(AlGa)As Semiconductor Lasers and Integr

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(AlGa)As Semiconductor Lasers and Integrated Optoelectronics
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Wilt, Daniel Paul
(1981)
(AlGa)As Semiconductor Lasers and Integrated Optoelectronics.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/f9jz-ez68.
Abstract
Five subjects related to monolithic integration of electronic and optical devices in the (AlGa)As material system are treated in this thesis. They are:
1. The Integrated Optoelectronic Repeater:
The design, fabrication, and testing of the first monolithic integrated optical repeater is described. This device consists of an optical detector, electronic gain stage, and current modulated semiconductor laser transmitter integrated in a single crystal chip to perform the function of regenerating an optical signal as might be found in an optical communication system. The device has a measured optical gain (light out to light in) of 10 dB.
2. Ion Implanted Lasers and Schottky Gate Field Effect Transistors:
The use of ion implantation as a technique to fabricate both lasers and field effect transistors is described. Devices fabricated include a beryllium implanted laser diode on N type GaAs substrate, a beryllium implanted laser diode on semi-insulating Cr doped GaAs substrate integrated with a field effect transistor driver, and sulfur implanted GaAs field effect transistors.
3. A Steady State Lateral Model of the Double Heterostructure Laser:
A theoretical model of the double heterostructure laser is described which treats the p-n junction in the device correctly by using fundamental semiconductor relationships and reasonable assumptions about the device heterointerfaces. The model treats both the electronic and optical properties self consistently, making the model valid above lasing threshold. Finite element formalism is adopted as a solution technique to enable the treatment of complicated diode geometries. An example is treated and theoretical and experimental results are compared.
4. The Effect of Lateral Carrier Diffusion on the Modulation Response of a Semiconductor Laser:
The effect of lateral carrier diffusion upon the modulation characteristics of the semiconductor laser is investigated. A self consistent analysis of the spatially dependent rate equations is performed using a finite element model. The transverse junction stripe laser is treated as an example and a comparison is made between lateral carrier diffusion and spontaneous emission as damping mechanisms for the resonance peak. Experimental results bear out the conclusion that the relaxation resonance in this device is damped mainly by lateral carrier diffusion. In addition, a simple analytic result is presented which illustrates qualitatively the effect of lateral carrier diffusion upon such devices. The conclusion from this result is that lateral carrier diffusion serves to damp the relaxation resonance in the semiconductor laser quite well, but probably will not serve to improve the upper limit on modulation frequency as might have been suspected.
5. Effective Permittivity Formalism and the Design of Buried Heterostructure Lasers:
An approach to effective permittivity formalism is presented which clarifies and extends the use of this technique particularly in the treatment of waveguiding in the semiconductor laser. The scalar wave equation is posed in a variational form, and the effective permittivity formalism is treated as a variational approximation technique. This approach shows clearly the nature and limits of the approximation involved. The formalism is applied to the case of the buried heterostructure laser and the results differ considerably from the conventional application of effective permittivity formalism to this device when a reasonable form is assumed for the variational modal profile.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
(Applied Physics)
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Yariv, Amnon
Thesis Committee:
Yariv, Amnon (chair)
Bridges, William B.
McGill, Thomas C.
Rutledge, David B.
Nicolet, Marc-Aurele
Defense Date:
12 May 1981
Funders:
Funding Agency
Grant Number
NSF
UNSPECIFIED
Office of Naval Research (ONR)
UNSPECIFIED
Record Number:
CaltechETD:etd-09182006-141505
Persistent URL:
DOI:
10.7907/f9jz-ez68
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3621
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(AIGa)As Semicondu ctor Lasers
and Integrated Optoelectr onics
Thesis by
Daniel Paul Wilt

In Partial Fulfillment of the Requiremen ts
for the degree of
Doctor of Philosophy

California Institute of Technology
Pasadena, California
1981

(Submitted May 12, 1981)

- ii-

AcknowledgeID.ents
I would like to express my sincere gratitude and appreciation to my advisor,
Dr. Am.non Yariv, for: his support and encouragement throughout my stay in his
quantum electronics:.group. I owe a great deal to his efforts and·kindness in constructing and encouraging this group of highly talented individuals.
I would like to thank also Dr. Nadav Bar-Chaim, Dr. Hussein Izadpanah, Dr.
Schlomo Margalit, and Dr. Moshe Yust for their aid and advice throughout the
course of the work presented in this thesis. Without their collective expertise
much of the work presented in this thesis could not have been undertaken. In
addition, I would like to thank Mr. Desmond Armstrong for his expert technical
assistance in construction and maintainance of experimental apparatus.
I would also like to express my appreciation to my fellow graduate students
for their help and discussions. In particular, I would like to thank Dr. C. P. Lee,
Dr. Israel Ury, Mr. P. C. Chen, Mr. Josef Katz, Mr. Kam Lau and Mr. Christoph
Harder for their participation in this work.
My wife Pamela and our parents also deserve my deepest thanks for: their
love, support, and patience:
My special thanks go to the group of people who introduced me to scientific
research. In particular, I would like to express my thanks to my advisor during
my first two years at Caltech, Dr. James Mercereau, for his support and
encouragement. I would also like to thank Dr. Harris Notarys for his enlightening discussions.

Mr. Edward Boud also deserves my appreciation for his

unselfish aid throughout the course of my research.
The financial support of the National Science Foundation and the Office of
Naval Research is gratefully acknowledged.

- ill-

Abstract
Five subjects related to m-onolithic integration of electronic and optical
devices in the (AlGa)As material. system are treated in this thesis. They are:
1. The Integrated Optoelectronic Repeater
The desiqn, fabrication, and testing of the first monolithic integrated optical
repeater is described~ This device consists· of an optical detector, electronic gain
stage, and current modulated semiconductor laser transmitter integrated in a
single crystal chip to perform the function of regenerating an optical signal as
might be found in an optical communication system. The device has a measured optical gain (light out to light in) of 10 dB.
2. Ion Im.planted Lasers
and Schottky Gate Field Effect Transistors
The use of ion implantation as a technique to fabricate both lasers and field
effect transistors is described. Devices fabricated include a beryllium implanted
laser diode on N type GaAs substrate, a beryllium implanted laser diode on
semi-insulating Cr doped GaAs substrate integrated with a field. effect transistor
driver, and sulfur implanted GaAs field effect transistors.
3. A Steady State Lateral Model
of the Double Heterostructure Laser
A theoretical model of the double heterostructure laser is described which
treats the p-n junction in the device correctly by using fundamental semiconductor relationships ·and reasonable assumptions about the device heterointerfaces. The model treats both the electronic and optical properties self consistently, making the model valid above lasing threshold. Finite element formalism is adopted as a solution technique to enable the treatment of complicated
diode geometries. An example is treated and theoretical and experimental

- ivresults are compared.

4-. The Effect of Lateral Carrier Diffusion,
on the Modulation Response of a Semiconductor Laser
The effect of lateral carrier diffusion upon the modulation characteristics of
the semiconductor laser is investigated. A self consistent analysis of the spatially dependent rate equations is performed using a :finite element model. The
transverse junction stripe laser is treated as an example and a comparison is
made between lateral carrier difiusion and spontaneous emission as damping
mechanisms for the resonance peak. Experimental results bear out the conclusion that the relaxation reson.ance in this device is damped mainly by lateral
carrier diffusion. In addition, a simple analytic result is presented which illustrates qualitatively the effect of lateral carrier difiusion upon such devices. The
conclusion from this. result is that lateral carrier diffusion serves to damp the
relaxation resonance in the semiconductor laser quite well, but probably will not
serve to improve the upper limit on modulation frequency as might have been
suspected.

5. Effective Permittivity Formalism
and the Design of Buried Heterostructure Lasers
An approach to effective permittivity formalism is presented which clarifies
and extends the use of this technique particularly in the treatment of waveguiding in the semiconductor laser. The scalar wave equation is posed in a variational form, and the effective permittivity formalism is treated as a variational
approximation technique. This approach shows clearly the nature and limits of
the approximation involved. The formalism is applied to the case of the buried
heterostructure laser and the results differ considerably from the conventional
application of effective permittivity formalism to this device when a reasonable
form is assumed for the variational modal profile.

-v-

Contents

Chapter I

1.1 Integrated Optoelectronics

1.2 Outline of Thesis

l.3

Chapter II

Chapter

Introduction

References for Chapter I

The Integrated Optoelectronic Repeater

11.1 Introduction

11.2 Design Considerations for the Repeater

11.3 Fabrication of the Repeater

II.4 Results

II.5 Conclusions

11.6 References for Chapter II

Ion Implanted Lasers
and Schottky Gate Field Effect Transistors

IlI.1 Introduction

IlI.2 Sulfur Implanted GaAs Schottky Gate Field Effect Transistors

III.3 Beryllium Implanted (AlGa)As Double Heterostructure Lasers

III.4 Conclusions

III.5 References for Chapter III

Chapter IV

A Steady State Lateral :Model
of the Double Heterostructure Laser

IV.1 Introduction

IV.2 Simplifying Assumptions used in the Model

IV.3 The Electrical Model

-vi IV.4 The Optical Model

N.5 Solution Technique

N.6 Sample Case and Comparison with Experiment

N. 7 Summary and Conclusions

IV.8 References for Chapter IV

Chapter V

The Effect of Lateral Carrier1Diffusion.
on the Modulation Response ·of a Semiconductor Laser

V.1 Introduction

V.2 The Spatially Dependent Rate Equations

V.3 Analytic Treatment of Lateral Carrier Diffusion Efiects

101

V.4 Modulation Response of the TJS Laser

108

V.5

Chapter VI

References for Chapter V

117

Effective Permittivity Formalism
and the Design of Buried Heterostructure Lasers

VI.1 Introduction

118

VI.2 Effective Per:rµittivity Formalism

119

VI.3 Design of the Buried Heterostructure Laser

123

VI.4 References for Chapter VI

139

- 1-

Chapter I
Introduction·
L1 lntepated Opt.oelect.ronics

Optical communication has become a subject of intense interest and
research recently. This is in large part due to the extremely large bandwidths
available and the small dimensions involved in the generation and guiding of
light waves as compared with conventional microwave transmission systems.
Of particular interest at present are systems based upon fused silica fibers as
the transmission medium, and semiconductor light sources and detectors.
These systems have a large number of advantages compared with other
transmission systems. Fused silica fibers are small, light weight, heat and.radiation resistant, free from electronic cross talk, have low losses, and. large
transmission bandwidths 1 . The semiconductor laser light sources and detectors used in these systems are small, high speed, easy to control (being solid
state devices) and highly efficient.
A communication s system based on these components will serve as an interface between purely elect.ronic media: high speed computers, television and
voice transmission systems, and analog microwave systems such as phased
array radars. In this case, the optical communication s link derives and delivers
its signals to and from electronic devices such as silicon based integrated circuits in an electronic environment. One can envision then that in the interest of
low cost and circuit simplicity the terminal sources and recievers in the optical
link may take on electronic processing involved with the communication link. As
an example, the source might include with its laser transmitter' an active device
to serve as an interface between standardized signal transmission levels and
impedances in the system and the drive requirements of the laser. On the .other
end of the optical link, again active devices may be included with the receiver

-2-

detector to interface. its characteristics to the requirements of: the system. and
might also include some signal.conditioning such as level detection; This is an
advantage unique to the use of semiconductor light sources and detectors,
where a monolithic single crystal chip can contain both electronic and optical
devices in an integrated design ..
A particular advantage of this concept ·is that simple optical and electronic
processing can be performed in a single ·chip, highly simplifying some of the
potential applications of such devices. A good example of this is in repeater links
for

long

distance

optical

communication

system,

where

detection,

amplification of the resulting electrical signal, and retransmission can all be
performed by the same integrated circuit.
These

devices

fall

under

the general

classification of

optoelectronic

integrated circuits, or OEICs. The integration of semiconductor· optical and electronic components in the OEIC was first suggested by Yariv 2 • A block diagram
of the OEIC is shown in Figure 1.1. In general, this device consists of three sections; a detector section, an electronic processing section, and a transmission
section. The detector section consists of detectors, preamps, and any other
components for signal preconditioning, and it feeds an electronic input to the
following electronic processing section which may also accept external electronic inputs. The processing section performs analog or digital processing of
these signals, and feeds electronic signals to both the external environment and
internally to a source section consisting of lasers which feed output optical
fibers and their associated driver electronics.
At present, the OEIC is envisioned to serve mainly as an interface between
electronic media and optical fiber communication links. However, there is no
fundamental reason why one cannot envision the OEIC eventually assuming a
role of its own, for example, as the fundamental component of a computer

• • •

Figure 1.1

Il
I\

• • •

electr onic inputs

analog processing

multip lexing , storag e

digita l logic, switch ing,

electr onic proce ssing

~ ·~

• • •

lasers

and

driver s

optica l fiber outpu ts

Block diagram of an optoelec tronic integrat ed circuit {OEIC).

preamps

optical fiber inputs

and

detectors

e I ectron ic outpu ts

Opto -Elec troni c Integrated Circuit

-4-

based upon optical data bussing;
The fundamental requirement for the OEIC is a material system and processing technology capable of producing light sources~ detectors. and electronic
components on the same single .crystal chip. Unfortunately the most well understood semiconductor material, silicon. is incapable at present of meeting this
requirement. This is because it is an indirect bandgap material and does not
generate light efficiently. At present, the optimum material system for the production of OEICs is the ternary, (AlGa)As. The use of this material as the basis
for the OEIC was first suggested by Yariv 2 • This material has the following properies:
1.

The ternary (AlGa)As is well lattice matched to the binary substrate GaAs for
the entire alloy composition range. This means that layered heterostructures can be grown with high quality material interfaces where a minimal
amount of recombination takes place, a requirement for efficient lasers and
detectors.

By varying the alloy composition, the bandgap energy of the material may be
varied over the range from 1.43 eV (GaAs) to 1.95 eV (Alo.45Gao.55As) in the
direct bandgap region, and from 1.95 eV to 2.17 eV (AlAs) in the indirect
bandgap region. This allows the construction of efficient light emitting
devices over the wavelength range from 0.9 µm. to 0.7 µm. and matching
detectors for these emitters.

Semi-insulating (Cr doped) substrates of GaAs are readily available; This
allows the fabrication of electronic devices such as Schottky gate field effect
transistors (MESFETS) with low parasitic capacitances and therefore good
high frequency characteristics.

-5-

There is a large number of epitaxial techniques available ·to produce high
quality layered beterostructures of (AlGa)As. Foremost among these is the
technique of liquid phase 'epitaxy (LPE). However, interest is growing in
molecular beam epitaxy (MEE) and metallorganic chemical vapor deposition
(MOCVD) as important futur:e technologies.

Well developed technologies ,exist for the fabrication of electronic and optical
devices separately in this material system. GaAs Schottky gate field effect
transistors are widely accepted microwave components, and a high speed
integrated circuit family utilizing this component as the active device is
developing 3 .4.5 • Semiconductor lasers using (AlGa)As are also in commercial
production, and state of the art devices are displaying single mode operation
and threshold currents below 10 mA 6 •
The problem to be overcome: in the construction of the OEIC in (AlGa)As, then,

is that of blending together these two separate technologies in a manner so that
the best features of each may be retained. This is a nontrivial problem, for the
two technologies have developed essentially independently. Field effect transistor technology has been developed upon semi-insulating Cr doped substrates
with strict control upon epilayer dimensions and dopings to obtain reproducible
device characteristics. However; little attention has been paid to the growth of
high quality (AlGa)As ternary material which is considerably more difficult due
to the reactivity of aluminum. On the other hand, laser and detector technology
has developed with primary attention to the growth of high quality ternary
material and secondary attention paid to the reproducibility of epilayer
thicknesses and dopings. In addition, lasers were developed first on conductive
substrates due to the high power dissipations and temperature sensitivity which
required the mounting of the devices with the epitaxial side embedded in high
thermal conductivity·material such as indium solder for heat sinking.

-6-

The first design choice to be· made is that of substrate material. Based upon
the choice of the MESFET as an. active device, which.needs semi-insulating substrate to operate at high frequencies, semi-insulating Cr doped GaAs has been
chosen. However, this choice requires the operation of the laser light sources far
away from the heat sink in a device-up configuration. In addition, GaAs has a
poor thermal conductivity at room temperature. To realize a practical OEIC
with a continuously operating laser requires careful attention to the power· dissipation requirement and particular attention to the development of low power
dissipation lasers on semi-insulating substrates.
From the standpoint of electronic component technology, the barriers· to be
overcome in OEIC technology are similar to those encountered in GaAs
integrated circuit technology. Reproducibility of device characteristics, and particularly the subject of substrate behavior under thermal cycling are important
subjects. During high temperature processing, semi-insulating Cr doped GaAs
has a tendency for lhe deep acceptor Cr atoms to redistribute leaving behind
conducting layers in the material 7 •

This is a disaster for many device

processes, and this problem is receiving a great deal of attention from workers
in the integrated circuit field. It should be noted that from the standpoint of
OEIC fabrication low ·temperature processing is to be considered a very important feature of new technology.
L2 Outline of Thesis

Throughout this thesis, a familiarity with both semiconductor devices and
double heterostructure lasers• is assumed. For background information, the
interested reader is ref erred to the standard texts a. 9 .io,11.12.1 3 •
Five subjects in the field of integrated optoelectronics will be presented and
discussed in this thesis. Chapter II will present a prototype OEIC fabricated from
(AlGa)As 14 . This circuit, a prototype of a repeater chip as might be found in an

-7-

optical communication system, detects optical signals, amplifies the resulting
electrical signals, and retransmits them through current modulation of a semiconductor laser. Some aspects of the design of this integrated circuit will be discussed, especially with regard to the expected performance. The fabrication and
testing of the device will then be described.
Chapter III will be concerned with ion implantation techniques and their use
in fabricating low threshold, well behaved lasers as well as reproducible GaAs
MESFETs for use in OEICs. The characteristics of beryllium ions implanted into
{AlGa)As as well as the fabrication and testing of beryllium implanted laser
diodes will be presented. Both a beryllium implanted laser on n type GaAs substrate 15 and a beryllium implanted laser on semi-insulating Cr doped GaAs substrate integrated with a field effect transistor driver 16 are described. These
devices are very simple and easy to fabricate, and have very attractive characteristics for integrated optoelectronics, such as single lateral mode operation
and low threshold currents which allow CW operation at room temperature;
Chapters IV, V, and VI are concerned with theoretical modelling of double
heterostructure laser diodes.
In Chapter IV a steady state model of the double heterostructure laser is
presented 17 which is fully consi~tent with the master equations for semiconductor materials, a significant departure from models popular in this field. In particular, it treats the diode junction of the device by using the continuity of
quasi-fermi levels across the heterojunction interfaces instead of the usual
diode equation. This introduces naturally into the theory the saturation of carrier quasi-fermi levels above lasing threshold, as has been measured experimentally 18 • This model is also valid above the lasing threshold, and shows quantitatively the behavior of the diode as it is driven into the regime where spatial
holeburning in the gain profile causes shifts between optical modes. The model

-8is formulated and solved using the finite element method as a solution technique so that it offers a large degree of flexibility in the choice of diode
geometry. A specific example is analyzed, and the results of the model are compared with experiment.
Chapter V treats the effects of lateral carrier diffusion upon the modulation
characterist ics of the laser diode 19 • Here lateral diffusion is taken to mean
diffusion parallel to the heterointerf aces on either side of the active region, a
direction in which the injected carriers are quite often not tightly confined~
The relaxation oscillation resonance in the semiconduc tor laser represents a
major obstacle to the use of this device in communicat ions systems. This resonance has a typical magnitude of 10 dB· over the low frequency value in the
small signal response of the laser, and degrades seriously the high frequency
modulation response of the laser. Several techniques have evolved to control
and damp this resonance, including the use of lateral carrier diffusion. The
diffusion of inverted carriers into and out of the optical mode region of the laser
represents a considerable damping to the exchange of power between the carriers and the cavity photons. In addition, one might suspect that lateral carrier
diffusion would serve to improve the upper modulation frequency limit of the
laser. This follows from the fact that in a device with a highly confined optical
mode, the inverted carriers may be considered to have a shorter effective lifetime based upon their diffusion away from the optical mode region.
In Chapter V, a simple general model of the effect of lateral carrier diffusion
upon the modulation characterist ics of the diode laser is presented and solved.
To aid the calculation, simplified spatial rate equations are used. A finite element model is used to determine both the steady state and small signal modulation characterist ics of the device. The model is applied to the case of a laser
where lateral carrier diffusion is known to affect the modulation response, the

-9 -

TJS (transverse junction stripe) laser, and the calculation is compared with an
experimental measurement of the modulation response for this device. In addition, an analytic relationship is derived which describes the maximum contribu1

tion of lateral carrier· difiusion to the modulation response of the laser.
The essential result of this calculation is that lateral carrier diffusion can
serve to damp the relaxation oscillations in the semiconductor laser quite well.
However, there is little improvement in the upper modulation frequency limitation from this technique.
Chapter VI presents a treatment of waveguiding in the semiconductor laser
that refines and extends the technique described as effective index formalism 20 • This technique involves the solution of a two dimensional waveguiding
eigenvalue problem through the use of one· dimensional solutions and averaging
techniques to form effect.ive refractive index profiles. This technique is often
presented as an approximate solution to the scalar wave equation, and the
approximations involved are rarely quantified. As a result, this technique is
quite often used in an incorrect manner. In this chapter, the effective index
method is derived as a variational technique for the solution of the two dimensional eigenvalue problem, and.the limits of the technique indicated. The refined
technique is applied to the problem of waveguiding in the buried heterostructure laser 6 • The usual approach to this problem is shown to be a poor approximation and a more accurate approach is treated as an example.

