Low threshold current strained InGaAs/Al

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Low threshold current strained InGaAs/AlGaAs quantum well lasers
Citation
Eng, Lars E.
(1993)
Low threshold current strained InGaAs/AlGaAs quantum well lasers.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/wxtc-4x91.
Abstract
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
Strained InGaAs quantum well lasers offer the prospect of lower threshold currents, higher modulation speed, and lower linewidths than lattice matched GaAs quantum well lasers. In addition, the useful wavelength region of the GaAs material system is extended from 0.87[...] to beyond 1[...] with the addition of indium to the quantum well.
The lasers are fabricated using Molecular Beam Epitaxy (MBE) for the semiconductor layer structure. Liquid Phase Epitaxy (LPE) is then used to provide lateral optical mode and current confinement. Broad area threshold current densities of [...] is the first demonstration of high quality MBE grown strained InGaAs laser material. Measured transparency currents of 25[...] are a factor of two lower than in GaAs, which is consistent with a lower valence band density of states in the strained material. Buried heterostructure lasers made from this material with 2[...] wide stripe widths lase with a minimum threshold of 1.0mA (CW), the lowest value for a single quantum well laser with as-cleaved mirrors in any material system. With high reflectivity coatings (R=0.9) the first sub milliampere strained InGaAS lasers are obtained, with [...]. Details of the material growth, device fabrication, and device optimization are presented.
The broad gain bandwidth of single quantum well lasers is used to tune the lasing wavelength of optimized GaAs lasers over 125 nm and InGaAs lasers over 170 nm in an external cavity configuration. The measured tuning curves obtained for the InGaAs lasers are qualitatively different, and the difference can be attributed to the modified strained valence band strucure.
Low temperature (5°K) performance of low threshold lasers is investigated. The decrease in threshold with temperature is found to be linear over a range of 200°K for both GaAs and InGaAs with a larger decrease in threshold for the GaAs case. This result agrees well with a lowered valence band effective mass in the strained laser.
Item Type:
Thesis (Dissertation (Ph.D.))
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Yariv, Amnon
Thesis Committee:
Unknown, Unknown
Defense Date:
14 May 1993
Record Number:
CaltechETD:etd-08272007-091655
Persistent URL:
DOI:
10.7907/wxtc-4x91
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Low Threshold Current Strained
InGaAs/AlGaAs Quantum Well Lasers

Thesis by

Lars E. Eng

In Partial Fulfillment of the Requirements
for the Degree of

Doctor of Philosophy

California Institute of Technology
Pasadena, California
1993

(Defended May 14, 1993)

To my Mother and Father

Acknowledgments

I would first like to acknowledge the support and encouragement of my advisor,
Professor Amnon Yariv. I appreciate the opportunity to have been part of his group
and interact with the first rate students, staff, and visiting scientists in the environ-
ment he has provided. I am very fortunate to have had the chance to work with
Professor Hadis Morkog who generously shared his wealth of MBE knowledge during
his stay at Caltech. I would also like to thank the following:

Dr. Steve Sanders for making laser processing more enjoyable, many helpful dis-
cussions and collaborations, and helping arrange a first date with Julie.

Dr. Anders Larson for helpful and open MBE discussions, and lending spare parts
when I needed them.

Dr. David Mehuys, Dr. Hal Zarem, and Dr. Kazumasa Mitzunaga for guidance
and collaborations in my first few years.

Dr. T.R. Chen, and Y.H. Zhuang for LPE fabrication, processing expertise, and
collaborations.

Han Grave, Dr. Michael Mittelstein, and Dr. Amir Sa’ar for collaborations.

Ali Shakouri and John O’Brien for stimulating discussions and proofreading of
this thesis.

Ali Ghaffari for dedicated work on the MBE system.

Former MBE group members Naohiro Kuze, Yoshihiro Yamada, Dr. Sidney Kan,
Dr. Howard Chen, and Dr. Pam Derry.

Thomas Schrans, Randal Salvatore, and Dr. Joel Paslaski for helpful discussions.

Desmond Armstrong and Kevin Cooper for building and trouble shooting elec-
tronics.

Jana Mercado for her day to day help and cheerful demeanor.

I would like to thank my family, Mom, Dad, Bjorn, Maria, Ingrid, Mats, and
Louise for supporting me through all the ups and downs in graduate school. Many
thanks also to my second family, Tom, Terri, Sandy, and Lindsey Soderstrom.

Finally, I want thank Julie Sheridan for her cheerleading and her unconditional

love all these years.

Abstract

Strained InGaAs quantum well lasers offer the prospect of lower threshold currents,
higher modulation speed, and lower linewidths than lattice matched GaAs quantum
well lasers. In addition, the useful wavelength region of the GaAs material system is
extended from 0.87pm to beyond lym with the addition of indium to the quantum
well.

The lasers are fabricated using Molecular Beam Epitaxy (MBE) for the semicon-
ductor layer structure. Liquid Phase Epitaxy (LPE) is then used to provide lateral
optical mode and current confinement. Broad area threshold current densities of
Jin = 144, is the first demonstration of high quality MBE grown strained InGaAs
laser material. Measured transparency currents of 25-4, are a factor of two lower
than in GaAs, which is consistent with a lower valence band density of states in the
strained material. Buried heterostructure lasers made from this material with 2m
wide stripe widths lase with a minimum threshold of 1.0mA (CW), the lowest value
for a single quantum well laser with as-cleaved mirrors in any material system. With
high reflectivity coatings (R=0.9) the first sub milliampere strained InGaAS lasers
are obtained, with [7" = 0.35mA. Details of the material growth, device fabrication,
and device optimization are presented.

The broad gain bandwidth of single quantum well lasers is used to tune the lasing
wavelength of optimized GaAs lasers over 125 nm and InGaAs lasers over 170 nm
in an external cavity configuration. The measured tuning curves obtained for the

InGaAs lasers are qualitatively different, and the difference can be attributed to the

modified strained valence band strucure.

Low temperature (5°) performance of low threshold lasers is investigated. The
decrease in threshold with temperature is found to be linear over a range of 200° A’ for
both GaAs and InGaAs with a larger decrease in threshold for the GaAs case. This

result agrees well with a lowered valence band effective mass in the strained laser.

Contents

1 Low Threshold Semiconductor Lasers 1
1.1 Introduction... 0... i
1.2 GaAs Quantum Well Lasers 2.0... 2 ee 2
1.3 Strained InGaAs Quantum Well Lasers .. 2... ..0.0.00004. 3
1.4 Outline of Thesis 2... 2.202000 00002 2 ee 4

2 Molecular Beam Epitaxy of GaAs, InGaAs, and AlGaAs for Opto-
electronic Devices 8
2.1 Introduction... . ee 8
2.2. MBE Growth: An Overview .. 2... 0.00.0 00002 pe eee 9
2.3 System Specifics. 2. 10
2.4 Precise Growth Ratesin MBE... 2... ee es 12
2.5 Cell Flux Stability 2... 0... 2 \7
2.6 Reproducible Growth Rates 2... 0.0.0.0. 0.0.0 00.22 00. 17
2.7 InGaAs Growth . 2... ee 21
2.8 Conclusions 2... 23

Vili

3 Model for Gain and Threshold Current in GaAs and Strained In-

GaAs Quantum Well Lasers 26
3.1 Introduction. 2... 26
3.2 Effects of Strained InGaAs on Quantized Energy Levels... . 2... 27
3.2.1 Strained InGaAs Subbands.........0..0.0...2.004. 27
3.2.2. Band Mixing in Strained InGaAs ................ 30

3.3 Spectral Gain 2... ee 40
3.3.1 Absorption and Transition Rates 2... ............. 40
3.3.2 K- Selection Rule 2... 2.002.020.0200... 00200008. 42
3.3.3 Parabolic Uncoupled Bands ........00..2...0.004. 43
3.3.4 Gain Equations... . 0... ee 44
3.3.5 Transition Broadening ... 2... ..0..0.0 00005550005 45

3.4 Calculated Gain Spectra 2... 0 : .. 46
3.4.1 Carrier Injection Efficiency. 2... 0. AT
3.4.2. Calculated Gain. 2... 50

3.5 Threshold current equations . 2... ee 54
3.5.1 Threshold Current 2.0.0.0 0000000 ee ee 55

3.6 Conclusions 2... 56
4 InGaAs Quantum Well Laser Performance 60
4.1 Introduction... 60

4.2 Structure and Growth 2.0.0.0... .0.0 0.0.0 00 0 eee ee es 61

4.3 Broad Area Single Quantum Well Devices 2... ......0..02, 63
4.4 Multiple Quantum Well Lasers... 2... 2. ee ee 6-4
4.5 Low Threshold Buried Heterostructure Lasers 2... .......0.. 68
4.6 Improvements in Performance .... 0.000000 00000002 eee 76
4.7 Conclusions 2... sO
Broadband Tuning of InGaAs Quantum Well Lasers 84
5.1 Introduction. 2... 0. S4
5.2. Quantum Wells for Tuning... 2... ee 85
5.3 Oxide Stripe Lasers and Tuning Results ......0.....0..0020. 86
5.4 Comparison of InGaAs and GaAs Tuning... ...........4.. 93
5.5 Conclusions . 2... Q7

Microampere Threshold Current Operation of GaAs and Strained

InGaAs Quantum Well Lasers at Low Temperatures 100
6.1 Introduction... 2... 100
6.2. Low Temperature Measurements... 0.0.02... 02 ee eee 102
6.3 Threshold vs. Temperature 2... 2.0... ee ee ee 106
6.4 Threshold Carrier Density... 0.0.0... 0.0 20... 000.200.0034 107
6.5 Threshold Current vs. Temperature... 0.2. ee es 110

6.6 Conclusions 2... lil

Chapter 1

Low Threshold Semiconductor

Lasers

1.1 Introduction

The needs of the telecommunications industry have been responsible for much of the
research on semiconductor diode lasers over the past two decades. Due to their small
size, high modulation speed, and wavelength tailorability, these lasers are ideal as
a source to send information as modulated light down optical fibers in high data
rate communication systems. The superior bandwidth of fibers over electrical coaxial
cables has resulted in communication links with data transmission rates above |
Gbit/s.

Optical data storage and retrieval, most notably within the audio and video com-

pact disc industry, is another area of active research in semiconductor lasers. Here,

speed is not as important as the small size and low cost. The main effort in this area
is to reduce the lasing wavelength in order to increase the data packing density.

The systems mentioned above use a single, or only a few, discrete laser diodes.
For future computer interconnects, however, thousands of lasers will be necessary for
parallel chip to chip optical interconnects. In large scale integration schemes using
lasers together with electronic devices, the current necessary to turn on the laser, the
threshold current, is the major source of power dissipation. For practicle large scale
systems, this current must be below 1 mA.

The approach to lowering laser threshold current amounts to reducing the number
of injected electrons necessary required for population inversion, as well as minimiz-
ing the losses experienced by the optical field. Optimum laser performance for low
threshold is achieved by shrinking the physical dimensions of the laser and lowering
the optical mode loss by increasing the mirror reflectivity and fabricating low loss

waveguides.

1.2 GaAs Quantum Well Lasers

With modern semiconductor crystal growth techniques, such as Molecular Beam
Epitaxy (MBE) and Metal Organic Vapor Deposition (MOCVD), it is possible to
grow semiconductor laser structures with active region thicknesses of less than 100
Angstroms. In this regime, quantum size effects can be observed and the electron

density of states is modified compared with the bulk material. Due mainly to the

thin active region, the threshold current can be reduced by over an order of magnitude
compared with bulk lasers. The predicted lower limit to the threshold current for an
optimized single quantum well laser with realistic losses is /7"" = 0.lmA [1]. Thresh-
old currents of 0.55 mA have been measured for a buried heterostructure (BH) GaAs
single quantum well laser with high reflectivity coatings [13,12]. Other geometries,

such as vertical cavity surface emitting lasers [5,5,6] and narrow stripe structures on

patterned substrates [7,9] have also produced thresholds below 1 mA.

1.3. Strained InGaAs Quantum Well Lasers

By adding indium to the GaAs active region, making an InGaAs alloy, the band gap is
lowered and increases the lasing wavelength range to beyond lym. This wavelength
region is particularly attractive since Ert doped fiber amplifiers used in state-of-
the-art communication systems are efficiently pumped at a wavelength of 980 nm.
Because of the lowered band gap, the barrier heights to the quantum well can be
made higher leading to more efficient current injection. By the same measure, the .
available index step for a given barrier height is larger, resulting in an increase in
optical confinement factor which increases the modal gain.

However, the lattice constant of InAs (ap = 6.0584A) is 7% larger than that of
GaAs (ay = 5.6533), so for InGaAs to be grown dislocation free on a GaAs lattice,
it will be under compressive strain. Matthews [16] has shown that a stable strained

crystal can be grown as long as the layer is thin enough. The first demonstration

of a current injected strained InGaAs laser structure was demonstrated in 1984 [11].
The threshold current densities for these devices were over 10004, more than 5
times higher than similar unstrained GaAs results. Yablonovich and Kane [1] and
Adams [2] calculated the energy band structure of strained layers and showed that
strain should actually enhance laser performance. The effect of strain is to split the
degenerate valence band energies and lower the effective hole mass to near that of a
conduction electron. This in turn will lead to a lower inversion condition, less loss
due to intervalence band absorption, and the prospect of lower threshold currents.
This thesis presents the theory, growth, and fabrication of sub milliampere thresh-

old current strained InGaAs/AlGaAs quantum well lasers. Threshold currents for

as-cleaved lasers of 1.65 mA are the lowest reported for a single quantum well laser

0.75 mA, which is a record for this material system and rivals the best results of

unstrained GaAs lasers [11,2].

