Oki_Thesis_Princeton2007

Ballistic Transport of AlAs
Two-Dimensional Electrons
Oki Gunawan

Ballistic Transport of AlAs
Two-Dimensional Electrons
Oki Gunawan
A dissertation
presented to the faculty of Princeton University
in candidacy for the degree
of Doctor of Philosophy
Recommended for acceptance
by the Department of Electrical Engineering
September 2007

iii
ABSTRACT
The success of our modern electronics age stems from our advanced technology to
process information by manipulating electrons in solid-state devices. Among the few
fundamental ways to manipulate the electrons, such as using their charge and spin, the
control and manipulation of the electron’s valley degree of freedom in semiconductors
remains practically unexplored. In this thesis we focus on some basic aspects of ballistic
transport in a two-valley two-dimensional electron system (2DES), realized in high
quality AlAs quantum wells.
We start by demonstrating valley-resolved ballistic transport in an experiment using a
Hall bar device with a surface grating. From the analysis of the frequencies of the
commensurability oscillations in the magnetoresistance at various densities we deduce
the mass anisotropy factor, namely the ratio of the longitudinal and the transverse
effective masses, / 5.2 0.5l tm m = ± , a fundamental parameter for the anisotropic
conduction bands in AlAs. We then present results from similar experiments in devices
with antidot lattices that reveal peaks in magnetoresistance. Through an analysis of the
positions of the peaks associated with the smallest commensurate orbit, we obtain a value
for the mass anisotropy factor, / 5.2 0.4l tm m = ± , consistent with the value deduced from
the surface-grating samples.
The anisotropy of the effective mass can be exploited to realize a simple "valley filter"
device using a quantum point contact (QPC), a one-dimensional quantum ballistic
channel. This device may play an important role in "valleytronics" or valley-based
electronic applications. Our experiments on the QPC in the AlAs 2DES reveal that the
conductance of this system is nearly quantized at multiples of 2
2 /e h , instead of 2
4 /e h
as expected from a valley and spin degenerate system. This observation indicates a
broken valley degeneracy due to the mass anisotropy as well as residual strain in the
QPC.
Finally, we demonstrate a novel giant piezoresistance effect in an AlAs 2DES with an
antidot lattice. Such a device may have potential applications as an ultra-sensitive strain
sensor. It exemplifies one of the many uses of manipulating the electron valley degree of
freedom in a solid-state device.

iv
ACKNOWLEDGEMENTS
The graduate school is such an arduous journey that has become possible because of the
following people. I am grateful to have Prof. Mansour Shayegan as my advisor, his
remarkable guidance and mentorship during my tenure in Princeton have made my
graduate school years an extraordinary life experience. It is a rare and true privilege to be
his student. I am thankful for Prof. Claire Gmachl and Prof. Stephen Lyon for their time
and attention in reading my thesis. I would like to acknowledge Army Research Office
and National Science Foundation for their generous support to our research.
Shayegan’s group (mashgroup) comes with an interesting mix of people that surely have
made my journey more enjoyable. The seniors: Tony Yau, Etienne, and Emanuel with
whom I learnt the ropes and many valuable skills and styles to survive in the group. My
contemporaries: Yakov, my lifestyle guru, a multi-talented person with a knack to code
various killer applications in Matlab, notably the MASHMEASURE. It is a real privilege to
have spent countless quality times with him discussing everything under the sun and to
absorb some of his Matlab expertise. Kamran, my spiritual guru, with whom I learnt to
develop critical eyes into experimental problems and data at hand and whose publication
record subliminally provides spiritual guidance to the rest of us. Babur, I am thankful for
his friendship and various help and for setting a new standard of grad student dress-code
thus entitling him my fashion guru; Nathan, for his contagious enthusiasm and his expert
guidance on American idioms, late night rides, warm family dinner and Grand Theft Auto
sessions at his place; Shashank for his insightful advice on various matters, from circuit
design to job hunting; and the younger members: Tayfun, Medini and Javad, I am
thankful for their friendship and assistance to me on various occasions. I should also
mention Eric Shaner of Lyon’s group who had given tremendous help in my early years.
I thank the Graduate College, and for the people who run it, for many memorable
moments that I spent in my first two years at Princeton: the awesome Gothic architecture,
the Sunday brunch, the mesmerizing winter, the Friday social, the spellbinding Procter
Hall’s organ tune on Sunday noon, and the wait for Vina. It was like living in a dream.
I should mention the following people, Maw Lin Foo, (a.k.a. hpy / “the good friend of
mine”), with whom I had shared many quality times in Graduate College talking about all
sort of things and who had provided a steady dosage of Singaporean culture at Princeton.

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His friendship has made my Princeton experience more enjoyable. It is a pleasure to have
known these people: Wang Chih Chun, Yuan Yu, Jian Zhang, I-Chun, Guillaume
Sabouret, and the Tsui’s group members: Gabor Csathy, Ravi, Amlan, Keji Lai, and
Wanli Li, I am thankful for their friendship and help on various matters, Edith and Brian
for their companionship in badminton courts, also my fellow graduate students in far
away places: Hendra Kwee and Wirawan Purwanto at William and Mary, Wahyu
Setyawan at Duke, Hery Susanto at NTU and then Lehigh, and Rizal Hariadi at Caltech.
For my friends in greater Princeton area whom I and Vina have come to know, we are
grateful for the times we spent together on various occasions and for the help that we
received in many ways: Iksan and Fiona, Lea and Daniel, Melany and Ngiap Kie, Yenty,
Rudy and Cisca, Olivia and Trogan, Esther and Robert, Wesin and Wenny, and not to
forget, Hari Intan of Philadelphia for his enjoyable companionship on many weekend
trips.
For my Mom, thanks for everything and for the great upbringing she gave and my Dad
for his support. My brothers and sister: Toto, Dede, Een, and Marlene, for the care and
wonderful times I had at home. I would like to mention my friends in my teenage years:
Johan Bulet, Melvyn, Ujuan, and Barnald in Jakarta; Way Kong, Tony Kuan and James
Tan in Singapore that surely had made my life very colorful. Special thanks for Bryan
Hoo and Tee Jong of the NTUCF who had made my journey to Princeton possible.
Looking back, I have to mention my high school physics teacher, Drs. Zaharah Ramli,
whose remarkable passion and enthusiasm had ignited a spark, the zeal for physics that
continues to this day. As such is the influence of amazing teacher, a rare jewel and I was
simply a very fortunate person that crossed her path. It is an honor to rest my
acknowledgment to her in a place where Einstein and Feynman once walked and talked. I
was again very fortunate to have met Prof. Yohanes Surya and Dr. Agus Ananda, who
with selfless dedication and tons of energy had devoted much of their time, when they
were graduate students, to teach me advanced level physics, a timeless rock-solid
foundation. They are practically the ones responsible in charting my future trajectory –
where here fourteen years later – I find myself defending my PhD thesis at Princeton. For
Prof. Ooi Boon Siew that had shaped my early interest in semiconductor physics and
provides valuable mentorship from time to time and also for Dr. Jurianto Joe that has
been an exemplar and had helped fuel my determination to go for graduate school in US.

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Mostly I am grateful for Vina, for her enduring companionship and love, her home-
cooked meals, many memorable moments we had during my graduate school years, and
for many trips together: the New Hope, the Six Flags, the Manhattan, and the APS trips.
More importantly for her total dedication in taking the most time in rearing baby Nael in
my final years at Princeton, also for my Mom-in-law for her generous help in Nael’s
early years. Looking back, I am filled with gratitude and finally would like to thank God
for His tremendous blessing and guidance to me in completing this journey.

vii
For Vina, Nael, and my teachers.

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CONTENTS
ABSTRACT iii
ACKNOWLEDGEMENTS iv
CONTENTS viii
LIST OF FIGURES x
1. INTRODUCTION 1
2. BACKGROUND
2.1 Multivalley Semiconductors 5
2.2 AlAs Two-Dimensional Electron System 7
2.3 Ballistic Transport 11
3. EXPERIMENTAL DETAILS
3.1 Sample Fabrication 13
3.2 Device Measurement 19
4. COMMENSURABILITY OSCILLATIONS IN ALAS 2DES
4.1 Introduction 22
4.2 Device Fabrication 25
4.3 Experimental Results 26
4.4 Analyses and Discussions 28
4.5 Summary 33
5. ALAS 2D ELECTRONS IN AN ANTIDOT LATTICE
5.1 Introduction 34
5.3 Experimental Results 37
5.4 Numerical Simulation 39
5.5 Data Analysis 42
5.6 Summary 46
6. QUANTUM POINT CONTACT IN ALAS 2DES
6.1 Quantum Point Contact 47
6.2 Quantized Conductance 49
6.3 Device Structure 51
6.4 Analysis and Discussion 52

ix
6.5 Source Drain Bias Spectroscopy 60
6.6 Summary 63
7. ANOMALOUS GIANT PIEZORESISTANCE IN ALAS 2DES WITH ANTIDOT
LATTICE
7.1 Introduction 64
7.3 Experimental Results and Discussions 68
7.4 Discussions 85
7.5 Device Characterization for Strain Sensor Application 86
7.6 Summary and Conclusion 91
8. SUMMARY AND FUTURE PROJECTS
8.1 Summary of the Results 93
8.2 Possible Future Projects 95
8.3 Conclusion 97
APP. A LIST OF PUBLICATIONS 98
APP. B HALL BAR LITHOGRAPHY MASK FOR ALAS 2DES SAMPLES 100
APP. C SUMMARY OF ALAS EFFECTIVE MASS DETERMINATIONS 104
APP. D PIEZO STRAIN FACTOR CALIBRATION
D.1 Piezo-actuator 108
D.2 Strain Gauge 110
D.3 Experimental Setup 112
D.4 Piezo Strain Factor Characterization 116
APP. E CIRCUIT DIAGRAMS OF CUSTOM INSTRUMENTATIONS
E.1 Bipolar Tunable Voltage Source 122
E.2 Active Voltage Adder 123
E.3 Programmable High Current Source 124
APP. F. A MODEL FOR THE ANOMALOUS GIANT PIEZORESISTANCE
EFFECT IN ANTIDOT LATTICE 126
BIBLIOGRAPHY 132

x
LIST OF FIGURES
1.1 Research perspective on the manipulation of electrons in solid-state
devices based on the electrons’ properties such as charge, spin and valley
degree of freedom. _________________________________________________ 2
2.1 Band structure of AlAs and the constant energy surface of a valley. ___________ 6
2.2 Constant energy surfaces in k-space for the conduction band edges of
AlAs, Si and Ge. ___________________________________________________ 6
2.3 The layer structure of AlAs MODFET wafers used in this thesis: M415
(15 nm wide QW) and M409 (11 nm wide QW). _________________________ 8
2.4 Valley occupation in AlAs quantum wells and the 2D Fermi contours for
narrow and wide wells. ______________________________________________ 8
2.5 Experimental setup for tuning the valley population in AlAs 2DES using
a piezo-actuator. __________________________________________________ 10
2.6 Electron trajectories in diffusive and ballistic transport regimes. _____________ 12
3.1 The main steps of the AlAs device fabrication process. ____________________ 13
3.2 Schematic diagram of the Astex ECR-RIE system and the etching test
pattern.__________________________________________________________ 17
3.3 A sample mounted on a piezo-actuator and the strain gauge to measure
the applied strain (sample M409K8). __________________________________ 19
3.4 A typical experimental setup for device characterization showing the
three main components._____________________________________________ 20
4.1 The original Weiss oscillations data and the conceptual description of the
commensurability oscillations. _______________________________________ 23
4.2 A micrograph of a surface grating device, its device schematic and the
diagrams for the X and Y valleys in k-space with their corresponding first
two resonant orbits in real space. _____________________________________ 24
4.3 Commensurability oscillations and Shubnikov-de Haas data from
M409N3 (a = 400 nm) and their corresponding Fourier spectra. _____________ 26
4.4 Commensurability oscillations at various densities (sample M409N3).________ 27
4.5 Density dependence of the frequencies of the commensurability
oscillations for X and Y valleys (sample M409N3). ______________________ 29