- 10 L3 References for Chapter I
1.

W. G. French, J. B. McChesney, P. B. O'Connor and G. W. Tasker, "Optical
Waveguides with Very Low Losses," BeUSyst. Tech. J. 53, 951 (1974).

A. Yariv, "Active :Integrated Optics," Proc. Esfahan Symposium on Fundamental and Applied Laser Physics, M. S. Feld, A. Javan, and N. A. Kurnit, eds.,
Wiley-Interscienc .e, New York (1973).

C. A. Ll.echti, "Microwave Field-Efiect Transistors - 1976," IEEE Trans.
Microwave Theory and Techniques MTT-24, 279 (1976).

C. A. Ll.echti, "GaAs Fet Logic," Proc. 6th International Symp. on GaAs and

Related Compounds, Pt. I, Edinburgh, Scotland. 1976, Inst. Phys., London
(1977).
5.

R. C. Eden, B. M. Welch, R. Zucca, and S. I. Long, "The Prospects for Ultrahigh
Speed VLSI GaAs Digital Logic," IEEE Trans. Electron Devices ED-26, 299
(1979).

K. Saito and R. Ito, "Buried-Hetero structure AlGaAs Injection Lasers," IEEE J.

Quant. Electron. QE-16, 205 (1980).
7.

C. A. Evans, V. R. DeLine, T. W. Sigmon, and A. Ll.dow, "Redistribution: of Cr

During Annealing of 80 Se-Implanted GaAs," Appl. Phys. Lett. 35, 291 (1979).
8.

A. Yariv, Quantum Electronics. 2nd edition, John Wiley and Sons, Inc., New
York {1975).

A. Yariv, Introduction to Optical Electronics, 2nd edition, Holt, Rinehart, and
Winston, New York ( 1976).

10. H. C. Casey, Jr., and M. B. Panish, Heterostructure Lasers: Part A, Fundamental Principles; Part B. Materials and Operating Characteristics, Academic

Press, New York (1978).

- 1111. H. Kressel and J. K. Butler, Semiconductor Lasers and HeterojunctioniLEDs.
Academic Press, New York (1977).
12. G. H. B. Thompson, Physics of Semiconductor Laser Devices. John Wiley &

Sons, Inc., New York (1980).
13. S. M. Sze, Physics of Semiconductor Devices. Wiley-Interscience, New York

(1969).
14. M. Yust, N. Bar-Chaim, S. H. Izadpanah, S. Margalit, I. Ury, D. Wilt, and A.
Yariv, "A Monolithically Integrated Optical Repeater," Appl. Phys. Lett. 35.
795 (1979).

15. N. Bar-Chaim, M. Lanir, S. Margalit, I. Ury, D. Wilt, M. Yust, and A. Yariv, "BeImplanted (GaAl)As Stripe· Geometry Lasers," Appl. Phys. Lett.

36 1 233

(1980).
16. D. Wilt, N. Bar-Chaim, S. Margalit, I. Ury, M. Yust, and A. Yariv, "Low Threshold
Be Implanted (GaAl)As Laser on Semi-Insulating Substrate," IEEE J. Quant.
Electron. QE-16•. 390 (1980).

17. D. Wilt and A. Yariv, submitted for publication.
18. T. L. Paoli and P. A. Barnes, "Saturation of the Junction Voltage in Stripe
Geometry (AlGa)As Double-Heterostructure Junction Lasers," Appl. Phys.
Lett. 28, 714 (1976).
19. D. Wilt, K. Y. Lau, and A. Yariv, submitted for publication.
20. D. Wilt and A. Yariv, submitted for publication.

- 12-

Chapter Il
The Integrated Optoelectronic Repeater
U.1 Introduction

Early integration experiments carried out by Lee et al. and Ury et al. involved
the integration of single electronic devices with lasers. The :first such experiment
involved the integration of a Ga.As bulk effect device, a Gunn device, electrically
in series with a laser diode 1 • High speed modulation of the ,laser diode was
demonstrated through the spontaneous transit time mode oscillations of the
series connected Gunn device.

The second such experiment involved the

integration of a single GaAs MESFET in series with a laser diode 2 and demonstrated modulation of the laser diode by varying the gate bias on the series connected FET. These were demonstrations concerned with the compatibility of the
device fabrication procedures.
As the next step in the development of OEIC technology, a working integrated
circuit was designed and fabricated 3 • This OEIC is fashioned as an optoelectronic repeater, a monolithic chip that will detect, amplify, and retransmit signals such as those that might be found in an operating fiber optic communications link.
D.2 Desi«n Considerations for the Repeater

The schematic of the circuit chosen for the repeater is shown in Figure 2.1.
This configuration was chosen as a compromise between simplicity and gain. The
device as shown consists of an optical detector, the MESFET labelled Q2, an
active load MESFET labelled Ql. a driver MESFET labelled Q3, and a laser diode.
In operation, the drain current of the active load transistor is trimmed by
adjusting its gate - source voltage so that it matches the drain current on the
detector transistor. This is the point where the pair has maximum voltage gain.
The load line diagram for this configuration is shown in Figure 2.2.

- 13 -

·-- - - -

- -.

......-~

...-----+-----~~ Q3

D3:

S3 '

~-------------

F"igure 2.1

___ ___ __ J

Sche mati c diag ram· of the repe ater inte grat ed

circu it.

- 14-

-........

;::, illuminated

ur--o.;;.;.;.~:..;..;;.;=--~-------

...•---~v--~..~

Figure 2.2

voltage

Load lin.e diagram for the detector - active load MESFET pair. The
voltage here appears on the gate of the driver transisto r. Q3.

- 15-

The operation of the repeater may be described as follows. When light falls on
the detector, Q2, the saturated: drain current of the transistor increases; causing the voltage at the gate of the driver transistor Q3 to rise in 'accordance with

the load line for the detector - active load pair. This causes theodrain current of
the driver transistor to increase, increasing the current flow through the laser
diode in series with its drain. Thus the light output from the laser diode
increases. The repeater circuit. can be seen to operate in a noninverting mode,
then, as is desired. Note that a current bias source external to the chip is used
to hold the laser near its threshold current. This bias current represents a large
power dissipation which is both wasteful and unnecessary to pass through the
driver transistor, Q3.
The gain of this simple circuit may be calculated in a straightforward
manner. When a flux of N photons per second falls on the detector MESFET, its
drain current increases by some amount 8Id. We can define a detector quantum
efficiency by the formula

(2.1)
where e is the electronic charge•. This represents the number of :additional channel carriers flowing per second due to the irradiation; The use of the MESFET as
an optical detector was first described and demonstrated by Gammel and Ballantyne 4 • The device is believed to operate as a photoconductor in a source follower configuration. This definition of a quantum efficiency is somewhat
misleading, then, since it can be greater than one and has been measured to be
as large as five. The response characteristics for this detector are quite good as
well, with response times on the order of 50 ps having been reported.
With the definition of quantum efficiency as above, the gain of the repeater

- 16 can be calculated as ·

(2.2)
where TJci represents1a coupling efficiency of the input light to the detector, 1Jd is
the quantum efficiency of the: detector as defined above, ~ is the detector active load pair dynamic load impedance, gm is the driver transistor transconductance, 171 is the laser quantum efficiency, and TJco is the output light collection
efficiency.
With regard to the operating speed of this IC, the major limitation is
represented in the circuit design. The MESFET transistors, if fabricated properly
on semi-insulating substrate, should respond well into the gigahertz regime. This
includes the operation of the detector MESFET. The laser can be expected to
operate at speeds up to approximately one gigahertz. based both upon measurements and theoretical modelling of the dynamic response of the laser diode.
With this circuit design, the limitation on the speed of the IC is based upon the
charging of the gate capacitance of the driver transistor, Q3, through the
dynamic impedance of the detector - active load pair; The time constant for this
mechanism is

(2.3)
where Rd is the dynamic load impedance, and C8 is the equivalent gate capacitance of the driver transistor, which should include the gate capacitance of t.he
detector and all stray capacitances in the device which must be charged
through Rd.
There is a design tradeoff to be made in the circuit, then, based on a
compromise between high speed and high gain. The parameters to be varied are
the load impedance, Rd, and the driver transconductance, gm, which is related to
the gate capacitance ..

- 17To get an idea of· what sort of limitation this represents, we will assume a
detector with 11ci 11d = 1, a load characteristic~= 10 kO , a driver MESFET with
transconductance and gate capacitance gm = 10 mmho , C11 = 0.3 pF , and a laser
with 171 1700

=0.5 . This gives as the gain 50 or 17 dB, and as the gate charging

delay time 3 ns. Thus the 3 dB cutoff frequency is expected to be approximately
50 MHz, and the gain - bandwidth product of the device 2.5 GHz. On the other
hand, with Rd = 1 k!l , we get 7 dB of gain and a cutoff frequency of 500 MHz.
High speeds are possible, but with a necessary reduction in gain.
The configuration of the repeater is shown in Figure 2.3. The laser in the
device was chosen to be the crowding effect laser of Lee et al. 5 •6 for simplicity.
This laser configuration makes use of the base crowding effect to confine
injected current in the active layer to a narrow region near the edge of the
etched mesa. Thus the carrier density _and optical gain are high in this region,
and the lasing filament forms there. The crowding is accomplished by making
the sheet resistivity of the lower n type layers as high as possible (consistent
with making good MESFETs). The lateral voltage drop in this layer along with the
exponential current - voltage distributed diode characteristics cause the
injected current to crowd to the mesa edge. This type of laser is extremely simple to fabricate but has a number of drawbacks. It places the optical gain region
near the etched mesa edge which serves both as a nonradiative recombination
center for injected carriers as well as ,a light scattering center for the optical
mode. These cause the device to have a high threshold current, and it is for this
reason that such a crowding effect laser has never been operated continuously
at room temperature.
The MESFETS forming the electronic portion of the repeater are fabricated
on the bottom n GaAs layer which also serves to connect the las.er diode with the
driver transistor. The MESFET gate pads are placed in an etched recess on the
semi-insulating substrate to minimize the gate capacitances. The gate width on

Figure 2.3

Schem atic drawing of the repeate r integra ted circuit.

p-GaA s
p-GaA I As
.,,,~ n-GaA s
n-GaA IAs
n-GaA s
SI-Ga As

.....
CD

-19-

the driver MESFET is made large to increase its transconductance, and Ure gate
width of Ql is made 'slightly larger than that of Q2 so that their drain currents
can be matched easily by applying negative bias to the gate of Ql with respect to
its source. The transitions from the recessed semi-insulating island to the FET
channels are all oriented in the same direction so that the gates of the FETs can
be made continuous over the slanted edge by performing an angled vacuum evaporation of the gate metal. Note that the gates of transistors Q2 and Q3 share a
common bonding pad, which must be externally connected .to the common
S2 - D1 metallization.
Il.3 Fabrication of the·repeater

The fabrication of the repeater begins with the LPE growth of five epilayers on
an (001) oriented Cr doped semi-insulating GaAs substrate approximately 300
µm thick. The approximate layer dopings and thicknesses are given in Table 2.1.

After growth, the wafer is clean.ed of any excess gallium and the Cr-Au p contact
is evaporated over the top surface. The laser mesa is then defined photolithographically and etching is performed. The contact metal is etched by conventional ·er and Au etchants, and the underlying epilayers by using 1:8:8

(H2S0 4 : H202: H20) as a nonselective etchant and HF as a selective etchant for
(AlGa)As. This leaves the bottom GaAs MESFET layer exposed. This layer is then
further etched with the nonselective 1:8:8 etchant to obtain the desired transistor characteristics. Then side metallization, AuGe eutectic alloy and Au, is then
shadow evaporated over the laser mesa to form the transistor source and drain
contact pads. Another lithographic step, etching of the n contact metal, and
nonselective etch is performed to define the gate pad semi-insulating recess. The
wafer must have been oriented as mentioned earlier so that the slope of this
etch allows the gate metal to be continuous over the step. The transistor channels and gate contact pads are defined photolithographically and the underlying

- 20 Table 2.1

Epilaye r charact eristics for the repeate r.

Layer

Compos ition.

Doping

Thickne ss

n-GaAs

2 · 10 18 cm-8

1.4µm

n-Gao.sAfo.~s

1017

2.1

n-GaAs

1017

0.3

p-Gao.5Alo.~s

1017

2.1

p-GaAs

1019

1.0

- 21-

n contact metal etched. The aluminum gate metal is then evaporated and the
gates formed by lifting otr the unused metal with the underlying photoresist.
This self aligned process produces gates typically 5 µm in length. The n metallization is then alloyed at a temperature of 380 °C and the wafer lapped and
cleaved into individual devices.
ll.4 Results

In the device reported, shown in Figure 2.4, the :final thickness of the MESFET
layer was 0.6 µm, and the measured transconductance of the driver FET was 12
mmho. Pinchotr voltages on the FETs were 5 V. The drain characteristics of the
three transistors in this device are shown in Figures 2.5, 2.6, and 2. 7. The laser,
whose length was 480 µm. had a threshold current of 400 mA and a differential
quantum efficiency of 10 percent.
The repeater was tested under pulsed conditions due to the high threshold
current of the laser diode which made continuous operation impossible. The
light source was an. external (AlGa}As laser coupled to a fiber, which was
oriented so that the light would fall on the detector MESFET. A silicon VMOS
transistor was used as an external current source to bias the laser near threshold. The Vdd contact of the chip was connected to an external pulse source,
which supplied a synchronization signal to the external light source. Tested
under these conditions, the repeater chip showed a gain (light output to light
input} of 10 dB. The operation of the repeater is shown in Figure 2.B. The upper
trace in these photographs shows the current passing through the external
laser light source, and the lower trace shows the light output from one facet of
the repeater laser diode. The scale on the lower trace is approximately 350 µW
per division. The measured power coupled into the detector was 140 µW.

- 22 -

Figure 2.4

Photograph of the completed repeater integrated circuit.

The

dimensions of the integrated circuit are approximately 1200 µm by
480 µm.

- 23 -

........ .........

- -

::J

Figure 2.5

,........,

Drain current versus voltage characte ristics of the detector MESFET
in the repeater . The gate of the device is shorted to its source. The
horizon tal scale is 1V/div and the vertical scale is 2 mA/ div.

- 24-

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
111111111111111111111
11119~······"

llllJ~imllllllllllll

---!!!!
~--!!~
~·1
r9~=
~~- z•••

Figure 2.6

Drain current versus voltage character istics of the active load MESFET in the repeater. Gate voltage is incremen ted at 1V/step from 0
to -6V. The horizonta l scale is 1V/div and the vertical scale is 2
mA/div.

- 25 -

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

1111111111~~!!~11

111111~~--··11

llllJ';~illllllllllll

llP:~illllllllllltl

-~---~-
F!!
!!!-!!!-!!~!J

Figure 2. 7

Drain curren t versus voltag e charac teristi cs of the driver MESFE

T in

the repeat er. Gate voltag e is increm ented at 1V/step from 0 to
-6V.
The horizo ntal scale is 1V /div and the vertica l scale is 10 mA/di v.

- 26-

Figure 2.B

Operation of the repeater integrated circuit. The top trace in each
photograph is the current through the external laser light source,
and the bottom is the light output from the repeater laser. The
scale on the lower traces is 350 µW /div. The horizontal time scale is
100 nS/div.

-27U.5 Conclusions

In this chapter we: describe an investigation whicb. demonstrates the feasibility of (AlGa}As based· optoelectronic integrated circuits. ThP limitations due to
fabrication problems. but not fundamental, of this first device are the poor performance of the laser diode and the nonreproducible.charac teristics of the MESFET layers. The laser that would go into a production OEIC should be capable of
at least 10 times lower threshold current and 5 times higher quantum efficiency.
One example of such an improved laser is the TJS laser on semi-insulating substrate, first demonstrated by Lee et al. 7 • This laser structure has been reported
to operate CW at heat sink temperatures as high as 110 °C 8 and in addition has
been reported to have extremely long lifetimes of 106 hours at room temperature 9 • Another example of an improved laser and an improved technique of
making reproducible MESFET characte_ristics is reported in the next chapter,
where ion implanted lasers and MESFETs are described.

- 28 ll.6 Referen ces for Chapte r n
1.

C. P. Lee, S. Margal it, I. Ury, and A. Yariv, "Integr ation of an Injectio n.Laser
with a Gunn Oscilla tor on a Semi-In sulating GaAs Substr ate," Appl. Phys.
Lett. 32, 806 {1978).

I. Ury, S. Margali t, M. Yust, and A. Yariv, "Monol ithic Integra tion of anlnjec

tion Laser and a Metal-S emicon ductor Field Efiect Transis tor," Appl. Phys.
Lett. 34, 430 {1979).
3.

M. Yust, N. Bar-Ch aim, S. H. Izadpa nah, S. Margal it, I. Ury, D. Wilt, and A.
Yariv, "A Monoli thically Integra ted Optical Repeat er," Appl .. Phys. Lett. 35,
795 (1979).