1.4 Outline of Thesis

Chapter 2 describes the MBE crystal growth system used to grow the epitaxial laser

structures used in this work. Considerations for high quality InGaAs growth are

discussed, as well as an approach to obtaining growth rates to within 2% accuracy.
Chapter 3 provides the theoretical background and calculations of energy levels,

density of states, and gain in a strained single quantum well with AlGaAs barriers.

Differences between GaAs and InGaAs gain are highlighted.

In chapter 4 the details of the InGaAs laser structure are given. The experimental
method for optimizing the laser for low threshold is given. Record low threshold
currents of of 1.65 mA (CW, R = 0.3) and 0.75 mA (CW, R = 0.9) are obtained.

In chapter 5 the broadband gain properties of a single quantum well is taken
advantage of to tune the lasing wavelength over a record wide range of 170 nm [17].
A comparison with optimized GaAS lasers tuning curves is made.

Chapter 6 describes measurements of laser threshold at low temperatures [18].

Thresholds are reduced to the microampere range, with a larger reduction in GaAs

than InGaAs.

References

(1] A.Yariv, Appl. Phys. Lett. 53, 1033 (1988)

[2] P. Derry, A. Yariv, K. Lau, N. Bar-Chaim, kK. Lee, and J. Rosenberg, Appl. Phys.

Lett. 50, 1773 (1987)
[3] K.Y. Lau, P.L. Derry, and A. Yariv, Appl. Phys. Lett. 52, 88 (1988)

[4] J. Jewell, J.P. Harbison, A. Scherer, Y.H. Lee, and L.T. Florez, J. Quant. Electr.

27, 6, 1332 (1991)

[5] K.Tai, R.J. Fischer, C.W. Seabury, N.A. Olsson, T. Hou, Y. Ota, and A. Cho,

Appl. Phys. Lett. 55, 2473 (1989)

[6] R.S. Geels, S.W. Corzine, J.W. Scott, D.B. Young, and L.A. Coldren, JEEE

Photn. Techn. Lett. 2, 234 (1990)

[7] E. Kapon, C.P. Yun, J.P. Harbison, L.T. Florez, and N.G. Stoffel, Electron. Lett.

24, 985 (1988)

[8] E. Marclay, D.J. Arent, C. Harder, H.P. Meier, W. Walter, and D.J. Webb,

Electron. Lett. 25, 892 (1989)

[9] D.J. Arent, L. Brovelli, H. Jackel, E. Marclay, and H.P. Meier, Appl. Phys. Lett.

56, 20, 1939 (1990)
[10] J.W. Matthews and A.F. Blakeslee, J. Cryst. Growth 27, 118 (1974)

(11] W.D. Laidig, P.J. Caldwell, Y.F. Lin, and C.K. Peng, Appl. Phys. Lett. 44, 653

(1984)
[12] E. Yablonovich and E.O. Kane, J. Lightwave Tech. LT-4, 504 (1986)
[13] A.R. Adams, Electron. Lett. 22, 249 (1986)

[14] I. Suemune, L.A. Coldren, M. Yamanishi, and Y. Kan, Appl. Phys. Lett 53, 1378

(1988).

(15) L.E. Eng, T.R. Chen, S. Sanders, Y.H. Zhuang, B. Zhao, H. Morkoc, and A.

Yariv, Appl. Phys. Lett. 55, 1378 (1989).

[16] T.R. Chen, L.E. Eng, Y.H. Zhuang, and A. Yariv, Appl. Phys. Lett. 56, 11, 1002

(1990)

[17] L.E. Eng, D.G. Mehuys, M. Mittelstein, and A. Yariv Electron. Lett. 26, 1675

(1990)

[18] L.E. Eng, A. Sa’ar, T.R. Chen, I. Grave, N. Kuze, and A. Yariv, Appl. Phys.

Lett. 58, 2752 (1991)

Chapter 2

Molecular Beam Epitaxy of GaAs,
InGaAs, and AlGaAs for

Optoelectronic Devices

2.1 Introduction

All epitaxial III-V semiconductor layers described in this thesis have been grown
by Molecular Beam Epitaxy (MBE). A brief description of the growth system and
process, and some key problems encountered are presented in this chapter. It is shown
that if care is taken, absolute layer thicknesses can be controlled to within 2%. A
complete review of the growth technique and process can be found in review articles

in references [1]-[4].

2.2. MBE Growth: An Overview

MBE is a crystal growth technique used to grow epitaxial layers of semiconductor crys-
tals. Effusion cells containing the constituent materials (Ga,Al,In,As) and dopants
(Si,Be) are pointing, in the horizontal plane, toward the heated substrate near the
chamber center. By heating the sources to their evaporation temperatures, molecular
beams are thus created which impinge on the GaAs substrate and recombine to form
crystalline III-V material at the substrate surface. Because the beams hit the sub-
trate at an angle, the substrate is rotated azimuthally during growth which results in
approximately 10% uniformity across the wafer. The growth parameters in an MBE
process include the choice of arsenic species (As) or As4), the ratio of As to group
III flux, and the substrate temperature. The alloy composition (for random alloys)
is controlled by the cell temperatures, and the beams are turned on or off using me-
chanical shutters in front of each cell. The alloy composition, x, for Al,Ga,.,As is

given by

_ Raias 2.1)
FRatas + Roads

where FR denotes the incorporation rate of the material into the crystal. Because
of the low fluxes, possible growth rates can vary from fractions of, to a couple of,
microns per hour. This translates to a rate on the order of one atomic layer per
second. This property makes MBE an ideal growth technique where layer thicknesses

with precision to atomic scales, with monolayer roughness, is desired.

2.3 System Specifics

The system used in our laboratory is a RIBER 2300 three chamber system with 2”
wafer capability. Many of the characteristics of crystal growth, such as uniformity,
vacuum, and cell temperature stability, are system dependent. All chambers are
under ultra high vacuum (UHV), pumped with ion pumps. Two of the chambers serve
primarily as sample introduction chambers, or load locks, so that the integrity of the
vacuum in the growth chamber is maintained. A cross section of the growth chamber
is shown in Figure 2.1. During growth, the pumping speed is increased by filling the
cryopanels, or shrouds, with liquid nitrogen (ZN2), and pressures in the 107!° Torr
range are achieved. The growth chamber is exposed to atmospheric pressure only
when the material in the cells are depleted or a crucial part of the hardware breaks
down and the system must be rejuvenated. After this, several weeks of baking,
degassing, and calibration are required to grow crystals with a purity demanded by
optoelectronic devices. In fact, a good measure of the cleanliness of the system is
the threshold current of a broad area "standard design” single quantum well laser,
which should be below 400-45. As long as no mechanical difficulties are encountered,
the time between venting can be longer than eight months, or after several hundred
pem of material growth. The system is equipped with a large volume (200 cc) arsenic
(As) cracking cell (Perkin-Elmer 06-200) which can be used to generate As2, Asy, or
a mixture of the two. The growth is always done with an excess of arsenic to prevent

group If island formation. ‘The growth rate is thus determined by the arrival rate of

LIQUID NITROGEN EED GUN

COOLED SHROUDS MAIN SHUTTER
EFFUSION ROTATING SUBSTRATE
CELL ct HOLDER
PORTS Th IONIZATION GAUGE
WA

" ak 5 FF GATE VALVE
aN / : \\
“AS easter
TE ‘f
qe U SAMPLE
Cc. EXCHANGE
— LOAD LOCK

\ oO
] VIEW PORT
EFFUSION —
CELL
SHUTTERS
FLUORESCENT TO VARIABLE
SCREEN SPEED MOTOR

AND SUBSTRATE
HEATER SUPPLY

Figure 2.1: A schematic of the RIBER 2300 MBE growth chamber.

group IIT atoms, and the excess arsenic is re-evaporated from the surface. Keeping
the arsenic overpressure to a minimum not only conserves source material but also
improves the crystal quality.

An optimum substrate temperature must also be found for each material or al-
loy. The choice of this temperature is often a trade off between sharp interfaces with
controlled growth rates and background impurity incorporation. To obtain high lu-
minescence efficiency, for instance, may require growing the material at a substrate

temperature higher than would be chosen to obtain the sharpest interfaces.

2.4 Precise Growth Rates in MBE

The trend in semiconductor devices is toward smaller and smaller dimensions to
lower power requirements. In the past six years, a new class of devices based on
microcavity resonators has emerged. This class includes Fabry Perot cavity devices
with epitaxially grown distributed Bragg reflectors (DBR), such as optical modulators
and vertical cavity surface emitting lasers (VCSEL) whose size is on the order of
a wavelength of light. In these devices the entire cavity is grown in the MBE. A
high finesse cavity (high R mirrors) requires a large number of mirror pairs of very
controlled thickness. Thus, the growth time is very long, often 6-8 hours, and growth
rates must be controlled to better than 2%. Previous to this class of device, this
amount of precision has not been necessary in MBE growth. Although MBE allows

interface sharpness to be made on the order of 1 atomic layer, a great deal of care

must. be taken in calibrations and growth rate monitoring to assure an absolute layer
thickness accuracy of better than 10%.

Our MBE, like most, is equipped with an ionization gauge which can be rotated
into the path of the molecular beams for flux measurements. The gauge sensitivity to
the different materials is known and the measured beam pressures can be converted to
beam fluxes with this data. It may seem that this would be a simple way to monitor
growth rates. In practice, however, we notice large variations in the measured fluxes
(due to gauge sensitivity variations) from one day to the next. Therefore, this method
only allows us to measure growth rates to within an order of magnitude.

The next simplest method is to grow several thick layers, measure the thicknesses
with a scanning electron microscope (SEM) and plot the growth rate vs. Toa ona
log scale. Using the vapor pressure curve to fit the data, it is possible to interpolate
or extrapolate growth rates with about 10% accuracy.

The most precise way to measure growth rates is to use Reflection High Energy
Electron Diffraction, or RHEED, intensity measurements. An electron gun is present
in the chamber and electrons are accelerated towards the sample at a shallow incident
angle and diffracted onto a phosphorous screen as shown in Figure 2.1. The intensity
of the specularly reflected electron beam is monitored with a Si photodiode followed
by a current amplifier. It has been shown [2] that this intensity oscillates with the
period corresponding precisely to one atomic layer of growth. Some typical RHEED
data is shown in Figure 2.2. Typically the oscillations are damped so it is only possible

to measure the growth rates for the first 10 - 30 monolayers.

Detector current [a.u.] =>

Figure 2.2: Measured oscillations in the specular reflected RHEED beam after initial
growth of AJAs (the aluminum shutter is opened at t=0). The corresponding growth

rate is approximately | monolayer per second.

The cells experience a different environment when the shutter is closed than when
it is open. This leads to a decrease in cell temperature, and therefore growth rate,
with time. A measurement of the aluminum and gallium cell fluxes versus time after
opening the shutter in front of the cell is shown in Figure 2.3. It is clear that the
flux undergoes a transient and is up to 15% higher at t=0 than it is in steady state.
Therefore, the RHEED oscillation measurement is very sensitive to the state of the
cell before opening. We have developed a technique to reproducibly measure growth
rates from RHEED. We open the shutter in front of the cell while keeping the main
shutter closed for two minutes. This gives the cell time to equilibrate and reproduce a
steady state flux. We then begin to record oscillations as the main shutter is opened.
In this way we can find the steady state growth rate for each cell at a given cell
temperature. By knowing the transients from the flux measurements, we therefore
know the growth rate at any time. The use of 1” spacers between the cells and growth
chamber increases the distance between the shutter and cell and has been shown to
reduce growth rate transients significantly [8,8]. In our system we have measured a
decrease in flux transients from 15% to 4% when using the spacers.

If we look at the gallium flux in Figure 2.3, we notice random fluctuations of a
few percent, which we don’t see in any of the other cells. Other researchers [9] have
seen this effect as well and attribute it to gallium “spitting.” They have been able to
eliminate this problem with a modern cell design. If the cell has not been modified,
it is important that these variations average out over the time it takes to grow one

layer. If they do not, the growth rate will vary from layer to layer, and no amount of

Cell} [~~ Al

Flux
(a.u.)

{ | | |
0 50 100 150 200

Time (sec)

Figure 2.3: Beam fluxes for a) aluminum and b) gallium cells vs. time after shutter

opening. Note the random fluctuations in the gallium flux.

calibration will be sufficient to grow high finesse Fabry Perot resonators.

2.5 Cell Flux Stability

Any variations in cell temperature will cause variations in the growth rate. Therefore,
since we have a strict limit how much the growth rate can vary, we need to determine
the allowable temperature variation. Within the temperature range of interest, the
cell flux, and therefore the growth rate R, can be related to the cell temperature
through the relation

R= Ae T (2.2)
where A and B are constants which depend on the material, and the temperature
is in degrees Kelvin. These constants can be found by knowing growth rates at two
different cell temperatures. As an example of the flux stability with temperature,

for our aluminum cell ( B = 3.8310*[? A] ) we require a cell temperature stablility

AT < 0.5°A at 1135°C for a flux variation an < 1%.

