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This is to certify that the
dissertation entitled

CHARGE ACCUMULATION IMAGING OF A TWO-
DIMENSIONAL ELECTRON SYSTEM

presented by

SUBHASISH CHAKRABORTY

has been accepted towards fulﬁllment
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6/01 cJClFiC/DateDue.p65«p. 15

CHARGE ACCUMULATION IMAGING OF A TWO-DIMENSIONAL ELECTRON
SYSTEM
By

Subhasish Chakraborty

A DISSERTATION
Submitted to
Michigan State University
In partial fulﬁllment of the requirements
For the degree of
DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

2003

ABSTRACT
CHARGE ACCUMULATION IMAGING OF TWO-DIMENSIONAL ELECTRON
SYSTEM
By

Subhasish Chakraborty

In this thesis, we studied GaAs/AlGaAs two-dimensional electron systems (2DES) and
quantum dots with a novel scanned probe method called charge accumulation imaging
(CAI), in a tunneling geometry. We imaged the 2DES near integer Landau level ﬁlling
and observed surprising density modulations, which gave rise to orientationally ordered
stripe-like structures. These stripe-like structures were not static, but appeared as a
function of the applied perpendicular magnetic field. We also imaged the 2DES in the
absence of a magnetic ﬁeld, to essentially image the disorder potential of the 2DES.
Although theories predict a spatial distribution of the ionized donors to be truly random,
micron size features were observed, consistent with similar experiments done in the
presence of a magnetic field. Progress was made towards electron wavefunction imaging
of quantum dots. Single-electron addition spectra of a disorder-induced quantum dot were

observed with the CA1 technique.

© Copyright by Subhasish Chakraborty, 2003

This thesis is dedicated to my parents.

iv

ACKNOWLEDGEMENTS

This work would not have been possible without the generous support and
assistance I received from my professors, colleagues, friends and family members. I
consider myself extremely fortunate to have the opportunity to work under the
supervision of my thesis advisor, Professor Stuart Tessmer. All through these years, he
has been an outstanding teacher, showing extraordinary patience and support. Not only
has he helped me countless time professionally; there have been several occasions when
he helped me personally. For that I am indebted to him and consider him my very good
friend.

Special thanks should go to Dr. Ilaari Maasilta, who taught me the technical
details of the experiment, and from whose insight I have gained immensely. Thanks
should also go to my lab-mates through the years — Sergei, Irma, Aleksandra and Cemil,
for their day-to-day assistance and fun-ﬁlled discussion sessions. I should thank Ms.
Debbie Simmons for her assistance on different occasions.

I should also thank Professor Norman Birge for his advice on the quantum dot
fabrication. I am grateful to Dr. Khalid Bid and Dr. Baokang Bi for helping me with the
fabrication process and scanning electron microscope. I am also grateful to Dr. Reza
Loloee for his help on numerous occasions. I am thankful to the people of the machine
shop, particularly James Muns, Thomas Hudson, and Thomas Palazzolo for their help
and assistance.

Last but not the least, I am thankful to my family for their support and

encouragement in pursuing a career in physics. My special gratitude goes to my wife,

Arunima. She has not only made countless sacrifices for my career, but has inspired and
motivated me through the years.

This work was supported by the National Science Foundation grant Nos. DMR99-
70756 and DMR00-7523 and the Alfred P. Sloan Foundation. The W. M. Keck
Microfabrication Facility provided valuable resources for fabrication and scanning

electron microscopy.

vi

TABLE OF CONTENTS

LIST OF TABLES ............................................................................... viii
LIST OF FIGURES ............................... ix
CHAPTER 1: Introduction ....................................................................... l
1.1: 2DES in perpendicular magnetic field ......................................... 2
1.2: Charge Density Waves ........................................................... 5
1.3: Quantum Dots ..................................................................... 8
CHAPTER 2: Heterostructures .................................................................. 11
2.1: Molecular-beam epitaxy ........................................................ 12
2.2: Mobility ........................................................................... 15
2.3: Formation of 2DES and modulation doping ................................. 16
2.4: The Sample ....................................................................... 20
CHAPTER 3: Experimental technique and apparatus ....................................... 22
3.1: Charge Accumulation Imaging ................................................ 22
3.2: The scan head ..................................................................... 31
3.3: The cryogenic charge sensor ................................................... 40
3.4: The cryostat ....................................................................... 46
CHAPTER 4: Imaging disorder in a2DES ................... 49
4.1: The simplest measurement ...................................................... 49
4.2: Imaging disorder .................................................................. 52
CHAPTER 5: Charge Density Waves: Theory and Experiment ............................ 59
5.1: Theory of CDWs ................................................................ 59
5.2: The CDW experiment ........................................................... 67
CHAPTER 6: Quantum Dots .................................................................... 79
6.1: Quantum dots: theory and motivation ......................................... 79
6.2: Quantum dot fabrication ......................................................... 84
6.3: Preliminary Experimental result ................................................ 90
CHAPTER 7: Summary and future direction .................................................. 94
7.1: Summary ........................................................................... 94
7.2: Future directions .................................................................. 95
REFERENCES .................................................................................... 98

vii

LIST OF TABLES

Table l. The cohesive energies of the Laughlin liquidEﬁm and the CDW ESE"! for r,

=J2 .107 is the optimal number of electrons per CDW bubble given approximately by

3vNN. The energy unit is hwc. The energy per electron in the uniform uncorrelated state
EUEL is also provided for reference [Fogler 1997]. ............................................. 63

viii

LIST OF FIGURES

Figure 1. The extended and localized states in the Landau bands. ............................ 4

Figure 2. Theoretically predicted CDW patterns [Koulakov]. (a) Stripe pattern. (b)
Bubble pattern. (c) Wigner Crystal, showing one cyclotron orbit. The spacing of 2.7RC
between the stripes is at vz 4. ..................................................................... 7

Figure 3. (a) Schematic diagram of an artiﬁcial atom located between two capacitor
plates. (b) Diagram of the sample used by Ashoori et al., the black disk is the artiﬁcial
atom in the quantum well. (0) Capacitance of the sample as a function of the top plate
voltage, each spike in the plot represents individual electrons entering the dot at a
particular gate voltage, which corresponds to the energy required to add an electron to the
system. Thus the plot is the addition spectra of the artiﬁcial atom [Ashoori 1996]. ...... 10

Figure 4. Schematic diagram of an MBE machine showing Knudsen cells for Al, As and
Ga with Shutters. The sample sits on a heated holder, which is rotated. The RHEED
apparatus is also shown [Davies]. ................................................................ 14

Figure 5. The energy-level diagram of AlGaAs and GaAs without taking into
consideration certain electronic processes [Prange]. .......................................... 17

Figure 6. Energy-level diagram of GaAs/AlGaAs heterostrueture. The donor electrons
tunnel through the potential barrier and occupy the ﬁrst conduction sub—band of the
potential well. The Fermi level is shown as 81:, The spacer layer is not shown [Prange]. .19

Figure 7. The sample and the tip. The typical scan area is 5pm x 5pm. .................... 21

Figure 8. Schematic of the CA1 technique. CAI is a capacitive technique in which the tip
and the 2DES can be thought of as the two plates of a capacitor, with a voltage being
applied to the 2DES that charges up the tip. .................................................... 23

Figure 9. Schematic of the 3D-to-2D CAI technique. Here an AC excitation is applied to
the 3D metallic substrate separated from the 2DES by a tunneling barrier, instead of a
direct Ohmic contact to the 2DES. The charge tunneling into the 2DES induces an image
charge on the tip, which is measured. ............................................................ 25

Figure 10. The circuit equivalent to the tip sample arrangement in the 3D-to-2D CAI
technique. ............................................................................................ 27

Figure 11. The in-phase and the out-of-phase charging as a function of logarithmic

frequency and tunneling resistance, as obtained from the model circuit. The dashed
curves are the charging when in-plane conductivity is taken into consideration. ......... 28

ix

Figure 12. The differential change in the in-phase and the out-of-phase charging with
respect to the change in the tunneling resistance plotted against the frequency. At the low
frequency limit, the out-of-phase component is more sensitive to changes in tunneling
resistance. ............................................................................................ 30

Figure 13. Schematic of the Besocke design. The sample sits on top of the three carrier
piezo tubes, which along with the scanning tube is attached to a bottom plate. The ramps
face the same side as the sample. ................................................................. 32

Figure 14. Schematic of a microscope design utilized by a handful of experiments. The
thermal-contraction mismatch between the metal body (hatched) and piezoelectric tubes
represents a major drawback. Essentially, in response to temperature ﬂuctuations, the
body expands and contracts much more than the piezo ceramic. These uncompensated
ﬂuctuations in height h result in signiﬁcant variations of the tip-sample separation. ..... 33

Figure 15. (a) Schematic of scanning head. Both the tip and the sample are supported by
identical piezoelectric tubes (dimensions: 7.6cm—long x 4.8mm-diam x 0.8 mm wall
thickness). As a result the unit is nearly perfectly thermally compensated. The displacer,
outlined in dashes, screens the tip from the electric ﬁelds of the piezo tubes. All the metal
parts are made of brass, shown hatched. (b) Sketch of support leg showing the rotating
foot assembly. (c) Bottom view of the microscope. The clamping screws ﬁx the ramps to
allow the positioning screw to be turned—the sample is attached to the other side of this
screw. In this way, we can adjust the vertical position of the sample by as much as 1 cm.
The diameter of the unit is 3.8 cm. ............................................................... 35

Figure 16. Photograph of the actual microscope. The shiny ruby balls at the end of each
foot help reduce friction during ﬁne approach of the sample. ................................ 36

Figurel7. (a) A 35 um x 35 um topographical image of a platinum sample with a grid of
5 pm x 5 pm pits that are 180 nm deep, obtained at room temperature. (b) Cross section
across a single pit measured both in the tunneling mode (solid line) and in capacitance
mode (dashed line). ................................................................................. 39

Figure 18. The cryogenic charge sensor circuit. C73 is the tip-sample capacitance, Cs is
the standard capacitor and Vin is the rrns value of the applied AC excitation to the sample.
......................................................................................................... 42

Figure 19. Typical cross section of a HEMT [Mimura]. ..................................... 43

Figure 20. Drain current- voltage characteristics, at different gate voltages VGS, of a
typical HEMT [Mimura]. .......................................................................... 45

Figure 21. The top loading He3 cryostat with the microscope directly immersed in liquid
He3. The microscope with the sample and the sensor chip is attached to a long stainless
steel stick, which is used to insert the microscope inside the cryostat. ..................... 47

Figure 22. Dependence of the in-phase capacitance signal Q," on the tip dc bias voltage.
The step feature of approximately l2aF marks the potential where the 2DES becomes
depleted locally. The data were acquired at 290mK. .......................................... 51

Figure 23. (a) A 3 mm x 3pm image of an intentionally created disorder to test the CA1
technique in imaging disorder. Bright regions imply grater electron accumulation. (b) A
line scan through the disk of electron accumulation shows the capacitance at the center of
the bright disk is approximately 100aF more than the dark surrounding area. ............. 53

Figure 24. A series of 3pm x 3pm images with decreasing dc tip bias voltage starting
from 0.5V, which is approximately equal to Vnuu, to —0.6V in decrements of 0.15. The
features coalesce and the images get darker as the tip depletes the 2DES. The difference
in capacitance from the brightest to the darkest region is approximately SOOaF. .......... 54

Figure 25. A series of 3pm x 3pm images processed from images in ﬁgure 24.
Topographical contribution has been subtracted. The images show intriguing micron
scale features. The middle image is enlarged, and the micron scale features are shown
with circles in the enlarged image. ............................................................... 56

Figure 26. Plot of relative energies (only ﬁrst terms in equations (5.1) and (5.2)) of
Wigner crystal (WC) and charge density waves (CDW) in units of hwc against vN. In this
example, for approximately vN > 0.1 the CDW has lower energy than WC ................ 62

Figure 27. (a) “Bubbles”. (b) “Stripes”. The black dots are the guiding centers of the
cyclotron orbits. The periodicity is ~ 3 RC. (c) The enlarged quasi-classical image of a
single bubble. The dark region is the guiding center of cyclotron orbits and the toroid is
the charge density distribution, which is created by electrons moving in the cyclotron
orbits centered inside the bubble. [Fogler 1997, Fogler 1996]. .............................. 66

Figure 28. (a) Fixed-tip measurement. As a function of perpendicular magnetic ﬁeld, Qm
and Qout display clear features at integer Landau level ﬁlling v. (b) Calculated in-phase
and out-of-phase charging based on the equivalent circuit. Rigorous modeling for the
system considers the distributed nature of the tip-sample capacitance and the effect of
charge motion within the 2D plane. The dashed curves show qualitatively enhancements,
which occur if the in-plane relaxation rates approach the tunneling rate, an effect that
may occur near integer ﬁlling. (Inset) A linear plot of the derivatives of Qin and Qout with
respect to tunneling resistance. In the low-frequency limit of this experiment (arrow), the
out-of-phase component is more sensitive to Rtun variations [Maasilta 2003, Maasilta
(rapids)]. ............................................................................................. 69

Figure 29. (a) Representative scanning images of Qm (left) and Qout (right) with an
applied magnetic ﬁeld far from integer ﬁlling of 3.89 T; scale bar length=lum. (b)
Topographical surface image showing the orientation of the growth features. The image
was acquired at room temperature without altering the orientation of the crystal. The
elongated mounds point along the[110] direction, as indicated by the orientation of the

scale bar, scale bar length: 2 pm. (0) Topography-subtracted out-of-phase image

xi

corresponding to the same data shown in (a). The images have been ﬁltered to remove
nanometer-scale scatter [Maasilta (rapids)]. .................................................... 72

Figure 30. (a) Topography-subtracted out-of-phase data near six ﬁlled Landau levels. The
ﬁxed-tip curve to the left shows the v=6 Qout peak with arrows indicating the ﬁelds of the
displayed scanning images. The displayed images have been ﬁltered to remove
nanometer scale scatter; scale bar length=2 pm. The crystal orientation is identical to
Figures 28(a) and 28(b). (b) Schematic of Fourier analysis procedure to identify
orientationally-ordered structure. The parallel lines in the direct image (indicated with

dashes) can be described with wave vector It, giving maxima in the Fourier transform

power spectrum at :tic. (shown as black dots). The orientation of the wave vector 00 is
seen as a peak in the angular spectrum, calculated by integrating the transform image
along a series of radial lines at angle 0 [Maasilta (rapids)]. .................................. 74