J. C. Gamme l and J. M. Ballant yne, "The OPFET: A New High Speed Optical
Detecto r," Proc. IEDM, 120 (1978).

C. P. Lee, S. Margali t, and. A. Yariv, "Doubl e Hetero structu re GaAs-GaAIAs

Lasers on Semi-In sulating Substr ates Using Carrier Crowdi ng," Appl. Phys.
Lett. 31, 281 (1977).
6.

C. P. Lee, S. Margal it, and A. Yariv, "Waveg uiding in an Expone ntially Decayin g
Gain Medium ," Optics Commu n. 25. 1 (1978}.

C. P. Lee, S. Margal it, I. Ury, and A. Yariv, "GaAs-GaAlAs Injectio n Lasers

Semi-In sulating Substr ates Using Lateral ly Diffuse d Junctio ns," Appl. Phys.
Lett. 32, 410 (1978).

H. Kumab e, T. Tanaka , H. Namiza ki, M. Ishii, and W. Susaki, "High Tempe rature Single Mode CW Operat ion with a Junctio n-Up TJS Laser," Appl. Phys.
Lett. 33, 38 (1978).

S. Nita, H. Namiza ki, S. Takami ya, and W. Susaki, "Single -Mode Junctio n-Up
TJS Lasers with Estima ted Lifetim e of 106 Hours, " IEEE J. Quant. Electro n.
QE-15, 1208 (1979).

-29-

Chapter ID
Ion lmplanted:Lasers and Schottky Gate Field Efrect Transistors
Dl.1 Introduction

One of the major problems- associated with the development of GaAs and
other III-V semiconductor materials for device applications is that of selective
doping. In particular, the attempted adaptation of diffusion masking and
diffusion processes from the silicon technology has failed to achieve satisfactory
results for many potential applications. The difficulty lies in two aspects of GaAs
technology: the lack of an easily grown thermally matched diffusion mask, and
the fact that one of the two components of the compound, arsenic, has a high
vapor pressure at normal diffusion temperatures (600 - 1000 °C). This high
vapor pressure requires that diffusions in this system be performed under partial pressures of arsenic vapor sufficient to prevent decomposition of the wafer
surf ace, or with a cap on the wafer to prevent the escape of the arsenic from the
surface. Either choice results in the introduction of further problems. The
diffusion coefficients of most dopants have some dependence on arsenic partial
pressure which represents additional control problems, and capping the wafer
with a poorly thermally matched dielectric layer leads to stress induced defects
and stress enhanced diffusion of the desired dopant, making pattern definition
more difficult. In addition, some desirable dopants (e.g. beryllium, a p type
dopant) have no compounds with sufficient vapor pressure to realize a satisfactory vapor phase diffusion process.
One of the techniques developed to alleviate some of the problems outlined
above is that of ion implantation of dopant atoms selectively into the semiconductor wafer. With this technique, an ion beam of atoms at moderate kinetic
energies (up to several hundred keV) is directed at the wafer and the atoms
penetrate the surface to a shallow depth {on the order of one µm). Usually the

- 30 wafer is oriented along a high index crystal direction or coated with an amorphous material such as Si3 N4 to minimize the effects of channeling, where high
energy atoms travel long distances along low index crystal directions. This process introduces the dopant atoms into the host crystal, but at the same time
introduces some lattice damage. A short duration high temperature anneal after
the implant repairs this damage and in addition incorporates the implanted
atoms into the host lattice, activating their electrical properties. These ion
implantations are usually done at low temperatures relative to diffusion
processes {up to a few hundred degrees Celsius) and can be masked with little
difficulty using conventional photolithographic techniques and non-thermally
matched dielectric films. The anneal step usually requires that the wafer be
under arsenic vapor· pressure or capped, but the short times involved and the
facts that dose has already been determined in the implant and the cap does
not have to serve as a diffusion mask make this a much easier process to control.
To minimize the damage created in the implanting step, one usually prefers
to implant light atoms. However, light atoms diffuse faster and are associated
with faster degradation rates for some device applications. Other criteria are
also applied to ion implanted dopants, such as annealing requirements, dose
activation percentage, and solid solubility limits. These are of course in addition
to such considerations as electrical behavior in the host lattice and the
difficulty of getting a good ion beam to implant.
At present, the most desirable p dopant for GaAs appears to be beryllium. It
has quite low mass and is easily implanted, and shows essentially complete
activation after anneals as low as 15 minutes at 800 °C. It has in addition
interesting diffusion behavior which will be discussed later. An important consideration, however, is its extreme toxicity.

Another important question

remains as well, regarding the degradation of beryllium implanted devices.

- 31-

A host of elements are available for n type implanted dopants. Sulfur, silicon,
and selenium have all seen use :in ion implanted devices; in particular , for MESFET applicatio ns. All have shown quite high activation s and good mobility in the
resulting layers, a requirem ent for high speed devices. This is a fortunate coincidence, for while zinc is considere d an acceptabl e p diffusant for GaAs, the n
diffusants are in compariso n much less controlled . This work has been spurred
mainly by the developm ent of GaAs MESFET integrated circuits, and most of this
work has involved the direct implantat ion of the selected dopant into Cr doped
semi-insu lating substrate s. The simplicity of the process for producing large
numbers of MESFETs on a common substrate is a prime considera tion in the
developm ent of OEIC technolog y.
In this chapter, the use of ion implantat ion as a technique in the fabricatio n
of OEICs is presented . A laser structure using beryllium as an ion implanted
dopant and sulfur implanted MESFETs are described . The resulting devices are
easier to fabricate and more reproduci ble in their character istics than their
correspon ding zinc diffused and LPE grown counterpa rts. In addition, they have
some advantage s in dimension al control associated with the simpler masking
technique s used for ion implanted dopants.
Ill.2 Sulfur Im.planted GaAs Schottky Gate Field Effect Transistor s

Other researche rs have already reported the use of sulfur implantat ion as a
method of forming MESFET layers in Cr doped semi-insu lating substrate s t.2 .
This method has been adopted to make MESFETs suitable for OEIC fabricatio n.
The process begins with a sulfur ion implantat ion at an energy of 200 keV and a
dose of 5 · 10 12 cm-2 into Cr doped semi-insu lating GaAs substrate . At this
implantin g energy, a projected. range of 0.15 µm with a standard deviation of
0.06 µmis predicted for sulfur in GaAs using the theory of Lindhard, Scharf, and

Schiott 3 . The wafer is then capped with approxim ately 2500 A of CVD Si02. and

- 32 -

annealed at 850 °C for 30 minutes. The resulting MESFET layers have mobilities
cm2
of approxim ately 3500 Vs""
and sheet resistiviti es of approxim ately 1000 0

a·

These layers have been used to fa bric ate ·MESFETs with character istics suitable
for OEICs.
This process is compatib le with the other processes used, for instance, in the
productio n of semicond uctor lasers. After the epitaxial step and etching
required to form the laser structures , a deep etch can be done to reveal the surface of the semi-insu lating wafer where it is desired to form MESFETs. Then a
selective implant can be done using the same mask that is used to perform the
etching, and the wafer capped and annealed. The remaining metallization~ steps
are easily performed together on both the MESFETs and the lasers.
Ill.3 Beryllium Implanted (AlGa)As Uouble Heterostru cture Lasers

Beryllium ion implantat ion has gained a great deal of interest lately as a
method of fabricatin g p type layers in GaAs. Beryllium is an extremely light ion
which can be implanted at low energy, and annealed easily at relatively low ternperatures .
The character istics of beryllium as an ion implanted dopant in GaAs have
been researche d by a number of investigat ors 4 •5 •6 •7 •8 • Annealing studies 6 •7 •8
have shown that full electrical activation can be achieved with as low as 700 °C
annealing temperatu re at large doses {correspo nding to beryllium concentra tions greater than 10 18 cm- 3 ), and have revealed an interestin g diffusion characteristic for beryllium implanted in GaAs. This is that at beryllium concentra tions
greater than approxim ately 10 18 cm-3 in GaAs, beryllium diffuses very quickly.
This allows the use of beryllium ion implantat ion as a predeposi tion step for
beryllium diffusion to form high quality p layers with relatively uniform concentrations near 10 18 cm-3 • The great advantage of this technique is that the junction depth depends most strongly upon the predeposi ted dose, which can be

- 33 controlled very precisely. No informati on· about beryllium implantat ion into the
ternary alloy {AlGa)As had appeared yet, an importan t considera tion in the production of optoelectr onic devices by ion implantat ion.
The use of ion implantat ion as a technique for fabricatin g semicond uctor
lasers had been reported by Barnoski et aL in 1974 9 • This involved a homostructure laser formed by implantin g zinc atoms at very low energy into a GaAs
substrate and subsequen tly diffusing the zinc in to form a shallow p-n junction
for the laser. Ion implantat ion is also used routinely as an isolation technique in
GaAs technolog y, utilizing the fact that damaged GaAs has high resistivity . Usually protons are implanted to produce this damage, and an effort is made to
keep the damaged region well away from the importan t recombin ation regions
in the device such as: the active layer of a laser.
In this section a double heterostr ucture laser formed using beryllium ion
implantat ion as a fabricatio n technique is described 10•11 • It has the desirable
features of low threshold current, well behaved modal properties , high quantum
efficiency, and easy fabricatio n that make it an attractive device for OEIC applications.
The lateral cross section of the device is shown in Figure 3.1. This device is
fabricated in a stripe geometry- which is closed in the sense that the p-n junction
in the device has a limited cross sectional area. With this geometry, the current
injection into the active layer of the device is limited to the narrow region where
the p stripe contacts it. The p-n junction formed in the upper cladding layer
draws a negligible current due to the difference in bandgap energy, and the surface leakage in this type of device is negligable .
This geometry has several advantage s from the standpoin t of laser structure.
The typical stripe geometry laser, formed with a broad area p-n junction and
current confining insulating stripe, relies upon sheet resistivity in the upper

F'igure 3.1

..-_

n -G a A IA s

n -G a As

n- G aA IA s

't'.

S i0 2

of a ber yll ium im pla nte d las er.

A u- G e- Au

S ub st ra te

... .. - ..... ... ... .. .

Au - Zn - C r- A u

Sch em ati c lat era l cro ss sec tio n

n+ G aA s

Be (P )

,pr.

CAl

- 35-

layers to provide confinem ent of injected: current. This depends strongly upon
growth condition s such as backgrou nd doping and gives a great deal of variation
in device behavior. In addition, it is difficult to achieve current confin-em ent
tight enough to limit the laser. oscillation to a single filament. This results in
multimod e oscillation and unstable character istics such as kink formation in
the light versus current character istics. An additiona l problem is the difficulty
of obtaining low threshold currents due to the presence of the two injected
current tails on either side of the stripe. This current is essentiall y lost.
In this closed stripe laser, filament stabilizati on is much easier to achieve due
to the reduction in the width of the p stripe. This results in stable near field patterns and linear light versus current character istics over a wider range of output power. In addition, the only current. lost is that which diffuses away from the
gain region of the laser, permittin g very low threshold currents to be attained.
The fabricatio n of the beryllium implanted laser begins with the LPE growth
of three layers on an n type GaAs substrate . Typical layer thickness es and dopings are given in Table 3.1. After the growth, the wafer is cleaned of excess gallium and coated with app~oximately 2500 A of CVD Si0 2 • The laser stripes are
then defined phototlith ographica lly in AZ-1350J photoresi st using stripe widths
of between 2 and 10 µm, and the underlyin g Si0 2 in these openings etched. A 100
keV beryllium implantat ion is then performed with a dose of 3 · 10 15 cm- 2 at
room temperatu re, using the photoresi st as a mask. The predicted projected
range and standard deviation in Alo.45 Ga0 .55As at this energy are 0.35 and 0.12
µm, respective ly 3 . The photoresi st is then stripped and a high temperatu re
anneal performed at 800 °C for 40 minutes. This anneal is uncapped but performed in an H 2 ambient in a semi-seale d graphite holder, so that arsenic loss
from the substrate is minimized .

The implanted beryllium diffuses down

through the structure in this step and contacts the active layer. This is related

- 36-

Table 3.1

Epilayer characteristics for the beryllium implanted laser.

Layer

Composition

Doping

Thickness

n -Ga 0 .s~lo.4sAs

10 17 cm-3

3µm

n-GaAs

101?

0.25

n-Ga0 _5~lo.45As

1016

- 37-

to the process in GaAs where :-the high beryllium concentration diffuses very
quickly. At this implanted dose, the exper.imental junction depth was 0.8 µm in
GaAs and 1.4 µ.m in Alo. 46 Gao.1515As. These junction depths are not strongly dependent upon anneal temperaturei as is shown in Table 3.2, again illustrating the
concentration dependence of the diffusion process, and providing some data on
beryllium implantation into {AlGa)As. The junction depth rises monotonically
with aluminum content up to an alloy composition of Al0•615 Ga0•35As, possibly indicating a decrease in the concentration required to initiate 'the fast diffusion
mechanism. The result of this is that the. diffusion front will often stop at the
heterojunction interface, a process easily controlled by regulating implanted
dose.
An SEM photomicrograph showing tne cross section of the finished device is
shown in Figure 3.2. The lateral diffusion of the Be can be seen to be minimal,
and the diffusion touches the active layer of the device, as desired.
An ohmic contact to the p region is formed by a shallow zinc diffusion followed by plating of AuZn alloy, and the evaporation of Cr and Au. The contact is
alloyed at 460 °C. The substrate is lapped and the back n contact of AuGe and Au
is formed by vacuum evaporation and alloying at 380 °C. The contact resistance
of the finished devices is typically 10 0 .
This technique has the virtue of having high device yields due to the simple
processing involved. Typical pulsed threshold currents for a 7.5 µ.m stripe width
were 55 mA for 250 µm cavity length, and 30 mA for 125 µm cavity length. For a
3.5 µm wide stripe, these were 40 mA and 25 mA for 250 µm and 125 µm c~vity
length, respectively. The lowest pulsed threshold current found was 21 mA. CW
operation was established, with the threshold current approximately 30 percent
higher than pulsed operation. The stable near field pattern of the device is

- 38 -

Table 3.2

Be diffusion data for Gai-xAlzAs. Implanted dose'.
energy

=100 keV.

Anneal

=3 · 10 1s cm-2,

Junction

Sheet

Depth

Resistance

{µm)

(fl..)

Temperature

Time

(oc)

(min)

800

0.8

100

850

0.9

800

1.4

370

850

1.6

800

1.9

1000

850

2.1

0.00

0.45

0.65

- 39-

'!<""'"" •

·~- '•...

1.0µ.

Figur e 3.2

SEM photo micr ograp h of the later al cross secti on
impl anted laser .

of a beryl lium

-40-

shown in Figure 3.3 .. The light versus current characteristics- were linear and
kink-free up to 10 mW power output. The measured difierential quantum
efficiency was 45 percent.
One of the interesting aspects of laser diode operation, and a .revealing test of

the understanding of these devices, is the guiding mechanism that determines
the optical modes of the laser. In a device such as the beryllium implanted
laser, it is not obvious whether the optical modes are determined by the gain
profile generated by the injected carriers or by a change in the real refractive
index of the difiused beryllium.doped region. Experimentally, a measurement of
the near field and far field patterns of the laser and a comparison of the two
shows whether the device is gain or real refractive index guided. To show how
this happens, we must look at the wave equation defining the optical modes.
As a starting point; we consider the vector wave equation,

(3.1)
where k is the vacuum wavevector, k =~.and e is the complex relative permitc
tivity of the medium. Assumjng a waveguide mode with a z dependence of e'P2
and substituting we obtain

(3.2)
where V t is the transverse gradient operator,

v t =ex a x + ey a y

(3.3}

The influence of the planar mirrors on the device is to introduce the round trip
phase and gain conditions

- 41-

r, ::: 35 mA

12::: 40mA

CJ)

13 ~ 50mA
I3

I4 ~ 60mA

::::>

0::

Be implanted stripe
width= 7µm

-><{

CJ)

r-z

~10µ~

DISTANCE
Figure 3.3

The near field pattern of the beryllium implanted laser.

-42-

N,,.

Re( fJ)=L

(3.4)

2 Im( p ) = a + L In{ R )-

(3.5)

where L is the length of the laser cavity, N is an integer, a is the distributed loss
in the cavity, and R is the mirror reflectivity. These relations~ equations (3.2)
through (3.5), define the electromagnetic modes of the laser cavity. The manner
in which this differs; from other types of lasers is that the dominant guiding
mechanism in the device is the internal permittivity instead of the mirror
geometry.

If the real part of permittivity is the dominant mechanism in determining the
laser optical modes, the operator in the eigenequation (3.2) is essentially real or
Hermitian. This forces the bound eigensolutions to the equation to have no
phase variation in planes of constant z, or in other words to have planar wavefronts. On the other hand, if the imaginary part of the permittivity is important
to the waveguide problem, the bound eigensolutions have a phase variation in
the planes of constant z or curved wavefronts.
Using the fourier ·transform relation between the near field distribution at
the mirror surface and the far field angular distribution 12 we find the imaginary permittivity ("gain") guided device to have a wider far field intensity distribution than the real permittivity ("index") guided device. given the same near
field intensity distribution. This can be quantified for a device with Gaussian
field distributions as
W®;?; .Xln4
7T

(3.6}

where W is the half power width of the near field, 0 is the half power width of the

- 43 far field, and X is the free space wavelength. Equality holds for the real permittivity guided mode.
In the typical semiconductor laser, the waveguide consists' of a perturbed
dielectric slab. This means that in one direction (normal to the p-n junction)
there is a large step in the real 1part of the permittivity, which essentially determines the modal profile in this direction. This modal profile is index guided. In
the perpendicular direction, parallel to the p-n junction, the guiding mechanisms are much weaker and can be either gain or index determined. We wish to
determine which controls the behavior of this laser. Accordingly, the near and
far field intensity distributions of the device parallel to the p-n junction were
measured, and the result of this measurement is shown in Figure 3.4. The measured near field half power beamwidth is 2.3 µm and the far field half power
beamwidth is 14°. The intensity distributions closely resemble gaussian distributions, and they are compared with the gaussian beam formula, equation (3.6).
For the wavelength of these devices, 0.683 µm,

>.. ln 4
1T

=22 µm-degrees: The

experimental result shows W 9 ·= 32 ,um-degrees so it is concluded that these
devices are gain guided. Equivalently, we can compare the location of the virtual
beam waists in the direction normal to the junction plane where it is index
guided to that in the; plane of the junction using an infrared microscope. Astigmatism in the output beam from the laser diode (different beam waist locations)
indicates the presence of a gain guiding mechanism in the lateral direction. This
qualitative measurement was performed and the output beam of the laser diode
was seen to be highly astigmatic, agreeing with the conclusion that the devices
are gain guided in the lateral direction.
This is not a surprising result since these devices have no lateral structure to
introduce a lateral perturbation in the real part of the permittivity. What the
measurement indicates is that the beryllium implanted region does not have a

- 44-

- ·-.

( J)

Be implanted

:::>

Stripe width= 4µ.m

a::

I =2 Ith

Far Field

(J)

IE--10°~

Cl
_J

IJ..

ANGLE

CJ)

:::>

a::

Near Field

~2µ.m~

(J)

Cl
_J

IJ..

DI STANCE
Figure 3.4

A comparison of the near and far field patterns of the beryllium
implanted laser.