2.6 Reproducible Growth Rates

We have found that the growth rates vary up to 10% from one day to the next. There-
fore, it is necessary to develop a procedure to determine the growth rate immediately
before growing a particular device. In order to accurately determine the rate, it is
necessary to grow structures which are sensitive to small fluctuations in layer thick-

nesses. Periodic reflective structures, Bragg reflectors, are sensitive to variations of

< 1%. Before growing a new device structure, we grow a Bragg reflector, measure the
spectral reflectance, and compare with the predicted curve. The difference between
the designed and measured data gives us a factor by which to multiply the RHEED
oscillation data. In Figure 2.4 we show measured reflectance, R(A), for a Fabry-Perot
cavity grown by the above procedure. The designed transmission peak was 950 nm,
and the measured peak was 937.5 nm, an error of less than 2%. The half width of the
transmission peak, which is a measure of the reflectivity, was also within experimental
error. The calculation of the reflectivity was done using the matrix method for wave
propagation in multilayered structures [11], and the refractive index data for GaAs
and AlGaAs was taken from reference [12]. In Figure 2.5 we show the reflectance
of a vertical cavity surface emitting laser grown by this technique. Note the narrow
dip at 881 nm. Losses in the cavity and nonuniformity over the sampled area limit
the depth of the dip. Also shown is the lasing peak of this structure under optical
excitation.

The above calibration techniques are only useful if the sticking coefficients of the
group III atoms are constant during growth. Gallium, indium, and aluminum all have
different critical temperatures, T.,;4, above which the sticking coefficient becomes less
than unity. Above this temperature, the growth rate becomes extremely sensitive
to substrate temperature as well as the presence of other group III elements at the
substrate surface [13], and is not easily controlled. Recent work [14,15] have experi-
mentally determined T.,;. = 540°C for InAs growth on GaAs, with desorption rates

of 10 — 25% at a substrate temperature of 600°C. If grown at too low temperature,

Reflectivity —->

800 937.5nm 1100
A [nm]

Con
am

. \ 4 :

Foo 900 1000 1100

Reflectivity

Wavelength [nm]

Figure 2.4: a) Measured and b) calculated reflectance spectra for a passive, i.e. no

gain medium, Fabry-Perot cavity grown by MBE.

Reflectivity =>
_—

700 881nm 1000

891.6

Figure 2.5: a)Measured reflectance spectra for a VCSEL structure cavity grown by

MBE. b) Lasing emission from the VCSEL pumped optically.

however, the luminescence efficiency of the crystal is drastically reduced. A compro-
mise has to be made in which a trade off between crystal purity and growth rate

accuracy is considered.

2.7 InGaAs Growth

The growth of InGaAs on GaAs places a restriction on the maximum layer thickness
while still retaining dislocation-free material, since the lattice constants of InAs (a9 =
6.0584A) and GaAs (ap = 5.6533A) differ by more than 7%. However, if the layers
are thin enough [16], the in-plane lattice miss-match is accomodated by strain, «,

defined as

InzgGay2As GaAs
ao ~ 4

€y =
il ageas

when the strained layer takes on the lattice constant of the substrate. The free-
standing lattice constant of InGaAs is taken as a linear interpolation of GaAs and
InAs lattice constants

agreCM—24® = 5 6533 + 0.4052.

Matthews and Blakeslee [16] have derived expressions for the maximum layer thick-
ness, the critical thickness, beyond which the energy of forming a dislocation is equal
to the amount of strain energy in the film. In practice [15} this limit is in reason-
able agreement with observed photoluminescence efficiency in strained InGaAs layers.
Figure 2.6 shows the calculated critical thickness for InGaAs grown on GaAs as a

function of indium concentration using this model. The practical limit of x for single

1000 7
>» 100 os
oO -4
10 it | i i L i i

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Figure 2.6: Calculated critcal layer thickness for strained InGaAs on GaAs using

Matthews and Blakeslee model [16] as a function of indium content.

InGaAs quantum wells is 40% for reasonable well widths (d > 40A).

2.8 Conclusions

We have shown that MBE can give very precise growth rates. For many structures,
such as edge emitting quantum well lasers, crystal purity is much more important
than accuracy. However, devices based on interference reflectors and filters require
thickness fluctuations of less than 2 % over growth times of 6 - 10 hours. In order
to obtain growth rate accuracy better than 10 %, it is essential to ensure long term
cell flux stability over the entire growth. It is also necessary to compensate for the
growth rate transients either by reducing them as much as possible, or by accounting

for them with averaged growth rates.

References

i] A.Y. Cho, Thin Solid Films 100, 291 (1983)
[2] C.T. Foxon, Acta Electronica 21, 2, 139 (1978)
[3] B. Joyce, Rep. Prog. Phys. 48, 1637 (1985)

[4] KK. Ploog, Ann. Rev. Mat. Sez., 171 (1981)

[5] J. Jewell, J.P. Harbison, A. Scherer, Y.H. Lee, and L.T. Florez, /. Quant. Electr.

27, 6, 1332 (1991)
[6] Riber operators manual
[7] A. Larsson, private communication
[8] J.S. Harris, private communication
(9] J. Miller, J. Vac. Set. Tech. B, 10, 803 (1992)

(10| W.T. Tsang, F.K. Reinhart, and J.A. Ditzenberger, Appl. Phys. Lett. 36, 118

(1980)

[li] P. Yeh, Optical Waves in Layered Media, Wiley (1988)
[12] Afromovitz, Solid State Comm. 15, 59 (1974)

[13] R. Fischer, J. Klem, T.J. Drummond, R.E. Thorne, W. Kopp, H. Morkoc, and

A.Y. Cho, J. Appl. Phys. 54, 2508 (1983)

[14] M.T. Emeny, L.K. Howard, K.P. Homewood, J.D. Lambkin, and C.R. White-

house J. Crystal Growth 111, 413 (1991)

[15] J.P. Reithmaier, H. Riechert, H. Schlotterer, and G. Weimann, J. Crystal Growth

111, 407 (1991)
(16] J.W. Matthews and A.E. Blakeslee, J. Cryst. Growth 27, 118 (1974)

[17] J.E. Schirber, LJ. Fritz, and L.R. Dawson, Appl. Phys. Lett. 46, 187 (1985)

Chapter 3

Model for Gain and Threshold
Current in GaAs and Strained

InGaAs Quantum Well Lasers

3.1 Introduction

In this chapter, the expression for gain as a function of photon energy is presented
for a quantum well laser structure. The equation is given in terms of the density of
states in the conduction and valence bands, the interband matrix element, and carrier
injection level, assuming that the k-selection rule holds. The addition of indium to a
GaAs quantum well introduces strain due to the lattice mismatch between InAs and
GaAs. The built-in strain has been shown to increase the band gap as well as split the

degenerate heavy and light valence band edges. This reduces the interaction between

valence bands, resulting in a lower valence band density of states compared with
unstrained GaAs quantum wells [14,13]. Calculations of a realistic laser structure,
assuming parabolic bands, an energy- and material-independent matrix element, and
a lorentzian broadening function are presented to illustrate important features of
quantum well laser gain spectra. From these calculations we see that the effect of a
lower hole mass leads to a lower transparency current, a larger gain bandwidth for

the same current level, and a larger differential gain for low threshold gain structures.

3.2 Effects of Strained InGaAs on Quantized En-

ergy Levels

3.2.1 Strained InGaAs Subbands

The effective band gap of a strained InGaAs quantum well sandwiched between Al-
GaAs barriers is given by the sum of the bulk InGaAs bandgap (including strain)
and the lowest quantized subband energies in the conduction and valence bands, as
illustrated in Figure 3.1. This section will focus on the calculation of the band
edge transition wavelength as a function of indium composition, x, and quantum well
width. Increasing the indium content has two competing effects which must be taken
into account when calculating the band gap. First, the effective band gap is decreased
due to the incorporation of lower band gap InAs into the well. Second, increasing

x also causes an increase in the effective gap due to the increased strain in the well

Ec(z}

~_e bee me eb — EIC

Elv

EV (2) ed

Figure 3.1: Lowest transition energy in strained InGaAs single quantum well.

(6). In practice, due to the maximum allowable layer thickness for a given strain, the
longest achievable wavelength in this material system is near 1.1m.

At the subband edge the component of k in the x-y plane, ky = 0, and the
valence and conduction band wavefunctions are decoupled and can be treated as
single isolated bands [15]. The spatial dependence of the wavefunction can be written
as

Wel?) = Youle eM ue y(t) (3.1)
where \..(<) is a slowly varying envelope function and u.,(r) is the Bloch function

periodic with the crystal lattice. The envelope function in each band then satisfies

the Schrodinger equation in each of the heterostructure layers :

with y and ~;\’(z) continuous at the interfaces. The potential V(z) is equal to 0

in the well and Vo in the barrier. The subband energies then satisfy the implicit

Vo m* f2m*,, Le a.
(ZG -vTe = tan( 72 E., 3) n= 1,3)... (3.3)

equations

Vo m~ L, ;
— on te = t enemy = 4,. : A
Ce an cot ( a BD zh n= 2, (3.4)

The masses m*,, refer to bulk carrier masses corresponding to motion perpendicular
to the quantum well in the well and in the barrier regions, respectively. Vo is the
barrier height and EF, is the subband energy, with the zero energy at the bulk band
edge, to be determined. Compressive strain increases the bulk band gap of the well

material which effectively lowers Vo. The increase in band gap is given by
AE, = aq (3.5)

where a is the hydrostatic deformation potential and e is the in-plane strain. The

barrier potential in the conduction and valence bands are then given by
Vo. = AER

Vou = AE, (1 — R)

with R the band offset ratio. For this ratio we use the model of reference [6]. Then

the transition energy between the first subbands in the conduction and valence bands

Ey; = Ey + E strain + Fy. + Ey (3.6)

and the transition wavelength is

he
Ey

Figure 3.2 shows calculated band edge transition wavelengths for InGaAs quan-
tum wells for various mole fractions and quantum well widths with GaAs barriers.
The material parameters used in this calculation are taken from Reference [4] and are
summarized in Table 3.1. These curves have been found to agree well with photolu-
minescence and TEM data [12], and are useful for estimating the indium content and

quantum well width in InGaAs quantum well laser material.

3.2.2 Band Mixing in Strained InGaAs

In order to understand the effect of strain and quantization on the valence band
density of states, it is necessary to consider the interactions between the bands. The
dominant interaction is between the heavy and light hole bands [12,14,11,13] and is
the only one considered here. In a quantum well, energies of both the heavy hole
and light hole bands are quantized in the z direction, and subbands consisting of a
mixture of heavy and light holes are formed. The wavefunction for the carrier in the
th

rn” subband can be written as

h=l

l=1

1.2

Oe ee ee ee ee ¥ ree

; a 0.35

1.1 b-

1.05
< —- 0.20 :
1 _ 7
ts
0.95 oto
0.9 be 0.05 4
0.85 Yada da pebao a bl { 1 cheb

40 50 60 70 80 90 100 110 120
Well width [A]

Figure 3.2: Calculated transition wavelength for a strained InGaAs single quantum

well with GaAs barriers as a function of well width, for various indium molefractions.

Lattice constant [A]

32
GaAs. InAs

5.6533 6.0584

Conduction band effective mass

mc
Luttinger parameters
YI

Y2
Y3

Yave

0.0665 0.027
6.85 19.67
2.1 8.37
2.9 9.29
2.5 9.0

Elastic Constants [* i9!l dyn/cm?]

Cll

C44
Deformation Potentials [eV]
Hydrostatic: av
Shear: b
Spin Orbit Energy [eV]
Alloy Bulk Bandgap [eV]
AlxGal-xAs

InxGal-xAs

11.88 8.33
5.37 4.526
5.94 3.959
- 8.0 - 6.0
- 17 - 1.8
0.34 0.38

1.424 + 1.247*x

1.424 - 1.601*x + 0.54 *x2

5.6605

3.45

0.68

1.29

0.99

Table 3.1: Material constants used in this chapter for calculating energy levels (from

Reference [4]).

with y,(h = 1,2,...,0 = 1,2,...) is the envelope function for the heavy (h) or light

(1) hole subband evaluated at ky = 0. The |HH > and [LH > Bloch parts of the

wavefunctions can be written in terms of the |J,m,; > basis as [11,12,14]

eiZ—%2) e-iGzZ-¥)
HH >= 3/2,3/2 > “8/2, -8/2 >
LH = 5, apes is p2.1p2
>= ———|3/2,-1/2 > -———|3//2, 1/2 >
| i (3/2, —1/ a

where @ = tan“ 1( 24), Solving the multiband effective mass equation in the envelope
function scheme has been shown to be an accurate method of calculating wavefunc-
tions and energy levels in heterostructures for finite in-plane wavevector ky [15,16].
We consider the simplest model which assumes infinite barrier heights so that band
edge wavefunctions and their interactions can be calculated analytically. The prob-
lem is further simplified by limiting the number of subbands to one light hole and
two heavy hole bands, i.e., h = 1,2 and | = | in the expansion above. The omission
of other bands gives less than 5% error {13] in calculating the highest valence band
energy dispersion. The resulting 3X3 Hamiltonian is diagonalized as a function of hy
to give the subband dispersion relations, and the eigenvectors quantify the amount of
mixing between the bands. From these calculations we find the valence band density
of states, the reduced density of states, and the polarization dependence of the matrix

element for the lowest e-hl transition.

