COULOMB ATTRACTION EFFECTS IN
SEMICONDUCTOR QUANTUM WELL LASERS
By
CHIAFU 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
CHIAFU HSU
DEDICATED TO MY PARENTS,
CHUNGMEI TSAI AND HUNGYI HSU
AND IN MEMORY OF MY DEAREST BROTHER,
CHIAHENG HSU
ACKNOWLEDGMENTS
Being able to have the strength to finish my dissertation, I am very grateful to my par
ents, ChungMei Tsai and HungYi Hsu, my sister, SungYa 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 stateoftheart
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. ChiLin "Kenny"
Young, Dr. ChihHung "David" Wu, Dr. Craig Largent, Bob Widenhofer, JeongSeok "Ja
son" 0, Carl Miester, John Yoon, Christian Keyser and JeeHoon "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, HsiHung 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. YuFu Hsieh, and his wife, HsuehChu 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, MingCheng "Benjamin" Liu,
his wife, ChunYi "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, HueiFeng "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 SINGLEPARTICLE 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 Inplane Subband Structures ............................................ 24
2.4.3 Numerical Results ................................................. ...................... 25
2.5 SingleParticle 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 MANYBODY 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 Gaincurrent 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 CARRIERINDUCED EFFECTS IN IIVI 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 NearField 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 lIVI TERNARIES ............................................ 157
A.1 Interpolation Method ............................. .............. .......................... 157
A.2 Binary Parameters ....................................................... ... 158
A.3 Ternary Parameters ............................... ............. 158
B TRANSITION MATRIX ELEMENT ............................................................ 162
B.1 PolarizationDependent Effects ................................... ..................... 162
B.2 AngleDependent 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
CHIAFU HSU
December, 1997
Chairman: Peter S. Zory
Major Department: Electrical and Computer Engineering
Theoretical and experimental studies of manybody effects in semiconductor quan
tum well (QW) lasers are presented. These manybody 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 electronhole plasma (EHP) theory with
manybody effects taken into account. Optical gain and spontaneous emission spectra are
calculated for the welldeveloped, infrared emitting, InGaAs/GaAs strained QW structure
and the relativelynew bluegreen CdZnSe/ZnSSe strained QW structure. The results show
that Coulomb attraction effects (CE) have a significantly stronger effect on the spectral
characteristics of the widebandgap CdZnSe QW structure than on the narrowbandgap In
GaAs QW structure. The roomtemperature 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
nancelike 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. Highpower diode lasers are also used in applications such as medical surgery,
freespace communication and diode laser pumped solidstate 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 builtin 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 highquality crystalgrowth 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 electronhole plasma (EHP) theory was used to model the laser devices
such as the welldeveloped infrared (IR) GaAsbased QW lasers [Asa93, Col95, Cor93].
Soon after the demonstration of the world's first bluegreen 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 socalled excitonic gain and laser emission in ZnSebased QWs and pro
posed a phenomenological model of optical gain based on partial phasespace filling (PSF)
of an inhomogeneously broadened exciton resonance [Din92]. This phenomenological
gain model, which includes the effects of excitonphonon interaction, was used to explain
their pumpprobe 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 resonancelike 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 bluegreen 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 manybody effects in the literatures. Haug et al. showed that gain
spectra of a semiconductor changed when manybody 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 widebandgap CdZnSe QW lasers
[Cho95b, Ree95a] than in the narrowbandgap GaAsbased 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 manybody 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 twodimensional (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 manybody effects in semiconductors.
In order to use these manybody 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 quasi2D QW, a number of subbands in both CB and VB become available from
solving the onedimensional (lD) Schridinger's equation. Second, effective mass anisot
ropy in the VB due to the large builtin strain the QW was considered, where the heavy hole
(HH) and light hole (LH) effective masses in the inplane 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 manybody effects really dealt only with the gain
function [Cho94, Hau90]. Four, the modified manybody formulas were integrated with
the stateoftheart 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 manybody 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 manybody physicists have been trans
lated to a form more userfriendly 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 welldeveloped IR InGaAs
strained QW lasers and relatively new bluegreen 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 singleparticle
model are discussed in detail. It begins by introducing the epitaxial structures for the two
QW lasers of interest: IlV InGaAs and IIVI 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 singleparticle model are calculated.
In Chapter 3, manybody effects, such as CS, BGR and CE, on the gain and sponta
neous emission functions are discussed sequentially. Semiderivations of BGR and CE are
presented. Calculation results of the optical gain and spontaneous emission spectra with
manybody effects included are shown and comments on how CS, BGR and CE modify the
singleparticle 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 carrierinduced antiguiding factor is calculated for the CdZnSe QW
lasers and its effect on the quality of laser beam is described. Nearfield patterns are mea
sured and compared qualitatively to the predictions.
