Download Discoveries of New Topological STates of Matter Beyond Topological Insulators

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
c Copyright by Su-Yang Xu, 2014.
All Rights Reserved
Abstract
The discoveries of new forms of matter have been so definitive that they are used
to name periods in the history of mankind, such as Stone Age, Bronze Age, and
Iron Age. Although all matter is composed of component particles, particles can
organize in various ways leading to different phases of matter. Finding all possible
distinct phases that matter can form and understanding the physics behind each
of them are fundamentally important goals in physics research and often lead to
new technologies, benefiting our society. A topological phase is an unusual type of
crystalline solid, characterized by a nontrivial topological number. This number is
a global quantity, which depends on the crystal’s bulk electronic wavefunctions. In
2007, the 3D Z2 topological insulator (TI) phase was discovered in bismuth-based
materials, marking the first realization of a topological phase in bulk crystals. A 3D
TI features spin-polarized Dirac electronic states on its surface, which enjoys a robust
protection against disorder. This discovery tremendously accelerated the field and led
to a surge of interest in searching for new topological phases of matter.
In this thesis, we present the experimental discovery of several new topological
phases and phenomena beyond the Z2 TI, including the topological quantum phase
transition in BiTl(S1−δ Seδ )2 , the topological crystalline insulator phase in Pb1−x Snx Te(Se),
the topological Dirac semimetal phase in Cd3 As2 and Na3 Bi (featuring Fermi arc surface states), evidence for the topological Kondo insulator state in SmB6 , and the
demonstration of superconductivity and magnetism in the surface states of the prototypic TI Bi2 Se3 . Each of these new states exhibits topological surface states with
unique protected properties. They may be useful in developing future technologies
such as fault-tolerant topological quantum computers and low-power spintronic devices, which will revolutionize our electronic and energy industries. The new topological states of matter presented here are currently being studied by many groups
worldwide. With our discoveries, the “topological world” has begun to unveil itself.
We believe that this is only the tip of the iceberg. Our ongoing work suggests that
there are many more with yet more exciting properties awaiting discovery.
iv
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 will 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 ran into difficulties with my experiments or when fighting in
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 tremendously benefited from his ability to understand and explain the essences of
physics without bringing many formulations, and his insights into condensed matter
physics, especially when it came to looking for the next exciting topics in our field.
This is one of the main driving forces to maintaining our group of 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 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 with 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
vi
not only a wonderful labmate, but also such a reliable friend. I thank him for his
generous help when I got into trouble.
During my Ph.D. studies, I was very lucky to have the unique opportunity to
work at ARPES beamlines at synchrotron radiation laboratories all over the world.
At these facilities, I learned a lot from many wonderful local scientists and staff.
I thank Alexei Fedorov, Sung-kwan Mo, Jonathan Denlinger and Zahid Hussain at
the ALS in Berkeley. I thank Makoto Hashimoto and Donghui Lu at the SSRL in
Stanford. I thank Hugo Dil, Fabian Meier, Bartosz Slomski, Gabriel Landolt and
Vladimir Strocov at the SLS in Switzerland. I thank Mats Leandersson, Thiagarajan
Balasubramanian, Johan Adell and Craig Polley at MAX-lab in Sweden. I thank
Jaime Sánchez-Barriga and Oliver Rader at Bessy II in Germany. I thank Yushiyuki
Ohtsubo, Bertran François and Amina Taleb-Ibrahimi at Soleil in France. I thank
Koji Miyamoto and Prof. Taichi Okuda-sensei at the Hisor in Hiroshima University
in Japan. I thank Ishida Yukiyaki, Takeshi Kondo and Prof. Shik Shin-sensei at the
ISSP at the University of Tokyo in Japan.
None of this work would be possible without our sample growth and first-principles
band structure calculation collaborators. I was truly lucky to work with a number of
sample growth groups, and their hard work produced so many high quality samples.
These samples were breakthroughs in solid state chemistry and physics. I thank Prof.
Bob Cava and his world-renowned team in Princeton Chemistry for single crystal
samples. I thank Anthony Richardella and Prof. Nitin Samarth in Penn state for the
MBE film samples. I also thank Raman Sankar and Prof. Fangcheng Chou in National
Taiwan University and Prof. Shuang Jia’s group in Peking University for single crystal
samples. I thank Hsin Lin at National University of Singapore, Tay-Rong Chang and
Horng-Tay Jeng at National Tsing Hua University, Cheng-Yi Huang and Wei-Feng
Tsai at National Sun Yat-Sen University, and Prof. Arun Bansil at Northeastern
University for their collaboration on first-principle calculations. They were always
ready to help and sent calculation results on short notices. The timely response from
our sample growth and theoretical calculation collaborators was crucially important
in our 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 Z2
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 sister Chujun. 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 grandparents Ruizhi and Ruilan, my mom Xiaobei, my dad Yuping, my aunt Hong, and
all family members for all their selfless love and care. Family is my forever source of
viii
happiness.
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
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 Z2 topological insulator - the quantum spin Hall insulator
7
1.2.3
The 3D Z2 topological insulator . . . . . . . . . . . . . . . . .
11
1.3 Theoretical formulations . . . . . . . . . . . . . . . . . . . . . . . . .
21
1.3.1
The Chern number . . . . . . . . . . . . . . . . . . . . . . . .
22
1.3.2
The Z2 topological invariant in two-dimensions
. . . . . . . .
22
1.3.3
The Z2 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 . . . . . . . . . . . . . . . . . . . . . . . . .
x
36
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 . . . . . . . . . . .
2.4.2
45
Surface Dirac crossing, surface-bulk connectivity, and spin-momentum
locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Topological Quantum Phase Transition in BiTl(S1−δ Seδ )2
47
52
3.1 Evolution of the electronic groudstate across a topological quantum
phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
3.2 Topological quantum phase transition in BiTl(S1−δ Seδ )2 . . . . . . . .
56
3.2.1
Identifying a suitable topological material system . . . . . . .
56
3.2.2
Topological quantum phase transition in BiTl(S1−δ Seδ )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−δ Inδ )2 Se3 82
4 Topological Crystalline Insulator Phase in Pb1−δ Snδ Te(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−δ Snδ Te . .
89
4.2.1
Band inversions and mirror symmetries in Pb1−δ Snδ Te . . . .
89
4.2.2
Topological surface states in Pb0.6 Sn0.4 Te . . . . . . . . . . . .
91
4.2.3
Mirror Chern number and mirror symmetry protection . . . .
98
4.3 Topological surface states in Pbδ Sn1−δ Se . . . . . . . . . . . . . . . . 103
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−δ Snδ Se . . . . . . . . . . . . 111
5 Topological Dirac semimetal state in Cd3 As2 and Na3 Bi
114
5.1 Theoretical concepts for a topological Dirac semimetal . . . . . . . . 116
5.2 3D Dirac semimetal state in high mobility Cd3 As2 . . . . . . . . . . . 121
5.3 Fermi arc surface states in topological Dirac semimetal Na3 Bi . . . . 132
5.3.1
Choice of the surface termination to observe FASS . . . . . . . 132
5.3.2
Observation of Fermi arc surface states in Na3 Bi . . . . . . . . 135
5.3.3
Topological invariant for the Dirac semimetal Na3 Bi . . . . . . 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 A route to 2D topological superconductivity
7.1 Hedgehog spin texture in a magnetic topological insulator
174
. . . . . . 175
7.2 Helical Cooper pairing in topological insulator/superconductor heterostructures
Appendices
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
232
xii
Chapter 1
Introduction
In this chapter, we give an introduction about how a topological phase can arise from
a conventional material via the band inversion process. We then briefly review the
research works that discovered the first topological phases (the quantum Hall states,
and the 2D and 3D Z2 topological insulator states). We try to avoid bringing in many
theoretical formulations or experimental details but explain in an intuitive way. After
all, there have already been a number of reviews and books that systematically describe the detailed theoretical formations or the experimental details. The spirit here
is to highlight the physical picture. In the third section, we provide the theoretical
formulations for various topological phases. In the last section of this chapter, we
elaborate how these previous works inspire us to search and discover new topological
phases of matter.
1.1
The band insulator state and the band inversion
The insulating state is one of the elemental groundstates for solids [1–18]. According
to band theory, an insulator is described by a set of completely filled electronic bands
1
1.1 THE BAND INSULATOR STATE AND THE BAND INVERSION
(known as the valence bands) that are separated from a completely empty set of bands
(the valence bands) by an energy gap. An insulator is electronically inert because it
takes a finite energy to dislodge an electron. Let us take a representative example an atomic insulator, such as a solid argon. Argon is a noble gas element characterized
by a “full shell” atomic structure 3s2 3p6 . As argon atoms crystalize into a solid at
low temperatures, the filled 3p6 orbital forms the lowest valence band, whereas the
empty 4s orbital forms the lowest conduction band, respectively. These two bands
are found to be separated by a large band-gap (∼ 12 eV), and therefore solid argon
is a nearly ideal conventional band insulator. Similarly, sodium chloride (NaCl) is a
well-known ionic salt. The one electron in the Na 3s1 orbital is lost to the Cl 3p7
orbital, making both Na+ and Cl− ions full shell. And in this case, the filled Cl− 3p
orbital and the empty Na+ 3s orbital form the lowest conduction and valence bands.
Although insulators can arise from various chemical bonding origins, all these well
known conventional band insulators belong to the same topologically trivial phase
as the vacuum. This can be conceptually visualized by the following: 1) Take any
conventional insulator, e.g. NaCl, and adiabatically change its Hamiltonian to an
atomic insulator (solid argon). This process can be viewed as merely changing the
energy gap value at each k point without closing the energy gap. 2) Furthermore,
with the atomic insulator solid argon, one could imagine increasing its lattice constant
(the distance between the nearest two argon atoms) to infinity, which results in a
number of unbound argon atoms (the atomic limit, or loosely speaking, the vacuum).
Following this conceptual picture, a very interesting question is ”Are all the insulating
states topologically equivalent to the vacuum (the atomic limit)?” The answer is no.
And those, which are topologically nonequivalent to the vacuum, are the fascinating
topological phases of matter.
How do we construct a topologically nontrivial phase? The above conceptual
picture inspires us to define a “band inversion”. A band inversion is a phenomenon in
2
Figure 1.1: The band insulator state and the band inversion . (a) The crystal
structure of a solid argon lattice. (b) The energy levels of a solid argon in the atomic
limit where the lattice constant goes to infinity resulting in a number of unbound argon
atoms. (c) As the lattice constant goes to the experimental value, the Ar3p and Ar4s
orbitals bceome Bloch bands where they gain energy dispersion as a function of the Bloch
momentum k. (d) First-principles calculation of the electronic band structure of a solid
argon, where the Ar3p valence band, Ar4s conduction band, and the normal band-gap are
seen. (e-h) The same as Panels (a-d) but for a topologically nontrivial band insulator
Bi2 Se3 . 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 in
Panels (f,g). This results in a topologically nontrivial state in Bi2 Se3 . In Panel (h), a
single Dirac cone surface state (red line) enclosing the Γ̄ within the bulk energy gap is seen.
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
4
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
1.2.1
Previously discovered topological phases.
The integer quantum Hall state and the Chern insulator 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
1.2 PREVIOUSLY DISCOVERED TOPOLOGICAL PHASES.
Figure 1.2: The interface between a quantum Hall state and an insulator has
chiral edge mode. (a) The skipping cyclotron orbits. (b) The electronic structure of a
semi-infinite edge described in a Chern insulator with a Chern number of n = +1. A single
edge 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
6
1.2.2 The 2D Z2 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 Z2 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−x Sbx )2 Te3 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 Z2 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-ofplane magnetization +M [Fig. 1.3(a,b)]. Under time-reversal operation T , quantities
1.2 PREVIOUSLY DISCOVERED TOPOLOGICAL PHASES.
Figure 1.3: A quantum spin Hall insulator can be constructed by two Chern
insulators that are related by the time-reversal symmetry. (a) A Chern insulator
with a Chern number of n = +1 has a chiral edge mode as a result of an out-of-plane
magnetization. (b) The bottom edge projection of the electronic structure the n = +1
Chern insulator. (c,d,) Same as Panels (a,b) but with a time-reversal operation onto the
system. (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 two
couterpropagating edge modes that cross each other at k = 0 (or k = π). The crossing is
protected by the time-reversal symmetry as a result of the Kramers theorem.
8
1.2.2 The 2D Z2 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 operation
because 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
~ − kKramers , where G
~ is any reciprocal lattice vector. The Kramers
as kKramers = G
theorem 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
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 Z2 invariant (ν) [14, 26, 29]. The Z2 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 Z2 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 heterostructures 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 Dirac
band 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.
10
1.2.3 The 3D Z2 topological insulator
1.2.3
The 3D Z2 topological insulator
In 2007, it was theoretically realized that the Z2 topological number in a QSH insulator 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 experimental 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 construct a 3D Z2 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 Z2 TI, which is the so
called 3D weak TI. A weak TI is not the most topologically protected Z2 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).
1.2 PREVIOUSLY DISCOVERED TOPOLOGICAL PHASES.
Figure 1.4: A 3D weak topological insulator can be obtained by stacking N copies
of identical 2D quantum spin Hall states along the out-of-plane direction. (a) A
2D 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 kz
direction and becomes a cube representing the 3D BZ. The time-reversal invariant momenta
in the 3D BZ are noted. (c) Before turning on the inter-slice coupling, different QSH slices
are completely identical and idenpendent (e.g. Cut2 and Cut3). This also means that there
is no energy dispersion along the out-of-plane Γ − Z direction (Cut1) (d) A finite (but
weak) inter-slice coupling means that the states in the QSH system gain energy dispersion
along the out-of-plane kz direction. The inter-slice coupling is weak enough, 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 (Cut1). Therefore, there are two band inversions at both the Γ and
the Z points in the 3D BZ (Cuts 2 and 3).
12
1.2.3 The 3D Z2 topological insulator
Figure 1.5: The surface of a weak TI shows even number of protected surface
states or no surface states depending on the surface termination. (a) A bulk BZ
of the weak TI. The green slices are chosen for detailed studies. (b) The edge projected
electronic structure for the green slices shown in Panel (a). Without any inter-slice coupling, these slices are identical and independent. The surface states for the kx − kz surface
projection is a straight line. (c) A bulk BZ showing the two band inversions at the Γ and
the Z points. (d) After turning on the inter-slice coupling, the edge states can hybridize and
open out gaps, except at the Γ and the Z points due to the protection of the time-reversal
symmetry. Therefore, the surface state Fermi surface is described by two dots at the Γ and
the Z points.
1.2 PREVIOUSLY DISCOVERED TOPOLOGICAL PHASES.
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 − kz
plane 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 the
inter-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 plane
becomes two dots at the Γ and the Z points. And if the EF 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 plane
are 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. This
is 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 Z2 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 Z2 TI
14
1.2.3 The 3D Z2 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
1.2 PREVIOUSLY DISCOVERED TOPOLOGICAL PHASES.
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 nontrivial way. (a) A 2D quantum spin Hall state where there is a band inversion at the
2D BZ center Γ point. (b) The 3D BZ. The time-reversal invariant momenta in the 3D
BZ are noted. (c) Before turning on the inter-slice coupling, there is no energy dispersion
along the out-of-plane Γ − Z direction (Cut1). (d) A strong inter-slice coupling can drive
a band inversion along the out-of-plane Γ − Z direction (Cut1). In this case, 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 atomic limit (no band inversion).
Therefore, there is only one band inversion at the Γ point of the 3D BZ.
16
1.2.3 The 3D Z2 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 Z2 topological numbers for a 3D bulk system [14–16,29]. In 2007, it was theoretically realized that in three-dimensions, there exist four Z2 topological invariants that
define the topological property of a 3D bulk material, namely (ν0 ; ν1 ν2 ν3 ), where ν0
is the strong topological invariant, and ν1 − ν3 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 ν0 = 0 but one or more than one of the weak
1.2 PREVIOUSLY DISCOVERED TOPOLOGICAL PHASES.
Figure 1.7: The surface of a strong TI features an odd number of protected surface
states at all surfaces irrespective of the surface termination. (a) A bulk BZ of the
strong TI. The green slices are chosen for detailed studies. (b) The edge projected electronic
structure for the green slices shown in Panel (a). Without any inter-slice coupling, these
slices are identical and independent. The surface states for the kx − kz surface projection
is a straight line. (c) A bulk BZ showing the one band inversion at the Γ point. (d) After
turning on the inter-slice coupling, in the case of a strong TI, only the band inversion at
the 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 is
described by one dots at the Γ point.
18
1.2.3 The 3D Z2 topological insulator
invariants is nonzero (ν1 + ν2 + ν3 6= 0), then the system is a weak TI. Finally, if
the strong topological invariant is nonzero (ν0 = 1), the system is a 3D strong Z2
topological insulator.
Experimentally, in 2007, the 3D strong TI phase was experimentally identified in
the Bi1−x Sbx semiconducting alloy system, marking the first realization of a topologically nontrivial phase of matter in 3D bulk materials [30, 31]. Shortly after Bi1−x Sbx ,
another class of strong TI - the Bi2 Se3 class - was experimentally discovered [32–36].
The Bi2 Se3 class was found to feature a single Dirac cone surface state due to a single 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 topological insulator phase in 2007 opened a new experimental era in fundamental topological
physics [30–128]. 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 magnetic field, bare surface), it is also possible to study the tunneling [88–95], electrical
transport [96–107], optical [108–115], and many other key properties of the topological surface states. (4) The 3D topological insulator materials can be doped or
interfaced to realize superconductivity or magnetism [53–61]. (5) Since its discovery
in 2007, there have been more than a hundred compounds identified as 3D Z2 topological insulators. These conditions not only make the 3D Z2 TI state a topic of huge
research interest worldwide, but also lay the experimental foundation for discover-
1.2 PREVIOUSLY DISCOVERED TOPOLOGICAL PHASES.
ing new topological phases of matter [37–45, 47–52], which is the main theme of this
thesis.
We note that time-reversal symmetry is required for a Z2 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 Z2 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 Z2 TI is a SPT state [129]. 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 Z2 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
20
a fractionalized Chern insulator [130], or a fractionalized topological insulator [131],
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-eletronlike quasiparticles. On the other hand, these difficulties do not exist in a SPT state
such as a Z2 TI state in Bi2 Se3 . Furthermore, the requirement of a symmetry in
a SPT state is usually a “given” in a solid state system. For example, in a solidstate crystal, the translational symmetry or sometimes also space group symmetries
(mirror, Cn ) 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 Z2 TI.
1.3
Theoretical formulations
In this section, we provide the theoretical formulations for the Chern number and the
Z2 invariants.
1.3 THEORETICAL FORMULATIONS
1.3.1
The Chern number
In the IQH effect, the quantized Hall conductivity can be written as σxy = ne2 /h,
where n is an integer. Thouless, Kohmoto, Nightingale and den Niks (TKNN) found
that the integer n in the transverse magneto- conductance σxy is precisely the Chern
number - a topological number calculated using the electronic wavefunction over the
entire Brillouin zone. Specifically, the Chern number in the IQH state in a periodic
potential is written as [21]:
1
n=
2π
Z
BZ
[∇k × A(kx , ky )]z d2 k
(1.1)
with
A = −i < u(k)|∇|u(k) >
(1.2)
One important consequence is that because A(k) is odd under time-reversal, σxy
must vanish if the 2D system respects time-reversal symmetry. This is consistent with
what we show in the previous section that both the IQH and the Chern insulator states
break time-reversal symmetry.
1.3.2
The Z2 topological invariant in two-dimensions
There are several mathematical formulations of the Z2 invariant ν, including a Pfaffian construction, as a topological obstruction, as a type of pumping, or in terms of
homotopy of Hamiltonian spaces [7]. We take the Pfaffian approach developed by Fu,
Kane and Mele [14]. We define a unitary matrix:
ωmn (k) =< um (k)|Θ|un (−k) >
(1.3)
where |um > is the occupied wavefunction, Θ is the time-reversal operator. Since
Θ is anti-unitary and Θ2 = −1, thus ω T (k) = −ω(−k). There are four special points
22
1.3.3 The Z2 topological invariants in three-dimensions
Λa in the bulk 2D BZ, where k and −k coincide. In the square 2D BZ as shown in
Fig. 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
p
δa = P f [ω(Λa)]/ Det[ω(Λa )] = ±1
(1.4)
Provided the Bloch wavefunction |um(k) > is chosen continuously throughout
the Brillouin zone (which is always possible), the branch of the square root can be
specified globally, and the Z2 invariant is
ν
(−1) =
4
Y
δa
(1.5)
a=1
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 ξm (Λa ) = ±1. The Z2 invariant then simply follows
from with
δa =
Y
ξm (Λa )
(1.6)
m
where the product is over the Kramers pairs of occupied bands. This has proven
quite useful for identifying Z2 topological insulators from numerical band structure
calculations.
1.3.3
The Z2 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 = (n1 b1 + n2 b2 + n3 b3 )/2, with n = 0, 1.
This leads to four independent Z2 topological invariants to define the Z2 topological
1.4 INSPIRATION FOR DISCOVERING NEW TOPOLOGICAL PHASES
properties of a 3D band insulator, namely [ν0 ; ν1 ν2 ν3 ], where ν0 is the strong invariant
and (ν1 , ν2 , ν3 ) are the weak invariants.
The strong invariant can be calculated by the pfaffian over all eight Kramers points
(−1)ν0 =
8
Y
δa
(1.7)
a=1
where again we have δa = P f [ω(Λa)]/
p
Det[ω(Λa)] = ±1.
Besides the strong invariant, it has been shown in theory that one can define
another 3 independent weak invariants (ν1 , ν2 , ν3 ) as
(−1)
νi =1,2,3
=
8
Y
δa .
(1.8)
ni =1;nj 6=i=0,1
Again, the conclusion from the above topological theory is that if all four invariants
are zero (ν0 = ν1 = ν2 = ν3 = 0), then the system is a conventional band insulator; if
ν0 = 1, then the system is a strong (Z2 ) topological insulator; if ν0 = 1 but at least
one of the weak invariants is nonzero, then the system is a weak (Z2 ) topological
insulator.
1.4
Inspiration for discovering new topological phases
Now that we have gone through all previous topological phases, we discuss the inspirations 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
24
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− 3p 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 an
electron 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 the
Kramers 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
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 there
is a large band-gap between the Cl− 3p and the Na+ 3s bands is consistent with that it is
energetically 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 way
so 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 for
this case. (g-i) In this case, 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 Bi2 Se3 belongs to this case.
26
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 Z2 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 [132], where a band inversion between the Sn5p and
Te5p 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 Bi2 Se3 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 it
is 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, Bi2 Se3 (Bi3+ ) and Na3 Bi (Bi3− ) 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
1.4 INSPIRATION FOR DISCOVERING NEW TOPOLOGICAL PHASES
to realize a topological state. In the case of the Z2 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−x Snx Te(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 degenerate
at 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 conduction and valence bands have finite overlap and there does not exist a full band-gap
28
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 questions, which lead to our experimental realization of the topological Dirac semimetal
phase in Cd3 As2 and Na3 Bi.
(3) In the case of Bi2 Se3 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 Bi2 Se3 ) are naturally superconductors 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 Bi2 Se3 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) topological phases beyond a Z2 TI, which means that they cannot be smoothly deformed
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 quantum 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−δ Seδ )2 system, which demonstrates the one of the
most basic topological quantum phase transitions - a topological transition between
a conventional band insulator and a Z2 topological insulator.
30
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 Z2 topological insulator state in
Bi1−x Sbx and Bi2 Se3 bulk materials.
2.1
Spin-integrated Angle-resolved photoemission
spectroscopy
Angle-resolved photoemission spectroscopy (ARPES) is a direct experimental technique to observe the distribution of the electrons (more precisely, the density of singleparticle electronic excitations) in the reciprocal space of solids [133, 134]. 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
2.1 SPIN-INTEGRATED ANGLE-RESOLVED PHOTOEMISSION SPECTROSCOPY
Figure 2.1:
sample.
The three-step model used to describe the photoemission process inside the
• 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
1p
2mEkin sin θ cos θ
~
1p
ky =
2mEkin sin θ sin θ
~
1p
2mEkin cos θ
kz =
~
kx =
(2.1)
(2.2)
(2.3)
The conservation of energy gives:
Ekin = hν − φ − |EB |
(2.4)
32
Figure 2.2: (a) The Scienta analyzer setup used to measure emitted photoelectrons from
the 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 detector
relative to the sample surface. The momentum of the electron inside the sample can be
extracted from the measured values of Ekin , θ, φ.
2.2 SPIN-RESOLVED ANGLE-RESOLVED PHOTOEMISSION SPECTROSCOPY
where φ is the work function of the sample and EB 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) Ef has an
energy offset of V0 with respect to the kinetic energy measured by the analyzer Ekin ,
namely
Ef = Ekin + V0
(2.5)
where V0 is usually referred as the inner potential.
Since the in-plane momentum is conserved, thus we know that ~kk =
Moreover, we know that
~2 k k 2
2m
+
~2 k ⊥ 2
2m
√
2mEkin sin θ.
= Ef . Therefore, from these relationships, we
get:
k⊥ =
1p
2m(Ekin cos2 θ + V0 )
~
(2.6)
where V0 is usually referred as the inner potential.
The value of V0 can be determined experimentally by measuring the periodicity
of the energy dispersion along the out-of-plane momentum k⊥ direction at the normal 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 spectroscopy
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
34
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 spinorbit 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 targets 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 Bi2 Se3 at a fixed energy E along a fixed momentum space direction 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 detector 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 magnetization. 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.
2.2 SPIN-RESOLVED ANGLE-RESOLVED PHOTOEMISSION SPECTROSCOPY
Figure 2.3: Mott polarimetry. Schematic of a Mott scattering geometry. Incident
electrons (red dot) with a polarization P~ on a high Z nucleus (yellow dot) are backscattered
to the left and right with a probability that is dependent on P~ . The figure is adapted from
Ref. [136].
2.2.1
Mott polarimetry
The interaction Hamiltonian between a photon and spin
1
2
electron can be described
by the Dirac equation:
Hint =
1
e~
e~ 2
~ + ie~ E
~ · p~ − e~ ~σ · (E
~ × p~) (2.7)
+ eΦ −
(~p − A)
~σ · (▽ × A)
2
2
2m
c
2mc
4m c
4m2 c2
~ is the photon vector potential, Φ is the scalar
where ~p is the electron momentum, A
~ is the electric field and ~σ is the electron spin. However by using linearly
potential, E
polarized photons in the UV to soft x-ray regime, it has been shown ( [135]) that
the spin dependent terms are greatly suppressed, and the photon electron interaction
e ~
A · p~,
Hamiltonian can be well approximated by the Schrodinger model Hint = − mc
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 ( [135]) is a method of separating electrons
of different spin from such a photoemitted beam based on the use of spin-orbit (Mott)
36
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 Ze.
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
~ of the nucleus can result
is large, the velocity ~v of the electron in the electric field E
~ in its rest frame given by:
in a considerable magnetic field B
~ = − 1 ~v × E
~
B
c
(2.8)
~ = (Ze/r 3 )~r, can be written as
which, using E
~ = Ze ~r × ~v = Ze L
~
B
cr 3
mcr 3
(2.9)
~ = m~r × ~v is the electron orbital angular momentum. The interaction of
where L
~ creates a spin-orbit (L
~ · S)
~ term in the
this magnetic field with the electron spin S
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 (< Sx >
, < Sy >, < Sz >), 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.
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 energy
and momentum analyzed using a hemispherical electrostatic analyzer and are alternately
deflected at a frequency of 1 Hz into two orthogonally mounted Mott polarimeters. The dual
polarimeter system is shown rotated by 90◦ for clarity. (b) The relationship between the
sample and Mott coordinate systems. When θ and φ are both zero, the sample coordinates
can 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. [136].
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 Pz = (N↑ − N↓ )/(N↑ + N↓ ). When the
scattering 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) =
NL − NR
NL + NR
(2.12)
where NL and NR are the number of electrons scattered to the left and right
respectively through an angle θ. Substituting the relations NL ∝N↑ [1+S(θ)]+N↓ ][1−
S(θ)] and NR ∝N↑ [1 − S(θ)] + N↓ ][1 + S(θ)] derived from Equation 5 to Equation 7
38
2.2.2 VLEED
yields:
Pz =
Az (θ)
S(θ)
(2.13)
which shows that given the Sherman function, measurement of Az (θ) yields Pz . In
a single Mott polarimeter, it is therefore possible to measure two orthogonal spin components 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 intensities 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 destructive
interference 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 enhance 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
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: Schematic
layout of hemispherical energy analyzer with VLEED spin polarimeter. A normal hemispherical electron analyzer (HA 50 are operated by the HAC 300 unit) is first used to detect
the energy and the momentum of the electronic states. The spin polarimeter requires three
additional voltage supplies EP , lens, and Vsc . The long dashed line shows light incidence,
the short dashed line shows an electron trajectory. Channeltron 1 (labeled ch. 1) is for
acquiring non spin-resolved data when the scattering target is removed. Channeltron 2 is
for 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 scattering target and photoemission source, empty symbols for antiparallel magnetization; (b)
asymmetry; and (c) figure of merit. The figure is adapted from Ref. [137].
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
40
2.2.2 VLEED
channeltron was replaced by a three-element electron lens, which transfers the energyanalyzed electrons to a drift chamber. The scattering target is located on the optical
axis in the drift chamber. The drift chamber is equipped with coils for magnetizing
the Fe target. The coils are made of self-supporting 1 mm Cu wire with four turns of
20 − 25 mm diameter. During data acquisition they are kept at the same potential
as the scattering chamber and the target.
The right panels of Fig. 2.5 demonstrate the ability of such Fe VLEED detector
for measuring spin polarization. Polarized electrons were excited from a magnetized
Fe sample by 90 eV photons. Such polarized electrons are injected to the VLEED
detector with the magnetization of the Fe target set for one direction. The counts
for the scattering at different binding energies are recorded by the empty triangles
in the top right panel of Fig. 2.5. Now we flip the magnetization of the Fe target
in the VLEED detector, and perform the same measurements, where the counts are
recorded by the solid triangles in the top right panel of Fig. 2.5. It can be seen
that a clear asymmetry between these two scattering geometries is observed, which
demonstrates that the scattering of a polarized electron beam is quite sensitive to
the magnetization direction of the VLEED target. Thus the spin polarization of the
incidnet electrons can be measured based on this principle.
In reality, although Fe (001)/Ag(100) surface has a very high spin dectection
efficiency as a VLEED detector, one complication is that the efficiency is strongly
dependent on the quality of the Fe (001) surface [137]. And furthermore, the Fe
(001)/Ag(100) surface quality is found to degrade quite fast even within a day [137].
Thus one has to re-prepare the surface once per 12 hours which is inconvenient. In
reality, the preoxidized Fe(001)-p(1 × 1)-O targets are often used because they are
found to be stable for a much longer time than a clean Fe surface, presumably for
weeks. For example, the VLEED spin-resolved ARPES endstation at the Hiroshima
Synchrotron Radiation Center (Hisor) in Hiroshima University in Japan uses the
2.3 SAMPLE PREPARATION FOR ARPES
preoxidized Fe(001)-p(1 × 1)-O targets as the spin detector [138].
2.3
2.3.1
Sample preparation for ARPES
Single crystal bulk samples
Figure 2.6: A schematic of the single crystal bulk sample geometry mounted inside the
ARPES measurement chamber. The ceramic top post was used to cleave the sample surface
in 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
42
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 a
copper 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-post
with 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 Prof. N. Sarmarth’s group in the Department of Physics in The Pennsylvania State University,
and Prof. S. Oh’s group in the Department of Physics & Astronomy, The State
University of New Jersey.
For MBE samples, they are usually too thin (≤ 40 nm) to be cleaved. Let us
take the example of thin Bi2 Se3 films grown on some substrates [e.g. GaAs(111)]. In
order to protect the surface from oxidation, a thick Se capping layer is deposited on
the Bi2 Se3 thin film immediately before taken out of the MBE growth chamber [see
Fig. 2.7(a)]. Such a Se capped sample is then loaded into the ARPES chamber and
pumped down to high vacuum. In order to reveal the clean Bi2 Se3 surface needed
for photoemission measurements, the MBE-grown thin films are heated up inside
the ARPES chamber to ∼ 250◦ C to blow off the Se capping layer on top of the
Bi2 Se3 film. During the annealing, the pressure goes up and typically remains better
than 1 × 10−9 torr. Fig. 2.7(b) demonstrates the decapping process by ARPES core
level spectroscopies measurement. Only selenium core level is observed before the
2.3 SAMPLE PREPARATION FOR ARPES
Figure 2.7: Demonstration of sample surface preparation (decapping) procedure
for the Bi2 Se3 MBE thin films in ARPES measurements. (a) Sample layout of
the MBE grown Bi2 Se3 films. (b) Core level spectroscopies on MBE thin film before and
after the decapping procedure. (c) typical ARPES dispersion mapping and corresponding
energy dispersion curves (EDCs) of Bi2 Se3 thin film along the M̄ − Γ̄ − M̄ momentum-space
cut-direction.
decapping process (blue curve), whereas both selenium and bismuth peaks are shown
after the decapping (red curve). ARPES measurements are then performed on the
clean thin film surface. Fig. 2.7(c) shows a typical ARPES measured dispersion
mappings after decapping of a Bi2 Se3 thin film, where the sharp ARPES spectrum
demonstrates our surface preparation procedure.
44
2.4
Probing the topological number in 3D bulk
materials
Since the topological number is defined by calculating a global quantity (an integral,
a Pfaffian) of the bulk electronic wavefunction over the entire BZ, it is not feasible
to measure its value in experiments based on this definition. Rather, the topological
number and topological properties are usually probed by studying the boundary of a
topological system. Such an approach takes advantage of a fundamental consequence
of the topological classification of gapped band structures, which is the existence
of gapless conducting states at interfaces where the topological invariant changes.
For example, in the case of 2D IQH effect, the quantized transverse conductance
provides a measure of the system’s topological number - the Chern number n. How
do we probe the topological number in a 3D bulk material? This is particularly an
important question to ask in three-dimensions since a 3D bulk topological material
is believed to exhibit no quantized transport response. In this section, we use the
example of the discovery of the 3D Z2 TI phase in Bi1−x Sbx and Bi2 Se3 classes of
materials, to elaborate the way of measuring a 3D topological number by the spinand angle-resolved photoemission spectroscopy (spin-ARPES). For the sake of the
simplicity of the presentation, we put the Bi2 Se3 class before Bi1−x Sbx although it
was discovered later, because Bi2 Se3 has a simpler surface state that is more suitable
for elaborating the methodology.
2.4.1
Separation of insulating bulk from metallic surface states
using incident photon energy modulated ARPES
In order to probe its topological number states in a 3D Z2 TI via studying its surface
states using spin-ARPES, the first thing is to separate the surface states from the
bulk bands, which both exist in the electronic structure near the Fermi level. This
2.4 PROBING THE TOPOLOGICAL NUMBER IN 3D BULK MATERIALS
is achieved by measuring the electronic dispersion along the out-of-plane kz direction
by varying the incident photon energy of the ARPES.
