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SCANNING PROBE STUDY OF A TWO-DIMENSIONAL
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HETEROSTRUCTURED SEMICONDUCTORS

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IRMA KULJANISHVILI

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SCANNING PROBE STUDY OF A TWO-DIMENSIONAL ELECTRON SYSTEM
AND DONOR LAYER CHARGING IN HETEROSTRUCTURED
SEMICONDUCTORS
By

Irma Kuljanishvili

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

2005

ABSTRACT
SCANNING PROBE STUDY OF TWO-DIMENSIONAL ELECTRON SYSTEMS
AND DONOR LAYER CHARGING IN HETEROSTRUCTURED
SEMICONDUCTORS

By

Irma Kuljanishvili

We have applied a novel scanned probe method called charge accumulation imaging
(CAI) to study two electronic systems in GaAs/AlGaAs heterostructured crystals: two-
dimensional electron systems (2DES), and dopant Si atoms confined to a single
monolayer. In a presence of magnetic field we have imaged the 2DES at half-filled
Landau level and observed evidence of intriguing density modulation induced structure
of wavelength 200 nm. The direction of this stripe-like structure is approximately 35
degree to the [110] crystalline axis, which is different from the easy direction of the
electronic transport based on measurements performed by other research groups. The
nature of this density variation is unclear; however, it may be related to a charge density
wave picture predicted theoretically. We have also probed the donor layer charging in
gallium arsenide as a function of bias voltage and magnetic field. In a tunneling regime,
we have studied the electronic energy spectrum of donor atoms and resolved individual
electrons entering the system. We interpret our experimental results in terms of a donor
molecule model. To the best of our knowledge , this is the first scanning probe study of

donor atom energy levels.

This thesis is dedicated to my parents.

iii

ACKNOWLEDGEMENTS

There is large number of people whom I would like to thank here for their support
and assistance. This work would be impossible without their presence in various stages of
my journey. First of all, I would like to thank my thesis adviser Professor Stuart Tessmer
who took me aboard his group several years ago. All through these years, he has been
very supporting and encouraging teacher, optimistic and directing supervisor. I am very
fortunate to have an opportunity to work with him. Personal qualities of his character and
love for science made my research experience in the group professional on one hand and
friendly on the other. I consider myself very lucky to have a chance to be working under
his supervision.

I am very grateful to Professors Carlo Piermarocchi and Tom Kaplan for their
interest in my research. Their expert theoretical input was very helpful in the process of
understanding the interesting physics of my latest experimental finding.

For the years spent in this Department, I have to thank all my teachers whom I
respect deeply and thank them for their help. I would like to start with Prof. Susan Simkin
who was the first person I met here in the department. She helped me with numerous
advice and motherly care during my very first interactions with different culture and
customs. I will always be deeply thankful to her for that. I would like to thank Prof.
Beers, with whom I worked on several occasions as an Astronomy teaching assistant and
from whom I have learned at lot of Astronomy: I deeply respect him as a person and a
very good scientist. I would also like to thank Prof. Stein at Astronomy department, with

whom I also worked as a teaching assistant, and whose pedagogical wisdom taught me

iv

how to be a better teacher. I would also like to thank all Professors who taught us all
classes and shared their expert knowledge of physics with usegraduate students: Prof.
Mahanti, Prof. Bass, Prof. Zelevinsky, Prof. Dykman, Prof. Birge, Prof. Stump, Prof.
Harrison, Prof. Pollack, Prof. Billinge, Prof. Repko, Prof. Kovacs and others. They
helped me learn how to be most successful in the classroom, which is a very important
part in every student’s life, especially the PhD students.

I would like to attribute my Special thanks to all members of our group starting
with Illari Maasilta, a former Post-Doc in our group. To all my lab-mates through the
years starting from the beginning: Sergei Urazhdin, the first graduate student in our group
who was a very original person and interesting and different in many ways, Subhashish
Chakraborty with whom we would discuss the lab issues, basketball games and life in
general. Aleksandra Tomic, my very good friend and colleague, who has this genuine
kindness that I admire so much, Cemil Kayis and Morewell Gasseller, my lab-mates and
good friends who were always ready to assist not only with the lab related tusks but any
day-to-day life surprises. I wish to every graduate student to have such nice colleagues in
the group with whom you may genuinely be honest even if their philosophical views of
the world sometimes differ from your own. It was a great pleasure to have these two
young men as my coworkers and friends.

I would like also to thanks all my friends in this Department, Post-Docs and
graduate students; some of them have already graduated and others that are still working
hard on their projects: Mykita Chubynsky, Tatyana Sharpee, Mikael Katkov, Emil Bozin,
Mikael Crosser, Mustafa Al-HajDArwish, Ion Mouraru, Mahdokht Behravan, HyunJeong

Kim, Chigusa Ohbuchi, Gassem Al-Zouli, Nikoleta Theodoropoulou, Peter Tobias and

other. It was very nice to have these people around, share opinions, professional
experience, discuss physics questions or simply cheer each other up occasionally. I wish
them all best of luck and great success in their professional carriers.

I should attribute my special thanks to Ms. Debbie Simmons for her constant
assistance on different occasions, always ready to give a timely advice or remind of
something important that one may forget to do. Her professionalism and responsibility
for her work are truly remarkable.

I should thank Dr. Reza Loloee and Dr. Baokang Bi for their help with sputtering
and all clean room equipment that I used. I am very thankful to all the stuff members of
the machine shop, particularly James Muns, Thomas Hudson, and Thomas Palazzolo for
their help and assistance and great sense of humor. The reason I could do such a precise
measurement it is because these people are precisely the experts in their work.

I would like to thank my dear fiiends Dali Giorgobiani and Maria Bailey for their
friendship through all the 17 years that we know each other. Dali herself is a PhD
graduate of Physics Department; she was the person who suggested I applied for the
Graduate school at MSU. I am very thankful for this great opportunity in life. I also
would like to thank my old childhood friends Liana and Asmara from Georgia whom I do
not see them often but I appreciate very much their friendship and support. I know how
much they are happy for me.

Finally, I would like to thank my family: my parents, Giorgi and Tsiala and my
sister Irina, whose love and support I feel every day. Lastly and very importantly, I would

like to thank my husband Juri who has an extraordinary patience with all things I do in

vi

life including my devotion to my work. He has been supportive and patient with me
during all these years, often encouraging me and always believing in me.

Here at MSU Physics Department, I grew up to become a mature and wise person
and I hope I am maybe approaching a boundary to become a good scientist. I believe that
the horizons in science are unlimited and despite all the great discoveries that were
already made we still live in a very young universe where in my view the scientific
knowledge has to be a main tool for understanding the nature.

This work was supported by the National Science Foundation grant No DMRO3-
05461 and the Alfred P. Sloan Foundation. The W. M. Keck Microfabrication Facility

provided valuable resources for fabrication and scanning electron microscopy.

vii

TABLE OF CONTENTS

LIST OF TABLES ................................................................................. ix
LIST OF FIGURES ................................................................................ x
CHAPTER 1: Introduction ....................................................................... l
1.1: 2DES in perpendicular magnetic field ......................................... 5
1.2: Charge density waves formation in high Landau Levels ..................... 9
1.3: Donor layer inside the semiconductor heterostructure ........................ 12
CHAPTER 2: Heterostructured samples ........................................................ 15
2.1: Molecular-beam epitaxy ........................................................ .16
2.2: Electron mobility ................................................................... 19
2.3: Formation of 2DES and modulation doping ................................. .21
2.4: Our sample ........................................................................ .26
CHAPTER 3: Experimental set up and the geometry of the technique .................... .28
3.1: Charge accumulation imaging .................................................. 29
3.2: The scan head ....................................................................... 33
3.3: The Cryostat ........................................................................ 37
3.4: The cryogenic charge sensor ..................................................... 40
CHAPTER 4: Numerical calculation for CAI spatial resolution ............................. 47
4.1: Electrostatic framework .......................................................... 48
4.2: Results and discussion ............................................................ 51
4.3: Comparison to measured data ................................................... 56
CHAPTER 5: Charge density waves: theory and observations .............................. 59
5.1: Description of the theory of charge density waves .......................... 61

5.2: Magnetocapacitance measurements and charge density variations. . . . . 69

CHAPTER 6: CPM study of the donor layer charging in GaAs heterostructure. . . 83

6.1: Resonant tunneling experiments ................................................. 87
6.2: Scanning probe capacitance tunneling spectroscopy ......................... 90
CHAPTER 7: Summary and future objectives ................................................ 104
7.1: Summary of the investigations .................................................. 104
7.2: Future directions .................................................................. 107
REFERENCES ................................................................................... 1 10

viii

LIST OF TABLES

Table l. The cohesive energies of the Laughlin liquid E cLoh and the CDW Egg", for r,

= 2 .117! 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
EU” is also provided for reference [Fogler 1997]. ............................................. 64

ix

LIST OF FIGURES

Figure 1. Three LLS broadened by disorder. Extended (conducting) and localized
(insulating) states exist in the center and on the flanks of the LLS, respectively ............ 8

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. .................................................................... 11

Figure 3. Simplified schematic diagram of an MBE apparatus showing Knudsen cells for
A1, AS and Ga with shutters. The sample sits on a heated holder, which is rotated during
the grth of the sample. The RHEED control system is also shown [Davies]............17

Figure 4. Conduction band around the heterojunction between n-AlGaAs and undoped
GaAs showing how electrons are separated from their donors and form 2D electron
system. Electrons can diffuse away from the donor ion and some cross into the GaAs.

There they lose energy and become trapped because they can not climb the barrier A E c,
......................................................................................................... 22

Figure 5. Energy-level diagram of a GaAs/AlGaAs heterostructure. The donor electrons
travel from their host Si atom and occupy the first conduction sub-band of the potential
well. The Fermi level is shown as 35, The spacer layer is not shown [Davis] ............... 25

Figure 6. Tip-Sample arrangement of our experiment .......................................... 27

Figure 7. Schematic of the CA1 technique. This is a capacitance technique where the tip
and the sample are coupled to each other through the electric field, like two plates of a
capacitor. To be more specific, we apply voltage to the bottom plate (sample) and
monitor the image charge that is induced on top plate (tip) ................................... 30

Figure 8. Schematic of the vertical tunneling geometry. (a) In the 3D-to-2D case we
probe the charging of the 2DES. (b) In the 2D-to-donor layer case we probe the charging
of the donor layer. In both cases an ac excitation is applied to the 3D metallic substrate
which resides below the tunneling barrier. The charge tunneling into the 2DES or to the
donor layer induces an image charge on the tip, which is measured by a cryogenic charge
sensor circuit. ....................................................................................... 32

Figure 9. Schematic of the Besocke type scanning head. Three carrier piezo tubes are
supporting the sample holder ramps. The sample sits upside-down rigidly mounted on the
sample support disk. The outside carrier tubes and the middle scanning tube are all
attached to a bottom plate. The scanning tube in the middle holds a small chip with the tip
and sensor circuit. (The circuit is not shown.) ................................................... 34

Figure 10. Photograph of the actual microscope. The front door of the microscope is
removed and the sample holder ramp is not shown ............................................. 36

Figure 11. Schematics of the cryostat (left) and top loading probe (right) are shown.
During the experiment probe resides inside the cryostat and scanning head is situated at
the bottom of the cryostat surrounded by superconducting magnet .......................... 38

Figure 12. Schematic picture of the cross section of a HEMT [Mimura,l982] ............. 41

Figure 13. Schematic of the charge sensor circuit. Cs is the standard capacitor and Vin is
the rms value of the applied AC excitation to the sample. The AC excitation applied to
Cs, is of the same amplitude but 1800 degree out of phase with the excitation voltage that
is applied to the sample. The charge sensitivity of the circuit is 0.01 electrons/VH2 ...... 43

Figure 14. Drain I-V characteristics, at different gate voltages VGs, of a typical HEMT
[Mimura,l982]. The circle marks the voltage range in the regime of experimental
operation ............................................................................................. 46

Figure 15. (a) Schematic picture of the finite element approach, showing the tip-sample
geometry used to model charge accumulation imaging measurements. The tip is
represented by discrete conducting points arranged to form a steep pyramid of half-angle
or. The apex is curved with radius r. The sample conductor is represented with points
arranged in a two-dimensional square grid imbedded in a dielectric. (b) Schematic of the

calculation of potential matrix element A ,1, for Case I , for which both i and k are points in
the tip and k is a distance s from the dielectric surface. The image charge at the
symmetric position with respect to the vacuum-dielectric interface k' accounts exactly for
the modification in potential due to the dielectric layer. R and R’ are the distances from i
to k and k', respectively. (c) Schematic for Case 2, for which i is in the tip and k in the
sample. Here the dielectric layer is accounted for using an image charge at the location
of the actual charge. ((1) Schematic for Case 3, for which for which both i and k are points
in the sample a distance d below the dielectric. In analogy to Case I, the dielectric layer
is accounted for by invoking an image charge at the symmetric position k' ................ 50

Figure 16. Calculated mutual capacitance function C(x, y) showing the characteristic bell
shape. The calculation used the tip geometry shown if Fig. 1(a) with a half angle on =8°,
radius r=50 nm, total height of 520 nm, and tip-surface separation of 5 nm. The modeled
sample was 1.06 x 1.06 pm, with a dielectric overlayer of d=60 nm. For both the tip and
sample the spacing between adjacent grid points was 20 nm. The half width at half
maximum is 92 nm, which determines the spatial resolution of our technique ............ 55

Figure 17. (a) A 3 x 3 pm subsurface charge accumulation image of a GaAs-AlGaAs
two-dimensional electron layer formed in a GaAs-AlGaAs heterostructure. The
contribution of surface topography has been subtracted so that the displayed signal arises
solely from disorder within the two-dimensional layer. The image is filtered to remove
nanometer scale noise. For comparison to modeling we focus on the data indicated by
the white line; here the signal increases abruptly, representative of the sharpest features

xi

imaged by the technique. (b) Cross section of the image along the white line, compared
directly to our model calculation as described in the text. In this case no filtering was
applied to the measured data, and no adjustable parameters were used in the calculation
to achieve the fit ..................................................................................... 57

Figure 18. (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, Chakraborty thesis] ........ 67

Figure 19. Three modes of operation of the microscope: tunneling mode (right), Kelvin
probe (middle) and charge accumulation (left) .................................................. 71

Figure 20. Fixed-tip measurement. Present plot shows measured charging as a function of
perpendicular magnetic field with tip fixed at a few nm above the surface. Clear features
at integer Landau level filling v are marked with arrows. (Here only the in-phase
component is shown) ............................................................................... 73

Figure 21. Both components of the signal plotted against inverse magnetic field. Different
set of data also taken with fixed tip similar to the measurement shown on Figure 19.

