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HIGH-POWER BIPOLAR AND UNIPOLAR QUANTUM CASCADE LASERS
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
O 2007 Arkadiy Lyakh
To the memory of my dear mother-in-law.
First, I would like to thank my advisor, Prof. Peter Zory, for his guidance and
encouragement throughout my research. I learned from him very diverse techniques on
semiconductor laser design, processing and characterization, and I am convinced that this
knowledge will be very helpful to me in the future. In addition, in his lab he has created an
atmosphere of trust and understanding. Therefore, the years that I spent working with him will
always remain a pleasant memory for me. It was a privilege to have Prof. Zory as my
supervisor. I would also like to thank Prof. Rakov, Prof. Xie and Prof. Holloway for being the
members of my supervisory committee.
Finally, I would like to thank my dear wife, mother, father and sister for their support
through the four years of my study at the University of Florida. Their encouragement and love
was very important to my success.
TABLE OF CONTENTS
ACKNOWLEDGMENT S .............. ...............4.....
LI ST OF T ABLE S ................. ...............7....____......
LIST OF FIGURES .............. ...............8.....
AB S TRAC T ........._. ............ ..............._ 1 1...
1 INTRODUCTION ................. ...............13......... .....
2 BIPOLAR CASCADE LASERS ............ ..... ._ ...............21...
2. 1 Light Polarization Rules for Bipolar and Unipolar QCLs ......____ ...... ...__ ...........21
2.2 Basic Operational Principles of Bipolar QCLs ................. ...............24........... ..
2.3 Review of Previous Work on Bipolar QCLs ................. ...............25..............
2.4 Experimental Results on Fabricated Bipolar QCLs ................. ................ ......... .27
3 SIMULATION OF CURRENT SPREADING IN BIPOLAR QCLs .............. ..................40
3.1 Influence of the Tunnel Junction Resistivity on the Current Spreading in Bipolar
Q C Ls ................ ....... ... .. .. ... .. ... ................4
3.2 Previous Work on Current Spreading Simulation in Diode Lasers .............. ..............43
3.3 Simulation of Current Spreading in SSL ......__....._.__._ ......._._. ..........4
3.4 Simulation of Current Spreading in DSL .............. ...............48....
4 UNIPOLAR QUANTUM CASCADE LASERS .............. ...............55....
4.1 Basic Operational Principles of Unipolar QCLs .............. ...............55....
4.2 Review of Previous Work of Unipolar QCLs ....._.__._ ..... ... .__. ........_........5
4.3 Low-Ridge Configuration Concept ................. ...............59...
4.4 Waveguide Structure for the Low-Ridge QCLs .............. ...............60....
4.5 Fabrication of the Low-Ridge QCLs .............. .... ........_.._ .............. ...........6
4.6 Experimental Results for Fabricated Unipolar Low-Ridge QCLs .............. ..................64
5 CURRENT SPREADING MODEL FOR UNIPOLAR QCLs .............. .....................7
5.1 Previous Work on Current Spreading Simulation in Unipolar QCLs ............... ...............79
5.2 Two-Dimensional Current Spreading Model .............. ...............80....
5.2.1 Active Region Transverse Conductivity .............. ...............80....
5.2.2 Two-Dimensional Finite-Difference Method.........._..._.._ .........................._82
5.3 M odel Results .............. ...............85....
6 SUMMARY AND FUTURE WORK .............. ...............93....
6. 1 Bipolar Quantum Cascade Lasers ........._._ ........... ...............93...
6.2 Unipolar Quantum Cascade Lasers .............. ...............94....
A CALCULATION OF ELECTRON ENERGY LEVELS AND WAVEFUNCTIONS IN
LAYERED STRUCTURES .............. ...............96....
LIST OF REFERENCES ................ ...............101................
BIOGRAPHICAL SKETCH ................. ...............106......... ......
LIST OF TABLES
2-1 Dependence of tunneling probability exponential factor D on indium composition........ .39
3-1 List of parameters used for simulation of current spreading in bipolar QCLs. .................54
3-2 Current spreading simulation results for bipolar QCLs. .................. ................5
4-1 List of parameters used to calculate transverse NF distribution in the low-ridge QCL
under the rid ge. ............. ...............78.....
4-2 List of parameters used to calculate transverse NF distribution in the low-ridge QCL
under the channels s ................. ...............78........... ....
LIST OF FIGURES
1-1 Semiconductor laser and its typical output power Po versus current I characteristic.
At threshold current Ith, laser action is initiated ................. ...............16..............
1-2 Interband mechanism ................. ...............17................
1-3 Intersubband mechanism .............. ...............18....
1-4 Illustration showing how an electron loses energy in a bipolar QCL ........._..._... ..............19
1-5 illustration showing how an electron loses energy in a unipolar QCL. .............. ...............20
2-1 Energy band diagram for SSL and DSL ................ ...._.._ ...............31. .
2-2 Ilustration of current spreading in bipolar QCLs. ............. ...............32.....
2-3 Schematic of TJ and QWT J. ............. ...............32.....
2-4 Bipolar cascade lasers with different waveguide configuration. A) Separate
waveguide for each stage. B) Single waveguide for all stages. ............. ....................33
2-5 Vertical cavity surface emitting bipolar QCL. ...._.._.._ .... .._._. ...._.._...........3
2-6 Schematic of TJ. ........._. ............ ...............34...
2-7 Quantum well tunnel junction ................. ...............35...............
2-8 Single-stage and double-stage structures ................ ...............35........... ...
2-9 Near-field pattern of the double-stage laser ................. ...............36........... ..
2-10 Far-field pattern of the double-stage taken at 2A ................. ...............36........... .
2-11 Voltage vs. current characteristics for the double-stage and single-stage lasers. ...........37
2-12 Power vs. current characteristics for the double-stage and single-stage lasers near
threshold and at high power levels............... ...............38.
3-1 Structure (comprising tunnel junction) used for illustration of influence of TJ
resistivity on current spreading. .............. ...............52....
3-2 Illustration of the current flow in stripe geometry SSL. ....._____ ... ....___ .............52
3-3 Voltage vs. current characteristics for the new double-stage and single-stage lasers. ......53
3-4 Near-field pattern of a new double-stage laser measured at 1A ................... ...............53
4-1 Schematic of electron transitions in unipolar QCLs ....._._._ ... ....... ........_.......67
4-2 Schematics of intersubband and interband transitions ................. .......... ...............67
4-3 Transverse near field distribution for the surface plasmon waveguide .............................68
4-4 High-ridge configuration for unipolar QCL. ............. ...............68.....
4-5 Schematic of the low-ridge laser (dimensions are given in microns). .............. .............69
4-6 Typical IV curve for a high-ridge unipolar QCL ................. ...............69........... .
4-7 Energy band diagram of the active region used in the low-ridge structure. ................... ...70
4-8 Transverse NF distribution in the low-ridge QCL under the ridge ................. ........._....71
4-9 Far Hield distribution corresponding to NF under the ridge. ............. .....................7
4-10 Transverse NF distribution in the low-ridge QCL under the channel. ............. ................72
4-11 Far Hield distribution corresponding to NF under the channels. ............. ....................72
4-12 SEM pictures of the fabricated low-ridge unipolar QCLs. ................ .......................73
4-13 Pulsed anodization etching setup. ............. ...............74.....
4-14 Etching rate of InP with GWA (8:4: 1) mixed with BOE in ration 600 to 7. .....................74
4-15 Power vs. current characteristics for the realized low-ridge QCL ................. ................75
4-16 Laser spectra measured at 4A and 20A taken at 80K. ............. ...............76.....
4-17 Voltage vs. current characteristics and corresponding power vs. current curves
measured at multiple temperatures (laser is without HR coating) .................. ...............76
4-18 Dependence of the threshold current on temperature. ...........__......_ ..............77
4-19 Far-Hield intensity distribution for a low-ridge QCL. ............. ...............77.....
5-1 Schematic of the low-ridge QCL. ........._. ...... .__ ...............86..
5-2 A) High-ridge IV curve. B) the corresponding dependence of the transverse active
region conductivity on the voltage across the active region. ............. .....................8
5-3 Rectangular mesh used for 2D Finite-Difference Method ........._.. ....... ._. ............88
5-4 A) Rectangular mesh used for the low-ridge QCL. B) Basic cell of the mesh. .................88
5-5 Schematic of the calculation procedure. ................ ...............89...............
5-6 List of parameters used for simulation. ................ ........................ ..............89
5-7 Current spreading model results. ............. ...............90.....
A-1 Model results for a GaAs-based layered structure. .............. ...............99....
A-2 Step sequence for the calculation procedure. ...._.._................. .. ......_..........10
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
HIGH-POWER BIPOLAR AND UNIPOLAR QUANTUM CASCADE LASERS
Chair: Peter Zory
Major: Electrical and Computer Engineering
High-power bipolar and unipolar quantum cascade lasers were designed, fabricated and
characterized. The importance of lateral current spreading is emphasized since it plays an
important role in operation of these devices.
Edge-emitting, gallium arsenide (GaAs) based bipolar cascade lasers were fabricated from
metalorganic chemical vapor deposition-grown material containing two diode laser structures
separated by a quantum-well tunnel junction (QWTJ). The QWTJ was comprised of a thin, high
indium content indium gallium arsenide layer sandwiched between relatively low-doped, p-type
and n-type GaAs layers. Comparison of near field data with predictions from a one dimensional
current spreading model shows that this type of reverse-biased QWTJ has a low effective
resistivity. As a consequence, current spreading perpendicular to the laser length in the plane of
the layers (lateral direction) is reduced leading to a relatively low threshold current for the
second stage. In addition, the differential quantum efficiency ~150% of these double stage lasers
is nearly twice that of single stage lasers.
Low-ridge unipolar quantum cascade lasers operating at 5.3Clm were fabricated from InP-
based MOCVD-grown material. Record-high maximum output pulsed optical power of 12W at
14A was measured from a low-ridge chip with a high reflectivity coated back facet at 80K. Also,
Far-Field measurements demonstrated current beam steering for this device. Modeling shows
that the lateral variation of transverse conductivity is essential for an accurate description of
current spreading in these devices.
Since their first demonstration in 1962, semiconductor lasers have become the enabling
components in many applications. Emission wavelengths for these devices now span the range
from the ultra-violet (0.4Clm for GaN-based interband lasers) to the far infrared (hundreds of
microns for intersubband lasers). Optical powers from single chips are now in the 100-watt
range in pulsed operation and the several-watt range in continuous operation. Conversion
efficiencies from electrical to optical power can be in the 50% range for red and near infrared
lasers with world record numbers now exceeding 70%.
Figure 1-1 is a diagram showing a typical semiconductor laser chip and its most important
characteristic, laser light output power Po versus current I. Electrons associated with current are
temporarily trapped in the active region where they lose energy either by photon emission or
some non-radiative process. Photons can be generated by either an electron-hole recombination
process between conduction and valence bands (interband mechanism, Figure 1-2) or a simple
electron transition between energy levels in the conduction band (intersubband mechanism,
Figure 1-3). Stimulated photons are confined and directed by a built-in waveguide and at a
sufficiently high current (threshold current Ith), laser action is initiated (Figure 1-1). As current
increases above Ith, Po continues to increase linearly to some high level that depends on the type
of laser chip used. Typical chip size as defined by L in (Figure 1-1) is about one millimeter.