We recall that the out-of-plane momentum value is obtained in ARPES by kz =
p
1
2m(Ekin cos2 θ + V0 ). We know that Ekin = hν − φ − |EB |. At the Fermi level
~
(EB = 0) and the normal emission kk = θ = 0, we have
kz =
1p
2m(hν − φ + V0 )
~
(2.14)
From the equation above, it can be immediately seen that the kz value of the electronic
state at the Fermi level and the normal emission is a function of the incident photon
energy value hν. Effectively, by varying the incident photon energy value, one can
effectively probe the electronic states at different kz values.
The topological surface state in 3D TIs is a type of 2D state that are localized
on the surface. Therefore, it is expected to be strictly non-dispersive along the kz
direction that is perpendicular to the sample surface normal. On the other hand,
the bulk conduction/valence bands are (3D) bulk bands. Thus they are expected to
show dispersion along all three momentum space directions including kz . Therefore,
by measuring the dispersion along the kz direction, one can effectively isolate the
topological surface states from the bulk bands.
Fig. 2.8 shows an ARPES study on Bi2 Se3 as a function of the incident photon
energy. As shown in Fig. 2.8(a). there are three major features in the measured
electronic structure: (1) a parabolic intensity continuum at 0≤EB ≤ −0.15 eV; (2) an
“X” shaped band that runs from 0 eV to -0.35 eV; (3) an “M” shaped band at higher
binding energies (EB ≥ 0.4 eV). As photon energy value is changed, It can be seen
that the “X” shaped band shows no change in its dispersion, whereas the a parabolic
intensity continuum at the Fermi level and the an “M” shaped band are dispersive.
The strong kz dispersion of the bulk valence band at binding energies higher than
46
2.4.2 Surface Dirac crossing, surface-bulk connectivity, and spin-momentum locking
Figure 2.8: Topological Surface States and electronic band dispersion along the
kz -direction in momentum space. (a) A schematic diagram of the full bulk threedimensional BZ of Bi2 Se3 and the two-dimensional BZ of the projected (111) surface. (b)
Electronic dispersion (EB vs kk ) of Bi2 Se3 at different incident photon energies. (c) The
energy distribution curves at the Γ̄ (kk = 0) at different photon energies, showing the strong
kz dispersion of the bulk valence band at binding energies higher than −0.4 eV. The figure
is adapted from Ref. [32].
−0.4 eV is better visualized in Fig. 2.8(c). Therefore, these incident photon energy
measurements demonstrate that the “X” shaped Dirac band is a 2D surface state,
whereas the other two features originate from the bulk (the bulk conduction and the
valence bands).
2.4.2
Surface Dirac crossing, surface-bulk connectivity, and
spin-momentum locking
Now that we have identified the surface states in Bi2 Se3 , we experimentally show
the following properties of the surface states to demonstrate the strong topological
insulator state with ν0 = 1 in Bi2 Se3 :
• The two branches (left-moving and right moving) of the surface states cross
each other, forming a surface Dirac point [Figs. 2.9(c,d)].
• The Dirac surface states span across the bulk energy gap, and connect the bulk
conduction and the bulk valence bands [Figs. 2.9(c,d)].
2.4 PROBING THE TOPOLOGICAL NUMBER IN 3D BULK MATERIALS
Figure 2.9: Detection of Z2 (symmetry protected) Topological number in Bi2 Se3 :
spin-momentum locking of spin-helical Dirac electrons in Bi2 Se3 and Bi2 Te3
using spin-resolved ARPES. (a) ARPES intensity map at EF of the (111) surface of
tuned Bi2−δ Caδ Se3 (see text) and (b) the (111) surface of Bi2 Te3 . Red arrows denote the
direction of spin around the Fermi surface. (c) ARPES dispersion of tuned Bi2−δ Caδ Se3
and (d) Bi2 Te3 along the kx cut. The dotted red lines are guides to the eye. (e) Measured y
component of spin-polarization along the Γ̄-M̄ direction at EB = −20 meV, which only cuts
through the surface states. Inset shows a schematic of the cut direction. (f ) Measured x (red
triangles) and z (black circles) components of spin polarization along the Γ̄-M̄ direction at
EB = -20 meV. (g) Spin-resolved spectra obtained from the y component spin polarization
data. (h) Fitted values of the spin polarization vector P. The figure is adapted from Ref. [34],
except Panel (g) which is adapted from Ref. [32].
48
2.4.2 Surface Dirac crossing, surface-bulk connectivity, and spin-momentum locking
• The surface state Fermi surfaces enclose the Kramers points of the surface BZ
[Figs. 2.9(a,b)].
• The surface state spin is locked with its momentum [Figs. 2.9(e,f)].
• There are an odd number of such spin-momentum locked surface state Fermi
surfaces [Fig. 2.9(g)].
We note that the first two points guarantee that the surface state cannot be
adiabatically removed from the bulk band-gap; the third and forth points reveal that
the surface states (therefore the topological phase) are protected by the time-reversal
symmetry; the last point shows that Bi2 Se3 is a strong not a weak TI.
Similarly, we can use ARPES and spin-ARPES to probe the Z2 topological state
in Bi0.91 Sb0.09 .
• The two branches (left-moving and right moving) of the surface states cross each
other, forming a surface Dirac point [Figs. 2.10(c,f)]. Now there are surface band
crossings both at the Γ̄ and the M̄ Kramers points.
• The Dirac surface states span across the bulk energy gap, and connect the bulk
conduction and the bulk valence bands [Figs. 2.10(c,f)].
• The surface state Fermi surfaces enclose the Kramers points of the surface BZ
[Fig. 2.10(d)].
• The surface state spin is locked with its momentum [Fig. 2.10(g)].
• There are an odd number of such spin-momentum locked surface state Fermi
surfaces [Fig. 2.10(d)].
We put Bi0.91 Sb0.09 after Bi2 Se3 , even though the experiments on Bi0.91 Sb0.09
were earlier than Bi2 Se3 , because the topological surface states in Bi0.91 Sb0.09 is relatively more complicated. There are multiple (5) pieces of surface state Fermi surface
2.4 PROBING THE TOPOLOGICAL NUMBER IN 3D BULK MATERIALS
Figure 2.10: Spin texture of the topological surface states in Bi0.91 Sb0.09 encodes
its Z2 topological number of the bulk (2008). (a) Schematic sketches of the bulk
Brillouin zone (BZ) and (111) surface BZ of the Bi1−x Sbx crystal series. (b) Schematic
of Fermi surface pockets formed by the surface states (SS) of a topological insulator that
carries a Berry’s phase. (c) Partner switching band structure topology. (d) Spin-integrated
ARPES intensity map of the SS of Bi0.91 Sb0.09 at EF . Arrows point in the measured
direction of the spin. (e) High-resolution ARPES intensity map of the SS at EF that
enclose the M̄1 and M̄2 points. Corresponding band dispersion (second derivative images)
are shown below. The left right asymmetry of the band dispersions are due to the slight
offset of the alignment from the Γ̄-M̄1 (M̄2 ) direction. (f ) Surface band dispersion image
along the Γ̄-M̄ direction showing five Fermi level crossings. (g) Spin-resolved momentum
distribution curves presented at EB = −25 meV showing single spin degeneracy of bands
at 1, 2 and 3. Spin up and down correspond to spin pointing along the +ŷ and -ŷ direction
respectively. (h) Schematic of the spin-polarized surface FS observed in our experiments.
This figure is adapted from Ref. [31].
50
2.4.2 Surface Dirac crossing, surface-bulk connectivity, and spin-momentum locking
[Fig. 2.10(d)]. The surface states connect the bulk conduction and valence bands in
a more complicated path [Figs. 2.10(c,f)]. The dispersion of surface states are bent
away from the surface band crossing. Therefore, the description of a Dirac crossing is
only valid at a very small k−space in the close vicinity of the Kramers points. However, despite all these complicities, it can be seen that by measuring the electronic
and spin groundstate of the surface electrons, the Z2 topological number (ν0 = 1) can
be experimentally demonstrated in Bi0.91 Sb0.09 .
The above methodology (using spin-ARPES to electronic and spin groundstate
for the surface states of a bulk material) can be more generally applied to detect
the other topological number in a bulk material, and therefore to discover new topological phases of matter. In general, there are two important logic steps: (1) To
show that the surface states have band crossings and they connect across the bulk
band-gap from the conduction to the valence bands. This basically demonstrates
the topological nontriviality of the surface states, because such kind of surface states
cannot be removed from the band gap unless one closes the bulk band-gap (requires
a band inversion) or opens up a gap at the surface band crossing (requires breaking
a symmetry). (2) To study the spin polarization texture as well as the Fermi surface
of the surface states. This reveals the symmetry that protects the topological phase
and its surface states are protected and also provides an experimental measure of the
associated topological number.
Chapter 3
Topological Quantum Phase
Transition in BiTl(S1−δ Seδ )2
A topological material is a distinct phase of matter since it features a nonzero topological number in the bulk electronic structure groundstate that can only be removed by
a topological quantum phase transition into a trivial phase. The topological number
guarantees the existence of protected surface states, which serve as the experimental
signature for the nontrivial topology. In this chapter, we present our experimental realization of a topological quantum phase transition (TQPT) from a conventional band
insulator to a 3D Z2 topological insulator in the BiTl(S1−δ Seδ )2 system. We systematically study the evolution of surface and bulk electronic and spin groundstate across
the topological phase transition. This study provides a general paradigm for understanding how various topological phases and their protected surface states arise from
a conventional material in experiments, and is of value for searching for new and engineered topological states in real materials. Our study is further suggestive for realizing
many fascinating physics such as higher dimensional Dirac fermions [120, 139], Weyl
fermions under magnetization [119, 120, 139–141], supersymmetry SUSY state [128]
and interacting topological states [142] predicted in the vicinity of the critical point
52
of a topological quantum phase transition.
3.1
Evolution of the electronic groudstate across a
topological quantum phase transition
As we have discussed above, if the system changes from a conventional band insulator
to a 3D Z2 topological insulator, it must go through a band inversion process. The
basic evolution of the bulk electronic structure across a topological quantum phase
transition from a conventional band insulator to a 3D Z2 TI have been theoretically
discussed in Ref. [120]. Here we state the conclusions in Ref. [120] without going into
the details of the formulations.
Figure 3.1: Phase transition in 3D between the conventional band insulator to the 3D Z2 TI
phases for inversion-symmetric and inversion-asymmetric cases. In the inversion symmetric
case, all bands are doubly degenerate.
Since time-reversal symmetry is indispensable for a Z2 TI phase, thus here we restrict ourselves on time-reversal-symmetric systems. Besides time-reversal symmetry,
another symmetry that plays an essential role is the space inversion symmetry. This
THE ELECTRONIC GROUDSTATE ACROSS A TOPOLOGICAL QUANTUM PHASE TRANSITION
Figure 3.2: Bulk band-gap across the topological phase transition in 3D between the
conventional band insulator to the 3D Z2 TI phases for inversion-symmetric and inversionasymmetric cases.
is because inversion symmetry can force the bulk bands to be doubly degenerate at all
k-points throughout the BZ, whereas for systems without inversion symmetry, their
bulk bands can be singly degenerate at all k-points except the Kramers point (doubly degeneracy forced by the Kramers theorem). We will discuss the band inversion
process for inversion symmetric and asymmetric cases separately. Finally, we note
that we do not consider any additional space group symmetries (such as mirror symmetries or rotational Cn symmetries). In the case of the presence of these additional
symmetries, then the conclusion can be different again. For example, please refer to
the topological Dirac semimetal phase discussed in Chapter 5.
Suppose we start from a conventional band insulator in 3D, and there exists some
tuning parameter δ (e.g. spin-orbit coupling, lattice constant, chemical composition,
temperature, etc.) that can effectively change the bulk band-gap.
For an inversion symmetric case, the conclusion is shown in Figs. 3.1, 3.2: (0) In an
inversion symmetric crystal, all bulk bands must remain doubly degenerate at all kpoints. (1) As the system goes through the band inversion, its band-gap decreases to
zero and then increases again. (2) There exists only a single value for the δ, where the
band-gap is zero. This is the critical point of the band inversion and the topological
phase transition. At this critical point, the bulk conduction and valence bands cross,
and the bulk band structure can be described by a 3D massless Dirac cone. (3) The
54
bulk band crossing (the 3D Dirac point) must occur at a Kramers point.
For an inversion asymmetric case, the conclusion is shown in Figs. 3.1, 3.2: (0)
In an inversion asymmetric crystal, bulk bands (with consideration of the spin-orbit
coupling) are usually singly degnerate except at the Kramers points. (1) As the
system goes through the band inversion, its band-gap decreases to zero and then
increases again. (2) There exists a range of δ values for the where the band-gap is
zero. Thus the gapless phase is not a single point by a region along the δ axis. The
gapless region is called the Weyl semimetal phase, because the low energy excitation
from the bulk bands are described by the Weyl equation. This is called a Weyl
semietal phase because one can actually identify a distinct topological number to it,
and therefore it is a new topological phase of matter. (3) The bulk band crossings
(the 3D Weyl points) will not occur at any Kramers points.
The Weyl semimetal phase in inversion asymmetric systems are of great current
interest, which remains experimentally elusive, and many groups are trying to realize
it. On the other hand, in the following sections in this chapter, we only consider
systems with inversion symmetry because it is directly relevant to our experiments
(BiTl(S1−δ Seδ )2 has inversion symmetry). In the next section, we present the experimental work that demonstrates the general frame of the band inversion and the
topological phase transition in BiTl(S1−δ Seδ )2 . Then in the following section, we focus on the vicinity of the critical point of the topological phase transition, and try
to reveal the formation mechanism of the topological surface states (how protected
surface states arise from a conventional material) by going through a topologicalcritical-point.
3.2 TOPOLOGICAL QUANTUM PHASE TRANSITION IN BITL(S1−δ SEδ )2
3.2
3.2.1
Topological quantum phase transition in BiTl(S1−δ Seδ )2
Identifying a suitable topological material system
A 3D Z2 topological insulator is a distinct phase of matter that cannot be adiabatically
deformed into a conventional band insulator in a time-reversal invariant condition
without going through a topological quantum phase transition. Within band theory,
the evolution of the bulk electronic groundstate across the topological phase transition
is effectively described the band inversion process. Therefore, realizing and studying
a band inversion and a topological quantum phase transition in real materials is of
critical importance for understanding the 3D Z2 topological insulator phase, and in
general how nontrivial topological phases arise from a conventional band insulator.
In order to realize a band inversion and a topological quantum phase transition
in experiments, there are two critical conditions: (1) one has to identify a real material system, where there exists a material parameter δ that can effectively tune
the bulk band-gap and the topological nature of the system, and (2) there has to be
an appropriate experimental probe, which can reveal the evolution of the electronic
groundstate across the topological transition at all δ values. While point (2) is solved
by the utilization of the spin-ARPES to measure the protected surface states and
their spin texture in bulk topological materials, point (1) had remained experimentally unachieved for quite a long time.
For example, Bi0.91 Sb0.09 is the first 3D topological insulator realized in experiments [30]. It has a single band inversion at the T point of the bulk BZ. However,
Bi1−δ Sbδ is not suitable for realizing a topological phase transition for several reasons:
(1) By going from Bi0.91 Sb0.09 to pure Sb (0.09 ≤ δ ≤ 1), the topological nature of
the system is unchanged. There remains only one band inversion at the T point, and
thus the system remains Z2 topologically nontrivial ν0 = 1. The only change is that
the energy position of the H band shifts up, see Fig. 3.3. Thus while Bi0.91 Sb0.09 is an
56
3.2.1 Identifying a suitable topological material system
Figure 3.3: (a) Bulk 3D Brillouin zone of Bi1−δ Sbδ showing the eight bulk Kramers points
(T , Γ, 3 L, 3 X). (b) Schematic evolution of the near EF band structure from Bi to Sb.
This figure is adapted from Ref. [136]
insulator, pure Sb is a metal. (2) By going from Bi0.91 Sb0.09 to pure Bi (0.09 ≥ δ0),
the topological nature is indeed changed, because it induces another three band inversions at the three L points of the bulk BZ [see Fig. 3.3(a)]. Thus Bi0.91 Sb0.09 is
Z2 topologically nontrivial ν0 = 1, whereas pure Bi is Z2 topologically trivial ν0 = 0.
However, the reality is much more complex. Since there do exist four band inversions
at the 1T and 3L points in pure Bi, and furthermore these band inversions coincide
with the mirror planes of the Bi crystal, therefore pure Bi does have surface states
within its partial energy gap (energy gap at each k point) and these surface states
can be protected by the mirror symmetries [117] (although pure Bi is Z2 topologically
trivial). Moreover, pure Bi is also a metal lacking a global energy gap. Furthermore,
the Bi0.91 Sb0.09 (ν0 = 1) is known to have multiple pieces of surface state Fermi surfaces (both at the Γ̄ and the M̄ points), but so is pure Bi (ν0 = 0). These facts
suggest that the Bi1−δ Sbδ is quite complicated both in terms of its evolution of the
topology and in terms of its surface state electronic structure as a function of δ. Thus
it is difficult to demonstrate a prototypic topological phase transition in the Bi1−δ Sbδ
system.
Another important topological insulator material is Bi2 Se3 discovered in 2009 [32,
33]. It has a layered rhombohedral structure with the space group of R − 3m. In Ref.
[33], Zhang et al. further predicted that Sb2 Se3 is topologically trivial. Thus it seems
3.2 TOPOLOGICAL QUANTUM PHASE TRANSITION IN BITL(S1−δ SEδ )2
like (Bi1−δ Sbδ )2 Se3 is a promising system for a topological phase transition. However,
it is important to note that Sb2 Se3 in fact crystalizes in a very different crystal
structure (an orthorhombic structure with the space group of P nma) [143]. Thus
by going from Bi2 Se3 to Sb2 Se3 , the system will go through a structural transition,
where the electronic structure is dramatically changed. Therefore, the (Bi1−δ Sbδ )2 Se3
system is also not ideal for the realization of a prototypic topological phase transition.
3.2.2
Topological quantum phase transition in BiTl(S1−δ Seδ )2
Figure 3.4: (a) Bulk 3D Brillouin zone of BiTl(S1−δ Seδ )2 showing the eight bulk Kramers
points (T , Γ, 3 L, 3 X), and their (111) surface projections. (b) Fermi surface map over
the entire first surface BZ of the (111) surface of BiTlSe2 (δ = 1). Only a single surface
state Fermi surface at the Γ̄ point is observed. (c) ARPES dispersion maps of the single
Dirac cone surface state at the Γ̄ point at three different incident photon energies.
Our interest on the BiTl(S1−δ Seδ )2 system starts from its selenium end compound.
BiTlSe2 (δ = 1) was identified as a 3D Z2 TI with a single Dirac cone surface state
at the surface BZ center Γ̄ point [144–148], which is very similar to Bi2 Se3 . However,
interestingly, we found that unlike in Bi2 Se3 where the crystal structure changes by going from Bi2 Se3 to Bi2 S3 or Sb2 Se3 , the BiTl(S1−δ Seδ )2 system remains the same crystal structure (Fig. 3.5) throughout the compositional range going from BiTl(S1 Se0 )2
to BiTl(S0 Se1 )2 . Thus by going from the sulfur end to the selenium end, the two
major changes are the change of the lattice constant [Fig. 3.5(b)] and the strength of
spin-orbit coupling (since Se is heavier than S). Both changes are adiabatic, leading
58
3.2.2 Topological quantum phase transition in BiTl(S1−δ Seδ )2
Figure 3.5: (a) The crystal structure of the BiTl(S1−δ Seδ )2 system. (b) In-plane lattice
constant a and out-of-plane lattice constant c determined by x-ray diffraction measurements.
The crystal structure remains the same, only the lattice constant value is found to change
as a function of the composition δ.
to an adiabatic evolution of the bulk electronic groundstate. These facts inspires us to
systematically study the BiTl(S1−δ Seδ )2 system as a function of chemical composition
in order to search for the experimental realization of a topological phase transition.
Fig. 3.6(a) presents systematic photoemission measurements of surface and bulk
electronic states that lie between a pair of time-reversal invariant points or Kramers
points (Γ̄ and M̄ ) obtained for a series of compositions δ of the spin-orbit material
BiTl(S1−δ Seδ )2 . As the selenium concentration is increased, the low-lying bands separated by a gap of energy 0.15 eV at δ = 0.0 are observed to approach each other
and the gap decreases to less than 0.05 eV at δ = 0.4. The absence of surface states
(SSs) within the bulk gap demonstrates that the compound is topologically trivial for
composition range of δ = 0.0 to δ = 0.4. Starting from δ = 0.4, a linearly dispersive
band connecting the bulk conduction and valence bands emerges which threads across
3.2 TOPOLOGICAL QUANTUM PHASE TRANSITION IN BITL(S1−δ SEδ )2
the bulk band gap. Moreover, the Dirac-like bands at δ = 0.6. The system enters
a topologically non-trivial phase upon the occurrence of an electronic transition between δ = 0.4 and δ = 0.6. While the system approaches the transition from the
conventional side (δ = 0.6), both energy dispersion and FS mapping (Fig. 3.6(a) for
δ = 0.4) show that the spectral weight at the outer boundary of the bulk conduction
band continuum which corresponds to the loci where the Dirac SSs would eventually
develop becomes much more intense; however, that the surface remains gapped at
δ = 0.4 suggests that the material is still on the trivial side. A critical signature
of a topological transition is that the material turns into an indirect bulk band gap
material. As δ varies from 0.0 to 1.0 [Fig. 3.6(c)], the dispersion of the valence band
evolves from a “Λ”-shape to an “M”-shape with a “dip” at the Γ̄ point (k = 0); the
δ = 0.0 compound features a direct band gap in its bulk, whereas the δ = 1.0 indicates a slightly indirect gap. These systematic studies demonstrate the existence of
the bulk band inversion and the topological phase transition between a conventional
band insulator and a Z2 topological insulator in the BiTl(S1−δ Seδ )2 system. This is
the first experimental realization of a topological phase transition in a tunable band
insulator system, where the evolution of the surface and bulk electronic groundstate
can be systematically tuned and visualized [37].
In order to reveal the band inversion process responsible for the topological transition, we track the energy positions of the bulk conduction band minimum and the
valence band maximum at various compositions. As shown in Fig. 3.7, an inversion
+
between the bulk conduction and valence bands (the Γ−
4 and Γ4 bands) is clearly vi-
sualized. Using linear interpolation, we estimate the critical point of this topological
phase transition to be at composition value δc = 0.5 ± 0.05. In the next section, we
will systematically study the surface electronic and spin groundstate near the critical
point with finer compositional steps. Nevertheless, the work presented in Figs. 3.6, 3.7
sets the foundation for the realization of the topological quantum phase transition in
60
3.2.2 Topological quantum phase transition in BiTl(S1−δ Seδ )2
BiTl(S1−δ Seδ )2 .
3.2 TOPOLOGICAL QUANTUM PHASE TRANSITION IN BITL(S1−δ SEδ )2
Figure 3.6: Topological quantum phase transition in BiTl(S1−δ Seδ )2 . (a) Highresolution ARPES dispersion maps along the Γ̄ − M̄ momentum space line, from a conventional band insulator (left panel) to a topological insulator (right panel). Band insulators
and topological insulators are characterized by a different Z2 topological number ν0 = 0,
and ν0 = 1, respectively. (b) ARPES Fermi surface maps at different compositional values.
(c) Left and right panels: Energy-distribution curves for stoichiometric compositions δ = 0
and δ = 1, respectively. Center panels: ARPES spectra indicating the bulk band-gap and
Dirac node for compositions δ = 0.2 to δ = 0.8. (c) Evolution of surface and bulk electronic
ground state imaged over as a function of energy (vertical axis), spin (arrows), and in-plane
momentum kx , ky (horizontal plane). Spin textures are indicated by yellow arrows above
the Dirac node and green arrows below the Dirac node. Each arrow represents the net
polarization direction on a k-space point on the corresponding Fermi surface. The figure is
adapted from Ref. [37].
62
3.2.2 Topological quantum phase transition in BiTl(S1−δ Seδ )2
Figure 3.7: Bulk band inversion process in BiTl(S1−δ Seδ )2 . A band inversion is revealed by tracking the ARPES measured energy positions of the conduction band minimum
and valence band maximum at various composition δ values. The figure is adapted from
Ref. [37].
3.2 TOPOLOGICAL QUANTUM PHASE TRANSITION IN BITL(S1−δ SEδ )2
3.2.3
Spin-orbit coupling vs. lattice constant
Figure 3.8: Comparing the role of spin-orbit coupling and lattice constant in the
topological phase transition in BiTl(S1−δ Seδ )2 . (a) Bulk band structure calculation of
BiTl(S0 Se1 )2 and BiTl(S1 Se0 )2 . Top row: Calculated Band structure will full SOC. Bottom
row: Calculated Band structure with S or Se SOC setting to zero. Black (parity −) and red
(parity +) represent orbital symmetries associated with the parity eigenvalue at the high
symmetry points. (b) Lattice constant change as a function of composition.
We try to determine what is the pivotal tuning parameter that drives the topological phase transition in BiTl(S1−δ Seδ )2 . As we have discussed above, as the composition δ is varied, the two major changes in the system are the spin-orbit coupling
strength (since Se is heavier than S) and the lattice constant. Here we evaluate which
one plays the decisive role in this topological transition. Fig. 3.8(a) shows the bulk
band structure calculation with the parity eigenvalues ± of the lowest conduction and
valence bands. Let us first focus on the top row of Fig. 3.8(a), which are calculations
with full spin-orbit coupling strength. It can be seen from Fig. 3.8(a) that the Se
end compound BiTl(S0 Se1 )2 (δ = 1) shows a parity inversion, consistent with a nontrivial TI phase, whereas the S end compound BiTl(S1 Se0 )2 (δ = 0) does not show
parity inversion, consistent with a topologically trivial phase. Now, in the bottom
64
3.2.3 Spin-orbit coupling vs. lattice constant
row of Fig. 3.8(a), we turn of the spin-orbit coupling in S and Se, only the spin-orbit
coupling of the Bi and Tl atoms are considered. However, as seen in the bottom
row of Fig. 3.8(a), the results remain the same. Therefore, these calculation results
demonstrate that the increase of spin-orbit coupling from S to Se does not play a
major role in the topological phase transition in BiTl(S1−δ Seδ )2 . The lattice constant
change, on the other hand, is more decisive.
3.2 TOPOLOGICAL QUANTUM PHASE TRANSITION IN BITL(S1−δ SEδ )2
3.2.4
3D Dirac semimetal state at the critical point
Figure 3.9: 3D Dirac semimetal state at the critical point of the topological
phase transition in BiTl(S1−δ Seδ )2 . (a) Theoretically calculated band inversion and
topological phase transition in the BiTl(S1−δ Seδ )2 system. Inset: a 3D Dirac semimetal
state at the critical point of this phase transition is seen in calculated bulk band structure
(b) ARPES results at the composition δ = 0.5, which corresponds to the critical point.
We also note that a 3D Dirac semimetal state is realized at the critical point of
the topological phase transition in BiTl(S1−δ Seδ )2 . This has been discussed in the
first section of this chapter (see section 3.1). Since BiTl(S1−δ Seδ )2 has inversion symmetric, according to the discussion in section 3.1, the bulk band-gap goes to 0 at
only one value of the tuning parameter δ, which corresponds to the critical point
of this topological phase transition. At the critical point, the bulk band structure
is exptected to be described by a 3D Dirac fermion state, where the band disperses
66
3.2.4 3D Dirac semimetal state at the critical point
linearly along all three momentum space directions. This is indeed seen in our theoretical calculation shown in Fig. 3.9(a). Furthermore, our ARPES measurements of
the critical composition δ = 0.5 also show a spectrum consistent with a 3D Dirac
cone [Fig. 3.9(a)]. But as we will systematically discuss in the next section, the real
situation is richer. While the bulk band structure is indeed described by a 3D Dirac
cone, the surface electronic ground state show exotic “preformed” behavior (see next
section).
3.3 TOPOLOGICAL-CRITICAL-POINT AND THE “PREFORMED” SURFACE STATES
3.3
Topological-critical-point and the “preformed”
surface states
After establishing the topological phase transition and band inversion in the BiTl(S1−δ Seδ )2 ,
in this section we systematically study the electronic and spin groundstate of the
BiTl(S1−δ Seδ )2 near the critical point of the topological transition. Studying the
topological-critical-point of a topological phase transition is of fundamental importance for the following reasons:
Understanding the physics of distinct phases of matter is one of the most important
goals in physics in general. For a new phase of matter, a powerful route toward such
understanding is to study the way it arises from an understood state by investigating
the nature of a phase transition. Studying the phase transition critical point not
only deepens our understanding of the new phase, but also sometimes even produces
surprising new phenomena. A well-known example is the study of the superconducting
phase transition between the high-Tc d−wave cuprate superconductor and the normal
state, which leads to the surprise of the pseudo-gap state with the concept of the
“preformed” Cooper pairing. In our context, a topological insulator is a distinct phase
of matter that cannot be adiabatically connected to a conventional material without
going through a topological quantum (T → 0 K) phase transition (TQPT), which
involves a change of the bulk topological invariant. Inspired by the discovery of 3D Z2
topological insulator state [2–9,11–13], many new topological phases of matter, such as
a topological crystalline insulator [39–41], a topological Kondo insulator [49–52, 118],
a topological Dirac/Weyl semimetal [43–45,119,120,139], etc, have just been predicted
or realized. All these phases are predicted to feature protected surface states, which
serve as the experimental signature for their nontrivial topology in the bulk, and they
are in fact formed via TQPTs and need to be understood in real materials. Therefore,
it is of general importance to study how protected surface state emerge from a trivial
68
material by crossing the topological-critical-point (TCP) of a TQPT.
As an example, for a Z2 topological insulator, it is well established that the odd
number of Dirac surface states and their spin-momentum locking are the signature
that distinguishes it from a conventional insulator. However, an interesting and vital question that remains unanswered is how topological surface states emerge as a
non-topological system approaches and crosses the TCP. One might imagine that
there are neither surface states nor spin polarization in the conventional insulator
(non-topological) regime. In this case, the gapless topological surface states and
spin-momentum locking set in abruptly and concomitantly at the topological-criticalpoint. However, here in this section, we report observation of an exotic, unexpected
, and yet intuitive phenomenon that there exists a form of novel precursor to the
topological surface states in the non-topological regime, and the spin-locking is nonconcomitant. Thus they go through a “preformed” state. These preformed surface
states are systematically enhanced and evolve into the actual topological surface states
across the TCP. This is particularly interesting because it can be viewed as a novel
“proximity effect” (in material compositional space) due to the adjacent topological
insulator phase. Therefore, in order to understand the formation of protected surface states in a Z2 TI, and also to establish a general guideline for such formation
in various topological phases, one needs to critically study the electronic and spin
groundstate in the close vicinity of the TCP of a TQPT, with spin and momentm
resolution, in a real topological material.
Understanding the nature of a TCP is also of broad interest because recent theories
have proposed a wide range of exciting quantum phenomena based on topological
criticality. It has been proposed that the TCP of various TQPTs can not only realize
new groundstates such as higher dimensional Dirac fermions [120,139], Weyl fermions
under magnetization [120,139–141], supersymmetry SUSY state [128] and interacting
topological states [142], but also show exotic transport and optical responses such
3.3 TOPOLOGICAL-CRITICAL-POINT AND THE “PREFORMED” SURFACE STATES
as chiral anomaly in magnetoresistence [149] or the light-induced Floquet topological
insulator state [150]. Realization of these very recent theoretical proposals can truly
open a new era in fundamental topological physics [128, 142]. In order to achieve
them in real materials, it is also quite suggestive to study the electronic and spin
groundstate in the vicinity of the TCP in some great detail.
In this section, in order to illuminate the nature of a topological-critical-point,
we systematically study the evolution of electronic and spin groundstate near the
topological-critical-point with a step finer than 2% in the prototype TQPT BiTl(S1−δ Seδ )2
system. The BiTl(S1−δ Seδ )2 system is known to host one of the most basic TQPTs between a conventional band insulator and a 3D Z2 topological insulator (TI) [37,38,75],
and is therefore an ideal platform for our goal.
We present in-plane electronic structure (EB vs kk ) of the BiTl(S1−δ Seδ )2 system
at varying compositions (δ). Fig. 3.10 shows that the two end compounds (δ = 0.0
and 1.0) are in clear contrast, namely, δ = 0.0 has no surface states and δ = 1.0 has
surface states connecting the bulk conduction and valence bands, which clearly reveals
the difference between the conventional semiconductor phase and the Z2 topological
band insulator phase, in agreement with previous studies [37, 38]. The conventional
semiconductor state is found to extend from δ = 0.0 to 0.4 [Fig. 3.10(a)], whereas the
topological state is clearly observed from δ = 1.0 to 0.6 [Fig. 3.10(c)]. A small but
observable bulk band-gap of about 30 meV is observed for δ = 0.45 in Fig. 3.10(e),
indicating that the system continues to belong to the conventional semiconductor
phase. Upon increasing δ to the region of 0.475 − 0.525, the bands are found to
further approach each other, and the linear dispersion behavior of the bands is observed to persist at energies all the way across the node (the Dirac point). Thus
based on the observed linear dispersion, the critical composition can be estimated
to be δc = 0.5 ± 0.03. At δ = 0.60 [Fig. 3.10(c)], a clear bulk conduction band is
observed inside the surface states’ upper Dirac cone. Moreover, the bulk conduction
70
and valence bands are separated by an observable bulk gap, which is traversed by the
gapless topological surface states. Thus, our data show that the system belongs to
the topological insulator regime for compositions of δ ≥ 0.60. As for the system lying
very close to the bulk inversion at δ = 0.50 or 0.525, based on the in-plane dispersion
data in Fig. 3.10(b) alone, the nature of the observed Dirac-like band cannot be conclusively determined, because it can be interpreted as two-dimensional topological
surface states or three-dimensional bulk Dirac states [139] expected near the bulk
band inversion. However, one of the two possibilities can be identified by measuring
the dispersion along the out-of-plane kz direction, since the three-dimensional bulk
Dirac states are expected to be highly dispersive [37, 43, 45] (the velocity along kz
−1
direction of the 3D bulk Dirac band at δc is estimated to be ∼ 2.5 eV·Å
[37]),
whereas the 2D surface states are not expected to show observable dispersion along
the kz direction.
Thus, in order to better understand the nature of the bands at compositions near
the TCP, we perform ARPES measurements as a function of incident photon energy
values [Fig. 3.10(e),(f) to probe their out-of-plane kz dispersion. Upon varying the
photon energy, one can effectively probe the electronic structure at different outof-plane momentum kz values in a three-dimensional Brillouin zone. Left panel of
Fig. 3.10(f) shows the incident photon energy (kz ) measurements at δ = 0.525 by the
Fermi surface mapping in kk vs kz momentum space. The straight Fermi lines that
run parallel to the kz axis show nearly absence of observable kz dispersion. Similarly,
incident photon energy measurements are also performed at compositions δ = 0.40
and 0.45, where a clear bulk band-gap is observed (ν0 = 0). Surprisingly, even for
the gapped electronic structure at δ = 0.40, 0.45, our data show clear absence of kz
dispersion. Therefore the observed bands cannot be interpreted as three-dimensional
bulk Dirac bands expected near the bulk band inversion. In fact, Our systematic kz
measurements [(Figs. 3.10(g),(h)]reveal that the electronic structure at δ = 0.40, 0.45
3.3 TOPOLOGICAL-CRITICAL-POINT AND THE “PREFORMED” SURFACE STATES
is largely dominated by the presence of gapped quasi-2D electronic states, which are
found to roughly disperse along the outer boundary of the bulk continuum. Such
anomalously strong quasi-2D states that dominate the low energy surface spectral
weight suggest a possible preformed nature as a result of their proximity to the topological insulator regime.
