Coulomb attraction effects in semiconductor quantum well lasers

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Coulomb attraction effects in semiconductor quantum well lasers
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Thesis (Ph.D.)--University of Florida, 1997.
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Includes bibliographical references (leaves 168-173).
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by Chia-Fu Hsu.
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COULOMB ATTRACTION EFFECTS IN
SEMICONDUCTOR QUANTUM WELL LASERS















By

CHIA-FU HSU










DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA






























Copyright 1997

By

CHIA-FU HSU















DEDICATED TO MY PARENTS,
CHUNG-MEI TSAI AND HUNG-YI HSU

AND IN MEMORY OF MY DEAREST BROTHER,
CHIA-HENG HSU














ACKNOWLEDGMENTS


Being able to have the strength to finish my dissertation, I am very grateful to my par-

ents, Chung-Mei Tsai and Hung-Yi Hsu, my sister, Sung-Ya Hsu, and my grandmother for

their unconditional love and support. I feel particularly indebted to my parents for not be-

ing able to spend time with them during these years of studying abroad. I hope at least my

academic accomplishment can make them proud.

I would like to express my sincere thanks to my advisor, Dr. Peter S. Zory. He has

always been very inspiring to me during my studies these years in the Department of Elec-

trical and Computer Engineering at University of Florida. He has the ability to explain

complicated subjects in a way that people can understand and that attracts me to the re-

search field of semiconductor diode lasers. I also thank him for offering me the financial

support as a research assistant and I have truly enjoyed working for him.

I would like to thank all the professors on my Ph.D. committee: Dr. Gijs Bosman, Dr.

Fredrik A. Lindholm, Dr. Ramakant Srivastava, and Dr. Kevin S. Jones. I have appreciated

the guidance they have provided for this dissertation and their contributions to my under-

standing of semiconductor physics and my experimental skills.

I particularly wish to thank Dr. Paul Rees at University of Wales, Bangor, UK, for

the very valuable discussions on Coulomb attraction effects in semiconductor lasers. I

would also like to thank Dr. M. A. Haase at the 3M Company and Dr. M. A. Emanuel at

the Lawrence Livermore National Laboratory for providing me with state-of-the-art

CdZnSe and InGaAs QW laser materials, respectively.

I would like to gratefully recognize all the students with whom I have worked during

the last four years of pursuing my Ph.D. Particular thanks go to Dr. Chi-Lin "Kenny"









Young, Dr. Chih-Hung "David" Wu, Dr. Craig Largent, Bob Widenhofer, Jeong-Seok "Ja-

son" 0, Carl Miester, John Yoon, Christian Keyser and Jee-Hoon "Steve" Han for their

professional opinions and friendships.

I also would like to gratefully acknowledge my friends at the Emmanuel Mennonite

Church in Gainesville, especially Mary and Paul Lehman for being my host family. It was

their kindness and caring that made me, as an international student and for the first time

being 13,000 miles away from home five and half years ago, feel comfortable in this coun-

try very quickly.

I would like to gratefully recognize a late friend, Hsi-Hung Kuo, for his generosity in

helping me and numerous new students from Taiwan during his two years of Ph.D. study

in Department of Electrical and Computer Engineering at University of Florida. I am also

grateful to my friend, Dr. Yu-Fu Hsieh, and his wife, Hsueh-Chu Chen, for helping me set-

tle down in Gainesville and treating me like their brother since then. I would like to thank

my dear friends, Margaret and Kathy Cheng, and their parents for inviting me to their house

during many holidays. I am also grateful to my best friend, Ming-Cheng "Benjamin" Liu,

his wife, Chun-Yi "Jean" Chen, for that they always reserved an extra pair of chop sticks

on their dining table for me.

Last, but certainly not least, I want to thank my girlfriend, Huei-Feng "Vicky" Lee,

for the companionship, love and encouragement that she gave me.















TABLE OF CONTENTS



an80

ACKNOWLEDGMENTS ....................................................... ......................... iv

A B STRA CT ................................................................................................................. ix

CHAPTERS

1 INTRODUCTION .............................................. ............................................. 1

2 SINGLE-PARTICLE MODEL ............................... ....................... 6

2.1 Introduction ........................................... .............................................. 6
2.2 Laser Structures ....................... ........................................ 7
2.2.1 InGaAs QW Laser ............................................................................. 7
2.2.2 CdZnSe QW Laser ............................................ ............... ......... 7
2.3 Strained Quantum Wells ................................... ............................ 8
2.3.1 Strained Bandgaps .................................. .............................. 8
2.3.2 Bandgap Lineup .............................................................. 15
2.3.3 Numerical Results .............................. .. ........................ 16
2.4 Active Region Model ............................... .................................... 21
2.4.1 Potential Wells and Quantized States ................................... .......... 21
2.4.2 Parabolic In-plane Subband Structures ............................................ 24
2.4.3 Numerical Results ................................................. ...................... 25
2.5 Single-Particle Gain and Spontaneous Emission Functions .......................... 26
2.5.1 Transition Energy .............................................................. 26
2.5.2 Gain and Spontaneous Emission ..................................... ......... 27
2.6 Simulation Results ............................ ......... ............... ....... 30
2.6.1 InGaAs QW ....................................................... 30
2.6.2 CdZnSe QW ................................................................. 31
2.6.3 Comparison ................................................................... 31
2.7 Sum m ary .................................................................. ............................ 36

3 MANY-BODY EFFECTS .................................... .............................. 39

3.1 Introduction ..................................................................... ..................... 39
3.2 Carrier Scattering (CS) ................................................. ....................... 39



vi









3.2.1 Gain and Spontaneous Emission Functions ............................................ 39
3.2.2 Simulation Results ....................................................... 42
3.3 Bandgap Renormalization (BGR) ................................... ...................... 45
3.3.1 Plasma Screening ............................................ 45
3.3.2 Bandgap Renormalization ............................. ........................... 49
3.3.3 Simulation Results ............................................... 52
3.4 Coulomb Enhancement (CE) ................................... ......................... 53
3.4.1 Coulomb Enhancement Factor ................................... ....................... 53
3.4.2 Gain and Spontaneous Emission Functions ......................................... 60
3.4.3 Simulation Results ................................... .............................. 63
3.5 Sum m ary ............................ ... .............................................................. 66

4 LASER THRESHOLD CHARACTERISTICS ................................................ 67

4.1 Introduction .................................................................... ...................... 67
4.2 Laser Devices ............................................................... ........................ 68
4.2.1 InGaAs QW .................................. ... ..... ........................ 68
4.2.2 CdZnSe QW ......................................................... ........................... 68
4.3 Cavity Length Dependence of Lasing Energy ........................................... 68
4.3.1 Experiment ............................................................... 71
4.3.2 Prediction ................................................................ ........................ 72
4.4 Cavity Length Dependence of Threshold Current ....................................... 89
4.4.1 Experiment ...................................................... 89
4.4.2 Prediction ................................... ................................. 90
4.4.2.1 Gain-current relation .......................... ....................... 96
4.4.2.2 Predicting threshold current ................................. .......... 97
4.5 Temperature Dependence of Lasing Energy ................................................. 106
4.5.1 Experiment .................................... ...... ....... ..... 106
4.5.2 Prediction .......................................................... ................................ 107
4.6 Sum m ary .................................................................... .......................... 110

5 ABSORPTION RESONANCES DUE TO COULOMB ATTRACTION ............ 114

5.1 Introduction ................................................................... 114
5.2 Claims of Evidence for Excitonic Gain ........................ ........ .............. 114
5.3 Calculation of Gain/Absorption Spectra ........................................................ 115
5.4 Comparison between Calculations and Experiments ..................................... 121
5.4.1 Spectral Correlation between Peaks of
Absorption and Spontaneous Emission ............................................. 121
5.4.2 Energy Arrangement of Some Key Spectral Features .......................... 121
5.4.3 Coexistence of Gain with Absorption ................................................... 125
5.4.4 Evolution of Gain/Absorption Spectra with Injection Level ................ 126
5.5 Sum m ary .................................. .... ........... .................................. 126









6 CARRIER-INDUCED EFFECTS IN II-VI QW LASERS ................................... 130

6.1 Introduction ............................................................. ................................... 130
6.2 Effect of Injected Carriers on the Refractive Index ....................................... 131
6.2.1 Complex Optical Dielectric Function .................................................. 131
6.2.2 Complex Refractive Index ...................................... .......................... 132
6.2.3 Simulation results ................................ ....... ........... ............ 133
6.3 Beam Q quality .................................................. ....................... ................ 134
6.3.1 Antiguiding Factor ................................... ........................ .. 134
6.3.2 A Qualitative Description ............................ ........................ .. 141
6.4 Near-Field Measurement ............................. ............................ 146
6.5 Sum mary ............................................. ................................................. 149

7 CONCLUSIONS AND FUTURE WORKS ....................................................... 152

7.1 Conclusions .............................. ... ................. ............................. 152
7.2 Future Works ................................................................. 154

APPENDICES

A MATERIAL PARAMETERS lI-VI TERNARIES ............................................ 157

A.1 Interpolation Method ............................. .............. .......................... 157
A.2 Binary Parameters ....................................................... ... 158
A.3 Ternary Parameters ............................... ............. 158

B TRANSITION MATRIX ELEMENT ............................................................ 162

B.1 Polarization-Dependent Effects ................................... ..................... 162
B.2 Angle-Dependent Effects ............................................ ........................ 165

REFERENCES ............................................. ................................................... 168

BIOGRAPHICAL SKETCH ............................ ............. ............................ 174














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for Degree of Doctor of Philosophy

COULOMB ATTRACTION EFFECTS IN
SEMICONDUCTOR QUANTUM WELL LASERS

By

CHIA-FU HSU

December, 1997

Chairman: Peter S. Zory
Major Department: Electrical and Computer Engineering



Theoretical and experimental studies of many-body effects in semiconductor quan-

tum well (QW) lasers are presented. These many-body effects arise from the Coulomb

force interactions among the charged particles trapped in the quantum well and their inter-

action with phonons. Expressions are derived for the dependence of optical gain and spon-

taneous emission on photon energy using an electron-hole plasma (EHP) theory with

many-body effects taken into account. Optical gain and spontaneous emission spectra are

calculated for the well-developed, infrared emitting, InGaAs/GaAs strained QW structure

and the relatively-new blue-green CdZnSe/ZnSSe strained QW structure. The results show

that Coulomb attraction effects (CE) have a significantly stronger effect on the spectral

characteristics of the wide-bandgap CdZnSe QW structure than on the narrow-bandgap In-

GaAs QW structure. The room-temperature dependence of lasing energy and threshold

current on cavity length for InGaAs and CdZnSe QW diode lasers are numerically simulat-

ed and compared to experiment In all cases, the comparisons are found to be in better

agreement when CE is included in the calculations. The importance of CE is also found









when comparing the experimental and calculated lasing energies for a CdZnSe QW laser

over the temperature range from 160 K to 300 K. For photon energies somewhat higher

than the lasing energy, the calculated optical gain spectra with CE included show reso-

nance-like absorption features usually attributed to excitons in the QW. A comparison of

these absorption spectra with published experimental spectra shows good agreement.

Since excitons do not exist in the CE model used, this success refutes published work which

says that free excitons play an important role in determining the magnitude of optical gain

in room temperature CdZnSe QW lasers.














CHAPTER 1
INTRODUCTION


Semiconductor diode lasers have become the key components in many commercial

products such as compact disk players, laser printers and optical fiber telecommunication

systems. High-power diode lasers are also used in applications such as medical surgery,

free-space communication and diode laser pumped solid-state lasers. Diode laser design

with a double heterojunction (DH) active layer of thickness about 100 to 200 nm was per-

fected in the late 1970s and early 1980s. When reducing the thickness of the active layer

down to the 10 nm regime, electrons and holes are confined in the active layer and the as-

sociated states become quantized in the potential wells along the direction normal to the ac-

tive layer. Quantum well (QW) diode lasers, devices with this very thin active layer design,

were realized in the 1980s and shown to have higher performance than their DH counter-

parts. By introducing built-in strain in the QW, these strained QW lasers were found to be

substantially superior to unstrained QW lasers. Although these improvements in diode la-

sers rely heavily on advances in high-quality crystal-growth techniques such as molecular

beam epitaxy (MBE), the ideas behind them came from thoroughly understanding the basic

mechanism of optical gain and spontaneous emission in diode lasers. Stimulated electron-

hole recombination in the active layer was believed to be responsible for the laser action in

diode lasers and an electron-hole plasma (EHP) theory was used to model the laser devices

such as the well-developed infrared (IR) GaAs-based QW lasers [Asa93, Col95, Cor93].
Soon after the demonstration of the world's first blue-green diode lasers fabricated using a

CdZnSe QW structure by 3M Company personnel in June 1991 [Haa91], confusion about

the gain mechanism in these new CdZnSe QW lasers began. J. Ding etal. reported the first









observation of so-called excitonic gain and laser emission in ZnSe-based QWs and pro-

posed a phenomenological model of optical gain based on partial phase-space filling (PSF)

of an inhomogeneously broadened exciton resonance [Din92]. This phenomenological

gain model, which includes the effects of exciton-phonon interaction, was used to explain

their pump-probe experiment at cryogenic temperature up to 220 K [Din93]. In 1994, J.

Ding et al. reported the measured gain/absorption spectra of CdZnSe QW diode lasers at T

= 300 K in which two resonance-like absorption peaks appearing at low excitation level

were attributed to absorption resonances from n = 1 HH and LH excitons [Din94]. The con-

clusion drawn was that the origin of optical gain in blue-green CdZnSe QW lasers is exci-

tonic in nature even at room temperature. Since the conventional EHP theory did not

include exciton effects, these claims attracted considerable attention in the diode laser com-

munity.

Setting aside the truth about the lasing mechanism in CdZnSe QW lasers for a mo-

ment, physicists, working for many years in the research field of fundamental semiconduc-

tor physics, have been studying the mutual interactions among charged particles (i.e.

electrons and holes) in semiconductors. These interactions in an EHP system include car-

rier scattering (CS), bandgap renormalization (BGR) and Coulomb enhancement (CE) and

are usually referred to as many-body effects in the literatures. Haug et al. showed that gain

spectra of a semiconductor changed when many-body effects were taken into account

[Hau90]. Although CS and BGR were usually included in the conventional EHP theory

[Asa93, Col95, Cor93], there seemed to be a tendency to ignore CE in the diode laser com-

munity since CE calculations are more complicated than those for CS and BGR. A theory

with such an ignorance of CE was shown to lead to predictions that were in worse agree-

ment with experiment than predictions of a theory with only CS included [Che93, Chi88].

