c© 2016 Guan-Lin Su

MODELING OF DEVICES FOR GALLIUM-NITRIDE-BASED INTEGRATEDPHOTONICS

BY

GUAN-LIN SU

DISSERTATION

Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Electrical and Computer Engineering

in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2016

Urbana, Illinois

Doctoral Committee:

Associate Professor John M. Dallesasse, ChairProfessor Weng Cho ChewProfessor Milton FengProfessor Jianming Jin

ABSTRACT

Credited as “the most important semiconductor since silicon (Si),” gallium

nitride (GaN) has received a tremendous amount of attention during the

past two decades due to its superb material properties. While the wide

band gap and high electron saturation velocity make it especially suitable

for high-power and high-frequency microelectronics, the ultra-wide spectral

coverage by the direct band gaps of aluminum gallium nitride (AlGaN) and

indium gallium nitride (InGaN) gives nitride semiconductors a central stage

in optoelectronic devices. Various applications such as medical inspection,

sterilization, optical storage, solid-state lighting, and the blue-green light

sources in full-color displays have made GaN-related devices influential in

people’s daily life.

Most of the research to date on III-nitride (III-N) semiconductors has

concentrated on the growth of novel nano-structures, the epitaxy of high-

quality materials, and the fabrication of devices targeting unprecedented

performance. Work on the underlying physics, device diagnosis, or even

the proposal and design of devices with new functionalities has been rela-

tively scarce. Even though the theory of III-N quantum wells (QWs) is fairly

mature, QW structures face overwhelming strain that limits the maximum

indium incorporation and generates a high density of threading dislocations.

These issues lead to difficulties in extending the emission wavelength to the

“green-yellow gap,” low device efficiencies and short lifetimes. These un-

desirable features have been motivating the pursuit of novel III-N active

materials, but in fact understanding the fundamental physics beneath these

nano-structures plays an equally important role, as it serves as an evaluating

or even predicting process for designing new optoelectronic devices.

While a great amount of research energy has been focused on exploring

GaN’s the effectiveness for photon-generating processes, its capability in

photon-manipulating processes has been significantly overlooked. The asym-

ii

metric wurtzite lattice actually implies strong optical effects, which, combin-

ing with the piezoelectric effect, make voltage-controlled optically-responsive

devices possible. Furthermore, taking advantage of the establishing GaN-on-

Si technology, such devices can be integrated with high-speed, high-power

GaN transistor driving circuitries or even be transferred to other material

platforms with a minimal amount of cost. These unique properties not only

make GaN a promising material for optical signal processing but also open

up an emerging field of “GaN integrated photonics.”

The first part of this work focuses on the theoretical study of InGaN/GaN

self-assembled quantum dot (SAQD) gain materials and corresponding lasers

emitting at visible wavelengths. Following a discussion of adopting III-Ns

as an active material in the telecommunications wavelengths, the second

part of the thesis considers the design of novel voltage-controlled photonic

components, such as polarization rotators (PRs) and gratings, for realizing

stable polarization-division multiplexing in coherent optical communication

systems. The last part of the thesis introduces experimental work on the

development of InP-based laser gain chips, which are the most efficient light

source in the optical telecommunications window and will be interfaced with

the GaN-based photonic elements. It is expected that the model of the In-

GaN/GaN SAQDs can be extended to study other nano-structures, and the

model of voltage-controlled PRs and gratings can contribute to conceptual-

ization and realization of GaN integrated photonics.

iii

To my Mom and Dad, for their love and support over almost thirty years.

To my girlfriend, Ran Zhang, for her encouragement and companionship.

iv

ACKNOWLEDGMENTS

First and foremost, I would like to thank my thesis advisor, Professor John

M. Dallesasse, for his guidance, support, and advice throughout the past two

and a half years. It would not be possible for me to complete this thesis

without the experiences, the critical and practical thinkings that he passes

down to me from Professor Nick Holonyak, Jr. I am also very thankful for

his kindness and willingness to offer help when I was experiencing Professor

Shun Lien Chuang’s passing. I also wish to thank Professors Weng Cho

Chew, Milton Feng, and Jianming Jin who served on my preliminary and

final exam committees. Their valuable inputs definitely made this thesis

richer and more rigorous.

I especially express gratitude to Professor Shun Lien Chuang for the as-

piration and encouragement that he gave me when I came to University of

Illinois for the first time in 2010 as an exchange student. I would not have my

faith without his words, let alone become a doctoral candidate. His mental

toughness and courage always keep me moving forward.

I appreciate the assistance and discussions from our group members; Ben-

jamin Kesler and Thomas O’Brien in the process development of the InP-

based gain chips, and Kanuo Chen in the measurement setup of InP-based

edge-emitting lasers.

Most important of all, I thank my Mother for raising me up through those

difficult days. She is always positive about my overseas study and is always

my biggest supporter. I am also grateful for the help, understanding, and

companionship from my girlfriend, Ran Zhang. Her help and warmth are

right beside me along this uneasy journey. Finally I would like to express

my gratitude to the Reverend J. Michael Smith and my fellows at Urbana

Grace Church for the happiness and belief which we share together all the

time. Their unending love means a lot and is always the driving force for me

to achieve this goal.

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TABLE OF CONTENTS

CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . 11.1 Basic material properties of III-nitrides . . . . . . . . . . . . . 11.2 InGaN-based nano-structures for laser applications . . . . . . 41.3 Proposals for GaN-based integrated photonic devices with

voltage-controlled functionalities . . . . . . . . . . . . . . . . . 71.4 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . 10

CHAPTER 2 MATERIAL MODEL FOR InGaN/GaNSELF-ASSEMBLED QUANTUM DOTS (SAQDS) . . . . . . . . . 142.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Strain distribution and piezoelectric effect in InGaN/GaN

SAQD structures . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Electronic states in InGaN/GaN SAQD structures . . . . . . . 192.4 Optical gain in the InGaN/GaN SAQD active material . . . . 25

CHAPTER 3 DEVICE MODEL FOR GaN-BASED RIDGEWAVEGUIDE LASERS . . . . . . . . . . . . . . . . . . . . . . . . 283.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Optical and electronic model of the InGaN/GaN SAQD-

based LD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Temperature-dependent rate equations . . . . . . . . . . . . . 303.4 Parameter extraction and discussions . . . . . . . . . . . . . . 333.5 Discussions of III-N light sources for telecommunications . . . 37

CHAPTER 4 GaN-BASED VOLTAGE-CONTROLLEDPOLARIZATION ROTATOR (VCPR) . . . . . . . . . . . . . . . . 404.1 Introduction and motivation . . . . . . . . . . . . . . . . . . . 404.2 Index ellipsoid, electro-optic and photo-elastic effects . . . . . 414.3 Design of GaN-based VCPR . . . . . . . . . . . . . . . . . . . 444.4 Voltage-controlled dispersion characteristics . . . . . . . . . . 494.5 Discussion and possible improvements . . . . . . . . . . . . . . 53

CHAPTER 5 DESIGN OF TUNABLE LASERS USINGGaN-BASED VOLTAGE-TUNABLE GRATINGS . . . . . . . . . . 545.1 Introduction and motivation . . . . . . . . . . . . . . . . . . . 54

vi

5.2 Grating design based on coupled-mode theory . . . . . . . . . 555.3 Wavelength tuning using sampled grating structures . . . . . . 595.4 Process development and preliminary results of InP-based

gain chips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.5 Discussions and possible improvements . . . . . . . . . . . . . 70

CHAPTER 6 SUMMARY AND FUTURE WORK . . . . . . . . . . 73

APPENDIX A FITTING PARAMETERS . . . . . . . . . . . . . . . 75

APPENDIX B EIGHT-BAND HAMILTONIAN FOR BULKWURTZITE CRYSTALS . . . . . . . . . . . . . . . . . . . . . . . . 78

APPENDIX C STRAIN AND HETEROSTRUCTURE EFFECTS . . 81

APPENDIX D EIGHT-BAND HAMILTONIAN UNDER THECHANGE OF BASIS . . . . . . . . . . . . . . . . . . . . . . . . . . 83

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

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CHAPTER 1

INTRODUCTION

1.1 Basic material properties of III-nitrides

Gallium nitride (GaN), once nearly ignored before the 1990s, has received

its biggest redemption thanks to the substantial improvement in material

quality. Several key technological breakthroughs, such as the development of

novel metal-organic chemical vapor deposition (MOCVD) reactor [1–3], the

incorporation of low-temperature aluminum nitride (LT-AlN) buffer layers

[4], and the activation of p-type doping [5,6], finally minimized the effects of

crystalline imperfections and allowed access to the true optical and electrical

properties that revolutionized the scientific and engineering fields. Today,

GaN has outperformed its early rival, the zinc selenide (ZnSe)-based alloys,

and GaN-based devices are viewed as the most efficient light sources in the

ultraviolet (UV)-blue spectral range.

The technological importance of group III-nitrides (III-Ns) can be easily

understood from Fig.1.1a, which plots the Γ-point band gap energies ver-

sus lattice constants of currently available III-N alloys in wurtzite crystal

structures. Compared with conventional III-V compound semiconductors

(gallium arsenide, indium phosphide systems), AlxGa1−xN alloys have sig-

nificantly larger band gaps, which make them especially suitable for high-

voltage, high-power electronic device applications. The direct nature of the

band gaps across all aluminum and indium composition ranges also provides

efficient light generation from AlxGa1−xN and InxGa1−xN alloys. Moreover,

the rediscovery of the fundamental band gap of InN in 2002 further extended

the spectral coverage of InxGa1−xN alloy down to telecommunications wave-

lengths (the 1310 nm and 1550 nm windows) and excited a great amount of

interest in growing high-quality InN-based quantum structures [7–10].

Due to the large mismatch in electron negativities between the metal an-

1

(a) (b)

Figure 1.1: (a) The Γ-point band gap as a function of lattice constant ofwurtzite III-N alloys at 300 K [11]. The lattice constants of technologicallyimportant substrates are also listed. (b) The wurtzite lattice structure andthe definition of the lattice constants [12].

ions and the nitrogen cation, III-Ns are typically formed in wurtzite crystal

structures, which is much more ionic but less symmetric than the zinc blende

structures commonly seen in conventional III-Vs. While the mismatch of

electron negativity creates a macroscopic internal polarization field (sponta-

neous polarization) pointing from the anions to the cations in the direction

of the c-axis, the lack of centro-symmetry in the lattice structure leads to

the piezoelectric effect, which produces another polarization field (piezoelec-

tric polarization) across the crystal upon being strained [13, 14]. Though

this can be utilized to provide “polarization doping” in AlGaN/GaN-based

transistor structures, the polarization fields actually distort the band edges

in III-N quantum wells (QWs) and spatially separate the electron and hole

wavefunction envelopes. This effect is known as the quantum-confined Stark

effect (QCSE), which is responsible for wavelength drift and low quantum

efficiency in c-plane III-N-based multi-quantum-well (MQW) LEDs [15,16].

Crystalline planes with a surface normal making a non-zero angle (θ) with

the c-axis suffer much less from the polarization fields [17,18]. For example,

QWs grown on nonpolar planes (m- and a-planes, θ = π/2) are completely

free of QCSE. Semipolar planes (0 < θ < π/2), such as 1122 and 2021

2

(a) (b) (c)

Figure 1.2: (a) Calculated band diagrams of two GaN/Al0.1Ga0.9N QWsgrown on c- (left) and m- (right) planes, respectively. The wavefunctionenvelopes and the effective band gaps are also shown [18]. (b) Thedefinition of c-, m- and other semipolar planes in the wurtzite latticecell [19]. (c) High-resolution transmission electron microscopy (TEM)images of GaN grown on (0001) SiC (left) and on (001) LiAlO2. Theimages are taken from the viewing angles specified in the insets. The latticeconstants and the symmetries in both cases can be seen easily [18].

planes, are also affected less by QCSE. Figure 1.2(a) shows a comparison

between two GaN/Al0.1Ga0.9N QWs grown on c- and m-planes, whose defi-

nitions are shown in Fig. 1.2(b). It is obvious that the band edges in the c-

plane growth case are highly distorted, the wavefunction envelopes are widely

separated, and the equivalent band gap is narrowed. Such a field-dependent

band gap changes with injection level due to the charge screening effect, as

will be described in Chapter 2. Figure 1.2(c) shows the high-resolution trans-

mission electron microscopy (TEM) images of the actual epitaxial structure

of c- and m-plane GaN thin films from the side of the samples. The structure

of the wurtzite lattice can be clearly seen in the images.

Although spontaneous emission (in the form of blue LEDs) has been seen

as the “ultimate lamp,” it is the stimulated emission (in the form of laser

diodes) that represents the most efficient electrical-to-optical energy conver-

sion. It is expected that III-N laser diodes (LDs) will be able to bypass the

injection-dependent secondary effects existing in LEDs, such as wavelength

blue shift, Auger recombination and electron overflow [20–23] by clamping

the carrier density at the threshold value [24]. The predicted higher power

conversion efficiency, wavelength stability and spectral purity make LDs es-

pecially suitable for high-brightness directional lighting, compact full-color

projecting and display applications [25,26].

3

1.2 InGaN-based nano-structures for laser applications

Despite the fact that III-N materials have raised high expectations in the

visible laser market, difficulties in obtaining high material quality and p-type

conductivity make their fate distinctly different from conventional III-Vs.

The first electrically-pumped GaN-based LD was demonstrated by Naka-

mura at Nichia in late 1995 [27], 24 years later than the first observation

of optically-excited stimulated emission in GaN by Dingle [28] and 33 years

later than the first electrically-pumped III-V LD by Holonyak [29]. Never-

theless, driven by the keen demand for blue-wavelength optical storage in

the late 1990s, the development of InGaN LDs emitting at ∼400 nm started

to soar. Continuous-wave (CW) operation at elevated temperatures (>50C), high output power (>10 mW) and long lifetime (>10000 hr) had been

demonstrated by the year 2000 [30–33]. In addition to the pursuit of better

device performance, the lasing wavelength started to redshift from the violet

(∼400 nm) to pure blue or even aquamarine (∼450 nm) to meet the need

from the projector market at that time. It is worth noting that molecular-

beam-epitaxy (MBE)-grown InGaN-MQW-based LDs also began to mature

Figure 1.3: The evolution of III-N LDs, categorized by operation modesand gain media. Several important technology breakthroughs are marked.

4

after their debut in 2004 [34–36]. However, it was not until 2012 that CW-

driven MBE-grown InGaN-MQW-based LDs were demonstrated [37], which

highlights the lower maturing level of MBE technology in the field of III-N

crystal growth.

The history of green InGaN-MQW-based LDs is relatively short. While

OSRAM insisted to grow high-indium-content (∼ 25%) MQWs on c-plane

GaN substrates [38, 39], Nakamura and his group at the University of Cali-

fornia, Santa Barbara continued to develop green LDs on non- and semipolar

substrates after their first demonstration of optically-pumped green stimu-

lated emission in m-plane InGaN MQWs [40–42]. Though long wavelength

(536.6 nm) and high power (> 100 mW) green LDs have been demon-

strated [43], strong QCSE in c-plane MQW structures and demanding re-

quirements on the quality and uniformity of the off-c-axis-grown material

still pose serious obstacles for improving device performance of green InGaN-

MQW-based LDs.

Extending the emission wavelength of InGaN MQW-based LDs into the red

portion of the spectrum is even more challenging. Figure 1.1 reveals that the

indium composition required to reach 600-nm emission in bulk InxGa1−xN

will be around 40%, implying that it is almost impossible to grow a uniform,

red-emitting InGaN thin film on GaN within the critical thickness [44]. In-

deed, there has been no report on stimulated emission in the red wavelengths

from InGaN MQW structures up to date, which has been more than 15 years

after the demonstration of amber and red LEDs [45]. As a result, drastically

different quantum structures must be adopted to break the fundamental limit

on the critical thickness of the epitaxial thin film.

Various three-dimensional nano-structures are viewed as excellent solutions

to alleviating the strain problem, as they have more interfaces that are free to

expand and relax the highly-distorted lattice. Among those, self-assembled

quantum dots (SAQDs) have emerged as an attractive way to preserve ma-

terial quality in highly lattice-mismatched systems, since they are exactly

formed as a result of the accumulation of compressive strain between the dot

and substrate materials. The “substrate” here refers to the material on which

the QDs are grown and it is not necessarily the same material as the one that

provides the mechanical support for the whole epitaxial structure. When the

dot material, which usually has a larger lattice constant, is forced to match

that of the substrate material during the crystal growth phase, strain starts to

5

build up in the crystal lattice in the form of mechanical energy. The stored

mechanical energy in the initial dot material thin film increases dramati-

cally when its thickness approaches the critical thickness, where dislocations

and crystalline defects start to form in order to release the exceedingly high

stored energy. However, what separates the growth of SAQDs and QWs is

that under very selective conditions, the surface morphology undergoes an

abrupt two-to-three-dimensional transition and the stored mechanical energy

is released through the formation of island-like structures at random sites.