In order to study the spin properties of the observed anomalous quasi-2D states,
we perform spin-resolved measurements on the system with compositions below and
near the TCP. We present spin-resolved data taken on the composition of δ = 0.40
[Fig. 3.11(a)] and focus on the vicinity of the Fermi level (EB = −0.02 eV). The momentum distribution curves (MDCs) for the spectrum are shown in Fig. 3.11(b), where
the highlighted curve is chosen for spin-resolved (SR) measurements. Fig. 3.11(d)
shows the in-plane SR-MDC spectra as well as the measured in-plane spin polarization along the Γ̄-M̄ and Γ̄-K̄ momentum space cuts. Clear in-plane spin polarization
is observed on the quasi-2D states from Fig. 3.11(d). Furthermore, the measured spin
polarization in Fig. 3.11(d) shows that the spin texture is arranged in a way that spins
have opposite directions on the opposite sides of the Fermi surface. In addition, the
out-of-plane component of the spin polarization along the Γ̄-M̄ and Γ̄-K̄ cuts is shown
in Fig. 3.11(e). No significant out-of-plane spin polarization [Fig. 3.11(e)] is observed
within our experimental resolution. The spin texture configuration can be obtained
from the spin-resolved measurements in Figs. 3.11(d),(e), as schematically shown by
the arrows in Fig. 3.11(c). Surprisingly, our spin-resolved measurements reveal that
these quasi-2D states are not only strongly spin-polarized, but their spin texture near
the native Fermi level resembles the helical spin texture on the topological surface
states as observed in Bi2 Se3 [34].
We present systematic spin-resolved studies to understand the way spin texture
of the quasi-2D states evolves as a function of binding energy EB and composition δ.
Figs. 3.12(a)-(d) show spin-resolved data at different binding energies for a sample
72
with δ = 0.40. The spin-momentum locking behavior is observed at all binding
energies from near the Fermi level (EB = −0.02 eV) to an energy near the conduction
band minimum (EB = −0.32 eV). While the magnitude of the spin polarization on
the Fermi level is found to be around 0.3, the spin polarization magnitude is found to
decrease to nearly zero while approaching small values of momenta near the Kramers’
point Γ̄ (the conduction band minimum). Furthermore, at energies cutting across the
bulk valence band at EB = −0.57 eV, EB = −0.72 eV, the measured spin polarization
profile is clearly reversed, where a right-handed profile is found for the quasi-2D states
on the boundary. In addition, the magnitude of the spin polarization is found to be
increased as the energy is tuned away from the bulk band-gap, which is consistent
with the gapped nature of the quasi-2D states. The observed reduction of net spin
polarization at small momenta and the absence of net spin polarization at the Γ̄ point
are important for the gapped nature of quasi-2D states in δ = 0.4 samples. As for the
gapless case with the system composition at δ = 0.50, the spin-resolved measurements
[Figs. 3.12(e)-(h)] reveal the same helical-like spin texture configuration on the Fermi
level, where the magnitude of the spin polarization is around 0.5 at the Fermi level
in this composition. However, in contrast to the δ = 0.4 case, it does not show any
obvious reduction in going to small values of momenta near the Kramers’ point Γ̄ (spin
polarization ∼ 0.45 for EB = −0.32 eV), which is consistent with its gapless nature.
The adequate energy-momentum resolution of our SR-ARPES instrument in order
−1
to resolve opposite spins at small momenta, such as k ∼ 0.05 Å , is demonstrated
by these SR measurements on δ = 0.50, which strongly supports that the observed
strong spin polarization reduction at the δ = 0.40 case reveals an intrinsic property
of the system relevant to the topological transition. Finally, we present the spin data
taken on the composition far into the topologically trivial side (δ = 0.0). Our spinresolved measurements [Figs. 3.12(i)-(l)] show only very weak polarizations (∼0.05),
which lie within the uncertainty levels of the measurements. The magnitude of the
3.3 TOPOLOGICAL-CRITICAL-POINT AND THE “PREFORMED” SURFACE STATES
spin polarization is too weak (comparable to the instrumental resolution) to obtain
the spin texture configuration around the Fermi surface for samples with δ = 0.0. The
observed weak polarization on δ = 0.0 suggests that the quasi-2D states are much
suppressed in going away from the TCP (such as the δ = 0.0 samples).
We model the semi-infinite system based on the 4×4 k·p model [151] and utilize the
Green’s function method to obtain the spectral weight as well as the spin polarization
near the surface region of the system as a function of bulk band-gap value in the model
(see details in Supplementary Methods). Indeed, our calculation shows that, in the
conventional semiconductor regime prior to the topological transition, the spectral
weight near the surface [left two panels of Fig. 3.13] is dominated by a single quasi2D band along the edge of the bulk band continuum, consistent with our ARPES
results. Moreover, the calculated spin polarization in the conventional semiconductor
region [left two panels of Fig. 3.13(b)] is found to be locked with momentum, also
consistent with our spin-resolved measurements. Furthermore, both the calculated
spectral weight and the spin polarization is found to become increasingly stronger
and predominate upon approaching the TCP from the conventional semiconductor
regime. A reasonable qualitative agreement between our experimental results and the
k · p model calculation is evident as seen in Fig. 3.13. The physical interpretation of
the observed agreement is discussed in the next section.
Although the observed quasi-2D states share important properties with actual
topological surface states, the following observations from our data clearly show that
they are still consistent with the non-topological bulk regime: (i) The experimentally
observed quasi-2D states are gapped, and disperse roughly along the edge of the bulk
continuum. Thus they do not connect or thread states across the bulk band-gap as
in a Z2 topological insulator. (ii) It is also possible to choose an energy value within
the bulk band-gap for samples lying in the conventional semiconductor regime (e.g.
δ = 0.4), so that no surface state is traversed, consistent with the topological triviality
74
of the sample. These experimental facts guarantee that the observed quasi-2D states
at δ . 0.5 are consistent with the conventional semiconductor phase of the system
(ν0 = 0, trivial Z2 index).
In order to better understand the spin texture of the quasi-2D states, we propose a phenomenological picture consistent with the basic topological physics for our
observation (Fig. 3.14): The quasi-2D states can be viewed as a Rashba-like state,
whose inner band is not observable because it is severely damped due to its strong
overlap with the bulk bands in E − k. As the system is tuned approaching the TCP
from the trivial side, the inner band completely loses its surface character, whereas
the outer band is systematically enhanced in terms of its surface spectral weight and
spin polarization, and evolve into the topological surface states (as clearly observed
in our data). We emphasize that we use the term “Rashba-like” for the observed perform surface states because there are two singly degenerate bands as in a real Rashba
2DEG. However, the Rashba surface states are due to a combined effect of atomic
spin-orbit coupling and the electrical field perpendicular to the surface, and follows
the Rashba Hamiltonian, whereas it is not fully applicable for the observed preformed
surface states. This issue needs further theoretical studies to illuminate the microscopic origin of the preformed surface states in theory. Regardless of this issue, our
systematic data, which is the main theme of this experimental paper, show: (1) these
preformed surface states show spin polarization texture that resembles a topological
insulator; (2) their surface spectral weight and magnitude of spin polarization are enhanced as we approach the TCP; (3) they evolve into the topological surface states.
These measurements with spin and mometum resolution clearly show that these preformed surface states are critically relevant to the bulk band inversion and TQPT in
the bulk. In the Supplementary Figure 8, we show our observation of preformed surface states near the TCP of another prototypical TQPT system (Bi1−δ Inδ )2 Se3 [75].
Therefore, these systematic and careful measurements on multiple systems suggest
3.3 TOPOLOGICAL-CRITICAL-POINT AND THE “PREFORMED” SURFACE STATES
that the observed preformed surface state is unlikely a special case due to material
details of the BiTl(S1−δ Seδ )2 system but an important proximity phenomenon that
describes the TCP in the electronic and spin groundstates in many TQPT systems.
Our observation of preformed surface states can also be applied to explain a number of recent experiments on some newly predicted topological matter, such as the
topological Kondo insulator phase predicted in SmB6 [118]. In SmB6 , the Kondo hybridization gap is believed to become significant below 30 K and the low temperature
resistivity anomaly occurs below 6 K [50]. However, ARPES experiments have observed quasi-2D low energy states without kz dispersion persisting up to temperatures
≥ 100 K [50, 51]. Therefore, our observation of preformed surface states is critically
important for developing a systemic theory in the vicinity of the TCP of a TQPT.
For example, a recent theoretical effort proposed that the preformed surface states
are due to the reversal of bulk Dirac fermion chirality across the TQPT [152], which
is consistent with our systematic experimental data.
Irrespective of their theoretical origin, our observation presents an interesting and
surprising critical topological phase transition phenomenon. Our systematic data in
this paper and hints from other works [49–51] suggest that the preformed surface
states we identified here is a universal feature about the topological-critical-point in
various topological phases. This important phenomenon has been largely missed (no
theoretical prediction) before. These results set an inspiring experimental methodological paradigm for understanding how protected surface states in various topological materials arise from a conventional material.
76
Figure 3.10: Observation of gapped quasi-two-dimensional states prior to the
topological-critical-point (TCP) of the topological quantum phase transition
(TQPT). (a)-(c) ARPES kk -EB maps of BiTl(S1−δ Seδ )2 obtained using incident photon
energy of 16 eV. The nominal composition values (defined by the mixture weight ratio
between the elements before the growth) are noted on the samples. For (a) conventional
band insulator (CBI), a band-gap is clearly observed for δ = 0.0 to 0.4; For (b) Compositions
near the TCP of the TQPT, δ = 0.45, 0.50, 0.525 and 0.55; And for (c) topological band
insulator (TBI), the conduction and valence bands are observed to be well-separated with
the surface states connecting the band-gap for δ = 0.6 to 1.0. (d) The energy levels of the
first-principles calculated bulk conduction and valence bands of the two end compounds
(δ = 0.0 and 1.0) are connected by straight lines to denote the evolution of the bulk bands.
The compositions selected for detailed experimental studies are marked by red arrows. The
+ and − signs represent the odd and even parity eigenvalues of the lowest lying conduction
and valence bands of BiTl(S1−δ Seδ )2 . (e) Incident photon energy dependence spectra for δ
= 0.45. (f ) kz vs kk Fermi surface maps for δ = 0.525 and 0.4. The kz range shown for δ
= 0.4 samples corresponds to the incident photon energy from 14 eV to 26 eV.
3.3 TOPOLOGICAL-CRITICAL-POINT AND THE “PREFORMED” SURFACE STATES
Figure 3.10: textbf(g),(h) ARPES dispersion maps at various incident photon energy values
for the compositions of δ = 0.4 < δc and δ = 0.525 ≃ δc . Some panels of this figure can also
be found in Ref. [69].
Figure 3.11: Observation of spin-momentum locking behavior on the native Fermi
level of the gapped quasi-2D states in the conventional semiconductor regime.
(a) ARPES kk -EB map of BiTl(S1−δ Seδ )2 for a δ = 0.40 sample. Dotted line shows the
binding energy where the spin-resolved measurements (d, e) are performed. (b) Momentum distribution curves (MDCs) of the dispersion map in panel (a). Highlighted MDC is
chosen for spin-resolved measurements. (c) Fermi surface mapping for δ = 0.40. The two
spin-resolved measurements are along the Γ̄-M̄ and Γ̄-K̄ cuts, respectively. Yellow arrows
represent the measured spin polarization vectors around the Fermi surface. (d) Upper
panel (left/right): measured in-plane spin-resolved momentum distribution spectra along
the Γ̄-M̄ /Γ̄-K̄ cut. Lower panel (left/right): measured in-plane net spin polarization along
the Γ̄-M̄ /Γ̄-K̄ cut. (e) same as panel (d) but for the out-of-plane component of the spin
polarization. Some panels of this figure can also be found in Ref. [69].
78
Figure 3.12
3.3 TOPOLOGICAL-CRITICAL-POINT AND THE “PREFORMED” SURFACE STATES
Figure 3.12: Evolution of the quasi-2D states’ spin polarization with binding energy and composition. (a),(e),(i), ARPES kk -EB maps with dotted lines indicating the
energy levels of spin-resolved measurements. Compositions of the samples are marked on the
top of each map. (b),(f ),(j), MDCs with highlighted curves chosen for spin-resolved measurements. (c),(g),(k), Spin-resolved momentum distribution spectra, and (d),(h),(l),
the corresponding net spin polarization measurements. Some panels of this figure can also
be found in Ref. [69].
Figure 3.13: Near-surface band structure and spin polarization of BiTl(S1−δ Seδ )2 .
(a), (b) Calculated electronic spectral weight distribution ((a)) and spin polarization ((b))
near the surface region of the constructed system are shown by the color maps in panels
((a) and (b)), respectively. The dispersion of the bulk bands are shown by the white
(black) dotted lines. Positive band-gap means that the system lies in the conventional
semiconductor (insulator) ν0 = 0 regime, whereas negative band-gap value means that the
system lies in the topological insulator ν0 = 1 regime. (c) ARPES measured electronic
dispersion on various δ values across the TQPT. The + and − signs in panels ((a) and (c))
represent the odd and even parity eigenvalues of the lowest lying conduction and valence
bands of BiTl(S1−δ Seδ )2 . The red ↑ and ↓ arrows schematically show the in-plane spin
polarization near the sample surface before and after the bulk band inversion (the TCP).
Some panels of this figure can also be found in Ref. [69].
80
Figure 3.14: A band-like schematic showing the phenomenological picture we proposed in the main text. The blue and green shaded areas represent the bulk conduction
bands (BCB) and the bulk valence bands (BVB), which go through a band inversion at
the critical point. The red lines represent electronic states that are strongly localized on
the surface due to their minimal overlap with the bulk band continuum. The inner band
(dotted lines) become bulk-like. One might worry about their singly degenerate nature,
which is inconsistent with the doubly degenerate nature of the bulk bands. However, it is
important to consider that there is another copy of these bands that has the opposite spin
texture one the opposite surface of the sample.
OLOGICAL PHASE TRANSITION AND “PREFORMED” CRITICAL BEHAVIOR IN (BI1−δ INδ )2 SE3
3.4
Topological phase transition and “preformed”
critical behavior in (Bi1−δ Inδ )2Se3
Figure 3.15: Topological phase transition and “preformed” surface states in
(Bi1−δ Inδ )2 Se3 . (a) ARPES dispersion maps at various In composition δ values. The
critical point of the topological phase transition is about δc ∼ 0.04. (b) Dispersion of
the composition δ = 0.06 sample repeated. The δ = 0.06 sample is found to be in the
conventional semiconductor (topologically trivial) phase. The blue dotted lines define the
momenta chosen for spin-resolved ARPES measurements. (c) Spin-resolved ARPES intensity and net spin polarization for the two spin-resolved cuts. Some panels can also be found
in Ref. [75].
We note that the topological phase transition from a conventional band insulator to a 3D Z2 TI can be realized in another material, namely (Bi1−δ Inδ )2 Se3 [75].
Here, again, the crystal structure remains the same throughout the whole range of
the chemical composition δ. Fig. 3.15(a) shows the ARPES dispersion maps of the
(Bi1−δ Inδ )2 Se3 system at various In composition δ values, where a topological phase
82
transition is clearly seen. The critical composition for this system is found to be
about δ = 0.04 [75]. In Figs. 3.15(b),(c), we show a composition (δ = 0.06) that
is in the trivial regime, where a small band-gap is found. Fig. 3.15(c) shows the
spin-resolved ARPES data near the Fermi level of the δ = 0.06 sample. Clear spin
polarization is observed. Furthermore, the spin polarization is found to reverse as
one goes from one side of the Fermi surface (cut1) to the other (cut2). These data
show the observation of preformed surface states in the (Bi1−δ Inδ )2 Se3 system. This
finding strongly suggests that the preformed surface state is not a special case in the
BiTl(S1−δ Seδ )2 system but an important proximity phenomenon that describes the
TCP in the electronic and spin groundstates in many TQPT systems.
In the following chapters, we will also discuss the topological phase transition in
a topological crystalline insulator (TCI) phase in the Pb1−δ Snδ Te(Se) system and
the associated phenomena in a topological Kondo insulator (TKI) candidate SmB6 .
This Chapter gives a general experimental guideline for how topological state and its
protected surface states can emerge from a conventional material by going through a
topological quantum phase transition.
Chapter 4
Topological Crystalline Insulator
Phase in Pb1−δ Snδ Te(Se)
A topological crystalline insulator is a new topological phase where space group symmetries replace the role of time-reversal symmetry in a Z2 (Kane-Mele) topological
insulator. In this chapter, we present our experimental identification of a topological crystalline insulator state (TCI) in the Pb1−δ Snδ Te(Se) systems, marking the
first realization of the topological crystalline insulator phase in real materials. In the
first section, we briefly review the key theoretical concepts for a TCI state. Then
we present the realization of a TCI state protected by the mirror symmetries in
Pb1−δ Snδ Te. Particularly, our surface state spin texture data provide a direct experimental measurements of its topological number (the mirror Chern number nM ).
Finally in the last section, we show systematic ARPES studies on the surface state
electronic structure of a similar TCI compound Pb1−δ Snδ Se, where a saddle point
surface electronic structure singularity is directly resolved.
Our ARPES and spin-resolved ARPES measurements identified the topological crystalline insulator (TCI) phase protected by the mirror symmetries in the
Pb1−δ Snδ Te, and further experimentally revealed its topological number, the mir84
ror Chern number, nM = −2. The observed topological surface states serve as a
new type of 2D electronic gas, which is distinct from both a regular 2DEG and the
surface states in a Z2 TI. The new crystalline surface states pave the way for many
uniquely-new quantum phenomena, including surface spin filtering, strain-induced
crystalline symmetry protected Chern currents [153–156]. As a result, these new
topological surface states that we observed in Pb1−δ Snδ Te have attracted interest by
many others and now studied by other experimental techniques (transport, STM,
etc.) [42,95,152,157–169], Our experimental work marks the first realization of a TCI
phase in realistic materials, and many other TCIs with yet more exotic properties
await their discoveries.
4.1
Key theoretical concepts for a topological crystalline insulator state
As we have elaborated before (see Chapter 1.4.2), to construct a new topological
state beyond a Z2 topological insulator, there are three steps (1) identify a material
with a finite number of band inversions between the bulk conduction and valence
bands, (2) identify a symmetry that can protect a surface state band crossing at
certain momentum space locations in the surface BZ, (3) identify the momentum
space locations for the band inversions and the symmetry protection coincide, so a
protected surface state with a surface band crossing is realized within the inverted
band-gap.
Thus the existence of band inversion (step 1) only provides a necessary but not
sufficient condition for giving rise to a topological phase. Steps 2 and 3, on the other
hand, determine the topological nature (the topological number) of a material system.
Let us take the BiTl(S1−δ Seδ )2 discussed in the previous section as an example: by
going from the sulfur end (δ = 0) to the selenium end (δ = 1), the bulk bands go
KEY THEORETICAL CONCEPTS FOR A TOPOLOGICAL CRYSTALLINE INSULATOR STATE
through a band inversion at the center of the BZ top (111) surface T point. At the
(111) surface, the T point projects onto the surface BZ center Γ̄ point, which is a
Kramers point protected by the time-reversal symmetry. Therefore, the inverted side
(δ > 0.5) of the BiTl(S1−δ Seδ )2 is a Z2 topological insulator.
Now a key question to ask is ”Can we identify other types of symmetries that
can also protect a surface state band-crossing (e.g. a surface Dirac point) within an
inverted band-gap?” That will certainly lead to a new topological state since the relevant symmetry is changed from time-reversal to others. Since we are working with
solid state crystals, an obvious choice of symmetry is the space group symmetries of a
crystalline system. The issue of what kind of crystalline space group symmetries can
protect surface state band-crossing has been theoretically addressed by the following
works [116, 117, 171]. Currently, theory has identified three-distinct crystalline symmetries that can protect topological surface states. However, we also note that no
one has proven that the existing three are the only possible symmetries for realizing
a TCI state. Rather, this is still an active on-going theoretical research. The three
known symmetries to protect surface state band-crossings in a TCI state are:
• Mirror symmetry M. This results in a Z classification, where the topological
number is a mirror Chern number that can be any integer [117, 170].
• C4 rotational symmetry + time-reversal symmetry without spin-orbit coupling
(spinless fermionic systems). This results in a Z2 classification, where the Z2
topological number can be 0 or 1 [116].
• Cnv symmetry + time-reversal breaking without spin-orbit coupling (spinless
fermionic systems). This results in a Z classification, where the topological
number can be any integer [171].
We briefly introduce the key theoretical concepts for the mirror symmetry type of
TCI state, which will be the focus of our experimental work. Let us take an example,
86
Figure 4.1: (a) A bulk BZ with a mirror (yellow) plane. (b) The surface (blue and red
solid lines) and bulk (grey shaded area) electronic structure alone the mirror plane.
in which a crystal has a bulk BZ as shown in Fig. 4.1(a), where the ky = 0 plane (the
yellow plane) is a mirror plane for the system. The fact that the kx − kz plane is a
mirror plane means the system is invariant under the mirror reflection with respect to
ky = 0, where M(ŷ) = P C2 (ŷ). This immediately implies that all the bulk electronic
states in the plane ky = 0 can be labeled with a mirror eigenvalue ±i [170]. Now
suppose that the bulk electronic structure within this 2D plane has a full energy gap,
then it is possible to assign a Chern number for the occupied energy bands for each
mirror eigenvalue:
n±i
−i
=
2π
Z
ky =0
[∇k × < u±i (k)|∇|u±i (k) > (kx , kz )]z d2 k,
(4.1)
where the bulk electronic wavefunction |u±i (k) > is labeled by its mirror eigenvalue. The total Chern number of the occupied bands is n = n+i + n−i . In a
time-reversal symmetric system, one has n = 0. However, one can define a mirror
KEY THEORETICAL CONCEPTS FOR A TOPOLOGICAL CRYSTALLINE INSULATOR STATE
Chern number as:
nM = (n+i − n−i )/2
(4.2)
Further theoretical analysis has shown that the mirror Chern number nM is a
topological invariant under the condition that the mirror symmetry is preserved in
the system [117, 170]. With these theoretical foundations, one could consider the
band structure at ky = 0 as two sets of band structures labeled by their different
mirror eigen values ±i. These two sets are related by the mirror symmetry operation
M(ŷ). For each set, let’s say mirror eigenvalue of +i, the band structure has a well
defined Chern number n+i . Thus we can conceptually view it has a Chern insulator,
where at the 1D boundary there are |n+i | altogether chiral edge states. Since we
assume time-reversal symmetry, we have n−i = −n+i . Therefore, for the sets of
bands with mirror eigenvalue −i, there will be the same number (|n+i | = |n−i |) of
chiral edge states but moving to the opposite direction since n−i = −n+i . These edge
states should have the same mirror eigen value (+i or −i) as their corresponding bulk
bands, which means the counterpropagating edge states must have the opposite mirror
eigen values. Therefore, the edge band-crossing between two counterpropagating edge
states at the mirror plane ky = 0 must be protected by the mirror symmetry. Because
if one can add a term that will gap out the band-crossing (degeneracy) between these
two counterpropagating edge states, that term must break the mirror symmetry. Any
term that respects the mirror symmetry only shifts the crossing in energy within the
mirror plane. Fig. 4.1(b) presents a case of n+i = 1 and n−i = −1. Thus one has one
right-going (red) edge state derived from the mirror eigenvalue +i bulk bands and
one left-going (blue) edge state derived from the mirror eigenvalue −i bulk bands.
Their edge band-crossing is protected by the mirror symmetry with a mirror Chern
number nm = +1. A similar argument applies to Fig. 4.1(c). We note that the mirror
symmetry at ky = 0 only requires the edge band-crossing to be within the ky = 0
88
plane. In principle, perturbations that respect the mirror symmetry can shift the
band-crossing along the energy axis and also only the momentum kx , as long as the
band-crossing stays within the ky = 0 mirror symmetry plane and the bulk bandgap along the energy-axis. Thus the schematics in Figs. 4.1(b),(c), where the edge
band-crossing is at the Γ̄ point kx = 0 is just a special case.
4.2
Discovery of mirror symmetry protected TCI
state in Pb1−δ Snδ Te
In this section, we present the realization of a TCI state protected by the mirror
symmetries in Pb1−δ Snδ Te. Particularly, our surface state spin texture measurements
provide a direct experimental measure for its topological number (the mirror Chern
number nM ).
4.2.1
Band inversions and mirror symmetries in Pb1−δ Snδ Te
The Pb1−δ Snδ Te pseudo-binary alloy system crystallizes in the sodium chloride crystal
structure, (space group Fm3̄m (225)). In this structure, each of the two atom types
(Pb/Sn, or Te) forms a separate face-centered cubic lattice, with the two lattices
interpenetrating so as to form a three-dimensional checkerboard pattern. The first
Brillouin zone (BZ) of the crystal structure is a truncated octahedron with six square
faces and eight hexagonal faces. The band-gap of Pb1−δ Snδ Te is found to be a direct
gap located at the L points in the BZ [172]. The L points are the centers of the eight
hexagonal faces of the BZ. Due to the inversion symmetry of the crystal, each L point
and its diametrically opposite partner on the BZ are completely equivalent. Thus
there are four distinct L point momenta. It is well established that the band inversion
transitions in the Pb1−δ Snδ Te take place at the four L points of the BZ [173–175].
As a result, an even number (four) of inversions exclude the system from being a
4.2 DISCOVERY OF MIRROR SYMMETRY PROTECTED TCI STATE IN PB1−δ SNδ TE
Figure 4.2: Band inversion transitions and mirror symmetries in Pb1−δ Snδ Te.
(a) The lattice of the Pb1−δ Snδ Te system is based on the sodium chloride crystal structure.
(b) The first Brillouin zone (BZ). The mirror planes are shown using green and light-brown
colors. These mirror planes project onto the (001) crystal surface as the X̄ − Γ̄ − X̄ mirror
lines (shown by red solid lines). (c) ARPES measured core level spectra (incident photon
energy 75 eV) of two representative compositions, namely Pb0.8 Sn0.2 Te and Pb0.6 Sn0.4 Te.
(d) The bulk band-gap of Pb1−δ Snδ Te alloy system undergoes a band inversion upon changing the Pb/Sn ratio at each L point in the bulk BZ [172–175]. (e), (f ) First-principles based
calculation of band dispersion (e) and iso-energetic contour with energy set 0.02eV below
the Dirac node energy (f ) of the inverted end compound SnTe. This figure is adapted from
Ref. [40].
Kane-Mele Z2 topological insulator [2]. However, it is interesting to note that the
momentum-space locations of the band inversions coincide with the momentum-space
mirror plane within the BZ. These facts suggest that the inverted Pb1−δ Snδ Te may
realize a distinct TCI state protected by mirror symmetry.
Before we go into further details, it is worth noting that before Pb1−δ Snδ Te was
discussed, many known Z2 TIs (such as Bi0.91 Sb0.09 , Bi2 Se3 ) also featured nonzero
mirror Chern number [31]. This is in fact quite intuitive. Let us take Bi2 Se3 as an
example. It has a single bulk band inversion at the BZ center Γ point. For a bulk
90
4.2.2 Topological surface states in Pb0.6 Sn0.4 Te
BZ corresponding to a rhombohedral crystal structure, it has 3 distinct mirror planes
along the (111) direction which goes through the Γ point. Therefore, Bi2 Se3 has a
mirror Chern number of |nM | = 1 (the sign of the mirror Chern number needs further
investigation based on the surface state spin texture). However, the key here is that
there are four (an even number of) bulk band inversions in Pb1−δ Snδ Te, which exclude
a nonzero Z2 topological number. Thus any nonzero mirror Chern number that might
realize in Pb1−δ Snδ Te is not “masked” by a finite Z2 number unlike in Bi2 Se3 . This
realizes a distinct TCI state without ambiguity.
Let us try to evaluate the possible TCI state and the associated mirror Chern
number in Pb1−δ Snδ Te before going into experimental data or theoretical calculations.
As shown in Fig. 4.2(b), each mirror plane contains two distinct L points and thus
two bulk band inversions per BZ. If we intuitively assume that each band inversion
changes the mirror Chern number by 1, then we immediately get that |nM | = 2 for
Pb1−δ Snδ Te.
4.2.2
Topological surface states in Pb0.6Sn0.4Te
In order to identify the mirror-nontrivial TCI state, we perform systematic ARPES
and spin-ARPES measurements to search for mirror symmetry protected surface
states in inverted compositions of the Pb1−δ Snδ Te system.
In order to capture the electronic structure both below and above the band inversion transition (theoretically predicted to be around δ ≃ 1/3 [173, 174]), we choose
two representative compositions, namely δ = 0.2 and δ = 0.4 for detailed systematic
studies. Fig. 4.2(c) shows the momentum-integrated core level photoemission spectra for both compositions. Photoemission peaks corresponding to tellurium 4d, tin
4d, and lead 5d orbitals are observed in Fig. 4.2(c). The energy splitting of the Pb
orbital is observed to be larger than that of the Sn orbital, which is consistent with
the stronger spin-orbit coupling of the heavier Pb nuclei. In addition, the spectral
4.2 DISCOVERY OF MIRROR SYMMETRY PROTECTED TCI STATE IN PB1−δ SNδ TE
intensity contribution of the Sn peaks in the δ = 0.4 sample (red) is found to be
higher than that of in the δ = 0.2 sample (blue), which is also consistent with the
larger Sn concentration in the δ = 0.4 samples.
We performed systematic low energy electronic structure studies on these two representative compositions. Since the low energy physics of the system is dominated
by the band inversion at X̄ points on the (001) surface, we present ARPES measurements with the momentum space window centered at the X̄ point, which is the
midpoint of the surface BZ edge. Figs. 4.3(a),(b) show the ARPES Fermi surface and
dispersion mappings of the Pb0.8 Sn0.2 Te sample (δ = 0.2). The system at δ = 0.2 is
observed to be gapped: No band is observed to cross the Fermi level in the Fermi
surface maps [Fig. 4.3(a)]. The dispersion measurements [Fig. 4.3(b)] reveal a single
hole-like band below the Fermi level. This single hole-like band is observed to show
strong dependence with respect to the incident photon energy [Fig. 4.4(a)], which
reflects its three-dimensionally dispersive bulk valence band origin. As a qualitative guide to the ARPES measurements on δ = 0.2, we present first-principles based
electronic structure calculation on the non-inverted end compound PbTe [Fig. 4.3(c)].
Our calculations confirm that PbTe is a conventional band insulator, whose electronic
structure can be described as a single hole-like bulk valence band in the vicinity of
each X̄ point, which is consistent with our ARPES results on Pb-rich Pb0.8 Sn0.2 Te.
The three-dimensional nature of the calculated bulk valence band is revealed by its
kz evolution in Fig. 4.3(c), which is in qualitative agreement with the incident photon
energy dependence of our ARPES measurements shown in Fig. 4.3(b).
Now we present comparative ARPES measurements under identical experimental
conditions and setups on the Pb0.6 Sn0.4 Te (δ = 0.4) sample. In contrast to the conventional band insulator (insulating) behavior in the δ = 0.2 sample, the Fermi surface
mapping [Fig. 4.3(d)] on the δ = 0.4 sample shows two unconnected metallic Fermi
pockets (dots) on the opposite sides of the X̄ point. The dispersion measurements
92
4.2.2 Topological surface states in Pb0.6 Sn0.4 Te
on the δ = 0.4 sample are shown in Fig. 4.3(e). The single hole-like bulk valence
band, which is similar to the δ = 0.2 conventional band insulator composition, is also
observed below the Fermi level. More importantly, a pair of metallic states crossing
the Fermi level on the opposite sides of the X̄ point is observed along the Γ̄ − X̄ − Γ̄
mirror line momentum space direction. These states are found to show no observable
dispersion upon varying the incident photon energy [Fig. 4.4(a)], reflecting its twodimensional character. On the other hand, the single hole-like band, similar to the
one seen in δ = 0.2 samples, is observed to disperse strongly upon varying the incident photon energy, suggesting its three-dimensional character. At a set of different
photon energy values (effectively probing different kz values), the bulk valence band
intensity overlaps (intermixing) with different parts of the surface states in energy and
momentum space: At a photon energy of 18 eV, the intermixing (intensity overlap)
is strong, and the inner two branches of the surface states are masked by the bulk
intensity. At photon energies of 10 eV and 24 eV, the surface states are found to
be relatively better isolated. These ARPES measurements shown in Figs. 4.3(d),(e)
suggest that the δ = 0.4 sample lies in the inverted composition regime and that the
observed surface states are related to the band inversion transition in Pb1−δ Snδ Te
system as predicted theoretically [117, 175]. As a qualitative guide, we present firstprinciples based electronic structure calculation on the inverted end compound SnTe
[Fig. 4.3(f)]. The calculated electronic structure of SnTe is found to be a superposition of two kz nondispersive metallic surface states and a single hole-like kz dispersive
bulk valence band in the vicinity of the X̄ point, which is in qualitative agreement
with the ARPES results on Pb0.6 Sn0.4 Te.
We now perform systematic measurements on the surface electronic structure of
the Pb0.6 Sn0.4 Te. Fig. 4.5(b) shows the wide range iso-energetic contour mapping
covering the first surface BZ. The surface states are observed to be present, and only
present, along the mirror line (Γ̄ − X̄ − Γ̄) directions. No other states are found along
4.2 DISCOVERY OF MIRROR SYMMETRY PROTECTED TCI STATE IN PB1−δ SNδ TE
any other momentum directions on the Fermi level. In close vicinity to each X̄ point,
a pair of surface states are observed along the mirror line direction. One lies inside
the first surface BZ but the other is located outside. Therefore, in total, four surface
states are observed within the first surface BZ, which is in agreement with the fact
that there are four band inversions in the Pb1−δ Snδ Te system. The mapping zoomedin near the X̄ point [Fig. 4.5(c)] reveals two unconnected small pockets (dots). The
momentum space distance from the center of each pocket to the X̄ point is about
−1
0.09 Å . Dispersion measurements (EB vs k) are performed along three important
momentum space cuts, namely cuts 1, 2, and 3 defined in Fig. 4.5(b), in order to
further reveal the surface states electronic structure. Metallic surface states crossing
the Fermi level are observed in both cuts 1 and 2, whereas cut 3 is found to be fully
gapped. In cut 1 Fig. 4.5(d), which is the mirror line (Γ̄ − X̄ − Γ̄) direction, a pair
of surface states are observed on the Fermi level. We also study the dispersion along
cut 2 Fig. 4.5(d), which only cuts across the surface states inside the first surface BZ.
Both the dispersion maps and the momentum distribution curves in cut 2 reveal that
the surface states along cut 2 are nearly Dirac-like (linearly dispersive) close to the
Fermi level. The surface states’ velocity is obtained to be 2.8±0.1 eV·Å (4.2 ± 0.2
×105 m/s) along cut 2, and 1.1±0.3 eV·Å (1.7 ± 0.4 ×105 m/s) for the two outer
branches along cut 1, respectively.
