Download Spin-Resolved Spectroscopic Studies of Topologically Ordered Materials

Spin-resolved spectroscopic studies of
topologically ordered materials
David Hsieh
A THESIS
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
June 2009
c Copyright by David Hsieh, 2009. All rights reserved.
°
Abstract
It has recently been proposed that band insulators with strong spin-orbit coupling can
support a new phase of quantum matter called a ‘topological insulator’. This exotic
phase of matter is a subject of intense research because it is predicted to give rise
to dissipationless spin currents, axion electrodynamic phenomena and non-Abelian
quasi-particles. However, it is experimentally challenging to identify a topological
insulator because unlike ordinary phases of matter such as magnets, liquid crystals
or superconductors, topological insulators are not described by a local order parameter associated with a spontaneously broken symmetry but rather by a quantum
entanglement of its wavefunction, dubbed ‘topological order’. Because conventional
experimental probes are designed to be sensitive to local order parameters, methods
of measuring topological order are relatively unknown.
Topologically ordered phases of matter are extremely rare, the most well known
example being the quantum Hall phase, which is realized in a cold two-dimensional
electron system subject to a large external magnetic field. Its topological order is
identified by measuring a quantized magneto-transport, which is carried by robust
conducting states localized along the one-dimensional edges of the sample. In topological insulators, intrinsic spin-orbit coupling simulates the effect of a spin-dependent
external magnetic field, leading to quantum Hall-like physics without any external
magnetic field. However, unlike quantum Hall phases, topological insulators exhibit
no quantized transport response, therefore its topological order cannot be detected
iii
by means of a transport measurement.
In this thesis, we show that topological insulators exhibit robust conducting states
that are localized on the two-dimensional surfaces of the sample. These surface states
have an unusual spin-polarized band structure that cannot be realized on the surfaces of ordinary insulators, nor in purely two-dimensional electron systems. By
measuring the spin-polarized band dispersion of surface states using a combination
of synchrotron based spin- and angle-resolved photoemission spectroscopy (ARPES),
we show that their topological order can be detected. In Chapter 1, we describe
the relationship between the topological orders found in quantum Hall systems and
in topological insulators. In Chapter 2, we provide a description of the experimental technique of spin-resolved ARPES and a general procedure for mapping spinpolarized surface state band structures. In Chapter 3, we apply this method to study
the Bi1−x Sbx alloy series and demonstrate that a topological insulator is realized in
a specific composition range. The work presented in this thesis constitutes the first
experimental evidence of a topological insulator in nature.
iv
Acknowledgements
First and foremost I would like to thank my advisor Zahid Hasan for his mentorship
and support. He has been a constant source of encouragement and advice throughout
my graduate career, and his enthusiasm and leadership are the driving forces of our
lab. I am also indebted to my colleagues for their help and friendship. Yinwan Li
introduced me to the theory and operation of neutron scattering instruments when I
first arrived at Princeton. Dong Qian taught me everything I know about the ARPES
technique, and has been an extremely generous and patient science teacher to me. I
thank Lewis Wray for his constant willingness to help, insatiable curiosity and unique
ability to keep me entertained during over-night experiments. I also thank Matthew
Xia for his team-first attitude and many insightful discussions.
Throughout the course of my Ph.D., I have had the unique opportunity of working with several different experimental techniques that has taken me to a number
of national facilities around the world. Over the course of this work, I benefitted
greatly from interactions with local scientists. I thank Jeff Lynn, Qing Huang and
William Ratcliffe from NIST for spending a great deal of time teaching me about
neutron scattering. I thank Alexei Fedorov, Sung Kwan Mo and Kiyohisa Tanaka
from the ALS for going above and beyond to ensure that our experiments always
went smoothly. I thank Donghui Lu and Rob Moore from the SSRL for their fantastic beamline support. I thank Sebastian Janowski and Hartmut Hochst from the
SRC for their always helpful support. Finally I thank Hugo Dil, Fabian Meier and
v
Jurg Osterwalder from the SLS for teaching me everything I know about SR-ARPES
and for generously hosting me during visits.
I have also been very fortunate to have been able to learn from world renowned
scientists at Princeton. I thank Bob Cava and members of his team including Yew
San Hor, Tyrel McQueen, Satoshi Watauchi, Garret Lau, Katie Holman and Emilia
Morosan for their generosity in providing pristine and innovative samples and teaching me about solid state chemistry. I thank Princeton theorists David Huse, Shivaji
Sondhi, Duncan Haldane and Phillip Anderson for giving me incredible insight into
the problems I have worked on. In particular, I thank Andrei Bernevig for spending countless hours discussing experimental results and introducing me to the most
current theories. I have also benefitted greatly from discussions with Princeton experimentalists including members of Ali Yazdani’s group and Phuan Ong’s group, who
are always willing to help with either sample characterization or manuscript reading.
I am grateful to Phuan Ong for agreeing to be second reader on my thesis. Finally I
thank Michael Romalis and members of his group from whom I learned a great deal
during my experimental project.
Of course, none of the work presented in this thesis would have been possible
without the architects of topological band theory, with whom I have had great discussions with over the years. I am especially thankful to Liang Fu, Charlie Kane,
Ashvin Vishwanath and Joel Moore for teaching me many of the theoretical aspects
of this work.
Pursuing a Ph.D. in physics can sometimes be a mentally trying task and it is
my friends who keep me sane. For this I owe them big time. I thank all the great
friends I’ve made at Princeton, my Stanford buddies, and of course my lifelong CIS
playmates who have all had to endure hours upon grueling hours of listening to me
ramble on about physics. I feel your pain and I truly appreciate it.
vi
Finally, I am so grateful to Jessica and my parents Jeff and Janet for always
keeping me on the right track in life. They are my steady source of inspiration and
happiness.
vii
Contents
Abstract
iii
Acknowledgements
iv
1 Introduction
1
1.1
TKNN topological order and the quantum Hall effect . . . . . . . . .
1
1.2
Z2 topological order and the quantum spin Hall effect . . . . . . . . .
4
2 Spin- and angle-resolved photoemission spectroscopy
2.1
2.2
10
ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.1.1
Basic principles . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.1.2
Experimental setup . . . . . . . . . . . . . . . . . . . . . . . .
14
2.1.3
Sample preparation for ARPES . . . . . . . . . . . . . . . . .
16
SR-ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.2.1
Basic principles of Mott polarimetry . . . . . . . . . . . . . .
18
2.2.2
Experimental setup . . . . . . . . . . . . . . . . . . . . . . . .
20
2.2.3
Fitting routine for vectorial spin analysis . . . . . . . . . . . .
23
3 SR-ARPES on Bi1−x Sbx
26
3.1
3D strong topological insulators and their surface states . . . . . . . .
26
3.2
Predicted bulk and surface electronic structure of Bi1−x Sbx . . . . . .
31
viii
3.3
Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.3.1
Topological surface states in insulating Bi1−x Sbx . . . . . . . .
38
3.3.2
Topological surface states in metallic Sb . . . . . . . . . . . .
57
3.3.3
Evolution of surface state spectrum from Bi to Sb . . . . . . .
71
4 Conclusions
Bibliography
76
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
78
Chapter 1
Introduction
1.1
TKNN topological order and the quantum Hall
effect
The insulating state is one of the most elementary quantum phases of matter [1]. It
is characterized by a completely filled set of electronic bands that are separated from
a completely empty set of bands by an energy gap, which, according to the semiclassical Boltzmann approach to electron transport [1], makes it electrically inert at
low energies. Naively, one might therefore expect the low energy physics of insulators
(or gapped systems in general) to be trivial, and expect interesting properties to occur
only in gapless systems where there are low energy degrees of freedom.
The discoveries of the integer [2] and fractional [3] quantum Hall effects in twodimensional electron systems in the early 1980s revealed a phenomenon that appeared
to violate this picture, namely a gapped system that exhibits highly non-trivial transport properties. In particular, transverse magneto-conductance σxy is quantized in
extremely precise (to nearly one part in a billion) rational multiples of e2 /h despite
an energy gap for electronic excitations. The solution to this paradox comes from
1
1.1. TKNN topological order and the quantum Hall effect
(a)
(b)
Trivial
2
Non-trivial
a
k2
k2
k1
k1
Figure 1.1: (a) Illustration of parallel transport of a vector along a closed path on
a sphere. (b) Schematic diagram of the phase of a wavefunction in the magnetic
Brillouin zone with and without the introduction of a topological defect.
the fact that in addition to their energy dispersion, the wavefunctions of electronic
states may also play a role in charge transport. As was first pointed out by Laughlin
[4], the quantum Hall current is contingent upon phase coherence of the many body
wavefunction. An explicit expression for how the phase is related to σxy was later derived by Thouless, Kohmoto, Nightingale and den Nijs (TKNN) [5, 6], which brought
to light a topological meaning to the quantum Hall effect.
