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Discoveries of New Topological States
of Matter Beyond Topological
Insulators
Su-Yang Xu
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Physics
Advisor: M. Zahid Hasan
January 2015
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c Copyright by Su-Yang Xu, 2014.All Rights Reserved
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Abstract
The discoveries of new forms of matter have been so definitive that they are used to
name periods of the history of mankind, such as the stone age, the bronze age, and
the iron age. Although all matter is composed of component particles, particles can
organize in various ways leading to different states or phases of matter. Finding all
possible distinct states that matter can form and understanding the physics behind
each of them are fundamentally important goals in physics research. Meanwhile, the
discoveries of new phases, such as superconductivity, ferromagnetism, also usually
lead to new technologies, which benefit our society. It had been long believed in
physics that all phases of matter (solid, liquid, gas, superconductivity, etc) are due
to the breaking of certain symmetries and can be described by local order parame-
ters. However, the discovery of the topological phase of matter provided a striking
exception that cannot be described by a local order parameter. A topological phase
fundamentally differs from a conventional material in that a topological phase of mat-
ter is described by a nontrivial topological number, which is a global quantity based
on the bulk electronic wavefunction throughout the momentum-space Brillouin zoneof a crystal. The nontrivial topological number guarantees the existence of protected
surface states, which are of interest because these surface states are insensitive to
smooth changes in material parameters and cannot be removed unless the system
passes through a topological quantum phase transition. In 2007, the 3D Z 2 topo-
logical insulator (TI) state was identified in bismuth-based materials. A 3D TI is a
band insulator in the bulk but features spin-polarized Dirac electronic states on its
surfaces. This discovery has tremendously accelerated the research field of topologi-
cal matter. Not only the 3D Z 2 TI phase itself has attracted huge research interest,
but also, more importantly, it has inspired theoretical predictions for a range of new
topological phases of matter. The primary examples include a topological crystalline
insulator, a topological Dirac/Weyl semimetal, a topological Kondo insulator, a topo-
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logical superconductor, etc. Each of them is predicted to exhibit protected surface
states with a unique set of topologically protected properties. Experimentally identi-
fying these new phases in real materials can tremendously deepen our knowledge of
the physics of topological matter. Moreover, utilization of the protected properties in
topological surface states can lead to future applications such as fault-tolerant topo-
logical quantum computers and low-power spintronic devices, which will revolutionize
our electronic and energy industry.
In this thesis, we present the experimental identifications of several new topolog-
ical phases of matter beyond the Z 2 TI. In Chapter 1, we present the key concepts
for topological phases and the relevant theoretical formulations. In Chapter 2, we
provide a description of the experimental technique of spin-resolved angle-resolved
photoemission spectroscopy (spin-ARPES) and the methodology of using it to probe
the topological number of a 3D topological phase. In Chapter 3, we present our exper-
imental realization of a topological quantum phase transition in the BiTl(S1−δSeδ)2
system, which demonstrates the way a topological phase of matter can arise from an
unremarkable band insulator. In Chapter 4, we demonstrate the observation of the
topological crystalline insulator phase in the Pb1−xSnxTe(Se) systems, where novel
topological surface states protected by the crystalline mirror symmetries and the as-
sociated mirror topological number are experimentally observed. In Chapter 5, we
describe the realization of a topological (Dirac) metal phase in Na3Bi and Cd3As2,
where certain topological state can be stabilized even if the bulk energy gap closes.
We report the observation of Fermi arc surface states, - a new type of 2D electron
gas, revealing the novel topological nature in a topological metal. In Chapter 6,
we present our studies on the topological Kondo insulator candidate SmB6, where
the bulk Kondo gap replaces a simple Bloch gap in a Z2 TI, potentially introducing
electronic correlation to the surface states. Finally, In Chapter 7, we present our
demonstration of superconductivity and magnetism in the topological surface states
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in a prototype Z 2 TI Bi2Se3, which are the keys to realizing a topological supercon-
ductor or a Chern insulator. The works in this thesis constitute the realization of
a number of new topological phases of matter beyond a Z 2 TI state that are being
studied by many groups worldwide. With these discoveries, the world of topologi-
cal matter gradually unveil itself. And we have many reasons to believe that this is
just the tip of the iceberg and there are many more of them with yet more exciting
properties that await their discovery.
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Acknowledgements
First and foremost, I would like to thank my advisor Prof. M. Zahid Hasan for
his mentorship and support. He has been a constant source of encouragement and
advice throughout my graduate studies. I always remember his words: “The idea
is to keep trying. Everyone will face failure. But if we keep trying, on average we
will do fine.” Every time I run into difficulties in my experiments or in fighting with
the reviewers to get a paper published, these words encouraged me. Indeed, now I
understand that there is no sure path to success, and the key is to “keep trying”. I
also benefited tremendously from his ability to understand and explain the essences
of physics without bringing many formations, and his insights in condensed matter
physics, especially when it came to looking out for the next exciting topics in our
field. This is one the main driving forces to maintain our group in the front line in
the fierce competition. Moreover, I am grateful for the freedom he gave me and also
every other member in our group to pursue our individual interests in our research
work. He allowed us to come up with our own research projects, design our own
experiments and consult experts in relevant areas such as ARPES technique, material
growth and condensed matter theory. I sincerely thank Zahid for every single aspect
of my graduate research.
I am also indebted to my colleagues and friends in my group. I wish to thank David
Hsieh, Dong Qian, and Matthew Xia for introducing me into the group and helping
me settle down. Andrew Wray taught me how to use ARPES and was extremely
generous as a teacher when I started to work on ARPES on topological insulators.
Madhab Neupane, Chang Liu and Nasser Alidoust have been wonderful teammates
in experiments, and Madhab and Chang have also been great friends in my personal
life. I thank Ilya Belopolski for his infinite curiosity when it came to new physics and
for always being unsatisfied with the answers that I provided to his questions. This
drove me crazy but lead to many new and exciting ideas. I thank Guang for being
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were always ready to help and sent the calculation results on short notices. The
timely response from our sample growth and theoretical calculation collaborators
was crucially important in the competitive research.
I have also been very fortunate to be able to learn from world-renowned scientists
at Princeton Physics. I thank Chen Fang for all the discussions we had in his office, at
lunch or dinner, while we were walking from the parking lot to the department or at
so many random places and random times where we suddenly had some inspiration.
Chen almost single-handedly taught me the physics of topological phases beyond Z 2
topological insulators. The discussions with him led to many fruitful ideas and new
results. I thank Prof. Phuan Ong for agreeing to be second reader on my thesis and
for his kind advice. And I thank Prof. Jason Petta and members of his group from
whom I learned a great deal during my experimental project.
Only people who have experienced the “synchrotron life” would understand the
meaning of “constant night shifts”. I cannot imagine myself going through all the
physical and mental hard times without my friends. I thank my Princeton friends
Jingke Xu, Bo Yang, Jun Xiong, Ke Wang, Zhizhen Zhao, and many others. I also
thank my friends back in China. Especially I want to thank Hei (Liang Zhang) for
being an honest, generous, reliable friend, for bearing my complaints, for comforting
my pains, and for sharing my happiness.
I thank my dear sister Chujun for being connected heart to heart. Wherever she
is and however the world changes, I wish her happiness in her life. I thank Cindy
for her wonderful cooking during “the SF time”. I thank Jingjing and Tian for their
accompany and education. I thank my angel girl momo for her warm care, for her
positive thinking and encouragement, and for always sticking with me to share my
thoughts and feelings.
Finally but most importantly, I am extremely grateful to my family, my grand-
parents Ruizhi and Ruilan, my mom Xiaobei, my dad Yuping, my aunt Hong, and
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all family members for all their selfless love and care. Family is my forever source of
happiness.