We highlight the following results from our data relevant to the topological nature
of the surface states: (1) there are in total 4 surface states within a surface BZ
[Fig. 4.5(b)], (2) none of the surface states enclose a Kramers point [Fig. 4.5(b)], (3)
all surface states locate alone the mirror symmetry (Γ̄ − X̄ − Γ̄ direction. The first
two points show that these surface states do not arise from a Z2 topological number.
Thus the Pb1−δ Snδ Te system is Z2 topological trivial. The three points are consistent
with a mirror symmetry protected TCI state.
94
4.2.2 Topological surface states in Pb0.6 Sn0.4 Te
Figure 4.3: Comparison of non-inverted and inverted compositions. (a),(b)
ARPES low energy electronic structure measurements on Pb0.8 Sn0.2 Te (δ = 0.2). (c)
First-principle calculated electronic structure of PbTe (δ = 0) as a qualitative guide to the
ARPES measurements on Pb0.8 Sn0.2 Te. (d),(e) ARPES low energy electronic structure
measurements on Pb0.6 Sn0.4 Te (δ = 0.4) under the identical experimental conditions and
setups as in (a and b). (f ) First-principle calculated electronic structure of SnTe (δ = 1)
as a qualitative guide to the ARPES measurements on Pb0.6 Sn0.4 Te.
4.2 DISCOVERY OF MIRROR SYMMETRY PROTECTED TCI STATE IN PB1−δ SNδ TE
Figure 4.4: Comparison of non-inverted and inverted compositions at different
incident photon energies. ARPES dispersion maps and the corresponding momentumdistribution curves (MDCs) at different incident photon energies for both the non-inverted
(δ = 0.2) and the inverted (δ = 0.4) compositions.
96
4.2.2 Topological surface states in Pb0.6 Sn0.4 Te
Figure 4.5: Mirror protected surface states in Pb0.6 Sn0.4 Te. (a) First-principle
calculated surface states of SnTe with energy set 0.02eV below the Dirac node energy are
shown in red. (b) Left panel: ARPES iso-energetic contour mapping (EB = 0.02 eV) of
Pb0.6 Sn0.4 Te covering the first surface Brillouin zone (BZ) using incident photon energy
of 24 eV. Right panel: Spectral intensity distribution as a function of momentum along
the horizontal mirror line (defined by ky = 0). (c) High resolution Fermi surface mapping
(EB = 0.0 eV) in the vicinity of one of the X̄ points. (d)-(f ) Dispersion maps (EB vs k)
and corresponding energy (momentum) distribution curves of the momentum space cuts 1,
2, and 3. The momentum space cut-directions of cuts 1, 2, and 3 are defined by the blue
dotted lines in panel (c). The second derivative image (SDI) of the measured dispersion is
additionally shown for (d). This figure is adapted from Ref. [40].
4.2 DISCOVERY OF MIRROR SYMMETRY PROTECTED TCI STATE IN PB1−δ SNδ TE
4.2.3
Mirror Chern number and mirror symmetry protection
In order to experimentally probe the topological number in the Pb1−δ Snδ Te system,
we measure the spin texture of the surface states as shown in Fig. 4.6. Our spinresolved measurements are performed along the mirror line (Γ̄ − X̄ − Γ̄) direction as
shown in Fig. 4.6(a), since the electronic and spin structure along this direction is
most critically relevant to the TCI phase [117]. Considering that the natural Fermi
level of our δ = 0.4 samples is very close to the Dirac point (which is spin degenerate),
the spin polarization of the surface states are measured at 60 meV below the Fermi
level in order to gain proper contrast, namely SR-Cut 1 in Fig. 4.6(a). As shown in
the net spin polarization measurement of SR-Cut 1 in Fig. 4.6(c), in total four spins
pointing in the (±) in-plane tangential direction are revealed for the surface states
along the mirror line direction. This is consistent with the observed two surface state
cones (four branches in total) near an X̄ point along the mirror line direction. To
compare and contrast the spin polarization behavior of the surface states (SR-Cut
1) with that of the bulk states, we perform spin-resolved measurement SR-Cut 2
at EB = 0.70 eV, where the bulk valence bands are prominently dominated. The
surface states are well-defined at lower binding energies (approximately below 0.3 eV
or so) where they emerge out of the bulk states and merge with the bulk band at
high binding energies. Indeed, in contrast to SR-Cut1 reflecting the surface states’
spin polarization, no significant net spin polarization is observed for SR-Cut2. This is
expected for the bulk valence bands of the inversion symmetric Pb1−δ Snδ Te system.
Now we use the obtained electronic and spin structure of the surface states along
the mirror line direction to obtain the topological number of the Pb1−δ Snδ Te system.
As shown in Figs. 4.3-4.6, along each mirror line, there are two Dirac surface states
within a surface BZ. Our spin polarization measurements in Fig. 4.6(c) further show
that these surface states are singly degenerate. These facts demonstrate that the
absolute value of the mirror Chern number is 2, |nM | = 2. The sign of the mirror
98
4.2.3 Mirror Chern number and mirror symmetry protection
Chern number can be determined by the handedness (chirality) of the surface state
spin texture as shown in Fig. 4.6(f). Specifically, at an energy below the surface Dirac
point, if the Fermi surface spin texture is left-handed, then nM = 2; If Fermi surface
spin texture is right-handed, then nM = −2. As shown in Fig. 4.6(c), our spin-resolved
measurements show that the spin texture profile is right-handed at EB = 0.06 eV,
which is below the surface Dirac point. Therefore, we conclude from our experimental
data that the TCI state in the Pb1−δ Snδ Te system has a mirror Chern number (its
topological number of nM = −2. We provide a conceptual picture to visualize the
mirror symmetry protection for the surface states observed in the TCI Pb0.6 Sn0.4 Te
system. As shown in Fig. 4.2(b), at the (001) surface, each X̄ point is in fact the
projection of two distinct L points (therefore two band inversions). If we intuitively
assume that each bulk band inversion gives rise to a surface Dirac cone whose Dirac
band crossing is at each X̄ point, then two bulk band inversions will lead to two
surface Dirac cones at each X̄ point, where the energies of the two Dirac points
have an offset, as shown in Fig. 4.2(c). The surface state Fermi surface (without
considering the hybridization between the two cones) is a circle [see first panel of
Fig. 4.2(c)]. These two surface Dirac cones can hybridize. Without, any additional
symmetry, hybridization will open up an energy gap everywhere on the circular-like
surface band-crossing and therefore the system will be fully gapped (topologically
trivial). However, if we consider a mirror plane along the Γ̄ − X̄ − Γ̄ direction, then
the surface state band-crossing is protected along the mirror line and the resulting
surface electronic structure is a “two Dirac cones along the Γ̄ − X̄ − Γ̄ mirror line
direction” state, which is consistent with our experimental data.
We present a comparison of the Pb0.6 Sn0.4 Te and a single Dirac cone Z2 topological
insulator (TI) system. As shown in Fig. 4.2(a), for a Z2 TI, a single surface Dirac
cone is observed enclosing the time-reversal invariant Kramers’ momenta Γ̄ in both
ARPES and calculation results, demonstrating its Z2 topological insulator state and
4.2 DISCOVERY OF MIRROR SYMMETRY PROTECTED TCI STATE IN PB1−δ SNδ TE
the time-reversal symmetry protection of its single Dirac cone surface states. On
the other hand, for the Pb0.6 Sn0.4 Te samples [Fig. 4.2(b)], none of the surface states
is observed to enclose any of the Kramers points. On the other hand, all surface
states locate along the mirror line directions. We note that one advantage of the
Pb1−δ Snδ Te system is that it can be easily doped with manganese, thallium, or indium
to achieve bulk magnetic or superconducting states [165–169]. The symmetry in
the Pb1−δ Snδ Te system (its nonmagnetic character) can be broken by magnetic or
superconducting doping into the bulk or the surface. In future experiments, it would
be interesting to explore the modification of our observed surface states brought out
by magnetic and superconducting correlations, in order to search for exotic magnetic
and superconducting order on the surface. Such magnetic and superconducting orders
on the surface states in Pb1−δ Snδ Te can be different from those recently observed
in theZ2 topological insulators [57, 58] due to its very distinct topology of surface
electronic structure. The novel magnetic and superconducting states to be realized
with this novel topology are not strongly related to the question of gapless or gapped
nature of the TCI phase. Therefore our observation of the spin-polarized surface
states presented here provides the much desired platform for realizing unusual surface
magnetic and superconducting states in future experiments.
100
4.2.3 Mirror Chern number and mirror symmetry protection
Figure 4.6: Experimental measurement of the mirror Chern number. (a) ARPES
dispersion map along the mirror line direction. The white dotted lines show the binding
energies chosen for spin-resolved measurements SR-Cut 1 at EB = 0.06 eV and SR-Cut
2 EB = 0.70 eV respectively. Inset: Measured spin polarization are shown by the green
and blue arrows on top of the ARPES iso-energetic contour at binding energy EB = 0.06
eV for SR-Cut1. (b),(c) Measured in-plane spin-resolved intensity (b) and in-plane spin
polarization (c) of the surface states (SR-Cut 1) near the Fermi level at EB = 0.06 eV.
(d),(e) Measured in-plane spin-resolved intensity (d) and in-plane spin polarization (e)
of the bulk valence bands (SR-Cut 2) at high binding energy at EB = 0.70 eV. f, Surface
spin texture configurations corresponding to mirror Chern number of nM = 2 and nM =
−2. g, Theoretically expected spin polarization configuration of the surface states which
corresponds to a mirror Chern number) nM = −2.
4.2 DISCOVERY OF MIRROR SYMMETRY PROTECTED TCI STATE IN PB1−δ SNδ TE
Figure 4.7: Mirror symmetry protection and a comparison of Z2 (Kane-Mele)
topological insulator and topological crystalline insulator (TCI) phases. (a)
ARPES and calculation results of the surface states of a Z2 topological insulator GeBi2 Te4 ,
an analog to Bi2 Se3 [32]. b, ARPES measurements on the Pb0.6 Sn0.4 Te (x = 0.4) samples
and band calculation results on the end compound SnTe. (c) A conceptual picture to visualize the mirror symmetry protection for the surface states observed in the TCI Pb0.6 Sn0.4 Te
system.
102
4.3
Topological surface states in Pbδ Sn1−δ Se
In this chapter, we systematically study the electronic structure of the topological
surface states in TCI phase in Pbδ Sn1−δ Se. We present our observation of a range of
novel electronic properties in these surface states, including Lifshitz transition, saddle
point singularities, temperature-driven topological phase transition, etc.
4.3.1
Lifshitz transition and saddle point singularities
Saddle point singularities in two-dimensional Dirac electron gas often results in electronic instabilities leading to exotic correlated quantum phenomena. A notable example is the extensive research efforts on the van Hove singularity in graphene, where
d-wave superconductivity and unusual magnetism or Kondo effect have been predicted to take place [176–179]. Very recently, striking Hofstadter butterfly spectrum
and fractional quantum Hall effect have been experimentally observed in twisted
graphene system due to the saddle point singularity in its Dirac band structure,
which further adds to the interest [180, 181]. Like graphene, the surface states of
topological insulators (TIs) also possess the light-like Dirac dispersion. What is more
interesting is that the Dirac surface states of TIs are spin-momentum locked, since
they are a consequence of the bulk band inversions and the nontrivial topology of the
bulk electronic band structure. As a result, saddle point singularities in topological
surface states (a spin-helical two-dimensional Dirac electron gas) are even more exotic
because they not only host correlated physics, but also allow one to explore the coexistence and the interplay between topological order and strong correlation. While none
of the surface states in any known Z2 TI materials satisfy the conditions to realize
a Lifshitz transition, the newly identified TCI surface states in the Pbδ Sn1−δ Se(Te)
system in fact provide an ideal platform.
As we have shown in the previous section, the (001) surface states in Pbδ Sn1−δ Se(Te)
4.3 TOPOLOGICAL SURFACE STATES IN PBδ SN1−δ SE
can be described by a pair of surface Dirac cones in the vicinity of each X̄ point. We
have also observed such type of surface states in the inverted composition (δ = 0.3) of
the Pbδ Sn1−δ Se system, as shown in Fig. 4.8. Moreover, in Pb0.7 Sn0.3 Se, the samples
are in fact n-type and thus the upper Dirac surface bands are also observed. Fig. 4.8(e)
shows the ARPES dispersion map along the Γ̄ − X̄ − Γ̄ mirror line direction, where
two Dirac surface states near an X̄ point is clearly visualized. The energy evolution
of the surface state constant energy contour is shown in Fig. 4.8(c). Two Dirac points
are observed sitting at the same energy EB = 70 meV [central penal of Fig. 4.8(c)]
near the X̄ point along the Γ̄ − X̄ − Γ̄ momentum space cut-direction in the surface
BZ. As the energy is tuned away from the Dirac points, the two Dirac points grow into
two unconnected pockets, and eventually “meet” each other, where they are found to
become two concentric contours both enclosing the X̄ point. Therefore a surface state
Lifshitz transition is observed in our data since the constant energy contour is found
to undergo a topological change. Thus there exist two Lifshitz transitions at EB = 40
meV and EB = 98 meV. These Lifshitz transitions are expected to lead to saddle
point band structure singularities at the same energies. In order to demonstrate the
singularity, we systematically study the electronic structure at the Lifshitz transition
energy (EB = 40 meV), as shown in Fig. 4.9. The two square markers in Fig. 4.9(a)
note the momentum space locations, where the two unconnected contours merge.
Their energy and momentum space coordinates are experimentally identified to be
−1
(EB , kx , ky ) = (40 meV, 0, ±0.02 Å ). To experimentally prove the saddle points in
momentum-space band structure, we focus on the upper blue square in Fig. 4.9(a)
and study the energy-momentum dispersion cuts along three important momentum
space cut-directions, namely cuts 1,2, and 3. Cuts 1 and 2 [Figs. 4.9(c),(d)] are along
the horizontal (kx ) and verticle (ky ) directions across the blue square. Interestingly,
the blue square is found to be a local band structure minimum along cut1 shown in
Fig. 4.9(c), whereas it is a local maximum along cut2 [Fig. 4.9(d)]. Observation of
104
4.3.1 Lifshitz transition and saddle point singularities
local minimum and local maximum at the same momentum space location (the blue
square) manifestly shows that it is a surface band structure saddle point. The observation of surface momentum-space saddle point immediately implies that there exist
certain intermediate cut-directions (between cuts 1 and 2), where the surface band
structure is completely flat in the vicinity of the blue square. Indeed, as shown in
Fig. 4.9(e), for cut 3, we found that the surface states are nearly flat near the saddle
point.
The observed flat band structure (along cut 3) may give rise to the divergence
of the surface density of states (DOS), leading to surface van Hove singularity. We
first present the theoretically calculated momentum-integrated DOS of the TCI surface states. As shown in Fig. 4.8(b), two DOS peaks are found, which correspond
to the van Hove singularities in the upper and lower parts of the Dirac cones, respectively. Additionally, three dips in the calculated DOS curve correspond to the
two Dirac points, as well as the upper and lower Dirac points (UDP and LDP)
[Fig. 4.8(a)]. In order to search for experimental evidence of van Hove singularities,
we study the momentum-integrated ARPES intensity as shown in Fig. 4.8(e). Indeed, a pronounced peak is observed at the energy corresponding to the saddle point,
namely EB = 40 meV, as labeled by “VH1” in Fig. 4.8(e). Additionally, we observe
a significant dip of ARPES intensity at the binding energy of EB = 70 meV, which
corresponds to the energy of the Dirac points. In order to better highlight other important features, we take the second derivative of the ARPES intensity [Fig. 4.8(f)],
which allows all five features to be visualized.
4.3 TOPOLOGICAL SURFACE STATES IN PBδ SN1−δ SE
Figure 4.8: Observation of the Lifshitz transition in the topological surface
states in Pb0.7 Sn0.3 Se. (a) ARPES dispersion maps upon in situ Sn deposition on the
Pb0.70 Sn0.30 Se surface. The dosage (time) for Sn deposition is noted. A different batch of
sample, which is p−type with the chemical potential below the Dirac points, is used for
the Sn deposition data shown in this panel. (b),(c) Schematics of surface band dispersion
of the TCI phase along the mirror line Γ̄ − X̄ − Γ̄ and the M̄ − X̄ − M̄ momentum space
cut-directions. Five important features of the surface states, including Dirac point of the
upper part of the Dirac cones (UDP), van Hove singularity of the upper Dirac cones (VH1),
two Dirac points along the Γ̄ − X̄ − Γ̄ mirror line (DP), van Hove singularity of the lower
part of the Dirac cones (VH2) and Dirac point of the lower part of the Dirac cones (LDP)
are marked. (c) Calculated density of state (DOS) for the surface states and the bulk bands
using the k · p model [151]. (d) Experimental observation of the Lifshitz transition - the
binding energies are noted on the constant energy contours.
106
4.3.1 Lifshitz transition and saddle point singularities
Figure 4.8: (e) ARPES measured dispersion plots along Γ̄ − X̄ − Γ̄ and M̄ − X̄ − M̄. (f )
Momentum (kx and ky ) integrated ARPES intensity as a function of binding energy (left).
2nd derivative of the ARPES intensity with respect to binding energy is presented to further
highlight the features. The upper Dirac point (UDP), upper van Hove singularity (VH1),
Dirac point (DP), lower van Hove singularity (VH2) and lower Dirac point (LDP) are
marked. This figure is adapted from Ref. [81].
Figure 4.9: Observation of the saddle point singularity. (a) ARPES constant energy
contour map in the vicinity of an X̄ point in the surface (001) BZ at binding energy 40 meV,
which corresponds to the surface Lifshitz transition and saddle point energy of the upper
part of the Dirac cones. The blue and green squares denote the momentum space locations
of the two surface saddle points. The blue dotted lines indicates the momentum space
cut-directions for cuts1, 2, and 3, which are centered at the blue square. (b) Calculated
surface state constant energy contour at the saddle point singularity energy (top) and a
three-dimensional schematic of a saddle point (bottom). (c)-(e) ARPES dispersion maps
(left) and their second derivative images along cuts 1, 2, and 3. The white and green arrow
point the saddle points [blue and green squares in Panel (a)]. This figure is adapted from
Ref. [81].
4.3 TOPOLOGICAL SURFACE STATES IN PBδ SN1−δ SE
4.3.2
Temperature-driven topological phase transition
Interestingly, we found that in the Pbδ Sn1−δ Se system, a topological phase transition
from a TCI phase to a trivial phase can be realized not only by changing the chemical
composition δ (as in BiTl(S1−δ Seδ )2 ), but also simply by varying the temperature T .
Fig. 4.10 shows the ARPES measurements of the TCI surface states in Pb0.70 Sn0.30 Se
at various tempeatures. At low temperature of T = 20 K, two Dirac cones are observed near the X̄ point along the Γ̄ − X̄ − Γ̄, which is evident in the nontrivial TCI
phase. Interestingly, as the temperature is raised to 100K, the two Dirac points are
found to move closer to each other. With further increase in the temperature, the two
cones are observed to merge at temperature around 250 K at the X̄ point. Finally
a gap at the Dirac point is opened for T > 250 K, which suggests that the system
enters the topologically trivial phase. In order to understand the interesting thermal
evolution of the surface state electronic structure, we perform first principle theoretical calculations at different lattice constant values [Fig. 4.10(b)]. A reasonably good
agreement between the temperature dependent ARPES data and the lattice constant
dependent calculation is found, where the essential features of the data, including
the two Dirac cones approaching, merging and eventually opening up a gap, are all
captured in the calculation. Thus our ARPES and calculation together suggest that
the observed thermally driven topological phase transition is a result of thermal expansion of the lattice. This scenario is further supported by our x-ray diffraction
measurements on the sample as a function of temperature. As shown in Fig. 4.10(c).
We now compare the observation with the topological phase transition found in
the Z2 TI system in BiTl(S1−δ Seδ )2 [37] shown in the previous chapter, in order to
highlight interesting properties of the topological phase transition in Pb0.70 Sn0.30 Se.
First, we observe that the momentum-space distance between the two Dirac points
near each X̄ point in Pb0.70 Sn0.30 Se can be systematically engineered. We note that
changing the surface Dirac point momentum space location is allowed in a mirror
108
4.3.2 Temperature-driven topological phase transition
protected TCI system [117], as long as the Dirac points are on the mirror lines. This
is, in contrast, not possible in BiTl(S1−δ Seδ )2 since it has only single Dirac cone at the
surface BZ center (a time-reversal invariant Kramer’s point) as required by the timereversal protected TI phase. Second, at the topological phase transition critical point
in Pb0.70 Sn0.30 Se, the two Dirac cones merge and also the bulk band-gap vanishes.
Thus both the surface and the bulk possess Dirac dispersion, and the system can be
viewed as a squared-version of graphene in three-dimension, potentially leading to
exotic transport and tunneling behaviors awaiting to be explored [158]. Third, it is
also interesting to note that similar temperature dependence study on BiTl(S0.4 Se0.6 )
does not show any dramatic temperature dependent effect. This is very likely due to
the fact that Pb0.70 Sn0.30 Se is in the cubic crystal structure so there exists only one
parameter a for the lattice constant.
4.3 TOPOLOGICAL SURFACE STATES IN PBδ SN1−δ SE
Figure 4.10: Temperature-driven topological phase transition in Pb0.70 Sn0.30 Se.
(a) Dispersion maps along the mirror line Γ̄− X̄− Γ̄ at different temperatures. The two Dirac
points are observed to approach, eventually merge into one node and then open up a gap as
temperature increases. (b) First-principles calculations of the TCI surface bands with varying lattice constants. SnSe assumed in the face-center cubic (FCC) structure is used. Red
lines and blue areas represent the surface and bulk bands, respectively. (c) Synchrotronbased temperature dependent X-ray diffraction (SXRD) measurements for Pb0.70 Sn0.30 Se.
The peak is observed to shifts towards the lower angles with increasing temperature, which
confirms the picture of the thermal expansion of lattice (see supplementary information for
details). (d) Surface Fermi surface plot measured at temperature of 20 K. This figure is
adapted from Ref. [81].
110
4.3.3 Topological phase diagram in Pb1−δ Snδ Se
4.3.3
Topological phase diagram in Pb1−δ Snδ Se
We study the TCI phase in Pb1−δ Snδ Se as a function of composition δ. Fig. 4.11(a)
shows the ARPES measurements of the low energy states of the δ = 0.3 sample, as
well as the two end compounds, namely PbSe and SnSe. For PbSe, low-lying bulk
conduction and valence bands with a clear band-gap of ∼ 0.15 eV is observed, which
proves that the system is topologically trivial for δ = 0. As δ is increased, the lowlying bulk bands are observed to approach each other, and eventually inverse with
the surface states spanning over the inverted band-gap. The band inversion critical
composition is found to be 0.2 ∼ 0.23 depending on the temperature. We note that
the fact that the increasing Sn concentration drives the system from trivial to TCI
has also been found in the Pb1−δ Snδ Se (see previous section and also Ref. [40]). This
is interesting since Pb is in fact heavier than Sn thus having a larger (atomic) spinorbit strength. However, in both Pb1−δ Snδ Se and Pb1−δ Snδ Te cases, shrinkage of
the lattice constant (with increasing Sn%) plays a more important role and therefore
makes the system topologically nontrivial on the Sn-rich side. We turn to the other
end compound SnSe, as shown in the third panel in Fig. 4.11(a). Surprisingly, at
the Fermi level, no electronic states are observed. On the other hand, a fully gapped
electronic structure with the chemical potential inside the band-gap is found. This is
because SnSe crystal is in the orthorhmobic phase [182, 183]. The change of crystal
structure, which denies both the band inversion and the mirror symmetries, completely destroys the TCI phase. And SnSe is found to be a trivial insulator by our
ARPES measurements and the band-gap is reported to be as large as ∼ 1 eV [184].
Based on our systematic ARPES studies, a rich topological phase diagram is found
in the Pb1−δ Snδ Se in Fig. 4.11(b). The blue and red lines represent the energy level
of the lowest lying bulk conduction and valence bands with assuming that the crystal
structure always remains in the FCC structure. Starting from PbSe (δ = 0), the system has a non-inverted band-gap of ∼ 0.15 eV. As δ increases, band inversion takes
4.3 TOPOLOGICAL SURFACE STATES IN PBδ SN1−δ SE
place and the system enters the TCI phase. the inverted band-gap increases until the
system enters the multi-(crystal structure)-phase regime at δ & 0.45, where cubic and
orthorhombic structures coexist. Finally, for δ & 0.75, the system becomes a large
band-gap trivial insulator in the single orthorhombic phase. Two distinct phase transition are observed (as labeled by δc1 and δc2 ), where the first transition is due the the
shrink of lattice constant (which increases the effective spin-orbit strength), whereas
the second is a result of a drastic structural transition. Therefore, our experimental
data reveal a delicate relationship among lattice constant, band gap, spin-orbit coupling strength and crystal structure associated with the topological phase transition
in Pb1−δ Snδ Se.
112
4.3.3 Topological phase diagram in Pb1−δ Snδ Se
Figure 4.11: Topological phase diagram Pb1−δ Snδ Se. (a) ARPES dispersion map for
PbSe (trivial insulator), Pb0.7 Sn0.3 Se (TCI) and SnSe (trivial insulator). (b) Topological
phase diagram of the Pb1−δ Snδ Se system. For composition range of 0 < x < 0.45, the
system is in the single crystalline FCC phase. The bulk band of Pb1−δ Snδ Se undergoes a
band inversion with Pb/Sn substitution. Topological crystalline insulator (TCI) phase is
observed in the band inverted region toward the Sn-rich side. The critical composition δc1
is ∼ 0.20 − 0.23 depending on the temperature. The conduction and valence band states
+
representing odd and even parity eigenvalues are marked as L−
6 and L6 , respectively. For
composition range of 0.45 < δ < 0.75, the system shows multi-structural-phase (cubic and
orthorhombic phases coexist. See the XRD data in the inset for δ = 0.5 and 0.6.). The
upper insets are schematic Fermi surface plots around the X̄ point. The inset in the bottom
right conner shows the resistivity measurements on SnSe, which proves its insulator nature.
BS FS and SS FS denote the bulk state Fermi surface and surface state Fermi surface,
respectively. The arrows at bottom notes the compositions where our ARPES studies have
been performed. For composition range of 0.75 < δ < 1, the system is in a single crystalline
orthorhombic phase. This figure is adapted from Ref. [81].
Chapter 5
Topological Dirac semimetal state
in Cd3As2 and Na3Bi
Unlike insulators, semimetals are materials, whose bulk conduction and valence bands
have small but finite overlap. Thus there does not exist a full band-gap irrespective
of the choice of the chemical potential. This is in contrast to all topological phases
discussed before this chapter (quantum Hall states, Chern insulators, quantum spin
Hall insulators, 3D Z2 topological insulators, and topological crystalline insulators),
where a band-gap is important both for the definition of a topological number and
for the existence of surface states. We ask the following questions: Can we construct
a topological state in a semimetal - a topological semimetal? What kind of surface
states it will have and how these surface states reflect the nontrivial topology in the
bulk of a semimetal?
In this chapter, we present our experimental realization of a topological Dirac
semimetal phase in Cd3 As2 and Na3 Bi. We show that the bulk has linearly dispersive
nodal crossings, realizing a 3D analog of graphene, whereas the surface features Fermi
arc surface states (FASS) that connect across the bulk Dirac nodes. Previously, Fermi
arc states were only found in the bulk bands of high Tc cuprate superconductors in
114
a strongly interacting condition. Our observation of novel surface states here thus
realizes disjoint Fermi arc states on the surface of a weakly interacting topological
system. The observed FASS serve as a new type of 2D electronic gas, which are distinct from both a regular 2DEG in semiconductor heterostructures and the surface
states in a topological insulator. The observed exotic surface and bulk electronic and
spin groundstate in topological Dirac semimetals Cd3 As2 and Na3 Bi here suggests
many interesting transport and tunneling phenomena, and is multiply-connected to
other novel groundstates under additional superconducting or ferromagnetic symmetry breakings [10, 119–124, 139–141, 149, 185–192].
In the first section, we discuss the theoretical works that are directly relevant
to the prediction the topological Dirac semimetal state. In the second section, we
present our ARPES observation of the 3D Dirac cone state in the bulk electronic
structure in Cd3 As2 . In the third section, we present our ARPES and spin-ARPES
data on Na3 Bi, not only showing the existence of 3D Dirac nodes in the bulk, but
also showing a spin-polarized double Fermi arc surface state (FASS) that connect
the bulk Dirac nodes. We further demonstrate how the observed FASS and its spin
texture provide an experimental measurement of the nontrivial topological number
in semimetal Na3 Bi. Finally in the last section, we go back to theory. We discuss the
general classification of topologically nontrivial semimetals, including the topological
Weyl semimetal [119, 188], the topological Dirac semimetal [122–124] and the topological nodal-line semimetal [193], and how they can transform into each other via
breaking or restoring certain symmetries.
5.1 THEORETICAL CONCEPTS FOR A TOPOLOGICAL DIRAC SEMIMETAL
5.1
Theoretical concepts for a topological Dirac
semimetal
In this section, we introduce the basic theoretical concepts relevant to the topological Dirac semimetal phase. We constrain ourselves within bulk materials that have
time-reversal T and space inversion symmetries I, and possess non-negligible spinorbit coupling. These conditions are needed for a topological Dirac semimetal. For
systems breaking T or I, and for spinless fermion systems (no spin-orbit coupling),
there are also theoretically predicted distinct topological semimetal phases, such as
a topological Weyl semimetal and a topological nodal-line semimetal. These phases
have not been experimentally realized to date. In the last section of this chapter, we
will briefly discuss them and their connection to a topological Dirac semimetal.
In a time-reversal and space inversion symmetric system, if the energy of the
top of a valence band goes above that of the bottom of a conduction band, without
considering spin-orbit interaction, these two bulk band will become degenerate (bandcrossing) at a finite number of momentum space locations. However, a finite spinorbit interaction is expected to lift the degeneracy at these band-crossing momenta.
Without considering any additional symmetry, there is no reason one would expect
certain band-crossing momenta are more “special” than others. Therefore, if spinorbit coupling open an energy gap, it should opens an full energy gap, meaning
the degeneracy at all band-crossing momenta are lifted. This is the case of the
BiTl(S1−δ Seδ )2 and the Pb1−δ Snδ Te(Se) systems [37, 40] that we present in Chapters
3,4.
However, the situation can become very different if additional symmetries are
considered, in particular if we consider a time-reversal and space inversion symmetric
system with an additional uniaxial rotational symmetry. Let us take the example of
Na3 Bi, where a band inversion occurs between the Na3s and the Bi6p bands in the
116
vicinity of the Γ point. Due to an additional C3 rotational symmetry along the kz axis
[Fig. 5.1(c)], the bulk electronic bands along the C3 (kz ) axis must be the eigenstates of
the C3 operation. In particular for the Na3s and the Bi6p bands in Na3 Bi, theory [123]
shows that these two bands have different C3 eigenvalues and therefore the bulk band
degeneracy is protected by C3 crystalline rotational symmetry even with a finite
spin-orbit coupling. Therefore, spin-orbit coupling opens an energy gap at all bandcrossing k-points except at two special momenta along the C3 (kz ) axis on opposite
sides of the Γ point. Similarly, in another tetragonal structured material Cd3 As2 ,
it has been theoretically shown that the band-crossing between inverted conduction
and valence bands is protected by an additional C4 rotational symmetry of the same
reason [122]. A systematic theoretical classification for all the possible topological
Dirac semimetal states with different rotational symmetries have been proposed in
Ref. [124] [also see Fig. 5.2].
Another important theoretical question is to define a topological number to characterize a Dirac semimetal state. Let us again take the example of Na3 Bi, in absence
of a full energy gap in the bulk BZ, one cannot define the Z2 invariants (ν0 ; ν1 ν2 ν3 )
as in a 3D Z2 topological insulator, even though there is a band inversion at the Γ
point in Na3 Bi. However, as pointed out in Ref. [124], since the bulk band-gap only
vanishes at two momenta along the C3 (kz ) axis on the opposite sides of the Γ point,
there exist many 2D k planes in the 3D BZ, where a full energy gap is present. In
particular, for Na3 Bi, the kz = 0 plane is invariant under time-reversal operation, and
also contains the Γ point where the band inversion happens. Therefore, Ref. [124]
shows that the kz = 0 plane can be viewed as a 2D quantum spin Hall system with
a distinct 2D topological number ν2D (kz = 0) = 1. The approach of assigning topological numbers to certain 2D k slices in a 3D BZ can be further generalized to other
Dirac semimetals (see Fig. 5.2 and Ref. [124]) and even Weyl semimetals [10,119,188].
Furthermore, the distinct 2D topological number (e.g. ν2D (kz = 0) = 1 in Na3 Bi) in a
5.1 THEORETICAL CONCEPTS FOR A TOPOLOGICAL DIRAC SEMIMETAL
Dirac semimetal guarantees the existence of double Fermi arc surface states (FASS),
as shown in Ref. [122–124]. About the FASS, we will elaborate in some detail in
section 3 with our ARPES data on Na3 Bi.
118
Figure 5.1: Comparison of bulk band inversion transitions in time-reversal and
inversion symmetric systems with or without additional symmetries. (a) A band
inversion process in a time-reversal and inversion symmetric system without the influence of
additional symmetries. In this case, as the conduction and valence bands are inverted, spinorbit coupling will open up a full energy gap. The BiTl(S1−δ Seδ )2 system [37] is an example.
(b) The critical point of the band inversion realizes a 3D Dirac semimetal. Such 3D Dirac
semimetal state requires fine tuning of the material composition δ. (c),(d) A band inversion
process in a time-reversal and inversion symmetric system with an additional rotational
symmetry. In this case, as the conduction and valence bands are inverted and cross each
other, spin-orbit coupling will open up an energy gap everywhere in the k-space where they
cross except at the momentum locations along the rotational axis. This is because the
bulk band crossings on the rotational axis are protected by the rotational symmetry of the
crystal. The groundstate in the inverted regime is described by a pair of Dirac nodes along
the rotational axis in the bulk and double Fermi arc surface states connecting the two bulk
nodes on the surface. Inset: an ARPES Fermi surface map of Na3 Bi showing the double
FASS and the two bulk Dirac nodes.
5.1 THEORETICAL CONCEPTS FOR A TOPOLOGICAL DIRAC SEMIMETAL
Figure 5.2: Classification table for 3D topological Dirac semimetals. This table is
adapted from Ref. [124].
120
5.2
3D Dirac semimetal state in high mobility Cd3As2
In this section, we present our ARPES results on the Dirac semimetal candidate
Cd3 As2 . We observe a highly linear bulk band crossing to form a three-dimensional
dispersive Dirac cone projected at the (001) surface BZ center by studying the (001)cleaved surface. These results demonstrate the existence of the 3D Dirac nodes (bandcrossing) in the bulk electronic structure of Cd3 As2 . Remarkably, an unusually inplane high Fermi velocity up to 1.5 × 106 m·s−1 is observed in our samples, where the
transport mobility is known up to 40,000 cm2 V−1 s−1 [194,195] suggesting that Cd3 As2
can be a promising candidate as an anisotropic-hypercone (3D) high spin-orbit analog
of graphene. Our observation of the 3D Dirac semimetal state in the stoichiometric,
stable and high mobility material Cd3 As2 opens the door for exploring the exotic
transport and tunneling phenomena based on 3D Dirac fermions [140, 141, 149, 186].