A useful measure of the topology of some manifold is the phase acquired by a
vectorial object after being parallel transported through a closed path in that manifold
[7]. An intuitive example is the parallel transport of a vector n̂(t)e−iα(t) along a
closed loop C (parametrized by t) on a sphere (Figure 1.1(a)), where the total phase
mismatch α(C) is given by
I
α(C) =
dα
(1.1)
dα = −in̂∗ · dn̂
(1.2)
C
with
1.1. TKNN topological order and the quantum Hall effect
3
The quantity α(C) is a topological invariant in the sense that it does not change under
continuous deformations of the sphere. Only through some drastic deformation that
changes its topology, such as the puncturing of a hole in the sphere, will α(C) change.
Because electronic states are represented by vectors in Hilbert space, Berry proposed that a similar geometric phase (coined a “Berry phase”) is accumulated when
a state is adiabatically transported through a closed path in parameter space [8].
By choosing this parameter to be the crystal wave vector k, TKNN showed that a
topological invariant n characterizes the Berry phase mismatch in momentum space.
The quantum Hall conductivity can be expressed as σxy = ne2 /h, where [5]
1
n=
2π
Z
[∇k × A(kx , ky )]z d2 k
(1.3)
BZ
with
A = −ihuk |∇k |uk i
(1.4)
where the integral is taken over the area of the magnetic Brillouin zone [5, 6] and
uk is the periodic part of the Bloch wavefunction [1]. Since the Berry curvature
B(k) = ∇k × A(k) is odd under time-reversal, for inversion symmetric systems,
σxy must vanish in the absence of an external magnetic field. However, because σxy
is proportional to a topological (TKNN) invariant, it cannot smoothly evolve from
zero as a function of magnetic field. Rather, it must jump from one discrete value
to another once the magnetic field is strong enough to change the topology of the
Hilbert space. Formally, a change in topology takes place through the introduction of
a topological defect, a magnetic monopole, in k-space (Figure 1.1(b)). The integral
nature of these local defects and their stability explain the precision with which σxy
is quantized and its experimental robustness.
Unlike an Ohmic current, which comes from the deviation of some distribution
function from equilibrium and is carried only by states at the Fermi level [1], the
1.2. Z2 topological order and the quantum spin Hall effect
4
quantum Hall current is a topological current that is carried by all occupied states
below the Fermi level [4]. Because there are no states below the Fermi level in which to
scatter, the bulk topological current is inherently dissipationless, which has made such
materials attractive for energy efficient electronics technologies. In practical devices
however, phase coherence of the bulk wavefunction is destroyed at low temperatures
by Anderson localization [9] due to the inevitable presence of random impurities.
Therefore no current carrying states exist in the sample bulk. Instead, building upon
Laughlin’s arguments [4], Halperin showed that for a system with Hall conductivity
σxy = ne2 /h, there must exist exactly n branches of gapless one-dimensional edge
states [10]. These edge states form a chiral Luttinger liquid, which means that they
cannot be localized by weak impurity disorder, and are the states responsible for
carrying the Hall current. As later shown by Hatsugai [11, 12], these edge states have
a topological invariant of their own, which he showed to be equivalent to the bulk
TKNN invariant. A von-Klitzing type measurement of the edge state conductance
thus represents a direct method of probing the bulk topological invariant.
1.2
Z2 topological order and the quantum spin Hall
effect
The spin Hall effect is the generation in a material of a spin current at right angles
to an applied charge current. The spin Hall effect was first proposed theoretically in
1971 based on an “extrinsic mechanism” [13], in which impurities in a conducting material deflect the spin-up and spin-down electrons in opposite directions through Mott
scattering. Interest in the early 2000’s shifted towards the possibility of an “intrinsic
mechanism” where spin currents arise from the spin-orbit field inherent to the band
structure of the material, independent of impurities. Like the quantum Hall current,
1.2. Z2 topological order and the quantum spin Hall effect
5
the spin current in the “intrinsic case” is generated by the Berry curvature of the
Bloch states. Although the charge current is still dissipative, the spin current is dissipationless. This effect was predicted to occur in a number of both three-dimensional
[14] and two-dimensional doped semiconductors [15], and has been indirectly observed
through the detection of edge spin accumulation by optical methods [16].
In contrast to doped semiconductors, Murakami, Nagaosa and Zhang proposed
in 2004 that certain band insulators could exhibit an intrinsic spin-Hall conductivity
SH
σxy
without any charge conductivity, making these systems completely dissipationless
[17]. Such “spin-Hall insulators” are realized in materials whose energy gap arises due
to spin-orbit coupling, and its mechanism is put briefly as follows. Spin-orbit coupling
gives rise to a splitting of bands into multiplets of the total angular momentum J =
L + S. Because all occupied bands contribute to a topological current, if the Fermi
level lies between two bands within the same J multiplet, the spin currents in the
totally occupied bands do not cancel.
The presence of a bulk energy gap in spin-Hall insulators raised the question of
whether a topological invariant that characterizes its many body wavefunction exists,
SH
which acts to quantize σxy
. To understand how this might arise, Bernevig and
Zhang, building upon earlier work by Haldane [18], presented the following model
for a quantum spin Hall effect in two-dimensions [19]. Landau level quantization in
the quantum Hall effect arises from a velocity dependent term in the Hamiltonian
A · p, where A is the vector potential of the applied magnetic field and p is the
particle momentum. By choosing the symmetric gauge A =
B
(−y, x, 0),
2
the velocity
dependent term is proportional to B(xpy − ypx ). In the absence of a magnetic field,
a spin-orbit term (p × E) · σ, where σ is the Pauli spin matrix and the electric field
E is chosen such that E ∼ E(x, y, 0), can give rise to a similar velocity dependent
term Eσz (xpy − ypx ). Such a system behaves as if a spin-up electron experiences an
1.2. Z2 topological order and the quantum spin Hall effect
6
Figure 1.2: Energy bands of a one-dimensional strip of graphene. Closely spaced lines
show the discrete bulk energy levels projected onto the edge Brillouin zone. The green
and red lines show the spin-filtered bands localized on either edge of the strip. (a)
Spectrum of the Z2 = 1 and (b) the Z2 = 0 phase achieved by tuning the parameter
λv (inest). [Figure adapted from [20]]
effective upward pointing orbital magnetic field while a spin-down electron experiences
a downward pointing one. This way, the spin-up electrons are described by a TKNN
invariant n↑ while spin-down electrons are described by n↓ = −n↑ , leading to an
overall vanishing charge Hall conductivity
conductivity
e
(n↑ −n↓ ).
4π
e2
(n↑
h
+ n↓ ) but a non-vanishing spin Hall
It was proposed that a measurement of the spin-filtered edge
state (in the sense that states with opposite spin propagate in opposite directions) in
such systems could reveal its time-reversal symmetric topological invariants.
Unfortunately, as Kane and Mele pointed out in 2005 [20], a topological characterization based on the spin TKNN invariant (n↑ − n↓ ) only holds true if σz is conserved,
which inevitably breaks down in real systems due, for example, to inter-band mixing,
disorder or interactions. Rather, they postulated that the topological properties of
the quantum spin Hall state are encoded in a new stable Z2 topological invariant that
is particular to time-reversal symmetric systems. Like the TKNN invariant, the Z2
1.2. Z2 topological order and the quantum spin Hall effect
7
invariant can be formulated as a defect in momentum space and becomes equivalent
to the parity of the spin TKNN invariant when σz is conserved [21, 22]. Its value,
either 0 or 1, determines whether the system is an ordinary or quantum spin Hall
insulator respectively.
Like Halperin’s edge state theory of the quantum Hall effect, spin filtered edge
states play a crucial role for spin Hall transport in the theory of Kane and Mele.