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Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Introduction 1
1.1 The band insulator state and the band inversion . . . . . . . . . . . . 1
1.2 Previously discovered topological phases. . . . . . . . . . . . . . . . . 5
1.2.1 The integer quantum Hall state and the Chern insulator state 5
1.2.2 The 2D Z 2 topological insulator - the quantum spin Hall insulator 7
1.2.3 The 3D Z 2 topological insulator . . . . . . . . . . . . . . . . . 11
1.3 Theoretical formulations . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.1 The Chern number . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3.2 The Z 2 topological invariant in two-dimensions . . . . . . . . 22
1.3.3 The Z 2 topological invariants in three-dimensions . . . . . . . 23
1.4 Inspiration for discovering new topological phases . . . . . . . . . . . 24
1.4.1 Band inversions and spin-orbit interactions . . . . . . . . . . . 24
1.4.2 Constructing new topological phases . . . . . . . . . . . . . . 27
2 Experimental techniques 31
2.1 Spin-integrated Angle-resolved photoemission spectroscopy . . . . . . 31
2.2 Spin-Resolved Angle-resolved photoemission spectroscopy . . . . . . . 34
2.2.1 Mott polarimetry . . . . . . . . . . . . . . . . . . . . . . . . . 36
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CONTENTS
2.2.2 VLEED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 Sample preparation for ARPES . . . . . . . . . . . . . . . . . . . . . 42
2.3.1 Single crystal bulk samples . . . . . . . . . . . . . . . . . . . . 42
2.3.2 MBE film samples . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4 Probing the topological number in 3D bulk materials . . . . . . . . . 45
2.4.1 Separation of insulating bulk from metallic surface states using
incident photon energy modulated ARPES . . . . . . . . . . . 45
2.4.2 Surface Dirac crossing, surface-bulk connectivity, and spin-momentum
locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Topological Quantum Phase Transition in BiTl(S1−δSeδ)2 52
3.1 Evolution of the electronic groudstate across a topological quantum
phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Topological quantum phase transition in BiTl(S1−δSeδ)2 . . . . . . . . 56
3.2.1 Identifying a suitable topological material system . . . . . . . 56
3.2.2 Topological quantum phase transition in BiTl(S1−δSeδ)2 . . . . 58
3.2.3 Spin-orbit coupling vs. lattice constant . . . . . . . . . . . . . 64
3.2.4 3D Dirac semimetal state at the critical point . . . . . . . . . 66
3.3 Topological-critical-point and the “preformed” surface states . . . . . 68
3.4 Topological phase transition and “preformed” critical behavior in (Bi1−δInδ)2Se3 82
4 Topological Crystalline Insulator Phase in Pb1−δSnδTe(Se) 84
4.1 Key theoretical concepts for a topological crystalline insulator state . 85
4.2 Discovery of mirror symmetry protected TCI state in Pb1−δSnδTe . . 89
4.2.1 Band inversions and mirror symmetries in Pb1−δSnδTe . . . . 89
4.2.2 Topological surface states in Pb0.6Sn0.4Te . . . . . . . . . . . . 91
4.2.3 Mirror Chern number and mirror symmetry protection . . . . 98
4.3 Topological surface states in PbδSn1−δSe . . . . . . . . . . . . . . . . 103
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CONTENTS
4.3.1 Lifshitz transition and saddle point singularities . . . . . . . . 103
4.3.2 Temperature-driven topological phase transition . . . . . . . . 108
4.3.3 Topological phase diagram in Pb1−δSnδSe . . . . . . . . . . . . 111
5 Topological Dirac semimetal state in Cd3As2 and Na3Bi 114
5.1 Theoretical concepts for a topological Dirac semimetal . . . . . . . . 116
5.2 3D Dirac semimetal state in high mobility Cd3As2 . . . . . . . . . . . 121
5.3 Fermi arc surface states in topological Dirac semimetal Na3Bi . . . . 132
5.3.1 Choice of the surface termination to observe FASS . . . . . . . 132
5.3.2 Observation of Fermi arc surface states in Na3Bi . . . . . . . . 135
5.3.3 Topological invariant for the Dirac semimetal Na3Bi . . . . . . 141
5.4 Topological semimetals: Dirac, Weyl, and nodal-line . . . . . . . . . . 150
6 Topological states in 4f Kondo systems SmB6 and YbB6 156
6.1 Observation of surface states in topological Kondo insulator candidate
SmB6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.2 Topological surface states in YbB6 . . . . . . . . . . . . . . . . . . . 166
7 Superconductivity and magnetism in topological insulators 174
7.1 Hedgehog spin texture in a magnetic topological insulator . . . . . . 175
7.2 Helical Cooper pairing in topological insulator/superconductor het-
erostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Appendices 231
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Figure 1.1: The band insulator state and the band inversion . (a) The crystalstructure of a solid argon lattice. (b) The energy levels of a solid argon in the atomiclimit where the lattice constant goes to infinity resulting in a number of unbound argonatoms. (c) As the lattice constant goes to the experimental value, the Ar3 p and Ar4sorbitals bceome Bloch bands where they gain energy dispersion as a function of the Blochmomentum k. (d) First-principles calculation of the electronic band structure of a solidargon, where the Ar3 p valence band, Ar4s conduction band, and the normal band-gap areseen. (e-h) The same as Panels (a-d) but for a topologically nontrivial band insulatorBi2Se3. In this case, the relative energy positions of the conduction and valence bands are
reversed with respect to their atomic limit (indicated by the blue and red color code inPanels (f,g). This results in a topologically nontrivial state in Bi2Se3. In Panel (h), asingle Dirac cone surface state (red line) enclosing the Γ̄ within the bulk energy gap is seen.
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1.1 THE BAND INSULATOR STATE AND THE BAND INVERSION
the electronic structure where the relative energy positions between the conduction
and valence bands in a solid are reversed with respect to those of in its atomic limit.
Therefore, within band theory, any topologically nontrivial state should have a finite
number of band inversions, because the ones that do not can be adiabatically tuned
into its atomic limit without closing the band-gap. Now we can understand both a
conventional band insulator and a topologically nontrivial insulator under the same
band inversion picture: Let us take an insulator and increase the lattice constant to
infinity. This will result in a number of unbound atoms or ions. We compare the
energy levels for all the filled and unfilled orbitals, and label the last filled orbital as
“V” and the first unfilled orbital as “C”, since these two will form the conduction
and valence bands in the actual compound. Now we gradually change the lattice
constant from infinity to the experimental value, each atomic level will form a Bloch
band and gain dispersion as a function of Bloch momentum k. For a conventional
band insulator, the energy of the valence band “V” will always remain lower than
that of the conduction band “C” at all k points throughout the Brillouin zone (BZ).
Therefore, a conventional band insulator is topologically trivial. In contrast, for a
topological insulator, the energy of “V” becomes higher than that of “C” at some
k points. In this case, in order to restore the relative energy order between “V”
and “C” as in the atomic limit, one has to close the bulk band-gap, which involves
a quantum phase transition that demonstrates the topological inequality between a
topological insulator and the vacuum. We note that band inversion is a necessary
but not a sufficient condition for topological phases. After all, a topological phase is
defined by a topological number - a global quantity calculated using the electronic
wavefunction throughout the BZ. Thus even if a system has nonzero band inversions,
one still needs to evaluate the system’s topological number to confirm its topological
nontriviality. Although band inversion cannot rigorously define any topological phase,
we found that it is in fact very suggestive in terms of identifying real compounds for
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new topological states and understanding the topological nature in various topological
materials, which are the main goals of the experiments in this thesis. For example in
numerical band structure calculations, which serves as a useful guide for experiments,
it is usually much easier to identify the band inversion in a material (by computing
the bulk band structure at different lattice constant) than calculating the actual
topological number (e.g. the Chern number involves an integral of the bulk electronic
wavefunction throughout the BZ, which is usually impractical to compute). Similarly,
experimentally driving a band inversion (by applying pressure, varying temperature
or chemical substitution) can drive a conventional material into a topological state
or transform an understood topological phase to an unknown and more exotic one by
going through a topological quantum phase transition. Therefore, it is in this context
that we highlight the importance of band inversion here.
1.2 Previously discovered topological phases.
1.2.1 The integer quantum Hall state and the Chern insula-
tor state
The experimental discovery of the 2D integer quantum Hall (IQH) state [19] in 1980
marks the first realization of a topological phase of matter. The IQH effect is a
quantum version of the Hall effect, which is achieved by applying strong magnetic
field in two-dimensional electron systems at low temperatures. In band theory, the
quantization of the electrons’ circular orbits leads to quantized Landau levels. The
IQH state is an insulator in the bulk because the Fermi level is located in the middle
of two Landau levels. On the other hand, the edges of the IQH state feature chiral
1D metallic states, leading to remarkable quantized charge transport phenomena
(σxy = ne2/h) [Fig. 1.2(a)]. It turns out that the key difference between an IQH state
and a 2D conventional band insulator is a matter of topology [20–22]. The quantized
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1.2 PREVIOUSLY DISCOVERED TOPOLOGICAL PHASES.
Figure 1.2: The interface between a quantum Hall state and an insulator haschiral edge mode. (a) The skipping cyclotron orbits. (b) The electronic structure of asemi-infinite edge described in a Chern insulator with a Chern number of n = +1. A singleedge state connects the valence band to the conduction band.
number n in the transverse magneto-conductivity σxy = ne2/h is a topological number
(the Chern number) that characterizes the nontrivial topology of the IQH state.