Such a growth mode is called Stranski-Krastanov (S-K) mode, featuring the

strain accumulation process in the initial atomically-thin layer, the so-called

“wetting layer.” Since the conditions for forming SAQDs are stringent, the

growth rate is usually kept slow so that the surface morphology can stay well

under control. As a result, although some MOCVD-grown SAQDs have been

reported [46–49], most of the SAQDs are grown using MBE technology.

Figure 1.4(a) shows a graphical presentation of the SAQD structure, and

Fig. 1.4(b) shows an atomic-force microscopy (AFM) image of uncapped

InGaN SAQDs during the growth phase. An optimized self-assembly pro-

cess normally gives a QD aerial density on the order of 1010 cm−2, and the

QD dimensions varies in the range of tens of nanometers in the lateral di-

rection and a few nanometers in height. Furthermore, SAQDs only form in

dislocation-free regions, where the strain energy is allowed to build up. Such

a fact implies that the SAQDs are not only excellent structures to keep the

chemical composition of the deposited material (as the larger the difference

(a) (b)

Figure 1.4: (a) A graphical presentation showing the structure of the SAQDand the wetting layer [50]. (b) An atomic-force microscopy image (1.0 µm× 1.0 µm × 7.0 nm) of uncapped InGaN/GaN SAQDs [51].

6

in lattice constant, the larger the strain) but also minimize the effects of

non-radiative recombination centers.

Besides the superior mechanical properties, SAQDs also provide several

quantum-mechanical properties that beneficiate LD designs. The smaller ac-

tive volume and the suppression of non-radiative recombination process at

defect centers lowers the threshold current, and the nearly delta-function-like

density of states (DOS) not only makes the optical gain spectra less sensi-

tive to temperature change but results in a higher differential gain [52, 53].

These strengths significantly promote the role of SAQDs as a gain medium

in InGaN-based LDs since all the difficulties, including the low indium in-

corporation capability, high dislocation densities and the strong piezoelectric

polarization field, found in the growth of MQW structures now find their cor-

responding solutions. Indeed, SAQD-based LDs emitting across the whole

visible spectrum have been demonstrated and have shown lower threshold

current densities and higher temperature stability compared to the MQW

counterparts [54–56].

1.3 Proposals for GaN-based integrated photonic

devices with voltage-controlled functionalities

Aside from the aforementioned piezoelectric effect, III-Ns also possess sev-

eral other optical effects due their lack of centro-symmetry. Among of them,

photo-elastic and electro-optic effects allow the manipulation of the permit-

tivity tensor (refractive index tensor) using external mechanical force and

electric field. Although subtle, these first-order effects may actually be the

keys to designing multi-functional, electrically-responsive novel devices for

integrated photonics.

Figure 1.5(a) and (b) show the block diagrams of an optical transmitter

and a coherent optical receiver, which are important elements in photonic

integrated circuits (PICs). Leveraging the optical effects in the constituent

materials, various functions meant to maximize the signal capacity of a sin-

gle channel, such as wavelength-division multiplexing (WDM), phase-shift

keying (PSK) and polarization-division multiplexing (PDM), can be imple-

mented. The basic principle of operation is stated in the following. On the

transmitter side, Fig. 1.5(a), the laser power (with a wavelength λi) is first

7

(a) (b)

Figure 1.5: (a) A schematic presentation of an optical transmitter. Mod:modulator, PS: phase shifter, PR: polarization rotator, PBC: polarizationbeam combiner. (b) A schematic presentation of a coherent optical receiver.PBS: polarization beam splitter, LO: local oscillator, PD: photodetector.

equally split with two arms for signal encoding into two orthogonal states

of polarization (SOPs). Two modulators are located at each arm to convert

two different data streams from the electrical to the optical domain using an

intensity modulation scheme, or on-off keying (OOK). One of the modulated

signal experiences a π/2 phase change as traveling through a phase shifter

and then merges with the other unshifted one (the PSK process). The re-

sultant signal pattern is unique as it is generated by the constructive and

destructive interferences between the two modulated signals. The other part

of the laser emission split in the very first stage undergoes exactly the same

process, except for being rotated from a TE-polarized state to a TM-polarized

one at the end of a polarization rotator. The two orthogonally polarized op-

tical signals are combined and sent out using a polarization beam combiner

at the end of the transmitter. Knowing how the input signal is encoded in

the transmitter end, a coherent optical receiver, Fig. 1.5(b), first splits the

polarization-diversified signal using a polarization beam splitter. The two

signals are separately beaten by a local oscillator, a tunable laser that also

emits at wavelength λi, to extract the phase-shift-keyed signal. The overall

signal capacity of the optical transceiver can be further multiplied by incor-

porating multiple laser sources whose emission wavelengths are separated by

proper spectral spacings, as such WDM or DWDM bands.

It is worth noting that multiple elements in current-generation optical

transceivers rely on resistive heaters and current injection to change the

refractive index of the material. Such approaches greatly increase static

power consumption and introduce p-n junction capacitances, which limit

8

the operating speed of the circuit. As a result, it is expected that if extra

degrees of freedom can be introduced to enhance the controllability over the

materials’ refractive indices, then the whole optical chip can be made less

complicated, faster and more energy efficient.

Table 1.1 summarizes the advantages of GaN, the most mature III-N ma-

terial, over other materials that are conventionally used in integrated pho-

tonics. It is obvious that the lower the crystal symmetry, the stronger the

optical effects since it is the anisotropy of the material that interacts with

the photon. Although the optical effects in GaN are not as strong as those

in lithium niobate (LiNbO3), its availability on (111) Si substrate makes it

highly attractive in terms of cost, fabrication complexity and compatibility

with GaN-based high-speed transistor technologies. More importantly, the

optical effects in GaN can be triggered simply by applying an external bias

Table 1.1: A comparison in terms of crystal structure, optical effects andmanufacturing technology between GaN and common materials (LiNbO3,InP and Si) used for integrated photonics. Images from [12,57–59].

MaterialsLiNbO3 GaN InP Si

Crystal symmetry/Optical effectsVery low/ Low/ High/ Very high/

Very strong Strong Low Very lowpiezoelectricphoto-elasticelectro-opticthermal-opticN.L.O. a

piezoelectricphoto-elasticelectro-opticthermal-opticN.L.O. a

piezoelectric b

electro-optic b

thermal-opticplasma effect c

thermal-opticplasma effect c

Manufacturing technologyBulk LiNbO3 sub-strate

GaN on (111) Si8 inch emerging

InP substrate6-inch emerging

CMOS technology

a N.L.O.: nonlinear optical effects.b Weak in zinc blende crystals.c Plasma effect: the change of the refractive index due to the high free carrier density.

9

field, which consumes significantly less power and generates much less heat

than the thermal-optical and plasma effects that change the refractive indices

of InP and Si. Consequently, incorporating GaN in the design of photonic

devices has the potential to give rise to more integratable, responsive and

energy-efficient PICs.

1.4 Thesis overview

Focusing on the modeling of GaN-based optoelectronic and photonic devices,

this thesis considers two major topics: theoretical investigation of novel In-

GaN/GaN nano-structure active media, and design of integrated photonic

elements using the macroscopic optical effects in bulk GaN. Experimental

studies on the development of more efficient InP-based gain material in the

telecommunications window is also briefly discussed at the end of this thesis.

In the first topic, a comprehensive model for In0.4Ga0.6N/GaN SAQD-

based LDs, which among the III-N LDs to date lase in record-long visible

wavelengths (∼630 nm) has been developed. The model contains three major

pieces, a material model, a laser cavity model and a device model. The ma-

terial model gives SAQD properties such as strain distribution, piezoelectric

potential, electronic states, and material gain spectra. The cavity model pro-

vides characteristics of the lasing mode, including modal profile, confinement

factor and threshold modal gain. The results generated from the two mod-

els directly enter the device model, which uses temperature-dependent laser

rate equations to simulate the laser light-current (L-I) characteristics. By

comparing the theoretical and experimental results, several physical param-

eters can be extracted. The model’s scalability with respect to the indium

composition x has been verified.

Chapter 2 details the material model, including the calculation of strain

distribution and electronic states in the SAQD-wetting layer system. The

SAQDs are modeled using truncated, hexagonal pyramids and the material

property variation across different QD sizes is considered using a Gaussian

distribution. The calculation of electronic states is formulated using a single-

band Hamiltonian with corrected effective masses. The correction for the

effective masses is to account for the change of band structure due to the

compressive strain in the whole structure. The screening of the polarization

10

Figure 1.6: Flowchart of the InGaN/GaN SAQD-based LD model.

fields is modeled using a Schrodinger-Poisson self-consistent solver. Subse-

quently, the gain spectrum is calculated using Fermi’s golden rule.

Chapter 3 covers the cavity model and the device model, which features

temperature-dependent rate equations. In the cavity model, information

about the actual epitaxial layer structure and the geometry of the ridge

waveguide is used to build the refractive index profile, which defines the

field distribution of the lasing mode. In the device model, current-voltage

(I-V) and L-I curves of the In0.4Ga0.6N/GaN SAQD-based ridge waveguide

laser are reproduced using the results from the previous two models. This

model contains a macroscopic picture of the electrical paths in the LD, which

is the diode-resistor model, and a microscopic picture of the carrier-photon

interaction, which is the temperature-dependent rate equation model. By

comparing the simulated and measured device characteristics, semi-empirical

device-level information that is useful for future device design is obtained. A

complete modeling flowchart is given in Fig. 1.6.

The second topic is on the development of a novel GaN-based voltage-

controlled integrated photonic devices for coherent optical transceivers. Two

types of devices are investigated: polarization rotators (PRs) and tunable

gratings. A PR is a waveguide device that alters the SOP of the incoming

light such that signals can be encoded into light waves with an identical wave-

length but different SOPs, as mentioned in Section 1.3. A tunable grating is

the key part for implementing a WDM system, as it not only selects out a

11

(a) (b)

Figure 1.7: (a) Proposed device structure of the GaN-based PR. (b) Topview of the GaN-based PR picturing the mode conversion process. V1 andV2 denote different biasing configurations.

single lasing wavelength that defines the optical channel at the transmitter

end but also ensures the matching between the source and local oscillator

at the receiver end. Via the piezoelectric, photo-elastic and electro-optic ef-

fects in wurtzite GaN, spatially varying and uniform refractive index contrast

can be created depending upon the distribution of the field line across the

device. Such operation mode makes the GaN-based voltage-controlled PRs

and gratings fundamentally different from the geometry-based dielectric PRs

and micro-heater-controlled gratings used in the current silicon photonics

platform.

Chapter 4 elaborates on the design of the voltage-controlled GaN PR.

The GaN PR adopts a rib-waveguide geometry and the electrodes are de-

posited on the top and sides of the waveguide. Transverse-electric (TE) and

transverse-magnetic (TM) modes in the PR are defined based upon the direc-

tion of the electric field in the propagating mode. The waveguide dispersion

is engineered using different rib aspect ratios, such that TE and TM modes

become degenerate (or the fundamental propagating mode becomes a hy-

brid mode) under a bias-free condition. The external electric field breaks

the mode degeneracy and, depending on the configuration of the voltage ap-

plied on the electrodes, the fundamental mode can be switched between TE-

and TM-SOPs. The hybrid mode serves as the bridge for the TE-to-TM and

TM-to-TE mode conversion process. Figure 1.7(a) shows a three-dimensional

device structure, and Fig. 1.7(b) shows a top view of the PR to illustrate

the mode conversion process.

Chapter 5 discusses the simulation of tunable lasers based upon standard

InP-based telecommunications gain chips and GaN-based voltage-controlled

12

Figure 1.8: Top view (top) and cross section of the InP-based gain,GaN-based phase shifter and grating regions (bottom) of thevoltage-controlled tunable laser. AR: anti-reflection, PS: phase shifter,MMI: multi-mode interference coupler, SOG: spin-on glass, M1: the firstmetal layer, M2: the second metal layer.

gratings, as illustrated in Fig. 1.8. A Y-branch replaces the high-reflection

(HR)-coated facet in conventional edge-emitting lasers (EELs) which splits

the optical power into two basically equal portions by a multi-mode inter-

ference (MMI) coupler. The split power is then injected into two grated

arms, where two contacts are deposited to change the effective indices of the

waveguide mode using the optical effects in GaN. Depending on the pitch

and the effective index contrast of its grating, each arm provides a unique

reflection spectrum, hence the lasing wavelength is defined by the overlap of

MQW gain spectrum and the spectral range where both arms give maximum

reflectivities simultaneously. The phase shifter section is to compensate any

potential difference between the resultant round-trip phase and 2π. Utilizing

the Vernier effect, the lasing wavelength of the laser source can be coarse- and

fine-tuned over a certain spectral range with a minimal power consumption.

Chapter 6 summarizes the achievement of this thesis and discusses possible

future directions in studying the physics behind novel III-N nano-structures,

and the design considerations of advanced GaN-based photonic devices.

13

CHAPTER 2

MATERIAL MODEL FOR InGaN/GaNSELF-ASSEMBLED QUANTUM DOTS

(SAQDS)

2.1 Introduction

As mentioned in Chapter 1, SAQDs grown in S-K mode are formed as a

result of the strain relaxation process. The strain relaxation process gives

InGaN/GaN SAQDs both structural and quantum-mechanical advantages

over their QW counterparts, as more indium can be incorporated in the nano-

structure without compromising the mechanical stability of the system. The

lower residual strain also reduces the QCSE, which lowers the wavefunction

envelope overlaps between the electron and hole states. These strengths

make InGaN/GaN SAQDs an exceptional gain medium for next-generation

III-N-based LDs.

To facilitate the development of SAQD-based III-N visible lasers, it is

crucial to have a comprehensive model that is able to describe the measured

data and extract physical parameters which are not readily available in the

literature or directly measured. However, modeling III-N SAQD structures is

not trivial. Assumptions regarding dot shape, the strain calculation method,

and the form of the Hamiltonian operator vary [60–63], and a systematic

comparison between theory and experiment has not yet been available. In the

following sections, a comprehensive model for the first-ever In0.4Ga0.6N/GaN

SAQD structure will be presented.

2.2 Strain distribution and piezoelectric effect in

InGaN/GaN SAQD structures

Because of the symmetry in the c-plane, a widely assumed, truncated hexago-

nal pyramid structure is adopted to model the InGaN/GaN SAQDs [60–62].

14

The material properties of the whole SAQD array can be treated using a

Gaussian distribution centered at “dominant dots,” which are the majority

in number and have characteristics that are representative for the whole QD

group [64]. The overall effect of dot size inhomogeneity can be modeled as

the broadening (the standard deviation in the Gaussian distribution) in the

energy spectra of the electron and hole states. The strain distribution in the

SAQD structure is formulated using linear elastic theory because the size of

the “dominant dots” is well above the atomic scale, as revealed by atomic

force microscopy (AFM) images [56]. In this model, the size parameters for

the “dominant dots” are chosen to be a = 10 nm, b = 25 nm, h = 3.5 nm

and t = 0.5 nm, as labeled in Fig. 1.6.

The most important feature of linear elastic theory is that the entire strain

distribution scales linearly with the amount of strain at the material interface,

defined as ε(0) in the model. The advantage of adopting linear elastic theory is

twofold: first, it provides a reasonable estimation of the residual strain at the

InGaN/GaN QD/matrix interface; second, the parameter extraction process

can be accomplished with the smallest amount of computational resources.

In linear elastic theory, the strain εεε(r) and stress σσσ(r) distributions are

connected via

εεε =1

2

[∇u + (∇u)t

]σσσ = ¯C : εεε (2.1)

∇ · σσσ + f = 0

where u is the macroscopic displacement field of the structure, ¯C the stiffness

tensor of the material, and f the external force exerted on the structure. Since

the SAQDs are formed as a result of the self-assembly process, the external

force f is zero in our SAQD model.

There is no closed-form solution to Eq. (2.1) for arbitrarily-shaped struc-

tures. However, reasonable approximations can be made to solve Eq. (2.1)

while preserving the geometric symmetry of the structure to the largest ex-

tent. Taking the advantage of the linear nature of Eq. (2.1), one can obtain

the global stress distribution by superimposing the local stress distributions

associated with embedded infinitesimal spheres at different locations [65].

The resultant strain distribution is found using the Hooke’s law σσσ = ¯C : εεε.

15

The displacement field associated with an embedded sphere of radius r0

can be found by solving

C44∇2u(r) + (C11 − C44)∇∇ · u(r) = 0 (2.2)

which can be derived from Eq. (2.1) by assuming a zinc blende symmetry.

The error arising from the difference between the assumed zinc blende and

the actual wurtzite symmetries will be minimized in the end by normalizing

the stress distribution by the volume of the sphere, and by using the correct

stiffness tensor in the following calculations.