The crystal structure of Cd3 As2 has a tetragonal unit cell with a = 12.67 Å and
c = 25.48 Å for Z = 32 with symmetry of space group I41 cd [see Figs. 5.3(a),(b)].
In this structure, arsenic ions are approximately cubic close-packed and Cd ions are
tetrahedrally coordinated, which can be described in parallel to a fluorite structure
of systematic Cd/As vacancies. There are four layers per unit and the missing CdAs4 tetrahedra are arranged without the central symmetry as shown with the (001)
projection view in Fig. 5.3(b), with the two vacant sites being at diagonally opposite
corners of a cube face [196]. The corresponding Brillouin zone (BZ) is shown in
Fig. 5.3(d), where the center of the BZ is the Γ point, the centers of the top and bottom
square surfaces are the Z points, and other high symmetry points are also noted.
Cd3 As2 has attracted attention in electrical transport due to its high mobility of 105
cm2 V−1 s−1 reported in previous studies [194,195]. The carrier density and mobility of
our Cd3 As2 samples are characterized to be of 5.2 × 1018 cm−3 and 42850 cm2 V−1 s−1 ,
respectively, at temperature of 130 K, consistent with previous reports [194, 195],
which provide evidence for the high quality of our single crystalline samples. In band
5.2 3D DIRAC SEMIMETAL STATE IN HIGH MOBILITY CD3 AS2
theoretical calculations, Cd3 As2 is also of interest since it features an inverted band
structure [197]. More interestingly, a very recent theoretical prediction [123] which
motivated this work, has shown that the spin-orbit interaction in Cd3 As2 cannot open
up a full energy gap between the inverted bulk conduction and valence bands due to
the protection of an additional crystallographic symmetry [124] (in the case of Cd3 As2
it is the C4 rotational symmetry along the kz direction [123]), which is in contrast
to other band-inverted systems such as Bi2 Se3 . This theory predicts [123] that the
C4 rotational symmetry protects two bulk (3D) Dirac band touching points at two
special k points along the Γ − Z momentum space cut-direction, as shown by the red
crossings in Fig. 5.3(d). Therefore, Cd3 As2 serves as a candidate for a spacegroup or
crystal structure symmetry protected C4 3D Dirac semimetal state.
In order to experimentally identify such a 3D Dirac semimetal state, we systematically study the electronic structure of Cd3 As2 on the cleaved (001) surface. Fig. 5.3(c)
shows momentum-integrated ARPES spectral intensity over a wide energy window.
Sharp ARPES intensity peaks at binding energies of EB ≃ 11 eV and 41 eV that
correspond to the cadmium 4d and the arsenic 3d core levels are observed, confirming
the chemical composition of our samples. We study the overall electronic structure
of the valence band. Fig. 5.3(e) shows the second derivative image of an ARPES dispersion map in a 3 eV binding energy window, where the dispersion of several valence
bands are identified. Moreover, a low-lying small feature that crosses the Fermi level
is observed. In order to resolve it, high-resolution ARPES dispersion measurements
are performed in the close vicinity of the Fermi level as shown in Fig. 5.3(f). Remarkably, a linearly dispersive upper Dirac cone is observed at the surface BZ center
Γ̄ point, whose Dirac node is found to locate at a binding energy of EB ≃ 0.2 eV. At
the Fermi level, only the upper Dirac band, but no other electronic state, is are observed. On the other hand, the linearly dispersive lower Dirac cone is found to coexist
with another parabolic bulk valence band, which can be seen from Fig. 5.3(e). From
122
the observed steep Dirac dispersion [Fig. 5.3(f)], we obtain a surprisingly high Fermi
velocity of about 9.8 eV·Å (≃ 1.5 × 106 ms−1 ). Compared to the much-studied 2D
Dirac systems, the Fermi velocity of the 3D Dirac fermions in Cd3 As2 is thus about
3 times higher than that of in the topological surface states (TSS) of Bi2 Se3 [32], 1.5
times higher than in graphene [198] and 30 times higher than that in the topological
Kondo insulator phase in SmB6 [50]. The observed large Fermi velocity of the 3D
Dirac band provides clues to understand Cd3 As2 ’s unusually high mobility reported
in previous transport experiments [194,195]. Therefore one can expect to observe unusual magneto-electrical and quantum Hall transport properties under high magnetic
field [140, 141, 149, 186]. It is well-known that in graphene the capability to prepare
high quality and high mobility samples has enabled the experimental observations of
many interesting phenomena that arises from its 2D Dirac fermions. The large Fermi
velocity and high mobility in Cd3 As2 are among the important experimental criteria
to explore the 3D relativistic physics in various Hall phenomena in tailored Cd3 As2 .
We compare ARPES observations with our theoretical calculations, which is qualitatively consistent with previous calculations [123]. The reason for the use of our
calculations is two fold: first, our calculations are fine tuned based on the characterization of samples used in the present ARPES study, second, sufficiently detailed cuts
are not readily available from Ref. [123] which is necessary for a detailed comparison
of ARPES data with theory. In theory, there are two 3D Dirac nodes that are expected at two special k points along the Γ − Z (kz ) momentum space cut-direction,
as shown by the red crossings in Fig. 5.3(d). At the (001) surface, these two k points
along the Γ − Z axis project on to the Γ̄ point of the (001) surface BZ [Fig. 5.3(d)].
Therefore, at the (001) surface, theory predicts one 3D Dirac cone at the BZ center
Γ̄ point, as shown in [Fig. 5.4(a)]. These results are in qualitative agreement with
our data, which supports our experimental observation of the 3D 3D Dirac semimetal
state in Cd3 As2 . We also study the ARPES measured constant energy contour maps
5.2 3D DIRAC SEMIMETAL STATE IN HIGH MOBILITY CD3 AS2
[Fig. 5.4(c),(d)]. At the Fermi level, the constant energy contour consists of a single
pocket centered at the Γ̄ point. With increasing binding energy, the size of the pocket
decreases and eventually shrinks to a point (the 3D Dirac point) near EB ≃ 0.2 eV.
A 3D Dirac semimetal is expected to feature nearly linear dispersion along all three
momentum space directions close to the crossing point, even though the Fermi/Dirac
velocity can vary significantly along different directions. In order to probe the 3D
nature of the observed low-energy Dirac-like bands in Cd3 As2 , we performed ARPES
measurements as a function of incident photon energy to study the out-of-plane dispersion perpendicular to the (001) surface. Upon varying the photon energy, one
can effectively probe the electronic structure at different out-of-plane momentum kz
values in a three-dimensional Brillouin zone and compare with band calculations. In
Cd3 As2 , the electronic structure or band dispersions in the vicinity of its 3D Dirac2
(kz − k0 )2 = E 2 , where k0 is the
like node can be approximated as : vk2 (kx2 + ky2 ) + v⊥
out-of-plane momentum value of the 3D Dirac point. Thus at a fixed kz value (which
is determined by the incident photon energy value), do we have an in-plane electronic
2
(kz − k0 )2 . It can be seen that only
dispersion takes the form: vk2 (kx2 + ky2 ) = E 2 − v⊥
at kz = k0 the in-plane dispersion that is a gapless Dirac cone, whereas in the case
for kz 6= k0 the nonzero kz − k0 term acts as an effective mass term and opens up
a gap in the in-plane dispersion relation. Fig. 5.5(a) shows the ARPES measured
in-plane electronic dispersion at various photon energies. At a photon energy of 102
eV, a gapless Dirac-like cone is observed, which shows that photon energy hν = 102
eV corresponds to a kz value that is close to the out-of-plane momentum value of the
3D Dirac node k0 . As photon energy is changed away from 102 eV in either direction,
the bulk conduction and valence bands are observed within experimental resolution
to be separated along the energy axis and a gap opens in the in-plane dispersion. At
photon energies sufficiently away from 102 eV, such as 90 eV or 114 eV in Fig. 5.5(a),
the in-plane gap is large enough that the bottom of the upper Dirac cone (bulk con124
duction band) is moved above the Fermi level, and therefore only the lower Dirac
cone is observed. We now fix the in-plane momenta at 0 and plot the ARPES data
at kx = ky = 0 as a function of incident photon energy. As shown in Fig. 5.5(b), a
E−kz dispersion is observed in the out-of-plane momentum space cut direction, which
is in qualitative agreement with the theoretical calculations [Fig. 5.5(c)]. The Fermi
velocity in the z-direction can be estimated (only at the order of magnitude level)
to be about 105 m/s. These systematic incident photon energy dependent measurements show that the observed Dirac-like band disperses along both the in-plane and
the out-of-plane directions suggesting its three-dimensional or bulk nature consistent
with theory.
In order to further understand the nature of the observed Dirac band, we study the
spin polarization or spin texture properties of Cd3 As2 . As shown in Fig. 5.5(f), spinresolved ARPES measurements are performed on a relatively p−type sample. Two
spin-resolved energy-dispersive curve (EDC) cuts are shown at momenta of ±0.1
Å−1 on the opposite sides of the Fermi surface. The obtained spin data shown in
Figs. 5.5(g),(h) show no observable net spin polarization or texture behavior within
our experimental resolution, which is in remarkable contrast with the clear spin texture in 2D Dirac fermions on the surfaces of topological insulators. The absence of
spin texture in our observed Dirac fermion in Cd3 As2 bands is consistent with their
bulk origin, which agrees with the theoretical prediction. It also provides a strong
evidence that our ARPES signal is mainly due to the bulk Dirac bands on the surface
of Cd3 As2 , whereas the predicted surface (resonance) states [123] that lie along the
boundary of the bulk Dirac cone projection has a small spectral weight (intensity) contribution to the photoemission signal. In other words, according to our experimental
data, the surface electronic structure of Cd3 As2 is dominated by the spin-degenerate
bulk bands, which is very different from that of the 3D topological insulators.
The distinct semimetal nature of Cd3 As2 is better understood from ARPES data
5.2 3D DIRAC SEMIMETAL STATE IN HIGH MOBILITY CD3 AS2
if we compare our results with that of the prototype TI, Bi2 Se3 . In Bi2 Se3 as shown
in Fig. 5.5(b), the bulk conduction and valence bands are fully separated (gapped),
and a linearly dispersive topological surface state is observed that connect across the
bulk band-gap. In the case of Cd3 As2 [Fig. 5.6(a)], there does not exist a full bulk
energy gap. On the other hand, the bulk conduction and valence bands “touch”
(and only “touch”) at specific locations in the momentum space, which are the 3D
band-touching nodes, thus realizing a 3D Dirac semimetal state. For comparison, we
further show that a similar 3D Dirac semimetal state is also realized by tuning the
chemical composition δ (effectively the spin-orbit coupling strength) to the critical
point of a topological phase transition between a normal insulator and a topological
insulator. Figs. 5.6(c)(d) present the surface electronic structure of two other 3D
Dirac semimetal states in the BiTl(S1−δ Seδ )2 and (Bi1−δ Inδ )2 Se3 systems. In both
systems, it has been shown that tuning the chemical composition δ can drive the
system from a normal insulator state to a topological insulator state [37, 38, 75].
The critical compositions for the two topological phase transitions are approximately
near δ = 0.5 and δ = 0.04, respectively. Fig. 5.6(c),(d) show the ARPES measured
surface electronic structure of the critical compositions for both BiTl(S1−δ Seδ )2 and
(Bi1−δ Inδ )2 Se3 systems, which are expected to exhibit the 3D Dirac semimetal state.
Indeed, the bulk critical compositions where bulk and surface Dirac bands collapse
also show Dirac cones with intensities filled inside the cones, which is qualitatively
similar to the case in Cd3 As2 . Currently, the origin of the filling behavior is not
fully understood irrespective of the bulk (out-of-plane dispersive behavior) nature
of the overall band dispersion interpreted in connection to band calculations (see,
Fig. 5.4). Based on the ARPES data in Figs. 5.6(c),(d), the Fermi velocity is estimated
to be ∼ 4 eV·Å and ∼ 2 eV·Å for the 3D Dirac fermions in BiTl(S1−δ Seδ )2 and
(Bi1−δ Inδ )2 Se3 , respectively, which is much lower than that of what we observe in
Cd3 As2 , thus likely limiting the carrier mobility. The mobility is also limited by
126
the disorder due to strong chemical alloying. More importantly, the fine control
of doping/alloying δ value and keeping the composition exactly at the bulk critical
composition is difficult to achieve [37]. For example, although similarly high electron
mobility on the order of 105 cm2 V−1 s−1 has been reported in the bulk states of
Pb1−δ Snδ Se (δ = 0.23) [158], the bulk Dirac fermions there are in fact massive due
to the difficulty of controlling the composition exactly at the critical point. These
issues do not arise in the stoichiometric Cd3 As2 system since its 3D Dirac semimetal
state is protected by the crystal symmetry, which does not require chemical doping
and therefore the natural high electron mobility is retained (not diminished). We
further note that Cd3 As2 is stable in ambient environment, whereas the other Dirac
semimetal candidate Na3 Bi reacts in seconds in air. These advantages are vital in
further studying the 3D Dirac electronic structure in transport.
5.2 3D DIRAC SEMIMETAL STATE IN HIGH MOBILITY CD3 AS2
Figure 5.3: Sample characterization on Cd3 As2 . (a) Cd3 As2 crystalizes in a tetragonal
body center structure with space group of I41 cd, which has 32 number of formula units in the
unit cell. The tetragonal structure has lattice constant of a = 12.670 Å, b = 12.670 Å, and
c = 25.480 Å. (b) The basic structure unit is a 4 corner-sharing CdAs3 -trigonal pyramid.
(c) Core-level spectroscopic measurement where Cd 4d and As 3d peaks are clearly observed.
Inset shows a picture of the Cd3 As2 samples used for ARPES measurements. The flat and
mirror-like surface indicates the high quality of our samples. (d) The bulk Brillouin zone
(BZ) and the projected surface BZ along the (001) direction. The red crossings locate at
(kx , ky , kz ) = (0, 0, 0.15 2π
c∗ ) (c∗ = c/a). They denote the two special k points along the
Γ − Z momentum space cut-direction, where 3D Dirac band-touchings are protected by
the crystalline C4 symmetry along the kz axis. (e) Second derivative image of ARPES
dispersion map of Cd3 As2 over the wider binding energy range. Various bands are wellresolved up to 3 eV binding energy range. (f ) ARPES EB − kx cut of Cd3 As2 near the
Fermi level at around surface BZ center Γ̄ point. This figure is adapted from Ref. [43].
128
Figure 5.4: Observation of in-plane dispersion in Cd3 As2 . (a) Left: First principles
2π
calculation of the bulk electronic structure along the (π, π, 0.15 2π
c∗ ) − (0, 0, 0.15 c∗ ) direction
(c∗ = c/a). Right: Projected bulk band structure on to the (001) surface, where the
shaded area shows the projection of the bulk bands. (b) ARPES measured dispersion
map of Cd3 As2 , measured with photon energy of 22 eV and temperature of 15 K along
the (−π, −π) − (0, 0) − (π, π) momentum space cut direction. (c) ARPES constant energy
contour maps using photon energy of 22 eV on Cd3 As2 growth batch I. (d) ARPES constant
energy contour maps using photon energy of 102 eV on Cd3 As2 batch II. In order to achieve
chemical potential (carrier concentration) control, we have prepared different batches of
samples under slightly different growth conditions (temperature and growth time). For the
two batches studied here, batch I is found to be slightly more n−type than batch II (e.g.
compare batch I in Fig. 5.3(f) with batch II in Fig. 5.5(a) rightmost panel). This figure is
adapted from Ref. [43].
5.2 3D DIRAC SEMIMETAL STATE IN HIGH MOBILITY CD3 AS2
Figure 5.5: Observation of out-of-plane dispersion in Cd3 As2 . (a) ARPES dispersion
maps at various incident photon energies are shown in the first and third rows. First
principle calculated in-plane electronic dispersion at different kz values near the 3D Dirac
node k0 is plotted in the second and forth rows. (b) ARPES measured out-of-plane linear
E − kz dispersion. (c) ARPES measured in-plane E − kx dispersion. The white dotted
lines are guides to the eye tracking the out-of-plane dispersion. (d) Theoretically calculated
out-of-plane E − kz dispersion near the 3D Dirac node shown over a wider energy window.
(e) Schematic (cartoon) of the 3D (anisotropic) Dirac semimetal band structure in Cd3 As2 .
(f ) Spin-integrated ARPES dispersion cut measured on the sample used for spin-resolved
measurements. The dotted lines indicate the momentum locations for the spin-resolved
EDC cuts. (g),(h) Spin-resolved ARPES intensity (black and red circles) and measured
net spin polarization (blue dots) for Cuts 1 and 2. This figure is adapted from Ref. [43].
130
Figure 5.6: Comparison of the surface electronic structure of 2D and 3D Dirac
fermions. (a) ARPES measured surface electronic structure dispersion map of Cd3 As2 and
its corresponding momentum distribution curves (MDCs). (b) ARPES measured surface
dispersion map of the prototype TI Bi2 Se3 and its corresponding momentum distribution
curves. Both spectra are measured with photon energy of 22 eV and at a sample temperature
of 15 K. The black arrows show the ARPES intensity peaks in the MDC plots. (c),(d)
ARPES spectra of two Bi-based 3D Dirac semimetals, which are realized by fine tuning
the chemical composition to the critical point of a topological phase transition between a
normal insulator and a TI: c, BiTl(S1−δ Seδ )2 (δ = 0.5), and (Bi1−δ Inδ )2 Se3 (δ = 0.04) (d).
Spectrum in panel (c) is measured with photon energy of 16 eV and spectrum in panel
(d) is measured with photon energy of 41 eV. For the 2D topological surface Dirac cone
in Bi2 Se3 , a distinct in-plane (EB − kx ) dispersion is observed in ARPES, whereas for the
3D bulk Dirac cones in Cd3 As2 , TlBi(S0.5 Se0.5 )2 , and (Bi0.96 In0.04 )2 Se3 , a Dirac-cone-like
intensity continuum is also observed. This figure is adapted from Ref. [43].
5.3 FERMI ARC SURFACE STATES IN TOPOLOGICAL DIRAC SEMIMETAL NA3 BI
5.3
Fermi arc surface states in topological Dirac
semimetal Na3Bi
The topology of a topological phase is encoded in its surface states in experiments.
The novel topological Dirac semimetal state has its bulk band-structure is similar to
that of a three-dimensional analog of graphene. This is indeed observed in Cd3 As2
and Na3 Bi (see the previous section and also Ref. [43–45]). However, none of these
experiments [43–45] revealed the signature of its topology in the Dirac semimetal
state. In this section, we report the experimental discovery of a pair of polarized
Fermi arc surface state modes in the form of a new type of two-dimensional polarized
electron gas on the surfaces of Dirac semimetals. These Fermi arc surface states
(FASS) are observed to connect across an even number of bulk band nodes and found
to have their spin uniquely locked to their momentum. We show that these states are
distinctly different from the topological surface states (TSS) seen in all topological
insulators. Our observed exotic two-dimensional states not only uncover the novel
topology of Dirac metals such as Na3 Bis but also opens new research frontiers for the
utilization of Fermi arc electron gases for a wide range of fundamental physics and
spintronic studies envisioned in recent theories [10, 119–124, 139–141, 149, 185–192].
5.3.1
Choice of the surface termination to observe FASS
We first elaborate the importance of choosing an appropariate surface termination in
order to observe the Fermi arc surface states (FASS) in a topological Dirac semimetal
candidate material.
Let us take Na3 Bi as an example. Na3 Bi is a semimetal that crystalizes in the
hexagonal P 63 /mmc crystal structure with a = 5.448 Å and c = 9.655 Å [199].
First principles bulk band calculations [123] show that its lowest bulk conduction and
valence bands are composed of Bi 6px,y,z and Na 3s orbitals. These two bands possess a
132
5.3.1 Choice of the surface termination to observe FASS
bulk band inversion of about ∼ 0.3 eV at the bulk BZ center Γ [123]. The strong spinorbit coupling in the system can open up energy gaps between the inverted bulk bands,
but due to the protection of an additional three-fold rotational symmetry along the
[001] crystalline direction, two bulk Dirac band-crossings (Dirac nodes) are predicted
to be preserved even after considering spin-orbit coupling, as schematically shown by
the blue crosses in Fig. 5.7(c). The two bulk Dirac nodes locate along the A-Γ-A [001]
direction. Therefore, at the (001) surface, the two bulk Dirac nodes project onto the
same point in the (001) surface BZ [Fig. 5.7(c)], making them difficult to separate,
isolate, and systematically study via spectroscopic methods. More importantly, Fermi
arc surface states are fundamentally not possible because a Fermi arc starts from one
bulk Dirac node and ends on another (therefore requires multiple bulk Dirac nodes
that are separated within the surface BZ). The situation in another topological Dirac
semimetal candidate Cd3 As2 is almost identical [see Fig. 5.3(d)], where the two bulk
Dirac nodes also locate on the [001] (kz ) axis. On the other hand, we note that
previous ARPES experiments on Na3 Bi and Cd3 As2 [43–45] were performed on the
(001) top surface, where the two Dirac nodes project onto the same point in the (001)
surface BZ. Therefore, considering these facts, it is intuitive to understand why FASS
were not observed in previous ARPES experiments [43–45].
On the other hand, at the (100) surface, the two bulk Dirac nodes are separated
on the opposite sides of the (100) surface BZ center Γ̃ [Fig. 5.7(c)]. Consequently,
the Fermi arc surface states that connect the bulk Dirac nodes are found in the (100)
surface electronic structure calculation [Fig. 5.7(e)]. The surface states are double
Fermi arcs because they exist everywhere on the Fermi surface contour except at the
two bulk Dirac points. In other words, if one travels around the Fermi surface contour,
the electronic states are localized at the surface everywhere, except at the locations
of the two bulk Dirac points where the states disperse linearly in all three dimensions.
We note that the existence of double Fermi arc surface states in Na3 Bi and cadmium
5.3 FERMI ARC SURFACE STATES IN TOPOLOGICAL DIRAC SEMIMETAL NA3 BI
arsenide has been theoretically predicted by a number of groups [122–124,186], which
strongly motivated our work to experimentally search for Fermi arc surface states in
Dirac metals.
Figure 5.7: Characterization of Na3 Bi system. (a) Crystal structure of Na3 Bi. The
two Na sites and Bi atoms are marked with different colors. (b) Projected crystal structure
at the (100) surface. (c) Structure of bulk and surface Brillouin zone at (001) and (100)
surfaces. Bulk Dirac nodes are marked by blue crosses. Note that the two bulk Dirac cones
project to the same Γ̄ point on the (001) surface, while they are separated in momentum
when studied at the (100) surface. (d), (e), First principles calculation of the (001) (d)
and (100) surface (e) electronic structure. Theoretical nontrivial surface state (green lines)
appears along the edge of the bulk 3D Dirac cone in the (001) surface, while it becomes
separated from the bulk continuum when seen from the (100) surface (e). (f ) First principles
bulk band calculation for Na3 Bi. It is clear from the calculation that the band touching
happens along the A-Γ-A direction close to the zone center. This figure is adapted from
Ref. [46].
134
5.3.2 Observation of Fermi arc surface states in Na3 Bi
5.3.2
Observation of Fermi arc surface states in Na3 Bi
In order to observe the FASS and to illuminate the topological nature of the Dirac
semimetal state in Na3 Bi, we systematically study the electronic and spin groundstate
of its (100) surface-surface. Fig. 5.8(a) shows the ARPES measured Fermi surface of
our Na3 Bi sample at its native Fermi level. Remarkably, the measured Fermi surface
is found to consist of two Fermi “points” along the ky (100) direction and two arcs
that connect the two Fermi points. The measured Fermi surface topology [Fig. 5.8(a)]
is in agreement with the theoretical prediction [123], where two surface Fermi arcs
connect the two bulk nodal touchings. In order to confirm that the observed Fermi
surface is indeed the double Fermi arcs connecting two bulk nodal points, we study the
evolution of constant energy contour as a function of binding energy EB . As shown in
Fig. 5.8(b), the energy contour area of the two bulk nodal touching points is found to
enlarge into contours as EB is increased (hole-like behavior), whereas the two surface
Fermi arcs shrink while increasing EB (electron-like behavior). The evolution of the
constant energy contour as a function of binding energy EB (electron or hole behavior
for different bands) is also consistent with the theoretical expectation [see Fig. 5.8(d)].
To further confirm, we study the energy dispersion for important momentum space
cut directions [Fig. 5.8(b)]. As shown in Fig. 5.8(c), surface states with a surface
Dirac crossing are clearly observed near the Fermi level in Cut β, consistent with the
theoretical calculation. The bulk valence band for Cut β is found to be away from
the Fermi level. On the other hand, for Cuts α and γ, no surface states are observed
but the bulk linear band is seen to cross the Fermi level, also consistent with the
calculation [see Fig. 5.8(e)]. The contrasting behavior that for Cut β only surface
states cross the Fermi level whereas for Cuts α and γ only the bulk linear bands cross
the Fermi level serves as a piece of critical evidence for the existence of double Fermi
arc surface states. We note that the two bulk nodal touchings observed in Fig. 5.8(a)
still expand in a finite area in momentum space, rather than being ideal “points”.
5.3 FERMI ARC SURFACE STATES IN TOPOLOGICAL DIRAC SEMIMETAL NA3 BI
This is because (1) electronic states in real samples have finite quasi-particle life time
and mean free path, and (2) it is also possible that the chemical potential of our
Na3 Bi sample is still slightly below the energy of the bulk nodes.
In order to further confirm the 2D surface nature for the double Fermi arc surface
states and the 3D bulk nature for the two bulk Dirac bands, we present Fermi surface
maps at different incident photon energies. Upon varying the photon energy, one
can study the electronic dispersion along the out-of-plane kz direction. Due to the
different 2D vs 3D nature of the Fermi arc surface states and the bulk Dirac bands,
the Fermi arc surface states are expected to be independent of kz (photon energy),
whereas the bulk bands are expected to show strong kz dependence. As shown in
Fig. 5.9(a), the double Fermi arcs are clearly observed irrespective of the choice of
the photon energy values, which confirms their 2D surface nature. On the other
hand, the two bulk nodal points are found to be quite pronounced at photon energies
of 58 eV and 55 eV, but they become nearly unobservable as the photon energy is
changed to 40 eV [e.g. the area highlighted by the red dotted circles in Fig. 5.9(a)].
This demonstrates the 3D bulk nature of the two bulk bands. To quantitatively
evaluate how much the bulk band becomes weaker as hν is changed from 58 eV to
40 eV (its kz dependence), we denote the middle point of the left Fermi arc as “S”
and the bottom bulk nodal touching point as “B” [Fig. 5.9(b)]. Fig. 5.9(c) shows
the relative ARPES intensity between “S” and “B”. The strong dependence of
I(B)
I(S)
upon varying photon energy shows that the bulk Dirac band becomes much weaker
relative to the surface states as hν is changed from 58 eV to 40 eV, due to the reason
that the kz value is moved away from the bulk nodal (Dirac) point. Furthermore,
Fig. 5.9(d) shows the dispersion along Cut β, where the surface states with a clear
Dirac crossing are observed at different photon energies. This confirms the 2D nature
of the Fermi arc surface states along Cut β. In Fig. 5.9(e), we present photon energy
dependent dispersion of the bulk nodal band [cut indicated by the white dotted line
136
5.3.2 Observation of Fermi arc surface states in Na3 Bi
in Fig. 5.9(a)]. It can be seen that the bulk Dirac band only crosses the Fermi level
at a photon energy around 58 eV and disperses strongly upon varying the photon
energy (kz ) value, which confirms the 3D bulk nature of the bulk Dirac band. These
systematic photon energy (kz ) dependent data, shown in Fig. 5.9, clearly demonstrate
that the electronic states at the Fermi level are localized at the surface (2D surface
state nature) everywhere around the Fermi surface contour, except at the locations
of the two bulk nodal touching points, where the states disperse strongly in all three
dimensions (3D bulk band nature). These data sets provide further support for the
observation of double Fermi arc surface states in bismuth trisodium.
We study the surface spin polarization along the Cut β. The white lines in
Fig. 5.10(a) define the two momenta chosen for spin-resolved studies (namely S-Cut1
and S-Cut2). Spin-resolved measurements are performed at these two fixed momenta
as a function of binding energy. Figs. 5.10(b),(c) show the in-plane spin-resolved intensity and net spin polarization. The magnitude of spin polarization reaches about
30% near the Fermi level. And the direction of spin polarization is reversed as one
goes from S-Cut1 to S-Cut2, which shows the spin-momentum locking property and
the singly degenerate nature of the Fermi arc surface states along the Cut β direction.
5.3 FERMI ARC SURFACE STATES IN TOPOLOGICAL DIRAC SEMIMETAL NA3 BI
Figure 5.8: Observation of Fermi arc surface states. (a) A Fermi surface map of the
Na3 Bi sample at photon energy of 55 eV. The BDP1 and BDP2 denote the two bulk Dirac
points’ momentum space locations. Two surface Fermi arcs are observed to connect these
two BDPs. (b) APRES constant energy contours as a function of binding energy at photon
energy of 55 eV. The dotted lines note the momentum space cuts for Panel (c). (c) ARPES
dispersion cuts α, β, γ as defined in Panel (b) at photon energy of 55 eV. Surface states for
the Fermi arcs are observed in cut β, whereas the two bulk Dirac bands are seen in cuts
α, γ. (d) Schematics of the constant energy contours drawn according to the theoretically
calculated band structure. The red shaded areas and the orange lines represent the bulk
and surface states, respectively. (e) Calculated band structure along Cut β and Cut α (γ).
138
5.3.2 Observation of Fermi arc surface states in Na3 Bi
Figure 5.8: (f ) Structure of bulk and surface Brillouin zone at (001) and (100) surfaces.
Bulk Dirac nodes are marked by blue crosses. Note that the two bulk Dirac cones project to
the same Γ̄ point on the (001) surface, while they are separated in momentum space when
studied at the (100) surface. The (100) surface was not studied in previous works due to the
challenge of such cleavage in ARPES experiments (g) First principles bulk band calculation
for Na3 Bi. (h) First principles calculation of the (100) surface electronic structure. This
figure is adapted from Ref. [46].
Figure 5.9: Systematic studies on the double Fermi arc surface states.
5.3 FERMI ARC SURFACE STATES IN TOPOLOGICAL DIRAC SEMIMETAL NA3 BI
Figure 5.9: Systematic studies on the double Fermi arc surface states. (a) ARPES
Fermi surface maps at different photon energies. The double Fermi arcs are observed independent of the photon energy value, whereas the two bulk Dirac band points along ky
become much weaker in intensity as the hν is changed from 58 eV to 40 eV, revealing their
surface state and bulk band nature, respectively. (b),(c) In order to quantitatively evaluate
how much the bulk band intensity gets weaker as hν is changed from 58 eV to 40 eV (its kz
dependence), we denote the middle point of the left Fermi arc as “S” and the bottom bulk
Dirac point as “B”. Panel (c) show the relative ARPES intensity between “S” and “B”.
The strong dependence of I(B)
I(S) upon varying photon energy shows the strong kz dependent
nature the bulk band Dirac points. (d) ARPES dispersion maps of the surface states at
two different photon energies. The surface states with a Dirac crossing are observed at
both photon energies, which further supports its 2D nature. (e) ARPES dispersion maps
of the bulk Dirac band at different photon energies [indicated by the white dotted lines in
Panel (a)]. The bulk Dirac band only crosses the Fermi level for excitation photon energies
around 58 eV, which demonstrates their three-dimensional dispersive (bulk) nature. This
figure is adapted from Ref. [46].
Figure 5.10: Spin-momentum locking in double Fermi arc surface states. (a) The
white dotted lines note the two momenta chosen for spin-resolved measurements. (b),(c)
Spin-resolved ARPES intensity and net spin polarization along the in-plane tangential direction for S-Cuts1 and 2 at photon energy of 55 eV. Clear spin polarization and spinmomentum locking are found in the data, which demonstrate the singly degenerate nature
of the Fermi arc surface states along the cut β k-space direction. This figure is adapted
from Ref. [46].
140
5.3.3 Topological invariant for the Dirac semimetal Na3 Bi
5.3.3
Topological invariant for the Dirac semimetal Na3 Bi
We use the obtained electronic and spin data for the (100) Fermi arc surface states to
gain insights to the predicted 2D topological number (ν2D = 1) in bismuth trisodium.
In Fig. 5.11, we present a series of ARPES dispersion maps. As schematically shown
in Fig. 5.11(a), these maps, which are perpendicular to the k[001] axis, intersect with
the axis at different k[001] values. As one goes from Slice1 to Slice7 along the k[001] axis,
the ARPES measured and schematic energy dispersion for different sides are shown
in Figs. 5.11(b),(c). Due to the existence of the two bulk nodes (Slices 2 and 6),
the bulk band gap closes and reopens as one goes across each bulk node. Therefore,
it is interesting to note that the k[001] axis serves as an effective axis for a bulk
mass parameter in a 2D system. It is further interesting to note that the bulk band
gap closings at Slices 2 and 6 correlated with the absence/appearance of the surface
states at the Fermi level. As clearly seen in Figs. 5.11(b),(c), there are no surface
states for Slice1. As one moves from Slice1 to Slice3, the bulk band gap closes and
reopens, and surface states appear at the Fermi level of bismuth trisodium. Similarly,
as one further goes from Slice3 to Slice7, the bulk band gap closes and reopens,
and the surface states do not exists anymore for the data in Slice7. We note that
the gap closing and reopening property that we observed here does not exist in the
Fermi surface of any known topological insulator system such as Bi2 Se3 , which again
highlights that the observed Fermi arc surface states (FASS) are different from the
topological surface states (TSS) in a topological insulator.
We focus on Slices 3-5, which are in-between the two bulk nodes. Interestingly,
as seen in Figs. 5.11(b),(c), although surface states exist at the native Fermi level for
any Slices that are in-between the two bulk nodes, they are in general gapped [e.g.
Slice3 in Figs. 5.11(b)(c)]. The gapped nature of the surface states means that, for
these Slices (k[001] 6= 0), there does not exist a nontrivial 2D topological number that
can protect a doubly degenerate Dirac crossing for the surface states. This is intuitive
5.3 FERMI ARC SURFACE STATES IN TOPOLOGICAL DIRAC SEMIMETAL NA3 BI
because these intermediate slices (k[001] 6= 0 and (k[001] 6= π) are not invariant under
time-reversal symmetry (because time-reversal operation will translate kz to −kz ).
Therefore, although the bulk bands are inverted, the two branches of the surface
states can hybridize and open up an energy gap (Slice 3). However, for Slice4, which
corresponds to k[001] = 0, the surface states are observed to be gapless with a surface
Dirac point. Furthermore, our spin-resolved data in Fig. 5.10 clearly show that the
two branches of the surface Dirac crossing carry opposite spin polarization. The
gapless surface Dirac crossing and the spin-momentum locking for Slice4 (k[001] = 0),
collectively, provide experimental evidence that the k[001] = 0 plane has a nontrivial
2D topological number, which is consistent with the ν2D = 1 for the time-reversal
invariant k[001] = 0 slice predicted in theory [124].