xi
4.6 Commensurability oscillations data (a = 300 nm) and the corresponding
Fourier spectrum (sample M415L3). __________________________________ 30
4.7 Inverse Fourier decomposition of the commensurability oscillations of
Fig. 4.3 for the X and Y valleys. _____________________________________ 32
5.1 The data of the original antidot lattice experiment by D. Weiss. _____________ 35
5.2 The antidot lattice experiment in AlAs 2DES showing a micrograph of
the antidot lattice region, the Fermi contours of the X and Y valleys in k-
space and their first four commensurate orbits. __________________________ 35
5.3 Magnetoresistance data from all four antidot regions (sample M415B1). ______ 37
5.4 Magnetoresistance data from the a = 0.8 µm antidot region at various
densities. ________________________________________________________ 38
5.5 Simulation snapshots showing various types of trajectories: chaotic,
pinned and skipping orbits for both X and Y valley electrons. _______________ 40
5.6 Magnetoresistance obtained from numerical simulations. __________________ 41
5.7 Summary of the density dependence of the main commensurability peaks
for all antidot regions. ______________________________________________ 43
5.8 Fourier analysis of the Shubnikov-de Haas oscillations from a = 0.8 µm
antidot region to deduce the density imbalance in the system. _______________ 44
5.9 Summary of the density dependence of the two sets of commensurability
peaks A and B for a = 0.6 µm and a = 0.8 µm antidot regions. ______________ 45
6.1 Schematic diagram of a split-gate quantum point contact (QPC) device
and the original quantized conductance data in the QPC.___________________ 47
6.2 Conductance vs. gate voltage in an AlAs QPC and its corresponding
transconductance ( / GdG dV ) trace. ___________________________________ 48
6.3 Device schematic of the shallow-etched QPC. ___________________________ 50
6.4 Schematic of the potential landscape surrounding the QPC and the
dependence of the Fermi energy on the gate voltage. ______________________ 53
6.5 Magnetoresistance traces for the AlAs QPC device._______________________ 55
6.6 Fourier spectrum of the longitudinal magnetoresistance and the dif-
ferential of the transverse magnetoresistance ( /xydR dB ) to deduce
various density components in the 2D reservoir and the QPC. ______________ 56
6.7 QPC channel electrical width, deduced from the kink in the magneto-
resistance. _______________________________________________________ 57

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6.8 A revised QPC energy level model with variable channel width, showing
the expected crossings of the Fermi energy and the quantized levels in
the QPC. ________________________________________________________ 59
6.9 Differential conductance / SDG dI dV= map of the QPC. __________________ 61
6.10 Full-width at half-maximum of the zero bias anomaly peak as a function
of gate voltage. ___________________________________________________ 62
7.1 An idealized (conventional) piezoresistance effect due to strain-induced
intervalley electron transfer in AlAs 2DES, a simple two-valley system. ______ 65
7.2 The experimental setup of the giant piezoresistance experiment in the
AlAs 2DES with antidot lattices. _____________________________________ 67
7.3 The giant piezoresistance effect in an AlAs 2DES from both the blank
and the antidot regions (sample M409K8). ______________________________ 69
7.4 The gauge factor vs. strain data calculated from Fig. 7.3.___________________ 70
7.5 Finite element simulation of the strain distribution in a 2D medium
perforated with an antidot lattice. _____________________________________ 71
7.6 Shubnikov-de Haas oscillations of the blank region at various piezo bias
(strain) values and their corresponding Fourier spectra (sample
M409K8). _______________________________________________________ 74
7.7 Shubnikov-de Haas oscillations of the a = 0.6 µm antidot region at
various piezo bias (strain) values and their corresponding Fourier spectra
(sample M409K8)._________________________________________________ 75
7.8 The low-field magnetoresistance data from all antidot regions showing
the commensurability peaks associated with a fundamental peak and their
subharmonics (sample M409K8). ____________________________________ 76
7.9 The strain dependence of the fundamental commensurability peak at
various gate voltages or densities (sample M409K8).______________________ 78
7.10 Magnetoresistance traces obtained from numerical simulations of the
transport through antidot lattice with variable channel width that
demonstrate the emergence of the sub-harmonic peaks. ___________________ 80
7.11 The strain (piezo bias) dependence of the low-field magnetoresistance for
all antidot regions showing variation of the subharmonic peak
amplitudes._______________________________________________________ 82
7.12 The variation of the commensurability peak amplitudes with strain (piezo
bias) for the 1 µm-AD (a = 1 µm) region. ______________________________ 84
7.13 The analogy between the giant magnetoresistance effect in a layered
magnetic metal sandwich structure and the giant piezoresistance effect in
the AlAs 2DES with an antidot lattice. _________________________________ 85

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7.14 Density dependence of the piezoresistance for the 1 µm-AD region.__________ 87
7.15 Temperature dependence of the piezoresistance from the blank and the 1
µm-AD regions and their corresponding gauge factors. ____________________ 89
7.16 Testing the strain detection limit by modulating the piezo bias and
monitoring the modulated resistance. __________________________________ 90
8.1 A more informative giant piezoresistance experiment in an AlAs 2DES
with an antidot lattice in a Hall bar with van der Pauw geometry. ____________ 95
8.2 A schematic diagram for a valley filter device employing QPCs. ____________ 97
B.1 A specially designed Hall bar mask for AlAs 2DES devices with various
new features. ____________________________________________________ 101
D.1 A single layer of piezoelectric element operating in the d33 mode and a
typical “piezo stack” piezoelectric actuator. ____________________________ 108
D.2 A “T-Rosette” strain gauge used in this thesis. _________________________ 110
D.3 Experimental setup for the piezo strain factor (PSF) measurement.__________ 113
D.4 Calibration of the bridge circuit. _____________________________________ 115
D.5 Piezo bias modulation and the corresponding output from the bridge
circuit showing a drift in the original signal.____________________________ 116
D.6 Temperature dependence of the PSF. _________________________________ 117
D.7 Strain gauge excitation current dependence of the PSF. ___________________ 118
D.8 Linearity between the piezo bias modulation amplitude and the bridge
output signal (∆R). _______________________________________________ 119
D.9 Piezo modulation frequency dependence of the PSF. ____________________ 119
D.10 Piezo series resistance dependence of the PSF. _________________________ 120
E.1 A bipolar tunable voltage source circuit._______________________________ 122
E.2 An active voltage adder circuit.______________________________________ 123
E.3 A programmable high current source circuit. ___________________________ 124
E.4 A closed-loop temperature control of the Oxford 3
He cryostat using the
programmable current source and a software proportional-differential
controller._______________________________________________________ 125
F.1 The stress situation in an antidot region showing the residual stress and
the applied tunable stress components. ________________________________ 127
F.2 The Fermi seas and the bottom of the conduction bands of the non-
uniformly strained antidot regions for X and Y valley electrons. ____________ 129

xiv
F.3 The channel-pinching effect seen in the X-valley Fermi sea with
increasing applied stress. __________________________________________ 130
F.4 Numerical simulation of the channel-pinching effect. ____________________ 131

1
1 Introduction
INTRODUCTION
Our modern time has witnessed the birth and explosive growth of the electronics
technology. It began with the invention of transistor that led to the development of
microcomputers, ushering in the era of information technology. These technologies have
tremendous impact to the world’s economic growth [1]. With pervasive influence and
applications, electronic devices and instruments are indispensable, often critical in our
everyday life. Needless to say, electronics technology has been the cornerstone of modern
civilization [2].
At the heart of the immensely successful electronics technology is the ability to control
and manipulate electron charge to process information. Starting from the bipolar junction
transistor [3], a host of other electronic devices were developed such as the diode, field
effect transistor (FET), thyristor, charge-coupled device, etc. In these devices one
basically uses the electric field to control the electron charge. Following suit, many
researchers recently started to look into another degree of freedom of electron, namely
spin, marking the birth of a new field called spintronics, or spin-based electronics. In
spintronics, the primary control over electron spin is achieved using magnetic field.
There is yet another electronic degree of freedom in certain types of semiconductors that
is relatively unexplored and may hold promise for future technology: valley occupation.
Valley is a local minimum point in the conduction band structure of a semiconductor
material where the electrons reside in k-space. Since the valley, or the band structure in
general, originates from of the crystal structure of the material, it is sensitive to the
deformation of the crystal. Therefore a means to control the valley occupation is by
manipulating the strain field in the crystal. Following similar nomenclature, the research
of valley-based electronics could be referred as “valleytronics”.
This research perspective in terms of which electron property one manipulates is
summarized in Fig. 1.1 below.
CHAPTER 1

2
Notes: CCD = charge coupled device
MRAM = magnetic random access memory
GMR = giant magnetoresistance
GPR = giant piezoresistance
Figure 1.1 Research perspective on the manipulation of electrons in solid-state devices based
on the electrons’ properties such as charge, spin and valley degree of freedom (in a
multivalley semiconductor). Ee, EZ and EV represent various energy scales namely, the
electron’s (kinetic) energy, Zeeman splitting and the valley splitting respectively.
The study of the electronic valley degree of freedom is of great importance and relevance
since after all Si, the most technologically important semiconductor, is a multi-valley
system (for n-type). Furthermore, the conduction-band valley is essentially a quantum
mechanical property of electrons in a solid, implying possible applications in the area of
quantum computation where the valley index might be utilized as a qubit [4]. As a
comparison, there have been intense efforts to utilize the spin degree of freedom to
realize a solid-state quantum computer [5,6].
To certain extent, the valley degree of freedom has been exploited by CMOS
manufacturers via a strain engineering technique to enhance the electrons’ mobility
[7,8].1
However such a technique does not allow an in-situ control of the valley
1
Another example of device exploiting the valley degree of freedom is the Gunn diode [9]. However in
this device, the carriers’ valley occupation is controlled by electric field through hot electron effect
instead of strain field.