The geometry of the sphere grants that the displacement only occurs in

the radial direction, u = urr. The shrink-fit boundary condition at the

dot/matrix material interface r = r0 requires [66]

ur(r−0 )− ur(r+

0 ) = ε(0)r0

σrr(r−0 ) = σrr(r

+0 )

(2.3)

where ε(0) denotes the residual strain between the two materials. The hexag-

onal symmetry of the wurtzite lattice enters the calculation through the

stiffness tensor ¯C. The resultant radial displacement is

ur(r) =

2C11 − C12 − C13

3C11

ε(0)r , r < r0

−C11 + C12 + C13

3C11

ε(0) r30

r2, r > r0

(2.4)

whose functional trend is shown in Fig. 2.1(a).

The contribution from each infinitesimal element at point r′ can be viewed

as a “stress density” σσσ(e)ij (r−r′), obtained by normalizing the local stress field

associated with the embedded sphere by its volume, 4/3πr03. The global

stress field σσσij(r) can be found using superposition principle, as depicted in

Fig. 2.1(b):

σij(r) = P. V.

∫QD

σ(e)ij (r− r′) d3r′

+ δij

∫QD

A

3(2C11 − C12 − C13) ε(0)δ (r− r′) d3r′ (2.5)

16

(a) (b)

Figure 2.1: (a) The radial displacement field in an embedded sphericalstructure (inset) as a function of radial distance. (b) A simple illustration

of the superposition principle. σσσ(e)ij (r− r′) denotes the stress density at

point r due to the presence of the point element at r′.

where P. V. is the Cauchy principal value of the integral, δij the Kronecker

delta, δ(r) the Dirac delta function to recover the singularity of the stress

density, A = (C11 + C12 + C13)/C11, and the stress density

σ(e)ij (r) =

ε(0)A

4π

C11(3x2 − r2) + C12(3y2 − r2) + C13(3z2 − r2)

r5, ij = xx

ε(0)A

4π

C11(3y2 − r2) + C12(3x2 − r2) + C13(3z2 − r2)

r5, ij = yy

ε(0)A

4π

3C13(x2 + y2) + 3C33z2 − (2C13 + C33)r2

r5, ij = zz

3ε(0)A

4π

C44yz

r5, ij = yz

3ε(0)A

4π

C44zx

r5, ij = zx

3ε(0)A

4π

C11 − C12

2

xy

r5, ij = xy

(2.6)

Equation (2.6) highlights the biggest feature of linear elastic theory: the

whole stress (or strain) distribution scales linearly with the residual strain

ε(0). As a fitting parameter, ε(0) reveals the actual amount of strain at the

17

(a) (b)

Figure 2.2: Spatial distribution of biaxial strain components (a) εxx and (b)εzz at different viewing angles.

dot/matrix interface, which is hard to measure in the real 3D nanostructure.

Figures 2.2(a) and (b) show the biaxial strain components εxx(r) at the

bottom of wetting layer and εzz(r) from the side of the SAQD structure re-

spectively. Although the SAQD region is compressively strained as expected,

the wetting layer is tensily strained in order to make the whole SAQD struc-

ture under a force-free condition, which is the nature of the self-assembly

process.

The connection between the mechanical and electrical properties of a mate-

rial is given by its piezoelectric tensor. In wurtzite crystals, the piezoelectric

tensor takes the form [64,67,68]:

Ppz,x

Ppz,y

Ppz,z

=

0 0 0 0 e15 0

0 0 0 e15 0 0

e31 e31 e33 0 0 0

εxx

εyy

εzz

εyz

εzx

εxy

(2.7)

where eij’s are the piezoelectric coefficients of the III-N alloy. While linear in-

terpolation between GaN and InN is acceptable for obtaining eij’s in InGaN,

bowing effect needs to be taken into account when evaluating the sponta-

neous polarization Psp [11]. The discontinuity in the total polarization field

18

(a) (b)

Figure 2.3: Spatial distribution of polarization charges (a) at the bottom ofthe wetting layer and (b) from the side of the SAQD structure.

Ppz+Psp across the dot/matrix interface gives rise to the piezoelectric charge:

ρpol.(r) = −∇ · (Ppz + Psp) (2.8)

which is visualized in Figs. 2.3(a) and (b). The positively-charged top surface

and the negatively-charged wetting layer form an effective electric dipole that

separates the envelopes of the electron and hole wavefunctions and lowers the

radiation efficiency, as will be seen in Section 2.3.

Temperature effects on SAQDs’ mechanical properties can be taken into

account by incorporating the coefficients of thermal expansion (CTEs) of

the materials [69–71]. The active region temperature may not significantly

change the amount of strain, but it actually has a much larger impact on the

calculation of electronic states in the SAQD system, as the band gaps and

Fermi-Dirac distribution are functions of temperature.

2.3 Electronic states in InGaN/GaN SAQD structures

One of the most sophisticated parts in the calculation of electronic states in

QD structures is the valence band coupling, which is generally described using

the k·p method [61,72,73]. Although realizable, applying the k·p method to

three-dimensional (3D) structures greatly increases the computational com-

plexity, and whether the resultant incremental accuracy is worthwhile needs

19

to be verified. Moreover, it is the geometry of the QD that weights over

the number of bands used in the k · p method [73]. As a result, instead of

using the k · p method to calculate the electronic states in the QD system,

a single-band effective mass Hamiltonian with corrected hole masses is used

in this model.

The hole effective masses in the strained, bulk In0.4Ga0.6N QD material

are corrected using a separate six-band k · p model [74]. Since the strain

components in the SAQD structure are now functions of position, spatially-

averaged values

〈εij〉 =

∫QD

εij(r)d3r

QD volume(2.9)

are used in the formulation of the Hamiltonian.

Figure 2.4 shows the valence band structures of unstrained and strained

bulk In0.4Ga0.6N alloy. While the compressive strain only shifts the position

of the heavy hole (HH) band, it drastically changes the energy dispersion

of the light hole (LH) and the crystal-field split-off hole (CH) bands. Not

only does the large energy separation (∼180 meV) between the LH and CH

band at zone center make the CH band almost irrelevant in the calculation

of SAQD states, the LH effective mass in the z direction is almost doubled

as listed in Table 2.1. As a result, only the HH and LH bands are considered

in the calculation of electronic states and the optical gain spectra.

Using the corrected effective masses, the single-band effective mass Hamil-

Figure 2.4: Valence band structures of strained (solid) and unstrained(dashed) bulk In0.4Ga0.6N.

20

Table 2.1: Heavy and light hole effective masses of strained and unstrainedInGaN, in terms of free electron mass m0.

m∗hh,t m∗lh,t m∗hh,z m∗lh,zstrained 0.209 0.217 1.742 1.720

unstrained 0.213 0.228 1.742 0.884

tonian can be written as

−~2

2∇ ·(

¯m−1 : ∇ψ)

+ V (r)ψ = Eψ (2.10)

where ¯m is the anisotropic effective mass tensor. The potential operator V (r)

consists of inherent band offsets in the heterostructure V0(r), strain-induced

deformation potential Vstrain(r), and polarization-induced electrostatic po-

tential Vpol.(r). The inherent band offsets are evaluated using Anderson’s

rule

∆Eg(T ) = ∆Ec(T ) + ∆Ev(T ) (2.11a)

∆Ev(T ) =900 meV

∆Eg(298 K)∆Eg(T ) (2.11b)

and Varshni’s parameters to include temperature effects [75]:

Eg(T ) = Eg(0)− αT 2

β + T(2.12)

The valance band offset (VBO) at the InN/GaN interface is chosen to be 900

meV at room temperature [76]. However, due to the heavy effective masses

of HH and LH bands and the high indium content, the exact value of VBO

does not significantly affect the calculation of hole states.

Spatially-averaged strain components, as defined in Eq. (2.9), are again

used in the calculation of deformation potential. Since the shear strain

components (εij, i 6= j) are linear combinations of shear stress components

(σij, i 6= j), they are odd functions of position, as can be seen from Eq. (2.6),

and the magnitudes of their averaged values vanish. Consequently, the defor-

mation potential can be simplified using only the biaxial strain components

Vstrain(r) =∑i

viεii(r) (2.13)

21

where vi’s are appropriate deformation potentials of either conduction or

valance band [11].

The calculation of the polarization-induced electrostatic potential Vpol.(r)

is relatively straightforward. Using the anisotropic permittivities ¯εεε in III-N

materials [77], Poisson’s equation gives

−∇ · (¯εεε : ∇Vpol.) = ρpol.(r) (2.14)

So far the potential V (r) = V0(r) + Vstrain(r) + Vpol.(r) represents the to-

tal potential of an “empty dot,” where no additional charge is present to

screen the polarization field. Figures 2.5(a) and (b) show the wavefunction

envelopes of the top three electron (C) and HH states associated with the

(a)

(b)

Figure 2.5: Spatial distribution of polarization charges (a) at the bottom ofthe wetting layer and (b) from the side of the SAQD structure.

22

“empty dot” potential at different viewing angles, respectively. The hexag-

onal symmetry in the c-plane makes the second and third states in either

conduction or valence band nearly degenerate in energy, thus the differential

gain at C2-HH2 and C2-LH2 transition wavelengths are expected to be dou-

bled. In addition, the polarization field distorts the band edges and makes

the electrons stay at the top of the SAQD and the holes stay at the bottom

of the wetting layer. The wide separation between the electrons and holes

renders poor wavefunction overlaps and hinders bimolecular recombination

processes, such as spontaneous and stimulated recombination.

When the carriers enter the system, either through optical or electrical

pumping, the spatially separated electron-hole pairs (EHPs) form another

electrical dipole in the opposite direction of the polarization field. As the

pumping level increases, this dipole starts to weaken the QCSE and restores

the slanted band edges. The screening effect can be taken into account by

adding a screening potential Vsc.(r) to Eq. (2.10). To calculate the screening

potential, it is necessary to determine the position of the quasi Fermi levels

(QFLs). The conduction band QFL is related to the injection via

n2D = 2N2D

M∑m=1

∫ ∞−∞

Gc (Ec − Eem) fc (Ec, T ) dEc

+

∫ ∞Ec,m

Leffρe3D (Ec − Ec,b) fc (Ec, T ) dEc (2.15)

where N2D is the aerial density of the SAQD, the upper limit M in the sum-

mation the total number of the bound states, Gc(Ec) the Gaussian distribu-

tion describing the size inhomogeneity of the QDs, fc (Ec, T ) the Fermi-Dirac

distribution function at temperature T , ρe3D (Ec − Ec,b) the 3D DOS of the

electron and Ec,m the conduction band minimum of the GaN matrix. It

should be noted that the wetting layer states are automatically included in

Eq. (2.15), as they are also solutions to Eq. (2.10) because there is no clear

physical boundary between the SAQD structure and the underlying wetting

layer in our model. The 3D-DOS of the matrix material is to ensure that

the QFLs move continuously with the injection level, as the bound states are

discrete in nature. The Leff in Eq. (2.15) is the effective height of the QD

23

structure, which is defined as

Leff = N2D ×QD volume (2.16)

in order to accurately convert the aerial density to the volume density. The

valence band QFL can be found using a similar expression to Eq. (2.15).

The screening charge produced by the injected electron-hole pairs can be

expressed as a function of QFLs:

ρsc.(r) = 2q∑

bound

|ψh,j(r)|2 [1− fv(Eh,j, T )]

− 2q∑

bound

|ψe,i(r)|2fc(Ee,i, T ) (2.17)

and the screening potential Vsc.(r) can be found using ρsc.(r) as the source

in Poisson’s equation analogous to Eq. (2.14). The electron and hole states

are obtained by solving Schrodinger and Poisson’s equations self-consistently.

Figure 2.6(a) shows the screened and unscreened energy band diagrams of

the SAQD structure, cutting along the c-axis, the top of the dot (left), and

the bottom (right, the red line) of the wetting layer (right, the pink line)

respectively. While the screening dipole formed by the EHPs barely changes

the potential profile in the transverse direction, it efficiently cancels part of

(a) (b)

Figure 2.6: (a) A comparison between the unscreened and screened bandedges of the SAQD at different viewing angles. The injection level isn2d = 37N2D and the active region temperature is 120 oC. (b) Theconduction-valence band wavefunction overlaps (C1-HH1 and C2-HH2) asfunctions of injection level.

24

the polarization field in the z direction at high injection levels. The alle-

viation of the QCSE leads to increased wavefunction overlaps, as shown in

Fig. 2.6(b). The saturation at high injection levels is attributed to the large

3D-DOS in the matrix material, which slows down the rate of change of the

QFLs.

The screening effect in SAQD structures in the z direction is actually the

same as that in QWs. However, it is the extra confinement in the transverse

direction that stops the carriers from leaving the 3D quantum structure and

recombining at defect centers. It is worth noting that the island-like geom-

etry also allows higher indium incorporation and strain relaxation, which

ultimately give rise to the formation of indium-rich, much-less-defective ac-

tive material.

2.4 Optical gain in the InGaN/GaN SAQD active

material

As in most of the III-V materials, compressive strain also leads to TE-

polarized emission in InGaN active materials. Such a polarization-dependent

gain in SAQD active materials with different indium compositions has been

confirmed experimentally [54, 56].

Since the effect of the CH band has been ruled out in this model, the

TE-polarized gain in the In0.4Ga0.6N/GaN SAQD is solely considered. The

TE-polarized material gain spectrum can be found using Fermi’s golden rule:

gTE(~ω) =2πq2N2D| 〈iS|px|X〉 |2

nr,tε0c0ωm20Leff

∑bound

Pij(~ω) (2.18)

where | 〈iS|px|X〉 | is the optical matrix element [74], nr,t the in-plane index

of refraction and Pij(~ω) the optical transition probability between the i-th

electron and the j-th hole states:

Pij(~ω) = |Ie,ih,j|2|wh,j|2

∫ ∞−∞

Gcv(E − Ee,ih,j) [fc(Ee,i, T )− fv(Eh,j, T )]

× L(E − ~ω)dE (2.19)

where |Ie,ih,j| is the wavefunction overlap between the i-th electron and the

25

j-th hole states, |wh,j| the in-plane (|X〉 or |Y 〉) component of the hole wave-

function, Gcv(E − Ee,ih,j) the linewidth broadening in the transition energy

due to the fluctuation in dot size and L(E − ~ω) the Lorentzian distribution

describing the homogeneous broadening around the photon energy ~ω.

While the inhomogeneous broadening in InGaN QWs is generally caused

by fluctuations in both alloy composition and well width, and its value ranges

from 8 to 48 meV [78–80], the inhomogeneous broadening in QD systems is

mainly due to the variation in dot size. However, because GaN-based ma-

terials generally have larger effective masses and higher band offsets than

traditional III-V semiconductors, we argue that the effect of size inhomo-

geneity tends to have less impact on the electron and hole energy states in

the QD structures. Given inhomogeneous broadening values generally around

30 meV in InAs QDs [81, 82], an inhomogeneous broadening FWHM in our

InGaN/GaN QD system is assumed to be 15 meV to reflect the less variation

in the energy spectrum.

The modal gain for a certain cavity mode is obtained by multiplying Eq.

(2.18) with the modal confinement factor Γ, which is defined as the fraction

of the modal field overlapping with the SAQD active material, as shown

in Fig. 2.7(a). Figure 2.7(b) shows a comparison between the measured

Hakki-Paoli gain spectrum right at threshold and the calculated modal gain

(a) (b)

Figure 2.7: (a) The calculated power Pz(x, y) of the fundamental mode inthe simulated laser structure. The optical confinement factor is calculatedusing the portion of the power overlapping with the SAQD layers, as shownin the zoomed-in picture. (b) Calculated (lines) and measured (points)modal gain spectra of an In0.4Ga0.6N/GaN SAQD-based ridge waveguidelaser. The experimental curve was obtained using the Hakki-Paoli methodat the lasing threshold.

26

spectra for the fundamental mode. The calculated gain curve follows the

experimental data fairly well, except for the portion in the long-wavelength

regime. The difference is mainly attributed to the size or indium content

fluctuation cross different SAQD layers. The homogeneous broadening, which

is half of the FWHM of the Lorentzian function in Eq. (2.19), is found

to be around 57.5 meV, which is considerably larger than that in InGaAs

QD active materials [82–84]. Limited reports on the direct measurement of

homogeneous broadening in InGaN alloys, nevertheless it can be inferred that

the temperature sensitivity, or more specifically the occurrence of phonon

scattering events, is more significant in such material. In fact, the large

optical phonon energy in the GaN material system (∼90 meV) and the dense

energy spectrum may facilitate carrier scattering between multiple energy

states and render a wider gain spectrum.

27

CHAPTER 3

DEVICE MODEL FOR GaN-BASED RIDGEWAVEGUIDE LASERS

3.1 Introduction

To further verify the effectiveness of our SAQD model, the gain spectra

calculated in Chapter 2 are incorporated into a device-level simulation to an-

alyze the performance of the In0.4Ga0.6N/GaN SAQD-based LD. The device

model contains an electronic part, the I-V curve, and an optoelectronic part,

the L-I characteristics of the device. Such a model allows the extraction of

system-level parameters such as leakage paths, diode ideality factor, ther-

mal impedance, and non-radiative recombination lifetime in the In0.4Ga0.6N

/GaN SAQD-based LD. The ridge waveguide model is shown in Fig. 3.1.