Due to their small size, low cost and ability to be directly pumped by electrical current,
low power semiconductor lasers are now widely used in applications such as printing, optical
memories and fiber-optic communications. For applications such as infra-red countermeasures,
free space optical communications, machining and range/ranging measurements, high power
lasers are required. In order to achieve very high output powers, semiconductor lasers are
operated in the pulsed mode using high current pulses, tens of nanoseconds wide. In order to get
even higher powers without increasing power supply current requirements, one can employ
designs using two or more optical gain (active) regions in the laser material. Since each active
region (AR) is sandwiched between a number of layers, this combination being called a stage,
electrons can produce photons as they move from stage to stage through the multistage device.
As a consequence, it's possible in principle to obtain laser power proportional to the number of
stages in the material without increasing drive current.
Semiconductor lasers with two or more stages, usually called quantum cascade lasers
(QCLs), fall into two major categories: bipolar (using interband transitions)  (Figure 1-4) and
unipolar (using intersubband transitions)  (Figure 1-5). In bipolar QCLs, light emission
occurs due to recombination of electron-hole pairs in the ARs. After the recombination process
in the first stage, electrons tunnel from the valence band of the first stage into conduction band of
the second stage. The tunneling process between stages takes place through a reverse-biased,
heavily doped p-n junction. The tunnel junction is a crucial element in bipolar QCLs since it
allows electron recycling. The number of stages in a bipolar QCL typically ranges from two to
For unipolar QCLs, electrons emit photons by making transitions between conduction band
states (subbands) arising from layer thickness quantization in the AR. After a radiative (or non-
radiative) transition is made, the electron "cools down" in a relaxation-inj section region and then
tunnels to the upper laser level of the next stage. Usually, the number of stages in a unipolar
QCL ranges from 20 to 35.
In both cases, tunneling to the next stage provides the electron with the opportunity to
generate another photon. This electron recycling process in the multistage device leads to
increased output power relative to that obtained from a single stage device operating at the same
In this work both types of QCLs are discussed. A large portion of the discussion is
dedicated to the analysis of current spreading in the plane of the layered structure (lateral current
spreading) since all devices studied were fabricated in either a stripe geometry as shown in
Figure 1-1 or in a low-ridge configuration (Figure 4-5). Since the conductivity of the layers
above the active region is usually quite high, lateral current spreading is substantial and has a
strong influence on laser operation.
In Chapters 2, 3, 4 and 5, various aspects of the high power QCLs that have been designed,
fabricated and characterized during this work are discussed. In Chapter 2, the relationship
between the design of the reverse-biased tunnel junction used in bipolar QCLs and lateral current
spreading is discussed. Experimental results for realized bipolar QCLs are also presented in this
chapter. Output power versus current curves show that the performance of double stage lasers
made using this design are close to the best possible. The lateral current spreading model
developed in Chapter 3 shows that resistivity of the quantum-well tunnel junction used in our
devices is not high enough to cause any additional current spreading. In Chapter 4, the
peculiarities of lateral current spreading in low-ridge unipolar QCLs and their influence on the
operation of these devices are qualitatively described. Output power versus current curves show
that this type of unipolar QCL has the unique feature of low threshold current combined with
very high peak output power. Results of the current spreading model for low-ridge unipolar
QCLs developed in Chapter 5 support the qualitative description of the spreading mechanism
described in Chapter 4. In Chapter 6, the thesis work is summarized and suggestions made for
future research directions.
Figure 1-1. Semiconductor laser and its typical output power Po versus current I characteristic.
At threshold current Ith, laser action is initiated.
: Guide Layer
Figure 1-2. Interband mechanism whereby conduction band (CB) electrons trapped in the
quantum well active layer recombine with valence band (VB) holes to produce
photons with energy hy.
Figure 1-3. Intersubband mechanism whereby conduction band electrons in a high energy state
in the active region quantum well make a transition to a lower energy state by
emitting a photon with energy hy.
r :~~J. I
Figure 1-4. Illustration showing how an electron loses energy in a bipolar QCL.
Figure 1-5. Illustration showing how an electron loses energy in a unipolar QCL.
BIPOLAR CASCADE LASERS
2.1 Light Polarization Rules for Bipolar and Unipolar QCLs
As discussed in Chapter 1, the radiative mechanism is different for bipolar and unipolar
QCLs. Using quantum-mechanical approach, in this section of Chapter 2 we demonstrate that
this distinction leads to different light polarization for these devices. In particular, for bipolar
QCLs emitted light is mostly TE-polarized, while for unipolar QCLs TM-polarization is
Using Fermi golden rule it can be shown that the absorption (emission) coefficient in the
both cases is proportional to:
(7, | H'| 7,l (2-1)
where ~, and cpyare electron wavefunctions corresponding to initial and final states involved in
7:f ,, r; (2-2)
withJ&- envelope functions and uv,v- periodic Bloch functions. The interaction Hamiltonian
H' is given by
H' = ~ )(A p +p ) (2-3)
where A is the vector potential, p is the momentum operator and m*" is the effective mass. Vector
potential A in its turn can be expressed as a polarization unit vector e multiplied by a scalar
function that slowly changes with the spatial coordinate r (its spatial variation is negligibly small
within each unit cell). Therefore, the matrix element in Eq. 2-1 is proportional to
where v v and n, n are the band and subband indices of the initial and final states. The first
right-hand side term of the equation corresponds to interband transitions, while the second one
corresponds to intersubband transitions.
Using the intersubband term it' s easy to show that light emitted by unipolar lasers is
mostly TM-polarized. In this case (21 u ,)=1 for all intersubband transition, since Bloch
functions are approximately the same for electrons within the same band. Using the fact that
there is no size quantization in the x-y plane (plane of semiconductor layers) the envelope
function can be expressed as
fnk, =7 iZ)8-zkir (2-5)
plane. As a consequence, dipole matrix element in Eq. 2-4 has the following form
~f~ e- pln fi'k' ~ d're-'*~zkirp (Z) exI~x ey~ p zkire" *,, (z) (2-6)
Terms proportional to ex and e, are nonzero only in case when n=n and k, = k's In other words
these components are always zero for photon absorption and emission. This is a consequence of
the fact that the conservation of energy and wavevector for a transition within the same subband
requires an electron interaction with the lattice (phonons). Therefore, absorption (emission)
coefficient for intersubband case is nonzero only when light has the z-component and it reaches
the highest value when the electric field is fully polarized along the z-direction (perpendicular to
semiconductor layers). This explains why light emitted by intersubband lasers is always TM-
The first term on the right-hand side of Eq. 2-4 can be analyzed in a similar way. Overlap
integral between the envelope wavefunctions determines selection rule for interband transitions,
which states that transitions are allowed only between states with the same quantum number in
contrast to intersubband transitions:
(n If = 0 if (ns n') (2-7)
Analysis of the dipole matrix element for the Bloch functions is more complicated than for
envelope functions (intersubband case). This analysis is based on the fact that hole Bloch
functions can be considered as linear combinations of so-called valence band basis functions.
The basis functions in their turn have symmetry of atomic orbitals of an isolated atom. Using this
fact and the fact that interband absorption in quantum well lasers is mostly determined by
conduction band to heavy hole band transitions, analysis of Bloch functions dipole matrix
element leads to the conclusion that light emitted by bipolar lasers is mostly TE-polarized
(electric field lies in the plane of semiconductor layers).
Due to different polarization, near field (NF) intensity distribution for intersubband lasers
has some peculiarities relative to interband lasers. For example, calculated NF shows presence of
intensity discontinues at boundaries between semiconductor layers in accordance with the
boundary condition eEEn= E2En,, while for bipolar lasers it is continuous at all interfaces. In
addition, since dielectric functions for a metal and semiconductor layers have different signs
dielectricc function is negative for the metal since imaginary part of the refractive index for the
metal is bigger than its real part) there is possibility of a surface plasma mode propagating at the
metal/semiconductor interface of unipolar lasers. This effect can be used to make unipolar lasers
based on so-called surface plasmon waveguide (see Section 4-1).
In the rest of the text we will always assume that the radiation corresponding to bipolar
lasers is TE-polarized, while it is TM-polarized for unipolar lasers.
2.2 Basic Operational Principles of Bipolar QCLs
The bipolar cascade laser project was dedicated to development and realization of a
double-stage laser (DSL) comprising a standard single-stage laser (SSL) as a recurrent stage.
Schematics of SSL and DSL are shown in Figure 2-1. After radiative recombination with holes,
electrons tunnel from valence band of the first stage to conduction band of the second stage
through the potential barrier in the tunnel junction (TJ). Therefore, in ideal case each electron
can give rise to two photons. As a consequence, DSL in principle can give twice as much power
as SSL at the same current.
Usually, TJs for bipolar QCLs are composed of two n and p heavily doped layers. If
doping of these layers is not high enough, potential barrier width increases. As a consequence,
the electron tunneling probability reduces and the effective resistivity of the TJ increases. High-
resistive layers can be the reason for a strong lateral current spreading in the laser (spreading
perpendicular to the laser axis in the plane of the layers) and, as a consequence, lower DQE and
higher threshold currents. The highest spreading in this case is expected in the N-layers (with
high mobility) above highly-resistive TJ (Figure 2-2). The most efficient way to get low TJ
resistivity is to dope both TJ layers above 5-1019 cm-3. In SOme material systems, the high carrier
concentrations required in one or both of the layers and/or dopant atom stability during the
growth of additional laser stages cannot be achieved. This problem can be reduced by
sandwiching an appropriate quantum well between the two tunnel junction layers (Figure 2-3)
since the tunneling probability exponentially depends on the barrier height. This possibility is
explored in this work. Esaki tunnel junctions are widely used in semiconductor device design
beyond bipolar QCLs. For example, TJs are employed to cascade solar cells. In these devices
each active region of the solar cell is optimized for light absorption in a particular wavelength
range. This increases conversion efficiency of the solar cell.
Bipolar QCLs can be designed to have either separate waveguides for each stage (Figure 2-
4a) or a single waveguide shared by all stages (Figure 2-4b). Due to reduced number of layers
the latter configuration benefits from suppressed current spreading in the structure and lower
strain. Also, such structures can be processed into distributed feedback lasers. However, overlap
between the optical mode and highly-doped TJs increases free-carrier absorption and as a
consequence decreases DQE for these devices. In this work we were mostly interested in getting
the highest possible DQE. Therefore, we employed the configuration with separate waveguides.
The bipolar cascade mechanism is also used for making Vertical-Cavity Surface-Emitting
(VCSE) QCLs (Figure 2-5). VCSEL emits light vertically (parallel to the growth direction)
rather than horizontally, which makes it easy to use them for 2D laser arrays. One key technical
advantage of VCSEL is its ability to produce a circular, low-divergence output beam. The active
regions of these quantum cascade lasers are placed in antinodes of the standing wave pattern to
increase the overlap with the optical mode, whereas TJs are located in the vicinity of a field null
to reduce the free-carrier absorption. The cascade configuration is used to increase roundtrip
gain, which is extremely low for these devices. Higher gain leads to lower threshold current and
as a consequence to higher optical power at the same current. Due to the low roundtrip gain DQE
for bipolar cascade VCSELs is also relatively low (~0.9 for triple-stage devices).