Figure 31. (a) Fourier power spectra and angular spectra corresponding to three Qoul
images of Figure 29(a), normalized to the peak value at 4.14 T. The power spectra are
displayed over the range i667 pm"; the magnitude of the wave vector corresponding to
the 4.14 T maxima is 2.4 pm", or a wavelength of 2.6 pm. (b) Angular spectra from three
different sample locations and magnetic ﬁelds. Similar peaks were observed near four
(6.75 T) and six (4.14 T) ﬁlled Landau levels. For comparison, a spectrum far from
integer ﬁlling (3.89 T, corresponding to Figure 28(c)) is included. In this plot, each curve
is scaled to have an average value of 1.0, which serves to enhance the weak 3.89 T curve.
The minimum at 0 and 180° is an experimental artifact arising from the scan direction. (c)
The topography-subtracted Qout images at 4.14 T and 6.75 T with arrows pointing to the
structure that yields the prominent Fourier peaks. Also included is the 6.75 T data for

which no ﬁltering was applied. The [I10] crystalline direction is indicated; scale bar
length=2 um [Maasilta (rapids)]. ................................................................. 76

Figure 32. Schematic of wavefuntion imaging for a quantum dot. A single electron
resonates between the bottom electrode and the dot. By mapping out the density
variations, quantum wavefunctions can be probed. As an example a 3D plot of Ill/I2 in the
single-particle approximation is shown. ......................................................... 83

Figure 33. Single layer resist process. (a) The resist is exposed to a beam of electrons. (b)
Resist after being exposed and developed. ..................................................... 86

Figure 34. Two layer resist process. (a) The resists are exposed to a beam of electrons. (b)
Resists after being exposed and developed, the top resist is underdeveloped and creates

an overhang. ........................................................................................ 88

Figure 35. (a) SEM micrograph of the gold gates on GaAs/AlGaAs sample. The circular
gates are 12um apart. (b-e) Further magniﬁed micrographs of individual dots of
diameters (b) 0.5 pm, (0) 1pm, ((1) 1.5 pm and (e) 2pm. ........................................ 89

Figure 36. STM image of a quantum dot metallic gate of size 0.5um. ..................... 91

xii

Figure 37. Addition spectra of a disorder induced quantum dot (left). The applied
excitation voltage was SmV. The arrows in the plot show the capacitance peaks, which
correspond to one electron entering the dot. There are no peaks seen above the tip voltage
of -0.45V, since the energy levels in the dot are too close to be resolved. ................. 92

xiii

Chapter 1

Introduction

The two-dimensional electron system (2DES) trapped at a doped heterojunction is
one of the most important systems for electronic transport. Not only is it at the heart of
the technologically important ﬁeld-effect transistors (FETS), but it is also a fascinating
quantum system for the exploration of fundamental electron interactions. Thus, it has
been an area of intense research for decades.

Most of the experiments on these systems have been what are called transport
measurements, i.e., a current is passed on one direction of the sample, and the voltage
both parallel and perpendicular to the current is measured. In the presence of a magnetic
ﬁeld perpendicular to the 2DES, these experiments have helped discover amazing
physics, like the integer quantum hall effect (IQHE) [Prange, von. Klitzing, Laughlin
1981] and the fractional quantum hall effect (FQHE) [Tsui, Laughlin 1983], the latter
being one of the best known many-body phenomena. Si/SiOz and GaAs/AlGaAs are the
most common two-dimensional electron system materials that have been studied over the
years. While Si/SiOz FETs are the most common electronic devices, they have been
largely superseded in the laboratory by GaAs/AlGaAs systems due to their very high
mobility.

With the discovery of scanning tunneling microscopy (STM) [Binnig 1982,
Binnig 1999] in the mid-eighties, and the advancement of other scanning techniques in
the recent years, there has been a new direction in the study of the 2DES. The interest in

imaging the 2DES with scanning techniques stems from some very exciting theoretical

work [Koulakov, Moessner]. Electronic transport measurements seem to verify some of
these theories, but do not conclusively settle the issues. In some of the cases transport
measurements are simply inadequate.

In this thesis, we shall focus on studying the 2DES with a scanning technique
called the charge accumulation imaging (CAI) [Tessmer 1998, Levi, Finkelstein]. This
Chapter continues with the introduction to physics of 2DES in perpendicular magnetic
ﬁeld, an introduction to charge density waves (CDWs) [Koulakov, Moessner] predicted
by theories, and concludes with an introduction to the physics of zero-dimensional
systems, popularly called quantum dots. The detailed theory along with experimental
results for the CDWs and the quantum dots will be presented in Chapters 5 and 6
respectively. Chapter 2 discusses the samples and how they are grown; it also has some
discussions of the physics of heterostruetures that are known to most experts in the ﬁeld.
Chapter 3 covers the details of the technique of CA1 and a subsection is devoted to the
description of the experimental apparatus. The physics of disorder in 2DESs along with
some interesting results will be described in Chapter 4, and Chapter 7 will summarize the

experimental effort and future directions.

1.1 2DES in perpendicular magnetic ﬁeld

The classical picture of the electronic motion in a transverse magnetic ﬁeld is that
the electrons move in circles, with the radius called the cyclotron radius R: getting tighter
with increasing magnetic ﬁeld. This means the angular frequency of revolution, called the
cyclotron frequency (ac, increases with the ﬁeld. Quantum mechanically, all the orbits are

not allowed, leading to the formation of discrete states. The quantum mechanical problem

was solved long ago by Landau for two-dimensional spinless electrons in a magnetic
ﬁeld B [Prange]. Schrddinger’s equation for such a system in the Landau gauge is given
by:

2
h2 l 6 e8 62 (11)
— -—-—y “—3 W=EW- '

2m idx h ﬂy

The solutions with n = 0, 1, 2, 3, deﬁne the energy levels,

1 1.2
Enk =hwc[n+§). ( )

These energy levels are called the Landau levels (LL’S). The spatial extent of the wave
function is of the order of the magnetic length E = (h/eB)'/2. These Landau levels are
highly degenerate and contain up to eB/h electrons and are separated by an energy of

hwc. . Now if we introduce impurities in the sample, many of the electrons in the 2DES

might be under the inﬂuence of an impurity potential and thus may suffer a Shift in
energy. In addition if we have electric ﬁeld, it causes some of the electrons to drift
through the array of impurities. Solving the problem of 2DES in an in-plane electric and
perpendicular magnetic ﬁeld under the inﬂuence of impurity potentials, we ﬁnd that the
degeneracy of LL'S is completely lifted. The density of states evolves from sharp LL'S to
a broader spectrum of levels as the impurity potential is turned on (see Figure 1). The
various quantum states in each energy band can be divided into two general classes. The
states near the top and bottom of each band, are localized in small regions of the sample.
These localized states occur in regions around impurity atoms that have an excess of

positive or negative charge. Near the center of each band are the so-called extended

EXTENDED STATES

 

LOCALIZED STATES

 

 

N(E)

Figure 1. The extended and localized states in the Landau bands.

states, each being spread out over a large region of space. The density variations shown
in Figure 1 also lead to compressibility variations. Compressibility is a measure of the
energy required to add an electron to the 2DES. Mathematically, we deﬁne
compressibility as dp/dp, where p is the electron density per unit area and p. is the
chemical potential. At integer ﬁlling, the system is less compressible. Here, the Fermi
level is in the region of lower density of states (the dip in the trace in Figure 1), so adding
an electron in this region will shift the Fermi level more than if we are at regions further

from integer ﬁlling. This fact has important consequences for our measurement.

1.2 Charge Density Waves

Before we proceed further, let us deﬁne a quantity called the ﬁlling factor
v = ph/eB, where h is the Planck constant and e is the electron charge. The quantum Hall
effect is different from its classical analog. If we plot the Hall conductance versus the
reciprocal of the magnetic ﬁeld, it shows distinct plateaus at certain values of v, instead
of a straight line. For integer values of v, the phenomenon is called the integer quantum
Hall effect (IQHE) [Prange, von Klitzing, Laughlin 1981], whereas plateaus at fractional
values indicate fractional quantum Hall effect (FQHE) [Tsui, Laughlin 1983]. The FQHE
was ﬁrst discovered at the lowest Landau level (v < 1), and typically occurs at odd
denominator values of ﬁlling factor, v = 1/3, 1/5, etc. This effect has been identiﬁed with
the formation of a uniform incompressible quantum state, called the Laughlin liquid. At
weaker magnetic ﬁelds, or at higher ﬁlling factors (v > 4), the Laughlin liquid fails to be
the ground state of the system. Hartree-Fock calculations show that in the case of higher

ﬁlling factors a phase called the “bubble” phase wins over the Laughlin liquid [Koulakov,

Fogler 1997, Fogler 1996]. Intuitively, there is a competition between the long-range
Coulomb repulsion and the short-range exchange attraction among the electrons in the
uppermost partially-ﬁlled Landau levels. The best compromise for the electrons is to
form clusters or “bubbles”, and calculations show that they would form a triangular
lattice. As we keep on adding electrons to the system, the bubbles grow in size, ﬁnally to
coalesce to form what is called the “stripe” phase [Koulakov, Fogler 1997, Fogler 1996].
On the other end, for the few electrons in the uppermost level, the “bubbles” contain only
one electron; this is a type of Wigner crystal. The three together are called the charge
density wave phase, shown in Figure 2. The goal of the experiment is to resolve these

charge density wave patterns with the CA1 scanning probe technique.

”'0 mo».

2°7Rc '~......}..

M’.’o'&‘o‘o‘o
Wear?
(a)

Figure 2. Theoretically predicted CDW patterns [Koulakov]. (a) Stripe pattern. (b)
Bubble pattern. (c) Wigner Crystal, showing one cyclotron orbit. The spacing of 2.7Rc

between the stripes is at vz 4.

1.3 Quantum Dots

Quantum dots are small structures in a solid, with sizes of the order of few tens of
nanometers to a few microns [Kastner, Mchen, Kouwenhoven 2001]. They consist of
103-109 atoms and an approximately equal number of elections. In a semiconductor, most
of these electrons are tightly bound to the nuclei except for a very small number. Current
nanofabrication technology allows us to precisely control the size and shape of these dots.
Metallic gates are patterned on the semiconductor with electron beam lithography to
deﬁne a quantum dot. Another common technique in use is etching, in which
semiconductor material is selectively removed to form a quantum dot. The conﬁning
potential in a quantum dot is usually bowl-like or parabolic, in which electrons tend to
fall in towards the bottom of the bowl. The conﬁnement of electrons is therefore, in all
three spatial directions, resulting in a quantized energy spectrum - much like an atom.
Hence they are also called artiﬁcial atoms. One can consider the quantum dot as a tiny
laboratory in which quantum mechanics and the effects of electron-electron interactions

can be studied.

Quantum dots have been an area of fruitful research for a long time. A lot of
research has gone into probing the energy levels of the individual quantum dots, mostly
by gated transport or capacitance spectroscopy [Reed, Meirav, Main, Gueret, Schmidt,
Tarucha]. For example, a classic experiment done by Ashoori et a1. [Ashoori 1996] is
shown in Figure 3. This experiment probed the transitions between energy levels of a
quantum dot by precisely controlling the number of electrons contained in the dots, down
to as few as one electron. While these measurements have yielded a wealth of

information about electronic transitions and the effect of electron-electron interactions,

many questions remain unanswered. Studying the spatial structure of the electron
probability density represents a new way to probe the eigenstates. However, such a study
would require new techniques. We use Charge Accumulation Imaging in a conﬁguration
that allows for the possibility of mapping out the spatial charge distribution of individual

electrons, and also studying the addition spectrum of electrons in quantum dots.

 

 

 

 

 

 

 

 

quantum well . GaAs
tunnel barrier ' ‘

 

   
 

 

 

  
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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-390 -360 -330 -300 -270

T0p plate voltage (mV)

Figure 3. (a) Schematic diagram of an artiﬁcial atom located between two capacitor
plates. (b) Diagram of the sample used by Ashoori et al., the black disk is the artiﬁcial
atom in the quantum well. (c) Capacitance of the sample as a ﬁmction of the top plate
voltage, each spike in the plot represents individual electrons entering the dot at a
particular gate voltage, which corresponds to the energy required to add an electron to the
system. Thus the plot is the addition spectra of the artiﬁcial atom [Ashoori 1996].

10

Chapter 2

Heterostructures

Heterostructures are semiconductors composed of more than one material
[Davies]. The high mobility 2DES is realized at the interface of two materials forming
the heterostrueture. One of the materials has a wider bandgap than the other, and a slight
doping of one layer produces an electric ﬁeld, essential to bend the bands forming a
quantum well at the interface and the electrons given up by the dopant atoms form the
2DES. The most common heterostruetures are made out of III—V semiconductors. The
conditions to be good candidates for the two materials to form a heterostruetures are that
they have the same crystal structure or at least the same symmetry, and that they have
nearly identical lattice constant to minimize strain in the ﬁnal structure. Gallium arsenide
(GaAs) and aluminium gallium arsenide (AlGaAs) meet these criteria, and form one of
the mostly commonly used III-V heterostruetures.

The 2DES are typically formed at or close to the interfaces of the heterostrueture,
as mentioned above. This puts a stringent condition on growth that the interface be nearly
perfect. Any imperfection would result in scattering of electrons as they travel along the
interface, decreasing the mobility. Thus heterostruetures should have an atomically sharp
interface if they are to perform well. Additionally, the layers are usually very thin and
require changing of composition very rapidly during growth. Also the interface must be
contamination free. Hence, it is a challenge to grow such structures. The most common

growing technique is the molecular-beam epitaxy [Davies]; with which it is possible

11

grow highly abrupt junctions between different materials. It also allows atomic scale
control over the thickness of layers.
2.1 Molecular-beam epitaxy

Samples used in this study were grown by molecular-beam epitaxy by Prof. M. R.
Melloch at Purdue University. Molecular-beam epitaxy or MBE is very simple in
principle. The substrate is heated on a holder, which sits inside an ultra-high vacuum
(UHV) chamber. The elements that compose the heterostrueture are vaporized in
individual furnaces with oriﬁces directed towards the substrate, but shielded from it by
shutters. These fumaces are called the Knudsen cells or K-cells. The reasons for the
substrate to be in UHV are twofold. Firstly, contamination has to be avoided during
growth. And more importantly, UHV ensures that the molecular ﬂow regime holds. In the
molecular ﬂow regime, the mean free path of a molecule is much longer than the length
of the chamber, so that it never suffers a collision in its path from the K-cells to the
substrate. Hence the molecules that emerge from the K—cells do not diffuse as in a gas at
higher pressure, but form a molecular beam traveling straight without collisions to
impinge on the substrate. Growth starts once the shutters of the K-cells are opened and
the temperature of the cells controls the ﬂux of each element. To ensure uniform growth
across the substrate, the sample holder is rotated during growth. Dopants are added using
additional cells.