The errors result from the fact that it is inconsistent to account for the Coulomb repulsion,

which gives rise to BGR, and at the same time ignore the Coulomb attraction, which gives

rise to CE. Since Coulomb interactions are stronger in materials with smaller dielectric









constants, CE is expected to be more important in the wide-bandgap CdZnSe QW lasers

[Cho95b, Ree95a] than in the narrow-bandgap GaAs-based QW lasers [Cho94, Hau90].
Now going back to the discussion about the nature of the gain mechanism in CdZnSe

QW lasers, two questions were thus raised: "Can the operation of CdZnSe QW lasers be
simulated using EHP theory with inclusion of CE?" and "Is optical gain in CdZnSe QW

lasers of excitonic nature?" The approach to answering these questions is twofold: theoret-

ical and experimental. In the theoretical part, theories of many-body effects available in

the literatures were studied extensively and understood thoroughly. These existing theoret-

ical works were all done under the assumption of a perfect two-dimensional (2D) QW with

zero QW thickness and infinite barrier height. In addition, the formula derived for many-

body effects are limited to the case that only the lowest conduction (CB) and valence band

(VB) subbands exist. The simple perfect 2D QW model has no physical meaning except

for easy demonstration of many-body effects in semiconductors.
In order to use these many-body formulas and implement them in the device simula-

tion for actual QW diode lasers, several modifications to the formulas had to be made.

First, the finite QW thickness and barrier height in real QW lasers were taken into account

In such a quasi-2D QW, a number of subbands in both CB and VB become available from

solving the one-dimensional (lD) Schridinger's equation. Second, effective mass anisot-

ropy in the VB due to the large built-in strain the QW was considered, where the heavy hole
(HH) and light hole (LH) effective masses in the in-plane direction become different from

those in the growth direction. Third, a model with CE included for the spontaneous emis-

sion function was developed in order to estimate the threshold current required for lasing,

since most of the existing literatures on many-body effects really dealt only with the gain
function [Cho94, Hau90]. Four, the modified many-body formulas were integrated with

the state-of-the-art EHP theory where CE has not been taken into account [Asa93, Col95,

Cor93]. Efforts were also placed on converting formulas from wavevector space to energy

space, from summations to integrals and from the C.G.S. units used by many-body physi-









cists to the M.K.S. units which are more familiar to diode laser engineers. As a result of

these works, the complicated formulas derived by many-body physicists have been trans-

lated to a form more user-friendly so that electrical engineers can implement them into

computer codes for device simulation of QW diode lasers.

In the experimental part of this work, a number of well-developed IR InGaAs

strained QW lasers and relatively new blue-green CdZnSe strained QW lasers were fabri-

cated. Since the laser threshold current and peak wavelength change with laser cavity

length, different models for the physical mechanism of optical gain and spontaneous emis-

sion were tested by investigating these length dependence. Since the emphasis in this

work has been placed on the CdZnSe QW laser, further measurements on the temperature

dependence of lasing wavelength were performed on the CdZnSe QW lasers.

This dissertation is organized such that the theoretical models are described and de-

veloped first, experiments and experimental results are then presented, predictions by using

the theoretical models and their comparisons with experiments are made and finally the

conclusions are drawn. Several comparisons and comments on the differences between the

InGaAs and CdZnSe QW lasers are made throughout each chapter.

In Chapter 2, the gain and spontaneous emission functions based on a single-particle

model are discussed in detail. It begins by introducing the epitaxial structures for the two

QW lasers of interest: Il-V InGaAs and II-VI CdZnSe QW lasers. Effect of strain on the

bandgap of the QW material are discussed and the active region model used in the theoret-

ical simulation for the QW laser structures described. Finally, the optical gain and sponta-

neous emission spectra in the single-particle model are calculated.

In Chapter 3, many-body effects, such as CS, BGR and CE, on the gain and sponta-

neous emission functions are discussed sequentially. Semi-derivations of BGR and CE are

presented. Calculation results of the optical gain and spontaneous emission spectra with

many-body effects included are shown and comments on how CS, BGR and CE modify the

single-particle spectra are made.






5


In Chapter 4, experimental measurements of laser threshold characteristics for In-

GaAs/GaAs QW and CdZnSe/ZnSSe QW lasers are presented and compared to predic-

tions. The first question mentioned above "Can the operations of CdZnSe QW lasers be

simulated by using the EHP theory with inclusion of CE?" is answered.

In Chapter 5, the other question "Is optical gain in CdZnSe QW lasers of excitonic

nature?" is answered by comparing several spectral features of the calculated gain/absorp-

tion and spontaneous emission spectra with the inclusion of CE to those of the experimental

observations.

In Chapter 6, the carrier-induced antiguiding factor is calculated for the CdZnSe QW

lasers and its effect on the quality of laser beam is described. Near-field patterns are mea-

sured and compared qualitatively to the predictions.

Finally, in Chapter 7, conclusions are drawn and future works are suggested.














CHAPTER 2
SINGLE-PARTICLE MODEL


2.1 Introduction


The chapter begins by introducing the epitaxial structures for the two QW lasers of

interest: LI-V InGaAs and II-VI CdZnSe QW lasers. Due to the mismatch of the in-plane

lattice constant between a thin QW layer and thick barrier layers, the thin QW layer is sub-

ject to lattice deformation and under biaxial strain. Then, effect of strain on the bandgap of

the QW material will be discussed. A common approach is used to determine the bandgap

lineup at the interface between the QW and barrier layers by assuming a conduction band

offset ratio. Strain calculation will be performed on both InGaAs and CdZnSe QW laser

structures. Next, we consider an active region model to be used in the theoretical simula-

tion for the QW laser structures. It consists of a QW layer and two barrier layers of infinite

thickness on each side. A flat-band approximation is assumed in constructing the potential

well diagram for the active region model. A typical quantum well problem is then solved

for the quantized energy levels in the CB and VB (including HH and LH bands) potential

wells. Numerical results from the calculation for the two material systems will be shown

and summarized. Finally, from the active region model developed, we will be able to ex-

plore the optical gain and spontaneous emission functions using the single-particle model.

A set of calculated gain and spontaneous emission spectra as a function of carrier density

in the QW for both InGaAs and CdZnSe QW lasers will be presented. Comparison between

two material systems will be made.

Several appendices will serve as the supplements for more in-depth discussions of









many relevant subjects in this chapter. In brief, derivation of material parameters for

CdZnSe and ZnSSe ternary alloys from the material parameters of the related binary alloys,

based on an interpolation scheme, can be found in Appendix A. In Appendix B, the tran-
sition matrix element, which determines the strength of interaction between two states, and
its polarization-dependent effects are discussed.


2.2 Laser Structures


2.2.1 InGaAs OW Laser


The InGaAs/GaAs graded-index separate-confinement heterostructure (GRIN-SCH)

single QW diode laser structure studied in this work, for which the schematic sketch of

cross-section epitaxial layers and the corresponding diagram of Al and In composition pro-
file are shown in Fig. 2.1, consists of a 25 nm graded AlxGal.As layer (x = 0.05 0.6)

grown on a n-GaAs substrate, a 1400 nm n-Alo.6Gao.4As cladding layer, a 200 nm graded

AlyGa1.As (undoped) guiding layer (y = 0.6 0.3), an 8 nm undoped InzGal.,As (z ~ 0.15)

compressively-strained quantum well centered between two 7 nm GaAs layers, a 200 nm

graded AlyGa1.yAs (undoped) guiding layer (y = 0.3 0.6), a 1300 nm p-Al0r6Gao0As clad-

ding layer, a 25 nm graded Al.Gal.xAs layer (x = 0.6 0.05) and a 100 nm p+-GaAs cap

layer. The guiding and cladding layers provide for electrical and optical confinement re-

spectively in the growth direction.


2.2.2 CdZnSe OW Laser


A cross-section diagram of the epitaxial layers of the CdZnSe/ZnSSe SCH SQW di-
ode laser studied in this work is shown in Fig. 2.2. It consists of a GaAs buffer layer grown

by MBE on an n-type GaAs (001) substrate [Gai93], a ZnSe buffer layer, a quaternary n-









ZnMgSSe cladding layer, a 4 nm CdxZnl.,Se (x ~ 0.3) compressively-strained quantum

well (QW) centered between n- and p-ZnSo.6Se0.94 guiding layers, a p-ZnMgSSe cladding

layer and a graded p+-ZnSexTel., (x = 1 to 0) contact layer. This contact scheme provides

a reasonably low resistance p-contact to the laser structure [Fan92].


2.3 Strained Ouantum Wells


2.3.1 Strained Bandgaps


When a very thin layer of QW is grown out of a material with a larger native lattice

constant than that of the surrounding barrier layers, the lattice compresses in the plane of

the well (in-plane direction denoted as II) to match that of the barrier layers and elongates

in the growth direction (normal to the plane of well and denoted as _) to keep the volume

of each unit cell the same, as shown in Fig. 2.3. In this case, QW is said to be under biaxial

compressive strain. If the native lattice constant of QW is smaller than that of the barrier

layers, QW is then under biaxial tensile strain. Degree of lattice-mismatch can be described

with a lattice-mismatch parameter e defined as follows [Col95]:

e= 1- ab, (2.1)
aw

where ab is the in-plane lattice constant of the barrier material and a, is the native un-

strained in-plane lattice constant of the strained QW material. From Eq. (2.1), we have e >

0 for biaxial compressive strain (a, > ab) and e < 0 for biaxial tensile strain (a, < ab). The

lattice distortion of the crystal due to the stress resulting from lattice-mismatch is usually

defined mathematically by a strain tensor e Assuming the shear component of stress (i *

j) can be ignored, which is valid in typical semiconductor application, we thus only need to

consider the three diagonal components of the strain tensor: e., ey and e, where x and y
subscripts indicate the in-plane directions (II) and z subscript is the growth direction (1).


















% In % Al
20 10 0 10 20 30 40 50 60 70
I I I I I I I I I I


cap 100 nm p-GaAs
buffer 25 nm p-AlGaAs


cladding 1300 nm p-AlGaAs


guiding 200 nm AlGaAs

bounding 7 nm GaAs
QW 8 nm InGaAs
bounding 7 nm GaAs

guiding 200 nmAlGaAs


cladding 1400 nm n-AlGaAs

buffer 25 nm n-AlGaAs


substrate


n-GaAs


Figure 2.1 Schematic diagrams of the cross-section epitaxial layers and the corre-
sponding aluminum and indium composition profile for the Ino.05Gao.BAs/
GaAs GRIN-SCH single QW laser.






















contact



cladding



guiding

QW

guiding



cladding


buffer
buffer
substrate


p+-ZnSeTe



p-ZnMgSSe



p-ZnSSe

CdZnSe

n-ZnSSe


n-ZnMgSSe


n+-ZnSe
n-GaAs
n-GaAs


Figure 2.2 Schematic diagram of the cross-section epitaxial layers for the Cdo.3Zno.Se/
ZnSo.06Seo. SCH single QW laser.








By symmetry, the strain in both x and y directions must be equal and we obtain

el exx = yy = -e. (2.2)

Using Eq. (2.1) in Eq. (2.2), we find that w, e& < 0 for biaxial compressive strain and e,
, > 0 for biaxial tensile strain. With no stress applied to the z direction and the cubic sym-
metry of the crystal, the strain in the growth direction is related to the strain in the in-plane
direction by [Col95]
C12
S= e = -2 iel, (2.3)

where CI1 and C12 are referred to as the Young's elastic moduli (usually described in the
units of 1010 N/m2). Since C11 and Ci2 are both positive in common semiconductors (usu-
ally Cn > C12), the lattice deformation along 1 direction will be opposite to the deformation
along II directions, as depicted in Fig. 2.3.

Because the bandgap of a semiconductor material is related to its lattice spacing, the
distortion in crystal lattice due to the strain should lead to modifications in the bandgap of
the strained QW layer. Putting aside the bandgap changes due to the quantum confinement
for the present moment, which is left to be discussed in Section 2.4.1, there are two modi-

fications in the strained bandgap. The first modification originates from the hydrostatic
component of the strain and under biaxial compressive strain, it produces an upward shift
in the conduction band (CB) as well as a downward shift in both heavy hole (HH) and light
hole (LH) valence bands (VB), as sketched in Fig. 2.4(a). All the energy shifts are in the
opposite directions under biaxial tensile strain. As a result of the hydrostatic strain, the
bandgap is changed by an amount SeH which is given by [Col95]
C11-C12
BeH = 2aE,,l C (2.4)

where a is the hydrostatic deformation potential and a < 0 in common semiconductors. We
find that the bandgap is increased (iSe > 0) for biaxial compressive strain and decreased
(6&H < 0) for biaxial tensile strain. The second modification originates from the shear com-












barrier -


+
QW a. .4
o f f -A- a -w


barrier 4-

ab
ab

biaxial compressive strain

(a)



barrier




a. +

barrier

ab
ab>aw au(= a)>a,>ai


biaxial tensile strain

(b)

Figure 2.3 illustration of the crystal lattice deformation resulting from the epitaxial
growth of a thin layer of QW with a native lattice constant a, between two
thick barrier layers with lattice constant (a) ab < a, (biaxial compressive
strain); (b) ab > aw (biaxial tensile strain).








ponent of the strain (the shear strain should not be confused with the shear stress which is
zero in this case [Col95]) and separates the LH band from the HH band, as sketched in Fig.
2.4(a). Under biaxial compressive strain, the shear strain produces an upward shift in HH
band by an amount Ses. and a downward shift in LH band by an amount 8es, t which can
be expressed as

Cl1 +2C12
8E5, hh = be C (2.5)


Be = ESh, (2.6)

where b is the shear deformation potential and b < 0 in common semiconductors and A is
the spin-orbit energy which separates the split-off (SO) band from both the HH and LH
bands. Under biaxial tensile strain, the shear strain produces an downward shift in HH band

( s, hh < 0) and an upward shift in LH band (Ses, th < 0). As a consequence of the shear
strain, the band edge degeneracy of the HH and LH bands is removed and two strained
bandgaps are now needed to be defined as follows:

Eg (HH) = E (bulk) +8eH-eS, hh
E (LH) = E (bulk) + 5et + 8es, Ih

where E, (bulk) is the unstrained bulk bandgap, E, (HH) is the C-HH strained bandgap and
E, (LH) is the C-LH strained bandgap. The splitting of the HH and LH bands is usually
quantified by the splitting energy S which can be defined as

S=Eg (LH) E (HH)
= ES, lh + ES, hh (2.8)

= 2S, hh( l ,hh

where Eqs. (2.7) and (2.6) are utilized in the last two expressions, respectively.










CB
CB
CB

shear hydrostatic hydrostatic shear
component component component component

SHH, LH
HH > HH, LH HH
HH, LH
LH


tensile strain --- unstrained -- compressive strain


(a)


barrier QW barrier


tensile strain


barrier QW barrier


compressive strain


Figure 2.4 Effects of the biaxial tensile and compressive strains on (a) the bulk band-
gap of QW material and (b) the bandgap lineup with two barrier layers.








2.3.2 Bandgap Lineup


The bandgap difference between QW and barrier materials causes band discontinui-

ties in both CB and VB at the interfaces of the heterostructure, as sketched in Fig. 2.4(b).

The CB offset is defined as AE, and the VB offsets are defined as AE" and AEI for HH and

LH bands since they are split by the shear strain. Defining AE, as the bandgap difference
between the C-HH strained bandgap of QW and the bandgap of barrier layers, we obtain

AEg = Eg (barrier) E (HH) (2.9)

where E, (barrier) is the bandgap of barrier material. As shown in Fig. 2.4(a), CB edge of
the strained QW, as well as VB edge, experiences a shift due to the hydrostatic strain. In
reality, it is difficult to experimentally separate the CB shift from the total bandgap shift
[Cor93]. A common approach is, first, not to worry about how the total shift is divided up
between the CB and VB, and then for heterostructures, the lineup of the strained QW band-

gap with the barrier bandgaps is typically determined by a CB offset fraction Q, which can
be experimentally measured. In this approach, the CB offset can therefore be expressed as

AEc = QcAES (2.10)

and the VB offset for HH band is given by

AEhh = (1- Q)AEg. (2.11)

Since the LH band is pushed away from the HH band by an amount S as defined in Eq.