2.3 Review of Previous Work on Bipolar QCLs
The first realization of an edge-emitting bipolar QCL was reported in 1982 . In this
work MBE-grown, GaAs-based, triple-stage cascade lasers were fabricated and tested. Each
active region of the structure had its own waveguide and cladding layers. Using Be and Sn as p
and n dopant atoms respectively the authors achieved doping densities in TJs above 1019cm-3
This allowed them to triple DQE for the cascade lasers relative to SSLs (0.8 vs. 0.27). However,
due to immature technology overall performance for these devices was quite low (nowadays
DQE close to 0.7 is routinely measured for SSLs). Edge-emitting bipolar QCLs with separate
waveguides for each active region reported later had substantially improved characteristics .
For example, DQE ~1.4 was reported for a double-stage 808nm lasers in this work.
Bipolar cascade lasers where all the stages share the same waveguide and cladding layers
were reported in [4, 5]. Due to increased Gamma factor these lasers have threshold currents
lower than SSLs. As discussed above, mode overlap with highly doped TJs reduces DQE. For
example, in  DQE for triple-stage lasers was reported to be 125%.
Realization of Vertical-Cavity Surface-Emitting bipolar QCLs was reported in [6-8]. Since
effective length of the gain region is increased, threshold current density for cascade VCSELs is
lower than for their single-stage counterparts. For example, in  comparison between triple-
stage and a single-stage VCSELs showed that threshold current density for the triple-stage
device was only 800A/cm2 COmpared to 1.4kA/cm2 foT Single-stage laser, while DQE was 60%
compared to 20%. Relatively low DQE in the both cases is explained by low roundtrip gain for
the vertical-cavity configuration.
The importance of high-doping of the layers composing TJ was emphasized in . It was
also mentioned there that the very high doping of the TJ layers is not always achievable. For
example when Si is used as the n-dopant, it becomes amphoteric as the doping concentration
increases beyond 5-10 scm-3. The idea on the reduction of TJ resistivity using QWTJ layers was
suggested in . In this work it was shown that resistivity of GaAs TJ can be decreased by one
order through sandwiching 120-thick Ino.15Gao.ssAs layer between the heavily doped layers.
However, in this work QWTJ were not used in the laser design due to expected high modal
losses. The first MOCVD-grown InP-based bipolar laser with QWTJ was reported in [1l].
Employment of QWTJ allowed the authors substantially reduce voltage drop across TJ and
improve laser performance.
2.4 Experimental Results on Fabricated Bipolar QCLs
It' s well known that depletion width for a p-n junction is given by:
2ck 1 1
q N, N,f 28
where N, and Nd are acceptor and donor densities and Yo is the contact potential. Since high
doping is used for the layers composing the TJ, the degenerate approach should be used to
calculate Yo (Figure 2-6):
Vo = Eg + AE,, + AE, (2-9)
Depletion region extends mostly in the n-layer since doping concentration for the p-layer is
higher by approximately one order of magnitude. Also, even though N,>Nd~ it is usually valid
that AEf>AEf, since density of states for conduction band is substantially lower than for valence
band. AEf, can be found from the following equation:
Nrl=2 Pe~(E-Ec)cdE (2-10)
where P, is the conduction band density of states. Similar equation should be applied to find
Using Eq. 2-9 and Eq. 2-10 it can be shown that Yo slowly changes with carrier density
concentration. Therefore, from Eq. 2-8 it can be concluded that the depletion width decreases as
the doping increases. As a consequence, potential barrier width (and electron tunneling
probability) seen by electrons increases.
As discussed in Section 2-2, very high doping for TJ layers is not always achievable. In
our case the highest possible doping level for GaAs-based TJ was 5-10lscm-3 (Si doping) for the
n-layer and 5-1019cm-3 (C doping) for the p-layer. Relatively low n-doping is the reason that a TJ
composed of these layers would have high effective resistivity. Therefore, other methods should
be used to reduce TJ resistivity. QWTJ concept can be employed in this case. Schematic showing
how QW transforms potential barrier seen by electrons is represented in Figure 2-7. It was
assumed in this figure that QW lowers potential barrier height in the vicinity of the interface
between the TJ layers. In quantum mechanics the probability of a particle tunneling through a
potential barrier is proportional to the following factor:
D = exp ~- 2m [Tx) E~d (2-11)
where U(x) is potential barrier and E is particle energy. Therefore, we should expect lower
effective resistivity for QWTJ due to lower barrier height in the vicinity of x = 0.
Rough estimation of influence of QW on TJ resistivity can be done using Eqs. 2-1, 2-2, 2-3
and 2-4 and Figure 2-6, 2-7. The following approximations are used: potential barrier for
electrons in depletion region has triangular shape with a deep around x = 0, electron effective
mass is constant through the structure and equal to 0.067mo (GaAs-based TJ), barrier width of
the TJ is not influenced by dimensions of QW, doping for the n-side GaAs is 5*"10' m and for
the p-side it is equal to 5*"1019 -m3, quantum well used is InxGal -xAs.
Results on D dependence on In composition are represented in Table 2-1 assuming 100A -
thick QW (critical thickness equal to 100A corresponds to In composition around 40% ).
As it mentioned above, introduction of QW in TJ with In composition 15% in  reduced
TJR by one order of magnitude which is roughly consistent with results presented in Table 2-1:
D increases approximately by factor of ten when In % increases from 0 to 15. Also, it can be
seen that further increase in In composition leads to further substantial increase of the tunneling
probability. For example, D is more than 2 orders higher for 40% than for 15%.
In this work  we explore the use of a QWTJ between two standard GaAs-based diode
lasers grown by MOCVD. The quantum well used is a 10nm thick, highly strained InGaAs layer
with a 25% indium content. The SSL and DSL structures are shown in Figure 2-8 respectively.
Doping of the n-AlGaAs cladding layer in the first stage of the DSL is reduced relative to the
corresponding region in the SSL (4-101 cm-13 COmpared to 101s cm-3) in order to decrease current
spreading. The QWTJ connecting the two stages is composed of two relatively low-doped GaAs
layers (5-101s cm-3 Si for n-doping and 5-1019 cm-3 C for p-doping) with a 10nm thick
In0.25Gao.75As layer sandwiched between them. The active regions in both structures have two
typical quantum wells sandwiched between standard barrier layers. The DSL structure is
designed such that the optical mode loss due to overlap with the QWTJ is negligible. The SSL
and DSL lasers used in the experiments were 750' long with 75' wide contact stripes. The
output facet reflectivity was about 5% and the rear facet reflectivity was about 95%. All
measurements were performed in pulsed mode at room temperature using 500ns wide current
pulses at a repetition rate of 1kHz.
The near field (NF) intensity distribution for the DSL measured at 1A is shown in Figure
2-9. Since the distance between the ARs is approximately 5' we used this scale to estimate
that the NF width at the second AR is around 100' 25' larger than the contact stripe width.
This demonstrates that the current spreading in the structure is non-negligible.
Far-field pattern for DSL measured at 2A is shown in Figure 2-10. Double-lobed lateral
profile predicted for some wide-stripe diode lasers due to V-shaped phase front  was
observed for DSLs.
IV curves measured in cw-mode at low current (Figure 2-11) show that the turn-on voltage
for the DSL is close to double that of the SSL. The additional voltage drop above turn-on is
attributed to the finite effective resistivity of the QWTJ.
The power vs. current (PI) characteristics near threshold and at high power levels for a
typical SSL and DSL are shown in Figure 2-12. The slope efficiency of the DSL (2W/A)
(corresponding DQE ~150%) is nearly twice that of the SSL (1.1W/A).
Maximum measured optical power is determined by generator maximum current (~7A).
Ratio of the threshold currents for DSL for the second AR (490mA) and the first AR (340mA) is
approximately 1.5. This is another indication of the current spreading in the structure.
Experimental results described above show that the goal to double slope efficiency for
DSL was achieved. However, as shown by NF measurements lateral current spreading is still
present in the structure. A possible reason for this effect is the finite resistivity of TJ
demonstrated in Figure 2-10. To determine the degree of influence of QWTJ resistivity on
current spreading, a current spreading model for DSL based on 1D spreading model for SSL was
developed. Its details and results are discussed in Chapter 3.
contact gold -
Figure 2-1. Energy band diagram for SSL and DSL
Figure 2-2. Illustration of current spreading in bipolar QCLs. Dashed line shows FWHM of
carrier density distribution. Highest spreading is expected in the N-layers with
mobility above highly-resistive TJ.
Figure 2-3. Schematic of TJ and QWTJ. Tunneling probability exponentially depends on the
potential barrier height. It is expected to be lower for QWTJ since in this case height
of the central part of the barrier is lower then for TJ.
Figure 2-4. Bipolar cascade lasers with different waveguide configuration. A) Separate
waveguide for each stage. B) Single waveguide for all stages. These figures
demonstrate transverse intensity distribution through the layers.
-r I --1
Figure 2-5. VCSE bipolar QCL. The quantum wells in these devices were placed in antinodes of
the standing wave pattern, whereas TJs were located in the vicinity of a field null to
reduce free-carrier absorption.
Figure 2-6. Schematic of TJ. Triangular potential barrier seen by electrons is confined by thick
p-GaaAs contact, 21019
p-AlGaAs cladding, l10
p-AlGaAs waveguide, 4 101
n-AlGaAs wnaveguide, 4 100
n-AlGaAs cladding, 101X
n-GaAs substrate, 2*10'8
Figure 2-7. Quantum well tunnel junction
Figure 2-8. Single-stage and double-stage structures
Figure 2-9. Near-field pattern of the double-stage laser. Distance between the active regions is 5pm measured at 1A.
Figure 2-10. Far-field pattern of the double-stage taken at 2A.
---- SSL voltage is doubled
(at the same current)
----- DSL .C...----
Figure 2-11. Voltage vs. current characteristics for the double-stage and single-stage lasers.
Turn-on voltage for the double-stage laser is approximately twice that of the
single-stage laser. The additional voltage drop above turn-on is attributed to the
finite effective resistivity of the QWTJ.
I I I I II III I I I I
0 1 2 3 4 5 6 7 8
0.4 0.8 1.2
* DSL with ~
* SSL with ~
Figure 2-12. Power vs. current characteristics for the double-stage and single-stage lasers near
threshold and at high power levels.