3
occupation thus hindering further exploration of the valley physics. This fact is further
complicated by the large (sixfold) valley degeneracy present in n-type Si.
This thesis presents a study of ballistic transport in a two-dimensional electron system
(2DES) with a two-fold valley degeneracy, the simplest multi-valley system. The system
is realized in a high-quality, wide AlAs quantum well (QW) [10,11]. The high mobility
attained allows experimentation in the ballistic transport regime where the dimensions of
the conducting channel are smaller than the electron mean-free-path so that electronic
transport (at low temperatures) is dominated by boundary scattering rather than by
scattering from impurities or phonons. We studied some basic properties of the ballistic
transport unique to this two-valley system in several mesoscopic devices such as a
surface-grating device, an antidot lattice device, and a quantum point contact.
In Chapter 2 we review some general information on multi-valley systems and the basic
properties of the 2DES confined to AlAs QWs. We discuss the valley degeneracies in
bulk AlAs and in AlAs QWs and briefly describe the technique we use to tune the in-
plane valley populations in AlAs wide QWs. We also present a general description of
ballistic transport. Chapter 3 deals with the experimental details such as the device
fabrication and measurement techniques. We highlight some original ideas and
developments which are particularly important for the AlAs 2DES device research.
In Chapter 4 we report a ballistic transport experiment in a surface-grating device that
exhibits commensurability oscillations in the magnetoresistance traces. We induce a one-
dimensional modulation potential in the 2DES with a surface-grating that leads to a
geometric resonance effect in the presence of a perpendicular magnetic field. Here we
demonstrate the first observation of a valley-resolved ballistic transport and, furthermore,
deduce the effective mass anisotropy factor /l tm m where lm and tm are the longitudinal
and the transverse effective masses respectively. Using inverse Fourier analysis we
disentangle the two transport components arising from the two valleys.
In Chapter 5 we present another type of ballistic transport experiment, i.e. transport in an
antidot lattice. These experiments reveal another remarkable ballistic transport
phenomenon associated with magnetoresistance peaks arising from the electron orbits
becoming commensurate with the antidot lattice. From the analysis of the peaks

4
associated with the shortest commensurate orbits we deduce the mass anisotropy factor
l tm m .
Chapter 6 covers our study of an AlAs 2DES quantum point contact (QPC) device. We
present a successful demonstration of quantized conductance in an AlAs QPC. Thanks to
the large value of the effective mass, the subband levels in the QPC constriction are very
closely spaced, making the observation of quantized conductance difficult. From our
analysis we deduce that it is the valley with larger mass (i.e. the longitudinal mass, ml)
along the QPC confinement potential that dominates the low-lying subband energies in
the QPC. This suggests the potential use of the QPC as a natural “valley-filter”. As
another interesting finding, we also observe a “0.7 structure” i.e., conductance quantized
at ≈ 2
0.7(2 / )e h , which is stronger than the other quantized plateaus, suggesting its
different origin.
In Chapter 7 we present a surprising finding, namely an anomalous giant piezoresistance
effect in AlAs 2DESs with antidot lattices. We demonstrate that it is possible to engineer
such devices to achieve very high and thermally stable piezoresistivity. Such a device
could be utilized as an ultra sensitive strain sensor. In this chapter we also present
extensive magnetoresistance data and a model that accounts for many of the features
observed in the experiment.
In Chapter 8 we summarize the work presented in this thesis and present a number of
ideas to pursue in the future. Finally in Appendices A through F we document some
additional information and important details of the experiments that have been originally
developed in this thesis work. Appendix A contains a list of publications originating from
this work. We then describe a new Hall bar design in Appendix B. Appendix C presents a
summary of various AlAs effective mass determinations found in the literature. Appendix
D describes an experimental technique to accurately calibrate the piezo strain factor. We
also document some circuit diagrams for custom-made instruments that are useful for
device characterization in Appendix E. To end with, in Appendix F we present an
extended model that describes the anomalous giant piezoresistance effect discussed in
Chapter 7 in more quantitative detail.

5
2 Equation Section 2Chapter 2
BACKGROUND
This chapter presents some background information and basic ideas for the work
presented in this thesis. We start with a discussion of multivalley semiconductors and
then describe the realization and some basic properties of the two-dimensional electron
system (2DES) confined to AlAs quantum wells (QWs). Finally we describe the ballistic
transport regime which is explored in this thesis.
2.1 MULTIVALLEY SEMICONDUCTORS
The single most important property of a semiconductor material is its energy band
structure as it governs many of its electrical and optical properties. The band structure
stems from a quantum mechanical description of the motion of electrons in the crystal.
The minima in the conduction band (or “valleys”) determine where the electrons reside in
the momentum space (k-space) and form “pockets” of electrons. Examples of multivalley
solids are n-type Si, Ge, AlAs and PbTe.
As an example, the band structure of AlAs is shown in Fig. 2.1 (a). It demonstrates an
indirect bandgap, where the conduction band minimum is located at the X point of the
Brillouin zone (BZ) (indicated by an arrow as X6), away from the maximum point of the
valence band at the Γ point. Being away from the symmetric Γ point, the conduction band
valleys in AlAs possess anisotropic constant energy surfaces, meaning that the electron
effective masses are different along the longitudinal and transverse directions due to
different energy dispersion curves. The constant energy surface is a prolate ellipsoid as
shown in Fig. 2.1 (b). This surface can be described by the following equation (for the
case of a valley along the [100] direction and centered at k0):
2 222
0( )
2
yx z
l t t
kk k k
E
m m m
 −
= + + 
 
 
(2.1)
The equation indicates that the ellipsoid is characterized by two important parameters: the
longitudinal (ml) and the transverse (mt) effective mass. The cubic crystal symmetry in
AlAs dictates that the X point is six-fold degenerate in the first BZ as shown in
CHAPTER 2

2.1 MULTIVALLEY SEMICONDUCTORS 6
Fig. 2.2 (a). However, for AlAs, the ellipsoid is right at the face of the BZ so effectively
there are three full ellipsoids (six halves). We label these valleys as X, Y and Z valleys
according to the direction of their principal axis: x, y and z.
Figure 2.1 (a) Band structure of AlAs [12,13]. The arrow indicates the conduction band
minimum at the X point of the Brillouin zone where the electrons reside. (b) The constant
energy surface of one valley with its principal axis along [100].
0 01.1 0.20tlm m m m= =
3vg =
0 00.92 0.19tlm m m m= =
6vg =
0 01.64 0.082tlm m m m= =
4vg =
(a) (b) (c)
Figure 2.2 Constant energy surfaces in k-space for the conduction band edge of three
multivalley semiconductors: (a) AlAs, (b) Si and (c) Ge. The values of the longitudinal (ml),
transverse effective masses (mt) and valley degeneracy (gv) are indicated.
(a) (b)

2.2 ALAS TWO-DIMENSIONAL ELECTRON SYSTEM 7
For comparison, Fig. 2.2 also presents the constant energy surfaces of other techno-
logically important semiconductors: Si and Ge. As we can see, AlAs is very similar to Si
except that Si has six degenerate valleys lying along the ∆-line (Γ→X line) of the BZ,
about 85% of the way to the zone boundary (X point). Ge has its conduction band minima
at the L points of the BZ. Like AlAs, Ge’s valleys are located right at the face of the BZ,
therefore Ge has a four-fold valley degeneracy.
If the electrons populate only a single valley, the electrical properties are highly
anisotropic. The electrons would have a high mobility in the direction where the effective
mass is small, and a lower mobility where the effective mass is large. This property has
been exploited to yield a large piezoresistance effect and can be utilized to realize a very
sensitive strain sensor [14-16]. However in bulk multivalley material, the electrons in the
whole set of valleys contribute to conduction and thus lead to an isotropic conductivity as
a consequence of the cubic symmetry of the crystal.
2.2 ALAS TWO-DIMENSIONAL ELECTRON SYSTEM
Modern crystal growth technology such as MBE (molecular beam epitaxy) has allowed
the fabrication of high purity material and heterostructures. One important device
structure is the MODFET (modulation-doped field-effect transistor), also known as
HEMT (high electron mobility transistor). MODFET is a field-effect transistor device
that typically has a 2DES trapped at a heterojunction interface, with dopants that provide
modulation doping located at separate locations [See, e.g., Fig. 2.3]. While in the Si/SiO2
MOSFET system the highest electron mobility achieved is around 4 m2
/Vs, in a
MODFET one can achieve values over 1000 m2
/Vs in GaAs 2DESs [17]. These are
mobilities measured at low temperatures, where they are limited by scattering from
impurities, defects and interfaces rather than phonons. The extremely high mobility in
GaAs 2DESs is attributed to the almost perfect crystalline quality of the GaAs/AlGaAs
heterostructures and the ability to separate carriers from the dopant impurities.
Our research has concentrated on the growth and characterization of AlAs QW structures.
Being closely related to GaAs, AlAs enjoys many advantages such as a lattice constant
which is closely matched to GaAs; this allows a high quality and dislocation-free AlAs
/AlGaAs interface. By adopting a single-sided doping structure, a record high electron
mobility of 31 m2
/Vs has been achieved [11].

Figure 2.3 The layer structure of AlAs MODFET wafers used in this thesis: (a) M415 (15
nm wide QW) and (b) M409 (11 nm wide QW). δ-Si indicates a delta-doped layer. (c) The
energy band diagram of M409 [11] showing the conduction band edges at the X and Γ points
of the Brillouin zone.
Figure 2.4 Valley occupation in AlAs QWs and the (in-plane) 2D Fermi contours for: (a)
Narrow well (w < 55 Å) and (b) Wide well (w > 55 Å).

When we confine electrons to an AlAs QW, two mechanisms lift the valley degeneracy:
the quantum confinement effect of the QW, and the strain effect arising from the lattice
mismatch between GaAs and AlAs. The quantum confinement causes the valley with
larger mass along the confinement direction, i.e. the out-of-plane (Z) valley, to have
lower energy. On the other hand, the slightly larger lattice constant of AlAs
( 5.6611AlAsa = Å) compared to the GaAs ( GaAsa = 5.6533Å) leads to a biaxial
compressive strain that lowers1
the energy of the in-plane valleys (X and Y) thus favoring
them to be the ground state [10,18-23]. These two effects compete with each other and,
depending on the thickness (w) of the QW, we have two cases of valley occupation in
AlAs QWs [10,18-23]:
1. Narrow AlAs QW (w < 55 Å):
The confinement effect dominates and the out-of-plane valley (Z) becomes the
ground state as shown in Fig. 2.4 (a).
2. Wide AlAs QW (w > 55 Å):
The strain effect dominates and the in-plane valleys (X and Y) become the
ground state as shown in Fig. 2.4 (b).
This thesis focuses on the wide AlAs QWs where the 2DES occupies the X and Y valleys.
This system has some unique properties that are very different from those of the more
commonly studied GaAs 2DES [11,24]: large and anisotropic effective mass, large Landé
g-factor and, most importantly, the possibility to tune the valley populations. The crystal
and band structures of the two wafers that we used in the experiments reported in this
thesis are given in Fig. 2.3.
We have discussed how the strain due to lattice mismatch breaks the degeneracy between
the Z and the X and Y valleys in AlAs QWs. One can further lift the degeneracy between
the X and Y valleys by applying symmetry-breaking strain along the [100] or [010]
directions [25]. This is achieved by gluing the sample on top of a piezo-actuator
[Fig. 2.5 (a)] and thus controlling the valley splitting that induces transfer of electrons
from one valley to the other as shown in Figs. 2.5 (b) and (c). This experimental setup
1
By convention, compressive strain has negative value and lowers the valley energy.

(a) (b) (c)
Figure 2.5 (a) Experimental setup for tuning the valley populations in AlAs 2DES using a
piezo-actuator. (b) Electron transfer from the X to the Y valley occurs with the application of
positive piezo bias (VP) to dilate the sample along [100] and shrink it along [010]. (c) Energy
diagram showing that the valley energies and populations are split by the symmetry-breaking
strain.
is utilized in Chapter 7 to demonstrate an anomalous giant piezoresistance effect in an
AlAs 2DES with an antidot lattice.
In the experimental setup as shown in Fig. 2.5 (a), the valley splitting VE∆ between the X
and Y valleys is given as:
( ), , [100] [010]V V X V YE E E E2∆ = − = −ε ε , (2.2)
VE E2∆ = ε , (2.3)
where 2 (5.8 0.1)E = ± eV is the AlAs shear deformation potential [26] and
[100] [010]= −ε ε ε is the symmetry-breaking strain (heretofore simply referred to as strain).
In the case of the piezo-actuated device as shown in Fig. 2.5 (a), at low temperature this
strain is proportional to the applied piezo bias, PV :
κ PV∆ = ∆ε (2.4)
where κ is the piezo strain factor.2
The value κ can be determined using the strain gauge
glued to the other side of the piezo [Fig. 2.5 (a)]. We have developed a reliable technique
to measure κ at low temperatures as described in Appendix D.
We would like to emphasize that among all multivalley semiconductor systems, the
2
Near room temperature the strain vs. piezo voltage exhibits significant non-linearity and hysteresis but
not at or below the liquid He temperature [25].