-45-

significantly dift'erent· real permittivity from the surrounding I.PE doped regions.
The active layer doping in the device described here was approximate ly
10 17 cm-8 • Devices with higher active layer dopings, up to 1018 cm-8 , had similar
behavior. This is characterist ic of semiconduc tor lasers, which operate at carrier densities of approximatel y110 18 cm- 8, so that the added dopant atoms play
a minor role in the device behavior.
One of the important features of this. device is its compatibilit y with semiinsulating substrates. This is by virtue of the closed structure used. The device
can be fabricated on semi-insulat ing substrate simply by incorporatin g one
more layer of epitaxial growth underneath the laser structure. This layer is n
doped GaA.s, which is required for good ohmic contact. In addition, by doping
this layer properly and controlling its· thickness, it can serve as the channel
layer for MESFETs. This device has been fabricated and is shown in Figure 3.5.
The bottom MESFET layer is doped approximate ly 10 16 cm-3 and grown approximately 0.8 µm. thick.
Fabrication of this device follows the procedure outlined above up to the
point where the p contact for the laser is alloyed. At this point, the laser mesa is
defined photolithogr aphically and etched with 1:8:8 (H 2S0 4 : H2 0 2 : H2 0). The n
side contact metal, AuGe and Au, is shadow evaporated onto the wafer, and the
MESFET drain and gate formed using the self aligned liftoff process described in
connection with the repeater integrated circuit. The n contact is alloyed at 380
°C and the wafer thinned and cleaved into individual devices.
The behavior of these lasers is slightly better than the devices on n type substrate. Pulsed threshold currents for 4 µm. stripe width and cavity lengths of 250
and 125 µm were 35 and 20 mA, respectively. The lowest threshold current was
15 mA for a 100 µm long cavity. The stable near field pattern of the device is

Fig ure 3.5
int eg rat ed MESFET dri ve

las er on sem i-il nsu lat ing
r.

Th e be ryl liu m im pla nte d

semi-insulatinQ GoAs

su bs tra te wi th an

o:i

.i:i-

-47-

shown in Figure 3.6.. The near field of this laser shows some asymmetry due to
the slight crowding of the injected carriers to the mesa side of the implanted
stripe. This occurs because of the sheet resistivity of the lower n type layers.
The light versus current characteristics were linear and kink free to 10 mW output power, and the external differential quantum efficiency was measured at 50
percent from both facets. The device was seen to operate in essentially one spectral mode as is shown in Figure·3.7.
The MESFET was used to modulate the light output of the laser by varying its
gate bias. This is shown in Figure 3.8.
Ill.4 Conclusions

Two devices important to the development of the OEIC have been presented
which make use of the unique advantages presented by ion implantation as a
fabrication technique. Particular attention has been paid to the questions of
process compatibility, reproducibility, device yield, and dimensional control.
These devices can represent significant improvement over their conventionally
fabricated counterparts in these aspects of device evaluation.
It is clear that ion implantation mu·st be seriously considered as an impor-

tant technique in fabrication of (AlGa)As optoelectronic devices.

-48-

I 1= I .10 Ith

I-

(/)

I 2= 1.27 Ith

I 3= 1.45 Ith

I4 = I .64Ith

I-

2µ

a::

DISTANCE
Figure 3.6

The near field pattern of the beryllium implanted laser on semiinsulating substrate.

- 49-

I =40mA
Ith= 22 mA

I-

-(/)

8770
8790
8810
8830
WAVELENGTH (A)
Figure 3.7

Emission spectrum for the beryllium implanted laser on semiinsulating substrate, showing single longitudinal mode operation.

- 50 -

Figure 3.8

Modulation of the beryllium implanted laser on semi-insulating substrate by the integrated MESFET. The laser current is shown on the
left side of the scale (10 mA per division) and the light output on the
right {arbitrary units) The gate voltage on the MESFET is varied at
0.5 V per step. The horizontal scale is 20 nS per division.

- 51-

DI.5 R.eferences for Chapter m

R. G. Hunsperger and N. Hirsch, "GaAs Field-Effect Transistors with IonImplanted Channels," Electron. Lett. 9, 577 (1973).

T. Mizutani, S. Ishida, and.:M. Fujimoto, "GaAs Field-Effect Transistors by
Sulfur-Ion Implantation," Electron. Lett. 12. 431 (1976).

3. W. S. Johnson and J. F. Gibbons {unpublished}.
4.

R. G. Hunsperger; R. G. Wilson, and D. M. Jamba, "Mg and Be Ion Implanted
GaAs," J. Appl. Phys. 43, 1318 (1972).

M. J. Helix, K. V. Vaidyanathan, and ..B. G. Streetman, "Properties of Belmplanted Planar GaAs p-n Junctions," IEEE J. Solid State Circuits SC-13, 426
(1978).

J. P. Donnelly, F. J. Leonberger, and C. 0. Bozler, "Uniform-CarrierConcentration p-Type Layers in GaAs by Beryllium Ion Implantation," Appl.
Phys. Lett. 28, 706 (1976).

W. V. McLevige, M. J. Helix, K. V. Vaidyanathan, and B. G. Streetman, "Electrical Profiling and Optical Activation Studies of Be-Implanted GaAs," J. Appl.
Phys. 48, 3342 (1977).

B. W. V. McLevige, K. V. Vaidyanathan, and B. G. Streetman, "Diffusion Studies of
Be-Implanted GaAs by SIMS and Electrical Profiling," Solid State Commun.
25. 1003 (1978).
9.

M. K. Barnoski, R. G. Hunsperger, and A. Lee, "Ion Implanted GaAs Injection
Laser," Appl. Phys. Lett. 24, 627 (1974).

10. N. Bar-Chaim, M. Lanir, S. Margalit, I. Ury, D. Wilt, M. Yust, and A. Yariv, "Belmplanted {GaAl)As Stripe Geometry Lasers," Appl. Phys. Lett.
(1980).

36, 233

- 5211. D. Wilt, N. Bar-Chaim, S. Margalit, I. Ury, M. Yust, and A. Yariv, "Low Threshold
Be Implanted (GaA.l)As Laser on Semi-Insulat ing Substrate," IEEE J. Quant.
Electron. QE-16, 390 (1980).
12. H. C. Casey, Jr., and M. B. Panish, Heterostruc ture Lasers: Part A. Fundamental Principles; Part B, Materials and: Operating Characterist ics, Academic

Press, New York (1978).

- 53 -

Chapter IV
A Steady State lateral Model of the Double Heterostructure Lase~
IV.I Introduction

It has become clear in recent.years that the semiconductor laser has a multitude of practical applications.; These include the field of optical communications, but also many others that can make use of the small size, high efficiency,
ruggedness, and simplicity of this component. With so many applications and
differing requirements for the devices needed, it can be seen that laser diode
modelling is potentially a tool of great value. Both the device designer, who tries
to optimize a device computationally before entering the laboratory, and the
diagnostician, who tries to understand a device he has fabricated, are potential
users for laser diode models.
As a comparison, the bipolar.transistor, the MESFET, the MOSFET, and a, large
number of other electronic devices all have complex finite element models available to the interested designer so that he can both optimize and analyze these
devices. With these models, an accurate estimation of the effects of geometry,
doping, contacts, and many other aspects of devices can be made with extremely
large savings in time and effort.
The laser diode has seen, perhaps, more variations in device geometry than
any other electronic component.

This is because the laser diode has many

aspects the designer can interact with. As an electronic component, a great deal
of effort has been made to confine the current flow through the device. This
results in low threshold current and high quantum efficiency. On the other
hand, as an optical component, a great deal of effort 1:las been made to build
dielectric waveguiding into the device lo define the lateral optical modes. The
advantage of this is that the modal characteristics of the laser are well defined,
and single lateral mode operation and linear light versus current characteristics

- 54-

are easier to achieve. In addition, to get a large power output from the. laser
diode, it is necessary to keep the lateral optical mode spread out to avoid
damaging the mirror face ts. This calls for the use of weak lateral guiding
mechanisms. A final aspect involves the fabrication limitations of the semiconductor material used.
In spite of the need, the status of laser dio\ie modelling is quite primitive. A
large number of authors have constructed highly simplified and idealized
models of the double·heterostructure laser to illustrate qualitatively the effects
of various. material. and structural parameters on device behavior 1 • These
models can be quite useful in correlating observed laser threshold currents with
experimental results, for instance, but are of littie use in understanding the
device behavior above lasing threshold. This, however, is perhaps the most
important regime of laser diode performance. In addition, these models are not
predictive; in practical application, they explain experimental results.
There are at present two general models of the laser diode above lasing
threshold 2•3 • However, both of these models make assumptions concerning the
electrical characteristics of the diode that are incorrect. Specifically, in each
model the diode p-n junction is assumed to behave according to the law

(4.1)
where j represents the injected electron and bole current densities (which are
assumed to be equal}, fo and n are material parameters, rp is the junction voltage, q is the electronic charge, k is Boltzmann's constant, and T is the absolute
temperature. This is not a fundamental relationship. It can be derived for a one
dimensional p-n junction from the master equations for semiconductors, and is
usually described as the Shockley equation after its discoverer. The use of this
relationship to describe the p-n junction in the laser diode, even as an approxi-

- 55 mation, neglects two very important effects: first, the effect on the electrical
characteri stics of lateral carrier drift and diffusion, and second, the saturation
of junction voltage (and carrier population s) associated with lasing threshold.
This effect, first described and measured by Paoli 4 , is associated with the
saturation of optical gain at lasing threshold. This causes the carrier populations to clamp, and the separation in carrier quasiferm i levels to clamp as well.
This causes a clamping of the junction voltage in the laser. A more reasonable
condition to apply to the diode junction in the double heterostru cture laser is to
assume the continuity of the carrier quasiferm i levels across the heterojunc tion
interface. This assumptio n leads naturally to the saturation of the diode voltage
as lasing threshold, and is consistent with the physics of the semicondu ctor.
However, the use of this model of the diode junction requires the use of a
different model and solution method from that of either of the models already
cited.
Another model specificall y designed to treat the behavior of a narrow planar
stripe laser introduces this type of assumptio n 5 • However, this model is constructed such a way that it does not lend itself to generaliza tion.
In this chapter, a steady state model of the double heterostru cture laser is
presented which treats the diode junction in the correct manner as described
above 6 • Fundamen tal relationsh ips that describe the device electrical and optical characteri stics are derived· and simultane ously solved in a self consistent
manner to yield both the electrical and optical behavior of the device. The model
is designed for use both above and below lasing threshold. To give as much freedom as possible in the treatment of device geometry, the finite element method
is adopted as a solution technique. A number of interesting geometrie s have
been examined and specific results for one such geometry are presented as an
example.

- 56 -

IV.2 Simplifyin£ Assumptions used·in the Model

In principle, the semiconductor laser is a well understood device. It obeys.
both the master equations for semiconductors and Maxwell's equations, and if
the material parameters for the device are known, its behavior can be calculated.
It is very important, however; to distinguish between understanding in principle and understanding in practice. If the relationships which describe the device
cannot be solved in a reasonable amount of calculation time, then in a sense the
device cannot be said to be understood. This statement describes the present
state of modelling of the semiconductor laser. To model the semiconductor
laser fully as both a semiconductor device and optical device would require
more calculating ability than is considered reasonable at present.
There are a number of physical approximations that can be made in the
equations that describe the semiconductor laser that still allow one to treat a
large class of semiconductor laser diodes with good calculating accuracy and
retain the desirable aspect of reasonable computational complexity. These
approximations are explained in this section.
Again, it is important to distinguish between approximations and ad-hoc
assumptions. This is in contrast to many popular laser diode models which begin
with such assumptions, leaving unclear the question of their validity or accuracy
for the structure analyzed. The advantage of approaching the problem from
this standpoint is that the limits of the validity of the model are clearly outlined, and if at some point these limits are crossed, the model can be changed as
necessary to remain valid.
The first approximation in this model is to average over longitudinal effects in
the laser. Since most semiconductor lasers are fabricated in a stripe geometry
with planar mirrors, they depend upon the lateral waveguiding in the device to

- 57 -

define the optical modes. From the longitudinal dimension we get the round
trip phase condition which defines the Fabry-Perot longitudinal modes, and the
round trip gain condition for stimulated emission, that gain balances loss. For
typical cavity lengths of the laser, say 250 µm, the Fabry-Perot modes are quite
closely spaced,

(4.2)

where A. is the center wavelength, n is the refractive index of the material,:and L
is the longitudinal length of the device. This distribution may be considered as a
continuum compared with the gain bandwidth of the semiconductor material
which is on the order of several hundred A. Thus for this model, the question of
longitudinal mode structure is ignored. Alllongitudinal modes are considered to
have the same lateral waveguide mode profile and the laser is assumed to oscillate at the peak of the gain distribution.
Due to the finite reflectivity of the laser mirrors, there is a longitudinal. variation in the stimulated power distribution and thus the stimulated recombination rate. These effects are not negligible but represent an uninteresting variation. Therefore, they are averaged over, and only a fictitious average lateral
cross section of the device is treated. This immediately turns the three dimensional problem of modelling the laser into a two dimensional problem of modelling the "average" cross section. The only averaging that need actually be done
is to determine the relationship between the average power flow in the laser cavity and the emitted power from the laser mirrors. This is done as follows.
Assume a mirror reflectivity R for the mirrors and a device length L. The power
distribution in the cavity is then

Pcavity =Po cosh(

~ ln( ~ ) )

(4.3)

- 58 where z is the variable representing the longitudinal dimension, and Po is an
unimportant scaling factor. The average power in the. cavity is then

Pave=

Po ( 1 - R)

vRln( ~ )

(4.4)

and the power emitted from the mirror facets is

Pemit

Po ( 1 - R)

YR"

(4.5)

so that

Pemit. = Pava In( R )

(4.6)

With the averaging over the longitudinal direction, th.e gain condition becomes
~ode = amode + - ln( ~)

(4. 7)

~"mode·

where gm.ode is the gain experienced by the mode, 1Xmode is the loss for the:mode
{for example, due to scattering)~ and Rmode is the mirror facet reflectivity for the
mode, all evaluated for the ·average cross section of the laser.
The second approximation is to assume that the active layer of the laser is
thin compared to the diffusion lengths of the carriers. Typically this condition is
met quite easily, for good practical reasons. To control the optical modal profile
in this direction, active layer thicknesses are usually required to be less than 0.5
µm. This is approximately a factor of ten less than a typical diffusion length.

With this assumption, the carrier densities in the active region can be assumed
to be constant across the active layer thickness. Accordingly, quantities which
vary across the active layer thickness, such as the stimulated recombination
rate, are averaged in this dimension. This reduces the modelling of the active
layer (but not the laser) to a one dimensional problem.

- 59 -

A third approximation is to assume that the cladding layers to the active
region which provide ·the optical and carrier confinement neces.sary for the double heterostructure laser have bandgap energies large enough so that only the
majority carrier conduction current fl.ow in these layers need be considered. A
double heterostructure laser that did not meet this requirement would be very
poor indeed. Note that this does not prohibit the inclusion of a minority carrier
leakage term in the treatment of the active region, it merely says that such
leakage is a minor perturbation to the ohmic conduction of majority carriers in
the cladding. This reduces the two dimensional lateral cross section modelling to
the solution of a linear ohmic conduction problem. Thus in the cladding layers

we must solve Laplace's equation
V -(uV 9')=0

(4.8)

where u is the conductivity of the material, and 9' is the electrostatic potential.
It should be noted that the word "cladding" is taken to mean all regions that
surround the active layer and conduct the majority current. This could include,
for instance, contact layers and the substrate of the device. The "large
bandgap" requirement applies only to the regions of the cladding which come
into contact with the active layer.
A fourth assumption used in this model is much harder to quantify without
presenting the mathematics of waveguiding in the semiconductor laser. This will
be treated in detail later in this chapter. Briefly, however, the method used to
find the optical modes of the semiconductor laser, (effective index formalism}
involves the solution of the waveguide eigenmode equation by a variational technique. This variational technique solves a reduced modal eigenequation in the
direction normal to the active layer, and uses this solution to derive simple
modal eigenequation. for the direction parallel to the active layer. It is assumed
that the tightly confined solution normal to the active layer is only slightly

- 60 -

perturbed by the lateral variation in the optical mode. Qualitatively, this
assumes that the lateral variations in refractive index are much less than the
normal variations in refractive index.
To treat now the problem of modelling the semiconductor laser, we break it
up into two coupled .subproblems. The first problem is that of the pumping process in the device, an electrical process involving the injection of electrons and
holes {at extremely high densities) into the active layer. This we label the electrical problem. The second problem involves finding the optical modes of the
device. This we label the optical problem. These two subproblems are coupled in
two ways. The optical modes of the device are in part determined by the injected
carriers through their modification of the complex dielectric constant. This
includes the effect of optical gain. In addition, the optical modes of the device
stimulate carrier recombination, which modifies the carrier density. Both of
these subproblems must be treated consistently with the coupling between
them.
IV.3 The Electrical Model

The typical geometry of the device we wish to model is shown in Figure 4.1. It
consists of two ohmic conduction regions, one p type semiconductor, the other
n type semiconductor, and a thin active region which is partially sandwiched
between them and partially surrounded by isotype cladding layers {in this case,
n type). The only cases that we exclude at this point are those where significant
injection occurs across a homojunction in the active layer or from a remote
junclion in the device. Using the assumptions in the previous section, the electrical modelling problem breaks up into four coupled subproblems: two ohmic
conduction problems in the cladding layers, and two continuity relationships in
the active layer.
In the cladding layers, we must solve Laplace's equation

active layer

5 2n

s,p
s2p

I I 1 \ \

DH Laser
lateral cross section

IF~~--~~--

S2n

Late ral cross secti on of the typic al doub le hete
rostr uctu re laser .
The left porti on of this figur e show s the regio ns
and boun dary surfaces defin ed in the calcu latio n, and the right
porti on show s the
curre nt flow throu gh the devic e.

\ sin

cr(x,y)

~·

s2p

Figu re 4.1

S2n

DH Laser
lateral cross section

.......

o:i

-62 -

V • ( u(x,y) V rpp } = 0

(4.9)

(4.10)

V · ( u(x,y) V 'f'n )

where u is the conductivity of the material, and 'Pp and 'Pn are the electrostatic
potential in the p and n regions. respectively. These equations are subject to the
following boundary conditions

rpp

=rfJpo on S tp

(4.11)

f/Jn

=f/JnO on Sin

(4.12)

(]' n . v f/lp =0 on s2p

(4.13)

(]' n . v rfJn =0

(4.14)

s2n

rpp

=f/Jp(y) on Ssp

(4.15)

'fin

=f/ln(Y) on Ssn

(4.16)

S1p and S1n represent equtpotential ohmic contacts, to the device, S2p and S2n

represent surfaces where no normal current is allowed to flow, and Ssp and San
represent the p-n heterojunction surfaces, where the potentialis assumed as a
function of the lateral coordinate to the interface. The outward pointing normal
to the surfaces is represented by n.
The solution to this problem yields the injected current density into the
active layer from each cladding layer

n . v rfJp on Sap

(4.17)

=- (]' ii . V fPn on Ssn

(4.18)

ip.inj = - CT

inJnj

- 63 -

as well as the potential distribution inside each of the claddinglayers, which for
self consistency must be related to the potential distribution along the active
layer.
This relationship is provided· in this model by the: boundary condition on the
heterojunction interfaces and the semiconductor continuity relationships. This
is in contrast to previous models where the Shockley relationship (4.1) is used
for this purpose. In comparison, the resulting relationship used here between
injected current and potential difi'erence across the p-n junction is both implicit
and nonlocal, making the solution of the relationships comparatively much
more difficult even though physically correct.
Referring to Figure 4.2, we have drawn a representative band diagram of the
p-n heterojunction interface under forward bias. The detailed spike structure at
the interfaces is assumed to be washed out by interfacial mixing as occurs in
liquid phase epitaxial material. In this diagram, the carrier quasifermi levels
appear as straight lines due to the assumption that the active layer is thin compared to the diffusion length. The actual variation in the quasifermi level in the
active region can be shown to be on the order of /J. 'ljJ

=k T ( tLaot ) where tact is
2,

the active layer thickness, 1if is the diffusion length, and the high injection
assumption has been used, which is negligible in this calculation. Outside the
active region, the minority carrier quasif ermi levels decay to the majority carrier fermi level in a distance approximately equal to the diffusion length, which
is much larger than the scale in this figure. In the case where the active layer is
surrounded by isotype cladding, again the continuity of quasifermi levels is
assumed.
With this assumption and Poisson's equation for the electrostatic potential in
the active layer,

Figure 4.2

n (AIGa)As

~o~ ~:s~erm~ level ~P

p (AIGa)As

~ctron quasifermi level '¥n

double heterost ructure laser.