Table 2-1. Dependence of tunneling probability exponential factor D on indium composition
SIMULATION OF CURRENT SPREADING IN BIPOLAR QCLS
3.1 Influence of the Tunnel Junction Resistivity on the Current Spreading in Bipolar QCLs
As discussed in Chapter 2, NF measurements clearly demonstrate that there is a substantial
current spreading between the 1st and the 2nd stages in realized DSLs. In this section using simple
1D model  it will be shown that tunnel junction resistivity can be the reason for this
For the demonstration of current spreading induced by TJ resistivity we'll use a structure
shown in Figure 3-1. The structure is comprised of a resistive TJ and a P-layer above it. The TJ
has the thickness h and the effective resistivity p '=ph [G2-cm2], where p is the resistivity of the TJ
and h is its effective thickness (p' instead of p is usually used to characterize TJ since depletion
width and p are usually unknown for a TJ, while p 'can be directly measured .). Thickness
for the p-layer is d and resistivity is p,. Stripe width and length are taken to be W and L
respectively. In this model it's assumed that voltage drop across the P-layers above the tunnel
junction is negligibly small and that current through TJ under the stripe (Ic) is constant. Leakage
current (2-lo) is defined as the current that doesn't flow under the stripe. Therefore, total current
is given by
frot = Ie + 210 (3-1)
For current density flowing through the TJ we have
E= pJ (3-2)
This expression can be modified as
V = pJh = p' J= Ly p' (3-3)
where V voltage across the TJ and -dI,- is the current across the TJ between y and y+dy. For
lateral current in the P-layer we have:
-dPy = p,ISdy (3 -4)
where p, is defined as p, /(Ld) and I,- is the current flowing in the y-direction. From Eq. 3-3
and Eq. 3-4 we get
--;L/ (p= L I (3-5)
Solution to Eq. 3-5 is
I, ,ex -, Ly+ C, expP y (3-6)
Using the fact that current can't grow exponentially and I,-(0)=Io we get
Iv = IO expl Iy(37
Current across the TJ between y and y+dy is obtained from Eq. 3-7
-dl= j~, (0 = I (3-8)
As a consequence, le is given by
I = WI, (3-9)
Therefore, total current through the device is
I,pp = : I,W + 21, (3-10)
From Eq. 3-10 for Io we get
Using Eq. 3-7 and Eq. 3-11 we get for current density flowing through the TJ
J(x) = 7toa exp -x rP x >0 (3-12)
2+ W I-
Weg (outptut) = -2 n (0.5) I+ W (3-13)
where Wegis the effective output stripe width (FWHM of the lateral current density distribution
right under the active region)
Typical values of p' for TJs are in the range from 10-3 to 10-' Ohms -cm2 .
Substituting p' = 10' Ohms cm2 t=200mA, W = 80pum and p, /d = 300hms we get from Eq.
3-11 and Eq. 3-13: I1otle = 27mA and Wfs(output) = 88um .
Eq. 3-11 and Eq. 3-13 can not be directly applied to our structure. However, they
demonstrate that a TJ with a typical resistivity in vertical direction can influence current
spreading and its contribution can be reduced by lowering resistivity. The question we are trying
to answer in this chapter is stated as follows: Does TJ resistivity contribute to current spreading
between the 1st and the 2nd stages of the DSL observed in the NF measurements? To answer this
question a current spreading model for DSL that assumes negligible TJ resistivity will be
presented. Validity of this assumption will be verified by comparing model results with
3.2 Previous Work on Current Spreading Simulation in Diode Lasers
A simple 1D current spreading model for diode lasers was presented in . Transverse
voltage drop across the p-layers above the active region was considered to be negligibly small
and current density under the wide contact stripe to be constant. In this work the diffusion
current in the active region wasn't taken into account.
In [16, 17] this model was extended to include the diffusion component. It was shown in
 that Ohmic current in the layers above the active region and the diffusion current in the
active region are coupled and can't be considered independently without losing self-consistency
of the problem. In  the following physical description of the lateral current in the active
region was suggested: holes are transported in the active region under the combined effect of
drift and diffusion, but the field causing diffusion is such that their motion is identical to that of
pure diffusion with an effective diffusion coefficient. Electrons on the other hand, are stationary
in the active region. Instead of moving there, the electrons are supplied from or to the N-layer at
just such a position-dependent rate that they maintain charge neutrality in the active region.
The model developed in [16, 17] is applicable only below threshold since it doesn't take
into account the stimulated emission term. Since the stimulated term is also involved in the scalar
wave equation, coupling between the optical mode and diffusion and Ohmic currents should be
considered in this case. A numerical model that takes into account the stimulated emission term
was reported in . In addition, it comprises Poisson equation and photon rate equation. This
model was used to predict an optimal ridge width for a diode laser.
In this work we adopt the subthreshold model developed in [16, 17]. The model (Section 3-
3) and its employment for simulation of the current spreading in DSL (Section 3 -4) are discussed
in the next two sections.
3.3 Simulation of Current Spreading in SSL
The exact solution to the lateral current flow problem in a laser requires the solving of the
2D current continuity equations. A maj or simplification used here is the reduction of what is
naturally a 2D problem to a problem in one dimension.
As discussed in the Section 3 -2, there are two maj or components to the lateral carrier flow
in a diode laser: a lateral current spreading in the layers away from the active region (Ohmic
current) and a lateral diffusion current in the active layer. It was also mentioned in Section 3-2
that these two currents are coupled and can't be evaluated separately without losing self-
consistency of the problem . Since it' s usually safe to assume that transverse voltage drop
across the P-layers above the active region is negligibly small, Ohmic current spreading is
characterized by normalized conductance E (Figure 3-2):
F = f(qny)h, = 0 (3-14)
where N is the number of P-layers and 0 is normalized resistance. When r is big ( 0 is small),
lateral current spreading is big.
In this work we assume that the lateral voltage gradient is small at the interface between
active layer and N-cladding layer compared to the lateral voltage gradient at interface between P-
cladding layer and active layer. This condition is usually satisfied for single stage semiconductor
lasers since the resistance of the layers below the active region is much smaller than the
resistance for the ones above it.
Effective current width is defined here as FWHM of spatial current distribution.
Following  we have for Ohmic sheet-current density C(x) (lateral Ohmic current per
unit of stripe length (A/cm) in layers above active region) (Figure 3 -2)
= -szC(x) (3-15)
= -J (3-16)
where v(x) is voltage drop across the active region at location x J is active region injected-
current density and 0Z = T1 (see Eq. 3-14) normalized resistance of the layers above active
region. Eq. 3-15 and Eq. 3-16 take into account that the current density inj ected in the active
region originates from the decrease in the lateral Ohmic current flowing above the active region.
It is assumed that the evolution of the electron concentration n in the active layer can be
described by a diffusion equation:
qDeg d = J d (3-17)
where Deff (see Section 3-2 ) is the effective diffusion coefficient  and JdIfs is a hole-diffusion
current density (A /cm2 ). The diffusion current has as its source the junction-current density J
and, as its sink, the concentration dependent recombination rate R(n)
df qR(n) (3-18)
where d -active layer thickness and R(n) = An + Bn2 Cn3 TOCOmbination rate with A = 0 ,
B = 0. 8*10-tocm3 / S and C = 3.5*"10-30 C6 / S taken from . Eqs. 3-15 through Eq. 3 -18 have
to be solved self-consistently taking into account that the voltage across the active region is a
function of the concentration n. For this dependence we use the following formula (see  and
kT n n
v(n) = -421n-n +A At (3-19)
where n = n, at x = x,, ns n,, A, 0.35 N,-effective density of states in the conduction
band. From Eq. 3-15 and Eq. 3-16 we can get
= 11 (3-20)
Solving Eq. 3-17 and Eq. 3-18 we have
D + R(n) (3 -21)
e dx 2 qd
First lets consider the case when x 2 W/2 Equations Eq. 3-20 and Eq. 3-21 are coupled through
J. Elimination of Jin these equations and some mathematical manipulations give (see )
dn dy(R 2 dy 1
-- R~) d = f (n) x >- W (3-22)
dlx dnl D, dn 2
where y = n + v(n) / qD~d Since it is assumed that current density under the stripe Jo is constant,
the first integral of Eq. 3-21 has the form
dn 2 J
where nO is the carrier concentration in the active region below the center of the stripe,
uniform current density under the stripe and W represents stripe width. Eq. 3-22 (or Eq. 3-23)
can then be integrated to yield as the solution x(n) for the carrier-concentration profile
x =- W (3-24)
where ne = n at x = xe W .
Calculation of current density distribution in the active region is done using the following
Carrier concentration at the stripe edge n, is used as an input parameter.
Using Eq. 3-22 and Eq. 3-23 f(n) is found for x 2 W and 0 I x < W
respectively. Also, f(n) under the stripe at this step is a function of two unknown parameters: no
Relation between Jo and no (Jo(no)) is found using the fact that Eq. 3-22 and Eq. 3-23 should be
equal under the stripe edge.
Using Jo(no), Eq. 3-23 and
1 "" dn
-W = (3-25)
no (and as a consequence Jo) is found as a function of ne.
Lateral carrier concentration n(x) is found using Eq. 3-24.
Using n(x) inj ected electron current density from the n-side of the p-n junction can be found from
the fact that it is equal to local recombination current density 
Jcx) = qd An (x) +Bn2 (tn3 (x)) (3-25)
where A is the coefficient corresponding to non-radiative recombination (for example through
interface states), B bimolecular radiative recombination coefficient and C the coefficient
corresponding to Auger non-radiative recombination.
According to definition output current effective width which is used as an effective stripe
width for the next stage is taken to be equal to:
W, = 2xJO 2 (3 -26)
where x,, is determined by:
J~y )_1,xJo/ > 0 (3 -27)
In addition, for diffusion current leakage we have:
Jdyr = -qD = qDf (n) (3 -28)
while for Ohmic sheet current density we get from Eq. 3-2 and Eq. 3-4:
1 dv dn dv /dn
C(x) Jdyd (3 -29)
0Z dn dx qD~d
Leakage current is sum of these two currents (multiplied by a factor of two) evaluated under the
stripe edge. Therefore, total leakage current is given by
ITotalLeakge = 2C(xe )L +2Jdy eX,)Ld = 2Jdyf ex)Ld [1+ z] (3-30)
z 2kT/nq2D~d (3-31)
is the relative importance of the Ohmic and diffusion current.
Total current corresponding to carrier density in the active region at the edge of the contact
stripe ne (which is used as an input parameter) is found integrating Eq. 3-25.
3.4 Simulation of Current Spreading in DSL
In order to determine the degree of current spreading at the two DSL active regions (ARs),
a model was developed based on the 1D discussed in the previous section. This model estimated
the effective current width (FWHM of the current density distribution) at each AR. It was
assumed that the QWTJ contribution to current spreading was negligibly small due to its low
effective resistivity. Comparison between the model and experiment was supposed to show
wither this approximation is valid.
It can be seen from Eq. 3-1 that spreading should be especially important for the bipolar
cascade lasers where several active regions are connected via tunnel junctions (TJs). Total
thickness of such structure is substantially bigger than for a common semiconductor laser and
therefore current leakage is expected to be enhanced in this case. Also, the resistivity of TJs
(TJR) in the current flow direction should contribute to total spreading since the current tends to
spread before entering a less conducting material.
In the discussion below by stage we mean active region and all the layers above it but up to
the active region of the previous stage. For example, according to this definition, second stage
comprises 2nd active region and all the layers between 1st and 2nd active regions.
For simplicity, current spreading in each stage is considered independently. This approach
is accurate when we can assume that the lateral voltage gradient is small at the interface between
active layer and N-cladding layer compared to the lateral voltage gradient at interface between P-
cladding layer and active layer. This condition is usually satisfied for single stage semiconductor
lasers since the resistance of the layers below the active region is much smaller than the
resistance for the ones above it. In our case we believe it's true for the both stages. Output value
of effective current width for the first stage (its value at the active region) is considered as an
effective width of the stripe for the second stage. This approximation substantially simplifies
Each layer above active layer for a stage under consideration is characterized by its
thickness, doping and mobility. Dependence of mobility on concentration and doping is taken
from  and  respectively. Parameters used for the calculations below are listed in Table 3-
The following parameters were fixed in the program: L = 750pum, T = 300K,
D, =19.35cm2/Vs, A=0, B=0.8*10-tocm3/iS, C.=3.5*10-30cm 6Sand d=140A. Also, n,
was adjusted before each calculation in such way that total current was around 200mA (below
threshold for the structure).
Simulation results are presented in Table 3-2. As expected, current spreading in the layers
of the first stage is low. However, current width at the second active region was calculated to be
97Clm, 22 Clm wider than the stripe width.