The Luttinger Hamiltonian [7,8]describing the interactions between the light and

heavy holes in strained InGaAs in the |HH >,|LH > basis is given by [10,14,13]

Hin tS C+iB
Hix = (3.8)
C-iB Hy-S

where

h dy 2
Ah, = By - mgt ~ 242) (t=) — (41 + ya) Aj]

hk id. |
Hy = Ey = 5——[(r1 + 27a)? = (91 7)

h d
= Bah) (i
B mv WO?
h V3

= 7 Yave

mo 2

Ch + Ci
el

S=—b
The y's are the Luttinger parameters of the material, the C;; are the elastic coeffi-
cients, and b is the shear deformation potential. At hy = 0 the coupling terms B,C

are zero and the decoupled heavy and light hole envelope functions each satisfy the

single band Schrodinger equation 3.2 with effective masses

. \
Try SS nn
1 242

. |
hy [ooo
1 9 + 292

In the infinite well approximation, the band edge wavefunctions and energies for light

and heavy holes become

The interactions between hh1, hh2, and lhl are calculated using the envelope functions

above and H;,. For the hhl-lh1l interaction
. V3
qq, H< xIC + Bly} = J Tavekij

and similarly for the coupling between hh? and lhl

and the Hamiltonian on the hhi, hh?, lhl basis is

En, - B; 0 Cy
Hax3 = 0 Eng — E; 24
an ch Eu — Ei
with
c h? ‘ ® 9 2
Ey = 5S me! - 2aNT) + (41 + y2) hj)
2m 4 9
Eng = 5 — dmg (1 ~ 2a) + (41 + Y2) hi)
and |
EB sf 2y)(—)? ke
n=—-S— Omg 7! + aT) + (41 — 72)hjj)

The valence band energies vs. k are then solutions of
(E — Bn )(B ~ Eno)(B — En) — eal(2 ~ En) + len\?(2 — £n2)) = 9 (3.9)

This equation has three roots E,(n = 1,2,3), corresponding to one of the three energy
bands. The normalized wavefunction for the n‘” (n=1,2,3) subband can be expressed

in terms of our reduced basis as

C1 (Ene _ E,)

(Ens - E,)( Eng ™ E,)

with the normalization constant

A= flea |? ( Bae _ E,) + lee? (Ena 7 E,)° + (Bnd _ E,)(En2 _ E,))*|7?

Calculated valence band dispersion curves and the density of states of the highest
band (h1), for GaAs and Ingo 2GaogAs 100 A quantum wells with infinite barriers, are
shown in Figure 3.3. The strain effectively splits the energy band edges in InGaAs
and reduces the coupling between the subbands compared with the GaAs case. The
density of states, proportional to the slope of On is therefore significantly lower
for the strained quantum well. This calculation estimates the in-plane hole effective
mass m;, == 0.0879 for InGaAs, a significant reduction compared with m, = 0.4577
for bulk GaAs. The effective hole mass in strained InGaAs has been measured, with
m, = 0.14mp by Jones et al. [17].

We can also calculate the matrix element once the expansion coefficients of the
electronic wavefunction in our reduced basis are known. For the lowest transition, the

lowest conduction state el to the highest valence band state hl, and a polarization

unit vector ¢, the matrix element is given by
M =< hy,leerlel >

The Bloch function in the conduction band has atomic orbital $ symmetry [7,9] and

we write the wavefunction for electrons in an infinite well as

op “7 50 SaRanaaee
r GaAs 4 ia j
F z=100 | F InGaAs (x=0.2)
-10 = + 40° lz= 100 4
\ 30 4
=—-20F 4 t
> oF = 20+ 3
“30> 4 a C
t \ 10 - 4
- \ - .
*405 \ 4 ' j
\ ° {
-50 : * te -10 “4 fl a f ;
oO - 9.02 0.04 0.06 0 002 0.04 0.06
k [174] k (A)
1: tT rrr T T 7 i t ai T T 4 T ;
Gaas ’ L }
Lz = 100 L InGaAs (x#0.2) |
0.8 - 4 0.8 F Lz = 100 4
0.6 — 4 0.6 | i
g g
7 \ 0.4 {
7 \ o2 | |
-35 -30 -25 -20 -15 -10 -5 10 15 20 25 30 35 40 45
Ev [meV]

Ev [meV]

Figure 3.3: a)Calculated E(k) and for a 100 A single quantum well with infinite
barriers, including coupling between two heavy hole bands and one light hole band.
for GaAs and Ing 2GaggAs. b) The density of states (DOS) normalized by a free

electron 2D DOS .po = 4%, for the highest hole band for the two structures.

rh’

For TE modes, the polarization vector is in the plane of the well, i.e., € = x, and the
matrix element, averaged over azimuthal angle ¢, can be found from the components
of Py = (Pia. Fro, Fag). After transforming the valence band wavefunctions back

from the |J,m,; > basis to the atomic p orbital functions |¥ >,|Y >,|Z > we find

for the lowest order transition el-hl

< Zlez|S > |? Fe

For TM modes the polarization is perpendicular to the well, « = z, and the matrix

element for the el-hl transition becomes

iMliay =| < Zlez|S > |?

The TE matrix element is larger than the TM matrix element in both GaAs and
compressively strained InGaAs, and, therefore, most important for gain. A plot of
the calculated TE matrix element is shown in Figure 3.4 as a function of in-plane
wavevector. The larger TE matrix element for the InGaAs is a result of the lowered
amount of mixing and, hence, larger heavy hole character of the highest valence band

in the strained material.

Te
L.
0.8 +
0.6 +
al
0.4 - ~
02 4 GaAs J
r InGaAs ;
0 0.01 0.02 0.03

0.04 0.05 0.06
k (1/A]

Figure 3.4: Calculated TE matrix element for the el-hl transition for the same

structures in Figure 3.3. The matrix element is normalized to its value at Ay = 0.

3.3. Spectral Gain

3.3.1 Absorption and Transition Rates

The electric field of a plane wave, polarized along z (TE), and traveling along y in
an amplifying medium, can be described by exponential gain and loss coefficients +
and a@ [18],

BE, = Ege D% eh) + ce, (3.11)
The optical loss, in em7!, due to absorption of a photon which excites an electron
from state | to state 2, can be calculated in terms of Bj, and the using Fermi’s Golden
Rule. The Einstein coefficients A2,, B2;, and By are the spontaneous and stimulated
transition rates between energy levels 1 and 2. To derive a, we consider the net rate
of upward transitions [s~'] due to a photon density P( £21) of frequency wa; = Ey— Ey
is
ry = Biol hi — fo) P(E a) (3.12)
where f;,2 are the fermi occupation factors for levels 1 and 2 given by

(Ey 2~E g)
€ FP +]

fiz = (3.13)

where we have used By. = Bo,. Now it remains to relate the net rate of upward
transitions [s~'] to the absorption coefficient [em7']. The net absorption rate is the
loss coefficient, a@, multiplied by the photon flux. But the photon flux is just the

photon density, P(£2,), multiplied by the group velocity, v,. Thus, we have

a= — 2 (3.14)

or, substituting equation 3.12 into 3.14,

_ Br

—(fi — fa) (3.15)

By. can be calculated using Fermi’s Golden Rule for the transition from one state
to a group of final states, p;, under the influence of an electromagnetic field. For a

time-dependent harmonic perturbation, the transition rate is given by
27 T 2 ¢ 2
Bry = |< UAW > PPps(Esinat = Emitiat + hw) (3.16)

where H! = A.p is the interaction Hamiltonian causing the transition. For a plane
wave with a vector potential A, we can find By, in terms of the matrix element
|M|? =

. If the plane wave is polarized along z and propagating along y, the
vector potential A can be written as A\ = e.[Ape'(*9-#9 4+ Aje*Ay- #9],

The initial and final states are

Vig = di glty, ze”. (3.17)

Writing the matrix element between the initial and final states with the momentum

operator as

IM|? =| < dy\peloi > | (3.18)
we obtain
Bra = (FEEL IM Eppa (3.19)

3.3.2 K - Selection Rule

When calculating the absorption due to valence-conduction band transitions, we con-
sider only the case where the crystal momentum, k, of the electron in the initial state
is equal to that of the electron in the final state; that is k. = k, . This is true since
k must be conserved in the overall process, and because the momentum of the emit-
ted photon is much less than the momentum of the electron in the conduction and
valence bands. This puts a restriction on the density of final conduction band states,
ps, included in the transition rate in equation 3.19. To find the rate of absorption in
the semiconductor, we relate the density of allowed transitions at the photon energy
En to the density of states in the conduction and valence bands. The photon energy
is given in terms of the band gap energy, ,, the conduction band energy, E, and

valence band energy, F,,, as

Ey, = Ey + Bo + By. (3.20)

A change in E,,,, therefore, corresponds to a change in E, and E,

dEy, = dE. + dE,y. (3.21)

valence bands

hey dk,

cv Lew =
Peal Hew) = 7p

(3.22)

we find that the density of allowed transitions, or the reduced density of states, 1s

given by

a (3.23)

In calculating the density of states in the conduction and valence bands, a factor of
two is included to account for two spin states at each energy. However, since spin
must be conserved in the transition, the number of allowed transitions above must be

calculated using 1/2 the total density of states in the valence and conduction bands.

3.3.3 Parabolic Uncoupled Bands

For parabolic subbands with effective masses m?,, the energy of the electron in the

conducton band, F’,, and hole in the valence band, F,, is

(3.24)

where £,, is the energy of the n™ subband and ie is the kinetic energy. Thus

the density of states for each subband for a 2D system with parabolic bands is

Pow = 2 (3.25)
This allows us to write
where
! == : + . (3.27)

44
3.3.4 Gain Equations

Now that we have an expression for the density of transitions and for B,,, we can

write the equation for gain, y . Substituting equation 3.19 into the expression 3.15

for a and recalling that 7 = —a , we obtain
2meEv, mese® —_

Thus we have a general expression for gain due to stimulated transitions in a semi-
conductor, characterized by a conduction and valence band density of states, under
k-selection rules. In this expression, f, is the probability to find an electron in the
conduction band and f, is the probability of finding a hole in the valence band at
the transition energy E. We have assumed quasi-equilibrium in the bands; that. is,
the occupation in the valence and conduction bands can be described by quasi Fermi

energies E;,, Fy, and

(Bev -E c »)
ear 4]

The factor f.(#.) + f,(/,)—1 represents the population inversion between the energy
FE. in the conduction band and FE, in the valence band. Thus the dependence of the

gain on carrier density is implied through the fermi energies. For a given carrier

injection, the fermi energies can be calculated via

[o*)

n= | p(B.) fdk.. (3.30)
E.=0

The valence band fermi level. £;,, can be found similarly assuming charge neutrality

n = p. For a constant density of states, this integral can be performed exactly;

however, in general it must be solved numerically or by approximation methods.

By inspection of equation 3.28, we see that the low energy spectrum is dominated
by the shape of p,(/), whereas the high energy side is cut off by the Fermi functions.
Therefore, for a step-like density of states, the gain will level off to a peak value, yo,
for energies with an inversion factor near unity, and fall off exponentially at energies

above the Fermi levels. The gain is equal to zero for a photon energy
Ey, = Epo t+ Ep t+ Ey (3.31)

This is the transparency condition. For energies greater than Ei), the material is

absorbing, and for lower energies (above the band gap) the material is amplifying.

3.3.5 Transition Broadening

The expression for gain in equation 3.28 assumes perfectly defined transitions as
well as a perfect two dimensional density of states. The model thus predicts sharp
features in the gain spectra due to the onset of higher subband transitions in the step-
like reduced density of states. In actual gain spectra, the sharp structure is smoothed
out due to carrier collisions and quantum well roughness. To model the uncertainty in
subband energy, each transition is given a lineshape B(E, Eo). The contributions to
the gain at energy Eo will then include all other transitions weighted by the lineshape

function. Mathematically, the gain is expressed as a convolution integral

g(Ea) =f 9(E)BUE. By)aB (3.32)

For a collision broadened two-level system with a collision time 7), the broadening

function is given by a Lorentzian centered at E,

T»

In addition to smoothing out sharp features, the broadening also decreases the
peak gain. Although the energy tails in the Lorentzian have been shown to be too
wide to quantitatively explain certain features of the gain spectra [6], we will use it

to show some of the main trends in the spectral gain curves.

3.4 Calculated Gain Spectra

As we have seen in Section 2, the main effects of strain on laser performance are a
shift in the bandgap and a decreased valence band effective mass. This latter effect
can be incorporated into the parabolic band approximation by simply assuming a
decreased effective hole mass in the InGaAs. To illustrate the effect of strain on
laser performance, we will use equations 3.28 and 3.32 to calculate the gain for two
different. cases: the GaAs case and the case of InGaAs where the electron mass is
the same as GaAS but the hole mass is decreased. In Section 2.2 the valence band
effective mass was calculated to be 0.08 mo using the envelope function formalism.
This is fairly close to the GaAs electron mass of 0.067 mo which we will use here to

illustrate differences between GaAs and strained InGaAs gain.