Figure 3.1: Graphical presentation of the macroscopic (electronic) andmicroscopic (optoelectronic) parts of the LD model.

28

3.2 Optical and electronic model of the InGaN/GaN

SAQD-based LD

The optical model produces the characteristics of the lasing mode, which

is the fundamental mode in the ridge waveguide structure according to the

far-field imaging measurement [56]. To solve the vector Maxwell’s equation,

the distribution of refractive index, or the epitaxial structure of the device,

is necessary.

The LD structure is grown on a bulk HVPE-grown c-plane n-GaN sub-

strate, followed by the growth of a 500-nm, n-doped GaN buffer layer. Two

additional lattice-matched In0.18Al0.82N layers are inserted between the upper

and lower Al0.07Ga0.93N claddings and In0.02Ga0.98N waveguide layers to en-

hance optical confinement in the visible range. Seven periods of In0.4Ga0.6N

QDs, embedded in and separated by 17-nm GaN barriers, are incorporated

in the active region as the gain medium. The aerial density of the QD is

estimated to be 3.9 × 1010 cm−2 from the AFM image, and an average in-

dium composition around 40% in the SAQD/wetting layer structure is read

from the energy dispersive X-ray spectroscopy (EDX) measurement. The

laser cavity is defined by coating two cleaved m-plane facets with SiO2/TiO2

distributed Bragg reflectors (DBRs) which provide reflectivities roughly 0.73

and 0.95 around the lasing wavelength of 630 nm. The ridge structure is

formed by a shallow etch down to the Al0.07Ga0.93N/In0.02Al0.98N interface in

order to minimize optical diffraction loss and parasitic leakage paths associ-

ated with the surface damages.

The electrical properties of the LD can be modeled using two parasitic

resistances and an ideal LD, which is characterized by diode ideality factor

nd, reverse saturation current I0, turn-on voltage V0. The series resistance Rs

is to account for the voltage drop external to the device, while the parallel

resistance Rp stands for the shunt leakage path. Combining the electrical

circuitry and the LD, one obtains the I-V curve of the device:

I =V − IRs

Rp

+ I0

exp

[q (V − IRs − V0)

ndkBT

]− 1

(3.1)

where kB is the Boltzmann constant.

The five parameters I0, nd, Vd, RS and RP can be extracted from the

experimental I-V curve in different intervals: the turn-on behavior gives in-

29

formation about the diode ideality factor and turn-on voltage, and the slopes

of the I-V curve at low and high voltages determine the shunt and series re-

sistors. The device current at low voltages is controlled by either the shunt

leakage or reverse saturation current, depending on which is larger.

3.3 Temperature-dependent rate equations

The optoelectronic part of the LD model is more involved. The flow of

carriers and photons can be described phenomenologically using a reservoir

model [85], as depicted in Fig. 3.2. In this model, the carriers (n2d) and

the photons of a certain lasing mode (s2d) are expressed in sheet densities

because the SAQDs are described using the aerial density. It should be noted

that only a fraction of the carriers that reaches the active region contribute

to recombination processes. The carriers entering the active region are con-

sumed by defect-related non-radiative recombination process, spontaneous

emission (the incoherent light output) and stimulated emission (coherent

light output). The photons in the lasing mode can be generated directly by

the stimulated emission process or indirectly by the coupling from part of the

spontaneously-emitted photons that share the same quantum characteristics

as the lasing mode. The photons in the lasing mode can be removed from

the cavity by the intrinsic cavity loss (material absorption, scattering, etc.)

and the transmission loss through the end mirrors.

The above phenomenological description can be formulated as the so-called

Figure 3.2: Graphical presentation of the macroscopic (electronic) andmicroscopic (optoelectronic) parts of the LD model.

30

laser rate equations:

dn2d

dt= ηi

J

q− n2d

τnr

− n2d

τr

− gTE(n2d, T )vgs2d (3.2a)

ds2d

dt= ΓgTE(n2d, T )vgs2d −

s2d

τph

+ Γβspn2d

τr

(3.2b)

where ηi is the injection efficiency describing the fraction of current density

entering the SAQD active layers, τnr and τr the non-radiative and radiative

recombination lifetimes, gTE(n2d, T ) the material gain as defined in (2.18) at

temperature T , vg the group velocity of the lasing mode, βsp the spontaneous

emission coupling factor and τph the cavity photon lifetime describing the

photon removal process by the intrinsic loss αi and the mirror loss αm:

1

τph

= (αi + αm)vg (3.3)

The light output can be expressed using both the spontaneous and stimu-

late emission processes:

L =

(βn2d

τr

+αmvg

Γs2d

)WLc~ω (3.4)

Since the photons generated by the stimulated emission process are highly di-

rectional (usually in the direction normal to the output mirror of the cavity),

only a small fraction of the broad spontaneous emission light which stays

within such a small solid angle contributes to the light output of the LD.

The parameters W and Lc in Eq. (3.4) are the width of the ridge and the

length of the cavity, and the optical confinement factor Γ in the denominator

is to convert the physical area where the carriers populate to the modal area

of the fundamental mode. β is the collection efficiency of the spontaneous

emission light in the direction of the lasing output power. It should be noted

that the current density J in Eq. (3.2a) refers to the current passing through

the contact of area W × Lc:

J =1

WLc

(I − V − IRs

Rp

)(3.5)

The active region temperature rise is caused by the heat dissipated by the

non-zero resistances in the diode structure. The increase of the active region

31

temperature relative to the substrate temperature Tsub is determined by the

thermal impedance Zth of the LD:

T = Tsub + Zth (IV − L) (3.6)

The dissipated heat not only broadens the Fermi-Dirac distribution in Eqs.

(2.15) and (2.18) but enhances the non-radiative recombination processes,

which are described by an activation energy ∆Ea in this model:

1

τnr

=1

τ0

exp

(−∆EakBT

)(3.7)

As the temperature approaches 0 K, the non-radiative recombination centers

become “frozen” and the recombination process in the device is intrinsically

dominated by the radiative recombination.

Because the step size of the increasing driving current in usual L-I-V tests

is much larger than the time scale of all the recombination processes (in

nanoseconds) in the active region, the physical quantities in the laser rate

equations can be well described by the steady-state solution (d/dt = 0).

Figure 3.3: Calculated and measured L-I-V curves of the In0.4Ga0.6N/GaNSAQD-based LD. The ridge width and cavity length of the laser are 5 µmand 1 mm respectively. The laser is operated under CW bias, and thesubstrate temperature is 15 oC.

32

Figure 3.3 shows a comparison between the L-I-V curves calculated using

our model and the experimental curves measured from a 5 µm × 1 mm,

CW-biased LD with a substrate temperature of 15 oC. The fitting processes

in Figs. 2.7(b) and 3.3 give a fairly good understanding of the studied

In0.4Ga0.6N/GaN SAQD structure and the LD device, as will be detailed

in Section 3.4.

3.4 Parameter extraction and discussions

In this model, only a small number of physical quantities are assigned as

unknowns to fit the experimental data. They can be divided into two groups:

material-related parameters that vary across different growth technologies,

and device-related parameters that are hard to measure directly due to the

limited lifetime of the unoptimized sample.

The extraction of material-related parameters depends largely on the mea-

sured peak electro-luminescence (EL) wavelengths and the Hakki-Paoli gain

curve deduced from the amplified spontaneous emission (ASE) spectrum.

The EL peak at the lowest excitation level readily reveals the effective band

gap of the In0.4Ga0.6N SAQD, which corresponds to the case of “empty dots.”

As the excitation level increases, the injected carriers start to screen out the

polarization field induced by the piezoelectric charges accumulated at the top

and bottom of the SAQD/wetting layer structure. As the screening effect can

be predicted using Eq. (2.17), the blue shift in the EL peak wavelength then

gives the amount of residual strain ε(0) that is directly related to the strength

of the polarization field. The whole process proceeds with the fitting of the

L-I curve in an iterative manner such that heating effect can be considered.

The left panel in Fig. 3.4(a) shows the impact of strain relaxation on

the band diagram of the SAQD/wetting layer structure cutting along the

c-axis. The strain percentages are quantified with respect to the original

lattice mismatch between bulk In0.4Ga0.6N and GaN:

ε(0) = s0

(ε(0)xx + ε(0)

yy + ε(0)zz

)(3.8)

33

(a) (b)

Figure 3.4: (a) Band edges and C1-HH1 transition wavelengths calculatedusing different amounts of residual strain at the dot/matrix interface. (b)The calculated and measured EL peak wavelengths.

where

ε(0)xx = ε(0)

yy =a(GaN)− a(In0.4Ga0.6N)

a(In0.4Ga0.6NaN)(3.9a)

ε(0)zz =

c(GaN)− c(In0.4Ga0.6N)

c(In0.4Ga0.6N)(3.9b)

where s0 is the strain percentage falling between 0 and 100%. In the un-

strained case, the band edges are still tilted because of the discontinuity in

the spontaneous polarization field across the dot/matrix interface; however,

such an electric field is relatively weak compared to the piezoelectric field,

which results in steep band edges. The right panel in Fig. 3.4(a) shows the

C1-HH1 transition wavelength as a function of the residual strain. The differ-

ence in emission wavelength between the unstrained and fully-strained cases

is as large as 160 nm, meaning that the high sensitivity of the emission wave-

length to the strain percentage can serve as a perfect indicator for the actual

amount of residual strain ε(0). Through the fitting of EL peak wavelength, as

shown in Fig. 3.4(b), a strain percentage s0 = 30% is found, which implies

that nearly 70% of the accumulated strain was released during the island

formation process. A differential gain (∂g/∂n) of 4.4× 10−17 cm2 at a lasing

wavelength of 634 nm is derived from our model, which is between a value

of 3.8 × 10−17 cm2 obtained from measuring the threshold current densities

in LDs with different cavity lengths and a value of 5.3× 10−17 cm2 measured

from the small-signal modulation response [56,86]. The fitting parameters of

34

the SAQD model are given in Table A.1 in Appendix A.

The temperature effect is investigated by adjusting the thermal impedance

Zth in Eq. (3.6) and fitting the measured L-I-V curve. As the active region

temperature increases, the Fermi-Dirac distribution is broadened and the

DOS of the GaN matrix becomes much larger than that in the room tem-

perature. Even though the DOS of the SAQD system is hardly affected by

temperature, the above two effects can significantly change the positions of

the QFLs and lead to the decrease of material gain and the red shift of the

gain spectra. Figure 3.5(a) shows the effect of active region temperature on

the modal gain spectra calculated using Eq. (3.6) at a fixed injection level

n2d = 40N2D with all the other parameters kept the same, and Fig. 3.5(b)

shows the measured and calculated L-I curves using difference values of ther-

mal impedance. The fitting gives a thermal impedance of 43 oC/W, which

falls between a value of 30 oC/W in a blue QW-based ridge-waveguide laser

grown on a GaN substrate and a value of 60 oC/W in another blue laser of

the same structure but on a sapphire substrate [87]. The high indium con-

tent in the In0.4Ga0.60N/GaN SAQD laser structure is presumably the factor

that increases the thermal impedance. Due to a very low wall-plug efficiency

(WPE = L/(IV )|I) of ∼0.15% at 400 mA, the large input power ∼ 6 W is

converted into a significant amount of heating and brings the active region

temperature to be over 250 oC. Such high temperature was also reported in

a simulation of high-power GaN LDs emitting at 400 nm [88].

A very high diode ideality factor nd = 38.5 found by our combined diode-

(a) (b)

Figure 3.5: (a) Modal gain spectra as functions of active regiontemperature under an injection level of n2d = 40N2D for every active regiontemperature. (b) Calculated L-I curves using different thermal impedances.

35

resistor model suggests a large resistance causing the self-heating in the LD.

In fact, as confirmed by the growers, the unoptimized p-doping in GaN ma-

terial system is the main reason for the low WPE. As a result, more carriers

need to be injected to further increase the gain and maintain laser operation.

The high injection level also strengthens the screening effect, increases the

peak gain and blue-shifts the peak gain wavelength. The lasing wavelength

at peak output power is estimated to be around 649 nm, almost 20 nm away

from the targeted 630 nm. The total loss seems to be dominated by the

intrinsic loss, which is higher than 30 cm−1. This is presumably due to both

the free-carrier absorption at high injection levels, the scattering of light at

imperfect device sidewalls or even at the 3D geometry of the SAQDs. A

characteristic temperature T0 = 271 K is derived from our model; however,

it deviates from a measured value of 236 K. The difference is attributed to

the fact that a constant injection efficiency ηi is used across all substrate

temperatures in the calculation, which is not necessarily true. Nevertheless,

it is possible to fill the gap by considering carrier transport effects in the

model. The fitting parameters for the L-I-V modeling are given in Table A.2

in Appendix A.

Our model also reveals the details of device performance and suggests

possible ways to obtain better devices. The shunt resistance, representing

parasitic leakage through the sidewall imperfections left by the dry etching

process, is found to be around 6 kΩ, meaning the shunt leakage component

is much less significant than the current entering the LD. Indeed, the shunt

leakage remains as low as a few milliampere over the entire operating range.

The shallow etch only going down to the waveguide layer instead of passing

through the active region explains the low leakage current.

Figure 3.6 shows a break-down of the rates, including injection, non-

radiative recombination, spontaneous and stimulated emissions in the simu-

lated LD. The heat generated by the highly resistive p-layers raises the active

region temperature and makes the non-radiative recombination process con-

sume most of the injected carriers and stay on the same order of magnitude as

the stimulated emission process, even beyond threshold. Though the SAQD

structure is supposed to provide tighter confinement in the transverse di-

rection, excessively high active region temperatures may thermally free the

carriers again and allow them to recombine non-radiatively. The residual po-

larization field also hinders the EHPs from recombining efficiently and thus

36

Figure 3.6: Simulated injection (black), non-radiative recombination (red)and spontaneous emission (blue) rates, together with stimulated emission(magenta) rate in the modeled device.

lowers the radiative recombination rates. Moreover, the call for more carriers

to re-pump the SAQD active material results in the carrier un-pinning effect.

To improve the device performance, it is suggested by the model that opti-

mizing the Mg doping and the including heat sinking structures are of top

priority in designing better high indium content devices, as the thermal cycle

is inevitable in every laser. The remedy to thermal issues cannot only avoid

the high carrier density to reach threshold but also reduce the free-carrier

absorption, which partly accounts for the high intrinsic loss in this specific

device.

3.5 Discussions of III-N light sources for

telecommunications

Although the narrow band gap of InN has brought an excitement of having

another direct-band-gap semiconductor material emitting in the telecommu-

nications window, growing high-quality InN has been proven extremely chal-

lenging. Aside from the low dissociation temperature (< 600 oC ) and the

subsequently excessively high nitrogen over-pressure in the reaction chamber

37

that are not aligned with typical growth conditions in the MOCVD pro-

cess [89–91], the lack of suitable substrates and the high tendency of forming

clusters greatly increase the difficulty in producing InN thin films with a

reasonable quality using any epitaxial technology. Three-dimensional nano-

structures, such as quantum dots, nano-column and nano-wires are found to

be feasible remedies as they have more interfaces to release the accumulated

strain and grow with low defect densities. Among them, nano-columns and

wires are highly favored as they can be grown on foreign substrates, which

mitigates the lattice matching problem [92–94].

It has been found that under certain growth conditions disk-like structures

can form in nano-wires and provide quantum confinement [95–97]. The size

and the aerial density of the disks are on an order similar to those of the

SAQDs (∼ 30 nm in diameter and ∼ 5 nm in height, and ∼ 1010 cm−2 re-

spectively) and they have been proven capable of emitting light in the green-

red spectral region under electrical pumping. The clean disk/barrier mate-

rial boundaries and the high uniformity of the wire arrays seem to indicate

that MBE-grown nano-wire structures could be an excellent platform to ac-

commodate high-indium-content InGaN alloys without generating too many

crystalline imperfections. Indeed, lasers based on In0.85Ga0.15N/In0.40Ga0.60N

disk-in-nano-wire (DINW) structures emitting close to the 1310-nm telecom-

munications window have been demonstrated under pulsed operation [98].

Using the advantage of being not selective to the substrate, these nano-wires

were grown on (001) Si and the resultant lasers showed lasing wavelengths at

around 1.17 µm with characteristic temperatures T0 ∼ 242 K. It is expected

that increasing the indium composition in InGaN alloys or even growing pure

InN DINW structures may extend the emission wavelength of the lasers into

the 1310- and 1550-nm windows so that III-N active materials can be merged

with silicon photonics; however, at this point, harnessing pure InN with the

presence of Ga atoms during MBE growth process still appears to be chal-

lenging and the resultant nano-wires are not uniform with respect to wire

length and radial geometry. Though the QCSE has been observed in the

system as the EL emission wavelength blue shifts with injection current, it is

still unclear where the quantum confinement comes about. Based upon the

TEM and EDX data available when this thesis is written, it is possible that

some QD-like indium segregations or superlattice-like compositional transi-

tion at material boundaries can trap the carriers and facilitate the radiative

38

recombination process, as calculated in Fig. 3.7. Nevertheless, more detailed

inspections of the InN/InGaN nano-wire structures, especially for identify-

ing the origin of quantum confinement and confirming the dimension of the

confining structure, are absolutely needed to help understand the material.