In-phase capacitance signal (top curve) and out-of-phase capacitance (bottom curve).
Dips in the in-phase curve and much smaller peaks in out-of-phase curve are marked my
numbers corresponding to integer LLS ........................................................... 74

Figure 22.(a) The circuit equivalent to the tip sample arrangement in the 3D-to-2D
charging case. (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 [Maasilta ,2003].
......................................................................................................... 76

Figure 23. Images of the out-of-phase charging signal are shown. Scan range for all
images is identical and equal to 730x730 nrn. Images are acquired at v=6, 6.1, 6.2, 6.3,
6.4 and 6.5 with corresponding B=l.38 T, 1.36 T, 1.34 T, 1.32 T, 1.30T and 1.28 T. The
images are filtered to remove the nanometer-scale scatter. The 5 aF contrast indicates ~7
% variation in density .............................................................................. 78

Figure 24. Direct images vs. two-dimensional Fourier transform. Top two rows represent
the out-of-phase images in progression from v=6 to v=6.5.with their corresponding 2D
Fourier transforms (FT) (same set of data as in Fig. 5.3.) Bottom row: enlargement of
v=6.5 direct image (left) with additional filtering and it’s FT (right). The circle indicate
the wavelength of about 160 nm predicted by theory for our sample. Peaks on this FT
image indicate the length of approximately 200 nm future in corresponding direct image
(left). ................................................................................................. 80

xii

Figure 25. Topography vs. charging image. 730 x 730 nm scan images of topography
(left) and out of-phase charging image (right) are shown on the top. Both images are
acquired at exact same location. Charging image shows the a stripe like structure of ~ 200
nm. Lefi upper comer of the topographic image shows some surface defect absorbate that
does not contribute to capacitances; the rest of the topographic image looks featureless.
Larger 4.3 pm image of the surface topography is also shown (bottom); typical growth
features of the GaAs surface of ~ lum long and ~ 5 nm in height are shown. The black

bar indicates the [I10] crystalline direction .................................................... 81

Figure 26. Schematic of the hydrogenic dopant atom. Wave function w(z) and the
potential V(z) are shown. R is the Rydberg energy and a B is the Bohr radius. ............ 85

Figure 27. Schematic diagram of the conduction-band profile of the RTD under the bias.
Tunneling occurs from 2DES through the ground state in the quantum well (for the main
resonance) or highly localized impurity levels at lower energies (Geim 1994) ............ 86

Figure 28. I(V) for the resonant tunneling structure 12 um across with the 5 doping in the
well at 1.3 K. The arrows indicate the main resonance (MR) and single donor resonance
(SDR). The dashed curve represents 20x magnification of the SDR. The inset shows the
schematics of the band diagram for the double barrier tunneling structure. Impurity
related states (IRS) also shown on the diagram not visible on the plot. This plot is
reproduced from reference [Geim] ............................................................... 88

Figure 29. (a) Schematic of the measurement for the study of the donor layer. (b)
Schematic of the conduction band of the system ................................................ 91

Figure 30. (a) Gated capacitance measurement. The three plateaus that correspond to
charge entering different layers in the sample. The area of the gate was ...(b)
Representative capacitance Spectroscopy measurement showing the charging spectrum of
a system of the dopant atoms below the tip. The black arrows in the plot show the
capacitance peaks. The applied excitation voltage was 16 mV and the temperature was
280 mK for both measurements. Due to different a—factors of the two measurements (see
text), plot (b) corresponds to probing the voltage range marked by the oval in part (a).
The voltages are measured with respect to the nulling voltage, where the gate or tip
effectively applies no electric field. Since the energy levels of individuals electrons are
too close to be resolved in these voltage ranges, we do not see individual single-electron
peaks in these plots ................................................................................. 95

Figure 31. Capacitance versus tip voltage. Three single tunneling events are marked on
the curve. The inset shows the comparison of the curve representative of the average of
these three peaks (filled dots) with the expected semi-elliptical function (solid curve), as
described in the text. Peak A is estimated to be due 10 unresolved electrons, as
indicated .............................................................................................. 97

Figure 32. Theoretical ground state electronic energy versus separation for various
numbers of electrons on a two-atom potential. These curves were calculated by Prof. J.F.

xiii

Chapter 1

Introduction

Semiconductor technology has revolutionized solid-state physics and its
applications. We use the products of these technological achievements in our every day
lives using computers, CD and DVD players, cellular phones, satellite television, and
much more. The most advanced semiconductors are based on atomic layer by layer
growth using molecular beam epitaxy (MBE); this technique allows the bandgap of the
material to be controlled with amazing precision. Using MBE, high-mobility two-
dimensional electron systems (2DES) can be created in GaAs and Si crystals. These
2DESS have been a focal point for both theorists and experimentalists, for more than few
decades. In addition to technologically important high mobility field-effect transistors,
2DESS have served as fascinating quantum system for the investigation of fundamentals
in electron interactions.

A 2DES can be produced at an interface of two distinct layers (called a
heterojunction) doped nearby with atoms that donate electrons. At low temperatures the
electrons at such a junction are confined to the lowest quantum state in the direction
normal to the interface. Biasing the gate electrodes on the top surface of the
heterostructure depletes the region below the gate. Similarly, electrons can be further
confined in the other directions to make dots, wires, and other shapes.

Many of the experiments on these systems have been what are called transport
studies. In these measurements, a current is passed in 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],
which became one of the most studied 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 F ETS are the most
common for electronic devices, GaAs/AlGaAs systems are often more attractive for
physics experiments due to their very high mobility. In recent years, many more
phenomena like single-electron transport, and conductance quantization in quantum point
contacts (QPCS) have led to firrther technological advancement [Sohn, Kouwenhoven,
SchOn]. The potential for exploiting these and many other quantum effects is leading to
new approaches to logic and quantum information processes. For example, Single
electron charges and spins can represent bits of data. Quantum information processing is
based on the coherent interactions of these “qubits” [Likharev, Bennett, DiVincenzo].

Despite the many experiments already performed on 2DESS and all that builds
upon the new science and phenomena made possible by these experiments, researchers
have been very curious to image the motion of electrons in 2D plane space, to see how
electrons actually move or interact [Topinka]. Obviously many details can be reviled and
understood by electron-transport measurements, yet most of the knowledge of electron
behavior in 2DEGS is still indirect when it is based on macroscopically averaged
quantities. Imaging techniques could be used to further revile the fascinating interplay
that occurs in the system on the nanometer scale.

First, the invention of the scanning tunneling microscope (STM) [Binnig 1982,

Binnig, 1999] in the mid-eighties, allowed scientists to directly view the pattern of atoms
on a material's surface. But, additional methods are needed to image the charge
distribution beneath the surface. The advancement of other scanning techniques in recent
years reviled new directions in the study of electron behavior in the confined
nanostructures in the heterostructured samples. The examples of those are: the 2D layer,
1D quantum wires, quantum dots QD and even the smallest quantum structure like a
single atom embedded inside the semiconductor.

Our interest in imaging the 2DES with scanning techniques was inspired by very
exciting theoretical work [Koulakov, Moessner]. This theoretical work predicts, based in
Hartree-Fock calculations, the formation of charge density wave like structure (CDW),
for the higher Landau level fillings at moderate magnetic fields. According to these
predictions the “bubble” or “stripe” phases are ground states of the 2DE system.
Electronic transport measurements seem to verify some of these theories, but do not
conclusively explain the nature of this interesting phenomenon. Thus, it makes the
problem very interesting for studying with a scanning probe.

A second experiment probes the quantum mechanics of semiconductor donor
impurities. Atomic scale electronics inside the semiconductor is one of the most popular
ideas proposed for the semiconductor-based quantum computer [Kane, Skinner,
Golding]. In particular, Kane [1998] and Golding [1999] have proposed quantum
computer systems based on the electronic states of defect atoms. Few recent tunneling
resonant experiments helped to understand the physics of donor-related tunneling
mechanism [Lok, Caro]. With regard to the donor layer study, we Shall investigate the

possibility of a successful application of our capacitance probe technique to learn about

the quantum states of the isolates donors as well as the interaction between the wave
functions of closely spaced dopant ions. Using the tunneling capacitance spectroscopy it
could be possible to measure the energy levels of the isolated dopant atom that binds an
extra electron to its proximity.

In present work, we shall focus on studying the GaAs/AlGaAs heterostructured
samples with a scanning technique called the Charge Accumulation Imaging (CAI)
[Tessmer 1998, Levi, Finkelstein]. The next section of this part of this chapter will
introduce the physics of 2DES in perpendicular magnetic field, and briefly describe the
charge density waves (CDWS) [Koulakov, Moessner, Aliegner, Glazrnan] theory. This
chapter will conclude with a brief introduction to the physics the donor layer system,
particularly the Si + 6-layer. More detailed theoretical background along with an
experimental findings for the CDWS will be presented in Chapters 5. Chapter 6 will
similarly present the theoretical background and experimental study of the donor layer
charging in the same sample. Chapter 2 discusses heterostructured samples and how they
are grown, it presents a sufficiently detailed discussion of the physics of heterostructures
that are essential for understanding many aspects of the physics presented in this work.
Chapter 3 mostly covers the details of the technique of CA1 and a subsection is devoted
to the description of the experimental apparatus. The detailed numerical calculation for
CAI spatial resolution will be presented in Chapter 4 [Kuljanishvili]. And finally, Chapter

7 will summarize the experimental efforts and present possible future directions.

1.1 2DES in presence of magnetic field and charge density modulations

The energy quantization of two-dimensional electrons in a perpendicular
magnetic field lies at the heart of the integer quantum Hall effect (IQHE). The regular
Hall effect for 3D bulk samples was discovered in the 19th century, when Edwin Hall
measured a voltage perpendicular to the direction of current flow in a conductor placed in
a magnetic field. This Hall voltage increases in proportion to the strength of the magnetic
field.

In the 2D case, 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 RC that gets tighter with increasing magnetic field B. This means the angular
frequency of revolution, called the cyclotron frequency (06: (eB)/m increases with the

field, where e is an electron charge and m is an electron mass. In 1980, it was discovered
by Klaus von Klitzing and coworkers that the Hall voltage of a 2D electron system is not
simply proportional to the applied magnetic field, but rather increases in distinct
quantized steps[von Klitzing]. These steps arise from the interplay of discrete quantum
states with magnetic field and disorder. These voltages were found to be related to
fundamental physical constants: Planck's constant and the electronic charge. Also, where
the Hall voltage is quantized, the 2D electrons carry current with no energy dissipation,
much like a superconductor does. These phenomena formulate the integer quantum Hall
effect for which their discoverers won the Nobel Prize in physics in 1985. A good
understanding of the integer quantum Hall effect is possible by considering how electrons
carry current when their energies are quantized into Landau levels (LL).

For the ideal two-dimensional system in a magnetic field B the quantum

mechanical problem was solved by Landau [Prange]. In the Landau gauge A= (0, Bx, 0)
and the magnetic field pointing in z direction the wave functions of the electron in the x
y-plane and the energy spectrum are given by equations (1.1) and (1.2) correspondingly

[Davis] .

 

_ _ 2
‘PNk(x,y)°cHN[x lkaCXP[-£ijzk—)]exp(iky), (1.1)

where N is the energy level index, the HN is the Hermite polynomial and the parameter

K is the magnetic length.
1
ENk = (N + Ejhcoc , for N=0, 1,2... (1.2)

where h Planks constant divided by 2n. These energy levels are called the Landau levels
(LL’S). The Spatial extent of the wave firnction is of the order of the magnetic length E =

OWE)” of electrons. Another important length scale is the cyclotron radius

RC = % = NZN +1, where V is the electron velocity. These LLS are equally spaced
C

by an energy given by Planck's constant times the cyclotron frequency th , and they are
labeled N =0, l, 2, etc. This spectrum is an example of the close analogy between a 2DES
in a magnetic field and a one-dimensional harmonic oscillator. Because EN,c is
independent of k, for a given N there may be many degenerate states. Similar quantization
of electron energy levels occurs in atoms, where an electron's closed orbit also requires
its wavelength to be only certain values. However, unlike atomic energy levels which can
accommodate only a few electrons each, a single Landau level in a sufficiently large

magnetic field may contain hundreds of millions. This degeneracy is given by eB/h, the

number of state per unit area per spin in each LL. At magnetic fields large enough to
resolve the LLS under realistic conditions, the electrons within each LL are generally
spin-polarized. The degeneracy of each Spin-resolved LL allows a 2DES with a density p
to fill v= ph/eB total LLS, where v is called the filling factor. For typical electron densities
on the order of 1015 m2, several Tesla are required to force all the electrons in the 2DES
into the same lowest LL, where v S 1. The filling factor can be adjusted either by varying
the electron density or the magnetic field.

In real samples, however, there is always a disorder, which broadens the delta-
function density of states of each LL. In addition, if we have electric field, it causes some
of the electrons to drift through the array of impurities. Solving the problem of a 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 lifted. Moreover, the states within each
LL may be classified into two categories. On the tails of the LLS, the states are localized;
they cannot carry electric current across a sample. In the center of the LLS, the states are
extended; they can carry current. See Figure 1. The formation of localized and extended
states can be understood by modeling the disorder as a smoothly varying potential
landscape of hills and valleys within the 2DES. In this background impurity potential
landscape electrons are suppose to move. At low temperature each electron trajectory can
be drawn as a contour in the landscape. Most of these contours encircle bills or valleys
and they do not transfer electrons from one side of the sample to another, they are
localized states. A few states in the middle, center of each LL will be extended across the
sample. At higher temperature, the electrons have more energy so more states become

delocalized and width of extended states increases. The density variations Shown in

Localized
-—— Extended

Energy

 

 

Density of states

Figure 1. Three LLS broadened by disorder. Extended (conducting) and localized
(insulating) states exist in the center and on the flanks of the LLS, respectively.

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/da, where p is the electron density per unit area and ,u 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 plays a significant role in certain aspects of our
measurement on 2D systems. The next section will describe the regime of higher LL

occupation in which a formation of CDW states is predicted.

1.2 Charge density wave formation in high Landau levels

In the previous section, we defined a quantity called the filling factor v = ph/eB.
The quantum Hall effect is qualitatively different from its classical analog. For 2D
systems, the Hall conductance plotted against the reciprocal of the magnetic field shows
distinct plateaus at certain values of v, instead of a straight line. For integer values of
v we have IQHE and in general fractional values indicate FQHE. The FQHE could only
be resolved at very high magnetic fields and very low temperatures and N_<_ 2. At weaker
magnetic field regime, or at higher filling factors (v > 4), the ground state of the system is
charge density wave (CDW). Based on Hartree-Fock calculations different groups of
theorists independently showed that at higher filling factors a competition between the
long-range Coulomb repulsion and the Short-range exchange attraction among the
electrons in the uppermost partially-filled Landau levels gives rise to a CDW sates

[Fogler, Koulakov, Shklovskii and Aliegner, Glazman].

There are two phases called the “bubble” phase and the “stripe” phase. These
phases have lower energy then a Laughlin liquid state which describes the FQHE.
Calculations Show that electrons lower their energy by forming clusters or “bubbles”
arranged on a triangular lattice. By 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]. This stripe CDW consists of alternating rows of filled and empty states of
the uppermost LL. Next the stripe CDW gives way to a triangular bubble CDW solution

as the partial filling of the valence LL moves away from one half again. The bubbles and
stripes both are separated by ~3Rc, which is approximately 150 nm for our sample at the

relevant magnetic fields.

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. All three phases together are called the
charge density wave phase, Shown in Figure 2. Our experimental investigation
was specifically targeting these ~150-160 nm scale structure. To probe directly for these

charge density wave patterns we employ the CAI scanning probe technique.

10

 

 

@tfifi it s- «it» . . .
233C e@?% '3' «It» .

I E“ fikfi .3 a' {é o . .

(a) (b) (c)

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

11

1.3 Donors layer inside the semiconductor heterostructure

Another system that could be conveniently studied using a non-scanning version
of our CAI technique is the donor layer of Si atoms that are present in the
heterostructured samples. The layer is 6-doped which means that the dopants are confined
in an atomically thin layer but randomly distributed within the layer as opposed to bulk
doping were dopants are everywhere in the material. More details about the fabrication
process will be described in Chapter 2.

Generally, our local probe technique could be considered as a new method for
locating and testing the basic properties of individual atoms or donor impurities. A single
dopant atom imbedded in a semiconductor crystal represents a smallest possible
semiconductor device. At low temperature, in general, the electron or holes can be
localized at the parent donor or acceptor. The quantum state of this charge has the
potential to be the basic building block of the future quantum computer as noted on page
2 of this chapter. Therefore, the ability to characterize the charge states of the dopant
atoms could be extremely important for building functioning semiconductor based
quantum computer.

The idea of quantum computing has generated great deal of interest in the
research community [Nielsen]. Solid state quantum computing systems are particularly
attractive because they allow for the possibility of scaling up the number of quantum bits
(qubits) to ~105, with the prospect of integration with the state-of-the-art semiconductor
device fabrication technology. With regard to candidate systems, qubits constructed from
semiconductors such as silicon or gallium-arsenide show considerable promise [Kane,

LOSS, Smelyansky]. Coherence in quantum state is a very critical issue. So naturally, the

12

parameter like spin (well isolated from the environment) will provide a long decoherence
times for the system but small length scales put limitations on such spin-based qubits.
For example, the Kane computer utilizes qubits based on the nuclear Spin of phosphorous
donors in Si [Kane]. Because of the fact that the exchange attraction between qubits falls
of exponentially the separation between donors is required to be 10-20 nm, which
represents challenges in process of making the device. An alternative approach makes use
of the electronic charge degree of freedom quantum dots or in dopant atoms [Chen,
Stieveter]. In this approach, the mechanism for qubit-qubit coupling is the direct
Coulomb interaction.

The properties of the dopant atoms in semiconductor could be well described by a
hydrogen like model. In gallium arsenide, the corresponding Bohr radius of about 10 nm
sets the size of the wave function. This size is much larger than for the hydrogen atom
and is due to the scaling for the effective mass of the carriers and the screening effects of
the dielectric constant of the semiconductor.

