �(7.15)

In Eq. (7.14), g is the photogenerated carrier pair generation rate.

In the doped layers:

0

( 0, ) ( ') ( 0, ') 't

overall m impulseresponse

V r t V t V r t t dt� ��

� �� (7.16)

where Voverall(r=0, t) is the overall voltage shielding at spot center and impulseresponse

V is the

voltage diffusion impulse response function for the p-i-n photodetector portion of the

device. Due to these simplifications, the simulation time (on the order of a minute) is

short compared to a complete coupled 3D model.

7.2.2 Simulation implementationNext, we describe how this reduced set of equations, Eqs. (7.11-7.16), was used in

a computer program to model the response of an OCOG-1 device to a pulse of light. The

core of the simulation calculated the voltage across the device, at spot center, as a

function of time ( )V t . That information, combined with the location of charge carriers in

the intrinsic region, enabled the electric field as a function of z to be determined. Finally,

the reflectivity of the device was calculated using empirical data of the absorption as a

function of electric field. More specifically, a time-iterative model was used to simulate

the dynamic behavior of the OCOG device, as shown schematically in Fig. 7.1. After

determining the generating function of the photogenerated carriers and electric fields, the

141

primary loop of the simulation was initiated whose internal structure consisted of the

following sections:

∙ determining the new carrier distribution as a function of vertical (z) position; newly

generated, freed, and trapped carriers were included in this step

∙ determining the electric field as a function of z due to space charges

∙ determining the resulting overall voltage, incorporating lateral voltage diffusion

∙ determining the total electric field as a function of z

∙ determining the absorption as a function of z

At the end of each loop period the reflectivity of the device was calculated. Each of these

sections is described in detail below.

Initial carrier distributionand electric fields

New carrier positions

New electric fields (z)

Voltage diffusion

Reflectivity (t)

Adjustment for final electric fields (z)

Figure 7.1: Schematic flow-chart of time-iterative large-signal computer simulation

7.2.2.1 Generating Function

The time and spatial dependence of the generating function, g, of Eq. (7.14) was

determined based on the particular experimental setup we used. The absorption length of

the incident 850 nm light near the heavy hole absorption peak of the 100 Å wide GaAs

QWs is about 1 �m. Given the intensity and duration of the control pulse, at each time

step the number of new photogenerated carrier pairs as a function of vertical position of

the QWs in the intrinsic region can be calculated. In determining the magnitude of the

incident light in the intrinsic region, it is assumed that there was no absorption in the

142

doped Al0.33Ga0.67As layers and the QW barriers, that there is no top reflection from the

antireflection coating, and that the reflectivity of the underlying DBR mirror is 100%.

7.2.2.2 Carrier Distribution

At each time period, after accounting for newly photogenerated carriers for each

QW, the number of carriers escaping for the first time and number escaping that had

already been recaptured were determined based on Eq. (7.14) and added to the free

carriers present at that location. Similarly, the number of free carriers at a point that were

recaptured by a QW was also computed using Eq. (7.15), reducing the number of free

carriers. A summary of the values used are presented in Table 7.1. As mentioned, the

small time needed for a recaptured carrier to escape resulted in little difference in the

modeled behavior of an OCOG-1 between simulations which included the trapping and

re-emission of carriers and those which did not. However, the initial escape (tunneling)

time constants, escapen� and escape

p� , were critical to device performance. For the quantum

wells in our device, the large mass of holes kept escapep� relatively constant as a function of

electric field; on the other hand, escapen� was quite sensitive. For small signals the electric

field is relatively constant, but as the control pulse intensity rises the electric field across

the wells and barriers shrinks, increasing escapen� .[2, 15] We did not model the electric

field dependence of the escape time. Instead, as a rough approximation, we used escapen�

as an “effective” electron escape time, making it a fitting parameter for larger-signal data.

143

Table 7.1: OCOG-1 Key Simulation Parameters

Parameter ValueIntrinsic region thickness 1 �m

Number of quantum wells 94

Capacitance per unit area, C A 0.11 fF/�m2

Resistance per square, Rsq 2500 ��v

Applied bias voltage varied

Pulse energy varied

Pulse wavelength varied

Spot size radius 7 �m

Max. drift velocity, electron, maxnv 1.5 107 cm/s

Max. drift velocity, hole, maxpv 0.6 107 cm/s

Initial escape time, electron, escapen� varied

Initial escape time, hole, escapep� 20 ps

Once the number of free carriers was determined, they were subject to drift due to

the local electric field. Carrier mobility and drift over MQWs and superlattices has been

studied (for example, Refs. [16-18]), though it is difficult to deconvolute the results from

other MQW effects such as trapping. Moreover, intervalley scattering becomes

important when the Al concentration of the barriers is greater than ~0.4 (becoming an

indirect bandgap material), complicating the picture. For OCOG-1s, carrier velocity was

very roughly approximated by multiplying the maximum drift velocity, max,n pv , by a

field-dependent term, ,( )n pE� , close to what would be expected in bulk GaAs.vi For

electrons,

� �

[ / ] [ / ]

1

[ / ] [ / ]

[ / ]

2.5 0.0 0.4

( ) 0.3 0.7 1 0.4 0.4 1.0

0.5 1.0

V m V m

n V m V m

V m

E E

E E E

E

� �

� �

�

�

�

� ��

� � � � � ��

��

(7.17)

v 4-pt probe measurements were only available for the top (p) layer and indicated 470 ��. The idealexpected resistance of the n layer was expected to be on the order of 10 ��, for a total (p layer + n layer)resistance per square of roughly 480 ��. However, the n layer consisted of several AlAs regions in theDBR that could potentially result in defects, significantly reducing the mobility. A value of 2000 �� forthe n layer was used in the simulation to fit the data.

144

while for holes,

[ / ] [ / ]

[ / ]

0.125 0 0.8( )

1.0 0.8V m V m

pV m

E EE

E� �

�

�

� ��

�� (7.18)

The drift velocity of free carriers determined how far they moved over the course of a

time interval, thereby redistributing them. Care was taken to ensure that the carriers

stopped their vertical movement at the edges of the intrinsic region (the doped layers).

7.2.2.3 Electric Field (z)

Having modeled the vertical motion of the carriers, the electric field as a function

of z in the intrinsic region due to space charge effects was calculated next. The thickness

of the intrinsic region typically is significantly smaller than the spot size width.

Consequently, the charge density at the spot center may be approximated as in an ideal

(infinitely extended) parallel plate capacitor so that the electric field due to the charge

density at a particular vertical location has constant magnitude between the doped layers,

though its direction flips at the charge location. This is what allows the use of the

integral on the right-hand side of Eq. (7.10), significantly simplifying Poisson’s equation.

At each vertical location in the center of the spot, summing the electric fields from the

charge densities along the z-axis determines the total field at that point due to the space

charge. When combined with the built-in and applied fields, this process provides a good

approximation of the solution to Poisson’s equation.

7.2.2.4 Overall voltage changes due to voltage diffusion

At this point, vertical motion and field calculations have been accounted for. To

account for lateral voltage decay and thus lateral carrier motion, at each time period the

overall change in voltage at spot center due to vertical carrier motion at each time period

was recorded. The remaining voltage change from a particular previous time step equals

its original magnitude multiplied by the diffusive conduction impulse response,

impulseresponse

V (r=0,t) given by Eq. (7.8), as a function of the time that elapsed between that

vi A better fit is later used for OCOG-2 and OCOG-3.

145

specific time step and the current moment. The overall voltage at a particular time

period, Voverall, can be calculated by summing the remaining voltage change of each

previous time step and adding it to the constant built-in and applied biases (a discrete

form of Eq. (7.16)).vii

7.2.2.5 Total Electric field

The electric field as a function of z was determined next. The difference between

the overall voltage and the voltage due to the space charge in the intrinsic region is due to

the carrier density in the doped layers. These carriers in the doped layers are not

explicitly handled – their presence is implicitly assumed and modeled via voltage

diffusion. Nevertheless, they provide a constant electric field across the intrinsic region

that balances the left- and right-hand sides of Eq. (7.19) below.

, &overall carriers space charge, biasdoped layers intrinsic layer built in

V V V V�

� � � (7.19)

The total electric field as a function of z is thus the sum of the electric field due to these

carrier densities in the doped layers, added to that of space-charges and the bias fields.

7.2.2.6 Absorption

The simulation determined the absorption, �(z), by using a measured empirical

relation between the reflectivity and the voltage applied across the device. The effective

absorption coefficient’s dependence on electric field was determined by measuring the

reflectivity, R, of the signal pulse as a function of voltage. In doing so, we used the

simplifying assumptions that the sum of the absorption and the reflectivity summed to

unity (i.e. disregarding the small amount of light transmitted through the DBR stack) and

that the signal absorption occurred only in the MQW region. Thus,

vii In these calculations we assume that the capacitance per unit area between the doped layers is constant.This is, however, only an approximation. As the photogenerated carriers separate and move towards the nand p regions, they may build-up at the edges and accumulate inside the intrinsic region, increasing theeffective CA. This is particularly significant at large intensities and strong shielding when much of theintrinsic region may actually be filled with a high density of charged carriers. In such circumstances, thevoltage diffusion coefficient changes (shrinks). The voltage diffusion impulse response function, therefore,becomes dependent on the configuration of the charges which change over time. This may be handled in astraight-forward manner with a simple p-i-n structure but becomes significantly more complicated as thenumber of layers in the device increases. Ultimately, a full 3-D finite difference approach would need to beimplemented instead.

146

� �� LVRV

2ln�

�� (7.20)

where L is the thickness of the bottom intrinsic (MQW) region.

