SCANNING PROBE STUDIES ON THE SURFACE OF BI2 SE3
By
Megan E. Romanowich

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Physics
2012

ABSTRACT
SCANNING PROBE STUDIES ON THE SURFACE OF BI2 SE3
By
Megan E. Romanowich
Topological insulators are an exciting material that are insulating in the bulk, but have
a metallic surface state where spin and momentum are locked perpendicular to each other
by strong spin orbit coupling. The simplest topological insulator, Bi2 Se3 , has a band gap of
0.3 eV and a single Dirac cone, a state where energy and momentum are linearly related, in
the gap. This special state is protected against scattering and small perturbations by time
reversal symmetry. Before any of the exotic effects predicted to be present in this material
can be fully realized, it is important to know what other electronic effects coexist with the
topological state on the surface of the material. To that end, I employed scanning tunneling
microscopy on the surface of Bi2 Se3 far from defect states. I probed the interplay of bulk
electrons with the Dirac cone that adds additional states. I also show that the Dirac cone can
coexist with a two-dimensional electron gas, although the amplitude is potentially reduced.

In memory of my father, Dr. Robert H. Knapp MD, who was responsible for fostering my
love of science and for my mother who helped me achieve my educational goals along the
way.

iii

ACKNOWLEDGMENTS

I could not go any further in writing my dissertation without first acknowledging the people
who helped me along the way. I first want to recognize my advisor, Stuart Tessmer, who
has been an invaluable resource in the lab. He helped me by answering questions sincerely,
no matter how simplistic they turned out to be, helped me thing through experimental
processes, and offered suggestions to troubleshoot the equipment. I always appreciated his
encouragement and his overly generous compliments about my scientific abilities. From
Stuart I not only learned how to be a thoughtful scientist, but also how to relax and be a
good mentor to others in the lab.
Stuart is not the only person to help support my research. Mal-soon Lee spent many
hours perfecting the calculations related in this work. She took the time to explain some of
the challenges with the program and the subtleties of the theoretical approach. S.D. Mahanti
likewise spent many hours in meetings helping with the analysis and offering his words of
wisdom.
There are many individuals who help the day-to-day activities of the department run
smoothly. The PA Machine Shop spent many hours talking through designs and making
the processing chamber that gave me the pristine samples for my experiments. I would
particularly like to thank Tom Palazzolo for his incredible machining knowledge. Baokang
Bi spent hours helping me learn and then troubleshoot the scanning electron microscope in
the clean room. Reza Lolee was an invaluable resource on cryogenic systems and always
helped find the tool I needed from somewhere in the basement. It never ceases to amaze
me just how much Reza knows about everything that goes on in the basement labs. My
time teaching labs would not have run nearly so smoothly without the knowledge and help
iv

of Mark Olsen and Richard Hallstein. Happily, Debbie Barratt and Kim Crosslan have been
invaluable resources and I am sure they have done more paperwork for me than I will know.
I would also recognize my committee: Professors Pengpeng Zhang, Remco Zegers, Kirsten
Tollefson and Carlo Piermarocchi. It was nice to have their support through this process.
I have very much appreciated those I shared a lab with throughout my time here. Josh
Veazey spent hours explaining what he was doing when he scanned and gave me helpful hints
on how to use the microscope. Kathy Walsh has been an amazing friend to me. I enjoyed
our conversations about science and Doctor Who over lunches and in the lab. Her undying
optimism, puns and dramatic flair helped me through many days where I thought nothing was
going to work. I also appreciate the hard work and creativity of three great undergraduates:
Mark Regalla, Andrew Belcher, and Eric Goodwin who perfected the process for and made
tips for my experiments. They all helped take a half-baked idea to make ultra-sharp tips
and make it a reality. Day to day lab work would not be nearly so interesting without the
day-to-day conversations with other lab-mates and others who also inhabit the basement
over the years: Matt DeNinno, Sean Wagner, Eric Gingrich and John Raguse. I have many
happy memories of time in the lab spent chatting about different scientifically irrelevant (and
sometimes relevant) topics with these individuals.
In addition to my work in the lab, I have greatly enjoyed participating in committees
and organizations over my time at Michigan State University. Women and Minorities in the
Physical Sciences gave me the opportunity to grow as a leader, learn about challenges facing
women in science, and take steps to address common problems locally. In particular, I have
loved my time in the Mentoring program, both as the Mentoring Program Coordinator and
as a mentor. I was thrilled with the department’s financial and graduate student mentors’
v

time commitments that enabled this program to be successful. Through mentoring, I was
able to meet some promising young undergraduates. Because I got to mentor them through
their entire undergraduate experience, the two that I particularly enjoyed were Monica Derris
and Caitlin Brecklin. These two mentees and some of the others impressed me with how
intentional they were with their future careers in physics and science. The other committee
that enabled me to not only become a better leader but also to change things within the
college was the Council of Diversity and Community in the College of Natural Science. Here,
I met some extremely motivated graduate students in the College outside my discipline and
dedicated staff members who truly want to address some key issues within the College to
make it a better place for everyone.
Were it not for my study group, I would not remember the very first years of graduate
school so fondly. Dan Olds has been a constant support from classes through research from
the very first day of orientation. He has provided some perspective on problems through
both verbal and alcoholic perspective. Carol Guess and Ania Kwiatkowski were both friends
and mentors. Their individual senses of poise and self-confidence are qualities I have tried
to emulate. These three made studying for subject exam classes enjoyable. Some of my
favorite study breaks were during girls nights with Amanda Prinke and Johanna Swanson.
Game nights with Dan Olds, Marilyn McDowall, Jenni Portman and her husband Guiseppe,
Connor Glosser and Shannon and Jim Demlow have been excellent at lightening the mood
after a lot of time in the lab or writing my thesis.
My life would not be complete without all the joy I get from my family. My parents
gave me so many advantages and emotional support throughout all my studies that helped
me to get where I am now. I count myself fortunate that my family has grown over my
vi

time in graduate school and that all of my family including my sister, Sarah, is nothing but
supportive of my sometimes demanding educational endeavors. I especially appreciate the
support of my husband, Tim, even when my path through school and ultimate employment
goals were unclear. He has been a particular support these last few months by taking on
much of the responsibility for our darling daughter, Amelia, so that I could focus on finishing
up. I love them all and I could not enjoy my work so much without the joy they bring me
at home.

vii

PREFACE

Having just left Dr. Mackey’s material science research lab in summer of 2008, I was very
excited to learn a new technique in the Tessmer group: scanning tunneling microscopy. Over
the next year, I rebuilt a scanning tunneling microscope, learning what makes scanning
tunneling microscopy work. At the same time, a hot new area of research was just coming to
prominence on the condensed matter scene. The earliest experiments on three-dimensional
topological insulators were published in 2008 and 2009, just after I began my research in the
Tessmer group. Due to a fascinating surface state, topological insulators are a natural pairing
with scanning tunneling microscopy and other surface probe techniques. Shortly after joining
the group, I proposed the experiments on the topological insulator Bi2 Se3 which made up
the experiments described in this work.
Over the next few chapters I will explain some scanning probe techniques and unusual
features in the local density of states in Bi2 Se3 . To ground the reader in the literature, I have
taken a historical approach in the introduction. As topological insulators are a rather specific
topic, I am motivating key aspects of their physics by first explaining the earliest experiments
in two dimensions. This early work shaped some of the terminology in topological insulating
literature today. Conversely, scanning probe microscopy has so many applications that to
do it justice, I very briefly elaborate on its scope before focusing on my Bi2 Se3 studies.

viii

TABLE OF CONTENTS

List of Figures

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xi

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Topological Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Scanning Probe Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
6

Chapter 2 Apparatus and Technique . . . . . . . .
2.1 Sample Processing Chamber . . . . . . . . . . . .
2.2 Scanning Tunneling Microscopy and Spectroscopy
2.2.1 Physics . . . . . . . . . . . . . . . . . . . .
2.2.2 Calibration . . . . . . . . . . . . . . . . .
2.2.2.1 Graphite . . . . . . . . . . . . .
2.2.2.2 Pt Pits . . . . . . . . . . . . . .
2.3 Experimental Constraints and Setup . . . . . . .
2.3.1 Besocke Type STM . . . . . . . . . . . . .
2.3.2 Noise Isolation . . . . . . . . . . . . . . .
2.3.3 Cryogenic System . . . . . . . . . . . . . .
2.3.4 Tips . . . . . . . . . . . . . . . . . . . . .
2.3.4.1 In situ Tip Cleaning Techniques

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25

Chapter 3 An Introduction to Topological Insulators
3.1 Background . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Graphene Model . . . . . . . . . . . . . . . .
3.1.2 HgTe/CdTe Wells . . . . . . . . . . . . . . . .
3.1.3 Three Dimensional Topological Insulators . .
3.1.4 Early Studies on Bi1−x Sbx . . . . . . . . . .
3.1.4.1 ARPES . . . . . . . . . . . . . . . .
3.1.4.2 Quasiparticle Interference Patterns .
3.2 Bi2 Se3 Background . . . . . . . . . . . . . . . . . . .
3.2.1 Structure . . . . . . . . . . . . . . . . . . . .
3.2.2 Dopants and Defects . . . . . . . . . . . . . .
3.2.3 Surface Potential Fluctuations . . . . . . . . .
3.2.4 Band Bending . . . . . . . . . . . . . . . . . .

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49

ix

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Chapter 4 Measurements of States on the Surface of Bi2 Se3
4.1 Sample Overview and Procedure . . . . . . . . . . . . . . . . .
4.2 Spectroscopy Measurements on Bi2 Se3 . . . . . . . . . . . . .
4.3 Analysis of the Bi2 Se3 Spectrum . . . . . . . . . . . . . . . .
4.4 Analysis of Two Dimensional Electron Gas on the Surface . .
4.5 Bi2 Se3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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54
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69

Chapter 5 Other
5.1 Bi2.02 Se2.98
5.2 Bi2.04 Se2.96
5.3 Bi2.08 Se2.92

Measurements of Bi2 Se3
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72
72
74
78

Chapter 6 Other Scanning Probe Methods
6.1 Capacitance and Kelvin Probe Circuit . .
6.1.1 High Electron Mobility Transistor .
6.2 Capacitance Measurement . . . . . . . . .
6.2.1 Physics . . . . . . . . . . . . . . . .
6.2.2 Demonstration . . . . . . . . . . .
6.3 Kelvin Probe Measurement . . . . . . . . .
6.3.1 Physics . . . . . . . . . . . . . . . .
6.3.2 Demonstration . . . . . . . . . . .
6.4 Tips . . . . . . . . . . . . . . . . . . . . .
6.4.0.1 Etched . . . . . . . . . . .
6.4.0.2 Cut . . . . . . . . . . . .
6.4.0.3 Ultrasharp . . . . . . . .

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94

Chapter 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97
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99

Bibliography . . . . . . . . . . . . . . .

x

. . . . . . . . . . . . . . . . . . . 100

LIST OF FIGURES

Figure 1.1

Figure 1.2

Figure 2.1

For interpretation of the references to color in this and all other figures, the reader is referred to the electronic version of this dissertation. Arrows in these diagrams represent spin. (a) Band diagram
of an insulator showing the filled valence states are separated from
empty conduction band states by a band gap. The bands are spin
degenerate. (b) In a topological insulator, a surface state connecting the conduction and valence band consists of a band where holes
traveling left (negative in k-space) are connected to electrons traveling right, and vice versa exists within the gap. In (b) the horizontal
scale represents momentum, but there is no physical meaning to the
horizontal scale in (a). . . . . . . . . . . . . . . . . . . . . . . . . . .

4

(a) Three quintuple layers of Bi2 Se3 . Inside each layer the atoms
form covalent bonds and each layer is held to its neighbor by Van
der Waals forces. This image is from of Dr. Mal-soon Lee. (b) My
Bi2 Se3 local density of states measurement. The dotted lines show
the Dirac cone. Specific features of the Dirac cone will be discussed
in Chapter 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

My processing chamber. (a) The back of the chamber showing the
exit to the microscope and two ports. The top port goes to a pressure
relief valve that prevents the chamber from exploding the top off and
exposing the contents to air. The lower port connects to a loading
chamber so that the main chamber is always under nitrogen. (b) The
top shows two access ports: one is for loading materials too big to
come in through the loading chamber. The other supports the tubes
that allow nitrogen to come in and out in a continuous flow through
the chamber. The two protrusions are the glove ports. All of these
tubes and lids are made of plexiglass to allow maximum light within
the chamber. The support structure on the right is made of aluminum
and provides the strength to support the loading fork. . . . . . . . .

10

xi

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

My experimental setup. The blue dewar in the lower left of the image
is my cryogenic system discussed in Section 2.3.3. The sample processing chamber is in the center of the image with yellow gloves. The
fork for transferring the sample is inside the steel tube to the right on
the image (and has the pink marker on its end). The sample loading
chamber is not visible as it is behind the nitrogen chamber itself. . .

12

(a) Schematic of the tip and the sample. The sample is in the x-y
plane and the tip is hovering above the sample in the z-direction.
(b) A model of tunneling barrier showing the wavefunction decays
exponentially in the gap [38]. . . . . . . . . . . . . . . . . . . . . . .

13

(a) Graphite atoms at a 1.0 nA tunneling current and 0.1 V bias. The
scale bar is 500 pm. (b) This is an image of step edge of graphite.
The spectra shown in (c) and (d) were taken on the smooth part of
the image. Because the scale of this image is so much larger than (a),
individual atoms are not resolved. The scale bar is 200 nm. (c) Shows
linear I-V curve on graphite. (d) Density of states measurement on
graphite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

Pt pits sample where the scale bar is 1 µm. The pit is 5 µm × 5 µm
and 180 nm deep. There is at least a double tip, possibly a triple tip
in this case giving rise to multiple apparent edges as indicated by the
arrows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

(a) Besoke design microscope where three piezoelectric tubes support
a sample and a central tube scans the tip. (b) My microscope outside
of its protective case. . . . . . . . . . . . . . . . . . . . . . . . . . .

21

Diagram of the cryostat. The microscope mounts on the probe entry
port and is isolated from the sample tube by a gate valve until it is
under vacuum. The microscope can then be lowered slowly into the
bottom of the dewar. The system is cooled by a main bath of nitrogen or helium. The transfer tube entry port is the site where liquid
is transferred into the system. Gas from liquid boil off is released
through the helium exhaust port [43]. . . . . . . . . . . . . . . . . .

24

xii

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure from Zhang et al. [44] shows the unit cell in a crystal lattice of
￿ ￿
￿
Bi2 Se3 . Primitive vectors t1 , t2 , and t3 show equivalent translation
vectors of the lattice. There are 3 inequivalent sites in the lattice, Bi,
Se1 and Se2 that stack in inequivalent layers C(Se1)-A(Se1)-B(Bi)C(Se2)-A(Bi)-B(Se1)-C(Se1). Viewed from above, this looks like a
hex lattice with inequivalent layers marking out triangular lattices. In
the bottom right, the three-dimensional unit cell is mapped out with
its two-dimensional projection showing high symmetry points in kspace, Γ, M and K that are time reversal invariant momentum points
for this crystal structure. Used under Creative Commons Attribution
3.0 Unported (CC-BY) license and all credit for creation of this figure
goes to [44]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

A calculation from Kane and Mele [48]. A single pair of edge states
traverse the bulk insulating gap due to spin orbit coupling in a band
structure calculation from a tight-binding model on a strip (in the
inset) of graphene. Reprinted figure with permission from Kane, C.L.
and Mele, E.J., Physical Review Letters, 95, 226801(2005). Copyright
(2005) by the American Physical Society. Abstract can be viewed at:
http://prl.aps.org/abstract/PRL/v95/i22/e226801. . . . . . . . . . .

33

A schematic band structure showing Kramers pairs traversing the
gap from Fu et al. [52]. (a) A schematic of a topological insulator
that has an odd number of Fermi level crossings. (b) A schematic of
a simple insulator that has an even number of crossings. There are
still degenerate points at Λa and Λb but they are not topologically
protected against deformation. Reprinted figure with permission from
Fu, Liang and Kane, C.L., Physical Review B, 76, 045302 (2007).
Copyright (2007) by the American Physical Society. Abstract can be
viewed at: http://prb.aps.org/abstract/PRB/v76/i4/e045302. . . . .

34

Figure adapted from Konig et al. [56]. Resistance measurement of
CdTe/HgTe quantum wells that support the quantum spin Hall state.
Sample I shows that the wells are insulating below the critical width.
Sample II shows disorder due to temperature effects. Samples III
and IV show that the conductance is precisely equal to 2e2 /h for two
different well widths above the critical dimension. From M. Koenig,
S. Wiedmann, C. Bruene, A. Roth, H. Buhmann, L. W. Molenkamp,
X.-L. Qi, S.-C. Zhang, Science, 318, 766-770 (2007). Reprinted with
permission from AAAS. . . . . . . . . . . . . . . . . . . . . . . . . .

36

xiii

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

(a) Adapted from [58]. Bi0.92 Sb0.08 dI/dV map with the Fourier
transform of the image inset. The snowflake pattern shows the sixfold symmetric scattering in k-space. Some scattering directions are
forbidden. (b) Adapted from Figure 4 in reference [58]. The scattering patterns are only duplicated when spin-dependent scattering is
accounted for in calculations. Reprinted by permission from Macmillan Publishers Ltd: NATURE (P.Roushan, J.Seo, C.V. Parker, Y.S.
Hor, D.Hsieh, D.Qian, A.Richardella, M.Z. Hasan, R.J. Cava, and
A.Yazdani. “Topological surface states protected from backscattering by chiral spin texture,” Nature, 460, 11061109, (2009)), copyright
(2009) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

Density functional theory calculation of the Bi2 Se3 bulk [gray] and
6 quintuple layer calculation of the Dirac cone in blue. The single
Dirac cone on the surface is circled in red. Figure courtesy of Dr.
Mal-soon Lee. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

Triangular defects on Bi2 Se3 where the bright triangles are attributed
to bismuth on a selenium site substitutions. The scan range is 65
nm×65 nm. Topograph from Urazhdin et al. [10]. Reprinted figure with permission from S.Urazhdin, D.Bilc, S.Mahanti, S.Tessmer,
T.Kyratsi, and M.Kanatzidis. Physical Review B, 69(8):085313, 2004.
Copyright (2004) by the American Physical Society. Abstract can be
viewed at: http://prb.aps.org/abstract/PRB/v69/i8/e085313 . . . .