2.3 BALLISTIC TRANSPORT 11
opportunity to have a strain tunable two-valley system is unique to the AlAs wide QWs,3
thanks to the slightly larger AlAs lattice constant compared to GaAs that leads to biaxial
compressive strain.4
This valley-tunability is a critical feature that allows us to unlock a
wealth of phenomena related to the role of the valley degree of freedom in 2DESs.
Among recent findings in our group are: spin susceptibility dependence on the valley
degree of freedom [28], observation of valley skyrmions [29], giant piezoresistance effect
[16], enhanced valley susceptibility values [4], spin-valley phase diagram of the two-
dimensional metal-insulator transition [30], and parallel field induced valley imbalance
[27]. A summary of some of these results can be found in Ref. [31].
2.3 BALLISTIC TRANSPORT
Electrical transport phenomena can be broadly categorized into two different regimes
based on the relative size of the carrier’s elastic mean free path (le) and the relevant
feature sizes of the sample such as width (W) or length (L). The elastic mean free path is
the distance over which the carrier can travel without experiencing elastic scattering so
that its momentum and energy are conserved. At low temperature and low bias excitation,
where the current is carried only by the electrons at the Fermi energy, the mean free path
is given by: e F el v τ= , where vF is the Fermi velocity and eτ is the elastic scattering time.
If the mean free path is much smaller than the relevant dimensions of the sample, electron
transport is in the diffusive regime. In this regime transport is dominated by the usual
scattering processes such as impurity, alloy or phonon scattering, as shown in Fig. 2.6 (a).
Transport is basically Ohmic and the usual definition of resistance applies. However,
when we shrink our device size or if we improve the mobility of the carriers, the elastic
mean free path becomes comparable or larger than the relevant dimensions of the sample
and transport is in the ballistic regime. Most of the scattering is now dominated by
boundary scattering as shown in Fig. 2.6 (b). The exploration of the ballistic transport
regime in semiconductors has become routine in recent years thanks to the realization of
high mobility 2DESs in modulation-doped GaAs/AlGaAs heterostructures (see, e.g.,
3
Strong perpendicular [24] or parallel [27] magnetic fields could also lift the valley degeneracy.
4
In contrast, in Si1-xGex/Si QWs for example, the lattice constant of the Si layer (the QW layer) is smaller
than Si1-xGex, thus the Si layer experiences an in-plane tensile strain, lifting the in-plane valley energies.
The confinement further enhances this lifting and, as a result, only the two out-of-plane valleys are
occupied.

2.3 BALLISTIC TRANSPORT 12
Figure 2.6 Electron trajectories in the two transport regimes: (a) diffusive and (b) ballistic.
Ref. [17]). Such studies have led to the discovery of several new phenomena such as
magnetic focusing [32], commensurability oscillations in lateral superlattice devices [33],
and quantized conductance in a quantum point contact [34].
The elastic mean free path for a 2DES is given by: 2π /el n eµ where µ is the
electron mobility and n is the electron density. For example, in our typical AlAs 2DES at
0.3 KT with a total electron density of Tn 6×1011
/cm2
and a corresponding mobility
of µ 10 m2
/Vs [11], we have an elastic mean free path of el 1 µm.5
In this thesis we describe various ballistic transport experiments in the AlAs 2DES:
commensurability oscillations in surface-grating devices, commensurability peaks in
antidot lattices, and quantized conductance in a quantum point contact. In these
experiments the relevant feature sizes of the sample are comparable to or smaller than the
elastic mean free path. These feature sizes are the grating period in the commensurability
oscillations experiment, the period of the antidot lattice, and the channel length (and also
width) of the quantum point contact constriction.
5
In calculating the mean free path we use / 2T
n n= to account for the valley degeneracy in the AlAs
2DES and a circular (instead of elliptical) Fermi contour as an approximation.

13
3 Chapter 3
EXPERIMENTAL DETAILS
In this chapter we describe the main experimental procedures: the sample fabrication
process and the measurement techniques. We highlight some new techniques and
improvements that have been developed in this thesis work.
3.1 SAMPLE FABRICATION
Figure 3.1 The main steps of the AlAs device fabrication process.
Figure 3.1 presents the main steps of the AlAs device fabrication process. A more
detailed, step-by-step account is documented in Appendix B of Ref. [23]. General
information on GaAs material processing techniques can be found in a book by R.
Williams [35]. We detail the procedures used in each step of Fig. 3.1 as follows:
CHAPTER 3

3.1 SAMPLE FABRICATION 14
1. Sample cleaving
We cleave samples from the MBE-grown wafer into square shapes of 4×4 mm2
to 5×5 mm2
to fit the arrangement of the Ohmic contacts dictated by a pre-
fabricated shadow mask. Since the sample is grown on a GaAs (001) substrate,
the principal crystal axes [100] and [010] lie along the diagonals of the cleaved
piece. After cleaving, we clean the sample using acetone and methanol.
2. Ohmic contact deposition
The quality of AlAs Ohmic contacts depends critically on the cleanliness of the
surface prior to contact metal deposition. Remnants of photoresist could easily
ruin the Ohmic contacts; therefore, we normally deposit the Ohmic contact
materials first using a shadow mask prior to Hall bar patterning that uses
photoresist.1
The shadow masks are fabricated from thin (0.4 mm thick) G10
composite plastic, instead of Al which was used previously. The G10 composite
is chosen for its semi-transparent property that makes it easy to align to the
sample. We coat the G10 shadow mask with poly-methyl-metacrylate (PMMA)
prior to contact evaporation so that it can be re-used by lifting off the PMMA
(and the metal on top) with acetone afterwards. The G10 plastic is quite strong
and resistant to acetone.
The Ohmic contact alloy consists of Au, Ge, Ni and Au with thicknesses of 20,
40, 10 and 40 nm respectively in order of the deposition steps. The alloy is then
annealed in a forming gas (10% H2 and 90% N2) environment at 470 °C for 11
minutes to allow it to diffuse into the sample and make contact to the electrons in
the quantum well (QW).
3. Hall bar patterning
The Hall bar mesa defines the device and contact terminals for longitudinal and
transverse resistance measurements typical in a quantum Hall experiment. We
designed a new set of Hall bar masks suited for AlAs devices and electron beam
lithography (EBL) process that incorporates alignment marks, EBL focusing
pads, and multiple Hall bar regions for redundancy. These features are critical to
achieve a successful and productive experiment. A complete description of this
1
Recently it was found that it is possible to pattern AlAs Ohmic contacts using a standard photo-
lithography technique. To ensure the complete removal of photoresist and a clean surface, one can use a
bilayer photoresist and a metal-ion free developer [36].

Hall bar design is given in Appendix B.
Hall bar mesas are defined on the sample using a standard UV photolithography
technique. We use a GaAs resist primer, Surpass 3000, prior to photoresist
deposition to enhance adhesion of the photoresist to the substrate; this leads to a
more faithful pattern transfer. The Hall bar is aligned along the [100] or [010]
direction so that the major axes of the two in-plane valleys are either parallel or
perpendicular to the Hall bar. For AlAs this means that the Hall bar has to be
oriented along the diagonal direction as shown in Fig. 3.1
We wet-etch the mesa using a H3PO4:H2O2:H2O solution with a ratio of 1:1:40
that gives a fast etching rate of 400 nm/min. We typically etch the mesa very
deep, down to 300 nm. (The QW is located at ~100 nm below the surface.) This
is done to produce a high contrast image of the mesa edges during the EBL step.
Typically a semiconductor surface is difficult to view using the scanning
electron microscope (SEM) due to a poor image contrast. After etching we strip
the photoresist completely using acetone.
4. Mesoscopic pattern fabrication
Most of the samples in this thesis contain mesoscopic patterns such as surface-
grating, antidot lattice, or quantum point contact. Since their feature sizes are
smaller than 1 µm, they have to be defined by EBL and, if necessary, followed
by wet or dry etching.
Electron beam lithography
We use 2.4% PMMA dissolved in chlorobenzene solution as a resist. The resist
is deposited using a standard spin-coating technique at 8000 rpm spinning speed
to achieve a film thickness of ~600 nm. The patterns are designed using standard
computer aided design (CAD) programs such as CorelDraw, AutoCAD or
DesignCAD. The EBL system is a JEOL 840, a modified SEM to perform
lithography. The control program is the Nanometer Pattern Generation System
(NPGS) from J. C. Nabity Lithography Systems. This program can perform soft-
ware pattern-alignment, pattern-writing, as well as image acquisition. To write
the pattern, the CAD design is first compiled into the machine code by
specifying the writing parameters such as dosage, current and magnification
settings. The software then executes the program to control the electron beam

position and dwell-time. After the lithography step, we develop the pattern for
~50 sec in methyl-isobutyl-ketone (MIBK) and isopropanol (IPAL) with a
volumetric ratio of 1:3.
Prior to writing on the real sample, we usually first test the lithography process
outcome by writing to dummy samples with varying dosages. Optimum pattern
demands correct dosage, good focusing, and appropriate beam current.
Wet etching
After pattern definition by EBL, one may need to perform etching to define the
pattern. A simple option is to do wet etching using a standard acid solution. The
advantages are simplicity, low cost, reasonable accuracy and reproducibility in
etching depth, and more importantly, less damage induced to the sample in
contrast to the dry etching technique. However, one major disadvantage is that
the wet etching technique cannot be used to etch very small patterns because of
the hydrophobic nature of the PMMA that repels the etchant. In this thesis, wet
etching is used to define quantum point contact constrictions (Chapter 6) where
we use a slow etchant: H2SO4:H2O2:H2O with a volumetric ratio of 1:8:160 that
gives an etch rate of 240 nm/min.
Dry etching
Dry etching, especially for GaAs materials, is an important processing step for
the fabrication of high-speed electronic and optoelectronics devices. In this
thesis, we use reactive ion etching (RIE) with a high density electron cyclotron
resonance (ECR) plasma, often referred to as the ECR-RIE process [37,38]. The
ECR-RIE technique performs etching using a high plasma density, low-pressure,
and low-temperature environment [39,40]. The high plasma density allows for a
low energy operation that introduces little damage to the sample.
The ECR plasma is created by a combination of absorption of microwave
radiation and a magnetic field that induces the electron cyclotron resonance at
low gas pressure. The substrate holder is negatively biased to attract the
positively-charged plasma that creates a bombardment of the substrate surface by
ions and free radicals. This bombardment provides both chemical and physical
etching of the sample surface.

(a) (b)
Figure 3.2 (a) Schematic diagram of the Astex ECR-RIE system. (b) The test pattern used in
a routine process calibration prior to every etching session.
We use an Astex ECR-RIE system whose schematic is shown in Fig. 3.2. The
main components are a 2.45 GHz microwave source, magnetic coils (that
provide a magnetic field of ~8.8 mT to induce the cyclotron resonance), an RF-
powered substrate holder built for 4-inch wafers, and a vacuum reactor chamber
connected to a turbo molecular pump and gas inlets.
The ECR-RIE technique has broad process windows and a variety of appli-
cations. The advantages are a highly anisotropic etching process, relatively low
damage (to the electronic properties of the 2DES) due to its low energy
compared to other RIE techniques, and the ability to etch small features (<0.5
µm) compared to standard wet etching. The disadvantages are high system
complexity, high maintenance and cost. Additionally, we found in our system
that it suffers from poor reproducibility in etching AlGaAs materials. We suspect
that this problem is attributed to a lack of temperature control in the substrate
holder, process sensitivity to the sample surface quality, and some inevitable
variations in various process parameters that influence the plasma stability, e.g.
the microwave power, RF power, RF bias and gas flow.