A better understanding of the physics is required to contribute to device

designs, as reported in this chapter. Incorporating InN as a laser source in

the telecommunications window faces two major concerns: the high threshold

current and the scattering of optical mode due to the irregular shapes of the

nano-wires. Grown along the c-axis, the nano-wires in principle suffer from

QCSE so that the optical gain may still remain low before carrier screening

takes place. This in turn gives rise to the self-heating process that ultimately

limits the laser to be operated in pulsed mode. Although device parameters

were taken from broad-area lasers meant to show the intrinsic behavior of

the nano-wires [98], it is the narrow-ridge geometry that gives better mode

confinement and decreases the threshold current. With III-N active materials

still under development in the telecommunications window, MQW structures

based on the InP system are chosen as the gain medium when designing GaN-

based integrated photonic devices in this thesis, as they are well-established,

relatively reliable and regarded as the most efficient.

Figure 3.7: Wavefunction and eigen-energies of the confined and injectorstates in superlattice-like structural features observed in InN/InGaNnano-wires. The calculated transition wavelength, 1.61 µm, close to themeasured PL peak at ∼1.65 µm.

39

CHAPTER 4

GaN-BASED VOLTAGE-CONTROLLEDPOLARIZATION ROTATOR (VCPR)

4.1 Introduction and motivation

The ever-increasing information traffic in optical communication networks

has called for next-generation technologies with a large data capacity, low

power consumption and high signal transmission fidelity. Silicon photonics

has been considered as a potential platform for realizing coherent optical

communications, as it is by nature cost-effective, scalable and processing-

compatible with the mature CMOS manufacturing technology [99–101]. Var-

ious photonic devices, such as tunable lasers, modulators, photodetectors and

many other passive elements have been demonstrated using the silicon pho-

tonics platform to achieve complex functionality [102–113]. Nevertheless,

the tuning parts in these devices have to rely on plasma and thermal-optic

effects, which introduce free-carrier absorption loss and extra power con-

sumption for the heater, since other optical effects are essentially missing

in Si, as mentioned in Section 1.4. As a result, it is difficult to implement

active devices able to efficiently manipulate photon properties on the silicon

photonics platform.

Polarization rotators (PRs) and polarization splitters (PSs), the very key

elements in polarization-division multiplexing (PDM), are some of the devices

that cannot be built on the silicon photonics platform with ease. Either ex-

traneous materials such as LiNbO3 or complicated 3D dielectric stacks have

to be introduced to build these polarization-specific devices [114–118]; how-

ever, incompatibility with the common silicon-on-insulator (SOI) technology,

small tolerance for fabrication errors, and high sensitivity to ambient tem-

peratures make these solutions not truly viable in the long run. To improve

the efficiency and robustness of PDM devices, new design strategies and new

electro-optically responsive materials have to be used.

40

In this chapter, a voltage-controlled PR (VCPR) based upon epitaxial GaN

is proposed. Due to the non-centrosymmetry, GaN has much stronger piezo-

electric, photo-elastic and electro-optic effects than GaAs and InP, which are

common materials for optoelectronic devices on the silicon photonics plat-

form. These effects not only make the refractive index distribution in the

PR more responsive to external electric stimuli, but allow feedback loops to

correct any refractive index drift caused by the change of surrounding tem-

perature. Furthermore, the emerging GaN-on-silicon growth technology not

only lowers the manufacturing cost, but avoids complicated substrate removal

processes such as laser liftoff [119–122]. In the following sections, device in-

formation, dispersion engineering and mode-switching characteristics in the

GaN PR are studied theoretically.

4.2 Index ellipsoid, electro-optic and photo-elastic

effects

Due to the hexagonal symmetry in the wurtzite lattice, III-Ns are biaxial

optical materials. Namely, they show two distinct refractive indices in the

directions parallel and perpendicular to the c-axis, which is also the optical

axis of the materials. However, the optical response of a biaxial material

actually depends upon the direction of incidence with respect to the optical

axis, as described using the index ellipsoid. Figure 4.1 shows a graphical

representation of the index ellipsoid, which can be viewed as equi-energy-

density surfaces in a biaxial material.

When an electromagnetic wave propagates through an anisotropic mate-

Figure 4.1: Graphical representation of index ellipsoid of a biaxial material.The propagation direction of the incoming wave is shown in κ [123].

41

rial, the electromagnetic energy density ue stored in that material is

ue =1

2D · E =

1

2D · ¯

εεε−1 : D =1

2ε0

∑i,j

DiDj

εij(4.1)

where i, j ∈ x, y, z and εij denotes the elements in the permittivity tensor

of the material. It can be easily seen that the anisotropy of the material

interacts with the incident wave in terms of stored electromagnetic energy.

Since biaxial materials can be characterized only using the ordinary (when

the electric field orients normal to the optical axis) and the extraordinary

(when the electric field lies parallel to the optical axis) permittivities (εo and

εe), the permittivity tensor takes the form

¯εεε =

εo 0 0

0 εo 0

0 0 εe

(4.2)

and the equi-energy surfaces can be simplified as

ue =1

2ε0

(D2x

εo+D2y

εo+D2z

εe

)=

1

2ε0

(D2x

n2o

+D2y

n2o

+D2z

n2e

)(4.3)

which is an ellipsoid as depicted in Fig. 4.1. It can be proven that the

extraordinary index changes with the angle θ between the wave vector of the

incident wave and the optical axis

1

n2e(θ)

=cos2 θ

n2o

+sin2 θ

n2e

(4.4)

while the ordinary index is independent of the direction of incidence. Since

wurtzite III-Ns are usually grown on the c-plane, which is the waveguide

plane in the design of the proposed PR, only θ = π/2 is considered.

The electro-optic (EO) effect describes the change of index ellipsoid in re-

sponse to the electric field. It is important to note that though the incoming

wave is able to interact with the material, the changes in the refractive in-

dices (∆n’s) due to its electric field is minute since the optical intensities

in common on-chip optical communication systems are not as high. In fact,

it is usually an external bias field applied on the EO device that produces

appreciable ∆n’s. The change of the material’s permittivity tensor due to

42

the applied bias field is [∆

(1

ε

)]ij

=∑k

rijkEk (4.5)

where rijk is the elements in the EO tensor of the material. In materials of

wurtzite lattice structures, the EO tensor is

¯r =

0 0 r13

0 0 r13

0 0 r33

0 0 0

0 0 0

0 0 0

(4.6)

Consequently, the ordinary and extraordinary indices under the effect of

applied bias field are

1

n2o(E)

=1

n2o(0)

+ r13Ex (4.7a)

1

n2e(E)

=1

n2e(0)

+ r33Ez (4.7b)

in normal EO materials, rijk’s are on the order of pm/V, which means that

no and ne can be approximated as

no(E) = no(0)− 1

2r13n

3o(0)Ex (4.8a)

ne(E) = ne(0)− 1

2r33n

3e(0)Ez (4.8b)

provided the bias field is low enough such that rijkEk 1. Indeed, the

bias field in real applications never exceeds the breakdown field of the EO

material, which is rarely above the order of MV/cm [124,125].

Similar to the EO effect, photo-elastic (PE) effect relates the change of

index ellipsoid with the applied mechanical stimulus on the material. The

∆n’s due to the PE effect is expressed using PE tensor ¯p:[∆

(1

ε

)]ij

=∑kl

pijklσkl (4.9)

43

where σkl is the mechanical stress components acted on the PE material. It

should be noted that the PE tensor can be expressed in either stress form

(pijkl in GPa−1) or strain form (pijkl dimensionless).

Again, the PE tensor in wurtzite materials has the form

¯p =

p11 p12 p13 0 0 0

p12 p11 p13 0 0 0

p13 p13 p33 0 0 0

0 0 0 p44 0 0

0 0 0 0 p44 0

0 0 0 0 0 p66

(4.10)

which results in

no(σσσ) = no(0)− 1

2n3o(0) [(p11 + p12)σxx + p33σzz] (4.11a)

ne(σσσ) = ne(0)− 1

2n3e(0) [2p13σxx + p33σzz] (4.11b)

nyz(σσσ) =1

√p44σyz

(4.11c)

nxy(σσσ) =1

√p66σxy

(4.11d)

Note that p44 and p66 generate non-zero off-diagonal elements nxy, nyz and

nzx in the refractive index tensor of the PE material.

4.3 Design of GaN-based VCPR

Figure 4.2 shows a proposed processing flow of the VCPR. The GaN epitaxial

layer is grown on (111) Si substrate, presumably on top of an AlGaN/AlN

buffer layer which absorbs most of the possible crystalline defects generated

during the growth phase. To transfer the GaN thin film on to standard SOI

substrates, a flat, mechanically stable bonding interface is highly required.

Such a bonding interface is prepared by depositing a thin layer of silicon

dioxide (SiO2) on top of the GaN and oxidizing the top of the SOI wafer

into SiO2, in order to have two chemically similar surfaces that facilitate the

bonding process. The (111) Si substrate and the AlGaN/AlN buffer layer on

the GaN side are removed from the bonded structure following wafer transfer.

44

Figure 4.2: Envisioned processing flow of the GaN VCPR.

The definition of the rib structure follows typical photo-lithography and dry

etching steps, and the electrodes where the external biases are applied are

formed by metallization. The following rib waveguide design is tailored for

the 1310-nm telecommunications window.

The modeling of the VCPR starts after the substrate/buffer layer removal

process. In principle, the device allows external voltages to modify the ordi-

nary and extraordinary refractive indices in the GaN layer, or equivalently,

the dispersion characteristics of the fundamental transverse electric (TE) and

transverse magnetic (TM) modes, such that the dominating mode in the rib

waveguide can be switched from one to the other. The modes in the GaN

VCPR are named using their electric field components and the number of

nodes in the electric field intensity profile. For instance, as shown in Fig.

4.3(a), the mode having a dominating x-component in the electric field with-

out any node is termed as the TE00 mode, whereas the mode with a node in

its dominating Ey intensity profile is termed as TM10. Figure 4.3(b) visual-

izes the model for the VCPR. Several design parameters are labeled, such as

the width (w) and the height (d) of the rib, the thickness (t) of the GaN layer

on the sides and the voltage configuration Vi’s. To simplify the design of the

VCPR, w and d are the only two parameters used to engineer the waveguide

dispersion under the bias-free condition. In fact, the dispersion curves of the

modes are solely controlled by the aspect ratio of the confining structure,

namely the rib, in this case. The thickness t of the GaN layer on the sides

is set to be 500 nm, and the thicknesses of the deposited oxide layer, the

45

(a) (b)

Figure 4.3: (a) The normalized electric field intensity profiles of the TE00

and TM10 modes in the rib waveguide structure. (b) The structure andmodeling parameters for the GaN VCPR.

sandwiched SOI Si, and the buried oxide (BOX) are set to be 500 nm, 220

nm and 2 µm, respectively. The (100) Si substrate is omitted in the model

as the modes are confined fairly well in the GaN rib waveguide. However, it

is worth noting that the Si layers actually have the highest refractive index

in the 1300-nm range, which means mode leakage may play an important

role in the propagation characteristics of the GaN VCPR. Special attention

has to be paid in this model since the optical axis of the GaN crystal now

aligns with y axis. As a result, the form of the refractive index, EO and PE

tensors, Eqs. (4.2), (4.6), (4.10), have to be modified accordingly.

The key part of device design is to create a mode-conversion process that

transfers the power of the TE00 mode at the input end to the TM00 mode at

the output end. This can be achieved by perturbing the waveguide device

using external bias fields such that the input end, the middle part, and the

output end of the waveguide predominantly support TE00, hybrid, and TM00

modes, respectively. The power-transferring process relies on the fact that

the power interaction integral between the dominant modes in the i- and j-th

sections of the waveguide ∫Ei ×H∗j · zdS (4.12)

is non-zero at the waveguide junction.

Figure 4.4(a) shows the dispersion of the ordinary and extraordinary re-

fractive indices of bulk GaN measured using prism coupling method [126].

Though the sample was grown on sapphire using MBE, the thick (> 2 µm)

46

(a) (b)

Figure 4.4: (a) The ordinary and extraordinary refractive indices of bulkGaN [126]. (b) The effective indices of TE and TM modes in a GaN slabwaveguide for λ = 1310 nm as functions of guide thickness. A crossoverpoint in the dispersion curve can be observed at ∼880 nm, where thedominant modes are two degenerate hybrid modes.

LT-AlN/AlN/AlGaN buffer layer between the epitaxial GaN and the sap-

phire substrate should yield reasonably good crystalline quality. It can be

foreseen that in a sufficiently thick GaN slab waveguide, a y-polarized mode

will have a higher effective index than its x-polarized counterpart, as the ex-

traordinary index in GaN is higher than the ordinary one for all wavelengths.

Nevertheless, changing the waveguide geometry can lead to changes in the

mode effective indices and create drastically different dispersion characteris-

tics. Figure 4.4(b) shows the effective indices of TE and TM modes in a GaN

slab waveguide for a wavelength of 1310 nm as a function of the guide thick-

ness. It is obvious that decreasing the guide thickness fundamentally changes

the optical property of the GaN layer, in other words, the GaN waveguide

seems more “TE-supportive” when the guide thickness is thin enough, which

is fundamentally different from what it is as a bulk. Moreover, the TE and

TM modes mix and become two degenerate hybrid modes at the crossover

point (guide thickness ∼ 880 nm). Such a simple dispersion engineering gives

a convenient way to generate hybrid modes, which are essential in the design

of the GaN VCPR.

Since the VCPR is of a rib-waveguide structure, either the rib width w

or the rib height d has to be fixed in order to simplify the dispersion engi-

neering process. Keeping a constant rib width is the most logical approach,

since, practically, varying w requires redesigning photo-lithography masks,

and physically, a ridge that is too wide tends to support higher-order trans-

47

verse modes. Consequently, taking the size and the separation between the

top two electrodes into account, a ridge width of 4 µm is chosen in the model

as a first step. Further modification on the ridge width, depending on the

configuration of the electrode, can be included in future designs. In fact, a

4-µm-wide ridge does support TE10 and TM10 modes; fortunately, the inci-

dent power, either generated by on-chip LDs or transmitted by optical fibers,

typically concentrates at the center of the corresponding guiding structure so

the power interaction integral Eq. (4.12) is minimized between either TE10

or TM10 and the incident mode.

Figure 4.5(a) shows the dispersion curves of TE00 and TM00 modes in the

GaN VCPR with different rib heights d. Note that the thickness of the GaN

layer on both sides of the rib is kept at 500 nm for all the rib heights. For a

given rib width of w = 4 µm, the dispersion curves of TE00 and TM00 modes

start to intersect within the 1310-nm window when the rib height is between

350 and 400 nm. Figure 4.5(b) shows the dispersion curves of TE00 and TM00

modes when the rib height is set to be 380 nm, and the crossover point is

∼1308.1 nm. It has been proven that the spectral position of the crossover

point is quite sensitive to the rib height, or effectively, the dry etching process

does not always guarantee a crossover point at a given wavelength under the

bias-free condition; however, a proper bias configuration is able to modify the

refractive index distribution in the GaN guide layer and correct the spectral

misalignment between the crossover point and the desired wavelength.

(a) (b)

Figure 4.5: (a) The TE00 and TM00 dispersion curves in the GaN VCPRwith different rib heights. Two crossover points are highlighted. (b) TheTE00 and TM00 dispersion curves in the GaN VCPR with a 380 nm height.

48

4.4 Voltage-controlled dispersion characteristics

Thanks to the piezoelectric, electro-optic and photo-elastic properties of

GaN, the effect of applied voltage is twofold: the external field can strain

the material through the piezoelectric effect, which leads to the change of

refractive indices due to the PE effect; the external field can also modify the

refractive tensor directly via the EO effect.

Modeling the seemingly simple GaN VCPR is actually sophisticated. The

whole model can be decomposed into two major blocks: a coupled mechanical-

electrostatic problem which describes the interaction between piezoelectric,

EO and PE effects when the device is biased, and a standard wave propa-

gation problem that determines the mode characteristics in the VCPR. The

coupled mechanical-electrostatic problem is described by the following dif-

ferential equations that are similar to those in the modeling of InGaN/GaN

SAQDs in Chapter 2:

σσσ − σσσ0 = ¯C : (εεε− εεε0)− ¯e t : E (4.13a)

D−D0 = ¯e : (εεε− εεε0) + ¯εεε : E (4.13b)

εεε =1

2

[∇u + (∇u)t

](4.13c)

∇ · σσσ + f = 0 (4.13d)

D = ¯εεε : E (4.13e)

−∇V = E (4.13f)

where σσσ is the stress induced by the strain in the material εεε, ¯C the stiffness

tensor, ¯e the piezoelectric tensor, D the electric displacement, ¯εεε the permit-

tivity tensor, and f the force exerted on the material. The subscripted D0,

σσσ0 and εεε0 denote the polarization, stress and strain inherent in the material

without external stimuli.