Represen tative band structur e diagram for the diode junction in a

Ga As

0.1µ.m typical

,po.

- 65 -

(4.19)

where rp is the electrostatic potential, p is the charge density,. e is the relative
permittivity, and e0 is the permittivity of free space, we can relate the electron
and hole densities in the active layer to the potential distribution along the p-n
heterojunction. Noting that the, typical Debye length for these devices is on the
order of !OOA, we will assume quasineutrality and write

p +Nd'= n + N;

{4.20)

2 NF (E.,-'l/lp
=Vrr
" 1le kT ) .

{4.21)

(4.22)

(4.23)

'I/In - 'I/Ip

=e { 'Pp - 'Pn )

{4.24)

where n and p are the electron and hole densities, Nt and N; are the ionized
donor and acceptor densities, N0 and N., are the etrective densities of states in
the conduction and valence bands, Fi;2 is the fermi function, 'I/In and 1/lp are electron and hole quasifermi levels, E 0 and E., are conduction and valence band
edges, E, is the bandgap of the active layer, and 'Pn and 'Pp are the electrostatic
potential on either side of the p·n heterojunction.
These equations completely define the electron and hole densities as a function of the potential difference across the p-n beterojunction, 'Pp - 'Pn·
To relate the injected current density to the potential distribution along the
active layer, we must consider the continuity relationships

- 66 -

.!!E.
....
dt =Gp - Up - -;;- V • Jp =0

(4.25)

+l.v
·Jn=O

{4.26)

dn =G -U

Gn and Gp are electron and hole generation rates, Un and Up are electron and

hole recombination rates, and ln {:tp) is the electron (hole) drift plus diffusion
current.
Injected current can most easily be included in these equations as a generation term. Thermal generation can be neglected for the laser diode under high
forward bias.
- jp,inj
Gp-

(4.27)

__
Gn-

{4.28)

e tact.
jn,inj
e tact

where jpJnJ and jnJnj are the injected current densities, e is the electronic charge,
and tact. is the active layer thickness.
The recombination terms consist of both nonrad.iative (trap, surface recombination} and radiative (spontaneous and stimulated) terms. The forms used for
these are

Up.trap = Un.trap = _

_...p_n_ _

P 1"n + n Tp

(4.29)

S 6( y - Ys) p n
Up.surface= Un.surface=------ p +n

(4.30)

Up.spent= Un.spent= BP n

(4.31}

U p,st.lm -u
n,st.lm -.EA
fu.J

(4.32}

- 67where '1"11 and 'rp are effective nonradiative minority carrier lifetimes, and may
include the effects of leakage over the confining heterojunction barriers. S is a
surface recombination velocity, y9 being the location of the surface interface. B
is a material constant, P is the:optical power density, g is the local optical gain
of the medium, and .fu..i is the. photon energy. In this model, the gain term is
assumed to be of the .form

=So + gip P + gin n + S2pn P n

(4.33)

These relationships are simplified forms of more general relationships 7 , making
use of the fact that the laser diode operates in the high forward bias regime. Of
course, to be consistent ·with the assumption that the active iayer is thin and
that the electron and hole densities are uniform across it, the relationship for
the stimulated emission recombination rate, ( 4.32), must be averaged over the
direction normal to the active layer.
The drift plus diffusion term that appears in the continuity equations
requires more elaboration. Using the degenerate Einstein relations, we have

=p µp v 'I/Ip

(4.34)

=n l4i. V 'I/In

(4.35)

where /4i. and /.1.p are the electron and hole mobilities. An additional complication that we wish to incorporate into this model is the case where the active
layer thickness is allowed to vary. Since we have already separated off the
injected current densities from this drift plus diffusion term, to be consistent we
must require the latter to be conservative, meaning that no drift plus diffusion
current may flow through the heterojunction interfaces, or equivalently to force
the drift and diffusion current to ft.ow parallel to the heterojunction interfaces.
We will assume that the magnitude of this current is constant across the active

- 68layer, consistent with our assumption that the injected carrier densities are
constant across the active layer, but the changing of the active layer thickness
introduces an additional term when we take the divergence in the continuity
equations {4.25) and (4.26). With the condition that the active layer thickness
varies slowly with respect to y and using a local system of cylindrical coordinates, we find for these terms

_ ..!. V . J = _ ..!. ( _1_ dtact + ..£__ ) (

tact

d'if/p )
p /1-p dy

(4.36)

(4.37)

These terms are seen to be conservative, as is desired. The derivatives of the
quasifermi levels that appear in these terms must of course be treated self consistently with the solution to the ohmic conduction problem. The identification
is provided by the assumption of continuity of quasifermi levels across the interfaces, as before. Neglecting the contribution of carriers that leak over the
confining heterobarriers, this allows us to identify 'fin with the fermi level in the
n cladding, and 'if/p with the fermi level in the p cladding. In the case where the
active layer is surrounded by isotype cladding, this identification is performed
for the majority carrier, and for the minority carrier we instead demand that
injected minority carrier current density be zero.
With these relationships, the electrical behavior of the device is completely
defined. It is interesting to note that at no point in the analysis was the assumption of equal injected current densities into the active layer or the assumption
of ambipolar diffusion required. These are. not necessarily bad assumptions, but
cannot be derived from the relationships above. If one could derive the equality
of injected current densities, one could also derive ambipolar diffusion and drift
coefficients. This stems from the assumption of equal generation and recombi-

- 69 -

nation rates in the derivation of these quantities.
The difficulty lies in the fact that the electron and hole populations are essentially in equilibrium with their isotype cladding layers. An interesting consequence of this is that mirror symmetric devices where p and n type layers are
interchanged but resistivities are held constant do not behave identically.
From the standpoint of solving the electrical behavior of the model, the ·problem is to find an electrostatic potential distribution and quasifermi levels in the
active layer which are consistent with all of the relationships set down above.
IV.4 The Optical Model.·

The optical model presented here is quite similar to that described elsewhere 2•3 • In brief, a two dimensional scalar modal equation for the laser
waveguide is derived· from the vector wave equation with the assumption of a TE
mode and uniformity in the longitudinal (z) direction. This scalar modal equation is approximately solved using a technique known as effective permittivity
formalism, presented here in a new variational form which makes clear the
approximations involved. If the exact permittivity profile is used, the modal gain
is available in the imaginary part of the mode propagation constant. Otherwise,
a first order perturbation calculation is used ("modal overlap") to find the
modal gain.
In this respect, semiconductor lasers fall into two equivalence classes, those
where injected carriers contribute significantly to the waveguiding in the device,
and those where injected carriers serve as only a minor perturbation upon the
waveguide modes. Roughly speaking, these two classes correspond to devices
with geometric structures that define the waveguide modes (e.g. buried heterostructure lasers

and channelled substrate lasers 9 •10 ) and devices that have

no built in geometric waveguide structure (e.g. beryllium implanted lasers

and oxide stripe lasers 12 ). This is a rough division because an important class

- 70 of laser diodes utilizes geometric antiguiding as a means of mode control 13 •
The optical model presented here. while for convenience limited to TE modes
and effective index formalism. is capable of treating both classes of semiconductor lasers.
The vector wave equation, which we take as our starting point, is

(4.38)
where k is the wavenumber for the frequency of interest (k =· .E.) and E is the

complex relative permittivity for the medium. This wave equation is derived with
the assumption that the scale of variation for

is much larger than a

wavelength so that a term involving the gradient of e can be dropped. i!: is of
course the electric fi.eld for the mode. assumed to vary as e1 ~t.
With the assumption that we are interested in a waveguide mode which varies
as e 1P2 where the z axis is chosen to be parallel to the guide, we get

( v f + k 2 E - {12 ) E =0
where "V f is a transverse Laplacian operator {V f

(4.39)

a2
aa
=--'2
+ ...2 ).
ax
ay

Semiconductor lasers are most often fabricated with a waveguide that is
tightly constricted in one direction and weakly constricted in the other. In the
tightly constricted direction, normal to the active layer, the waveguiding is dominated by the discontinuity in permittivity between the active layer and the
larger bandgap cladding layers. In order to achieve modal control in this direction, the active layer is made quite thin {on the order of 0.2 µm} so that the
lowest modes are much more tightly confined than the higher modes. In the
other direction, parallel to the active layer, the waveguiding is usually weak.
Thus, the waveguide modes split in a manner similar to the uniform planar
waveguide, into pseudo-TE and pseudo-TM modes. The pseudo-TE modes are

- 71 -

known to dominate the behavior of the semiconductor laser. This is because
they are more tightly confined to the gain region (the active layer) and also have
higher reflectivities otr of the cleaved mirrors.
For this reason, we will assume a TE mode for the laser and solve a scalar
modal eigenequation.
(4.40)

where u represents the TE electric field, E1 • We have taken the y coordinate, as
earlier, to be parallel to the active layer. This eigenequation can be put into
variational form as

j dx dy ( - ( Bu
) 2 - ( ~ ) 2 + k2 e u2 )
ax
ay

-m

...

J dx dyu

) =O

(4.41)

To apply effective permittivity formalism to this equation, a variational form will
be assumed and this variational principle will be used to derive an Euler equation for the lateral modal field.
Keeping in mind the situation that the mode is tightly confined in one dimension (normal to the active layer) and weakly confined in the other, we might
imagine that the profile of the mode in the direction normal to the active layer
is quite similar to the profile for a uniform planar guide. Accordingly, we solve
the one dimensional eigenequation in the direction normal to the active layer

( -

ax 2

+ k2 e(x,y) - 7f(y) ) X(x,y) =0

(4.42)

for an effective variation in the direction normal to the active layer (x). The
lateral coordinate y is considered here to be a parameter, and we use the solution with the largest eigenvalue (the lowest mode}. Consistent with the use of

- 72 complex permittivities. the normalization condition on this field is taken to be

fdxx2=1

(4.43)

We would now like to find the best possible approximation to the true modal field
using this field X to represent the variation in the perpendicular (x) direction.
To do this, we assume a variational form to substitute into the variational equation (4.41)

u(x,y)

=X(x,y) Y(y)

(4.44}

where Y(y} is the function we will allow to vary. Since we are no longer interested
in the variation in the x direction, we may integrate over x in the variational
equation and get as a result

j dy ( _ ( dY ) ~ + ( 'i'r + 'i'~ } y2 )
0 { -m

dy •

(4.45)

f dyY2

-·
where

(4.46)

The normalization condition (4.43) and the field equation for X{x.y) (4.42) have
been used in the simplification of the expression (4.45).
The Euler equation for this variational expression is then

( dy2 + k

where the effective permittivity is

£eft -

fJ 2 ) Y =0

(4.47)

- 73 -

(4.48)
The second term in this expression, 71, is usually quite small and is neglected
here. However, it is important to keep·. in mind when the structure to be
analyzed has abrupt changes in geometry in the y direction. In this case, it is
wiser to use a slightly different variational form than ( 4.44). This leaves us with
the expression

(4.49)
for the effective permittivity. The field Y will be assumed to be normalized
according to

f dyY =1

(4.50)

so that the normalization on the field u is then

J dx dy u =1

(4.51)

The advantage of approaching the effective permittivity problem from the standpoint of the variational principle, aside from the term (4.46} which we have
chosen to neglect, is that it assures in some sense that the best possible approximation to the true modal field has been obtained~ If first order perturbation
theory is applied to the modal profiles found, {assuming for the moment that
the extra term has not been neglected) the lowest nonzero correction to the
modal field involve overlaps of the field X with higher order modes in the x direction. This means that corrections associated with overlaps of X with itself are
not present. This quantifies the approximation in effective index formalism, in
contrast to the literature on this subject where the form for u (4.44) is substituted directly into the modal equation 2•3 •

- 74The modal gain is made ava:ilable as the imaginary part of the propagation
constant for the mode.

(4.52)

gmode = 2 Im Pmode

If the proper permittivity is used in the calculation. this result is exact. Otherwise, a first order perturbation calculation gives

d ( (32 )

=J dx dy u 2 k 2 de
--

(4.53)

this simplifies to
2 ..

d(3

=1L
J dx dy u 2 dt
2 (3 _,.

(4.54)

To treat a laser, one must of course include the effects of the longitudinal
cavity. As mentioned earlier, this amounts to the condition that modal gain is
equal to modal losses,

gmode

=Clmode + L1 In( ~. 1

~"mode

(4.55)

where imode is the modal gain, Clmode is the. modal loss, Lis the device length, and
Rmode is the mode mirror reflectivity.

In this model, the loss term Clmode and the

mirror reflectivity Rmode are assumed to be constants. It is important to note
that there is a difference between gain and loss. This difference is that the gain
term appears in the stimulated emission term in the electrical model. The loss
term here represents nonretrievable losses in the optical mode due to such
mechanisms as scattering. Assuming that all modes have the same mirror
reflectivity is a good approximation since we have taken them all to have the
same modal profile in the x direction where the large variation in permittivity is.

- 75 The optical power density in the device is

p = P1 ..

I u 12

(4.56)

f dx dy I u 12

_..,

where P1 is the total power flowing in the cavity, the average over the length of
the device of the forward and backward travelling waves. As before, this power is
related to that emitted from the mirrors by

(4.57)

where P 0 is the power emitted from both mirrors of the device.
If the device under consideration is judged to be a member of the geometri-

cally guided class, the lateral mode profiles need be calculated only once and
perturbation theory can be applied to ·determine modal gains for any value of
the injected carrier ·populations. If the device is judged to be of the carrier
guided class, the lateral mode profiles must be recalculated for every value of
the carrier populations, in particular, while the solution to the electrical problem is being iteratively se6:I"ched for. If the device is carrier guided, the dependence of both the real and imaginary parts of the permittivity on the carrier
density must be included in the calculation. This model is capable of treating
both types of device, including the more reasonable procedure for the carrier
guided device of using a combination of exact and perturbation methods.
It is an unfortunate fact, however, that the dependence of real permittivity
upon carrier concentration is not well known. This means that the carrier
guided device represents not only a larger investment in computation time, but
also throws doubt on the validity of the calculated results, as the mode profiles
in this device can depend strongly upon the magnitude of the change in the real
part of the permittivity caused by the injected carriers. It is interesting to note

- 76 that the geometrically guided devices are not only easier to analyze, they are
better behaved in general than the carrier guided devices in terms of their
characteristics such as modal behavior. For this reason, the example of the
analysis program presented here is an analysis of a geometrically guided device.
IV.5 Solution Technique

As a first step, the functional dependence between junction voltage and
injected carrier concentrations in the active layer is solved, equations (4.20)
through (4.24). This is done using a nonlinear root:finding technique. Since this
is only dependent upon the material used for the active layer, it need be done
only once for a given material and doping density.
The remaining relationships to be solved (the ohmic conduction problems,
the continuity relationships in the active layer, the waveguide modal problem)
all have spatial dependences which are influenced by the geometry of the laser
diode. One of the major goals in the formulation of the laser diode model
presented here was to include the effect of diode geometry on performance. With
the proliferation of diode geometries, the advantage of using a general modelling
technique was obvious. Thus, a very general technique was chosen to model
these spatially dependent problems. This technique, known as the finite element
method,

has

been · applied

both

semiconductor . devices

and

optical

waveguides, but never before to the problem of the laser diode.
The finite element method encompasses a large class of approximation
methods for the solution of differential equations. In the most general sense, it
consists of the use of a class of functions which appear somewhat like "patchwork quilts" to approximate the true solution to the differential equation. To
define these patchwork functions, the solution domain is broken up into many
smaller subdomains, called elements. Local functions, called interpolation functions, are defined in each element in terms of local parameters. In one dimen-

.;. 77 sion, an element could be a line segment, and the local parameters could be the
function values at the endpoints. The interpolation functions could then be
functions which interpolate linearly between the endpoint values. A patchwork
function of this sort has the obvious advantage of continuity over the domain. A
more complex set of parameters {values and slopes at the endpoints) and cubic
interpolation functions {the first order Hermite polynomials) would yield a
patchwork function with continuity of both the function and its derivative.
In two dimensions, the simplest element one can imagine would be a triangle,
with parameters being the function values at the corners. The interpolation
functions could again be linear and the function values inside the element would
correspond to passing a plane through the three points defined by the corner
values. This choice has the advantage of continuity for the resulting patchwork
function.
Once we have a class of these patchwork approximation functions which are
determined by a finite set of parameters {e.g. the function values at the nodes of
the elements) we need only find an error criterion to choose the best approximating function in the class. This can be relatively easy in some cases, and quite
hard in others. The easiest case is one where a variational form exists for the
equation whose solution is sought. In this case, one need only substitute the
patchwork function into the variational form, differentiate it with respect to its
parameters, and set the result to zero. Of course, the patchwork functions in
this case must have sufficient continuity as demanded by the variational principle. This requirement is usually less strict than the differential equation that the
function must satisfy. The result gives exactly as many conditions on the
parameters as are needed. In addition, this procedure leads to the best approximating function available in the class.
If. on the other hand, a variational principle does not exist or cannot be

- 78 found, one needs to derive enough error criteria in some other fashion to define
the solution to the problem. Usually the approach taken to this problem is
Galerkin's criterion. Suppose that the patchwork function can be defined as follows:

f a .... ·8n -1

~ aI fI
L.J
1=1

(4.58)

where a 1 are the parameters, and f1 are the local interpolation functions, defined
to be zero outside the element they are associated with. We desire to find the
solution of an equation which we will represent by

Af-g=O

(4.59)

where A is a generaloperator , not necessarily linear, and g is a driving function.
If a function which is not a solution of this equation is substituted, in general
the right hand side of this equation will not be zero. One possible choice of
error criteria, then, is to choose n suitable weightingfunctio ns w1 , i

=1, ... ,n and

define the errors

. e1 =f(Afa1, .... ,an -g)w1

(4.60)

The variational parameters are then chosen by demanding that these errors
e1 , i

= 1, ... ,n be zero.

The remaining problem is to choose these weighting functions w1• Galerkin's
method consists of using the variational functions f1 for this purpose. This
choice has the benefit of yielding the same finite element equations as the variational principle if it exists and the operator A is linear. However, the continuity
requirements in this approach to the problem are usually much more strict
than the variational approach.