44

Quasiparticle scattering data from Kim et al. [95]. The six-fold symmetry gets smeared by the bulk valence band when the voltage is
less than -500 meV and by the bulk conduction band when it is
greater than -250 meV. Scattering is enhanced in the Γ-M direction
and suppressed in the Γ-K. At some voltages, they 6-fold symmetry is less clear. (a) and (b) show the smearing of the topological
state due to the valence band and (g)-(i) show the smearing due to
the conduction band. Reprinted figure with permission from S.Kim,
M.Ye, K.Kuroda, Y.Yamada, E.Krasovskii, E.Chulkov, K.Miyamoto,
M.Nakatake, T.Okuda, Y.Ueda, K.Shimada, H.Namatame, M.Taniguchi, and A.Kimura. Physical Review Letters, 107:056803, (2011).
Copyright (2011) by the American Physical Society. Abstract can be
viewed at: http://prl.aps.org/abstract/PRL/v107/i5/e056803. . . .

48

xiv

Figure 3.9

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

ARPES measurements on Bi2 Se3 showing effects of the top quintuple layer separating from the crystal [102] E vs. k of the Dirac cone
is shown with the 2DEG evident as the bright rim surrounding the
diffuse conduction band edge. The Dirac cone is the outer x shaped
feature with the Dirac point at the center of the x. Reprinted by
permission from Macmillan Publishers Ltd: NATURE COMMUNICATIONS (M.Bianchi, D.Guan, S.Bao, J.Mi, B.B. Iversen, P.D.C.
King, and P.Hofmann. “Coexistence of the Topological State and a
Two-Dimensional Electron Gas on the Surface of Bi2 Se3 ”. Nature
Communications, 1:128, 2010.), copyright (2010) . . . . . . . . . . .

53

Characteristic triangles from bismuth substitutions on selenium sites
in Bi2 Se3 doped with 4% excess Bi. Image is at 4.2K with -0.8 V
bias and 0.50 nA set current. The scale bar is 3 nm. . . . . . . . . .

56

(a) The calculated I-V curve on Bi2 Se3 doped with 4% excess Bi at
4.2K. (b) The same I-V curve expanded in the region of the Dirac
cone. The slope does not go to zero at zero voltage, indicating there
are states in the gap. Due to an analog to digital issue in the electronics, the current was calculated from the dI/dV curve in Figure 4.3 (a).
The dI/dV curve was measured directly from using the lock-in amplifier as described in Section 2.2.1 and therefore did not experience
the same analog to digital issue. . . . . . . . . . . . . . . . . . . . .

57

(a) Density of states on Bi2 Se3 doped with 4% excess Bi at 4.2K
where the linear region of the Dirac cone is between +/-0.15 V. The
Fermi level is in the center of the gap. There are additional states
that lift up the bottom of the Dirac cone. The curve is an average
of 150 curves taken successively to reduce the signal to noise ratio.
(b) The density of states only in the region of the Dirac cone showing
parameters α and β, described in the text. . . . . . . . . . . . . . .

58

Density of States where each of the curves is an average of 50 measurements. Data set A was taken before Data set B without manually
moving the tip. Data set A has been offset by one tick mark to ease
visual comparison. A blue bracket has been drawn to highlight the
enhancement. We speculate the tip drifted during the measurement,
causing the feature at +0.2 V to change. . . . . . . . . . . . . . . .

60

xv

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

(a) Valence band calculation with increasing numbers of QL. The
shoulder state in the Γ-M direction increases in energy and eventually
crosses the Dirac point. (b) Graph of the change in energy with QL.
Eventually the height of the shoulder will reach a bulk value as the
number of QL increases. Figure courtesy of Dr. Mal-soon Lee . . . .

62

Partial density of states (PDOS) calculations performed by Dr. Malsoon Lee on a 6 QL slab. The inset shows the parts of k-space that
contribute to Region I and Region II. The contribution to PDOS from
the Dirac cone is depicted in red. It resides primarily in the top layers
within the bulk band gap. The contribution from the shoulder states
(blue) have the largest contribution in the middle QL, however it has
a finite contribution in the top QL Se and Bi atoms. In this case, the
peak is just to the left of zero energy (the Fermi level) for this rather
thin slab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

Comparison of bulk and slab calculations. This contrasts the partial
charge density from bulk band structure calculations (gray in the
band structure) on the left side of the image with those from the slab
calculations (blue in the band structure calculations). The Se are
yellow, the Bi atoms are blue-gray and the the intensity of the red
halos depicts how strong the charge density is at that region. No halo
means there is no significant charge density. The bulk calculations
show the partial charge density is mostly distributed on the Se atoms
in all layers equally. The partial charge density on the slab calculation
on the right is primarily located on Se atoms as well, but has the
highest density in the middle layers (the bulk-like area of the slab).
However, there is some charge density on Se atoms in the outermost
quintuple layers shown by the green arrows. Figure courtesy of Dr.
Mal-soon Lee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

Band structure calculations of a 6 QL slab of Bi2 Se3 for which the
top QL is separated from the others. The percentages refer to how
much additional separation is given to the top QL. We see as the
top layer detaches, the bottom of the conduction band drops into the
gap and forms a two-dimensional electron gas on the surface. Figure
courtesy of Dr. Mal-soon Lee. . . . . . . . . . . . . . . . . . . . . .

70

STM data at 77K. Both the (a) I-V curve and the (b) density of
states are an average of 5 curves. (c) Atoms on the surface of Bi2 Se3
at -0.20 V and 0.1 nA. Scale bar is 1.6 nm. . . . . . . . . . . . . . .

71

xvi

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6

Figure 6.1

Data taken at 77K without the chamber. (a) Atoms on the surface
of Bi2.02 Se2.98 at a set current of 0.11 nA and -0.5 V. Scale bar is 3
˚. (b) Dark pits seen all over the Bi2.02 Se2.98 potentially caused by
A
surface contamination. Scale bar is 17 ˚. . . . . . . . . . . . . . . .
A

74

Defects on the surface of Bi2.02 Se2.98 at 0.5 nA set tunneling current
and 77K. Scale bars are 7 ˚. Each image is taken in sequence at a
A
bias voltage of (a) -0.5 V (b) -0.55 V (c) -0.50 V and (d) -0.4 V.
The bright spot in the upper right corner of every image is surface
contamination. Impurities in the crystal are visible in the upper left
in (a) and (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Defects on the surface of Bi2.04 Se2.96 at 0.5 nA set tunneling current
and liquid helium temperature. Each image is taken in sequence at
a bias voltage of (a) -0.80 V, 36 pA (b) -0.49 V, 73 pA and (c) -0.29
V, 52 pA. Dark impurity spots become more diffuse with changing
voltage. Bright specs of surface contamination also visible in every
image. Scale bars are 3.3nm. . . . . . . . . . . . . . . . . . . . . . .

76

Data taken at 4.2K. Scale bars are 3.3 nm.(a) Bright triangular defects
on the surface of Bi2.04 Se2.96 at a set current of 60 pA and bias
voltage of -0.80 V. (b) Clusters of triangular defects arranging in
strings at 60 pA and -0.79 V. . . . . . . . . . . . . . . . . . . . . . .

77

Data taken at 77K. (a) Bright triangular defects on the surface of
Bi2.04 Se2.96 at a set current of 24 pA and bias voltage of -0.48 V.
Scale bar is 17 nm. (b) Pits on the surface at 118 pA and -0.38 V.
The scale bar is 67 nm. . . . . . . . . . . . . . . . . . . . . . . . . .

78

Data at 77K, scale bar is 3.3 ˚ and set current 1.02 nA. (a) -0.6 V
A
(b) -0.3 V and (c)-0.6 V. There is both surface contamination, atomic
resolution and a voltage-dependent defect in the upper right corner.

79

HEMT circuit diagram: The standard capacitor (line drawn in blue)
is only included for capacitance measurements. Oscillations are applied either to the bias voltage for capacitance measurements or by
oscillating the tip above the sample in Kelvin probe. The lock-in
amplifier detects the change in capacitance on the tip due to these
oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

xvii

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

Figure 6.6

Figure 6.7

Figure 6.8

Figure 6.9

The plot shows the drain current vs. the drain-source voltage of a
HEMT at different gate voltages. Varying the gate voltage changes
the effective resistance of the source-drain channel. The HEMT is
operated in the linear region circled in red to minimize power output
[43]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A photograph of the Fujutsi model FHX35X HEMT. The size of the
device is ∼ 500 µm across [43]. . . . . . . . . . . . . . . . . . . . . .
(a) Topography image of the corner of a pit. (b) Capacitance image
in the same region as (a). Both are imaged at 77K, the scale bars are
330nm long and were taken with an etched tip. . . . . . . . . . . . .
Nulling voltage measurement on Bi2 Se3 . The arrow shows the voltage
where all the curves intersect (0.9 V). Because there is a -0.43 V
applied to the tip relative to the sample, the nulling voltage here is
0.9 − (−0.43) = 1.23 V. . . . . . . . . . . . . . . . . . . . . . . . . .

84

84

87

89

Bi2 Se3 topography (a) and Kelvin probe (b). In this case, the Kelvin
probe image is nearly featureless because the signal is dominated by
the high plateau in the lower left. The topography image shows two
step edges, where each step is 1 nm high. The Kelvin probe scan
must be within half a nanometer of the sample – even one nanometer
beyond does not contribute. The scale bar is 170 nm on both images.
The amplitude of oscillation is 1 nm. . . . . . . . . . . . . . . . . . .

90

Bi2 Se3 topography (a) and Kelvin probe (b), Here we see a scan
showing a trench indicated by an arrow. The trench is three times
as wide in Kelvin probe as topography. Th Kelvin probe scan had a
1 nm amplitude of oscillation on the tip at the nulling voltage and
was 1 nm above the sample. The scale bars are both 33 nm. The tip
may have drifted over the four hours between the topography scan
and the Kelvin probe scan . . . . . . . . . . . . . . . . . . . . . . .

92

SEM image of an etched tip made by Agilent with a tip radius of
curvature of 100 nm. Both (a) and (b) are the same tip with different
scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

Cut tips with a radius of curvature of 10 nm and 25 nm. . . . . . . .

94

xviii

Figure 6.10

Figure 6.11

Figure 7.1

(a) The micropositioner used to assemble the ultrasharp tips. It finely
positions a Pt(.8)Ir(.2) wire with respect to a AFM cantilever. (b) A
view through the microscope used to see tip placement on the epoxy.
The triangular cantilever is touching the conducting epoxy. . . . . .

95

Ultrasharp tip with radius of curvature of 10 nm. (a) All components of the tip are visible: the cantilever, conducting epoxy, and
Pt(.8)Ir(.2) wire. This tip has nice cylindrical symmetry. (b) The
apex of the tip is the highest point. . . . . . . . . . . . . . . . . . .

96

Schematic of the proposed experiment. An STM tip will probe vortices in an superconducting antidot patterned on a topological insulator to search for Majorana modes [51]. Figure from Dr. Stuart
Tessmer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

xix

Chapter 1
Introduction
This thesis is organized as follows: After I briefly introduce the major topics covered in
my thesis experiment, topological insulators, I will begin to discuss my apparatus and the
scanning tunneling microscopy technique in Chapter 2. Chapter 3 will cover a historically
motivated background of the physics of topological insulators and the topological insulator I
studied, Bi2 Se3 . Chapter 4 covers both my thesis experiment, spectroscopy on the surface of
Bi2 Se3 , and the analysis. Then I go on to discuss other observations on scanning tunneling
microscopy studies on Bi2 Se3 samples in Chapter 5 before I completely change topics to
discuss other scanning probe techniques I performed in Chapter 6. Finally, I will conclude
with a summary of my experimental observations and analysis and discuss future directions
in Chapter 7.

1.1

Topological Insulators

Undoped semiconductor materials are normally insulators. Figure 1.1 (a) shows the standard
electronic structure, consisting of a filled valence band and an empty conduction band. The
1

bands are separated in energy by a band gap as shown in Figure 1.1 (a). The reason
a semiconductor cannot support a current is because of the filled valence band. Every
electronic state in the band has a twin state with the opposite sign in momentum-space
(k-space). In other words, for every electron that could transport current to the right, there
is a twin that carries current to the left. Since all of the states are filled (i.e. occupied), there
is no way to build a quantum state that will carry an overall current in any one direction.
The quantum Hall effect was discovered in 1980. When a large magnetic field is applied
perpendicular to the two-dimensional conducting plane, a quantized current is established
on the edges, with the interior becoming an insulator [1]. This current is robust against
backscattering; in other words, a charge cannot rebound from a defect in a way that would
change its direction. A topological invariant, called a Chern number, distinguishes the
quantum Hall insulator from a simple insulator and helps to explain the robust nature of the
edge states in the integer quantum Hall effect [2]. It would be years before another boundary
state that is protected by a topological invariant would be predicted and discovered.
A topological insulator is a material that is insulating in the bulk but conducting on the
surface; it has edge or surface states that are conducting (gapless) that lie inside the bulk
gap as depicted in Figure 1.1 (b). Moreover, the surface state, that arises due to strong
spin-orbit coupling, traverses this gap is composed of a single (more generally it just needs
to be an odd number) pair of eigenstates on each (conduction and valence) band edge which
are symmetric under time reversal. These states are known as Kramers pairs or Kramers
doublets [3, 4]. However, if there are even numbers of Kramers pairs on each band, elastic
backscattering destroys any conducting edge state and it is the topological equivalent of a
simple insulator [4]. As long as time reversal symmetry holds, the topological surface state
2

where electron spin is locked perpendicular to its momentum is robust against scattering.
The interplay of momentum and spin is amazing and simple: charge moving to the right
must have opposite spin as charge moving to the left.
Much of the research in the last five years has focused on 3D topological insulators.
Here, the surface state has an odd number of massless Dirac cones around high symmetry
points [5] that are protected by a topological invariant. A Dirac cone is a state that can
be described by Dirac’s equation where energy is linearly related to momentum. Figure
1.1 (b) shows the topological insulator’s Dirac cone where states of opposite sides of the
cone (opposite momentum) have opposite spin due to the spin-orbit interactions. Spin orbit
interactions are crucial in creating a topological insulator as it gives rise to band inversion
that forms the cone [6]. Due to the magneto-electric effect, surface states of topological
insulators are described by Maxwell’s equations with the addition of a topological term [7].
This gives rise to fascinating properties. For example, theory predicts that bringing an
electric charge in close proximity above the surface produces an image magnetic monopole
inside the material [8]. Moreover, the interface between a superconductor and a topological
insulator can support a particle analogous to a Majorana fermion, a particle which is its own
antiparticle. The state created in the topological insulator is actually a particle that is its
own hole that could be useful in quantum energy storage [9]. The prospect of these effects is
thrilling, but before serious experimental progress can be made in creating and discovering
these exotic effects, all the fundamental aspects of the material including bulk effects need
to be characterized to avoid experimental complications later.
For example, Bi2 Se3 is a narrow band gap semiconductor that has been studied for
years. Previous interest in these narrow gap semiconductors came from their thermoelectric
3

(a)

(b)

Figure 1.1: For interpretation of the references to color in this and all other figures, the reader
is referred to the electronic version of this dissertation. Arrows in these diagrams represent
spin. (a) Band diagram of an insulator showing the filled valence states are separated from
empty conduction band states by a band gap. The bands are spin degenerate. (b) In a
topological insulator, a surface state connecting the conduction and valence band consists of
a band where holes traveling left (negative in k-space) are connected to electrons traveling
right, and vice versa exists within the gap. In (b) the horizontal scale represents momentum,
but there is no physical meaning to the horizontal scale in (a).

4

(a)

(b)
Figure 1.2: (a) Three quintuple layers of Bi2 Se3 . Inside each layer the atoms form covalent
bonds and each layer is held to its neighbor by Van der Waals forces. This image is from
of Dr. Mal-soon Lee. (b) My Bi2 Se3 local density of states measurement. The dotted lines
show the Dirac cone. Specific features of the Dirac cone will be discussed in Chapter 4.
5

properties [10, 11, 12]. It was not until recently that Zhang et al. predicted Bi2 Se3 , Bi2 Te3
and Sb2 Te3 to be strong topological insulators while their sister material, Sb2 Se3 , was
predicted to be a trivial insulator [6]. The topological surface state arises from strong spin
orbit coupling that induces a p-orbital from Bi and Se to invert giving the surface state a
finite skin depth [6]. Bi2 Se3 has a single, nondegenerate Dirac cone at the Γ point (where
￿ = ￿ in k-space), making it a very simple topological insulator [6, 13].
k 0
Bi2 Se3 forms a quintuple layer of three Se nets sandwiching two Bi layers as shown in
Figure 1.2 (a). The quintuple layers are held together weakly with Van der Waals forces;
therefore, these materials are very attractive for STM because they cleave easily with scotch
tape and the surface stays relatively clean. Prior work in our group showed that a Bi
impurity five lattice sites down from the surface can show up in topography scans [14]. My
research took a different approach to study point spectroscopy far from defects to see what
unexpected interactions appear in the electronic structure seen in Figure 1.2 (b). I find that
bulk states propagate to the surface and that the Dirac cone coexists with a two-dimensional
electron gas. Neither of these effects have previously been described by scanning tunneling
spectroscopy measurements.