7.2.2.7 Reflectivity

Finally, the reflectivity of the device was determined by determining how much

of an incoming pulse would be absorbed as it passed through the MQW region, bounced

off the DBR mirror, and once again passed through the MQW:

� �0 exp 2 ( )bottom

reflectedz top

I I z z�

�

� � �� (7.21)

7.3 LARGE-SIGNAL MODEL: OCOG-2Modeling the response of OCOG-2 devices was simpler than modeling OCOG-1

devices in some respects; the OCOG-2 device operation did not depend on carrier escape

time from, and transport across, quantum wells since the control (top) diode made use of

a bulk intrinsic region rather that MQWs. Equations (7.14) and (7.15) simplified to just

nG g� and 0nR � (7.22)

The resulting behavior is similar to that of a p-i-n photodetector. The finite difference

model used to simulate the large-signal described below is similar in concept to the

related methods that have been used to model large signal photodetector dynamics.[12,

13, 19, 20] An interesting alternative based on the use of matrices, a ‘state space’ model,

has also been reported.[10]

Vertical motion of the carriers is due to drift and diffusion. For large-signal

response, regular carrier diffusion along the z-axis cannot be ignored. Although such

diffusion effects are small when the carriers are moving at saturated drift velocities, large

pulse intensities result in strong shielding that can significantly reduce the drift

component of vertical motion. As a result, the vertical carrier diffusion becomes an

important factor in carrier dynamics. Electron current density of Eq. (7.12) is therefore

expressed as:

n nq n qD n�� nj E ∇ (7.23)

The overall simulation model was similar to that used for OCOG-1. One

147

significant difference was that because regular carrier diffusion along the z-axis was not

ignored, the carrier densities in the doped layers whose magnitudes were controlled by

voltage diffusion, had to be explicitly included in the model. The vertical diffusion of

these carriers plays an important role in large-signal behavior. Another difference was

the use of a multilayered structure; there was strong voltage coupling between the control

and modular to structures. These and other changes are described in detail below as the

different sections of the simulation are presented.

7.3.1 Generating FunctionThe time and spatial dependence of the generating function, g, of Eq. (7.22) was

determined based on the particular experimental setup we used. The absorption length of

the incident 427 nm light in Al0.08Ga0.92As is short, about 33 nm.[21] The control signal,

as a consequence, was absorbed primarily in the top n-doped layer and the top of the

intrinsic layer beneath it. Given the duration of the control pulse, at each time step the

number of new photogenerated carriers as a function of vertical position in these top two

layers may be calculated.

It was important to account for the injection of photogenerated carriers from the

thin (50 nm) top n-doped layer into the intrinsic region. In our simulations this was

modeled by assuming that injection was due to thermionic emission.[22] The expression

for thermionic emission in a homojunction (appropriate for the OCOG device used here)

is rather simple since there is no barrier to prevent carrier motion. By assuming that the

relatively small number of photogenerated carriers created in the intrinsic region may be

ignored compared to the number created in the top n-layer due to the large absorption

coefficient, the thermionic current density is just

photogenththermionic nqvJ � (7.24)

where q is unit charge, nphotogen is the photogenerated carrier density, and the expected

thermal velocity in one direction (across the doped/intrinsic boundary) is given by

Cth qN

ATv2

� (7.25)

148

A is Richardson’s constant, T is temperature, and NC is the density of states. We can thus

write the impulse response of an instantaneous pulse that creates 0photogenn carriers.

� �0 expphotogen th thdn n v v tdt

� � (7.26)

in which �th=0.6x106 cm/s in our simulation. We use Eq. (7.26) to determine how many

photogenerated carriers initially enter the intrinsic region in the first “bin” near the

n-layer. From this point the carriers become subject to drift and diffusion, Eq. (7.23), as

described in the Carrier Distribution section below. The electrons tend to quickly move

back into the doped n layer while the holes cross the entire intrinsic region.

7.3.2 Carrier DistributionIn order to determine the new carrier distribution compared to the previous time

step, the numbers of photogenerated carriers present in the doped layers (excluding the

initial electron-hole pairs) were determined first. The voltage across the intrinsic region

due to carriers in the doped layers, ,carriersdoped layers

V was determined using Eq. (7.27). The

carrier number is proportional to ,carriersdoped layers

V divided by the capacitance of the intrinsic

region, Eq. (7.28).

,overall carriers space charge,doped layers intrinsic layer

V V V� � (7.27)

,doped layers carriersdoped layers

Q C V� (7.28)

Interestingly, it is possible to require “negative” carrier density in the doped

layers (i.e., a local reduction) if particular circumstances arise. For example, if

photogenerated charge carriers were to separate from, say, the center of the intrinsic

region until they are halfway to the doped layers (Fig. 7.2), there is a change in voltage

across the entire intrinsic region. If these carriers then stop moving, the field between

them due to their separation continues to affect the overall voltage between the doped n

and p layers. This change in voltage between the doped layers, however, diffuses

laterally away until the initial overall voltage is restored in spite of the ongoing shielding

due to the photogenerated carriers that we have kept fixed in the intrinsic region.

149

n pE(z,t=0)

n pE(z,t=t1)

Voverall=V0

Voverall=V0-�V

n pE(z,t=t2) Voverall=V0

free electrons due to doping

electrons and holesdue to photogeneration

free holes due to doping

Figure 7.2: Conceptual illustration of the behavior of a p-i-n device at a particular lateral (x,y) point inwhich voltage diffusion occurs results in the overall voltage between the doped layers recovering from thechange in voltage due to the separation of photogenerated carriers before those carriers have been swept outof the intrinsic region. The density of free carriers in the doped layers is reduced at that (x,y) point due tolateral voltage diffusion, creating an effect similar to image charges, until the photogenerated carriersthemselves actually reach the doped regions.

In this process, the final resulting number of carriers in the doped regions at this

vertical location is less than their starting values since some have laterally diffused away.

This “negative” carrier density compared to the steady-state situation balances the

shielding between the separated photogenerated carriers still in the intrinsic region so that

the overall voltage is the same as its initial value. What has occurred is that the free

carriers that result from doping the n and p layers have moved laterally away, behaving

analogously to a conductor with an image charge.

In our simulations, however, the free carriers due to dopants were not explicitly

modeled. As a result, when such “negative” carrier densities arose, the overall voltage

was properly accounted for, but the actual “excess” (photogenerated) carrier density in

the doped layers-- important for vertical diffusion calculations -- was assumed to be

simply zero.

It was also assumed that there was no vertical field inside the doped regions and,

therefore, these carriers did not experience drift; they did, however, diffuse. Vertical

diffusion within the doped regions was not explicitly modeled. Instead, it was assumed

that these electrons immediately equalized their distribution across the top n-layer

150

because it was so thin. In the p-layer, holes were arbitrarily assumed to immediately

evenly diffuse throughout the first 100 nm but then stop moving, even though that layer

was significantly thicker (1 �m). These assumptions determined the density of

photogenerated carriers at the edges of the intrinsic region. This information was

important for the later calculations of vertical diffusion of carriers into or out of the

doped layers. The relatively slow rate of hole transport to the p-region (unlike the rapid

electron transport due to the near negligible distance electrons had to move to reach the

n-layer) and the relatively fast voltage diffusion minimized the importance of the hole

concentration at the intrinsic/p-layer boundary. The arbitrary choice of 100 nm instant

diffusion was therefore not a critical detail. While inexact, these assumptions simplified

the modeling. More accurate models would explicitly include diffusion calculations

within the doped layers.

Next, carrier motion in the intrinsic region was determined with Eqs. (7.12),

(7.22), and (7.23). The drift velocity’s field-dependence for both carrier types was

modeled based on [23, 24].

( ) 1 expsathole hole sat

hole

Ev E vE

� ��

� �� (7.29)

4

4( )

1

sate elec peak

elecelec

peakelec

EE vE

v EE

E

��

� � ��

� ��

(7.30)

using satholev =0.7x107 cm/s, sat

holeE =2x104 V/cm, satelecv =1.1x107 cm/s, and peak

elecE =4x105 V/cm.

To determine the new positions of the carriers for a particular time step, n(z,t) and

p(z,t), a finite difference method approach was used to solve the vertical drift-diffusion

equations for the electrons and holes based on the Crank-Nicholson method. For

numerical stability of the simulation, a 50-50 average between explicit and implicit

151

FTCM (forward time-centered space) functions[25] was used with 10 fs time steps and

10 nm vertical spatial steps.viii

7.3.3 Overall voltage changes due to voltage diffusionAt this point, vertical motion and field calculations have been accounted for; the

lateral voltage decay and thus lateral carrier motion has not. To do so, Eq. (7.16) is used.

At each time period the overall change in voltage at spot center due to vertical carrier

motion at each time period, Vm(t), is recorded. The impulse response of the top n-i-p

layers was coupled with the behavior of the lower p-i(MQW)–n layers, precluding the use

of Eq. (7.8) for a simple p-i-n device as an expression for impulseresponse

V (r=0,t).ix Instead, the

top and bottom layers’ diffusive conduction impulse response functions,top

impulseresponse

V and bottomimpulseresponse

V , were calculated by solving the coupled multilayer diffusion equations

as outlined in Chapter 4.x Thus, the overall voltage for a particular layer, Voverall, for a

specific time period can be calculated by summing the remaining voltage change (using

the correct impulse response function for that layer) of each previous time step and

adding it to the constant built-in and applied biases (a discrete form of Eq. (7.16)).

In these calculations we simply assume that the capacitance per unit area CA

between the doped layers is constant. As the photogenerated carriers separate and move

towards the n and p regions, however, they may build-up at the edges and accumulate

inside the intrinsic region. This is particularly significant at large intensities and strong

shielding because much of the intrinsic region may actually be filled with a high density

of charged carriers. In such circumstances, the width of the intrinsic region shrinks, the

effective CA rises, and the voltage diffusion coefficient changes (decreases). The voltage

diffusion impulse response function, therefore, becomes dependent on the configuration viii The stability condition of the finite difference method is given by 1

v tz�

��

.[25] To avoid instability for

the parameters and step size we used electron vertical diffusion coefficients with values < 50 cm2/s wererequired. Although the electron diffusion coefficient is approximately 175 cm2/s, to ensure stability a valueof only 40 cm2/s was used.ix This coupling behavior is what is responsible for the induced voltage across the bottom diode. Theresponse of the top diode does not differ significantly from its isolated behavior except that the voltageshielding buildup magnitude is reduced (as if placed across a voltage divider).