AT
3.4.1 Carrier Injection Efficiency

The lower bandgap of InGaAs, compared with GaAs, in the quantum well will in-
crease the barrier height for both electrons and holes for a given Al,Ga,_,As barrier
material. As the height increases, fewer carriers will populate the Al,Ga,_,As for
a fixed carrier density in the quantum well [5,23]. Since the barrier region is much
thicker (dparrjer > 1000A) than the well, the number of ” wasted” carriers in the barrier

can be significant. We define the electron injection efficiency as

n € cps
Ne = ——te— (3.33)
Now + Nobarrier

for Nguws Nbarrier Clectrons in the well and barrier regions respectively. The efficiency

can be calculated by integrating the conduction band density of states given by

4 The m* ..
n=l ‘

for k, < V, and

J2me ..

Peau( Ee) = lea Ga VE — Ve (3.35)
for B. > Vz.

In the barrier material (F, > V.), the density of states is written as

J2 me?

Pebarrier( Ee) = Abarrier ZT E. _ V. (3.36)

The summation in p,,, is over all the bound states in the well. The valence band
density of states is similar with the summation including all bound light and heavy

hole states.

Electron injection efficiency

Hole injection efficiency

0.4

80 A GaAs

Carrier density [cm

0.8 -

0.6 -

0.4 -

0.2 -

0.4

0.3

0.2

80 A GaAs

Carrier density [cm 2

10'3

Figure 3.5: Calculated electron and hole injection efficiency for an S0AGaAs quantum

well as a function of total carrier density for various aluminum concentrations in the

barrier,

Electron injection efficiency

Hole injection efficiency

0.8

0.6

0.4

0.2

80 A Ing »Ga

o.g/s

10'?

Carrier density [cm

0.1

80 A In, ,Ga

0.2

0.0

0.8

Carrier density [cm

409'3

Figure 3.6: Calculated electron and hole injection efficiency for an S0AIng 2Gag gAs

quantum well as a function of total carrier density for various aluminum concentra-

tions in the barrier.

Figures 3.5 and 3.6 show the calculated efficiency of injected electrons and holes
for 80AGaAs and I no.2GaogAs quantum wells, respectively, with various barrier alu-
minum concentrations. For the same aluminum content in the barrier, the efficiency

can be as much as 5X larger for the InGaAs case.

3.4.2 Calculated Gain

The gain is calculated for the n=1 and n=2 transitions using a lorentzian convolution
with a relaxation time T, = 0.1 ps. Figure 3.7 shows calculated gain spectra at
two different injection levels, for equal electron and hole masses as well as for the
GaAs case where m? = 7m*. In both cases the peak gain shifts to higher energies
with higher pumping. At a certain level, the gain at the n = 2 level becomes larger
than the gain at the n = 1 state. For the same low level injection, the equal mass
case gives a larger peak gain. The maximum available gain from a given subband,
however, is larger in the nonequal case due to the larger reduced density of states.
This condition is reached at high current levels. Another interesting feature is that
the bandwidth of positive gain is significantly larger for the equal mass case which is
useful for tunability purposes.

The transparency current is also lowered by lowering the valence band effective
mass. Figure 3.8 shows the transparency current as a function of the ratio of valence
band effective mass to conduction band effective mass. F igure 3.9 shows the peak
gain vs. current in a simple model [12] for the two cases. The differential gain is

larger for low currents but levels off and saturates easily for the low hole mass.

GaAs QW gain, parabolic bands

100
80 - :
5 60 =.
3 7 7
20 ~ \ ~
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
E-Eg [eV]
InGaAs gain, parabolic bands
100 pete i | °
80 +
& 60;
& r
g 40°
20

E-Eg [eV]

Figure 3.7: Calculated gain spectra for a) m, = 0.45 (GaAs) b) m, = 0.07 for carrier

densities from 10!” to 10!9,

2e418

tests

Transparency density [cm-3]

leet?

Figure 3.8: Transparency carrier density vs. valence to conduction band mass ratio

R=Mv/Mc,

150 T is " T sa s Y T ¥ T v T T ¥ vr T

=~ 100
c=
7)
Oo
oO
oO
a 6650

0 400 800 1200 1600 2000
J [A/em?]

Figure 3.9: Calculated peak gain vs. current for various hole masses,

3.5 Threshold current equations

Once we have an expression for the gain in the material for a given current injection,
we can use the laser oscillation condition to find the threshold current. The conditon

for laser oscillation is that the gain equals the cavity losses, or

| l

7 (9% (3.37)

Pain = a; +

where

R= /R, Rp

a; is the internal loss, L is the cavity length, and Rj» are the mirror reflectivites. T

is the optical confinement factor and can be calculated for a given structure by

a 2
7s, [Bola a
and L, is the active region width. For very thin active regions, which is certainly

the case for a single quantum well, the optical field is approximately constant in the

overlap integral and thus the confinement is given by

T=
Wir ode

where Wiode 18 the width of the optical mode.
The internal losses include the optical losses due to waveguide imperfections, free
carrier absorption, and inter-valence band absorption all grouped together in a dis-

tributed loss constant. The loss term can be measured, together with the internal

quantum efficiency n;, by measuring the external quantum efficiency for various cavity

lengths
1 Ll al
— = —(—— +1 3.39)
Nd ni logh (

3.5.1 Threshold Current

A simple, but useful, expression for the threshold current can be obtained by lineariz-

ing the gain about the transparency carrier density. Let

where g;;, is the threshold gain, N is the carrier density, Nj, is the transparency carrier
density, and g’ = oo isa constant. To translate the carrier density to a current density,

it is customary to introduce a radiative recombination constant T :

NA at)

rT ‘em?

This gives an explicit expression for the threshold current

eW d l .
where
Im = Tg!

and d is the active region thickness, and W the width of the device.
We have an expression for the threshold current in terms of measurable parame-

ters. We use this equation to experimentally compare laser structures, and to optimize

cavity lengths and mirror reflectivity for low threshold. The above expression will be
used in the next chapter to analyze our strained layer lasers. It is important to keep
in mind that we have linearized the gain to obtain this result. This is especially
important in quantum well lasers as the gain becomes sublinear at reasonable carrier

levels.

3.6 Conclusions

The heavy hole density of states in GaAs quantum wells is largely due to interactions
with light hole bands. This valence band mixing is reduced in InGaAs quantum
wells due to the strain induced splitting of the heavy and light hole energy levels.
The lowest order transition, el-hl, will thus retain a heavy hole character even as
the wavevector k increases from k=0. More importantly, however, the valence band
heavy hole mass is dramatically reduced from an average of 0.4577 to 0.087n9 close
to that of an electron in the conduction band. This in turn should lead to a lower
transparency current, a larger gain bandwidth, and lower threshold current in strained

layer quantum well lasers.

References

E. Yablonovich and E.O. Kane, J. Lightwave Tech, LT-4, 504 (1986)
A.R. Adams, Electron. Lett. 22, 249 (1986)

I. Suemune, L.A. Coldren, M. Yamanishi, and Y. Kan, Appl. Phys. Lett. 53,

1378 (1988)

Landolt-Bornstein, Numerical Data and Function Relationships in Sctence and

Technology 17 a-b, Springer, New York (1982)

R.M. Kolbas, N.G. Anderson, W.D. Laidig, Y. Sin, Y.C. Lo, K.Y. Hsieh, and

YJ. Yang, JERE J. Quant. Electr. 24, 1605 (1988)

C. Van de Walle, Mat. Res. Soc. Symp., Boston (1987)
J.M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955)
J.M. Luttinger, Phys. Rev. B 102, 1030 (1956)

E.0O. Kane, J. Phys. Chem. Solids. 1, 249 (1957)

G.E. Pikus and G.L. Bir, Sov. Phys. Solid State 1, 1502 (1960)

(11] D.A. Broido and L.J. Sham, Phys. Rev. B, 31, 888 (1985)

(12) A. Twardowski and C. Herman, Phys. Rev. B 35, 8144 (1987)
[13] E. P. O’Reilly, Semicond. Sci. Technol. 4, 127 (1989)

[14] D. Ahn and S. Chuang, [FEE J. Quant. Electr. , 24, 2400 (1988)

[15] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, Hal-

stead Press (1988)

[16] M. Altarelli, Heterojunctions and Semiconductor Superlattices, ed. G. Allen, G.

Bastard, N. Boccara, M. Lanoo, and M. Voos (Springer, Berlin), 12 (1986)

117] E.D. Jones, H. Ackermann, J.E. Shirber, T.J. Drummond, L.R. Dawson, and

T.J. Fritz, Solid St. Comm. 55, 525 (1985)
[18] A.Yariv, Optical Electronics 4" edition, Saunders Publishing (1991)
{19} H. Casey and M.B. Panish, Heterostructure Lasers, Academic Press (1978)

[20] R.H. Yan, S. Corzine, L. Coldren, and I. Suemune, JEEF J. Quant. Electr. 26,

213 (1991)

[21] M. Asada, A. Kameyama, and Y. Suematsu, IEEE J. Quant. Electr. 20, 745

(1984)
[22] J. Nagle, M. Hersee, T. Weil, and C. Weisbuch, Appl. Phys. Lett. 49, 719 (1985)

[23] H. Hirayama, Y. Miyake, and M. Asada, JERE J. Quant. Electr. 28, 68 (1992)

[24] S.R. Chinn, P. Zory, and A.R. Reisinger, JEEE J. Quant. Electr. 24, 2191 (1988)

[25] K. Lau, S. Xin, W. Wang, N. Bar-Chaim, and M. Mittelstein, Appl. Phys. Lett.

55, 1173 (1989)

[26] J.B. Schirber, IJ. Fritz, and L.R. Dawson, Appl. Phys. Lett. 46, 187 (1985)

Chapter 4

InGaAs Quantum Well Laser

Performance

4.1 Introduction

This chapter will describe the structure, fabrication, and possible future improvements
of broad area and buried heterostructure InGaAs lasers [11,2].

It is interesting to investigate InGaAs lasers for several reasons. First, the small
band gap allows lasing at longer wavelengths than GaAs; reliable operation has been
shown out to l.lum [6]. It is of particular interest to the communications industry
to have high power sources at 980 nm to efficiently pump Er* doped fiber amplifiers.
Second, as we have seen in the previous chapter, InGaAs exhibits a larger gain than
GaAs, yielding lower transparency currents and a potential for lower threshold cur-

rents. Third, the ability to fabricate deeper wells in InGaAs allows less leakage of the

carriers to the confining layers. Finally, the narrower band gap of InGaAs allows us to
use a narrower band gap confining region. This leads to a larger available refractive

index step, and hence larger confinement of the optical mode for a larger modal gain.

4.2 Structure and Growth

The laser consists of an InGaAs quantum well active region surrounded by higher band
gap, and therefore lower refractive index, GaAs and AlGaAs. The larger band gap
confines the carriers to the quantum well, and the lower index provides a waveguide
for the optical mode.

The epitaxial layers as shown in figure 4.1 consist of the following: a 1 jum
silicon doped n-GaAs buffer layer, a 1.5 um n-Alo.sGao.3As cladding layer, an 0.2 pom
n-Al,Ga,_,As graded confinement region with z ramped parabolically from 0.5 to
0.2, a 4 nm GaAs spacer, and the 5 nm InyGa,_,As quantum well. The beryllium
p-doped upper half of the structure is the mirror image of the n - side except for
the p* contact layer. This layer was doped as high as possible, 2 x10 '? cm7*, to
minimize the contact resistance. The Al,Ga,_,As regions provide transverse optical
mode confinement as well as electronic barriers for efficient current collection. The
substrate was (100) GaAs tilted 4° towards (111) A. This choice was made since we
have found growth conditions to be more tolerant on this substrate type [4]. Apart
from the active quantum well region, the device structure is identical to optimized

GaAs quantum well lasers [7] grown in our laboratory. The substrate temperature was

1.5um p-GaosAl, / 0.2um graded undoped

Gay gAly oAS — Gay sAly sAS
-{ 0 GaAs

50A In,Ga,_As
40A GaAs

“\ 5 Nt .2um graded n-

Gao Al) sAS>Gay cAly oAs
NY SUM N-Gap sAly AS

<— 0.5um n*-GaAs Buffer

«<— n*-GadAs substrate

Litiit.» Al content in Ga, AlAs
00 05

Figure 4.1: The epitaxial layer structure in the InGaAs quantum well lasers.

held at 600°C for the GaAs buffer and cap layers, 720°C in the cladding and graded
Al,,Gay_,As regions, and ramped down to about 620°C for the InGaAs quantum well,
where the temperatures quoted are pyrometer readings. To minimize incorporating
impurities into the material, the growth is continuous throughout with no growth
interruption; that is, the substrate temperature is ramped down, and then up again
during the GaAs spacer growth. To quickly ramp the substrate temperature, the
substrate was mounted to a direct heating holder [8]. It should be noted that the
structure uses a parabolically graded confinement region. The reason for this is
actually more historical than for device optimization. It has been shown theoretically
[9.5] that a linearly graded profile is a better current confining structure over a wide

range of threshold gains.