Near field for the DSL measured at 1A is shown in Figure 2-7. FWHM of the intensity
distribution at the second active region was estimated to be 100 lm. It was also observed that NF
width didn't substantially change with current and it was approximately the same at current close
to the threshold of the second active region. Therefore, assuming that NF width is close to the
width of the current density distribution, simulation and experimental results are in good
agreement. As mentioned above, the current spreading model did not include the effective
resistivity of the QWTJ under the assumption that it should be small. This assumption is
validated by the good agreement between the model used and the measured NF of the second
AR. Therefore, we showed that even though TJR is finite (Figure 2-8) it's not high enough to
cause any additional current spreading in the bipolar cascade laser.
This model conclusion was consistent with the following experimental results. The DSL
structure was grown again using the design described above. However, QWTJ used in the new
structure was different: GaAs (n-doped with Te above 1019cm-3) 0.15lGao.ssAs (10nm thick
quantum well)/ GaAs (p-doped with C above 5-1019cm-3). Measured voltage vs. current
characteristics and near field intensity distribution for a laser fabricated from this structure are
shown in Figure 3-3 and Figure 3-4 respectively. The IV curve shows that QWTJ effective
resistivity is very low (due to higher n-doping of the GaAs QWTJ). However, near field
measurements demonstrate that despite this fact there is still a considerable lateral current
spreading between the first and the second active regions.
In conclusion, in this work it has been demonstrated that the employment of a deep QW
inserted between TJ layers with relatively low doping densities can be used to fabricate DSLs
with slope efficiencies and DQEs close to twice that of SSLs. It was also demonstrated that a 1D
model can be used to accurately calculate the current spreading in DSLs provided that a QWTJ
with low effective resistivity is used. It is expected that this type of QWTJ should be of use in
any device requiring monolithically stacked diodes where material growth limitations require
that the doping level densities in the TJ layers be kept relatively small. Future work in this area
could be related to understanding how width of QW in the TJ influences its effective resistivity.
Such information would be very useful for future device design.
h p'= p-h V TJ
Figure 3-2. Illustration of the current flow in stripe geometry SSL.
| 10 I
Figure 3-1. Structure (comprising tunnel junction) used for illustration of influence of TJ
resistivity on current spreading.
Voltage gradient is negligibly small
3.5 j~voltage twice that for SSL
0.0 0.5 1.0 1.5 2.0
2.5 3.0 3.5 4.0
Figure 3-3. Voltage vs. current characteristics for the new double-stage and single-stage lasers.
Figure 3-4. Near-field pattern of a new double-stage laser measured at 1A.
Table 3-1. List of parameters used for simulation of current spreading in bipolar QCLs.
Al composition Thickness, nm Doping, 1017cm-3 Mobility, cm2 Vs
P+ 0 200 200 70
P 0.3 50 10 67
P 0.6 1300 10 25
P 0.3 700 4 78
P 0.1 20 0.05 280
N 0.1 20 0.05 5900
N 0.3 300 4 950
N 0.6 20 4 70
N 0.3 2500 10 630
N 0.2 50 10 1100
n 0 50 50 1200
p 0 50 1000 40
p 0.3 50 10 67
p 0.6 1300 10 25
p 0.3 700 4 78
p 0.1 20 0.05 280
Table 3-2. Current spreading simulation results for bipolar QCLs.
Effective Output Leakage current, mA 0, Ohms
w width, p~m IOhmic if total
1st stage 75 80 35 2 37 185
2nd Stage 80 97 73 1 74 38
UNIPOLAR QUANTUM CASCADE LASERS
4.1 Basic Operational Principles of Unipolar QCLs
Operation of unipolar (intersubband) QCLs relies only on one type of carrier in contrast to
bipolar QCLs where light emission occurs due to radiative recombination between holes and
electrons. The best performance for intersubband QCLs was demonstrated for n-doped devices.
In these devices light is generated due to radiative electron transitions between energy levels
localized in conduction band (Figure 4-1). Electrons tunnel through the injector barrier from the
inj section region to the upper laser level. The radiative transitions occur between the 3rd and the
2nd energy levels. Calculation procedure of electron energy levels and wavefunctions in layered
structures is discussed in Appendix. Initial and final states of the intersubband transitions have
approximately the same curvature of the energy vs. wavevector dependence (Figure 4-2). As a
consequence, joint density of states corresponding to these transitions and gain spectrum are
substantially narrower than for interband transitions. In addition to radiative transitions between
the 3rd and the 2nd levels, there are parallel non-radiative transitions between these levels through
emission of longitudinal optical phonons. These transitions are very fast (~5ps) and as a
consequence strongly increase laser threshold current density. To create population inversion
between the upper and lower laser levels, energy separation AE21 between the 2nd and 1st energy
levels is usually designed to be equal to the energy of the longitudinal optical phonon (~34meV).
In this case transition time 221 between these levels is very short (~0.5ps) and T21< 232. However,
if AE21 1S Smaller than the energy of the longitudinal optical phonon, this fast process is
prohibited and transition between the 2nd and the 1st levels occurs through emission of acoustical
phonons which is a much slower process (~100ps). As a consequence, lasing can be
unachievable in this case. Electrons get recycled through tunneling from the 1st energy level to
the miniband (multitude of closely spaced (in energy) levels) of the inj section region. A reverse,
undesirable effect (so-called backfillingg') that reduces population inversion between the laser
levels occurs when the quasi Fermi level of this miniband is located close to the lower laser
As discussed in Section 1-1 light emitted by unipolar QCLs has TM polarization in
contrast to TE polarization typical for interband transitions (diode lasers). Therefore, to avoid
high losses usually cladding layers in unipolar QCLs are designed to decouple the guided mode
from the plasmon mode propagating at the metal/semiconductor interface. The opposite design
approach is to employ surface plasmon waveguide. In this configuration there is no need to use
cladding layers which can be helpful to improve heat dissipation in these devices. To illustrate
corresponding mode the following 3-layer structure is used. One micron thick active region
characterized by refractive index equal to 3.2 is sandwiched between a metal with n = 2.0 + 32i
and substrate with n=2.8. Calculated transverse intensity distribution for this structure is shown
in Figure 4-3. Gamma factor (active region (including inj ector layers) confinement factor) was
found to be 93%, much higher than typical values ~60-70% (including inj ector layers).
Calculated effective refractive index and intensity loss were 3.13 and~-100cml respectively.
Therefore, disadvantage of this configuration is high loss and as a consequence high threshold
current density. Figure 4-3 shows discontinuity at the interface between the active region and the
substrate, the consequence of the boundary conditions particular to TM-polarization.
Unipolar QCLs are usually fabricated in the high-ridge configuration since surface
recombination is not present in unipolar devices. This helps to reduce threshold current densities
for these devices. Typical length for unipolar QCLs is in the range of several millimeters.
Unipolar QCLs are used in atmospheric sensing, medical breath analysis, process
monitoring and food production. Future possible applications for high-power devices include
infra-red countermeasures and free space optical communications.
4.2 Review of Previous Work of Unipolar QCLs
Concept of the intersubband cascade configuration for light amplification was suggested in
1971 . However, the first quantum cascade laser was demonstrated more than twenty years
later at Bell Labs in 1994 . It became possible due to high growth precision of molecular beam
epitaxy (MBE) and development of band structure engineering.
InP-based structures were used for fabrication of the first QCLs. This choice of material
allows employment of heterojunctions based on In0.53Ga0.47As-Al0.48In0.52As layers lattice-
matched to InP. High conduction band discontinuity (~0.5eV) of this composition makes it
possible to fabricate QCLs emitting at relatively low wavelength (below 5 Cm). In addition, InP
has low refractive index and as a consequence can be effectively used as a cladding layer. Strain-
compensated InP-based QCLs were reported in . In these structures barrier height can be
increased relative to unstrained In0.53Ga0.47As-Al0.48In0.52As composition. However, In and Al
percentage in the barrier and quantum-well layers should be changed simultaneously to avoid
strain build-up in the structure. InP-based QCLs were demonstrated to operate at room
temperature in continuous mode  with hundreds mW of output optical power .
GaAs-based QCLs were realized for the first time in 1998 . Since AlGaAs-layers are
almost lattice-matched to the GaAs-layer independent of Al composition, this structure allows
more design flexibility compared to InP-based material. However, it should be taken into
account that AlxGal-xAs-structure becomes indirect when x>0.45. In this case, scattering to X-
valleys can be harmful for laser performance . GaAs-based QCLs performance at 80K was
demonstrated to be as good, if not better, as for InP-based lasers and it is steadily improving .
It was also shown that employment of active region with deep quantum wells (In0.3Gao.7As) can
be used to substantially reduce carrier leakage from the inj ector region to continuum and as a
consequence increase tunneling inj section efficiency to the upper laser level .
Gas spectroscopy applications for unipolar QCLs require laser linewidth to be below Icm~
,substantially less than typical linewidth for edge-emitting QCLs (>10cm '). Distributed
feedback (DFB) configuration, where a grating is introduced in the structure, proved to be very
efficient for reduction of the linewidth for diode lasers. First DFB unipolar QCL was
demonstrated in . Linewidth for DFB QCLs was reported to be below resolution of FTIR
spectrometer (0.125cm- ). Wavelength in this case can be adjusted with temperature and current
variation since refractive indices of the layers composing the structure depend on these
parameters. Typical adjustment rates are 0.5nm/K and 20nm/A respectively and wavelength
adjustment usually lies in the range 30nm 100nm. Further increase of the scanning range can
be achieved through employment of the external cavity configuration and bound-to-continuum
active region design . In these devices emission wavelength can be controlled with position
and angle of the external grating. Wavelength for external cavity QCLs can be varied by ~ 100 -
400nm. Output power in cw-mode in this case can be as high as several hundreds of mW [32,
33]. Second order DFB (surface-emitting) configuration can be used to reduce strong beam
divergence typical for edge-emitting QCLs from ~60ox150 to ~ lox150 . Maximum output
optical power for these devices working in cw-mode on a Peltier cooler is in the range of tens of
New type of intersubband QCLs emitting in far-infrared range (~100Clm and above),
terahertz QCLs, was reported in 2002 . In this work lasing was based on radiative transitions
between minibands. Also, guided mode was confined by two metallic claddings, which
decreased mode losses and increased confinement factor for the active region. Terahertz QCLs
are still limited to low-power, low-temperature operation.
4.3 Low-Ridge Configuration Concept
High-peak-power, pulsed-operated quantum cascade lasers (QCLs) operating in the first
and the second atmospheric windows are being developed for use in application areas such as
infra-red countermeasures, free space optical communication and laser detection and ranging
(LADAR). Previous work on such devices [37, 38] employed optimized structural designs in a
narrow width, standard high-ridge configurations (Figure 4-4). Another approach of getting high-
power QCLs is to increase width of the high-ridge.