(2.8), the VB offset for LH band can be shown to be

AEt = AEhh-S, (2.12)

which means AElh < AEhh under biaxial compressive strain (S > 0) and AErh > AE, under
biaxial tensile strain (S < 0), as depicted in Fig. 2.4(b).








2.3.3 Numerical Results


InGaAs OW. The material parameters as functions of In and Al mole fractions for

InGaAs/AlGaAs system have been summarized in Table III of [Cor93]. As shown in Fig.

2.5(a), for the unstrained Ino.15Gao.g5As, we have E, (bulk) a 1.194 eV and for the strained

Ino15Gao.sAs QW/GaAs barrier, we obtain E, (barrier) 1.424 eV, E, (HH) 1.267 eV, E,
(LH) = 1.329 eV and S = 0.062 eV. The CB offset in the strained InGaAs/GaAs system is

uncertain at present. Various values for Qc have been reported anywhere in the range from

0.4 to 0.8 [Cor93]. We will assume that Qc = 0.55 for the InGaAs/GaAs system in this work

[Wu94, Hsu97a]. As a consequence, with AE,= 0.157 eV, we obtain AEc 0.087 eV, AE,

a 0.071 eV and AEh, 0.025 eV, as sketched in Fig. 2.5(b). Numerical values of the mate-

rial parameters used in this work for the strained Ino.5sGao.5sAs QW/GaAs barrier system

are summarized in Table 2.1.

CdZnSe OW. For II-VI CdZnSe/ZnSSe system, the ternary material parameters are

less well known. In this work, we derive some material parameters used in the calculation
for CdZnSe and ZnSSe ternary alloys from the material parameters of the related binary al-

loys based on an interpolation scheme. Given the present focus on how strain modifies the

bandgap of CdZnSe QW, we leave the derivation of the ternary material parameters to Ap-

pendix A. Then plugging the II-VI ternary material parameters derived in Appendix A into

Eqs. (2.1) through (2.8), we have E, (bulk) 2.373 eV for the unstrained Cdo.3Zno.7Se and

for the strained Cdo.3Zno.Se QW/ZnSoo.Se.94 barrier, we obtain E, (barrier) a 2.729 eV,

EH = 0.081 eV, ASs. h = 0.057 eV, Es, h = 0.041 eV, E, (HH) 2.396 eV, E, (LH) = 2.494
eV and S s 0.098 eV, as depicted in Fig. 2.6(a). Assuming Qc = 0.6 for the CdZnSe/ZnSSe
heterostructure [Hsu97b] and with AE, = 0.333 eV, we obtain AEc 0.200 eV, AE = 0.133

eV and AEa = 0.035 eV, as sketched in Fig. 2.6(b). Numerical values of the material pa-

rameters used in this work for the strained Cdo3Zno.tSe QW/ZnSo.0Seo.94 barrier system are
summarized in Table 2.2.













Ino.i5Gao.ssAs Ino.s1Gao.s5As
(bulk) (strained QW)
CB CB

1.194 eV
1.267 eV
1.329 eV
HH, LH
HH
0.062 eV
LH



(a)


GaAs Inoi5Gao.s5As GaAs

8 nm
CB 0.087 eV



1.424 eV 1.267 eV


HH

0.025 eV LH 0.071 eV



(b)

Figure 2.5 The strain modified (a) bulk bandgap and (b) potential wells of the 8 nm
Ino.isGao.s5As/GaAs QW under biaxial compressive strain.

















Parameters Ino. 5Gao.85As GaAs
bulk bandgap E, (eV) 1.194 1.424
Luttinger Parameter y, (x h2/2mn) 8.773 6.85
Luttinger Parameter y2 (x h2/2mo) 3.041 2.1
electron effective mass me (x mo) 0.061 0.067
growth-direction HH effective mass mn (x mo) 0.372 0.377
growth-direction LH effective mass mkL (x mo) 0.067 0.091
background refractive index n 3.6
static dielectric constant e, 13.33
In o.1Gao.sAs/GaAs strained parameters
Ino.5Gao.s5As HH strained bandgap E,(HH) = 1.267 eV
Ino.5Gao.ssAs LH strained bandgap E,(HH) = 1.329 eV
Ino.5Gao.85As splitting energy S = Eg(LH) Eg(HH) = 0.062 eV
Bandgap difference AE, = 0.157 eV
CB offset fraction Qc AAE/AE, = 0.55
CB offset AE, = 0.087 eV
HH band offset AEh = 0.071 eV
LH band offset AE, = 0.025 eV
Ino.15Gao.ssAs in-plane HH effective mass m;hn = 0.085 m,
Ino.05Gao.ssAs in-plane LH effective mass mth, = 0.174 m,



Table 2.1 List of material parameters for the Ino.15Gao.85As/GaAs strained QW.












hydrostatic
component


Cdo.3Zno.7Se
(bulk)
CB
2.373 eV


VB


shear
component

Cdo.3Zno.Se
(strained QW)


- I 0.057 ev HH

0.080 eV -- LH
0.041 eV


(a)


ZnSo.Seo.94 Cdo.3Zno.7Se ZnSo.06Seo.94


Figure 2.6 The strain modified (a) bulk bandgap and (b) potential wells of the 4 nm
Cdo.3Zno.7Se/ZnSo.o6Seo94 QW under biaxial compressive strain.















Parameters Cdo.3Zno7Se ZnSo.o6Sea94
bulk bandgap E, (eV) 2.373 2.729
Luttinger Parameter y (x h2/2mo) 4.495 4.3
Luttinger Parameter y2 (x t2/2m,) 1.206 1.14
electron effective mass m, (x mo) 0.139 0.152
growth-direction HH effective mass mm (x m,) 0.480 0.571
growth-direction LH effective mass mlL (x mo) 0.145 0.157
background refractive index zi 2.947 2.726
static dielectric constant e, 9.524 9.23
Cdo.3Zno.Se/ZnSo.o6Se.94 strained parameters
Cdo3Zno.7Se HH strained bandgap E,(HH) = 2.396 eV
Cdo.3Zno.7Se LH strained bandgap E,(HH) = 2.494 eV
Cdo.3Zno.Se splitting energy S Eg(LH) Eg(HH) = 0.098 eV
Bandgap difference AE, = 0.333 eV
CB offset fraction Q, = AEJAE, = 0.6
CB offset AE, = 0.200 eV
HH band offset AEhh = 0.133 eV
LH band offset AEl = 0.035 eV
Cdo3Zno.7Se in-plane HH effective mass mhhll = 0.175 mo
Cdo.Zno.7Se in-plane LH effective mass mthll = 0.304 m,


Table 2.2 List of material parameters for the Cdo.3Zno.Se/ZnSoo.6Se.94 strained QW.








2.4 Active Region Model


So far, we have determined the bandgap lineups at interfaces of the strained hetero-

structures. The next step, before going ahead to introduce the gain and spontaneous emis-

sion functions, is to define an active region model for the laser structures to be simulated.

For the QW laser structures studied, as shown earlier in Figs. 2.1 and 2.2 for the InGaAs

QW and CdZnSe QW respectively, they both have an active region structure consisting of

a strained QW material sandwiched between two barrier layers. It is conventional to model

the actual active region structure with the barrier layers on each side of QW extending to

infinity. It is also assumed that the potential wells associated with the active region are

symmetric and rectangular, like the ones shown in Figs. 2.5(a) and 2.6(a). As discussed in

[Blo96], there is good reason to believe that this so-called flat-band approximation is valid

for QW lasers under typical operating conditions.

With the active region model described above, in the following subsections we first

will solve the quantized electron and hole states for the one-dimensional potential wells in

CB and VB respectively. Then we will discuss the in-plane subband structures in a para-

bolic band approximation.


2.4.1 Potential Wells and Ouantized States


Fig. 2.7(a) shows a generic potential well diagram along the growth direction of the

active region model used in the simulation. For the CB electrons and VB holes moving

along the growth direction, they will experience potential wells with barrier heights equal

to the CB offset AEc and VB offset AE,, respectively. As a result, energies are quantized

in the potential wells. We define Eq and Ej as the energies of thejth quantized states in the

CB and VB potential wells, respectively, and are measured relative to the bottoms of their

corresponding potential wells. In the VB, there are HH and LH bands in our active region








model. Therefore, there is the need to clarify that the HH quantized energy EnHH and LH
quantized energy ELj are measured from the bottoms of the HH and LH potential wells of
barrier heights AE, and AEi,, respectively. Finding Eq and Ej now becomes solving a
quantum well problem.
Details of solving this quantum well problem can be found in many basic quantum

mechanics textbooks, here we refer it to the treatment in [Cor93]. Basically, first it is to
solve the Schr6dinger's equation for each section of the active region (two barrier and one
QW). Then by matching proper boundary conditions across two well-barrier interfaces, a
pair of characteristic equations are obtained

tan( k nid, m = 0 for j is odd, (2.13)
-- 2 mnb knjl


"cot( k. )!-n- = 0 for jis even, (2.14)
nj 2 mnb knj.

with


kn = F., (2.15)

2mnb (AEn-E .)
a n = t (2.16)

where d, is the thickness of QW, mj is the growth-direction effective mass of the QW ma-
terial, m, is the effective mass of the barrier material, kjnl is referred later in Chapter 3 to
as the equivalent growth-direction component of the wavevector in QW associated with

E,y, aj is the magnitude of the complex wavevector in the barrier layer associated with Ej
and AE, is the barrier height. The n subscript indicates CB for electrons and HH and LH
bands in VB for heavy and light holes, respectively. Eqs. (2.13) and (2.14) can only be sat-
isfied by a set of discrete energies Ej which represent the bound solutions of the cone-
sponding quantum well.





















electron energy








k,



0 -------- -



hole energy



(a) (b)


Figure 2.7 Generic diagrams of (a) the flat-band potential wells and (b) the parabolic
subband structure of the QW active region model.








2.4.2 Parabolic In-plane Subband Structures


As discussed above, the potential wells are one-dimensional along the growth direc-
tion. Within the plane of the well, carriers (electrons and holes) are not confined and still
behave like "free." Thus, for each quantized energy level in the QW, an in-plane energy
subband exists, as shown in Fig. 2.7(b). The subband structure is usually characterized by
defining an in-plane effective mass as [Kit86]
[ I d2E ,1-1
m,, = [ d (2.17)

where mnfi is the in-plane effective mass, Enjk is the in-plane energy measured relative to
the bottom of the well and kl is the in-plane wavevector associated with Enj, Thej sub-
script indicates the jth subband in the well and the n subscript indicates CB for electrons,
HH and LH bands for heavy and light holes, respectively. If the parameter m^, is a constant
with respect to Enjk the subband structure is said to be parabolic. In a nondegenerate
band, such as the CB (excluding spin degeneracy), all the subbands are assumed to be par-
abolic and the electron effective masses along the I and II directions are identical (mj.e =

mqt). However, the in-plane subband structure in a degenerate band, such as the VB, is
more complicated, since band coupling between the HH and LH bands can be very strong.
In that case, the VB subbands become nonparabolic band due to this so-called valence band
mixing effect. Since the quantum wells are under biaxial compressive strain for the laser
structures studied in this work, the LH band is pushed away from the HH bands due to the
shear component of the strain. This greatly reduces the valence band mixing effect between
the HH and LH bands and allows us to assume that the subband structures in the VB are
also parabolic. In any event, the growth-direction HH and LH effective masses are not af-
fected by the strain and are given by [Cor93]

1 h2 1 h2
mhh = T- m' = +2 (2.18)








where the material constants yj and y are referred to as the Luttinger parameters [Lut56].
In the high strain approximation, the in-plane HH and LH effective masses can be ex-

pressed in close forms with in terms of Luttinger parameters yj and t2 (in the units of

h2/2m.) [Cor93]

1 h2 1 h2
mt = -+-2 2' m"y- T, -- 2 (2.19)
hhll Y1+722 1 Y-722

From the preceding discussion of strain and by comparing Eqs. (2.18) and (2.19), we find
that not only do the HH and LH bands split apart, but also the HH band becomes "lighter"
and the LH band becomes "heavier" in the plane of compression. Thus, this is contrary to
the bulk material where the effective masses along I and II directions are identical.


2.4.3 Numerical Results


InGaAs OW. For the InOGaossAs QW, we obtain electron effective mass m, = mc.

= my = 0.061m,, Luttinger parameters y, = 8.773, y2 = 3.041 and for the GaAs barrier, we
have me = mb = 0.067m,, Ti = 6.85 and T2 = 2.1 using the material parameters for the In-

GaAs/AlGaAs system summarized in Table III of [Cor93]. Plugging the numerical values

for yi and 2 in Eqs. (2.18) and (2.19), we obtain mm = 0.372m,, ma" = 0.067m,, mhl =

0.085m, and m1ry = 0.174m, for the In.15Gao.g5As QW and m,,. = 0.377mo and mk =
0.091m, for the GaAs barrier. Having all the values of effective masses plugged into Eqs.
(2.13) through (2.16), we solve the quantized levels for the CB and VB (HH and LH) po-

tential wells of width d, = 8 nm, as shown in Fig. 2.5(b). The calculation yields one electron
subband in the CB (Ec1 = 31.6 meV relative to the bottom of CB potential well), two HH

subbands (EnHH = 9.2 meV and Enm = 35.4 meV relative to the bottom of HH potential

well) and one LH subband (ELU, = 6.6 meV relative to the bottom of LH potential well) in
the VB. Numerical values of the material parameters used in this work for the strained

Ino isGao.s5As QW/GaAs barrier system are summarized in Table 2.1.








CdZnSe OW. For the Cdo.3Zno.7Se QW, we obtain electron effective mass m,= ml.

= mru = 0.139mo, Luttinger parameters yi = 4.495, y2 = 1.206 and for the ZnSo.06Seo.9 bar-
rier, we have mc = mcb = 0.152m0, Yi = 4.3 and a2 = 1.14 using the ternary material param-

eters for CdZnSe/ZnSSe system derived in Appendix A. Using the numerical values for t

and y2 in Eqs. (2.18) and (2.19), we obtain m = 0.480mo, m.L = 0.145mo, mkq = 0.175m,

and mu = 0.304m, for the Cdo.3Zn.7Se QW and m;,_ = 0.571m, and mta = 0.157m0 for

the ZnSo.06Seo.94 barrier. Plugging all the values of effective masses into Eqs. (2.13)

through (2.16), we solve the quantized levels for the CB, HH and LH potential wells of

width d, = 4 nm, as shown in Fig. 2.6(b). The calculation yields one electron subband in

the CB (Ec- = 62.6 meV), two HH subbands (Enm- = 23.9 meV, EHH = 90.3 meV) and one

LH subband (EmI = 23.1 meV) in the VB. Numerical values of the material parameters

used in this work for the strained CdoZ.Zno.7Se QW/ZnSo.o6Se.94 barrier system are summa-

rized in Table 2.2.