Finally, in Chapter 7, conclusions are drawn and future works are suggested.
CHAPTER 2
SINGLEPARTICLE MODEL
2.1 Introduction
The chapter begins by introducing the epitaxial structures for the two QW lasers of
interest: LIV InGaAs and IIVI CdZnSe QW lasers. Due to the mismatch of the inplane
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 flatband 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 singleparticle 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 indepth 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 polarizationdependent effects are discussed.
2.2 Laser Structures
2.2.1 InGaAs OW Laser
The InGaAs/GaAs gradedindex separateconfinement heterostructure (GRINSCH)
single QW diode laser structure studied in this work, for which the schematic sketch of
crosssection 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 nGaAs substrate, a 1400 nm nAlo.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)
compressivelystrained 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 pAl0r6Gao0As 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 crosssection 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 ntype GaAs (001) substrate [Gai93], a ZnSe buffer layer, a quaternary n
ZnMgSSe cladding layer, a 4 nm CdxZnl.,Se (x ~ 0.3) compressivelystrained quantum
well (QW) centered between n and pZnSo.6Se0.94 guiding layers, a pZnMgSSe cladding
layer and a graded p+ZnSexTel., (x = 1 to 0) contact layer. This contact scheme provides
a reasonably low resistance pcontact 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 (inplane 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 latticemismatch can be described
with a latticemismatch parameter e defined as follows [Col95]:
e= 1 ab, (2.1)
aw
where ab is the inplane lattice constant of the barrier material and a, is the native un
strained inplane 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 latticemismatch 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 inplane 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 pGaAs
buffer 25 nm pAlGaAs
cladding 1300 nm pAlGaAs
guiding 200 nm AlGaAs
bounding 7 nm GaAs
QW 8 nm InGaAs
bounding 7 nm GaAs
guiding 200 nmAlGaAs
cladding 1400 nm nAlGaAs
buffer 25 nm nAlGaAs
substrate
nGaAs
Figure 2.1 Schematic diagrams of the crosssection epitaxial layers and the corre
sponding aluminum and indium composition profile for the Ino.05Gao.BAs/
GaAs GRINSCH single QW laser.
contact
cladding
guiding
QW
guiding
cladding
buffer
buffer
substrate
p+ZnSeTe
pZnMgSSe
pZnSSe
CdZnSe
nZnSSe
nZnMgSSe
n+ZnSe
nGaAs
nGaAs
Figure 2.2 Schematic diagram of the crosssection 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 inplane
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]
C11C12
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 spinorbit energy which separates the splitoff (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) +8eHeS, hh
E (LH) = E (bulk) + 5et + 8es, Ih
where E, (bulk) is the unstrained bulk bandgap, E, (HH) is the CHH strained bandgap and
E, (LH) is the CLH 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 CHH 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 = AEhhS, (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 IIVI 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 IIVI 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
growthdirection HH effective mass mn (x mo) 0.372 0.377
growthdirection 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 inplane HH effective mass m;hn = 0.085 m,
Ino.05Gao.ssAs inplane 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
growthdirection HH effective mass mm (x m,) 0.480 0.571
growthdirection 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 inplane HH effective mass mhhll = 0.175 mo
Cdo.Zno.7Se inplane 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 socalled flatband 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 onedimensional potential wells in
CB and VB respectively. Then we will discuss the inplane 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 wellbarrier 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 (AEnE .)
a n = t (2.16)
where d, is the thickness of QW, mj is the growthdirection 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 growthdirection 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 flatband potential wells and (b) the parabolic
subband structure of the QW active region model.