After systematically studying the electronic structure of the Fermi arc surface
states, we study the electronic structure of the two bulk Dirac cones. As shown in
Figs. 5.8(b),(d) the two bulk nodes at the Fermi level expand into two contours as
one goes to high binding energies (0≤EB ≤ 50 meV). Now we show the constant
energy contour maps at even higher binding energies (EB ≥ 50 meV). As shown in
Fig. 5.12(a), the two contours along the ky axis that correspond to the two bulk
Dirac bands expand as one goes to higher binding energies. At the binding energy
of ∼ 150 meV, the two separated Dirac band contours are found to just merge with
each other. The merging of the two contours locates at two special momenta, (kx , ky )
−1
= (±0.15, 0) Å , which lead to the occurrence of saddle point singularity structure
at these two special momenta. To directly identify the saddle point band structure
as a result of the Lifshitz transition in experiments, we center our ARPES detector
at one of the special momenta at (kx , ky ) = (+0.15, 0) Å
−1
and study two E-k cuts.
As shown in Fig. 5.12(b), the hybridized band reaches its local energy maximum at
−1
(kx , ky ) = (+0.15, 0) Å
along kx , while arriving a local energy minimum along ky .
This behavior defines a saddle point singularity of band structure. In Fig. 5.12(c)
142
5.3.3 Topological invariant for the Dirac semimetal Na3 Bi
we compare and contrast the saddle point singularity electronic structure observed
here with that observed in the surface states of a topological crystalline insulator
Pb0.7 Sn0.3 Se. In the n-typed sample of Pb0.7 Sn0.3 Se, the two surface Dirac cones
hybridizes at EB = 23 and 96 meV, signified also by saddle point singularities that
are reported from ARPES as well as STM studies. The critical difference between
these two cases is that for bismuth trisodium, bulk Dirac quasiparticles, instead of the
surface Dirac electron gas in the case of TCI [81], merge and hybridize. Existence of
singularities in the Dirac metals suggest that magnetic or superconducting dopants
can potentially materialize correlated electron phenomena in these materials in future
studies.
The significance of the observation of Fermi arc surface states in metallic samples is
analogous to the observation of Dirac surface states in 3D topological insulators [2, 6]
since it is the boundary modes that carry the signature of a topological material
[124, 186]). In fact, the topological property of Dirac semimetals was theoretically
understood only very recently (see, Ref. [124, 186]), which took place after several
numerical calculations and photoemission works showing that some of these materials
possess bulk band structure that is analogous to 3D graphene [37, 43–45, 75, 122,
123]. It is theoretically shown that the most interesting or exotic physics of these
materials are that of their surfaces not the bulk. Our observation of double Fermi
arc surface states and their spin momentum locking demonstrates the topology of a
Dirac semimetal. The observed surface states represent a new type of 2D electron
gas, which is distinct from that of the surface states in a topological insulator. In
a typical topological insulator such as Bi2 Se3 , the surface states’ Fermi surface is a
closed contour. In sharp contrast, the Fermi surface in bismuth trisodium consists of
two arcs, which are bridged by the two bulk nodes [Figs. 5.13(e),(f)]. Therefore, in this
novel Fermi surface topology, as one goes along the surface Fermi arc and reaches a
bulk node, the wavefunction of the surface state gradually loses its surface nature and
5.3 FERMI ARC SURFACE STATES IN TOPOLOGICAL DIRAC SEMIMETAL NA3 BI
becomes a bulk band. Such exotic behavior, shown by our systematic ARPES data
in Figs. 5.7-5.13, does not exist in surface states of Bi2 Se3 or any known topological
insulators. This highlights the distinct topological nature of the nontrivial semimetal
phase in a Dirac semimetal, which is fundamentally different from the known 3D
TI phase [2, 6]. In surface electrical transport experiments at high magnetic field, it
would be interesting to study how the surface electrons wind along the arc and enter
the bulk singularity [186]. Moreover, if superconductivity can be induced by bulk
doping in a Dirac semimetal, the surface state superconductivity can be topologically
nontrivial, which could possibly lead to double Majorana-arc states [125,192]. Finally,
the topological Dirac semimetal phase in a Dirac semimetal can be visualized as
two copies of Weyl semimetals, where each bulk Dirac node is a composite of two
degenerate Weyl nodes Fig. 5.13(e) that can be further split by additional symmetry
breaking. There is no analogous picture for the surface states of strong topological
insulators. At most, symmetry breaking in materials such as Bi2 Se3 will gap out the
surface states. Additionally, our observation of multiple bulk nodal touching points
realize a saddle point singularity in a strongly spin-orbit coupled material with linear
dispersion in three-dimensions, which potentially paves the way for correlated electron
physics [200, 201] with 3D Dirac or Weyl electrons.
In conclusion, our systematic studies of bismuth trisodium surfaces have demonstrated a topological surface state analog for a 3D Dirac semimetal. The double
surface Fermi arcs that we have observed connect the two bulk linear band touching points and exhibit spin polarization thus these results have identified the first
topological phase in a Dirac semimetal, where the arc surface states and their unique
spin momentum locking further evaluates a topological quantum number of ν2D = 1
for the k[001] = 0 2D (momentum slice) that uniquely defines the particular topological invariant realized in bismuth trisodium. Additionally, our data show that the two
bulk linear bands exhibit a saddle point band structure, a form of singularity that can
144
5.3.3 Topological invariant for the Dirac semimetal Na3 Bi
induce electronic instability in Dirac fermions upon magnetic and superconducting
doping of the metallic state in future studies. The observed Fermi arc surface states,
which represent a new type of 2D electron gas, not only open the door for studying new
fundamental physics phenomena, but also offer a new class of topological materials
for further engineering into the nanoscience world [10,119–124,139–141,149,185–192].
5.3 FERMI ARC SURFACE STATES IN TOPOLOGICAL DIRAC SEMIMETAL NA3 BI
Figure 5.11: 2D topological number: Surface spin polarization and Momentumdependent band gap closing. (a) A schematic view of the band structure of the topological Dirac semimetal phase. Seven slices/cuts that are taken perpendicular to the k[001] axis
are noted. (b),(c) ARPES measured [Panel (b)] and schematic [Panel (c)] band structure
for these slices are shown. This figure is adapted from Ref. [46].
146
5.3.3 Topological invariant for the Dirac semimetal Na3 Bi
Figure 5.12: Evolution through the bulk singularity. (a) ARPES constant energy
maps obtained on the (100) surface. At EB = 50 meV and higher binding energies, the
electronic states are mainly from the two bulk Dirac bands. The Constant energy contour
at EB = 150 meV, where the two bulk disconnected (bulk) Dirac band contours just merge
(the Lifshitz transition energy). (b) Dispersion of Cuts x and y, which are defined in Panel
(c). The hybridization of the two bulk Dirac cones results in a saddle point singularity at
the intersection of Cut X and Cut Y, signified by band maximum (lowest binding energy)
along cut X and band minimum (highest binding energy) along cut Y. (c) Lifshitz transition
of two bulk Dirac bands in bismuth trisodium with the Lifshitz transition of two surface
Dirac bands in a topological crystalline insulator Pb0.7 Sn0.3 Se.
5.3 FERMI ARC SURFACE STATES IN TOPOLOGICAL DIRAC SEMIMETAL NA3 BI
Figure 5.13: Comparison between the fine-tuning of a 3D Dirac semimetal and
the topological Dirac semimetal. (a) A band inversion process in a time-reversal and
inversion symmetric system without the influence of additional symmetries. In this case,
as the conduction and valence bands are inverted, spin-orbit coupling will open up a full
energy gap. The BiTl(S1−δ Seδ )2 system [37] is an example. (b) The critical point of the
band inversion realizes a 3D Dirac semimetal. Such 3D Dirac semimetal state requires fine
tuning of the material composition δ. (c),(d) A band inversion process in a time-reversal
and inversion symmetric system with an additional rotational symmetry. In this case, as
the conduction and valence bands are inverted and cross each other, spin-orbit coupling will
open up an energy gap everywhere in the k-space where they cross except at the momentum
locations along the rotational axis. This is because the bulk band crossings on the rotational
axis are protected by the rotational symmetry of the crystal.
148
5.3.3 Topological invariant for the Dirac semimetal Na3 Bi
Figure 5.13: The groundstate in the inverted regime is described by a pair of Dirac nodes
along the rotational axis in the bulk and double Fermi arc surface states connecting the two
bulk nodes on the surface. Inset: an ARPES Fermi surface map of Na3 Bi showing the double
FASS and the two bulk Dirac nodes. (e),(f ) Schematic Fermi surface topology of bismuth
trisodium and Bi2 Se3 . In bismuth trisodium, each Dirac node (the blue shaded area) can be
viewed as a composite of two degenerate Weyl nodes (yellow and green areas). The orange
arrows note the spin polarization according to our spin-resolved ARPES measurements. The
inset of Panels (c),(e) shows the ARPES measured Fermi surface. This figure is adapted
from Ref. [46].
5.4 TOPOLOGICAL SEMIMETALS: DIRAC, WEYL, AND NODAL-LINE
5.4
Topological semimetals: Dirac, Weyl, and nodalline
Now that we have presented the topological Dirac semimetal state and its double
FASS using a concrete example of Na3 Bi, in this section we discuss different classes
of topological semimetals. We note that topological semimetal is an actively ongoing research area both in experiments and in theories. The three classes (Dirac,
Weyl, and nodal-line) listed here are the currently theoretically known ones (within
band theory), but there might be other types being proposed in the future. We
also note that our observation of 3D Dirac nodes and FASS in Na3 Bi is the first
concrete realization of a topological Dirac semimetal (in the sense that the topological
surface states (FASS) are observed). The other two classes, namely the topological
Weyl semimetal [119, 188] and the topological nodal-line semimetal [193], remain
completely experimentally elusive. Again, our discussion will only cover the concepts
of these phases. While the detailed formulations can be found in theory papers
[10,119,188,193], here the idea is to point out the key concepts that serve as important
guidelines for us to experimentally search these new phases in real materials.
Let us start from a Weyl semimetal. We have briefly mentioned the Weyl semimetal
state in Chapter 3.1 (see Fig. 3.1). There we said that in a time-reversal invariant
but space inversion symmetry breaking bulk material, as the system is tuned to go
through a band inversion and a topological phase transition from a conventional band
insulator to a topological insulator, there exists a range of tuning parameter δ values,
where the band-gap goes to zero. Within this a range of tuning parameter δ values,
the system is a Weyl semimetal. We note that the key is that in an inversion asymmetric system, the bulk electronic bands are singly degenerate except at the Kramers
points. Therefore, as the bulk conduction and valence bands are inverted and cross
each other, these crossings are between two singly degenerate bands. This is in sharp
150
contrast to the case in a Dirac semimetal (see the previous two sections for Cd3 As2
and Na3 Bi), where the crossings are between two doubly degenerate bands.
For a band crossing between two singly degenerate bands, we can write down an
effective Hamiltonian that describes the vicinity of the crossing point as
H = ±v(px σx + py σy + pz σz ),
(5.1)
where ~p is the momentum from the crossing point, ~σ is the pseudo-spin. It can be
immediately seen that this Hamiltonian satisfy the Weyl equation, and therefore the
low-energy excitation of the bulk bands near the crossing is described by the Weyl
equation. The ± sign here defines the “handness” or “chirality” of the (pseudo)spinmomentum locking. In theory, the “chirality” is also proportional to a “topological
charge”, as one integrates the Berry’s phase around a Weyl point [10]. Now let us
consider the simplest Weyl semimetal system where there are two Weyl nodes with
the opposite “chiralities”, as shown in Fig. 5.14. This is a similar situation as in
Na3 Bi (see Fig. 5.11) in the sense that there are only two points in the whole bulk BZ
where the band-gap vanishes. Therefore, we consider the surface and bulk electronic
structure on different 2D k-slices, similar to what we did in Fig. 5.11. As shown in
Fig. 5.14, except at Slides 2 and 4, the bulk band structure has a full energy gap at all
other 2D k-slices. Due to the existence of the Weyl nodes, the bulk bands go through
two band inversions (gap closing and reopening) as one scans from Slide 1 to Slide 5.
It is theoretically shown [10] that the 2D k-slide’s Chern number n changes by (±)1
as one crosses a Weyl point with the ± chirality. Therefore, in a Weyl semimetal, each
k-slice (except the slices that contain the Weyl points where the system is gapless)
can be treated as a 2D Chern insulator. This is shown in Fig. 5.14. Therefore, similar
to a topological Dirac semimetal, a Weyl semimetal is described by a number of Weyl
nodes in the bulk and Fermi arc surface states (FASS) connecting the Weyl nodes
5.4 TOPOLOGICAL SEMIMETALS: DIRAC, WEYL, AND NODAL-LINE
on the surface. However, there is a key difference, which is the following: For an
individual 2D k-slice that contains the FASS in a Weyl semimetal (e.g. see Slide 3
in Fig. 5.14), the surface state is a Chiral mode that is always gapless as long as
the 2D k-slice cuts across the FASS (between Slides 2 and 4 in Fig. 5.14). On the
other hand, for a topological Dirac semimetal, there is only one 2D k-slice (e.g. see
Slide4 Fig. 5.11), where the FASS are gapless because they are protected by timereversal symmetry. In other words, the FASS in a Weyl semimetal share the same
topological properties as the chiral edgestates in a Chern insulator, and such FASS
are protected as long as lattice translational symmetry and the charge conservation
(U(1) symmetry) hold. Whereas in a topological Dirac semimetal, besides translation
and charge conservation, its FASS needs additional time-reversal or mirror symmetry
to protect.
Let us discuss the connection between a topological Weyl semimetal and a topological Dirac semimetal. It is well-known that each Dirac fermion can be viewed as a
composite of two degenerate Weyl fermions with the opposite chiralities. Therefore,
under time-reversal or space inversion symmetry breaking, a Dirac semimetal can be
tuned into a Weyl semimetal. Let us again take the topological Dirac semimetal state
in Na3 Bi as an example. Fig. 5.15 shows the evolution of the Na3 Bi groundstate as
one breaks space inversion symmetry. It can be seen that each Dirac node splits into
two Weyl nodes. And the system turns into a Weyl semimetal. It is interesting to
see the surface states at different 2D k-slices, as there are in total 4 Weyl nodes, the
Chern number changes its value 4 times. Additionally, since time-reversal symmetry
is preserved, the 2D k-slice of kz = 0 is still protected by the time-reversal symmetry,
and thus features a 2D Z2 topological number ν2D = 1. Similarly, one can figure out
the evolution as one breaks time-reversal symmetry in Na3 Bi.
Lastly, we briefly mention the topological nodal-line semimetal phase [193]. So
far in Dirac and Weyl semimetals, the conduction and valence bands only cross each
152
other at discrete (0D) points in a bulk BZ. In a topological nodal-line semimetal, the
band-crossing between the conduction and valence bands forms a 1D-closed-loop in a
bulk BZ. And the surface states can be described as a “drum surface” whose boundary
is the closed-loop of the bulk band crossing (see Fig. 5.16). Theoretically, such type
of novel topological semimetal can be realized in a time-reversal and space-inversion
symmetric spinless fermion bulk material, or a time-reversal or space-inversion breaking system with additional mirror symmetries.
Since the experimental progress on these semimetals are preliminary, we do not go
into further details. Nevertheless, it can be seen that topological semimetal is a truly
exciting field. Realization of these new nontrivial semimetals can open a new era in
topological insulator research. And breaking symmetries in the known topological
Dirac semimetals Na3 Bi and Cd3 As2 is a promising route toward these new phases.
5.4 TOPOLOGICAL SEMIMETALS: DIRAC, WEYL, AND NODAL-LINE
Figure 5.14: Electronic groundstate of a Weyl semimetal. Top: The Fermi surface
of a Weyl semimetal is described by two Weyl nodes with opposite chiralities in the bulk and
a FASS that connects the Weyl nodes on the surface. It has been theoretically shown that
each k-slice (except the slices that contain the Weyl points where the system is gapless)
can be viewed as a 2D Chern insulator. And as one scans across a ± Weyl point, the
system’s Chern number changes by ±1. Bottom: Bulk and surface electronic dispersions
on different 2D k-slices.
154
Figure 5.15: Tuning the Dirac semimetal Na3 Bi into a Weyl semimetal via space
inversion symmetry breaking. Each Dirac node in Na3 Bi is a composite of two degenerate Weyl nodes with the opposite chiralities. Under inversion symmetry breaking, the
Weyl nodes are separated in momentum space, realizing a Weyl semimetal.
Figure 5.16: Bulk and surface electronic groundstate in a topological nodal-line
semimetal. In a topological nodal-line semimetal, the conduction and valence bands are
degenerate (cross each other) at a closed-1D-loop in a bulk BZ. And the surface states
can be described as a “drum surface” whose boundary is the closed-loop of the bulk band
crossing. This figure is adapted from Ref. [193].
Chapter 6
Topological states in 4f Kondo
systems SmB6 and YbB6
Understanding the physics of strongly correlated systems is one of the most challenging tasks in physics. In solid states, strong electron-electron interaction is known to
lead to many exotic groundstates ranging from Mott insulation, unconventional superconductivity, heavy fermion behavior, to a fractional quantum Hall state. Studying
the interplay between nontrivial topology and strong correlation is of fundamental
importance. In fact, the definition of a topological order (not a symmetry-protected
topological state) intrinsically requires strong electron-electron interaction [24], and
the only topological order realized in experiments to date is that of in a fractional
quantum Hall state. Moreover, even within the framework of symmetry-protected
topological states, introducing strong electronic correlation is also believed to dramatically enrich the possible topological classification as proposed recently [142, 202].
On the other hand, studying topological physics in a strongly correlated system is
very challenging. Since the surface states in a strongly correlated system becomes
non-electron-like quasi-particle excitations, currently there is no direct probe to systematically study the metallic surface modes including their energy-momentum dis156
persion and spin polarization, as what has been achieved via spin-ARPES in weakly
interacting electronic systems.
In this chapter, we present our ARPES studies on the surface and bulk electronic
groundstate of certain Kondo systems with 4f electrons, in order to search for possible
topologically nontrivial states in these correlated materials. The fact that we can still
use ARPES to measure their surface states means that the correlation effects are
not extremely strong so that band theory completely breaks down in all aspects
in these systems. However, we show that electron interaction plays a vital role in
the formation of the topological states and properties of the resulting surface states
such as the surface Fermi velocity are quite different from a nearly non-interacting
topological insulator such as Bi2 Se3 . Our study servers an important initial step
toward realizing strongly correlated topological insulator states. The identification
of topological surface states in 4f Kondo materials is particularly interesting due
to a variety of symmetry breaking states such as heavy fermion superconductivity,
ferromagnetism, antiferromagnetism, hidden order state, etc., which are known to
occur in the rare-earth Kondo materials.
6.1
Observation of surface states in topological Kondo
insulator candidate SmB6
Materials with strong electron correlations often exhibit exotic ground states such as
the heavy fermion behavior, Mott or Kondo insulation and unconventional superconductivity. Kondo insulators are mostly realized in the rare-earth based compounds
featuring f -electron degrees of freedom, which behave like a correlated metal at high
temperatures whereas a bulk bandgap opens at low temperatures through the hybridization [203–205] of nearly localized-flat f bands with the d -derived dispersive
conduction band. With the advent of topological insulators [2] the compound SmB6 ,
SERVATION OF SURFACE STATES IN TOPOLOGICAL KONDO INSULATOR CANDIDATE SMB6
often categorized as a heavy-fermion semiconductor [203–205], attracted much attention due to the proposal that it may possibly host a topological Kondo phase (TKI) at
low temperatures where transport is anomalous [118, 206, 207]. The anomalous residual conductivity is believed to be associated with electronic states that lie within the
Kondo gap [208–217].
Following the prediction of a TKI phase, there have been several surface-sensitive
transport measurements, which include observation of a three-dimensional (3D) to
two-dimensional (2D) crossover of the transport carriers below T ∼ 7K [106,107,218].
However, due to the lack of the critical momentum resolution for the transport probes,
neither the existence of in-gap surface states nor their Fermi surface topology (number of surface Fermi surfaces and enclosing or not enclosing the Kramer’s points) have
been experimentally studied. By combining high-resolution laser- and synchrotronbased angle-resolved photoemission techniques in Ref. [50], we present the surface
electronic structure identifying the in-gap states that are strongly temperature dependent and disappear before approaching the coherent Kondo hybridization scale.
Remarkably, the observed Fermi surface for the low-energy part of the in-gap states
keeping the sample within the transport anomaly regime (T ∼ 6 K) reveals an odd
number of pockets that enclose three out of the four Kramers’ points of the surface
Brillouin zone, consistent with the theoretically calculated Fermi surface topology of
the topological surface states. Concurrent ARPES studies on SmB6 are also reported
in Refs. [49, 51].
SmB6 crystallizes in the CsCl-type structure with the Sm ions and the B6 octahedra being located at the corner and at the body center of the cubic lattice, respectively
[Fig. 6.1(a)]. The bulk Brillouin zone (BZ) is a cube made up of six square faces. The
center of the cube is the Γ point, whereas the centers of the square faces are the X
points. Due to the inversion symmetry of the crystal, each X point and its diametrically opposite partner are completely equivalent. Therefore, there exist three distinct
158
X points in the BZ, labeled as X1 , X2 and X3 . It is well-established that the low
energy physics in SmB6 is constituted of the non-dispersive Sm 4f band and the dispersive Sm 5d band located near the X points [106,211,212,218,219]. Figs. 6.1(d),(e)
show ARPES intensity profiles over a wide binding energy scale measured with a
synchrotron-based ARPES system using a photon energy of 26 eV. The dispersive
features originate from the Sm 5d derived bands and a hybridization between the Sm
5d band and Sm 4f flat band is visible especially around 150 meV binding energies
confirming the Kondo features of the electronic system in our study [Figs. 6.1(d),(e)].
In order to search for the predicted in-gap states within 5 meV of the Fermi level,
a laser-based ARPES system providing ∆E ∼ 4 meV coupled with a low temperature
(T ≃ 5 K) capability is employed in Ref. [50]. Since the low-energy physics including
the Kondo hybridization process occurs near the three X points [Fig. 6.1(f)] in the
bulk BZ and the X points project onto the X̄1 , X̄2 , and the Γ̄ points at (001) surface
[Fig. 6.1(b)], the Kramers’ points of this lattice are X̄1 , X̄2 , Γ̄ and M̄ and one needs
to systematically study the connectivity (winding) of the in-gap states around these
points. Fig. 6.2(c) shows experimentally measured ARPES spectral intensity inte−1
grated in a narrow (±0.15 Å ) momentum window and their temperature evolution
around the X̄ point. At temperatures above the hybridization scale, only one spectral
intensity feature is observed around EB ∼ 12 meV in the ARPES EDC profile. As
temperature decreases below 30 K, this feature is found to move to deeper binding
energies away from the chemical potential, consistent with the opening of the Kondo
hybridization gap while Fermi level is in the insulating gap (bulk is insulating, according to transport, so Fermi level must lie in-gap at 6 K). At lower temperatures,
the gap value of hybridized states at this momentum space regime is estimated to be
about 16 meV. More importantly, at a low temperature T ≃ 6 K corresponding to the
2D transport regime, a second spectral intensity feature is observed at the binding
energy of EB ∼ 4 meV, which lies inside the insulating gap. Our data thus experi-
SERVATION OF SURFACE STATES IN TOPOLOGICAL KONDO INSULATOR CANDIDATE SMB6
mentally shows the existence of in-gap states. Remarkably, the in-gap state feature
is most pronounced at low temperature T ≃ 6 K in the 2D transport regime, but
becomes suppressed and eventually vanishes as temperature is raised before reaching
the onset for the Kondo lattice hybridization at 30 K. The in-gap states are found
to be robust against thermal cycling, since lowering the temperature back down to
6K results in the similar spectra with the re-appearance of the in-gap state features
[(Re 6K in Fig. 6.2(c)]. The observed robustness against thermal recyclings counts
against the possibility of non-robust (trivial) or non-reproducible surface states. We
further performed similar measurements of low-lying states focusing near the Γ̄ point
(projection of the X3 ) as shown in Fig. 6.2(d). Similar spectra reveal in-gap state
features prominently around EB ∼ 3 − 4 meV at T ≃ 6 K which clearly lie within the
Kondo gap and exhibit similar (coupled) temperature evolution as seen in the spectra
obtained near the X̄ point.
We further study their momentum-resolved structure or the k-space map for investigations regarding their topology: 1) The number of surface state pockets that
lie within the Kondo gap; 2) The momentum space locations of the pockets (whether
enclosing or winding the Kramers’ points or not). Fig. 6.1(f) shows a Fermi surface
map measured by setting the energy window to cover EF ± 4 meV, which ensures the
inclusion of the in-gap states (that show temperature dependence consistent with coupling to the Kondo hybridization) within the Fermi surface mapping data as identified
in Fig. 6.2(d),(e) at a temperature of 6 K inside the 2D transport anomaly regime
under the “better than 5 meV and 7 K combined resolution condition”. Our Fermi
surface mapping reveals multiple pockets which consist of an oval-shaped as well as
nearly circular-shaped pockets around the X̄ and Γ̄ points, respectively. No pocket
was seen around the M̄ -point which was measured in a synchrotron ARPES setting.
Therefore the laser ARPES data captures all the pockets that exist while the bulk is
insulating. This result is striking by itself from the point of view that while we know
160
from transport that the bulk is insulating, ARPES shows large Fermi surface pockets
(metallicity of the surface) at this temperature. Another unusual aspect is that not
all Kramers’ points are enclosed by the in-gap states. Our observed Fermi surface
thus consists of 3 (or odd number Mod 2 around each Kramers’ point) pockets per
Brilluoin zone and each of them wind around a Kramers’ point only and this number
is odd (at least 3). Therefore, our measured in-(Kondo) gap states lead to a very
specific form of the Fermi surface topology [Fig. 6.1(f)] that is remarkably consistent
with the theoretically predicted topological surface state Fermi surface expected in
the TKI groundstate phase despite the broad nature of the contours.
Since for the laser-ARPES, the photon energy is fixed (7 eV) and the momentum
√
window is rather limited (the momentum range is proportional to hν − W , where hν
is the photon energy and W ≃ 4.5 eV is the work function), we utilize synchrotron
based ARPES measurements to study the low-lying state as a function of photon
energy as demonstrated in Bi-based topological insulators [2]. Figs. 6.2(e),(f) show
the energy-momentum cuts measured with varying photon energies. Clear E − k
dispersions are observed within a narrow energy window near the Fermi level. The
dispersion is found to be unchanged upon varying photon energy, supporting their
quasi-two-dimensional nature [see, Fig. 6.2(g)]. The observed quasi-two-dimensional
character of the signal within 10 meV of the gap where surface states reside does
suggest consistency with the surface nature of the in-gap states. Due to the combined
effects of energy resolution (∆E ≥ 10 meV, even though the sample temperature, 7
K, is near the anomalous transport regime) and the intrinsic self-energy broadening
coupled with the higher weight of the f -part of the cross-section and the strong band
tails, the in-gap states are intermixed with the higher energy bulk bands’ tails. In
order to isolate the in-gap states from the bulk band tails that have higher crosssection at synchrotron photon energies, it is necessary to have energy resolution (not
just the low working temperature) better than half the Kondo gap scale which is about
SERVATION OF SURFACE STATES IN TOPOLOGICAL KONDO INSULATOR CANDIDATE SMB6
7 meV or smaller in SmB6 . Our experiment reports of Fermi surface mapping covering
the low-energy part of the in-gap states keeping the sample within the transport
anomaly regime reveals an odd number of pockets that enclose three out of the four
Kramers’ points of the surface Brillouin zone strongly suggesting a topological origin
for the in-gap state.
162
Figure 6.1: Brillouin zone symmetry, and band structure of SmB6 . (a) Crystal structure of SmB6 . Sm ions and B6 octahedron are located at the corners and the
center of the cubic lattice structure. (b) The bulk and surface Brillouin zones of SmB6 .
High-symmetry points are marked. (c) Resistivity-temperature profile for samples used
in ARPES measurements. (d), (e) Synchrotron-based ARPES dispersion maps along the
M̄ − X̄ − M̄ and the X̄ − Γ̄− X̄ momentum-space cut-directions. Dispersive Sm 5d band and
non-dispersive flat Sm 4f bands are observed, confirming the key ingredient for a heavy
fermion Kondo system. (f ) A Fermi surface map of bulk insulating SmB6 using a 7 eV
laser source at a sample temperature of ≃ 6 K (Resistivity= 5 mΩcm), obtained within
the EF ± 4 meV window, which captured all the low energy states between 0 to 4 meV
binding energies, where in-gap surface state’s spectral weight contribute most significantly
within the insulating Kondo gap. Intensity contours around Γ̄ and X̄ reflect low-lying
metallic states near the Fermi level, which is consistent with the theoretically predicted
Fermi surface topology [207] of the topological surface states. This figure is adapted from
Ref. [50].
SERVATION OF SURFACE STATES IN TOPOLOGICAL KONDO INSULATOR CANDIDATE SMB6
Figure 6.2: Temperature dependent in-gap states and its two-dimensional nature. (a) Cartoon sketch depicting the basics of Kondo lattice hybridization at temperatures above and below the hybridization gap opening. (b) Partially momentum-integrated
−1
ARPES spectral intensity in a ±0.15 Å window (∆k defined in Panel (a)] above and below the Kondo lattice hybridization temperature [TH ). (c) Momentum-integrated ARPES
spectral intensity centered at the X̄ point at various temperatures.
164
Figure 6.2: (d) Analogous measurements as in Panel (c) but centered at the Γ̄ pocket
−1
(∆k = 0.3 Å ). ARPES data taken on the sample after thermally recycling (6 K up to
50 K then back to 6 K) is shown by Re 6K, which demonstrates that the in-gap states
are robust against thermal recycling. (e) Synchrotron based ARPES energy momentum
dispersion maps measured using different photon energies along the M̄ − X̄ − M̄ momentum
space cut-direction. Incident photon energies used are noted on the plot. (f ) Momentum
distribution curves (MDCs) of data shown in a. The peaks of the momentum distribution
curves are marked by dashed lines near the Fermi level, which track the dispersion of the
low-energy states. (g) Momentum distribution curves in the close vicinity of the Fermi level
(covering the in-gap states near the gap edge) are shown as a function of photon energy
which covers the kz range of 4π to 5π at 7 K. This figure is adapted from Ref. [50].
6.2 TOPOLOGICAL SURFACE STATES IN YBB6
6.2
Topological surface states in YbB6
In a topological Kondo insulator (TKI), it is the Kondo hybridization between the
d and f bands that leads to an inversion of the band parity and therefore realizes a
topologically nontrivial insulator phase. A well-known example is the prediction of
the TKI state in SmB6 [118, 207], and the following experimental works which identified the existence of in-gap surface states in SmB6 , consistent with the theoretically
predicted TKI phase [49–52, 106, 107]. However, the surface states in TKI phase of
SmB6 only exist at very low temperatures [50] and their Fermi velocity is expected to
be low due to a strong f -orbital contribution [50, 207], limiting its future utilization
in devices. In order to search for other novel correlated topological phases even without a Kondo mechnism, it is quite suggestive to systematically study the electronic
groundstates of other rare-earth materials that are closely-related to SmB6 . SmB6
belongs to a class of materials called the rare-earth hexaborides, RB6 (R = rare-earth
metal).
In general, rare-earth hexaboride compounds are known to feature three types of
electronic bands in the vicinity of the Fermi level, namely the the rare-earth 5d orbital
band, the rare-earth 4f orbital band, and the boron 2p band. The low energy physics
for a rare-earth hexaboride is collectively determined by the relative energies between
these bands and the Fermi level, which depends on a delicate interplay among the
key physical parameters including the valence of the rare-earth element, the lattice
constant, the spin-orbit coupling (SOC), etc. Therefore, interestingly, the rare-earth
hexaborides can realize a rich variety of distinct electronic ground states as seen in
SmB6 , YbB6 , EuB6 and the superconducting state in LaB6 (Tc ∼ 0.5 K) [220, 221].
Motivated by these aspects, we study the surface and bulk electrnic structure of of
another representative member of the rare-earth hexaboride family, ytterbium hexaboride (YbB6 ). Fig. 6.3 shows the theoretical calculated bulk band structure of both
SmB6 and YbB6 . For SmB6 , calculation shows that a hybridization between the Sm
166
5d and Sm 4f bands near the Fermi level leads to a Kondo insulating gap, consistent
with previous findings [207]. Interestingly, our calculation results [Figs. 6.3(a)-(c)]
show that the groundstate is insensitive to the strength of the on-site coulomb repulsion U (qualitatively the same results for U = 0 and U = 8 eV), which demonstrates
the robustness of the Kondo insulating state in SmB6 . In sharp contrast, in YbB6 ,
a Kondo insulating state is also seen without including coulomb repulsion [U = 0,
see Fig. 6.3(d)]. However, as one increases the strength of the coulomb repulsion
U, our calculation shows that the energy position of the Yb 4f bands are pushed
to much higher energies below the chemical potential [Figs. 6.3(e),(f)], and therefore
a Kondo insulating mechanism becomes irrelevant to the low energy physics.On the
other hand, the B 2p bands become closer to the Fermi level and at certain intermediate U values [U = 5 eV, see Fig. 6.3(e)], an inversion between the Yb 5d and the
B 2p bands is observed. This calculation suggests a novel scenario that a topological
insulator state can be realized in the rare-earth hexaborides even without a Kondo
insulating mechanism.
In order to reveal the electronic state of YbB6 , we systematically study its electronic structure at the (001) natural cleavage surface. As shown in Fig. 6.4(c), the
lowest 4f flat band in YbB6 is 1 eV below the Fermi energy. This is in sharp contrast
to the ARPES data on SmB6 , where the flat 4f band is found to be only ≤ 15 meV
away from the Fermi level [50]. Therefore, our data reveal the physical origin for
the absence of the Kondo insulating state in YbB6 , which also negates the recent
theoretical work that predicts the existence of the topological Kondo insulator phase
in YbB6 [222]. Apart from the intense 4f band, our data in Fig. 6.4(c) also reveal an
electron-like pocket centered at the X̄ point that crosses the Fermi level.
In order to systematically resolve the Fermi level electronic structure, we present
high-resolution ARPES measurements in the close vicinity of the Fermi level. It is
important to note that at the studied (001) surface, the three X points (X1 , X2 , and
6.2 TOPOLOGICAL SURFACE STATES IN YBB6
X3 ) in the bulk BZ project onto the Γ̄ point and the two X̄ points on the (001) surface
BZ, respectively. Since the valence band maximum and conduction band minimum
are at the X-points, at the (001) surface one would expect low energy electronic
states near the Γ̄ point and X̄ points. The Fermi surface map of YbB6 is presented
in Fig. 6.4(a). Our Fermi surface map reveals multiple pockets, which consist of an
oval-shaped contour and a nearly circular-shaped contour enclosing each X̄ and Γ̄
points, respectively. No pocket is seen around the M̄-point. YbB6 shows metallic
behavior in transport [221] which is consistent with our data.