4.3. Broad Area Single Quantum Well Devices

After the growth, to get an estimate of the indium content and quantum well width,
room temperature optical absorption of the material was measured. The absorption
edge was at 1050 nm, which correponds to a 50 - 60 Angstrom well width containing
35% indium. To test the MBE material quality, we fabricate broad area lasers with
100m wide stripes. This consists of evaporating Cr/Au stripes on top of the material,
thinning the wafer to 125 ym, depositing AuGe/Au on the back side, and annealing
at 360 °C’ for 30 seconds. Lasers of various lengths are then cleaved and light vs.

current is measured. We use 200 ns current pulses at 10 kHz repetition rate to avoid

heating effects. The lasing wavelength of long devices is 990 nm. Threshold current
density vs. 1/L for this material is shown in 4.2. From these measurements we see
that for cavity lengths greater than 400m, or(Ji, < 300—4;),the threshold current is
linear with 1/L. For these lasers we find a good fit to the data to be

A, 10[4]
em?" Llem|-

(4.1)

The lowest threshold current density measured was Jj, = 114-4, for a 1540um long
cavity. These threshold current densities were the lowest results published, nearly
50% less than the previous best material grown by MOCVD, a factor of 5 lower than
the best MBE material, and only slightly higher than the best GaAs quantum well
laser material [2]. The internal loss of this structure was found to be 9 cm7! with an
internal quantum efficiency n; = 0.6. Using this, we estimate a transparency current
density in the material of J,, = 20-25-45. Recent progress has been made by Choi et
al. [5] using OMVPE, and Williams [6] using MBE, both using a structure with higher
optical mode confinement, they have been able to obtain threshold current densities in
InGaAs below that of the best GaAs material. They measure a transparency density

of 25 ~4;, similar to our result. Transparency densities of the best GaAs lasers {7]

emé*

are found to be nearly twice this value at JG*4* = 50

4.4 Multiple Quantum Well Lasers

If the number of quantum wells is increased to N, we can still use the linearized

threshold equations from chapter 2. Now, however, the confinement factor, I’, is N

400

300 fF
Rd
“i 200 F

100 F

y = 31.180 + 12.504x R%2 = 0.995

1/L [cm-1]

Figure 4.2: Threshold current density vs. 1/L for broad area InGaAs SWQ lasers.

times larger, and the transparency current is increased by the same factor. Also, for
the same current injection, the carrier density in each well is lower by a factor N.
resulting in lower quasi fermi energies and less population of confining region states.
Neglecting the population of confining region states, the modal gain for the laser with
N wells, compared with the single well, at a given current injection can be written as

[16]

IS y PlR)+ AG) =
ght f.(n) + film) 1

For devices with high threshold gain requirements, the threshold current may be lower
for a MQW than a SQW laser.

A double quantum well laser with the same confining structure as the SQW men-
tioned in the previous section was grown, except that as an active region we used
two 5 nm quantum wells spaced by 8 nm. Figure 4.3 shows Jj, vs. 1/L for the
DQW broad area laser superimposed on the SQW laser data. The transparency cur-
rent is about 50% higher; however, the slope ay is less, so for short cavity lasers
the threshold current will be less. Because the gain requirement for high frequency
passive mode-locking experiments are high, multiple quantum well material is pre-
ferred. From this DQW material we have made two section lasers producing passively

mode-locked pulses at 120 GHz repetition rate [1].

Jth [A/cm?]

800 / i | ; j T ] y i T T 7 T
700 DQW

600 SQW
500
400
300
200
100

tipitrrtibriritiriitiirrlirrir terri tira

x 0

PYYTPDPPPPErr rp rrrrprrrryrrrryprrr ry rrr

i | | i

11.25 22.9 33.75 45
1/L [cm-1]

Figure 4.3: Threshold current density vs. 1/L for broad area InGaAs DQW lasers

shown together with SQW data,

68
4.5 Low Threshold Buried Heterostructure Lasers

To achieve ultra-low threshold current devices, we need to minimize the active region
volume while maintaining the integrity of the material. The structure must provide
index guiding of the optical mode in the lateral direction, as well as confine the current
to the quantum well. The buried heterostructure shown in Figure 4.4 is the best
structure to date in accomplishing this goal. The fabrication involves surrounding
the active region with higher bandgap, and hence lower refractive index, material
epitaxially regrown by Liquid Phase Epitaxy (LPE). The fabrication is as follows:
First, mesas with active regions of 2 zm are chemically etched down to the substrate.
Then, the stripes are oriented along the (110) direction and the etched cross section
has a “dove tail” shape. This cross section maximizes the contact area for a given
active region width. Immediately prior to loading into a Liquid Phase Epitaxy (LPE)
system for regrowth, the GaAs cap layer is removed to avoid growing on top of the
ridge. The regrowth is done in two steps. First p — AlpsGao5As to the junction
plane is grown, and then n — Alp sGao,5As to the top is grown, both at 800 °C’. This
provides a reverse P-N junction in the AlGaAs when the diode is forward biased which
blocks leakage current in these layers. Later, a contact region must be formed with a
Zn diffusion in the top layer. After regrowth, the devices are cleaved and mounted,
junction side up, on a copper heatsink. The lasing wavelength of the finished lasers
was 950 nm, a shift of 40 nm from the unprocessed devices. We believe that the high

temperature in the LPE furnace causes an interdiffusion of the InGaAs and the GaAs

GRIN SCH SQW

structure

LPE regrowth

; p-AlGaAs,
n-AlGaAs | | n-AlGaAs
‘n-AlGaAs
p-AlGaAs \(——\__ P-AlGaAs

~--7 n*-GaAs

n+-GaAs substrate

Figure 4.4: Schematic of buried heterostructure laser.

spacers. Similar observations have been reported elsewhere{12]. As with the broad
area devices, we characterize the lasers with threshold current density vs. 1/L. The
results are shown in Figure 4.5. Comparing this data with the broad area devices in
figure 4.2, we see that threshold current densities in the regrown structure is still very
low, and has increased by around 50% compared with the broad area lasers. This
is a small price to pay for the 50X reduction in active region volume. A minimum
Jin, = 167-4, for a 16764m long cavity is compared with J,, = 114, for a 1540
ym long broad area laser. Again we measure the differential quantum efficiencies

to extract the internal loss constant and find a; = 3.lem7}.

Using the linearized
threshold current equations from chapter 2 and data from our devices, we find that

our BH devices obey

A A ol 1
J}, = 80— B———(—)] 4.3
and thus
A A 1
= —— + 5.8 Wl 4.4
fen OWL re em? 08( FR,” (44)

where W is the active region width. The linear gain regime for these lasers is seen to

be for cavity lengths of

5.8log( aR)
400

The lowest threshold current for these lasers is 1.65 mA for a 338 ym long device,
which is still the lowest threshold current to date for these structures. The cavity

length at which the threshold reaches a minimum is longer than for similar GaAs lasers

(Lope = 200p0m) . This observation is consistent with a lower available maximum gain

1500

Threshold current densityJ,, (A/cm?)

i j t i it i L i be
24 6 8 10 20 30 40 50
Inverse cavity length 1/L (em7’)

Figure 4.5: Measured threshold current density vs. 1/L for buried heterostruture

InGaAs SQW lasers.

37 Ri=R2=.3
24

1 th (mA)

Ri«R2=.7

17
ae

0 100 200 300 400 500

Cavity length (um)

Figure 4.6: Predicted threshold current for BH lasers vs. cavity length.

in InGaAs material.

To further minimize the threshold current, we apply high reflectivity coatings
to the mirrors in order to lower the mirror loss term. Plotting equation 4.4 as a
function of cavity length and for various mirror reflectivities, we can obtain values

of I, we expect for our devices. This plot is shown in figure 4.6. We predict a

0.3
R= 0.3
alpha = 3.1 cm-1
wet Transparency [mA]
——e-—— Internal loss [mA] q
= 02+ wmmafeeeeeeee Mirror loss [mA]
cs)
Kove
oe
0.1
0.0 mn J L i 4 L i i rs L rs
0 50 100 150 200 250 300

Figure 4.7: Estimated current components at threshold vs. cavity length for R = 0.3

minimum threshold current of Itamin = 0.4mA for a cavity length of 100um and mirror
reflectivities of Ry = Ry = 0.9. For the given values of internal loss, transparency
current, and differential gain, it is interesting to examine the various components of
I, as a function of cavity length. Figure 4.7 shows the /,, components of an uncoated
laser. It is clear that the mirror loss term is the dominant loss for all cavity lengths
of interest. Therefore, it does not matter how much the transparency or internal loss
is reduced, the threshold current will be virtually unchanged. Note, however, that the
mirror loss component can be reduced by an increased differential gain in addition
to increasing the mirror reflectivity. The /,, components of the same laser with 90%

reflectivity coatings are shown in figure 4.7. The mirror loss is of the same order as

0.5
R=09
+ 6alpha = 3.1 cm-1
———tk—— == Transparency [mA]
0.4 - ——e— internal loss [mA]
———t-—— Mirror loss [mA] >
— 0.3 fF
jn
be
é) 0.2
0.0 4 i rn 1 L i n i "
0 100 200 300 400 500

Figure 4.8: Current components at threshold vs. cavity length for R = 0.9.

the other loss components, and is actually less than the other losses for cavity lengths
greater than 300 um.

Figure 4.9 shows the measured L-I curves for a 200m long device without mirror
coatings, with one side coated, and with both sides coated to 0.9. For the high reflec-
tivity structure of this cavity length we predict a threshold of 0.5 mA. Experimentally
we have obtained a minimum threshold current of [7/'” = 0.75mA This threshold is
comparable to the best devices in GaAs[{13,12] and is the lowest value reported for
strained layer single quantum well material. Recently [15], we have improved this
result to 17"'" = 0.35mA which is record low for any single quantum well laser in any

material system.

Optical Power (mW)

5k
4 R,=0.3

0 | \ {

0 1 2 3 4 5 6 7 8

Injection Current (mA)

Figure 4.9: Light output vs current input for 200 um long laser,

4.6 Improvements in Performance

As stated earlier in this chapter, the device structure grown was identical to our GaAs
structure, and is by no means optimized for the InGaAs quantum well. Due to the
lower bandgap of InGaAs, the AlGaAs barriers in the laser structure can have the
aluminum mole fraction ramped from 0 (pure GaAs) to 70% in the cladding while still
providing the same current barrier height as in the GaAs quantum well case. To see
the improvement in the optical confinement we can calculate the optical confinement
factor by approximating the parabolic waveguide by a three layer waveguide of indices
n; and ng as in figure 4.10. It has been shown [17,6] that the shape of the index
waveguide does not significantly alter the magnitude of the confinement factor, but
affects the optimum width. Thus, the step index waveguide is a good approximation
for determining possible enhancements in the confinement factor.

For a TE mode polarized along y, propagating in the z direction with a propaga-
tion constant 8, we can write the lowest order transverse mode within the high index

region as

Ey = Egcos(nx el?) (4.6)
and in the low index region as
Dy = Ege tet), (4.7)

Using the boundary conditions at |r| = d/2, we find that the propagation constant 3

nex)

a od ~~ ae med nee, _ i —
al — — _—- — — — ee

lad

Figure 4.10: Three layer waveguide used to approximate the optical confinement in

GRINSCH lasers.

is related to & via the transcendental equation

Kd. kd yd

with y defined by
k= ((nky)® — 8?)?-y = (8? = (miko)?)?, (4.9)

where ko = a and Ag is the free space wavelength. For the three layer waveguide,
the overlap of the optical power with the active quantum well region is then given by

the analytic expression

_ t + 4sin(kt)
dt 2sin(Kd) + 2cos*(“t)’

(4.10)

where d is the waveguide width and t is the quantum well width. A plot of I vs.
waveguide width d is shown in figure 4.11 for the two waveguides: the already
grown structure, consisting of index steps corresponding to aluminum molefractions
of 0.2/0.5, and the improved structure of 0.1/0.7. We see that by increasing the index

step, we can, for an optimized waveguide width, increase the confinement factor by

40%. If we Jet the relative confinement factor T, equal Fimproved | the effect of the

Pactual
improved confinement factor can be illustrated by rewriting the threshold equation

4.4 in terms of [,.

oT, 8 RR,

ORC

Ty, = W{L( Jo + )+ ). (4.11)

Notice that the transparency, Jo, is unaffected by T,. A plot of ;, vs. T,. for uncoated

and coated (R = 0.9) mirrors is shown in figure 4.12. The projected decrease in

f rs ———— gamma (.5/.2)
F196 % rete gamma (.7/.1)

0.0 0.2 0.4 0.6 0.8 1.0
d [um]

Figure 4.11: Confinement factor vs. waveguide width for the three layer waveguide.

threshold is much less for the coated lasers since the mirror loss term, where the main
contribution to the threshold reduction comes, is already small. Thresholds will be
reduced by 30% for an uncoated laser and 15% for the laser with high reflectivity

coatings.

4.7 Conclusions

In conclusion, we have demonstrated the first sub milliampere threshold current in
buried heterostructure InGaAs lasers. Threshold current densities in broad area
(we = 100pm) lasers of Jy, = 114A/em? for cavity lengths of L = 1540um . Trans-
parency currents are approximately a factor of two lower than for GaAs lasers in
agreement with calculations. Buried heterostructure devices lase at a threshold of
Iy, = 1.0mA with as-cleaved facets, and [,, = 0.35mA with high reflectivity mirror
coatings (R = 0.9), which is record low for a single quantum well laser in any material
system. Further improvements in the confinement structure are proposed to take full

advantage of the InGaAs quantum wells.

we=2um
L = 200 um

Ith [mA]

Ratio, Up

Figure 4.12: Projected improvement of threshold current with increased optical con-

finement.