Comparison. As shown above, electron and hole effective masses of the CdZnSe/
ZnSSe system are much larger than those of the InGaAs/GaAs system. In Section 2.6.3,

the difference in the effective masses will be shown to play an important role in the differ-

ence of the transparency carrier densities between the two QW diode laser systems.


2.5 Single-Particle Gain and Spontaneous Emission Functions


2.5.1 Transition Energy


In Section 2.4, we have shown how the strain modifies the bandgap of a QW and how

the quantum effect modifies its strained bandgap. As illustrated in Fig. 2.7(a), we define

Ej as the "modified" energy gap between the jth conduction and valence subbands

E = E + Ec + E, (2.20)

where E, is E,(HH) for the C-HH strained bandgap or E,(LH) for the C-LH strained band-








gap, Eq and E, are the quantized energy levels in the QW. For an electron-hole-pair par-
ticipating in a transition with energy E,,, as depicted in Fig. 2.7(b), we obtain

Ec = E + Ecjk + Evjk, (2.21)

where Ecjk and Evjk, are the energies of electrons and holes respectively participating in
this transition and are measured relative to the bottoms of CB and VB potential wells. In
terms of the in-plane wavevector k1, the expressions for Ejk, and EVk, are given by

h2k h2k2
Ecjk =E Ejk == EVj 2m ,1 (2.22)

where mcl and mj, are the in-plane effective masses and both conduction and valence sub-
bands are assumed to be parabolic as discussed in Section 2.4.2.


2.5.2 Gain and Spontaneous Emission


In simulating the operating characteristics of standard QW diode lasers, it is custom-
ary to utilize two functions which describe the dependence of QW gain and spontaneous
emission on photon energy hAw. In deriving these functions, defining g (ha) as the spectral

gain function and R, (ha) as the spontaneous emission function, approximations with var-

ious levels of accuracy can be made. In the first level approximation, one ignores many-

body effects and assumes that electrons trapped in the QW interact only with the electro-
magnetic field. In this "single-particle" model, where hcw is equal to the transition energy
E, we have g (hew) = g (E_) and R, (ha) = R,p (E_). g (E,) and R,, (E,) are calculated by
summing over the contributions from all the allowed subband transition pairs as given by

g (Ec,) = ,gj (Ev), (2.23)


Rv (Ec) = Rs (Ecv), (2.24)

where gj(E,) and R,, (E,) are the spectral gain function and spontaneous emission function








for an allowed subband transition pair, respectively. Since there is only one subband in the
CB in our structure, there are two allowed transition pairs, viz. Cl-HH1 and C1-LH1. In
the single-particle model, one ignores many-body effects and assumes that momentum is
conserved (k-selection holds) in the photon emission and absorption processes. In this
case, it can be shown that gj (E_) and Rj (E,) are given by [Cor93]

g (Ec) = (ML) 2P dM[f(Ecjk) fh(Evjk) -1] (2.25)
0 0
e2i -2
Rs (Ec) = (lo) I Jt2eCp3i d (Ecjk,) f (EVk,)] (2.26)

where e is the electron charge, eo is the vacuum permittivity, c is the vacuum light speed,
m, is the electron rest mass, i is the background refractive index of the QW material,

IMT2 is the transition matrix element, lM]2 is the average transition matrix element, p,r
is the reduced density of states,fe is the Fermi-Dirac distribution for CB electrons andfh -
1 -f, is the distribution for VB holes. The transition matrix element IMA2 is given by

2+ cos20 for C-HH TE gain

IMT12 1-cos20 for C-HH TM gain
S- 5 1 2 ,(2.27)
IM2 -cos28 for C-LHTE gain

S+ cos20 for C-LH TM gain

where

MI2 (E +- 2 mEg, (2.28)


(E C+Ej)m

coS20 = mrjI. (2.29)
(E, E Ecj- E) + (Ecj + E ,)

For the InGal,As QW, MIM2 can be simplified and expressed as a function of indium mole








fraction [Cor93]

M 2 (28.8 6.6x) (emo) (2.30)

The parameters m,n and m,j are the reduced effective masses along the growth and in-plane
directions, respectively and are defined as follows:

1 1 1 1 1 1
1 +- = -+-. (2.31)
mrjl mcjl mvjl mrj mcjill m yjll
The average transition matrix element IRT12 can be shown to be

M 2 = 2 for both C-HH and C-LH spontaneous emission. (2.32)
IM 3

Derivations of Eqs. (2.27) and (2.32), as well as Eq. (2.29) which shows that 0; is a function
of transition energy, are given in Appendix B. The reduced density of states p,, (spin not
included) is given by [Cor93]

I imrji 1 for E, E
Pred = 20th dzE (2.33)
0 for Ec
The Fermi-Dirac functions, andfh are given by

fC(Eckl)= 1 + exp[ (Ecjk Efc)/kT] (234)


fh (E ) = (2.35)
(Ej 1 + exp [ (Ejk -E)/ kT] (2.35)

where Efc and Ef are the quasi-Fermi levels in the CB and VB respectively and are mea-
sured from the CB and VB edges into the bands, kg is the Boltzmann constant and T is the
temperature of the structure. For parabolic conduction and valence subbands, we can write
the electron concentration N and hole concentration P in the quantum well as

k-T
N = h2d mjIm ln { 1 + exp [-(Ecj -Efc)/kBT] } (2.36)









P= k vll In {1 + exp [-(EVj-Efv)/kT] } (2.37)
Icthd, i

The summation is over all subbands which includes HH and LH subbands forP in the VB.
Under the condition of overall charge neutrality in the quantum well, we have N = P. For
a given value of N, we can solve for the values of Ef and Ef which satisfy Eqs. (2.36) and
(2.37) at a given T.


2.6 Simulation Results


The general procedure for calculating the single-particle g (E,) and R, (E,) func-
tions is summarized as follows: (1) Specify N and T; (2) Input numerical values of material
parameters discussed in Section 2.4.3; (3) Input the quantized energy levels solved in Sec-
tion 2.4.3; (4) Solve EIf and E, from a plot of Ef and Ef as a function ofNusing Eqs. (2.36)
and (2.37); (5) Calculate g (E,) and R, (E,) using Eqs. (2.23) through (2.26).


2.6.1 InGaAs OW


Fig. 2.8 shows plots of Ef and Ef vs. N for the 8 nm Ino.sGaO.sAs strained QW with

GaAs barriers. Because the QW is under biaxial compressive strain, the first VB subband
to be filled up with holes is a HH subband. From Eq. (2.27), it indicates that the TE tran-
sition matrix element is stronger than its TM counterpart for a C-HH transition. As a result,
TE gain dominates over TM gain for the InGaAs strained QW lasers. Typical TE gain
spectra for N = 1.3 x 1018 and 1.7 x 1018 cm3 at T = 300 K, calculated from Eqs. (2.23) and
(2.25), are shown in Fig. 2.9(a). It shows that gain increases with increasing N. The spec-
tral position of the peak gain occurs at Eg, h, which corresponds to the energy gap between
Cl and HH1 subbands defined in Eq. (2.20), and remains there as N increases. Since no
transition is allowed for E, smaller than E,;, ,, the gain for photons with energies below









ES,. is zero. Fig. 2.9(b) shows the spontaneous emission spectra for the correspondingN,
calculated from Eqs. (2.24) and (2.26). Two unrealistic discontinuous peaks occurring at

E, and Ej in the spontaneous emission spectra are due to the step-like density of states
in a QW structure [Cor93]. For the gain spectra shown in Fig. 2.9(a), only the discontinuity

at E, m is obvious. The one at Egi, should also show up, but it is in the absorption region

which is beyond the scale of the plot.


2.6.2 CdZnSe OW


Fig. 2.10 shows plots of Ef, and Ef vs. N for the 4 nm Cdo.3ZnO7Se strained QW with

ZnS.06Se.94 barriers. Typical TE gain spectra for N = 4.6 x 1018 and 5.4 x 1018 cm3 at T
= 300 K, calculated from Eqs. (2.23) and (2.25), are shown in Fig. 2.11(a). Similar to the

gain spectra in Fig. 2.9(a), only the discontinuity at Eg, is obvious. In Chapter 5, when
we discuss the absorption spectra in the CdZnSe QW lasers, we will show clearly that both

discontinuities exist within single-particle model. Fig. 2.11(b) shows the spontaneous
emission spectra for the corresponding N, calculated from Eqs. (2.24) and (2.26). They also
show unrealistic spectral features of discontinuity due to the step-like density of states for
C-HH1 and C-LH1 transitions.


2.6.3 Comparison


From Eq. (2.25), we see that when

fc(Ecjk,) + fh (Evjk) 1 > 0, (2.38)

gj(E,) is positive and an incoming light wave with photon energy E, will be amplified by
the material. Using Eqs. (2.34) and (2.35) in Eq. (2.38), one can show that this inequality
is equivalent to saying that



























N (x 1018 cm3)
(a)


4 6
N (x 1018 cm-3)
(b)


Figure 2.8 Quasi-Fermi level as a function of carrier density for (a) CB electrons and
(b) VB holes of the 8 nm Ino 5Gao.gAs/GaAs QW.



























ho (eV)
(a)


2.0 10l


1.5 102 si


. 1.0 1028


S5.0 1027 '


0 E .1 LLL.* .L
1.2 1.3 1.4
ho (eV)


1.5 1.6


Figure 2.9 Single-particle (a) TE gain and (b) spontaneous emission spectra for the 8
nm Ino.sGao.ssAs/GaAs QW at various indicated carrier densities (in units
of 1018 cm3) at room temperature.











0.15 .
T= 300 K
4 nm CdZnSe QW

0.10 -




0.05




0 2 4 6 8 10
N (x 1018 cm-3)
(a)


0.08 .. I
ST=300K
4 nm CdZnSe QW
0.06


0.04


0.02


0
0 2 4 6 8 10
N (x 1018 cm3)
(b)


Figure 2.10 Quasi-Fermi level as a function of carrier density for (a) CB electrons and
(b) VB holes of the 4 nm Cdo3Zno7Se/ZnSo6Seo.94 QW.




























2.45 2.55
hco (eV)


2.45 2.55
hoa (eV)


2.65 2.75


Figure 2.10 Single-particle (a) TE gain and (b) spontaneous emission spectra for the 4
nm Cda3Zn.7Se/ZnSo6Seo.94 QW at various indicated carrier densities (in
units of 10"1 cm3) at room temperature.


T= 300 K
4 nm CdZnSe QW
single-particle model





4.6

i-------------------
.. I ..". l l .


0


-1000
2.:


35


8.0 1028


6.0 128



, 4.0 1028



t 2.0 10o



0
2.3


T= 300K
4 nm CdZnSe QW
single-particle model



\- I 5.4
4.6




;


5


................................. lisle1


I I









Ea< < Ec E < E+E + E AEf (2.39)

is the requirement for gain at a photon energy E,. In other words, the quasi-Fermi level
separation AEf must be greater than the QW subband energy gap E, (or AEf Ed > 0) to
achieve optical gain in the material. Eq. (2.39) is similar to the one for bulk semiconductors

given in [Cas78, Cor93] where Ey is replaced by bulk E,. For a QW under biaxial com-
pressive strain, Eg indicates the smallest C1-HH1 subband energy gap Es, u. Plots of (AEf

- E8, ,) vs. N are shown in Fig. 2.12 for both the InGaAs QW and CdZnSe QW structures
studied. The cross-over of the (AE E,1. M) vs. N curve on the N-axis is defined as the trans-
parency carrier density N, which is the electron-hole plasma density required to provide the
quasi-Fermi level separation so that the material will become transparent for photon energy

equal to Ei according to Eq. (2.39). Optical gain in the material is attained when we inject
a carrier density N > N, such that AE > E,;, h,. From the curves, we find that N, = 0.9 x
1018 cm-3 for the strained InGaAs QW which is smaller than NMr = 3.1 x 108 cm-3 for the

strained CdZnSe QW. This is mainly due to that the density of states for the CdZnSe QW
is larger than that for the InGaAs QW. From Eq. (2.33), the density of states is found to be
proportional to the carrier effective masses of the material system and, as discussed in Sec-

tion 2.4.3, the effective masses of the CdZnSe QW are much larger than those of the In-
GaAs QW. Another observation is that the slope of the curve for the InGaAs QW is larger

than that for the CdZnSe QW. This means that carrier fills up the subbands in the InGaAs

QW faster than in the CdZnSe QW. Effect of density of states on the transparency condi-

tion is discussed in great detail in [Cor93]. In Chapter 4, we will see how the difference in
N, contributes to the difference in transparency current densities for the two material sys-
tems.


2.7 Summary


Optical gain and spontaneous emission spectra are calculated for both InGaAs and







































0 2 4 6
N (x 1018 cm-3)


8 10


Figure 2.12 Difference between the quasi-Fermi level separation and C1-HH1 QW sub-
band energy gap as a function of carrier density for both the 8 nm
Ino15Gao.7As/GaAs and the 4 nm Cdo.3Zno.7Se/ZnSo.0Seo94 QW at room
temperature.






38


CdZnSe QW laser structures using the single-particle model. Since the measured spectra

do not have sharp features, it is clear that this single-particle model is incomplete. In Chap-

ter 3, we show that when many-body effects are incorporated into the single-particle model,

realistic gain and spontaneous spectra are obtained. However, it is interesting to note that

in the limit of very thick QWs, the single-particle model gives realistic-looking spectra.

This is due to the fact that the stair-step density of states function gradually smooths out as

QW thickness increases [Cor93]. As a consequence, the single-particle model applied to

thick active layer lasers generates spectra which look realistic and it is not obvious that the

model is incomplete.














CHAPTER 3
MANY-BODY EFFECTS


3.1 Introduction


In Chapter 2, we have discussed the optical gain and spontaneous emission functions

based on a single-particle model where mutual interactions among carriers are ignored.

However, upon current injection into the semiconductor QW diode lasers, the carrier den-

sities in the QW are typically on the order of 1017 10"l cm". In the presence of such high

density of electron-hole plasma, mutual interactions among the carriers are expected to be

important These mutual interactions are usually referred as "many-body effects" and in-

clude carrier scattering (CS), bandgap renormalization (BGR) and Coulomb enhancement

(CE). In this chapter, effects of CS, BGR and CE on the single-particle g (E,) andR,. (Ec)

functions will be discussed sequentially.