MBE is a very slow process, and grows material about 1pm per hour. The growth
is monitored using a technique called reﬂected high-energy electron diffraction or
RHEED. With RHEED, growth can be counted precisely in monolayers, and it also

reveals the structure of the surface. A very simpliﬁed schematic of a MBE machine is

12

shown in Figure 4. MBE machines produce very high quality GaAs/AlGaAs samples.
The quality of a semiconductor sample is determined by its electron mobility. The next

section describes what is meant by mobility.

l3

 

Electron gun

 

 

 

 

 

Shutter Substrate
Knudsen cell -lr> As Jlrri; , .
' Heated rotatrng
\ ,5" substrate holder
Molecular beam » g.
; Screen for viewing

I RHEED

Figure 4. Schematic diagram of an MBE machine showing Knudsen cells for Al, As and

Ca with shutters. The sample sits on a heated holder, which is rotated. The RHEED
apparatus is also shown [Davies].

14

2.2 Mobility

Mobility is a very important quantity in characterizing semiconductor materials. It
describes the ease with which electrons drift in materials in response to an electric ﬁeld.
Following the Drude model [Ashcroft], the electrons in a solid are in constant random
motion. Since the motion is random, there is no net displacement of the electron over a
period of time. Of course, this is only true for a large number of electrons, and not any
individual one. Hence in the absence of an electric ﬁeld, there is no net current. Now if an
electric ﬁeld E x is applied in the x-direction, then each electron experiences a force —eEX,
and there is a net motion in the x-direction. If px is the x-component of the total

momentum, then for n electrons/cm3, the force of the ﬁeld is

_neEX zdpx. (2.1)

dt

 

However, the net acceleration of equation (2.1) is just balanced by the decelerations due
to collision, for steady state current ﬂow. The probability that any electron has a collision
in the time interval dt is given by dt/r, where ris the relaxation time. Thus the differential

change in momentum px in time dt, due to collisions is given by:

dPx = ‘Px ?- (22)
Hence rate of change of momentum of electrons due to collisions is

dPx z - at, (2.3)
dt 2'

 

For steady state current, equations (2.1) and (2.3) should add to zero, so that the average

momentum per electron is

15

PX

<px >=—=—erEX. (2'4)
n

The mobility ,u is deﬁned as the average drift velocity per unit electric ﬁeld or

_ <le > _ <Px > _ er (2.5)
#-"—E—---—*—-—*-
X m Ex In

Where ”2‘ is the effective mass and vx the average drift velocity of the electron. The
minus Sign in the deﬁnition of ,uin equation (2.5) ensures that ,u is positive, since
electrons drift opposite to the ﬁeld. The unit of mobility derived from its deﬁnition is

cmz/ V-sec.

2.3 Formation of 2DES and modulation doping

As mentioned earlier 2DES is formed at the interface between two
semiconductors. In this section, we shall discuss the formation of the 2DES in the GaAs/
AlGaAs heterostrueture [Davies, Prange]. The principle is quite simple. An atomically
perfect interface of GaAs and AlGaAs is grown with MBE. Because the conduction band
has an abrupt step at the interface, an electric ﬁeld perpendicular to the interface attracts
electrons into a quantum well created by the ﬁeld and the interface. The motion
perpendicular to the interface iS quantized and is therefore frozen out, if all electrons are
in the lowest state. This results in a two- dimensional electron system. The electric ﬁeld
perpendicular to the interface is achieved through doping. With MBE atomically perfect
layers of GaAs are grown, followed by a sequence of AlGaAs. The two materials have
nearly the same lattice and dielectric constants. Without taking into consideration the

effect of the dopants, the band would be as shown in Figure 5.

16

CONDUCTION BAND

5? """""" IT BOTTOM
”0481/ I

 

 

 

 

AlGaAs 2.28V 1.5‘6/ GaAs
‘ 7 I + EF
“ 032v ' I
W , VALE’T‘gg BAND

Figure 5. The energy-level diagram of AlGaAs and GaAs without taking into
consideration certain electronic processes [Prange].

l7

The AlGaAs is deliberately doped n-type, so that it has mobile electrons in its
conduction band. GaAs is typically p-type, with few holes in its valence band. The
electrons in the conduction band of AlGaAs migrate to ﬁll in these holes, but usually end
up at the bottom of the conduction band of GaAs. However, positive charge is left on the
donor impurities due to migration, which attracts these electrons back toward the
interface and bends the band in the process, forming a quantum well. This is the source of
the electric ﬁeld in this system. The transfer of electrons from AlGaAs to GaAs will
continue until the dipole layer formed from the positive donors and the negative inversion
layer is sufﬁciently strong. The dipole layer thus formed makes the Fermi level of the
GaAs equal to that of AlGaAs. The density of electrons in the quantum well is
determined by the dopant density. Figure 6 shows the energy-level diagram for the
formation of the quantum well described above.

An important part of this technique is to implant the donors in the AlGaAs
physically as far away from the interface as possible. This reduces the scattering of
electrons in the well by the positive donors and increases the mobility of the 2DES many
fold. This kind of doping is called remote or modulation doping. A reﬁnement made by
leaving a spacer layer of undoped AlGaAs between the n-AlGaAs and GaAs, increases
the separation between the electrons and the donors. This further reduces the scattering
and increases the mobility of the 2DES, but at the cost of a lower density of electrons.

High mobility is of utmost importance while studying the physics of 2DES.

18

AlGaAs I ‘ GaAs

 

CONDUCTION‘
BAND

  
  

con oucnou
BAND

 

 

VALENCE
.BAND

 

VALENCE
BANO'

 

Figure 6. Energy-level diagram of a GaAs/AlGaAs heterostrueture. The donor electrons
tunnel through the potential barrier and occupy the ﬁrst conduction sub-band of the
potential well. The Fermi level is shown as s}; The spacer layer is not shown [Prange].

l9

2.4 The Sample

The sample and the tip used in the experiment, is shown in Figure 7. The sample
is a GaAs/AlGaAs heterostrueture grown with MBE technique as described in section
2.1. The 2DES is formed at the interface of GaAs and AlGaAs. There is no ohmic contact
to the 2DES, instead there is a metallic substrate below the 2DES and separated from it
by a tunnel ban’ier. The metallic substrate is a heavily doped n-type GaAs, and the tunnel
barrier is composed of layers of undoped AlGaAs. A sinusoidal voltage is applied to the
metallic substrate, in response to which electrons tunnel into the 2DES just below the
apex of the tip. This tunneling produces an image charge on the tip, which is measured.
The details of the technique are discussed in the next Chapter. The 2DES sits
approximately 60 mm below the surface of the sample and 40 nm above the metallic
substrate. The mobilty of the sample was 105 cmZ/V-sec, and the average density was 6 x

10ll cm'z.

20

tipﬁ‘!
6 tip

 

 

 

 

 

 

 

 

* scan\
* area \~5Ilm
GaAs d1:
+ + -l- + + + +
AlGaAs 60 “m
S d2:
AlGaAs 40 nm
n+ GaAs

 

Figure 7. The sample and the tip. The typical scan area is 5pm x 5pm.

21

Chapter 3

Experimental technique and apparatus

Two-dimensional electron systems are formed in GaAs, tens of nanometers below
the surface of a heterostrueture sample. Hence, scanning tunneling microscopy (STM)
[Binnig 1982, Binnig 1999] and other techniques that probe the surface are not adequate
to probe the interior of a 2DES. Charge accumulation imaging (CAI) [Tessmer 1998,
Levi, Finkelstein] is a cryogenic technique that senses the electric ﬁeld emanating from a
sample. Hence, it can locally measure the accumulation of mobile charges within a
conducting system like the 2DES. An AC excitation is applied to the sample, and a
metallic tip is scanned over it. The charge is sensed by a cryogenic sensor constructed
from a high electron mobility transistor (HEMT) [Mimura], which is mounted very close
to the tip. The sample and the cryogenic sensor chip with the tip‘are mounted on a
modiﬁed scanning tunneling microscope, made of piezo-electric tubes. The microscope
sits at the end of a very long stick, with the help of which the microscope is dipped into a
liquid He3 cryostat, which operates at a temperature of 270 mK. The rest of this Chapter

describes in detail the CA1 technique, the scan head, the charge sensor and the cryostat.

3.1 Charge Accumulation Imaging

Charge accumulation imaging is essentially a capacitive measurement. As shown
in Figure 8, the tip and the sample can be thought of as the two plates of a parallel plate
capacitor. As a voltage is applied to the sample, an image charge is induced on the tip.

The charge sensor attached to the tip measures this image charge. This technique was

22

 

q.

T

Figure 8. Schematic of the CA1 technique. CAI is a capacitive technique in which the tip
and the 2DES can be thought of as the two plates of a capacitor, with a voltage being
applied to the 2DES that charges up the tip.

23

ﬁrst developed by S. H. Tessmer et al [Tessmer 1998]. They imaged random dr0plet
patterns that mirrored the static disorder potential of the 2DES in GaAs/AlGaAs
heterostrueture. The important difference from the earlier experiment is that, they had a
direct Ohmic contact to the 2DES, as mentioned in Chapter 2. Instead, in the
measurements described here, our sample has a metallic substrate approximately 40nm
below the 2DES. This 3D-to-2D tunneling arrangement is shown in Figure 9. An AC
excitation of 8mV at a frequency of 20KHz is applied to the metallic substrate. In
response to the AC excitation, electrons tunnel from the metallic substrate to the 2DES.
This induces an image charge on the tip, which is measured by the sensor circuit
(described in section 3.3). The advantage of this technique over its predecessor [Tessmer
1998, Levi, Finkelstein] mentioned above is signiﬁcant. If the 2DES is mostly non-
conducting, which happens if the magnetic ﬁeld is such that the 2DES is close to integer
Landau level ﬁlling, a direct contact to the 2DES would prevent the electrons from
reaching the region below the apex of the tip, and the signal would vanish. Thus the
technique is not useful in imaging of 2DES structures when its bulk is nonconducting.
The technique employed in our measurement, on the other hand, is local in the sense that
even if the 2DES is non-conducting as a whole electrons are able to enter the conducting
pockets in the 2DES by tunneling from the metallic substrate. Hence we are able to
resolve structures in the 2DES at magnetic ﬁelds, when the system is close to integer
ﬁlling, and is mostly non-conducting. Another important feature of this technique is that
we are sensitive to electron correlation effect, which can only be measured in tunneling
experiments. Ashoori et a1. [Ashoori 1990] and Eisenstein et a1. [Eisenstein] have found

that electron correlations give rise to a Coulomb psuedogap in the tunneling density of

24

 

7 3D metal

Figure 9. Schematic of the 3D-to-2D CAI technique. Here an AC excitation is applied to
the 3D metallic substrate separated from the 2DES by a tunneling barrier, instead of a
direct Ohmic contact to the 2DES. The charge tunneling into the 2DES induces an image
charge on the tip, which is measured.

25

states of the 2DES. A more detailed description and experimental results pertaining to the
psuedogap physics will be discussed in Chapter 5.

As we apply the AC excitation to the metallic substrate, the tunneling into the
2DES is not instantaneous; there is a time delay between the measured charge on the tip
and the applied AC excitation. Thus the measured signal has two components, one in
phase with the applied excitation and the other 900 out of phase with it. We call the in-
phase capacitance signal Q;,,, and the out-of—phase signal QM. Figure 10 shows the actual
tip-sample arrangement and the equivalent circuit. For simplicity, the capacitance
between the tip and 2DES can be thought of as a parallel plate capacitor Cup. Similarly
the capacitance between the 2DES and the metallic substrate is another parallel plate
capacitor CT, which is in parallel to the tunneling resistance RT. These form an RC circuit.
The plot of Q;,, and Q0“, with respect to the logarithm of the applied frequency f or the
tunneling resistance RT is shown in Figure 11.

The plots can be understood qualitatively as follows. For low frequencies, the charge has
enough time (compared to the time constant of the RC circuit) to tunnel into the 2DES
and thus stays in phase with the applied AC excitation. Hence the in-phase signal, which
is purely capacitive, dominates, while the out-of-phase signal is almost zero. As the
frequency is increased, the excitation period becomes comparable to the RC time
constant, and the measured signal goes out of phase with the applied excitation. Hence
the signal shows up mostly in the QW, which peaks at a certain value of the frequency f0,
while the Q" rolls off. AS the frequency is further increased, the excitation period is very
short compared to RC and the charge has almost no time to tunnel into the 2DES. Hence

both the in-phase and the out-of-phase signal go down.

26

tip
Ctip

 

 
 
 
 

+++++

 

 

 

 

AlGaAs /2D
GaAs CT— RT
AlGaAs \
3D
n+ GaAs -3D

Figure 10. The circuit equivalent to the tip sample arrangement in the 3D-to-2D CA1
technique.

27

 

 

 

 

 

 

 

-2 { log(f/fo) } 2
'°Q(RT/Ro)
Figure 11. The in-phase and the out-of-phase charging as a function of logarithmic

frequency and tunneling resistance, as obtained from the model circuit. The dashed
curves are the charging when in-plane conductivity is taken into consideration.

28

A Similar argument holds for the in-phase and out-of-phase signal versus the
tunneling resistance. Thus the charging components have identical functional dependence
on the frequency f and the tunneling resistance RT, with characteristic values of

1 l
= and R0 =
MAC} +C,.,.) 27m, + C...)

 

 

respectively. The magnitude of the curves

f0

is set by the tip-sample capacitance difference AC between a fully charging and locally
non-charging 2DES. More rigorous modeling [Maasilta, personal communications] for
the system considers the distributed nature of the tip-sample capacitance and the effect of
charge motion within the 2D plane. Such models show an enhancement in the charging
characteristics as the in-plane relaxation rates approach the tunneling rate, an effect that
may occur near integer ﬁlling, as indicated qualitatively by the dashed curves. Figure 12
shows a plot of the differential change of the in-phase and the out-of-phase components
with respect to the tunneling resistance versus frequency of the applied excitation. As can
be seen, the out-of-phase signal is more sensitive to the change in tunneling resistance
than the in-phase, at low frequencies. In Chapter 5, we shall apply this model of the tip

and sample circuit to explain some of our experimental observations.