In this thesis, the ECR-RIE process is used to etch antidot lattices with
submicron feature size (Chapters 5 and 7). We use Ar and Cl2 [41] gasses, with a
typical microwave power of 200 W, RF substrate bias of –100 V, and 2 mTorr
process pressure. We achieve a typical etch rate of 55 nm/min for our AlAs
samples. PMMA is used as the resist that can sufficiently withstand the harsh
environment of the etching process for a brief time (< 3 min).
Due to the rather poor reproducibility of the etching rate, we always calibrate the
process using dummy samples made of the same material as the real sample. We
etch a test pattern as shown in Fig. 3.2 (b) and characterize the result by
inspecting the surface quality and measuring the etching depth using a surface
profiler machine (DekTak).
5. Front gate deposition
The front gate (or top gate) is used to control the density of the 2DES. It is
deposited using an electron beam evaporator. Our typical front gate consists of
10 nm Ti and then 30 nm of Au. The Ti layer serves to enhance the adhesion of
the front gate to the sample. The front gate is thin enough so that it does not
completely block the light necessary to induce persistent photo conductivity
effect to populate the QW with electrons [42].
6. Wiring and packaging
By this point the sample fabrication is practically finished. In case we need to
apply tunable strain to the sample, we can mount it on a piezo-actuator as shown
in Fig. 3.3 (a). For that purpose we need to thin the sample to 150 - 200 µm
using a lapping machine and evaporate the back gate (100 nm Ti and 20 nm Au)
on the back of the sample. The back gate is used to screen out the stray electric
field from the piezo and also during the illumination to induce carriers in the QW
[42]. Finally, we glue the sample to the piezo using epoxy. To monitor how
much strain we induce, we glue a metal foil strain gauge on the other side of the
piezo as shown in Fig. 3.3 (b). Detailed procedures can be found in Ref. [25] and
Appendix B of Ref. [23]; also, for details of our calibration process, see
Appendix D.
Next, we wire all the contacts using thin (25 µm diameter) Au wire and In
solder. We then mount the sample on a dual-in-line-pin (DIP) header that fits in
the cryostat as shown in Fig. 3.3 (a). To store the samples, it is important to keep

3.2 DEVICE MEASUREMENT 19
them in a dessicator under vacuum to prevent possible degradation. A potential
problem arises from the oxidation of Al and AlGaAs at the exposed areas such as
the mesa walls.
Prior to low temperature measurements, we first test all the contacts in ambient
light. Typical good Ohmic contacts show two-point resistances of 0.5 to 2 MΩ at
room temperature; at 4 K the two-point resistance of good contacts is around 5 to
20 kΩ (following illumination). Front and back gates should show Schottky
behavior (i.e. the resistance should be smaller when we bias the gate positively
with respect to the 2DES than vice versa).
Figure 3.3 (a) A sample (M409K8) mounted on a piezo-actuator. (b) The strain gauge
mounted on the back of the piezo to measure the applied strain.
3.2 DEVICE MEASUREMENT
The three main components of our device measurement setup are shown in Fig. 3.4; they
are the cryostat, instrumentation rack and control PC.
1. 3
He Cryostat
The cryostat is used to cool down the sample by immersing it in liquid 3
He, as
well as housing the magnet that delivers a high magnetic field. Low temperatures
near absolute zero are necessary to reduce phonon scattering and also to quench
(a) (b)

the kinetic (thermal) energy of the 2DES. For the experiments presented in this
thesis we used two cryogenic systems, a home-made diffusion-pumped 3
He
system and a sorption-pumped 3
He system made by Oxford Instruments. Both
systems can reach a base temperature of 300 mK and are equipped with 8 T and
12 T superconducting magnets, respectively.
Figure 3.4 A typical experimental setup for device characterization showing the three main
components.
2. Instrumentation Rack
The instrumentation rack hosts a switch box (contact terminal panel) and various
instruments. The switch box provides connections to the sample and is equipped
with switches that enable the connections to be grounded to the rack chassis
when necessary. This is particularly important for high impedance connections
such as gate terminals. The various instruments used are lock-in amplifiers,

digital multimeters (DMMs), voltage or current sources, and digital to analog
converters (DAC) and the magnet power supply.
Most of the data in the form of resistance are obtained using lock-in amplifiers
operated at low frequency (<50 Hz) in a four-point configuration. We pass a
very small AC current (typically 10 nA to 50 nA to avoid Joule heating),
obtained from the reference channel of the lock-in amplifier. The voltages across
the device, typically on the order of 10 µV, are measured by the lock-in
amplifiers, whose analog outputs are measured by digital multimeters (Keithley
2000 or HP34401A). These instruments are connected to the control PC through
GPIB connections.
In the course of projects we have also developed a number of custom-made
instruments to facilitate efficient data acquisition, for example, a tunable bipolar
voltage source, a voltage adder, and a programmable high-current source for the
3
He sorb to control the temperature of the sample space. For reference, their
circuit schematics are presented in Appendix E.
3. Control PC
Most of the instruments are controlled by a PC using a program called
“MASHMEASURE”, originally developed by Yakov Shkolnikov. This program
runs in MATLAB environment. Compared to previous programs written in
LabView, this program is far superior. It is more robust, more user-friendly and
more versatile in controlling the instruments. The users can write their own
measurement sequence in command-line fashion just like any ordinary
MATLAB code (m-file) thus making it a very flexible platform to yield
productive experiments. For example, it is easy to program a data acquisition
sequence that controls many parameters such multiple gate voltages, piezo
voltages, magnet power supply, and even temperature.2
2
Temperature control can be implemented in the sorption-pumped 3
He cryostat using the programmable
current source that powers the sorb heater and a software control system written in MATLAB (see
Appendix E, Sec. E.3).

22
4 CO in AlAs 2DES
COMMENSURABILITY OSCILLATIONS IN ALAS
2DES
In this chapter we describe the results of valley-resolved ballistic transport measurements
in the AlAs two-dimensional electron system (2DES). We use surface grating devices
which exhibit commensurability oscillations (COs) in the magnetoresistance (MR) traces
at low magnetic field. Through Fourier and partial inverse Fourier analyses of the
oscillations, we disentangle and study the COs of each valley component, and obtain their
amplitude, phase and scattering time. More importantly, from an analysis of the CO
frequencies, we directly determine /l tm m , the ratio of the longitudinal and transverse
electron effective masses, a fundamental parameter of the AlAs conduction-band.
4.1 INTRODUCTION
Commensurability oscillations (also known as Weiss oscillations) are MR oscillations
observed in a 2DES modulated by a periodic, one-dimensional potential, ( )V x , as shown
in Fig. 4.1 [33,43,44]. This phenomenon can be understood in terms of a classical
geometric resonance effect where the cyclotron orbits become commensurate with the
periodic potential. Since these oscillations are periodic in 1/B they resemble Shubnikov-
de Haas oscillations except that they typically occur at low magnetic fields (B < 1 T) and
often exhibit only a few oscillations.1
The resistance maxima occur at the “commensurate” orbit conditions when:
1
42 ( )cR a p= + , (4.1)
where Rc is the cyclotron radius and p is an integer. The enhanced resistance at this
commensurate condition can be understood in terms of a resonant ×E B drift of the
cyclotron orbit guiding center as illustrated in Fig. 4.1 (b) [45]. The electric field E
originates from the modulating potential, ( )V x= −∇Ε , which oscillates along the
1
For a typical two-dimensional carrier system with density 1011
–1012
/cm2
subjected to a submicron
grating.
CHAPTER 4

4.1 INTRODUCTION 23
longitudinal (x) direction. Such a drift enhances the diffusivity, or the conductivity ( yyσ ),
in the transverse (y) direction that leads to an enhanced longitudinal resistance xxρ .2
Figure 4.1 (a) The original Weiss oscillations (thick trace) [33] and a theoretical curve (thin
trace) derived from Eq. (4.2) [45]. (b) The conceptual description of the origin of the
commensurability oscillations [45]. Shown are a resonant ( 1
4
2 2C
R a= ) and a non-resonant
orbit for the electron motion in the presence of a periodic potential, ( )V x , and perpendicular
magnetic field, B. A resonant orbit experiences a significant ×E B drift along the transverse
(y) direction that enhances the longitudinal resistance xxρ .
2
This follows directly from the inverse relationship: /( )xx yy xx yy xy yxρ σ σ σ σ σ= − .
(a)
(b)

4.1 INTRODUCTION 24
Figure 4.2 (a) A typical micrograph taken from a surface grating device with a = 0.8 µm. (b)
Schematic cross section of the device used to measure COs. Application of a bias to the
Ti/Au surface gate with respect to the 2DES produces a potential modulation which is
periodic in the [100] direction and has period a. (c) The AlAs in-plane valleys X and Y in k-
space (left), and their corresponding first two resonant CO orbits in real space (right).3
The
Fermi wave vectors kF,X and kF,Y relevant for the COs of the X and Y valley are also indicated.
Here we report measurements of COs in a high-mobility AlAs 2DES. Figure 4.2
highlights the basic principle of our study. Using a grated surface gate, we apply a lateral
periodic potential with period a to the 2DES, and measure the low-field magneto-
resistance ( xxρ ) along the potential modulation direction as a function of a perpendicular
magnetic field B. If transport is ballistic, xxρ oscillates with B as the classical electron
cyclotron orbit diameter becomes commensurate with a. These oscillations in xxρ can be
expressed as:
( )cos 2π / πxx COf Bρ∆ ∝ − /2 , (4.2)
2 /CO Ff k ea= , (4.3)
where fCO is the oscillation frequency and kF is the Fermi wave vector perpendicular to
the modulation direction (parallel to the grating stripes). Note that the oscillations are
periodic in 1/B.
3
More precisely, as Eq. (4.1) indicates, a maximum in xx
ρ is seen whenever the cyclotron orbit diameter
equals 1
4( )a p + . In Fig. 4.2 (c), for simplicity, we schematically show the conditions for the first two
resonances as when the diameter is equal to a and 2a.

4.2 DEVICE FABRICATION 25
In our AlAs 2DES, there are two in-plane valleys occupied: X and Y. Their cyclotron
orbits in real space have the same shape as their k-space orbits but rotated by 90° as
shown in Fig. 4.2 (c). If both valleys participate in the ballistic transport independently,
we expect two superimposed sets of COs whose frequencies are related to the Fermi
wave vectors parallel to the grating stripes as indicated in Fig. 4.2 (c):
2
, 2π /F X X t lk n m m= , (4.4)
2
, 2π /F Y Y l tk n m m= , (4.5)
where nX and nY are the 2D electron densities for the X and Y valleys respectively. These
relations can be combined to yield:
( )
2
, ,/l t CO Y CO X X Ym m f f n n= , (4.6)
implying that, if the valley densities are known, the frequencies of the COs can be used to
directly determine the mass anisotropy ratio /l tm m , independent of a.
4.2 DEVICE FABRICATION
We performed measurements in two surface-grating samples on 11 nm (M409N3) and 15
nm-wide (M415L3) AlAs QWs with grating periods equal to 400 nm and 300 nm
respectively. The results presented in this chapter are primarily taken from M409N3. A
Hall bar mesa was defined on each sample using standard photolithography and wet
etching techniques. The Hall bar was aligned along the [100] direction so that the major
axes of the two in-plane valleys were either parallel or perpendicular to the Hall bar. To
fabricate the grating patterns, we spun 150 nm of poly-methyl-metacrylate (PMMA) on
top of the sample, and used electron beam lithography to define an array of PMMA
ridges. We then deposited 10 nm Ti and 30 nm Au to form a top gate. Biasing this top
gate with respect to the 2DES results in a periodic potential modulation in the 2DES.
Using illumination at low temperatures and front/back gate biasing, we varied the 2DES
density between 5 to 9×1011
/cm2
, with maximum mobility around 9.3 m2
/Vs prior to
patterning the grating on top of the sample. We measured xxρ in a 3
He cryostat with a
base temperature of 0.35 K, and used a standard lock-in technique.