- 79 The finite element method .is applied 'to this model of the laser diode in
several ways. For the ohmic conduction problem the equations {4.9) and (4.10)
may be converted to a variational form

o( ff dx dy ~ { v , )2 + f dS ·1, ) =o

(4.61)

Here the quantity dS is an oriented surf ace element on the boundary of the
solution region, in the direction of the outward pointing normal, n. The current
flowing out of the region at any point is just n . j.
For this problem, a system. of triangular elements and linear interpolation
functions as described earlier is used. The finite element equations take the
form of a linear algebraic system. The result of this calculation can be stated in
the form of a Green's function for each ohmic conduction region which relates
the nodal values of the potential along the active layer to the normal current
flow into the active layer,

J dy jPJni f 1 =TKlj ( 'po - 'Pl )

(4.62)

{4.63)

where the potentials on the contacts are ,po and ~no as before, and 'Pl and V'nj
are the nodal potentials along the heterojunction interface.
For the optical mode profile, the variational form X(x,y) is obtained from the
treatment of the slab waveguide in the direction normal to the active layer. The
remaining modal equation for' Y, which has a variational form appearing in
equation (4.45), is formulated as a finite element problem with a one dimensional grid and first order Hermite interpolation functions. The finite element
equations take the form of a generalized eigenvalue problem, which is solved by
the use of an efficient iterative technique 14 .

- 80 -

The remaining continuity relationships in the active layer are also solved
using a finite element technique. Again, a one dimensional grid is used and the
nodal variables are defined as the electron and hole quasifermi levels. Li.near
interpolation is used between nodes. These equations are highly nonlinear, and
no variational principle could be found to define the finite element equations.
Thus Galerkin's criterion was adopted for this problem, and the resulting finite
element equations take the form of a system of nonlinear simultaneous equations. This system is .solved through the use of a modified Newton's method technique. It should be noted that the mesh in this problem corresponds to the
boundary mesh for the ohmic conduction problem. With the use of linear. interpolation functions for the continuity relationships, as in the ohmic conduction
problem, the results. of the calculation in equations (4.62) and (4.63) may be
substituted directly.
The solution of the model then reduces to finding the nodal values of the electron and hole quasifermi levels as functions of a global boundary condition,
which may be formulated as either the device voltage V'po - ~nO• or the total
device current. With the inclusion of stimulated emission, additional variables
{stimulated power in each mode) and additional relationships (modal gain condition, equation {4.55)) must be considered. In this calculation, up to four
lateral modes are included for stimulated emission.
IV.6 Sample Case and Comparison with Experiment

Several device structures have been analyzed, including both .cases where carriers are treated as a perturbation and where carri.ers define the lateral .mode
structure. Lasers of the first type analyzed include both embedded lasers 16 and
channelled substrate lasers 9 •10 • Only one laser structure of the second type has
been analyzed, the beryllium implanted laser of Chapter Ill 11 . Specific results
are presented here for the structure of Burnham et al. 10 which has been

- 81 -

analyzed in simplified form by Streifer et al. 16•17 • Unfortunately. that analysis
neglects the effect mentioned in connection with equations (4.36) and (4.37) and
as a result the solution of the diffusion equation in this analysis is incorrect, as
it does not conserve carriers.
The structure of this device is shown in Figure 4.3. The material and structural parameters assumed for the device are listed in Tables 4.1 and 4.2 respect_ively. The n GaAs top layer in the structure is used only as a blocking layer
which is shorted by the zinc dittused stripe so the electrical model om.its the top
n layer and considers the zinc diffusion as a 2 µm. wide stripe contact. Refractive
indices are given instead of permittivities, where £

=n 2• The substrate and con-

tact layer can be omitted from the waveguiding problem, with the result that the
effective permittivity is real. Gain and recombination parameters are chosen
compatibly with both direct experimental measurements and measurements of
broad area lasing threshold current density.
The solution of the equations for electron and hole densities as a function of
voltage difference across the heterojunction is shown in Figure 4.4. Note that
since the fermi functions appropriate to degenerate semiconductors are used,
the curves begin to saturate at high injection levels.
This device has an obvious mirror symmetry, and this will be exploited to ease
the calculation. However, it must be remembered that with this simplification,
all currents and output powers should be doubled.
The finite element model used for the calculation of the Green's functions
(4.62) and (4.63} is shown in Figure 4.5. The use of a large number of elements
for the modelling of the substrate is not necessary, but does give the device a
reasonable series resistance. In most situations, assuming the substrate - epilayer interface to be equipotential is a good approximation.

Figure 4.3

2.5µ.m ~

n GaAs

device is charact erized as a nonpla nar large-optical~cavity laser.

Lateral cross section of the exampl e device treated . This type of

n GaAs {substrate)

-4

n Ga _65 Al _ As
0 35

P Gao.aoA 10.20 As
"'-- p

p Ga0.65 AI0.35 As

Zn diffusion

- 83 -

Table 4.1

Material parameters used in the modelling program.

Parameter

Value

4. 7 · 10 17 cm-3

7 · 10 18 cm-3

N;-Nt

3 · 10 17 cm-3

Tn, Tp

1 . 10-? s

0.9' 10- 10 ~

Jl-n

4000 ..£!!!__

300~
Vs

-180 cm- 1

g1J> •gin

g2pn

7 · 10-35 cm5

12 cm-1

Eg, n CJ

1.43 eV

2.5 · 10-2 cm

0.32

-84Table 4.2

Structural parameters used in the modelling program.

Layer

Conductivity

Refractive Index

n GaAs substrate

1000 ohm- 1 cm- 1

3.64-0.0528i

n Gao.65 Alo.35 As

200

3.39

n Gao.ea Alo.20 As

200

3.50

p Gao.95 Alo. 05 As

3.64 + dncarriers

n Gao.ea Alo.20 As

3.50

n Gao.e5 Alo.35 As

3.39

n GaAs isolation

3.64-0.0528i

- 85 -

22
10

20
10
18

N- =3x10 17 cm- 3

Ne =4. 7Xl0 cm- 3

Nv =7Xl0 18 cm- 3
12

IOo.~80----1-.0-0----1.2~0----l~A-0----IE~O----l~E-0---2_,DO

Junction
Figure 4.4

Voltage

(Volts)

The electron and hole densities in the active layer as a function of
the voltage across the heterojunction.

- 86 -

Figure 4.5

The finite element model constructed for the ohmic conduction
problems. Use has been made of the device symmetry.

- 87-

The geometric model of the ;·device, Figure 4.3, is used for the calculation of
the effective permittivity and the lowest four lateral optical modes (Y) are calculated as described earlier. The active layer thickness is assumed. to vary as
tact= 0.08 + 0.12 e-0 •0732

(4.64)

where tact is the active layer thickness and y is the lateral distance measured
from the center of the stripe, both measured in µm. The effective permittivity
profile for the device is shown in Figure 4.6, and the lowest four lateral modes
and their corresponding far field patterns are shown in Figure 4.7. Since the
waveguiding properties are geometrically determL11ed, equation {4.54) is used to
determine modal gains for the device.
The solutions for the steady state device behavior with pump current as a
parameter are shown in Figures 4.8 through 4.11. Figure 4.8 shows the current
versus voltage characteristics of the diode, and clearly shows the saturation of
junction voltage associated with lasing threshold, which can be seen to occur at
approximately 31 mA of total device current. Above threshold, further increases
in device voltage are due to the finite device resistance, here approximately 2.4
0. The carrier profiles for the device with pump current as a parameter are
shown in Figure 4.9. The total device current varies in this figure at 4 mA per
step from 4 to 100 mA. The saturation of the carrier populations at lasing
threshold and the effects of spatial holeburning can be seen. This is a different
effect from the "diffusion focussing" described by Streifer et al. 16•17 • The light
versus current characteristics of the device are shown in Figure 4.10, where
stimulated power output to each mode as well as modal gains are plotted as a
function of pump current. The total power output as a function of pump current
is shown in Figure 4.11. The effect of spatial holeburning in this device can be
seen to eventually let higher order lateral modes of the structure emit

- 88 -

Effective Permittivity Profile
{Im E'ef f ~ 0)

12.6

::: 12.5

Cl>
\II
Q.>

a::

12.4

y {µ. m)
Figure 4.6

The effective permittivity profile for the device. Use has been made
of the device symmetry.

- 89-

Far Field Intensity

Far Field Intensity
Near Field Intensity

Near Fie Id Intensity

0th order

2nd order
Symmetry
axis

axis

I \,3

Y (µ.m)

Far Field Intensity

Near Field Intensity

I st order

3rd order

Symmetry
axis

y (µ.m)
Figure 4.7

Y (t.Lm)

The four lowest lateral modes (Y) of the device and their
corresponding far field patterns. Use has been made of the device
symmetry.

- 90-

000
00

00
00
00

CL>
''-

::I

\_lasing threshold

C'I

....J

~r~~~~~'~
~~~-'-~~~
---JL....-~~~~'
1.48
1.58

1.38

1.68

1.78

Device Voltage (volts)
80

00
00

CL>
''::I

lasing

threshol d\
20

00
00

Figure 4.8

1.38

1.48

1.68
Device Voltage (volts)
1.58

1.78

The current-volta ge characteristi cs of the device. To obtain total
device current, the scale should be doubled.

- 91-

3.2

(X)

2.4

E 1.6

a.
0.8

Y(µm)

2.4
(X)

rt)

1.6

0.8

Figure 4.9

12
Y(µ.m)

The lateral carrier density profiles for the device in operation. Total
device current is varied as a parameter from 4 to 100 mA with a
step of 4 m.A.

- 92 -

00000

ooooOJ0

....

Current ( m A)

~~~
e:.A ++

o+++

·-0

-10

(!)

"'O

D = 0th order mode

o = I st order mode
= 2nd order mode
+ = 3rd order mode

Ill

t:;,.

-50

-90

Current ( m A)
Figure 4.10 The light versus current characteristi cs and modal gains for the
device. Each lateral mode is plotted separately.

To obtain total

device current and power, the scales should be doubled.

- 93 -

E 30

::J

:::s

Q,)

a..

I-

o"--

0·

kink

40
Current ( mA)

Figure 4.11 The light versus curren t charac teristi cs for the devke,

with output

powers in the lateral modes summe d. To obtain total device curren
and power, the scales should be double d.

- 94-

stimulated power. The kink associated with the onset of oscillation of the first
order mode at approximately 52 mA of total current and 20 mW of total power
output is clearly visible. These:results are in agreement with the experimental
results for this device, which claim threshold currents as low as 32 mA and output powers into the lowest mode of up to 25 mW.
To compare with the results presented in the references {16) and {17), the
sheet resistance for the p layers assumed here fS approximately 500 0 . The calculated threshold current in reference {16) for this sheet resistance and a 2 µm
wide stripe contact is 53.7 mA. In this model, the injected carrier profile at
threshold falls to haif of its value at the center of the stripe at a lateral distance
of 10 µm. In comparison, reference (16) yields 6 µm. for this distance. In addition, the above threshold analysis in reference {17), although for a different
structure, shows a different type of spatial holeburning than this model. In that
calculation, spatial holeburning was found to significantly lower the carrier
population at the center of the stripe under lasing conditions. In this model, the
carrier population at the center of the stripe is nearly constant above threshold,
and lateral mode switching results from the increase in the carrier population
outside the lasing mode. This difference can be attributed directly to the p-n
junction boundary conditions applied in the two models.
IV.7 Summary and Conclusions

A model of the double heterostructure laser has been presented which
correctly treats the diode junction of this device. It is valid above lasing threshold and is capable of treating a large number of the device geometries in popular use. With this model, the quantitative behavior of devices can be investigated
independently of experimental variations and compared and possibly optimized.
The channelled substrate structure of Burnham et al. 10 has been treated as an

- 95 -

example, and the experimentaLand theoretical results are in good agreement.

- 96 IV.8 References for Chapter IV

For some examples and more references, see:
H. C. Casey, Jr. and M. B. Panish, Heterostructure Lasers: Part B. Materials
and Operating Characteristics. Academic Press, New York ( 1978 ).

J. Buus, "A Model for the Static Properties of DH Lasers," IEEE J. Quant.
Elect. QE-15. 734 (1979).

K. A. Shore, T. E. Rozzi, and G. H. in't Veld, "Semiconductor Laser Analysis:
General Method for Characterising Devices of Various Cross Sectional
Geometries," IEE Proc. pt. I 127, 221 {1980).

T. L. Paoli and P. A. Barnes, "Saturation of the Junction Voltage in Stripe
Geometry (AlGa}As Double-Heterostructure Junction Lasers," Appl. Phys.
Lett. 28. 714 (1976).

P. M. Asbeck, D. A. Cammack, J. J. Daniele, and V. Klebanoff, "Lateral Mode
Behavior in Narrow Stripe Lasers," IEEE J. Quant. Elect. QE-15. 727 (1979).

D. Wilt and A. Yariv, submitted for publication.

S. M. Sze, Physics of Semiconductor Devices, Wiley-Interscience, New York
( 1969).

T. Tsukada, "GaAs-GaxAli-xAs Buried Heterostructure Injection Lasers," J.
Appl. Phys. 45, 4899 (1974).

K. Aiki, M. Nakamura, T. Kuroda, and J. Umeda, "Channelled-Substrate
Planar Structure (AlGa}As Injection Lasers," Appl. Phys. Lett. 30, 649 (1977).

10. R. D. Burnham, D. R. Scifres, W. Streifer, and S. Peled, "Nonplanar Large Optical Cavity GaAs/GaAlAs Semiconductor Laser," Appl. Phys. Lett. 35, 734
(1979).
11. N. Bar-Chaim, M. Lanir, S. Margalit, I. Ury, D. Wilt, M. Yust, and A. Yariv, "Be

- 97 Implanted {GaAl)As Stripe· Geometry Lasers," Appl. Phys. Lett.

36, 233

(1980).
12. H. Yonezu, I. Sakuma, K. Kobayashi, T. Kamejima, M. Ueno, and Y. Nannichi,
"A GaAs-AlicGa 1-icAs Double. Heterostructure Planar Stripe Laser," Jpn. J.
Appl. Phys. 12. 1.585 (1973).
13. D. Botez, "CW High-Power Single-Mode Lasers Using Constricted Double Heterostructures with a Large Optical Cavity (CDH-LOC)," Topical Meeting on
Integrated and Guided Wave Optics, paper MC2, Incline Village, Nevada, January 1980.
14. S. B. Dong, J. A. Wolf, Jr., and F. E. Peterson, "On a Direct-Iterative Eigensolution Technique," International J. for Numerical Methods in Engineering 4,
155 (1972).
15. J. Katz, S. Margalit, D. Wilt, P. C. Chen, and A. Yariv, "Single Growth Embedded

Epitaxy AlGaAs

Injection

Lasers

with

Extremely

Low

Threshold

Currents," Appl. Phys. Lett. 37, 987 {1980).
16. W. Streifer, R. D. Burnham, and D. R. Scifres, "Analysis of Diode Lasers with
Lateral Spatial Variations in Thickness." Appl. Phys. Lett. 37, 121 (1980).
17. W. Streif er, D. R. Scifres, and R. D. Burnham, "Above Threshold Analysis of
Double-Heterostructure Lasers with Laterally Tapered Active Regions," Appl.
Phys. Lett. 37, 877 (1980).

-98-

ChapterV
The Effect of Lateral Carrier Diffusion on the
Modulation Response of a Semiconductor Laser
V.1 Introduction

Semiconductor lasers are potentially devices of great use in optical communications systems. Part of their attractiveness for this application stems from
their high efficiency and high upper modulation frequency limit. Compared with
conventional light emitting diodes, the laser offers much more in terms of both
of these criteria, due to the presence in the laser of high densities of stimulated
photons which have very short lifetimes (on the order of picoseconds). These
afiect very strongly the behavior of the laser diode.
The modulation response of the laser diode at present still leaves something
to be desired. The upper modulation frequency limit, now approximately 2 GHz,
is still unacceptable for some application·s. In addition, most laser diodes have a
strong resonance peak in the range of 1-2 GHz in their small signal response
which proves to be a considerable impediment to their use in communications
systems. This resonance manifests itself in the time domain as damped oscillations in the light output of the device when driven with a step input current.
Many lasers exhibit this type of behavior, which results from the exchange of
energy between the inverted medium and the photons in the resonant cavity of
the device.
As a result of these problems, methods for improving the modulation
response of these devices such as damping of this relaxation resonance are
topics of active current interest.
Several authors have investigated the effect of lateral carrier diffusion upon
the modulation response of the semiconductor laser l-G. These investigations
have centered mainly upon the time domain response of the laser to a step

- 99 -

change in input current 2•3 , or the small signal frequency response 1.4.5 , and
have indicated that .the transverse diffusion of carriers can improve considerably the modulation response of the laser. This improvement comes mainly in
the damping of the relaxation oscillations of the laser which usually occur
within a time scale of nanoseconds after the current step, or in the damping of
the resonance peak in the small signal frequency response at approximately 1-2
GHz.

The purpose of this chapter is to present a simple, self consistent model of
the effect of lateral carrier diffusion upon the semiconductor laser to illustrate
both its theoretical and practical limits for improving the modulation response
of these devices 6 • This model is formulated in the frequency regime so that the
analog small signal response of the device is made available. The general model
developed is applied to the case of the TJS {transverse junction stripe) laser 7 as
an example, and the effect of both the carrier diffusion and the spontaneous
emission factor in the damping of the relaxation resonance of this device are
compared with experiment. The TJS laser is chosen as an example since its
injection can be modelled by a 6 - function spatial dependence, eliminating the
question of the lateral distribution of injected current and clarifying the contribution of lateral carrier diffusion to the device.
V.2 The Spatially Dependent Rate Equations

The starting point of this analysis is the spatially dependent rate equations,
using the assumption that only a single carrier need be considered (for example,
holes). We also assume that the active layer in the laser is thin so that all the
physical variables are averaged over this dimension. In addition, the laser is
assumed to oscillate in a single lateral mode.