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Morkog, J. Vac. Sei. Technol. B3, 536 (1985)

83
[9] H. Hirayama, Y. Miyake, and M. Asada, JEEE J. Quant. Electr. 28, 68 (1992)
[10] J. Nagle, M. Hersee, T. Weil, and C. Weisbuch, Appl. Phys. Lett. 49, 719 (1985)
[11] S. Sanders, L.E. Eng, and A. Yariv, Electron. Lett. 26, 1087 (1990)

[12] R. M. Kolbas, Y.J. Yang, and K.Y. Hsieh, Superlattices and Microstructures 4,

603 (1988)

[13] P. Derry, A. Yariv, K. Lau, N. Bar-Chaim, K. Lee, and J. Rosenberg, Appl. Phys.

Lett. 50 1773 (1987)
[14] K. Lau, P. Derry, and A. Yariv, Appl. Phys. Lett. 52, 88 (1988)
[15] To be published
[16] A. Yariv, Quantum Electronics, 3" edition, Wiley (1989)
[17] W. Streiffer, R.D. Burnham, and D.R. Scifres, Opt. Lett. 8, 283 (1983)

[18] S. Chinn, Appl. Opt. 23, 3508 (1984)

Chapter 5

Broadband Tuning of InGaAs

Quantum Well Lasers

5.1 Introduction

Tunable compact semiconductor laser sources are an attractive alternative to solid
state and dye laser systems pumped with high power gas lasers. Semiconductors can
exhibit broadband gain and can emit radiation over a wide spectrum under proper
operating conditions. Quantum well lasers, in particular single quantum well lasers,
can take better advantage of this bandwidth than bulk double heterostructure lasers
due to their small active region volume and, therefore, larger band filling at the same
current density. This large bandwidth can be useful in creating short mode locked
pulses [1], and for applications where broadband tunability is desired [2]. The use of

InGaAs for the active region of the laser permits the lasing to be extended beyond

lym which includes the 980 nm wavelength of current interest for pumping Er*
doped fiber amplifiers. In this chapter, we report on the broadband tuning properties
of strained InGaAs quantum well lasers and compare them with GaAs quantum well
lasers optimized for this purpose. Our InGaAs lasers have been tuned over 170 nm,
a much wider range than in previous reports [3], and we observe a difference in the

tuning characteristics compared with GaAs lasers.

5.2 Quantum Wells for Tuning

An optimal laser for tunability has a perfectly flat gain spectrum so that a wavelength
dependent loss mechanism can select the desired lasing wavelength without a large
increase in bias current. As discussed in chapter 2, single quantum well lasers, with
parabolic conduction and valence bands, have a step-like reduced density of states vs.
energy profile. The width of the first flat step is given by the separation of the el-hl
(n=1) and e2-h2 (n=2) transition energies in the quantum well, and it is the width
of this step which we would like to maximize for broadband tuning. Although it isn’t
necessary to limit the lasing to the first subband, the resulting threshold currents will
be much lower due to the lower number of states to fill to reach inversion. A narrow,
deep quantum well is desirable to separate the two subbands as far as possible. The
limit is reached when the second state is pushed out of the well and into the confining
region, which drastically increases the threshold current. So, the tuning is limited on

the short energy side by the bandgap of the well material, and on the high energy side

by the bandgap of the confining material. Once optimized, the laser is then operated
under conditions where most of the states between these levels are populated with
carriers, and lasing can then occur at any of these energies given the proper feedback.
The operating condition for maximum tuning range in GaAs quantum wells has been
found to be a threshold gain such that the laser just operates in the second quantized
state [4,5] before adding any feedback. Calculated [4] gain vs. energy curves [4] for a
single GaAs quantum well laser with parabolic energy bands is shown in Figure 5.1.
The laser cavity is cleaved short enough to lase in the n=2 transition and without
feedback lases at Ag. With feedback the lasing occurs at A; with a threshold current
of J,. The resulting threshold current vs. tuned wavelength is illustrated below in

the same figure.

5.3 Oxide Stripe Lasers and Tuning Results

The optimized GaAs lasers have a 60.4 quantum well with Al,Ga,_,As graded barri-
ers, where x is graded parabolically from 0.4 to 0.7. The cavity length is cleaved just
short enough to lase free running (without feedback) in the n = 2 state. A diffraction
grating is used to provide feedback in an external cavity configuration as shown in
Figure 5.2. The grating feeds back one wavelength, and therefore raises the effective
facet reflectivity there. The lowered loss fixes this lasing wavength and this selective
feedback provides the mechanism for tuning. The wavelength range over which we

can tune the GaAs laser is from 730 to 855 nm, a record 125 nm.

T ieee T T “t T ui a T T om 9 T v me 3 T T t ce ee ames | rr ¥ ’
100+ md
r 4
q Ao
BO }-
~ +
— 60
(53
Oo
40
20
1.4 1.5 1.6
i Photon Energy (eV)
1500 :
fod ' Y
Ow 1250 free-running +
Le
23 0
oO — 1000
SB >
L = n=2
Ge
® @ 750 n=l
— O
om
500 . ‘ .
B60 840 B20 800 780 760

Tuned Wavelength (nm)

Figure 5.1: Above: Calculated gain vs. photon energy for a GaAs quantum well laser
assuming parabolic bands for various pumping levels (from Reference [2]). Below:

The resulting threshold current vs. grating tuned wavelength.

To Diagnostics

Output A
ro ro +AXr 4
R R i ran
1 2
GaAs/GaAlAs 1.0 BS 750 nm blaze
SQW Laser 8mm R=B% 600mm!

Figure 5.2: Configuration for wavelength tuning in an external cavity.

To extend the tuning range to shorter wavelengths, it is necessary to confine the
carriers with a higher barrier than is available in Al,Ga,_,As. To access longer wave-
lengths, however, an active region of lower band gap material must be used. With
a lower bandgap material, the relative barrier heights are now larger, permitting the
use of a narrower well. This increases the energy subband separation and hence, the
absolute tuning range. /n,Ga,~,As has a lower band gap than GaAs and, when
used as the active region in quantum well lasers, has been shown to emit at wave-
lengths as long as 1.lum [6]. Although the In,Ga,_,As is not lattice-matched to
the GaAs substrate, high quality devices, i.e., devices with low threshold currents
[11] and high quantum efficiencies [8], can still be fabricated since the thin wells in
these structures are less than the critical layer thickness [9] for this material system.
Figure 5.3 is a calculation of the two lowest. transitions in a 50A strained quantum
well with Alo ,Gap¢As barriers as a function of indium content. The difference in
these energies is a good indication of the possible tuning range of strained InGaAs
quantum well lasers. We have examined the tunability of an In,Ga,_,As quantum
well structure bounded by a GaAs spacer and an Al,Ga,_,As (x=0.2 - 0.5) graded
region for confinement of the optical mode.

The 50A In4sGagsAs single quantum well strucure was grown on a GaAs sub-
strate by molecular beam epitaxy (MBE) and has a 40A GaAs spacer layer on each
side. Details of the growth conditions have been given in a previous chapter. From
this low threshold current material, 10jm wide stripe ridge waveguide lasers were fab-

ricated and tested. We cleaved lasers of various lengths and measured the emission

1.1 T l T T T

0.9

bua itetliriar tua

0.8

e2-hh2

| Ll | | ! 7

i ae De ee a We a

0.7

0.6
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Figure 5.3: Calculated el-h2 and e2-h2 band edge transition wavelengths for a

strained InGaAs quantum well with Alp.4Ga0.6As barriers

990 4
9704 *
~ 2
E 950
ij
Peal
x=
3s 930
Fa *
9104
890 5 » *
870 . r r .
9 100 200 300 400 500 600 700

Cavity length (um)

Figure 5.4: Measured lasing wavelength vs. cleaved cavity length for InGaAs 10pm

wide ridge waveguide lasers used for tuning.

wavelength versus cavity length, ZL, which is shown in Figure 5.4. A sharp transition
to shorter wavelengths is apparent for 1 < 150m, which we attribute to the n = 2
lasing state [4]. For the tuning experiments, the lasers were uncoated, and operated
pulsed at low duty cycle and placed in the external cavity with the grating. Results
from the tuning measurements are presented in figure 5.5, where we have plotted
threshold current as a function of tuned wavelength.

As a comparison, we also show the tuning characteristic of an optimized GaAs
quantum well laser. The InGaAs laser oscillates over 170nmm, which is to be compared

with 125nm for the best GaAs laser. To our knowledge these are the widest published

In (MA)

300
r 1
200 + 4
e 4
ry
wor ¢ 4
0 " i i re i i
720 780 840 900 960 1020
A (nm)

Figure 5.5: InGaAs and GaAs broadband tuning results.

tuning ranges for these material systems. The threshold currents are comparable for
the two lasers and are low enough to permit CW operation. Also, in agreement
with other work [3] we see that we are able to tune into GaAs wavelengths without

problems.

5.4 Comparison of InGaAs and GaAs Tuning

Our measurements indicate, however, that the tuning characteristics of InGaAs lasers
are qualitatively different from the GaAs lasers. For GaAs lasers we have seen, from
calculations of gain spectra and previous tuning measurements, that the spectral
gain saturates at high current injection, and is relatively flat just at the onset of
second quantized state lasing, in agreement with the simple theory [4]. When the
gain requirement of the resonator is set too high, an abrupt reduction in tuning range
is experienced since the n=1 transitions are no longer accessible [5]. However, for
InGaAs lasers, we observe a gradual loss of tuning on the long wavelength side upon
increasing the gain requirement. We have measured the tunability of InGaAs lasers
of various cavity lengths, shown in figure 5.6, and found that as we shorten the
cavity length the tuning range on the long wavelength side decreased. This indicates
that the maximum available gain from the n = 1 transitions in the strained lasers is
not as flat as for unstrained GaAs quantum well lasers. The results suggest that the
maximum gain from n = | transitions is significantly lower at the longer wavelengths,

and the feedback from the grating is not sufficient to overcome the losses.

Irn (MA)

100 F

0 m7 4
820 860 900 940 980
A (nm)

Figure 5.6: InGaAs tuning for various cavity lengths,

1020

Our experimental results, in particular the difference in tuning characteristics of
InGaAs and GaAs, can be explained by considering a simple model for the maximum
achievable gain. When operating at high current densities, the gain at a given wave-
length reaches a maximum value and, to a first approximation, follows the spectral
distribution of the density of interband transitions. From energy band calculations
by Suemune et al. [14], the valence band density of states (D.O.S.) is an increasing
function of hole energy in strained InGaAs quantum wells, and can increase by as

much as a factor of 3 in the first quantized state. The reduced D.O.S. is given by

Pv
= 5.1
par (5.1)

2 |F

and thus we see that the maximum gain is increased by a factor of 1.5 when oa

increases from a value near unity at the first sub band edge compared with its value
near 3 just below the second sub band edge. If we scale the gain constant of GaAs
with effective mass, we see that the absolute gain difference is on the order of 35em™?,
which can explain our observations. A more detailed calculation of the modal gain
constant vs. wavelength is shown in Figure 4. Here we have used the data published
by Suemune et al. [14] for the valence band D.O.S., assumed a constant electron
effective mass, and a matrix element equal to that of GaAs. These calculations

' no matter how strong

suggest that the modal gain at 1000 nm is limited to 50 cm7
the pumping, while the maximum available gain at 900 nm is more than 1.5 times

larger. Consequently, the gradual loss in tuning at long wavelengths can be due to

the nonparabolicity in the valence band.

100

y(cm™)

870 920 970 1020
A (nm)

Figure 5.7: Calculated maximum gain using the strained valence band density of

states given in Reference [{14]

5.5 Conclusions

In conclusion we have used a grating in an external cavity to tuned InGaAs quantum
well lasers over 170 nm, more than 40 nm greater than previous reports. Together
with an optimized GaAs laser operated in the same configuration, we see that the
entire region between 740 nm and 1010 nm can be spanned. In the course of these
measurements, we have detected a different behavior between the strained and un-
strained lasers, corresponding to a different shape of spectral gain curves for the two
types of lasers. A possible explanation is an energy dependent valence band effective

mass in the strained laser.

References

[1] 5. Sanders, L.E. Eng, J. Paslaski, and A. Yariv, Appl. Phys. Lett. 56, 310 (1990)

[2] M. Mittelstein, D. Mehuys, A. Yariv, J. Ungar, and R. Sarfaty, Appl. Phys. Lett.

54, 1092 (1989)

[3] D.C. Hall, J.S. Major, N. Holonyak, P. Gavrilovic, K. Meehan, W. Stutius, and

J.E. Williams, Appl. Phys. Lett. 55, 752 (1989)

[4] M. Mittelstein, Y. Arakawa, A. Larsson, and A. Yariv, Appl. Phys. Lett. 49,

1689 (1986)

[5] D. Mehuys, M. Mittelstein, A. Yariv, R. Sarfaty, and J.E Ungar, Electron. Lett.