152

of the charges, which changes over time. This may be modeled in a straightforward

manner for a simple p-i-n structure but becomes significantly more complicated as the

number of layers in the device increases. Ultimately a full 3-D finite difference approach

would need to be implemented instead.

7.3.4 Total electric fieldThe electric field as a function of z was determined in a manner similar to that for

OCOG-1 devices. For both top and bottom intrinsic layers, Eq. (7.19) holds sway, and

the total electric field as a function of z can be determined as before. Actually,

determining the voltage across any intrinsic layer except the top-most one is particularly

simple. Because no photogenerated carriers are present in these layers, no carrier motion

needs to be modeled nor are there any space-charge effects. The voltages present are

those due only to the voltage changes in the top diode and their resulting dynamic

coupling with the other layers as expressed by the impulse response functions.

Therefore, the electric fields in these layers are constant: the overall voltage across them

divided by the width of the intrinsic layer.

7.3.5 Absorption and reflectivityAbsorption, �(z), and reflectivity are calculated in the same manner as that for

OCOG-1 devices.

7.4 SMALL SIGNAL FFT MODELS: OCOG-3Modeling the small-signal response of OCOG-3 devices enabled significant

simplification of OCOG-2 simulations. The device response was assumed to be linear,

independent of history or signal intensity for small signals. As such, the electric fields

across the device were assumed to be constant. Additionally, as in OCOG-1 simulations,

regular carrier diffusion in the z direction was ignored. Four impulse functions, G(t),

R(t), D(t), and S(t), were convolved to model device behavior and are described below.

x To determine the voltage across the device, it is assumed that there is no field across the doped regionsthemselves in spite of local space charge effects.

153

This small-signal simulation method is analogous to a Green’s function approach that has

been used in p-i-n models elsewhere.[1, 14, 26-28]

G(t):

To determine the device behavior, first the intensity of the incident light as a

function of time, G(t), was determined. We assume that the incident light has a Gaussian

time evolution, parameterized by specifying the full-width half-maximum, �FWHM, of the

control pulse (see Fig. 3).

� �2

0 2exp/ ln(2)FWHM

tG t I�

� ��

� �(7.32)

It was also assumed that the spatial shape of the pulse was Gaussian, Eq. (7.6), whose

spot size (1/e2) radius, w0, was an adjustable parameter. This value was used later when

the diffusive conduction response was determined. Intensity dependence based on the

spot size was not modeled since small signal response was assumed to be independent of

the intensity.

Figure 7.3: G(t), the normalized incident light pulse that creates photogenerated carriers. In thissimulation, �FWHM=2 ps.

R(t):

As the control pulse was incident on the device, the light was absorbed during one

instantaneous moment – in a sense, a very short “time slice” of the control pulse –

creating photogenerated carriers that began to separate and shield the voltage across the

154

top intrinsic region. R(t) was the normalized change in voltage (rise-time behavior)

across the top diode as a function in time due solely to the separation of photogenerated

carriers of an arbitrary “time slice.” This function was calculated using a simplified

form of the OCOG-1 simulation. The parameters used were the overall reverse bias from

which the constant electron and hole drift velocities were determined, the intrinsic region

thickness, and, to determine the initial charge distribution, the absorption coefficient.

Fig. 7.4 provides a sample function for R(t).

Figure 7.4: R(t), the change in voltage shielding across the top intrinsic layer due to the absorption of ainstantaneous light pulse. Parameters used in this simulation were a 0.3 �m intrinsic layer thickness andthe absorption coefficient for 425 nm light (~2x105 cm-1) in the intrinsic region.

D(t):

Next, the voltage decay, D(t), of an instantaneous amount of new voltage

shielding was calculated by solving the multi-layer diffusion eigenvector problem

described in Chapter 4 and specifying for which layer in the structure the resulting

diffusive conduction time dependence was determined (such as the MQW bottom

intrinsic layer). For example, in a two-layer device D(t) is described by Eq. (4.48). For

this type of calculation, the resistance per square and the capacitance per unit area for

each layer of the device needed to be specified as well as the spot size radius. The initial

voltage distribution used was normalized to 1 V in the top layer and zero volts across the

other layers.

155

Figure 7.5: D(t), the voltage decay in the bottom layer of a two-layer device due to diffusive conduction ofa “time slice” of voltage shielding. Simulation parameters were: Rsq=[100; 1000; 10] ��,Carea=[0.3; 0.1] fF/�m2, w0=3.5 �m)

S(t):

The read-out probe (signal) pulse was assumed to have the same functional form

as G(t).

The voltage dynamics of an OCOG-3 device were calculated using the first three

of these temporal responses. For each “time slice” of the initial control pulse,

photogenerated carriers were created whose voltage build-up was determined. And each

“time slice” of voltage build-up decayed due to the calculated diffusive conduction.

Overall device response, therefore, was determined by convolving the temporal pulse

shape with the voltage build-up function and the diffusive conduction response. Finally,

since the data collected used a read-out (signal) beam, the temporal shape of this second

light pulse also needed to be accounted for. Simply convolving the device response with

the temporal shape of the read-out pulse, which was assumed identical to the control

pulse, provided the measured device behavior, as Eq. (7.33) expresses.

Voltage (t) = 0 0 0

( ') ( ' '') ( '' ''') ( ''') ''' '' 'S t t D t t R t t G t dt dt dt� � �

� � �� (7.33)

In the actual simulation program, the convolution was actually over a finite, albeit

relatively long, time period and is solved numerically, not analytically. By assuming a

1 V total initial buildup in the top diode, and using normalized laser pulse and voltage

156

build-up response functions (the integral of each was unity), the final output provided the

voltage as a function of time across the layer of interest.

Figure 7.6: Convolution of G(t), R(t), D(t), and S(t) from Figs. 7.3, 7.4, and 7.5 above, providing thesmall-signal voltage response of the bottom layer of a two-layer device.

Multiple pulses

Multiple control pulses were simple to model using this simulation. By

convolving the input control pulse with a series of time-separated delta functions, overall

device response to such burst-logic sequences was readily obtained as shown in Fig. 7.7.

The pulse repetition rate limit is strongly influenced by the decay tail, which falls off

much more slowly than an exponential response, particularly for just single or

double-layered devices. New pulses rest on the tails of previous ones

As the pulse repetition rate increases, the base level of the voltage across the

devices at spot center (the dashed lines) shifts and the relative contrast ratio -- the voltage

difference between the toughs and valleys -- shrinks. As each dotted line indicates, the

rise in the base bias voltage quickly reaches a limiting value for given repetition rate.

Pattern dependence thus may become important as the bit rate increases.

157

Figure 7.7: Small signal simulation of multiple pulses with various repetition periods. The simulationshave been offset for ease of viewing. The simulation parameters were the same as those used in Fig. 7.3 –7.4 except that the resistances here: Rsq=[10;100;10] ��. Spacing between pulses was 25, 12.5 and 8 psfor the bottom, middle, and topmost simulations, respectively. Note that as the spacing between the pulsesdiminished, the next pulse starts to rise before the previous pulse finishes. Consequently, the effectivecontrast ratio of a pulse shrinks and, as the dashed lines indicate, a voltage offset develops.

Off-center simulations

These small-signal simulations were also used to model the off-center response of

OCOGs (e.g., Fig. 6.8) To accomplish this, the impulse diffusive conduction function

(the multilayer version of Eq. (7.8)), D(t), was calculated using a specified radial

distance, �, from the spot center for the particular layer in the device of interest.

158

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160

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161

Chapter VIII: Discussion and Future Directions

Motivation to study and use OCOGs was initially established throughout this

study followed by an overview of a representative selection of different types of optical

devices. The intervening chapters presented a detailed description of optically-controlled

optical gates (OCOGs) and data showing low-powered 50 GHz burst-logic operation.

This brings us to our current state of knowledge. In this final chapter we draw the

conclusion of this work and discuss future possible directions.

8.1 ULTIMATE LIMITS OF OCOG DEVICES

We are now in a position to answer the questions that were posed in Chapter 1:

∙ What are the underlying physical principles that govern the operation of

OCOG devices and how do these principles constrain its performance?

∙ What switching energies are required?

∙ How fast can OCOGs operate and over what wavelength range?

These questions are addressed throughout this section.

There are four ways to reduce the gating time of OCOGs. (1) Reduce the

thickness of the photodetector intrinsic region. As this layer thins, the transit time for

photogenerated carriers diminishes, reducing the turn-on time of the gate. Moreover,

there is a reduction in the bias voltage needed to provide a particular electric field. Two

disadvantages arise, however. First, the break-down voltage is similarly reduced,

limiting the total possible change in voltage. Second, the thinner layer increases the

capacitance and so slows down the diffusive conduction recovery. (2) The device may be

grown with greater doping of the n and or p layers to reduce Rsq. This increases the

diffusive conduction decay, speeding up device recovery. However, diffusive conduction

also eats away at the voltage change as it is rising during the “turn-on” stage. The result

is a smaller overall change in voltage. (3) The spot size of the incident light can be made

smaller. Not only does this provide faster diffusive conduction (with the same pros and

cons as for a lower resistance), but at the same time it reduces the required energy needed

for switching – a significant advantage. The flip-side is that the slower, non-linear

large-signal response occurs at lower total incident energies. (4) The functional form of

162

the decay of diffusive conduction can be altered by using multi-layered devices. As

described in Chapter 4, the signature hyperbolic decay, �/(�+t), may be made to fall at

significantly faster rates as a function of time due to the coupling of the voltage behavior

between layers. The result can be a much sharper decay, allowing for simultaneously

lower-power and faster devices. The initial voltage is, however, divided among the many

layers. Consequently, the magnitude of the voltage changes is reduced – the price of the

faster decay. It should be noted that although the gating time can be significantly

shortened for particular layers, the overall repetition rate is limited by the response of the

slowest layer which may not improve as quickly.