Two years after the discovery of the IQH effect, the fractional quantum Hall (FQH)
effect was observed [23]. The fractional quantum Hall effect (FQHE) is a physical
phenomenon in which the Hall conductance of 2D electrons shows precisely quantized
plateaus at fractional values of e2/h. Unlike the IQH effect, the FQH effect relies
fundamentally on electron-electron interactions and therefore cannot be understood
in the frame of band theory. It is interesting to note that the FQH effect is not only
the second topological state realized in experiments, but also it is believed to have
topological order [24]. We will discuss the difference between the topological order
and the symmetry-protected topological state later in this section.
Let us get back to the IQH state. Since the generators of translations do not
commute with one another in a magnetic field, electronic states cannot be labeled
with momentum k. Can we construct a quantum Hall like state with a nonzero
Chern number but without an external magnetic field so that the system can be
understood by the Bloch band theory? The answer is the Chern insulator. The first
theoretical model example is the Haldane model [25], where a magnetic field that is
zero on the average, but has all of the spatial symmetries of the honeycomb lattice, is
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1.2.2 The 2D Z 2 topological insulator - the quantum spin Hall insulator
applied to graphene. And Haldane showed that although the average magnetic field is
zero, it opens up a gap at the Dirac nodes of graphene, and furthermore this gapped
state is not an ordinary insulator, but rather has a quantized Hall conductivity (a
nonzero Chern number). More recently, the Chern insulator state is also theoretically
predicted in the magnetically doped thin film of a 3D Z 2 topological insulator [9],
where no external magnetic field is required and the time-reversal symmetry breaking
is provided by the magnetization of the sample itself. We note that such type of Chern
insulator is also referred as a quantum anomalous Hall insulator, and it was recently
experimentally realized in Cr-doped (Bi1−xSbx)2Te3 thin films [9]. One important
consequence of the Chern insulator is that now we can understand it in a band
theory picture again. As shown in Fig. 1.2(b), a Chern insulator can be understood
as a single edge state that connects the valence band to the conduction band.
1.2.2 The 2D Z 2 topological insulator - the quantum spin
Hall insulator
Although a Chern insulator does not require external magnetic field, its internal
magnetization still breaks time-reversal symmetry. Can we construct a topological
phase that respects time-reversal symmetry? This is particularly motivated by the
fact that there are many more time-reversal invariant materials than the time-reversal
breaking ones in nature. Such motivation leads to the theoretical prediction of the
third type of topological insulator - the quantum spin Hall (QSH) insulator [26,27].
In a QSH insulator, the bulk is insulating and features a pair of counter-propagating
edgestates, which are related by time-reversal symmetry. And the whole system,
unlike a Chern insulator, respects time-reversal symmetry. We provide a physical
picture to build a quantum spin Hall insulator from two Chern insulators: Let us take
a Chern insulator with a Chern number of n = +1, which is achieved by an out-of-
plane magnetization +M [Fig. 1.3(a,b)]. Under time-reversal operation T , quantities
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1.2 PREVIOUSLY DISCOVERED TOPOLOGICAL PHASES.
Figure 1.3: A quantum spin Hall insulator can be constructed by two Cherninsulators that are related by the time-reversal symmetry. (a) A Chern insulatorwith a Chern number of n = +1 has a chiral edge mode as a result of an out-of-planemagnetization. (b) The bottom edge projection of the electronic structure the n = +1Chern insulator. (c,d,) Same as Panels (a,b) but with a time-reversal operation onto thesystem. (e,f,) A quantum spin Hall state can be obtained by combining the above-two
Chern insulator states. The resulting electronic structure on the edges is described by twocouterpropagating edge modes that cross each other at k = 0 (or k = π). The crossing isprotected by the time-reversal symmetry as a result of the Kramers theorem.
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1.2.2 The 2D Z 2 topological insulator - the quantum spin Hall insulator
including momentum k, angular momentum (including spin σ), magnitization M will
change sign. The resulting system after time-reversal transformation is shown in
Fig. 1.3(c,d), where a Chern insulator with a Chern number of n = −1 is obtained.
Now if one imagines combining these two Chern insulators into a single system, the
total magnetization goes to zero, and the system is invariant under time-reversal
operation. On the edges of this new system, there are two edge states that are
counter-propagating. We draw the electronic structure that is projected to the bottom
edge. Since the two Chern insulators are symmetric over time-reversal operation, in
the combined system, the two edge modes must “meet” each other at k = 0. In
general, as two states cross, they will hybridize and open up a gap. However, in this
case, these two edge states that have opposite quantum numbers (Left moving: +k
and ↑; Right moving: −k and ↓;) are directly linked by the time-reversal operationbecause they cross each other at k = 0, a time-reversal invariant momentum (TRIM)
(also referred as a Kramers point) [Fig. 1.3(e,f)]. A Kramers point kKramers is defined
as kKramers = G − kKramers, where G is any reciprocal lattice vector. The Kramerstheorem states that the electronic states have to remain doubly degenerate at the
Kramers points in a time-reversal invariant system. And therefore the edge state
crossing at k = 0 is protected by the time-reversal symmetry, giving rise to the
quantum spin Hall phase. From the above pictures, it is evident that the quantum
spin Hall phase is topologically distinct from the vacuum. This is because it is not
possible to remove the metallic edge states from the band-gap as long as the edge
band-crossing remains intact. The topological nontriviality indicates that there exist
a finite number of bulk band inversions at the Kramers points of the BZ. In a real
QSH system, no magnetization is present unlike in our conceptual picture. Thus
one has to identify a physical interaction that can lead to an inverted band-gap in a
time-reversal invariant condition. It turns out that the spin-orbit interaction in heavy
elements plays such a critical role. On the other hand, since spin-orbit interaction
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1.2 PREVIOUSLY DISCOVERED TOPOLOGICAL PHASES.
mixes the spin and orbital degrees of freedom, the physical spin is not a quantum
number. Thus it is more correct to think of the “spin” in the quantum spin Hall state
in strongly spin-orbit coupled systems as the pseudo-spin rather than the real spin.
In 2005, theoretical advances realized that the quantum spin Hall phase is indeed
topologically nontrivial [26]. In fact, it is characterized by a new topological number,
the Z 2 invariant (ν ) [14, 26, 29]. The Z 2 invariant can only take two values, 0 or
1, where ν = 0 (1) is topologically trivial (nontrivial). In 2007, the QSH phase was
experimentally demonstrated in the (Hg,Cd)Te quantum wells using charge transport
by measuring a longitudinal conductance of about 2e2/h (two copies of quantum Hall
currents) at mK temperatures [28]. However, no spin polarization was measured in
this experiment thus spin momentum locking, which is essential for the Z 2 topological
physics, was not experimentally observed [28].
It is important to note that the 2D topological (IQH, FQH, and QSH) insulators
have only been realized at buried interfaces of ultraclean semiconductor heterostruc-
tures at very low temperatures to date. Furthermore, their metallic edge states can
only be probed by the charge transport method. These facts hinder the systematic
studies of many of their important properties, such as their electronic structure, spin
polarization texture, tunneling properties, optical properties, as well as their responses
under heterostructuring or interfacing with broken symmetry states. For example,
the two counter-propagating edge modes in a QSH insulator are predicted to feature
a 1D (usually Dirac) band crossing in energy and momentum space. And edge mode
moving along the +k direction is expected to carry the opposite spin polarization
as compared to that of one moving in the −k direction. However, neither the Diracband crossing nor the spin-momentum locking of the edge modes in a QSH insulator
are experimentally observed, due to the lack of experimental probe that can measure
these key properties for a 1D edge mode at a buried interface at mK temperatures.
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1.2.3 The 3D Z 2 topological insulator
1.2.3 The 3D Z 2 topological insulator
In 2007, it was theoretically realized that the Z 2 topological number in a QSH insula-
tor can be generalized to three-dimensions, thus realizing the first three-dimensional
topological phase of matter [14–18]. This is again a critical breakthrough in experi-
mental and material physics because there are more 3D bulk materials than 2D films
in nature and the preparation of bulk materials is usually easier. How do we con-
struct a 3D Z 2 TI? The most straightforward approach is to simply stack a number
of uncoupled 2D quantum spin Hall states to form a 3D bulk. It turns out that such
simple stacking of the 2D QSH states does lead to a type of 3D Z 2 TI, which is the so
called 3D weak TI. A weak TI is not the most topologically protected Z 2 nontrivial
states in bulk materials. But let us start from a weak TI.