- 100 -

an= - -n+ Da- n
s r a ( n-nt ) + -i at
T 11
ox2 - -0·c..l- F
e d

(5.1)

-dS
=v S r f-•• F a ( n - nt ) dx - -S + (J v n d f • ..!L dx
dt

(5.2)

r.;

-•Ts

In these equation s, n is the inversion density, Ts is the spontane ous lifetime, D is
the lateral diffusion constant , S is the power in the single lasing mode, fl r.; is the
photon energy, a is the gain coefficie nt and nt is the inversion required for
transpar ency, so that the optical gain is a { n - nt ), F is the lateral intensity
profile of the optical mode, assumed normaliz ed, r is the intensity confinem ent
factor normal to the active region, j is the injected current density, e is the electronic charge, d is the active layer thicknes s, v is the group velocity of the optical mode, Tp is the photon lifetime, and (:J is a coupling coefficie nt for spontane ous emission into the optical mode.
The spontane ous emission factor is treated in a very simple manner here. To
treat spontane ous emission into the lasing mode properly would involve a much
more complex treatme nt which is unneces sary for this calculati on. This is
because the major contribu tion of spontane ous emission to the modulat ion
characte ristics comes from the backgro und it adds to the optical power. Spon-

taneous emission could also be consider ed by adding a constant term
~ to
Tp

equation (5.2).
These equation s are normaliz ed in the following manner. Time is normaliz ed
to T,, distance is normaliz ed to the diffusion length, ~ n is normaliz ed to
nt. S is normaliz ed to

nc:.Jd~

, and j is normaliz ed to ednt . With this

choice of normaliz ation, the rate equation s appear as

- 101 -

on
at =- n
dS
dt

o2n
+ ax2 - S F ( n - 1 ) + J

(5.3)

=a1 a2 S J_.... F ( n - 1 ) dx - a 1 S + P a 1 a 2 J_•.. n dx

(5.4)

The two parameters a1 and a 2 are

(5.5)

(5.6)

W"th
reasonable values o f th e various
paramet ers, ,-9 = 3 ns, D -- 10 _cm ,
IlCJ = 1.43 eV, r = 0.5, d = 0.2 µ;rn, a= 10-18 cm 2, ni = 2 · 10 18 cm-3 , v = 10 10 .£!!!..,

sec

=2 ps, the values for these parameters are approximately 1.5 · 10 for a and

2 for a 2 • In addition we have the condition

V Tp

=_____,L___,__
a L + ln( ~ )

(5. 7)

where L is the diode length, a is a distributed loss, and R is the mirror
reflectivity of the device.
V.3 Analytic Treatment of Lateral Carrier Diffusion Effects

In a qualitative sense, one might argue that lateral carrier diffusion in a
device with a narrow optical mode and a very restricted current injection would
increase the maximum modulation speed of the laser. This argument claims
that the effective carrier lifetime in the device is shortened by the fact that carriers that diffuse away from the optical mode region are lost to stimulated emission. This, and the fact that the maximum modulation rate depends upon carrier lifetime, would seem to indicate that such a diffusion dominated laser would
be a faster device. Jn addition, one might expect that lateral carrier diffusion

- 102 -

serves to damp the relaxation oscillations,in the laser diode, since the exchange
of energy between the photons and inverted carriers responsible for these oscillations is damped strongly by the gain and loss of carriers that diffuse into and
out of the optical mode region.
An analytic result can be derived from the rate equations that indicates to
what extent these qualitative arguments are true. One simple limiting case is
that where the optical mode and current injection have no spatial dependence.
Another limiting case, the one of most interest here, is that where the current
injection and optical mode are both o functions in the lateral direction. In
either case, we will derive small signal modulation transfer functions, and a
comparison reveais the extent to which the above assertions are true. To simplify the analytic results and to clarify the contribution of diffusion, we will
assume the spontaneous emission coupling, (J, to be zero.
The steady state solution to the spatially uniform case is then

(no - 1)

=-a21

(5.B)

S0 F=a2 (fo-jt.h)

(5.9)

=1 + -a21

(5.10)

Jth

where zero subscripts indicate steady state values. Note that a power density,
S0 F, is given, where F is a constant, and the threshold current is defined as jth.
The small signal equations for the spatially uniform case are, assuming

i r:.i ni

=- n1 - So F nt - S1 F (no - 1 ) + h

(5.11)
(5.12)

- 103 Using the steady state relationships, equations {5.B) through {5.10), these simplify to

(5.13)

(5.14)
Solving these relationships yields

81 F

=a2 h 1+.!.E..(1+
1+i "

(5.15)

S0 F

This transfer function represents a two pole lowpass response, with a characteristic frequency of

(5.16)
With reasonable values of ai. a 2 , and the power density S 0 F, this resonance lies
in the microwave range of 1-2 .GHz. However, the resonance also has a typical
amplitude of 10 dB over the de value. This represents a considerable problem to
the use of this device in a !J.igh speed communications system. This response is
plotted as a function of frequency in Figure 5.1 for the values of the parameters
as specified above. S 0 F takes the values of 0.1, 0.3, 1, and 3 mW in these curves.
µm

In the case where the current injection and the optical mode have 6 function
profiles,

j(x} =Ao( x}

(5.17)

=6( x - x

(5.18)

F(x)

0 )

the steady state solution to the equations above, neglecting spontaneous .emission, is

-10

Figure 5.1

-30.

-20

-+·c

-0

..........

-0

-en

IOI-

µ.m

10 10

. . . . . * > >. ' ( ' < ' ( '

Frequency (Hz)

] .

A/\ I\ /\

?.> mW

Small signal transfer function for the spatially uniform laser.

)lo

So = 0.1 0.3 I

10 11

(f)

...c

(f)

..........

'"O

Q)
Q)

I - 180

-90

I90

-1180

.......

..,..

- 105 -

nc{xo)= l+ae

{5.19)

no { x ) = -~ e- I z I - -So- e - I z - Zo I
2 ae

(5.20)

(5.21)
Ath

=2 el Zo I ( 1 + ...!... )
ae

(5.22)

where the lasing threshold A.th has been defined. Note that this threshold is just
e 1zo 1 tin1es the threshold for a two - diffusion - length wide uniform iaser. The
small signal equations for this case are then
d 2n

i"' n1

=- n1 + -dx-21 - S1 ( no - 1 ) o( x - xo )

{5.23)

- So n1 o( x - Xo ) +Ai o( x)

(5.24)

Again, using the steady state relationships , equations (5.19) through (5.22), we
can reduce these relationships to

d 2n1

-2- - n1 ( 1 + i"') =Ai o{ x) - So ( 1 + -.ai-·) n1( Xo) o( x - Xo)
dx
le.>
A solution to these equations which must be made self consistent is

(5.26)

- 106 -

(5.27)

So ( 1 + _a_i )
ic..i

) -fx-x 0 1~

_2_J_,1_+_i-"'- ni Xo e
The requirement of self consistency yields

(5.28)

With some simplificatio n and the substitution of equation (5.25) this gives

(5.29)

The resemblance between equation (5.29) and equation (5.15) is remarkable.
The behavior of this transfer function is plotted in Figure 5.2, again for the
parameters chosen earlier, and with the offset x 0 chosen to be zero. The
parameter S 0 takes on the values 0.1. 0.3. 1, and 3 mW. This transfer function
has slightly improved frequency response over the uniform case, manifested
mostly in the reduced resonance peaks. This is achieved, however, at the price
of slightly worse phase response in the region near the resonance.
This transfer function represents the maximum contribution of diffusion to
the modulation behavior of the semiconduct or laser. Any other case should lie
in the intermediate region between the spatially uniform case and this o - function laser. It is most interesting to note that this limiting case has an analytic
solution which does not show pathological behavior such as an infinite frequency response. In fact, even with the infinite power density represented by the

o function optical mode, this laser has an upper limit on modulation frequency

- 107 -

(saa..15ap)

o())

aso~d

enI

..:
Cl)

-c
«S

....0
......
C.>

.....::s

'Cl
Cl)

.c:
......

""
.....

..........0

(])

.....::t

:::)

CJ)

(])
\,,._

.....rn""
cltS
Cl)

......

«i
ctill

....rn

=:
r-

-I

(8P) apn+!u50V'J

ltS

Cl)

- 108 -

which is quite similar to the spatially uniform case~,Thus it appears that while
lateral carrier diftusion may be expected to improve·.the damping of the relaxation resonance, it cannot be expected to greatly improve the upper modulation
frequency limit of the semiconductor laser.
V.4: Modulation Response of the TlS Laser

To analyze the case where the injected current and the optical mode are
allowed to assume arbitrary profiles, numerical analysis of the rate equations
(5.3} and (5.4) is required. First, the steady state solution of these equations is
found, and using this solution, small signal equations for the system are derived.
Since the steady state equations are nonlinear, their solution can be quite tedious. However, if we assume the optical mode profile F to be fixed and the profile
of the injected current j to be fixed so that j =A G(x) where G is a fixed function
and A is a scalar, this calculation can be reduced to the solution of a linear systern.
With the output power S 0 assumed, the equations to be solved for the steady
state solution are

a2n - S F ( n - 1 ) + AG =O

- n + -

ox 2

ai a2 So

J F ( n - 1 ) dx - ai So + (3 a 1 a 2 J n dx =0

(5.30)

(5.31)

These equations are both linear in the unknowns, n and A. With the assumption
of a finite element variational form for n involving a one dimensional grid where
the values of n at the nodes are the variational parameters and using linear
interpolation between nodes, these two equations are transformed into a linear
algebraic system which may be solved quite simply to yield both the nodal values
of ri and the scalar current A.

-109 In the small signal analysis, the equations are already linear in all the
unknowns, and we simply assume a value for the small signal output power 8 1,
assume a finite element form for the small signal carrier distribution ni. and
solve the resulting linear algebraic system for the nodal values of n 1 and the
smail signal scalar current A 1•
This calculation has been performed for the TJS laser, and the results are
shown in Figures 5.3 through 5.6. The TJS laser was taken as the test device
because of its simple electrical and optical structure~ Since it is a lateral homojunction device, the current injection profile G can be taken to have a ofunction
form. This eliminates the problem of determining how the current distribution
varies with steady state optical power output or modulation frequency. In addition, this device has a built in index profile in the lateral dimension that defines
the lateral optical mode profile F. In the calculation, the parameters for the
device were taken to be a= 50 cm- 1, R = 0.3, r
d = 0.2 µ,m,

L = 250 µ,m,

=2 · 10 18 cm-3 ,

=0.6, I:ic.; = 1.43 eV, a= 10-15 cm 2,
,/'ff'T;= 3 µ,m,

T 9 = 3 ns,

and

=0.83 · 1010 ..£!!!...
This gives for the parameters a 1 = 2500 and a 2 = 1.2. The
sec

lateral mode intensity profile assumed for this device is

F( x)

= v'2 1 w e
71'

(5.32}

where Xoft is the offset between the center of the optical mode and the diode
junction, and w is the width of the lasing mode, both assumed to be 0.6 µ,m. The
origin is taken to lie at the diode junction. The small signal modulation transfer
functions calculated for these parameters, a power output from the device of
1 mW, and various values of the spontaneous emission coupling factor are shown
in Figures 5.3 and 5.4. The interesting feature of these curves is the suppression
of the relaxation oscillation resonance caused by the lateral diffusion of carriers
in the device. Compared with the spatially uniform case, the height of the

CT>

-+-

::J

"'O

-40 6

-20

-m 201

Figure 5.3

..,.I '

dependence upon the spontaneous emission coupling factor.

Magnitude of small signal transfer function for TJS laser, showing

Frequency (Hz)

f3=10
f3=10

u-v

{3=0, no diffusion~•

S0 = I mW

40-----~------------------------·--------~------~

.....
.....

...c

Q)
(/)

-0

CJ)

(/)
Q)
Q)

-180
10

-90"

Figure 5.4

10
10
Frequenc y (Hz)

dence upon the spontaneous emission coupling factor.

Phase of small signal transfer function for TJS laser, showing depen-

/3 =I 0

{3=16 _ /

''~¥/3 =O, no diffusion

-= -------f'

S0 = I mW

180-----------~----------~-------~,-----~--,

.....
.....
.....

-112 -

resonance is lowered by approximately 9 dB by the action of lateral diffusion.
The additional contribution of spontaneous emission into the lasing mode can
be seen to be appreciable only for relatively large spontaneous emission coupling factors of 10-4 • Jn Figures 5.5 and 5.6, the behavior of the transfer function as a function of optical output power.is calculated for a spontaneous emission coupling of zero~ The optical output powers are 0.1 mW for the lowest curve,
1 mW for the center curve, and 10 mW for the upper curve.
Figure 5. 7 shows the measured small signal transfer function for a TJS laser.

It should be noted that the vertical scale in this measurement is twice as large
as in the theoretical calculations. This is due to the fact that the measurement
was done with a square law detector, a silicon avalanche photodiode (APD). The
resonance peak in the amplitude response of this device has a magnitude of
approximately 5 dB on this scale, or 2.5 dB on the scale of the theoretical
curves. This corresponds well with the calculations if a spontaneous emission
factor of approximately 3 · 10-5 is assumed. This is a reasonable value to assume
for spontaneous emission coupling in this type of device. It is quite difficult to
draw any more information from this experimental measurement as it contains
also the frequency response of the measurement system, including the APD
response which has a 3 dB cutoff frequency of approximately 1.8 GHz.
The measured phase response of the system also supports the diffusion
damped model of the laser, since it displays a gradual rolloff rather than the
sharp transition of the spatially uniform laser.
The model of the TJS laser, then, as a diffusion damped device can be seen to
be verified quite well. This device should be quite useful in both analog and digital information transmission systems where its well damped resonance would
cause minimal interference with modulated information.

CJ)

·c

::J

'"'O

-m

-40

-20

20 1

Figure 5.5

"" ""

Magnitud e of small signal transfer function for TJS laser, showirg

Frequency (Hz)

= 0.1

depende nce upon the steady state output power.

40--~~---~~~-----------,------~~~
--

c.J

.....
.....

en
...c

-0

Q)
Q)

-90

Figure 5.6

IC>

Frequency (Hz)

0.1

dence upon the steady state output power.

Phase of small signal transfer function for TJS laser, showing depen-

So =

l80r---~--,--------,-------1r-----~---------

.p.

- 115 -

Ith;::;30mA

-en

- -20
-0
Q)

-0
:J

......

37mA
-40

31.5mA

Frequency (GHz}
0------------------.~--...---.,.---..,....--~

-60
Q)
(/)

..c

a...

-120

-180L----JL.---'----'---~2~--'---~3~----~4

Frequency (GHz)
Figure 5.7

Measured amplitude and phase response of a TJS laser.

- 116 -

In conclusion, a model of diffusion eff'ects on the small signal modulation
characteristics of the semiconductor laser bas been presented which shows both
the theoretical limits and practical behavior of the diffusion dominated laser. It
can be seen that lateral carrier diffusion is a useful mechanism for the control
of the relaxation resonance in this device. It has also been shown that the use
of lateral carrier diffusion to improve the upper limit on modulation frequency
is not practical. In addition, the model of the TJS laser as a diffusion damped
laser has been shown to account for the experimental small signal modulation
function of this device.

-117V.5 References for Chapter V

T. Ikegami, "Spectrum Broadening and Tailing Effect in Directly Modulated
Injection Lasers,'' First European Conference on Optical Fibre Communication, IEE London, September, 1975.

J. Buus and M. Danielsen, "Carrier Diffusion and Higher Order Transversal
Modes in Spectral Dynamics of the Semiconductor Laser," IEEE J. Quant.
Electron. QE-13, 669 (1977}:

N. Chinone, K. Aiki, M. Nakamura, and R. Ito, "Effects of Lateral Mode and
Carrier Density Profile on Dynamic Behaviors of Semiconductor Lasers,"
IEEE J. Quant. Electron. QE:'."14-. 625 (1978).

K. Furuya, Y. Suematsu, and T. Hong, "Reduction of Resonance - Like Peak in
Direct Modulation due to Carrier Diffusion in Injection Lasers," Appl. Opt. 17.
1949 (1978).

G. H. B. Thompson, Physics of Semiconductor Laser Devices, Section: 7.3.4,
John Wiley & Sons, Inc., New York, (1980).

D. Wilt. K. Y. Lau. and A. Yariv, submitted for publication.

H. Namizaki, H. Kau, M. Ishii, and A. Ito, "Transverse-;.Junction-St ripeGeometry Double· Heterostructure Lasers with Very Low Threshold Current,"
J. Appl. Phys. 45• 2785 (1974).

-116 -

Chapter VI
Effective Permittivity Formalism
and the Design of Buried Heterostructure Lasers
VI.1 Introduction

One of the most important aspects in the design and understanding of the
semiconductor laser is the waveguiding that defines its optical modes, since this
impacts strongly upon the modal behavior of the lasing device. Well behaved
semiconductor lasers oscillate in a single spatial mode, and this phenomenon is
based upon gain discrimination between the optical modes of the waveguide
structure.
The optical guiding in these devices has often been treated theoretically by a
technique known as the effective permittivity formalism 1- 7 • This technique is
usually presented as an approximate solution method for the eigenvalue equation defining the optical modes of the laser. This approximation takes note of
the fact that the waveguide in a typical semiconductor laser resembles quite
closely that in a uniform planar waveguide. The active region in the laser is usually quite thin and is sandwiched between two cladding layers of considerably
lower permittivity, so that the optical mode in the direction normal to the active
layer is quite strongly confined. In the lateral dimension, the variations in permittivity are usually much smaller, so that one can make the approximation
that the optical mode in the laser has a separable form with the profile in the
direction normal to the active layer unperturbed from the mode shape for the
uniform planar guide. One then uses this mode shape to define an effective permittivity in the lateral dimension by forming an average of the permittivities in
the normal direction. This approach to the definition of effective permittivity
formalism leaves open the question of what weighting form to use for the
effective permittivity.

- 119 -

A more sophisticated approach to the problem includes error criteria which
aid in the choice of. this weighting function 7 • The best choice is to choose a
weight which minimizes the correction to the lateral modal field in a perturbalion analysis of the original waveguide equation. This criterion leads to the
choice of the intensity profile as a weighting function, and prefers to weight the
relative permittivity (as opposed, for instance, to the refractive index).
This paper presents an even more general approach to the treatment of
effective permittivity formalism which treats this technique as a variational
approximation, and allows considerably more freedom in the choice of the
modal profiles used in the approximation.; With this approach, one is no longer
limited to the use of simple planar waveguide profiles in the direction normal to
the active region, and one can even use a normal profile which is not an eigensolution in this direction. This may be a wise choice in certain types of lasers such
as the buried heterostructure laser. In this case, the choice of a mode profile in
the direction normal to the active layer which is an eigensolution leads to the
use of plane wave modes in the regions of regrown cladding. This is clearly a
poor approximation. to the true modal profile in this device. A much better
approximation would be to use the same modal profile normal to the active
layer in the regrown clad as is used inside the buried mesa. The value of the
effective permittivity thus obtained is quite different from that resulting from
the usual technique, and shows that the guiding in this device is much stronger
than might be suspected from the usual effective permittivity formalism.
VI.2 Effective Permittivity Formalism

The starting point in this analysis is the vector wave equation,
(6.1)

here k is the wavenumber for the frequency of interest (k = £.) and e is the

- 120 complex permittivity of the medium. This wave equation is derived with the
assumption that the scale of variation for e is much larger than a wavelength so
that a term involving the gradient of £ can be dropped. E is the electric field,
assumed to vary as e1r.it.
With the assumption that we are interested in a waveguide mode which varies
as e1P2 where the z axis is chosen to be parallel to the guide, we get

( v f + k 2 e - p2 ) E = o
where v t is the transverse gradient operator ( v t

(6.2)

=e:s; :x + ey ;y ).

Semiconductor lasers are most often fabricated with a waveguide that is
strongly confining in one transverse direction and weakly confining in the other.
In the strongly con.fined direction, normal to the active layer, the waveguiding is
dominated by the discontinuity in permittivity between the active layer and the
larger bandgap cladding layers. In order to achieve modal control in this direction, the active layer is made quite thin {on the order of 0.2 µm.) so that the low
order modes are much more tightly con.fined than the higher modes. In the
lateral direction, parallel to the active layer, the waveguiding is usually weak.
Thus, the waveguide modes split, in a manner similar to that in the uniform
planar waveguide, into pseudo-TE and pseudo-TM modes. The pseudo-TE modes
are known to dominate the behavior of the semiconductor laser. This is because
they are more tightly confined to the gain region (the active layer) and also have
higher refiectivities from the cleaved mirrors. For this reason, we will assume a
TE mode for the laser and solve a scalar modal eigenequation

(6.3)
where u represents the TE electric field, Ey. We have taken the y coordinate to
be parallel to the active layer, and the x: coordinate normal to the active layer.

- 121 -

This eigenequation can be cast in a variational form as

....