We present ARPES energy and momentum dispersion maps, where low-energy
states consistent with the observed Fermi surface topology are clearly identified. As
shown in Fig. 6.4(b), a “V”-shaped nearly linearly dispersive band is observed at each
Γ̄ and X̄ points. We further study the photon energy dependence of the observed “V”shaped bands. As shown in Figs. 6.4(b) and Figs. 6.5(a), the “V”-shaped bands are
found to show no observable dispersion as the incident photon energy is varied, which
suggests its quasi two-dimensional state nature. Therefore, our systematic ARPES
data has identified three important properties in the YbB6 electronic structure: (1) A
odd number of Fermi surface pockets are observed to enclose the Kramers’ point; (2)
The bands at the Fermi level are found to exhibit nearly linearly (Dirac like) in-plane
dispersion; (3) No observable out-of-plane (kz ) dispersion is observed for these Dirac
like (“V”-shaped) bands. All these properties that we observed seem to suggest a
possibility for a topological insulator state in YbB6 . Nevertheless, it is important
to note that if these “V”-shaped bands are indeed the upper Dirac cone part of the
topological surface states, then one would also expect the lower Dirac cone part and a
well-defined Dirac crossing (the Dirac point of the surface states). In our data, some
hole-like band [e.g. E ≃ −0.4 eV in the first panel of Figs. 6.5(a)] is seen that can
correspond to the low Dirac cone. The observed hole-like band features are not sharp
enough to draw any definitive conclusion for a clear Dirac crossing. We further test
168
the temperature dependence of these “V”-shaped bands. As shown in Fig. 6.5(b),
the “V”-shaped bands are observed to be quite robust as one increases temperature.
Finally, in order to confirm that what observed in ARPES is indeed the topological
surface states, we perform First-principles slab calculation of the (001) surface with
U = 4 eV as shown in Fig. 6.6. Dirac-cone surface states are clearly seen in the bulk
energy-gap at the Γ̄ and the X̄ points, both of which correspond to the projection of
the X points in the bulk BZ. Taken the ARPES and the calculation results collectively,
we have demonstrated the topological surface states in YbB6 and further shown that
the topological insulator state in YbB6 is irrelevant to a Kondo insulating gap since
the lowest 4f flat band in YbB6 is 1 eV below the Fermi energy. Therefore, our studies
of the surface and bulk electronic structure in YbB6 present a novel scenario that a
topological insulator state is realized in YbB6 due to d − p hybridization without a
Kondo insulating mechanism.
6.2 TOPOLOGICAL SURFACE STATES IN YBB6
Figure 6.3: Bulk band structure (GGA+U) calculation of Sm6 and YbB6 . (a),
(b) Band structure of SmB6 from GGA calculations. Sizes of yellow, blue, and green dots
denote weights of Sm-4f , Sm-5d, and B-2p orbitals in various bands. Zoomed in view in
Pane (b). (c) Low-energy band structure around EF based on GGA and GGA+U (U = 8
eV) computations. (d)-(f ) Band structure calculation of YbB6 at various U values. Top:
large energy range. Middle: Zoomed-in view. Bottom: Scematic.
170
Figure 6.4: Topological surface states in YbB6 . (a) ARPES measured Fermi surface of
YbB6 . Circular shaped pockets are observed at Γ̄ and X̄ points. This spectrum is measured
with photon energy of 50 eV at temperature of 15 K. (b) ARPES dispersion maps measured
with different photon energy. (c) ARPES measured dispersion maps along the Γ̄ − X̄ − Γ̄
momentum-space cut-directions. Dispersive cone like pocket and non-dispersive flat Yb 4f
bands are observed. The measured photon energy are noted on the plots. This figure is
adapted from Ref. [86].
6.2 TOPOLOGICAL SURFACE STATES IN YBB6
Figure 6.5: Photon energy and temperature dependent dispersion maps. (a)
Photon energy dependent dispersion map with wider binding energy. The measured photon
energy are noted on the plots. The 4f flat bands are about 1 eV below the Fermi level. These
spectra are measured along the Γ̄ − X̄ momentum space cut direction with temperature of
15K . (b) Temperature dependent ARPES spectra. The measured temperature are noted
on the plots. The slight variation of the pocket with temperature is likely due to the thermal
expansion of the sample. These spectra are measured along the Γ̄ − M̄ momentum space
cut direction. This figure is adapted from Ref. [86].
172
Figure 6.6: Theoretical calculation showing the topological surface states on the
(001) surface of YbB6 . First-principles slab calculation of the (001) surface with U = 4
eV. Dirac-cone surface states are clearly seen in the bulk energy-gap at the Γ̄ and the X̄
points, both of which correspond to the projection of the X points in the bulk BZ.
Chapter 7
A route to 2D topological
superconductivity
The interplay between a topological insulator state and a symmetry-breaking (superconducting or ferromagnetic) state is of interest in both fundamental physics and
applications [2, 6, 9, 53–61, 125–128, 223–226]. Specifically, inducing superconductivity
in the Dirac surface states of a topological insulator is believed to realize Majorana
fermion excitation [125] and supersymmetry phenomenon [128] in a condensed matter setting. A Majorana fermion is a non-Abelian quasi-particle, which is believed
to be the key building block of a topological quantum computer. Supersymmetry is
a fascinating high energy physics concept, which remains elusive in particle physics
experiment to date. On the other hand, a ferromagnetically ordered topological insulator can realize a Chern insulator in its 2D thin film geometry [127], which means that
the boundary of the 2D film will host chiral electronic edge-modes with a quantized
Hall conductance even without an external magnetic field. This is of much interest
in building low-power electronics. Furthermore, a magnetic topological insulator is
predicted to show quantized Kerr rotation [126, 223], which serves as an independent
measure of the topological number besides the spin-ARPES measurements on the
174
topological surface states. The same effect is also theoretically known to simulate the
electrodynamics of the Axion [224], a particle candidate for the dark matter. Despite
the interest, none of the major TI materials (such as Bi2 Se3 ) are natural superconductors or ferromagnets. Therefore, one has to dope a TI material or interface with
a superconductor or ferromagnet, in order to induce these symmetry breaking-states.
In this chapter, we present our systematic ARPES and spin-ARPES measurements
on magnetically doped TIs and heteostructure samples between a TI thin film and an
s-wave superconductor. Since ARPES is a probe that measures the electronic quasiparticle spectrum without nanometer spatial resolution, thus it is not practically
possible to directly measure the Majorana exciations (non-electron quasi-pariticles)
in a superconducting TI or the chiral edge states (a 1D state) in a magnetically
doped TI thin film. The goal of our experiments is to experimentally demonstrate
the electronic and spin groundstates that are required for these theoretically proposed
phenomena.
7.1
Hedgehog spin texture in a magnetic topological insulator
Understanding and control of spin degrees of freedom on the surfaces of topological
materials are the key to future applications as well as for realizing novel physics such
as the axion electrodynamics associated with time-reversal symmetry breaking on the
surface. We experimentally demonstrate the momentum space spin texture in Bi2 Se3
surface states, where a surface band-gap is observed to open due to magnetic Mn
doping. The resulting electronic and spin groundstate on the surface of Mn-doped
Bi2 Se3 exhibits unique hedgehog-like spin textures at low energies which directly
demonstrates the mechanics of time-reversal symmetry breaking on the surface. We
further show that an insulating gap induced by quantum tunneling between surfaces
7.1 HEDGEHOG SPIN TEXTURE IN A MAGNETIC TOPOLOGICAL INSULATOR
exhibits spin texture modulation at low energies but respects time-reversal invariance.
These spin phenomena and the control of their Fermi surface geometrical phase first
demonstrated in our experiments pave the way for future realization of many predicted
exotic magnetic phenomena of topological origin.
Interplay between ferromagnetism and the Z2 topological insulator state is believed to give rise to novel physics, such as a quantum anomalous Hall state [127] and
the topological magneto-optic effect [126]. These phenomena rely on breaking timereversal symmetry in a topological insulator. It is known that the a gap at the Dirac
point will open in the surface states as time-reversal symmetry is breaking. However,
a surface band-gap can also open due to other mechanisms such as the tunneling between the top and bottom surfaces as a TI becomes thin. Here we use spin-ARPES to
measure the momentum space spin texture of gapped surface states in Bi2 Se3 samples,
where the Dirac point gap is opened by two distinct ways, namely doping magnetic
(Mn) impurities or reducing the thickness of the Bi2 Se3 sample. We show that the
spin texture near the edge of the surface gap behavior distinctly different in these two
cases due to the different gap opening mechanisms. In particular, we show that Mndoping in Bi2 Se3 leads to a unique hedgehog-like spin texture at low energies near the
edge of the magnetic gap, which directly demonstrates the mechanics of time-reversal
symmetry breaking on the surface. The spin-resolved measurements demonstrated
here can be utilized to probe quantum magnetism on the surfaces of other materials
as well.
In order to study the evolution of topological surface states upon magnetic doping,
magnetically (Mn%) and (chemically similar) non-magnetically (Zn%) doped Bi2 Se3
thin films are prepared in high quality using the molecular beam epitaxy (MBE)
growth method. A sample layout and a photograph image of a representative MBE
grown film used for experiments are shown in Figs. 7.1(a),(b). Using standard Xray magnetic circular dichroism [227, 228], we characterize the magnetic properties of
176
doped Bi2 Se3 films [Figs. 7.1(c)]. In Mn-doped Bi2 Se3 , a hysteretic behavior in the
out-of-plane magnetic response is observed consistently which suggests a ferromagnetically ordered groundstate. The observation of ferromagnetic character and its absence in Zn-Bi2 Se3 motivate us to systematically compare and contrast the electronic
density of state behavior in the vicinity of the surface Dirac node of these samples.
Fig. 7.1(d) shows the measured electronic states of Mn(Zn)-doped Bi2 Se3 using highresolution (spin-integrated) angle-resolved photoemission spectroscopy (ARPES). In
the undoped Bi2 Se3 film [leftmost panel of Fig. 7.1(d)], a map of spectral density of
states reveals a bright and intact Dirac node (signaled by the red spot located at
the Dirac crossing in the panel), which indicates that in undoped Bi2 Se3 the Dirac
node is gapless, consistent with the previous studies of pure Bi2 Se3 [32]. In samples
where Mn atoms are doped into the bulk [first row Fig. 7.1(d)], we observe that the
corresponding bright (red) spot at the Dirac node gradually disappears, revealing a
clear systematic spectral weight suppression (SWS) with increasing Mn concentration. In contrast, the spectral intensity at the Dirac node is observed to survive
upon systematic Zn doping except for the Zn=10% sample where some suppression
of intensity is observed. This suggests that the Dirac node remains largely intact
upon Zn doping. The observed systematic behavior of spectral evolution motivates
us to quantitatively define an energy scale, ESWS , associated with the SWS observed
at the Dirac node. The ESWS is taken as the energy spacing between the upper
Dirac band minimum and the Dirac node location along the energy axis as illustrated
in Fig. 7.1(e), which roughly corresponds to half of the surface gap magnitude. The
value of the energy scale can be quantitatively determined by fitting the ARPES measured energy-momentum distribution curves. The doping dependence of the ESWS on
samples measured at T = 20 K is shown in Fig. 7.2(c). The ESWS is observed to
increase rapidly with Mn concentration but it remains nearly zero with Zn doping.
The effect of temperature dependence on ESWS is shown in Fig. 7.2(d). The temper-
7.1 HEDGEHOG SPIN TEXTURE IN A MAGNETIC TOPOLOGICAL INSULATOR
ature induced decrease of ESWS is consistent with gradual weakening of magnetism.
These observations collectively reveal direct correlation between X-ray magnetic circular dichroism measured ferromagnetic character and the ARPES measured SWS
(or gap) on the Mn-Bi2 Se3 films.
Although the predominant trend in the doping evolution of surface states suggests
a correlation between magnetism and SWS, we do notice a non-negligible ESWS at
high Zn concentration (10%). This is likely due to increasing chemical disorder on
the surface of the film since disorder degrades the surface quality. Similarly, magnetization measurements show that ferromagnetism vanishes before reaching T = 300 K
[inset of Fig. 7.2(b)], but nonzero SWS [Fig. 7.2(d)] is still observed perhaps, in this
case, due to thermal disorder of the relaxed film surface at high temperatures. Therefore, the correlation between ferromagnetism and the ARPES gap is only clear at low
temperatures in samples with reduced disorder. The momentum width (∆k) of the
surface electronic states can be taken as a rough relative measure of surface disorder,
sample to sample, which is found to significantly increase upon both magnetic and
non-magnetic dopings [Fig. 7.2(c)]. Strong spatial fluctuations of the surface electronic states in doped Bi2 Se3 has been observed in a recent STM work by Beidenkopf
et al. [93], where the authors suggest the observation of gap-like feature at the Dirac
point without breaking TR symmetry. These ambiguities associated with the observed gap-like feature across many different experiments strongly call for critically
important spin-resolved measurements which also serve as a collective method, as we
show, to unambiguously identify the correct nature of the gap.
In order to study the evolution of spin texture upon magnetic doping, we perform
spin-resolved measurements on Mn-Bi2 Se3 topological surface states. We present two
independent but representative spin-resolved ARPES measurements on Mn(2.5%)Bi2 Se3 film I and film II. Films I and II, both containing same nominal Mn concentration, are measured and analyzed using two different spectroscopic modes, namely,
178
spin-resolved momentum distribution curve (spin-resolved MDC) measurement mode
and spin-resolved energy distribution curve (spin-resolved EDC) measurement mode,
in order to exclude any potential systematic error in the spin measurements. Figs. 7.3(a)(d) show measurements on film I. Our data shows that out-of-plane spin polarization Pz is nearly zero at large momentum k// far away from the Dirac point energy
[0 < EB < 0.1 eV in Figs. 7.3(c), (d)]. While approaching the Dirac point (0.1
eV< EB < 0.3 eV), an imbalance between the spin-resolved intensity in +ẑ and −ẑ
is observed [Fig. 7.3(c)]. The imbalance is found to become systematically more pronounced in the data set where scans are taken by lowering the energy toward the
Dirac point. This systematic behavior observed in the data reveals a net significant
out-of-plane spin polarization in the vicinity of the “gapped” Dirac point or near the
bottom of the surface state conduction band. More importantly, the out-of-plane
spin component Pz does not reverse its sign in traversing from −k// to +k// . Such
behavior is in sharp contrast to the spin textures observed in pure Bi2 Se3 [34] where
spins point to opposite directions on opposite sides of the Fermi surface as expected
from TR symmetry. Therefore, our Pz measurements on film I near the gap edge
reveal the TR breaking nature of the Mn-Bi2 Se3 sample where magnetic hysteresis
was observed using X-ray magnetic circular dichroism technique. In order to directly
measure the spin of the surface state at Γ̄ (the Kramers’ momentum, k// = 0), we
perform spin measurements on Mn-Bi2 Se3 film II (same Mn concentration as film I)
working in the spin-resolved EDC mode. The measured out-of-plane spin polarization
(Pz ) is shown in Fig. 7.3(g). We focus on the Pz measurement at Γ̄, the Kramers’
momentum k// = 0 (red curve): the surface electrons at TR invariant Γ̄ are clearly
observed to be spin polarized in the out-of-plane direction. The opposite sign of Pz for
the upper and lower Dirac band [red curve in Fig. 7.3(g)] shows that the Dirac point
spin degeneracy is indeed lifted up (E(k// = 0, ↑)6=E(k// = 0, ↓)). Such observation
directly counters the Kramers’ theorem and therefore manifestly breaks the TR sym-
7.1 HEDGEHOG SPIN TEXTURE IN A MAGNETIC TOPOLOGICAL INSULATOR
metry on the surface. Next we analyze Pz measurements at finite k// [green curves
in Fig. 7.3(g)] to extract the detailed configuration of the spin texture. In going to
larger k// away from the Γ̄ momenta, the measured Pz is found to gradually decrease
to zero. Moreover, the constant energy momentum space plane at the Dirac point
(EB = ED ) is observed to serve as a mirror plane that reflects all of the out-of-plane
spin components between the upper and lower Dirac bands. Thus both spin-resolved
MDC (film I) and spin-resolved EDC (film II) measurement modes result in the consistent conclusions regarding the spin configuration of the films. These systematic
measurements, especially at the vicinity of the gap, reveal a hedgehog-like spin configuration for each upper(or lower) Dirac band separated by the magnetic gap, which
breaks TR symmetry, as schematically presented in the inset of Fig. 7.3(f).
Spin texture measurements on non-magnetically doped films Zn(1.5%)-Bi2 Se3 are
presented in Figs. 7.4(a)-(d). The out-of-plane polarization Pz measurements reveal
a sharp contrast to the magnetically doped Mn-Bi2 Se3 films, specifically, the near absence of finite Pz component around Γ̄ within our experimental resolution [Fig. 7.4(d)].
A very small Pz , however, at large k// is observed, which is expected due to surface state warping also observed in other topological insulator (TI) compounds [229]
[Fig. 7.4(d)]. The signal being associated with warping is further confirmed in our
data due to their TR symmetric nature, that Pz is observed to reverse its sign in
traversing from −k// to +k// . Moreover, our in-plane spin measurements [Fig. 7.4(c)]
show that Zn-Bi2 Se3 film retains the helical spin texture protected by the TR symmetry, as observed in pure Bi2 Se3 and Bi2 Te3 single crystals [34]. Therefore we conclude
that nonmagnetic Zn doping does not induce observable spin reorientation on the
topological surface. The contrasting behavior observed between Mn-Bi2 Se3 and ZnBi2 Se3 samples as presented in Figs. 7.3, 7.4 provide clear evidence for TR symmetry
breaking in Mn-Bi2 Se3 .
A surface band gap at the Dirac point can also be generated in Bi2 Se3 in its
180
ultra-thin film limit. In this case, the top and bottom surfaces couple together and
open up a gap as electrons can tunnel from one to the other [Figs. 7.4(e)-(h)]. Such
a gap in the surfaces is not related to magnetism. It is important to know the spin
configuration associated with such a gap. In Figs. 7.4(e)-(h), we utilize Spin-resolved
ARPES to measure the spin configuration on the very top region (within 5 Å) of
a Bi2 Se3 film whose thickness is three quintuple layers (3 QL≃ 28.6Å). At large
−1
parallel-momenta far away from Γ̄ [e.g. -0.10 Å
in Fig. 7.4(g)], we observe clear
spin polarization following left-handed helical configuration with the magnitude of
the polarization around 35 − 40%. However, in going to smaller k// , the magnitude
of the spin polarization is observed to be reduced. At the TR invariant Γ̄ momenta,
spin-resolved measurements [Fig. 7.4(g) red curve] show no net spin polarization. This
reduction of the spin polarization at small momenta near the gap is an intrinsic effect.
These observations can be understood by considering the scenario where the surfaceto-surface coupling dominates, and the two energetically degenerate surface states
from top and bottom that possess opposite helicities of the spin texture cancel each
other at Γ̄ [62]. This results in strong suppression of spin polarization in the vicinity
of this gap, whereas upon probing momenta to large k// away from Γ̄, the finite
kinetic energy of the surface states (∝vk// ) naturally leads to the spatial decoupling
of two Dirac cones. These spin measurements on the ultra-thin Bi2 Se3 film reveal the
interplay between quantum tunneling (coupling) and the spin texture modification,
which is of importance in spin-based device design with thin films. The observed
spin texture however does not break TR symmetry, since the spins remain doubly
degenerate at the TR invariant momenta Γ̄. This is in clear contrast to the spin
texture observed in Mn-Bi2 Se3 .
The magnetic contribution to the gap of the Mn-Bi2 Se3 film can be quantitatively identified using the spin texture data. The simplest k·p Hamiltonian that
describes topological surface states under TR symmetry breaking can be written as
7.1 HEDGEHOG SPIN TEXTURE IN A MAGNETIC TOPOLOGICAL INSULATOR
H = v(kx σy − ky σx ) + bz σz , where σ and k are the spin and momentum operators
respectively, bz corresponds to half of the magnetic gap and v is the velocity of the
surface Dirac band. We specify the out-of-plane polar angle θ of the spin polarization
vector [inset of Fig. 7.3(g)] as θ = arctan PP//z . The magnitude of the polar angle θ
reflects the competition between the out-of-plane TR breaking texture (∝bz ) and the
in-plane helical configuration component (∝vk// ). Using the measured spin-resolved
data sets (θ, k), we fit the magnetic interaction strength bz within a k·p scenario. As
an example, we fit bz based on spin-resolved data sets in Fig. 7.3(g) on Mn(2.5%)Bi2 Se3 film II, as shown in Fig. 7.5(c) and obtain a value of 21 meV. This is a
significant fraction of the SWS energy scale observed on the same sample, ESWS > 50
meV (see Fig. 7.2(c) for Mn=2.5%) obtained from the spin-integrated measurements
in Fig. 7.1(c). Thus we identify the magnitude of the magnetic contribution (bz ) to
the observed spectral weight suppression using spin-sensitive measurements, which
suggests that the magnetic contribution is significant to ESWS .
As demonstrated recently [37], the geometrical phase (GP) defined on the spin
texture of the surface state Fermi surface [14] (also known as the Berry’s phase)
bears a direct correspondence to the bulk topological invariant realized in the bulk
electronic band structure via electronic band inversion [34, 37]. We experimentally
show that a GP tunability can be realized on our magnetic films which is important to
prepare the sample condition to the axion electrodynamics limit. On the Mn-Bi2 Se3
film, spin configuration pattern can be understood as a competition between the outof-plane TR breaking component and the in-plane helical component of spin. The
in-plane spin that can be thought of winding around the Fermi surface in a helical
pattern contributes to a nonzero GP [34], whereas the out-of-plane TR breaking spin
direction is constant as one loops around the Fermi surface hence does not contribute
to the Berry’s phase (GP). As a result, the GP remains almost π if the chemical
potential lies at energies far away from the Dirac point, whereas it starts to decrease
182
and eventually reach to 0 as one approaches the TR breaking gap by lowering the
chemical potential as discussed in theory [230], at least within the magnetic energy
scale bz (Fig. 7.5). We show that this theoretical requirement can be experimentally
achieved on the Mn-Bi2 Se3 surface via surface NO2 adsorption at various dosage levels.
Fig. 7.5(a) shows the Mn(2.5%)-Bi2 Se3 surface states with in situ NO2 adsorption.
The chemical potential is observed to be gradually shifted and finally placed within
the magnetic gap. The associated phase (GP) at each experimentally achieved sample
chemical potential [noted at the top-right corner of each panel in Fig. 7.5(a)] is found
to gradually change from π to 0. The GP= 0 is the experimental condition for
realizing axion electrodynamics with a topological insulator [223, 224].
With the chemical potential moved into the magnetic gap, the time-reversal breaking in-gap state features a singular hedgehog-like spin texture [Fig. 7.5(d)]. Such
spin configuration simultaneous with the chemical potential placed within the magnetic gap [Fig. 7.5(d)] is the fundamental requirement for most of the theoretical
proposals relevant to the utilization of magnetic topological insulators in novel devices [126,127,223,224,226]. Particularly, if Mn impurities form clusters on the surface
of a 2D topological superconductor, the edges of these magnetic (time-reversal breaking) islands can exhibit helical Majorana edge states. We will further elaborate this
exciting theoretical possibility in the next subsection.
7.1 HEDGEHOG SPIN TEXTURE IN A MAGNETIC TOPOLOGICAL INSULATOR
Figure 7.1: Magnetic (Mn%) and non-magnetic (Zn%) doping on Bi2 Se3 films.
(a),(b) Schematic layout and photograph of MBE grown Bi2 Se3 films used for the experiments. (c) Magnetization measurement at T = 45 K using magnetic circular dichroism
shows out-of-plane ferromagnetic character of the Mn-Bi2 Se3 film (111) surface through the
observed hysteretic response. Inset shows the measurement geometry. L(R)CP represents
left(right)-handed circularly polarized light. (d) Electronic band dispersion of Mn(Zn)doped Bi2 Se3 MBE thin films along the M̄ − Γ̄ − M̄ momentum space cut. (e) Energymomentum distribution curves of Mn(Zn)-doped Bi2 Se3 samples. The energy scale associated with the spectral weight suppression (SWS) ESWS is observed as the energy spacing
between the upper Dirac band minimum and the Dirac point location along the energy axis.
This figure is adapted from Ref. [58].
184
Figure 7.2:
Temperature and doping dependence of magnetically induced
changes on Mn-Bi2 Se3 surface. (a) The Mn atoms on the surface of the film are
out-of-plane magnetically ordered, serving as a local magnetic field which results in the
spin texture reorientation. (b) Two independent hysteresis measurements at T = 45 K using X-ray magnetic circular dichroism reveal the ferromagnetic character of the Mn-Bi2 Se3
film surface. The lower inset shows the remanent surface magnetization as a function of
temperature. The out-of-plane magnetic hysteresis and ARPES gap were found to be correlated with each other. The upper inset shows the gap at the Dirac point in Mn(2.5%)-Bi2 Se3
film. (c) The spectral weight suppression energy scale ESWS and inverse momentum width
1/∆k of the surface states are shown as a function of Mn and Zn concentration measured at
T = 20 K. (d) Temperature dependence of spectral weight suppression energy scale around
the Dirac point of Mn(2.5%)-Bi2 Se3 film (as noted in Panel (c) by the dotted square). The
ESWS decreases as temperature is raised signaling gradual weakening of magnetism. This
figure is adapted from Ref. [58].
7.1 HEDGEHOG SPIN TEXTURE IN A MAGNETIC TOPOLOGICAL INSULATOR
Figure 7.3: Out-of-plane spin configuration measurements. Panels (a)-(d) present
spin-resolved measurements on film I using 20 eV photons in the momentum distribution
curve (MDC) mode. Panels (e)-(g) present spin-resolved measurements on film II using
9 eV photons in the energy distribution curve (EDC) mode. (a),(b) Spin-integrated data
and corresponding MDCs. (c) Spin-resolved MDC spectra for out-of-plane direction as
a function of electron binding energy. (d) Measured out-of-plane component of the spin
polarization, presented in terms of respective out-of-plane polar angles (θ) defined in the
inset of Panel (g). (e)-(f ) Spin-integrated dispersion and EDCs. The EDCs selected for
detailed spin-resolved measurements are highlighted in green and red (Γ̄ momenta). (g)
Measured out-of-plane spin polarization of the EDCs corresponding to Panel (f ). Inset
~ and the out-of-plane polar angle θ. The
defines the definition of spin polarization vector P
momentum value of each spin-resolved EDC is noted on the top. The polar angles (θ) of
the spin vectors obtained from measurements are also noted. The 90◦ polar angle observed
at Γ̄ suggests that the spin vector at Γ̄ momenta points in the vertical direction. The spin
behavior at Γ̄ and its surrounding momentum space reveals a hedgehog-like spin configuration for each Dirac band separated by the gap, which breaks time-reversal symmetry
(E(~k = 0, ↑)6=E(~k = 0, ↓)), as schematically shown in the inset of Panel (f ). This figure is
adapted from Ref. [58].
186
Figure 7.4: Spin configurations on non-magnetic samples. (a)-(d) Spin-resolved
measurements on 1.5% non-magnetic Zn-Bi2 Se3 film. The in-plane polarization measurements (c) reveal the helical spin configuration, as in pure Bi2 Se3 topological insulator [34],
suggesting that non-magnetic impurities do not induce spin reorientation on the topological surfaces. Out-of-plane measurements (d) show that no significant out-of-plane spin
polarization Pz is induced near the Γ̄ point (a Kramers point), leaving the system timereversal invariant overall. (e)-(g) Spin-resolved ARPES measurements on ultra-thin undoped Bi2 Se3 film of three quintuple-layer thickness. The net spin polarization is found
to be significantly reduced near the gap edge around the Γ̄ momenta. This is consistent
with the fact that in ultra-thin films electrons tunnel between the top and bottom surfaces.
(h),(i) A schematic of the two types of spin textures observed in our data. This figure is
adapted from Ref. [58].
7.1 HEDGEHOG SPIN TEXTURE IN A MAGNETIC TOPOLOGICAL INSULATOR
Figure 7.5: Chemical potential tuned to lie inside the magnetic gap. (a) Measured
surface state dispersion upon in situ NO2 surface adsorption on the Mn-Bi2 Se3 surface. The
NO2 dosage in the unit of Langmuir (1 L = 1 × 10−6 torr·sec) and the tunable geometrical phase (see text) associated with the topological surface state are noted on the top-left
and top-right corners of the panels, respectively. The red arrows depict the time-reversal
breaking out-of-plane spin texture at the gap edge based on the experimental data. (b)
Geometrical phase (GP) associated with the spin texture on the iso-energetic contours on
the Mn-Bi2 Se3 surface as a function of effective gating voltage induced by NO2 surface adsorption. Red squares represent the GP experimentally realized by NO2 surface adsorption,
as shown in Panel (a). GP= 0 (NO2 =2.0 L) is the condition for axion dynamics [224]. (c)
The magnetic interaction strength bz (see text for definition), which corresponds to half
of the magnetic gap magnitude, is obtained based on spin-resolved data sets (polar angle
θ, momentum k) for Mn(2.5%)-Bi2 Se3 film II [see Figs. 7.3(e)-(g)]. (d) The time-reversal
breaking spin texture features a singular hedgehog-like configuration when the chemical
potential is tuned to lie within the magnetic gap, corresponding to the experimental condition presented in the last panel in Panel (a). (e),(f ) Spin texture schematic based on
measurements of Zn-doped Bi2 Se3 film (60 QL), and 3 QL undoped ultra-thin film with
chemical potential tuned onto the Dirac point energy or within the tunneling gap. This
figure is adapted from Ref. [58].
188
7.2
Helical Cooper pairing in topological insulator/superconductor heterostructures
Realization of novel superconductivity is one of the central themes in condensed
matter physics in general [2, 6, 53, 55, 125, 128, 225, 231–247]. Superconductivity is a
collective phenomenon, where electrons moving to the opposite directions (±k) form
dynamically bound pairs, resulting in a Cooper pair gas. In an ordinary superconductor, the conduction electrons that move along a certain direction have both spin-up
and spin down electrons available for the Cooper pairing. The superconductivity observed so far, including in the conventional s-wave BCS superconductors as well as
the cuprate or heavy fermion d-wave superconductors, all share this property. Recently, the discovery of 3D topological insulator in bismuth based semiconducting
compounds have attracted much interest in condensed matter physics. In these TI
materials, the bulk has a full energy gap while the surface exhibits an odd number
of Dirac-cone electronic states, where the spin of the surface electrons is uniquely
locked to their momentum [2, 6]. Therefore, at any given surface of a TI, the surface
electrons moving to one direction (e.g. +k) will have only spin up electrons available
whereas those of moving to −k only have spin down available. This is in contrast
to the case in an ordinary superconductor, where at any given direction the conduction electrons will have both spin up and spin down available for the Cooper pairing.
Such distinction can give rise to a wide range of exotic physics. Recently, a number of
theories have highlighted these possibilities from both the fundamental physics and
application point of view [125, 128, 225, 231–234]. For example, supersymmetry and
Majorana fermions are both very interesting physics phenomena predicted in high
energy theories that remain unobserved in particle physics experiments. And it has
been theoretically predicted, very recently, that such new physics can be realized in
a condensed matter setting [125, 128], if superconductivity can be induced in a spin-
OOPER PAIRING IN TOPOLOGICAL INSULATOR/SUPERCONDUCTOR HETEROSTRUCTURES
helical gas. Moreover, a low energy realization of these phenomena can also be utilized
to build the topological qubit for a topological quantum computer, which therefore is
also of value in device applications. The first step towards the realization of any of the
fascinating theoretical proposals requires a clear demonstration of the helical-Cooper
pairing. Helical-Cooper pairing is defined as the superconducting Bose condensation
of a spin-momentum locked Dirac electron gas, independent of the bosonic character of the pairing glue [2, 6]. To date, a direct experimental demonstration of the
helical-Cooper pairing and their magnetic response remain elusive.
Here we use spin- and momentum-resolved photoemission spectroscopy with sufficiently high resolution and at sufficiently low temperature to allow direct evidence
for the helical Cooper pairing in a spin-momentum locked Dirac electron gas. We
achieve this through the observation of momentum-resolved Bogoliubov quasi-particle
spectrum of a topological insulator (Bi2 Se3 ) in proximity to a superconducting NbSe2
substrate. We further systematically investigate the dependence of the helical Cooper
pairing in the Dirac electrons upon varying the TI film thickness or doping magnetic
impurities. Our data show that the helical superconductivity in the Dirac surface
states can be suppressed by introducing time-reversal symmetry breaking magnetic
elements. Our observation of helical-Cooper pairing and superconductivity in spinDirac electronic gas serves as an important platform for realizing many exotic physics
including emergent supersymmetry [128] physics. We also demonstrate a systematic methodology using the combination of spin- and momentum-resolved ARPES
and interface transport that can be more generally applied to discover, isolate, and
systematically optimize exotic superconductivity in engineered materials. Previous
studies of superconductivity in topological insulator settings have been limited to
transport and STM [53, 55, 235–243]
High quality Bi2 Se3 / 2H-NbSe2 interface-heterostructures [Figs. 7.6(a),(b)] are
prepared using molecular beam epitaxy growth (MBE). The growth conditions are
190
systematically optimized to enhance the superconductivity signals in our ARPES
measurements. In order to protect the Bi2 Se3 surface from exposure to atmosphere,
an amorphous selenium layer is deposited on top of the TI surface. This layer can be
removed in situ in our angle-resolved photoemission spectroscopy (ARPES) experiments by annealing the samples. Fig. 7.6(c) shows the momentum-integrated ARPES
intensity curves over a wide energy window (core-level spectra) taken on a representative 3 quintuple layer (QL) film before and after removing the amorphous selenium
capping layer (decapping). High-resolution ARPES measurements on the Bi2 Se3 surface are then performed [Fig. 7.6(e)]. A sharp spectrum for the Dirac surface states
is clearly observed, indicating a good surface/interface quality of our heterostructure.
Consistent with previous studies of ultrathin TI films [58, 62, 80], we observe a gap at
the Dirac point because of the hybridization between the top and bottom surfaces.
Furthermore, we perform spin-resolved ARPES measurements (photon energy 50 eV)
on the 4QL sample [Fig. 7.6(f)]. Our spin-resolved measurements confirm that the
surface states are indeed singly degenerate near the Fermi level, which is at an energy
level far away from the hybridization gap (v·kF > ∆hybr ) [58, 80]. At the Fermi level,
a left-handed spin-momentum locking profile is observed, which is one of the critical
ingredients for the helical-Cooper pairing as we will show in later sections.
In order to study the possible proximity induced superconductivity in the Dirac
surface states, we perform systematic ultra-low temperature (T ∼ 1 K) and ultra-high
energy resolution (∼ 2 meV) ARPES measurements on these TI/superconductor heterostructures. We start with the 4QL sample using incident photon energy of 18 eV.