To illustrate the different behavior of the edge states when Z2 = 0 or 1, they constructed a model of graphene with spin-orbit coupling that could be switched between Z2 = 0 and 1 by tuning the strength of the staggered sublattice potential λv
[23]. In bulk graphene, all bands must be doubly spin degenerate owing to a combination of inversion symmetry [E(k, ↑) = E(−k, ↑)] and time-reversal symmetry
[E(k, ↑) = E(−k, ↓)]. Because inversion symmetry is broken at a terminated edge,
the spin degeneracy of the one-dimensional edge states is generally lifted. However,
by Kramers theorem [24], they must remain doubly degenerate at the center (k = 0)
and boundary (k = π) of the edge Brillouin zone, since those momenta are invariant
under time-reversal. The topological distinction between Z2 = 0 and Z2 = 1 insulators is manifest in the number of times, even and odd respectively, that their edge
state Fermi surface encloses these time-reversal invariant momenta (TRIM). The edge
state spectrum for the Z2 = 1 case (Figure 1.2(a)) shows a single pair of bands that
cross the Fermi level, resulting in a Fermi surface that encloses an odd number of
TRIM, namely once around k = π. In contrast, the ordinary insulating phase (Figure 1.2(b)) shows a Fermi surface that encloses an even number of TRIM, namely
none at all.
The gapless nature of the edge states in the quantum spin Hall insulator is guaranteed by time-reversal symmetry. Moreover, because there is no counter-propagating
channel in which a spin polarized electron at the edge can backscatter, weak dis-
1.2. Z2 topological order and the quantum spin Hall effect
8
order will not lead to localization and transport is dissipationless [20, 23]. Due to
the presence of these robust metallic edge states, dubbed a helical liquid [25], the
spectrum of a quantum spin Hall insulator cannot be adiabatically deformed into an
ordinary insulator with no edge states. Unlike the quantum Hall insulator, the topological properties of the ground state wavefunction of the quantum spin Hall insulator
are manifest in its energy dispersion. This provides a great experimental advantage
for measuring topological invariants because the relevant temperatures scales are no
longer set by phase coherence lengths but rather by spin-orbit gap energies.
The first realistic proposal for a quantum spin Hall insulator was made in 2006
by Bernevig, Hughes and Zhang for the HgTe/(Hg,Cd)Te quantum well system [26].
However, there are several practical limitations to a direct imaging of the edge state
energy dispersion. First, the edge states lie at a buried interface between two semiconductor films, which makes them difficult to access with scattering probes. Second, the
spatial extent of the edge state is typically small because they are localized to within
a few atomic layers near the sample edge, which is far smaller than the focused spot
sizes of most scattering probes. In 2007, Konig et al. gave the first indirect evidence
for helical edge states through a remarkable set of charge conductance measurements
in HgTe/(Hg,Cd)Te quantum wells [27]. However, the Z2 invariant was not directly
measured because the spin degeneracy and the energy dispersion of the edge states
were not imaged.
The prospect for a direct measurement of Z2 topological invariants arose between
2006 and 2007 following a series of theoretical predictions [22, 28, 29, 30] showing
that three-dimensional insulators can also be characterized by Z2 invariants. Rather
than a single Z2 invariant, three-dimensional time-reversal symmetric band structures
are described by four Z2 invariants ν0 ; (ν1 ν2 ν3 ), of which only ν0 is robust in the
presence of disorder. While the ν0 = 0 phase describes ordinary three-dimensional
1.2. Z2 topological order and the quantum spin Hall effect
9
insulators, the ν0 = 1 phase describes an intrinsically three-dimensional topological
phase of matter called the “strong topological insulator” [22, 28], which exhibits
unusual spin-filtered surface states. This is akin to the predicted, but never observed,
three-dimensional quantum Hall insulator [31].
In this thesis, I will detail an experimental method of directly measuring the
Z2 invariant ν0 in strong topological insulators. Chapter 2 will explain the basic
principles of spin and angle-resolved photoemission spectroscopy that can be used to
measure the spin-polarization and energy dispersion of surface states. Chapter 3 will
first describe how to identify candidate strong topological insulator materials, and
then discuss the photoemission based experimental method we used to measure the
Z2 invariant in the Bi1−x Sbx alloys.
Chapter 2
Spin- and angle-resolved
photoemission spectroscopy
2.1
2.1.1
ARPES
Basic principles
Angle-resolved photoemission spectroscopy is a method of studying the electronic
structure of solids by using the photoelectric effect [32, 33]. The basic principles of an
ARPES experiment are as follows. A monochromatic beam of light with energy hν,
typically from a synchrotron radiation source, impinges on a sample and photoexcites
electrons into the vacuum. These photoelectrons are then collected in an electrostatic
analyzer that measures their kinetic energy (Ekin ) as a function of emission angles
(ϑ,ϕ) relative to the sample surface. This way, the wave vector (K = p/h̄) of the
photoelectrons in vacuum is completely determined via
10
2.1. ARPES
11
(a)
E kin
(b)
Spectrum
EF
z
hn
E
hn
e-
J
Sample
Evac
EF
y
N(Ekin )
W
j
EB
E0
hn
x
Core level
N(E)
Figure 2.1: (a) Relation between energy levels in a solid and the electron energy
spectrum produced by photons of energy hν. The electron energy distribution inside
the solid is expressed in terms of the binding energy (EB ), which is referenced to
the Fermi level (EF ), whereas the photoelectron kinetic energy is referenced to the
vacuum level (Evac ). (b) Geometry of an ARPES experiment. The emission direction
of the photoelectrons is specified by the polar (ϑ) and azimuthal (ϕ) angles.
2.1. ARPES
12
1p
2mEkin sinϑcosϕ
h̄
1p
2mEkin sinϑsinϕ
Ky =
h̄
1p
2mEkin cosϑ
Kz =
h̄
Kx =
(2.1)
By exploiting energy and momentum conservation, it is straightforward to relate
the measured kinetic energy of the photoelectron to its binding energy (EB ) while
inside the sample
Ekin = hν − W − |EB |
(2.2)
where W is the work function, which is typically 4-5 eV for metals. The photon
momentum has been neglected since it is much smaller than a typical Brillouin zone
dimension at VUV photon energies. Because of translational symmetry in the plane
of the sample surface, the parallel component of momentum is conserved in a photoemission process and thus Kk = (Kx , Ky , 0) can be related to the parallel component
of the electron crystal momentum h̄kk via
|kk | = |Kk | =
1p
2mEkin sinϑ
h̄
(2.3)
The perpendicular component of momentum h̄k⊥ , on the other hand, is not conserved due to the surface potential barrier. Although there exist several experimental
methods to measure this quantity absolutely, under certain circumstances a simpler alternative is to invoke a three-step model of the photoemission process and make some
a priori assumption about the dispersion of the electron final states. The three-step
process consists of 1) an optical excitation of an electron in the solid from a low to
high energy Bloch state, followed by 2) transport of the electron to the surface, and
finally 3) transmission of the electron from a high energy Bloch state beneath the
2.1. ARPES
13
(a)
E
optical
transport
excitation to surface
escape
to vacuum
(b)
E kin
Ef
1
2
3
G
Ef
hn
hn
E vac
EF
Ei
E0
Ei
W
K
V0
-p
a
0
p
a
z
0
Figure 2.2: (a) Illustration of the three-step model of the photoemission process. (b)
Kinematics of the photoemission process within the three-step nearly-free-electron
final state model.
surface into a free electron state in vacuum (Figure 2.2(a)). If the final Bloch state is
assumed to be free electron like, then electron escape will take place at the surface via
transmission between parabolic bands inside and outside the sample that are offset
by some energy V0 known as the inner potential. For the band structure of a nearly
free electron model (Figure 2.2(b)), the final state energy is then given by
h̄2 (k2k + k2⊥ )
− |E0 |
Ef (k) =
2m
(2.4)
where Ef and E0 are both referenced to the Fermi level. Substituting Ef = Ekin + W
and Equation 2.3 into Equation 2.4, one arrives at the expression
|k⊥ | =
1p
2m(Ekin cos2 ϑ + V0 )
h̄
(2.5)
where V0 = |E0 | + W , which relates |k⊥ | to the measured values Ekin and ϑ once
V0 is known. In this work, two independent methods of determining V0 are used: (i)
2.1. ARPES
14
agreement between experimental and theoretical band structures is optimized and (ii)
the periodicity of the dispersion E(k⊥ ) measured at normal emission (ϑ = 0◦ ) is used
to extract V0 . Although the free electron final state approximation is most accurate for
materials with weak crystal potentials or at high excitation energies where the kinetic
energy of the electron far exceeds the lattice potential, it has proven successful for
mapping bands in even in high Z materials such as bismuth [34] and using VUV
photon energies.