As shown in Fig. 1.4, let us assume that we start from a 2D QSH state that has an
inverted band-gap at the Γ point. The blue and green colors in Fig. 1.4(a) show the
orbital nature of the bulk bands near the Γ point, where a band inversion is evident.
Now let us have N copies of such QSH state and stack them along the out-of-plane
ẑ direction. This will form a bulk material. And in momentum space, it means that
the 2D BZ will gain periodicity in the kz direction and becomes a cube representing
the 3D BZ [Fig. 1.4(b)]. In the 3D BZ, the Kramers points can be sorted by their
origins from the 2D BZ before stacking. For example, both the Γ and the Z points
in the 3D BZ results from the Γ point in the 2D BZ. We assume that the physical
coupling between these 2D QSH slices are extremely weak (e.g. in real materials,
imagine they are coupled by a very weak van der Waals interaction). This means
that the energy dispersion along the kz direction is sufficiently weak, so that it does
not change the orbital nature of the conduction and valence bands by going from the
Γ(3D) point to the Z point. Therefore, the one band inversion in the 2D Γ point in
the QSH slice becomes two band inversions at both the Γ and the Z points in the 3D
BZ after stacking (here only the band inversions at the Kramers points are counted).
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1.2 PREVIOUSLY DISCOVERED TOPOLOGICAL PHASES.
Figure 1.4: A 3D weak topological insulator can be obtained by stacking N copiesof identical 2D quantum spin Hall states along the out-of-plane direction. (a) A2D quantum spin Hall state where there is a band inversion at the 2D BZ center Γ point.(b) stacking N copies of such quantum spin Hall slices along the out-of-plane direction form
a bulk material. In momentum space, it means that the 2D BZ will gain periodicity in the kzdirection and becomes a cube representing the 3D BZ. The time-reversal invariant momentain the 3D BZ are noted. (c) Before turning on the inter-slice coupling, different QSH slicesare completely identical and idenpendent (e.g. Cut2 and Cut3). This also means that thereis no energy dispersion along the out-of-plane Γ − Z direction (Cut1) (d) A finite (butweak) inter-slice coupling means that the states in the QSH system gain energy dispersionalong the out-of-plane kz direction. The inter-slice coupling is weak enough, so that it doesnot change the orbital nature of the conduction and valence bands by going from the Γ(3D)point to the Z point (Cut1). Therefore, there are two band inversions at both the Γ andthe Z points in the 3D BZ (Cuts 2 and 3).
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1.2.3 The 3D Z 2 topological insulator
Figure 1.5: The surface of a weak TI shows even number of protected surfacestates or no surface states depending on the surface termination. (a) A bulk BZof the weak TI. The green slices are chosen for detailed studies. (b) The edge projectedelectronic structure for the green slices shown in Panel (a). Without any inter-slice cou-pling, these slices are identical and independent. The surface states for the kx − kz surfaceprojection is a straight line. (c) A bulk BZ showing the two band inversions at the Γ andthe Z points. (d) After turning on the inter-slice coupling, the edge states can hybridize andopen out gaps, except at the Γ and the Z points due to the protection of the time-reversalsymmetry. Therefore, the surface state Fermi surface is described by two dots at the Γ and
the Z points.
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What about the edge states of each QSH slice? Let us look at a few representative
QSH slices at different kz values as shown by the green 2D slices in Fig. 1.5(a). As
seen in Fig. 1.5(b), before we turn on the coupling between these slices, each slice
is a QSH that features two counterpropagating edge modes at its edge. We fix the
chemical potential at the crossing point of the edge modes and investigate the kx −kzplane where all the QSH edges are stacked to form a surface. Thus all the 1D edge
states from the QSH slices are stacked and become a 2D surface state. The surface
state Fermi surface is a straight line that goes from Z − Γ − Z . Now we turn on theinter-slice coupling (hybridization). As we have mentioned above, in general if two
edge states cross, they will open up an energy gap. However, at and only at the Γ and
the Z points, the edge state crossings are preserved because these two points are the
Kramers’ points. Consequently, the surface state Fermi surface of the kx − kz planebecomes two dots at the Γ and the Z points. And if the E F is shifted slightly away
from the crossings, then the two Fermi dots will evolve into two circles that enclose
the Γ and the Z points, respectively. Therefore, the surface states at the kx −kz planeare described by two surface states that enclose the Γ and the Z points, respectively.
In contrast, at the top kx − ky surface, no protected surface state is expected. Thisis because both Γ and Z points project onto the Γ̄ point of the top kx − ky surface.The two surface states that originate from the band inversions at Γ and Z points
can hybridize and open up a gap. Therefore, it can be seen that a weak TI features
protected 2D surface states only at certain crystalline surface terminations.
In fact, from the topology point of view, the 3D weak TI state is essentially
equivalent to a number of independent 2D quantum spin Hall slices, because there
involves no band inversion by decomposing a weak TI into uncoupled 2D QSH slices.
Can we construct a 3D Z 2 TI state that cannot be adiabatically reduced into a bunch
of stacked QSH states? Also, from the surface state point of view, a 3D weak TI
only has protected surface states at certain surface terminations. Is there a 3D Z 2 TI
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1.2.3 The 3D Z 2 topological insulator
state that shows protected surface states at all surfaces irrespective of the choice of
the termination? The answers for both questions lead to the 3D strong TI phase.
Let us again start by stacking QSH slices (Fig. 1.6). It is obvious that a weak
inter-slice coupling only leads to the weak TI phase. Now we examine the strong
coupling scenario. We know that at the 2D Γ point the conduction and valence
bands are inverted with respect to the atomic limit. Without interlayer coupling, it
means that there exist two band inversions at both the Γ and the Z points of the
3D BZ. Now we turn on the inter-slice coupling. And in this case, the inter-slice
coupling is strong enough and affects the electronic structure in a way so that the
band inversion at the Z point is in fact removed [Cut1 in Fig. 1.6(d)]. As shown
in the Cuts 2 and 3 in Fig. 1.6(d), while at the Γ point (Cut2) the conduction and
valence bands are still inverted with respect to the atomic limit, at the Z point (Cut3)
these two bands restore the relative energy positions as in the atomic limit (no band
inversion). Therefore, in this case, there is only one band inversion at the Γ point
throughout the 3D BZ. This is an example of a 3D strong TI state. It is important to
note that the 3D strong TI state is topologically distinct from both the weak TI and
the conventional band insulator (the atomic limit) states. This is because (1) it is not
possible to reduce the strong TI to uncoupled 2D QSH slices without going through
a band inversion at the Z point. Therefore a strong TI is topologically inequivalent
to a weak TI. (2) it is also not possible to change a strong TI to a conventional band
insulator without going through a band inversion at the Γ point.
We now investigate the possible existence of protected surface states in the strong
TI state (Fig. 1.7). We note that there is only one band inversion at the Γ point for
the strong TI state, and that at any surface termination the Γ point projects onto
the surface BZ center Γ̄ point. Therefore, one expects a single surface state enclosing
the surface BZ center Γ̄ point at any surface termination. We also note that the case
presented here in Figs. 1.6 and 1.7 is only one of the possible ways to generate a strong
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Figure 1.6: A 3D strong topological insulator can be obtained by coupling N copies of 2D quantum spin Hall states along the out-of-plane direction in a non-trivial way. (a) A 2D quantum spin Hall state where there is a band inversion at the2D BZ center Γ point. (b) The 3D BZ. The time-reversal invariant momenta in the 3DBZ are noted. (c) Before turning on the inter-slice coupling, there is no energy dispersionalong the out-of-plane Γ − Z direction (Cut1). (d) A strong inter-slice coupling can drivea band inversion along the out-of-plane Γ − Z direction (Cut1). In this case, while at the Γpoint (Cut2) the conduction and valence bands are still inverted with respect to the atomiclimit, at the Z point (Cut3) these two bands restore the atomic limit (no band inversion).Therefore, there is only one band inversion at the Γ point of the 3D BZ.
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1.2.3 The 3D Z 2 topological insulator
TI. In general, a strong TI should always possess an odd number of band inversions
at the Kramers’ points in a 3D BZ, but the exact number of band inversions (1, 3, 5,
or other odd numbers) depends on the exact form of the inter-slice coupling as well
as the properties of the 2D QSH state to start with (e.g. one can also imagine to have
a QSH state with a band inversion at the 2D BZ corner M point not the Γ point).