1.2

Scanning Probe Methods

In 1982, Binnig and Rohrer [15, 16] revolutionized surface science bringing surface probes
to the angstrom scale by inventing the scanning tunneling microscope (STM). A conducting
tip is attached to a piezoelectric material that can expand and contract in all three spatial
dimensions. This not only helps to bring the tip to within a nanometer of the surface,
but also facilitates raster scans for topographical images. Without the ceramic piezoelectric
6

materials, the microscope would not have had the ability to scan precisely at the atomic scale.
Once in tunneling range, controller electronics keep the tip from running into the surface by
maintaining either a constant current or constant height using a feedback mechanism. The
tunneling current, usually around 1 nA, is amplified and processed by a computer. By taking
a series of these measurements an image of the electronic state of the surface is generated.
Tunneling between two metals separated by an insulating barrier had been understood
for years before the advent of STM [17, 18, 19]. However, the popularity of the scanning
tunneling microscopy technique fueled further interest in the area experimentally and theoretically [20, 21, 22]. Early research not only focused on studying different materials but
also on creating different STM designs including the louse [23], single tube scanner[24], the
beetle[25], and, a decade later, the walker [26]. Many different materials were scanned including metals [27], semiconductors [15], and biological materials [28]. Since its inception,
it has been used to study many different areas of condensed matter physics. Topics of interest include superconductivity[29], charge density waves[30], atomic manipulation[31], and
surface lattice structures[32].
Other techniques using the same mechanical principles as STM have splintered off to
form many different sorts of measurements. Moreover the principles behind the scanning
probe techniques had been the subject of research for years prior to the advent of precisely
controlled piezoelectric devices. For instance, Kelvin probe measurements were well understood [33], and performing them in tandem with STM was a natural extension. As another
example, scanning capacitance methods can measure single dopant molecules under a surface
[34, 35, 36, 37], often used in conjunction with STM. Needless to say, over the past 30 years,
scanning probe techniques have become a scientific staple.

7

Chapter 2
Apparatus and Technique
In this chapter, I will explain my apparatus and the scanning tunneling microscopy (STM)
technique. I will begin by detailing the design parameters of the sample processing chamber
that I made to achieve better sample surface quality for my thesis experiment. Then I will
explain the physics of STM before explaining the details of my experimental setup.

2.1

Sample Processing Chamber

Early in the experimental process as I started to attempt measurements on Bi2 Se3 , I determined that exposing the sample to air for any length of time was detrimental to the surface
quality. It is likely this contamination results from water or hydrocarbons in the air attaching
themselves to the surface. I solved this problem by designing a nitrogen gas sample-cleaving
chamber. To be successful, the chamber needed to maintain a nitrogen environment at all
times. The sample must be loaded directly from the chamber onto the STM using a loading fork so it could be immediately placed under vacuum. These constraints dictated the
geometry of the chamber. In order to ensure cleaving was easy to do and see, we used a tall
8

plexiglass tube. Tools needed to be easily loaded into the chamber. To that end, I included
standard size ports on the back to which additional apparatus could be attached without
breaking the nitrogen environment. The chamber included a removable lid and within the
lid there were capped loading ports that could similarly be repurposed.
Figure 2.1 shows the chamber design. In addition to its excellent optical properties,
plexiglass is easy to glue and is more durable than other materials with similar optical
properties. All access port supports were likewise of plexiglass to prevent any leaks that
might occur from trying to glue two unlike materials together. The entire central chamber
tube is 14 inches tall and 10 inches in diameter to provide adequate work space. The
arm access tubes in the front of the chamber are 4 inches in diameter to accommodate
large hands. The lid and its inserts are sealed with vacuum grease to allow easy access
for removing apparatus (for example: tape dispensers). Extremely large materials could
be transferred through the main lid if required. The other small access port has two holes
that allow nitrogen gas to continuously circulate through the chamber. Finally, the entire
chamber tube is mounted on an aluminum support that provides adequate strength to hold
up the transfer fork without compromising the integrity of the chamber.
After it was built by the machine shop, the chamber was attached to the cryogenic STM
system shown in Figure 2.2. To minimize the cleaving chamber’s air exposure, a sample
loading chamber was added to one of the ports in the back of the nitrogen cleaving chamber.
Nitrogen was then continuously circulated through both chambers and vented into the main
laboratory to ensure all air remains purged from the system at all times. The only time the
main chamber gets exposed to laboratory environment is when the gloves are changed or
large items need to be removed or added. The quality of the sample surfaces and subsequent
9

(a)

(b)
Figure 2.1: My processing chamber. (a) The back of the chamber showing the exit to the
microscope and two ports. The top port goes to a pressure relief valve that prevents the
chamber from exploding the top off and exposing the contents to air. The lower port connects
to a loading chamber so that the main chamber is always under nitrogen. (b) The top shows
two access ports: one is for loading materials too big to come in through the loading chamber.
The other supports the tubes that allow nitrogen to come in and out in a continuous flow
through the chamber. The two protrusions are the glove ports. All of these tubes and lids
are made of plexiglass to allow maximum light within the chamber. The support structure
on the right is made of aluminum and provides the strength to support the loading fork.

10

data improved greatly with the addition of the cleaving chamber!

2.2

Scanning Tunneling Microscopy and Spectroscopy

Scanning tunneling microscopy operates on the principle of quantum tunneling through two
conducting materials separated by a vacuum or air barrier. If the tip-sample separation is
∼ 5 ˚ and a voltage is applied to the sample relative to the tip, a picoamp or nanoamp
A
scale tunneling current develops that is sensed with the help of a preamplifier. Without an
applied voltage, electrons would freely tunnel from one side to the other and no net current
would develop. In its most common operation mode, the tip can raster over a surface with
the help of a feedback loop that keeps the tunneling current constant. By monitoring how
far the tip is extended or contracted to maintain the current at each position in the scan
range a computer builds up a topograph of the surface.
The simplest quantum-mechanical model of STM is a trapezoidal one-dimensional barrier of width d separating the sample and tip. A visual representation of this model can
be seen in Figure 2.3 where the surface of the sample is taken to be at z = 0. The electronic wavefunction, ψ(z), is governed by the one-dimensional time-independent Schr¨dinger
o
equation:
−

￿2 ∂ 2 ψ(z)
+ U ψ(z) = Eψ(z)
2m ∂z 2

(2.1)

where m is the electron mass, ￿ is Plank’s constant divided by 2π, U is the average barrier
height and E is the electron energy. U − E is an effective barrier to overcome for tunneling.
While outside the barrier, the solutions of the Schr¨dinger equation are sinusoidal. However,
o
inside the barrier the wavefunction decays exponentially. The tunneling current is then
11

Figure 2.2: My experimental setup. The blue dewar in the lower left of the image is my
cryogenic system discussed in Section 2.3.3. The sample processing chamber is in the center
of the image with yellow gloves. The fork for transferring the sample is inside the steel tube
to the right on the image (and has the pink marker on its end). The sample loading chamber
is not visible as it is behind the nitrogen chamber itself.

12

(a)

(b)
Figure 2.3: (a) Schematic of the tip and the sample. The sample is in the x-y plane and the
tip is hovering above the sample in the z-direction. (b) A model of tunneling barrier showing
the wavefunction decays exponentially in the gap [38].

13

proportional to the transmission probability through the classically forbidden gap; as ψ
decays exponentially, so does the transmission probability which turns out to be:
￿
2m(U − E)
I∝
|ψ(0)|2 e−2κd ; κ =
.
￿
n
￿

(2.2)

This result highlights the importance of the width of the gap on the magnitude of the
√
tunneling current. The constant, κ = 0.51 U − E ˚−1 . Assuming the effective barrier (or
A
work function) for platinum is 5 eV [39, 40], retracting the tip 1 ˚ reduces the tunneling
A
current by an order of magnitude. The success of STM at producing informative topographs
is largely due to this exponential dependence on distance.

2.2.1

Physics

The coplanar tunneling barrier model of STM is particularly good at highlighting the importance the tunneling gap has on the current; however, it leaves out a lot of information
about the surface and tip. In reality, the pointed shape of the tip affects the tunneling and
while a realistic picture is perhaps too complex for this thesis, other models exist that relate
the tunneling current to intrinsic material properties with few assumptions. In most cases,
tip sample coupling is weak, and first-order perturbation theory can capture more of the
picture. In other words, the proximity of the tip induces only a small change in the sample
electronic wavefunctions, and vice versa. In this approach first put forward by Bardeen in
1961 on metal-insulator-metal systems[19], the tunneling current, I, is related to a tunneling
14

matrix element, M , between the tip and sample wavefunctions:
￿
4πe +∞
I=
[f (EF − eV + ￿) − f (EF + ￿)]ρtip (EF − eV + ￿)ρsample (EF + ￿) | M |2 d￿.
￿ −∞
(2.3)
Here, V is the bias voltage (the applied potential difference between the tip and sample),
EF is the Fermi energy, f (E) is the Fermi distribution function, ρ(E) is the density of
states, and ￿ is the energy of the electrons. Equation 2.3 can be simplified when the thermal
contribution, kB T , in the Fermi distribution function is smaller than any significant features
in the density of states. Then the Fermi function can be approximated by a step function
and the tunneling current can be approximated as:
￿
4πe eV
I=
ρtip (EF − eV + ￿)ρsample (EF + ￿) | M |2 d￿
￿ 0

(2.4)

So we see that the tunneling current is related to the density of electronic states of the tip
and the sample. Provided the tunneling matrix does not change much at low energies around
the Fermi level [17, 18], it can be assumed constant. Tersoff’s and Hamann’s [40] further
simplification of Equation 2.4 by assuming that the tip’s density of states is constant over
the energy range of interest, which is reasonable for low voltages applied to metals [39, 40].
If the conduction electrons at the apex of the tip have a spherically symmetric wavefunction
(an s-orbital on the tip) then Equation 2.4 becomes
￿ eV
4πe
2
I≈
ρ |M |
ρsample (EF − eV + ￿)d￿.
￿ tip
0

15

(2.5)

Therefore, a more accurate perspective on STM measurements is that all the electronic states
from the Fermi level up to eV in the sample contribute to the tunneling current:

I∝

￿ eV
0

ρsample (EF + ￿)d￿.

(2.6)

In other words, each position on the topograph is composed of all the electronic density of
states that exist between the material’s Fermi level and the applied bias energy, eV . When a
surface feature, like an atom, has more accessible states than its surrounding environment, it
will appear brighter. Varying the bias voltage is a good way to distinguish between features
that are actually topography and those that are due to additional electronic states in that
region. Suppose there is a localized defect atom with defect states at a particular energy.
Once you scan below a voltage where the defect states are not active, the enhancement will
disappear. Atomic protrusions, step edges, and other physical features remain.
Semiconductors samples are a bit more complicated. Tunneling out of the valence band
is not necessarily the same as tunneling out of the conduction band. This can lead to voltage
dependent images if the tunneling electrons are part of the conduction band or valence band.
In GaAs [21, 20, 41] at positive voltages only the gallium atoms show up as enhancements in
the scan because conduction band electrons are more localized on gallium. Negative voltages
reveal the arsenic atoms. You only get the full picture of topography if you look at both
positive and negative voltages. However, Equation 2.6 still holds provided the bias is much
less than the work function [42].
In addition to surface topography, STM can be applied to probe the electronic spectrum
of the sample. Scanning tunneling spectroscopy measurements at low temperatures essentially show the local density of states as a function of energy. We can see this by taking
16

the derivative of the proportionality in Equation 2.6 with respect to voltage produces an
expression for dI/dV , also known as the tunneling conductance:

dI
∝ ρsample (EF + eV )
dV

(2.7)

dI/dV can be found by measuring I vs. V at a single sample location (i.e. while the tip
is stationary) and numerically differentiating the result to find the slope at all voltages.
However, to improve the signal to noise ratio, the local density of states can be measured
directly using a lock-in technique and applying an oscillation to the voltage as the voltage is
ramped over the energy range of interest. A comparison of different dI/dV curves at various
spots on the same material is necessary to look at local changes in the density of states.
Because the magnitude of the resulting curve depends directly on the tunneling gap, dI/dV
is usually plotted with arbitrary units.
If the bias energy, eV , does become a significant portion of the tunneling barrier, then the
simple picture in Equation 2.7 no longer holds. At these large voltages, the tunneling matrix
elements becomes exponentially dependent on voltage and cannot be considered constant
[39]. Fortunately, there is a painless and simple solution to normalize the dI/dV spectrum
by I/V to get d ln I/d ln V and remove the exponential dependence on voltage (some authors
refer to this quantity as the logarithmic density of states). In this work, the relevant voltages
are low and this normalization is not required.

2.2.2

Calibration

After a piezoelectric tube cracked, I rebuilt the microscope and it was necessary for me to
calibrate it to determine the piezoelectric tubes’ sensitivity. To that end, I used two standard
17

samples. Graphite’s well-established atomic spacing provided an excellent way to calibrate
the scanning tube in the x-y dimensions. The z-direction calibration was performed on a
sample composed of a grid of conducting platinum pits of known depth. Figure 2.3 (a) shows
the x, y and z directions with respect to the sample.

2.2.2.1

Graphite

Highly oriented pyrolytic graphite is a layered material of carbon atoms arranged in a hexagonal honeycomb pattern. Layers of graphite are bonded via weak Van der Waals forces,
allowing a fresh layer to be easily cleaved by scotch tape. As seen in Figure 2.4 (a), only
every other atom in the hexagonal lattice is visible in an STM image. Thus the carbon
atoms appear to be in a triangular pattern. Step edges are a good way of determining if
you have a very blunt tip that only has a few atoms protruding. A smeared step edge image
means the tip is less than ideal. Spectroscopy on graphite can indicate if there is anything
electronically active on the tip and demonstrate that it is a good metallic conductor. My
routine was to use the graphite spectrum like those in Figure 2.4 to determine the quality
of the tip before taking measurements on Bi2 Se3 . The I-V curve for graphite is not quite
linear and the density of states is a characteristic u-shape typical of the material as shown
in Figure 2.4 (c).

2.2.2.2

Pt Pits

The platinum pit calibration sample consists of a grid of evenly-spaced square pits of 180 nm
depth. Each pit is 5 µm x 5 µm and separated from the next pit by 5 µm in each direction.
My microscope’s maximum scan range is just over 5 µm, so it is rare to be able to image
both edges of the pit at one time. Nevertheless, a corner is sufficient for calibrating the
18

(a)

(b)

(c)

(d)

Figure 2.4: (a) Graphite atoms at a 1.0 nA tunneling current and 0.1 V bias. The scale bar
is 500 pm. (b) This is an image of step edge of graphite. The spectra shown in (c) and (d)
were taken on the smooth part of the image. Because the scale of this image is so much
larger than (a), individual atoms are not resolved. The scale bar is 200 nm. (c) Shows linear
I-V curve on graphite. (d) Density of states measurement on graphite.

19

Figure 2.5: Pt pits sample where the scale bar is 1 µm. The pit is 5 µm × 5 µm and 180 nm
deep. There is at least a double tip, possibly a triple tip in this case giving rise to multiple
apparent edges as indicated by the arrows.
vertical (z) extension of my scanning tube. A calibration image is shown in Figure 2.5. The
tip used to create this topography image is at least a double tip, and possibly has more tips
around the side. The edge of the pit comes in pairs, indicating a double tip. We can see
this in the image. For example, the upper-left part of the pit shows two edges as indicated
by the arrows in Figure 2.5. This indicates that the left part of the tip has a double apex.
Even with the multi-tip, the pit is confirmed to be 180 nm deep comparing the center of the
pit to the edge.

2.3
2.3.1

Experimental Constraints and Setup
Besocke Type STM

The STM I rebuilt was of the Besocke type design(also known as the beetle), which consists
of three carrier piezoelectric tubes that support the sample and one central scanning tube
20

(a)

(b)

Figure 2.6: (a) Besoke design microscope where three piezoelectric tubes support a sample
and a central tube scans the tip. (b) My microscope outside of its protective case.

seen in Figure 2.6. The three carrier piezoelectric tubes are responsible for fine positioning
the sample and bringing it into tunneling range. During approach a sawtooth waveform is
applied to these carrier tubes which slowly extend in the direction of twisting necessary to
bring the sample into tunneling range and then quickly snap back to a neutral position.
Friction keeps the sample in place during the slow movement and the fast return leaves the
tubes supporting a slightly lower part of the sample ramps in what is called a stick-slip type
approach [25]. After each cycle of the sawtooth, the scanning tube retracts during this fine
approach and extends to determine if it is within tunneling range. The control program
halts the approach as soon as tunneling current is detected.
The main advantage of the design is that all identical tubes are attached to a rigid baseplate, giving excellent mechanical stability [25]. Scanning range is determined to a large
extent by the tube’s geometry. A tube with thicker walls is more robust against vibrations,
21

but has a smaller range than a longer, thinner tube. To counter thermal drift during measurements, all four tubes are identical. The Besocke design has a stable construction that
helps produce quality STM images.

2.3.2

Noise Isolation

Moreover, because the tunneling current is exponentially dependent on distance, vibrations
on the scale of angstroms can have a large effect. When imaging something on such a small
scale like atoms, even small vibrations can impair the ability to get a sensible image. There
are two main sources of vibrations: sound (acoustic) and building vibrations (mechanical).
To ensure minimal interaction with sound vibrations, the room is covered in polyurethane
acoustic foam. More constant sources of mechanical vibration are from the building, which
tends to shake at low frequencies (10-20 Hz). To isolate the system from the building
vibrations, the microscope is mounted on four pressurized air springs and four optical table
legs to decouple it from the floor. Lead bricks weigh the entire platform to decrease its lowest
normal-mode frequency to about 1 Hz. Because this is far from the building frequencies, we
see very little noise in the system due to building vibrations.

2.3.3

Cryogenic System

My STM is attached to the end of a stainless steel probe so that it can be lowered into a low
temperature environment. The cryogenic system I used for my experiments is a standard
Oxford Instruments refrigerator capable of operating at liquid 4 He and 3 He temperatures.
My primary thesis data was acquired at liquid 4 He temperatures. The dewar itself consists
of an inner sample tube sealed off from the probe entry port by a gate valve, as seen in Figure
22

2.7. Surrounding the tube is an inner vacuum chamber that can serve to insulate the sample
tube from a surrounding main cryogen bath. As I always used the main bath to cool the
sample tube directly, the inner vacuum chamber always had a small amount of exchange gas
to ensure adequate thermal contact between the sample tube and the main bath. A transfer
tube entry port on the top of the dewar allows easy access to the main bath through which
to transfer the cryogen. To ensure efficient cooling, the main bath is thermally insulated by
a outer vacuum chamber that also has mylar batting to help reflect heat from the outside.
In typical operation, the sample space is purged of all air to reach pressures of around
2 × 10−6 torr. The vacuum not only ensures the cleanliness of the surface of the sample,
but also preserves the purity of the sample space within the dewar from contaminants. A
4 He exchange gas in the sample tube within the dewar helps the microscope equilibrate to
the temperatures quickly.
Cooling the microscope is a process that takes several hours. It is important not to
thermally shock the fragile ceramic piezoelectric tubes. To monitor the microscope’s temperature, an Allen-Bradely resistor acts as method of gauging the temperature. The carboncomposite resistor has a resistance that changes significantly with temperature and has
therefore been calibrated to act as a low temperature thermometer. Once the microscope
is firmly seated in the bottom of the dewar, it is left to equilibrate overnight to avoid any
thermal drift.