In this work an alternative approach for achieving high-peak-pulsed power QCLs is
described, that uses a narrow width, low-ridge configuration (Figure 4-5). Figure 4-5 includes
low-ridge laser dimensions used in this work
To understand operation of low-ridge unipolar QCLs it' s first necessary to understand
physical mechanisms responsible for the shape of IVs for high-ridge QCLs. Typical IV curve for
a high-ridge unipolar QCL is shown in Figure 4-6. At low bias the inj ector and the upper laser
levels (Figure 4-1) are misaligned. In this range the active region is in the high differential
resistance mode (low effective conductivity). As the applied voltage increases (8V 12V) these
levels line-up and differential resistance substantially decreases. At voltages above 12V this
alignment breaks again and the active region becomes resistive. In this work it is suggested that
this mechanism can be used to design high power, low-ridge unipolar QCLs with relatively low
The basic mechanism that allows narrow width, low-ridge QCLs provide high output
power with relatively low threshold currents is lateral current spreading (spreading perpendicular
to the laser axis in the plane of the layers). As shown in , lateral current spreading mainly
occurs in layers above the active region and decreases when the active region transverse
conductivity increases. At low bias, the active region conductivity is low and the current density
distribution is wide. At higher bias, the conductivity of the central part of the active region
begins to increase (as the inj ector and upper laser levels align). As a consequence, the width of
the current density distribution (characterized by its full width at half maximum, FWHM)
decreases. This narrowing of the current density distribution is the reason for the relatively low
threshold current of these devices. When the applied voltage exceeds the voltage value that
causes misalignment between energy levels, the conductivity of the central part of the active
region diminishes causing current to spread laterally and the current density width to increase.
This additional lateral current spreading effect, not taken into account in a previous current
spreading model of low-ridge QCLs , allows higher peak powers than expected to be
achieved. The previous model  and the modifications required to include the additional
lateral current spreading effect are described below.
Low-ridge unipolar QCLs were reported in several previous works. In [40, 41] it was
demonstrated that low-ridge QCLs can to give substantially higher output optical power than
high-ridge lasers fabricated from the same wafer (for the same ridge width and length). However,
low-ridge lasers in this case were treated just as broad-area devices. In  low-ridge
configuration helped to improve heat dissipation, which substantially increased characteristic
temperature of the laser. In this work proton implantation through the active region was used to
suppress the lateral current spreading.
4.4 Waveguide Structure for the Low-Ridge QCLs
Low-ridge quantum cascade structure realized by our group comprised the active region
design reported in  (Figure 4-7) embedded in the waveguide discussed below. In  using
this active region design it was demonstrated that MOCVD-grown InP based QCLs are capable
to demonstrate as good performance as IVBE-grown devices. The sequence of layers for the low-
ridge QCL was the following: InP low-doped substrate (S, 1-2-107cm-3), 2Clm InP cladding layer
(Si, 1017cm-3), 300A InGaAsP graded layer (Si, 1017cm-3), 3000A InGaAs waveguide layer (Si,
3-1016cm-3), 1.5Clm active region comprising 30-stage AllnAs-InGaAs QC strain-balanced
structure [43, 44], 3000A InGaAs waveguide layer (Si, 3-1016cm-3), 300A InGaAsP graded layer
(Si, 107cm-3), 2Clm InP cladding layer (Si, 1017cm-3), 0.2Clm InP contact layer (Si, 1017cm-3,
100A InGaAs top layer (Si+).
Input parameters required for transverse waveguide calculations include imaginary and real
parts of the refractive index for each layer. At low photon energy limit (comparable with thermal
electron energy koT) these parameters can be obtained using classical Drude theory. Theory
e = 8,-ie, = N (4-1)
N =n -ik (4-2)
where El and E, are real and imaginary parts of the complex dielectric constant E n and k are
real and imaginary parts of the complex refractive index N.
e &(-25 09 (4-3)
82 __ (,/2 29)1/W (4-4)
where E, is the high frequency dielectric constant, r is the electron scattering time
nr = 47tn e2 ,, (4-5)
m /e (4-6)
q=_1+ 1/02 2) (4-7)
For the real and imaginary parts of the refractive index we get
n= (e+e,)/2 1/2 (4-8)
k= 62 /2nZ= (E-e//2 1/2 (4-9)
e= e2 1/2,
Quantum-mechanical extension of this theory described in [45, 46] gives
-[e_/(1-X)] 1-8 u02o) (4-10)
X = Aco/E, (4- 11)
Also, relaxation time z in Eq. 4-4 was found to be a function of both photon energy and layer
doping concentration. Plots of z for InP and GaAs at 300K can be found in .
Near field (NF) transverse intensity distribution calculated based on the parameters listed
in Table 4-1 (calculated based on Eq. 4-1 Eq. 4-11) is shown in Figure 4-8. Gamma factor for
the active region (including inj ector layers) was found to be ~78% with real and imaginary part
of the refractive index equal to 3.3 5 and 1.4E-5 respectively. Losses corresponding to the
imaginary part of the refractive index can be calculated using
47r Im n7*
a = (4-12)
In our case mode loss is below Icm l. It is important to mention that calculated losses are usually
significantly smaller than losses obtained from experiment. The reason for this effect is still not
completely understood . Full width at half maximum (FWHM) of the corresponding Far-
Field (FF) transverse intensity dependence on the emission angle (Figure 4-9) was calculated to
be approximately 68o. Calculated Gamma Factor for the active region (including inj ector layers)
was found to be ~80% with real and imaginary parts of the refractive index equal to 3.3 5 and
1.46E-5 respectively. Also, FWHM of the FF was found to be ~700. Therefore, transverse NFs
(and corresponding FFs) are almost the same under the ridge and under the channels despite
substantially thinner cladding layer thickness in the latter case (0.8Clm vs. 2.0plm). The reason for
this effect is the low refractive index of the Si3N4 USed as an insulator for the low-ridge QCL.
Basically this layer acts as a cladding layer separating mode from lossy gold contact above it.
4.5 Fabrication of the Low-Ridge QCLs
The QC wafer was grown by low pressure MOCVD at a slow rate (0.1nm/sec) in the same
reactor as in  and under essentially the same conditions (except for growth uncertainties).
1.4Clm-high, 25Clm-wide ridges were etched in the wafer using pulsed anodization etching (PAE)
. The channel width on the both sides of the ridge was defined to be 50Clm. The surface was
then passivated by 300nm of Si3N4, deposited by plasma-enhanced CVD at 3000 C. Metal
contact-windows 12Clm wide were opened on top of the ridges by photo-lithography. The Si3N4
in the openings was etched by RIE. The substrate was then thinned to approximately 120Clm by
mechanical lapping. Non-alloyed contact metals of Ge(1 2nm)/Au(27nm)/Ag(50nm)/Au( 100nm)
were deposited on the substrate side of the wafer and Ti(10nm)/Au(400nm) were deposited on
the top by metal evaporation. Finally, the wafer was scribed into chips of dimensions 2.5mm by
SEM pictures of the fabricated low-ridge QCLs are shown in Figure 4-12. Both insulator
and contact layers are smooth and etching quality is good.
As discussed above etching was done using so-called pulsed anodization etching
technique. This is a fast, inexpensive and safe procedure. PAE setup is shown in Figure 4-13.
Solution is composed of glycol (40): water (20): 85% phosphoric acid (1) (GWA). Generator
drives 50V, 100Hz and 0.7ms-wide pulses through the solution. When the pulse is on, OH- ions
are attracted to the positive polarity applied to the sample. As a consequence, due to chemical
reaction native oxide is growing on the sample surface. When pulse is off GWA solution mildly
etches the native oxide. In result, the oxide slowly propagates through the structure. When
desired etching depth is achieved, native oxide can be removed with BOE or KOH solutions.
Etching rate can be increased by adding a small percentage of BOE (GWA (8:4: 1) 650ml: BOE
(7:1) 7ml) (Figure 4-14).
4.6 Experimental Results for Fabricated Unipolar Low-Ridge QCLs
Operating parameters for all testing procedures were 60ns pulse width at 5kHz repetition
rate. The lasing wavelength measured with Fourier Transform Infrared Spectrometer (FTIR)
equipped with Mercury-Cadmium-Telluride (MCT) detector cooled to 80K was found to be
5.3Clm. Figure 4-15 shows the power vs. current (P-I) curves for the low-ridge QCLs. The laser
was placed in cryostat and light was focused using two Ge lenses on a room-temperature MCT
detector. At 80K threshold current, maximum optical power per facet and slope efficiency per
facet were measured to be 2A, 6.7W and 730mW/A respectively.
The same chip with high-reflectivity coated back facet demonstrated at 80K a threshold
current of about 1.3A and a peak output power of about 12W. This is the record high power
reported at this temperature. At 300K these characteristics were 4A and 2.2W respectively.
Spectral measurements performed at 80K at currents 4A and 20A didn't reveal substantial
spectral changes with current increase (Figure 4-16).
I-V characteristics and corresponding P-I curves measured up to 5A at multiple
temperatures are shown in Figure 4-17. At higher currents precision of the I-V measurements
decreased due to impedance mismatch between the laser and a transformer used to extend the
driving range of the current generator. Measurements of threshold current temperature
dependence based on P-I data (Figure 4-18) showed that the characteristic temperature To was
Far-Field intensity dependence on lateral angle is shown in Figure 4-19.
Electroluminescence below threshold current is symmetrical. However, it becomes asymmetrical
above threshold. The angle corresponding to intensity maximum slowly increases as current
increases. Also, spectrum broadens with current increase. As discussed in  the beam steering
in high-power QCLs can be explained using the concept of interference between the two lowest
order lateral modes. A small difference in the effective refractive index between these modes
causes beating along the stripe length. As a consequence, the angle corresponding to the
maximum of the FF lateral intensity distribution shifts from one lateral side to another depending
on the phase shift between the modes at the output facet (depending on the position within
beating period at the output facet). This phase shift is influenced by the current since it changes
the effective refractive indices of the modes. As a consequence, the lateral angle corresponding
to maximum of the FF distribution shifts with current change. Maximum steering angle in 
was reported ~100, approximately the same as measured for our low-ridge lasers. For low-ridge
QCLs however there is a possibility of existence of several lateral modes at high current since
effective stripe width (current width) is substantially bigger in this case (>70Clm instead of 13Clm
The experimental data presented above demonstrate that low-ridge QCLs are capable of
giving very high peak-pulsed optical powers with relatively low threshold currents. Also, the
qualitative mechanism on current spreading responsible for operation of these devices was
suggested. Far-field measurements demonstrated beam steering. This indicates presence of
several lateral modes. Spectrally resolved near-field measurements would be very useful for
better understanding of the FF behavior.
High-ridge lasers from the same wafer were made with the goal of comparing their
performance with the low-ridge lasers. However, due to processing issues, reliable lasing was not
achieved for the high-ridge lasers. Since we couldn't repeat the experiment due to lack of
material, we modeled current spreading in the low-ridge lasers based on high-ridge data
presented in . In the next section a 2-D current spreading model quantitatively supporting the
current spreading mechanism suggested in this work will be developed.
Electric field across the active layers is in the range of 10Ot of kV/cm
Figure 4-1. Schematic of electron transitions in unipolar QCLs
Figure 4-2. Schematics of intersubband and interband transitions. For intersubband transitions
joint density of states is substantially narrower.
0.0 0.5 1.0 1.5 2.0
Transverse direction (clm)
Figure 4-3. Transverse near field distribution for the surface plasmon waveguide.
Figure 4-4. High-ridge configuration for unipolar QCL.
,--- 1 .0
Figure 4-5. Schematic of the low-ridge laser (dimensions are given in microns).
Figure 4-6. Typical IV curve for a high-ridge unipolar QCL.
S~i 1 ;111
Figure 4-7. Energy band diagram of the active region used in the low-ridge structure.