However, regardless of these details, a strong TI will always have an odd number of
band inversions in the bulk and feature an odd number of surface states enclosing the
Kramers points irrespective of the surface termination. It is also important to note
that the surface state of 3D TIs are usually referred as “Dirac cone” surface states
in many research works. “Dirac cone” means that the surface states cross at the
Kramers point with a linear dispersion. This is, however, not always true. In fact,
the Kramers theorem only requires the surface states to remain doubly degenerate at
a Kramers point, or, in other words, the surface states have to cross each other at the
Kramers point. But it does not require a specific fashion of the crossing in terms of
the energy dispersion of the surface states. It is actually also possible to have surface
states with cubic or even higher order crossings in principle. But a cubic or high
order crossing means that the linear term is somehow forbidden due to the presence
of certain additional symmetry. This is quite hard and it has not been experimentally
achieved yet. Therefore, “Dirac” surface state is usually used to describe the metallic
boundary mode of 3D strong or weak TIs.
The above physical pictures can be mathematically formulated, which results in
the Z 2 topological numbers for a 3D bulk system [14–16,29]. In 2007, it was theoreti-
cally realized that in three-dimensions, there exist four Z2 topological invariants that
define the topological property of a 3D bulk material, namely (ν 0; ν 1ν 2ν 3), where ν 0
is the strong topological invariant, and ν 1 − ν 3 are the weak topological invariants,respectively [14]. If all four invariants are zero, then the system is a conventional
band insulator. If the strong invariant ν 0 = 0 but one or more than one of the weak
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Figure 1.7: The surface of a strong TI features an odd number of protected surfacestates at all surfaces irrespective of the surface termination. (a) A bulk BZ of thestrong TI. The green slices are chosen for detailed studies. (b) The edge projected electronicstructure for the green slices shown in Panel (a). Without any inter-slice coupling, theseslices are identical and independent. The surface states for the kx − kz surface projectionis a straight line. (c) A bulk BZ showing the one band inversion at the Γ point. (d) Afterturning on the inter-slice coupling, in the case of a strong TI, only the band inversion atthe Γ point remains whereas the band inversion at the Z point is removed. Furthermore,for k−space region near the Γ point where the conduction and valence bands are inverted,the edge states can hybridize and open out gaps, except at the Γ point itself due to the
protection of the time-reversal symmetry. Therefore, the surface state Fermi surface isdescribed by one dots at the Γ point.
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1.2.3 The 3D Z 2 topological insulator
invariants is nonzero (ν 1 + ν 2 + ν 3 = 0), then the system is a weak TI. Finally, if the strong topological invariant is nonzero (ν 0 = 1), the system is a 3D strong Z 2
topological insulator.
Experimentally, in 2007, the 3D strong TI phase was experimentally identified in
the Bi1−xSbx semiconducting alloy system, marking the first realization of a topologi-
cally nontrivial phase of matter in 3D bulk materials [30,31]. Shortly after Bi1−xSbx,
another class of strong TI - the Bi2Se3 class - was experimentally discovered [32–36].
The Bi2Se3 class was found to feature a single Dirac cone surface state due to a sin-
gle bulk band inversion at the Γ point (same as the example given in Figs. 1.6 and
1.7) and therefore it serves as the prototype 3D TI, which has been the most widely
researched topological insulator even to date.
More importantly, it turns out that the experimental discovery of the 3D topologi-
cal insulator phase in 2007 opened a new experimental era in fundamental topological
physics [30–127]. In contrast to its 2D analogs, (1) a 3D topological insulator can be
realized at room temperatures without magnetic fields. Their metallic surface states
exist at bare surfaces rather than only at buried interfaces. (2) The electronic and
spin groundstate of the topological surface states can be systematically studied by
spin- and angle-resolved photoemission spectroscopy (spin-ARPES), which provides a
unique and powerful method for probing the topological number in three-dimensional
topological phases. (3) Due to the accessible conditions (room temperature, no mag-
netic field, bare surface), it is also possible to study the tunneling [87–94], electrical
transport [95–106], optical [107–114], and many other key properties of the topo-
logical surface states. (4) The 3D topological insulator materials can be doped or
interfaced to realize superconductivity or magnetism [52–60]. (5) Since its discovery
in 2007, there have been more than a hundred compounds identified as 3D Z 2 topo-
logical insulators. These conditions not only make the 3D Z 2 TI state a topic of huge
research interest worldwide, but also lay the experimental foundation for discovering
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1.2 PREVIOUSLY DISCOVERED TOPOLOGICAL PHASES.
new topological phases of matter [37–51], which is the main theme of this thesis.
We note that time-reversal symmetry is required for a Z 2 topological insulator
phase. This can be visualized in the surface state point of view. If time-reversal
symmetry is broken, the surface state degeneracy at the Kramers point can be lifted.
And in that case, the distinction between a Z 2 topological insulator and a conventional
band insulator no longer exists. In the topological theory, a state that is topologically
nontrivial only if a symmetry is respected is called a symmetry protected topological
(SPT) state. A Z 2 TI is a SPT state [128]. On the other hand, a state whose
topological nontriviality does not rely on any additional symmetry is defined as a
state that features topological order [24]. In general, the realization of a topological
order inherently requires strong electron-electron interaction, which is beyond band
theory [24]. Strong interaction leads to fractionalized non-electron-like quasi-particle
boundary excitations, which is believed to be one of the key properties for a system
with a topological order. A fractional quantum Hall (FQH) state, whose edges feature
gapless excitations with fractional charges, is an example of a state with a nontrivial
topological order. This is in contrast to the Z 2 TI phase (a SPT state) where the
metallic surface states are electron quasi-particles. In theory, it is believed that the
boundary modes in a state with topological order can be robust against any local
perturbation, whereas the gapless boundary modes in a SPT state are robust against
local perturbations that do not break the corresponding symmetry. It is in this
sense that a state with topological order is more robust than a SPT state. However,
let us also view this issue based on experimental and material facts. Due to the
requirement of very strong electron-electron interaction, topological order is more
difficult to realize and its existence is rarer in real solid-state materials as compared
to a SPT state. As a matter of fact, the FQH state is the only state with topological
order that has been experimental realized so far. Other theoretical proposals, such as
a fractionalized Chern insulator [129], or a fractionalized topological insulator [130],
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etc., remain elusive due to the difficulty of identifying a real system that has the
necessary material parameters. Moreover, the experimental study of a topological
phase relies on measuring its gapless boundary states. In a topologically ordered state,
the boundary modes are fractionalized non-electron like quasi-particles. Although in
the case of FQH, their existence can be demonstrated by charge transport. But it is in
general extremely challenging to systematically study the energy dispersion, angular
momentum (spin polarization), and other key properties of fractionalized non-eletron-
like quasiparticles. On the other hand, these difficulties do not exist in a SPT state
such as a Z 2 TI state in Bi2Se3. Furthermore, the requirement of a symmetry in
a SPT state is usually a “given” in a solid state system. For example, in a solid-
state crystal, the translational symmetry or sometimes also space group symmetries
(mirror, C n) is a given. Furthermore, time-reversal symmetry is also a very common
property that exists in most non-magnetic materials. In fact, in experiments, it is
quite often that one has to make considerable effort to break a certain symmetry (e.g.
to apply strong magnetic field to break time-reversal symmetry for a FQH state)
in order to realize certain topological order. Therefore, by these (experimental or
material) considerations, the realization, the systematic study and the utilization of
SPT states are of importance and interest. In this thesis, we focus on the experimental
discoveries of new symmetry-protected topological states (mostly weakly interacting
systems within the band theory) beyond a Z 2 TI.
1.3 Theoretical formulations
In this section, we provide the theoretical formulations for the Chern number and the
Z 2 invariants.
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1.3.3 The Z 2 topological invariants in three-dimensions
Λa in the bulk 2D BZ, where k and −k coincide. In the square 2D BZ as shown inFig. 1.4(a), these four points are the Kramers points, namely 1 Γ, 2 X , and 1 M .
At these four points, ω(Λ) is antisymmetric. The determinant of an antisymmetric
matrix is the square of its pfaffian (P f ), which allows us to define
δ a = P f [ω(Λa)]/
Det[ω(Λa)] = ±1 (1.4)
Provided the Bloch wavefunction |um(k) > is chosen continuously throughoutthe Brillouin zone (which is always possible), the branch of the square root can be
specified globally, and the Z 2 invariant is
(−1)ν =4
a=1
δ a (1.5)
If the crystal has inversion symmetry, there exists a shortcut to obtain ν [29]. At
the Kramers points Λa the Bloch states um(Λa) are also the eigenstates of the parity
operator with the eigenvalue ξ m(Λa) = ±1. The Z 2 invariant then simply followsfrom with
δ a =
m
ξ m(Λa) (1.6)
where the product is over the Kramers pairs of occupied bands. This has proven
quite useful for identifying Z 2 topological insulators from numerical band structure
calculations.