There is a fundamental trade-off between switching times and switching energy.

In each of the above options, switching speed can be increased but only at the expense of

a reduction in the possible voltage swing. The answers to the questions of “How fast can

it switch?” and “How much will it cost (in energy)?” are thus entangled. Also, their

relationship is not always linear: i.e., reducing the time by a factor of two means that

more energy will be needed, but not necessarily twice as much. Many factors are

involved in this complex interaction. Establishing what constraints are most important

for a particular application can significantly limit the range of some of these variables and

simplify the optimization to be determined.

The practical gating speed of OCOG is constrained by three factors: (1) the

separation of the photogenerated carriers, (2) the diffusive-conduction relaxation of

voltage change, and (3) the pulse length of the optical beams. A reasonable system-level

restriction might be limiting control power to 10 mW, which corresponds to 100 fJ per

pulse at 100 GHz repetition rates. For significant absorption modulation, voltages on the

order of a few volts across a micron of QW material are typically needed. Spot sizes on

the order of 2 �m radius are obtainable, ultimately limited by the diffraction limit of the

light. Taken together, this suggests that the energy needed for OCOG switching at

100 GHz will require energy greater than about 10 fJ per pulse. This is significantly less

than the 100 fJ limit assigned above. However, this value does not account for diffusive

conduction decay even as the change in voltage is growing, nor the percentage of voltage

that will fall across other layers, nor parasitic light loss. Finally, the pulse length

available must be considered. Inexpensive short-pulse lasers with high repetition rates

163

are not yet commercially available. Given the current state of research, it is conceivable

that in the near future sources such as mode-locked fiber or on-chip lasers would be able

to provide pulses in the 1-2 ps range. This value is important when, for 100 GHz

operation, the total switching time must be on the order of 10 ps or less.

With these constraints, the challenge becomes designing a system in which the

diffusive conduction is able to close the switching window quickly while also allowing

the voltage to change significantly. In a proposed next-generation OCOG device,

OCOG-4, these constraints have been addressed in order to achieve useful ultra-fast

gating (see Fig. 8.1). The top intrinsic region is made thin so that the build-up time for

the voltage is small and multiple p and n regions are used to decrease the diffusive

recovery time of the bottom MQW region in an n-i-p-n-p-n-i(MQW)-n structure.

Although it comes at a price of low efficiency (the voltage shielding in the top layer is

divided among the many layers), the result is a 2 ps optoelectronic gate.

A small-signal simulation is presented in Fig. 8.1. The change in voltage of 3 V

across a 1 �m MQW region is obtained with a switching window that is 7 ps FW10%M

(3 ps FWHM), sufficient for 100 GHz operation. The energy required for this operation

is predicted to be 150 fJ (not including parasitic light-loss), which is close to the 100 fJ

goal. It was important in designing this structure to restrain the voltage diffusion

enhancement of the middle layers so that the voltage was not eaten away too quickly.

Obtaining a large capacitance (20 fF/�m2) with highly resistive layers (20 k��) was

essential to achieving this object but not easy to accomplish. Large capacitance is

possible with very thin depletion regions. However, a thin depletion region requires high

doping of the p and n layers, providing highly conductive instead of resistive layers.

These competing effects practically cancel each other, making large RC p-n junctions

difficult to grow. Future studies may try to accomplish this by using low-mobility

material such as Al0.3Ga0.7As in AlxGa1-xAs systems for n-doped layers. Deliberately

disordered material may also satisfy these opposing requirements.

164

0 5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

7 ps FW10%MCha

nge

in R

efle

ctiv

ity (

a.u.

)

Time (ps)

Figure 8.1: Small-signal simulation of next-generation OCOG device. This is ann-i-p-i-p-i-p-i-p-i-p-i (MQW)-n structure with paramaters of R=[300; 5 k; 20 k; 20 k; 20 k; 20 k; 10] ��,C=[0.6; l2.0; 2.0; 2.0; 2.0; 0.1] fJ/�m2 using a 3.5��m spot size radius. The FW10%M=7 ps.

Pushing for even faster gating times without significantly increasing the energy

for switching will likely prove to be quite difficult for OCOG devices. Spot size will be

constrained by diffraction effects, turn-on time is limited by the speed that electrons and

holes can separate (about 0.1 �m per picosecond), and material constraints might limit

the resistance and capacitance available for the layers. But it is possible that more

carefully tailored coupled layers may be able to provide some relief to the required

switching energies, and absorption-sensitive layers may emerge that require significantly

less electric field changes to alter their absorption and hence the signal pulse.

The wavelength range of OCOG devices is typically on the order of 10 nm, the

shift of exciton peak with the change in electric field. This value depends heavily on the

specific QW design used. Placing OCOGs within a resonant structure, such as an

asymmetric Fabry-Perot cavity, could be used to significantly reduce its control energy

165

but at the price of a reduced wavelength operating range and more complex design.i

At sufficiently large input control pulse energies, the device response deviates

from its linear response: the change in reflectivity approaches a limit and the switching

time slows significantly. As described in Chapter 6, if the incident power is sufficient to

significantly shield the reverse bias in the control diode, device recovery slows due

primarily to the reduction of the vertical drift current. The pulse energy at which these

effects begin to dominate device response determines the dynamic range of the input

control pulse and is strongly dependent on the spot size (energy density), absorption

length, and the magnitude of reverse bias that can be applied without breakdown.

The signal energy is also important to consider. Although it is possible to use

signal energies larger than the control pulse energies (recall the effective gain of the

OCOG-1 device, demonstrated in Fig. 6.4), the absorption of the signal pulse creates

photogenerated carriers in the MQW modulator diode. As these carriers separate, they

shield the reverse bias in the modulator diode section that may effect subsequent signal

pulses. This behavior tends to limit the signal energies. The particular pulse energies at

which this effect becomes dominant and the resulting changes in switching time depend

on the specific sweep-out and voltage diffusion dynamics of a particular device. In

practice, we found it difficult to achieve greater than a gain of a factor of 2 (signal

energies about 5 times the control energies) while the fastest results were obtained with

signal energies typically one-tenth the control energy.

Ignoring the matter of more practical switching devices, it is interesting to also

consider the fundamental limitations of diffusive conduction itself. This issue was

discussed in Chapter 4; the conclusions are reiterated here. When a large number of

photogenerated carriers are created within a small spot size and separate, creating a

shielding voltage with a large lateral gradient, large lateral electric fields are present. In

such circumstances the voltage diffusion tends not to be as large as expected. Voltage

diffusion also slows, limited by the thermal velocities of the carriers, if the lateral device

dimensions are small (e.g., approximately 0.1 �m) or if the diffusive conduction time

constant, �, is shorter than the momentum scattering time. Finally, if strong magnetic

i There have been some clever designs in surface-normal optical switches that use asymmetric Fabry-Perotresonances but which manage to have a wide operating wavelength range.[1]

166

fields are created due to large electric fields and fast diffusion times, voltage decay

becomes LC limited, analagous to a 2D lossless transmission line -- the voltage decay is

limited by the speed of light in the medium.

8.2 HOW DO OCOGS COMPARE TO OTHER OPTICAL

SWITCHES?This dissertation has provided a firm background for understanding the

capabilities of OCOG devices. With this knowledge we can more fully consider in what

practical systems, if any, OCOGs should be used and how OCOGs compare to other

optical switches in this regard. As described in Chapter 1, data in today’s world is

commonly transferred across long distances encoded as optical bits in fiber optic cable.

The bit rate of individual data channels in a fiber has steadily risen over the past decade.

Over 1 Tbps has been demonstrated for a single channel, as has 20 wavelength channels

each at 160 Gbps and 160 wavelength channels each at 10 Gbps, as Fig. 8.2 (excerpted

from Ref. [2]) shows. The current section addressed the following question: What role is

there for optical switches in a global network of interlaced optical datastreams?

When one endpoint in an optical network sends data to another endpoint, the

system that enables this communication provides two functions. (1) First, the system

routes the data from one endpoint to another. If necessary, the characteristics of the

bitstream may be altered to facilitate this service. For example, the bit rate of the data

may be changed, the wavelength of the data shifted, or the fiber down which the data

travels may be switched. (2) Second, the system ensures that the data sent is still in a

readable form by the time it reaches its destination. Distortion, attenuation, and

cross-talk among other factors culminate to reduce the signal-to-noise ratio of the data

stream. The system must provide 3R functionality (regeneration, reshaping, and

retiming) to maintain the datastream’s viability. To date, all aspects of these two

functions have been handled electronically using OEOs. As mentioned in Chapter 1, the

logic required for determining where to direct an incoming data signal -- the routing

look-up -- is complex and is likely to remain as an electronic process for the foreseeable

future. For many of the other functions that the network management system provides,

167

however, the required logic is minimal and optical switches may challenge traditional

electronics. It is not hard to envision a network in which a datastream, once sent, remains

in an optical form from start to finish. Whether or not this is a vision of the near or

distant future remains uncertain.

Figure 8.2: Reported demonstrations of data transmission in a fiber optic cable. The right-most (bolded)diagonal line represents transmission with only one data channel (wavelength) in the fiber. Moving to theleft in the chart, the number of data channels per fiber increases, typically with reduced bit rates perchannel. The total bit rate per fiber is represented by the y-axis.