The ability to map band dispersions perpendicular to the surface plane provides
an elegant way to distinguish surface from bulk electronic states in three-dimensional
systems. This is especially useful for systems whose bulk states disperse strongly with
k⊥ and have small c-axis lattice parameters because the electron mean free path (the
average distance that an excited state electron travels in the solid with no change in
energy and momentum) is limited to about 5Å for VUV photon energies [32, 33].
2.1.2
Experimental setup
Our non spin-resolved ARPES measurements were performed using linearly polarized synchrotron radiation and hemispherical electron analyzers from Scienta. Experiments took place at beamlines 10.0.1 and 12.0.1 of the Advanced Light Source
in Berkeley, California, at beamline 5-4 of the Stanford Synchrotron Radiation Laboratory in Stanford, California, and at beamlines PGM(A) and U1 NIM at the Synchrotron Radiation Center in Stoughton, Wisconsin.
The spectrometer setup at all of these end stations is shown in Figure 2.3. Due
to the surface sensitivity of ARPES, the samples are cleaved and maintained in ultra
high vacuum at pressures less than 5 × 10−11 torr to minimize the adsorption of
atoms to its surface. Electrons are photoemitted from the sample in situ and are
decelerated and focused onto a w by l sized entrance slit of the hemispherical analyzer
2.1. ARPES
15
V2
R1
V1
R2
w
lens
z
J
hn
sample
x
E
2D
detector
y
Figure 2.3: Schematic of a hemispherical electron analyzer. The path of photoelectrons within some narrow energy range ∆Ekin centered about Epass is marked in
gray.
via an electrostatic input lens. The hemispherical analyzer consists of two concentric
hemispheres of radius R1 and R2 maintained at a potential difference of ∆V = V2 V1 . Therefore only those electrons reaching the entrance slit within a narrow kinetic
energy range ∆Ekin centered at the value Epass = e∆V /(R1 /R2 − R2 /R1 ) will be
able to travel around the analyzer and onto a multichannel 2D detector. Because
the electrons are spread apart along the detector y axis as a function of their kinetic
energy, a ∆Ekin (typically around ±0.1Epass ) slice of phase space can be imaged
simultaneously. At the same time, photoelectrons emitted along a finite angular
range (either 9◦ , 14◦ or 38◦ depending on the analyzer model) defined by the entrance
slit length l are spread apart along the detector x axis. Therefore a 2D snap shot of
phase space of dimension ∆Ekin by ∆ϑ is captured at once.
To obtain an ARPES spectrum near EF , the first step is to determine the kinetic
energy that corresponds to EF by taking an energy distribution scan of polycrystalline
2.1. ARPES
16
gold, and fitting this to a convolution of a Gaussian resolution function with a FermiDirac distribution. A detailed scan near EF can then be obtained by setting the input
lens voltage so as to decelerate photoelectrons from this particular kinetic energy to
Epass , and sweeping the input lens voltage in incremental energy steps in case energy
ranges greater than ±0.1Epass need to be covered. The majority of our experiments
were taken with an Epass of either 5 eV or 10 eV. At these values of Epass , the
total energy resolution, which accounts for the resolving power of both the beamline
monochromator and electron analyzer, is around 15 meV.
2.1.3
Sample preparation for ARPES
The samples we use for the work presented in this thesis are single crystals of both
doped and undoped Bi1−x Sbx . These were cleaved using a razor blade from a boule
grown from a stoichiometric mixture of high-purity elements, resulting in shiny flat
silver surfaces. The boule was cooled from 650 ◦ C to 270 ◦ C over a period of five days
and was annealed for seven days at 270 ◦ C. X-ray diffraction patterns of the cleaved
single crystals, collected on a Bruker D8 diffractometer using Cu Kα radiation (λ
= 1.54 Å), exhibit only the (333), (666), and (999) peaks, showing that the cleaved
surface is the (111) plane. Powder X-ray diffraction measurements were also taken to
check that the samples were single phase (rhombohedral A7 crystal structure, point
group R3̄m).
The following steps were taken to prepare the samples for an ARPES measurement
(Figure 2.4). (1) Single crystals were typically cut to around 2 mm × 2 mm × 0.5 mm
in size, and were mounted onto a copper sample post using silver epoxy. Because of
the relatively poor electrical conductivity of the samples (Chapter 3), silver epoxy was
used to ensure good electrical contact with the copper post so as to prevent sample
charging from photon exposure. (2) The Laue back-reflection technique was used
2.1. ARPES
17
(b)
(a)
ceramic post
epoxy
sample
silver epoxy
(c)
copper post
Figure 2.4: . (a) Sample mounting schematic. (b) Example of a Laue back-reflection
image of bismuth, adapted from [35]. (c) Example of a LEED image of bismuth.
to determine the in-plane orientation of the crystal axes. A three-fold rotationally
symmetric diffraction pattern is typically observed, consistent with the three-fold
rotational symmetry of the A7 structure around the [111] axis. (3) A ceramic toppost is mounted onto the exposed face of the sample using an epoxy resin and the
entire object is then covered with a layer of silver paint for improved electrical contact
and then a layer of colloidal graphite for masking purposes. The copper post is finally
affixed to the inside of the ARPES chamber, where it is cooled and pumped down,
after which the sample is cleaved by knocking the ceramic top-post in situ. After
the experiment, the crystal structure of the cleaved surfaces was examined using
low energy electron diffraction (LEED) to ensure a good quality. Detailed analyses of
LEED [36] and photoelectron diffraction measurements [37] of the surfaces of bismuth
and antimony show negligible surface structural reconstruction or relaxation effects,
which provides assurance that the surface state spectra measured using ARPES are
representative of states arising from the terminated bulk crystal.
2.2. SR-ARPES
18
P
ki
NL k f
Ù
nL
Ù
nR
NR
kf
q
q
Figure 2.5: 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.
2.2
2.2.1
SR-ARPES
Basic principles of Mott polarimetry
The interaction Hamiltonian between a photon and spin 1/2 electron can be described
by the Dirac equation
Hint =
e ´2
eh̄
eh̄
1 ³
ieh̄
p − A +eΦ−
σ ·(∇×A)+
E·p−
σ ·(E×p) (2.6)
2m
c
2mc
4m2 c2
4m2 c2
where p is the electron momentum, A is the photon vector potential, Φ is the scalar
potential, E is the electric field and σ is the electron spin. However by using linearly
polarized photons in the UV to soft x-ray regime, it has been shown [38] that the
spin dependent terms in Equation 2.6 are greatly suppressed, and the photon electron
interaction Hamiltonian can be well approximated by the Schrodinger model Hint =
−(e/mc)A · p, which conserves spin.
Provided the photoemission process is spin-conserving, the spin of the initial state
of an electron in a solid can be determined by measuring its spin after it has been
photoemitted. Mott electron polarimetry [39] is a method of separating electrons of
2.2. SR-ARPES
19
different spin from such a photoemitted beam based on the use of spin-orbit (Mott)
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
is large, the velocity v of the electron in the electric field E of the nucleus can result
in a considerable magnetic field B in its rest frame given by
1
B=− v×E
c
(2.7)
which, using E = (Ze/r3 )r, can be written as
B=
Ze
Ze
r×v=
L
3
cr
mcr3
(2.8)
where L = mr × v is the electron orbital angular momentum. The interaction of
this magnetic field with the electron spin S creates a spin-orbit (L · S) term in the
scattering potential and introduces a spin dependent correction to the Rutherford
cross section
σ(θ) = σR (θ)[1 + S(θ)P · n̂]
(2.9)
where S(θ) is the asymmetry or Sherman function, P is the polarization
(2/h̄)(hSx i, hSy i, hSz i), and n̂ is the unit normal to the scattering plane defined by
n̂ ≡
ki × kf
|ki × kf |
(2.10)
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. SR-ARPES
20
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 +ẑ and N↓ of them
polarized along −ẑ, 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 − N R
NL + NR
(2.11)
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 (2.9) into Equation
(2.11) yields
Pz =
Az (θ)
S(θ)
(2.12)
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
Experimental setup
Our spin-resolved ARPES measurements were performed using the COmplete PHoto
Emission Experiment (COPHEE) spectrometer [41] with linearly polarized VUV synchrotron photons generated at the Swiss Light Source in Villigen, Switzerland. In the
COPHEE spectrometer, energy and momentum analysis of the photoelectrons again
takes place using a hemispherical electrostatic analyzer (Figure 2.6). Electrons at
some selected energy and momentum are then accelerated to high energy (typically
2.2. SR-ARPES
(a)
21
(b)
+J
y
+j
z
x
x’
y’
z’
Polarimeter 1
y’
x’
z’
Polarimeter 2
Figure 2.6: (a) Schematic of the spin-resolved ARPES spectrometer COPHEE
adapted from [40]. 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.