25, 143 (1989)

[6] R.G. Waters, P.K. York, K. Beernik, and J.J. Coleman J. Appl. Phys. 67, 1132

(1990)

[7] L.E. Eng , T.R. Chen, S. Sanders, Y.H. Zhuang, B. Zhao, H. Morkoc, and A.

Yariv, Appl. Phys. Lett. 55, 1378 (1989)

[8] A. Larsson, J. Cody, and R. Lang, Appl. Phys. Lett. 55, 2268 (1989)

[9] LJ. Fritz, $.T. Picraux, L.R. Dawson, T.J. Drummond, W.D. Laidig, and N.G.

Andersson, Appl. Phys. Lett. 46, 967 (1985)

[10] I. Suemune, L.A. Coldren, M. Yamanishi, and Y. Kan, Appl. Phys. Lett 53, 1378

(1988)

100

Chapter 6

Microampere Threshold Current
Operation of GaAs and Strained

InGaAs Quantum Well Lasers at

Low Temperatures

6.1 Introduction

Although the effect of temperature on the operation of semiconductor lasers has been
studied extensively, most papers are concerned with temperatures near and above
300° K [1]-[8]. In these cases, a laser with a low sensitivity to temperature variation is
desirable for stability purposes. In some systems, for example antennas used in space

applications [9], electronic devices and circuitry may be at cryogenic temperatures.

101

For the potential use of optical interconnects in this circuitry, it is desirable to know
the laser properties at these temperatures. Here, a large sensitivity to temperature
is an advantage since the lowest achievable 1, will be lower.

Low temperature operation, in particular down to temperatures as low as 5° Kk,
has to our knowledge previously not been addressed. It has been shown that for bulk
active regions, the threshold current increases exponentially with increasing temper-
ature, whereas for quantum well lasers this relation only holds in limited regions. In
either case, we expect a significant reduction in the threshold current when reducing
the temperature to 5° kK. In this chapter, we concentrate on the temperature depen-
dence of threshold current and lasing wavelength in regions at and well below room
temperature. We use a simple model to explain the observed differences in threshold
behavior between GaAs and InGaAs lasers. Specifically, expressions for the trans-
parency and threshold carrier densities as a function of temperature are derived for
a single quantum well laser with carriers limited to one quantized state. At low tem-
peratures, the higher subbands become less populated and these simple expressions
will become more valid.

Early reports on the lasing threshold current density, J;;,, as a function of tem-

perature, found an exponential increase with T characterized by Jo and To [1],

Jin = Joe? (6.1)

However, more recently it has been found, both experimentally [8] and theoretically

[3], that this relation only holds in a limited range of temperature, and that the pa-

102

rameters Jo, 7p are strong functions of temperature and threshold gain. An expression

which has been found to better characterize the temperature behavior is
Jn (PT) = KT’, (6.2)

where x has been calculated for the non-degenerate case [3] to be 3/2, 1, 1/2 for bulk,

quantum well, and quantum wire active regions.

6.2 Low Temperature Measurements

The lasers used in this experiment are buried heterostructures (BH) made from single
quantum well graded index separate confinement heterostructure (GRINSCH) layers
grown by Molecular Beam Epitaxy (MBE). For specifics on the laser structure and
growth, we refer to Reference [11]. High reflectivity mirror coatings, Rl = R2 =
0.9, were applied to lower the end losses and to maximally reduce the threshold as
discussed in chapter 3. The lasers were mounted junction side up using In solder, with
Au wire bonds making the electrical connections. The devices were then attached
to a larger copper block, acting as a heat sink, with a thermocouple attached for
temperature measurement. Lasing threshold was defined as the extrapolated current
value of the sharp turn-on of the light output vs. current curves (L-I) under continuous
wave (CW) operation. These threshold current values, for both GaAs and InGaAs
lasers, were then recorded as a function of mounting block temperature and are shown
in figure 6.1.

The lasing wavelength was also measured, and the corresponding lasing energy

Ith(mA)

103

i x InGaAs ?
ak @® GaAs
3r
r e
ar
er e x
e ° x * x
xe x x
0 lL i i A
0 100 200 300
T (K)

Figure 6.1; Threshold current vs. temperature for GaAs and InGaAs lasers.

104

1.6

bd GaAs

Lasing photon energy (ev)

1.47
x x
x Xx x y x
1.37 x x
1.2 L 1 i. 4 nM
0 100 200 300
T (K)

l-jonre 6.2: Lasing energy vs. temperature for GaAs and InGaAs BH SQW lasers.

105

ith [mA] dith/dT [pA/K] dElaser/dT [meV/k]
SK 300K SK S3OOK
GaAs 0.120 45 8.32 -0.14 -0.38
ING@AS 0.165 1.6 3.29 -0.18 -0.10

Table 6.1: Summary of low temperature measurements

as a function of temperature is displayed in Figure 6.2, along with the band gap
variation of GaAs with temperature. The lasing energy of both lasers increases with
decreasing temperature at a rate similar to the GaAs band gap. Some of the main
properties of interest are assembled in table 6.1.

From Figure 6.1 we see that the laser with the GaAs active region starts out
with a higher threshold current (1;, = 4.5m A) at room temperature than the InGaAs
(I., = 1.6mA), but decreases at a faster rate with temperature. Below 200°A’, both
laser thresholds fall nearly linearly with decreasing temperature. Also, we note a
cross-over point at about 20° K below which the GaAs threshold is lower than the

InGaAs. The ultimate achievable threshold current is lower for the GaAs laser, which

106

reaches a minimum value of 120A. The InGaAs threshold levels off to a value of
165A at 15° K and becomes constant. This effect is not seen in the GaAs lasers,

and may be due to a residual leakage current .

6.3. Threshold vs. Temperature

We employ a simple model to explain the different slopes of the curves for the two
types of lasers. At elevated temperatures and for high gain laser structures, the
higher quantized energy levels in the active region and even in the barrier regions
play a significant role in determining I,,(7) [5,6]. Also, the quality of the barrier
region has been shown to have a pronounced effect on 1,,(7) due to nonradiative
recombination [7]. The effect of broadening, both in the transitions[6,7] and in the
density of states [8], has been treated and has shown to influence 7 in Equation (1)
above significantly, but leaves the slope ,K, in Equation (2) unchanged. Since our
analysis results in Equation 2, broadening should not have an effect on our model.
Also, at low temperatures and for low loss lasers, it is reasonable to neglect the higher
energy levels and include the n = 1 transitions only. Next, we model the addition
of In to the active region of a GaAs laser by using a lowered valence band effective
mass [12]. Finally, we derive an expression for the threshold carrier density, with the

valence/conduction band effective mass ratio as a parameter.

107

6.4 Threshold Carrier Density

The threshold gain is given by

os toa!
gn = a+ Floge

(6.3)

and clamps the carrier density at threshold to n;,. Next, we assume that the losses
are independent of temperature. This will especially be true at low temperatures since
the internal losses due to free carriers and intervalence band absorption scale with nip,
and n%, respectively, will become much less than the constant mirror loss term. From
chapter 2, the gain written in terms of the fermi factors and valence to conduction

band mass ratio R is given by

= wallet fe = 1). (6.4)

The fermi functions at the band edge, B, = 0 and E,, = 0, including only one
quantum well sub band, become

for a carrier injection level n (= p). The term n,., is the familiar two dimensional
density of states multiplied by kT

MewkyT

ahh? -

Substituting into the threshold condition in equation 6.4, we arrive at an equation

for the carrier density as a function of threshold gain:

Gth "th Mth.

(1 - )-€ % —e Rn = 0), (6.7)

Imax

where

Imax = WR (6.8)

. Since Gmax depends on R, this equation will be solved numerically. First we see,
however, that if the threshold gain is independent of temperature, each of the other
terms in the equation above must also be constant with respect to T. This means
that the threshold carrier density has the same temperature dependence as ng, 1.€., it

is linear. We can solve for the ratio @ defined by

ga ve

as a function of R.

3 is a measure of the temperature sensitivity of the threshold carrier density and
depends on the ratio of valence to conduction band density of states, R, through
Equation (7). 9 increases with R and threshold gain. Intuitively, we can see this
by examining the number of populated states above the fermi level which do not
contribute to gain. The larger the density of states above E,., the larger the number of
"wasted” carriers will be, as far as gain is concerned. Hence, a decrease in temperature
causes a larger reduction in the number of carriers at threshold in a material with a
larger density of states. Therefore, we expect the GaAs lasing thresholds to decrease
faster with temperature than the InGaAs lasers.

We have plotted 3 in figure 6.3 only for cases such that the threshold gain is less
than 70% of the maximum gain. # increases sharply beyond this point due to the

saturation of the gain with current. In addition, the model will break down since this

109

TR
Ww
oT yt T Lt oe ae i | | ie eee aes | ] ToT TT ] TOT TF | rT Ty

Ww
Oo

Nm
oO

a |
Ooo

pootirrirtrrirtirirrtiirrr dia;

Figure 6.3: Calculated threshold temperature sensitivity factor 3 for various threshold

gain as a function of valence/conduction band effective mass.

110

corresponds to a high injection level and higher subband population can no longer
be neglected. G(R) is shown for threshold gains of 10, 20, 30, and 50 cm~!. The
transparency value (gain = 0) is also shown and coincides with the plot of a(R),

which is just 3(F) calculated for the transparency condition, in reference [10].

6.5 Threshold Current vs. Temperature

The threshold current density, Jj, is related to ny, and can be expressed in terms of

the recombination coefficients

At low temperatures and carrier densities, the radiative component of the current
density is dominant [13] and we approximate (9) by neglecting A and C. We arrive
at an expression for the threshold current density as a function of effective mass ratio

and bimolecular recombination constant B,
Jy, = eL,8?(R)n2B(T) (6.10)
or using the expression for n, ,
Jin = e(LzpcksT)? 8° R)B(T). (6.11)

Now, B(T) is a function of temperature, density of states, and the dipole matrix
element describing the transition. For the case of non-degenerate bands, using k-

selection rules, B(T) has been solved for analytically [3], and for the quantum well

lll

case, was shown to be inversely dependent on temperature. Also, experimental results
[13] indicate a T~! dependence of B. Substituting this form of B into (11), and

neglecting any band structure dependence of B, we obtain
Jin = C1 B?(R)T (6.12)

which is our final result. Cy is a constant (we neglect the band structure dependence
of B) and the slope of the transparency current with temperature, ate is determined
by the valence/conduction band mass ratio through the factor 3(R).

If we use this model of laser threshold, we find a good agreement between the
ratio of measured aaa Calculations [14,15] predict, for strained InGaAs, a valence
band effective mass of 0.08 — 0.09m, whereas measurements [15] indicate a value of
0.14mpo. Using R = 2 for InGaAs and R = 8 for unstrained GaAs we calculate a slope
ratio, ( faaae)? Table 6.2 shows the value of the slope ratio for various threshold

gains. we see that for low threshold gain, the slope ratio is calculated to be 3, which

agrees well with the measured value of 2.5.

6.6 Conclusions

In conclusion, we have demonstrated 120A and 165A threshold currents in GaAs
and InGaAs lasers, respectively, at. cryogenic temperatures. We have measured the
lasing threshold current and wavelength of GaAs and strained InGaAs lasers for tem-
peratures between 5 — 300°A’ and, consistent with existing models[3,7], the threshold

currents increase linearly with increasing temperature. The InGaAs laser is less sen-

112

2th [cm-!] 0 10 20 30 50

B(8)/B(2) 3.01 3.11 3.28 3.62 4.4

Table 6.2: Calculated values of 3?(R) for various threshold gains

sitive to temperature variations at these low temperatures, which we attribute to a
reduced valence band effective mass. We have proposed a simple model to explain

the difference in afeh for the two laser types which agrees well with our findings.

113

References

[1] J. Pankove, Optical Processes in Semiconductors, Dover, New York (1975)
[2] N.K. Dutta, Plectron. Lett. 18, 451 (1982)
[3] A. Haug, Appl. Phys. B 44, 151 (1987)

[4] M.M. Leopold, A.P. Specht, C.A. Zmudzinski, M.E. Givens, and J.J. Coleman,

Appl. Phys. Lett. 50, 1403 (1987)
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[6] S.R. Chinn, P.S. Zory, and A.R. Reisinger, JEEE J. Quantum Electron. 24, 2191

(1988)

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J. Quantum Electr. 25, 1459 (1989)
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(9] T. Bartolac, Aerojet Corp., private communication

114

[10] L.E. Eng, A. Sa’ar, T.R. Chen, I. Grave, N. Kuze, and A. Yariv, Appl. Phys.

Lett. 58, 2752 (1991)

[11] L.E. Eng, T.R. Chen, S. Sanders, Y.H. Zhuang, B. Zhao, H. Morkoc, and A.

Yariv, Appl. Phys. Lett. 55, 1378 (1989)

[12] K. Lau, S. Xin, W.I. Wang, N. Bar-Chaim, and M. Mittelstein, Appl. Phys. Lett.

55, 1173 (1989)

[13] A.P. Mozer, S. Hausser, and M.H. Pilkuhn, JERE J. Quantum Electron. 21, 719

(1985)

(14] I. Suemune, L.A. Coldren, M. Yamanishi, and Y. Kan, Appl. Phys. Lett. 53,

1378 (1988)

[15] J.E. Schirber, [.J. Fritz, and L.R. Dawson, Appl. Phys. Lett. 46, 187 (1985)