2.3.4

Tips

The quality of the tip on the STM is probably the biggest variable in the entire experiment.
An important experimental consideration is the material and electronic structure of the tip;
23

Figure 2.7: Diagram of the cryostat. The microscope mounts on the probe entry port and is
isolated from the sample tube by a gate valve until it is under vacuum. The microscope can
then be lowered slowly into the bottom of the dewar. The system is cooled by a main bath
of nitrogen or helium. The transfer tube entry port is the site where liquid is transferred
into the system. Gas from liquid boil off is released through the helium exhaust port [43].

24

when performing spectroscopy the resulting spectrum is a convolution of the density of states
of the tip with the sample. I chose to use a metallic PtIr tip which has a flat density of
states near the Fermi level and is chemically inert to avoid those problems. This choice is not
critical: gold or tungsten tips would have been adequate. While performing measurements,
I took great care to keep the tip free of any other material to ensure the integrity of the
measurements. I used manually cut tips most often because they are convenient to re-cut
and renew the apex. A more detailed discussion of tip preparation is in Section 6.4.

2.3.4.1

In situ Tip Cleaning Techniques

Even when I took great care to ensure the integrity of the apex of the tip, sometimes the
tip became unstable. This can happen spontaneously during a scan or there may be a
precipitating event, such as a crash. It is therefore extremely important for STM to have a
variety of tools to rid the tip of the offensive particle or flake of foreign material on the apex.
The first tool I generally employed was shaking the tip. While the tip is far enough above
the sample that there will be no chance of interaction between them, the tip is moved as
quickly as possible over a large area in a raster pattern in hope that the offensive particle
will be shaken loose. This technique is least damaging to the tip and the sample and has a
good success rate for loose particles. For slightly more stubborn particles on the tip, I found
a flat, clean spot on the sample and slowly dragged the tip over the surface with a high set
tunneling current to reduce the tip sample separation. Both of these in situ approaches are
safe to use during STM and capacitance measurements.
The most firmly attached particles might require field emission or evaporation to pull
them off the tip. Some of the most clingy particles attach themselves to the tip during
spectroscopy as I continuously ramp the voltage over a fixed point on the sample. The
25

smaller the voltage, the less aggressive the approach. For the most persistent particles, I
would typically position the sample just out of tunneling range to widen the gap so that
there is no possibility that the tip will smash the sample. With the tip grounded, I would
then apply 150 V to the sample for five seconds before coming back into tunneling range at
a new location on the sample. The only complication with this technique is that the fields
involved can be strong enough to loosen a flake off a layered sample thus interacting with
the tip and making the situation worse. This technique is so successful at removing particles
from the tip that it also has a tendency to blunt the apex. However, a blunt tip is often
better for spectroscopy, even if it is worse for resolving small length scales. When there is a
lot of resources and time invested in a particular data run, including cooling the microscope,
trying several methods to recover a tip in situ is justified.
If all else fails, the procedure to change the tip first involves slowly warming up the
microscope. Then the entire microscope can be moved to the bench top to unsolder the
mounted tip. It is possible that the existing tip can simply be recut with wire cutters,
soldered back on, and retested for suitability. However, if the radius of curvature of the tip
is important (and it was not for the thesis experiment described in Chapter 4) then it may be
necessary to attach a tip that has been characterized by Scanning Electron Microscopy (SEM)
to a new chip with conducting low-temperature epoxy. Before re-cooling the microscope,
every tip must be tested at room temperature on a freshly cleaved sample in nitrogen to
ensure its stability. If the tip has additional features in the density of states or cannot readily
be used to produce high-quality images, then it is discarded and replaced before ever being
used at low temperatures. Under ideal circumstances, warming the microscope, replacing
the tip and re-cooling takes three days.

26

Chapter 3
An Introduction to Topological
Insulators
In this chapter, I will set up the physics necessary to understand the conclusions in the next
two chapters. I have taken a historical approach to the topic and will therefore include some
discussion of materials like graphene that are not what I probed. However, these topics will
help to explain key ideas related to topological insulators. I will then explain topological
insulator physics and give an overview of the literature of a prototypical topological insulator,
Bi2 Se3 , to help you understand how my main result fits in with the overall picture.

3.1

Background

Over the last 80 years, band theory of solids has been a fruitful area of study in Condensed
Matter Physics. One of the ways to model the static electronic structure of a crystal is
using the tight binding model. This model captures the quantum mechanics of the lattice by
treating electrons as tightly bound to the atoms in the lattice, analogous to atomic orbitals
27

Figure 3.1: Figure from Zhang et al. [44] shows the unit cell in a crystal lattice of Bi2 Se3 .
￿ ￿
￿
Primitive vectors t1 , t2 , and t3 show equivalent translation vectors of the lattice. There
are 3 inequivalent sites in the lattice, Bi, Se1 and Se2 that stack in inequivalent layers
C(Se1)-A(Se1)-B(Bi)-C(Se2)-A(Bi)-B(Se1)-C(Se1). Viewed from above, this looks like a
hex lattice with inequivalent layers marking out triangular lattices. In the bottom right,
the three-dimensional unit cell is mapped out with its two-dimensional projection showing
high symmetry points in k-space, Γ, M and K that are time reversal invariant momentum
points for this crystal structure. Used under Creative Commons Attribution 3.0 Unported
(CC-BY) license and all credit for creation of this figure goes to [44].

28

in isolated atoms. It is useful for situations when there is some overlap between atomic
orbitals in a lattice, but the states are not extended enough that treating the atomic orbitals
is irrelevant. As such, interactions with other electrons is limited; often only nearest neighbor
hopping is considered [45].
The atomic positions in the crystal are mapped out in what is called a unit cell–the
fewest number of atoms arrangements needed to make one repeating unit of the crystal.
This structure can be transformed to find the reciprocal lattice called the Brillouin zone.
Instead of atoms being mapped into real space, the structure is converted to momentum (k)
space [45]. Momentum space highlights the importance of certain high symmetry directions
in the crystal and these points are assigned letters. An example of a crystal and its reciprocal
lattice is given in Figure 3.1. Its unit cell and primitive vectors of translation are drawn on
the lattice in black. On the left part of the figure is a rhombohedral lattice supercell and
its momentum space with 2D projections of both the lattice and the Brillouin zone [44].
Because of its layered structure, one unit cell must cover multiple layers to fully reproduce
a crystal.
In the simplest case, ignoring spin for now, the tight-binding Hamiltonian consists of a
hopping term to capture how electrons move between nearest-neighbors:

H = −t

￿

n,n￿

†
[Ψn Ψ ￿ + h.c.],
n

(3.1)

where Ψ and Ψ† are the annihilation and creation operators respectively. The indices n
and n￿ refer to nearest-neighbors only, and the sum is therefore between nearest-neighbor
pairs. The coefficient t dictates the strength of nearest neighbor hopping arising from lattice
29

potential effects. If it is 0 then no hopping will occur [45]. The strength of this hopping
term may vary depending on which atomic orbitals are involved. d and f orbitals have
strongly bound electrons that would make hopping less probable. Because they are usually
the orbitals participating in the conduction and valence bands respectively, s and p orbitals
would have a higher probability to hop to the nearest neighbor [46]. Electron wavefunctions
are mapped out in to what are called Bloch wavefunctions to capture the lattice periodicity.
Bloch waves have a plane wave component that captures the electrons’ movement through
free space and a component that captures the periodic potential from the lattice, u ￿ (￿).
r
nk
The wave functions appear as:
￿˙
￿r
ψ ￿ (r) = expık u ￿ (￿)
r
nk
nk

(3.2)

where subscript n stands for the n-th electron state, ￿ are real-space vectors, and ￿n (￿ =
r
k)
￿n (￿ + K). Here, ￿n (￿ is the eigenenergy of the Hamiltonian defined by ψ ￿ H = ￿n (￿
k ￿
k)
k)H.
nk
￿˙
￿
Bloch’s theorem dictates that ψ ￿ (R) = ψ ￿ (￿)eık and is one full translation in k space
r ￿R
nk
nk
through a 2D band. Unless spin orbit coupling is explicitly added into Equation 3.1, any
band calculated will be doubly degenerate [45].

3.1.1

Graphene Model

While my thesis experiment only encompasses the material Bi2 Se3 , I have taken a historical
approach to motivate the physics of topological insulators necessitating an explanation of a
graphene model and the quantum spin Hall effect. First there will be a brief explanation of
the quantum Hall effect and its topologically protected electronic surface state.
Electrons in the quantum Hall effect form edge states due to the large magnetic field that
30

are then impervious to backscattering. In the presence of a magnetic field, Hall conductance
becomes quantized in unit conductance of σxy = νe2 /h [47, 2] where ν is an integer. The
quantum Hall conductance is fundamentally different from conduction through a normal
metal and therefore represents a state of matter that is topologically different than other
conduction states. To distinguish materials exhibiting the quantum Hall effect from normal
insulators, Thouless, Kohmoto, Nightingale, and den Nijs developed a topological invariant,
ν, called a Chern number or Chern invariant. This number is a way of distinguishing
band structures that can be deformed into one another from those that cannot. It is a
quantity that is robust against perturbations. From a theoretical perspective, perturbations
are smooth deformations of the Hamiltonian. From an experimental perspective, materials
of dissimilar Chern numbers have some conserved quantity that can be measured to high
precision. A material that exhibits the quantum Hall effect, for example has a quantized
Hall conductance that remains quantized even when defects are introduced into the system.
However, the world is full of materials that do not exhibit quantized conductance and never
can be made to do so. Another example of a Chern number is given in the following pages.
More recently, there arose interest in the quantum spin Hall effect, in part due to the
scientific interest in spintronics applications. Unlike the quantum Hall effect, the quantum
spin Hall effect does not require a magnetic field and is time reversal symmetry invariant
due to spin-orbit interactions [4, 48, 49]. The quantum spin Hall effect has spin split edge
states that counter-propagate on the edges. While the total current through this state is
s
zero, the spin current has unit spin conductance σxy = e/2π [50].
Because graphene is a 2D sheet of carbon atoms arranged in a hexagonal lattice, the discussion will focus on the 2D tight binding model of the quantum spin Hall effect in graphene.
31

These material bands require relativistic effects to calculate key band structure features.
Spin-orbit coupling can result in time reversal invariant (T -invariant) spin-dependent forces
in a lattice. One such tight binding Hamiltonian which includes spin and spin-orbit couplingwas introduced by Kane and Mele [48] for graphene.
Graphene is a two-dimensional material that has a hexagonal Brillouin zone where there
are two inequivalent, K and K ￿ points that form two triangular sublattices. When Kane and
Mele [48] solved the tight binding model with spin-orbit interactions for a strip of graphene,
an edge state emerges from the bands. Edge states with opposite spins travel in opposite
directions, a sort of “spin filtered” state. The bands in k-space shown in Figure 3.2 (a) show
the edge state in the bulk insulating gap of graphene that traverses the gap. These states
persist so long as the spin-orbit term in the calculations dominates. Two states at kx = π/a
form a Kramers doublet that is protected from any time reversal symmetric perturbations
and are thus protected from elastic backscattering [4]. The linear relationship of the states
that traverse the gap is relativistic.
For a band insulator, the surface states connect edges of like spins and no state traverses
the gap. However, for the topological insulator in Figure 3.2 the spins switch partners across
the gap while preserving spin. Therefore, the edge states are spin split. In fact, the state
supported is predicted to be the quantum spin Hall state that carries a quantum of spin
conductance. For this spin split, or “helical,” state to be a topologically distinct from an
insulator, a single pair of spin-split edge states crosses the gap [50]. In a sense, the resulting
edge state is half a metal in that only half the spins of a normal metal can travel in a
direction. It is far more robust than a metal because electron’s states are extended and not
impacted in the presence of disorder by Anderson localization. Moreover, it turns out that
32

PRL 95, 226801 (2005)

1

PHYSICAL REVIEW LETTERS

actions and disorder distinguishes th
states from ordinary one-dimension
localized by weak disorder.
E/t
A parallel magnetic field Hk brea
leads to an avoided crossing of the
0
reduces the symmetry, allowing term
X
which provide a continuously gapped
states generated by z z sz and z .
gapping the edge states Hk elimin
distinction between the QSH phase a
-1
The spin filtered edge states have i
π/a kX
2π/a
0
ces for both the transport of charge an
low temperature we may ignore the
FIG. 1. (a) One-dimensional energy bands A strip of graFigure 3.2: A calculation from Kane and Mele [48]. for a single pair ofing processes, and describe the ball
edge states traphene
¨
verse the bulk insulating(shown in inset) arespin filtered edge states. 0:03. The
gap the gap modeled orbit with t2 =t in a band edge states within a Landauer-Butti
due to spin by (7) coupling
structure calcubands crossing
For a two terminal geometry [Fig.
lation from a tight-binding model on a strip (in the inset) of graphene. Reprinted
ballistic two terminal
figure with permission from 0Kane, C.L. and Mele, E.J., Physical Review Letters, 95, charge cond
For the spin filtered edge states the
of K and K are both at k 0. It is interesting to note that
226801(2005). Copyright (2005) bythe edge states persist for Society. Abstractrelatedbe viewed density, since b
the American Physical ! 0, where
is can to the spin
for zigzag edges
so
at: http://prl.aps.org/abstract/PRL/v95/i22/e226801. to an enhanced
nL# . Thus the charge current is accom
they become perfectly flat [16]. This leads
mulation at the edges. The interplay
density of states at the Fermi energy associated with zigzag
spin can be probed in a multitermin
edges. This has been recently seen in scanning tunneling
multiterminal spin conductance by
spectroscopy of graphite surfaces [17].
an insulator can be one ofhave also considered a a band neighbor Rashba topological insulator requires Gs
We two distinct things, nearest insulator or a
reversal symmetry
ji
d cy cj . This violates the
term, of the form iz s
^
e=4
Fig. 2(b) it is clear that Gs
i
ij
[4, 51]. (This relates conservation of s , so that the the “Z2 argument no longer
to a classification called Laughlin index”).
z
i and j. In the four terminal geometr
applies. Nonetheless, we find that the gapless edge states
current I s eV=4 flows into the
< so , of graphene is band gap
remain, provided R surface so that the bulk part of the states appeal. As used to measu
geometry can also be
The nature of the electrons on the
remains intact. The crossing of the edge states at the
spin current incident from the left (i
=a in Fig. 1 (or at k 0
Brillouin zone boundary kx
the states cross in Figure 3.2 they edge) is protected by time reversal symfor the armchair form an “x” where energy and momentum of the band
=a form a Kramers doublet
metry. The two states at kx
are linearly related as is the case for cannot be lifted by any time reversal For this reason, the
whose degeneracy massless particles in free space.
symmetric perturbation. Moreover, the degenerate states at
electrons are often called “relativistic.” At the center of the cross is a point called the “Dirac
=a q are a Kramers doublet. This means that
kx
I
V
elastic backscattering from a random potential is forbidden. coexist with scattering energy and absolute
point” where both spins More generally, the same from a region of disorder value of momentum.
can be characterized by a 2 2 unitary S matrix which
(a)
relates both inversion and time reversal symmetry , [51].
This point is protected by the incoming and outgoing states: out S in
where is a two component spinor consisting of the left
V/2
and right moving edge states L" , R# . Under time reversal
Time reversal symmetry necessitates the existence of high symmetry points where ￿ =
k
in;out ! sy out;in . Time reversal symmetry therefore imposes the constraint S sy ST sy , which rules out any off
diagonal at ￿ = 0
−￿ This typically happens elements. ￿ and on the band edge [5] and are called “Kramerss
k.
k
I
0
Electron interactions can lead to backscattering. For
y
y
instance, the term u L" x L" R# @x point when ka =
degenerate points.” Figure 3.2 has a high@symmetry R# , does not violate π. Away from these
time reversal, and will be present in an interacting
Hamiltonian. For weak interactions Consequently, the
points, spin-orbit coupling causes the bands to split. this term is irrelevant topological state has
under the renormalization group, since its scaling dimen-V/2
sion is
4. It thus will not lead to an energy gap or to
an odd number of band crossings, while a simple insulator has an even number. The number (b)
localization. Nonetheless, it allows inelastic backscattering. To leading order in u it gives a finite conductivity of
FIG. 2. Schematic diagrams showing
33
the edge states, which diverges at low temperature as
(b) four terminal measurement geome
u 2 T 3 2 [18]. Since elastic backscattering is prevented
current I
2e2 =h V flows into the ri
s
by time reversal there are no relevant backscattering proe=4 V flows into the righ
current I
the right indicate the population of the e
cesses for weak interactions. This stability against inter-

d odd
leads
h we
on a
WTI
udied

n the
tered
way
ransng a
aded
The
versal
of a

i a2 ,

which satisfy
pendicular to G.

a1

G=2, onto the line per-

a2

E (a)

E (b)

ε

a2

ε

ε

ε

b2

a2

b1

ε

ε

a1

ε
b3
ε
b1

a1

k
Λa

k

Λb

Λa

Λb

Figure 3.3: A schematic band structure showing Kramers pairs traversing the gap from Fu
FIG. 1. A schematic of a topological insulator that has spectra as of Fermi level
et al. [52]. (a)Schematic surface (or edge) state an odd numbera function
crossings. (b) A schematic of a simple insulator that has an even number ofto
crossings. There
of momentum along a line connecting
a
b for
are still degenerate points at Λa and Λb but they are not topologically protected against
1 and (b) a b
1. The shaded region shows
(a) a b
deformation. Reprinted figure with permission from Fu, Liang and Kane, C.L., Physical
the bulk 045302 In (a) the (2007) by the American Physical a and b ,
Review B, 76,states.(2007). CopyrightTRP changes between Society. Abstract
can be viewed at: http://prb.aps.org/abstract/PRB/v76/i4/e045302.

while in (b) it does not.

106803-1

© 2007 The American Physical Society

of band crossings is the Chern number [53, 5]. Figure 3.3 illustrates the difference between
simple insulators which have an even number of doubly degenerate pairs in the gap in (b)
and a topological surface state with a single crossing in (a). Because these crossings cannot
be deformed into one another, they are topologically distinct [52].