2.2. SR-ARPES
22
around 40 keV) and are alternately deflected into two Mott polarimeters that are
mounted perpendicular to one another so that a total of four (three independent)
components of spin are measured. Each Mott polarimeter consists of a gold foil target
with silicon diode detectors positioned to its left, right, top and bottom. To account
for unequal sensitivities between a detector pair, we applied a small multiplicative
factor to the intensity from one detector to ensure that the unpolarized background
intensity yields zero polarization. The two Mott polarimeters share a common coordinate (primed) frame, with polarimeter 1 sensitive to the y 0 and z 0 components of
spin and polarimeter 2 sensitive to the x0 and z 0 components. The sample coordinate
frame (unprimed) is positioned such that when ϑ and ϕ are both equal to zero, it
is related to the Mott coordinate frame by a 45◦ rotation about z 0 (Figure 2.6(b)).
A spin polarization vector Px0 measured in the Mott coordinate frame can then be
expressed in sample coordinates Px via a matrix transformation T




T =


cosϑcosϕ+sinϕ
√
2
−cosϑcosϕ+sinϕ
√
2
−cosϑsinϕ+cosϕ
√
2
cosϑsinϕ+cosϕ
√
2
sinϑ
√
2
−sinϑ
√
2
−sinϑcosϕ 

sinϑsinϕ 


cosϑ
where Px = T Px0 . A typical in-plane Fermi wave vector (kF ∼ 0.1 Å−1 ) can be
accessed at a tilt angle ϑ ∼ 2◦ using 30 eV photons. At such small tilt angles, the
effect of ϑ contributes only a very small correction to P (∼ 1%) and thus setting ϑ
to zero is often a good approximation.
The accuracy of the spin polarization measurement depends critically on knowledge of the Sherman functions of both Mott polarimeters. While the Sherman function for electron scattering from a single atomic nucleus can be calculated, scattering
from a solid state target is complicated by multiple scattering and inelastic scattering
events that reduce the analyzing power. In practice, the Sherman function of both
polarimeters are experimentally calibrated using a source of electrons with known po-
2.2. SR-ARPES
23
larization, from a magnetized sample for instance [40], to yield an effective Sherman
function (Sef f ), which can be cross checked by measuring the spin polarization along
the common (z 0 ) axis shared between them. Typical values of Sef f in our experiments
lie in the range 0.07 to 0.085.
An additional source of inefficiency comes from the fact that only a fraction N/N0
of the total number of incoming electrons are backscattered into the detectors. The
overall efficiency of a Mott polarimeter is thus usually quantified by the figure of merit
2
² = (N/N0 )Sef
f
(2.13)
which typically ranges between 10−3 to 10−4 . The statistical error of a polarization
measurement is given by
1 δAα0
1
δPα0
=
=√
Pα0
Sef f Aα0
²N0
(2.14)
Due to the low efficiency of detection, spin-resolved ARPES measurements are usually taken with increased photon beam sizes and analyzer slit widths w to increase
the photoelectron flux to the Mott polarimeters, which compromise the energy and
momentum resolution of the measurement. Nevertheless, typical hours long counting
times are still required to achieve a high signal to noise ratio scan, which is the reason gold is used as the scattering target because it presents a compromise between
high atomic number and material inertness. Typical electron counts on the detector
reach 5 × 105 , which places an error bar of approximately ±0.01 for each point on our
measured polarization curves.
2.2.3
Fitting routine for vectorial spin analysis
In a typical SR-ARPES experiment, the electron beam polarization at some fixed
energy is measured as a function of the manipulator angle ϑ to yield a continuous trace
2.2. SR-ARPES
24
P(ϑ). However because of the overlap of several bands due to intrinsic broadening
or resolution effects and the presence of a background, the polarization data do not
directly reveal the spin polarization vector of each band individually. To overcome
this issue, we employ the following fitting routine developed by Meier et al. [42] to
perform quantitative spin analysis.
The spin-averaged intensity Itot (ϑ), defined as the sum of the intensities measured
by each left (L) right (R) pair of Mott detectors over all three spin components
¡
¢
P
α0 = x0 , y 0 , z 0 Itot = α0 IαL0 + IαR0 , is first fit to a sum of Lorentzians (I i ), one for
each band, and a background B,
Itot =
n
X
Ii + B
(2.15)
i
where n is the number of bands traversed by the ϑ scan. Each band is then assigned
a polarization vector
Pi = (Pxi , Pyi , Pzi ) = ci (cosθi cosφi , cosθi sinφi , sinθi )
(2.16)
where 0 ≤ ci ≤ 1 is the magnitude of the spin polarization vector of band i and θi
and φi are the polar and azimuthal angles measured relative to the sample coordinate
frame. The polarization is then transformed into the Mott coordinate frame via T −1 ,
which can be well approximated as ϑ independent for small ϑ. This is used to define
a spin-resolved spectrum for each peak i
Iαi;↑,↓
= I i (1 ± Pαi 0 )/6 , α0 = x0 , y 0 , z 0
0
(2.17)
where the + and - correspond to ↑ and ↓ respectively, which mean spin parallel or
antiparallel to the α0 direction. The entire spin-resolved spectrum for some component
α0 can then be constructed from spin-resolved spectra of the individual bands via
2.2. SR-ARPES
25
Iα↑,↓
0
=
n
X
Iαi;↑,↓
+ B/6
0
(2.18)
i=1
where the background is divided equally between the different spatial directions. The
spin polarization for each spatial component α0 can be expressed as
Pα0 =
Iα↑0 − Iα↓0
Iα↑0 + Iα↓0
which is used to fit the measured polarization spectra Pα0 = Aα0 /Sef f .
(2.19)
Chapter 3
SR-ARPES on Bi1−xSbx
3.1
3D strong topological insulators and their surface states
When an infinite three-dimensional crystal is terminated at a two-dimensional surface, new states that are localized near the surface can emerge, which are formed by
matching evanescent Bloch wave states into the crystal to evanescent plane waves
into the vacuum [1]. In the presence of spin-orbit coupling, these two-dimensional
dispersing surface states are generally spin-split. Because the spin-splitting is induced
entirely electrically, the surface states of strongly spin-orbit coupled materials have
been heavily investigated for their potential in spintronics technologies [43] such as
the proposed Das-Datta spin transistor [44].
Recent theoretical works [22, 29, 30] have shown that on the surfaces of spin-orbit
coupled insulators, topologically distinct varieties of surface states can be formed,
which are related to the topological properties of the bulk band structure. In general,
time-reversal symmetric band structures in three dimensions are characterized by four
topological Z2 invariants ν0 ;(ν1 ν2 ν3 ) [22, 29, 30], which gives rise to 16 distinct bulk
26
3.1. 3D strong topological insulators and their surface states
27
kz
(0,p)
(0,0)
(p,p)
(p,0)
(0,0,p)
(0,p,0)
ky
(p,0,0)
kx
Figure 3.1: Bulk Brillouin zone of a three-dimensional cubic lattice and its projected
(001) surface Brillouin zone. The eight bulk TRIM are located at the corners of the
bold cube and the four surface TRIM are located at the corners of the bold square.
3.1. 3D strong topological insulators and their surface states
28
topological classes. However, of these four topological invariants, only ν0 is robust in
the presence of disorder. Therefore only two classes, the ν0 = 0 ordinary insulator
and the ν0 = 1 strong topological insulator (STI) are expected to be observable in
nature.
The relationship between the topology of the surface states on an arbitrary crystal
face and the bulk Z2 invariant was first established by Fu, Kane and Mele in 2007 as
follows [28, 45]. In a general three-dimensional Brillouin zone, there are eight independent time-reversal invariant momenta (TRIM) ki that satisfy ki = −ki modulo
G, where G is any reciprocal lattice vector. For a surface perpendicular to G, the
surface Brillouin zone has four independent TRIM Λ̄a . These are located at the projections of pairs ki=a1 , ki=a2 , that differ by G/2, onto the plane perpendicular to G
(Figure 3.1).