For the sample we probed, the Si donor layer resides 60 nm below the surface of
the sample and 20 nm above the 2D conducting layer. We can study the system of the
dopant atoms with the capacitance tunneling spectroscopy technique as will be described
in details in Chapter 6. This donor assisted tunneling technique can give us chance to
study the donor atoms energy spectra, their interactions with each other and the effects of
correlation between the electrons and screening effects of the dielectric environment.

Chapter 6 will describe the scanning probe measurements that have been
performed on GaAs/AlGaAS sample with Si being a donor atom. We will present the first

attempts to study the capacitance spectroscopy of the donor layer inside the

13

GaAs/AlGaAs heterostructure in the tunneling regime and to locally investigate the

energy levels of the dopants atoms and the effects of the proximity of nearby neighbors.

14

Chapter 2

Heterostructured samples

Semiconductor heterostructures are semiconductor materials that are made of
layers with different compositions. Variations in compositions are used to control the
motion of electrons and holes through the band engineering [Davis]. In all later
discussions, we Shall assume few simple approximations. First, we shall use the standard
Simplification that conduction electrons are free particles with an effective mass. It means
that an electron at the interface of the heterostructure can be treated using an elementary
problem of a particle in a potential well. Secondly, the random nature of alloys is usually
neglected which allows treating them as crystals.

Many III-V materials have been studied for their properties but only a few are
eventually used in a heterostructure design. Since the active regions of heterostructure are
usually at or very close to the interfaces, they must have nearly perfect interfaces; hence,
interface properties are critical in forming a heterostructure. For example, it is essential
for materials in adjacent layers to have a similar crystal structure and their lattice
constants to be as close as possible. Moreover, surface roughness scattering will
compromise the mobility of the electrons. These conditions are met in gallium arsenide,
GaAs, and aluminum gallium arsenide AlGaAs (these are not chemical formula but
abbreviations). The sample that was use in the studies described in this thesis belongs to
this type of heterostructure.

AlGaAs has a wider bandgap than the GaAs, and a slight doping of the AlGaAs

layer produces an electric field, essential to bend the bands, forming a quantum well at

15

the interface; the electrons that are given up by the dopant atoms form the 2DES.
Heterostructures 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 to 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 these studies were grown by the molecular-beam epitaxy
technique by Loren Pfeiffer and Ken West at Bell Labs, Lucent Technologies [Pfeiffer,
West]. Molecular-beam epitaxy, or MBE, technique is very simple in its essentials. Here
we briefly describe the principles of this method.

Figure 3 shows a schematic picture of the MBE grth chamber. The substrate is
heated on a holder, which is situated inside an ultra-high vacuum (UHV) chamber with a
pressure of about ~5x10’H mbar. The elements that compose the heterostructure 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 UHV environment are the following: First, it is important to prevent
contamination of the substrate. Secondly and more importantly, low pressure is

advantageous for molecular flow without collisions with other molecules. In this regime,

16

  
   
  

 

Electron gun

Shutter Substrate

 

substrate holder

Screen for viewing
RHEED

Figure 3. Simplified schematic diagram of an MBE apparatus showing Knudsen cells for
Al, AS and Ga with shutters. The sample sits on a heated holder, which is rotated during
the grth of the sample. The RHEED control system is also shown [Davies].

17

the mean free path of a molecule is much larger than the length of the chamber, so
molecules do not suffers collisions in their path until they meet the substrate [Davis].
This is the Knudsen or molecular-flow regime of gas, and the furnaces are called
Knudsen or K-cells. 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 grth
across the substrate, the sample holder is rotated during growth. If dopants need to be
added, additional cells are used. A common dopant for III-V systems, such as GaAs is Si.
Atoms of Si are group IV in periodic table and could act in principle as donor, or
acceptor, in a group III-V compounds. In practice, when grown on (100) surface, Si
atoms act as donors, giving up electrons. Our sample is grown with Si by modulation
doping, also called remote doping, which will be described in Section 2.3. Moreover, the
dopants in our sample are in a “delta” configuration. For 8-doped materials, the dopant
layer is very thin, close to a monolayer. We will come back to the discussion of the Si
donor layer in our sample in Chapter 6.

Few important steps need to be perfected for ensuring growth of high quality
samples. The wafers must be ground with extreme purity, which requires starting
materials not to be polluted by the K-cells. The background pressure needs to be kept low
to reduce contamination as well as ensure molecular-flow regime. The flux from K-cells
must be uniform to minimize the variations in composition across the wafer. The
temperature of the furnaces as well as substrate temperature needs to be closely

controlled. For example, defects will not have time to be removed by annealing at low

18

temperatures, while unwanted diffusion will occur and blur the interfaces if temperatures
too high.

MBE is a very Slow process, and grows material about 1 pm per hour. The growth
is monitored using one of the common UHV diagnostic techniques called reflected high-
energy electron diffraction, or RHEED. With this technique, growth can be counted
precisely in monolayers, and it also reveals the structure of the surface. MBE machines of
the type shown in Fig. 3 produce very high quality GaAs/AlGaAs samples. The quality of
a semiconductor sample is determined by its electron mobility. In the next section we will

describe the meaning of this “parameter”.

2.2 Electron mobility

Any motion of free carriers in a semiconductor leads to a current. This motion can
be caused by an electric field due to an externally applied voltage. This mechanism is
usually referred to as carrier drift. Mobility describes the ease with which electrons move
in materials in response to an electric field [Ashcroft, Davis]. For completeness, we
should mention that in addition, carriers also move from regions of high density to the
regions where the density of the carriers is lower. This transport mechanism is called
carrier diffusion or thermal diffusion; which is associated with random motion of the
caniers. The total current in a semiconductor equals the sum of the drift and the diffusion
current. For the rest of this section we will concentrate only on carrier drift.

When electric field is applied to a semiconductor, the electrostatic force causes
the carriers to first accelerate. However, they soon reach a constant average velocity, 1),

due to collisions with the impurities and lattice vibrations (phonons). The ratio of the

19

velocity to the applied filed is called the mobility. In absence of an external electric field
and considering a large numbers of carriers, the overall net motion of the carriers is zero.
When an electric field is applied, there is a net motion along the direction of the field. Let
us now analyze the carrier motion considering only the average velocity, u of the carriers.

According to Newton’s second law, the acceleration of the carriers is proportional to the

d6?)

total forceF = m5 = m—Er. The net force consists of the sum of the electrostatic force

qE , and the scattering force due to the loss of momentum at the time of collision. The

scattering force on average equals the momentum divided by the average time I between

the scattering events (collisions) and has the opposite direction of the electrostatic force.

Therefore, the total force isF = qE — fl . Here q is the charge of the carrier particle.
r

Combining this equation with the Newton’s second law, we get the following result:

 

-_ d6?) m6?)
qE—m dt + z' . (2.1)

For the steady state situation where the particle has already accelerated and has
reached a constant average velocity, the first term in the right part of the equation (2.1)
equals zero. Therefore, the velocity is proportional to the applied electric field and the

mobility is defined as the velocity to field ratio.

I(VH CIT
= _. = _. 2.2
x1 l El m ( )
Thus, according to the equation (2.2), the mobility of the carrier in a
semiconductor is expected to be large if mass is small and the time between scattering

events is large. Lastly, we have to take into account the effect if the periodic potential of

the atoms in a semiconductor crystal and use the effective mass m. rather then a fiee

20

particle mass. The unit of mobility is cmz/ V-sec. All these approximations were made

according to the Drude model.

2.3 Formation of 2DES and modulation doping

AS mentioned earlier a 2DES is formed at the interface between two
semiconductors grown in a UHV chamber under precisely controlled conditions using
MBE. In this section, I shall present a brief discussion of the formation of the 2DES in
the GaAs/AlGaAs heterostructure with the modulation doping technique [Davies,
Prange]. Using MBE, an atomically perfect layer of GaAs is grown, followed by a layer
of AlGaAs. The two materials have nearly the same lattice and dielectric constants.
Because of the difference in a band gap of the two compounds, the conduction band has a
step at the interface. An electric field perpendicular to the interface attracts electrons into
a quantum well created by the field and the interface, which will result in an
accumulation of the electrons trapped in the quantum well. The electric field
perpendicular to the interface is achieved through doping. Let us describe these processes
in more details.

The obvious way to introduce the carriers into the semiconductor compound is to
dope the regions where electrons or holes are desired. If electrons are needed, the
semiconductor compound is doped with n-type material. The question is how to minimize
the effects of ionized dopants that are left behind. They can scatter electrons due to the
Coulomb interaction and thus spoil the propagation within carefully grown structure. The
solution is remote or modulation doping, where the doping is grown in one region but

carriers (electrons in our case) migrate to another. This is Shown on Figure 4 for a

21

IFAMSaAs

 

 

 

 

 

 

 

6 EB OED
undoped
GaAs
ABC
2.-
w/ave function
5%
______ 4/ e
(96963969969 \
2DES

Figure 4. Conduction band around the heterojunction between n-AlGaAs and undoped
GaAs showing how electrons are separated from their donors and form 2D electron
system. Electrons can diffuse away from the donor ion and some cross into the GaAs.
There they lose energy and become trapped because they can not climb the barrier A E c.

22

heterojunction between n-AlGaAs and undoped GaAs. Here the AlGaAs is deliberately
doped n-type, so that it has mobile electrons in its conduction band. GaAs is intrinsically
p-type, with few holes in its valence band. The electrons in the conduction band of
AlGaAs migrate and eventually fill these holes on the top of the GaAs valence band.
However, some electrons usually end in states near the bottom of the GaAs conduction
band. The, positive charge left on the donor ions attracts these electrons to the interface
and bends the energy 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 Fermi level of the 2DES is aligned with the Fermi level of the AlGaAs.
Due to the abrupt step in the conduction band the electric field can only squeeze the
electrons against the interface where they are trapped in a roughly triangular well. This
well is typically about 10 nm wide at the energy of the electrons, and the energy levels
for motion along 2 direction are quantized Similarly to those in a square well. The density
of electrons in the quantum well is determined by the dopant density. For sufficiently low
doping, only the lowest level is occupied. All electrons then reside in the same state for
motion in z direction and move freely in other two directions x and y .This is how 2DES
is formed. Figure 5 Shows the energy diagram for the resulting quantum well.

An important part of the remote or modulation doping is to place the donors in the
AlGaAs physically as far away fiom the interface as possible. This will further reduce the
scattering Of electrons in the well by the positive donors and increase the mobility of the
2DES, therefore the refinement was made by leaving a spacer layer of undoped AlGaAs
between the n-AlGaAs and GaAs. But there is trade-off: an increased mobility of the

2DES in this case reduces the density of the electrons in the 2D system. Thus, the

23

optimum spacer thickness depends on the purpose of the heterostructure. For example,
high mobility is vital in many physics experiments including present work. On the other
hand, for very high performance electronic devices based on heterostructures, it is most
valuable to have very high density of electrons in the system; in that case a spacer layer
placed between n-AlGaAS and GaAs is extremely thin. A good example for such devices

is modulation doped field-effect transistors (MODFETS).

24

 

Figure 5. Energy-level diagram of a GaAs/AlGaAs heterostructure. The donor electrons
travel from their host Si atom and occupy the first conduction sub-band of the potential
well. The Fermi level is shown as a}; The spacer layer is not shown [Davis].

25

2.4 Our sample

The sample and the tip geometry used in our experiments is shown in Figure 6.
The sample is a GaAs/AlGaAs heterostructure grown with MBE described in section 2.1.
The 2DES is formed at the interface of GaAs and AlGaAs using a modulation doping
technique. The “8-doped” layer of Si was fabricated with MBE and has an averege
density of 1.2 x 10'2 cm'z.

There is no ohmic contact to the 2DES; instead the bottom of the sample has a
metallic substrate which separates 2DES from it by a tunneling barrier. The metallic
substrate is a heavily doped n-type GaAs, and the tunneling barrier is composed of layers
of undoped AlGaAs. An excitation voltage is applied to the metallic substrate, in
response to which electrons tunnel into the 2DES just below the apex of the tip. When tip
is biased elecrtons are forced to tunnel further into the donor layer above the 2DES. As a
result we have an image charge capacitavely induced on the tip, which is measured. The
details of the technique are discussed in the next Chapter 3. The 2DES sits approximately
80 nm below the surface of the sample and 40 nm above the metallic substrate. The

mobilty of the sample is 105 cmZ/V-sec, and the average density was 2 x 10H cm'z.

26

”\x

 

Figure 6. Tip-Sample arrangement of our experiment.

27

Chapter 3

Experimental set up and the geometry of the technique

Several techniques can be used to image local electronic structure. To examine
conducting surfaces, the most powerful technique is scanning tunneling microscopy
(STM) [Binnig 1982, Binnig 1999]. However, the STM cannot be used in our
experiments because we study electronic systems that are buried inside the sample. The
fact that for GaAs/AlGaAs the 2DES typically exists below the surface 40nm-80nm
represents a major difficulty to STM and other standard scanned probe techniques.
Nevertheless, we are able to probe this system with our technique called charge
accumulation imaging (CAI) [Tessmer 1998, Levi, Finkelstein]. The measurement
mainly consists of monitoring the induced charging of a sharp metallic tip that is
equipped with a charge sensor in response to an ac excitation applied to the sample. The
electric field lines emerging from the sample are terminated on the apex of a sharp tip. In
this way, charge flowing in and out of the sample capacitatively induces charge to flow in
and out of the tip. Therefore, by monitoring this signal, we can locally map out the
mobile charges accumulated right under the tip inside the conductive 2D layer. The donor
layer situated above the 2DEG, closer to the surface of the sample, could also be studied
using this technique.

A detailed description of our technique will be presented in section 3.1 of this
chapter. The design of our scanning head is very similar to a standard scanning tunneling
microscope. The details of the scanning head and the brief description of the cryostat

operation will be presented in sections 3.2 and 3.3 correspondingly.

28

The essential part of our experimental set up is a cryogenic charge sensor constructed
from high electron mobility transistors (HEMTS) [Mimura], which is attached directly to

the tip. This circuit description will be presented in section 3.4.

3.1 Charge accumulation imaging

Charge accumulation imaging is basically a measurement Of a capacitance signal
translated into voltage by a charge amplifier. One can think of the tip-sample
combination as two conducting plates of a capacitor. As shown schematically in Fig. 7, as
we apply voltage to the sample we immediately induce an image charge on the tip. This
charge is detected by a censor circuit, which is directly attached to the metallic tip. Our
technique builds upon one that was first developed and implemented by S. H. Tessmer
and coworkers [Tessmer 1998] at MIT, with a few significant differences. For instance,
one of the most important differences from the earlier experiment is that we do not have a
direct ohmic contact to the 2DES. Instead, we use vertical tunneling geometry from a 3D
substrate into the 2D layer. For typical samples, the 3D metallic substrate is located about
40 nm below the 2D layer. Fig. 8(a) shows the schematic picture of the 3D-to-2D
tunneling arrangement. The advantage of not having an ohmic contact to 2D is that our
measurement becomes insensitive to bulk conductivity of the 2DES. This allows us to
probe the system even when it is mostly non-conductive. For example, an insulating state
results at certain values of magnetic field when the 2DES is close to an integer Landau
level filling. In these cases, having the direct contact to the 2D layer would not allow
electrons to enter the bulk of the system and subsequently accumulate right beneath the

apex of the tip; therefore, our local signal would disappear.

29

 

Figure 7. Schematic of the CA1 technique. This is a capacitance technique where the tip
and the sample are coupled to each other through the electric field, like two plates of a
capacitor. To be more Specific, we apply voltage to the bottom plate (sample) and
monitor the image charge that is induced on top plate (tip).

30

AS Shown in Fig. 8(a) our method of measurement with a vertical tunneling geometry
does not suffer from this problem. Electrons can enter the 2DES vertically without
traveling horizontally through the bulk of the 2DES. We can apply the method in two
ways: (1) we can scan the tip to acquire the map-like images of the electron density
variations; or (2) while we sweep the sample voltage or magnetic field, the tip could be
fixed at a few nanometers away from the surface to acquire a capacitance profile. In
addition, our technique is also sensitive to electron correlations which where found to
give rise to a so-called Coulomb psuedogap that is pinned to a Fermi level of 2D system
[A.H. MacDonald 1997]. This effect can be measured only in tunneling experiments as
demonstrated by Ashoori et a1. [Ashoori 1990] and Eisenstein et a1. [Eisensteinl992].
Our method can be considered as locally resolved version of those pioneering
experiments [Tessmer 1998, Levi, Finkelstein].