29

 

 

0

 

 

Figure 12. The differential change in the in-phase and the out-of-phase charging with
respect to the change in the tunneling resistance plotted against the frequency. At the low

frequency limit, the out-of-phase component is more sensitive to changes in tunneling
resistance.

30

3.2 The scan head

The goal of a scanning tunneling microscope (STM) [Binnig 1982, Binnig 1999]
is usually to get atomic resolution surface topography or high quality spectroscopy.
Achieving that goal would require very high stability of the scan head, which leads to
small piezo lengths and hence small scan ranges. The physics that was studied in our
experiment not only requires hours of stable operation, but also a micron-scale scan
range. Hence conventional STM is not suitable for our experiment. It is actually a
challenge to build a piezo-electric based microscope with high stability, without
compromising on its scan range. Since we use a direct immersion He3 cryostat to cool
down the sample, there is an additional issue. In conventional Besocke (see Figure 13)
design [Besocke], the sample rests on top of piezo tubes. Therefore using long piezo
tubes to achieve big scan ranges not only compromises the stability, but also keeps the
sample at a high position relative to the bottom of the scan head. As a result, for direct
immerSion in liquid He3, the liquid level may soon drop down below the level of the
sample, causing the sample to warm up before any meaningful data can be acquired.
Alternatively, a scheme [Tessmer 1998] in which the sample Sits relatively low on the
scan head, with a long scanning piezo coming from the top as shown in Figure 14, seems
to be a good idea. The drawback of this design is its thermal instability. Since the thermal
contraction of a metal body is much greater than that of the piezo-electric tubes,
temperature changes can cause the tip to crash into the sample. Even minute thermal
instabilities at the base temperature can cause signiﬁcant drift in tip-sample separation.

To fulﬁll all the three requirements, namely high thermal stability, high scan

range and sample sitting as close to the bottom of ascan head as possible, we have

31

 

Figure 13. Schematic of the Besocke design. The sample sits on top of the three carrier
piezo tubes, which along with the scanning tube is attached to a bottom plate. The ramps
face the same side as the sample.

32

 

$ , xix .» ' body

scanning
‘ piezo

 

 

 

 

 

 

 

 

\RS\\\‘R?:~N\V:~\\\NQ~R\
\p
m
m
3
'9.
(D

 

 

 

 

 

 

 

 

 

(I) piezos

Figure 14. Schematic of a microscope design utilized by few experiments. The thermal-
contraction mismatch between the metal body (hatched) and piezoelectric tubes
represents a major drawback. Essentially, in response to temperature ﬂuctuations, the
body expands and contracts much more than the piezo ceramic. These uncompensated
ﬂuctuations in height h result in signiﬁcant variations of the tip-sample separation.

33

designed and built a novel scan head [Urazhdin 2000]. A schematic of the design is
shown in Figure 15, and the actual microscope is shown in Figure 16. The novelty of the
design is that both the three piezo tubes holding the sample and the one scan tube are
soldered to a top plate. This essentially solves two of the three problems — thermal
instability and He3 level dropping below the sample too soon. The sample and the tip are
attached to identical piezo tubes. Thermal contraction of the three carrier piezo tubes,
which tend to move the sample towards the tip, is compensated by the tip moving
upwards, away from the sample as the scanning tube to which the tip is attached contracts
by the same amount. Thus temperature changes do not affect the tip-sample separation.
Also, as seen from the schematic, the sample sits almost at the bottom of the microscope;
hence the microscOpe only warms up when the sample space in the cryostat runs out of
liquid He3. Lastly high scan range can be achieved by having all the piezo tubes very
long, without compromising on any of the above requirements. The piezo tubes are 7.6
cm long and are soldered to a brass base plate, which is rigidly attached to a very long
stainless steel stick. The sample rests on the small feet that are part of the cylindrical legs,
as indicated in Figure 15(a). A mechanism inside the legs allows for the rotation of each
supporting foot by 180°. This allows for the brass displacer, used for screening electric
ﬁeld of the piezo tubes, to be installed and removed. Each leg consists of a piston rigidly
attached to the foot, and a small Be-Cu compression spring. The resulting upward
pressure on the piston presses the supporting piece against the leg, without limiting its
rotational motion. This is shown in Figure 15(b). A groove made in the leg ﬁxes the piece
in two positions, directed inward or outward. The sample holder diameter is slightly

bigger than the opening between two legs. Hence two out of the three legs are notched

34

(a) (b)

top-loading ‘ ‘ PISIO"
probe _ Be—Cu
spﬁng

 

 

 

 

 

 

 

 

// ’4 {scanning

(C) sample support
with ramps

  
  

 

 
 
    

positioning
screw

 

 

 

 

 

 

 

 

 

 

_\clamping screw

Figure 15. (a) Schematic of scanning head. Both the tip and the sample are supported by
identical piezoelectric tubes (dimensions: 7.6cm-long x 4.8mm-diam x 0.8 mm wall
thickness). As a result the unit is nearly perfectly thermally compensated. The displacer,
outlined in dashes, screens the tip from the electric ﬁelds of the piezo tubes. All the metal
parts are made of brass, shown hatched. (b) Sketch of support leg showing the rotating
foot assembly. (c) Bottom view of the microscope. The clamping screws ﬁx the ramps to
allow the positioning screw to be turned—the sample is attached to the other side of this
screw. In this way, we can adjust the vertical position of the sample by as much as 1 cm.
The diameter of the unit is 3.8 cm.

35

 

Figure 16. Photograph of the actual microscope. The shiny ruby balls at the end of each
foot help reduce friction during ﬁne approach of the sample.

36

longitudinally, and the sample is mounted through the gap between them.

Figure 15(c) shows a bottom view of the assembly. The sample holder is similar
in design to the Besocke ramps (shown in Figure 13) [Besocke] except that the sloped
ramps and the sample are situated on opposite sides. We chose a slope of 2°, yielding a
coarse vertical positioning range of 0.4 mm. Moreover, the sample sits on a positioning
screw that allows manual vertical positioning in the range of almost 1 cm. This is
important in preventing the tip from being accidentally crashed during the insertion of the
sample. In practice, the sample is typically 0.5 cm from the tip after initially inserting the
sample holder. Of course, before we can turn the positioning screw to bring it closer, the
position of the ramps must be ﬁxed. This is accomplished by pinching the sample holder
from the sides using the three clamping screws, shown in Figure 15(c). The clamping
screws are attached to the brass displacer that also serves to screen the electric ﬁeld
generated by operating the piezo tubes. After the clamping screws are removed, an end
cap is put in place. The interior of this cap provides a smooth surface for contact with the
sides of the sample holder. We ensure low friction by applying MoSz powder to the
interior surface of the cap in addition to the sides and the ramps of the sample holder. The
microscope is cooled by slowly lowering the probe into the cryostat -- achieving a
cooling rate of l K/min. There is little risk of mechanical vibrations causing a crash
during this procedure, as gravity tends to pull the sample away from the tip. Finally, the
ﬁne sample approach is achieved by the standard inertial walking procedure utilized in
Besocke-type designs. We ﬁnd our scanning piezo tube (7.6-cm—long x 4.8-mm-diam x
0.8 mm wall thickness) to have a sensitivity of 1300 AN laterally and 73A/V vertically

at liquid-helium temperatures. Using commercially purchased electronics, which provide

37

a voltage range of $130 V, we thus achieve an impressive cryogenic lateral scan range of
34 pm. The same large range of motion of the carrier piezo tubes also enables us to do
the ﬁne sample approach very quickly. Typically, the full range of the ramps is traversed
in approximately 2 min at liquid-helium temperatures. We ﬁnd the microscope to be
sufﬁciently stable to function without a low-temperature vibration isolation stage. With
regard to the entire cryostat, we have taken care to minimize building vibrations using a
bungee cord suspension system. Mechanical pumps, which are part of the cryogenic
system, represent a major source of additional vibrations. We have decoupled those
modes by using tubing that has a section made of ﬂexible rubber, connected to a section
of stainless steel. That produces an impedance mismatch so that part of the vibrations is
reﬂected at the junction [Movsovich]. We have also connected the tube rigidly to a heavy
lead block at one point and immersed it into a sandbox at another.

Figure 17(a) Shows a 35 um x 35pm topographic image of a platinum sample with
a grid of 5pm x 5pm pits that are 1800A deep, obtained in tunneling mode with the
microscope at room temperature. This is how we calibrate the piezo tubes of our
microscope. The 35 um scan range is approximately the maximum range at liquid-helium
temperatures. Figure 17(b) shows line scans across a single pit measured in both
tunneling and capacitance modes. For the capacitance scan no feedback was used; the tip
was simply scanned above the sample in a straight line. The tunneling mode clearly has
better resolution, comparing the sharpness of the walls. This is natural because tunneling

takes place only at the apex of the tip on atomic scale, whereas the resolution in the

38

 

 

 

 

 

0
X0111)

Figurel7. (a) A 35 mm x 35 um topographical image of a platinum sample with a grid of
5 pm x 5 pm pits that are 180 nm deep, obtained at room temperature. (b) Cross section
across a single pit measured both in the tunneling mode (solid line) and in capacitance
mode (dashed line).

39

capacitance mode is determined by distance between the tip and the sample and by the
radius of curvature of the tip’s apex, ~50nm for our chemically etched tips. Nevertheless,
Figure 17(b) also shows that the capacitance mode can give us more information than just
the topography, even for simple metallic samples. As is clearly visible, capacitance has
local maxima at the edges of the pits, as expected from electrostatics (because electric

ﬁeld is higher around sharp comers).

3.3 The cryogenic charge sensor

Bipolar junction transistors (BJTS) are useless for low temperature applications
because at temperatures below 100K there are almost no charge carriers in the conduction
band to carry current. Field effect transistors (FETS) perform better than BJTS at low
temperatures, but the power dissipated by such devices is typically in the mW range,
which is too large a heat load for He3 based cryostats. High-electron mobility transistors
(HEMTS) [Mimura] are GaAs/AlGaAs based modulation doped heterostrueture devices,
which are well suited for cryogenic operations. These are very similar to the commonly
used FETS, except that the source-drain channel is a two-dimensional electron system,
just like the system studied in the experiment. Since the 2DES is the channel for the
HEMTS, they do not have to depend on thermally activated carriers like the BJTS, and
hence do not suffer carrier freeze out at low temperatures. Moreover, since the only
source of scattering for its charge carriers is the dopant atoms, as temperature is lowered,
these dopants freeze out, increasing the mobility of the charge carriers. These devices

can be operated in the Ohmic region, far away from saturation, reducing the power

40

dissipation by 2-3 orders of magnitude compared to saturation mode operation [Urazhdin
2002]

The schematic of the cryogenic charge sensor circuit is shown in Figure 18. It
consists of two HEMTS. The one on the right is the measurement transistor. The gate of
the measurement transistor is biased through the HEMT on the left. The bias HEMT
simply acts as a very high resistor, preventing the charge induced on the tip from going to
the left. The standard capacitor Cs is used to subtract away the stray capacitance, arising
mostly out of the electric ﬁelds from the 2DES not terminating at the very apex of the tip.
This is accomplished by applying an AC excitation to C3, of almost the same amplitude
but exactly 1800 out of phase with the applied sample excitation.

The cross-sectional structure of a HEMT is shown in Figure 19. The epilayers
consisting of undoped GaAs, Si-doped n-type Alea|-xAs and n-type GaAs are grown on
a (100) semi-insulating GaAs substrate by molecular-beam epitaxy (MBE). Some
epilayers are grown with a 6 nm-thick undoped Alea;-xAs spacer layer inserted between
the undoped GaAs and n-AleaHAs layers to enhance electron mobility. The AlAs mole
fraction, denoted as x, is usually about 0.3.

The input capacitance C,-,, is the capacitance between the gate and the source-drain

channel of the HEMTs. It is very important to have a small C,-,,, since the voltage

measured by the lock-in ampliﬁer VmaS is given by Vmeas =G Vin M. G is the gain

in
of the circuit, which is usually ~l, V,-,, is the input voltage, which in our case is 8mV, and
Cmeas and Cm are the measured and input capacitance respectively. Thus low input

capacitance means higher sensitivity. The input capacitance of the HEMT is 0.3pF

41

 

HI-

 

 

 

—ll

 

MI...
a;

{A

 

HEMT
_I-

‘1

——-)To LocIrJn

V

in

 

 

 

 

I;

Ime'hlbiémh—

 

Figure 18. The cryogenic charge sensor circuit. CTs is the tip-sample capacitance, Cs is
the standard capacitor and Vm is the rrns value of the applied AC excitation to the sample.

42

Source Gate Drain
/ Schottky contact lTi-Pt-Au)

-~-='.--:-=-..-.-:«—Ohmic contact lAu-Ge/Au)

 

 

 

n-Aleo.-xAs

 

 

 

 

l

Undoped GaAs \\
" Electron layer (channel)

Semi-insulating
(IOOIGoAs substrate

Figure 19. Typical cross section of a HEMT [Mimura].

43

ultimately allowing a charge sensitivity of 0.01 electrons/VH2. The charge sensitivity
number comes from the Johnson noise. If Cm is too large, VmeaS is reduced and other
sources of noise will dominate. This will lead to a decrease in the sensitivity. The drain
current —voltage characteristic curves for different gate voltages of a typical HEMT are
shown in Figure 20. Before we start our experiment, we look for the characteristic fan-
shaped family of curves for our measurement HEMT on a curve tracer, to make sure the

HEMT is working.

44

 

 

   
 
    

 

 

 

 

 

 

 

so
4
E sz- 0.7V
33° 30
S
g ,_I oev
8
C 20L
E — 05v
0
l0 0.4 V
__ 0.3 v
02v
0 0.5 1.0 1.5 2.0 2.5 so
DraIn Voltage VDSW)

Figure 20. Drain current- voltage characteristics, at different gate voltages VGS, of a
typical HEMT [Mimura].

45

3.4 The cryostat

The scan head with the sensor chip and the sample is attached to the end of a long
stainless steel stick. The stick is then mounted on a top loading He3 cryostat, and is
inserted slowly into the cryostat. The cryostat has a superconducting solenoid
surrounding the sample space, with which a magnetic ﬁeld up to 12 T can be achieved.
The arrangement is shown schematically in Figure 21. The operation of the H63 cryostat
will be described brieﬂy in this section.