The connectivity of the spin polarized surface state bands between Λ̄a and Λ̄b is
determined by the quantities δ(ki ) = ±1, which in turn are related to ν0 via
(−1)ν0 =
8
Y
δ(ki )
(3.1)
i=1
In particular, whether a surface band crosses EF an even or odd number of times
between Λ̄a and Λ̄b depends on whether the product of their surface fermion parities
π(Λ̄a )π(Λ̄b ) equals +1 or -1 respectively, where the surface fermion parities are given
by π(Λ̄a ) = δ(ka1 )δ(ka2 ) and π(Λ̄b ) = δ(kb1 )δ(kb2 ). It follows that the surface state
Fermi surface forms a boundary in the surface Brillouin zone separating two regions,
one containing the surface TRIM with surface fermion parity +1 and the other containing the surface TRIM with surface fermion parity -1. It was proven in general
[28] that when ν0 = 0, each region must contain an even number of TRIM and when
ν0 = 1, each region must contain an odd number of TRIM (Figure 3.2).
3.1. 3D strong topological insulators and their surface states
(a)
(c)
n0 = 0
n0 = 1
kz
kz
+
+
+
+
-
+
+
+
ky
ky
kx
kx
(b)
(d)
E
E
EF
EF
La
29
ky
Lb
La
ky
Lb
Figure 3.2: (a) Schematic of the surface state Fermi surface for ν0 = 0 and (b) ν0 = 1
cases. The surface fermion parities at the surface TRIM are labeled by + and − signs.
(c) Schematic of the surface band dispersion for the ν0 = 0 case along the direction
marked by the red arrow in (a). The shaded grey regions represent the projection of
the bulk bands onto the (001) surface. (d) Analogous schematic for the ν0 = 1 case
illustrating the partner switching.
3.1. 3D strong topological insulators and their surface states
30
The topological distinction between the ν0 = 0 and ν0 = 1 surface states is similar
to that between the edge states of Z2 = 0 and Z2 = 1 planar insulators [23]. Because
the surface bands must be doubly spin degenerate at both Λ̄a and Λ̄b by Kramers
theorem, a pair of spin-polarized bands that intersect at Λ̄a must either intersect
with one another again at Λ̄b , which is the case for ν0 = 0, or must separately
intersect with different spin-polarized bands at Λ̄b , which is allowed only when ν0 = 1
(Figure 3.2). The latter phenomenon is dubbed “partner switching” [28].
The prediction of a two-dimensional surface state with a Fermi surface enclosing an odd number of TRIM represents a fundamentally new and exotic metallic
system. Firstly, its gapless electrons are protected against time-reversal symmetric
perturbations. Secondly, because the wave function of a single electron spin acquires
a geometric phase factor of π [24] as it evolves by 360◦ in momentum space along a
Fermi contour enclosing a TRIM, an odd number of Fermi pockets enclosing TRIM
in total implies a π Berry’s phase, which prevents these gapless electrons from being localized by disorder [28]. Such an electronic spectrum is well known to not be
realizable in purely two-dimensional electronic systems with Rashba or Dresselhaus
spin-orbit coupling because of the fermion doubling theorem [46]. To predict a real
material that exhibits these surface states, it is essential to be able to calculate the
δ(ki ) from its bulk many body wavefunction. This can in principle be done by calculating Brillouin zone integrals of Berry phase terms or finding zeroes of Pfaffian
functions [22, 28], however these are very computationally intensive. In 2007, Fu
and Kane found that for inversion symmetric systems [22], the expression for δ(ki )
is greatly simplified, and depends only on the parity eigenvalues ξ2m (~ki ) = ±1 of the
occupied bulk states at the eight bulk TRIM via the relation
δ(ki ) =
N
Y
m=1
ξ2m (~ki )
(3.2)
3.2. Predicted bulk and surface electronic structure of Bi1−x Sbx
31
where N is the number of bulk Kramers degenerate states below EF . By applying this
formalism to the calculated band structures of a number of inversion symmetric spinorbit insulators, they identified the compounds Bi1−x Sbx , α-Sn, HgTe, Pb1−x Snx Te
as candidates for being an STI [28]. With the exception of Bi1−x Sbx however, all
of these compounds require the application of an external strain to realize the STI
phase, which is the reason we focus on Bi1−x Sbx .
3.2
Predicted bulk and surface electronic structure of Bi1−xSbx
The crystal structure of bismuth and antimony has a rhombohedral A7 symmetry
(space group R3̄m) typical of the group V semimetals, which is inversion symmetric
and consists of two interpenetrating trigonally distorted FCC lattices with two atoms
per unit cell (Figure 3.3(a)). Using a rhombohedral Bravais lattice, the three primitive
translation vectors of the lattice are
µ
1 1
1
√ a, − a, c
a1 =
2 3
2 3
¶
µ
1 1
1
√ a, a, c
a2 =
2 3 2 3
¶
µ
1
1
a3 = − √ a, 0, c
3
3
¶
(3.3)
with the relative position of the two basis atoms given by d = (0, 0, 2µ)c. Measured
values of the lattice parameters for (Bi, Sb) are a = (4.5332 Å, 4.3007 Å), c = (11.7967
Å, 11.2221 Å) and µ = (0.2341, 0.2336). The structure can be viewed as a set of puckered triangular lattice bilayers stacked along the rhombohedral [111] direction, with
3.2. Predicted bulk and surface electronic structure of Bi1−x Sbx
z
(a)
32
(b)
1st layer
2nd layer
3rd layer
4th layer
y
a3
a1
x
a2
(c)
1.59 Å
2.35 Å
y
z
-x
x
Figure 3.3: (a) Crystal structure of group V semimetals such as bismuth and antimony. Volume enclosed by the shaded planes denotes the rhombohedral unit cell.
(b) Top view of the first three atomic layers after cleavage along the (111) plane.
Each layer consists of a triangular lattice. (c) Side view of the first four atomic layers
along the mirror (yz) plane, showing the first and second neighboring layers of each
triangular lattice plane. The distances shown are for bismuth.
each atom having three nearest-neighbor atoms within the bilayer and three nextnearest-neighbor atoms in the adjacent bilayer (Figure 3.3(b) and (c)). Because of a
much weaker van der Waals type inter-bilayer bonding, the natural cleavage plane is
along the (111) plane.
3.2. Predicted bulk and surface electronic structure of Bi1−x Sbx
33
The reciprocal lattice vectors are given by
¶
1 1
1
b1 = 2π √ , − ,
3a a c
¶
µ
1 1 1
b2 = 2π √ , ,
3a a c
¶
µ
1
2
b3 = 2π − √ , 0,
c
3a
µ
(3.4)
and throughout our work, the (hkl) plane denotes the surface that is perpendicular
to the reciprocal lattice vector hb1 + kb2 + lb3 . The bulk Brillouin zone is shown in
Figure 3.4(a), and consists of eight TRIM, Γ, T , 3 × L and 3 × X.
The bulk band structures of Bi and Sb, derived from a tight binding model [47],
are shown in Figure 3.4(b) and (c). Eight doubly spin degenerate bands are shown,
which originate from the atomic s, px , py and pz orbitals located on the two atoms
in the unit cell. Because of the large atomic weight of the Bi and Sb atoms, a
large atomic spin-orbit coupling mixes the p orbitals such that the six bands nearest
to the Fermi energy have strongly mixed px , py and pz character. In bismuth, the
valence band crosses EF near the T point, forming a small hole pocket, while the
conduction band crosses EF at the three L points, forming small electron pockets. In
this tight binding calculation, the bottom of the conduction band at L is composed
of a symmetric linear combination of p-orbitals (Ls symmetry) and is separated by
a small energy gap from the top of the valence band at L, which is composed of an
anti-symmetric linear combination of p-orbitals (La symmetry) (Figure 3.4(e)). In
antimony, on the other hand, the hole pocket is near the H point instead of near the
T point. Although the electron pocket is still near the L point, the symmetry of the
valence and conduction bands at this momentum are now switched relative to those
in bismuth (Figure 3.4(g)).