The most likely point-of-entry for optical switches is use in restoration switch

fabrics; indeed, this has already begun to occur commercially. Restoration networks

require the ability to redirect tens or hundreds of light beams from fiber optic cables,

whether single or multi-wavelength channels, in case a fiber breaks or needs repair. They

are also used to adjust to slow changes in traffic patterns by reprovisioning network

bandwidth so that it is efficiently allocated. These have been the prime motivators for the

use of optical MEMS switches in telecommunications networks even though their

switching times are relatively slow (microseconds to milliseconds) compared to other

switches. MEMS switches can be bit rate, format, and wavelength independent,

providing much useful functionality. Using these switches avoids duplication of

expensive OEOs. Moreover, improvements in speed and flexibility are constantly being

168

made. For example, the electro-holographic switch of Trellis Photonics mentioned in

Chapter 2 [3] demonstrates switching at time scales about 1000 faster (nanoseconds)

combined with wavelength filtering, and research on MEMS-based tunable arrayed

waveguide gratings on nanosecond time scales may open the door to wide incorporation

of all-optical switches inside networks.[4]

Individual computers’ bitstreams are interconnected across the globe in a

hierarchical manner. Data is aggregated from the slower channels at the edges into

higher bitrate datastreams at progressively higher levels in the network, and the ability to

accomplish this multiplexing is essential (as is demultiplexing for the opposite operation).

Such time division multiplexing (TDM) is currently accomplished using OEOs. As

Fig. 8.2 shows, the ability to transport data at hundreds of GHz is feasible. It becomes

quite difficult to manage this using electrical switches at bit rates of 40 GHz and

higher.[5] Fortunately, several types of optical switches have demonstrated TDM

functionality at hundreds of GHz and may find themselves useful for this network

requirement.[5-8]

Another difficulty may arise when there are hundreds of different wavelength

channels that arrive at a switching node. It is useful to be able to switch data that is on

one particular wavelength channel onto a different wavelength. This increases the

connectivity of the routing node and helps avoid wavelength contention. Wavelength

conversion, too, is relatively easily accomplished with OEOs since the wavelength and

spatial location of the output is disjoint from the incoming data. This process is

discouragingly expensive and becomes increasingly difficult to scale as both the number

of wavelengths and bit rate rise. Although the choice of wavelength may change slowly

(e.g., at the packet or frame rate, if not slower), the actual conversion occurs at the bit

rate. This requires minimal logic and fast switching, allowing subsystems that use

all-optical switches to provide a viable alternative to OEOs. There is an excellent review

in Ref. [9], and research has been extensive with several hundred papers written in the

year 2000 alone. References [10-13], for example, provide a flavor of current

experimental work.

There is a significant caveat in this discussion about wavelength conversion and

TDM. Since wavelength conversion and TDM are relatively easy to implement if OEOs

169

are used in the system, all-optical implementations will likely be widely useful only when

the other functionalities of OEO switches, such as retiming, reshaping, and

reamplification are also done all-optically. This is possible because 3-R also does not

require extensive logic. While optical 3-R devices have been demonstrated,[11, 14-16]

whether or not all-optical buffering is also needed is an open question. However, their

incorporation into the network would require not only proven functionality but also a

significant redesign of the system, subsystem, and perhaps even network architecture,

dramatically increasing the effective cost of such a switch. It is not clear if there is a

progressive incremental path for incorporating these types of changes or if an abrupt,

more revolutionary path would be needed.

Optical switches come in wide varieties, as the number of them described in

Chapter 2 shows. The energies and time required for switching provide a sense of the

efficiency of a device. But the fastest switch with the lowest energy is not necessarily the

one that is best to use. The type of switch to use is application-dependent. Each switch

has it pros and cons and that matching changes, too, as technologies are introduced and

refined. Table 8.1 presents a short listing of some optical switches and the applications

for which they might be particularly suited. Presently, XPM in SOA interferometers

seem to offer good switching characteristics that can be integrated with a wide variety of

optical and electronic systems and stand out (in this author’s eyes) from the crowd.

These devices are relatively large, often millimeters in length and hundreds of microns

wide, and their interferometric nature makes them sensitive to fluctuations such as

temperature. Advances in non-linear materials continues to advance, as the switching

capability of PPLN waveguides demonstrates. OCOGs are neither the lowest energy nor

fastest switch. However, when its switching requirements are combined with its

two-dimensional scalability, it becomes an attractive option for applications that need an

array of NxN switches. The cost of the insertion loss of OCOGs due to their

absorption-based switching might be worth the price if the resulting electronic signal (the

average power for a given input datastream) is used for data-monitoring functionality.

170

Table 8.1: Comparison Between Selected All-Optical SwitchesType of Switch Energy Speed Pros Cons Applications Refs.

Soliton Gates 1-10 pJ 200 GHz fast, size, TDM, [17-19]all-fiber latency optical logic

SOAsXPM (fiber) 0.5-1 pJ 100 GHz integration & [8]

fast, size (fiber) TDM,integrable, wavelength

XPM (waveguide) 0.05-1 pJ 160 GHz 1-D complexity, conversion [11](1-10 pJ) input power

range

Coherent Wave Mixing 5-10 pJ 100 GHz fast, phase match, TDM, [20, 21] (PPLN Waveguides) 1-D, fabrication wavelength

format conversion independent,

multi-�

OCOGs 1-2 pJ 50 GHz 2-D, contrast ratio, NxN [22, 23]integrable insertion loss switch fabrics

8.3 FUTURE RESEARCH DIRECTIONS

There are several directions that would be fruitful to explore in future research.

More optimized multi-layer OCOG devices, such as the OCOG-4 described previously in

this chapter (see Fig. 8.1), provide an opportunity to both enhance the viability of OCOG

devices in general and also offer a chance to study the behavior of charge transport and

electromagnetic interactions with semiconductor material at a picosecond time scale,

perhaps faster. Designing a device with very low resistance may prove to be a good test

bed for examining how voltage diffusion changes from being RC constrained to being

limited by electromagnetic propagation (LC behavior).

171

pin

Modulator diode, Vm

Control diode, Vc

CW signal

Control data stream

Figure 8.32: Scematic illustration of an dual-diode optically controlled waveguide switch (OCWS). Whenproperly biased, the surface-normal incident control data stream is imprinted, through cross-absorptionmodulation, onto the c.w. signal light that travels along the length of the waveguide.

Creative solutions to the low contrast ratio of OCOG devices can also be

explored. There is an effort now underway at Stanford that seeks to accomplish this and

is briefly described here.[24] The primary limitation to achieving a large contrast ratio in

an OCOG is that the active region is only a micron or so thick, providing insufficient

length to absorb most of the signal light near the heavy-hole exciton frequency.

However, by changing the geometry of the device, this problem may be overcome.

While still maintaining the surface-normal incident control light, the signal light may

instead propagate along a waveguide onto which the control light shines (Fig. 8.3). In

this configuration, referred to as an optically controlled waveguide switch (OCWS), the

signal light now sees an effective active region as long as the width of the spot size

which can be tens or hundreds of microns long, as desired. Although there are several

difficulties that must be addressed regarding the switching behavior of such a device and

its construction is complex, it holds significant promise to provide not only high bit-rate

switching with contrast ratios of 10 or 20 dB but also switching over a broad wavelength

range (e.g., 30 nm). This is possible because the extended interaction region allows the

device to be operated relatively far from the exciton peak with smaller changes of

absorption. Moreover, because the control and signal beams are not co-propagating, in a

dual-diode configuration the top diode may be made completely absorbing to both signals

(since only the control is incident on it). This allows more efficient light collection and

the ability to provide effective wavelength conversion from a larger wavelength control

stream to a lower wavelength signal and vice-versa.

172

�2

Vc

Input data streams:

�a

�1 �3 Modulatedoutput signals:

�b

Vc

�c

c.w. signals Vc

OCWS, disabledtransparent for signallight in bottom diode(waveguide)

OCWS, enabledmodulating signal light inbottom diode (waveguide)due to top-illumination

Figure 8.4: Conceptual view of a reconfigurable NxN wavelength converter based on OCWS devices. c.w.light is launched into each of the waveguides. The default bias condition of the OCWS devices(n-i-p-i(MQW)-n) can be set so that they are relatively transparent (illustrated by the grey boxes) to thesignal light travelling along the waveguide and no modulation occurs regardless of the presence or absenceof surface-normal incident light (the data streams). An OCWS can be biased, however, (illustrated bygreen boxes) so that the light in the bottom diode (the waveguide) is strongly absorbed unless light isincident from the top. In this configuration the data of the incident surface-normal light is imprinted ontothe light stream travelling along the waveguide.

A single waveguide structure can be processed so that several OCWS devices lie

along its length. Each OCWS device can be biased so that it is either “enabled” or

“disabled.” In a “disabled” device, the top diode’s bias is set close to zero.

Consequently, no voltage change may be induced across the bottom diode regardless of

the presence or absence of control (surface-incident) light. At the same time, the bottom

diode may be biased so that it is relatively transparent, allowing the signal light to pass by

unimpeded.ii In an “enabled” device, on the other hand, the situation is different. The

bottom diode is biased so that the signal light is fully absorbed. If there is an incident

control pulse from the top, however, a change in voltage is induced in the bottom diode,

making it transparent. Thereby, the data stream of the control light is transferred to the

signal light propagating along the waveguide. By properly enabling or disabling the

OCWS devices along a waveguide, any one of the surface-normal incident beams,

ii See Fig. B.1 for an example of the relationship between the voltage and the absorption in a p-i(MQW)-ndiode.