Trasnverse distance (pm)
Figure 4-8. Transverse NF distribution in the low-ridge QCL under the ridge.
0.0 g g g g g g g .
-80 -60 -40 -20 0 20 40 60 80
Figure 4-9. Far field distribution corresponding to NF under the ridge.
j 08 lk =1.46E-5 3.
'r 0.6~ I 3.0
C 0.4- J1 2.5
S0.2 -1 -1 2.0
0.0 I 1.5
0 2 4 6
Transverse direction (pLm)
Figure 4-10. Transverse NF distribution in the low-ridge QCL under the channel.
I FWHM ~700
S0.0 6 ~ ~ ~ g g g g g
Fiur 41. arfel dsriuto crrsonin o F nerth hanes
Figure 4-12. SEM pictures of the fabricated low-ridge unipolar QCLs.
'i'cRan f9ver J i
Figure 4-13. Pulsed anodization etching setup.
0 2 4 6 8 10
Figure 4-14. Etching rate of InP with GWA (8:4:1) mixed with BOE in ration 600 to 7.
Etching rate through InP cladding
layer ~ 0.3pLmmimn (linear region)
110 -80K, HR Coated
80K, No HR Coating
Sm 300K, HR Coated
0 5 10 15 20
Figure 4-15. Power vs. current characteristics for the realized low-ridge QCL.
Figure 4-16. Laser spectra measured at 4A and 20A taken at 80K.
0.0 1.0 2.0 3,0 4.0
Figure 4-17. Voltage vs. current characteristics and corresponding power vs. current curves
measured at multiple temperatures (laser is without HR coating).
Figure 4-18. Dependence of the threshold current on temperature.
- 0.95 x Ith (x10)
- 2.05 x Ith
- 3.1 x Ih
- 4.43 x Ith
-30~ ~ -20 -1 0 0 0
Far Field Angle (~deg.)
Figure 4-19. Far-field intensity distribution for a low-ridge QCL.
To = 172K
calculate transverse NF distribution in the low-ridge QCL
Layer um n 2*nr*k/A
Gold inf 1.83 42.3
InP, 1E19 0.2 2.63 0.005
InP, 1E17 2 3.1 0.0000362
InGaAsP, 1E17 0.03 3. 18 0.0000471
InGaAs, 3E16 0.3 3.26 0.000008058
AR 1.53 3.49 0.000013
InGaAs, 3E16 0.3 3.26 0.00000806
InGaAsP, 1E17 0.03 3. 18 0.0000471
InP, 1E17 2 3.1 0.0000362
InP, 2E17 inf 3 0.0000697
Layer Thickness, um N 2*nr*k/A
Gold Inf 1.83 42.3
Si3N4 0.3 2 0
InP, 1E17 0.8 3.1 0.0000362
InGaAsP, 1E17 0.03 3. 18 0.0000471
InGaAs, 3E16 0.3 3.26 0.000008058
AR 1.53 3.49 0.000013
InGaAs, 3E16 0.3 3.26 0.00000806
InGaAsP, 1E17 0.03 3. 18 0.0000471
InP, 1E17 2 3.1 0.0000362
InP, 2E17 Inf 3 0.0000697
List of parameters used to calculate transverse NF distribution in the low-ridge QCL
under the ridge
Table 4-2. List of parameters used to
under the channels.
CURRENT SPREADING MODEL FOR UNIPOLAR QCLS
5.1 Previous Work on Current Spreading Simulation in Unipolar QCLs
Current spreading model for unipolar, stripe-contact QCLs was developed in .In this
work it was suggested that this configuration can be useful for optimal heat dissipation and
single spatial optical mode operation. It was also explained that current spreading in these
devices is substantially different from current spreading in diode lasers. In particular, since
carrier concentration in each stage of the active region is fixed by inj ector doping, there is no
diffusion spreading component in the active region. As a consequence, there is only Ohmic
current spreading component given by
J(x, y) = e V V(x, y) (5-1)
a= (x,,"y) (5-2)
Also, the effective charge separation in each stage is negligible compared to typical scale of the
current spreading, it can be assumed that there is no space charge in the active region. As a
consequence, using charge conservation law we get
Substitution of Eq. 5-1 into Eq. 5-3 gives for each layer
82V(x, y) 1 82V(x, y)
+ = 0 (5-4)
8x2 2 ~2
Solution to Eq. 5-4 was obtained in  extending solution given in . For each layer of
the QC structure (excluding the active region), the conductivities ax and oy were taken to be
isotropic ( ox = o, ) and equal to the bulk conductivity of the layer material. Lateral conductivity
x,, for the active region was calculated as a weighted average of the bulk conductivities of all of
the layers composing the active region. Transverse conductivity oy for the active region was
assumed to be constant and was found by fitting simulated IV curve to the measured one. Using
this approach, it was found in  that current spreads mostly in the layers above the active
region and current width decreases as current increases.
In our work we extended this model by taking into account lateral variation of the active
region effective transverse conductivity. Also, we extended simulation above voltage
corresponding to roll-over point of the PI characteristic.
5.2 Two-Dimensional Current Spreading Model
5.2.1 Active Region Transverse Conductivity
Schematic diagram of the low-ridge QCL is shown in Figure 5-1. X-axis was chosen to be
along the lateral direction and y-axis along the transverse direction and pointing downwards.
These axes intersected at the center of the top of the contact stripe. It was assumed that voltage
applied to the ridge was Vo and voltage applied to the substrate was zero. Also, since there was
no electrical current flowing through the rest of the boundaries, corresponding spatial voltage
derivatives -or taken at these boundaries were also assumed to be zero (see Figure 5-1).
As mentioned above, each layer of the structure is characterized by its transverse and lateral
conductivities and thickness.
Eq. 5-4 is valid in a layer with constant transverse and lateral conductivities. In the general
case, both of these quantities can have spatial dependence. Therefore, Eq. 5-4 should be modified
e + a = 0 1 ; (5-6)
In this form it is applicable to the whole layered structure. Following , ax for the AR was
calculated as a weighted average of the bulk conductivities of all of the layers composing the
AR. However, a, for the active region depends on the active region design and the voltage
across the active region:
o, =f (V,) (5-7)
The dependence of the active region transverse conductivity on the voltage across the active
region o,, (V,) (Eql. 5-7) can be calculated from the active region IV for a high-ridge laser
fabricated from the same wafer as the low-ridge laser. The wafer discussed in [44, Fig.2a] was
grown in the same MOCVD reactor as our structure under similar conditions and has the same
active region design. Since cladding layers, waveguide layers and substrate have high
conductivities we believe that the IV measured for the high-ridge laser reported in  is a good
approximation to its active region IV (For example, voltage across the high-ridge QCL in Figure
2a at 0.75A is equal to 12V. Using typical conductivity values for waveguide layers, cladding
layers and substrate it can be shown that total voltage drop across these layers at 0.75A is less
than 0.3V. Therefore, if voltage drop across the contacts is small, the laser IV is basically
determined by its active region IV). As a consequence, we believe it is appropriate to use the IV
reported in [44, Figure 2a] for simulation of current spreading mechanism in realized low-ridge
QCLs. This IV and corresponding dependence of active region transverse conductivity on the
voltage drop across active region is shown in Figure 5-2a, b. Part of the IV curve in the range
from 13V to 18V was obtained by the extrapolation (dashed line) to emphasize the effect of
current spreading when transverse conductivity of central part of AR gets low. Figure 5-2b was
obtained assuming a uniform current density flowing through the active region in the high-ridge
lasers reported in [44, Figure 2a]. Using o- (91) shown in Figure 2b, Eq. 5-6 and Eql. 5-7 were
solved self-consistently for a low-ridge QCL using the 2D Finite-Difference Method. Calculation
procedure is described in the next section.
5.2.2 Two-Dimensional Finite-Difference Method
The two dimensional Finite-Difference method is a numerical type of solution to a
82 u2 8 u du
a C2+bZ+c-d+d -+ eu + f = 0 (5-8)
dUx2 ;,idUx2 hy~+
d221 d221 21 +1 2uI~ +ulg+
I-I I- I -1~(5-11)
\ x I,\ x I, 2h
-I= -I, 2 (5-12)
Usually the calculation procedure is done as follows. The structure is divided by a rectangular
mesh (Figure 5-3) into cells. Boundary u-values corresponding to the physical problem are set at
the boundary points of the mesh and initial 'best guess' u-values are set at the all inner points.
Boundary conditions are left unchanged through the calculation procedure. U-values at each
point P are found from the u-values of the four neighbor points of the mesh using Eq. 5-8
through Eq. 5-12. This procedure (calculation of u-values at all the points) is repeated iteratively
until the convergence condition is achieved (convergence rate strongly depends on the 'initial
guess' of the u-values at the inner points). Using found values at each point of the mesh all
related quantities such as can befound.
\ x P,
In our case we had to solve Eq. 5-6. Using 2D Finite-Difference Method this equation can
be modified as
+- = 0 (5-13)
(e)' 28 ex' 8x0 26y -1 2 y =0 (5-14)
Making the transformations 26x 3 x and 23y S y then gives
Using Eq. 5-15 we can find V-values at each point P(i, j) using V-values at neighbor points (i+1,j;
i-lj; i3j+1; i3j-1) Figure 5-4. Also, conductivity values should be set at points C (i+1/2,j; i-1/2,j;
Figure 5-5a, b illustrate the rectangular mesh used for the numerical calculation.
Increments along x and y-directions, Sx and 6y we chosen to be 1 Clm and 0.2 Clm
respectively. As discussed above, voltage on the top of the ridge and at the bottom of the
substrate we fixed at Vo and 0 respectively. Since there is no current through all the other
boundaries we can assume that voltage for the neighbor points at each boundary should be the
same. For example, since there is no current flow through the left vertical edge of the laser, at
this edge the following equation should hold
= 0 (5-16)
Employing the discrete approach used here we get
0, "= (5-17)
Therefore, at the left vertical boundary the following condition is true
Vo~ = V~~ (5-18)
Calculation procedure is shown in Figure 5-5. As mentioned above, initial voltage values
can strongly influence convergence rate. However, in our case convergence rate wasn't very
important. Therefore, for simplicity initial voltage values at the all inner points were taken to be
equal to zero. Initial value for o~ is taken as 1R Omv Each cycle of the solution of Eq. 5-6
(using Eq. 5-15) involves calculation of the voltage value at each point of the structure using
voltage and conductivity values at the neighbor points (Figure 5-4b). This cycle is repeated until
the convergence condition is satisfied. After that the active region transverse conductivity values
are updated using Eq. 5-7 that reflects the dependence given in Figure 5-2b. If new values are
close enough to the old ones, current density distribution can be calculated using obtained
voltage distribution. If not, Eq. 5-6 should be solved again using the new active region transverse
5.3 Model Results
Parameters used for calculation are listed in Figure 5-6. Since substrate conductivity is
high it is appropriate to neglect by voltage drop across the substrate. Therefore, its thickness was
taken to be ~7Clm instead of 150Clm defined by the wafer processing. Active region lateral
conductivity was borrowed from . In this work it was shown that model results aren't very
sensitive to the choice of this parameter.