3.2. Predicted bulk and surface electronic structure of Bi1−x Sbx
(a)
(b) Bismuth
(c) Antimony
(f)
0.07 < x < 0.22
34
(d)
T
H
L
G
X
(e)
Pure Bismuth
EF
La
Ls
La
H
Pure Antimony
La
Ls
T
(g)
L
T
H
L
Ls
T
H
L
Figure 3.4: (a) Bulk 3D Brillouin zone of Bi1−x Sbx showing the eight bulk TRIM (T ,
Γ, 3×L, 3×X). (b) Theoretical bulk band structure of Bi and (c) Sb based on the
tight binding model of Liu & Allen [47]. (d) Table of symmetry labels of the five
occupied valence bands of Bi and Sb, the right column displays the product δ(ki ).
Tables adapted from [22]. (e)-(g) Schematic evolution of the near EF band structure
from Bi to Sb.
3.2. Predicted bulk and surface electronic structure of Bi1−x Sbx
T (K)
r (mW-cm)
´
Bi 1-x Sb x
r (mW-cm)
r (mW-cm)
Bi 1-x Sb x
35
T (K)
T (K)
Figure 3.5: Resistivity versus temperature curves of undoped and doped Bi1−x Sbx .
The resistivity curve of the x=0 sample in the left panel has been multiplied by a
factor of 80.
Although both Bi and Sb are semimetals, because there is a finite direct energy
gap separating the valence and conduction band at every point in k-space, they can
both be adiabatically connected to an insulating state by pulling the conduction band
above EF and pushing the valence band below. Therefore Equation 3.2 can be applied
to calculate their Z2 invariant. Such a calculation was carried out in [22] and the δ(ki )
are displayed in the rightmost columns of Figure 3.4(d). The only difference between
Bi and Sb is in their value of δ(L), which arises from the La to Ls symmetry inversion.
By substituting these values of δ(ki ) into Equation 3.1 , Fu and Kane concluded that
Bi has trivial topological properties (ν0 = 0) while Sb is adiabatically connected to
an STI (ν0 = 1).
A true STI phase was postulated to exist in a Bi1−x Sbx alloy [22] provided there
is a composition range x where the material is both in an insulating state and in the
inverted band regime where E(La ) > E(Ls ). Based on extensive quantum oscillation,
magneto-optic and transport data previously collected on the Bi1−x Sbx series [48],
it is widely accepted that the band inversion transition takes place at x ∼ 0.04,
3.3. Experimental results
36
and that the alloy exhibits an insulating trend between 0.07 < x < 0.22. The
highly insulating trend of samples near x = 0.1 is supported by our transport data
(Figure 3.5), although we note that the resistivity of these samples saturates at low
temperatures suggestive of either conduction in an impurity band or through surface
states. Therefore the latter regime represents a true STI, which is the main subject
of this thesis.
Based on the calculated values of δ(ki ), the topology of the surface state Fermi
surface of insulating compositions of Bi1−x Sbx can be predicted. In the (111) surface
Brillouin zone, the four TRIM are located at Γ̄ and 3×M̄, which are the projections
of the (Γ,T ) and 3×(L,X) bulk TRIM pairs onto the (111) surface (Figure 3.6(a)).
It can be checked that each pair of bulk TRIM are indeed separated by G[111] /2. The
surface fermion parities at each of these four surface TRIM can be calculated from
the table in Figure 3.6(b), which show that for Bi1−x Sbx , π(Γ̄) = +1 and π(M̄) = −1.
The Fermi surface shown in Figure 3.6(b), which encloses an odd number of TRIM
(7), is consistent with these surface fermion parities (Figure 3.6(d)).
3.3
Experimental results
In this section, we will show that a measurement of ν0 for spin-orbit insulators (and
in some instances metals) can be made using a combination of two techniques: 1)
Incident photon energy modulated angle-resolved photoemission is first used to map
and distinguish between the bulk and surface state band structures of a material;
2) Having isolated the surface state Fermi surface, spin-resolved ARPES is then employed to measure the spin degeneracy and chirality of each piece of the Fermi surface
in order to determine the surface fermion parities and in turn the bulk ν0 number.
3.3. Experimental results
37
(111)
(a)
(b)
K
G
M
M
M
M (L,X)
T
L
X
G
1
L
2
G (G,T)
M (L,X)
3
L
X
X
(d)
(c)
BiSb and Sb
d(G) d(L)
d(T) d(X)
n0
Path
p(La)p(Lb)
Fermi level
crossings
Bi
-1
-1
-1
-1
0
1
-1
odd
Sb
-1
+1
-1
-1
1
2
+1
even
BiSb
-1
+1
-1
-1
1
3
-1
odd
Figure 3.6: (a) Bulk 3D Brillouin zone of Bi1−x Sbx showing the projection of the eight
bulk TRIM (T, Γ, 3×L, 3×X) onto the four surface TRIM (Γ̄, 3×M̄) on the (111)
surface Brillouin zone. (b) Schematic of a Fermi surface formed by the surface states
of a Z2 topological insulator showing an odd number of Fermi level crossings along Γ̄M̄ and an even number along M̄-M̄. (c) Bulk parity invariants of the Bi1−x Sbx system
and their corresponding Z2 topological number ν0 . (d) Product of surface fermion
parities between two surface TRIM on a Z2 topological insulator.
3.3. Experimental results
3.3.1
38
Topological surface states in insulating Bi1−x Sbx
The Bi1−x Sbx alloy series has been investigated intensively since the 1960s for its thermoelectric properties [48]. The high thermoelectric figure-of-merit has been theoretically attributed to the highly non-parabolic electronic structure of pure bismuth near
the L point, which is described by the massive (3+1)-dimensional relativistic Dirac
p
equation [49]. The resulting dispersion relation, E(k) = ± (v · k)2 + ∆2 ≈ v · k, is
highly linear owing to the combination of an unusually large band velocity v and a
small inter-band gap ∆ (such that |∆/v| ≈ 5 × 10−3 Å−1 ), which has successfully explained various peculiar properties of bismuth [50, 51, 52]. Substituting bismuth with
antimony is believed to change the critical energies of the band structure according to
Figure 3.8(e). At an Sb concentration of x ≈ 4%, the gap ∆ between La and Ls closes
and a massless three-dimensional Dirac point is realized. As x is further increased
this gap re-opens with inverted symmetry ordering, which leads to a change in sign of
∆ at each of the three equivalent L points in the Brillouin zone. For concentrations
greater than x ≈ 7% there is no overlap between the valence band at T and the
conduction band at L, and the material becomes an inverted-band insulator. Once
the band at T drops below the valence band at L, at x ≈ 7 − 8%, the system evolves
into a direct-gap insulator whose low energy physics is dominated by the spin-orbit
coupled Dirac particles at L [22, 48]. Although this Dirac electronic structure has
been experimentally inferred via transport, quantum oscillation and magneto-optical
measurements [48], no direct observation has ever been made.
We used incident photon energy modulated ARPES to measure the electronic
band dispersion along various momentum space trajectories in the 3D Brillouin zone
of single crystals of Bi0.9 Sb0.1 . Examples of such trajectories are shown in Figure 3.7,
which are constructed by invoking the free-electron-final state approximation with an
inner potential V0 = -10 eV [34]. ARPES spectra taken along two orthogonal cuts
3.3. Experimental results
39
k z (Å-1)
3
29 eV
2
18 eV
1
X
L
0
-3
-2
-1
0
1
2
3
-1
k x (Å )
Figure 3.7: Location of the L (black circles) and X (red circles) points in the bulk
Brillouin zone in the kx − kz plane together with the constant energy contours that
can be accessed by changing the angle ϑ. The two contours correspond to hν = 29
eV and 18 eV and span an angular range of ±80◦
3.3. Experimental results
40
through the L point in the third bulk Brillouin zone using 29 eV photons are shown
in Figures 3.8(a) and (c). A Λ-shaped dispersion which intersects within 50 meV
below the Fermi energy (EF ) can be seen along both directions. Additional features
originating from surface states (SS) that do not disperse with incident photon energy
are also seen. Owing to the finite intensity between the bulk and surface states, the
exact binding energy (EB ) where the tip of the Λ-shaped dispersion lies is unresolved.