173

regardless of their wavelength (assuming they are absorbed in the top diode), may be

imprinted onto the signal wavelength passing through the waveguide. As Fig. 8.4

illustrates, multiple waveguides may be used to create an NxN, non-blocking wavelength

space switch.[25] In Fig. 8.4 we have assumed that each control bitstream is incident on

all of the waveguides. Such a configuration requires at least a 1/N power loss per

OCWS. Other designs, however, may be used which avoid this limitation.[23]

Telecommunications has been the focus for optical switches in this dissertation.

There are additional areas of research where OCOGs may also find use. OCOGs can be

used as components in optical interconnects, for instance between chips in a computer.

They can be used as gated photodetectors in optoelectronic sampling devices such as

analog-to-digital converters. An OCOG could also be used as an interesting mode-locker

in a laser, simultaneously providing a feedback mechanism for laser stabilization control.

8.4 CONCLUSIONS (SUMMARY)Optically controlled optical gates are interesting and potentially useful devices.

This dissertation has presented the theory behind their operation and experimental results

demonstrating their capabilities. The large electric fields created due to carrier

separation, combined with the quantum confined Stark effect of MQWs, enables large

changes in absorption that require only small optical control energy. Device recovery is

based on diffusive conduction and may occur on a picosecond time scale with proper

device design when the control and modulation functions are spatially distinct. The use

of multiple layers in the device can provide particularly sharp and fast recovery, although

at the cost of reduced intensity modulation. 50 GHz burst-logic operation has been

demonstrated, as have contrast ratios of 2-to-1 using 1.5 pJ pulses.

We began with a motivation for the use of optical switches in telecommunications

and ended with a discussion of their potential roles as, principally, ultrafast time division

multiplexing and demultiplexing components and wavelength conversion devices. A

variety of optical switches were described and OCOGs placed within their context. Low

energy optoelectronic switching and scalability in two-dimensional arrays make OCOGs

intriguing and worthy of further study and consideration as practical devices.

174

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3. Agranat, A.J., et al., The Electroholographic Optical Switch (Trellis Photonics,2000).

4. Yu, K. and O. Solgaard, Personal Communication (2001).

5. Takara, H., "High-speed optical time-division-multiplexed signal generation,"Optical and Quant. Elec., vol. 33, pp. 795-810 (2001).

6. Saruwatari, M., "All-Optical Signal Processing for Terabit/Second OpticalTransmission," IEEE J. Selected Top. Quant. Elec., vol. 6, pp. 1363-1374 (2000).

7. Kawanishi, S., "Ultrahigh-Speed Optical Time-Division-MultiplexedTransmission Technology Based on Optical Signal Processing," IEEE J. Quant.Elec., vol. 34, pp. 2064-2079 (1998).

8. Runser, R.J., et al., "Interferometric ultrafast SOA-based optical switches: Fromdevices to applications," Optical and Quantum Electronics, vol. 33, pp. 841-874(2001).

9. Yoo, S.J.B., "Wavelength Conversion Technologies for WDM NetworkApplications," J. Lightwave Tech., vol. 14, pp. 955-966 (1996).

10. Durhuus, T., et al., "All-Optical Wavelength Conversion by SemiconductorOptical Amplifiers," J. Lightwave Tech., vol. 14, pp. 942-954 (1996).

11. Tajima, K., S. Nakamura, and Y. Ueno, "Ultrafast all-optical signal processingwith Symmetric Mach-Zhender type all-optical switches," Optical and QuantumElectronics, vol. 33, pp. 875-897 (2001).

12. Geraghty, D.F., et al., "Wavelength Conversion for WDM CommunicationSystems Using Four-Wave Mixing in Semiconductor Optical Amplifiers," IEEEJ. Selected Topics Quant. Elec., vol. 3, pp. 1146-1155 (1997).

13. Yu, J., et al., "40-Gb/s All-Optical Wavelength Conversion Based on a NonlinearOptical Loop Mirror," J. Lightwave Tech., vol. 18, pp. 1001-1006 (2000).

14. Fischer, S., et al., "Optical 3R regenerator for 40 Gbit/s networks," Elec. Lett.,vol. 35, pp. 2047-2049 (1999).

175

15. Otani, T., T. Miyazaki, and S. Yamamoto, "Optical 3R Regenerator UsingWavelength Converters Based on Electroabsoprtion Modulator for All-OpticalNetwork Applications," IEEE Phot. Tech. Lett., vol. 12, pp. 431-433 (2000).

16. Sartorius, B., 3R regeneration for all-optical networks in International Conferenceon Transparent Optical Networks, 2001 (2001).


18. Islam, M.N., C.E. Soccolich, and J.P. Gordon, "Ultrafast digital soliton logicgates," Optical and Quantum Electronics, vol. 24, pp. S1215-S1235 (1992).

19. Friberg, S.R., "Demonstration of colliding-soliton all-optical switching," Appl.Phys. Lett., vol. 63, pp. 429-431 (1993).

20. Kawanishi, S., et al., "All-optical modulation and time-division-muliplexing of100 Gbit/s signal using quasi-pasematched mixing in LiNb03 waveguides," Elec.Lett., vol. 36, pp. 1568-1569 (2000).

21. Parameswaran, K., Personal Communication (2001).


23. Demir, H.V., D.A.B. Miller, and V. Sabnis, "Surface-Normal OpticallyControlled, Lumped RC, Waveguide Switch," Stanford University Patent no. S01-210 (2001).

24. Sabnis, V., et al., Observation of Wavelength-Converting Optical Switching at 2.5GHz in a Surface-Normal Illuminated Waveguide in IEEE Lasers and Electo-Optical Society 2001 (2001).

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176

Appendix A: Example of Multilayer DiffusiveConduction Calculation

This section describes in more detail how to determine the coupling diffusion matrix,

D�

, for an arbitrary multilayered structure.

Ji,x

Ii,xIi, x-1

Figure A.1: Illustration depicting Ii and Ji.

The variables used are shown below in Table A.1 and some are illustrated in Fig. A.1.

Recall that for N (capacitive) layers, there are N+1 resistive planes. Some of the variables

are per layer, while others are per resistive plane; care should be taken in relationships

involving both N and N+1 indices.

Table A.1: Multiple-Layer Variables

Ci i=1 to N Capacitance

Vi i=1 to N Voltage across a layer

(“vertical”, between resistor planes)

Ri i=1 to N+1 Resistance

Ii i=1 to N+1 Current along a resistor plane

Ji=Ii+1-Ii i=1 to N “Vertical” current (current between resistor planes)

177

The N J’s are related to the N+1 I’s by:

11

21 2

1

1

for 1< i <N+1ii i

NN

dI JdxdI J JdxdI J JdxdI Jdx

�

�

� �

� �

� �

�

(A.1)

This relationship may be rearranged to show the dependence of the J’s on the I’s:

� �

� �

1 1

2 1 2

1 2 for i Ni i

dJ IdxdJ I IdxdJ I I Idx

� �

� � � �

� � � � � � ��

(A.2)

The (N+1)th equation involving IN+1 is simply

11

N

N ii

d dI Idx dx�

�

� �� (A.3)

We can rewrite the first NxN relationship of J in terms of I in matrix notation as shown:

[ ] [ ]

1 0 0 0 01 1 0 0 01 1 1 0 0

01 1

N NdJ Idx

��

� ��

� ��

�

� �

� � � �

� �

(A.4)

or, more succinctly,1

1[ ][ ] [ ]NN NdJ M Idx

��

�� (A.4b)

where [ ]NI� is a vector of the current along only the first N resistor planes. Note that 1[ ]NM�

is,

similarly, of dimension NxN.

178

The relationship between the V’s and I’s is given by

1 1i i i i id dV I R I Rdx dx� �

� �2

∇ (A.5)

Again, (N) Vi are described by (N+1) Ii and Ri. However, we can once more use Eq. (A.1) to

write

11

N

N i i N Ni

dV R I R Idx �

�

� ��

� ��

2∇ (A.6)

allowing us to use the vector [ ]NI� . In matrix form this becomes

[ ] [ ]

1 1 1 1

1 2 1

1 1 0 00 1 1 00 0 1 1

00 0 0 1 1

N N

N N N N N

N N

dV I Rdx

R R R R RR R R R

� � � �

�

��

��

��

�

�

� ��

� � �

�

� �

2∇ (A.7)

or

2 [ ] [ ] [ ]N N NdV M I Rdx

�

�� 2

∇ (A.7b)

where it is understood [ ] [ ]N NI R� � is an element-by-element multiplication, not matrix

multiplication. Since 2 [ ]NM�

is an NxN matrix, finding the inverse of M2 is straightforward.

Putting it all together, we have three distinct equations:1

1[ ][ ] [ ]

2 [ ] [ ] [ ]

NN N

N N N

dJ M Idx

dV M I Rdx

JVt C

��

�

�

�

��

�

� �

� � �

��

�

2∇ (A.8)

(Eqs. (A.5) and (A.2) are reprinted here for clarity) where the multiplication (of R� ) and

division (of C� ) mean element-by-element operations instead of matrix operations.

179

Combining these three equations, we can solve for the coupling diffusion matrix, D�

,

resulting in Eq. (4.50).

As mentioned earlier, from this point forward the method described in Chapter 4 for

determining the voltage dynamics using eigenvalues and eigenvectors becomes

straightforward to apply, providing a complete solution to the voltage dynamics of a

multilayer structure.