A convenient way to interpret the model results is to examine the evolution of the lateral
distribution of the active region o, as a function of bias change (Figure 2a), and by doing so,
determine its influence on current density distribution (Figure 2b), current width and the low-
ridge IV curve (Figure 2c). When voltage across the low-ridge device is below approximately
6V, transverse conductivity of the AR is low and almost constant. Due to the low conductivity,
current density distribution at the AR for these voltages is broad and has large tails. When the
device bias increases above 6V, the conductivity of the central part of the AR increases rapidly
with increasing voltage, as can be seen in Figure 2b. The high transverse conductivity of the AR
central part and the peak shape of the transverse conductivity lateral distribution lead to a strong
narrowing of the current density profile. Further voltage increase leads to decrease of the
transverse conductivity of the central part of the AR. Due to the decrease in conductivity, current
starts to spread in the lateral direction and the effective current width goes up (Figure 2c),
extending the laser dynamic range. Spreading is expected to be stronger at lower temperatures
since mobility increases rapidly as the device temperature decreases.
The IV curve for the low-ridge laser does not show a strong increase in differential
resistance at high current (Figure 2c), the effect shown in Fig.1la for the high-ridge laser. This
difference is a consequence of lateral current spreading in the low-ridge structure when the
transverse conductivity of the central part of the AR goes down.
The simulation results show that low-ridge QCLs have relatively low threshold current due
to current narrowing when the central part of the active region enters high-conductivity mode
(it' s usually by factor of 2-3 higher than threshold for high-ridge lasers processed from the same
wafer). It is also demonstrated that high powers achievable by these lasers may be due to lateral
current spreading at higher voltages. The work discussed here on the low-ridge, unipolar QCLs is
summarized in .
Figure 5-1. Schematic of the low-ridge QCL.
0.0 0.2 0.4 0.6 0.8
> 0 10.5-
U 4 8 12 16 20
Figure 5-2. A) High-ridge IV curve. B) the corresponding dependence of the transverse active
region conductivity on the voltage across the active region.
1~ l -1 1 +1
0 h 1, -
Figure 5-3. Rectangular mesh used for 2D Finite-Difference Method.
a) ... x
-- --I 1----r----I -r -
(i-lJ j)P(i3j) (i1)
Figure 5-4. A) Rectangular mesh used for the low-ridge QCL. B) Basic cell of the mesh.
Solve Eq. (5-6)
Find new values for oy in the
active region using Eq.(5-7)
Are new values for a, equal to
the old ones?
Calculate current density
InP 1E19 0.2um
InP 1E17 2.0um
InGaAsP grade 300OA
InGaAs 3E16 0.3um
InGaAs 3E16 0.3um
InGaAsP grade 300OA
InP 1E17 2.0um
Figure 5-5. Schematic of the calculation procedure.
Parameters for the model
2.2um, 9600 0 nzm
0.3um, 6400 0 Z'm
1.5um, lateral 400 0 Z'm
0.3um, 6400 0 nzm-
7.0um, 9600 0 Z'm
Figure 5-6. List of parameters used for simulation.
Set initial values for ox, ay, V and the
boundary conditions at each point of the mesh
2.8-1 I 1 8V
C E 2.4
:.o 0 .
03O 0.4 6V
U 100 200 300 400 500
Lateral distance (pm)
Figure 5-7. Current spreading model results. A) Lateral dependence of the active region
transverse conductivity at different current values. B) Corresponding lateral
distribution of the current density at the active region. C) Low-ridge IV curve and
current width dependence on the current calculated based on (B).
3.0 ] .
2.0- 1 2V
SO 10200 300 400 500
Lateral distance (pFm)
Figure 5-7. Continued
1 8 140 a.
-12 0 -o -
e 10- -o 7 1~10~
U 2 4 6 8 10 12
Figure 5-7. Continued
SUMMARY AND FUTURE WORK
6.1 Bipolar Quantum Cascade Lasers
It was demonstrated in Chapters 2 and 3 that deep quantum wells can be effectively used
for designing tunnel junctions with relatively low effective resistivity. Employment of the
quantum well tunnel junctions is especially important when very high doping density is not
possible for the layers composing the tunnel junction. In this work this approach allowed us to
achieve double stage laser slope 2.0 W/A twice that of the single stage laser even though
maximum doping level for the n-GaAs tunnel junction layer was relatively low (5-10lscm-3.
Results of the developed current spreading model for the double stage laser showed that the
substantial current spreading between the first and the second active regions in our structure (and
corresponding relatively high threshold current for the second active region) was not a
consequence of the quantum well tunnel junction resistivity but rather caused by presence of the
n-layers with high mobility and relatively large total thickness of the layers between the active
Further experimental and theoretical study of the influence of quantum well parameters
(such as quantum well thickness and depth) on the tunnel junction resistivity at different doping
levels of the layers composing the tunnel junction would be very helpful. This information would
make the future design of semiconductor devices incorporating quantum well tunnel junctions
(bipolar QCLs, multiple junction collar cells, etc.) much more flexible since it would
substantially relax the requirement of the very high doping in the tunnel junction layers. Also,
wall-plug efficiency could be reduced due to lower voltage drop across the tunnel junction at the
same current. Since the tunnel junction comprises the highly strained InGaAs layer, reliability
testing is necessary to verify wither the strain in the structure reduces the lifetime of the double
6.2 Unipolar Quantum Cascade Lasers
In Chapters 4 and 5 we demonstrated that low-ridge unipolar quantum cascade lasers can
deliver very high peak pulsed optical powers (12W at 80K for the laser with HR coated back
facet) with relatively low threshold current (~1.5A at 80K for the same laser). Developed 2D
numerical current spreading model takes into account lateral dependence of transverse active
region effective conductivity, in contrast to the original model. Also, the new model was
extended to the high differential resistance range at high bias caused by the coupling breaking
between the inj ector and upper laser electron energy levels. It was shown that the low threshold
current was a consequence of the current density distribution narrowing at voltages
corresponding to a good alignment between the lowest inj ector energy level and the upper laser
level. It was also suggested that the high optical power achieved by the low-ridge laser was due
to the additional current spreading in the structure caused by the increase in the differential
resistance of the central part of the active region at voltages when this alignment breaks. Far-
field measurement results demonstrated current beam steering for the low-ridge lasers. Possible
reason for this effect is the co-existence of several lateral modes with slightly different effective
refractive indices. Far-field intensity distribution is determined by the interference between these
modes and, as a consequence, depends on the phase shift between them at the output laser facet.
The current beam steering occurs as a result of the effective refractive indices dependence on the
current since the phase shift also changes in this case.
Further study of this effect can include Near-Field measurements at different currents. In
particular, spectrally resolved Near-Field observation would allow thorough analysis of the Far-
Field behavior. Also, numerical simulation of the Near-Field and Far-Field intensity distributions
would be very useful for understanding the current steering. This simulation should take into
account the lateral variation of the current density distribution and coupling between the current
density and the optical modes. Finally, since the low-ridge devices operate at high voltages (up
to 40V), reliability testing has to be done to verify whether these lasers can be used in
practical high-power mid-infrared applications. If performance degradation with time is
observed, structural changes improving device reliability should be made. For example, increase
of the contact window width on the top of the ridge to reduce current density flowing through
this region could be helpful.
CALCULATION OF ELECTRON ENERGY LEVELS AND WAVEFUNCTIONS IN
Calculation procedure described here is to a large degree based on the approach discussed
in . In general case it is required to find electron energy levels and wavefunctions in a
layered structure with doped layers taking into account band non-parabolicity. Also, the electron
effective mass can change at the interfaces between the layers.
The governing equations in this case are coupled Schroidinger and Poisson equations:
,,sIy(z)+ V(z)-E yWz)= (A-1)
V2V = _P(A-2)
where nz* is the electron effective mass, V(z) is the potential profile and p is the spatial charge
First let' s consider calculation procedure for Eq. A-1 and then extend the solution to
general case by taking into account its coupling with Eq. A-2. Eq. A-1 can be modified using the
expansion given in Eq. 5-10 (taking into account effective mass discontinuity)
m ( z1 87(z)d I ( z1 87(z)d
=+"= A2" [V(z)-E y(z) (A-3)
1( + z y(z + 23z) ry(z) 1 "( z y(z) ry(z 23z)
which can be modified as
ry(z + z) 23
[V(z)-E]+ + yz
nz(z3:2)A2 n*(z + :/ 2) n *(z :/ 2)
n *(z :/ 2)
Therefore, p(z+5z) can be found using p(z), p(z-5z) and m(z+5z/2), m(z-6z/2).
Solution to Eq. A-1 is found as follows. The (left) boundary condition used for Eq. A-5 is
@(0) = 0 and y(5z) = 1. Using it, p(z) at all the other points is found using Eq. A-5. When the
energy E is not a solution of Eq. A-1, p(z) diverges as : approaches the right boundary
(accumulation of the error). The sign of this divergence changes when the energy passes its
solution value. During the calculation process the energy is varied in a particular range (usually
from the energy corresponding to the bottom of the quantum well layers up to the energy
corresponding to the top of the barrier layers). When the sign of the divergence changes an
iterative procedure is used to find the solution.
When external electric field EEF(z) is applied to the structure, V(z) should be taken as
V(z) 4 V(z) + q EEF(z) (A-6)
Figure A-1 shows results of the calculation procedure described above for a GaAs -based
structure with AlGaAs barriers and InGaAs quantum wells. Applied electric field was assumed
to be 80kV/cm. Interface grading (composition doesn't change abruptly at the interfaces) was
taken into account to create the potential profile for this structure (transition layer thickness was
assumed to be equal to two monolayers).
Inclusion of Poisson equation can be done using found electron energy levels and
wavefunctions. The approximation that is typically used in this case is that all donor atoms are
fully ionized and electrons from these atoms are distributed among the found electron energy
levels. Solution of the rate equations is necessary to find the electron population on each of these
levels. Potential created by positive ionized donor atoms and electrons distributed among the
energy levels can be found using the following formulas.
Assuming that the layered structure is infinite in the x-y plane, electric field (EF(z))
created by electrons and ionized dopant atoms (with doping density d(z)) can be found using
EFz)= sign(z -z ') (A-7)
CN = N= d(z)dz (A-9)
where o-(z) is the areal charge density given by Eq. A-8 and N, is the electron population on the i
electron energy level (found using the rate equations). The latter equation reflects charge
neutrality in the structure. Using found EF(z) corresponding potential can be calculated using
V (z)=- EF(z')dz' (A-10
Therefore, the full calculation procedure for solution of the coupled Eqs. A-1 and A-2 can
be described by the scheme presented in Figure A-2. Eq. A-1 is solved using the expansion given
by Eq. A-5. Found electron energies and wavefunctions are used in the rate equations to find
electron population on each energy level. Using Eqs. A-7 through A-10 potential profile is
updates and compared to the previous profile. If convergence is achieved, all desired quantities
can be calculated using obtained energy values and corresponding wavefunctions. If no, the new
potential profile should be used in the next iteration. Example of an employment of this
procedure for doping optimization in quantum cascade lasers can be found in .
Graded structure transitionn layer thickness is 2ML)
Figure A-1. Model results for a GaAs-based layered structure.
Using found Wi and Ei solve rate equations,
which gives Ni
Using found 9;i, Ni and d(z) find V,(z)
V(z)[new] = V(z)w/o spatial charge]+ V,(z)
Solution of Eq.(A1I) using Eq.(A5) (1st iteration
is done neglecting by spatial charge)
V(z) = V(z)[new]
V (z) [new ] = ? V(z)[ol d]
Figure A-2. Step sequence for the calculation procedure.