In summary, the tight binding model with spin-orbit coupling predicts many of the hallmarks of 2D topological insulators. (It predicts a Z2 classification for topological insulators).
There is a spin split edge [51] that is protected by time reversal symmetry that is robust
against perturbation [48, 4, 49, 5, 52, 54, 7, 8]. Inside the insulating gap are relativistic edge
states. The resulting quantum spin Hall effect has quantized spin currents on the surface
where the currents of opposite spins move in opposite directions [4, 51].
34

3.1.2

HgTe/CdTe Wells

Subsequent experiments on graphene showed that its spin-orbit coupling was too weak to
support a quantum spin Hall state [51]. Instead, the focus shifted to heavy elements known
to have high spin-orbit coupling. Bernevig et al. [55] successfully predicted the existence
on the quantum spin Hall state in HgCdTe quantum well structures that would be the
first realization of the quantum spin Hall phase. They first predicted that placing HgTe,
which has an inverted band structure with the p level above the s level, between a “normal”
semiconductor band structured material, CdTe, would result in the quantum spin Hall state
[55, 51]. Spin-orbit coupling in the Hg atoms make the band inversion happen. The state
would only exist as long as the well width was above a critical well width of 6.3 nm. Below
this critical width the normal band structure associated with the CdTe would persist through
the well. However, above the width, the bands invert resulting in a quantum phase transition
from a trivial insulator to a topologically protected quantum spin Hall insulator [51].

Konig et al. [56] performed transport experiments that first realized the quantum spin
Hall state experimentally by measuring the conductance due to the edge states. Figure 3.4
shows the resistance measurement of their results. Below the critical width in sample I the
resistance shows insulating behavior with the resistance on the order of 107 Ω. However at two
well widths above the critical dimension, the quantum spin Hall state with a conductance G =
2e2 /h associated with the top and bottom edges is realized. This was the first measurement
of the quantum spin Hall state.
35

Figure 3.4: Figure adapted from Konig et al. [56]. Resistance measurement of CdTe/HgTe
quantum wells that support the quantum spin Hall state. Sample I shows that the wells are
insulating below the critical width. Sample II shows disorder due to temperature effects.
Samples III and IV show that the conductance is precisely equal to 2e2 /h for two different
well widths above the critical dimension. From M. Koenig, S. Wiedmann, C. Bruene, A.
Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, S.-C. Zhang, Science, 318, 766-770 (2007).
Reprinted with permission from AAAS.

36

3.1.3

Three Dimensional Topological Insulators

While 2D topological insulators can be distinguished by a single Chern number, three dimensions adds three more integer Chern numbers, one for each of the xy, yz, and xz planes
[5]. While this leads to several different possibilities, they can really be divided into three
categories: simple insulator, weak topological insulator, and strong topological insulator.
Strong topological insulators have a Chern number ν0 = 1 and corresponds to an odd number of Kramers pairs on the surface. A trivial and weak topological insulators have a Chern
number ν0 = 0 [5, 57].
Edge states no longer have the same meaning once you extend a discussion to three
dimensions. Although there are three other Chern numbers that only arise in three dimensions, only ν0 is independent of the reciprocal lattice primitive vector, and therefore does not
depend on the reciprocal lattice direction. If ν0 = 0 the material can be a trivial insulator
or a weak topological insulator. A trivial insulator has either 0 or 4 number of Dirac points
enclosed by the Fermi surface. The states in a trivial insulator are extremely localized and
not topologically protected.
One way to extend a two-dimensional material into three dimensions is to create a stack
of the two-dimensional material. This does create a state that has topological protection in
one dimension. However, because it is not topologically protected in the direction the stack
was built, this state is “weak” such that disorder will destroy it [52, 57]. A two-dimensional
Fermi surface modeling the stack would enclose two Dirac cones, one from each 1D edge
state. However, there is another way to extend to three dimensions. If the Fermi surface
encloses a single Dirac cone there will be an odd number of band crossings in any direction
in k-space [52]. Thus it is topologically protected because a single band crossing cannot be
37

transformed into any other state without first closing the gap. Such a state can arise due to
spin-orbit coupling and is called a strong topological insulator [57]. The strong topological
insulator has an odd Chern number and ν0 = 1. Like a quantum spin Hall state, a strong
topological insulator is robust against defect scattering due to its topological protection.
It forms a two-dimensional layer on the surface of a strong spin-orbit coupling material.
Amazingly, momentum and spin are always perpendicular to each other in this topologically
protected state [57].
In summary, a three-dimensional strong topological insulator is characterized by a twodimensional metallic state on the surface of the material. A key idea is that due to spinorbit coupling, momentum is locked perpendicular to the spin [52, 51, 58, 59]. To form
a topologically protected state that is robust against small perturbations, time reversal
symmetry must be preserved [52, 59]. The strong topological state supports an odd number
of Dirac cones, which means that there can only be an odd number of crossings of the Fermi
level [51, 60].

3.1.4

Early Studies on Bi1−x Sbx

Bi1−x Sbx was one of the first well studied three-dimensional topological insulators. Bismuth
had long been known to be a good candidate for supporting a quantum spin hall phase [50].
The spin-orbit coupling in bismuth is very strong, which helps it support a spin current [61].
Antimony is likewise an element with strong spin orbit coupling roughly about one third of
bismuth and serves to tune the spin-orbit coupling strength so that the alloy will be a strong
topological insulator by perturbing the bands just enough that they invert [52, 59].
38

3.1.4.1

ARPES

Hsieh et al. experimentally realized the three-dimensional Dirac cone in Bi0.9 Sb0.1 . In
their work they resolved the five band crossings that distinguished the predicted topological
insulating regime where the conduction and valence bands had inverted [60]. The topological
insulating phase in a bismuth alloy is significant because when there is less antimony in the
crystal, the bands do not invert and the alloy remains a weak topological insulator with an
even number of band crossings [59, 52].

3.1.4.2

Quasiparticle Interference Patterns

STM measurements showing snowflake-like scattering patterns really sparked excitement in
topological insulators. Modulations in the density of states appear when electrons elastically
scatter from the impurities, mixing eigenstates of different momentum, k. These eigenstates
mix along quasiparticle contours of constant energy in momentum space [62]. When different
momentum states, k1 and k2 , interfere after scattering, modulations in the local density of
states arise with wavelength λ = 2π/ | q | where q = k1 − k2 . This can be measured with a
scanning tunneling microscope; differential tunneling conductance scans in real space expose
these modulations called Friedel oscillations. Taking the Fourier transform of the differential
conductance image yields the momentum-space information on the wavefunctions. This
technique is called Fourier-Transform STS (FT-STS) [62]. In other words, when quasiparticle
interactions with the bulk states are sufficiently strong, bulk states can scatter off surface
impurities in a way that can be imaged directly by the STM.
Bi1−x Sbx forms a disordered crystal alloy with plenty of opportunities for the topological
surface state to scatter off of Sb impurities. The conductance map of Figure 3.5 (a) appears
39

(a)

(b)
Figure 3.5: (a) Adapted from [58]. Bi0.92 Sb0.08 dI/dV map with the Fourier transform of
the image inset. The snowflake pattern shows the six-fold symmetric scattering in k-space.
Some scattering directions are forbidden. (b) Adapted from Figure 4 in reference [58].
The scattering patterns are only duplicated when spin-dependent scattering is accounted
for in calculations. Reprinted by permission from Macmillan Publishers Ltd: NATURE
(P.Roushan, J.Seo, C.V. Parker, Y.S. Hor, D.Hsieh, D.Qian, A.Richardella, M.Z. Hasan,
R.J. Cava, and A.Yazdani. “Topological surface states protected from backscattering by
chiral spin texture,” Nature, 460, 11061109, (2009)), copyright (2009)

40

to be quite messy and disordered, but the Fourier transform of the image into momentum
space reveals elegant complex snowflake patterns. These intricate scattering patterns cannot be simulated without performing a calculation that use a matrix connecting like spins,
suppresses rotations of spins and forbids scattering into opposite spin states in what they
refer to as spin-dependent scattering probability (SSP). Figure 3.5 (b) shows that a similar
calculation of the joint density of states (JDOS) that does not include spin in any way fails
to capture theses intricate patterns. The strong topological insulating nature of the material
is therefore quite robust against disorder because the spin selective rules hold over distances
larger than the disorder [58].

3.2
3.2.1

Bi2Se3 Background
Structure

Bismuth Selenide, Bi2 Se3 , is a strong topological insulator with a rhombohedral lattice
structure. A layered material, Bi2 Se3 has covalent bonds inside and each quintuple layer
(QL) is held together by Van der Waals forces. Each quintuple layer is approximately 1 nm
high [63]. As seen in Figure 1.2, a quintuple layer consists of Se-Bi-Se-Bi-Se bonds. This
layered structure makes them ideal for surface experiments, as a new surface can be cleaved
with scotch tape to reveal a clean black mirror-like surface, with a single Dirac cone on
the surface [6]. This was shown schematically in Chapter 1, Figure 1.1. Figure 3.6 shows
the calculated band structure of Bi2 Se3 including the single Dirac cone that makes it a
model topological insulator calculated by Dr. Mal-soon Lee. The computational technique
is density functional theory introduced in Section 4.3.
41

Figure 3.6: Density functional theory calculation of the Bi2 Se3 bulk [gray] and 6 quintuple
layer calculation of the Dirac cone in blue. The single Dirac cone on the surface is circled in
red. Figure courtesy of Dr. Mal-soon Lee.

42

Bi2 Se3 and Bi2 Te3 had long been studied for their thermoelectric properties [14, 10, 64].
Because of its interest as a thermoelectric material, many effects that would later be examined
for their interactions with the surface state were already known before topological insulators
had even been predicted. In 2004, Dr. Sergei Urazhdin was the first to characterize cloverlike and triangular defects from Se vacancies and Bi substitutions [14, 10]. Angle-resolved
photoemission spectroscopy (ARPES) studies identified the time dependence of the surface
state forming on Bi2 Te3 and attributed it to band bending at the surface [64].
In 2009, Zhang et al. [6] did the first theory calculations predicting that Bi2 Se3 , Bi2 Te3
and Sb2 Te3 were strong topological insulators. When spin orbit coupling is added to band
calculations, the pz bonds from the conduction and valence band hybridize, inverting the
bands and creating a Dirac cone. Since then, there has been keen experimental interest on
investigating the topological insulator surface state on these materials.
Bi2 Se3 has been studied by countless groups trying to realize aspects of the topological
insulating state and how it interacts with other physical effects. The helical spin texture
in Bi2 Se3 was realized using circular dichroism ARPES measurements which identified that
opposing momentum states also have opposite spin structure [65]. That spin structure persists even at room temperature [66], an important characteristic for spintronic applications.
Consistent with its topological protection, the weak electron-phonon coupling experiments
show the surface state persists unperturbed [67, 68]. Because there is potential to mix with
the bulk, measurements of the Landau levels show that they are suppressed on the left,
valence side arm of the Dirac cone and follow uneven spacing consistent with Dirac electrons
on the other arm [69, 70, 71, 72] potentially from mixing with the bulk. Transport studies
have attempted to reduce the bulk contribution and realize the dissipationless spin current
43

Figure 3.7: Triangular defects on Bi2 Se3 where the bright triangles are attributed to bismuth on a selenium site substitutions. The scan range is 65 nm×65 nm. Topograph
from Urazhdin et al. [10]. Reprinted figure with permission from S.Urazhdin, D.Bilc,
S.Mahanti, S.Tessmer, T.Kyratsi, and M.Kanatzidis. Physical Review B, 69(8):085313,
2004. Copyright (2004) by the American Physical Society. Abstract can be viewed at:
http://prb.aps.org/abstract/PRB/v69/i8/e085313
for years by various means [73, 74, 75, 76]. Recently, a research group revealed they had realized metallic, ambipolar gapless conduction without any bulk contribution [77]. The charge
carriers of the surface state have been examined by optical conductivity measurements [78]
and transport measurements [73, 77, 75, 74, 76] have been found to agree. Some groups have
focussed their studies on thin films (i.e. [63, 79]) and others on bulk crystals (i.e. [14, 80, 81])
and show weak anti-localization effects. Even ignoring the studies involving dopants, the last
four years have been extremely fruitful for research on the Bi2 Se3 surface states.

3.2.2

Dopants and Defects

Many experiments use doped crystals to look at electronic interactions with the defect states.
Non-magnetic elements like Cd [73] and Ca [80, 81, 82, 83, 75] are added to make the sample
more p-type to shift the Fermi level toward the Dirac point. This characteristic is desirable
in any transport-like measurement and for spintronic applications to realize a completely
44

spin split state. The dopants can help to realize a dissipationless spin current because local
disorder does not destroy the topologic state. Amazingly, copper dopants intercalated into
the Van der Waals gap have been shown to create a superconducting state [84, 65].
In order to test how robust the surface state is to magnetic perturbations, magnetic
dopants like Mn [80], Cd [73], Cs, Gd, Rb [85], and Co [86] all show the surface state to
persist when the magnetic order is weak. However, enough Fe doping can open up a gap
and destroy the Dirac cone [51]. Bi2 Se3 nanoribbons doped with iron exhibit a minimum in
resistance associated with Kondo effect [87] but often otherwise leave the topological state
undisturbed [88, 81, 89] unless there is magnetic order to open the gap [88].
STM measurements showed how these defect states influence the topography. Dr. Sergei
Urazhdin pioneered STM studies on Bi2 Se3 . He attributed triangular defects seen in Figure
3.7 to bismuth substituted on a selenium lattice site (BiSe ). These triangular defects appear
as clovers in differential conductance maps. The shape of the electronic enhancement is
attributed to p bonds lining up from the lowest Se layer in the top quintuple layer terminating
on the surface with three bright spots and are not of topographic origin [14]. Triangular
defect states that do not change the local density of states (LDOS) in their vicinity are
attributed to interstitial bismuth between the top two layers locally creating a strain on the
lattice [10]. Because the triangular defect states are voltage dependent, they are known to be
part of the electronic structure of the material and not topography. These conclusions have
been accepted as the most likely explanation of triangles that are just part of the surface
and mark the locations of subsurface dopants. STS studies of dopants show they create a
resonance within the bulk gap that can obscure the Dirac cone close to the defect [10, 90].
These resonances depend on the location with respect to the defect state’s center [10, 90, 89];
45

the impurity strength diverges as the Fermi level approaches the Dirac point and a pure spin
split state is realized [90]. These resonance states can occur at a variety of different voltages
relative to the Dirac point depending on how a dopant goes in and its electronic contribution
to the crystal.
A resonance state in the spectroscopy measurements relates to the voltage. The triangular
defects may appear bright or dark. Se vacancies appear as bright triangles at positive voltages
relative to the Dirac point when the state is empty [82, 91] and dark triangles at negative
relative voltages [92]. It acts like a double donor [93]. Native Bi2 Se3 tends to be n-type
due to selenium vacancies. Bi vacancies are far less common. Density functional theory
calculations on common dopants show that Bi vacancies act like a triple acceptor [93]. These
same calculations show thatbismuth on a selenium site acts as an amphoteric dopant that is
an acceptor for n-type materials, which is consistent with bright triangular defects described
in reference [14]. Other dopants like Ca [82] and Cu [84, 94] also show bright triangles at
positive relative voltages and are likewise viewed as substitutional dopants.
Unlike Bi1−x Sbx and Bi2 Te3 , Bi2 Se3 quasiparticle interference patterns are partially
suppressed on the surface and often do not reveal quite as rich patterns. Conductance maps
of defects on the surface reveal clover patterns [14], but very few wave patterns. Some groups
have speculated that quasiparticle scattering is suppressed by mixing with the bulk state in
the helical Dirac fermion regime [95, 96]. These interference patterns can only arise from
scattering off defects within the layers that overlap with the topological state [97, 86]. As
such, adsorbates on the surface are not responsible for scattering the topological insulating state and topographic features like flakes or warped surfaces can impact quasiparticle
scattering [97]. Conductance maps on the surface of related material, Bi2 Te3 , support this
46

conclusion by showing standing waves that result from scattering off step edges that show
strongly damped oscillations from a step [98]. Once scattered, topological insulating states
can decay into bulk states via low energy electron-electron interactions and defect scattering
[96] .

3.2.3

Surface Potential Fluctuations

On the surface, deep defect states influence the topological insulating state by creating
an energy landscape, i.e. potential fluctuations, that influence the surface [80]. While the
disorder may locally shift the Dirac point relative to the applied voltage, the topologic surface
state persists under the influence of this bulk disorder [80]. Moreover, magetic impurities
break time-reversal symmetry and ought to allow backscattering. However, Beidenkopf et
al. showed that quasiparticle interference patterns are not altered by the magnetic impurity,
Mn [80]. They also showed that quasiparticle interference patterns are suppressed when
the average disorder length is approximately that of the topological insulator’s interference
pattern wavelength [80]. The topological insulating state screens the bulk disorder over
approximately 26 nm in Bi2 Te3 doped with Mn. The minimum of the local density of
states, e.g. the Dirac point, shifts in space locally on the surface due to the underlying
defect pattern. The Dirac cone and bulk bands are otherwise unchanged by the bulk doping.
However, there is an offset of the Dirac cone in tunneling spectroscopy from zero energy.
The vertical offset of the Dirac cone has been observed in thin films [79] and bulk crystals
[80, 71, 95, 10], but its origin was either ignored [80, 95, 14, 10] or explicitly stated as of an
unknown origin [71].
Because the bulk provides other decay paths, it is important to know when the surface
47

To gain deeper insight into the origin of the unexpected
interference pattern, the cross-sectional FFT patterns
" "
traced along À M direction are shown in Fig. 4(j). For the
whole range of sample bias voltages, we observe the
largest intensity at q ¼ 0, which might originate from

-0.5

#
0:5 AÀ
intensi
The o
#
0:3 AÀ
the Dir

0

0

Figure 3.8: Quasiparticle scattering data from Kim et al. [95]. The six-fold symmetry
FIG. bulk valence band when the voltage less than -500 meV and by of
gets smeared by the 4 (color online). (a)–(i) Fastis Fourier transformation the the dI=dV
bulk conduction band when it is greater than -250 meV. FFT images for (a)–(i) along À M . (k)
(j) Cross-sectional profiles of the Scattering is enhanced in the Γ-M " "
direction and suppressed in the Γ-K. At some voltages, theyarea). symmetry is shows possible surfa
projected bulk band structure (shaded 6-fold An arrow less clear.
(a) and (b) show the smearing of the topological state due to the valence band and (g)(i) show the smearing due to the conduction band. Reprinted figure with permission from
S.Kim, M.Ye, K.Kuroda, Y.Yamada, E.Krasovskii, E.Chulkov, K.Miyamoto, M.Nakatake,
056803-3
T.Okuda, Y.Ueda, K.Shimada, H.Namatame, M.Taniguchi, and A.Kimura. Physical Review
Letters, 107:056803, (2011). Copyright (2011) by the American Physical Society. Abstract
can be viewed at: http://prl.aps.org/abstract/PRL/v107/i5/e056803.