180

Appendix B: Analysis of Photocurrent Spectra

One of the most interesting effects of thin film deposition (e.g. MBE growth) is

that quantum mechanical effects may dominate the wavefunctions of electrons and holes

due to the very short physical dimensions – on the scale of nanometers – made possible

by these technologies. Quantum wells, for example, strongly affect the band-edge

absorption in semiconductor materials. As described by the quantum-confined Stark

effect [1], exciton absorption peaks undergo a strong shift as an electric field is applied

across the QWs. The wave nature of particles also allows coupling between quantum

wells -- an effect enhanced as barrier thicknesses and well spacing shrink due to greater

overlap of wavefunctions in nearby wells. In a p-i(MQW)-n semiconductor device,

photocurrent spectra may be used to investigate the details of the discrete, quantized

energy levels available to electrons and holes and to see how these levels shift as the

electric field changes. For the work described in this dissertation, photocurrent spectra

provide an important means for checking that the basis for optical switching, the change

in absorption, is understood. This was particularly important for OCOG-1. In that

device, ultrathin barriers – just 5 angstroms thick – were used, raising questions about

whether exciton effects would be observed and, if they were, the coupling behavior

between the wells. Other groups had previously reported that MQW structures with thin

barriers ranging from about 30 Å down to 10 Å exciton behavior was still present.[2, 3].

Only one reference, [2], reported excitonic absorption for 6 Å barriers, but they did not

show their data and added that it faded quickly with field.

B.1 MEASUREMENT PROCEDURES

Photocurrent spectra were taken by reverse-biasing the p-i(MQW)-n sample,

shining light on it, and measuring the resulting current. Absorption spectraversus energy

and electric field were made by changing the applied voltage and wavelength. In our

experiments a programmable voltage supply was placed across an OCOG device in series

with a 1 k� resistor. The incident light (3 �W), provided by a tunable c.w. Ti-Sapphire

laser, was chopped at 500 Hz. The current was taken from across the resistor and fed to a

181

lock-in amplifier. The lock-in, the voltage supply, and the tuning control for the laser

were all controlled by a local computer system. Laser power was monitored using a 3 V

reverse biased photodetector and 1 k� resistor and used to normalize the data for laser

power fluctuations. Typical photocurrent spectra are shown in Figs. B.1a and B.1b.

0 2 4 6 8 10 12 14 16 18 20

1410

1415

1420

1425

1430

1435

1440

1445

1450

1455

1460

1465

1470

1475Photocurrent (a.u.)

Bias Voltage (V)

Pho

ton

Ene

rgy

(meV

)

1330 -- 1400

1260 -- 1330

1190 -- 1260

1120 -- 1190

1050 -- 1120

980.0 -- 1050

910.0 -- 980.0

840.0 -- 910.0

770.0 -- 840.0

700.0 -- 770.0

630.0 -- 700.0

560.0 -- 630.0

490.0 -- 560.0

420.0 -- 490.0

350.0 -- 420.0

280.0 -- 350.0

210.0 -- 280.0

140.0 -- 210.0

70.00 -- 140.0

0 -- 70.00

Figure B.1a: OCOG-1 photocurrent (arbitrary units).

182

846

848

850

852

854

-1 0 1 2 3 4 5

Photocurrent (a.u.)W

avel

engt

h (n

m)


1545 -- 1600

1490 -- 1545

1435 -- 1490

1380 -- 1435

1325 -- 1380

1270 -- 1325

1215 -- 1270

1160 -- 1215

1105 -- 1160

1050 -- 1105

995.0 -- 1050

940.0 -- 995.0

885.0 -- 940.0

830.0 -- 885.0

775.0 -- 830.0

720.0 -- 775.0

665.0 -- 720.0

610.0 -- 665.0

555.0 -- 610.0

500.0 -- 555.0

Figure B.1b: OCOG-1 photocurrent, expanded view.

Photocurrent may also be studied in a slightly different manner in order to more

easily distinguish peaks and valleys; specifically, differential photocurrent can be

measured. A graph of the first or second derivatives of the photocurrent with respect to

voltage can be made by adding an oscillatory component to the applied voltage at a

frequency that is also fed to the lock-in. If the lock-in uses that frequency as its

reference, the first derivative is mapped. If the lock-in uses twice that frequency as its

reference, the second derivative is tracked; any magnitude peak or trough value gets

passed through twice in the oscillatory voltage cycle and hence has a component at twice

the oscillatory frequency. The first derivative data is somewhat difficult to interpret since

the original peaks and troughs are now both reported as zeros. On the other hand, with

the second derivative, the original peaks may be clearly distinguished.

To implement a differential spectra measurement, a relatively small 100 mV,

500 Hz modulation voltage was capacitively coupled (0.15 �F) to the circuit and the

chopper was removed. Two 10 k� resistors were used: one placed between the device

183

and ground, the other between the DC bias voltage and the connection between the

capacitor and the device. The current was taken across the second resistor. As can be

seen by comparing Fig. B.2 to Figs. B.1a and b, the spectra are much more vivid and

descriptive.

0 2 4 6 8 10 12 14 16 18 20

1410

1420

1430

1440

1450

1460

1470

Photocurrent (a.u.)

Bias Voltage (V)

Ene

rgy

(meV

)

95.00 -- 100.0

90.00 -- 95.00

85.00 -- 90.00

80.00 -- 85.00

75.00 -- 80.00

70.00 -- 75.00

65.00 -- 70.00

60.00 -- 65.00

55.00 -- 60.00

50.00 -- 55.00

45.00 -- 50.00

40.00 -- 45.00

35.00 -- 40.00

30.00 -- 35.00

25.00 -- 30.00

20.00 -- 25.00

15.00 -- 20.00

10.00 -- 15.00

5.000 -- 10.00

0 -- 5.000

Figure B.2: Differential photocurrent (second derivative) of OCOG-1 (a.u.).

B.2 SIMULATION

With the above data in hand, a question of interpretation arose. What absorption

resonances do the various curves represent? This was answered by simulating energy

resonance (absorption) dependence of the device. The simulation of the OCOG-1

absorption spectrum is described below.

B.2.1 DescriptionThe simulation method used finds resonances for electrons or holes via a

transfer-matrix method [4, 5] and then adds a manual adjustment for Coulomb attraction,

a rough approximation of the QCSE. The electron wavefunction, �, can be modeled as a

184

wave with left and right propagating components which undergo either exponential decay

(if the energy of the electron is less than the potential barrier) or sinusoidal phase

oscillation (if the energy of the electron is greater than the potential barrier). At

interfaces where the potential changes, such as between types of material, certain

boundary conditions must be met: both � and 1 dm dz

��

must be continuous, where z is

the direction perpendicular to the interface (often the direction the wave is travelling).

Each boundary condition relates the both the left and right waves on one side of the

interface to the left and right waves on the other side; in other words, a pair of coupled

equations. One way to handle this mathematically is to use a 2x2 matrix to represent the

transformation of the wave crossing such an interface. Similarly, a 2x2 matrix may also

be used to express the propagation of the wave as it travels between interfaces. These

matricies may all be multiplied together to describe the propagation of a wavefunction

through an entire device. The output at one end may be calculated for a given energy by

assuming that at the input there is only, say, a right-going wave. Resonances are

determined by finding at what energies a local maximum in overall ‘transmission’ occurs.

Five coupled GaAs quantum wells, each 100�Å wide and separated by 5 Å AlAs

barriers were simulated and their resonances found as a function of field

(0-200,000 V/cm). By looking at the resulting wavefunctions as a function of position, it

was possible to identify which if any wells were coupled together for a specific choice of

parameters. Optical absorption resonances were calculated as the differences in the

electron and hole energies added to the bulk bandgap energy and modified (reduced) by

Coulomb attraction. In this model, Coulomb attraction was estimated to reduce

absorption energy by a blanket 4.2 meV. This value was based on a hydrogenic-like

binding energy4

2 2 208B

r

eEh�

� �� , where

11 1

elec holem m�

�

� ��

, melec and mhole are the effective

masses of the electron and hole, respectively, and �r is the dielectric constant.[6]

B.2.2 ResultsThe simulation results are presented in the next four graphs. Table B.1 describes

the labeling convention used in these figures below. The primary absorption resonance is

185

due to the lowest energy heavy-hole exciton with both the electron and hole in the ground

state. The QCSE predicts a quadratic shift in energy resonance level as a function of

applied electric field, and as may be seen in Fig. B.3 the primary resonance (blue dots) is

roughly quadratic. The first graph also clearly shows the effect of two resonances

intersecting, in this case that of the heavy hole exciton and the resonance due to the same

hole with the second-lowest energy electron of the adjacent well. When the electron

energy level of the adjacent well approaches that of the electron in the same well with the

hole, energy splitting occurs. The simulations also correctly show the symmetric and

anti-symmetric waveforms with lower and higher energy, respectively.

Table B.1: Legend Notation

Symbol Description Details

e# electron energy level 1=ground state, 2=first excited state, …

h# hole energy level 1=ground state, 2=first excited state, …

w# quantum well number 1 through 5, from left to right

S symmetric wavefunction

AS anti-symmetric wavefunction

1400

1410

1420

1430

1440

1450

1460

1470

1480

1490

0 50000 100000 150000 200000

Electric Field (V/cm)

w3e1, w3h1S w3e1/w2e2, w3h1AS w3e1/w2e2, w3h1

Figure B.3: Simulation of heavy hole exciton and 1st excited electron-state exciton resonance splitting

186

Figure B.4 shows simulation results of intra-well bandgap resonances: electrons

in one well overlap with holes in adjacent wells. Absorption resonance energy in such

cases is primarily determined by the relative energy spacing between the centers of the

wells under consideration. This linear field dependence, a Stark ladder, manifested

clearly in the simulation. In this graph, the resonance level of the ground-state electron

and the hole of an adjacent well is plotted, as is a similar resonance for an electron and

hole both in their first-excited state.

14001410142014301440145014601470148014901500

0 50000 100000 150000 200000 250000


w3e1, w4h1w3e2, w4h2

Figure B.4: Simulation of Stark ladders for adjacent wells, ground state electron and hole and 1st excitedstate electron and hole.