All the concepts of our technique that have been described above could also be
applied to image the donor layer, above the 2D system. A schematic picture of the
arrangement including the donor layer is shown on Fig. 8 (b). The O-doped Si donor layer
is located at 60 nm below the surface of the sample and 20 nm above the above the 2D
layer. At sufficiently positive voltages on the tip electrons enter the 2D system and then
tunnel further into positively charged Si ions of the donor layer. Thus, we can study the

tunneling spectroscopy of the system of donor ions.

31

a)

 
  
 
  

image
charge

3D metal

Figure 8. Schematic of the vertical tunneling geometry. (a) In the 3D-to-2D case we
probe the charging of the 2DES. (b) In the 2D-to-donor layer case we probe the charging
of the donor layer. In both cases an ac excitation is applied to the 3D metallic substrate
which resides below the tunneling barrier. The charge tunneling into the 2DES or to the
donor layer induces an image charge on the tip, which is measured by a cryogenic charge
sensor circuit.

32

3.2 The scan head

There is a variety of scanning probe microscopes that are used nowadays for
studying a wide range of problems. For example, a scanning tunneling microscope (STM)
[Binnig 1982, Binnig 1999] can achieve atomic resolution surface topography or high
quality spectroscopy. These days STM is also used to manipulate atoms [Eigler, 2000].
Achieving these goals requires very high stability of the scan head, which leads to small
piezo lengths and therefore small scan ranges.

The goal of our experiment is to study physics of nanometer-to-micron size range
structure. We use the standard Besocke design STM head [Besocke] schematically
pictured is in Fig. 9. The method of the kinetic approach system is used here. The main
principle of its work lays in so-called inertial translation or slip-stick mechanism. The
approach of the sample is achieved by utilizing a sawtooth-shaped or parabola waveform
to drive the piezo tube. The waveform of the signal forces carrier-piezo tubes to slowly
expand and then quickly contract their length by which they produce the translations in
one or more directions. Repeating this hundreds of times allows the sample holding
ramps to move with low fiiction along the slope. This results in the rotation of the ramps
clockwise and moves sample in z direction closer to the tip. Similarly, sample can be
moved away from the tip when ramps move in the counter clock direction. With the use
of the same mechanism sample can be moved in x and y lateral direction for the course
positioning purposes. In scanning mode the scanning piezo-tube is moved over the
sample in x-y direction. This is achieved by applying voltages to different quadrants of
the scanning piezo tube which causes it to bend in x and y direction during the scanning

process.

33

 

Figure 9. Schematic of the Besocke type scanning head. Three carrier piezo tubes are
supporting the sample holder ramps. The sample sits upside-down rigidly mounted on the
sample support disk. The outside carrier tubes and the middle scanning tube are all
attached to a bottom plate. The scanning tube in the middle holds a small chip with the tip
and sensor circuit. (The circuit is not shown.)

34

In order to increase the scan range of our microscope (scan head) we used very
thin 0.125 inch piezo electric tubes that enhance the piezoelectric response. The length of
one inch (2.54 cm) was chosen for stability reasons. This design fulfills all our
requirements: extreme thermal stability during almost 100 hours of experiment and
large scan range up to 7 um at He3 temperature. Due to the compact design of the scan
head, the working area is very small which requires extra caution and adds some
challenges while assembling the microscope and the circuit.

The actual picture of the microscope is shown on Figure 10. It has three main
stainless steel parts: bottom plate, body and door. The bottom plate holds three carrier
piezo-tubes and one scanning piezo-tube. All piezo tubes are made of the same lead-
zirconium-titanate material of equal length and outer diameter. The only difference is the
scanning piezo tube’s inner diameter that is larger, or in other words it has thinner walls.
This gives our otherwise short scanning tube relatively large scan range of ~38 um at
room temperature and ~ 7 urn at He3 temperature. Equal length piezo-tubes are important
for thermal stability of the microscope, so that they expand and contract identically as the
temperature changes. All four piezo tubes are soldered to the bottom plate. Three carrier
piezo tubes hold copper feet with stainless steel balls attached to them. The balls are
highly polished for low fiiction performance during the approach of the sample. The
center of the scanning piezo-tube has a miniature L-shaped copper cap soldered to it. This
copper piece holds the sensor chip with the tip rigidly attached to it. Details of the sensor
circuit will be presented later in this chapter in Sec.3.4. Two other stainless steel parts of
the microscope are mostly used for the protection purposes and for filling the inside

spaces of the probe for maximum liquid He3 displacement. Finally, when all parts are in

35

 

Figure 10. Photograph of the actual microscope. The front door of the microscope is
removed and the sample holder ramp is not shown.

36

place the microscope gets rigidly attached to an eight foot long probe for subsequent

lowering of the unit inside the cryostat.

3.3 The Cryostat

Once the scan head is attached to the end of a long stainless steel probe, we place
the sensor chip with a cryogenic circuit inside the scan head, mounted on the L shaped
copper holder on the top of the scanning piezo tube. It needs to be handled very carefirlly,
because the microscopic Sharp tip is already epoxied on the chip. Lastly, the sample gets
placed inside the microscope. The probe is then loaded onto the top of the He3 cryostat
and is inserted Slowly. The process of lowering the probe takes approximately 5-6 hours.
It is important to protect the fragile ceramic material of the piezo tubes from thermal
stress; the piezo-tubes may brake or become depolarized. The cryostat is equipped with a
superconducting magnet surrounding the sample space. A magnetic field up to 10-12 T
can be achieved. Once the microscope is all the way down, it is directly in the middle of
the magnetic solenoid. Thus, the magnetic field lines inside are lined up exactly
perpendicular to the sample. The schematic picture of this arrangement is shown in
Figure 11.

The principle of the operation of a He3 cryostat is relatively simple. The whole
cryostat is cooled in stages from room temperature to liquid He4 temperature of 4.2 K.
First, we cool the cryostat down to 77 K with liquid nitrogen. Then we evacuate the
nitrogen liquid and fill the cryostat with liquid He4 (LHe4). The sample space is kept
under vacuum during the cool-down procedure. The cryostat can be maintained at liquid

He4 temperature of ~4.2 K for months, by filling it up with LHe4 weekly.

37

Probe entry port

 

 

 

 

 

 

 

 

Top loading
Gate Trantsyfegotube probe
valve
, Sliding
:4 seal port
0
O
0
Helium
exhaust Valve
port
Ma net -. Scannin
g g \ headg
uid delivered to
\ blo rn i01‘ the LHe

 

reservo

Figure 11. Schematics of the cryostat (left) and top loading probe (right) are shown.
During the experiment probe resides inside the cryostat and scanning head is situated at
the bottom of the cryostat surrounded by superconducting magnet.

38

Below I Shall describe, briefly, the procedure of the cooling down from 4.2 K to
the base temperature of 0.270 K. Outside the sample space and attached to its outer wall
is a small chamber called the l K pot. First, the 1 K pot is filled with LHe4 and pumped
to maintain it at around 1.2 K. The sample space is equipped with a big charcoal
adsorption pump, called the sorb.

We assume that He3 gas has been released already from the storage cylinder into
the sample space. At this point and sorb temperature is in equilibrium with the
surrounding temperature of liquid He 4. AS a result, gaseous He3 gets trapped inside the
cold massive surface area of the porous sorb. We then start heating the sorb up to 32 K,
causing it to release the He3 gas. The He3 condenses onto the cold walls of the 1 K pot
and drips down to the bottom of the sample space. In order to avoid overheating the sorb
and balance the temperature, we use a heat exchanger which cools the sorb with LHe4.
This also helps us to maintain temperature below 1.5 K. The process of condensing the
He3 gas takes about 45 minutes to 1 hour. Finally, the base temperature between 0.270 K
and 0.290 K is reached by removing the sorb beat; this allows the sorb to pump on the
liquid He3. The design of our experiments requires long hours of stable base temperature.
Therefore, we run our system in “continues fill” mode where the l K pot helium input
valve is slightly open for continues flow of LHe4. With practice, the level of opening in
the 1 K pot input valve can be adjusted according to its effect on the temperature of the 1
K pot to achieve the lowest base temperature. The sample Space remains at the base
temperature for about 100 hours, until it runs out of liquid He3. At this point all the He3

molecules are stuck inside the adsorption pump; but they are recycled for next data run.

39

3.4 The cryogenic charge sensor

There are several types of transistors that could be used for constructing a charge
sensor, but most of them are not suitable for our experiments due to incompatibility with
low temperatures. For example, bipolar junction transistors (BJTS) depend on thermally
activated carriers. These BJTS would be impossible to use because their charge carriers
will be frozen out. In general, field effect transistors (FETS) are used for cryogenic
applications. However, the power dissipated by such devices is typically in the miliwatt
range. This would be large enough to heat up the sample surroundings, shorten the He3
hold time of the experiment, and may cause thermal instability. To overcome these issues
we use high-electron mobility transistors (HEMTS). These HEMTS are a kind of field
effect transistors that were invented at Fujitsu Laboratories in Japan by Takashi Mimura
[Mimura]. These transistors are GaAs/AlGaAs based modulation doped heterostructure
devices.

For our charge sensor, the key advantage is the low input capacitance of HEMTS
in comparison with the standard FETS, as will be discussed in the following paragraphs.
Moreover, these HEMTS can be operated in the ohmic region, far away from saturation.
As a result, the power dissipation is decreased by 2-3 orders of magnitude compared to
saturation mode operation [Urazhdin, 2002].

Figure 12 Shows the schematic picture of the cross section of the HEMT. High
electron mobility transistors are fabricated using a heterojunction of a highly Si doped n-
type AleaI-xAs layer, n—type GaAs and the undoped GaAs. The value of x is
representative of a fraction of AlAs in the compound, which is usually 0.3. The epilayers

are grown by molecular-beam epitaxy (MBE) on a (100) GaAs substrate. The electrons

4o

Source Gm Drain Schottky contact (Ti- Pt-Au)
. .-.._.:-,.-..;-.::1~—Ohmic contact (Au - GelAu)

 

 

 

 

 

 

 

Semi-Insulating
I (100) GaAs substrate

 

Figure 12. Schematic picture of the cross section of a HEMT [Mimura,l 982].

41

are generated in the n-type AlGaAs drop into the next GaAs layer form a depleted
AlGaAs layer. This happens due to the band-gap difference between AlGaAs and GaAs.
This results in a steep canyon in the conduction band on the GaAs side where the
electrons can move without colliding with the ionized dopants from which they have
originated. As you may have noticed already, this is a brief description of how 2D system
is usually formed in semiconductor heterostructure. Indeed, in this case the 2D system is
the conducting channel of the HEMT.

The schematic of our cryogenic charge sensor circuit is Shown in Fig.13.(a). 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 role of the bias
HEMT is to act as a very high resistor, preventing the charge induced on the tip from
leaking to the left. The standard capacitor Cs is used to subtract away the stray (or
background) capacitance, arising mostly from the electric fields from the sample which

do not terminate on the apex of the tip. This is accomplished by applying an AC,

excitation Vbalance to Cs of the same amplitude but exactly 180° out of phase with the
applied sample voltage Venue. The typical value of the Cs is about 20 fF (femtoFarad).

To describe our circuit in a little more detailed way let us use a simple model shown in
Fig. 13.(b). We have already assumed that bias HEMT is tuned to insure the big resistance

preventing current from leaking away. We concentrate now on measurement HEMT.

The circuit acts like a Simple voltage divider. Here Vin is the input voltage, in our case
Vacue, and V0”, is proportional to the measured voltage Vmea, / G. Here G is the gain

factor. In addition, C,-,, is the input capacitance, in our case it is a capacitance

42

 

 

 

( C. ‘ vbeianoe VI" ' Vexdte
I L
| , ,
"emu-i 1
o- H E MT +To lock-In

 

 

Vout —'l Vexdro VouE’ Vmeas
O Z2

 

 

 

 

 

 

 

 

 

[ sample }

 

Figure 13. Schematic of the charge sensor circuit. Cs is the standard capacitor and Vin is
the rms value of the applied AC excitation to the sample. The AC excitation applied to
Cs, is of the same amplitude but 1800 degree out of phase with the excitation voltage that
is applied to the sample. The charge sensitivity of the circuit is 0.01 electrons/VH2.

43

between the gate and the source-drain channel of the HEMT. Lastly, Thus, Cmea, is the

tip-sample capacitance. As it is proportional to the charge, Cmeas is the quantity that we

need to measure. Thus, we can write the following equation:

Vout = Vmeas = 22
Vin GVexcit ZI+ZZ

 

 

 

where Z1=.—l— and Z2 =_1
‘ meas “0cm

Typical value for Z 2is much smaller than Z1 therefore we can neglect Z 2 in the
denominator of above equation. The voltage measured by the lock-in amplifier mm is

given by the following relationship.

C
__ meas
Vmeas "GVexcit °
C in

 

Here G is the gain of the circuit, which is usually ~1.The key point, is that low input
capacitance means larger Signal. The input capacitance of the HEMT is only 0.3 pF. It is
important to keep in mind that the stray capacitance adds to this parameter. In other
words, the self-capacitance of the entire device must be ~0.1 pF. This sets the 1 mm as
the size limit of the circuit.

Our circuit is ultimately limited in sensitivity by the Johnson noise in the bias
HEMT which gives a charge sensitivity of 0.01 electrons/VH2. However, if input
capacitance is too large it directly reduces the value of the signal so that other sources of
noise will start to dominate, decreasing the sensitivity. The drain current-voltage
characteristic curves for different gate voltages of a typical HEMT are shown in F ig.14.

We operate our circuit in the low voltage regime far from saturation, indicated by a

44

circle. Because HEMTS are susceptible to damage from static charge, we check our
HEMTS before we even start room temperature testing. Using a device called a curve
tracer, we check directly for the characteristic fan-shaped family of curves shown in

Figure 14.

45

 

    
 
   
  

&

 

3

 
 

Drain current I, (mA)
I
O
O
<

— 05V

 

 

 

 

 

 

I0 I _ — 0.4V
_o.3v
0.2V
0.5 to It: 2.0 25 30
DraanoItageVDsM

Figure 14. Drain I-V characteristics, at different gate voltages VGs, of a typical HEMT

[Mimura,l982]. The circle marks the voltage range in the regime of experimental
operation.

46

Chapter 4

Numerical calculation for CAI spatial resolution.

The tip geometry can be a critical factor for electric-field—sensitive scanning probe
microscopies. This is true for techniques such as scanning capacitance microscopy
[Martin, Abraham, Huang], scanning single-electron transistor microscopy
[Yoo,Yacoby], and charged-probe atomic force microscopy [Erikkson,Topinka], and in
addition to our own subsurface charge accumulation imaging technique. The samples of
interest typically consist of conducting layers or nanostructures buried beneath a
dielectric. To interpret the data and the influence of the tip geometry, researchers often
rely on modeling in which the tip is taken to have an ideal shape such as a perfect cone or
Sphere [Belaidi,l997]. However in many cases the tip may be better described by a less
regular shape. For example, the end of the tip may break due to contact with the surface
resulting in a truncated cone. Moreover, the tip may be bent or mounted in a non-
perpendicular angle so that any model that relies on cylindrical symmetry would be
inappropriate. We have developed a straightforward numerical method that can
conveniently model a tip of arbitrary shape [Kuljanishvili, 2003].

Our method is based on a boundary element approach — inspired by the numerical
method used to model the charging patterns within a GaAs-AlGaAs two-dimensional
electron layer probed with the CAI technique [Tessmer, 2002]. As shown in Figure
15.(a), we approximate continuous tip and sample electrodes as arrays of point-like
conducting elements. By considering the Coulomb interaction among all the elements
and using image charges to account exactly for a dielectric overlayer, we directly solve

for the complete self-capacitances and mutual capacitance for the tip-sample system.

47

Here, we introduce the method and demonstrate its utility by calculating the mutual
capacitance of a realistic tip and sample used for a CAI measurement. We then compare
the resulting modeled spatial resolution to experimental measurement. Below we will

describe in details the architecture of our modeling.