48

state interacts with the bulk. Quasiparticle interference patterns from Kim et al. suggest
that bulk mixing may obscure the quasiparticle interference. Figure 3.8 shows that the 6-fold
symmetric pattern in k-space gets smeared out; this effect was attributed to overlap with
bulk states [95]. (To be clear, this picture for bulk mixing is very different from my analysis
presented in Chapter 4). The scattering patterns are enhanced along the Γ-M direction and
repressed along Γ-K in k-space in both ARPES patterns and QPI patterns. The Reference
[95] study also identified other quasiparticle interference pattern measurements on Bi2 Se3 ,
showing that the central maximum in the Fourier transform is enhanced by the surface state
mixing with the bulk, particularly in the valence band below the Dirac point [95].
Photocurrent measurements show that bulk and surface current mixing also occurs in
measurements that use a spin-polarized laser to measure the spin dependent photocurrents.
The photon drag effect allows spin-polarized excitations to relax through bulk valence channels instead of just inducing a spin-dependent current when the incident photon interacts
with the opposite spin/momentum channel through the surface helical Dirac cone [99]. Band
structure calculations show an overlap with the Dirac cone and some bulk states [100]. It
is speculated that this not only interferes with quasiparticle scattering, but that it also
suppresses Landau level formation in Bi2 Se3 [101].

3.2.4

Band Bending

Subsurface charges or the application of electric fields can cause shifts in the surface electronic
structure; this effect is called band bending. In both Bi2 Te3 [64] and Bi2 Se3 [102], band
bending effects due to contamination or layer decoupling can make the surface state shift
with time. Some measurements show that the Dirac cone forms and shifts over a period of
49

a few hours. This localized band bending is credited with shifts in the Dirac point energy
between ARPES and STM measurement of up to 120 meV [69].
Band bending is responsible for quantum well formations in the top layer of the material. When the surface is exposed to different chemicals, be it incidental environmental
contamination or intentional (i.e. [77]), band bending on the surface forms a two-dimensional
electron gas (2DEG) over the period of a few minutes to hours depending on concentration
[103, 102] and temperature [102, 101]. ARPES and transport measurements have identified
a variety of contaminants that can cause the 2DEG formation.
Other contamination effects, including water [104], show a split band to appear out of
the conduction band, perhaps by chemically altering the surface. High levels of carbon
monoxide can confine both the conduction and valence bands due to extreme band bending
[103]. ARPES measurements show the Dirac cone shifts upon exposure to molecules of N2
and air [101]. Brahlek et. al. showed that in transport measurements O2 and water exposure
decreases the number of two-dimensional carriers where N2 and vacuum can nearly restore
the original transport characteristics given a week. Other studies on 2DEG formation show
that in extreme cases of band bending where the bending exceeds the width of the valence
band, both the valence and conduction bands can become confined simultaneously [103].
Band bending on the surface is no small matter.
Figure 3.9 shows Bianchi et al.’s ARPES measurements of a two-dimensional electron
gas coexisting with the topological surface state on Bi2 Se3 . In Figure 3.9 there is the energy
vs. momentum ARPES data of the linear topological insulating Dirac cone and a parabolic
band coming down from the region of the conduction band. The sharpness of the rim is an
indication of a two-dimensional state, and the three-dimensional bulk state above it appears
50

diffuse. To determine the source of the 2DEG, the authors confirmed this band bending was
due to extrinsic defects and not a crystal reorganization using LEED. Because of the time
dependence of band bending, quantum well forms over time and this data was collected after
the crystal was three hours in a vacuum during which time the Dirac point shifting to lower
energies. While the 2DEG can coexist with the topological state, the Dirac cone must become
wider in k-space to accommodate the 2DEG and has more scattering channels into which it
can decay [102]. Elsewhere in their work, Bianchi it al. show the inner circular 2DEG state
is clearly distinguishable from the hexagon topological insulating state [102]. The hexagonal
shape not only becomes more pronounced with atmospheric exposure [101] but it is related to
the six-fold symmetry of the Bi2 Se3 system and is one of the characteristics that is different
from the closely related topological insulating material, Bi2 Te3 , whose Fermi surface has
a hexagram shape [105, 13]. The topological state deviates from its hexagonal state and
evolves with energy through the Dirac point, where it is a bright focused spot [102] before
it joins the valence band at -0.5 eV.

The mechanism of the 2DEG formation is something that is debated. Some groups
advocate that a chemical modification of the surface shifts the bands [106]. Indeed, it is
standard practice in transport to treat the surface with molecules to prevent Se vacancy
formation [107, 77, 100]. Others advocate that the molecules intercalate into the Van der
Waals gap and the expansion alone drives 2DEG formation [101]. STM measurements which
are sensitive to defects on the surface or intercalated defects have never reported indications
of a 2DEG formation. Theory calculations reject the chemical modification explanation and
instead advocate that the chemicals intercalate between the layers with causes the Van der
Waals gap to expand [108]. They show that as the 2DEG forms in the top quintuple layer, the
51

topological surface state is displaced to the second quintuple layer in stoichiometric Bi2 Se3 .
Because of the sensitivity of tunneling on distance in STM measurements, the implications
of these calculations are that as a 2DEG forms, the Dirac cone would not be as prominent
in the density of states.
Clearly, Bi2 Se3 has a lot of rich physics. Band bending can confine the conduction
band and sometimes even the valence band. Therefore, the topological surface state may
coexist with a 2DEG. Quasiparticle scattering experiments show the spin split nature of
the surface as scattering is suppressed in the Γ-K direction. Bulk mixing obscures the
interference patterns and contributes to the muted nature of pattern development. Despite
these potential non-topological effects, to the simplicity of the Dirac cone, Bi2 Se3 is a model
topological insulator.

52

Figure 3.9: ARPES measurements on Bi2 Se3 showing effects of the top quintuple layer
separating from the crystal [102] E vs. k of the Dirac cone is shown with the 2DEG evident
as the bright rim surrounding the diffuse conduction band edge. The Dirac cone is the outer
x shaped feature with the Dirac point at the center of the x. Reprinted by permission from
Macmillan Publishers Ltd: NATURE COMMUNICATIONS (M.Bianchi, D.Guan, S.Bao,
J.Mi, B.B. Iversen, P.D.C. King, and P.Hofmann. “Coexistence of the Topological State
and a Two-Dimensional Electron Gas on the Surface of Bi2 Se3 ”. Nature Communications,
1:128, 2010.), copyright (2010)
53

Chapter 4
Measurements of States on the
Surface of Bi2Se3
The topological surface state has exciting characteristics described in Chapter 3 that may
be exploited for device applications. It is therefore important to know what other states on
the surface are competing with the topological state as they may limit applications. In the
next sections, I will show that there is a bulk contribution to the electronic structure on the
surface of Bi2 Se3 in the spectroscopy. There is also an indication that a two-dimensional
electron gas contributes a feature in the density of states.

4.1

Sample Overview and Procedure

The crystals I studied were grown in the Kanatzidis group by Duck-Young Chung and
Jung-Hwan Song by slowly cooling an elemental mixture from 850◦ C, which is a common
preparation technique for bulk crystals. Samples of Bi2 Se3 with 2%, 4% and 8% excess
bismuth were prepared and the purity was established by powder x-ray diffraction. The 2%
54

samples showed no impurity phases, while the 4% doped sample had some Bi8 Se9 and the
8% doped sample had BiSe impurity phases. Although they could not guarantee that all
the additional bismuth was taken up by the crystal, unit cell analysis on the crystals showed
that the lattice expanded a bit as more excess bismuth was added, indicating that more was
taken in by the lattice.
To take measurements, a crystal of 1 mm or less in thickness and a few mm wide of
Bi2 Se3 was taken from the ingot. In general, parts of the ingot that were in the curved
bottom of the growth tube did not cleave well and were avoided. A sample disk was cleaned
with acetone and isopropyl alcohol to remove any dirt and oil. Once cut with a razor to
size, a tiny dot of silver paint was placed in the center of a sample disk to dry for half an
hour. Once dry, the mounted sample was inspected to ensure it still lay flat on the sample
disk. Any edges of the sample that had been engulfed in silver paint were trimmed to
ease the cleavage process. The sample was cleaved with scotch tape to expose a mirror-like
surface for initial testing. It is easiest to remove any huge steps at this time. After bench
top testing, the molybdenum disulfide powder was applied to the sample ramps to aid the
walking process before the entire sample was loaded in a two stage process into the sample
processing chamber discussed in Section 2.1. The microscope was evacuated of all impurities
by pumping it to a hundred mtorr. The crystal was cleaved in the chamber, being careful to
ensure a mirror-like surface without any part of the flake sticking up much higher than the
surrounding area to prevent tip crashes. Finally, the sample was loaded into the microscope
in its N2 gas environment, isolated from the chamber and the microscope was pumped out
overnight to a pressure of 3 × 10−6 torr.
One of the characteristic features of Bi2 Se3 doped with excess bismuth is bright triangular
55

Figure 4.1: Characteristic triangles from bismuth substitutions on selenium sites in Bi2 Se3
doped with 4% excess Bi. Image is at 4.2K with -0.8 V bias and 0.50 nA set current. The
scale bar is 3 nm.
enhancements discussed in Section 3.2.2. Figure 4.1 shows triangles I measured on the
surface of Bi2 Se3 with 4% excess bismuth. This finding confirms that the samples exhibit
the qualities that are expected, and that the sample processing chamber does result in a
sufficiently clean surface.

4.2

Spectroscopy Measurements on Bi2Se3

I applied STM at 4.2K to perform spectroscopy on Bi2 Se3 doped with 4% excess Bi. All
spectroscopy measurements were consistently made on sample locations far from defects to
ensure the spectrum was of the pure Bi2 Se3 state. In a material with a band gap, the slope
of I-V curves goes to zero in the gap. Figure 4.2 (a) shows the I-V curve. We see that the
slope described in Section 2.2.1, dI/dV , is never zero at any bias voltage. By focusing on
the low voltage region (+/-0.15 V) in Figure 4.2 (b), it is even more clear that the slope is
56

(a)

(b)
Figure 4.2: (a) The calculated I-V curve on Bi2 Se3 doped with 4% excess Bi at 4.2K. (b)
The same I-V curve expanded in the region of the Dirac cone. The slope does not go to
zero at zero voltage, indicating there are states in the gap. Due to an analog to digital issue
in the electronics, the current was calculated from the dI/dV curve in Figure 4.3 (a). The
dI/dV curve was measured directly from using the lock-in amplifier as described in Section
2.2.1 and therefore did not experience the same analog to digital issue.

57

(a)

(b)
Figure 4.3: (a) Density of states on Bi2 Se3 doped with 4% excess Bi at 4.2K where the
linear region of the Dirac cone is between +/-0.15 V. The Fermi level is in the center of the
gap. There are additional states that lift up the bottom of the Dirac cone. The curve is an
average of 150 curves taken successively to reduce the signal to noise ratio. (b) The density
of states only in the region of the Dirac cone showing parameters α and β, described in the
text.
58

nonzero. The nonzero slope indicates that there are states in the gap. If the basic picture of
Bi2 Se3 is completely described by only a single Dirac cone that resides inside an insulating
band gap, there should be a point where there are truly no states. At the point the slope
of the I-V curve, dI/dV would have to be zero. By examining only the I-V curve, it is clear
that other physics is in play.
Stoichiometric Bi2 Se3 is n-type, meaning the current is predominantly carried by electrons. The Fermi level tends to be pinned to the conduction band edge due to Se vacancies
in the crystal [109]. We suspected that the excess Bi would tend to shift the Fermi energy,
EF . In Figure 4.3, we see from the horizontal axis that the excess Bi almost perfectly compensates and the Fermi level is at zero volts. While this is not essential to the analysis, it
is still a nice feature of the data that several research groups have strived to achieve. The
clear V-shaped Dirac cone is in the region of +/-0.15 V. With respect to the vertical axis,
we see that the Dirac point is lifted up from zero. This is just another visual depiction of
how dI/dV never goes to zero; there is always an offset of the Dirac cone.
To determine that the offset of the Dirac cone, or “background”, was intrinsic to the
system and not an artifact, I compared spectroscopy in different spots on the sample and
different doping levels of Bi. In Bi2 Se3 spectroscopy, I compared the ratio of the height of
the offset, α, to the height of the Dirac cone at -0.1 V, β, as drawn in Figure 4.3(b). Because
the scale on density of states measurements is arbitrary, it is necessary to compare the ratio
of feature heights when comparing different spectroscopy curves. In this case, the choice of
β is arbitrary; it could have been chosen to be anywhere in the Dirac cone so long as the
comparison is at the same cone heights between spots on the sample. For the data shown in
Figure 4.3, β/α = 2.87. I repeated the same measurement at different locations within the
59

Figure 4.4: Density of States where each of the curves is an average of 50 measurements.
Data set A was taken before Data set B without manually moving the tip. Data set A has
been offset by one tick mark to ease visual comparison. A blue bracket has been drawn to
highlight the enhancement. We speculate the tip drifted during the measurement, causing
the feature at +0.2 V to change.
sample at 4.2K and the ratio β/α is consistent within 15%, which is the error expected due
to scatter in the data. Given the consistency between these measurements, I concluded the
offset is an intrinsic feature of Bi2 Se3 .
While the Dirac cone and the background states do not depend on the measurement
location within the sample, the spectroscopy features above +0.2 V is location dependent.
Figure 4.4 shows the density of states on Bi2 Se3 far from defects. Above +0.2 V the curves
show progressively enhanced density of states. Each of the curves plotted is an average of
fifty curves. There were another 50 curves between Data Set A and Data Set B that show an
intermediate step between the enhancement and depression at +0.2 V that has been removed
for clarity. The apparent dip at +0.2 V arises where the Dirac cone is no longer lifted up by
the shoulder states. Calculations in the following discussion showed only that the shoulder
60

extended past the Dirac point, but how far beyond the Fermi level the Dirac cone extended
was unable to be determined.
I believe that any differences in each of the three successive curves arise from thermal
drift during the measurement. In other words, the spot on the sample the measurement
began is not the same point the measurement ends. Similar enhancements and depressions
have been observed at other locations on this sample. However, this data set is unique in
that I see an evolution between them.

4.3

Analysis of the Bi2Se3 Spectrum

Density functional theory (DFT) is a computational technique to find the electronic structure of periodic quantum mechanical systems. It is a way of approximately solving the
Hamiltonian using only the density of electrons in space. To analyze my data, Dr. Mal-soon
Lee working with Professor S.D. Mahanti performed DFT calculations. The calculations
were performed using the projector augmented wave (PAW) methods as implemented in
the commercially available VASP package. The Perdew-Burke-Ernzerhof (PBE) generalized
gradient corrected exchange-correlation functional were used in the calculation. To fully
solve the band structure, spin orbit interactions were included. Theoretical details of the
calculation can be found in Reference [110].
Figure 3.6 shows the bulk calculation in gray and a 6 quintuple layer (QL) slab calculation
in blue with the Fermi energy at the Dirac point. While a single Dirac cone transverses the
gap, the bulk has a shoulder state in the Γ-M direction similar to calculations by Xia et
al.[100], Yazyev et al.[111], and Kim et al.[95]. The same state appears in the calculations
performed by Dr. Mal-soon Lee shown in Figure 4.5 (a) and Figure 4.7 marked by the red
61

(a)

(b)
Figure 4.5: (a) Valence band calculation with increasing numbers of QL. The shoulder state
in the Γ-M direction increases in energy and eventually crosses the Dirac point. (b) Graph
of the change in energy with QL. Eventually the height of the shoulder will reach a bulk
value as the number of QL increases. Figure courtesy of Dr. Mal-soon Lee

62

arrows.
Prior to this work, nobody had studied systematically how increasing the number of
quintuple layers in a slab calculation increases the shoulder’s energy. Slab calculations of
Bi2 Se3 seen in Figure 4.5 (a) from 6-18 QL show how the energy of the shoulder increases
with the number of QL which indicates that they probably originate in the bulk. However,
the Dirac cone states do not change as the number of quintuple layers increases which
demonstrates that the topological surface state is well established by 6 QL. To match the
energy scales, the Dirac points of each slab were aligned with the Fermi level at zero voltage.
The Dirac cone itself is due to the presence of the surface and is not present in any bulk
calculations.
We see from Figure 4.5 (a) that the shoulder in the Γ − M direction of k-space is getting
higher in energy as the number of QL increases. Figure 4.5 (b) shows the trend of the change
in energy, ∆E, which is the difference between the top of the Dirac cone and the maximal
shoulder energy. ∆E becomes more negative with increasing QL. This is consistent with the
energy of the Dirac cone becoming comparable with the height of the shoulder. By an 18 QL
slab, the energy height of the shoulder exceeds the Fermi level and ∆E becomes negative.
Extrapolating from the graph, the energy will eventually level out at some sort of constant
“bulk” value if the number of quintuple layers is increased further. Extra states at the Fermi
level would indeed add to and lift the Dirac cone. Hence, these shoulder states are then
excellent candidates for explaining the background in STS measurements as an interplay of
surface and bulk states.
Figure 4.5 (a) shows that shoulder states exist at the right energy to mix with the
topologic surface state in the region of the Dirac cone. To fully explain my measurements,
63

these electronic states need to reside in the top layer of the material. Because of the tunneling
current’s exponential dependence on the tunneling gap distance, described in Section 2.2,
STS measurements are most sensitive to the topmost atom, Se1, and to a lesser extent the
Bi1 atom immediately below it. In other words, these shoulder states need to have spectral
weight on the surface for STM to measure them. Only if the shoulder state resides on the
surface can it explain why dI/dV never goes to zero at the Dirac point in Figure 4.3.
Figure 4.6, also calculated by Dr. Mal-soon Lee, contrasts the partial density of states
(PDOS) from the Dirac cone (Region I) with the shoulder states (Region II). These calculations are for a 6QL slab because anything thicker would be computationally prohibitive.
As expected, the contributions to the partial density of states from the Dirac cone resides
primarily in the Se1 and Bi1 on the topmost quintuple layer within the gap. In the middle
QL, there are no contributions from Region I in the gap but Region II dominates. There is a
big contribution to the density of states in the middle QL Se1 from the shoulder states and
a lesser contribution from the middle QL Bi1. The fact that the shoulder states dominate
in the middle, bulk-like, region of the slab agrees with the bulk calculations found in Xia et
al.[100] and Kim et al.[95]. To determine if the shoulder states are present on the surface,
one needs to look for a peak in the top layer Se1 at approximately -0.3 V, the energy of
the shoulder state for a 6QL slab. In Figure 4.5 (a), there is indeed a big peak in Se1 on
the topmost layer and it is even evident in the Top QL Bi1. While the shoulder states may
originate in the bulk-like part of the slab, they are still present in the top quintuple layer
that is probed in STM measurements.
To highlight these results pictorially, Dr. Lee calculated the partial charge density contribution from just the shoulder states for a 6 QL slab and from the bulk calculations in Figure
64

4.7. The shoulder states in from bulk calculations are selenium-like, meaning that they have
a higher spectral weight near Se atoms, depicted with red halos throughout the quintuple
layers. While most of the charge from shoulder states in the slab calculation is centered
around the Se atoms in the middle of the slab, we see that there is some finite charge density
near the top of the top quintuple layer which would be accessible to tunneling. (Figure 4.6
is better at showing the relative intensities of the Dirac cone and the shoulder states on the
surface).
In summary, the calculations show how there are bulk states that reside on the surface
that have the potential to contribute to STM measurements. STM is not sensitive to the
momentum of the electrons, so any states available at the Fermi level will contribute to
tunneling. Hence, the shoulder states sum with the Dirac cone states, giving rise to the
offset, α in Figure 4.5 (a), seen in my density of states measurement. In other words, we
believe there is an interplay between the bulk and topological surface state in Bi2 Se3 that
STM probes.