2.4.2 Parabolic Inplane Subband Structures
As discussed above, the potential wells are onedimensional 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 inplane energy
subband exists, as shown in Fig. 2.7(b). The subband structure is usually characterized by
defining an inplane effective mass as [Kit86]
[ I d2E ,11
m,, = [ d (2.17)
where mnfi is the inplane effective mass, Enjk is the inplane energy measured relative to
the bottom of the well and kl is the inplane 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 inplane 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 socalled 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 growthdirection 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 inplane 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 Y722
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 SingleParticle 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 CHH strained bandgap or E,(LH) for the CLH strained band
gap, Eq and E, are the quantized energy levels in the QW. For an electronholepair 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 inplane 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 inplane 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 "singleparticle" 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. ClHH1 and C1LH1. In
the singleparticle model, one ignores manybody effects and assumes that momentum is
conserved (kselection 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 FermiDirac distribution for CB electrons andfh 
1 f, is the distribution for VB holes. The transition matrix element IMA2 is given by
2+ cos20 for CHH TE gain
IMT12 1cos20 for CHH TM gain
S 5 1 2 ,(2.27)
IM2 cos28 for CLHTE gain
S+ cos20 for CLH 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 inplane
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 CHH and CLH 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 FermiDirac 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 quasiFermi 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
kT
N = h2d mjIm ln { 1 + exp [(Ecj Efc)/kBT] } (2.36)
P= k vll In {1 + exp [(EVjEfv)/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 singleparticle 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 CHH 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 steplike 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 singleparticle 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 steplike density of states for
CHH1 and CLH1 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 cm3)
(b)
Figure 2.8 QuasiFermi 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 Singleparticle (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 cm3)
(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 QuasiFermi 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 Singleparticle (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
singleparticle 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
singleparticle 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 quasiFermi 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 C1HH1 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 crossover of the (AE E,1. M) vs. N curve on the Naxis is defined as the trans
parency carrier density N, which is the electronhole plasma density required to provide the
quasiFermi 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 cm3 for the strained InGaAs QW which is smaller than NMr = 3.1 x 108 cm3 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 cm3)
8 10
Figure 2.12 Difference between the quasiFermi level separation and C1HH1 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 singleparticle model. Since the measured spectra
do not have sharp features, it is clear that this singleparticle model is incomplete. In Chap
ter 3, we show that when manybody effects are incorporated into the singleparticle model,
realistic gain and spontaneous spectra are obtained. However, it is interesting to note that
in the limit of very thick QWs, the singleparticle model gives realisticlooking spectra.
This is due to the fact that the stairstep density of states function gradually smooths out as
QW thickness increases [Cor93]. As a consequence, the singleparticle model applied to
thick active layer lasers generates spectra which look realistic and it is not obvious that the
model is incomplete.
CHAPTER 3
MANYBODY EFFECTS
3.1 Introduction
In Chapter 2, we have discussed the optical gain and spontaneous emission functions
based on a singleparticle 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 electronhole plasma, mutual interactions among the carriers are expected to be
important These mutual interactions are usually referred as "manybody 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 singleparticle 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 singleparticle gj (E,) and R (E,) functions
for a given subband transition pair with a spectralbroadening 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 spectralbroadening 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 (ee), electronhole (eh) and electronLO phonon (eLO) and those contributing
to ,j are holehole (hh), holeelectron (he) and holeLO phonon (hLO). Details concern
ing the calculation of the these scattering times can be found in [Asa93].
The nonLorentzian 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)EE kE /KreK[EEjkffAT"
jk (E) 2, i 1 + eE/T (37)
where
K e4t T (0, kj kj,) m ,,,kT
487e2E2hkTd2 1 +eE',/kT h2 ) '
with
2 1 2
T(0, k'4,kj) = { + +4k2j d ,2 1[lexp(d.,d,) ]
X= (2+4k[. )2
Sdvkj (djkv ) (X4k ) 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 growthdirection 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 singleparticle 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 C1HH1 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 singleparticle 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 blueshifted 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 singleparticle counterparts (dotted curves) for the 8 nm
Ino.15Gao.85As/GaAs QW atN = 1.7 x 1018 cm3 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 singleparticle model
w/ CS
. . .
El. hh
4000 ... .. i I .
T= 300K
3000 i 4 nm CdZnSe QW
N = 5.4 x 101 cm3
2000 singleparticle 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 cm3
2 ., singleparticle 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 singleparticle 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 manybody interactions in an electronhole plasma (EHP)
is the screening of the Coulomb potential. In a twodimensional (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 inplane area of the quantum well. The parameter q, is the magnitude of a
qll vector which is defined as the difference between inplane 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 inplane reduced effective mass discussed in Chapter 2 to take
the effectivemass 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 socalled 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 plasmonpole 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 multisubband 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,
aaoc, 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 multisubband QW system. But for
a singlesubband 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 multisubband 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 multisubband 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 8n2L 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 singleparticle energies and,
as a consequence, a reduction in bandgap. In this case, the total bandgap renormalization
(BGR) AEBGR is the sum of Coulombhole (CH) selfenergy AECH and screenedexchange
(SX) selfenergy 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) selfenergies 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
weakmomentum 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 C1HH1 and C1LH1 transitions are shown as the dashdotted, 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 BGRshifted 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 C1HH1 and C1
LHI transitions are shown as the dashdotted, dashed and solid lines in Fig. 3.6, respective
ly. The corresponding BGRshifted 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 onethird 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 multisubband 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 C1HH1 and
C1LH1 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 Coulombhole components (CH) and the dashed
curves are the screenexchange (SX) components.