3.2 Carrier Scattering (CS)


3.2.1 Gain and Spontaneous Emission Functions


In high quality QW laser material, CS is the main contributor to spectral broadening

and is taken into account by convolving the single-particle gj (E,) and R (E,) functions

for a given subband transition pair with a spectral-broadening lineshape function. The con-

volved gain function can be expressed as [Asa93]









g((ho) = Real{I g (Ec,)
(E ECV -ho i (3.1)
x L (ho Ec) 1l- Ecy

and the convolved spontaneous emission is given by

RP (ho) = Real { -j Rpj (E)

xLj(hE Ec)[l- ( Ecy)j

where the notation Real refers to the real part of a complex quantity. The subscript B indi-
cates the spectral broadening due to carrier scattering. Since gj (E,), R,,j (E,) and, as
shown later, L,(hto E_) are all real quantities, Eqs. (3.1) and (3.2) are often rewritten as

gB (ho) = CT gj(E) L (ho E,) dEc, (3.3)
j 8

RB ( 0o)) = YE RsPj (Ec) Lj (h Ecv) dEc, (3.4)

by dropping the square brackets which are the only complex terms in these equations and
thus dropping the notations Real. Eqs. (3.3) and (3.4) mean that, for a given subband tran-
sition pair, the transitions with energy Ec, h(o contribute to optical gain and spontaneous
emission at ho) with the weight Lj (hto Ec,) and the total optical gain and spontaneous emis-
sion are the summation over the contributions from all the allowed subband transition pairs.
The width of the spectral-broadening lineshape function is closely related to the time
constants associated with various carrier scattering processes [Asa93, Yam87]. The param-
eters Tcj and tj denote the electron scattering time and hole scattering time in the jth sub-
bands of CB and VB respectively. Various CS processes contributing to Tcj are electron-
electron (e-e), electron-hole (e-h) and electron-LO phonon (e-LO) and those contributing
to ,j are hole-hole (h-h), hole-electron (h-e) and hole-LO phonon (h-LO). Details concern-
ing the calculation of the these scattering times can be found in [Asa93].








The non-Lorentzian form used for the lineshape function is given by [Asa93, Hsu97a,
Yam87]

1 F cv (h o, Ec) (3.5)
L7 (hA Ec) E 2 + r2 (hA, Ec()'

where
Tcv (hA, Ec) = Frj, (ho) Ec + Ecjk) + Fjk (ho- Ec, + Eik) (3.6)

This lineshape function differs from a Lorentzian function in that F, depends on photon
energy ho. Fjk, for thejth VB subband can be approximated as [Asa93]

h 1+e)E-E k-E /Kre-K[E-EjkffAT"
jk (E) 2, i 1 + e-E/T (37)

where

K e4t T (0, kj kj,) m ,,,kT
487e2E2hkTd2 1 +e-E',/kT h2 ) '

with
2 1 2
T(0, k'4,kj) = { + +4k2j d ,2 1--[l-exp(-d.,d,) ]


X= (2+4k-[. )2
Sdvkj (djkv ) (X-4k ) i 2
4( +24k [)2[exp(d j ,) -1]

The parameter \ is the inverse screening length and is given by [Asa93]
S e2 mc fc (Ec) mjll fV (E,.)
) 2 e = e2 Y + d (3.10)
S Osh2oC j dC) d J

where e, is the static dielectric constant of the quantum well material, d, is the effective
well width for thejth subband in the CB and VB (n = c and v, respectively). The summa-
tion over means that the screening effect is considered with inclusion of all the subbands
in CB and VB. dn is defined by [Hsu97a]









d = j h .x, (3.11)
d ____ = iJknjil'

where the parameter knj" is the equivalent growth-direction component of the wavevector
associated with the quantized energy level E,. For the jth CB subband, Fic can be cal-
culated by replacing all the subscripts v with c in Eqs. (3.7) through (3.9). Details concern-
ing the derivation of Eq. (3.7) and the associated parameters are given in [Asa93].


3.2.2 Simulation Results


InGaAs OW. The convolved TE gain and spontaneous emission spectra with inclu-
sion of spectral broadening due to carrier scattering are plotted as the solid curves in Figs.
3.1(a) and (b), respectively, atN = 1.7 x 1018 cmt and T = 300 K. It is evident that these
spectra are broadened and smoothed relative to their single-particle counterparts which are
replotted as the dotted curves in Figs. 3.1(a) and (b) for easy comparison. The peak gain is
reduced significantly and its spectral location is shifted away from the C1-HH1 subband
energy gap E,.1, to higher photon energy (in this case by about 11 meV). As discussed
later in Chapter 4, this shift is important in determining the dependence of lasing wave-
length on cavity length.
CdZnSe QW. The solid curves in Figs. 3.2(a) and (b) are the convolved TE gain and
spontaneous emission spectra, respectively, of their single-particle counterparts plotted as
the dotted curves for the 4 nm Cdo.3Zno.7Se/ZnSo.o6Sea94 QW at N = 5.4 x 1018 cm3 and T
=300 K. In this case, peak gain is also reduced and its spectral location is blue-shifted from

E,1, i to higher photon energy by about 11.5 meV.











3000 -


2000


1000


0 -


-1000
1.15


2.0 102


" 1.510(


. 1.0 102


S 5.0 1027


1.35
Aho (eV)


1.45 1.55


1.15 1.25 1.35 1.45 1.55
ho (eV)
(b)

Figure 3.1 The convolved (a) TE gain and (b) spontaneous emission spectra (solid
curves) of their single-particle counterparts (dotted curves) for the 8 nm
Ino.15Gao.85As/GaAs QW atN = 1.7 x 1018 cm-3 and T = 300 K. The convo-
lution takes into account the spectral broadening due to carrier scattering.


T= 300K
S 8 nm InGaAs QW
: N = 1.7 x 1018 cm3
S single-particle model

w/ CS




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









El. hh

4000 ... .. i I .
T= 300K
3000 i 4 nm CdZnSe QW
N = 5.4 x 101 cm-3

2000 single-particle model

l w/CS
1000 -

0 ---- ------------------


-1000 '.... .... i...
2.35 2.45 2.55 2.65 2.75
tCo (eV)
(a)


81028 .*.I.*I
T= 300 K
4 nm CdZnSe QW
S 6 10C N= 5.4 x 1018 cm-3

2 ., single-particle model
s 410 -
S w/CS 1.W\

$ 21028



2.35 2.45 2.55 2.65 2.75
ha (eV)
(b)

Figure 3.2 The convolved (a) TE gain and (b) spontaneous emission spectra (solid
curves) of their single-particle counterparts (dotted curves) for the 4 nm
Cdo.3Zno.7Se/ZnSo.oSeo.94 QW at N = 5.4 x 1018 cm- and T = 300 K. The
convolution takes into account the spectral broadening due to carrier scat-
tering.








3.3 Bandgap Renormalization (BGR)


3.3.1 Plasma Screenine


One of the most important many-body interactions in an electron-hole plasma (EHP)

is the screening of the Coulomb potential. In a two-dimensional (2D) quantum well sys-

tem, the Fourier transformation of the unscreened Coulomb potential in the real space to

the momentum space (or wavevector k space) is given by [Cho94, Hau90]
e2
Vq= 2e (3.12)


where A is the in-plane area of the quantum well. The parameter q, is the magnitude of a
qll vector which is defined as the difference between in-plane wavevector ki and k l, that

is qll = k kl. Given the angle 0 between klI and kl,, qll can be given from the law of

cosine

q2 = k2 +k 2k- k,,cos0. (3.13)

Defining the exciton Bohr radius a, and exciton Rydberg constant ER as (in M.K.S. units)

[Cho94, Hau90]

4nh2oEs
ao = (3.14)
e2mrj,


ER 2 (3.15)
2m,jllpa2'

we obtain


ao = 8e (3.16)

where aoER depends only on the static dielectric constant E, of the material. Unlike in

[Cho94, Hau90], we use the in-plane reduced effective mass discussed in Chapter 2 to take

the effective-mass anisotropy into account in Eqs. (3.14) and (3.15). Using (3.16) in Eq.








(3.12), the unscreened Coulomb potential can be expressed in terms of ao and ER as

4naoER
V, 4 a (3.17)
Vq, = qllA

The screened Coulomb potential is given by [Cho94, Hau90]

Vq = (3.18)
sQ Ell (ql, to)

where ell (q11, 0) is the longitudinal dielectric function and is frequency dependent (dy-
namic). At high temperature where screening is governed by free carriers (not excitons),
screening effect can be described within the so-called random phase approximation (RPA)
[Cho94, Hau90]. Replacing a continuum of poles in RPA by a single effective plasmon-
pole and neglecting the frequency dependence of screening effect, one obtains the much
simpler static plasmon-pole approximation (SPPA) [Cho94, Hau90]. In this case, the lon-
gitudinal dielectric function can be expressed as

1 = 1 to- (3.19)
ell (q,0) 02 '

where oL is the plasma frequency and q, is the effective plasmon frequency. For a gen-
eral multi-subband QW system, cop can be expressed as [Cam94]


PI = AVq,, mn+ mPj
f (= qlJ (3.20)

S4.namERqI m+il

where the summation is overall all the conduction and valence subbands. In the last form

of P21, the expression for Vq, in Eq. (3.17) is used. The effective plasmon frequency Oq1
is related to cop by [Cho94, Hau90]

02 = 1 +~j 0 +2 i( ) (3.21)

where the inverse screening length X, is given in Eq. (3.10) with inclusion of all the sub-








bands. Dividing both sides of Eq. (3.21) by w)2 and taking the inverse, we obtain

o= 1 (3.22)
I = q 11 1 2ft
S 1+- ----
(p 2mrjII
and therefore,
qi + _I ( Ihq ] q+II 1 q2
!t- I + 1 2muII + tjL I (3.23)


Before further expanding Eq. (3.23), we introduce several new parameters which, in anal-
ogy to a. and ER defined Eqs. (3.14) and (3.15), are defined as

4h2ftes, 4xh2ele,
aa-oc, aov --- (3.24)
e '- m, e2mv

f2 f2
E h E = (3.25)
Rc 2mcjua2c 2mVla2

These parameters at, a,, ERe and ER, don't have any physical meaning except for the con-
venience of mathematical manipulation. Similar to Eq. (3.16), we obtain

aocERc = aovER = =aER (3.26)
8neoeb

Using Eq. (3.26) in Eq. (3.24) to solve for the effective masses, we have
1 2 12
mcji (aoER) a m i 2 aoER) aov. (3.27)

For the terms containing co2 in both square brackets of Eq. (3.23), we use Eq. (3.20) for
op2 and Eq. (3.27) for mq, and mv, to obtain

1 ( hql 2 1 ( Aq2 2
"2 2mrjI) 41taoER L P 2mrj )
1 mcjI mvj) (3.28)
q3a
87tX (n iao + Pjao.)
J








We have generalized the expression of Eq. (3.28) for a multi-subband QW system. But for
a single-subband QW system in both the CB and VB, which is an assumption always made
in [Cho94, Hau90], we have ni = pj = n2D and therefore the summation over can be sim-
plified as

(njac + Pja,) = n2D (aoc + av)
S, (3.29)
= n2Da,,

where it can be shown that

4th2Eos 1 1i
a0c +a0v -

4xh2EOse 1 (3.30)
e2 mr'jl
=ao

Finally with the use of Eq. (3.29), Eq. (3.28) reduces to

1 (hq 2 q3a
l 2rJ 8 2D (3.31)

For a multi-subband QW system, which is the real case of the two QW laser systems dis-
cussed in Chapter 2, it can be shown that Eq. (3.31) still gives a close numerical result to
that Eq. (3.28) gives. To simplify the calculation, Eq. (3.31) will be used hereinafter for
the multi-subband system as well. Plugging Eq. (3.31) into Eq. (3.23), we have

[q_ + aoq,,3^ F +q, aq,,3 32)
1 L-+ 8 1"2zJ 8 + n2oJ (3.32)
W q2 X, 2D I 23D]

Finally, using Eq. (3.32) in Eq. (3.19) for e, and then plugging eu into Eq. (3.18), the
screened Coulomb potential can be expressed as
4nao, q,, ao!q131 [ q1l a0q,1311
Vsq qA 8n-2L 8 n2 (3.33)
Ss n s s s f bnd n
With this expression for Vq, we can discuss the effects of bandgap renormalization due to








plasma screening next.


3.3.2 Bandgap Renormalization


As carrier density increases, more unoccupied states become available to the VB
electrons. As a consequence, the VB electron distribution can change in such a way that
the Coulomb repulsion with the CB electrons is more effectively screened. This slight re-
arrangement of the charge carriers results in decreases in the single-particle energies and,
as a consequence, a reduction in bandgap. In this case, the total bandgap renormalization
(BGR) AEBGR is the sum of Coulomb-hole (CH) self-energy AECH and screened-exchange
(SX) self-energy AEx. The first contribution AEcH describes the energy reduction of elec-
trons (holes) with different spin by avoiding each other because of the mutual Coulomb re-
pulsion and is the difference between screened and unscreened Coulomb potential [Cho94,
Hau90],

AEcH = (Vsq, -Vq,) (3.34)

Using Eqs. (3.17) and (3.33), we obtain

4xtaoE,, qr anoql3
V -v r -- 1 [ 0 + (3.35)
sq* q= q11A Xs 8,n2Di

Eq. (3.34) can be then rewritten as

4xaoER [I+11 + 013 a,3 -1
AECH = -_ 4 R + q + a 3 (3.36)
qCH LA 87n2D

To evaluate the summation over q11, we first must be able to count the available q1l states.
For a QW, within the plane of QW, a 2D vector q1l sweeps out a circle of radius q, in q,
space, as shown in Fig. 3.3. One q1 state occupies a square of area in q1 space equal to
4x2/A, which is just like that a kgl state takes up a square area of 4g2/A in k1 space
[Cor93]. Therefore, number of ql states within the area defined by increments in q1 and






























-----------q dq
q.
















Figure 3.3 Distribution of qul vector states in q, space for a QW. One qll state takes up
a square area of 4:x/A indicated by a mesh of dashed lines. The shaded area
of qfddq1, represents the infinitesimal area defined by an increment of dqu
in the magnitude of q11 and an increment of do in its sweep angle 4. The
number N, of ql1 states within this shaded area is just the number of the
squares within it. Every qll state can be taken into account by an integration
of N, over the whole q, space.








sweep angle 0 (shown as the shaded area in Fig. 3.3) can be written as

q,ddq (3.37)
4n2
A
where the spin degeneracy is not included. The summation over ql can be replaced by an
integration of N, over the whole q,1 space

F dq" (3.38)
A
Since the function to be summed in Eq. (3.36) is independent of 0, its integration over 4 is
equal to 2t and Eq. (3.36) becomes

AEc = -2aoEF 4I + dq11 (3.39)

To perform the integration in Eq. (3.39), we evaluate the integrand at two extreme cases:


q1 afor q,, 11
+1 !- + (3.40)
S8n2D 0 forq1 I r.a
fSao

Using the evaluation in Eq. (3.40), Eq. (3.39) can be approximated as


Af 1.,,
AEC -2aoE, 1J dq,
S(3.41)

=-2aoE sln[1+ 82D

The second contribution AEsx results from the change in electron (hole) self-energies due
to the fact that electrons (holes) with equal spin avoid each other (Pauli exclusion principle)
and is given by [Cho94, Hau90]

AEsx= Vsq (fcq, + fq,) (3.42)
411








We substitute Eq. (3.33) for the screened Coulomb potential VqM and Eq. (3.38) for the
summation over q1l in Eq. (3.42) to obtain

^ E [q1,, aoq3 1 ][, aoq11]-
AEs= -2aoE + 8xn,2Di + X 81n2 (fcq + fhq,)
(3.43)
2a ERf, s aoqll21 + q1I+ aq,113 -1(.
s qldq 1 + 8ntn2DJ I I 8nn2D (fcq+ fhq,)
where a factor of ql/X, is pulled out of the first bracket in last form of AEgs. Often, the
weak-momentum dependence of AEs is neglected and at the subband edge qll = kl' with
kl1 = 0 in Eq. (3.13), then we have

2aoE ,. + saok2"[ k', aok3 "]-I
AEsx = kR dk 1 + 1 1 + + -
Asx = o 8nn2D [X, 8n2DJ (3.44)
x [fc (Eck) + fh(Ejk,)]


3.3.3 Simulation Results


InGaAs OW. Plots of AEcH, AEE and AEBGR as a function of volume carrier density
N for C1-HH1 and C1-LH1 transitions are shown as the dash-dotted, dashed and solid lines
in Fig. 3.4, respectively. Often, if the weak momentum dependence of BGR is neglected,
BGR can be taken into account by rigidly shifting the gain and spontaneous emission by an
amount of AEBGR, where the negative sign on the values of AEBGR means a red shift of spec-
tra to lower photon energy. The corresponding BGR-shifted TE gain and spontaneous
emission spectra of Figs. 3.1(a) and (b) are shown in Figs. 3.5(a) and (b). As discussed later
in Chapter 4, the red shift in gain spectra gives an opposite direction in predicting the de-
pendence of lasing wavelength on cavity length.
CdZnSe OW. Plots of AEcy, AEs and AEBGR as a function of N for C1-HH1 and C1-
LHI transitions are shown as the dash-dotted, dashed and solid lines in Fig. 3.6, respective-
ly. The corresponding BGR-shifted spectra of Figs. 3.2(a) and (b) are shown in Fig. 3.7(a)








and (b) where the resulting peak gain location is well below E.1 A by about 55 meV. This
situation, which will be discussed later in Chapter 5, was often mistakenly elucidated as the
evidence for existence of excitonic gain in CdZnSe QW lasers even at room temperature
[Ding94].
Comparison. By comparing Fig. 3.4 with Fig. 3.6, we find that AEGR vs. N curves
are about the same for the InGaAs QW and CdZnSe QW, although their AECH and AE,
components are different. This could be an coincidence, however, it agrees with an univer-
sal formula commonly used for AEsGR that goes with one-third power of N [Chi88, Col95].