4.4

Analysis of Two Dimensional Electron Gas on the
Surface

Figure 4.4 shows how over the course of a long measurement, there is a feature above +0.2
V that begins as an enhancement and then becomes a depression in the density of states. In
Section 4.2, I speculated that the tip may have drifted during the course of the measurement
so that the area of the sample the tip was probing at the beginning was not the same as the
one it was probing at the end. Due to reasons about to be expanded upon, the tip may have
65

Figure 4.6: Partial density of states (PDOS) calculations performed by Dr. Mal-soon Lee on
a 6 QL slab. The inset shows the parts of k-space that contribute to Region I and Region
II. The contribution to PDOS from the Dirac cone is depicted in red. It resides primarily in
the top layers within the bulk band gap. The contribution from the shoulder states (blue)
have the largest contribution in the middle QL, however it has a finite contribution in the
top QL Se and Bi atoms. In this case, the peak is just to the left of zero energy (the Fermi
level) for this rather thin slab.
66

Figure 4.7: Comparison of bulk and slab calculations. This contrasts the partial charge
density from bulk band structure calculations (gray in the band structure) on the left side
of the image with those from the slab calculations (blue in the band structure calculations).
The Se are yellow, the Bi atoms are blue-gray and the the intensity of the red halos depicts
how strong the charge density is at that region. No halo means there is no significant charge
density. The bulk calculations show the partial charge density is mostly distributed on the
Se atoms in all layers equally. The partial charge density on the slab calculation on the right
is primarily located on Se atoms as well, but has the highest density in the middle layers
(the bulk-like area of the slab). However, there is some charge density on Se atoms in the
outermost quintuple layers shown by the green arrows. Figure courtesy of Dr. Mal-soon Lee

67

drifted from an area of the sample where the top QL was less connected to where it was more
connected. The readily cleavable surface on Bi2 Se3 leaves the possibility for the top layer to
be less bound in some areas. The Van der Waals gap between the top two layers may widen
during the cleaving process or later when a strong field is applied to the crystal. Impurities
may also contribute to layer separation. Clearly, if one QL were completely separated from
the crystal, the extended electron wavefunction may resemble a two-dimensional electron
system.
Few groups have studied the formation of a 2DEG on the surface of topological insulators.
The explanations from ARPES studies focussed on the effects of impurities. Some studies
have shown that separating the top layer from the subsurface layer increases causes the
conduction band edge to move into the gap and localizes the 2DEG in the top layer [102,
103, 112]. Common impurities that are believed to cause layer separation are oxygen[106],
CO [103], water [104], and other adsorbates [112, 102, 81], many of which are common
vacuum contaminants. Theoretical calculations support the possibility that the layer is
more loosely bound in some areas by predicting a two-dimensional electron gas sets up on
the top quintuple layer.
To explore these ideas more carefully, Dr. Lee performed similar DFT calculations to
those in Section 4.3. Figure 4.8 shows her results that support the idea that a 2DEG may
have set up on the surface of the material. We see that an increase in the physical separation
gap between the top two quintuple layers of thirty to fifty percent causes the lowest band of
the conduction band to extend into the gap. The shape of this band is parabolic like a 2DEG.
For the data shown in Figure 4.4, as the STM measurement progresses I believe the tip is
drifting from a more separated region to a less separated region removing the enhancement
68

in the density of states by the end of the measurement. The enhancement to the LDOS
cannot be from other sources like Se vacancies because not only was there no evidence in
the topography, but also the enhancement appears at a higher energy than Se vacancies [91].
Moreover, there is no other electronic evidence of an impurity of the surface of any kind even
though we were probing with a blunt tip. As further spectroscopic evidence, the Dirac cone
is not as sharp of a V shape between +/-0.1V in Data Set A as Data Set B.
Menshcikova et al. [108] calculated the partial charge density of the Dirac cone with an
increasing layer separation. Their partial charge density calculations showed that the 2DEG
pushes the Dirac cone into the second quintuple layer, which would reduce the Dirac cone’s
contribution to my STM measurement. Interestingly, the background states are not effected
by 2DEG formation.
In summary, there is some evidence that a 2DEG may have locally formed on the surface
due to layer separation. Dr. Lee’s calculations show that when the Van der Waals gap
between the topmost quintuple layers expands, a 2DEG comes down from the conduction
band and resides at +0.2 eV above the Fermi level. This voltage is same energy that the
enhancement appears in my measurement in Figure 4.4 leading to the possibility that there
is a 2DEG in the top quintuple layer of some of my Bi2 Se3 sample.

4.5

Bi2Se3

As a comparison, I looked at an undoped crystal of Bi2 Se3 previously used by Dr. Sergei
Urahzdin. Figure 4.9 (a) and (b) show the current-voltage and density of states curves. In
this case, the curves are averages of only five curves, and the higher noise level reflects it.
The Dirac cone is not very pronounced, but the background that lifts up the center of the
69

Figure 4.8: Band structure calculations of a 6 QL slab of Bi2 Se3 for which the top QL is
separated from the others. The percentages refer to how much additional separation is given
to the top QL. We see as the top layer detaches, the bottom of the conduction band drops
into the gap and forms a two-dimensional electron gas on the surface. Figure courtesy of Dr.
Mal-soon Lee.

gap is still clearly there. The whole curve is shifted to more negative voltages than the 4%
doping in Section 4.2 making it more n-type than that sample. This shift is consistent with
the previous conclusion that Bi tends to restore the Fermi level to the center of the gap. Its
n-type nature is probably from Se vacancies [14]. The enhancement described at +0.2 V in
Section 4.4 associated with the 2DEG has also shifted with all the bands. Consistent with
previous conclusions, a 2DEG formation obscures the Dirac cone. Undoped Bi2 Se3 shows
many of the same spectral hallmarks of the pristine Bi2 Se3 I probed in Sections 4.3 and 4.4.
Excess bismuth doping only serves to shift the band structure all at once and not relative
to each other as was described in reference [101]. Doping only fills in the selenium vacancies
and shifts the bands, but does not alter the topological state. Moreover, these measurements
on the undoped sample take place at 77K instead of 4.2K in Section 4.2 demonstrating that
temperature effects do not impact any of the previous conclusions. Figure 4.9 (c) shows an
70

(a)

(b)

(c)
Figure 4.9: STM data at 77K. Both the (a) I-V curve and the (b) density of states are an
average of 5 curves. (c) Atoms on the surface of Bi2 Se3 at -0.20 V and 0.1 nA. Scale bar is
1.6 nm.
atomic resolution image of the surface where there is evidence of the dark smudginess I have
come to associate with all of these samples, which will be expanded upon in Chapter 5. In
short, neither temperature effects or doping alter characteristic features of Bi2 Se3 samples.
There is an interplay of bulk states with the topological surface state between samples which
is not altered by doping and a 2DEG can potentially set up on the surface of the material.

71

Chapter 5
Other Measurements of Bi2Se3
This chapter gives an overview of different samples I examined, but does not directly impact
the analysis of my primary results in Chapter 4. This chapter should be seen as supplemental
information as it is focussed on empirical observations. It covers additional information about
my primary sample of Bi2 Se3 doped with 4% excess bismuth, as well as samples with 2%
and 8%. Details about the sample quality can be found in Section 4.1.

5.1

Bi2.02Se2.98

All my earliest studies on topological insulators were on Bi2.02 Se2.98 . Most of the measurements took place before the processing chamber was installed so there was more surface
contamination. The final cleave took place in the laboratory environment before the sample
was placed in the microscope, pumped down to a pressure of 3×10−6 torr overnight and
cooled to a temperature of 77K. The biggest challenge for these samples was finding a spot
that was clear enough to get any kind of meaningful image. Many scans had debris that
often attached to the tip, necessitating tip recovery techniques fairly often. Nevertheless, I
72

was able to scan many topographic features that appeared in every doping level of Bi2 Se3
studied.
For instance, Figure 5.1 (a) shows the top selenium layer of Bi2.02 Se2.98 . There is no
electronic impurity signature in this scan range. The dark spots in Figure 5.1 (b) seem
to have a slight triangular shape and originate from a selenium vacancy [92, 91]. Dark
triangular features are often imaged in Bi2 Se3 and can appear right alongside bright triangles
originating from interlayer Bi substitutions [10].
Impurities should have an electronic signature on a sample. If they are truly part of
the sample and not some debris on the surface, the apparent topography of impurities will
change with the voltage. Figure 5.2 shows both surface contamination and electronic defect
states. These images were taken in succession with only the bias voltage changing between
them. Rows of atoms are also visible across the images. The upper right hand corner of all
the images has a bright blob that is independent of voltage. In many of the scans, there
is streakiness that extends down the image from that corner. This smearing is common in
scanning probe measurements when there is something unstable that is not securely attached
to the sample or tip. This observation leads to the conclusion that it originates with the
surface contamination and not interlayer Bi, which creates triangular topographic features
that are independent of voltage [10]. The tip and the sample are interacting that causes
artifacts in the topography image. Elsewhere in the image in the scans at -0.50 V in Figure
5.2 (a) and (c) there are bright blobs in the upper left corner and close to the middle of the
image. Unlike the surface debris, these two enhancements disappear at -0.55 V in Figure
5.2 (b) and at -0.4 V in Figure 5.2 (d). The reason the defect states do not appear as
triangular is unknown but could be an artifact of the tip, but the defects are behaving like
73

(a)

(b)

Figure 5.1: Data taken at 77K without the chamber. (a) Atoms on the surface of
Bi2.02 Se2.98 at a set current of 0.11 nA and -0.5 V. Scale bar is 3 ˚. (b) Dark pits seen all
A
over the Bi2.02 Se2.98 potentially caused by surface contamination. Scale bar is 17 ˚.
A
Bi substitutions from reference [14].

5.2

Bi2.04Se2.96

Here I give additional measurements on what turned out to be my primary sample, Bi2 Se3
doped with 4% excess Bi. Even with a processing chamber, there tended to be a little bit of
debris on the surface of Bi2.04 Se2.96 . Many times images were so streaky that there are no
meaningful features on the surface. Defect states can be hard to discern when the surface
is shifting. The tip was compromised and dragging the tip to refurbish it was not very
helpful. Field emission tended to work best, but was only reliable at high voltages. After
a few recoveries at 150 V, the tip became extremely blunt. Topography images were then
featureless but the spectroscopy was excellent. Consequently, some of the best spectroscopy
curves, like that in Figure 4.3 had a completely featureless topography image taken in tandem. Nevertheless, many topography features were evident on the surface of Bi2.04 Se2.96 .
74

(a)

(b)

(c)

(d)

Figure 5.2: Defects on the surface of Bi2.02 Se2.98 at 0.5 nA set tunneling current and 77K.
Scale bars are 7 ˚. Each image is taken in sequence at a bias voltage of (a) -0.5 V (b) -0.55
A
V (c) -0.50 V and (d) -0.4 V. The bright spot in the upper right corner of every image is
surface contamination. Impurities in the crystal are visible in the upper left in (a) and (c).

75

(a)

(b)

(c)
Figure 5.3: Defects on the surface of Bi2.04 Se2.96 at 0.5 nA set tunneling current and liquid
helium temperature. Each image is taken in sequence at a bias voltage of (a) -0.80 V, 36 pA
(b) -0.49 V, 73 pA and (c) -0.29 V, 52 pA. Dark impurity spots become more diffuse with
changing voltage. Bright specs of surface contamination also visible in every image. Scale
bars are 3.3nm.
Because it seemed to routinely cleave with fewer step edges than other doping levels, this
sample was the most extensively studied material.
Dark blobs appear all over the sample. They definitely have some some electronic origin
because they are sensitive to voltage. As Figure 5.3 shows, the dark blotches become darker,
less diffuse and better resolved as the voltage changes. There is also debris in the upper
right corner of this sequence. Some of the bright spots are electronic features because they
are active at the same bias as the triangular defects in Figure 5.4 (a). Because the bright
irregular blobs are much larger than the triangular defects, the large features, while voltage
76

(a)

(b)

Figure 5.4: Data taken at 4.2K. Scale bars are 3.3 nm.(a) Bright triangular defects on the
surface of Bi2.04 Se2.96 at a set current of 60 pA and bias voltage of -0.80 V. (b) Clusters of
triangular defects arranging in strings at 60 pA and -0.79 V.

independent, are likely from some surface debris and not the interlayer Bi described in
reference [10].
The characteristic bright voltage dependent triangular defects [14] appear in scans of
Bi2.04 Se2.96 . Figure 5.4 (a) shows relatively isolated defects. Each triangle is about 7nm
across, similar to Urazhdin et al.. Sometimes the defects cluster in a way that makes them
appear as lines in the images instead of just isolated triangles. These lines in Figure 5.4 (b)
are parallel because triangles all point the same direction due to their perturbation on the
lattice electronic structure[10].
The same features of triangular defects and pits persist at liquid nitrogen temperatures.
Figure 5.5 (a) shows several triangular defects all over the image. While there are clusters
of defects, they do not seem to arrange themselves into strings of defects in the same way as
Figure 5.4 (b). Unfortunately, the lack of concentrated regions of defects limited my ability
77

(a)

(b)

Figure 5.5: Data taken at 77K. (a) Bright triangular defects on the surface of Bi2.04 Se2.96
at a set current of 24 pA and bias voltage of -0.48 V. Scale bar is 17 nm. (b) Pits on the
surface at 118 pA and -0.38 V. The scale bar is 67 nm.
to image these defects with Kelvin probe. The dark spots also persist at 77K. They are
quite large, often tens of nanometers across. While these are electronically different from
the surrounding area, they do not locally change the capacitance signal. If they do represent
any difference in the electric field from the sample, they are electronically screened over a
distance of approximately 20-30nm. This length scale is comparable to the screening length
calculated for Bi2 Te3 of 26 nm in reference [80].

5.3

Bi2.08Se2.92

I spent the least amount of time on the eight percent doped sample. Most of the time I
spent studying these samples also took place before the cleaving chamber was installed. Of
all the samples I worked with, this one was the most difficult to get a good cleave where
large areas of the sample were step edge free. This sample would flake up quite a bit more
78

(a)

(b)

(c)

Figure 5.6: Data at 77K, scale bar is 3.3 ˚ and set current 1.02 nA. (a) -0.6 V (b) -0.3 V and
A
(c)-0.6 V. There is both surface contamination, atomic resolution and a voltage-dependent
defect in the upper right corner.

than the others. Perhaps due to the high doping level, that Bi2.08 Se2.92 leaves a lot more
surface residue from the last cleave than any of the other crystals. It is fairly common to see
a few tiny grains on the mirror like surface after cleaving any crystal with scotch tape. I have
always interpreted these bits as coming from the fracture that occurs around the step edges
as the scotch tape is peeled away. In 8% doping, the fine dust is more wide spread (so are
the step edges) and is adhered to the surface well enough that it cannot be dislodged with
N2 gas. A data run on the sample after the chamber was installed was short in part because
of all the particles that still remained on the sample that contaminated the tip. As you can
see from Figure 5.6, there is quite a bit of surface contamination leading to a streaky image.
Nevertheless, there is still a visible electronic lattice in each of these images. In the upper
right corner of Figure 5.6 (a) and (c), there is not only a tiny piece of surface contamination,
but a larger bright spot that disappears in (b). While 8% excess bismuth does show features
characteristic of Bi2 Se3 , it is not the best sample to probe the Bi2 Se3 state.
In summary, all of the samples of doped Bi2 Se3 show similar features. Dark spots, most
likely from Se vacancies, appear on all of the samples, but so do voltage-dependent bright
79

spots from bismuth on a selenium site substitutions. These electronic structure signatures
are distinguishable from the debris that covers the surface.