1.3.3 The Z 2 topological invariants in three-dimensions
In three dimensions there are eight Kramers points, which are expressed in terms of
primitive reciprocal lattice vectors as Λa = (n1b1 + n2b2 + n3b3)/2, with n = 0, 1.
This leads to four independent Z 2 topological invariants to define the Z 2 topological
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1.4 INSPIRATION FOR DISCOVERING NEW TOPOLOGICAL PHASES
properties of a 3D band insulator, namely [ν 0; ν 1ν 2ν 3], where ν 0 is the strong invariant
and (ν 1, ν 2, ν 3) are the weak invariants.
The strong invariant can be calculated by the pfaffian over all eight Kramers points
(−1)ν 0 =8
a=1
δ a (1.7)
where again we have δ a = P f [ω(Λa)]/
Det[ω(Λa)] = ±1.Besides the strong invariant, it has been shown in theory that one can define
another 3 independent weak invariants (ν 1, ν 2, ν 3) as
(−1)ν i=1,2,3 = 8ni=1;nj=i=0,1
δ a. (1.8)
Again, the conclusion from the above topological theory is that if all four invariants
are zero (ν 0 = ν 1 = ν 2 = ν 3 = 0), then the system is a conventional band insulator; if
ν 0 = 1, then the system is a strong (Z 2) topological insulator; if ν 0 = 1 but at least
one of the weak invariants is nonzero, then the system is a weak (Z 2) topological
insulator.
1.4 Inspiration for discovering new topological phases
Now that we have gone through all previous topological phases, we discuss the in-
spirations that are brought by these previous works for discovering new topological
matter.
1.4.1 Band inversions and spin-orbit interactions
First of all, it is evident that the band inversion is a key (a critical necessary condition)
that gives rise to the topological insulator state. More specifically, by band inversion,
we mean two phenomena in the bulk electronic structure: (1) At certain k points, the
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1.4.1 Band inversions and spin-orbit interactions
relative energy positions between the conduction and the valence bands are reversed
with respect to those of in the atomic limit. (2) An insulating gap is opened once the
conduction and valence bands are inverted.
How do we realize a band inversion with these two phenomena in real materials?
Is there any clue or hint regarding how to search for materials with band inversions?
Let us again start from a very simple case, which is NaCl (an ionic insulator). We
write it as Na+Cl− to highlight the distinct ionic states of sodium and chlorine in
this insulator. As shown in Fig. 1.8, the lowest valence and conduction bands near
the Fermi level are the Cl−3 p and the Na+3s orbitals, respectively. As the system
goes from the atomic limit [Fig. 1.8(a)] to the actual lattice constant [Fig. 1.8(b)],
these bands gain energy dispersion as a function of Bloch momentum k. In the case
of Na+Cl−, the relative energy positions between the conduction and valence bands
remain the same as the atomic limit at all k points throughout the BZ. Thus a large
full band-gap is formed. The physical meaning of the band-gap can be understood
in the picture that it costs a finite amount of energy (∼ the band-gap) to excite anelectron from the valence band to the conduction band, or more loosely speaking,
to give the electron that is transferred to Cl− back to the Na+ [Fig. 1.8(c)]. Such a
process is highly energetically unfavorable because the system gains much energy as
Na atom transfers its 3s1 electron to the Cl atom to form Na+Cl−.
Now let us imagine another compound A+B−, where we label the lowest valence
and conduction bands as B− and A+, respectively as shown in Fig. 1.8(d). Now as we
tune the lattice constant to the actual value, the valence and conduction bands gain
energy dispersion in a way so that these two bands actually cross at certain Kramers
point, as shown in Fig. 1.8(e). Now in this scenario, at the k−space vicinity to theKramers point where band-crossing occurs, the system actually gains energy to give
the electron that is transferred to the B atom back to the A atom. Furthermore, it
is important to note that as the conduction and valence bands cross, they become
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1.4 INSPIRATION FOR DISCOVERING NEW TOPOLOGICAL PHASES
Figure 1.8: Band Inversions and the spin-orbit interaction. (a-c) In the case of sodium chloride Na+Cl−, the system does not have any band inversion. The fact that thereis a large band-gap between the Cl−3 p and the Na+3s bands is consistent with that it isenergetically costly to give the electron that is taken by the Cl− back to the Na+. (d-f)
In a material A+
B−
, the valence and conduction bands gain energy dispersion in a wayso that these two bands actually cross at certain Kramers point. Further consideration of the spin-orbit interaction opens a full insulating gap. SnTe is a real material example forthis case. (g-i) In this case, without spin-orbit coupling, there is no band crossing, and thespin-orbit interaction is responsible for both inverting the conduction and valence bandsand for opening up the full insulating gap. We note that Bi2Se3 belongs to this case.
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1.4.2 Constructing new topological phases
degenerate at certain k points in the BZ [see Fig. 1.8(e)]. Thus there does not exist
a full insulating gap as required by the Z 2 topological insulator state. However, as
we further consider the effect of the spin-orbit interaction, it causes the bands that
cross to hybridize and open up a full band-gap. In terms of real materials, SnTe is
a famous example of this kind [131], where a band inversion between the Sn5 p and
Te5 p orbitals is found even without the including the spin-orbit interaction at each
L point of its BZ. And the spin-orbit interaction is responsible for opening up a full
band-gap. There is another slightly different case, as shown in Figs. 1.8(g-i), where
without spin-orbit coupling, there is no band crossing, and the spin-orbit interaction
is responsible for both inverting the conduction and valence bands and for opening up
the full insulating gap. We note that Bi2Se3 belongs to this case [33]. From the above
physical picture, it is quite clear that we need to work with materials with heavy
(high Z ) elements for a strong spin-orbit coupling, which is necessary to open up a
full band-gap. Furthermore, it can be seen that we cannot have A+B− materials like
NaCl, where Na strongly prefers to loose an electron and Cl strongly prefers to take
an electron. In this case, it is hard to imagine that one could get a band crossing at
certain k− points because a band crossing creates a local k− space regime, where itis energetically favorable to return the electron taken by the B− back to the A+. This
means that we should look for materials that are composed of heavy “semimetal”
elements. Therefore, bismuth (Bi) is a good candidate because it can form both
a positive ion as Bi3+ or Bi5+, and a negative ion as Bi3−, whose energies are not
significantly different. In fact, Bi2Se3 (Bi3+) and Na3Bi (Bi
3−) are two good examples
for the above argument in searching for materials with band inversions.
1.4.2 Constructing new topological phases
How do we construct new topological phases? From the discussions in the previous
sections, we know that the band inversion is a necessary but not a sufficient condition
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to realize a topological state. In the case of the Z 2 TI state, the argument is that
the bulk band inversions occur at the Kramers points of the BZ. Therefore, a surface
(edge) state band crossing can be protected by the Kramers theorem.
(1) Inspired by this logic, can we find other types of symmetries that can also
protect a surface (edge) state band crossing (a doubly degeneracy of the surface
bands)? If a symmetry can protect a surface state band crossing at certain momentum
space locations in the surface BZ, then we just need to construct a material that
has a number of bulk band inversions that locate at these special momentum space
locations. In a solid state crystal system, the most common symmetries besides the
time-reversal symmetry are the various space group symmetries. In Chapter 4, we
will systematically discuss such possibilities, leading to our experimental identification
(one of the three concurrent ARPES works) of the first topological crystalline insulator
(TCI) phase in the Pb1−xSnxTe(Se) system.
(2) In general, in a 3D bulk crystal, if two bands cross each other as shown in
Fig. 1.8(e), spin-orbit interaction can cause hybridization between the two bands that
cross and open a full energy gap. As two 3D bulk bands cross, they become degenerateat multiple k points in the BZ. Thus a full energy gap means spin-orbit interaction
lifts up the degeneracy at all k points where they cross. This is actually quite intuitive
because without consideration of extra symmetries, there is no reason a certain k point
is more special than other k points where the two bands are degenerate. Therefore,
if spin-orbit interaction breaks the degeneracy, it should do that at all bulk band
crossing k points.