3.4 Coulomb Enhancement (CE)


3.4.1 Coulomb Enhancement Factor


The Coulomb attraction between electrons in the CB and holes in the VB tends to
keep electrons and holes within the vicinity of each other. This increases the radiative re-
combination rate and results in an enhancement in optical gain as well as spontaneous emis-
sion rate. To investigate the effects of Coulomb enhancement on optical gain g (hoA) and
spontaneous emission R,, (hao), we start with the complex optical susceptibility X (hw),
since both g (hmo) and R, (hen) are related to X (ho). In a multi-subband system, the total
optical susceptibility (7ho) is the summation over the contributions j (hco) from the al-
lowed subband transition pairs and is expressed as

(2 (h) = 2~ (ho) (3.45)

where the subscripts indicates thejth subband transition pair. With inclusion of Coulomb
enhancement, j (hoA) is given by (in C.G.S. units) [Cho94, Hau90]

4 d (3.46)
j (hC) = 1 (3.46)
E V k I qk'




























-0.020


-0.040


-0.060


-0.080


-0.100


0 2 4 6
N (x 10's cm3)


8 10


Figure 3.4 Bandgap renormalization as a function of carrier density N for C1-HH1 and
C1-LH1 subband transitions in the 8 nm Ino.5Gao.ssAs/GaAs QW at T = 300
K. The solid curves are the total bandgap renormalization (BGR), the dash-
dotted curves are the Coulomb-hole components (CH) and the dashed
curves are the screen-exchange (SX) components.


""AECH, C -HHI
: '~:" z .- AECH, ClI-A

AEsx, cS.LI
AE SX.Cl-HHI


AEBGR.Cl-LHI
AEBGR, CI-HHI
T=300K
8 nm InGaAs QW
! I I [ l ] t I i .




























1.25 1.35
Aco (eV)


1.45 1.55


1.1:


5


1.35
ho (eV)


1.45 1.55


Figure 3.5 The rigidly shifted (a) TE gain and (b) spontaneous emission spectra (solid
curves) for the 8 nm Ino.lsGao.ssAs/GaAs QW at N = 1.7 x 10'" cm'3 and T
= 300 K by the bandgap renormalization due to plasma screening. The
dashed curves are the spectra without BGR shifts.


T=300 K
8 nm InGaAs QW


N = 1.7 x 10" cm-

w/CS+BGR w/CS




----% -' -- -------------- 7
** -- -' -- '- ** *' *' *


I


. I


15


0



-1000
1.1


2.0 102



1.5 1028



1.0 1028



5.0 10"2


T=300K
8 nm InGaAs QW
N = 1.7 x 10"8 cm-


w/CS+BGR 4 w/CS






1 "
































-0.020


-0.040


-0.060


-0.080


-U. IUU


0 2 4 6
N (x 1018 cm3)


8 10


Figure 3.6 Bandgap renormalization as a function of carrier density N for Cl-HHI and
C1-LH1 subband transitions in the 4 nm Cdo.3Zno.7Se/ZnSao6Seo.94 QW at T
= 300 K. The solid curves are the total bandgap renormalization (BGR), the
dash-dotted curves are the Coulomb-hole components (CH) and the dashed
curves are the screen-exchange (SX) components.


IAEmrI c lIII
AEsx.I Hn


--
AEcH, CI-HHI
-AECH, Cl-Wll

AEBGR, CI-LHI

T=300K EBGR, Cl-HHI
4 nm CdZnSe QW
= l ,











4000 i I| i 1i | T .i' 1 .... I ..
T=300 K
3000 4 nm CdZnSe QW
N = 5.4 x 101s cm"3

2000
w/CS+BGR w/CS

1000 ...


0 ----- '.'---------I-----


2.25 2.35 2.45 2.55 2.65
(o (eV)
(a)


8102s ....
T= 300K
4 nm CdZnSe QW
610 N= 5.4 x 10'" cm3


4 102s w/CS+BGR w/CS


S2102 -

0 ... ....

2.25 2.35 2.45 2.55 2.65
ho (eV)
(b)

Figure 3.7 The rigidly shifted (a) TE gain and (b) spontaneous emission spectra (solid
curves) for the 4 nm Cdo.3Zno.Se/ZnS.o6Seo.4 QW at N = 5.4 x 1018 cm3
and T = 300 K by the bandgap renormalization due to plasma screening.
The dashed curves are the spectra without BGR shifts.








with

o (1- fk-fhk)
S h ( + iS e eh) (347)
(3.47)
(fek fhk-1)
= -id + i (e + ehk )

and

q = d~ ~-Tk'k., (3.48)

where V is the volume of QW, e. = i2 is the background dielectric constant of QW, d,
is the square root of the dipole moment matrix element at transition involving in-plane
wavevector kli, 1/(1 qk) is the Coulomb enhancement factor in Pad6 approximation,

Vsk,k' is the screened Coulomb potential,fLk and fh are the Fermi-Dirac distributions of
electrons and holes with energies ek and ek in the CB and VB respectively and h8 is the
dephasing or broadening factor due to carrier scattering. Originally, Eqs. (3.45) to (3.47)
are derived under the assumption of a single-subband system and solved from the Bethe-
Salpeter equation for the only one existing subband transition pair [Cho94, Hau90]. Since
in a multi-subband system a set of Bethe-Salpeter equations for various allowed subband
transition pairs can be solved independently for each allowed subband transition pair as
long as the screening effect is considered with inclusion of all the subbands (as we have
done in Section 3.3.1) [Cam94], we are allowed to generalize Eqs. (3.46) to (3.48) to be
used in Eq. (3.37) of a multi-subband system.

Now we want to expand Eqs. (3.47) and (3.48) in terms of the parameters that we are
familiar with. We begin with the dipole moment matrix element. The relationship between
the transition matrix element and the dipole moment matrix element in M.K.S. units is giv-
en by [Col95]

e21M 2 = 2m2c(2e2j12, (3.49)

where e21x12 is the dipole moment matrix element and x is the position operator. The factor







of 2 in the front counts for the spin degeneracy buried in IM| 2 and is not included in the
reduced density of states pd given in Eq. (2.33). On the contrary, in [Col95] the spin de-
generacy is taken into account in p,,d instead of MT 2. From Eq. (3.49), we can relate the
dipole moment matrix element d2 in C.G.S. units to IMT2 in M.K.S. units as follows

d e21x12 e IMd2
dc 4' (3.50)
cV 4Eo 87E om2W2'

where the factor of 1/(4xeo) is the converting constant between C.G.S. and M.K.S. units.
By the following substitutions in Eqs. (3.47) and (3.48)
k -~ kll, k' -- k;
elM elM-|
de C 7C d1.8,c-MO
fek fc (Ecjk), fhk fh(Ecjk,) (3.51)
V, k k' -+ Vsq
h8 r,,(hw, EcY)
eek +eh -k Ec
we obtain

S eM lM c [c(Ej) + fh (Evjk)-]
8k f mo oT cv (A, Ecy) + i (Ec, tto)

and

'omom 4tao ER X, isaoqr411 lr q a,3i -1
q-j(ho, Ec) = 1e+ Sl l +
e M k; A 8n2DL 8Tnr 2 J
(3.53)
S elM [Ifc (Ecjk)+f h (Evjk,)- (353)
1 8*Iem^o r7, (ho, EV) + i (E;Y ho)
where Eq. (3.33) is used for Vsq The superscript' indicates the parameters involving tran-
sition energy Ec, and the associated wavevector k,. We also use the hfe dependent broad-
ening factor F, defined in Eq. (3.6) in the CE expression above instead of the ho)








independent broadening factor hS usually used in previous literatures discussing many-
body effects [Cho94, Hau90]. Similar to Eq. (3.38), we replace the summation over k in
Eq. (3.53) with

-+ 2 f k2 ddk (3.54)

to obtain

aoE,
qlj (W 9, E ,) = -i a
(3.55)
[fc (Ecjki) +h (Evjk) 1]
0d MTI F(hco, E) + (Ec, -hCo)0 1 )

where

2' sao I laq2 I q+11 ao113 1dO
e(k11,k1 ) =0 1+ D + XdO. (3.56)

Since q1, is 0 dependent as defined in Eq. (3.13), the integrand can not be pulled out of in-
tegration over 0. So far, we have successfully derived the Coulomb enhancement factor to
be in a form such that we can use. Next, we will show how it can corporate into the optical
gain and spontaneous emission functions.


3.4.2 Gain and Spontaneous Emission Functions


The optical gain g (ho) is related to the complex optical susceptibility X (ha) by
[Cho94, Hau90]

g(ho) = Imag[ (hc)]
C
(3.57)
= -nReal[i (hr )]

where the notations Imag and Real refer to the imaginary and real parts of a complex pa-
rameter, respectively. Plugging Eq. (3.47) into Eq. (3.46) for j (hW), we can rewrite Eq.








(4.45) as
41 dv (fek4h 1
(h( ) = -i[(f + f 1) (3.58)
-C P Lk -qlkh8+i(eek+eh -hO ".

Applying Eq. (3.58) to Eq. (3.57) and substituting e_ with h2, we have
F dv 47o (kf +fhk-1)
g(h(O) = Real -q c +i(k+ehk--) (3.59)
j k r 1k C ll*+(ek+ hk-ho)
The summation over k can be replaced by

+ -- 2Pred dEcv (3.60)

where the lower limit of the integral is chosen to be Egj because p,d is zero for Ec, < Eg for
a given transition pair. Using the substitutions in Eqs. (3.51) and (3.60), Eq. (3.59) be-
comes

g(w) = Real J Ee2i [fc (Ecjkr,) + fh(Ev, ) 1
1 0qj (ho, E) (3.61)
1 1
x r, (hco, Ec) + i (Ec- ho)Ec

It can be shown that

1 1 =L(h (E -h(o) (3.62)
xrF (hO, Ec) + i (Ec h) c (h, E) (3.62)

where L (hco Ec) is defined in Eq. (3.5). Using Eqs. (2.25) and (3.62) in Eq. (3.61), fi-
nally the optical gain with Coulomb enhancement can be expressed as

gE (o) = Realo g(E-,)
E JE l-qlj(hto, E,)
S (E h_-o' ) }', (3.63)
xL.(1-Ec) [1-i(cv idE
S F ic (h o, Ey,) I cv

where the E subscript indicates the Coulomb enhanced gain. If qy is zero, the optical gain
function with CE, gE (how), in Eq. (3.62) will reduce to its counterpart without CE, gB (hwo),
in Eq. (3.1).








Recalling Eq. (3.58), j (hA) contains the factor associated with the Fermi-Dirac dis-
tributions which can be subdivided into two components as

(fe + fhk- 1) = [fekfhk] [ (1 fek) (1 ) (3.64)
The first square bracket is referred as the emission component, since it is the joint proba-
bility of finding an occupied state in the CB and an unoccupied state in the VB. The second
square bracket is referred as the absorption component, since it is the joint probability of
finding an occupied state in the VB and an unoccupied state in the CB. Using this subdi-
vision in Eq. (3.58), (ho)) can be subdivided into two components as follows

m(i, O) =i 4x dl, (fekfhk)
S V k I-qtlkh +i(ek+e hk-(o)
(3.65)
4o dCV ( fek) (1 fh)
a(V k, [1k q +i(e +ehk-h o)j
where ( (h() = j, (h(o) Xab (hfi). The subscripts em and ab indicate emission and ab-
sorption, respectively. The spontaneous emission function is related to the imaginary of
j,. (Ato), which can be seen as an analogy to Eq. (3.57). Similar to the derivation of Eq.
(3.63), the Coulomb enhanced spontaneous emission function REp (ha) can be expressed as

R (A) = Real Rj (E,)
RspE(h) =Real JE 1l-q1(ho, Ec,)
Lyi -EE)[1 V Ic (3.66)
xL (ho) E,) 1i ( ((c, Ec)I dE .

As we recall, the emission component of Eq. (3.64) appears in the expression for single-
particle Rp (ho) function given in Eq. (2.26). For the spontaneous emission function, the
Coulomb enhancement factor 1/(1 q,) is slightly different from that of the gain function
by replacing JM] in the denominator and IM I in the numerator in Eq. (3.55) with IMT
and AM S As a consequence, MTI and |Mj cancel with each other since they are both
equal to 2/3~IM as shown in Eq. (2.15). Again, the spontaneous function R,,E (ho) with
CE included in Eq. (3.66) reduces to its counterpart Rsp (ho)) without CE included in Eq.








(3.2) if qj = 0.


3.4.3 Simulation Results


InGaAs OW. Fig. 3.8(a) shows plots of gB (ha) and g9 (hc) vs. htw, with inclusion of

the rigid shift due to BGR, forN = 1.7 x 1018 cm-3. In this case, CE increases the peak ma-
terial gain g by a factor of 1.4 and blue shifts its spectral position E by 4 meV. This blue

shift is very important in predicting the slope of cavity length dependence of lasing energy

as would be discussed in Chapter 4. Fig. 3.8(b) shows plots of R,pS (th) and RpE (hw) vs.

Ah, with inclusion of the rigid shift due to BGR, for N = 1.7 x 1018 cm-. The increase in

spontaneous emission due to CE is evident and is also important in predicting the threshold

current of diode lasers (see Chapter 4).