The principle of a He3 cryostat is very simple. Gaseous He3 is liqueﬁed and is
pumped upon to reach the base temperature of 270mK. The whole cryostat is cooled in
stages from room temperature to liquid He4 temperature of 4.2K. It is usually maintained
at this temperature for months, by ﬁlling it up with liquid He4. The procedure for cooling
down from 4.2K to the base temperature is as follows. Just outside the sample space and
attached to its outer wall is a small pot called the 1K pot. The 1K pot is ﬁlled with liquid
He4 and pumped upon to maintain it at around 1.2K. The He3 stored in a storage cylinder
in gaseous form is allowed to enter the sample space, and starts to liquefy around the 1K
pot and drips down into the bottom of the sample space. The sample space has a big
charcoal adsorption pump, also called the sorb, which starts pumping on liquid He3,
when the adsorption pump is below 10K. It is usually maintained at around 5K by using a
He4 heat exchanger. The base temperature of 270mK is reached by pumping on the
liquid He3 by the adsorption pump. The sample space remains at the base temperature for
about 96 hours, until it runs out of liquid He3. At this point virtually all the He3
molecules are stuck to the adsorption pump, but they are recycled for subsequent data

11.1115.

46

scanning
head

  
   

cryostat

Figure 21. The top loading He3 cryostat with the microscope directly immersed in liquid
He3. The microscope with the sample and the sensor chip is attached to a long stainless
steel stick, which is used to insert the microscope inside the cryostat.

47

Inevitably the cryostat is allowed to warm up to room temperature from time to
time. It must then be re-cooled from room temperature to 4.2K in an efﬁcient way. The
procedure for doing so is as follows. The cryostat has an outer vacuum chamber, which is
ﬁrst pumped down to pressures of 2-3 x 10'6 torr, and remains under vacuum throughout
the cryostat operation. The cryostat also has another vacuum chamber called the inner
vacuum chamber (IVC), between the main bath, which holds the liquid H64, and the
sample space where the microscope is inserted. It is ﬁrst evacuated of air and a tiny bit of
dry nitrogen gas is introduced, to act as an exchange gas for the cool down. Liquid
nitrogen is transferred very slowly into the main bath, to cool down the cryostat along
with the magnet to 77K. The liquid is kept overnight, and is drained the next day. The
IVC is evacuated once again and is ﬁlled up with very little dry helium gas. The helium
gas acts as an exchange gas during liquid He4 transfer, which is done right after
removing the liquid nitrogen, to cool down the cryostat to 4.2K. Finally the IVC is

evacuated to 2-3 x 10'6 as well.

48

Chapter 4

Imaging disorder in a 2DES

The integer quantum Hall effects (IQHEs) [Prange, von Klitzing] and the
fractional quantum Hall effects (FQHES) [Tsui, Laughlin1983] are among the most
remarkable phenomena to be discovered in solid-state physics in the last two decades. It
is well known that disorder plays a major role in the formation of localized states in the
Landau levels (LLs), the existence of which is ultimately responsible for the quantum
Hall plateaus. Disorder in the GaAs heterostruetures is due mostly to the randomly
situated ionized donor atoms between the quantum well and the surface. Studying
disorder in the two dimensional electron systems (2DES) is thus an important step
towards understanding the quantum Hall states and other exciting phenomena such as
metal-insulator transitions [Das Sarma], skyrmions [Das Sarma], and the predicted charge
density waves (CDWS) [Koulakov, Moessner]. We use the charge accumulation imaging
technique, to image the potential due to the random distribution of the ionized donor
atoms at zero magnetic ﬁeld. We observe micron-scale features, consistent with the

previous work of Tessmer et a1 [Tessmer 1998] and F inkelstein et al [F inkelstein].

4.1 The simplest measurement

The simplest measurement that can be performed with the CA1 technique is
shown in Figure 22. Until now we have assumed implicitly that the tip is only a
measurement device, and we do the capacitance measurements non-invasively. But we

can also study how the tip can perturb the 2DES locally by applying a DC bias voltage.

49

Figure 22 shows the tip-2DES in-phase capacitance signal Q;,. as a function of the DC tip
voltage Vtip at a temperature of 290mK. In this measurement, the tip was ﬁxed over a
certain place in the sample, and its DC voltage was varied while the corresponding in-
phase capacitance was measured. The capacitance curve shows a clear step feature. The
step feature can be easily explained. Initially when Vtip is positive, the capacitance
measured is between the tip and the fully charged 2DES, which is the case at the right
Side of the plot in Figure 22. AS we decrease the tip voltage, it starts depleting the 2DES.
At around a tip-voltage of ~ -0.25V the 2DES is completely depleted, and the signal
drops abruptly, forming the step. The measured capacitance now is that between the tip
and the 3D metallic substrate. Thus the 12aF step height is the difference in capacitance
between a fully charged 2DES layer and the tip, and the back electrode and the tip —
where the 2DES is locally depleted. As we decrease the tip bias voltage further the
depleted hole in the 2DES grows in size. The voltage Vnuu shown in the plot is the work
function difference between the tip material, which is an alloy of platinum and iridium,
and the GaAs sample — called the contact potential. It is determined by a technique called
the Kelvin probe method. We set Vup= Vnu“ when we want to acquire data non-

invasively.

50

 

I ] I l I I I

.azm-tsgazsaatww
: .. T"? .O_ ‘i
l . . r, -,
72. ;: f} if

5.."

 

 

 

 

 

 

Vtip(v)

Figure 22. Dependence of the in-phase capacitance signal Q“ on the tip dc bias voltage.
The step feature of approximately 12aF marks the potential where the 2DES becomes
depleted locally. The data were acquired at 290mK.

51

4.2 Imaging disorder

While describing the sample in Chapter 2, it was mentioned that the donor atoms
in the AlGaAs give up their electrons, which migrated to the quantum well to form the
2DES. Although the modulation doping technique reduces the scattering, the ionized
donors still cause disorder within the 2DES. Also in the quantum Hall regime, the width
of the Landau levels comes from the ﬂuctuations of the density of charged donors. Hence
it is very important to know the distribution of these charged donor atoms in the dopant
layer in AlGaAs.

We would like to image the disorder potential due to these ionized donors
[Chakraborty]. To test that CAI works in imaging this potential, we can intentionally
introduce disorder by rearranging the ionized donors at a certain place in the sample.
Figure 23 Shows the image of such a place. The disorder was introduced by applying a
large negative voltage (~ -10V) to the tip when it was very close to the sample. The large
negative voltage rearranges the charge among the ionized donors in the dopant layer, so
that the positively charged ions are accumulated below the apex and the surrounding
region of the tip. The large voltage was then removed, and the image was obtained by
scanning the tip over the sample at zero effective potential (Vup=Vnun). The lighter region
in the image corresponds to greater electron accumulation. Therefore as evident from the
ﬁgure, there was electron accumulation in the 2DES below the region where we
introduced the excess positive charge. This data was taken at 290 mK.

Now to image the intrinsic disorder potential we scan the tip while the tip is

depleting the 2DES. Also, we move to a different location away from the region where

52

1 OF

 

egn-phase ChargeuF)

3.51.01.51.25”
distancemm)

 

Figure 23. (a) A 3 pm x 3 pm image of an intentionally created disorder to test the CA1
technique in imaging disorder. Bright regions imply greater electron accumulation. (b) A
line scan through the disk of electron accumulation shows the capacitance at the center of
the bright disk is approximately 100 aF more than the dark surrounding area.

53

   

v,,= 0.50v v..,= 0.2ov

 

         
 

' v,= 0.05v v..,,= -0.1ov

   

v..,= '0-40V vrp= -0.55v v,,,= -o.7ov

Figure 24. A series of 3pm x 3pm images with decreasing dc tip bias voltage starting
from 0.5V, which is approximately equal to Vnu", to —0.6V in decrements of 0.15. The
features coalesce and the images get darker as the tip depletes the 2DES. The difference
in capacitance from the brightest to the darkest region is approximately 500aF.

54

we intentionally introduced disorder. The scan images are shown in Figure 24. We start
with the tip at 0.5V, and we make the tip voltage more and more negative with every scan
image. As can be seen from the ﬁgures, the tip depletes the 2DES. The micron size
features exhibit reduced charging, which is displayed as darker shades and coalesce. Thus
the ﬁnal image at tip voltage —0.6V is very dark throughout. There is an important
distinction in this measurement from the previous one, which is that, this —0.6V is not
large enough to rearrange charge in the donor layer. In this measurement the depletion is
indeed caused only by the tip. A depleted “bubble” follows below the tip’s apex as it is
scanned. This experiment then is the scanning version of the data shown in Figure 22,
except that these images were taken at about 1 °K.

In order to extract the distribution of the ionized donors, we need to process these images.
Particularly, we need to get rid of the topography contribution which couples with the in-
phase charging Q1", but has nothing to do with the signal from the 2DES. Also it is
important to set the gray scale to highlight the relative changes in charging for each
image. The former was achieved by subtracting the image containing the most
topography from the rest. The image that would have the most topography is easy to ﬁnd.
The structure in the image for which Vtip=Vnun should be exclusively due to topography,
and hence we can consider this to represent the purely topographical contribution. In this
case, the topographic image is the very ﬁrst image in Figure 24, for which the Vup= 0.5V.
The latter was done by line-by-line averaging. Line-by—line averaging means Shifting
each line of an image so that its average value is zero. Naturally the average value of the

entire scan is also zero, so that we have removed the offset due to the depleting voltage.

55

 

”guru

Figure 25. A series of 3m x 3pm images processed from images in ﬁgure 24.
Topographical contribution has been subtracted. The images show intriguing micron-
scale features. The middle image is enlarged, and the micron-scale features are shown
with circles in the enlarged image.

56

Moreover, this line-by-line subtraction serves to correct for tip-sample drift effects. The
result is shown in Figure 25. Not all the processed images are shown since they are
mostly similar. We interpret the features in these processed images to reﬂect the
distribution of the ionized donors in the sample. The most intriguing part of the images is
the micron—size features. According to theorists [Efros 1988, Efros 1999, Shklovskii], the
amplitude of potential ﬂuctuations in the donor layer is expected to vary as the square
root of the donor density, and is assumed to be no larger than the donor layer-2DES
distance. Turning our attention to the spatial distribution of the donors, it is widely
believed that the MBE growth results in a truly random distribution. The average inter-
donor distance in our case turns out to be approximately 10 nm, which is calculated from
the dopant density of 1018 cm”, bulk doped over an AlGaAs thickness of 10 nm. This
number is smaller than the donor layer-2DES distance of 20 nm, which is the key length
scale. With regard to larger length scales, a random distribution means that all other
length scales larger than the donor layer-2DES distance should be equally present. Hence
present theories do not explain any mechanism that would lead to the distribution of
ionized donors into these micron-sized islands. Tessmer et al [Tessmer 1998] and
Finkelstein et al [Finkelstein] have observed similar micron-scale features using a
technique, which employed a magnetic ﬁeld to put the 2DES near integer ﬁlling. In their
paper Finkelstein et al argue that screening by residual electrons in the donor layer
smoothes the potential to micron lengths. But, any screening that might be present due to
residual electrons in the donor layer should preferentially smooth out the larger length
scales rather than the smaller ones. Hence we do not believe screening actually explains

the micron length scales. It is possible that the micron-scale length scale is somehow tied

57

with the growth mechanism, although existing theories predict otherwise [Efros 1988,

Efros 1999, Shklovskii].

58

Chapter 5

Charge Density Waves: Theory and Experiment

Two-dimensional electron systems formed in GaAs/AlGaAs heterostruetures
represent an ideal laboratory to study many—particle physics in lower dimensions. Under
various conditions, electron-electron interactions may result in ordered inhomogeneous
states. The Wigner crystal represents a classic example, arising from direct Coulomb
repulsion at sufﬁciently low density, temperature, and disorder. In an applied
perpendicular magnetic ﬁeld, more general charge density wave (CDW) [Koulakov]
ground states are possible. At sufﬁciently high Landau level ﬁlling, Hartree-Fock based
theories predict a CDW ground state of length scale on the order of the cyclotron radius,
resulting from the competition between Coulomb repulsion and exchange attraction
[Koulakov, Moessner]. Recent transport measurements of high-mobility samples show
that highly anisotropic dissipation occurs for more than four ﬁlled spin-split Landau
levels, with the uppermost level near half ﬁlling [Lilly]. Unidirectional CDW stripes are
believed to lie at the heart of these observations. In contrast to transport measurements,
scanned probe techniques sensitive to electric ﬁelds can provide direct images of the
GaAs/AlGaAs electronic structure. In this Chapter, I shall present a detailed theory of
the formation of the CDWs, and discuss our effort in trying to resolve them with our CAI

technique.

5.1 Theory of CDWs
Prior to 1982, physicists believed that the ground state of the 2DES in a

perpendicular magnetic ﬁeld was that of charge density waves (CDWS). But with the

59

discovery of the fractional quantum Hall effect (FQHE), the CDW picture, which does
not exhibit FQHE, was replaced by the ad hoc Laughlin liquid picture at the lowest
Landau levels (N=0,l). Laughlin liquid [Laughlin 1983] is a uniform incompressible
quantum state. In the 19903 it became increasingly evident, both from calculations and
experiments, that FQHE was absent for N>l. In 1996, Koulakov, Fogler, Shklovskii
[Koulakov, Fogler 1996] and Moessner and Chalker [Moessner] independently proposed
an alternative to the Laughlin liquid picture. The Laughlin liquid ground state was still
lower in energy than the conventional Wigner crystal [Wigner] for N >1. But Koulakov,
F ogler, Shklovskii, based on Hatree-Fock calculations, predicted that the ground state of
a couple of CDW phases do actually have lower energies than the Laughlin liquid ground
state for N >1. There has been a growing body of experimental evidence [Cooper, Lilly],
albeit indirect, for the existence of at least one of the two predicted CDW phases. In the
next few paragraphs, I shall present the essential elements of this CDW picture and show
that for N >l CDWs do win over the Laughlin liquid and Wigner crystal.

Before we begin let us deﬁne some of the important length scales and parameters.

For a 2DES in a perpendicular magnetic ﬁeld B, the spatial extent of the wavefunction of

an electron is of the order of the magnetic length 3: 1,—2— . Another relevant length scale
e

is the classical cyclotron radius RC given by k; [2, where k; is the Fermi wave vector.
Other parameters of importance are vN = v - 2N, where v is the spin-split ﬁlling factor,
and N is the number of completely ﬁlled Landau levels. The factor 2 comes from the spin

splitting of the Landau levels.