4.3 EXPERIMENTAL RESULTS 26
4.3 EXPERIMENTAL RESULTS
A typical xxρ vs. B trace, taken at a total density nT = 8.7×1011
/cm2
, is shown in
Fig. 4.3 (a). It exhibits both COs, in the low field range –1 < B < 1 T, and Shubnikov-de
Haas oscillations (SdHOs), at B > 1.7 T. The COs are more clearly seen in the second
derivative ( 2 2
/xxd dBρ ) plot shown in the inset of Fig. 4.3 (a). Fortunately, the COs and
SdHOs are well separated in their field range, thus simplifying their analysis.
Figure 4.3 CO and SdH data for M409N3 (a = 400 nm): (a) Magnetoresistance trace showing
COs at low fields (-1 < B < 1 T) and SdHOs at high fields (B > 1.7 T). Inset: numerically
determined second derivative 2 2
/xx
d dBρ . (b) Fourier power spectra of COs from both
xx
ρ (solid curve) and 2 2
/xx
d dBρ (dotted curve). (c) Fourier power spectrum of SdHOs.

Figure 4.4 CO traces from M409N3: (a) Series of low field MR traces exhibiting COs at
various densities. The traces are offset for clarity. (b) The corresponding Fourier power
spectra for the original (solid line) and the second derivative (dotted line) of the CO signal in
the range 0.1 T to 1 T. The dashed line is a guide to the eye.
The SdHOs provide information regarding the electron densities of the 2DES and the
valleys. In Fig. 4.3 (c) we show the Fourier power spectrum of the SdHOs. To calculate
this spectrum, we used the xxρ vs. 1/B data for B >1.7 T, subtracted a second-order
polynomial background, and multiplied the data by a Hamming window [46] in order to
reduce the side-lobes in the spectrum. The spectrum exhibits three peaks, marked in
Fig. 4.3 (c) as nT, nT/2 and nT/4.
The peak frequencies multiplied by /e h give the 2D density (e is the electron charge and
h is Planck's constant). We associate the nT peak with the total density, as this peak's
frequency multiplied by /e h indeed gives the total 2DES density which we independent-
ly determined from the Hall coefficient. For Fig. 4.3 data, we deduce nT = nX + nY =

4.4 ANALYSES AND DISCUSSIONS 28
8.7×1011
/cm2
. The presence of the nT/4 peak indicates the spin and valley degeneracy of
the 2DES.4
Figure 4.3 (b) shows the Fourier power spectra of COs calculated using xxρ and
2 2
/xxd dBρ vs. 1/B data in the 0.1 < B < 1 T range. Both spectra exhibit two clear peaks
at fCO,X and fCO,Y, which we associate with the CO frequencies of the X and Y valleys,
respectively. If we assume that the two valleys have equal densities, we can use Eq. (4.6)
to immediately find / 4.4l tm m = . This value, however, is inaccurate because there is a
small but finite imbalance between the X and Y valley densities in our sample. Such
imbalances can occur because of anisotropic strain in the plane of the sample and are
often present in AlAs 2DESs. Note that the Fourier spectrum of the SdHOs cannot
resolve small valley density imbalances.
Figure 4.4 presents several CO traces and their Fourier spectra at different densities
achieved by varying the top gate bias. As detailed in the next paragraph, we analyze the
dependence of CO frequencies on density to deduce the imbalance between the valley
densities, and also to determine the /l tm m ratio more accurately.
4.4 ANALYSES AND DISCUSSIONS
4.4.1 Determination of Mass Anisotropy Ratio / tlm m
Figure 4.5 summarizes the density dependence of our CO frequencies. Denoting the
difference between the valley densities by Y Xn n n∆ = − , we rewrite Eqs. (4.3) to (4.5):
( )2
, 2
l
CO Y T
t
mh
f n n
e a mπ 2
= + ∆ , (4.7)
( )2
, 2
t
CO X T
l
mh
f n n
e a mπ 2
= − ∆ . (4.8)
The slopes and intercepts of the 2
COf vs. nT plots give the /l tm m ratio and n∆ .
Concentrating on the Y valley, a least-squares fit of 2
,CO Yf data points (circles in Fig. 4.5)
to a line leads to values / 5.2 0.5l tm m = ± and 11 2
( 0.6 0.4) 10 cmn −
∆ = − ± × . Note that
4
As detailed in Ref. [24], the spin and valley degeneracies are lifted at higher B, leading to the presence of
the nT/2 and nT peaks in the Fourier spectrum.

Figure 4.5 Density dependence of the CO frequencies for the Y (circles) and X valleys
(squares) for sample M409N3 (a = 400 nm). The line through the circles is a least-squares fit
to the data; its slope determines the ratio /l tm m and its intercept the density difference n∆
of the two valleys. The dashed line is described in the text.
such a small value of n∆ is consistent with the nearly valley-degenerate picture deduced
from the existence of the / 4Tn peak in the SdH frequency spectrum [Fig. 4.3 (c)].
The above determination of the /l tm m ratio is based on the density dependence of ,CO Yf
only and does not use the measured ,CO Xf . As a consistency check, we use Eq. (4.8) to
predict COf for the X valley using /l tm m and n∆ deduced from the above analysis
of 2
,CO Yf above. This prediction, shown as a dashed line in Fig. 4.5, agrees well with the
measured 2
,CO Yf (solid squares), and confirms that we are indeed observing COs for both
valleys.
We repeated similar experiments in a sample (M415L3) from a different wafer,
containing a 2DES confined to a 15 nm wide AlAs quantum well. The data for this
sample are summarized in Fig. 4.6. In this sample only the COs of the Y valley could be
reliably determined. By performing similar analysis using Eq. (4.7), in the density range

Figure 4.6 CO data for M415L3 (a = 300 nm): (a) Density dependence of the CO
frequencies, similar to Fig. 4.5. (b) The CO trace at n = 6.4×1011
/cm2
. The trace in red is the
second derivative 2 2
/xx
d dBρ . (c) The corresponding Fourier power spectrum of the CO taken
from xx
ρ (black) and 2 2
/xx
d dBρ (red).
from 5 to 8.5×1011
cm-2
, we deduce / 5.4 0.5l tm m = ± , in good agreement with the results
for M409N3.
At this point it is worthwhile emphasizing that the COs described here uniquely probe the
/l tm m ratio.5
Conventional experiments that probe the effective mass, such as cyclotron
resonance or measurements of the temperature dependence of the amplitude of the
SdHOs, lead to a determination of the cyclotron effective mass, CRm . In a 2DES with an
elliptical Fermi contour, CRm is equal to l tm m , and therefore provides information
complimentary to the /l tm m ratio, so that lm and tm can be determined. In fact, using
the measured 0.46CR em m= in AlAs 2DESs,6
we use the / 5.2 0.5l tm m = ± ratio to
5
Faraday rotation experiments can also determine the /l t
m m ratio, but such determination requires
knowing ml or mt and the
2 2
/τ τ〈 〉 〈 〉 ratio where τ is the scattering time. In fact, B. Rheinländer et al. [47]
used Faraday rotation measurements in bulk AlAs and, assuming 00.19t
m m= (determined from a ⋅k p
calculation) and
2 2
/ 1τ τ〈 〉 〈 〉 = , deduced a ratio / 5.7l t
m m = .
6
The most accurate cyclotron resonance (CR) measurements in AlAs 2DESs so far were reported by T. S.
Lay et al. [10], and yielded (0.46 0.02)CR em m= ± . This value is in very good agreement with the results

deduce ( )1.1 0.1l em m= ± and ( )0.2 0.02t em m= ± . These values are in good agreement
with the (theoretical) value of 0.19t em m= that is calculated in Ref. [47] and 1.1l em m=
that is deduced from the Faraday rotation measurements [47];5
they also agree well with
the results of the majority of theoretical and experimental determinations of the effective
mass in AlAs. A summary and discussion of AlAs effective masses can be found in
Ref. [50] and is also summarized in Appendix C.
4.4.2 Resolving the Ballistic Transport in Individual Valleys
We proceed to extract more information, such as the amplitude, phase, and scattering
time from the COs of each valley by performing partial inverse Fourier analysis. Figure
4.7 summarizes the results of such analysis. The Fourier power spectrum shown in
Fig. 4.3 (b) is separated into two ranges7
chosen to isolate the two CO peaks. The range
for COs of Y valley (0.57 < fCO < 1.21 T) is inverse Fourier transformed and divided by
the original window function. The result is shown as the solid curve in Fig. 4.7 (a). The
range for the COs of X valley (0.29 < fCO < 0.57 T) is analyzed in a similar manner and
the result is the solid curve in Fig. 4.7 (b). We fit the deduced COs for each valley to a
simple expression that assumes the amplitude of the COs decreases exponentially with
1/ B :
exp( π / ω ) cos(2π / )xx C CO COf Bρ ρ τ θ0 0∆ ∝ − − , (4.9)
where , ,CO COfρ τ0 , and θ0 are the fitting parameters; ω /C CReB m= is the cyclotron
frequency with 0.46CR l t em m m m= = . The exponential term in Eq. (4.9) is analogous to
the Dingle factor used to describe the damping of the SdHOs' amplitude with increasing
1/ B , and has been used successfully to fit COs in GaAs 2D electrons [51] and holes [52].
In Fig. 4.7 the results of the best fits are shown as dotted curves along with their fitting
parameters. The best-fit θ0 for the COs of Y and X valleys are 0.37π and 0.47π
respectively, in excellent agreement with the expected value of 0.5π (the relative phase
errors are 6 % and 2 % of 2π). This consistency affirms that the reconstructed oscillations
of CR measurements by N. Miura et al. [48] on n-type AlAs layers (0.47 0.01)CR em m= ± , and by T. P.
Smith III et al. [19] on 2DESs in multiple AlAs quantum wells ( 0.5~ )CR em m ; the latter data, however,
show a very broad CR. There was also a CR study of GaAs/AlAs short-period superlattices by H.
Momose et al. [49], where 1.04 elm m= and 0.21 etm m= were deduced.
7
Varying these ranges by reasonable amounts (±10 %) does not lead to significant changes in parameters
that are deduced from the inverse Fourier transform curves.

Figure 4.7 Results of the inverse Fourier decomposition of the COs of Fig. 4.3 for the Y and
X valleys. The dotted curves show the best fits of Eq. (4.9) using the indicated parameters.
The fits are done only for fields smaller than those marked by square points which indicate
the positions of the first CO resonant orbits. The arrows indicate where the first two CO
resonances occur.
faithfully represent the COs of the two valleys. The amplitude of the oscillations for the Y
valley is larger than for the X valley as expected from the shorter real-space, resonant
orbital trajectories for this valley [Fig. 4.2 (b)]. On the other hand, the scattering times,
COτ , that we deduce from the fits are comparable for the two valleys, suggesting that
scattering is nearly isotropic.8
We also deduce two other scattering times: the quantum lifetime SdHτ and the mobility
scattering time µτ , and compare them with COτ . From fitting the B dependence of the
amplitude of the SdHOs in the patterned region to the damping factor exp( π ω )cτ− / , we
8
For a circular cyclotron orbit trajectory, or for an elliptical orbit if we use the average Fermi velocity
along the trajectory, the exp( π ω )C
τ− / term in Eq. (4.9) is equivalent to exp( / 2 )L l− where L is the orbit
length and l the mean-free-path. For the data of Fig. 4.7, a τ ~ 10 ps corresponds to l ~ 0.5 µm.