4.1 Electrostatic framework

Following basic electrostatics [Jackson], for a system of n conductors each with

potential V,- and charge Q), the voltage is a linear function of charge:

’1
Vi = 241'ka , (4.1)
k=l

where the elements A ,7, represent a potential matrix A: and give the voltage of point i due
to unit charge at conductor k and all other charges equal to zero. The matrix is
symmetric: A”, = Aki. In our typical calculations, roughly 25% of the n conductors

correspond to tip points and 75% correspond to sample points. Inverting Eq.4.1 yields an
expression for the charge on each conductor, which depends highly on the surrounding

conductors and dielectric medium:
n " " I
Q; = Zcika, C: A‘ (4.2)

k=l

where the C”, are the coefficients that make up the capacitance matrix C.

48

Our scheme follows directly from equations (4.1) and (4.2). We first construct If

by considering the Coulomb interaction among all pairs of points and invoking image
charges to account for bound charges at the dielectric surface, as described below. C is
then found by inverting 14A. The tip-sample mutual capacitance is a quantity of
considerable importance for electric-field-based microscopies. Once we have solved for
all Cik, we can use Eq. 4.2 to calculate the mutual capacitance. Clearly, the calculation
must treat tip points differently from sample points. In constructing A: three cases arise

depending on the identity of each of the pair of points, as detailed in the following

subsections.

4.1.a. Case I: tip-tip
Here we consider both i and k to be conductors in the tip. The potential at i due to
unit charge at k must also consider the image charge at k' to account for the dielectric, as

shown in Fig. 15.(b). In this case the image charge solution [Jackson, New York 1975] is
, { 1c — I]
q = — q
K +1 ,

where K is the dielectric constant, q is a unit charge located at point k and q' is it’s image

located at the symmetric position about the vacuum-dielectric interface, k'. The potential

matrix element then becomes

1 K—1 1
A- =—-— — —,
'k R (KHIR'

where R and R' are distances from point i to points k and k' respectively.

49

(b) "

 

dielectric

 

 

   

 

 

   

R’ s
conductin . ,
layer _" .. q k -_.--
sample sample
R/Id
d
O O O O I 0 O I I I O I O . ' - - - .
Cik I R Q k
sample sample

Figure 15. (a) Schematic picture of the finite element approach, showing the tip-sample
geometry used to model charge accumulation imaging measurements. The tip is
represented by discrete conducting points arranged to form a steep pyramid of half-angle
or. The apex is curved with radius r. The sample conductor is represented with points
arranged in a two-dimensional square grid imbedded in a dielectric. (b) Schematic of the
calculation of potential matrix element A ,1, for Case 1, for which both i and k are points in
the tip and k is a distance 3 from the dielectric surface. The image charge at the
symmetric position with respect to the vacuum-dielectric interface k' accounts exactly for
the modification in potential due to the dielectric layer. R and R’ are the distances from i
to k and k', respectively. (0) Schematic for Case 2, for which i is in the tip and k in the
sample. Here the dielectric layer is accounted for using an image charge at the location
of the actual charge. (d) Schematic for Case 3, for which for which both i and k are points
in the sample a distance d below the dielectric. In analogy to Case I, the dielectric layer
is accounted for by invoking an image charge at the symmetric position k‘.

50

4.1.b. Case 2: tip-sample
Similarly we calculate the coefficients of potential for point i in the tip and point k

in the sample. In this case, point k is immersed in the dielectric and the image charge

solution results in a Simple reduction of the potential:

2 1
A' = _ 9
'k (“JR

where R is the distance between points i and k, as shown in Fig. 15.(c).

 

4. 1. c. Case 3 .° sample-sample
Lastly, if both i and k are conductors in the sample, the dielectric surface is

accounted for with an image charge term analogous to Case I.

where R is the distance from conductor i to k in the sample, and R' is the distance from
conductor i to k', located at the symmetric position about the vacuum-dielectric interface,

as shown in Fig. 15.(d).

4.2 Results and discussion
4.2.1. CAI measurements

We have employed charge accumulation imaging to study the local structure of
the interior of a two-dimensional electron system embedded within a GaAs/AlGaAs

heterostructure. For the data we will model here, the sample contained a 2D layer 60 nm

51

below the exposed surface; the dielectric constant of the overlayer is approximately K =
12.2. The tip was a chemically-etched PtIr wire with a half angle or z 8° for the conical
section, and a nominal apex radius of curvature of r = 50 nm [Materials Analytical
Services]. Below the 2D layer the sample contained a heavily doped metallic substrate
(3D) separated from the 2D layer by a tunneling barrier. As described in detail in Section
4.3, ac excitation voltage (20 kHz, 8mV rms) applied to the 3D substrate causes electrons
to tunnel from the 3D to the 2D. The charge that enters the 2D layer induces ac image

charge on the apex of the tip, which is positioned a few nm from the surface. We measure
this capacitively induced image charge Q,,-p using a cryogenic high electron mobility
transistor attached directly to the tip.

The local measured charge induced on the tip can be expressed as
Qtip (xo . yo) = ]¢(x.y)cmut (x - xo . y - yo )dxdy , (4.3)

where ¢(x, y) is the effective local ac potential in the 2D layer and cmu,(x, y) is the tip-2D

mutual capacitance per unit area of the 2D layer [Tessmer,2002]; cmu,(x, y) is a crucial

function that sets the spatial resolution of the technique. Below we describe the

application of our method to calculate this firnction.

4.2.2. Mutual capacitance calculation

We modeled the tip as a set of point conductors forming a steep pyramid

52

terminated with a rounded apex, as Shown in Fig. 14(a), with or = 8° and r = 50 nm. The
apex was positioned directly above the center of the sample a distance of 5 nm from the
surface. The sample model consisted of a square array of grid points to form the 2D
layer, fixed at a position of d=60 nm below a K = 12.2 dielectric layer. We used a
spacing of 20 nm between adjacent points for both the points that represent the tip and the
grid points that form the 2D layer to approximate continuous conductors. Of course,
neither the tip nor sample can be infinite in extent; we took the total tip height and sample
width to be 0.52 um and 1.06 pm, respectively. With respect to length scales ~100 nm,
these sizes are sufficiently large so that edge effects do not contribute Significantly to the
charging near the center of the sample and near the tip’s apex. The total number of
conductors (tip+sample) in the calculation was 3,746.

Here we use a standard programming software, Matlab 6.0 version (Matlab-short

for Matrix Laboratory). This software is specifically convenient for manipulations of
large matrices. We first solve for C =21“l and then we find Cmu,(x, y) by calculating the

charge on each grid point of the 2D layer per unit potential difference applied between
the tip and sample. This is most conveniently accomplished by considering the tip to be
at potential V with the sample grounded. The summation in Eq. (2) then involves only
points in the tip. We can further rewrite Eq. 4.2 by dividing each term by voltage and

area:

tip
Cmut (xisyi) = Qt /(Va) = ZCik “1 ,
k

where (x,-, y,-) refer to the coordinates of point i in the sample, and a is sample area per

grid point.

53

Figure 16. shows the resulting calculated mutual capacitance firnction cmu,(x, y).

We see that it is a bell-shaped function peaked directly below the tip’s apex at the center
Of the sample. The breadth of this peak determines the Spatial resolution of the
measurement, which we characterize using the half—width at half-maximum, w=92 nm.
To check the errors introduced by the discrete grid spacings and truncated tip and
sample, we performed similar calculations using spherically Shaped tips with no dielectric

layer, and then compared the calculations directly to the analytical solutions. We estimate
the plotted cm“,(x, y) function is accurate to within 10%, with the exception of points near

the edges.

54

Tip-2D Mutual Capacitance

' qt'i
'5 .1“

cm..(x. y) <1 0'4 F/mz)

1: ' “chi—:— ;!d!;3q'é€fé?fi'égfflugiqu‘ -.".--:—-- . if,“ g .
. _ ‘ ~ ‘r , ,ifm‘yfis 9’3 .1I 15113:"; ;, 5 - ‘5‘}; 'f, ~~
" .‘ 1- ' I : I):“I‘vAg‘arféfg3$?E§€g}“fi‘ ‘ i ::::-' If: -- 3 ~ .

 

 

Figure 16. Calculated mutual capacitance function c(x, y) Showing the characteristic bell
shape. The calculation used the tip geometry Shown if Fig. 1(a) with a half angle or =8°,
radius r=50 nm, total height of 520 nm, and tip-surface separation of 5 nm. The modeled
sample was 1.06 x 1.06 pm, with a dielectric overlayer of d=60 nm. For both the tip and
sample the spacing between adjacent grid points was 20 nm. The half width at half
maximum is 92 nm, which determines the Spatial resolution of our technique.

55

4.3. Comparison to measured data

To compare these calculations to measured CAI images, it would be instructive to
select an image exhibiting an especially Sharp feature so that we can assume the true
physical extent of the feature is much smaller than w. In that case, the measured width
would be mostly determined by the experimental resolution. Figure 17 shows such a
comparison. Here, image was acquired by colleagues, Maasilta and Chakraborty, on a
sample similar to one studied in this thesis. The measurement was performed at 1° K in a
top loading helium-3 cryostat. A depleting voltage of -0.23 V was applied to the tip
which serves to significantly enhance the effect of density variations to produce contrast
in the measurement. The imaged features simply reflect sample disorder. To calculate the
model charging profile we assume that the effective potential ¢(x, y) varies as an

arbitrarily sharp step function along the direction of the white line. The step firnction is
smeared out by convolving it with cmu,(x,y) following Eq. 4.3. As shown in Fig. 4.3(b),

the resulting curve reasonably approximates the experimental spatial resolution with no
adjustable parameters.

The common picture for the spatial resolution of electric field sensitive scanning
probe measurements holds that the radius of curvature of the tip and the tip-sample
conductor separation set the resolution length scale. Because the radius of curvature is
the smaller length for our modeled system, we can estimate roughly that the length scale
Should equal 65 nm, the depth of the 2D layer (d) plus the additional tip to surface
distance. This simple estimate is supported by calculations of Eriksson et al. based on a
numerical routine that solves the Laplace equation in an axially symmetric geometry. For

a similar tip-sample system with the tip in direct contact with the dielectric surface, the

56

A
m
v

Charge (arb.)

 

 

A
)3
N
I
l

 

 

9b

 

 

 

Charge (arb
O
3
CD
8

O

 

- o
9 W o
to g9 659 -
5’ «9
-500 0 500
Distance (nm)

 

 

Figure 17. (a) A 3 x 3 um subsurface charge accumulation image of a GaAs-AlGaAs
two-dimensional electron layer formed in a GaAs-AlGaAs heterostructure. The
contribution of surface topography has been subtracted so that the displayed signal arises
solely from disorder within the two-dimensional layer. The image is filtered to remove
nanometer scale noise. For comparison to modeling we focus on the data indicated by
the white line; here the signal increases abruptly, representative of the sharpest features
imaged by the technique. (b) Cross section of the image along the white line, compared
directly to our model calculation as described in the text. In this case no filtering was
applied to the measured data, and no adjustable parameters were used in the calculation
to achieve the fit.

57

calculations Showed that the half-width at half-maximum of Cmu,(x, y) was approximately

equal to the depth of the 2D layer, w zl.0d. (The agreement between w and d was to
within 5% as long as the area of contact was less than about 25 nm.) In contrast both our
calculation and our experimental CAI measurements, in which the tip is not directly
touching the surface, Show a significantly larger Spatial resolution length scale of w=l .5d.

In summary, we have developed a numerical method to model electric field
sensitive scanning probe microscopy measurements. It is a finite element approach that
uses image charges to exactly account for a sample dielectric overlayer. The elements can
be arranged in three dimensions, with no restrictions on the symmetry of the tip geometry
-hence it is straightforward to model a tip of arbitrary shape. We have applied method to
calculate the expected spatial resolution of a subsurface charge accumulation imaging
system. The model predicts a spatial resolution length scale 50% greater than the
thickness of the dielectric overlayer, in reasonable agreement with the sharpest features
seen in the images.

This calculation showed the resolution of our experimental techniques to be well
within 100 nm scale limit. If is an important factor in our measurements. In next two
chapters and I will present our main experimental findings and their possible

interpretation.

58

Chapter 5

Charge density waves theory and observations

In this chapter, I will present some background for the charge density wave
(CDW) theory based on Hartree-Fock approximation and also Show our experimental
findings related to the observation of about 200 nm length scale structure that could be
related to CDW structure predicted by this theory. Two-dimensional electron systems
formed in GaAs/AlGaAs heterostructures represents an ideal environment to study many-
particle physics in lower dimensions. Since the discovery of the integer (IQHE) and
fiactional (FQHE) quantum Hall effects the problem of the ground state of an interacting
2D system is one of the most interesting in the field of condensed matter physics. In
presence of the magnetic B field, the system is highly degenerate and Coulomb
interaction play a significant role. Depending on various conditions, electron-electron
interactions can result in ordered inhomogeneous states. The most extreme example is
Wigner crystal (WC) [Fukuyama, Platzman, Anderson]. Certain conditions of course are
necessary for the WC formation: low enough temperature, density and disorder. At higher
densities though, there is a competition between so called Laughlin liquid state and
CDWS.

In the quantum limit, where only one Landau level is occupied (N51), a uniform
uncorrelated spin-polarized electron liquid forms a charge density wave structure. This
picture is realized only in high magnetic field. It was also later calculated with HF

approximation that the optimal period of the CDW coincides with that of the classical

59

WC but the difference was that electrons were smeared over a distance of the order of

magnetic length l = h \/—_ around the sites of the WC [Fukuyama, Yoshioka, Lee].
mare

Let us now consider moderate to week magnetic fields, or in other words high
Landau level numbers . In the presence of moderate or week perpendicular magnetic field
more general picture arises. Under such conditions the ground state of the electronic
system is represented by various phases of CDWS [Koulakov]. The essential assumption

in this theory is that Landau levels are still present in such magnetic field ranges. This

simply means that even in week magnetic fields, where the cyclotron gap hare is small

the electron-electron interactions do not destroy the Landau quantization.

At higher 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].
For typical samples with densities ~10ll cm'z, and magnetic fields of ~ 2 T the predicted
length scale is ~100-150 nm. Several transport measurements of high-mobility samples
Show that highly anisotropic behavior occurs for more than four filled spin-split Landau
levels, with the uppermost level near half filling [Lilly]. Unidirectional CDW stripes
oriented along the [110] crystalline direction are believed to be able to explain these
observations. Of course the transport measurements give only an indirect way of testing
the spatial structure of the 2D electronic system. Our scanning probe technique indeed
provides us with a direct way of studying electronic behavior of 2D system in
heterostructure sample by imaging it in real space. Moreover, previous experiments
performed in our group, [Maasilta] showed first direct images of an ordered structure

within the interior of a GaAs/AlGaAs 2DES. Structures on these images exhibit clear

60

orientational order roughly along the [110] crystalline direction. The features were
suggestive of surprisingly large micron-scale stripe-like states. However, because the
Spacing between features were a substantial fraction of the experimental scan range, only
the approximate value for the characteristic length scale was given without discussion to
the periodicity of the structure.

In this chapter, I am going to present only some theoretical details about the
formation of the CDWS states and discuss our experimental findings related to a smaller
length scale structure of about 150-200 nm. Challenges associated with our
measurements will also be discussed in this chapter along with the suggestions for future

improvements.

5.1 Description of the theory of CDW

Let us introduce a bit of history on this subject. Prior to 1982, physicists believed
that the ground state of the 2DES in a perpendicular magnetic field was described in a
form of charge density waves (CDWS). Yet this theory based on HF state picture, failed
to explain the FQHE at the time of its discovery. The explanation was made possible

when Laughlin [Laughlin 1983] suggested the non-HF state of uniform density for the

fractional numbers of v = ,g, (v = K%'[ 2 is a spin-Split filling factor), which happened to

halt—-

be lower in energy by a few percent. This was shown rigorously for the lowest Landau
levels (N=0,1). However, Laughlin liquid theory does not describe important electron-
electron correlations at a partially filled LL. In the years to come in 1990s it became more
evident, both from calculations and experiments, that FQHE was likely to be absent for

N>1. In 1996, Koulakov, Fogler, and Shklovskii [Koulakov, Fogler 1996] and Moessner

61

and Chalker [Moessner, Chalker 1997] independently proposed theory alternative to the
Laughlin liquid theory of the uniform uncompressible state. Of course one should be
clear in that the Laughlin liquid ground state was Shown to have lower energy than the
conventional Wigner crystal [Wigner] for N >1. However, the Hartree—Fock based theory
by Koulakov, F ogler and Shklovskii, predicted that the ground state of various phases of
CDWS, in the uppermost Landau levels have lower energies than the Laughlin liquid
ground state for N >1. Soon after these theories were introduced transport measurement
[Cooper, Lilly] was indirectly suggested the existence of the predicted CDW phases.
Here, below we describe the key theoretical arguments that show why the CDW phases
are favorable for N >1.