60

Assuming Wigner crystal (WC) [Wigner] to be the ground state of the 2DES in a
perpendicular magnetic ﬁeld, one can calculate the cohesive energy of the WC for the
case when vN is not too small (vN >> l/N), i.e., when thecyclotron orbits at neighboring
lattice sites may overlap. For the case when vN is indeed small, the concept of WC is
natural. Cohesive energy is deﬁned as the energy per particle at the upper Landau level
with respect to that in the uncorrelated electron liquid (UEL) of the same density. The

WC cohesive energy for not too small vN is given by [Koulakov]

 

 

 

(5.1)

::;2:_ hare £+iln(NvN) _ l—vN ha) ln(Nrs).
167W rs 27r 2

The factor rs is deﬁned by r, = \[2 /kF a3, where a3 is the effective Bohr radius. In realistic
samples r, ~ 1. Let us now compare the cohesive energy of the CDW state, in the same

regime, i.e., for vN >> l/N. It is given by [Koulakov]

 

 

, 0.3 1— v , ln(Nr ) (5.2)
Ewwz— v rho) ln1+—— -— Aha) ‘ .
f( .I . . [ . j 2 . ,NH

The function f (vN) is proportional to vN for UN << vN << ‘/2. The second term in both
cases is the same, and one has only to compare the ﬁrst terms. So in the regime when vN
>> l/N, the ﬁrst term for the CDW case is more negative than the WC case and thus the
CDW state has lower energy and wins over the WC state. The plot in Figure 26 Shows the
qualitative behavior of the WC and the CDW energies. Only the ﬁrst terms of the
equations (5.1) and (5.2) are plotted in units of hwc. The plot shows the transition of the
system from WC to CDW phase. Here we have used f (vN) = 0.15vN, N = 5 and r, = 1.
For the Laughlin liquid state, comparing the cohesive energy with the CDW state,

it is not very obvious from the expressions that the CDW state wins over the Laughlin

61

0.000 I
-0.002

8" -o.004 '
5

3 aces

-0.008 /

-0.010 CDW
-0.012

-0.014
0.0 0.1 0.2 . 0.3

VN

WC

I l l l I

Relative Ener

 

 

Figure 26. Plot of relative energies (only ﬁrst terms in equations (5.1) and (5.2)) of
Wigner crystal (WC) and charge density waves (CDW) in units of hwc against vN. In this
example, for approximately vN > 0.1 the CDW has lower energy than WC.

62

state for N > 1. But the cohesive energies can be calculated numerically, and in the
following table we present the results for rs= J2 . From the table one can see that for vN
= 1/3 and N = O and 1, the Laughlin liquid is lower in energy. However for N > 1 CDW

wins. The same is true for vN = 1/5 and N > 2.

 

 

 

 

‘3]:1/3 L CDW
"" CDW
N M E UEL Eco}: Em}! 63/1320}:
0 1 -0.1206 -0.1159(1) -O.1037 -11.8%
I l -0.l297 —O.1519(3) --O.l424 -6.7%
2 2 -0.1136 -0.1141(3) -0.1188 4.0%
3 3 -0.1034 —0.0946(3) -0.1018 7.1%
4 4 -0.0965 -0.0824(3) -0.0896 8.0%
5 5 -0.0914 -0.0733(3) -0.0805 8.9‘3/o
V =I/5
~ N r. CDW
N M E UEL Eco}: Ecol: 515/53}? W
0 1 -0.0560 -0.0903(2) -0.0880 -2.6%
l 1 -0.0765 —O.l 72 7(7) -0.1692 -1104,
2 1 -0.0677 -0. 1420(9) —0. 1396 - 1.7%
3 2 -0.0618 -0.1 139(9) -O.1202 5.29/0
4 2 —0.0577 -0.096319) -O.1050 8.39211
5 3 -0.0547 -0.0849( 9) -0.0946 10.3%

 

 

 

L and the CDWECDW for (ﬁx/2.

coh coh

Table 1. The cohesive energies of the Laughlin liquid E

M is the optimal number of electrons per CDW bubble given approximately by 3vNN.
The energy unit is ﬁrm. The energy per electron in the uniform uncorrelated state EUEL is
also provided for reference [Fogler 1997].

Having established theoretically that CDW states for higher Landau levels (N >1)
are preferred, let us try to understand the physics intuitively. It was brieﬂy discussed in
Chapter 1 (Section 1.2) that there was a competition between the long-range Coulomb

repulsion and short-range (~E ) exchange attraction between the electrons, which led to

63

the formation of clusters of electrons, called “bubbles” and “stripes” with a periodicity of
~3RC. But this does not answer the question of why there is no CDW for low Landau
level ﬁlling (N = 0 and 1). The answer lies in the all important length scale RC, the
cyclotron radius, which appears in the exponent of the CDW wavefunction. The

wavefunction of a single bubble with M electrons at the lowest Landau level is given by

 

M Ia-I 2
LI’0{r,(}=1_[(z,—zj)xexp ”ZkF . (5.3)
K} (:1 4R(.'

Here 2,- = x, + iyj- is the complex co-ordinate of the jth electron. The wavefunction of a

bubble at the Nth Landau level centered at R can be constructed from the above

wavefunction as

 

91011:“

.5 W

exp[b—'+—§—\/:§b'—R:] ‘I’O {rk }, (5-4)

where at? is the inter-ladder operator, raising the ith electron to the next Landau level, and
bf is the intra-ladder Operator. Finally, the wavefunction of the CDW, with an anti-

symmetric combination of bubbles centered at the triangular lattice sites R1 is given by,
WC!)W =ngn(P)l—I LII/1P(rk) 1 (5.5)
P I

P’s are the permutations of electrons between bubbles.

The intuitive reason for the absence of CDW for low Landau level ﬁlling (N = 0
and 1) is as follows. In all these arguments we are talking only about electrons in the
upper partially ﬁlled Landau level. Laughlin liquid can be thought of as CDW state
melted by zero point vibrations. The melting can occur only if the amplitude of the zero
point vibrations is comparable to the lattice constant. This amplitude never exceeds the

magnetic length 6 . There is an important difference between low and high Landau level

64

ﬁlling. At low Landau level ﬁlling, Coulomb repulsion dominates, and the electrons can
lower their energy by staying as far apart as possible. Hence the CDW contains only one
electron per unit cell. The lattice constant is of the order of RC, and it is small at low
Landau level ﬁlling, since RC is inversely proportional to the ﬁeld. The lattice constant
decreases with increasing ﬁlling factor, since now the system accommodates more
electrons in the same amount of space. At some value of vN, the lattice constant becomes
of the order of f , the spatial extent of the electron wavefunction in a perpendicular
magnetic ﬁeld, and the crystal melts into a Laughlin liquid. At higher Landau level ﬁlling
though, increasing the ﬁlling factor, adds more electrons to the lattice sites of the CDW
crystal, so that the lattice constant does not change much. The lattice constant, which is of
the order of RC, is much greater than 6 , at higher Landau level ﬁlling. Therefore it is
unlikely that at higher Landau level ﬁlling the CDW would be melted by the zero point
ﬂuctuations.

The CDW state in equation (5.5) forms patterns shown in Figure 27. Each black
dot in the ﬁgure represents the guiding center of the cyclotron orbits. An enlarged view of
a “bubble” is shown in Figure 27(c). The dark region is the guiding centers of the
cyclotron orbits of the electrons in the bubble, while the toroid is the charge density
distribution. The aggregation of many particles into large domains helps the system
achieve a lower value of the exchange energy, while the Coulomb repulsion keeps the
aggregates separate. The actual charge density variation of the uppermost Landau level is
of the order of 20%. At higher Landau level ﬁlling, when the system is very close to
integer ﬁlling (vN ~ l/N), each bubble consists of one electron. As we move away from

integer ﬁlling more electrons are added to the lattice sites forming the bubbles. This

65

          

 

 

: 1 r I‘I_I‘E‘I;r‘r',r:r r z, ‘r z 1
a) % w b) I:1Jr’rﬁﬁxﬁrﬁrfr‘rjrxxlzngrlr

      

 

Figure 27. (a) “Bubbles”. (b) “Stripes”. The black dots are the guiding centers of the
cyclotron orbits. The periodicity is ~ 3 RC. (c) The enlarged quasi-classical image of a
single bubble. The dark region is the guiding center of cyclotron orbits and the toroid is
the charge density distribution, which is created by electrons moving in the cyclotron
orbits centered inside the bubble. [Fogler 1997, F ogler 1996].

66

happens as long as vN < 0.3. As the system moves further from integer ﬁlling, the bubbles
coalesce to form the stripes.

The motivation for this experiment is to resolve these predicted bubbles and
stripes. We have calculated the periodicity of these CDW structures, for the magnetic

ﬁeld we use and the density of our sample. The calculation goes as follows:
The periodicity of a CDW lattice according to prediction 2 3Rc. RC = kaz , and kp in 2D

= 27rp = 1.94 x 108 m‘1 for 2DES density p = 6 x 10‘5 m'z. While 22 =18: 1.64 x 1046
e

m2 for magnetic ﬁeld B = 4T. Hence 3Rc = 3 x 1.94 x 1.64 x 10‘8 m z 95 nm, comparable

to our spatial resolution of ~60 nm.

5.2 The CDW experiment

The measurement described in the previous Chapters is reiterated here as follows:
A tunneling barrier separates the 2D layer and a parallel 3D substrate. An AC excitation
voltage applied between the substrate and a sharp metal tip locally induces charge to
tunnel back and forth between the 3D and 2D layers. A Schottky barrier blocks the
charge from tunneling directly onto the tip. The measured signal is the resulting AC
image charge on the tip electrode, which is proportional to the number of electric ﬁeld
lines terminating on it. In this way, the experiment provides a local measurement of the
ability of the 2D system to accommodate additional electrons. The method can be
considered as a locally resolved version of the pioneering work of Ashoori et al. [Ashoori
1990], and Eisenstein et al. (in a 2D-2D tunneling geometry) [Eisenstein], which
demonstrated that the tunneling signal into a 2DES is sensitive to both gaps in the energy

spectrum at integer Landau level ﬁllings, and a Coulomb pseudogap that is pinned to the

67

Fermi level of the 2D system. At particular magnetic ﬁelds, we observe micron-scale
features, corresponding to ordered electronic density modulations. To the best of our
knowledge, these data represent the ﬁrst tunneling images of the interior of a
GaAs/AlGaAs 2DES.

The sample used for these measurements was an A103 Gao_7As / GaAs (001) wafer
grown by molecular beam epitaxy (MBE), discussed in Chapter 2. The average electron
density in the 2DES is 6x10ll cm'z, the low-temperature transport mobility is ~10S
cmz/Vs, and the zero magnetic ﬁeld 3D-2D tunneling rate is approximately 200 kHz.
Cryogenic temperatures are achieved by direct immersion in liquid helium-3 at 270mK.
We position the tip to within a few nanometers of the sample surface using a scanning
head discussed in Chapter 3 as well. The image charge signal is detected using a sensor
circuit discussed in detail in Chapter 3. Most of the signal (~99.8%) corresponds to
electric ﬁeld emanating from areas macroscopically far from the location under the
probe. We subtract away this background signal using a bridge circuit. To acquire
images, the tip is scanned laterally across the surface without the use of feedback. We
apply an excitation voltage of 8 mV rrns at a frequency f==20 kHz. In addition, we apply a
DC voltage of 0.6 V to the tip to compensate for the tip—sample contact potential, as
measured in situ using the Kelvin probe technique [Tessmer 1998, Yoo]; hence the
effective DC voltage is zero. The spatial resolution of the measurement is roughly 60 nm
[Tessmer 2002], limited by the radius of curvature of the chemically etched PtIr tip (50
nm) and by the tip-2DES distance (60 um). All images presented here were acquired and
are displayed with identical sample orientation. To establish the technique’s sensitivity to

the 2D electron system, Figure 28(a) shows the measured charging as a function of

68

 

 

 

 

 

 

 

 

 

 

 

(a)? I T
25aF[
v=26
I
ilé's‘4-2 “(f/f) 2
Magnetic ﬁeld (T) {IOEFRTIRQ}

Figure 28. (a) F ixed-tip measurement. As a function of perpendicular magnetic ﬁeld, Qin
and Qout display clear features at integer Landau level ﬁlling v. (b) Calculated in-phase
and out-of—phase charging based on the equivalent circuit. Rigorous modeling for the
system considers the distributed nature of the tip-sample capacitance and the effect of
charge motion within the 2D plane. The dashed curves show qualitatively enhancements,
which occur if the in-plane relaxation rates approach the tunneling rate, an effect that
may occur near integer ﬁlling. (Inset) A linear plot of the derivatives of gin and Qout with
respect to tunneling resistance. In the low-frequency limit of this experiment (arrow), the
out-of-phase component is more sensitive to Rtun variations [Maasilta 2003, Maasilta

(rapids)].

69

magnetic ﬁeld with the tip position ﬁxed (i.e. not scanned), acquired using the same tip as
for the subsequently displayed images. Both the in-phase and out-of-phase curves
(acquired simultaneously) show clear structure with I/B periodicity, corresponding to
integer Landau level ﬁllings. To understand the origin of the in-phase dips and out-of-
phase peaks, we invoke the charging characteristics of the model equivalent circuit
described in detail in Chapter 3. Figure 28(b) plots the calculated variation of Q" and Q0“,
with frequency f and resistance RT. This is the same plot shown in Chapter 3, and is
shown here again for convenience. The charging components have identical functional

dependence on f and RT, with characteristic values of f0=[21tR7(Cr+Cti,r))]'l and
R0=[27If(Cr+Ctip)]'l, respectively. The magnitude of the curves is set by the tip-sample

capacitance difference AC between a fully charging and locally non-charging 2DES. For
the actual measurements, the characteristic zero magnetic ﬁeld tunneling rate iS f0 ~200
kHz, a factor of ten greater than the applied excitation (i.e., f/fo ~0.1). Therefore we focus
on the model’s low-frequency behavior, f/fo<<1; as shown in the inset, the out-of-phase
component is more sensitive to variations in the local tunneling resistance. In contrast,
the in-phase component is mostly sensitive to capacitance variations (not shown),
including the compressibility contribution to the capacitance [Tessmer 2002, Smith]. In a

parallel plate capacitor model, the measured capacitance is given by [Stern, Smith]

 

 

l = l + l (5.6)
dn'
Cmeas Cgeom 82 __
d/1

where Cmm is the measured capacitance and Cgeom is the geometric capacitance. Hence,

we conclude that the dips in the measured in-phase curve reﬂect a reduction in

capacitance caused by the diminished compressibility Z—nof the 2D system at integer
,u

70

ﬁlling. With respect to the out-of-phase curve, the peaks at integer ﬁlling likely reﬂect an
increase of the pseudo gap, resulting in an increased tunneling resistance RT [Deviatov];
variations in the in-plane conductivity may also contribute.