CdZnSe OW. In Fig. 3.9(a), g8 (ho) and gE(ho) vs. ho), which include the rigid shift

due to BGR, are plotted as the dotted and solid curves forN = 5.4 x 10' cm-3, respectively.

It is shown that CE increases g by a factor of 2.35 and blue shifts E by 6 meV. Fig. 3.9(b)

shows plots of Rsp (hA) and Rsp (ht) vs. ho, with inclusion of the rigid shift due to BGR,
forN = 5.4 x 108 cm-3.

Comparison. Comparing the TE gain spectra in Figs. 3.8(a) and 3.9(a), it is clear that

CE is stronger in the wide bandgap II-VI CdZnSe QW than in the III-V InGaAs QW. A

stronger Coulomb attractive effect can be indicated by a larger exciton binding energy AE.

in the semiconductors. For a 2D QW structure, AE, is equal to 4ER where ER is the Ryd-

berg constant defined in Eq. (3.12). By plugging the material parameters listed in Tables
2.1 and 2.2, AE, of the n = 1 HH excitons is found to be about 11 meV in an Ino. iGaO.85As

QW while it is about 46 meV (about a factor of 4) in a Cdo.3Zno.7Se QW.

Comment. In Eq. (3.46), the Coulomb enhancement factor is derived in the Pad6 ap-

proximation which is a good approximation for carrier densities above the so-called Mott
density Nm,o,. At the Mott density, the exciton binding energy vanishes and excitons cease
















"w/CS+BGR+C


T= 300 K
8 nm InGaAs QW
'E N=1.7 x108 cmr3


S w/CS+BGR


1.35
hA (eV)


1.25 1.35
hco (eV)


1.45 1.55


Figure 3.8 The inclusion of Coulomb enhancement on (a) TE gain and (b) spontaneous
emission spectra (solid curves) for the 8 nm Ino.s1Gao.sAs/GaAs QW at N =
1.7 x 10'8 cm- and T = 300 K. The dashed curves are the spectra without
CE.


5


-1000 t
1.1:


2.0 102



^ 1.5 102



. 1.0 1028



5.0 1027


''''''''''''''''' "'~' '''''''''''''''''


L


- -












4000


3000


2000


1000


0


-1000
2.2


81028



6108



4102



2 1028


5


2.35 2.45
o (eV)


2.25 2.35 2.45 2.55 2.65
ko (eV)
(b)

Figure 3.9 The inclusion of Coulomb enhancement on (a) TE gain and (b) spontaneous
emission spectra (solid curves) for the 4 nm Cdo.3Zno.Se/ZnSoa Seo.94 QW
at N = 5.4 x 1018 cm3 and T = 300 K. The dashed curves are the spectra
without CE.


. 1 .. I. .. I I I .. ..
T =300 K
4 nm CdZnSe QW
N = 5.4 x 10ts cm3
w/CS+BGR+CE

w/ CS+BGR



------ ------- --------- --
.. .. ., ..._ I. I. 1. .


n









to exist due to the screening of charged carriers in semiconductors [Hau90, Zim88]. The

value for NM, can be simply found from a plot of AEBGR vs. N at which IAEBGRI is equal to

AE, [Zim88]. For an 8 nm Ino.15Gao.ssAs QW, with AE, = 11 meV, we thus find Nm, <<

5 x 1017 cm3 from Fig. 3.4. For a 4 nm Cdo.3Zno.Se QW, with AE, = 46 meV, NM,, is

found to be about 2 x 1018 cm-3 (or 8 x 1011 cm2 in sheet density) from Fig. 3.6. Because

of this large value for Nmo in II-VI CdZnSe QW, it has always been a suspicion that lasing

in CdZnSe QW lasers is excitonic in nature [Ding92, Ding93, Ding94]. In Chapter 4, we

will show that even for the CdZnSe QW lasers, the carrier densities required to achieve las-

ing thresholds at room temperature are higher than the Mott density.


3.5 Summary


In this chapter, we have discussed many-body effects, such as carrier scattering,

bandgap renormalization and Coulomb enhancement, on the optical gain and spontaneous

emission spectra of the semiconductor quantum wells. We have also demonstrated that CE

is in fact stronger in the CdZnSe QW than in the InGaAs QW structures.

In next chapter, we will use these gain and spontaneous emission functions developed

in Chapters 2 and 3 to make predictions on laser characteristics at threshold and compare

them with experiments using both InGaAs and CdZnSe QW lasers.














CHAPTER 4
LASER THRESHOLD CHARACTERISTICS


4.1 Introduction


In Chapter 2 and 3, models of optical gain and spontaneous emission based on elec-

tron-hole plasma (EHP) were described. In the so-called conventional EHP theory, only

carrier scattering (CS) and bandgap renormalization (BGR) are taken into account

[Asa93,Col95, Cor93] and Coulomb enhancement (CE) is left out mostly due to its involve-

ment with complicated computation. In the case of the wide bandgap lasers, such ignorance

of CE can be a mistake since it has been reported that CE should be more important in blue-

green CdZnSe and InGaN quantum well (QW) lasers [Cho95a, Cho95b, Ree95a, Ree95b,

Ree96] than in infrared InGaAs QW lasers. In fact, the calculations in Chapter 3 have

shown that CE has larger effects on the shapes and magnitudes of the optical gain and spon-

taneous emission spectra for the 4 nm Cdo.3Zno.7Se QW than for the 8 nm Ino.15Gao.ssAs

QW.
In this chapter, we will investigate effects of CE on the simulation of laser threshold

characteristics for both the well-developed narrow-bandgap InGaAs QW lasers and the rel-

atively new wide-bandgap CdZnSe QW lasers. In order to see if the inclusion of CE in the

simulation yields improved agreement with experiment, a number of InGaAs/GaAs QW

and CdZnSe/ZnSSe QW lasers were fabricated, characterized and compared to predictions.

In Section 4.2, the laser devices characterized are described. Comparisons between

experiments and predictions on cavity length dependence of lasing energy and threshold

current are discussed in Section 4.3 and 4.4, respectively. Lasing energy dependence on








temperature is investigated in Section 4.5 and the work is summarized in Section 4.6.


4.2 Laser Devices


4.2.1 InGaAs OW


The epitaxial layers of the InGaAs/AlGaAs GRIN-SCH single QW diode lasers are

described in section 2.2.1. Using standard photolithographic techniques, 100 gm stripes on

500 pim centers were defined on the p+-GaAs cap layer. The p-side of the wafer was then

metalized using Au pulsed electroplating technique. After thinning the n-substrate down

to a wafer thickness of about 100 p.m, the n-side was also Au-electroplated, as shown in

Fig. 4.1. The wafer was cleaved into bars with three cavity lengths (L= 500, 1000 and 1500

pm) and then characterized.


4.2.2 CdZnSe OW


The epitaxial layers of the CdZnSe/ZnSSe SCH single QW diode laser are described

in section 2.2.2. Fig. 4.2 shows the 5 (tm wide buried-ridge index-guided devices fabricat-

ed from this structure where the p-contact metal stack is Pd/Pt/Au annealed at 200"C for 15

minutes in forming gas and the n-contact metal stack is Pd/Ge/Au annealed at 200 "C up to

an hour in forming gas [Hab97]. These state-of-the-art CdZnSe QW laser devices were

provided by the 3M Company. Laser bars with three cavity lengths (L = 740, 1220 and

2010 umn) were characterized.


As discussed in Section 3.4.3, CE has effects on the spectral location and the magni-


4.3 Cavity Length Dependence of Lasing Energy















500 Rm

100 gm
SI- Au --
cap 100 nm p*-GaAs
buffer 25 nm p-AlGaAs


cladding 1300 nm p-A1GaAs


guiding 200 nm A1GaAs

bounding 7 nm GaAs
OW 8 nm InGaAs
bounding 7 nm GaAs

guiding 200 nm A1GaAs


cladding 1400 nm n-AlGaAs


buffer 25 nm n-A1GaAs
substrate n-GaAs

meal Au


Figure 4.1 Cross-section sketch of the InGaAs/AlGaAs GRIN-SCH single QW diode
lasers. Au stripes with widths of 100 gm on 500 gm centers were deposited
on the p+-GaAs cap layer using pulsed electroplating technique. After thin-
ning the n-GaAs substrate down to a wafer thickness of about 100 gm, Au
was deposited everywhere on the n-GaAs substrate.
















5 gm

metLl Pd'Pt/Au
/p-ZnSeTe\

ZnS cladding ZnS
Sp-ZnMgSSe


guiding p-ZnSSe

QW CdZnSe

guiding n-ZnSSe



cladding n-ZnMgSSe


buffer n'-ZnSe
buffer n-GaAs
substrate n-GaAs

melal PdiGe/Au


Figure 4.2 Cross-section sketch of the 5 pm wide buried-ridge index-guided CdZnSe/
ZnSSe SCH single QW laser devices provided by the 3M Company. The p-
contact metal stack is Pd/Pt/Au annealed at 2000C for 15 minutes in forming
gas and the n-contact metal is Pd/Ge/Au annealed at 2000C up to an hour in
forming gas [Hab97].









tude of the peak of a gain spectrum. As a consequence, the lasing wavelength X (or lasing

energy E) for conventional cleaved facet lasers (Fabry-Perot lasers) should depend on CE,

since they lase at the peak of the gain spectra. Due to the fact that an accurate prediction

of E depends on a precise knowledge of various QW bandgap determining parameters such

as QW material composition and degree of biaxial compression, QW thickness, band off-

sets etc., it is not practical to test the importance of CE by comparing measured and calcu-

lated E values. However, it can be shown that the dependence of E on carrier density N is

relatively insensitive to the choice of these bandgap-determining parameters but is sensitive

to CE. Since this dependence can be determined easily by measuring the dependence of E

on laser cavity length L, it is practical to determine the importance of CE by comparing cal-

culations of the E vs. L dependence with experiment.


4.3.1 Experiment


InGaAs OW. The arrangement of experimental setup is sketched in Fig. 4.3. The

lasers were characterized in the bar form on a Cu probing stage at T = 300 K using 2 psec

pulses at 1 kHz repetition rate to avoid heating effect. The laser beam from the laser facet

was focused using a spherical lens on to the front slit of the monochromator which in addi-

tion has two plane mirrors, a grating and a rear slit, as depicted in Fig. 4.4. The prime mir-

ror brings the laser beam to the grating and the secondary mirror images the diffracted beam

from the grating to the rear slit. A silicon PIN-10D photo-detector is placed after the rear

slit and plugged into a digital voltmeter to measure the photo voltage. The reading of the

voltmeter reaches a maximum once the laser beam is exactly imaged to the rear slit by ro-

tating the grating to a certain angle for a given lasing wavelength X. Since the angle has

been calibrated to the corresponding wavelength by the manufacture of the monochroma-

tor, the value of K is easily read out. At each cavity length L, X was therefore recorded for

several lasers and converted to an average emission energy E for that L value using the fol-









lowing relation

E = (4.1)

where h is the Plank constant. The measured dependence of E on 1/L for the InGaAs QW

lasers is plotted in Fig. 4.5. As shown, E increases by about 3 meV as L decreases from

1500 to 500 pm (i.e. 1/L increases from 6.7 to 20 cm-').

CdZnSe OW. The arrangement of experimental setup is sketched in Fig. 4.6. The

lasers were characterized in the bar form on a Cu probing stage at T = 300 K using 2 psec

pulses at 1 kHz repetition rate to avoid heating effect. The edge emission spectra from the

laser facet were imaged using a microscope objective on to the front slit of the optical mul-

tichannel analyzer (OMA). Basically OMA is similar to a monochromator which has an

array of 1024 photodetectors (1024 channels) at the rear image plane instead of a rear slit.

Photon counts received by each channels are sent to the computer and reconstructed as a

spectrum with each channel corresponding to a specific wavelength. Spectra obtained from

a typical CdZnSe QW laser, below and just above lasing threshold, are shown in Fig. 4.7.

The lasing wavelength X, defined at the peak of the spectrum just above lasing threshold,

was recorded for several lasers at each cavity length L and converted to average emission

energy E for that L value using Eq. (3.1). Fig. 4.8 shows the measured dependence of E on

1/L for CdZnSe QW lasers. In this case, E increases by about 6 meV as L decreases from

2010 to 740 pm (i.e. 1/L increases from 5 to 13.5 cm-).


4.3.2 Prediction


For typical cleaved-facet lasers (Fabry-Perot lasers) with a cavity length L, the fre-

quency spacing Av between longitudinal modes is given by [Ver89]

c
Av = (4.2)
2nLwhere n is the refractive index of the cavity. For example, a GaAs laser cavity with n = 3.6
where n is the refractive index of the cavity. For example, a GaAs laser cavity with n = 3.6

























photo-detector


monochromator


laser diode



lens


voltmeter


Figure 4.3 Illustration of experimental setup for measuring the lasing wavelength for
the InGaAs QW diode lasers. The lasers were tested in a pulsed mode and
edge emission from the laser facet was imaged to a monochromator using a
spherical lens.


oscilloscope




















































Figure 4.4 Sketch of the components inside the monochromator used in Fig. 4.3. The
prime mirror brings the laser beam imaged at the front slit to the grating and
the secondary mirror images the diffracted beam to the rear slit by rotating
the grating to a proper angle for a given lasing wavelength.









































0 5 10 15
1/L (cm-')


20 25


Figure 4.5 Measured lasing energies E of various laser cavity lengths L for the 8 nm
Ino. jsGao.85As QW lasers. An increase of about 3 meV in E is found as 1/L
increases from 6.7 to 20 cm-1 (i.e. L decreases from 1500 to 500 gtm).


T=300 K
8 nm InGaAs QW




_A

A


























OMA


computer


oscilloscope


Figure 4.6 Illustration of experimental setup for measuring the edge emission spectra
for the CdZnSe QW diode lasers. The lasers were tested in a pulsed mode
and edge emission from the laser facet was imaged to an optical multichan-
nel analyzer (OMA) using a microscope objective.























150 1
T=300K
4 nm CdZnSe QW

100 above threshold




50 -
below threshold
x 10


460 500 540 580
(nm)











Figure 4.7 Edge emission spectra obtained from a typical 4 nm Cdo.3Zno.-Se QW diode
laser. The spectrum below the threshold, which is magnified 10 times in the
plot, is wide. Just above the threshold, the spectrum becomes very narrow
and the lasing wavelength X is defined at the peak of the spectrum.

























T = 300 K
4 nm CdZnSe QW
-

- -



- -


0 5 10
1/L (cm')


15 20


Figure 4.8 Measured lasing energies E of various laser cavity lengths L for the 4 nm
Cdo.3Zno7Se QW lasers. In this case, E increases by about 6 meV as L de-
creases from 2010 to 740 pm (i.e. 1/L increases from 5 to 13.5 cmrl). The
error bars show the variation of E obtained from each laser bar and the open
circles are the corresponding average values of E.