The splitting in Fig. B.3 is actually a splitting of the primary heavy-hole exciton

when it overlaps the Stark ladder resonance due to the first-excited state electron coupled

to the adjacent ground-state hole. Other Stark ladder resonances also intersect the

heavy-hole exciton, resulting in various degrees of splitting. In Fig. B.5, coupling

between the first-excited state electron and the ground state hole in the next-to-adjacent

well is shown as well as the resonance splitting due to a similarly coupled second-excited

state electron. Note that the slopes of these next-to-adjacent Stark ladders are twice as

steep as the ones for directly adjacent wells (compare to Fig. B.4).

187

1400

1420

1440

1460

1480

1500

1520

0 50000 100000 150000 200000 250000


w3e1, w3h1

S 321 (w1e3/w3e1?), w3h1

AS 321 (w1e3/w3e1 ?),w3h1S w3e2, w5h1

AS w3e2, w5h1

Figure B.5: Simulation of next-to-adjacent well Stark ladder splittings with heavy hole exciton resonance

Figure B.6 combines all of the simulated resonances and their splittings described

above as well as a few others. The result is a rather busy parameter space.

1400

14101420

14301440

1450

14601470

14801490

1500

0 50000 100000 150000 200000 250000


w3e1, w3h1S w3e1/w2e2, w3h1AS w3e1/w2e2, w3h1S 321 (w1e3/w3e1?), w3h1AS 321 (w1e3,w3e1?), w3h1w3e1, w4h1w3e2, w4h2S w3e2, w5h1AS w3e2, w5h1

Figure B.6: Simulation of various Stark ladders and the heavy hole exciton resonances

188

B.3 COMPARISON OF SIMULATION TO DATA

The simulation results shown in Fig. B.6 can be compared to the data. As can be

seen in Fig. B.7, the simulation results match the data remarkably well, following several

of the resonance splittings quite closely.

0 5 10 15 201400

1450

1500

Photocurrent (a.u.)

Data and Simulation

Bias Voltage (V)

Pho

ton

Ene

rgy

(meV

)

0 -- 5.000

5.000 -- 10.00

10.00 -- 15.00

15.00 -- 20.00

20.00 -- 25.00

25.00 -- 30.00

30.00 -- 35.00

35.00 -- 40.00

40.00 -- 45.00

45.00 -- 50.00

50.00 -- 55.00

55.00 -- 60.00

60.00 -- 65.00

65.00 -- 70.00

70.00 -- 75.00

75.00 -- 80.00

80.00 -- 85.00

85.00 -- 90.00

90.00 -- 95.00

95.00 -- 100.0

w3e1w4h1

w3e1w3h1

AS3e12e2w3

S3e1e2w3h1

Sw3e2w5h1

ASw3e2w5h1

Figure B.7: OCOG-1 photocurrent data and simulation

One of the limitations of the simulation program is that identifying individual well

wavefunction components becomes difficult that when multiple resonances overlap or are

weak (for example, the resonances around –10V bias). Most of the splittings are due to

Stark Ladders overlapping the main heavy-hole exciton absorption. We can make use of

this by finding clear, isolated Stark Ladders using the photocurrent data, trace them back

189

to their origins (zero-internal bias) using their linear relationship to voltage.1 Such an

origin point should correspond to a basic electron and hole energy level resonance of a

quantum well and can then be compared against expected values provided by simulation

of a single quantum well. Moreover, each origin point acts as the source for one-, two-,

three-, (and so on) well Stark ladders. It is straight-forward to then check if these

assumed Stark ladders themselves correspond to other resonances in the data as well.

This procedure has been carried out in Fig. B.8 and is able to account for almost all of the

resonances seen! Table B.2 compares the origin points of Fig. B.8 to the simulation’s.

Considering that the simulation used only a blanket 4.3 meV adjustment for Coulomb

attraction, Fig. B.8’s origin points and those of the simulation are reasonably

well-matched.

(2,1h),(1,2l)

(1,1h)

(2,2l),(2,2h)

(3,1h)

(1,3l)

(1,1l),(1,2h)

(1,3h)

(2,3h)

(2,2l)

1460

1480

1515

1578

1598

1630

1687

1740

1785

1500

1400

Ener

gy (m

eV)

9 5 . 0 0 - - 1 0 0 . 0

9 0 . 0 0 - - 9 5 . 0 0

8 5 . 0 0 - - 9 0 . 0 0

8 0 . 0 0 - - 8 5 . 0 0

7 5 . 0 0 - - 8 0 . 0 0

7 0 . 0 0 - - 7 5 . 0 0

6 5 . 0 0 - - 7 0 . 0 0

6 0 . 0 0 - - 6 5 . 0 0

5 5 . 0 0 - - 6 0 . 0 0

5 0 . 0 0 - - 5 5 . 0 0

4 5 . 0 0 - - 5 0 . 0 0

4 0 . 0 0 - - 4 5 . 0 0

3 5 . 0 0 - - 4 0 . 0 0

3 0 . 0 0 - - 3 5 . 0 0

2 5 . 0 0 - - 3 0 . 0 0

2 0 . 0 0 - - 2 5 . 0 0

1 5 . 0 0 - - 2 0 . 0 0

1 0 . 0 0 - - 1 5 . 0 0

5 . 0 0 0 - - 1 0 . 0 0

0 - - 5 . 0 0 0

1-well separation2-well separation3-well separation

Stark Ladders:

-1.4 0 5 2010 15Applied reverse bias (V)

Photocurrent (a.u.)

(e#, h#)

Figure B.8: Theoretical Stark ladders overlapped against photocurrent spectra

1 The slopes of the various Stark ladders determined by tracing match theoretical slopes.

190

Table B.2: Comparison Between Simulated and Graphically-Determined

Exciton Resonances

Exciton Resonance Graph (meV) Simulation (meV) Error: Simulation.-Graph (meV)e1, hh1 1460 1461 1

e1, lh1 / e1, hh2 1485 1480 -5e1, hh3 1535 1515 -20

e2, hh2 / e1, lh2 1570 1578 8e2, lh 1 / e2, hh2 1595 1598 3

e2, hh3 1630 1633 3e2, lh2 1675 1687 12e1, lh3 1725 1740 15

191

REFERENCES

1. Miller, D.A.B., et al., "Electronic Field Dependence of Optical Absorption nearthe Bandgap of Quantum Well Structures," Phys. Rev. B, vol. 32, pp. 1043-1060(1985).

2. Goossen, K.W., J.E. Cunningham, and W.Y. Fan, "Electroabsorption inultranarrow-barrier GaAs/AlGaAs multiple quantum well modulators," Appl.Phys. Lett., vol. 64 (1994).

3. Fujiwara, K., K. Kawashima, and T. Imanishi, "Tunneling escape time ofelectrons from the quasibound Stark localized states in ultrathin barrierGaAs/AlAs superlattices," Phys. Rev. B, vol. 54, pp. 17724-17729 (1996).


5. Macleod, H.A., Thin-Film Optical Filters, 2nd Ed (McGraw-Hill, New York,1989).

6. Landolt-Bornstein, Semiconductors. Numberical Data and FunctionalRelationships in Science and Technology, Editor O. Madelung, Vol. 17, pp. 219(Springer-Verlag, Berlin, 1982).

192

Appendix C: Optically-Controlled Optical Gate Device Designs

OCOG-1

Table C.1: OCOG-1 Structure DesignDescription Material Thickness Dopant

(angstroms) (cm-3) p layer GaAs 50 Be 1019

Al0.10Ga0.90As 10,000 Be 3x1018

Al0.10Ga0.90As 100 -X94 AlAs 5 -

i (MQW) layer X94 GaAs 100 -AlAs 5Al0.10Ga0.90As 100 -

n layer Al0.10Ga0.90As 1000 Si 1018

AlAs 710 Si 1018

DBR mirror X18 Al0.10Ga0.90As 595 Si 1018

X18 AlAs 710 Si 1018

AlAs 10,000 Si 1018

n+ GaAs (001) substrate

193

OCOG-2


(angstroms) (cm-3) top n layer GaAs 50 Si 1018

Al0.07Ga0.93As 450 Si 1018

top i layer Al0.07Ga0.93As 3,000 -

Al0.07Ga0.93As 6,000 Be 1018

top p layer AlAs 50 Be 1018

Al0.07Ga0.93As 6,000 Be 1018

Al0.07Ga0.93As 500 -bottom i X66 AlAs 50 - (MQW) layer X66 GaAs 95 - AlAs 50 -

Al0.07Ga0.93As 500 -

Al0.07Ga0.93As 5,000 Si 1018

bottom n layer AlAs 50 Si 1018

Al0.07Ga0.93As 5,000 Si 1018

superlattice X30 AlAs 20 -cleaning layer X30 GaAs 20 -

Al0.07Ga0.93As 605 -DBR mirror X25 AlAs 723.6 -

X25 Al0.07Ga0.93As 605 -

undoped GaAs substrate

194

OCOG-3


(angstroms) (cm-3)

GaAs 50 Be 1019

top p layer Al0.34Ga0.66As 50 Be 1019

Al0.34Ga0.66As 400 Be 1018

top i layer Al0.34Ga0.66As 1,000 -

Al0.34Ga0.66As 2,000 Si 1018

top n layer AlAs 300 Si 1018

Al0.07Ga0.93As 3,000 Si 1018

Al0.34Ga0.66As 3,000 Be 1018

bottom p layer AlAs 300 Be 1018

Al0.34Ga0.66As 7,000 Be 1018

Al0.34Ga0.66As 100 - bottom i X60 AlAs 40 - (MQW) layer X60 GaAs 120

Al0.34Ga0.66As 100 -

bottom n layer AlAs 200 Si 1018

Al0.34Ga0.66As 5,000 Si 1018

AlAs 711 Si 1018

DBR mirror X25 Al0.34Ga0.66As 628 Si 1018

X25 AlAs 711 Si 1018

AlAs 10,000 Si 1018

undoped GaAs substrate

doctor of philosophy harris (coach) and olav solgaard for reading this dissertation, especially in...

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