Figure 29(a) presents typical in-phase and out-of—phase charging images far from
integer ﬁlling. The in-phase data are representative of hundreds of similar images which
show charging patterns that are insensitive to magnetic ﬁeld. We attribute this structure to
surface topography, which modulates the geometric capacitance and couples
preferentially to Q,,,. This conclusion is supported by comparison to a surface topography
measurement (at a different location) acquired by operating the microscope in a standard
scanning tunneling microscopy mode [Urazhdin 2000]. As shown in Figure 29(b), the

surface consists of elongated mounds qualitatively similar to the in-phase structure. These
surface features point along the [110]direction, as indicated, reﬂecting an MBE growth

anisotropy [Ballestad]. In contrast to the in-phase images, we ﬁnd the out-of-phase
images can show signiﬁcant changes with magnetic ﬁeld. The Q0“, images also Show
features arising from surface topography, although this effect is signiﬁcantly smaller than
for the Q0“, component (as expected in the low-frequency limit). For example, in Figure
29(a) the out-of-phase gray scale is an order of magnitude smaller than the in-phase scale.
Because the in-phase component does not change signiﬁcantly with ﬁeld, we ﬁnd that a
convenient method to remove the small topographical contribution to Q0“, is to scale
down the corresponding Q... image and then subtract it from QM. Figure 29(c) shows the
same out-of—phase data as Figure 29(a), where we have subtracted the topography in this
way. We see that in this case no signiﬁcant features remain in Q0“, discernible above the

background noise, as was typical for the out-of—phase images far from integer ﬁlling. As

71

(a) ln-phase Out-of-phase (b)Topography (c)Out-of-phase

0—250 0-25 m
Charge (aF) Height (nm) Charge (aF)

   

0-10

Figure 29. (a) Representative scanning images of Q». (left) and Qou! (right) with an
applied magnetic ﬁeld far from integer ﬁlling of 3.89 T; scale bar length=lum. (b)
Topographical surface image showing the orientation of the grth features. The image
was acquired at room temperature without altering the orientation of the crystal. The
elongated mounds point along the [110] direction, as indicated by the orientation of the
scale bar, scale bar length= 2 pm. (c) Topography-subtracted out-of-phase image
corresponding to the same data shown in (a). The images have been ﬁltered to remove
nanometer-scale scatter [Maasilta (rapids)].

72

the magnetic ﬁeld approaches integer ﬁlling, while the Q" images remain unchanged, the
simultaneously acquired Q0“, images show completely different behavior, clear and
reproducible features appear that are not correlated with topography or experimental
parameters such as scan direction.

Figure 30(a) shows a series of topography-subtracted Q0“, images acquired
sequentially near ﬁlling factor v=6 at the ﬁelds indicated by the small arrows; very
similar behavior was observed near v=4. Similar to the out-of-phase peaks of Figure
28(a), the imaged Q0“, structure likely results from local variations of the pseudo gap
and/or in-plane conductivity. In either case, we can identify the structure as originating
from modulations in the 2DES density ~5% which bring the system slightly closer and
further from integer ﬁlling as a function of lateral position [Tessmer 2002]. Some of the
Q0“, features appear as randomly situated micron scale droplets, such as the bright feature
in the upper right corner of the 4.38 T image. These are consistent with droplets and
potential contours observed in references [Tessmer 1998] and [Finkelstein], which
utilized direct ohmic contacts to the 2DES (i.e. non-tunneling experiments). Following
this work, we interpret the droplets as originating from static density modulations
induced by the disorder potential. This view is also supported by observations we have
made of random features that reappear and evolve in a similar manner at successive
integer ﬁllings. In contrast to references [Tessmer 1998] and [Finkelstein], the images
exhibit additional structures with orientational order, superimposed with the random

droplets. To clearly demarcate these structures, we apply a standard approach based on

the 2D Fourier transforms: F (kvky) = F[Qou, (x, y)], where k. and ky are the wave vector

coordinates. The topography has been subtracted from the Q0“, data as described above.

73

 

 

 

 

 

(:12) 8131qu aseud-Io-Ino

3.8‘ '42‘ '45 .
Magnetic ﬁeld (T)

 

 

 

Fourier transform Angular spectrum
yi v . .

    

 

 

 

 

 

Figure 30. (a) Topography-subtracted out-of-phase data near six ﬁlled Landau levels. The
ﬁxed-tip curve to the left shows the v=6 Qout peak with arrows indicating the ﬁelds of the
displayed scanning images. The displayed images have been ﬁltered to remove
nanometer scale scatter; scale bar length=2 pm. The crystal orientation is identical to
Figures 29(a) and 29(b). (b) Schematic of Fourier analysis procedure to identify
orientationally-ordered structure. The parallel lines in the direct image (indicated with
dashes) can be described with wave vector k, giving maxima in the Fourier transform
power spectrum at ik(shown as black dots). The orientation of the wave vector 00 is
seen as a peak in the angular spectrum, calculated by integrating the transform image
along a series of radial lines at angle 0 [Maasilta (rapids)].

74

Because we are interested in the spatial patterns and not the shift in the average value
(this information is already given by the ﬁxed-tip curves), we have subtracted a constant
from each image to set the average value to zero. No additional processing or ﬁltering

was performed on the data prior to taking the Fourier transforms. To further emphasize
orientational order, we have also calculated the angular spectrum A (0) by integrating the

Fourier power spectrum along radial lines oriented at angle 0,
A(0) = f "'“| F (k cosO,k sin 0) |2 dk as Shown in Figure 30(b). The integration limits, km,"

and kmax, set the range of wave vectors to be included; of course, they can be expressed in
terms of cutoff wavelengths: kmm=27r/Amax and kmax=27t/Amm. Figure 31(a) presents three
sets of unﬁltered Fourier transforms and angular spectra corresponding to Q0“, images of
Figure 30(a). We see that the transform at 4.14 T has developed clear maxima at
relatively short wave vectors. To highlight the corresponding range of wavelengths, the
angular spectra are calculated choosing cutoffs of Amm=lum and [Amm=3pm, displayed
directly below each transform image. We see that the 4.14 T angular spectrum shows the
strongest structure at these short wave vectors; the prominent peak indicated by the arrow
results directly from the maxima in the transform image.

Figure 31(b) compares the 4.14 T spectrum (near v = 6) to a similar observation
obtained at a different location and at a ﬁeld of 6.75 T (near v = 4). Figure 31(c) shows

the corresponding Q0“, images, where the arrows point to the stripe-like modulations that
form the prominent peak in the angular spectra. As an example of the raw data, we also
Show the 6.75 T out-of—phase image with no ﬁltering. These lines roughly point along the

[110] direction. The images Show evidence of weaker modulations in other directions.

We have acquired dozens of Q0“, images near v = 6 and v = 4 that show similar micron-

75

 

 

 

 

 

E
<(
B ‘ ' 1.4. I -
8 0 It ‘ . ..
a) .89T _ r”: a ..
O . . , I ’\ \ ’\ \ \
o 6 (deg) 180 4.14 T 6.75 T Raw Data

Figure 31. (a) Fourier power spectra and angular spectra corresponding to three Qoul
images of Figure 30(a), normalized to the peak value at 4.14 T. The power spectra are
displayed over the range 21:6.67 pm"; the magnitude of the wave vector corresponding to
the 4.14 T maxima is 2.4 pm], or a wavelength of 2.6 um. (b) Angular spectra from three
different sample locations and magnetic ﬁelds. Similar peaks were observed near four
(6.75 T) and six (4.14 T) ﬁlled Landau levels. For comparison, a spectrum far from
integer ﬁlling (3.89 T, corresponding to Figure 29(c)) is included. In this plot, each curve
is scaled to have an average value of 1.0, which serves to enhance the weak 3.89 T curve.
The minimum at 0 and 180° is an experimental artifact arising from the scan direction. (c)
The topography—subtracted Qout images at 4.14 T and 6.75 T with arrows pointing to the
structure that yields the prominent Fourier peaks. Also included is the 6.75 T data for

which no ﬁltering was applied. The [110] crystalline direction is indicated; scale bar
length=2 pm [Maasilta (rapids)].

76

scale structure. In contrast to the droplet features that mirror static potential variations,
the stripe-like structure appears to form spontaneously. Unlike the droplets, these
structures shift position and evolve dramatically with magnetic ﬁeld, as can be seen in
Figure 30. The high sensitivity to magnetic ﬁeld precludes tunneling barrier non-
uniformity as the origin. In addition, we note that scanning tunneling microscopy and
atomic force microscopy measurements show no evidence of atomic step edges oriented

along [110]. We have not observed such structure near v =5 nor v =3. (The 10 T limit of

our superconducting solenoid prevents probing Landau level ﬁllings less than v =3 for
this sample.) We have not found evidence of ordered structure on the predicted cyclotron
radius length scale, ~100 nm.

In summary, we have obtained the ﬁrst direct images of ordered structure within
the interior of a GaAs/AlGaAs 2DES using a novel scanned probe method. At magnetic
ﬁelds near four and six ﬁlled Landau levels, we observe density modulations that exhibit
clear orientational order, roughly along the [110] crystalline direction. The features are
suggestive of micron-scale stripe-like states. However, because the spacing between
features is a substantial fraction of the experimental scan range, we can only give a rough
value for the characteristic length scale, and cannot speak to the periodicity of the
structure. It is tempting to equate the observations with states similar to theoretical high-
Landau-level charge density wave states [Koulakov] discussed in this Chapter. However,
the length scales represent a major discrepancy: our calculations from theories predict a
characteristic length set by the cyclotron radius ~100 am, much smaller than the observed
structures ~2 pm. It is possible that the physical origin of the modulations is similar to the

CDW theories, but that disorder causes a larger length scale to be selected. Alternatively,

77

the structures may arise from an entirely different mechanism. Clearly, more theoretical

work is needed.

78

Chapter 6

Quantum Dots

Quantum dots are small structures in solids containing electrons conﬁned in all

three directions [Kastner, Kouwenhoven 1997, Kouwenhoven 2001]. This conﬁnement
leads to discrete energy levels for the electrons, much like an atom. Although the
conﬁning potential in a quantum dot can be chosen at will, the one kind that will be
studied in the present experiment is bowl-like or parabolic. In this Chapter we shall
describe the theory of quantum dots conﬁned by the parabolic potential, discuss the

motivation behind the experiment and ﬁnally show some preliminary results.

6.1 Quantum dots: theory and motivation

Although quantum dots are very similar to real atoms in that they form discrete
energy levels, there is one very important difference between them [Ashoori 1996]. The
quantum dots are typically much larger in size than a real atom. The electron orbits of the
dots do not scale in size. Imagine an atom containing many electrons whose size is
continuously variable. As it is made larger the Coulomb energy due to repulsion between
electrons orbiting around the nucleus decreases because the average spacing between the
electrons increases. However, as the atomic size increases, the separation in energy of the
different orbits of electrons decreases faster than the Coulomb energy. Hence in quantum
dots the electrons are effectively much closer, and Coulomb interactions are much more
signiﬁcant than in a real atom. What all these mean is that although the two are very

similar, and we have extensive knowledge about real atoms, quantum dots are very

79

interesting systems for studying basic quantum mechanics and electron-electron
interactions.
Electronic states in quantum dots are described by their energy eigenvalues and

corresponding eigenstates. The standard model for these dots assumes N electrons

II! I. , _ .
conﬁned by a parabolic potential gm (002,2 (m 15 the effective electron mass, (001$ the

angular frequency and r is the radial distance) in a 2D plane interacting via Coulomb
repulsion. In a perpendicular magnetic ﬁeld the exact solutions for non-interacting
electrons for the energy eigenvalues, with n, l, 0', being the radial, angular momentum

and spin quantum numbers for the electrons are given by [Fock, Darwin]

 

 

2
En”r =h (0:) +60: (2n+|l|+l)—m:)c +g'yBBO', (6.1)

n = 0,1, 2 etc. l=:I:0,i1,:t2 etc. and 0'= i%

where a), ze—Zfis the cyclotron frequency, [13 is the Bohr magneton, and g* is the
m

effective g-factor. The eigenstates are given by [Maksym]

2 2
III”, (r,0)=r"' exp(- i16)L',j' [2:7] exp(— Era—21’ (6.2)

 

80

where LIZ] is a Laguerre polynomial. The quantum number It gives the number of nodes in

the radial direction and I gives the number of nodes circumferentially.

This is a very Simple picture and does not take into account the Coulomb
interactions. Many observations can already be explained with this simple picture, by
assuming the Coulomb interaction to be a small perturbation, so that the energy levels are
the usual quantum levels due to the parabolic potential plus the additional Coulomb
charging energy that varies smoothly with particle number. This picture accounts for the
shell structure observed by Tarucha et al [Tarucha, 1996], which arises from the (2n+|l|
+1) factor in equation (6.1). Addition Spectra also show transitions, which are not
explained by this simple model but can still be understood in terms of single-particle
levels subject to the exchange interaction. However a thorough understanding of the
transitions requires precise calculations. Recent calculations [Tarucha, 1998] show
deviations from the single-particle model. Although some indications of this structure
may be present in the addition spectra results reported by Tarucha et al [Tarucha, 1996],
a new technique capable of identifying these quantum states is needed.

Another important aspect is the interplay between disorder and electron-electron
interactions in a quantum dot. The disorder would most likely distort the parabolic
potential in a quantum dot into an irregular shape, leading to eigenstates much different
from equation (6.2). Moreover the shape would depend on the exact location of the
impurities and donor density variations. The interplay between disorder and electron-
electron interactions can lead to remarkable behavior. For example in some cases
electrons seem to be violating Coulomb repulsion, and two of them enter the dot at the

same energy [Zhitenev 1997, Zhitenev 1999, Brodsky]. Most of the experiments done

81

until now have focused on the addition spectra of the system. With our CAI technique we
will not only be able to study the addition spectra, but also CAI in principle has the
profound capability to image the spatial characteristics of the quantum states and thus test
both many-body effects and the effects of disorder on the eigenstates.