J J dx dy ( - ( V t u ) 2 + k 2 e u 2 )

0 (3 2

=0 ( --------..- ..- - - - - - - ) =0

J J dxdyu

(6.4)

Keeping in mind the situation that the mode is tightly confined in one dimension (normal to the active layer) and weakly confined in the other, we will
assume a variational form for u with a fixed mode profile in the direction normal to the active layer and allow the lateral mode profile to vary. The resulting
Euler equation for the lateral mode profile contains the effective permittivity.
One choice for the variation in the direction normal to the active layer is to
solve an eigenmode equation in this direction, ignoring the partial derivative in
the y direction.

ax 2 + k2 e(x,y) - 1NY) ) X(x,y) =0

( -

(6.5)

The lateral coordinate y is considered here to be a parameter, and we choose
the solution with the larges.t eigenvalue (the lowest mode). This eigenvalue, 7 1 , is
not the propagation constant for the mode, but is related to the effective permittivity.

Alternatively, any functional form X{x,y) desired can be used to

represent the variation in the normal direction, if for some reason the eigenmode solution above is felt to be a poor approximation. Consistent with the use
of complex permittivities, the normalization condition on this field is taken to
be

...

J dx X2 = 1

(6.6)

We next need to find the best possible approximation to the true modal field
using the field X to represent the variation in the perpendicular (x) direction. To

- 122 this end, we assume a variational form to substitute into the variational equation (6.4)
u(x,y}

=X(x,y} Y(y}

(6.7)

where Y{y) is the function we will allow to vary. Since the variation in the x
direction has been determined, _we may integrate over x in the variational equation and get as a result

j dy ( - ( ~y ) 2 + ( k2 Eetr ) y2 }
.r (

y•

-•

) =O

(6.8)

f dyY2
where

k 2 E:ea{y)

= J dx ( - ( Vt X ) 2 + k 2 e X2 )

{6.9)

The normalization condition (6.8) has been used in the simplification of the
expression {6.8). This effective permittivity can be simplified in the case where X
is a solution of the eigenmode equation {6.5} to

k 2 eea(Y)

=l'r(y) - j dx <
--

:x )

(6.10)

where 7[{y) is the eigenvalue of the equation (6.5). Apart from the second term
involving

:~ . this is the same result as that of the conventional effective index

formalism. In many cases, this extra term is small enough to neglect. However,
it represents an important consideration in some cases, especially those where
the variational form assumed for X bas an abrupt variation in the lateral direction. This is the case, for instance, in the conventional analysis of the buried
heterostructure laser 11 •

- 123 The Euler equation for this variational expression is then

(6.11)

If the normalization is taken as

f"' dyY = 1

(6.12)

then the normalization of the field u is

••

J J dx dyu 2 = 1

-- --

(6.13)

The advantage of approaching the effective permittivity problem from the standpoint of the variational principle is that it assures in some sense that the best
possible approximation to the true modal field has been obtained. This statement can be quantified further if the profile taken for X is an eigensolution of
the equation (6.5). In this case, if first order perturbation theory is applied to
the modal profiles found, all corrections to the modal field u involving overlap
integrals of X with itself are zero. The lowest nonzero terms involve overlap
integrals of X with higher order modes in the x direction. However, it is important to keep in mind that the choice of eigensolutions for X may not necessarily
be the optimal one from the standpoint of approximating the field u.
VI.3 Design of the Buried Heterostructure Laser

The design of the buried heterostructure laser 8•9

and its optimization

represent an interesting and illustrative example of the use of this effective permittivity formalism. The typical buried heterostructure laser is illustrated in
Figure 6.1.

Its fabrication involves the growth of a double heterostructure

wafer, the subsequent etching of narrow mesas, and the regrowth of cladding on
the sides of these mesas. This device has a waveguide that consists of the

Figure 6.1

regrown

n-GaAs

E 2 n-AIGaAs

regr own

n-AIGaAs

laser.

\__ __

&It~tt\ltt\t\ttt~\t~\lt\tI\\f:1tttttt\~\~\tI%ttttitKt

Latera l cross sectio n of a typica l buried hi~teroslructure

n-A IGa As

i~fritIIrr~rrmirmrmmmmmmmmmrmmmmimmmim~mmmmm& .-

Zn diffusion

Cr- Au
/ Si0 2

"'"

.....

[\)

- 125 -

rectangular active layer and surrounding lower permittivity cladding layers.
The smallness of the lateral dimension of the active layer, typically less than
2 µ.m, greatly simplifies the electrical characteristics of this device. Uneven
current injection, spatial holeburning, and other considerations of paramount
importance in other laser structures are of minor importance in this device
since the lateral carrier diffusion tends to smear out any nonuniformities along
the active layer. A self consistent analysis 10 of this type of laser yields the
result that nonuniformities in pumping rarely exceed 5 percent of the total, and
this occurs only at high levels of stimulated emission.
Thus, the design criteria for the buried heterostructure laser center upon the
optical mode design of the device. Clearly, to achieve single lateral mode operation in such a laser, one must include a gain discrimination between lateral
modes that exceeds the expected nonuniformity in pumping. A design value of 5
percent represents a reasonable choice. A good way to achieve this gain discrimination is the use of mode confinement. To do this, the waveguide in the device is
constructed in such a manner as to be near cutoff, so that the fundamental
mode ( TE00 ) has an appreciably larger confinement to the high permittivity
gain region (the active layer) than higher order modes.
Another important design consideration involves scattering from the dominant lasing mode. In a real laser, there is considerable scattering from nonuniformities in the guide of the device. The scattered radiation may be lost ( coupled into radiation modes} or may continue to travel down the guide in a higher
order confined but nonlasing mode. This confined scattered radiation is emitted
from the laser mirror facets in the same manner as the radiation in the lasing
mode, and the interference of these fields causes the far field of this device to
display undesirable speckle-type patterns. This effect can be reduced consider-

- 126 -

ably by cutting off as many of the lateral modes in the device as possible, and
making the confinement of any remaining modes other than the lowest (lasing)
mode as small as possible.
Thus, the design criteria for the buried heterostructure laser may be stated
as the design of an optical waveguide with the largest possible confinement of
the lowest lateral mode (for low threshold current) and the lowest possible
confinement of any higher order lateral modes, consistent with electrical and
fabrication considerations.
For the purpose of this paper, we will analyze the structure shown in Figure 1.
We will employ two approximations, first the popular one that uses plane wave
modes in the regrown cladding of the device, and second what is felt to be a
more accurate approximation that uses the same normal mode profile in the
regrown cladding as in the mesa region of the device. The first approach resembles the one taken by Saito and lto 11 •
Inside the mesa, we solve the one dimensional eigenmode equation (6.5). This
is just the familiar symmetric three layer slab waveguide problem. With the
definition of the dimension~ess parameter

a = _k_t v~e-___e_2_

where k is the wavevector, k

=£,
tis the active layer thickness,

(6.14)

E:t is the permit-

tivity of the active layer, and e2 is the permittivity of the upper and lower cladding material (assumed identical), the eigenequations for the normal mode

profile reduce to

a 1 tan a 1
ai cot ai

(even modes)
(odd modes)

(6.15)

- 127 -

(6.16)

where we look for solutions with positive a1 and ao. which are defined by

(6.17)

(6.18)

The solution of equations (6.15) through (6.18) yields the effective permittivity
inside the mesa· region. In addition, we have the following interesting relationships
(6.19)

where the factor b is
{even modes)
(6.20}

(odd modes)
For the even modes, we also have

= active layer
j-

dxX

{6.21)

{even modes)

dxX

In addition. we define the intensity confinement factor

I sin a1 I ( a + I sin a1 I )

dxX2
1 +a I sin a1 I
active layer
r =-------- = cos a1 I ( a + I cos a1 I )
f dxX2
1 +a I cos ai I

{even modes)
(6.22)

(odd modes)

In the popular approximation, the eigensolutions for X outside the mesa are

- 128 -

plane waves. However, we should note that the normal field profiles inside the
mesa and outside the mesa do not match. This introduces a singularity in the
effective permittivity at the interface between the two regions, according to
equation (6.9}. This singularity represents a considerable problem if it is
included in the lateral waveguiding, as it demands that the lateral field Y be zero
at the interface. In previous treatments of the waveguiding in buried heterostructure lasers 11 , this singularity has been ignored, and the lateral waveguiding treated as a slab waveguide problem. The effective permittivity in the
regrown cladding is then
(6.23)

and thus the dimensionless parameter for lateral waveguiding is

a1ateraJ

kw
=TV
b £1 + { 1 - b ) £2 - es

(6.24)

where w is the lateral width of the buried mesa. However, it should be pointed
out that this approach to the problem no longer makes use of the V!'iriational
principle. By ignoring the singularity at the interface between the two modal
profiles used, the benefits and advantages to the use of the variational principle
are lost. One possible means of fixing this problem is to smear out the interface
between the two regions. This approach leads to an effective permittivity without
singularities and well behaved solutions. In some situations, this fix represents a
minor correction to the modal profiles found by ignoring the singularity. A good
example of this is the channelled substrate structure 12 where a smearing on
the order of a fraction of a micron will give a smooth .effective permittivity variation and nearly identical profiles to those obtained by ignoring the singularity.
On the other hand, the large changes in permittivity and small dimensions associated with the buried heterostructure laser make the smearing approach a
poor one. It is for this reason we adopt the following approximation using the

- 129 same X modal profile in the regrown cladding as in the laser mesa. An additional incentive to the use of this type of profile for the variational form is that
it matches more closely the true modal pattern of this device. For this choice of
a variational mode profile, we are using a non - eigensolution profile in the
regrown cladding. However, we avoid the singularity in effective permittivity
associated with the previous approximation.
For this profile we must resort to the expression (6.9) to calculate the
effective index in the .regrown cladding

k 2 &etf,out =

J dx ( - { .V X ) + k t X2 )

(6.25)

This expression can be simplified by using the fact that the profile X is an eigensolution of the slab waveguide inside the mesa,

(6.26)
In the lateral direction we again solve .a slab waveguide problem, but with a
different effective permittivity for the regrown cladding layer than for the previous approximation. This e~ective permittivity in the regrown clad is always less
than that in the previous approximation.

In this case, the dimensionless

parameter for lateral waveguiding is

aiateral

= k2w .Jr

+ ( 1 - r ) &2 - &3

(6.27)

where w is again the lateral width of the buried mesa.
To decide which of the two approximations is the more accurate, we will make
use of the variational principle, equation (6.4). This variational principle always
yields a {J for the lowest mode that is lower than the true {J in the case of a
bound, real index guided mode. Thus, approximations can be judged simply by
which gives the largest {J for the lowest mode.

If we keep the singularity

- 130 associated with the first approximation, the resulting lateral eigenmodes are
sinusoidal inside the mesa, zero at the interfaces, and zero outside the mesa.
The lowest mode in this case can be shown rigorously to posess a fJ which is
smaller than that produced by the second approximation. On the other hand, if
the singularity in effective permittivity is ignored, negating the effect of the variational principle, the fJ resulting from the lateral eigenmode equation (6.11) and
the fJ resulting from the variational principle (6.4) are no longer in agreement.
In fact, the fJ from the variational principle, which we compare with, is (3

=- "".

The approximation which results from ignoring the singularity is not actually as
poor as this indicates, but it cannot be compared in a reasonable manner
through the use of the variational principle. Another comparison that can be
made, however, is to point out that when the a variational approximation yields
a bound mode to the scalar eigenequation, such a bound mode must exist,
based upon the fact that the true fJ for the mode must be larger than that from
the variational approximation. Noting that the dimensionless parameter aie.teral
for lateral guiding in the second case is always larger than that for the first case
where the singularity in the effective permittivity is ignored, and that it can yield
bound modes when the first approximation does not, leads us to the conclusion
that the second case is the better approximation.
In the design of the buried heterostructure laser, we need only concern ourselves, then, with solutions of the three layer symmetric slab waveguide. These
can be presented in convenient form as graphs of the parameters b and r as
functions of the dimensionless parameter a for the various mode orders. These
graphs are shown in Figures 6.2 and 6.3.
To design a buried heterostructure laser, one could proceed by first choosing
an upper and lower cladding layer aluminum content and an active layer thickness. Then the effective permittivity inside the mesa can be calculated. The
aluminum content of the regrown cladding can then be chosen to get the

- 131 -

0.8

0.6

0.4

0.2

4.0

Figure 6.2

6.0

8.0

10.0

The parameter b as a function of the dimensionless parameter a,
for various mode orders.

- 132 -

0.8

0.4

0.2

.....

o--~~----~--~~--~~--~-----~~-------~--~~ ~~~~~~--10.0
8.0
6.0
4.0
2.0

Figure 6.3

The parameter r as a function of the dimensionless parameter a.
for various mode orders.

- 133 -

highest possible lateral mode confinement { r) for the lowest mode and cut off
as many of the higher order lateral modes as possible.
As an example of this procedure we choose a GaAs active layer with

Alo.4 Ga 0 .6 As cladding. The optimum active layer thickness in terms of threshold
current density for this choice is approximately 0.15 µm.. The lasing wavelength
is taken to be A.
and e2

=0.683 µ, m, and we also assume the values e =3.64 =13.25

=3.32 =11.02. This gives for the direction normal to the active layer

r =0.563

and

=0.356.

Thus

inside

the

laser

mesa

have

=3.44 =11.83. If Al Ga As material is regrown as cladding, we have
Eefi.out =3.25 2 =10.56 for the second approximation. This corresponds. to a

Eetr.1n

0 •4

0 •8

strongly guided case in the lateral dimension. To have aiateral less than 5, as
would be required for reliable single lateral mode operation, requires the lateral
width of the guide to be less than 1.25 µ,m. Considerable advantage can be had,
however, by the regrowth of much lower index cladding. If one were to choose

=3.45 =11.90 ) the effective
permittivity in the regrown cladding would be tetr.out =3.38 2 =11.42, again using

A10 •2 Ga 0•8As as the regrown cladding material ( e8

the second approximation.· corresponding to a laser with weak lateral guiding.
To get a 1ateral less than 5, the lateral width must be less than 2.30 µ,m, a much
less stringent requirement on device processing. Of course, there is an advantage to reduce a1ateral even more, down to a lower limit of approximately 1. Below
this value, the confinement of the lowest mode becomes objectionably small.
When a1ateral is less than ~, the lateral waveguide has only one bound mode.
To illustrate the difference between the two approximations to the design of
buried heterostructure laser waveguides, the curves in Figures 6.4 through 6.7
are presented. These curves show the relationship between refractive index of
the regrown aluminum cladding and the mesa width of laser for different active

- 134 -

3.6

n1 = 3.64
n2 = 3.32

= 0.883,um

3.5

3.4

3.3

Figure 6.4

The relationship between refractive index of the regrown cladding
and lateral width of the buried mesa, using the first approximation
(singularity ignored) and a lateral waveguiding parameter of 5,
corresponding to 5 percent gain discrimination between the zeroth
and first order lateral modes.

- 135 -

3.6

3.5

n 1 = 3.64
n2 = 3. 3 2

= 0.883µ.m

t =0.3µ.m
0. 25
0.2
0.15~

0. I
3.4

3.3

W (µ. m)
F"i.gure 6.5

The relationship between refractive index of the regrown cladding
and lateral width of the buried mesa, using the second approximation and a lateral waveguiding parameter of 5, corresponding to 5
percent gain discrimination between the zeroth and first order
lateral modes.

- 136 -

3.6

n 1 = 3.64
n2 = 3.32

a lateral =kw~b
+(1-b) €2 - €3 --:II..
El

\ = 0.883,um
t = 0. 3,um

0.25

3.5

0.2

r<)

0.15
3.4

0.1

F"igure 6.6

The relationship between refractive index: of the regrown cladding
and lateral width of the buried mesa, using the first approximation
1r

(singularity ignored) and a lateral waveguiding parameter of 2'
corresponding to cutoff of the first order lateral mode.

- 137 -

3. 7 ------...---__,..--....------...---.------,.---...........--..----.

n1 = 3.64
n2 =3.32

3.6

A. = 0.883 µ. m~_ _t_=r::0~.351µ.:m--------,
.25
0.2

3.5

0 15

0.1
3.4

3.3

3.2

W (µ.m)
F"l.gure 6. 7

The relationship between refractive index of the regrown cladding
and lateral width of the buried mesa, using the second approximation and a lateral waveguiding parameter of ; , corresponding to
cutoff of the first order lateral mode.

- 138 -

layer thicknesses and lateral waveguiding param.ete!"s. The lateral wavego.id.ing
parameters in these curves are 5, corresponding to approximately 5 percent
gain discrimination between the zeroth and first order lateral modes, and ; ,
corresponding to the cutoff cf the first order lateral mode. In these curves the
active layer is assumed to be GaAs, the upper and lower cladding layers are
assumed to be A10 .4 Ga 0 .6 As, and the emission wavelength is assumed to be 0.883
µ.m. The difference between the_two approximations can be seen to be consider-

able, with the second·approximation yielding much stronger confinement for the
modes.
In conclusion, a new form of effective permittivity formalism has been
derived, ""hich illustrates clearly both the nature of the approximation and its
limits. The technique has been extended to cases where it may be desirable to
use non - eigensolution profiles for the variation in the strongly guided direction, and this technique has been applied to the buried heterostructure laser as
an example. This technique shouid be extremely useful with new type~ c:>f laser
structures which depend upon strong geometric effects for waveguiding in the
lateral direction.

- 139 Vl.4 References for Chapter VI

W. V. McLevige, T. Itoh, and R. Mittra. ..New Waveguide Structures for
Millimeter-Wave and Optical Integrated C-Lrcuits," IEEE Trans. Microwave
Theory Tech. M'l'T-23, 788 (1975).

P. A. Kirkby and G. H. B. Thompson, "Channeled Substrate Buried Heterostructure GaAs-{GaAl)As Injection Lasers," J. Appl. Phys. 47, 4578 (1976).

T. Itoh, "Inverted Strip Dielectric Waveguide for Millimeter Wave Integrated
Circuits," IEEE Trans. Microwave Theory Tech. M'IT-24 821 (1976).

T. Rozzi, T. Itoh, and L. Grun, "Two Dimensional Analysis of the GaAs Double
Hetero Stripe Geometry Laser," Radio Science 12. 543 (1977).

G. B. Hocker and W. K. Burns, "Mode Dispersion in Diffused Channel

Waveguides by the Effective Index Method," Appl. Opt. 16, 113 (1977).
6.

J. Buus, "A Model for the Static Properties of DH Lasers," IEEE J. Quant. Electron. QE-15, 734 (1979).

7. W. Streifer and E. Kapon, "Application of the Equivalent-Index Method. to DH
Diode Lasers," Appl. Opt. 18, 3724 (1979).
8.

T. Tsukada, "GaAs-Ga 1-sAlxAs Buried-Heterostructure Injection Lasers," J.
Appl. Phys. 45, 4899 (1974).

Kajimura,

Saito,

Shige,

and

Ito,

"Leaky-Mode

Buried-

Heterostructure AlGaAs Injection Lasers," Appl. Phys. Lett. 30, 590 (1977}.
10. D. Wilt and A. Yariv, submitted for publication.
11. K. Saito and R. Ito, .. Buried-Heterostructure AlGaAs Injection Lasers," IEEE J.
Quant. Electron. QE-16, 205 (1980).
12. K. Aiki, M. Nakamura, T. Kuroda, J. Umeda, R. Ito, N. Chinone, and M. Maeda,
"Transverse Mode Stabilized Al:xGa 1-xAs Injection Lasers with Channeled-

- 140 -

Substrate-Planar Structure," IEEE J. Quant. Electron. QE-14, 89 (1978).