The refractive index (permittivity) is obtained by solving Eq. (4.13) self-

consistently, with boundary conditions defined by the applied voltages Vi’s for

the electrostatic problem and a fixed-end condition u = 0 for the static me-

chanical problem. Solving the wave equation with updated refractive indices

is relatively straightforward. As the goal is to design the voltage-controlled

characteristics in the GaN PR, a commercial two-dimensional finite-element

solver is used for optimized speed and accuracy. The material parameters

49

Figure 4.6: Flowchart of the GaN VCPR model. The relative permittivitiesof the materials without any external stimuli are denoted as ε

(0)r .

of GaN are taken from [67, 126–128], and a complete modeling flowchart is

shown in Fig 4.6.

Figure 4.7(a) shows the electrostatic potential and electric field distribu-

tions in the GaN VCPR with a bias voltage of 10 V on the top two electrodes

(a) (b)

(c) (d)

Figure 4.7: (a) The electrostatic potential and electric field distribution inthe GaN VCPR when the top two electrodes are biased at 10 V, and thetwo side electrodes at −10 V. And the change in the ordinary (b) andextraordinary (c) and off-diagonal (d) permittivities in the VCPR underthe same bias condition.

50

and −10 V on the two side electrodes. An optimal design should yield a max-

imum amount of overlap between the electric field with the highest strength

and the mode profile with the peak intensity such that the propagation mode

can perceive the change of the refractive indices to the largest extent. Figures

4.7(a) and (b) show the spatial distribution of the change in the ordinary (εxx)

and extraordinary (εyy) permittivities. It can be observed that the change

in permittivities in the region right beneath the electrodes, where the field

is strong, is dominated by the EO effect, whereas in the region outside the

shadow of the electrodes is dominated by the PE effect. The PE effect also

renders a change in the off-diagonal permittivity εxy, which promotes power

conversion between orthogonal modes [129], as shown in Fig. 4.7(d).

In the proximity of the crossover point, 1308.1 nm in the case, there are

two nearly degenerate modes with very similar effective indices. Figure 4.8(a)

shows the effective indices of these two modes as functions of applied voltage

on the electrode. The bias configuration is symmetric about the center of the

PR structure, as shown in the inset. “Mode 1” in this case refers to mode

having a lower effective index among the two nearly degenerate modes, and

“Mode 2” refers to the one with a higher index, the dominant mode in the

waveguide. The effective indices of both Mode 1 and Mode 2 decrease with

an increasing bias; however, with different slopes below and above a voltage

∼ 0.74 V. The SOPs of these two degenerate modes can be characterized

(a) (b)

Figure 4.8: (a) The voltage-dependent effective indices of the two nearlydegenerate modes in the GaN VCPR. The inset shows the biasconfiguration used in this plot. (b) The “X-percentages” of the two nearlydegenerate modes as functions of applied voltage.

51

using “X-percentages,” defined as the fraction of the power carried in the

x-component of the mode

X-percentage =

∫|Ex(x, y)|2dS∫

|Ex(x, y)|2dS +∫|Ey(x, y)|2dS

(4.14)

As shown in Fig. 4.8(b), the X-percentage in the dominant mode (Mode

2) remains approximately the same before ∼ −2.5 V, but drops precipitously

beyond that point. In other words, the SOP of the dominant mode changes

from x-polarized to y-polarized when the bias sweeps from −10 V to 10

V. The same bias configuration has an opposite effect on Mode 1, which

switches from y-polarized to x-polarized across the bias window. It is clear

that the dominant mode in the GaN VCPR can be changed from TE00 to

TM00 mode solely using a proper bias configuration. A summarizing plot of

the operation and the evolution of the dominant mode of the GaN VCPR

is given in Fig. 4.9. The electric field intensities are plotted at −10 V, 0.74

V and 10 V, respectively. To receive the TE-polarized wave, which most of

the semiconductor lasers emit, at the input port, the VCPR will be biased at

−10 V and the voltages will be gradually increased from −10 V at the input

port to ∼ 0.74 V at the hybrid section in order to transfer the power in the

TE00 mode to the dominant hybrid mode. By the same token, the bias will

Figure 4.9: The electric field intensity of the dominant mode at differentbias points.

52

then be increased again from ∼ 0.74 V at the hybrid section to 10 V at the

output port for power transferring from the dominant hybrid mode to the

TM00 mode. The polarization rotation is thus completed.

4.5 Discussion and possible improvements

The distance over which the polarization rotating process can be completed

is the most important figure of merit of PRs. While the length of reported

pure-dielectric-based PRs are generally on the order of several hundreds of

micrometers [116–118], the length of the proposed GaN-based VCPR should

be, in principle, decided by the lengths of the TE-supportive, hybrid, and the

TM-supportive sections. However, as the first step to simulate the propaga-

tion characteristics of the VCPR, obtaining the three-dimensional refractive

index profile was proven to be very challenging because a very dense mesh

needs to be used in order to catch the local variation in the refractive indices

in the cross section plane of the waveguide and the fringing field around each

segmented electrode. Such calculation is computationally expensive and does

not converge when device length becomes large compared to the size of the

discretization elements.

Another obstacle comes from the lack of precise material parameters. Data

obtained from earlier studies suffered from low material quality, as charged

crystalline imperfections, free carriers and scattering at rough material in-

terfaces can easily change the behavior of the electromagnetic waves trav-

eling through the material [130–133]. Some systematic measurement of

photo-elastic and electro-optic coefficients of GaN have been reported re-

cently [126, 127]. Although the measurements were performed on different

GaN samples, the general methodology can be applied to any GaN epitax-

ial thin film of interest. More definite values should come with improved

material quality and it is expected that the optical constants could actually

be larger than the reported ones in high-quality GaN, which will help the

simulation and device performance.

53

CHAPTER 5

DESIGN OF TUNABLE LASERS USINGGaN-BASED VOLTAGE-TUNABLE

GRATINGS

5.1 Introduction and motivation

Lasers emitting at a single wavelength with high side-mode suppression ratios

(SMSR) are absolutely required in the transmitter end of WDM systems.

Distributed feedback (DFB) lasers are viable choices as grating structures are

embedded to provided optical feedback and select a particular longitudinal

mode that matches the WDM channel wavelength [134]. However, while

keeping a few DFB lasers in the transmitter end is possible in coarse WDM

(CWDM) applications with a channel spacing of 20 nm [135], doing so in

dense WDM (DWDM) is problematic since the channels are tightly spaced

and the number of channels is large. Tunable lasers turn out to be a plausible

solution, as their wide range of wavelength tunability eliminates the need of

packaging and aligning multiple DFB lasers in the optical system [136].

Since the first demonstration of tunability in semiconductor lasers in the

early 1980s [137, 138], different tuning mechanisms based upon carrier in-

jection [139–142], heating elements [143, 144] and microelectromechanical

systems (MEMSs) [145, 146], have been realized on sampled gratings, Y-

branches and micro-ring resonators to enable wavelength selection [147,148].

Nevertheless, these tuning mechanisms generally generate excessive heat and

intrinsically slow due to the slow temporal characteristics of the heat trans-

ferring process and the response of the mechanical system. In this chapter,

following the idea of incorporating GaN in the field of integrated photonics, a

voltage-controlled tunable laser is proposed. The index change in this specific

proposal, as opposed to the aforementioned methods, leverages the EO and

PE effects in GaN and is solely dependent on external bias fields across the

Y-branch grating arms. Controlling the index using electric fields avoids the

slow heat process and does not involve additional MEMS structures which

54

complicates the fabrication process.

In the following sections, a Y-branch with sampled gratings is modeled in

a general way, that is, without knowledge of the waveguide material, and a

process development of InP-based gain chip is presented. Once measured,

additional information about the effective index of the mode in the GaN

waveguide can be easily folded into the model as guidance for designing and

fabricating actual devices in the future.

5.2 Grating design based on coupled-mode theory

While the propagation matrix method is sufficient to handle calculations of

reflection and transmission spectra of a given layered structure, coupled-mode

theory (CMT) deals with more complicated structures where interactions

between modes are considered. In CMT, the interaction between any given

pair of modes is defined generally by a unique coupling coefficient κ, which

in general carries structural information of the optical system. Knowing the

coupling coefficient κ, one can utilize CMT to model structures as simple

as distributed Bragg reflectors (DBRs) to multi-ring add-drop optical filters

with ease [149]. When designing the grating arms of the GaN-based tunable

laser only the fundamental waveguide mode is concerned, as the presence

of higher-order modes is undesirable because these modes generally compete

with the fundamental mode for incoming power.

To analyze wave propagation in a grating structure, shown as a solid black

line in Fig. 5.1 for instance, it is generally assumed that the periodic variation

in permittivity ∆ε(r) only acts as a perturbation to the propagation mode,

namely, ∆ε(r) does not significantly change the mode profile. Two waves,

forward- and backward-propagation modes, are considered here and they are

coupled by the grating’s Bragg vector, 2π/Λ. Their amplitudes are now

spatially-varying functions A(z) and B(z) due to the grating structure and

their propagation constants are no longer the same as those in a unperturbed

waveguide owing to the coupling between them.

The coupled-mode equations can be set up by incorporating the permit-

tivity variation into the wave equation

∇2 + ω2µ0 [ε1 + ∆ε(r)]E′t(r) = 0 (5.1)

55

Figure 5.1: Graphical representation of a digital grating with a period Λand a permittivity variation ∆ε(r). The amplitude functions A(z) and B(z)describe the field distribution resulted from the grating structure.

where the new field distribution E′t is simply the original one Et modulated

by the amplitude functions A(z) and B(z)

E′t(r) =[A(z)eiβ0z +B(z)e−iβ0z

]Et(r) (5.2)

with Et(r) the original field distribution satisfying

(∇2 + ω2µ0ε1

)Et(r) = 0 (5.3)

Assuming that the two modes are coupled via the grating Bragg vector

β0 = −β0 +2πP

Λ, P ∈ N (5.4)

and the permittivity can be written as its Fourier series

∆ε(r) =∑P

εP (x, y)ei2πP

Λz (5.5)

one can transform Eq. (5.1) into the coupled-mode equation by matching

the phase terms in the exponential functions

dA

dzeiβ0z = iKbaBe

−iβ0zei2πP

Λz (5.6a)

dB

dze−iβ0z = iKabAe

β0ze−i2πP

Λz (5.6b)

where Kij’s are the coupling coefficients between the two modes and defined

by the spatial overlap between the unperturbed mode profile and the P -th

56

order Fourier component of the grating:

Kab =ω

4

∫εP (x, y)|Et(x, y)|2d2r (5.7a)

Kba = −ω4

∫ε−P (x, y)|Et(x, y)|2d2r (5.7b)

Approximation based upon the effective index method can be used to cal-

culate the coupling coefficient of the grating. Using the fact that ∆ε =

[neff + ∆neff(z)]2 − n2eff ' 2neff∆neff(z) and the normalization condition

1

2η

∫|Et(x, y)|2dS = 1 (5.8)

one arrives at a general expression of the coupling coefficient associated with

the P -th order grating vector:

κ = k0 ×∆n

(2πP

Λ

)(5.9)

where k0 is the free-space vector at the wavelength of interest and ∆n is the

Fourier transform of the index variation ∆n(z).

Although Eq. (5.6) directly defines the spatial evolution of the amplitude

functions, it is more concise to express it in the grating’s eigen-space by

defining a detuning parameter

∆β = β0 −πP

Λ(5.10)

such that

A(z) = A′(z)e−i∆βz (5.11)

B(z) = B′(z)ei∆βz (5.12)

and the new amplitude functions A′(z) and B′(z) satisfy

d

dz

[A′

B′

]= i

[∆β Kba

Kab −∆β

][A′

B′

](5.13)

Equation (5.13) is essentially an eigen-value system whose eigen-values are

57

the actual propagation constants of the two modes under mutual interaction.

Solving the above equation yields[A′(z)

B′(z)

]=

[cos(βz) + i∆β

βsin(βz) iKba

βsin(βz)

iKabβ

sin(βz) cos(βz)− i∆ββ

sin(βz)

][A′(0)

B′(0)

](5.14)

where β2 = (∆β)2 +KabKba is the dispersion relation in the grating.

With appropriate boundary conditions, different physical information can

be extracted from Eq. (5.14). For instance, if the structure is used as the

grating in a distributed-feedback (DFB) laser with a device length L, A′(0) =

B′(L) = 0 shall be imposed as the boundary condition as there is no incident

wave at both ends of the cavity while lasing. This gives rise to the lasing

condition of the DFB laser:

β = i∆β tan βL (5.15)

Note that Eq. (5.15) does not hold for real β; instead, β has to be complex

to carry the information of modal gain that compensates for the diffraction

loss of the grating at the threshold.

When using the grating as a reflector, only B′(L) vanishes because there

is no incident wave from the end of the grating. The reflection coefficient of

the grating can be expressed as

rg =B′(0)

A′(0)|B′(L)=0 =

iKab sin(βL)

β cos(βL)− i∆β sin(βL)(5.16)

Figure 5.2 illustrates the effect of the coupling coefficient on the power

reflection spectra |rg|2, assumed that |Kab| = |Kba| = κ. It is obvious that

the magnitude of the coupling coefficient determines the effectiveness of the

reflection spectra, as when κ drops to zero there is no reflection because the

grating reduces to a waveguide with a smooth interface. It should be also

noted that for any given coupling coefficient the reflection peak occurs at

∆β = 0, where the half of the effective wavelength of the light in the un-

perturbed structure matches the grating period. Consequently, one should

carefully design the grating geometry such that the coupling coefficient makes

the reflectivity reach the desired value at the target wavelength. As a nu-

merical example, to realize κL = 4 with a grating length L = 500 µm at

58

(a) (b)

Figure 5.2: Power reflectivity spectra of a grating with L = 500 µm,neff = 3.5 and Λ = 187.14 nm, plotted against (a) the normalized detuningparameter, and (b) wavelength.

the 1310-nm window, one needs a index variation ∆n ∼ 5× 10−3 in a digital

grating.

5.3 Wavelength tuning using sampled grating

structures

Although a proper choice of grating period Λ and coupling strength κL can

provide a high enough reflectivity for a certain wavelength, the bandwidth

of the reflectivity spectra remains narrow (∼1 nm for κL = 1 and ∼2 nm

for κL = 4) and only enough to accommodate a small number of cavity

modes of the gain chip, whose free spectral range (FSR) is around the order

of 5 A. Such a narrow bandwidth is not sufficient for CWDM applications,

which require a channel spacing of 20 nm as defined by the International

Telecommunication Union [135], and not enough for compensating for the

red shift of lasing wavelength due to the self-heating in the gain chip either.

Sampled gratings, Fig. 5.3(a), turn out to be an elegant solution. The orig-

inal dense grating, characterized by a period of Λ, is truncated periodically.

The dense grating only appears in the “burst” region and its spatial repetition

occurs with a much larger “burst period” Z. Following the aforementioned

analysis, one can easily show that while only one Fourier component is enough

for describing the interaction between the two counter-traveling fundamental

modes in the original denser grating, it is now necessary to include multiple

59

(a)

(b)

Figure 5.3: (a) (Left) a schematic of a sampled grating structure. Inside thered dashed-line box is the zoomed-in features of the burst part. The dutycycle of the burst is denoted as r. (Right) Mathematical model for thesampled grating structure. (b) Phase matching diagram of the gratingstructure. The magnitudes of the “allowed” wave vectors have to liebetween the two half circles to satisfy the guidance condition.

Fourier components to cover the interactions between all possible modes that

satisfy the phase-matching condition, as illustrated in Fig. 5.3(b), since the

burst period Z is in general much larger than Λ. The burst grating Bragg

vector 2π/Z is much smaller in magnitude, hence waves with slightly longer

(shorter) wavelengths are now allowed to exist in the sampled grating struc-

ture with the interaction of the burst grating vectors 2π/Λ± 2πm/Z, where

m is an integer.

The mathematical formulation of the sampled grating’s reflection spectrum

is actually concise. Knowing that in the real space the sampled grating is the

product of the dense grating and the coarse bursts, the coupling coefficient

of the sampled grating is approximately given by the convolution of their

separate coupling coefficients κg and κb:

κsg ' κg

(2π

Λ

)⊗ κb

(2πm

Z

)(5.17)

60

and for a digital burst grating, the overall coupling coefficient is given by∫κgδ

(k′z −

2π

Λ

)∑m

re−i(kz−k′z)rZsinc

[1

2(kz − k′z)rZ

]dk′z∣∣kz= 2πm

Z

= rκg∑m

e−i2πmrsinc (πmr) =∑m

Ksg,m (5.18)

Equation (5.18) reveals that the overall reflectivity spectrum is found by

duplicating that of the original dense grating at different spectral locations

where the phase-matching condition is satisfied.

rsg =∑m

iKsg,m sin(βmLsg)

βm cos(βmLsg)− i∆βm sin(βmLsg)(5.19)

where Lsg is the total length of the sampled grating. ∆βm and βm are the

detuning parameter and propagation constant associated with the m-th burst

grating order.