80

Chapter 6
Other Scanning Probe Methods

6.1

Capacitance and Kelvin Probe Circuit

For scanning capacitance measurements, the apex of the tip is essentially a nanometer-size
capacitor plate and the nanometer scale piece of the sample below represents the other
capacitor plate. Just like a large capacitor, if charge enters the sample, image charge of the
opposite sign appears on the tip. Another way to say this is that electric field emanating
from the sample induces image charge on the tip.
Largely the same circuit is used in both Kelvin probe and capacitance measurements. It
consists of a tip and two high electron mobility transistors (HEMT) on a gallium arsenide
chip. Gold pads sputtered on the GaAs allow the circuit connections to be made with a
wire bonder. Low temperature conducting epoxy adheres the HEMTs and tip to the chips.
I assemble all the chips in the lab.
The two HEMTs in Figure 6.1 serve very different purposes. The bias HEMT serves to
tune the measurement HEMT to its most sensitive region for the measurement. In addition,
81

Figure 6.1: HEMT circuit diagram: The standard capacitor (line drawn in blue) is only
included for capacitance measurements. Oscillations are applied either to the bias voltage
for capacitance measurements or by oscillating the tip above the sample in Kelvin probe.
The lock-in amplifier detects the change in capacitance on the tip due to these oscillations.

82

the bias HEMT has a large impedance on its source-drain channel that ensures all the charge
that oscillates on the tip goes to the gate of the measurement HEMT. Any charge oscillating
on the tip then changes the effective resistance of the source-drain channel to be measured
√
by the lock-in amplifier. The charge sensitivity of the circuit is .16e/ Hz at 77 K [113].
The measurements the circuit is able to make are limited by other sources of capacitance
and noise in the system. Warming of the system is minimized by keeping the power dissipated
by the circuit low, in the microwatt range. As seen in Figure 6.2, this requirement puts the
device in the linear region of the source-drain characteristics.
Part of measuring anything with this HEMT circuit is a background capacitance. The
largest source of capacitance is the capacitive coupling between the gate and source-drain
channel for the measurement HEMT. This capacitance, Cin , is around 0.3pF and impacts
the voltage signal that reaches the lock-in amplifier. There is also a self-capacitance from
the circuit’s size, which is minimized by keeping the tip-HEMT distance as small as possible.

6.1.1

High Electron Mobility Transistor

A high electron mobility transistor (HEMT) is a field effect transistor mediated by a twodimensional electron gas. The surface of the transistor has gold pads that provide ohmic
contact to the sub-surface material for integration into a circuit. A layer of electron-rich
Alx Ga1−x As donates some of its electrons to the undoped GaAs layer below, creating a
highly mobile two-dimensional electron layer. The conducting properties of the electron
layer are mediated by voltage applied to the gate. Therefore, any voltage applied to the gate
changes the effective resistance of the source-drain channel.
83

Figure 6.2: The plot shows the drain current vs. the drain-source voltage of a HEMT
at different gate voltages. Varying the gate voltage changes the effective resistance of the
source-drain channel. The HEMT is operated in the linear region circled in red to minimize
power output [43].

Figure 6.3: A photograph of the Fujutsi model FHX35X HEMT. The size of the device is
∼ 500 µm across [43].
84

6.2

Capacitance Measurement

The capacitance measurement uses the HEMT circuit to detect charges entering the sample, which induces charge on the tip. The capacitance measured depends on the geometric
capacitance, Cgeo , which is due to physical features of the tip and sample and on the compressibility or thermodynamic density of states of the sample, dn , relating to the electronic
dµ
states through the equation:

A
Cmeas

=

A
Cgeo

+

1
e2 dn
dµ

(6.1)

The tip-sample distance is fixed about 1nm above the sample. The electric field from the
sample lands on the tip as it is scanned over the surface. The capacitance measurement can
be used to detect defects within the surface provided they are charge sinks [34] .

6.2.1

Physics

The HEMT circuit measures charge oscillating on the tip, q, in response to charge entering
the sample due to a voltage oscillation, VAC applied to the sample. The charge oscillations
are sensed by the tip as an image charge oscillating in phase with the voltage oscillation,
VAC – finally causing an oscillating voltage on the gate of the measurement HEMT. Not all
of the electric field lines emanating from the sample land on the apex of the tip; many land
on other areas of the macroscopic tip wire. These contribute to a background capacitance
which is constant (independent of the tip’s apex position). To subtract away this background
signal, a signal oscillating at the same frequency but 180◦ out of phase as the VAC is applied
to a standard capacitor, depicted as CS in Figure 6.1.
85

The gate and source-drain channel of the measurement HEMT have an effective capacitance Cin which is approximately 0.3 pF which limits the sensitivity of the circuit. The
self-capacitance can be minimized by making the tip-circuit distance as small as possible, so
that its contribution is approximately 0.1 pF.
The total capacitance, including the background signal is usually Ctip−sample = 20
fF (2 ×10−15 F. Voltage on the gate in Figure 6.1 is related to the voltage at the lock-in
amplifier, Vlock−in , by a gain factor, G. Therefore, Vlock−in /G is related to VAC and the
capacitive impedance by the voltage divider equation:

Vlock−in
G

= VAC

Z2
Z 2 + Z1

(6.2)

where Z1 = (iωCtip−sample )−1 and Z2 = (iωCin )−1 . G depends on electronics external
to this HEMT circuit and is approximately unity. The frequency of oscillation is ω. Equation
6.2 can be further simplified by realizing that Cin ￿ Ctip−sample and will only become
increasingly greater if you account for the circuit’s self-capacitance which only adds to Cin .
Ctip−sample
Therefore, Vlock−in =
VAC with respect to the tip’s charge, and the signal
Cin
is q = Vlock−in Cin = Ctip−sample VAC . To maximize the signal on the lock-in amplifier,
Ctip−sample should be as large as possible. Maintaining the smallest gap between the tip
and sample achieves this end because capacitance varies inversely with the distance between
the electrodes. The transistor was chosen to minimize Cin . Increasing VAC increases the
signal, but any increase of VAC will reduce the energy resolution of the measurement. The
typical frequency of the sample voltage excitation is 20 kHz. The typical amplitude of
oscillation is 0.1 V peak to peak. With regard to the noise level, at 0.3 K the circuit achieves
√
an amazingly low charge noise of 0.01e/ Hz, where e is the electron charge
86

(a)

(b)

Figure 6.4: (a) Topography image of the corner of a pit. (b) Capacitance image in the same
region as (a). Both are imaged at 77K, the scale bars are 330nm long and were taken with
an etched tip.

6.2.2

Demonstration

In addition to charging signals, the capacitance measurement is sensitive to changes in topography. The Pt pits sample serves as a good proof-of-principle sample. In Figure 6.4, the
edge of a pit is clearly visible in the upper left corner of the scan range. This same edge
appears in the capacitance. In this case, the contrast is due to a geometric capacitance.

6.3

Kelvin Probe Measurement

The Kelvin probe method employs an oscillating tip connected to the charge sensing circuit.
It measures electric field patterns emanating from the sample and is sensitive to any field
that is not screened away. Local measurements can measure the work function of the metal
[114] and can also be applied to image surfaces [37].
87

6.3.1

Physics

In Kelvin probe measurements, the voltage oscillations the lock-in amplifier detects originate
from physical oscillations of the tip. As the tip extends, more electric field lines from the
sample terminate on the tip than when it retracts. The resulting signal on the HEMT circuit
is proportional to the change in capacitance (or charge) with distance. However, there is no
electric field between the tip and the sample at a particular tip voltage, called the nulling
voltage or contact potential (Vnull ). If Vtip = Vnull , the voltage on the lock-in is not
sensitive to distance. The charge the lock in senses in not only sensitive to the oscillation
amplitude of the scanning tube, ∆z, but also its height above the sample, z0 . Moreover, the
signal charge on the lock-in consists of the actual charge signal, ∆q, in addition to an offset
signal, q0 , that arises from stray AC electric field landing on the HEMT gate.

q0 + ∆q =

A￿0
(V + Vsample − Vnull )
z0 + ∆z tip

A￿0
1
(V + Vsample − Vnull )
z0 z + ∆z tip
0
z0
A￿0
∆z
≈
(1 −
)(Vtip + Vsample − Vnull )
z0
z0
=

(6.3)

where A is the area of the measurement, Vtip and Vsample are the voltage of the tip and
sample respectively and ￿0 is the permittivity of free space. Only the oscillating part of this
equation will be sensed by the lock-in amplifier, so Equation 6.3 simplifies to:

qlock−in =

A￿0
∆z(Vnull − Vtip − Vsample ).
2
z0
88

(6.4)

Figure 6.5: Nulling voltage measurement on Bi2 Se3 . The arrow shows the voltage where
all the curves intersect (0.9 V). Because there is a -0.43 V applied to the tip relative to the
sample, the nulling voltage here is 0.9 − (−0.43) = 1.23 V.

(For simplicity, Equation 6.4 assumes q0 = 0.) By this relation, the signal is insensitive
to the tip distance whenever Vsample = Vnull − Vtip . In practice, the tip is at a voltage
set by the source of the bias HEMT to keep the measurement HEMT in its sensitive region
and only the voltage applied to the sample can be varied. To find the nulling voltage the
Kelvin probe signal versus voltage is measured at a mean tip distance above the sample,
z0 . This measurement is then repeated at two or more other different z0 above the sample.
All these are plotted on the same graph. Because only the nulling voltage is insensitive to
distance, the voltage at which the curves intersect is Vnull . The nulling voltage in Figure
6.5 is actually 1.2 V when one accounts for the offset from a voltage applied to the tip as
well as the sample.
89

(a)

(b)

Figure 6.6: Bi2 Se3 topography (a) and Kelvin probe (b). In this case, the Kelvin probe
image is nearly featureless because the signal is dominated by the high plateau in the lower
left. The topography image shows two step edges, where each step is 1 nm high. The Kelvin
probe scan must be within half a nanometer of the sample – even one nanometer beyond
does not contribute. The scale bar is 170 nm on both images. The amplitude of oscillation
is 1 nm.

6.3.2

Demonstration

The tip is held at a constant height above the sample for scanning Kelvin probe images
and the oscillation is applied on top of it. Due to the limitations of an RC charging time
constant from my cables, the largest frequency I can perform measurements at is 1 kHz,
which limits the sensitivity of the HEMT. The only way to increase the signal is to increase
the amplitude of the oscillation. However, it is extremely important to limit the amplitude
of the oscillation so that the tip does not repeatedly tap the sample anywhere in the area of
the scan. All Kelvin probe measurements were performed at 77K.
The relevance of the distance to the sample in Kelvin probe mode of operation is most
clearly illustrated in Figure 6.6. In this case, the tip is only measuring changes due to a
geometric capacitance. The topography image in Figure 6.6 (a) shows a double step on the
90

layered material, Bi2 Se3 , where each step is one nanometer tall. While the tip is closest to
the sample, a mere 0.5 nm away, the step shows up in Kelvin probe scans. There is not even
a hint of a step just one and a half nanometers below the tip. Hence we see that Kelvin
probe imaging has limitations for surfaces that have height variation of more than a few
nanometers.
The distance to the sample is not the only relevant dimension in Kelvin probe measurements. Unlike STM, where the atom protruding farther than the rest does the imaging,
the radius of curvature of the tip limits Kelvin probe resolution. In the topography image
in Figure 6.7, a trench crosses the Bi2 Se3 sample that is 1nm deep and 4nm wide and is
denoted by an arrow. Other dark blotches in the image are likely due to defects. A few hours
later, a Kelvin probe image of the same location shows the electric field emanating from the
edges of the trench as a bright line. However, the trench’s field is 13nm across, three times
as large! This discrepancy is actually a good indication that the tip used was large compared
to the size of the trench and demonstrates that the tip’s radius matters more for capacitance
type measurements [115]. To be sure, Figure 6.7 does illustrate that Kelvin probe imaging
is useful to map out the patterns of electric field; such as the higher field emanating from
the edges of the trench.

6.4

Tips

For capacitance and Kelvin probe measurements, the shape of the tip determines the resolution of an experiment. A blunt tip will not be able to resolve fine features, so it is important
to have a tip whose radius of curvature is smaller than any feature of interest on a sample.
The ideal tip is one that is cylindrically symmetric with as small of a radius of curvature as
91

(a)

(b)

Figure 6.7: Bi2 Se3 topography (a) and Kelvin probe (b), Here we see a scan showing a trench
indicated by an arrow. The trench is three times as wide in Kelvin probe as topography. Th
Kelvin probe scan had a 1 nm amplitude of oscillation on the tip at the nulling voltage and
was 1 nm above the sample. The scale bars are both 33 nm. The tip may have drifted over
the four hours between the topography scan and the Kelvin probe scan
possible. In my experiments, it was desirable to resolve electronic variations on the sample
of approximately 10nm, so I paid a great deal of attention to the tips. I examined a couple
hundred tips with the Scanning Electron Microscope (SEM not to be confused with the STM
I did in my thesis measurements) to determine their suitability for the capacitance or Kelvin
probe experiment. For previous discussion about tips see Chapter 2.3.4.

6.4.0.1

Etched

Prior members of the group had repeated success with etched tips. Etched tips are made
of Pt(.8)Ir(.2) wire that is electrochemically etched to a fine point and then rinsed with
detergent to get rid of residual chemicals. We purchased etched tips from Agilent. After
reviewing thirteen etched tips with a scanning electron microscope, I determined that it was
unlikely that I would find one that had a 20 nm radius of curvature. Most tip’s radius of
92

(a)

(b)

Figure 6.8: SEM image of an etched tip made by Agilent with a tip radius of curvature of
100 nm. Both (a) and (b) are the same tip with different scales.

curvature was well over 100 nm as seen in Figure 6.8. Many tips visually had white debris
even around the apex, however, surprisingly, they consistently produced stable tunneling
images.

6.4.0.2

Cut

Cut tips produced the most variation of all the types of tips I examined. While all were cut
from the same Pt(.8)Ir(.2) wire, some were extremely blunt and others had a large aspect
ratio. Many tips could be determined unsuitable by visual inspection. Over one hundred cut
tips were examined with SEM and about 40% had a radius of curvature of 25 nm or more.
An additional 20% were used for STM measurements, where resolution and tip shape are
less directly related. In STM, for example, a step edge will be better resolved with a sharp
tip, whereas a blunt tip might smear the edge out. Figure 6.9 shows tips with a 20 nm or
less radius of curvature.
93

(a)

(b)

Figure 6.9: Cut tips with a radius of curvature of 10 nm and 25 nm.

6.4.0.3

Ultrasharp

With tip size being extremely important for capacitance or Kelvin probe measurements, I
wanted a way to make tips with a radius of curvature 20 nm or less reliably. It needed to be
something that was inexpensive compared to tips already on the market. Furthermore, the
tips needed to be able to be fabricated with apparatus already existing within our laboratory
or for common use in the Keck Microfabrication facility (which excluded fabricating our own
nanotube tips). Because undergraduates were going to ultimately develop, optimize and
execute the ultrasharp tip fabrication, no multi-step technique requiring harsh chemicals
was appropriate.
Ultrasharp tips were therefore fabricated from ultrasharp silicon nitride AFM tips attached to a PtIr wire with conducting epoxy, a technique developed by Mark Regalla and
perfected by both Andrew Belcher and Eric Goodwin. To make the tips conducting, a 2nm
layer of gold was evaporated onto the apex of the tip. Each tip was examined with SEM
to determine the radius of curvature and ensure that the apex of the repurposed AFM tip
was higher than the epoxy. We did not want to tunnel though epoxy because it would
94

(a)

(b)

Figure 6.10: (a) The micropositioner used to assemble the ultrasharp tips. It finely positions
a Pt(.8)Ir(.2) wire with respect to a AFM cantilever. (b) A view through the microscope
used to see tip placement on the epoxy. The triangular cantilever is touching the conducting
epoxy.
cause instabilities and have a larger radius of curvature than the AFM tip. Many tips ended
up not having cylindrical symmetry, which made placing them on the HEMT circuit more
challenging. Of the tips that were not engulfed by epoxy, many had a radius of curvature
of 20-25 nm and there were some that was just over 10nm. Ultimately, the ultrasharp tip
fabrication cost did not justify continuing to make them.

95

(a)

(b)

Figure 6.11: Ultrasharp tip with radius of curvature of 10 nm. (a) All components of the
tip are visible: the cantilever, conducting epoxy, and Pt(.8)Ir(.2) wire. This tip has nice
cylindrical symmetry. (b) The apex of the tip is the highest point.

96

Chapter 7

Conclusions

7.1

Summary

Even with the Bi2 Se3 topological simplicity, there are other effects that coexist in the surface
layers that may have an impact on future experiments. My measurements show bulk-like
wavefunctions overlap with the topologically protected Dirac cone on the surface. The interplay between the shoulder states and the Dirac cone states presents potential competing
channels for scattering to take place. As the top quintuple layer separates from the crystal,
a two-dimensional electron gas can form on the surface and likewise overlap with the Dirac
cone. Menshcikova et al. showed that a 2DEG can push the topological state into the second
highest quintuple layer, which might reduce the amplitude of the Dirac cone that overlaps
with the surface, perhaps complicating applications [108].
97

Figure 7.1: Schematic of the proposed experiment. An STM tip will probe vortices in an
superconducting antidot patterned on a topological insulator to search for Majorana modes
[51]. Figure from Dr. Stuart Tessmer.

98

7.2

Future Directions

The first experiment planned by my successor, Matthias Muenks, will probe small superconductor islands on a large Bi2 Se3 crystal to explore the superconducting/topological insulator
interface. When a superconductor is interfaced with a topological insulator many exotic effects are predicted, the most exciting of which is the Majorana fermion [9]. The Majorana
fermion in this context is a particle that is its own hole. Because inversion symmetry is broken at the surface of a topological insulator, putting a topological insulator in close proximity
to a superconductor induces unconventional p-wave order [9, 116].
The proximity effect occurs when a normal metal is interfaced with a superconductor
[117]. Cooper pairs leak into the normal metal leaving a reduction of Cooper pairs in the
superconductor. This effect should also be present when interfacing with topological insulator, but at present nobody has successfully obtained local density of states measurements to
confirm it. Now that I have characterized the surface of Bi2 Se3 , the next focus in the group
will be to explore these effects. The plan for the first measurements is to evaporate a superconducting material like Nb, Pb or PbBi until it forms random superconducting islands on
the surface of the Bi2 Se3 crystals to study the superconducting local proximity effect in this
hybrid system. Once these experiments are successful, future experiments are planned which
require the superconductor to be patterned on the topological insulator, like that in Figure
7.1. By probing the holes in these patterns, Matthias Muenks hopes to find the predicted
Majorana Fermion [9, 118, 119].

99

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100

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