2.425


2.420


2.415


2.410


2.405








and L = 1000 gm, Av is 42 GHz calculated using Eq. (4.2) which corresponds to an energy

spacing hAy of 0.17 meV. This spacing is relatively small compared to the spectral width
of the gain function g(ho) which is typically in the order of 100 meV at threshold. As a

consequence, the Fabry-Perot lasers will lase at the peak of g(h ) and the lasing energy E

is essentially determined by the spectral location of the peak of g(Ah(). The value of E is

associated toL by the fact that the magnitude of the peak gain g of g(hA) required for lasing
at L is given by the expression [Chi88, Col95]

g = gth = In 1+ ai] (4.3)

where r is the transverse mode overlap factor which can be calculated using the software

MODEIG, R is the modal reflectivity at the facets and a, is the mode loss coefficient. The

value of ai can be determined from an experimental plot of the inverse differential quantum
efficiency 1/rd vs. L [Col95, Hsu96].

The procedures for obtaining E as a function of L, depicted in Fig. 4.9, are summa-
rized as follows: (1) Specify N and T; (2) Calculate and plot the spectral gain function g(co)

using Eqs. (2.3) and (2.5) for single-particle model, Eq. (3.1) for the inclusion of CS, Eqs.

(3.41) and (3.44) for the addition of BGR, and Eq. (3.63) for the inclusion of CE; (3) De-

termine the spectral location E of the peak of g(hto) and the peak gain g of g(ho); (4) Use

this value of g in Eq. (4.3) to solve for L; (5) Plot this E vs. 1/L pair, (6) Vary N and repeat

steps (1) through (5). Finally, a plot of E vs. 1/L curve is obtained.

InGaAs OW. For the wide-stripe InGaAs QW lasers used in our experiment, the lat-

eral mode overlap factor is assumed to be unity. Using F = 0.022, R = 0.32 and ai = 1.7
cm-' in Eq. (4.3), the estimated gh are 422, 595, 1113 cm-1 for lasers with L = 1500, 1000

and 500 pm, respectively. Using the CS-modified gain function gB (hco) given in Eq. (3.1),
Fig. 4.10(a) shows the calculated TE gain spectra for N = 1.2, 1.4, 1.6, 1.8 x 1018 cm-3

which produce g in the range of interest (~ 300 to 1200 cm-'). The plots indicate that E blue

shifts as N increases. Using the procedures described in Fig. 4.9, we translate these plots








to an E vs. 1/L curve in Fig. 4.10(b). An increase in E (blue shift) of about 5 meV is ob-
tained as L decreases from 1500 to 500 im (i.e. 1/L increases from 6.7 to 20 cmr'). The

corresponding change in the experimental data is about 3 meV, a factor of 1.7 smaller. If

the BGR shown in Fig. 3.4 is added to the gain spectra of Fig. 4.10(a), the resulting rigidly
shifted gain spectra in Fig. 4.11 (a) show essentially no shift of peak gain position as N in-

creases. As a consequence, the corresponding E vs. 1/L curve in Fig. 4.11(b) shows that
the BGR shift essentially cancels the CS shift. It is interesting to note that this CS/BGR

cancellation-type effect has been reported previously in the modeling of the dependence of

E on L in GaAs SQW lasers [Che93, Chi88]. It is suggested in those works that BGR is

overestimated or some other many-body effect needs to be taken into account. We expect

that CE may be the answer to this problem. Therefore, we use the CE-modified gain func-
tion gE(hoi) given Eq. (3.63) to calculate the TE gain spectra forN= 1.1, 1.2, 1.3, 1.4 x 1018

cm"3 as shown in Fig. 4.12(a). It clearly shows that E increases with N and the correspond-

ing E vs. 1/L curve in Fig. 4.12(b) shows a blue shift in E of about 3 meV as L decreases

from 1500 to 500 im, in good agreement with the experimental shift. Consequently, we

conclude from this comparison that CE needs to be included in predicting the dependence

ofE on L in InGaAs QW lasers. The importance of CE in determining the dependence of

operating wavelength on temperature in InGaAs/GaAs vertical-cavity surface-emitting la-

sers (VCSEL) has been reported previously [Cho95].

CdZnSeOW. For the CdZnSe QW lasers used in our experiment, we have r 0.01,

R 0.243 and ai = 10 cm-'. With these numbers, the estimated gt are 1703, 2159 and 2910

cm-' for L = 2010, 1220 and 740 pim, respectively. With inclusion of CS, the calculated TE

gain spectra forN = 6.5, 7.4, 8.3, 9.2 x 1018 cm3 produce g in the range of interest (~ 1700
to 3000 cm-') as shown in Fig. 4.13(a). The corresponding E vs. 1/L curve in Fig. 4.13(b)

shows an increase in E (blue shift) of about 7 meV as L decreases from 2010 to 740 pm.

While this is close to the 6 meV blue shift obtained experimentally (see Fig. 4.8), the pre-

dicted values of E are about 85 meV higher than the measured ones. If the BGR shown in









Fig. 3.6 is added to the gain spectra of Fig. 4.13(a), the resulting rigidly shifted gain spectra
in Fig. 4.14(a) show a red shift of peak gain position as N increases. As a consequence, Fig.

4.14(b) shows a red shift of the predicted E with decreasing L, opposite to the observed blue

shift With inclusion of CE, the TE gain spectra for N= 5.0,5.4, 5.7, 6.0 x 1018 cm-3 in Fig.

4.15(a) show that E increase with N. As shown in Fig. 4.15(b), one obtains a blue shift in

E of about 4 meV as L decreases from 2010 to 740 pm, in good agreement with the exper-

imental shift. It also shows that the predicted absolute values of E are close to the measured

values, differing only by about 10 meV. Consequently, we conclude from this comparison

that CE needs to be included in predicting the dependence of E on L in CdZnSe QW lasers.

Comment #1. With respect to the discrepancy between the measured absolute value
of E and the prediction with CE included, we believe that this is due to a lack of precise

knowledge of various QW bandgap determining parameters as stated in the beginning of

this section. Since the dependence of E on L is relatively insensitive to the choices of these

parameters, we believe that the above conclusions about the importance of CE in both the
InGaAs and CdZnSe QW lasers is justified.

Comment #2. In Section 3.5, we discussed a little bit on the definition of Mott den-
sity Nmo where bound excitons cease to exist and the Pad6 approximation for CE factor

works better. For the 8 nm In0o.1Gao.sAs QW lasers, N,,t is much less than 5 x 1017 cm3

and for the 4 nm Cdo0.Zn0.Se QW lasers, it is about 2 x 1018 cm-3. As we have just shown

in Figs. 4.12 and 4.15, the carrier densities required to achieve lasing action in the InGaAs

and CdZnSe QW lasers are in the order of 1 x 1018 cm3 and 5 x 1018 cm3, respectively,

which are higher than the their associated N.,, values. This rules out the possibility of ex-

citonic gain at room temperature even in CdZnSe QW lasers for which excitons were said

to be responsible for providing material gain [Ding94]. In Chapter 5, we will show more
evidences that electron-hole plasma, not excitons, is the source for gain/absorption in the

CdZnSe QW lasers by comparing the theoretical calculations using model with CE taken

into account to the experiments.




















g(ho)
g
00)


E \ wo


i E

Plot an E vs. 1/L pair x

1/L
Do you want to
Yes repeat for a new N?

No


END

i/L

Figure 4.9 Flow chart of the procedures for obtaining a theoretical E vs. 1/L curve.











1500 . .. .. . .. .
T= 300 K
8 nm InGaAs QW
1000 w/CS 1.8
1.6
1.4
500 1.2


0 --- --


-500 ......
1.25 1.30 1.35 1.40
hw (eV)
(a)


1.325I ..
T= 300 K
8 nm InGaAs QW
1.320
w/ CS

S 1.315


1.310


1.305
0 5 10 15 20 25
1/L (cm 1)
(b)

Figure 4.10 With CS included, (a) the calculated TE gain spectra for the 8 nm
Ino~lGao.ssAs QW lasers at various N (x 108 cm3) indicated by the num-
bers. The arrow indicates that E increases with N; and (b) the corresponding
E vs. 1/L curve. An increase in E (blue shift) of about 5 meV is obtained as
1/L increases from 6.7 to 20 cm-1 (i.e. L decreases from 1500 to 500 gm).











1500 ......
T=300K
8 nm InGaAs QW
1000 w/CS+BGR 1.8
1.6
f 1.4
500 1.2


0 --------- ----


-500
1.20 1.25 1.30 1.35
hto (eV)
(a)


1.290 ,
T= 300K
8 nm InGaAs QW
1.285


1.280 w/CS+BGR


1.275


1.270
0 5 10 15 20 25
1/L (cm-')
(b)

Figure 4.11 With CS and BGR included, (a) the calculated TE gain spectra for the 8 nm
Ino.IsGao.85As QW lasers at various N (x 1018 cm-3) indicated by the num-
bers. The arrow indicates essential no shift in E with N; and (b) the corre-
sponding E vs. 1/L curve. BGR shift essentially cancels the CS shift.











1500 ....
T= 300K
8 nm InGaAs QW
1000 w/CS+BGR+CE 14
1.4
1.3
Sh 1.2
5001.1


0 ------ --------


-500 .
1.20 1.25 1.30 1.35
ho (eV)
(a)


1.290 | .. .
T=300 K
8 nm InGaAs QW
1.285

w/ CS+BGR+CE
1.280


1.275


1.270 I I
0 5 10 15 20 25
1/L (cm-')
(b)

Figure 4.12 With CS, BGR and CE included, (a) the calculated TE gain spectra for the
8 nm In0o.1Gao.sAs QW lasers at various N (x 1018 cm3) indicated by the
numbers. The arrow indicates that E increases with N; and (b) the corre-
sponding E vs. 1/L curve. A blue shift in E of about 3 meV is obtained as
1/L increases from 6.7 to 20 cm-1 (i.e. L decreases from 1500 to 500 gm).


























2.45 2.50
hco (eV)


0 5 10
1/L (cm-1)


2.55 2.60


15 20


Figure 4.13 With CS included, (a) the calculated TE gain spectra for the 4 nm
Cdo.3Zna~Se QW lasers at various N (x 1018 cm-3) indicated by the numbers.
The arrow indicates that E increases with N; and (b) the corresponding E vs.
1/L curve. An increase in E (blue shift) of about 7 meV is obtained as 1/L
increases from 5 to 13.5 cml' (i.e. L decreases from 2010 to 740 gm).


001


-1000 -
2.40


T= 300 K
4 nm CdZnSe QW
Sw/ CS


2.510


2.505


2.500


2.495


2.490











4000.
T= 300 K
3000 4 nm CdZnSe QW 9.2
w/ CS+BGR 8.3
7.4
2000 6.7


-1000


0 -- -----------------

-1000 ...
2.30 2.35 2.40 2.45 2.50
hto (eV)
(a)


2.425 ,
T=300K
4 nm CdZnSe QW
2.420


S 2.415


2.410 w/ CS+BGR


2.405 i I *
0 5 10 15 20
1/L (cm-')
(b)

Figure 4.14 With CS and BGR included, (a) the calculated TE gain spectra for the 4 nm
Cdo.ZnoTSe QW lasers at various N (x 1018 cm-3) indicated by the numbers.
The arrow indicates that E decreases with N; and (b) the corresponding E vs.
1/L curve and the measured data. The predicted shift in E with 1/L is in the
opposite direction to the measured shift.











4000 .
T=300K K6
3000 4 nm CdZnSe QW 6.
5.4
w/CS+BGR+CE "/

2000 5.0


1000

0 -- ----------- ------

-1000 I .. I .
2.30 2.35 2.40 2.45 2.50
ho) (eV)
(a)


2.430
T= 300ok'
4 nm CdZnSe QW w/CS+BGR+CE
2.425


2.420


2.415


2.410 ..
0 5 10 15 20
1/L (cm-1)
(b)

Figure 4.15 With CS, BGR and CE included, (a) the calculated TE gain spectra for the
4 nm Cdo.3ZnoiSe QW lasers at various N (x 108 cm-3) indicated by the
numbers. The arrow indicates that E increases with N; and (b) the corre-
sponding E vs. 1/L curve with the measured data. A blue shift in E of about
4 meV is obtained as 1/L increases from 5 to 13.5 cm'.









Comment #3. In previous work on the theory of CdZnSe QW lasers [Cho95b], it was
predicted that E should shift to lower energies (longer wavelengths) as N increases from 4

to 5 x 1018 cm-3 in an 8 nm thick QW. In this work, we report on E vs. L measurements

using 4 nm thick CdZnSe QW buried ridgeguide lasers [Hsu97b], which indicate that E

shifts to higher energies (shorter wavelengths) as N increases. In addition, we use a single

particle model [Cor93] modified by the addition of CE [Hau90, Cho94], carrier scattering

[Asa93, Yam87] and bandgap renormalization [Cho94, Hau90] to calculate a dependence

of E on N which is in good agreement with the experimental results.


4.4 Cavity Length Dependence of Threshold Current


Since the threshold gain increases with decreasing cavity length, the threshold cur-

rent should also change as a function of cavity length. By investigating the cavity length
dependence of threshold current, we are able to demonstrate the true mechanism behind the
optical gain and spontaneous emission of semiconductor lasers.


4.4.1 Experiment


The experimental geometry is shown schematically in Fig. 4.16. The laser bar is put

on probing stage with p-side up. The probe and stage are connected to "+" and "-", respec-
tively, of a HP 214A pulse generator via co-axial cables with a current-varying pot in the

circuit. A PIN-10D photo-detector is used to measure the laser output power from one fac-

et. The measured voltage and current pulses in the circuit alone with output power pulse

are sent to three separate channels of a Stanford Research System (SRS) where a pulse sig-

nal is averaged over a very short sampling gate placed in the middle of the pulse duration

and converted to a DC signal representing the averaged value. Then a set of DC signals,

representing voltage, current and output power, are sent to a personal computer (PC) for









further data processing and a real-time output power Po vs. current I curve is displayed on

the PC monitor. By extrapolating the linear region of P, vs. I curve back to the I axis, the

cross-over is defined as the threshold current I,,. The voltage, current and power pulse sig-

nals can also be monitored simultaneously on an oscilloscope and compared to the reading

shown on PC monitor.

InGaAs OW. Fig. 4.17 shows the measured I,h vs. L for the 8 nm Ino.15Gao.s5As single

QW lasers with 100 Wm stripe width. The open circle is the averaged I,h for the good laser

devices tested from the same laser bar and the error bars represent the span of variation in

lh.
CdZnSe OW. Fig. 4.18 shows the measured Ih vs. L for the 4 nm Cdo.3Zno.7Se single

QW lasers with 5 pm wide buried-ridge.


4.4.2 Prediction


In Section 4.3, we have shown that a certain value of Nin QW is needed to provide

a peak material gain g required for lasing at a given cavity length L. For diode lasers, N is

provided by injecting current through metal contacts, as depicted in Fig. 4.19, where elec-

trons and holes are injected to the active layer (or QW in the laser structures studied) from

n- and p-side, respectively, and then recombine with each other. Under steady state condi-

tion, rate of injected electrons into QW is equal to the rate of recombining electron in QW

which can be expressed mathematically as

ml
-- = RecwLd, (4.4)

where r, is the injection efficiency which is defined as the fraction of current being injected

into QW, I is the injected current, w is the stripe width, L is the cavity length, d, is the QW

thickness and R,.c is the rate of recombining electron per unit volume (in units of s-cm3).

The recombination processes include a spontaneous recombination rate R,, a nonradiative




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