Let us add few more important parameters before we continue further. One more

relevant length scale is the classical cyclotron radius RC given by k}? £2, where k; is the
Fermi wave vector. Other important parameters are vN = v — 2N, where v is the spin-split
filling factor, already mentioned above and N is the number of completely filled Landau
levels. The factor 2 comes from the spin splitting of the Landau levels. If we follow the
calculation assumptions from the calculations of Koulakov, Fogler, Shklovskii work we
get this picture.

Assuming that Wigner crystal (WC) [Wigner] is a 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 >> 1/N), i.e., when the cyclotron orbits at neighboring
lattice sites may overlap [AG]. Of course, in the case when vN is indeed small, the
cyclotron orbits do not overlap, and 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

62

uncorrelated electron liquid (UEL) of the same density. The WC cohesive energy for not

too small vN is given by [Koulakov]

 

WC hwc ./2 3 l—vN 1n(Nr,)
E =-—— —+—ln NV - ha) —. 5.1
00h Isms/I r, 27r ( 1”] 2 ° 2N+1 ( )

The factor r, is defined by r, = J2 /kp a3, where as 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, Fogler], also see Chakraborty

Thesis.

 

CDW ~ 0.3 1— vN ln(NrS) (5.2)
ECO}! ~—f(VN)rS hwc 111(14' ;]— 2 hwc—Z—I'er.

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. 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 state for N > 1. But the cohesive

energies can be calculated numerically. Results are presented in Table 1, below [F ogler,

1997] for rs =x/2. From the Table 1 it is easy to see that for vN = 1/3 and N = 0 and l,

the Laughlin liquid is lower in energy. On the other hand for N > 1 CDW sate is the
winner. The same is true for vN = 1/5 and N > 2.

So, it has been predicted theoretically that CDW states for higher Landau levels
(N >1) are preferred, let us try to understand the physics intuitively. As it was mentioned

earlier in the Chapter 1 that there is a competition between the long-range Coulomb

63

 

 

v =I/3

 

 

and N
N M EUEL 32:. £5? W 61:7 5;}? W
0 I -0.1206 -0.I I59”) -0.IO37 -II.8%
I I -0.1297 -0.I 5 19(3) '0. I424 -6.7%
2 2 -0.I I36 -0.l I4II3) -0.1188 4.0%
3 3 -0.1034 -0.0946I3) -0.I‘018 7.1%
4 4 -0.0965 -0.0824(3) -0.0896 8.0%
5 5 -0.0914 -0.0733I3,l -0.0805 8.9%
v =r/5
an N
N M 15mL 5.5:.» ES? W 675/331? W
0 I -0.0560 -0.0903(2) -0.0880 -2.6%
I I -0.0765 ’0.1 72717 I -0.1692 -2. I “III
2. I -0.0677 -0.I420(9) -0.I 396 - I.7"/o
3 2 -0.0618 -0.1 ”9(9) -0.1202 5.2%
4 2 —0.0577 -0.0963I9I -0.1050 8.3%
5 3 -0.0547 -0.0849( 9) -0.0946 10.3%

Table]. The cohesive energies of the Laughlin liquid Efoh and the CDW 153le for r,

= 2 . 117 is the optimal number of electrons per CDW bubble given approximately by
3vNN. The energy unit is hare. The energy per electron in the uniform uncorrelated state
E UEL is also provided for reference [Fogler 1997].

64

repulsion and Short-range (~ E) exchange attraction between the electrons, which leads to
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 on this question is related to the length scale RC the

cyclotron radius, which appears in the exponent of the CDW wavefimction. The

wavefunction of a single bubble with M electrons at the lowest Landau level is given by
M IZII 2 (5 3)
LI’0{rk} =n(zi—zj)xexp —ZkF_—— . '
i< j ‘—
Here 2,» = x, + iyj is the complex coordinate 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

 

M (aIYv bIR—b-R (5 4)
RirkI=H ' eXP[J————l‘]q’0{rk}. '
i=1 ./NI NE

where air is the inter-ladder operator, raising the ith electron to the next Landau level, and

bi 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,

‘PCDW =§sgr1(P)l:[‘P1{P(r/c) }- (5.5)

P’s are the permutations of electrons between bubbles. The intuitive reason for the
absence of CDW for low Landau level fillings (N =0 and l) is as follows [Fogler
Koulakov, also see S.Chakraborty thesis]. In all these arguments we are talking only
about electrons in the upper partially filled Landau level. Laughlin liquid can be thought

of as a CDW state melted by zero point vibrations. The melting can occur only if the

65

amplitude of the zero point vibrations is comparable to the lattice constant. This
amplitude never exceeds the magnetic length- i. There is an important difference
between low and high Landau level 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 l , 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 i , 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 18(a) and (b).
Each black dot in the figure represents the guiding center of the cyclotron orbits. An
enlarged view of a “bubble” is Shown in Figure 18(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

66

 

 

Figure 18. (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, Chakraborty thesis].

67

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
happens as long as vN < 0.3. As the system moves further from integer filling, the bubbles
merge together 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 periodicity of a CDW lattice according to

prediction z 3Rc. Simple calculation Shows:

RC = kptz, and k. in 2D = ./2np = 1.12 x 108 m" for 2DES density p = 2 x 1015 mi.
- 2 h 16 2 . -
While 6 = —- = 3.3 x 10' m for magnetic field B 13 2 T, Wthh corresponds to
eB
v between 4 and 6 for our sample. Hence 3Rc = 3 x 1.12 x 3.30 x 10'8 m z 110 nm, (for

v = 4) and it is comparable to our spatial resolution of ~92 nm. (See Chapter 4).

Similarly one can estimate the length scale for v = 6.5 which is approximately 150 nm.

In the following parts of this chapter I will present our observation on the charge
density variations near v = 6.5 in a presence of perpendicular magnetic field. And I will
also discuss the magnetocapacitance data with the tip fixed at a few nanometers distance

to the sample.

68

5.2 Magnetocapacitance measurement and charge density variations

In this section, I will describe two types of measurements that were performed on
a heterostructured sample. First, I shall discuss a magnetocapacitance measurement that
was performed with the tip fixed at a constant distance above the sample without
scanning. I will then describe our scanning measurements. The charge is driven into the
2DES by an AC excitation applied to the 3D substrate. Capacitive coupling between the
2DES and the probe allows us to detect charge accumulated in the 2DES. The CAI
technique has been proven to be very successful in imaging local compressibility features
in 2DES [Tessmer, Finkelstein] and ordered micron scale features [Maasilta] in an
integer quantum Hall regime. Here I will show our attempt to image smaller scaled
structure in a similar regime. The essentials of the measurement were already described
briefly in Chapter 2 of this thesis. More detailed picture of the measurement is as follows.

Our sample, as already stated before, has a 3D metallic substrate which we use for
our ohmic contacts. Thermal diffusion of indium from the back of the sample makes
direct contact to a 3D bottom substrate. A tunneling barrier separates the 2D layer and a
parallel 3D layer. The standard 2DES is formed at A103 GamAs / GaAs interface 80nm
below the sample surface. AS already discussed before in Chapter 2, our (001) wafer is
grown by molecular beam epitaxy (MBE). It has an electron density of z 2x10ll cm'2 and
low temperature mobility u ~106cm2/V sec. At zero magnetic field 3D-2D tunneling rate
is approximately 200 kHz. This value is extrapolated from the analysis on a similar
sample. 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. The

surface Schottky barrier blocks the charge from tunneling directly onto the tip. The

69

resulting measured signal is the 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. Our observation Show clear evidence of the quantum Hall behavior
in the 2DES system. Characteristic dips in the capacitance signal are observed every time
system enters the incompressible state. Our Observations also Show evidence of charge
density variations in 150-200 nm scale range. We believe that these are hints of dynamic

and unstable CDW type structure.

Let us start by describing the recipe for performing these measurements. All our
measurements are performed at cryogenic temperatures that are achieved by direct
immersion of whole microscope in liquid He-3, typically at 270 mK. Chemically etched
Ptlr tip of ~50nm radius is typically positioned within a few nanometers of the sample
surface. Positioning of the tip is first achieved in “tunneling mode” regime, where
constant tip sample distance is controlled by tunneling current feedback loop. This way
tip is positioned at a distance of a few angstroms above the sample surface in “tunneling
mode”, after which the feedback is switched off. Figure 19 shows the schematic of
different modes of operation of our microscope. Usually, the tip is pulled away from
sample about 30-40 nm and moved laterally sufficiently far from the tunneling location to
unperturbed location. After that, the tip is brought closer to the surface using manual fine
approach and left at a fixed distance of 3-4 nrn above the surface. The image charge
signal is detected on the tip 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. Normally, this background signal

70

Modes Of Operation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 19. Three modes of operation of the microscope: tunneling mode (right),
Kelvin probe (middle) and charge accumulation (left).

71

gets subtracted away using a bridge circuit. To acquire images of the interior of the

2DES, the tip is scanned laterally across the surface without the use of feedback loop.

Let’s concentrate now on fixed tip measurements in the presence of a

perpendicular magnetic field. We apply and excitation voltage of 4 mV ms 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 [Tessmer 1998, Yoo] to ensure the effective zero DC
voltage. To determine this compensating voltage, we perform Kelvin probe
measurements, where we mechanically vibrate the tip in the vertical direction with a
small 2 kHz frequency and amplitude of ~2 to 4 nm. We then measure the oscillating
charge induced on the tip in response to the charge oscillations in the sample. This charge
results from the electrostatic potential difference AV between the metallic probe and the
GaAs sample. It is proportional to AV( dC/dz), where C is the probe sample capacitance
and z is their separation. We measure the capacitance signal vs DC voltage between
probe and the sample. We repeat this measurement at different distances between tip and
sample AZ and from these measurements we deduce the nulling voltage (Vnull) necessary
to compensate the electric field created by the work function difference between tip and

the sample (tip-sample contact potential).

Magnetocapacitance data is shown in Figure 20. Here, we plot the measured
Charging as a function of magnetic field with the tip position fixed. The plot on this figure
exhibits clear structure with I/B periodicity if plotted against inverse magnetic field,
Corresponding to integer Landau level fillings. Usually, two components of the signal are
detected for fixed tip data and for scanning images. One is in phase with the applied

excitation and the other 900 out of phase with it. We call the in-phase capacitance signal

72

 

G)

 

Capacitance (aF)
A ' O)
_.
—o
\l

. -8 v=6 .

N
I
<
II
0"

 

 

0 n l a I l a l 1
08 LG 12 L4 L6 L8

Magnetic Field (T)

 

Figure 20. Fixed-tip measurement. Present plot shows measured charging as a ftmction of
perpendicular magnetic field with tip fixed at a few nm above the surface. Clear features
at integer Landau level filling v are marked with arrows. (Here only the in-phase
component is shown).

73

 

80

T
I

(II
D
I
<
ll
4:.
<
II
\J
I

 

A
D
I
l

m
C)
l
I

 

v=5

 

Capacitance (aF)

M
C)
I

l

I
1

1E]

 

 

I 1 1 I l I 4 L ‘

0.24 I 0.3 0.48 l 0.80 I 0.72 0.84 I 0.86 1.08
Inverse magnetic field (1/T)

 

Figure 21. Two components of the signal plotted against inverse magnetic field. Different
set of data also taken with fixed tip similar to the measurement shown on Figure 20.
In-phase capacitance signal (top curve) and out-of-phase capacitance (bottom curve).
Dips in the in-phase curve and much smaller peaks in out-of-phase curve are marked my
numbers corresponding to integer LLS.

74

Q)", and the out-of-phase signal QM. In Figure 20, we show only the in-phase component

of the Signal and the dips are representative of the usual behavior of the 2DES as it passes
through the incompressible states at integer Landau level (LL) filling. The out-of-phase
component usually resembles peaks at the corresponding integer LL fillings. Figure 21
shows similar magnetocapacitance data from a different data set that has both
components of the signal plotted with respect to inverse magnetic field. In order to
explain the nature of the in-phase dips and out-of-phase peaks, we present the charging
characteristics of the model. equivalent circuit. Figure 22(a) shows the actual tip-sample
geometry and the equivalent circuit. For simplicity, the capacitance between the tip and
2DES is considered as a parallel plate capacitor Cup. Similarly, the capacitance between
the 2DES and the metallic substrate is assumed to be another parallel plate capacitor C1,
which is in parallel to the tunneling resistance RT. These form an RC circuit. The plot of
Qm and Q0“, with respect to the logarithm of the applied frequency f or the tunneling
resistance RT is shown in Figure 22(b).The charging components have identical
functional dependence on f and RT, with characteristic values offo = [21rRr (CT+Ctip)]'l
and R0 = [21tf(CT+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 fl; ~200 kHz, a factor often greater than the applied excitation (i.e.,f/fo ~0.1).

If we focus on the model’s low-frequency behavior, f/fo<<l, 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, including the

compressibility contribution to the capacitance [Tessmer 2002, Smith]. In a parallel plate

75

 

 

 

 

 

 

 

 

 

up' 0 Q l I I
v in
C
U
+ + + + + 20
AlGaAs 12D
GaAs Rs
AIGaAs
: D Q... _
n+ GaAs :80 ,2 { 1091‘” a) } 2
i A WIRr’Ro)

Figure 22.(a) The circuit equivalent to the tip sample arrangement in the 3D-to-2D
charging case. (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-
sarnple 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 [Maasilta ,2003].

76

capacitor model, the measured capacitance is given by following equation [Stern, Smith].

1 1 + —1—— (5.6)

=-_ 2 dn
Cmeas Cgeom e /d#

 

. . . . . dn .
where Cm“, IS the measured capacrtance and Cgeom Is the geometric capacrtance; :1— 18
,u

the density of states with respect to the chemical potential. This quantity is often referred
to as the compressibility. Thus, one could conclude that the dips in the measured in-phase
curve reflect a reduction in capacitance caused by the diminished compressibility of the
2D system at integer filling. With respect to the out-of-phase charging curves (not
Shown), the peaks that usually appear at corresponding integer filling likely reflect an

increase of the pseudo gap, resulting in an increased tunneling resistance RT [Deviatov].

Let us now move to the charging images. Figure 23 shows a series id the out-of-
phase charging images between fully filled and exactly half-filled upper LL, namely
between v = 6 and v = 6.5. The in-phase component of the signal, (not shown), is
typically dominated by geometric capacitance Signal and does not reflect the local density
variations. These images were acquired on an undisturbed area with tip scanned very
close, about 4-5 nm, from the surface. The scan range of ~750 nm was chosen in order to
search for the predicted ~110 tolSO nm periodic density variations near half-integer

fillings.

In this data set we start at the magnetic field values corresponding to the integer

filling v=6. Charging images were acquired at six different values of magnetic field that

77

 

Figure 23. Images of the out-of-phase charging signal are shown. Scan range for all
images is identical and equal to 730x730 nm. Images are acquired at v=6, 6.1, 6.2, 6.3,
6.4 and 6.5 with corresponding B=l.38 T, 1.36 T, 1.34 T, 1.32 T, 1.30T and 1.28 T. The
images are filtered to remove the nanometer-scale scatter. The 5 aF contrast indicates ~7
% variation in density.

78

correspond to incrementing the filling factor by 0.1. We expect to observe the local
charge density variations by their coupling to the out-of-phase component of the signal.
as demonstrated by Maasilta and coworkers. This density variation occurs due to the
variation of the RT parameter as the system gets closer of further from integer filling.

Although it is difficult to see from the direct images, as we approach the half integer

filling v = 6.5, the charging pattern becomes somewhat structured. Figure 24 shows the
same set of images (upper row) with their corresponding Fourier transform (bottom
row). The bright peaks in the transform indicated ordered structure in the diagonal

direction for the image for v=6.5.

The comparison of the charging image for v=6.5 and the topographic image of the
same location at zero magnetic field is shown on Figure 25. The larger scale image of
surface topography is also shown on this figure, which was acquired at room temperature
without alternating the sample orientation. The [I10] direction is indicated. Hence we
see that the orientational preference in the charging images is about 35 degree with the
[110] crystalline direction which is often referred to as the “easy direction” for current
transport [Lilly, Cooper]. It was suggested by Cooper and coworkers however that at
temperatures higher then 150 mK the anisotropic behavior in electronic transport was
largely suppressed. Thus, it is believed that the fragments of the CDW structure do not

preserve their preferential orientation and thus are more likely to be isotropic.