4.5 SUMMARY 33
obtain 0.76psSdHτ = . The mobility scattering time is 24 psµτ = , determined from the
mobility of the same sample prior to patterning. Similar to CO experiments in other 2D
carrier systems [51,53], we observe SdH CO µτ τ τ< < . This observation can be qualitatively
understood considering the sensitivity of these τ to the scattering angle [51]: µτ is the
longest since the mobility is least sensitive to small-angle scattering, while SdHτ is the
shortest because the SdHOs are sensitive to all scattering events (both small- and large-
angle).
4.5 SUMMARY
In summary, we have demonstrated valley-resolved ballistic transport in an AlAs 2DES
using a surface grating device that introduces a lateral, one-dimensional, periodic
modulation potential in the 2DES. We observe COs in the MR traces. The Fourier spectra
of the oscillations reveal two distinct peaks associated with the transport from the two
valley components. Using partial inverse Fourier analyses of the oscillations, we
disentangle and study the COs of the electrons in the two valleys. Furthermore, this
experiment allows us to probe the Fermi contour and deduce the effective mass
anisotropy ratio / 5.2 0.5l tm m = ± .

34
5 AlAs 2DES in an Antidot Lattice
ALAS 2D ELECTRONS IN AN ANTIDOT LATTICE
We describe in this chapter ballistic transport experiments on the AlAs two-dimensional
electron system (2DES) in the presence of an antidot (AD) lattice where we observe
peaks in the low-field magnetoresistance (MR). We present numerical simulations to
elucidate the transport in this system and explain the resulting MR peaks. Similar to the
commensurability oscillations experiments of the previous chapter, the data also provide
a direct measure of the anisotropy of the Fermi contour, or equivalently /l tm m , the ratio
of the longitudinal and transverse electron effective masses.
5.1 INTRODUCTION
The commensurability oscillations presented in the previous chapter essentially
demonstrate a geometrical resonance effect of the electrons’ cyclotron motion with a one-
dimensional periodic potential, i.e. a surface grating, that can be observed in transport
measurements. Extending our study of this phenomenon, we perform similar experiments
on samples with a two-dimensional periodic potential, i.e. an AD lattice.
An AD lattice is a periodic array of holes, typically defined by etching. If imposed on a
2DES, the AD holes completely exclude the electrons thus creating a lateral superlattice
in the 2DES. Such a structure alters the transport properties significantly and exhibits
distinctive features in the MR. The system allows studies of classical chaos dynamics in
condensed matter physics [54,55]. Early experiments in GaAs 2DESs with square
(isotropic) AD lattices showed pronounced peaks in the low-field MR traces [56]. These
peaks have been attributed to the pinned orbits around groups of ADs [56,57], as shown
in Fig. 5.1, as well as runaway trajectories that skip from one AD to another [58,59].
These effects occur when the cyclotron diameter is commensurate with the lattice period.
For example, in the GaAs 2DES where the Fermi contour is circular, the MR peak of the
first commensurate orbit is observed at 2 CR a= , where RC is the radius of the cyclotron
orbit and a is the AD lattice period.
CHAPTER 5

5.1 INTRODUCTION 35
Figure 5.1 The results of the original AD lattice experiment by Weiss et al. [56] where the
MR exhibits peaks associated with the commensurate orbits.
Figure 5.2 The AD lattice experiment in AlAs 2DES: (a) Micrograph of the AD lattice
region with period a = 0.8 µm. (b) The Fermi contours of AlAs in-plane valleys X and Y in k-
space. The Fermi wave vectors ,F Xk and ,F Yk are indicated. (c) The first four commensurate
orbits for the X and Y valleys that give rise to peaks in MR for current along the x-direction;
these orbits have diameters in the y-direction that are equal to a multiple integer of the AD
lattice period (see text); this integer is given by the index i in Xi and Yi.

5.2 DEVICE FABRICATION 36
Here we report a similar experiment in the AlAs 2DES. The system provides a rather
unique situation where we have a 2DES with anisotropic (elliptical) Fermi contours [Fig.
5.2 (b)] in an isotropic AD lattice as shown in Fig. 5.2 (a). We also perform an analysis
of the MR peaks associated with the shortest commensurate orbit to determine the mass
anisotropy factor /l tm m . To highlight the significance of this measurement, it is shown
in the next chapter that this mass anisotropy can be exploited to realize a simple "valley-
filter" device using a quantum point contact structure [60]. Such a device may play an
important role in "valleytronics" or valley-based electronics applications [61], or for
quantum computation where the valley state of an electron might be utilized as a qubit
[4].
5.2 DEVICE FABRICATION
We performed experiments on 2DESs confined in a 15 nm-wide AlAs quantum well
(sample M415B1) whose structure is shown in Fig. 2.3 (a) in Chapter 2. We patterned a
Hall bar sample, with the current direction along [100], using standard optical
photolithography. We then deposited a layer of PMMA and patterned the AD arrays
using electron beam lithography. The PMMA layer served as a resist for a subsequent dry
etching process used to define the AD holes. We used an electron cyclotron resonance
etching system (see Chapter 3) with an Ar/Cl2 plasma [41], at an etch rate of ~55 nm/min,
to obtain small feature sizes without a degradation of the 2DES quality. The AD pattern
was etched to a depth of 80 nm, thus stripping the dopant layer and depleting the
electrons in the AD regions. The micrograph of a section of one of our AD arrays is
shown in Fig. 5.2 (a). Each AD array is a square lattice and covers a 20 µm × 30 µm area.
There are four regions of AD lattice with different lattice periods: a = 0.6, 0.8, 1.0 and
1.5 µm as schematically shown in the inset of Fig. 5.3. The aspect ratio /d a of each AD
cell is ~1:3, where d is the AD diameters. Finally, we deposited a front gate, covering the
entire surface of the active regions of the sample to control the 2DES density. Following
an initial back-gate biasing and brief illumination [42], we used the front gate to tune the
total density (nT) from 2 to 5×1011
/cm2
. This density was determined from both
Shubnikov-de Haas oscillations and Hall coefficient measurements that agree with each
other. From measurements on an unpatterned Hall bar region in a different sample but
from the same wafer, we obtain a mobility of ~10 m2
/Vs at a typical density of 3×1011
/cm2
and T = 0.3 K. This gives a typical mean-free-path of ~1 µm.

5.3 EXPERIMENTAL RESULTS
Figures 5.3 and 5.4 summarize our main experimental results. Figure 5.3 shows the low-
field MR traces, measured as a function of perpendicular magnetic field (B), for all the
AD regions. We observe two peaks, A and B, which are symmetric with respect to B =
0 T. Peak A, whose position is higher in field than peak B, is seen in the traces from all
the AD regions. Peak B, on the other hand, is not observed in the a = 1.5 µm trace. In
general, we observe that, as the AD lattice period becomes smaller, the positions of both
peaks A and B shift to higher field values (as indicated by the dashed lines). Figure 5.4
captures the gate-voltage ( GV ) dependence of the MR traces for the a = 0.8 µm AD
region. As we increase GV to increase the 2DES density, peak A shifts to higher field
values while peak B does not appear to shift. We have made similar observations in other
AD regions as GV is varied.
Figure 5.3 Low-field MR traces for all four AD regions (sample M415B1) with periods
equal to a = 0.6, 0.8, 1.0 and 1.5 µm (from top to bottom). The Hall bar with the different
AD regions is schematically shown on the right. For clarity, traces are shifted down (from
top to bottom) by: 2885, 390, 350, and 0 Ω.

Figure 5.4 Low-field MR traces (sample M415B1) for the AD region with period a = 0.8 µm
for VG = -0.1 V to 0.15 V (from top to bottom), corresponding to a linear variation of the
density nT from 2.27 to 3.53×1011
/cm2
. For clarity, traces are shifted down (from top to
bottom) by: 1410, 980, 645, 385, 175 and 0 Ω.
In order to analyze and understand the data of Figs. 5.3 and 5.4, we first briefly review
what is known about ballistic transport in AD arrays for GaAs 2DESs where the Fermi
contour is isotropic. Low-field MR traces for such systems typically exhibit
commensurability peaks at magnetic fields where the classical cyclotron orbit fits around
a group of ADs [54,56,62]. Although there are subtleties associated with the exact shape
of the AD potential and also the possibility of chaotic orbits that bounce from one AD
boundary to another, the peak observed at the highest magnetic field corresponds to the
shortest period that fits around the smallest number of ADs; for an isotropic Fermi
contour, this would correspond to a circular orbit, with a diameter equal to the AD period,
encircling a single AD. There have also been studies of ballistic transport in 2DESs with

5.4 NUMERICAL SIMULATION 39
isotropic orbits in an anisotropic (rectangular) AD lattice.1
Experimental results [63],
followed by theoretical analysis [64], have indicated that the commensurability peaks are
observed only when the orbit diameter matches an integer multiple of AD lattice period
along the direction perpendicular to the current.2
Based on the above considerations, we
can predict the first four (smallest) commensurate orbits of the X and Y valleys that may
give rise to MR peaks in our system; these are shown in Fig. 5.2 (c). To evince this
conjecture we performed numerical simulations as described in the following section.
5.4 NUMERICAL SIMULATION
To elucidate the transport mechanism in our samples we performed a kinematic,
numerical simulation for our system, a 2DES with elliptical Fermi contours in an
isotropic AD lattice in the presence of a perpendicular magnetic field B. We simulate the
kinematic of a large number of electrons and calculate the MR based on a classical linear
response theory using the Kubo formula [54,67]. The simulation details are similar to
those described in Ref. [68]. We calculate two separate cases: X and Y-valley electron
transport by assuming equal electron densities in each case.
The kinematic of the electrons is governed by the following equations:
xx v= , yy v= , (5.10)
x y
x
eB
v v
m
= − , y x
y
eB
v v
m
= , (5.11)
where xm and ym are the effective masses along the x-direction and y-direction. The
electrons are constrained to the elliptical constant energy (Fermi) contours. The Fermi
velocity of the electron varies depending on its location on the Fermi contour. We use a
large number of electrons (NP), typically 10,000, and have verified that the calculated
resistances converge to within ±3.5 % of the asymptotic value for NP > 5,000. The AD
boundaries are represented by hard-wall potentials so that the electrons are scattered
elastically upon collision. The geometry of the AD lattice is based on the experimental
parameter i.e. / 1/3d a = . Figure 5.5 shows snapshots of the simulation at magnetic
fields when the commensurate orbits X1 and Y1 occur (peaks X1 and Y1 in Fig. 5.6).
1
The problem of ballistic transport for a 2DES with a circular Fermi contour in a rectangular AD lattice is
equivalent to transport in a 2DES with an elliptical Fermi contour in a square AD lattice.
2
This situation resembles the magnetic electron focusing effect in a system containing multiple, parallel
one-dimensional channels. See, e.g., Ref. [65] and [66].