However, this inspires us to ask a question whether we can identify additional
symmetries that can protect the bulk band crossings at certain k points even with
the consideration of the spin-orbit coupling? If that scenario can be realized, then an
important consequence is that the system becomes a semimetal, where the conduc-
tion and valence bands have finite overlap and there does not exist a full band-gap
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1.4.2 Constructing new topological phases
irrespective of the choice of the chemical potential. Then an interesting question is
whether the system can still be topological if the bulk band gap closes (meaning that
there is no full energy gap)? If so, how do we define its topological number and what
would the surface state be like? In Chapter 5, we will deal with these intriguing ques-
tions, which lead to our experimental realization of the topological Dirac semimetal
phase in Cd3As2 and Na3Bi.
(3) In the case of Bi2Se3 as shown in Figs. 1.8(g-i), the inverted band gap is solely
opened due to the spin-orbit coupling. The effect of electron-electron correlation is
negligible. Thus the low energy physics within the inverted band-gap is described
by the single-particle picture. Can we find a material where the electron-electron
interaction plays a non-negligible role in opening up the inverted bulk band-gap? In
Chapter 6, we present our experimental studies on a well-known f -electron system
SmB6. In SmB6, a narrow (∼ 15 meV) band-gap opens at low temperatures ( 30 K)due to the coherent hybridization between the conduction d electrons and the localized
f electrons near the the Fermi level. We show experimental evidence consistent with
the existence of topological surface states within the Kondo gap, supporting the
existence of a topological Kondo insulator phase in SmB6.
(4) None of the known TI materials (represented by Bi2Se3) are naturally super-
conductors or magnets. It is of interest to study the interplay between a topological
state and a broken-symmetry (superconducting or ferromagnetic) state. In Chapter 7,
we present systematic studies of the electronic and spin groundstate of the prototype
TI Bi2Se3 as it is doped into or in proximity to a symmetry-broken (superconducting
or magnetic) state. We identify the key experimental signatures in our experiments
that are considered as the keys to realizing a topological superconductor or a Chern
insulator.
(5) Finallly, all the states mentioned above are new (symmetry-protected) topo-
logical phases beyond a Z 2 TI, which means that they cannot be smoothly deformed
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1.4 INSPIRATION FOR DISCOVERING NEW TOPOLOGICAL PHASES
into each other without going through a band inversion (a topological quantum phase
transition). Can we experimentally realize a band inversion and a topological quan-
tum phase transition? This will bring valuable insights in understanding the way
how two states with distinct topological numbers can be turned into each other. In
Chapter 3, we present our our experimental realization of a topological quantum
phase transition in the BiTl(S1−δSeδ)2 system, which demonstrates the one of the
most basic topological quantum phase transitions - a topological transition between
a conventional band insulator and a Z 2 topological insulator.
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Chapter 2
Experimental techniques
In this chapter, we introduce the basic principles of the experimental technique used
in this thesis (the spin-resolved angle-resolved photoemission spectroscopy) and the
review how such technique is used to reveal the 3D Z 2 topological insulator state in
Bi1−xSbx and Bi2Se3 bulk materials.
2.1 Spin-integrated Angle-resolved photoemission
spectroscopy
Angle-resolved photoemission spectroscopy (ARPES) is a direct experimental tech-
nique to observe the distribution of the electrons (more precisely, the density of single-
particle electronic excitations) in the reciprocal space of solids [132,133]. ARPES is
one of the most direct methods of studying the electronic structure of the surface of
solids. The photoelectric effect was first observed by Hertz. When photons of energy
hν , strike a sample surface, electrons absorb the energy and escape into the vacuum
with certain kinetic energies. A good approximation of the photoemission process
can be described by the three-step model, as shown in Fig. 2.1:
• An electron is first excited by the photon inside the bulk.31
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2.1 SPIN-INTEGRATED ANGLE-RESOLVED PHOTOEMISSION SPECTROSCOPY
Figure 2.1: The three-step model used to describe the photoemission process inside thesample.
• The excited electron then travels to the sample surface.
• The electron escapes from the sample into the vacuum.
After a photoelectron comes out of the sample, it is being collected by an electron
analyzer, as a function of its kinetic energy as well as the surface emission angles θ
and φ (Fig. 2.2). The excited photoelectron then follows the kinematics below which
describes the momentum p = k
kx = 1
2mE kin sin θ cos θ (2.1)
ky
= 1
2mE kin sin θ sin θ (2.2)kz =
1
2mE kin cos θ (2.3)
The conservation of energy gives:
E kin = hν − φ − |E B| (2.4)32
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Figure 2.2: (a) The Scienta analyzer setup used to measure emitted photoelectrons fromthe sample surface. The two hemispheres probe the electrons to produce a 2D image of energy (E ) vs. momentum (k) in one measurements. (b) The geometry of the detectorrelative to the sample surface. The momentum of the electron inside the sample can b eextracted from the measured values of E kin, θ , φ.
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2.2 SPIN-RESOLVED ANGLE-RESOLVED PHOTOEMISSION SPECTROSCOPY
where φ is the work function of the sample and E B is the binding energy of the
electrons. Due to the existence of a surface potential experienced by the escaping
electron, the component of the momentum perpendicular to the sample surface is not
conserved. To capture the effect of the surface potential, we assume that the energy
of the “real” final state (free from the influence of the surface potential) E f has an
energy offset of V 0 with respect to the kinetic energy measured by the analyzer E kin,
namely
E f = E kin + V 0 (2.5)
where V 0 is usually referred as the inner potential.
Since the in-plane momentum is conserved, thus we know that k =√
2mE kin sin θ.
Moreover, we know that 2k
2
2m +
2k⊥2
2m = E f . Therefore, from these relationships, we
get:
k⊥ = 1
2m(E kincos2θ + V 0) (2.6)
where V 0 is usually referred as the inner potential.
The value of V 0 can be determined experimentally by measuring the periodicity
of the energy dispersion along the out-of-plane momentum k⊥ direction at the nor-
mal emission angle. However, for samples where the dispersion perpendicular to the
sample surface is small, such a measurement can be difficult.
2.2 Spin-Resolved Angle-resolved photoemission spec-
troscopy
The Spin-Resolved Angle-resolved photoemission spectroscopy not only measures the
energy E and the Bloch momentum k but also the spin polarization of an electronic
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2.2.1 Mott polarimetry
state. In this section, we review two currently used techniques for spin-resolved
photoemission experiments. To perform spin-resolved electron studies, an electron
spectrometer has to be combined with a suitable spin polarimeter. To determine
the spin of an ensemble of electrons, one needs to utilize a spin-dependent scattering
process.
The traditional one is the Mott polarimetry. Such a technique arises from spin-
orbit interaction of heavy elements (such as gold) that are used as the detector target
where a spin polarized electron beam is injected, causing spin-dependent scattering.
However, since the spin-orbit interaction is usually a small effect compared to the
Coulomb interaction between electrons, the spin dependence for most scattering tar-
gets and a wide range of parameters is small. The relative influence of the electron
spin on the scattering cross section only becomes significant under conditions, where
the mean scattering cross section is small. Thus in general, the efficiency of the Mott
polarimetry method is not so high. For example, measuring the spin polarization of
the surface state of TI Bi2Se3 at a fixed energy E along a fixed momentum space di-
rection cut takes about 6 - 8 hours to gain good statistics using the Mott polarimetry
with synchrotron radiation light source.
A newer technology, the very low energy electron diffraction (VLEED) spin de-
tector was recently applied. A VLEED uses a magnetic scattering target. In electron
diffraction on ferromagnets, the intensities of diffracted beams in general depend on
the spin polarization of the primary electron beam relative to the sample magnetiza-
tion. If the incident electron beam has a very low energy (usually < 10 eV), then the
efficiency of measuring spin polarization is dramatically improved as compared to a
Mott polarimetry.
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2.2 SPIN-RESOLVED ANGLE-RESOLVED PHOTOEMISSION SPECTROSCOPY
Figure 2.3: Mott polarimetry. Schematic of a Mott scattering geometry. Incidentelectrons (red dot) with a polarization P on a high Z nucleus (yellow dot) are backscatteredto the left and right with a probability that is dependent on P . The figure is adapted fromRef. [135].
2.2.1 Mott polarimetry
The interaction Hamiltonian between a photon and spin 12
electron can be described
by the Dirac equation:
H int = 1
2m( p −e
c A)2
+ eΦ − e
2mcσ · (▽ × A) + ie
4m2c2 E · p − e
4m2c2σ · ( E × p) (2.7)
where p is the electron momentum, A is the photon vector potential, Φ is the scalar
potential, E is the electric field and σ is the electron spin. However by using linearly
polarized photons in the UV to soft x-ray regime, it has been shown ( [134]) that
the spin dependent terms are greatly suppressed, and the photon electron interaction
Hamiltonian can be well approximated by the Schrodinger model H int =
− emc
A
· p,
which conserves spin.