∆βm = β0 −π

Λ− πm

Z(5.20)

β2m = (∆βm)2 − |Ksg,m|2 (5.21)

Different burst grating orders have to be considered as long as the paired

modes satisfy the guidance condition. Also, the reflections from the burst

structure need to be phase-synchronized with the dense grating.

Figure 5.4(a) shows the power reflection spectra of two sampled gratings

with an identical dense digital grating cell (Λ = 285.25 nm) but different

burst periods Z. It is clear that the overall reflection spectra contain a good

number of copies of Fig. 5.4(b), the power reflection spectrum of the dense

digital grating, within a sinc-function-like envelope. While keeping the dense

grating period Λ the same locks the spectral location of the 0-th order Fourier

component (the main channel), changing the burst period changes the FSR

of the reflection comb.

If sampled grating structures are implemented in the two arms of the Y-

branch, as shown in Fig. 1.8, lasing occurs at wavelengths where the two

combs overlap assuming that they are within the bandwidth of the gain

section. For instance, for a Y-branch tunable laser adopting the design de-

scribed in Fig. 5.4(a), its lasing wavelength will be predominantly around

61

(a) (b)

Figure 5.4: (a) Power reflection spectra of two sampled gratings with anidentical dense grating cell but different burst periods. (b) Power reflectionspectrum of a single dense grating cell of length 5 µm.

1310 nm as the two combs align with each other only at that wavelength. It

should be noted that though increasing the coupling coefficient of the dense

grating κ increases the peak reflectivity and lowers the threshold current, it

also broadens the high-reflectivity band and deteriorates SMSR as multiple

longitudinal modes can lase at the same time, as shown in Fig. 5.2(b).

Because the lasing wavelength is defined by the comb overlap, tuning the

wavelength away from the initial spectral position can be achieved by either

moving the two combs together by a same amount such that their main chan-

nels still align with each other at a different spectral location, or moving one

comb at a time such that the two side channels coincide at another spectral

region. The former one results in a fine-tuning of the lasing wavelength, as

both main channels only move by

∆λ = 2Λ∆neff (5.22)

and the latter one, the coarse-tuning process, gives a much larger shift in the

lasing wavelength via the Vernier effect:

∆λ =FSR1

|FSR1 − FSR2|δλ (5.23)

where δλ is the shift of Comb 2 relative to Comb 1, and ∆λ is the shift of

the wavelength where Comb 1 and 2 overlap.

62

(a) (b)

(c) (d)

Figure 5.5: Power reflection spectra of (a) the two arms, and (b) the overallstructure of a Y-branch laser described in Fig. 5.4 using fine-tuning. Powerreflection spectra of (c) the two arms and (d) the overall structure of aY-branch laser described in Fig. 5.4 using coarse-tuning.

Figure 5.5(a) illustrates the fine-tuning by introducing an effective index

shift of −0.2% in both arms. As shown in Fig. 5.5(b), the sum of the two

power reflection spectra peaks at 1308.8 nm, 1.2 nm away from 1310 nm,

where the two main channels aligned initially. On the other hand, Fig. 5.5(c)

pictures the comb shifting of a single arm whose effective index is changed

by only −0.1%. Since the dense grating period is fixed at 285.25 nm, the

main channel of the shifted arm now locates at Λ/(2neff) = 1309.4 nm, 0.6

nm away from 1310 nm. Following Eq. (5.23), one obtains a new reflectivity

peak at around 1303 nm, as shown in Fig. 5.5(d). Even though subtle, an

index shift of −0.1% in only one arm actually induces a much larger hopping

in the lasing wavelength than applying an index shift of −0.2% on both arms.

63

(a) (b)

Figure 5.6: (a) Top view and (b) power reflection spectra of devicesadopting the additive (top) and the multiplicative (bottom) Vernier effects.The differential phase shifter (PS) φd is to compensate for the phasedifference between the two arms and the common PS φc is to ensure theround-trip phase is an integer multiple of 2π.

It is worth noting that two types of the Vernier effect can be implemented

in tunable laser structures: the additive and multiplicative Vernier effects.

Figure 5.6(a) shows the top view of the device based upon these two effects.

In devices adopting the additive Vernier effect, only the back end of the

cavity is replaced by a Y-branch with two grating arms, whereas in devices

adopting the multiplicative Vernier effect, gratings are set on both ends of the

cavity. According to Fig. 5.6(b), it is clear that the additive Vernier effect

gives a much higher SMSR than the multiplicative one, since the destructive

interference occurring between the side channels can only be captured using

an additive operation. It should be noted that the differential phase shift φd

in the additive case measures the phase difference between the two overlap-

ping reflection peaks and does not significantly change the phase difference

between side channels that were already out-of-phase before phase-shifting.

The common phase shift φc in both cases only ensures the round-trip phase

meets an integer multiple of 2π such that resonance can occur in the cavity,

hence it does not affect the overall power reflectivity. Furthermore, using

the additive Vernier effect allows anti-reflection (AR) coatings on the output

facet of the device so that the output power can be maximized.

64

5.4 Process development and preliminary results of

InP-based gain chips

Despite the capability of being an active material in short-wavelength opto-

electronics and electrically responsive enough for building tuning photonic

elements, GaN has not shown any promise in substituting the InP-based ma-

terial system in the telecommunication window. In fact, in the 1310- and

1550-nm spectral region, the InP system still dominates over any other ma-

terial in terms of epitaxial quality, processing technology and device perfor-

mance. Devices such as 56-GHz directly-modulated DFB lasers, high-power,

high-linearity photodiodes, vertical-cavity surface emitting lasers (VCSELs)

and systems such as electroabsorption-modulator-integrated DFB lasers and

large-scale transmitter PICs have been demonstrated [150–154]. As a result,

lasers based upon InP substrates naturally turn out to be an ideal source for

our proposed GaN-based tunable laser.

The InP-based gain chip is essentially a semiconductor optical amplifier

(SOA). With two as-cleaved facets and a ridge structure, an SOA can be

turned into a Fabry-Perot (FP) ridge waveguide laser. The processing of

the InP-based lasers introduced below was developed in the clean room lab-

oratory of the Micro and Nanotechnology Laboratory at the University of

Illinois at Urbana-Champaign. The L-I-V and spectral measurements were

performed in the laboratory of the Advanced Semiconductor Device and In-

tegration Group at the University of Illinois at Urbana-Champaign. Figure

5.7 shows a complete process flow of the InP-based ridge-waveguide lasers.

In the very first step, silicon dioxide (SiO2) was deposited as the hard mask

for the following ridge waveguide etching. The ridge waveguide patterns were

Figure 5.7: Processing flow of the InP-based ridge-waveguide laser.

65

(a) (b)

Figure 5.8: (a) An SEM image of a SiO2 hard mask defined byfluorine-based RIE. (b) An SEM image of the ridge-waveguide structureafter the ICP-RIE and SiO2 remvoal process.

defined using a dark-field mask with stripe widths varying from 1 to 6 µm.

Extra care has to be taken to align the waveguide direction normal to the

cleaved crystalline planes, as the two cleaved facets have to be parallel to

each other in order to sustain laser oscillation. A nickel (Ni) thin film was

deposited on top of the SiO2 to serve as a hard mask material for transferring

the photo-lithographically defined waveguide patterns down to the SiO2 layer

in a fluorine-based reactive-ion etcher (RIE). A scanning electron microscope

image of the SiO2 hard mask after the RIE process is shown in Fig. 5.8(a).

The rough top surface of the SiO2 hard mask was due to the Ni removal

process in a sulfuric-acid-based solution. The surface of the wafer has to be

clean as any form of redeposition resulted from the RIE process will serve as

a hard mask in the following waveguide etching.

The ridge waveguide structure was created using an inductive-coupled-

plasma RIE (ICP-RIE) with a chlorine/methane/hydrogen chemistry (Cl2

/CH4/H2). As a starting point, the etching process was stopped at a ridge

height of 1.2 µm. Unlike etching through the DBR pairs in VCSEL processing

technologies, there is no abrupt material interface in this specific InP-based

laser structure so monitoring the etching process via laser reflectrometry was

not feasible. Careful characterization of etching rate was performed such

that the ridge height can be well controlled by timing the etching process.

It is worth noting that adding CH4 to the chemistry helps passivate the

sidewalls during the fast Cl2 etching [155], so that scattering loss due to

66

sidewall imperfections can be minimized. The remaining SiO2 mask was

removed using the same RIE process as the first etching process.

Planarizing the top of the device is the most critical step when fabricating

the ridge-waveguide laser. Decent metal contacts to the p-doped top layer can

only be achieved with a relatively flat surface, as any metal not deposited in

the heavily-doped top surface results in a Schottky contact high series resis-

tance. In this work, Honeywell ACCUGLASSr T-12B Spin-on Glass (SOG)

512 B was used since it is thick enough to cover the ridge structure after be-

ing cured. The curing process is to bake out the solvent in the SOG solution

such that the whole material is hardened. The curing process was performed

in a vacuum furnace at 250 oC for more than 15 hours. Figure 5.9(a) shows

optical images of the laser samples planarized with SOG. Fluorine-based RIE

was used to etch the top portion of the SOG layer in order to expose the

p-doped contact layer. Cleanness of the exposed p-type surface is highly re-

quired as any residual (unetched SOG or redeposition) can cause poor metal

adhesion or high series resistance and degrade device performance. Figure

5.9(b) shows the clean top surface and the desired sloped sidewall profile of

the device after the RIE process.

Contact metallizations were done in a CHA SEC-600 e-beam evaporator.

A multi-layer titanium/platinum/gold (Ti/Pt/Au) metal film was deposited

on top of the SOG as the p-contact. The sample was then manually thinned

down to around 5 mil (around 125 µm) using aluminum-oxide (Al2O3) slurry

(a) (b)

Figure 5.9: (a) Optical images of cured and etched (inset) SOG onInP-based laser samples. (b) SEM images of an exposed ridge structure andits cross section (inset) after SOG etch back.

67

and polished to ensure good n-side metal adhesion. The thickness of the

finished sample is critical, as a thinner structure not only facilitates heat

conduction through the substrate but makes the cleaving process much easier.

The n-type contact consisted of germanium-gold alloy/nickel/gold layers and

was annealed at 380 oC in a nitrogen-purged furnace for 3 minutes.

The L-I-V and spectral measurements of the finished device were taken us-

ing an HP 4144A semiconductor parameter analyzer (SPA) and an HP/Agilent

70951B optical spectrum analyzer (OSA). Single-facet optical output power

was collected using a Newport 818-IR Ge detector and butt-coupled to a

multi-mode fiber in these two scenarios respectively. Figure 5.10(a) shows

the L-I-V curves of a 6 µm × 440 µm FP ridge waveguide laser operating

under CW bias at room temperature. No additional heat sink was used to

enhance heat dissipation in this set of measurement. The threshold current

(Ith) of this specific device, around 68 mA as revealed by Fig. 5.10(b), was

fairly high and the I-V curve revealed a series resistance of roughly 16 Ω.

Since devices narrower than 6 µm did not show any lasing characteristic be-

low a bias current of 100 mA, it is speculated that the ridge etch (around 1.2

µm in this set of devices) is not deep enough for lateral mode confinement.

Higher ridge structure may be required to further lower the threshold cur-

rent, as the optical confinement factor increases with ridge depth. However,

with a derived differential efficiency of 0.047 W/A, it is projected that the

output power will be able to reach 3 mW at 100 mA provided that Ith could

be made as low as 30 mA. The output power can be further improved by

AR-coating the output facet of the laser. Figure 5.10(c) shows the emission

spectra of the aforementioned device at room temperature under different

CW bias currents. Lasing occurred at around 1270 nm, which agreed well

with the fact that the MQW structure was grown to have a PL wavelength

of 1280 nm. The lasing peak started to red shift at higher bias currents

due to self-heating. An ILX Lightwave LDP-3811 precision pulsed current

source was used to deliver bias currents higher than 100 mA and the rate

of red shift was deduced to be around 20 nm over a course of the 130 mA

change in the bias current. Though the peak wavelength did not reach the

1310-nm window even at a bias current of 200 mA, at an ambient tempera-

ture of 85 oC the peak may further move by almost 20 nm using a red shift

rate of 0.3 nm/oC. More detailed temperature-dependent measurement can

be conducted with a thermoelectric cooler (TEC) stage.

68

(a) (b)

(c) (d)

Figure 5.10: (a) Single-facet L-I-V curves of a 6 µm-wide ridge-waveguideFP laser. (b) Emission spectra of the device under CW operation at roomtemperature around the threshold. (c) Emission spectra of the device underCW operation at room temperature at different bias currents. (d) The redshift of lasing wavelength as a function of the bias current.

The effective index of the lasing mode and Hakki-Paoli gain spectra can

be deduced from the ASE spectra of the laser, shown in Fig. 5.11(a). The

effective index of the lasing mode can be found simply by identifying the

FSR, namely the spectral separation between two adjacent peaks of the ASE

spectrum

∆λ =λ2

2neffL(5.24)

Knowing that the device length is 440 µm, an effective index of around 3.44

is shown in the left panel of Fig. 5.11(b). While FSR gives the effective

index of the lasing mode, the intensity contrast between adjacent peaks and

69

(a) (b)

Figure 5.11: (a) ASE spectra of the laser under different bias currents. (b)(left) Derived effective index of the lasing mode and (right) Hakki-Paoligain spectra of the laser.

valleys is related to the round-trip net modal gain of the laser through

√R1R2e

(Γg−αi)L =

√Imax −

√Imin√

Imax +√Imin

(5.25)

Since the two facets were not coated and remained as-cleaved, the reflectiv-

ities R1 and R2 are estimated based upon the derived effective index. The

right panel of Fig. 5.11(b) shows the net modal gain spectra under different

bias currents. The threshold modal gain was estimated to be 23 cm−1.

5.5 Discussions and possible improvements

As the threshold current Ith was high for this specific device and the over-

all yield was low, it is clear that the ridge height had not been optimized.

Simulation on how ridge height affects the optical confinement factor was

performed, as shown in Fig. 5.12, based upon the refractive index data

of the InP/InGaAsP material system available when this thesis was writ-

ten. Though the optical confinement factor is not a strong function of ridge

height for wider waveguides, it almost doubles for a 2-µm-wide device as-

suming the ridge height could be increased from 1.2 to 1.5 µm. Halving

the threshold material gain greatly helps in reducing the threshold current

density, as the peak gain of a MQW structure can be estimated using an

70

(a) (b)

Figure 5.12: (a) The change of fundamental mode profile with differentridge heights. (b) Calculated optical confinement factor as a function ofridge height, for different waveguide widths.

empirical formula [85]

gmax(λ) = g0(λ) ln

(J + JsJtr + Js

)(5.26)

where Jtr is the current density when the material reaches transparency, and

Js is an additional fitting parameter ensuring Eq. (5.26) does not fail with

no current injection.

As one of the ultimate goals of GaN photonics is to integrated the opti-

cal components with the high-speed GaN-based transistors, harvesting the

strong built-in field in the AlGaN/GaN interface of the HEMT structures

could be a reasonable solution. The built-in field, typically around a few

MV/cm, will be able to create a refractive index shift at least on the order of

10−4 and such change could be modulated by controlling the two-dimensional

electron gas (2DEG) density using the gate voltage as well. Moreover, the

depletion and accumulation of the 2DEG below the AlGaN/GaN interface

may also introduce another index manipulation mechanism as the imaginary

part of the structure’s permittivity is related to the density of 2DEG. Fur-

ther investigation on the change of refractive index due to the change of

2DEG density and on the design of incorporating AlGaN/GaN interfaces in

the waveguide arms with minimal scattering loss and free-carrier absorption

are absolutely required; however, these processes are expected to ultimately

71

nourish the development of GaN photonics.

72

CHAPTER 6

SUMMARY AND FUTURE WORK

In the first part of this thesis, a comprehensive model for InGaN/GaN SAQD

and SAQD-based ridge-waveguide lasers was developed. Linear elastic the-

ory and a single-band Hamiltonian approximation were used to calculate the

strain distribution and electronic states in the truncated hexagonal pyra-

midal QD structure. Temperature effects were considered in the laser rate

equation model in order to describe the self-heating process of the device.

The calculated result showed a fairly good agreement with the measurement,

and several important material- and device-level parameters were extracted.

Though the work on modeling the SAQD active material was successful,

device-level simulation could only be completed on macroscopic properties

of the laser. Current spreading, carrier transport, spatial hole burning, gain

compression and other higher-order effects need to be taken into account

when developing devices with high efficiencies, good beam qualities and long

lifetimes. Models for MQW-based lasers have been developed and optimized;

however, how novel quantum structures change the electrical and optical

properties of the device remains a nontrivial issue and needs further study.