Figure 5.5 Simulation snapshots showing various types of trajectories: (i) chaotic, (ii) pinned
and (iii) skipping orbits for: (a) X-valley electrons at 4
0/l tB m m B= (X1 orbit), and (b) Y-
valley electrons at 4
0/t lB m m B= (Y1 orbit). B0 is the magnetic field of the first
commensurate orbit if the Fermi contour were circular.
The conductivity of the system can be calculated based on classical linear response theory
where the Ohmic conductivity is proportional to the electrons’ diffusivity and is given by
the Kubo formula [67]:
/
0
( ) (0)t
ij i jc e v t v dtτ
σ
∞
−
0= ∫ , (5.12)
where 2
/ Fc ne E0 = , n is the electron density, EF is the Fermi energy, τ is the electron
mean scattering time for the system without the AD, i and j subscripts represent the
directions x and y. The term ( ) (0)i jv t v is the velocity-velocity correlation function
averaged over all the particles, where (0)jv is the electron’s initial velocity in direction j.
We run the simulation from 0t = to 10τ, divided into 10,000 discrete time intervals, and
perform the numerical integration of Eq. (5.12). Once we obtain the conductivity tensor
components ijσ , we can calculate the resistivity through the inverse relationship, 1−
=ρ σ .
The longitudinal MR xxρ is given by:
yy
xx
xx yy xy yx
σ
ρ
σ σ σ σ
=
−
. (5.13)
The results of our simulations of xxρ for the X and Y valleys are presented in Fig. 5.6.
The MR traces indeed show peaks at or near the expected values, namely orbits X1, X2,
X3 for the X-valley and Y1 for the Y-valley. This observation evinces the conjecture
outlined in the preceding section that the commensurability peaks are observed only
when the orbit diameter matches an integer multiple of the AD lattice period along the
direction perpendicular to the current.

Figure 5.6 MR obtained from numerical simulations (smooth curves are guides to the eye).
Vertical lines indicate the expected positions of the peaks for orbits X1, X2, X3 and Y1. We
assumed equal densities for the two valleys and a current along the x-direction. Inset:
Schematic of the commensurate orbits. B0 is the magnetic field of the first commensurate
orbit if the Fermi contour were circular.
Furthermore, the simulation elucidates the transport processes that give rise to the peaks
in the MR traces. At the commensurate conditions, we can classify the electron orbits into
three types as shown in Fig. 5.5. The pinned orbits tend to localize the electrons in space,
practically removing them from the conduction process, and thus increase the resistance.
The chaotic orbits at the commensurate condition too tend to localize the electrons around
one or more ADs and can also lead to enhanced resistance.
The skipping orbits (or runaway trajectories) clearly increase the conductivity in the
transverse direction ( yyσ ) as shown in Figs. 5.5 (a) and (b). Since xx yyρ σ∝ , this effect
enhances the longitudinal resistivity further and its contribution may be the most
significant. This explains our previous conjecture that commensurability peaks are
determined by the lattice period along the direction perpendicular to the current.
The behavior of the skipping orbits could also explain the difference in the relative

5.5 DATA ANALYSIS 42
strengths of the X1 and Y1 peaks in Fig. 5.6. For orbit Y1, the skipping orbits are harder
to occur since the electrons come close to colliding with the ADs in the adjacent column,
thus breaking the skipping trajectories. In the case of skipping orbits for X1, on the other
hand, the electrons practically skip along in a free space due to their skinny orbits. Since
more electrons can follow such trajectories they therefore enhance the conductivity along
the y-direction leading to a stronger peak in the longitudinal resistivity.
5.5 DATA ANALYSIS
After presenting numerical simulations that provide insight into the transport in our
system, we proceed with the analysis of our experimental data. We associate peak A in
our data of Figs. 5.3 and 5.4 with the shortest orbit X1 in Fig. 5.2 (c). From the field
position of this peak, and if we assume that the electron density of the X valley is half the
total density, we can directly obtain a value for the anisotropy (ratio of the major to minor
axes diameters)3
of the elliptical orbits in our system, thus obtaining the effective mass
ratio /l tm m . However, there is a finite imbalance between the X and Y valley densities in
our sample. Such imbalances can occur because of anisotropic strain in the plane of the
sample and are very often present in AlAs 2DESs [11,69]. Therefore, we present here an
analysis to determine the /l tm m ratio independent of the density imbalance.
Consider a primary, commensurate orbit whose diameter in the direction perpendicular to
the current is equal to the AD lattice period [orbits X1 and Y1 in Fig. 5.2 (c)]. These
would give rise to MR peaks at fields 2 /P FB k ea= where Fk is the Fermi wavevector
along the current direction. For the X and Y valleys, these wavevectors are ,F Xk and ,F Yk ,
respectively, as shown in Fig. 5.2 (b). For an elliptical Fermi contour, they are related to
the densities of the X and Y valleys, nX and nY, via the following relations:
2 2
, ,2π / , 2π /F X X l t F Y Y t lk n m m k n m m= = . (5.14)
Note that the total density T X Yn n n= + and the valley imbalance X Yn n n∆ = − . We can
obtain nT from the Shubnikov-de Haas oscillations of the MR at high magnetic fields or
from a measurement of the Hall coefficient. Now consider orbit X1 as shown in
Fig. 5.2 (c). Its associated MR peak position , 1P XB is given as:
( )
2
2
,X1 2 2
/
π
P l t T
h
B m m n n
e a
= + ∆ . (5.15)
3
The ratio of major to minor axes of the elliptical orbits is given by /l tm m .

We use this expression to analyze our data.
We assign peak A in our data to orbit X1 and plot the square of its field position 2
,P AB as a
function of the total density nT in Fig. 5.7. It is clear that for all four AD lattice regions,
2
,P AB varies linearly with nT as expected from Eq. (5.15). Moreover, we obtain the slopes
β , and the intercepts of the lines in Fig. 5.7 by performing a least-squares fit of each data
set. Note that according to Eq. (5.15), 2 2 2
/ / πl th m m e aβ = , and the intercept is equal to
nβ ∆ .4
Finally, we plot β as a function of 2
a−
in the inset of Fig. 5.7. This figure shows
that, consistent with the prediction of Eq. (5.15), β indeed depends linearly on 2
a−
and
the line has a zero intercept.
Figure 5.7 Summary of the density dependence of 2
,P AB for all four AD regions in sample
M415B1; ,P AB is the position of peak A observed in MR traces. The straight lines are linear
fits using Eq. (5.15). The error bars reflect the uncertainty in the peak positions which were
determined by subtracting second-order polynomial backgrounds from the MR traces. Inset:
Slope (β) of the 2
,P AB vs. nT lines of the main figure are plotted as a function of 2
a−
. The
dashed-line is a linear fit to the data.
4
Here we assume that n∆ is fixed in the density range of interest.

From the slope, 2
/ ( )aβ −
∆ ∆ , of the line in Fig. 5.7 inset, we can deduce the effective
mass anisotropy ratio:
24
4 2
/
π
l t
h
m m
e a
β
2 −
 ∆
=  ∆( ) 
. (5.16)
Note that this mass anisotropy ratio is related to the slope of the line in the inset of
Fig. 5.7 by a pre-factor containing only physical constants ( 4 2 4
/ πh e ). Our data analysis
and determination of this ratio is therefore insensitive to parameters such density
imbalance between the two valleys. From data of Fig. 5.7 we obtain / 5.2 0.4l tm m = ± , in
very good agreement with the ratio / 5.2 0.5l tm m = ± determined from the ballistic
transport measurements in AlAs 2DESs subjected to one-dimensional, periodic potential
modulations [69] as described in Chapter 4.
Figure 5.8 Density imbalance deduced from the Fourier analysis of the Shubnikov-de Haas
oscillations for the 0.8 µm AD region (sample M415B1). Inset: The Fourier spectrum with
peaks associated with half total density and half (X and Y) valley densities. The half valley
density peaks nX/2 and nY/2 at f = 5.9 and 3.3 T correspond to a valley imbalance of
∆n = 1.3×1011
/cm2
.

A few other features of the data presented here are noteworthy. From the intercepts of the
linear fits in Fig. 5.7 we can determine the valley density imbalance for each AD lattice.
Such analysis gives n∆ = 1.5, 1.2, 1.0, and 1.3 ×1011
/cm2
(± 0.2×1011
/cm2
) for the AD
regions with a = 0.6, 0.8, 1.0 and 1.5 µm, respectively. Such a variation of valley
imbalance for different AD regions may come from non-uniform residual strain across
the sample. (Fortunately, the values of ∆n or their variations from one region to another
do not affect the /l tm m value as determined from our analysis in Fig. 5.7). Note that,
because of the close proximity of the different AD lattice regions, we expect this
variation to be small, consistent with the n∆ values deduced from the above analysis. We
can also deduce the valley imbalance from the Fourier analysis of the Shubnikov-de Haas
oscillations measured across the AD regions [24], provided that the valley density peaks
in the Fourier spectrum are well developed and well separated. We obtained such data for
Figure 5.9 Summary of the density dependence of 2
PB of peak A and B for the AD regions
in sample M415B1: (a) a = 0.6 µm and (b) a = 0.8 µm. The lines X1 are linear fits to peak A
position using Eq. (5.15). Lines X2, X3 and Y1 are the predicted peak positions calculated
using equations similar to Eq. (5.15).

5.6 SUMMARY 46
the a = 0.8 um AD region at high density as shown in Fig. 5.8,5
where we deduce
∆n = 1.3×1011
/cm2
, consistent with ∆n = 1.2×1011
/cm2
deduced from the intercept of the
linear fit in Fig. 5.7.
As for peak B, it is tempting to associate it with orbits X2 or Y2 in Fig. 5.2 (c). This is
qualitatively consistent with the data of Fig. 5.3, which indicate that peak B moves to
smaller values of magnetic field as the period of the AD lattice is made larger. Moreover,
peak B becomes weaker with increasing AD lattice period and disappears for the largest
period a = 1.5 µm. This is also consistent with the larger size of the X2 and Y1 orbits
(compared to the X1 orbit), and the fact that for a = 1.5 µm, the lengths of these orbits
become large compared to the electron mean-free-path. Quantitatively, using the values
of /l tm m and n∆ obtained above, we can modify Eq. (5.15) and determine the expected
peak positions associated with the X2 and Y1 orbits.6
As illustrated in Fig. 5.9, we find
that the predicted peaks for X2, X3 and Y1 orbits are quite close to each other in field7
and approximately straddle the observed positions of peak B in Fig. 5.3. It is possible
then that peak B may originate from a superposition of X2, X3 and Y1 peaks that cannot
be resolved in our experiment. We cannot rule out, however, that peak B may be strongly
influenced by non-linear orbit resonances in the system. Such resonances are known to
occur for orbits with long trajectories in the presence of a smooth AD potential [54].
5.6 SUMMARY
In summary, we performed ballistic transport experiments in AD lattices imposed on an
AlAs 2DES where the electrons occupy two valleys with anisotropic Fermi contours. The
low-field MR traces exhibit two sets of peaks. From the analysis of the positions of the
peak associated with a commensurate orbit with the shortest trajectory [X1 orbit in
Fig. 5.2 (c)], we deduced the effective mass anisotropy ratio / 5.2 0.4l tm m = ± , a
fundamental parameter of the AlAs conduction-band structure that cannot be directly
measured from other transport experiments. This ratio is consistent with the ratio deduced
from the measurements of the commensurability oscillations described in the previous
chapter.
5
We have repeated similar measurement for lower densities and for other AD regions, unfortunately this is
the only data set where we observe well-developed and well-separated valley density peaks.
6
To calculate peak positions for orbits X2 and X3, simply replace a with 2a and 3a in Eq. (5.15). For orbit
Y1, replace / tlm m with /t lm m and n∆ with n−∆ .
7
In Fig. 5.9 the expected BP for Y1 orbits are lower than BP for X3 because nY < nX in our experiments.

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