Provided the photoemission process is spin-conserving, the spin of the initial state
of an electron in a solid can be determined by measuring its spin after it has been
photoemitted. Mott electron polarimetry ( [134]) is a method of separating electrons
of different spin from such a photoemitted beam based on the use of spin-orbit (Mott)
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2.2.1 Mott polarimetry
scattering of electrons from nuclei.
The physical principle of Mott scattering can be understood from the classical
picture of a moving electron scattering off of a stationary bare nucleus of charge Z e.
At low incident energies, the electron interacts with the nucleus predominantly via its
charge, and scattering is described by the Rutherford cross section σR(θ), where the
scattering angle θ is typically small. At high incident energies and in cases where Z
is large, the velocity v of the electron in the electric field E of the nucleus can result
in a considerable magnetic field B in its rest frame given by:
B = −
1
c v × E (2.8)
which, using E = (Ze/r3)r, can be written as
B = Ze
cr3r × v = Ze
mcr3 L (2.9)
where L = mr × v is the electron orbital angular momentum. The interaction of this magnetic field with the electron spin S creates a spin-orbit ( L
· S ) term in the
scattering potential and introduces a spin dependent correction to the Rutherford
cross section:
σ(θ) = σR(θ)[1 + S (θ) P · n̂] (2.10)
where S (θ) is the asymmetry or Sherman function, P is the polarization 2
(< S x >
, < S y >,< S z >), and n̂ is the unit normal to the scattering plane defined by
n̂ = ki × kf | ki × kf |
(2.11)
Where ki and kf are the initial and final wave vectors of the electron respectively.
The direction n̂ depends on whether scattering to the left or right is being considered.
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2.2 SPIN-RESOLVED ANGLE-RESOLVED PHOTOEMISSION SPECTROSCOPY
Figure 2.4: Spin-resolved ARPES (Mott polarimetry) experimental setup. (a)Schematic of the spin-resolved ARPES spectrometer COPHEE. Photoelectrons are energyand momentum analyzed using a hemispherical electrostatic analyzer and are alternatelydeflected at a frequency of 1 Hz into two orthogonally mounted Mott polarimeters. The dualpolarimeter system is shown rotated by 90◦ for clarity. (b) The relationship between thesample and Mott coordinate systems. When θ and φ are both zero, the sample coordinatescan be transformed into the Mott coordinates via a 45◦ rotation about their common z axis.The Mott axes marked red denote the spin components that the polarimeter is sensitive to.The figure is adapted from Ref. [135].
This spin-orbit scattering relation allows for the measurement of the component of
spin polarization perpendicular to the scattering plane in the following way. Consider
a beam of N incident electrons with N ↑ of them polarized along +z and N ↓ of them
along -z, which leads to a net polarization P z = (N ↑ − N ↓)/(N ↑ + N ↓). When thescattering of this beam from a nucleus takes place in the xy plane, there results a
left-right scattering asymmetry Az(θ) defined as:
Az(θ) = N L − N RN L + N R
(2.12)
where N L and N R are the number of electrons scattered to the left and right
respectively through an angle θ. Substituting the relations N L∝N ↑[1+S (θ)]+N ↓][1−S (θ)] and N R∝N ↑[1 − S (θ)] + N ↓][1 + S (θ)] derived from Equation 5 to Equation 7
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2.2.2 VLEED
yields:
P z = Az(θ)
S (θ) (2.13)
which shows that given the Sherman function, measurement of Az(θ) yields P z. In
a single Mott polarimeter, it is therefore possible to measure two orthogonal spin com-
ponents of an electron beam by arranging four detectors in two orthogonal scattering
planes in front of a target.
2.2.2 VLEED
The concept of VLEED spin detector is based on a simple and intuitive phenomenon,
which is that as electrons are injected to a ferromagnetically ordered material, the in-
tensities of diffracted beams in general depend on the spin polarization of the primary
electron beam relative to the sample magnetization.
Obviously, the efficiency of such detector depends on the ratio of electrons that
are being reflected with respect to other processes. In a simplified kinematic picture,
the intensity modulation can be described in terms of constructive or destructiveinterference of reflections from different lattice planes. As the energy of the primary
beam is lowered, the radius of the Ewald sphere decreases until it is smaller than the
smallest reciprocal lattice vector. In this situation there are no higher-order diffracted
beams, and only the specularly reflected beam can be observed. This leads to an
increase of the intensity of the specular beam in LEED at very low energies VLEED,
which sets one of the foundation for using VLEED as an efficient spin detector.
So far in the above-paragraph, the discussion only contains the issue how to en-
hance the the simple reflection of an electron beam, but it has not covered the spin
sensitivity. Let us consider a VLEED process on a ferromagnet, such as an iron
(Fe) crystal. In a ferromagnetic material the spin degeneracy of the electronic states
is lifted by the exchange interaction, such that, e.g., a critical point in the band
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2.2 SPIN-RESOLVED ANGLE-RESOLVED PHOTOEMISSION SPECTROSCOPY
structure occurs at lower energy for the majority states than for the minority states.
Therefore, the features in the energy dependence of the absorption and reflection
probability occur at different energies for majority and minority states. This may be
used to realize a spin-dependent scattering for electron spin analysis. Of the three
elemental 3d ferromagnets, Fe has the largest exchange splitting of about 2.5 eV.
Therefore Fe is a good candidate for this type of spin polarimeter.
Figure 2.5: Spin-resolved ARPES (VLEED) experimental setup. Left: Schematiclayout of hemispherical energy analyzer with VLEED spin p olarimeter. A normal hemi-spherical electron analyzer (HA 50 are operated by the HAC 300 unit) is first used to detectthe energy and the momentum of the electronic states. The spin p olarimeter requires threeadditional voltage supplies E P , lens, and V sc. The long dashed line shows light incidence,the short dashed line shows an electron trajectory. Channeltron 1 (labeled ch. 1) is foracquiring non spin-resolved data when the scattering target is removed. Channeltron 2 isfor counting the electrons scattered back under 10◦ from the magnetized Fe surface. Right:Spin-dependent specular electron scattering off a Fe(100) film at about 10◦ incidence angle.(a) Number of scattered electrons; filled symbols are for parallel magnetization of scat-tering target and photoemission source, empty symbols for antiparallel magnetization; (b)asymmetry; and (c) figure of merit. The figure is adapted from Ref. [136].
Fig. 2.5 (left panel) shows schematically the layout of the Fe VLEED spin detector.
The spectrometer is a commercial hemispherical analyser (HA 50, 50 mm mean radius,
with power supply HAC 300 by VSW). Behind the exit slit of the analyzer, the
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2.3 SAMPLE PREPARATION FOR ARPES
preoxidized Fe(001)-p(1 × 1)-O targets as the spin detector [137].
2.3 Sample preparation for ARPES
2.3.1 Single crystal bulk samples
Figure 2.6: A schematic of the single crystal bulk sample geometry mounted inside theARPES measurement chamber. The ceramic top post was used to cleave the sample surfacein situ, to ensure a clean surface for measurements.
For single crystal bulk samples, they are grown by our collaborators including
Prof. R. J. Cava’s group in the Department of Chemistry in Princeton University,
Prof. F. C. Chou’s group in the Center for Condensed Matter Sciences in National
Taiwan University, Prof. Y. Chen’s group in the Physics Department in Purdue
University, Prof. S. Jia’s group in the International Center for Quantum Materials,
Peking University, Beijing 100871, China.
In order to measure them in ARPES, one needs a very clean and flat surface
because ARPES is a very surface-sensitive probe. To do so, the sample is then
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2.3.2 MBE film samples
mounted in a geometry as depicted by Fig. 2.6. The crystals are first cut into pieces
of approximately 2 mm × 2 mm × 0.5 mm. A small piece is then mounted onto acopper post, using the commercial epoxy resin Torr seal. For other samples with poor
electrical conductivity, silver epoxy is used instead to ensure good electrical contact
with the sample holder. A ceramic top post is then attached onto the sample with
another layer of epoxy. After the sample is mounted inside the ARPES measurement
chamber, it is cooled and pumped down for a few hours. The sample was then cleaved
insitu at pressures of less than 1 ×10−10 torr, by knocking down the ceramic top-postwith a cleaver. The exposed surface was typically visually shiny, indicating a flat
surface.
2.3.2 MBE film samples
For MBE film samples, they are grown by our collaborators such as