The research interest in III-N materials and related devices has shifted

in recent years from blue-green wavelengths to the extrema of the optical

spectrum, namely telecommunication and deep-UV regions. A tremendous

amount of effort has been put into synthesizing high-quality indium- and

aluminum-rich quantum structures and utilizing them as active materials in

optoelectronic devices. A significant amount of indium not only increases

the difficulty in the growth process but introduces droplets and clusters into

the resultant material that complicates the study of the underlying physics.

High aluminum content, in contrast, triggers complexity in the physics of the

structure, as the crystal-field splitting ∆cr is negative for AlN. As a result,

challenges in developing surface-emitting or normal-incident devices come

about because of the change of the polarization state of the optical process.

73

It is worthwhile to investigate how strain or structural properties of the nano-

structure can delay or minimize the valence band mixing effect in designing

surface-emitting or normal-incident deep-UV devices.

A theoretical study on the working principle of GaN-based photonic devices

was presented in the second half of this thesis. Due to the electro-optic (EO),

piezoelectric (PZ) and photo-elastic (PE) properties of GaN, changes in the

refractive index can be achieved by introducing external bias fields across the

material. GaN-based voltage-controlled polarization rotators (VCPRs) and

tunable gratings are proposed in the realm of this topic. A voltage-controlled

PR, designed as a rib-waveguide structure with a set of segmented electrodes

on its top and sides, relies on local changes in the distribution of the ordinary

and extraordinary refractive indices to switch the state of polarization of its

dominant waveguide mode. Device simulations were carried out based upon

the best available EO, PZ and PE coefficients of GaN by the time this thesis

was written, and it was shown that the waveguide can be programmed to

support TE modes at the input end, funnel the incoming optical power to

the hybrid modes as the wave travels to the middle section of the device

and transfer the power to the fundamental TM mode at a TM-supportive

output end. While a flip-chip bonding process was envisioned in this thesis,

it is also feasible to fabricate the device directly using GaN-on-Si wafers.

More precise measurements on the ordinary and extraordinary indices and

the electro-optic response of the GaN sample will help in the design and

optimization of the VCPR.

Extracting the optical constants of GaN also benefits the design of the

proposed Y-branch tunable laser. In this case more pronounced and uniform

change of the ordinary index is required when building the grating arms.

Introducing AlGaN/GaN heterojunctions to provide strong electric fields is

a preferable approach to enhance the index change, which allows the two

reflection combs to move farther away from the main channel and gives rise

to a wider tuning range of lasing wavelength with a lower power consumption.

At the end of this thesis, it is speculated that upon the availability of GaN’s

optical constants there will definitely come an opportunity for realizing the

proposed “GaN integrated photonics.” It is expected that the numerical

simulation, device analysis and process development described in this thesis

can serve as keys to construct novel photonic components and provide a good

reference for blazing the trail in this exciting area.

74

APPENDIX A

FITTING PARAMETERS

Table A.1: Material parameters for the In0.4Ga0.6N/GaN SAQD model.

In0.4Ga0.6N GaN

Lattice constants [11]

a(T = 298 K) (A) 3.331 3.189

c(T = 298 K) (A) 5.392 5.185

CTEs [69–71]

αa (10−6/K) 3.90 3.95

αc (10−6/K) 2.78 2.80

Band parameters [11]

Eg(T = 0 K) (eV) 1.91∗ 3.51

α (meV/K) 0.634 0.909

β (K) 748 830

∆cr (meV) 22 10

∆so (meV) 12.2 17.0

Stiffness constants [11]

C11 (GPa) 323.2 390

C12 (GPa) 133.0 145

C13 (GPa) 100.4 106

C33 (GPa) 328.4 398

C44 (GPa) 48 105

Dielectric constants [77]

εz (ε0) 12 10.4

εt (ε0) 10.94 9.5

75

Table A.1 (cont).

Deformation potentials [11]

az (eV) −4.34 −4.9

at (eV) −8.18 −11.3

D1 (eV) −3.7 −3.7

D2 (eV) 4.5 4.5

D3 (eV) 8.2 8.2

D4 (eV) −4.1 −4.1

Polarization [11,67,68]

Psp (C/m2) −0.029∗ −0.034

e15 (C/m2) −0.301 −0.326

e31 (C/m2) −0.522 −0.490

e33 (C/m2) 0.826 0.730

Mass parameters [11]

m‖e (m0) 0.148 0.2

m⊥e (m0) 0.148 0.2

A1 −7.61 −7.21

A2 −0.54 −0.44

A3 7.04 6.68

A4 −4.17 −3.46

A5 −4.08 −3.40

A6 −5.32 −4.90∗ fitting parameters used in this model.

76

Table A.2: Fitting parameters used in L-I-V modeling.

Symbol Value

Reverse saturation current I0 42 µAIdeality factor nd 38.5Turn-on voltage V0 4.3 VShunt resistor (leakage path) RP 6 kΩSeries resistor (external resistance) RS 2 Ω

Injection efficiency ηi 0.3Non-radiative recombination lifetime parameter τ0 1.2 nsNon-radiative recombination activation energy ∆Ea 30 meVGroup index ng 3.08Spontaneous emission coupling factor βsp 0.00404Spontaneous emission collection efficiency β 0.016

77

APPENDIX B

EIGHT-BAND HAMILTONIAN FOR BULKWURTZITE CRYSTALS

The eight-band Hamiltonian can be formulated using Lowdin’s perturbation

theory [156], where the outermost s- and p-like orbitals are chosen as “class

A” and the rest of the inner orbitals “class B.”

H8×8 = H0 + H(k) + Hso (B.1)

The component H0 in Eq. (B.1) describes the orbital’s eigen-energies, which

are the solutions to the crystal’s periodic Hamiltonian, H(k) the coupling

between the orbitals due to the dispersion in k-space, and Hso the interaction

between the orbitals carrying different spins.

The presence of the spontaneous polarization field along the c-axis breaks

the reflection symmetry about the x-y plane of the wurtzite crystal. As a

result, the eigen-energy of pz orbital is no longer the same as those of px

and py. This broken degeneracy results in the crystal-field split-off hole band

(CH), which is a unique quantity in wurtzite crystals and quite different from

the spin-split-off hole band (SO) in traditional III-V materials. Since the spin

state does not affect the eigen-energies, the full eight-band Hamiltonian can

be diagonalized into two identical four-band ones whose elements, using the

orbital functions |S〉, |X〉, |Y 〉, |Z〉, read

〈Z|H0|Z〉 = 0 (B.2a)

〈X|H0|X〉 = 〈Y |H0|Y 〉 = ∆1 = ∆cr (B.2b)

and depending on the sign of the crystal-field splitting ∆cr, the eigen-energy

of the s-like orbital is

〈S|H0|S〉 = max 0,∆cr+ Eg (B.3)

where Eg is the energy band gap.

78

The band-coupling Hamiltonian H(k) can also be obtained by recognizing

the asymmetry of the pz orbital. Using the orbital functions as the basis and

following Kane’s formulation for zinc-blende crystals [157], one can arrive at

the wurtzite equivalence

H(k) =

[H4×4(k) O

O H∗4×4(k)

](B.4)

where the diagonal element H4×4(k) is given byA′

2(k2x + k2y) +A′1k

2z iP2kx iP2ky iP1kz

−iP2kx L′1k

2x +M1k

2y +M2k

2z N ′

1kxky N ′2kxkz −N ′

3kx

−iP2ky N ′1kykx M1k

2x + L′

1k2y +M2k

2z N ′

2kykz −N ′3ky

−iP1kz N ′2kzkx +N ′

3kx N ′2kzky +N ′

3ky M3(k2x + k2y) + L′2k

2z

(B.5)

and the operation A∗ denotes the complex conjugate of matrix A.

In Eq. (B.5), the parameters P1 and P2 refer to the optical momentum

matrix element associated with the optical transition between the s-like to

pz-like and s-like to px- (and py-) like orbitals respectively. The coefficients

A′i, L′i,Mi, N′i carry the information of the effective masses (the Luttinger-

like coefficients Ai’s) and coupling between bands. They are defined as

P1 =~m0〈iS|pz|Z〉 =

√~2

2m0Ep1 (B.6a)

P2 =~m0〈iS|px|X〉 =

√~2

2m0Ep2 (B.6b)

A′1 =

~2

2m‖0

− P 21

Eg(B.6c)

A′2 =

~2

2m⊥0

− P 22

Eg(B.6d)

L1 =~2

2m0(A2 +A4 +A5) +

P 21

Eg(B.6e)

L2 =~2

2m0A1 +

P 22

Eg(B.6f)

M1 =~2

2m0(A2 +A4 −A5) (B.6g)

M2 =~2

2m0(A1 +A3) (B.6h)

79

M3 =~2

2m0A2 (B.6i)

N ′1 =

~2

2m02A5 +

P 21

Eg(B.6j)

N ′2 =

~2

2m0

√2A6 +

P1P2

Eg(B.6k)

N ′3 = i

√2A7 (B.6l)

The asymmetry of the pz orbital continues to play a role in spin-orbital

interaction. Although the spin-orbital Hamiltonian gives different effects on

the px-py and px-pz orbital pairs, different values associated with those two

pairs have not been reported experimentally. Thus far it is generally assumed

that the two spin-orbital couplings are the same [74].

~4m2

0c20

〈X|∇V × p · σσσ|Y 〉 = −i∆2 (B.7a)

~4m2

0c20

〈Y |∇V × p · σσσ|Z〉 = −i∆3 (B.7b)

~4m2

0c20

〈Z|∇V × p · σσσ|X〉 = −i∆3 (B.7c)

∆2 = ∆3 =∆so

3(B.7d)

where V is the potential of the wurtzite lattice and σσσ is the Pauli matrix.

The eight-by-eight spin-orbital coupling Hamiltonian Hso under the orbital

function basis hence reads

Hso =

[Gso Γ

−Γ∗ G∗so

](B.8)

where

Gso =1

3∆so

0 0 0 0

0 0 −i 0

0 i 0 0

0 0 0 0

(B.9)

Γ =1

3∆so

0 0 0 0

0 0 0 1

0 0 0 −i0 −1 i 0

(B.10)

80

APPENDIX C

STRAIN AND HETEROSTRUCTUREEFFECTS

The incorporation of heterostructures, the core of optoelectronic devices, gen-

erally introduces lattice-mismatch-induced deformation potential that alters

the electronic band structure of the material. Such a strain-related effect can

be considered in the eight-band Hamiltonian following Bir and Pikus as a

correction term [158], which takes a form very similar to the band-coupling

Hamiltonian H(k). Under the orbital functions basis, the full strain-related

Hamiltonian Hstrain can be diagonalized in the same way as Eq. (B.4), and

the element Hstrain,4×4 is given bya2(εxx + εyy) + a1εzz 0 0 0

0 l1εxx +m1εyy +m2εzz n1εxy n1εxz

0 n1εyx m1εxx + l1εyy +m2εzz n1εyz

0 n1εzx n1εzy m3 (εxx + εyy) + l2εzz

(C.1)

Similar to the Luttinger-like coefficients in Eq. (B.5), the coefficients li,mi, ni’s

carry the information of the deformation potential constants

l1 = (D2 +D4 +D5) (C.2a)

l2 = D1 (C.2b)

m1 = (D2 +D4 −D5) (C.2c)

m2 = (D1 +D3) (C.2d)

m3 = D2 (C.2e)

n1 = 2D5 (C.2f)

n2 =√

2D6. (C.2g)

Recognizing that the k vector in Lowdin’s perturbation theory comes from

the spatial derivative of the Bloch wave function, uk(r)eik·r, one should re-

81

place the ki components in Eq. (B.5) by the differential operator −i∂/∂xiwhen solving the effective-mass Hamiltonian of the quantum structure. The

finite difference method (FDM) is usually adopted for simulation, as the im-

plementation of the differential operators ∂/∂xi is difficult on finite element

method (FEM) grids. Nevertheless, it is very critical to preserve the physics,

namely, the mathematical properties of the operators, under the realm of

numerical methods.

The most basic requirement for any numerical method is to guarantee

the Hermitian property of the physical observable operators. As a result,

under the scheme of FDM, any differential operator ∂/∂xi associated with

the spatial-dependent quantities of the heterostructure A(xi) must be written

as [159]

A(xi)∂2ψ

∂x2i

→ ∂

∂xiA(xi)

∂

∂xiψ (C.3a)

A(xi)∂ψ

∂xi→ 1

2

[A(xi)

∂

∂xi+

∂

∂xiA(xi)

]ψ (C.3b)

and under the FDM scheme the second- and first-order derivatives can be

approximated as

(A[n− 1] +A[n])ψ[n− 1]− (A[n− 1] + 2A[n] +A[n+ 1])ψ[n] + (A[n] +A[n+ 1])ψ[n+ 1]

2∆xi2

(C.4)

and(A[n] +A[n+ 1])ψ[n+ 1]− (A[n− 1] +A[n])ψ[n− 1]

4∆xi(C.5)

82

APPENDIX D

EIGHT-BAND HAMILTONIAN UNDERTHE CHANGE OF BASIS

Although the eight-band Hamiltonian, Eq. (B.1), is directly solvable, the

solutions do not necessarily give enough physical information about the bands

of interest. In particular, while the eigen-energies remain the same under the

change of basis, the true probably density functions can only be found by

summing all the envelopes of each basis function, which makes the analysis

not as straightforward. Selecting a proper basis not only extracts physical

information directly but also benefits the numerical process as the resultant

matrix, made close to diagonal, makes operations more efficient.

If the relation between the new and old bases is described by a unitary

transform U such that

[|φ〉] = Ut[|ψ〉] (D.1)

then the matrix form of an observable operator A under the new basis is

given by

A′ = U†AU (D.2)

The most effective basis is the one that best diagonalizes the zone-center

(Γ-point) Hamiltonian, Eq. (B.8), as heavy hole (HH), light hole (LH) and

crystal-field split (CH) hole bands are named based upon the effective mass

at the zone center when the material is not strained. A reasonable choice of

the new basis is [74]

|u1〉 = |iS ↑〉 (D.3a)

|u2〉 = − 1√2|X + iY ↑〉 (D.3b)

|u3〉 =1√2|X − iY ↑〉 (D.3c)

|u4〉 = |Z ↑〉 (D.3d)

|u5〉 = |iS ↓〉 (D.3e)

83

|u6〉 =1√2|X − iY ↓〉 (D.3f)

|u7〉 = − 1√2|X + iY ↓〉 (D.3g)

|u8〉 = |Z ↓〉 (D.3h)

thus using the unitary transformation matrix

U =

i 0 0 0 0 0 0 0

0−1√

2

1√2

0 0 0 0 0

0−i√

2

−i√2

0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 i 0 0 0

0 0 0 0 01√2

−1√2

0

0 0 0 0 0−i√

2

−i√2

0

0 0 0 0 0 0 0 1

(D.4)

the new zone-center Hamiltonian becomes

H′so =

[G′so ΓΓΓ′

ΓΓΓ′ G′so

](D.5)

where

G′so =

Ec 0 0 0

0∆so

3+ ∆cr 0 0

0 0 −∆so

3+ ∆cr 0

0 0 0 0

(D.6a)

ΓΓΓ′ =

0 0 0 0

0 0 0 0

0 0 0

√2∆so

3

0 0

√2∆so

30

(D.6b)

The new coupling Hamiltonian H′4×4(k), as defined in Eq. (B.4), is now

84

A′

2

(k2x + k2

y

)+A′

1k2z −

P2√2

(kx + iky)P2√

2(kx − iky) P1kz

−P2√

2(kx − iky)

L′1 +M1

2(k2x + k2

y) +M2k2z

M1 − L′1

2(k2x − k2

y) + iN ′1kxky

−N ′2kz +N ′

3√2

(kx − iky)

P2√2

(kx + iky)M1 − L′

1

2(k2x − k2

y)− iN ′1kykx

L′1 +M1

2(k2x + k2

y) +M2k2z

N ′2kz −N ′

3√2

(kx + iky)

P1kz−N ′

2kz −N ′3√

2(kx + iky)

N ′2kz +N ′

3√2

(kx − iky) M3

(k2x + k2

y

)+ L′

2k2z

(D.7)

and similarly the strain-related Hamiltonian under the new basis H′strain,4×4 reads

a2 (εxx + εyy) + a1εzz 0 0 0

0l1 +m1

2(εxx + εyy) +m2εzz

l1 −m1

2(εxx − εyy) + in1εxy −

n2√2

(εxz − iεyz)

0l1 −m1

2(εxx − εyy)− in1εxy

l1 +m1

2(εxx + εyy) +m2εzz

n2√2

(εxz + iεyz)

0 −n2√

2(εxz + iεyz)

n2√2

(εxz − iεyz) m3 (εxx + εyy) + l2εzz

(D.8)

85

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