It must be emphasized, however, that we consider these data as evidence for the
charge density variations but not a firm observation of CDWS. Similar quality charging

images were not acquired near v = 4.5. Moreover, we did not see any evidence of a

79

 

Direct Charging Image (arb.) Fourier Magnitude (arb.)

    

730 x 730 nm

 

Figure 24. Direct images vs. two-dimensional Fourier transform. Top two rows represent
the out-of-phase images in progression from v=6 to v=6.5.with their corresponding 2D
Fourier transforms (FT) (same set of data as in Fig. 5.3.) Bottom row: enlargement of

=6.5 direct image (left) with additional filtering and it’s FT (right). The circle indicate
the wavelength of about 160 nm predicted by theory for our sample. Peaks on this FT
image indicate the length of approximately 200 nm future in corresponding direct image
(left).

80

 

4.3 x 4.3 pm

Figure 25. Topography vs. charging image. 730 x 730 nm scan images of topography
(left) and out of-phase charging image (right) are shown on the top. Both images are
acquired at exact same location. Charging image shows a stripe like structure of ~ 200
nm. Left upper comer of the topographic image shows some surface defect absorbate that
does not contribute to capacitances; the rest of the topographic image looks featureless.
Larger 4.3 um image of the surface topography is also shown (bottom); typical growth
features of the GaAs surface of ~ lum long and ~ 5 nm in height are shown. The black
bar indicates the [I10] crystalline direction.

81

“bubble”- like phase near the integer filling during our experiment. Only a stripe like
orientational tendency near v = 6.5 filling was observed. The length scale of this
structure is in relatively good agreement with the predictions of the charge density wave
picture. Estimated length scales for our sample is about 110-150 nm. The length scale of
the observed structure is approximately 200 nm.

In summary, we have obtained images suggesting an ordered structure within the
interior of a GaAs/AlGaAs 2DES using a novel scanning capacitance imaging technique
At magnetic fields near six and a half filled Landau levels, we observe density
modulations that exhibit orientational order, approximated in 35 degree to the [110]
crystalline direction. The structure is not very robust and, therefore, does not allow us to
conclude with the certainty about its nature. The structure is suggestive of the high-

Landau-level charge density wave structure predicted by theory [Koulakov].

82

Chapter 6

SPM study of the donor layer charging in a GaAs
heterostructure

In this chapter the donor layer charging in a GaAs heterostructured sample will be
discussed. AS was already described in previous chapters, the 2D system in our sample is
formed with a use of the O-doping technique, with Si being used as the dopant atom. This
allows for the creation of monolayer wide Sheet of Si+ ions randomly distributed above
the 2D layer. The density of these Si atoms is 1.25 x1012 cm'z. One can study the effects
of charging inside the O-doped layer by allowing electrons to tunnel into it from a 2D
reservoir. This tunneling will happen when a sufficiently positive voltage is applied to a
top gate, or in our case, an effective positive voltage to the tip. This way, electrons fill up
the energy levels available in the donor system which is made of the ionized Si.

The basic model for an isolated ionized donor in a semiconductor considers the
system as hydrogen ion [Davis]. The mass of the electron is rescaled to the effective mass
of the electrons in the host material, and the charge is screened by a dielectric constant of
the host. Both effects reduce the ionization energy R substantially fi'om the 13.6 eV for
real hydrogen to only about 5 meV for the donors in GaAs and AlGaAs. R called the

Rydberg energy, which can be written as follows:

 

 

 

2 2 2 2
R: e ”‘2: '12:1 e , (6.1)
47:3 271 2maB 247E138
47thzé‘
a3 = 2 , (6.2)
me

83

Here a a is the Bohr radius, which is scaled up to about 10 nm for electrons in GaAs and
AlGaAs, compared to a B = 0.053 nm for real hydrogen. To describe the binding of an

electron to an isolated donor ion, to a good approximation we can use the textbook
hydrogen atom solutions. Thus for the hydrogenic model, the wave function for the
lowest state has an exponential form for 3D case [Davis].

‘P(R) = mg)“ exp(-R/aB). (6.3)
The Rydberg energy and Bohr radius are defined by equations (6.1) and (6.2), and the
energy states are an: - R / n", The schematic picture of the wave function for the
hydrogenic atom is shown on Figure 26.

Can these hydrogenic ions bind a second electron? For many systems the answer
is yes. The state is known as the D"; these D' centers are produced by attachment of an
extra electron to a neutral donor B". These states could play an important role in testing
and understanding many-body phenomenon; this subject has attracted a significant
interest for many decades since the beginning of quantum mechanics. The behavior of the
D' has been well characterized for well isolated (weakly doped) systems in confined
geometry and strong magnetic field. In addition to several optics experiments [Huant,
Holmes, Shi], experiments using the resonant tunneling spectroscopy method have
observed the D' states in double barrier resonant tunneling device (RTD) [Lok, Geim].

AS the resonant tunneling experiments motivated this part of my thesis work, it is
worthwhile to describe briefly the RTD measurements. The idea behind the resonant
tunneling is this: the sample is biased and the current flows when the energy of an
electron in two-dimensional electron system formed at the interface is resonant with the

states in the quantum well. See schematic of this arrangement on Figure 27. As a bias

84

Y’rZ)
/

 

/V(Z)

 

Figure 26. Schematic of the hydrogenic dopant atom. Wave function w(z) and the
potential V(z) are shown. R is the Rydberg energy and a B is the Bohr radius.

85

DC voltage is increased the energy levels in the well pass through the resonant conditions
and are showing up as peaks in the I(V) characteristics. These, resonant tunneling
experiments will be described in more details in section 6.1. The results from our
capacitance measurements and the attempts to explain our observations will be described

in section 6.2.

 

 

 

 

 

Figure 27. Schematic diagram of the conduction-band profile of the RTD under the bias.
Tunneling occurs from 2DES through the ground state in the quantum well (for the main
resonance) or highly localized impurity levels at lower energies (Geim 1994).

86

6.1 Resonant tunneling experiments

The heterostructure interface between AlGaAs/GaAs represents a simple example
of the tunneling barrier, similar to one shown on Figure 27. AS we know classically, an
electron would not be able to pass through the barrier unless it has sufficient kinetic
energy to pass over the top, but quantum mechanically it is able to tunnel through the
barrier. The opposite of the barrier is a quantum well, for example a relatively thin layer
of the GaAs sandwiched between two thick layers of AlGaAs. In this important case
where GaAs is surrounded by AlGaAs the electrons reside in bound states and the energy
levels can be measured by optical techniques. AS we know well from quantum mechanics
if we make a quantum well that has thin baniers on both sides, then the electron in the
state is not truly bound anymore but quasi-bound or resonant. There is not a true bound
state because the electron can tunnel through and escape from the well. But if the walls of
the well are thick enough then electron can remain in the well a long time in a resonant
quasi-bound state.

In resonant tunneling experiments [Lock, Geim], the resonant tunneling structure
was constructed inside the AlGaAs double barrier with intentional Si 6 doping in the
quantum well. The thickness of both barriers and AlGaAs was 5.7 nm and the width of
the quantum well 9 nm. The structure was fabricated in square mesas of sides of length
varying from 5-100 um and there was a 20 nm undoped spacer layer between the barriers
and the doped contact regions. The average donor separation was between 0.1 and 0.5 pm
for different samples; this is sufficient so that the dopant atoms could be considered as
isolated. But because there is always some statistical probability of finding some of the

donors at distances much smaller then average the binding energies of the This is an

87

 

2080

 

 

 

 

 

  

 

   

I (M) g

 

 

 

 

o - .. J
:' I
11R
I
o 120
-ao~ ~
1’
...80 .-
-0.4 -0.2 0.0 0.2 0.4

V (V)

Figure 28. I(V) for the resonant tunneling structure 12 um across with the 6 doping in the
well at 1.3 K. The arrows indicate the main resonance (MR) and Single donor resonance
(SDR). The dashed curve represents 20x magnification of the SDR. The inset shows the
schematics of the band diagram for the double barrier tunneling structure. Impurity

related states (IRS) also shown on the diagram not visible on the plot. This plot is
reproduced from reference [Geim].

88

interesting aspect of the work described by these groups [Geim, Lok]. Tunneling through
the paired donors could become dominant when other states remain out of resonance. The
structure and I(V) plot for the device with 8x109 cm'2 donor concentration is shown on
Figure 28. Tunneling occurs when a state in quantum well comes in resonance with the
2DES formed in an accumulation layer near the emitter barrier as the bias is increased
showing as the main resonance peak (MR) in I( V) plot. The lowest 2D subband in the
well gives rise to the main resonance and the 8 doping gives rise to an additional
resonance at smaller biases which originates form tunneling through the localized ground
state (D0) of shallow donors. There is also additional peak observed in high magnetic
field indicating the tunneling through D’ states [Lok].

The RTD observation of D’ centers has a few important aspects to it.Tunneling
spectroscopy gives directly the binding energy of the donor state. In addition, the shallow
donors in this and other similar experiment have been found to give rise to resonant
features described as “donor molecule” levels and donor state coupled with random
potential fluctuations, which are not visible on the plot. However, such states were
observed at biases below the threshold for the main resonance a broad peak, as shown in
Fig. 28. The authors model the observation of deeper levels as being caused by a pair of
Si donors situated in a close proximity by random chance; the two dopants are modeled
as a simple hydrogenic donor molecule, which we will refer to as “H2”. Binding energies
were estimated for a donor pair separated by one Bohr radius and for a triangle of three
Shallow donors [Geim]. As separation decreases the binding energy increases; in the limit
of zero separation it reaches the expected values for ZZR, where Z is number of donors in

a cluster and R is the Rydberg energy. This approach will become relevant for our own

89

capacitance tunneling spectroscopy experiments, which will be described in the next

section.

6.2. Scanning probe capacitance tunneling spectroscopy measurements

In this section, we will describe the new scanning probe approach to study
quantum states of the donor atom inside the semiconductor. The concept of the
measurement is almost identical to one already described in Chapter 3. The imaging
technique will be applied to probe the behavior of the Si dopant atoms in the 8 layer of
AlGaAs/GaAs sample. Here we will present our first tunneling capacitance
measurements that study the system of the quantum states of the donor layer and effects
of the proximity to neighboring atoms. This new approach has indeed proved to be a
powerful tool for study such systems.

Let’s repeat the basic idea of the measurement. With the probe being immersed in
liquid helium-3 at a temperature of 270-280 mK we position the tip few nanometers fiom
the surface of the sample. As was already described, the essential part of the
measurement consists of monitoring the tip’s AC charging in response to an AC
excitation voltage Venue applied between the metallic substrate and the tip. By adjusting
the bias voltage applied to the tip we can find the potentials for which the dopant energy
levels are in resonance with the chemical potential of the substrate. So, generally
speaking, at these voltages the electron is forced to tunnel locally on and off between the
substrate and the dopants below the tip. Figure 29 shows the schematic of the
measurement set up and the conduction band diagram of our sample, including the Si+ in

AlGaAs/GaAs heterostructure as described in Chapter 5.

90

a)

   
 

image
charge

tunneling

1
systems ‘4 charge

 

Figure 29. (a) Schematic of the measurement for the study of the donor layer. (b)
Schematic of the conduction band of the system.

91

An accepted picture for a single isolated dopant ion inside a semiconductor at low
temperature is a hydrogen ion, as described earlier in this chapter. In samples with the
relatively high average density of dopants this picture is incomplete. Moreover, as shown
in tunneling resonant experiments, additional features besides the main resonance and D-
and D0 states could be explained by the existence of other impurity related states. In our
measurement we investigate the physics of similar effect with the capacitance
spectroscopy technique.

Before describing the main measurements, we Show a more basic capacitance
measurement performed by my colleague Cemil Kayis on a sample from the same wafer.
Instead of a tip, the gate electrode fabricated onto the surface plays a role of the probe.
This is an illuminating test because it is very similar to the scanning probe measurement.
It has the additional advantages of well-defined parallel plate geometry, and a precisely
known measurement area, defined by the 5.7x10'7 m2 area of the gold gate. The result is
shown in Fig. 30(a). At sufficiently negative gate voltage, the 2D layer itself is depleted
of electrons; hence the capacitance signal corresponds to the gate-substrate separation,
which is relatively low. As the gate voltage increases to about -0.5 V, the 2DES becomes
populated with electrons. Hence, the two plate effectively get closer together and the
capacitance increases. At still higher voltages, the capacitance rises steeply. Here the
electrons are entering another layer — the cap layer formed at the GaAs-AlGaAs interface
near the gate electrode.

Apparently, there are no clear features due to electrons entering the dopants below
the gate electrode. The micron-size area probed in gated measurements such as this have

a disadvantage of probing a relatively large area. The vast number of dopants and

92

impurities below the gate likely allow relatively rare defects to dominate the
measurement — effectively shorting out the measurement of the donor layer structure.

We now turn to our CAI technique to probe locally the voltage range where we
expect to see evidence of donor layer charging. In order to probe the characteristics of the
delta-doped layer we use the similar approach to the one that was already described in
Chapter 5 for the magnetocapacitance data. We situate the tip in a close proximity to the
sample in capacitance mode. This means we fix the tip at few nanometers from the
surface of the sample without the feedback response. In this experiment, we use the
presence of the 2D layer below the donor layer as a reservoir of free electrons. Electrons
can be induced to tunnel from the 2D into the donor layer when the bias voltage is being
swept positive. In the actual measurement we apply negative DC voltage to our sample
which creates an effective positive voltage in the tip. We monitor the capacitance signal
as a function of bias voltage and tunneling events are manifested as peaks in a
capacitance signal.

It is important to realize that the effective voltage applied by the tip is scaled-
down compared to the gated capacitance measurement in Figure 30(a). The relevant
voltage is the 2DES-to-donor layer potential difference. This is proportional to the tip

voltage or gate voltage. The proportionality constants are often called the lever-arm

factor and denoted as or. The two constants relevant for Figure 30 are at“, =
Ctip/(Cdp+CZD) and agate = Cgate/(Cgate'I'Czp), where Ctip is the tip-to-dopant layer
capacitance per unit area, Cgate is the gate-to-dopant layer capacitance per unit area, and

C21) is the dopant layer-to-ZDES capacitance per unit area. For the gated measurement,

the capacitances are well defined and give agate = 1A. For the scanned probe

93

measurements, Ctip is small compared to Cgate, as the tip is never in full contact with the
surface. Hence, we expect chip to be less than ‘A and the effective voltage to be scaled
down considerably. To estimate Cap we have performed detailed numeric calculations, as
described in Chapter 4. [Kuljanishvili, 2004]. These calculations, together with

calculations performed by Eriksson et al., imply ortip zl/ 15. Lastly, it should be

emphasized that we find that Ottip changes by as much as 30% when comparing

measurements acquired at different locations. This is likely due to variations in the tip-
surface separation.

The results of the CA1 measurement are shown in Figure 30(b) for which the tip
position was fixed (not scanned). Three clear peaks are present as a function of voltage,
labeled as A, B and C. Similar peaks were observed in all locations we probed that
exhibited a stable signal. To improve the Signal to noise ratio, the data shown in the
figure represent the average of more than a two hundred hours of data acquisition,
sampled at three different locations. (All the data shown in subsequent plots were
acquired at a single location.)

Are these peaks related to the D0 and D- states? We must remember that the
donors in our system are not isolated. Also, the area probed by the measurement is set by
the tip-donor layer spacing, which is about 60 nm. Therefore, we are probing a circle of
roughly 120 nm diameter. The fact that the average spacing between the dopant in our
sample is about 9 nm implies that we simultaneously are probing many dozens of the
donor atoms. More importantly, the average spacing is approximately equal to the Bohr
radius. Therefore, we cannot evaluate the energies or quantum states of the single dopant

atom; our model must include interactions among the dopants. The interactions should be

94

 

(a) cap layer (b) dopant

     
 

 

 

 

 

 

 

 

m resonances
8
S
g 1nF 2DES
Q.
(U
0
11.0 30.5. o '0.5' o lo.4 ' 0.3 ' 1.2 I
gate voltage (V) tip voltage (V)

Figure 30. (a) Gated capacitance measurement. The three plateaus that corres nd to
charge entering different layers in the sample. The area of the gate was 5.7x10' m2 (b)
Representative capacitance spectroscopy measurement showing the charging spectrum of
a system of the dopant atoms below the tip. The black arrows in the plot Show the
capacitance peaks. The applied excitation voltage was 16 mV and the temperature was
280 mK for both measurements. Due to different a-factors of the two measurements (see
text), plot (b) corresponds to probing the voltage range marked by the oval in part (a).
The voltages are measured with respect to the nulling voltage, where the gate or tip
effectively applies no electric field. Since the energy levels of individuals electrons are
too close to be resolved in these voltage ranges, we do not see individual single-electron
peaks in these plots.