Fig. 7.7(b) shows the measured dispersion of the Bi2 Se3 film. Both the topological
surface states (TSSs) and the bulk conduction bands are observed. Six representative momenta, namely ±~k1 , ±~k2 , and ±~k3 are chosen for detailed studies, where
k1 = 0.12 Å
−1
−1
corresponds to the TSSs, and k2 = 0.08 Å
−1
and k3 = 0.04 Å
corre-
spond to the outer and inner parts of the bulk band states, respectively. In order to
OOPER PAIRING IN TOPOLOGICAL INSULATOR/SUPERCONDUCTOR HETEROSTRUCTURES
search for possible superconductivity signals, we study the ARPES energy-spectra at
various momentum space locations in close vicinity to the Fermi level (EB = EF ± 5
meV). Fig. 7.7(c) shows the ARPES spectra at the momentum of k1 (TSSs) at different temperatures. Clear leading-edge shifts (superconducting gap) and coherence
peaks are observed at low temperatures. The observed superconducting signals as
temperature increases disappear at higher temperatures such as T = 7 K and 12
K. In order to better visualize the superconductivity gap in our data, the ARPES
spectra are symmetrized with respect to the Fermi level, where the temperature evolution of the full (symmetrized) superconducting gap and the coherence peaks are
clearly seen in Fig. 7.7(d). These measurements show the existence of induced superconductivity in the helical Dirac electrons occurring in the Bi2 Se3 TSSs, which
is not possible in conventional momentum-integrated experiments that lacking spin
resolution [53,55,235–243]. We compare and contrast the proximity induced superconductivity in the Dirac surface states to that of the bulk band states. Fig. 7.7(e) shows
the ARPES spectra at k2 , where the bulk conduction bands are identified. Superconducting signals including leading edge shifts and coherence peaks are also observed
at k2 . In order to obtain the magnitude of the superconducting energy gap, we fit
the bulk state (±~k2 , and ±~k3 ) data by the Dynes function [248] [black curves in
Fig. 7.7(g)], which is widely used in s-wave superconductors, whereas the surface state
data (±~k1 ) is fitted by a BCS function with consideration of the spin-momentum
locking and Dirac dispersion properties of the TSSs [blue curves in Fig. 7.7(g)].
Since the surface states and the bulk conduction bands co-exist at the chemical
potential [Fig. 7.7(b)], we need to examine whether the observed superconducting
proximity signal at k1 has contribution from the bulk bands. To further isolate the
signals of Dirac surface states from the bulk bands, we choose another incident photon
energy of 50 eV where we utilize the photoemission matrix element effect to suppress
the spectral weight of the bulk conduction states. As shown in Fig. 7.7(i), at photon
192
energy of 50 eV, the bulk conduction band is almost completely suppressed and the
only dispersive band near the Fermi level is the Dirac surface state. We subsequently
study the spectra at the momentum k1 where the ARPES signal is dominated by the
contribution from the surface states. Leading edge shifts and coherence peaks are
clearly observed from k1 , which confirms the superconductivity in the Dirac surface
states using a different photon energy. These systematic momentum-resolved measurements clearly show the existence of a superconducting helical electron gas, which
is realized on the top surface of Bi2 Se3 grown on top of an s-wave superconductor
NbSe2 .
We perform ARPES measurements around the surface state Fermi surface as a
function of Fermi surface azimuthal angle θ, to study the extent of anisotropy of
the surface state superconducting gap. To isolate the surface state signal from the
bulk, photon energy of 50 eV is used, where only surface states are observed near the
Fermi level as seen in Fig. 7.8(a). Five representative momentum space cut-directions
(θ1−θ5) are chosen as indicated by the dotted lines in Fig. 7.8(a). The helical-surface
state superconducting gap observed by ARPES at different θ angles and their fits are
shown in Fig. 7.8(b). The reasonably good surface state fitting results [blue curves in
Figs. 7.7(g) and Fig. 7.8(b)] indicate that the obtained surface state superconductivity
is consistent with its spin-helical and linear dispersive properties, which supports its
helical-Cooper pairing nature. The obtained magnitude of superconducting order
parameter is then plotted as a function of Fermi surface angle in Fig. 7.8(c). The
superconducting gap is found to be nearly isotropic, which is also consistent with
the time-reversal invariant helical nature of the surface state superconductivity as
expected theoretically [125, 225]. We note that the helical Cooper pairing in the
topological surface states as observed here is also different from that of in other singly
degenerate (spin-momentum locked) but non-topological systems. For example, in
a Rashba-2DEG, although the electrons are also singly degenerate, but along any
OOPER PAIRING IN TOPOLOGICAL INSULATOR/SUPERCONDUCTOR HETEROSTRUCTURES
k−direction, there are still both spin up and spin down electrons available for Cooper
pairing [see Fig. 7.8(e)]. Therefore, only in the topological surface state, there are
electrons with only one spin available for the Cooper pairing along any k−direction
[Fig. 7.8(d)]. This unique property as demonstrated here (which we refer as the helical
Cooper pairing) is indispensable for all the fascinating phenomena as predicted in
theories [125, 128, 225, 231–234]. In Fig. 7.8(f), we present model calculation analysis,
which shows a px ±ipy superconducting order parameter in the Dirac surface states,
supporting the helical Cooper pairing nature. Therefore, through our systematic
ARPES measurements with simultaneous energy, momentum and spin resolution, we
observe superconductivity in an odd number of spin-momentum locked Dirac surface
states, which serves as the direct experimental evidence for the helical-Cooper pairing.
We study the observed superconductivity in the surface states as a function of
surface-to-surface hybridization strength (effectively as a function of the TI film thickness), in order to experimentally prove its proximity-induced nature. As shown in
Fig. 7.9(a), a sample with a Dirac point hybridization gap as large as ∼ 200 meV is realized in a 3QL film sample. Clear leading-edge shifts, coherence peaks and their temperature evolution are observed in the TSSs near the Fermi energy [Figs. 7.9(b),(c)]
evidence for the helical-Cooper pairing and helical superconductivity in the 3QL film
samples. The observed surface state superconducting gap value is about 0.7 − 0.8
meV. We now turn to a gapless (7QL) sample as shown in Fig. 7.9(d). The absence
of hybridization gap at the Dirac point reveals that surface state wavefunctions from
the top and the interface surfaces are completely separated in real space. Finite superconductivity signals are observed in the ARPES spectra at both momenta of k1
(Dirac surface states) and k2 (bulk conduction states), which are found to be weaker
than the gapped samples. The Bi2 Se3 top surface’s superconducting gap as a function of Dirac point gap value (surface-to-surface hybridization strength) is shown in
Figs. 7.9(g),(h) in a (surface and bulk) band-resolved fashion. It can be seen that the
194
induced superconducting gap near the top surface increases with a larger the Dirac
point hybridization gap, which is realized in thinner TI films. This observation is
qualitatively consistent with the theoretical description of the superconducting proximity effect, where the Cooper pair potential on the top surface is enhanced with the
decreasing thickness of the normal metal. More interestingly, it can be seen that the
surface state superconducting gap increases at a faster rate than that of the proximity gap on the bulk band. Such contrast reveals that stronger surface-to-surface
hybridization significantly enhances the helical pairing in the surface states on the
top surface. These microscopics of the superconducting proximity effect observed in
our data will be a valuable guide in properly interpreting the vast complexity of the
transport data addressing the proximity effects in TI films and heterostructures.
In order to test the time-reversal invariant character of surface state superconductivity of Bi2 Se3 required by its helical nature, we study manganese (Mn) doped
Bi2 Se3 grown on top of NbSe2 . Mn atoms are introduced into Bi2 Se3 throughout the
film during the MBE growth. ARPES studies on 4QL Bi2 Se3 films with two different
Mn doping (4% and 10%) levels are presented in Figs. 7.10(a)-(f). Bulk manganese
doping is found to hole dope the system, thus bringing the chemical potential closer
to the Dirac point. In principle, Mn impurities can affect the induced superconductivity in several different channels, all leading to the suppression of proximity
superconductivity: The major effect is that Mn impurities introduce (either random
or ordered) magnetism into the TI, which is destructive to the helical pairing; Another minor effect, which can also contribute, is that impurities generate random
disorder reducing the electron mean free path. As shown in Figs. 7.10(a)-(f), the
superconducting coherence peak is strongly suppressed in the heavily Mn-doped samples. These momentum-resolved measurements allow us to isolate the strength of
the effect on helical surface states and bulk bands, thus directly demonstrating that
magnetic impurities lead to strong pairing breaking in both conventional and helical
OOPER PAIRING IN TOPOLOGICAL INSULATOR/SUPERCONDUCTOR HETEROSTRUCTURES
pairing channels. The complete suppression of superconductivity in the helical Dirac
surface states upon strong Mn doping effectively drives a topological phase transition
from a helical superconductor to a normal Dirac metal state as seen in our data. A
sample fabricated to lie near the critical point of this transition can host many exotic
phenomena (which will be discussed later in this paper).
In order to check that the top surface superconductivity is indeed a proximity effect, we use the well-established methodology of point-contact Andreev reflection measurements to compare the induced gap values at the NbSe2 /Bi2 Se3 heterointerface.
The point contract probe is characterized by measurements on pure NbSe2 [249]. The
point-contact spectra on a NbSe2 /Bi2 Se3 (16QL) heterostructure sample are shown
in Fig. 7.12(a). At temperatures below the Tc (∼ 7.2 K) of NbSe2 , the differential
conductance (dI/dV ) around zero bias increases as a result of the Andreev reflection
process, similar to the bare NbSe2 spectra. Interestingly, a second differential conductance increase appears below ∼ 5 K. The sharp rise of differential conductance
corresponds to the energy gap of the superconducting layer. From the data, we obtain
the larger gap (∆1 ) changing from 0 to 1.3 ± 0.2 mV from 7.5 K to 3.0 K. The second
gap (∆2 ) feature changes from 0 to 0.8 ± 0.2 mV from 5.0 K to 3.0 K. Similar two gap
features have been observed in point-contact studies of Ag/Pb [250] and Si/Nb [251]
interface samples and were attributed to the superconducting energy gap of the superconducting layer and the proximity induced gap in the normal metal layer at the
N/S interface, respectively. Injected electrons from the point-contact are Andreev
reflected inside the superconducting Bi2 Se3 proximity layer if their energies are lower
than the induced superconducting gap in Bi2 Se3 . When their energies are above the
induced gap ∆TI but below the NbSe2 superconducting gap ∆SC , injected electrons
are not affected by the order parameter in the Bi2 Se3 layer but Andreev reflected in
the NbSe2 region. Therefore, the edge around ∆1 is likely due to the NbSe2 gap, while
the sharp edge around ∆2 in the conductance spectrum is likely to reflect the induced
196
gap in Bi2 Se3 near the interface. It is worth noting that unlike ARPES, which is
mostly sensitive to the top surface, the point-contact transport probes deeper into
the superconductor, similar to the electron mean free path, which is estimated to be
∼16 nm in our films. The induced gap in Bi2 Se3 at the Bi2 Se3 /NbSe2 interface is ∼0.8
meV at 3 K from the point-contact measurement, which is in reasonable agreement
with the fitted gap value extracted from the ARPES measurement (see data on 3QL
sample). Our results thus suggest that the combination of ARPES and point-contact
transport together provides a powerful method for probing superconducting proximity effect which can be used to correlate the proximity gap on the top surface and the
buried interface if film thickness is not too large (not larger than the superconducting
coherence length).
In contrast to idealized theoretical models [125, 225] of topological superconductivity where only Dirac surface states cross the Fermi level, real samples exhibit a
complex phenomenology due to the coexistence of multiple bands at the chemical
potential, as demonstrated in our data above. Thus, the interpretation of experimental studies must take into account both the desirable Cooper pairing from the
Dirac surface states and conventional superconductivity from the bulk, trivial surface
states and impurity surface states. The coexistence of multiple bands at the Fermi
level means that any superconductivity realized in actual TI materials consist of not
only the desirable helical Cooper pairing from the Dirac surface states but also conventional superconductivity from the bulk states, as shown above in our data. We
note that although progress has been reported by using conventional transport and
STM experiments [53,55,235–243], those studies do not have the spin and momentum
resolution necessary to distinguish the helical Cooper pairing from that of the conventional superconductivity from other bands that intermix at the interface making
interpretation of Majorana fermions unreliable or complex. Hence a direct experimental demonstration of the existence of superconductivity in the helical Cooper
OOPER PAIRING IN TOPOLOGICAL INSULATOR/SUPERCONDUCTOR HETEROSTRUCTURES
pairing channel remained elusive before our momentum and spin space observations
reported here. In fact, it has been recently shown both theoretically and experimentally [244–247] that the conventional superconductivity in the bulk and impurity
bands at the interface or surface lead to ambiguous interpretations of the transport
and STM data. In order to achieve a clear case for Majorana zero mode, the helical
component of the Cooper pairing must be isolated, as demonstrated here. Therefore,
it is in this context that our observation of helical Cooper pairing and superconductivity in a half Dirac gas is of critical importance. Additionally, the overall methodology
employed here can be applied to isolate helical Cooper pairing in other systems and in
connection to a feedback loop for material growth for the optimization of the helical
channel.
We also note that our systematic studies (by observing the superconducting gap
(leading edge shift), the clear coherence peak, as well as their systematic dependence
upon varying temperature, TI film thickness and doping magnetic impurities) are in
contrast to the debatable ARPES results on Bi2 Se3 /BSCCO samples [83, 84, 252]. In
that case, no superconducting coherence peak was observed [83,84,252], and the claim
of a ≥ 15 meV leading edge shift [252] in the Dirac surface states in Bi2 Se3 /BSCCO is
in contrast to the absence of any observable leading edge shift in the other two studies [83, 84]. In fact, a strong superconducting proximity effect is inconsistent with
important facts including the severe mismatch of both Fermi momenta and crystal
symmetries between Bi2 Se3 and BSCCO, very short out-of-plane superconducting
coherence length of high-Tc superconductors, as well as the different superconducting pairing symmetries between a TI and a d−wave cuprate superconductor. Thus,
our data strongly supports the view that TI/NbSe2 is a more ideal platform than
TI/BSCCO for the proposed novel physics if the system can be further optimized
increasing helical Cooper pairing channel by tuning the material parameters.
We discuss the emergent topological phenomena that can be enabled by our iden198
tification of helical-Cooper pairing. One exciting scenario is to realize supersymmetric phenomenon in our experimental setup [128] by further improvement of the film
quality and magnetic doping process. As shown in Fig. 7.12(d), magnetic doping
or an external in-plane magnetic field is necessary to drive the system to the critical
point between the helical superconductivity and the normal Dirac gas states, with the
chemical potential tuned to the Dirac point [demonstrated in Fig. 7.12(b)]. Under this
condition, theory predicts that topological surface states (a fermionic excitation) and
the fluctuations of superconducting order (a bosonic excitation) satisfy supersymmetry relationship, and therefore, strikingly, possess the same Fermi/Dirac velocity and
same lifetime or self-energy [128]. While the superpartners of elementary particles
in high energy physics have never been experimentally observed, the experimental
methodologies, artificial sample fabrication control and experimental observations reported here pave the way for simulating and testing supersymmetric physics concepts
in future sub-Kelvin nanodevices fabricated out of sample configurations discussed
here. Another very exciting proposal is to deposit magnetic impurities on the surface of our TI/SC heterostructure samples [2]. Since the superconducting topological
Dirac surface states naturally realize a 2D helical topological superconductor [61,125],
a ferromagnetic island (or a cluster) can create a local non-superconducting region
(Fig. 7.11). In this case, the edge of the magnetic island corresponds to the boundary
between a topological superconductor and a normal (non-topological) state and is
therefore expected to feature helical Majorana edge modes, which can be readily detected by advanced spatial-resolved spectroscopic probes such as scanning tunneling
spectroscopy.
OOPER PAIRING IN TOPOLOGICAL INSULATOR/SUPERCONDUCTOR HETEROSTRUCTURES
Figure 7.6: Topological superconductivity via proximity effect. (a) A schematic
layout of ultra-thin Bi2 Se3 films epitaxially grown on the (0001) surface of single crystalline
s-wave superconductor 2H-NbSe2 (Tc = 7.2 K) using the MBE technique. (b) High resolution transmission electron microscopy (TEM) measurements of the Bi2 Se3 /NbSe2 interface
at 200 keV electron energy. An atomically abrupt transition from NbSe2 layered structure
to the layered quintuple layer structure of Bi2 Se3 is resolved, showing a good atomically
flat interface crystal quality. (c) Momentum-integrated ARPES intensity curves over a
wide binding energy window (core-level spectra) taken on a representative 3QL Bi2 Se3 (≃ 3
nm) film grown on NbSe2 before and after removing the amorphous selenium capping layer
(decapping). (d) A low-energy electron diffraction (LEED) image on a 4QL Bi2 Se3 film
shows six-fold pattern providing evidence that the thin Bi2 Se3 film is well-ordered. (e)
High-resolution ARPES dispersion map of a 4QL Bi2 Se3 film on NbSe2 after decapping
using incident photon energy of 50 eV. The white circle and cross schematically show the
measured direction of the spin texture on the top surface of our 4QL Bi2 Se3 film shown in
Panel (f ). (f ) Spin-resolved ARPES measurements on 4QL Bi2 Se3 as a function of binding
energy at a fix momentum which is indicated by the white dotted line in Panel (e). (g)
High-resolution ARPES dispersion map of a 6QL Bi2 Se3 film on NbSe2 at T = 12 K. The
white arrow indicates the momentum for the temperature dependent EDC in Panel (h).
(h) Temperature dependence of the ARPES spectra in Panel (g). This figure is adapted
from Ref. [61].
200
Figure 7.7: Momentum resolved helical 2D topological superconductivity. (a)
ARPES dispersion maps of Bi2 Se3 /NbSe2 as a function of Bi2 Se3 film thickness. All dispersion maps are measured with photon energy of 50 eV, except the 7QL sample which is
measured by 18 eV. The blue arrows quantitatively depict the spin texture configuration
in the ultra-thin limit. The length of the arrow is proportional to the magnitude of the
spin polarization. (b) ARPES dispersion map of a 4QL Bi2 Se3 film measured at T = 12
K using incident photon energy of 18 eV. (c) ARPES spectra at the fixed momentum of
k1 (the topological surface states). (d) Symmetrized ARPES spectra at k1 . (e) ARPES
spectra at the fixed momentum of k2 (bulk band states). (f ),(g) Symmetrized ARPES
spectra at ±k1 , ±k2 , and ±k3 at T ∼ 1 K. The surface state gap (±k1 ) is fitted by a BCS
function considering its spin-momentum locking and Dirac dispersion properties, where as
the bulk gap is fitted by the Dynes function [248]. (h) ARPES spectrum of the in situ
evaporated gold film, where the kinetic energy of the Fermi level and the energy resolution
are determined. (i) ARPES dispersion map of a 4QL Bi2 Se3 film measured with incident
photon energy of 50 eV. (j),(k) ARPES spectra at the fixed momentum of k1 . The insets
of Panels (c),(j) show an ARPES dispersion map near the Fermi level at low temperature
of T ∼ 1 K at photon energies of 18 eV and 50 eV, respectively. This figure is adapted from
Ref. [61].
OOPER PAIRING IN TOPOLOGICAL INSULATOR/SUPERCONDUCTOR HETEROSTRUCTURES
Figure 7.8: Topological superconducting gap and helical pairing magnitude. (a)
Fermi surface map taken at incident photon energy of 50 eV. The white dotted lines indicate
the momentum-space cut-directions chosen to study the surface state superconducting gap
as a function of Fermi surface angle θ around the surface state Fermi surface. (b) Symmetrized and normalized ARPES spectra along θ1 through θ5 respectively (red) and their
surface gap fittings. (c) Fermi surface angle dependence of the estimated superconducting
gap around the surface state Fermi surface. (d) Illustrations for helical-Cooper pairing in a
spin-momentum locked helical electron gas and (e) The conventional s-wave Cooper pairing
in an ordinary superconductor. Note that the superconductivity in a Rashba-2DEG can be
visualized also in the same schematic [Panel (e)] but with the length of the k-vector for
spin up and spin down electrons being different (spin-split). However, for a Rashba-2DEG,
it is still true that along any k-direction, both spin up and spin down electrons cross the
Fermi level and are available for the Cooper pairing. (f ) Model calculation results of a
topological insulator film in proximity to an s-wave superconductor shows the calculated
e S ) and triplet (∆
eT)
total superconducting pairing amplitude and its decomposed singlet (∆
components on the top surface of a 4-unit-cell thick TI interfaced with an s-wave superconductor, which further confirms the helical (topologically nontrivial) nature of the induced
e is a dimensionless, we denote
surface state superconductivity in our Bi2 Se3 films. Since ∆
e to differentiate from the superconducting gap ∆ measured in experiments. This
a tilde ∆
figure is adapted from Ref. [61].
202
Figure 7.9: Hybridization dependence of bulk superconducting gap vs. topological superconducting gap. (a) ARPES dispersion map of a 3QL Bi2 Se3 film measured
at T = 12 K using incident photon energy of 50 eV. (b) ARPES spectra at the fixed
momentum of k1 at different temperatures. (c) Symmetrized ARPES spectra at different
temperatures. (d) ARPES dispersion map of a 7QL Bi2 Se3 film measured at T = 12 K
using incident photon energy of 18 eV. (e),(f ) ARPES spectra at the fixed momenta of k1
and k2 . (g) ARPES measured superconducting gap for topological surface states (k1 ) and
for the bulk conduction states (k2 ) as a function of Dirac point gap value (surface-to-surface
hybridization strength). The dotted lines are guides to the eye. (h) Surface-to-surface hye TI in calculation is a
bridization dependence in calculation. Since the pairing amplitude ∆
dimensionless number, we normalize it by the pairing amplitude of the substrate supercone Substrate (a constant). This figure is adapted from Ref. [61].
ductor ∆
OOPER PAIRING IN TOPOLOGICAL INSULATOR/SUPERCONDUCTOR HETEROSTRUCTURES
Figure 7.10: Destruction of the helical Cooper pairing via time-reversal symmetry breaking magnetic doping. (a) ARPES dispersion map of a 4QL Mn(4%)-doped
Bi2 Se3 film measured at T ∼ 1 K. (b),(c) ARPES spectra at the fixed momenta of k1 (topological surface states) and k2 (bulk conduction states). (d)-(f ) Same as Panels (a)-(c) but
for 10% Mn doping. The Mn concentrations indicated here are nominal, which means they
are estimated by the flux ratio of Mn flux : Bi flux during the MBE growth. This figure is
adapted from Ref. [61].
204
Figure 7.11: A proposal to realize Majorana edge-modes by creating ferromagnetic islands on the surface of a 2D helical topological superconductor.
OOPER PAIRING IN TOPOLOGICAL INSULATOR/SUPERCONDUCTOR HETEROSTRUCTURES
Figure 7.12: Point-contact interface transport and conditions for theoretically
predicted emergent supersymmetry. (a) Point-contact transport (dI/dV vs bias voltage) as a function of temperature. For T = 3.5 K, below the Tc of NbSe2 , as the bias
voltage is swept from ±3mV to 0mV, dI/dV first increases, then levels off at |V | ≃ 1.3 mV,
and then increases again, reaching a maximum at |V | ≃ 0.8 mV. dI/dV for other T < Tc
exhibits a similar behavior. This indicates two Andreev reflection channels with different
sizes of superconducting gap. The inset illustrates the two Andreev reflection processes via
the induced superconducting gap in Bi2 Se3 and intrinsic superconducting gap in NbSe2 ,
respectively. (b) Measured surface state dispersion upon in situ NO2 surface adsorption
on the surface of a 7QL Bi2 Se3 /NbSe2 sample using incident photon energy of 55 eV at
temperature of 20 K. The NO2 dosage in the unit of Langmuir (1L = 1 × 10−6 torr·sec) is
noted on the top-right corners of the panels, respectively. The white dotted lines in the last
panel are guides to the eye. (c),(d) Theoretically proposed [128] unusual criticality related
to supersymmetry phenomena can be realized in the topological insulator/s-wave superconductor interface as the chemical potential is tuned to the surface state Dirac point and an
in-plane magnetization drives the system to the critical point between superconducting and
normal Dirac metal phases. Fermions and bosons are expected to feature the same band
velocity and quasi-paticle lifetime [128], where fermion velocity is estimated to be around
5 × 105 m/s from our ARPES result. This cartoon refers to a future possibility. This figure
is adapted from Ref. [61].
206
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Appendices
232
A List of Publications Relevant to This Thesis
A. Principal Publications
1. Su-Yang Xu et al., Fermi Arc Surface States in a Topological Metal: A New
Type of 2D Electron Gas. DOI:10.1126/science.1256742 Science (2014).
2. Su-Yang Xu et al., Topological Phase Transition and Texture Inversion in a
Tunable Topological Insulator. Science 332, 560-564 (2011).
3. Su-Yang Xu et al., Momentum space imaging of Cooper pairing in a half
Dirac gas topological superconductor. Nature Phys. doi:10.1038/nphys3139 (2014).
4. Su-Yang Xu et al., Hedgehog spin texture and Berrys phase tuning in a
magnetic topological insulator. Nature Phys. 8, 616-622 (2012).
5. L. A. Wray, Su-Yang Xu et al., A topological insulator surface under strong
Coulomb, magnetic and disorder perturbations. Nature Phys. 7, 32-37 (2011).
6. L. A. Wray, Su-Yang Xu et al., Observation of topological order in a superconducting doped topological insulator. Nature Phys. 6, 855-859 (2010).
7. Su-Yang Xu et al., Observation of a topological crystalline insulator phase
and topological phase transition in Pb1−x Snx Te. Nature Commun. 3, 1192 (2012).
8. Su-Yang Xu et al., Su-Yang Xu et al., Unconventional transformation of
spin-Dirac phase across a topological quantum phase transition. in review in Nature
Commun. (2014).
9. M. Neupane*, Su-Yang Xu* et al., Observation of a topological 3D Dirac
semimetal phase in high-mobility Cd3 As2 . Nature Commun. 5, 4786 (2014).
10. H. Lin, L. A. Wray, Y. Xia, Su-Yang Xu et al., Half-Heusler ternary compounds as new multifunctional experimental platforms for topological quantum phenomena. Nature Mater. 9, 546-549 (2010).
11. Su-Yang Xu et al., Fermi-level electronic structure of a topological-insulator/cupratesuperconductor based heterostructure in the superconducting proximity effect regime.
Phys. Rev. B, 90 085128 (2014).
B. Full List
1. Su-Yang Xu et al., Fermi Arc Surface States in a Topological Metal: A New
Type of 2D Electron Gas. DOI:10.1126/science.1256742 in Science (2014).
2. Su-Yang Xu et al., Su-Yang Xu et al., Unconventional transformation of
spin-Dirac phase across a topological quantum phase transition. in review in Nature
Commun. (2014).
3. Su-Yang Xu et al., Momentum space imaging of Cooper pairing in a half
Dirac gas topological superconductor. Nature Phys. doi:10.1038/nphys3139 (2014).
4. M. Z. Hasan, Su-Yang Xu, M. Neupane et al., Topological Insulators, Topological Crystalline Insulators, and Topological Kondo Insulators. Preprint at http://arXiv:1406.1040
(Invited Book Chapter) (2014).
5. M. Neupane*, Su-Yang Xu* et al., Observation of non-Kondo-like electronic
structure in strongly correlated electron system YbB6 . Preprint at http://arXiv:1404.6814
(2014).
6. M. Neupane, A. Richardella, J. Snchez-Barriga, Su-Yang Xu et al., Observation of Quantum-Tunneling Modulated Spin Texture in Ultrathin Topological
Insulator Bi2 Se3 Films. Nature Commun. 5, 3841 (2014).
7. Anjan A. Reijnders, Y. Tian, L. J. Sandilands, G. Pohl, I. D. Kivlichan, S.
Y. Frank Zhao, S. Jia, M. E. Charles, R.J. Cava, Nasser Alidoust, Su-Yang Xu et
al., Optical evidence of surface state suppression in Bi based topological insulators.
Phys. Rev. B 89, 075138 (2014).
8. Chang Liu*, Su-Yang Xu* et al., Spin correlated electronic state on the
surface of a spin-orbit Mott system (Layered Iridates). Phys. Rev. B 90, 045127
(2014).
9. Su-Yang Xu et al., Fermi-level electronic structure of a topological-insulator/cupratesuperconductor based heterostructure in the superconducting proximity effect regime.
Phys. Rev. B, 90 085128 (2014).
234
10. M. Neupane*, Su-Yang Xu* et al., Saddle point singularity and topological phase diagram in a tunable topological crystalline insulator (TCI). Preprint at
http://arXiv:1403.1560 (2014).
11. M. Neupane*, Su-Yang Xu* et al., Observation of a topological 3D Dirac
semimetal phase in high-mobility Cd3 As2 . Nature Commun. 5, 4786 (2014).
12. N. Alidoust, G. Bian, Su-Yang Xu et al., Observation of monolayer valence
band spin-orbit effect and induced quantum well states (QWS) in MoX2 . Nature
Commun. 5, 4673 (2014).
13. Su-Yang Xu et al., Observation of a bulk 3D Dirac multiplet, Lifshitz transition, and nestled spin states in Na3 Bi. Preprint at http://arXiv:1312.7624 (2013).
14. M. Z. Hasan, Su-Yang Xu, D. Hsieh, L. A. Wray & Y. Xia, Topological
Surface States: A New Type of 2D Electron Systems (Book chapter). in Topological
Insulators edited by M. Franz and L. Molenkamp (Elsevier, Oxford 2013).
15. M. Neupane, S. Basak, N. Alidoust, Su-Yang Xu et al., Oscillatory surface
dichroism of an insulating topological insulator Bi2 Te2 Se. Phys. Rev. B 88, 165129
(2013).
16. Y Okada, M. Serbyn, H. Lin, D. Walkup, W. Zhou, C. Dhital, M. Neupane,
Su-Yag Xu et al., Observation of Dirac node formation and mass acquisition in a
topological crystalline insulator. Science 341, 1496-1499 (2013).
17. M. Neupane, N. Alidoust, Su-Yang Xu et al., Surface electronic structure of
a topological Kondo insulator candidate SmB6 : insights from high-resolution ARPES.
Nature Commun. 4, 2991 (2013).
18. L. A. Wray, Su-Yang Xu et al., Chemically gated electronic structure of
a superconducting doped topological insulator system. J. Phys.: Conf. Ser. 449
012037 (2013).
19. C. Fang, M. J. Gilbert, Su-Yang Xu, B. A. Bernevig, M. Z. Hasan, Theory
of quasiparticle interference in mirror symmetric 2D systems and its application to
surface states of topological crystalline insulators. Phys. Rev. B 88, 125141 (2013).
20. H. Lin, Tanmoy Das, L. A. Wray, Su-Yang Xu et al., An isolated Dirac cone
on the surface of ternary tetradymite-like topological insulators. New J. of Phys. 13,
095005 (2011).
21. Y. J. Wang, W.-F. Tsai, H. Lin, Su-Yang Xu et al., Nontrivial spin texture
of the coaxial Dirac cones on the surface of topological crystalline insulator SnTe.
Phys. Rev. B 87, 235317 (2013).
22. Hsin Lin, Tanmoy Das, Yung Jui Wang, L. A. Wray, Su-Yang Xu et al.,
Adiabatic transformation as a search tool for new topological insulators: Distorted
ternary Li2 AgSb-class semiconductors and related compounds. Phys. Rev. B 87,
121202 (2013).
23. Su-Yang Xu et al., Observation of a topological crystalline insulator phase
and topological phase transition in Pb1−x Snx Te. Nature Commun. 3, 1192 (2012).
24. Su-Yang Xu et al., Hedgehog spin texture and Berrys phase tuning in a
magnetic topological insulator. Nature Phys. 8, 616-622 (2012).
25. M. Brahlek, N. Bansal, N. Koirala, Su-Yang Xu et al., Topological-Metal
to Band-Insulator Transition in (Bi1−x Inx )2 Se3 Thin Films. Phys. Rev. Lett. 109,
186403 (2012).
26. M. Neupane, Su-Yang Xu et al., Topological surface states and Dirac point
tuning in ternary topological insulators. Phys. Rev. B 85, 235406 (2012).
27. D. Zhang, A. Richardella, D. W. Rench, Su-Yang Xu et al., Interplay between ferromagnetism, surface states, and quantum corrections in a magnetically
doped topological insulator. Phys. Rev. B 86, 205127 (2012).
28. Su-Yang Xu et al., Dirac point spectral weight suppression and surface
”gaps” in nonmagnetic and magnetic topological insulators. Preprint at http://arXiv:1206.0278
(2012).
29. Su-Yang Xu et al., Anomalous spin-momentum locked two-dimensional
236
states in the vicinity of a topological phase transition. Preprint at http://arXiv:1204.6518
(2012).
30. Helin Cao, Su-Yang Xu et al., Structural and electronic properties of highly
doped topological insulator Bi2 Se3 crystals. physica status solidi 7, 133-135 (2012).
31. M. Neupane, C. Liu, Su-Yang Xu et al., Fermi-surface topology and lowlying electronic structure of the iron-based superconductor Ca10 (Pt3 As8 )(Fe2 As2 )5 .
Phys. Rev. B 85, 094510 (2012).
32. H. Ji, J. M. Allred, N. Ni, J. Tao, M. Neupane, A. Wray, Su-Yang Xu et
al., Bulk intergrowth of a topological insulator with a room-temperature ferromagnet.
Phys. Rev. B 85, 165313 (2012).
33. Su-Yang Xu et al., Topological Phase Transition and Texture Inversion in a
Tunable Topological Insulator. Science 332, 560-564 (2011).
34. L. A. Wray, Su-Yang Xu et al., A topological insulator surface under strong
Coulomb, magnetic and disorder perturbations. Nature Phys. 7, 32-37 (2011).
35. L. A. Wray, Su-Yang Xu et al., Spin-orbital ground states of superconducting
doped topological insulators: A Majorana platform. Phys. Rev. B 83, 224516 (2011).
36. S. Basak, H. Lin, L. A. Wray, Su-Yang Xu et al., Spin texture on the warped
Dirac-cone surface states in topological insulators. Phys. Rev. B 84, 121401(R)
(2011).
37. Su-Yang Xu et al., Realization of an isolated Dirac node and strongly modulated Spin Texture in the topological insulator Bi2 Te3 . Preprint at http://arXiv:1101.3985
(2011).
38. M. Z. Hasan, D. Hsieh, Y. Xia, L. A. Wray, Su-Yang Xu, C. L. Kane, A new
experimental approach for the exploration of topological quantum phenomena: Topological Insulators and Superconductors. Preprint at http://arXiv:1105.0396(2011).
(Review article).
39. Su-Yang Xu et al., Discovery of several large families of Topological Insu-
lator classes with backscattering-suppressed spin-polarized single-Dirac-cone on the
surface. Preprint at http:// arXiv:1007.5111 (2010).
40. L. A. Wray, Su-Yang Xu et al., Observation of topological order in a superconducting doped topological insulator. Nature Phys. 6, 855-859 (2010).
41. H. Lin, L. A. Wray, Y. Xia, Su-Yang Xu et al., Half-Heusler ternary compounds as new multifunctional experimental platforms for topological quantum phenomena. Nature Mater. 9, 546-549 (2010).
42. Y. S. Hor, P. Roushan, H. Beidenkopf, J. Seo, D. Qu, J. G. Checkelsky, L. A.
Wray, D. Hsieh, Y. Xia, Su-Yang Xu, Development of ferromagnetism in the doped
topological insulator Bi2−x Mnx Te3 . Phys. Rev. B 81, 195203 (2010).
43. W. Al-Sawai, H. Lin, R. S. Markiewicz, L. A. Wray, Y. Xia, Su-Yang Xu et
al., Topological electronic structure in half-Heusler topological insulators. Phys. Rev.
B 82, 125208 (2010).
44. M. D. Schroer, Su-Yang Xu et al.,Development and operation of researchscale IIIV nanowire growth reactors. Rev. Sci. Instrum. 81, 023903 (2010).
238