As shown in Figure 32, the same tunneling geometry that was employed to image
the CDWS can be employed to image the wavefunctions of a quantum dot and to do
spectroscopy of the dot. A gold gate is fabricated on the surface of the sample. By
applying a negative potential to the gate, the 2DES is completely depleted, except in the
hole region, which forms the quantum dot. By applying a negative voltage further,
electrons can also be pushed from the dot region. While raising the gate voltage would
add electrons to the dot, which would tunnel from the metallic substrate into the dot.
With the tip ﬁxed and varying the gate voltage, we can do spectroscopy of the dot, i.e.,
electrons will be added to the dot when the gate potential is equal to the energy level
spacing, and would Show up as enhanced capacitance (a peak in a capacitance versus gate
voltage plot). On the other hand to image the amplitude of the wavefunction, we would

scan the tip at a particular gate voltage that corresponds to a peak in the addition spectra.

82

I

I N
,’1 l‘l’l‘ of
elecrron

 

Figure 32. Schematic of wavefuntion imaging for a quantum dot. A single electron
resonates between the bottom electrode and the dot. By mapping out the density
variations, quantum wavefunctions can be probed. As an example a 3D plot of lull2 in the
single-particle approximation is shown.

83

6.2 Quantum dot fabrication

To achieve lateral conﬁnement of electrons in the 2DES we fabricate metallic
gates on the surface of the sample. Application of a negative voltage on the gates depletes
the 2DES of electrons below the gate region, leaving a conﬁned puddle of electrons in the
un-gated region. The method of fabricating the gates is quite similar to the ones used in
the semiconductor industry. First a resist stencil of the pattern is made. Then the
particular metal is deposited. Finally the stencil is removed by soaking the sample in a
solvent. Broadly, there are two methods of forming the resist stencil — one is optical
lithography, in which the resist is exposed to light (usually UV) and the other is electron-
beam lithography, in which the resist is exposed to a controlled beam of electrons. The
exposure chemically alters the resist. Subsequent immersion in a developer results in the
removal of the exposed areas (for positive resist), forming the stencil.

The smallest feature that can be patterned with optical lithography is
approximately 1 pm. As discussed at the beginning of the previous section (Section 6.1),
the level spacing of the quantum dots decreases faster than the Coulomb energy as the
size of the dot is increased. This means that it is harder to resolve the energy levels of a
bigger dot compared to a smaller one. Hence one needs to fabricate small dots, and lum
dots may not be small enough. Although it is possible to improve on the resolution of
optical lithography, it is much easier to use electron-beam lithography, which can easily
achieve resolutions of sub-100 nm. Ideally the resolution of e-beam lithography is limited
by the beam diameter, which is typically 10 nm, but is extremely difﬁcult to achieve in
practice. In the following paragraph I shall describe the fabrication of the metallic gates

with e-beam lithography.

84

Before the procedure can be started, it is very important to clean the surface of
the sample of dirt and grease. For relatively clean substrate, ultrasonic cleaning in
acetone, isopropyl alcohol (IPA) and de-ionized water in that order is sufﬁcient. For a
really dirty substrate, scrubbing with Alconox should be done ﬁrst, before proceeding to
acetone. The substrate is ﬁnally dried by blowing dry nitrogen gas. To aid the adhesion of
the resist, the substrate may be treated with hexa-methyl disilizane (HMDS) vapor for 10-
15 minutes, but this was not always done. The resists used to form the stencil in the e-
beam process are based on methyl-merthacrylates (PMMA or MMA). The usual
technique used to form a smooth layer is to spin the substrate rapidly while adding the
liquid PMMA/MMA. A layer of PMMA/MMA 6% was spun on the sample at 3000 rpm
for 50 seconds. It was then baked at 200 0C in an oven for an hour to remove the solvent
and to harden the resist. This layer of resist can be exposed to a beam of electrons and
developed. PMMA/MMA is a long chain polymer. When it is exposed to a beam of
electrons, the polymer chain is broken. The coating is then dipped in a developer — which
is generally a solution of MIBK (Methyl-iso-butyl Ketone) and IPA. The developer
removes the exposed resist, to form the pattern as shown in Figure 33(b). The problem
with this procedure is that when the metal is deposited, a thin layer of it tends to stick to
the sidewalls of the resist and forms a continuous coating. This makes it very hard to lift-
off the resist stencil.

One way to circumvent the problem is to spin another layer of a different resist on
top of the 6% PMMA/MMA. A layer of PMMA 2% was thus spun at 4000 rpm for 50

seconds on top of the ﬁrst layer. It was also baked in an oven at 200 0C for an hour to

85

(a) electrons

llll

 

(b)

 

 

Figure 33. Single layer resist process. (a) The resist is exposed to a beam of electrons. (b)
Resist after being exposed and developed.

86

remove its solvent and to harden it. It was then exposed to electron beam in a JEOL840 e-
beam writer. The dose (deﬁned as the amount of charge incident on an unit area of the
sample) used was 250uC/cm2 with a current of 20pA. What this does is produce an
overhang of the top resist when developed, as shown in Figure 34(b). This overhang
produces a natural break in the metal ﬁlm during deposition. The reason the overhang
forms is very simple: the top layer of resist responds more slowly to the developer than
the bottom layer, and hence the opening on it is smaller than the bottom layer for the
same amount of soaking time. Finally 0.2um thickness of gold was evaporated and lifted
off in acetone to produce the pattern shown in Figure 35.

Four different sizes of circular gates were fabricated with diameters of 0.5 pm,
1.0 pm, 1.5 pm and 2 pm. The patterned gates are shown in Figure 35. The line thickness

is 0.2 pm. The total pattern occupies a square of approximately 0.6 mm x 0.6 mm in a
sample of approximately 0.5 cm x 0.5 cm size. Hence there are several thousand quantum
dots fabricated on the sample; this approach makes it more likely to ﬁnd at least one dot
by doing topography with STM. Still the magnitude of the square is just about enough to
have a realistic chance of ﬁnding a dot by STM. This is because the coarse positioning of
our scanning microscope is about 1 mm, and hence to ﬁnd a dot by doing topography, the
tip would need to land at worst 1 mm (laterally) from the pattern, after it approaches the
sample. The reason the gates are fabricated in this particular pattern instead of circular
holes within deposited metal is that it would have required immense e-beam writing time
to write such a huge pattern, with a lower dose. Using higher dose shortened the time, but

destroyed the holes due to proximity effect. Negative e-beam resist (N E322) was tried in

87

electrons

llll

 

 

   

(b) Top resist /Overhang

. [7. ”c .7, “‘32., .J.

Substrate . ,

 

Figure 34. Two layer resist process. (a) The resists are exposed to a beam of electrons. (b)
Resists after being exposed and developed, the top resist is underdeveloped and creates
an overhang.

88

 

Figure 35. (a) SEM micrograph of the gold gates on GaAs/AlGaAs sample. The circular
gates are 12pm apart. (b-e) Further magniﬁed micrographs of individual dots of
diameters (b) 0.5um, (c) 1 mm, (d) 1.5um and (e) 2pm.

89

order to circumvent this effect, to produce circular holes, but usually failed to adhere to

the GaAs surface even with HMDS procedure.

6.3 Preliminary Experimental Result

Although no experimental result has been obtained as yet on the wavefunction
imaging of the quantum dots, we have achieved two very important initial goals. First, we
were able to ﬁnd a quantum dot by doing STM. One such STM image is shown in Figure
36. Second, we were able to do addition spectra of a disorder induced quantum dot, to
test the validity of using the CA1 technique for studying quantum dots in the vertical
tunneling geometry. The result is shown in Figure 37. The disorder was induced by
applying a large negative voltage (~ -10V) on the tip, to form the circular shaped
quantum dot, as described in Chapter 4, Section 4.2. The large voltage was then removed
and with the tip ﬁxed over the dot region, its DC voltage was varied from —O.6V to 0.3V.
The peaks shown in the plot corresponds to one electron being added to the quantum dot.

This experiment is similar to the pioneering work by Ashoori et al [Ashoori 1992,
Ashoori 1996], and mentioned brieﬂy in Chapter 1, in the quantum dot section. The
difference of course is that we have a tip-induced quantum dot and a tip, instead of a
metallic gate, to measure the addition spectra. The width of the peaks is set by the AC
excitation, if the nns value of the excitation is greater than kT; otherwise the temperature
T sets the width. In our case the excitation amplitude was SmV rms and indeed the width
of the peaks are exactly equal to SmV. Ideally the peak spacing should be equal. The
reason that it is not can be attributed to that fact that the quantum dot potential is not truly

parabolic, but has wiggles in it due to disorder. This will lead to unequal spacing of the

90

 

Figure 36. STM image of a quantum dot metallic gate of size 0.5um.

91

 

In-phase Charge

 

 

 

I l l l
0.5) -0.50 -0.40
Tip Bias Voltage (V)

 

Figure 37. Addition spectra of a disorder induced quantum dot (left). The applied
excitation voltage was 5mV. The arrows in the plot show the capacitance peaks, which
correspond to one electron entering the dot. There are no peaks seen above the tip voltage
of -0.45V, since the energy levels in the dot are too close to be resolved.

92

energy levels of the quantum dots, which thus leads to unequal peak spacing in our data.
Another aspect of the plot is the unequal heights of the peaks. In the ideal case again they
should be equal, since either one electron enters the quantum dot or none at all. The
heights are not equal because not all the electrons entering the dot, enter below the apex

of the tip, and thus exhibit decreased capacitance.

93

Chapter 7

Summary and future directions

7.1 Summary

In summary, several important goals have been achieved in the course of the

research described in this thesis. First, we were able to image the GaAs/AlGaAs 2DES
with the CA1 technique in the tunneling geometry. This is extremely important since
techniques that utilize direct Ohmic contact to the 2DES are not capable of resolving
2DES structure near integer ﬁlling [Tessmer 1998, Finkelstein]. The tunneling geometry
also provides us information about the tunneling density of states of the 2DES and is
sensitive to the psuedogap physics [Eisenstein, Deviatov, Dolgopolov]. With the
tunneling geometry, we have observed the charging of the 2DES as a function of the
magnetic ﬁeld. The in-phase charging showed clear dips with 1/B periodicity at integer
ﬁlling, arising out of quantum Hall physics. The out-of—phase peaks, also observed at
integer ﬁlling were the signature of the tunneling density of states of the 2DES. We
believe they arise from correlation effects that modulate the tunneling density of states,
which then result in increased tunneling resistance at integer ﬁlling.

We have also imaged the 2DES in this tunneling geometry, and have seen
structure near integer ﬁlling, which brings us to the second important goal achieved. The
observation of orientationally ordered structure in the 2DES near integer ﬁlling is the ﬁrst
of its kind to be observed in any geometry. Although the intention of the experiment was
to search for periodic structures (“bubbles” and “stripes”) with periodicity of 100 nm,

stripe like structures were observed near ﬁlling factors v = 4 and 6 with the length scale

94

an order of magnitude too large. Fourier transforms of the images showed clear
orientational order with a length scale of approximately 2 pm. We speculate that the
physical origin of the modulations is similar to the CDW theories, but that disorder
causes a larger length scale to be selected. Alternatively, the structures may arise from an
entirely different mechanism. Clearly, more theoretical work is needed.

The 2DES was also imaged in the absence of a magnetic ﬁeld to study disorder.

Although theories [Efros 1988, Efros 1999, Shklovskii] predict a spatial distribution of
the ionized donors to be truly random, observation were made to the contrary. Micron
size features were observed, consistent with similar experiments done in the presence of a
magnetic ﬁeld [Tessmer 1998, Finkelstein].

Finally, progress was made to image the wavefunction amplitude of electrons and
to study the addition spectra of a quantum dot. The addition spectra of a disorder induced
quantum dot was measured, which showed peaks in the in-phase charging at certain
values of the tip bias voltage. This measurement was similar to the work of Ashoori et al
[Ashoori 1992, Ashoori 1996], except a local probe was used for the measurement

instead of a ﬁxed metallic gate on the sample.

7.2 Future directions

The CAI technique in the tunneling geometry, which was successfully
implemented in the work described in this thesis, holds great promise for future research
with 2DES. The future directions of this research can be divided into two broad
categories — imaging a 2DES with higher mobility (hence lower density and lower

impurities) and imaging electron probability density of single and coupled quantum dots.

95

With regard to imaging a higher mobility GaAs/AlGaAs sample, it is important to
explore the relationship of the stripe-like modulations with disorder. A sample with
mobility an order of magnitude higher is already available, provided by Loren Pfeiffer of
Lucent Technologies. A study on this sample will provide a critical test for the degree to
which disorder alters the stripe-like structure.

The next key issue is to resolve the 2DES structure to smaller length scales,
particularly in the length scale predicted to be the periodicity of the CDW patterns. The
higher mobility (or lower density) sample has a distinct advantage here. Since the new
sample has almost 1/3 the density of the previous sample, the particular Landau level
ﬁlling factor is reached at about 1/3 the magnetic ﬁeld. This lower magnetic ﬁeld value
makes the cyclotron radius R; larger, which ﬁnally sets the periodicity of the CDW
patterns. In the particular case of ﬁlling factor v = 6 the periodicity is almost 300nm
instead of the 95am calculated in Chapter 5. This value of 300nm is much larger than the
spatial resolution of 60am of our system.

Apart from these two immediate future goals, “spin bottleneck” phenomena can
be studied with the 3D-2D tunneling CAI technique. Recent 3D-to-2D tunneling
measurements at magnetic ﬁelds near v = 1 have shown that there are two different
tunneling rates for charge entering the 2DES [Chan, Deviatov, Dolgopolov]. The fast rate
is the usual RC charging time. The slower rate is believed to be the “spin bottleneck”
effect, which is a many body phenomenon arising from skynnion formation in the system
[MacDonald]. It essentially reﬂects a relatively slow electron relaxation time of
approximately lms. By imaging the 2DES at two different excitation frequencies — one

close to the characteristic RC charging time (~ lOOkHz) and the other at a much lower

96

characteristic spin bottleneck frequency (~lkHz) — the Spin bottleneck effect and the
effect on it due to disorder can be studied [Murthy, Brey, Green, Nazarov].

With regard to quantum dots, the next step is to image the electron wavefunction
amplitude and the addition spectra of the individual dots. A future project would be to
study coupled quantum dots. When two quantum dots are sufﬁciently close, tunneling
can occur between them. The discrete energy levels of each dot are altered by the
presence of the other. This coupled quantum dot system is also called “artiﬁcial
molecules” [Austing, Brodsky]. It is a very interesting system for studying the
competition between kinetic energy and Coulomb interactions in the regimes inaccessible
in real molecules. Moreover, such systems may prove to be relevant for quantum

computing applications.

97

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