95

more important than in the experiments by Geim and coworkers as the density is more
than two orders of magnitude higher. To describe our data, we will use both a statistical
approach and a Simple donor molecule model, for which calculations are within reach.

Figure 31 shows remarkable single-electron tunneling events covering the voltage
range of the first electrons to enter the system to peak A. To observe the single electrons,
it was necessary to increase the voltage resolution of the measurement, compared to
Figure 30(b). This was accomplished by employing smaller voltage increments and a
smaller excitation voltage of 3.6 mV ms. The width of the single-electron peaks is set by
the AC excitation, if the rrns value of the excitation is greater than kT; otherwise the
temperature T sets the width. For the low-temperature of 280 mK employed here, we are
in the regime where AC amplitude sets the scale.

The expected shape of the peaks is a semi-ellipse [Ashoori]. The inset of Figure
31 compares the measured peak Shape directly with the elliptical peak expected for a 3.6
mV rms excitation. The model peak also includes the additional broadening caused by the
output filter of the lock-in amplifier. This causes the slight asymmetry in the theoretical
peak. Indeed, the measured peaks are very close to the expected shape, consistent with
the interpretation that we are probing individual electrons entering the donor layer below
the apex of the tip. Based on the magnitude of the individual electron peaks, we can
estimate the number of electrons that comprise resonance A, for which individual
electrons are not well resolved. This is accomplished by comparing the area of a single-
electron peak with the area between peak A and a background line, as shown in Fig. 31.
This estimate yields a value of approximately 10. In other words, peak A consists of 10

electrons which tunnel at nearly the same voltage.

96

 

 

.1:
I

 

 

 

 

N
I
4

 

Capacitance (aF)

~10 electrons.

 

single electrons
l

A a I a l l A l n l a l

o 100 ‘200 ' 300 400
Tip Voltage (mV)

Figure 31. Capacitance versus tip voltage. Three single tunneling events are marked on
the curve. The inset Shows the comparison of the curve representative of the average of
these three peaks (filled dots) with the expected semi-elliptical function (solid curve), as
described in the text. Peak A is estimated to be due 10 unresolved electrons, as indicated.

97

To analyze our data quantitatively, we use a very simple model: we assume that
for each dopant atom the energy levels are shifted due to the interaction with its nearest
neighbor. Essentially this approach considers the system to be made up of many two-
atom “molecules”. Clearly, this is a drastic simplification. It ignores larger groupings of
atoms and the presence of impurities. Nevertheless, it is a usefirl approach as a first
model. Prof. James F. Harrison at the Michigan State Department of Chemistry applied
this model using hydrogenic dopants; the energy levels for different numbers of electrons
bound by the two-atom pair were calculated as a function of separation using the
configuration interaction method [Sabin&Brandas] to solve ShrOdinger’s equation. The
results are shown in Figure 32; all energies are plotted with respect to the free electron
energy. The inset of Figure 32 Show a statistical calculation we performed to find the
average distance between nearest neighbor dopants. The calculation considered 200
random two-dimensional arrays of dopants, each with the same area as the area probed by
our technique and the same density as our actual sample. We see that the average nearest-
neighbor separation is 4.65 nm=0.465ao.

To understand how to relate the calculated energy levels to our CAI capacitance
measurements, let’s consider the tip situated above a single pair of dopant atoms with a
separation of 0.465ao (the average nearest-neighbor separation). If the tip voltage is
effectively zero, we expect no electrons to tunnel onto the pair. AS the tip voltage is made
more positive, eventually an electron will tunnel and we will see a peak in the
capacitance. This occurs when the electron in the 2DES at the Fermi energy II has the
same energy of the ground state of the molecule. Reading the data off of the “first

electron” curve in Fig. 32, we see the predicted energy is E(1)=-3.5 R. A second electron

98

electronic energy (R)

Figure 32. Theoretical ground state electronic energy versus separation for various
numbers of electrons on a two-atom potential. These curves were calculated by Prof. J .F.
Inset: statistical histogram of the nearest neighbor separations. This was
calculated using 200 random two-dimensional arrays of dopants, each with the same area
as the area probed by our technique and the same density as our actual sample. To make
the comparison to the electronic energy calculation more transparent, the results are given

Harrison.

-1 p 58; ------ E(§;:.: 1st electron '
_2 2nd
'2' ”2..-.-. ._.._ttn ‘
.. f,::Of_:::.7.::.H's-EECT-a-oq
H2
-3r .
.. 40..-.-rvTfi
I [‘3 9 ' “9.30.465” - 4
i"; '5 '5 " 14.65 nm -
'3 a, a . ‘
'4' I! a e . . ‘
i’ g 3 - .
a" . g _ .
. :‘I g .g r .. ‘
'IIH : Ol- ‘
-5- : 2 l . 1 A 1 L l . l -
s" 0 1 2
- 7 separation (a0)
0 2 4 5 8 10

 

 

 

 

 

 

   

 

 

 

 

interrns ofao= 10 nm.

99

separation (a0)

will be trapped by the potential when u+E(l)=E(2); i.e. for u=E(2)-E(l), where E(2) is
the ground state for two electrons in the molecule. Here the calculations predict that the
second electron enters at an energy of 5.1 R-3.5 R = -l.6 R. Likewise, the third electron
enters when u=E(3)-E(2). In this case the theory predicts that the second and third
electrons have nearly the same energy. In other words the third electron Should be barely
bound. Hence, for our measurement it will enter near the free electron energy. Using this
same reasoning, the fourth electron is not bound. Therefore, we should observe three
peaks as electrons enter the dopant layer. This indeed matches our observation.

If we assume that this model applies, then the voltage corresponding to the free
electron energy is given by the voltage of the third peak (peak C), which is 0.75 V on Fig.
30. Next, to compare the predicted peak spacing directly to the observations, we must
remember that R = 5 meV and that there is a lever-arm factor that scales the tip voltage
by approximately 1/15. Therefore, we expect the first peak at -3.5 R x 5 mV/R x 15 =
-0.26 V, relative to the free electron voltage. Likewise, we expect the second peak at
-0. 12 V, relative to the free electron voltage. In comparison, we observe peak A (the first
peak) on Fig. 30(b) at 0.43 V, which is -0.32 V relative to peak C. Peak B (the second
peak) occurs at 0.57 V, which is -0.18 V relative to peak C. Hence, peak A occurs at a
voltage 60 mV lower than predicted and peak B occurs 60 mV lower than predicted.
Scaling these differences back to Rydbergs gives -0.80 R for peak A and 0.80 R for peak
B. This agreement is surprisingly good considering the highly simplified nature of the
model. Figure 33(3) compares the width of peak A with theory. To generate the theory

curve, we used the histogram of nearest-neighbor distances shown in Fig. 32 and scaled

100

 

l ' l ' l
a — measurement,
_ ( ) - - - theory

 

 

 

capacitance (arb.)

 

300 1 ' 400
tip voltage (mV)

 

' I ' I

-(h) _

Ezts'r \;\\ -

0

capacitance (arb.)

 

 

I J r I

0 0.5 1.0
tip voltage (V)

 

Figure 33. (a) A comparison of measured data (peak A from Fig.3l) and the theoretical
calculation generated from model shown on Fig. 32 is shown. The widths at half max
(FWHM) of these curves agree reasonably well with each other. (b) Capacitance addition
spectra at 0 T and at 4.5 T are shown. The shifts that occur at the places of the resonances
imply an intriguing behavior of the system.

101

the horizontal axis using the slope of the E(l) curve at 0.465 a0. The measured curve is
peak A from our high precision data shown in Fig. 31, where we have subtracted the
background line, as described above. We see that the width of the two peaks agree
reasonably well. Figure 33(b) shows how the peaks shift with magnetic field. Although a
theory for this measurement has not yet been developed, it is very interesting to see the
behavior of the third peak. It effectively splits into two peaks! In the context of our
nearest-neighbor model, this implies that we actually have four electrons that can enter
the molecule.

Although our proposed model works surprisingly well, the amplitude of peak A
does not agree with simple expectations. In the area probed by the tip, we expect about
140 dopant atoms, or 70 nearest-neighbor pairs. If each pair contributes to the observed
peaks, then the peak area Should be consistent with 70 electrons. However, as Shown in
Figure 31, only 10 electrons comprise the peak. This implies that only 10 nearest-
neighbor pairs contribute in a way consistent with our model. We speculate that the
majority of dopants have significant Shifts in their energies due to the close proximity of
more than one neighbor. Impurity atoms may also play a role. Another possibility is the
presence of deeply bound electronic states known as Dx centers. Dopants in this
configuration will not contribute to our measurement as they are effectively permanently
occupied by an electron.

In conclusion, we have shown that our scanning probe technique can be a
powerful and sensitive method to study a local capacitance of dopants in semiconductors.
Our result compare favorably with many of the predictions of a simple model. Clearly

more work to be done both experimentally and analytically. For example, to further

102

investigate the effect of magnetic field on individual resonance peaks magnetic field B
could be incremented with small ranges. This is the first probe study of the donor system

in semiconductor in tunneling regime with very encouraging and intriguing results.

103

Chapter 7

Summary and future objectives

7.1 Summary of the investigations

In summary, I would like to point out several important goals have been achieved
as a result of this research. First experiments on AlGaAs/GaAs 2D system described in
this thesis were building upon the previous work done in our lab [Maasilta, 2003]. It was
important for the further understanding of the electronic behavior of the 2D system to
study it in a specific regime, searching for the ordered CDW structure and further
investigating the effects seen in previous work. The work by Maasilta and coworkers first
reported the direct imaging of interior of 2DES the GaAs/AlGaAs with the CA1
technique in the tunneling geometry; this effort proved the technique to be a powerful
new extension of the original CAI technique that utilize direct ohmic contact to the 2DES
[Tessmer 1998, Finkelstein]. This is extremely important since this new technique allows
us to resolve 2DES structure near integer filling. The tunneling geometry also provides us
with information about the tunneling density of states of the 2DES and is sensitive to the
pseudogap physics [Eisenstein, Deviatov, Dolgopolov]. The main focus of the first
experiment presented in this thesis was to investigate the density modulation of the 2DES
in presence of moderate magnetic fields. In the tunneling geometry, we have observed a
characteristic charging behavior of the system near integer fillings. The in-phase
component of the capacitance signal Showed a diminished signal at integer fillings with
the HE periodicity which is the quantum mechanical signature of a low-disorder 2DES

with a fixed density. The out-of-phase peaks observed at the integer fillings are typically

104

much smaller features. They do confirm however the variations in 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 variations in the tunneling resistance at
integer fillings.

We have also imaged the density modulations of the 2DES in the tunneling
geometry. In this measurement we have concentrated on imaging of ~ 150 nm scale
CDW structure that was theoretically predicted to be spontaneously arising near integers
and half-integer fillings [Fogler, Koulakov, Shklovskii, Moessner]. To image the bubble
and the stripe phases was the primarily goal for us in this experiment. We have observed
the evidence of stripe-like structure in 2DES for the magnetic field corresponding to

=6.5. The length scale of this structure is about 200 nm that is in good agreement with
the calculated length scale for this sample. However, we did not observe similar structure
for magnetic field corresponding to v = 4.5 as it was also predicted. The orientation of
observed structure at v=6.5 was approximately 35 degree to the [110] crystalline
direction of the host sample. This direction does not support widely accepted concept
based on transport measurements, which implies that CDWS tend to orient themselves in
the “easy” direction. This direction is a perpendicular to [110] crystalline direction [Lilly,
Cooper]. Nevertheless, we find our observation to be a very interesting result that raises
several points. For example, an open question is whether the CDW orientation persists to
the temperatures at which we performed our measurements, or only fragments of the
CDW structure survive. Such fiagments do not necessarily exhibit a strong anisotropy but
become more isotropic with the increasing temperatures. This idea was also suggested by

a transport experiment [Cooper, 2000]. It is still a debated subject and more experimental

105

and theoretical work could allow for better understanding of the origin of these density
modulations.

The topic of our next experimental investigation was to study the charging
processes in the delta-donor layer in the same AlGaAs/GaAs heterostructure sample with
scanned probe technique. With the first success of this experiment, we have demonstrated
an extremely sensitive new way of studying the quantum states of the system. More
specifically, we have studied the quantum behavior of the Si +dopant atoms randomly
distributed in the 6—doped monolayer. The local capacitance signal from the area below
the tip was monitored and the energy levels were evaluated fi'om the capacitance
tunneling resonances. This resonances Show up as peaks in the capacitance signal. We
also resolved remarkably well series of single electron tunneling events at low biases. We
attribute the resonances to tunneling through states similar to the D0 and D- states of the
isolated hydrogen atom model. Our measurements repeatedly Show a relatively tall
resonance peak and two Shorter peaks that occur at lower voltages. Our interpretation of
this data is based on the picture of two-atom donor molecules. This is an extreme
simplification. For example, clusters of three or more atoms would give rise to deeper
states.

Tunneling resonant experiments done by other groups studied the single isolated
dopant atom states with resonance tunneling measurements on AlGaAs/GaAs samples
have also reported additional features below the threshold of zero bias [Lock, Geim].
However, in most of those experiments the authors were studying an isolated donor atom
systems with the donor densities more then two orders of magnitude lower. The features

that we observed in our measurements at lower voltages indicate the existence of deeply

106

bound states that could originate fi'om the presence of impurities in the system or maybe
other electronic effects also contribute. Values for the electron binding energies and the
width of our capacitance features (peaks) are in a surprisingly good agreement
considering the highly simplified nature of our model.

We have also observed the evidence of the magnetic field dependence of our
capacitance peaks. It has not been studied fully at present time, but it raises an interesting
point: for example, what is the total number of electrons that could be bound in our
system, what are their quantum states and how they evolve with applied magnetic field?
To continue this discussion more measurements and additional theoretical work is

required.

7.2 Future directions

In this work, we have demonstrated that our CAI technique is a powerful method
for studying the 2DES and other nanoscale structures. The future directions in this
research can be categorized into two main themes. First would be to fiirther continue the
study of the 2DES in a quantum Hall regime and explore more about the bubble phase
structure, which was not observed in our measurement near integer fillings. Most recent
reports indicate a strong resonances in diagonal conductivity oxx in the microwave regime
[Lewis, Chan] near v=4 1/4 and v=4 3/4. The authors interpret the resonance as due to a
pinning mode of the bubble phase. There is one more very interesting phenomenon that
could be studied by imaging the 2DES at high magnetic field on high mobility samples:
the “spin bottleneck” phenomenon. Several 3D-to-2D tunneling measurements at

magnetic fields near v = 1 have shown that there are two different tunneling rates for

107

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 skyrmion formation in the system [MacDonald]. It
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 (~ 100kHz) and the other at a much lower characteristic spin bottleneck
frequency (~lkHz) —- the spin bottleneck efféct and the effect on it due to disorder can be
studied [Murthy, Brey, Green, Nazarov]. This effect can be resolved in principle with the
3D-2D tunneling CAI technique

There are also new directions building upon the success of our capacitance
Spectroscopy measurements of donor atoms in the AlGaAs/GaAs samples. The obvious
first goal would be to investigate further the magnetic filed dependence of the resonances
in the capacitance addition spectra, and deveIOp a complete theory of the effect. There is
plenty of motivation to characterize dopant atoms locally. In the context of developing a
solid sate quantum computer based on Si atoms, the single dopant atom embedded in a
semiconductor crystal represents the smallest possible semiconductor electronic device.
At low temperature, the electron or hole can be localized at the parent atom and the
quantum state of this charge could play a role of the building bock in the quantum
machine — the qubit. For charge based qubits, the qubit-qubit coupling is set by Coulomb
interaction, and thus allows for separation of ~100 nm; whereas spin-based qubits have
only 10-20 nm coherence length dictated by rapid exponential fall of the exchange

interaction [Golding , Kane]. Therefore, the next immediate goal would be to apply the

108

same scanned probe approach for studying single, isolated atoms in both Si and GaAs

based systems. In fact, such project has already been planned in our lab.

109

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