""AECH, C HHI
: '~:" z . AECH, ClIA
AEsx, cS.LI
AE SX.ClHHI
AEBGR.ClLHI
AEBGR, CIHHI
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 ClHHI and
C1LH1 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
dashdotted curves are the Coulombhole components (CH) and the dashed
curves are the screenexchange (SX) components.
IAEmrI c lIII
AEsx.I Hn

AEcH, CIHHI
AECH, ClWll
AEBGR, CILHI
T=300K EBGR, ClHHI
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 fkfhk)
S h ( + iS e eh) (347)
(3.47)
(fek fhk1)
= 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 inplane
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 FermiDirac 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 singlesubband system and solved from the Bethe
Salpeter equation for the only one existing subband transition pair [Cho94, Hau90]. Since
in a multisubband system a set of BetheSalpeter 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 multisubband 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,cMO
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
qj(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 +fhk1)
g(h(O) = Real q c +i(k+ehk) (3.59)
j k r 1k C ll*+(ek+ hkho)
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 lqlj(hto, E,)
S (E h_o' ) }', (3.63)
xL.(1Ec) [1i(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 FermiDirac 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 Iqtlkh +i(ek+e hk(o)
(3.65)
4o dCV ( fek) (1 fh)
a(V k, [1k q +i(e +ehkh 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 1lq1(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 cm3. 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' cm3, 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 cm3.
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 IIVI CdZnSe QW than in the IIIV 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 socalled 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 cm3 (or 8 x 1011 cm2 in sheet density) from Fig. 3.6. Because
of this large value for Nmo in IIVI 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 manybody 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
tronhole plasma (EHP) were described. In the socalled 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 welldeveloped narrowbandgap InGaAs QW lasers and the rel
atively new widebandgap 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 GRINSCH 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 pside of the wafer was then
metalized using Au pulsed electroplating technique. After thinning the nsubstrate down
to a wafer thickness of about 100 p.m, the nside was also Auelectroplated, 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 buriedridge indexguided devices fabricat
ed from this structure where the pcontact metal stack is Pd/Pt/Au annealed at 200"C for 15
minutes in forming gas and the ncontact metal stack is Pd/Ge/Au annealed at 200 "C up to
an hour in forming gas [Hab97]. These stateoftheart 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 pAlGaAs
cladding 1300 nm pA1GaAs
guiding 200 nm A1GaAs
bounding 7 nm GaAs
OW 8 nm InGaAs
bounding 7 nm GaAs
guiding 200 nm A1GaAs
cladding 1400 nm nAlGaAs
buffer 25 nm nA1GaAs
substrate nGaAs
meal Au
Figure 4.1 Crosssection sketch of the InGaAs/AlGaAs GRINSCH 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 nGaAs substrate down to a wafer thickness of about 100 gm, Au
was deposited everywhere on the nGaAs substrate.
5 gm
metLl Pd'Pt/Au
/pZnSeTe\
ZnS cladding ZnS
SpZnMgSSe
guiding pZnSSe
QW CdZnSe
guiding nZnSSe
cladding nZnMgSSe
buffer n'ZnSe
buffer nGaAs
substrate nGaAs
melal PdiGe/Au
Figure 4.2 Crosssection sketch of the 5 pm wide buriedridge indexguided 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 ncontact 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 (FabryPerot 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 bandgapdetermining 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 PIN10D photodetector 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 cleavedfacet lasers (FabryPerot 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
photodetector
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 cm1 (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 FabryPerot 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 singleparticle 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 widestripe 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 cm1 for lasers with L = 1500, 1000
and 500 pm, respectively. Using the CSmodified 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 cm3
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
cancellationtype 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 manybody effect needs to be taken into account. We expect
that CE may be the answer to this problem. Therefore, we use the CEmodified 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 verticalcavity surfaceemitting 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 cm3 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 cm3. 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 electronhole 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 cm1 (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 cm3) 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 cm1 (i.e. L decreases from 1500 to 500 gm).
2.45 2.50
hco (eV)
0 5 10
1/L (cm1)
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 cm3) 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 cm3) 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 (cm1)
(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 cm3) 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 cm3 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 pside up. The probe and stage are connected to "+" and "", respec
tively, of a HP 214A pulse generator via coaxial cables with a currentvarying pot in the
circuit. A PIN10D photodetector 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 realtime 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
crossover 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 buriedridge.
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 pside, 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 scm3).
The recombination processes include a spontaneous recombination rate R,, a nonradiative