The linearity of the bulk Λ-shaped bands is observed by locating the peak positions
at higher EB in the momentum distribution curves (MDCs), and the energy at which
these peaks merge is obtained by extrapolating linear fits to the MDCs. Therefore
50 meV represents a lower bound on the energy gap ∆ between La and Ls . The
extracted band velocities along the kx and ky directions are 7.9 ± 0.5 × 104 ms−1
and 10.0 ± 0.5 × 105 ms−1 respectively, which are similar to the tight binding values
7.6×104 ms−1 and 9.1×105 ms−1 calculated for the La band of bismuth [47]. Our data
is consistent with the extremely small effective mass of 0.002me observed in x = 0.11
samples by magneto-reflection measurements [53]. The Dirac point in graphene, coincidentally, has a comparable band velocity (|vF | ≈ 106 ms−1 ) [54] to what we observe
for Bi0.9 Sb0.1 , but the spin-orbit coupling in graphene is several orders of magnitude
weaker [23] and the only known method of inducing a gap in the Dirac spectrum of
graphene is by coupling to an external chemical substrate [55]. The Bi1−x Sbx series
thus provides a rare opportunity to study relativistic Dirac Hamiltonian physics in a
3D condensed matter system where the intrinsic (rest) mass gap can be easily tuned.
Studying the band dispersion perpendicular to the sample surface provides a way
to differentiate bulk states from surface states in a 3D material. To visualize the
near-EF dispersion along the 3D L-X cut (X is a point that is displaced from L
by a kz distance of 3π/c, where c is the lattice constant), in Figure 3.9(a) we plot
energy distribution curves (EDCs), taken such that electrons at EF have fixed in-
3.3. Experimental results
41
®
k = (0.8, ky, 2.9) = L ± dky
0.1
0.1
ky
0.0
0.0
EB (eV)
(c)
(b)
(a)
Low
0.1
2
ky
L
-0.1
-0.1
kx
L
-0.1
-0.05
-0.3
-0.3
-0.3
-0.4
-0.4
-0.4
-0.2
0.0
-1
kdeg
y (Å )
SS
1
kx
SS
SS
-0.10
-0.15
-0.5
-0.5
-0.5
SS
0.00
kx
-0.2
-0.2
-0.2
ky
0.05
L
SS
0.0
SS
®
k = (kx ,0, 2.9) = L ± dkx
High
-0.20
L-0.1
0.2
L
L+0.1
0.4 0.6 0.8 1.0 1.2
-1
k xdeg
(Å )
-1
k (Å )
kz
M
(e)
(d)
G
T
2
1
T
(f)
ky
kx
E
K
X
L
L
X
LS
La
Bi
-1
k y (Å )
4% 7% 8%
x
-1
k x (Å )
Figure 3.8: Selected ARPES intensity maps of Bi0.9 Sb0.1 are shown along three ~kspace cuts through the L point of the bulk 3D Brillouin zone (BZ). The presented
data are taken in the 3rd BZ with Lz = 2.9 Å−1 with a photon energy of 29 eV.
The cuts are along (a), the ky direction, (b), a direction rotated by approximately
10◦ from the ky direction, and (c), the kx direction. Each cut shows a Λ-shaped
bulk band whose tip lies below the Fermi level signalling a bulk gap. The surface
states are denoted SS. (d), Momentum distribution curves (MDCs) corresponding
to the intensity map in (a). (f), Log scale plot of the MDCs corresponding to the
intensity map in (c). The red lines are guides to the eye for the bulk features in the
MDCs. (e) Schematic evolution of bulk band energies as a function of x is shown.
The composition we study here (for which x = 0.1) is indicated by the green arrow.
3.3. Experimental results
42
plane momentum (kx , ky ) = (Lx , Ly ) = (0.8 Å−1 , 0.0 Å−1 ), as a function of photon
energy (hν). There are three prominent features in the EDCs: a non-dispersing, kz
independent, peak centered just below EF at about −0.02 eV; a broad non-dispersing
hump centered near −0.3 eV; and a strongly dispersing hump that coincides with the
latter near hν = 29 eV. To understand which bands these features originate from, we
show ARPES intensity maps along an in-plane cut K̄M̄K̄ (parallel to the ky direction)
taken using hν values of 22 eV, 29 eV and 35 eV, which correspond to approximate kz
values of Lz − 0.3 Å−1 , Lz , and Lz + 0.3 Å−1 respectively (Figure 3.9(b-d)). At hν =
29 eV, the low energy ARPES spectral weight reveals a clear Λ-shaped band close to
EF . As the photon energy is either increased or decreased from 29 eV, this intensity
shifts to higher binding energies as the spectral weight evolves from the Λ-shaped into
a ∪-shaped band. Therefore the dispersive peak in Figure 3.9(a) comes from the bulk
valence band, and for hν = 29 eV the high symmetry point L = (0.8 Å−1 , 0 Å−1 ,
2.9 Å−1 ) appears in the third bulk BZ. In the maps of Figure 3.9(b) and (d) with
respective hν values of 22 eV and 35 eV, overall weak features near EF that vary in
intensity remain even as the bulk valence band moves far below EF . The survival of
these weak features over a large photon energy range (17 to 55 eV) supports their
surface origin. The non-dispersing feature centered near −0.3 eV in Figure 3.9(a)
comes from the higher binding energy (valence band) part of the full spectrum of
surface states, and the weak non-dispersing peak at −0.02 eV reflects the low energy
part of the surface states that cross EF away from the M̄ point and forms the surface
Fermi surface.
As an additional check that we have indeed correctly identified the bulk bands
of Bi0.9 Sb0.1 in Figures 3.8 and 3.9, we also measured the dispersion of the deeper
lying bands well below the Fermi level and compared them to tight binding theoretical calculations of the bulk bands of pure bismuth following the model of Liu and
3.3. Experimental results
43
®
k = L ± dk z
Bulk state
(b)
G
Intensity (arb. units)
M
EB(eV)
(a)
hn = 22eV
00
High
-1-1
-2-2
-3-3
-4-4
Low
-0.1
0.0
0.1
0.2
-0.10
0.00
0.10
0.20
deg
(c)
EB(eV)
SS
hn = 29eV
0
0
-1-1
-2-2
-3-3
-4-4
-0.10
0.00
0.10
0.20
-0.1
0.0
0.1
0.2
deg
(d)
kZ
EB(eV)
M
G
K
M
K
ky
kX
T
X
3
0
-1
-2
-3
-4
L
L
X
hn = 35eV
-1.0
-0.5
0.0
-0.10
0.00
0.10
0.20
-0.1
0.0
0.2
0.1
deg
-1
EB(eV)
-1
k y (Å ) ; kx = 0.8 Å
Figure 3.9: (a) Energy distribution curves (EDCs) of Bi0.9 Sb0.1 with electrons at the
Fermi level (EF ) maintained at a fixed in-plane momentum of (kx = 0.8 Å−1 , ky =
0 Å−1 ) are obtained as a function of incident photon energy to identify states that
exhibit no dispersion perpendicular to the (111)-plane. Selected EDC data sets with
photon energies of 28 eV to 32 eV in steps of 0.5 eV are shown for clarity. The
energy non-dispersive (kz independent) peaks near EF are the surface states (SS).
(b-d) ARPES intensity maps along cuts parallel to ky taken with electrons at EF
fixed at kx = 0.8 Å−1 and with photon energies of hν = 22 eV, 29 eV and 35 eV are
shown. The faint Λ-shaped band at hν = 22 eV and hν = 35 eV shows some overlap
with the bulk valence band at L (hν = 29 eV), suggesting that it is a resonant surface
state which is degenerate with the bulk state in some limited k-range near EF . The
flat band of intensity centered about -2 eV in the hν = 22 eV scan originates from
Bi 5d core level emission from second order light.
3.3. Experimental results
44
Allen [47]. A tight-binding approach is known to be valid since Bi0.9 Sb0.1 is not a
strongly correlated electron system. As Bi0.9 Sb0.1 is a random alloy (Sb does not
form a superlattice [48]) with a relatively small Sb concentration (∼0.2 Sb atoms per
rhombohedral unit cell), the deeper lying band structure of Bi0.9 Sb0.1 is expected to
follow that of pure Bi because the deeper lying (localized wave function) bands of
Bi0.9 Sb0.1 are not greatly affected by the substitutional disorder, and no additional
back folded bands are expected to arise. Since these deeper lying bands are predicted
to change dramatically with kz , they help us to finely determine the experimentally
probed kz values. Figure 3.10(f) shows the ARPES second derivative image (SDI) of
a cut parallel to K̄M̄K̄ that passes through the L point of the 3D Brillouin zone, and
Figure 3.10(h) shows a parallel cut that passes through the 0.3 XL point (Figure 3.7),
which were achieved as follows. By adjusting ϑ such that the in-plane momentum
kx is fixed at approximately 0.8 Å−1 (the surface M̄ point), at a photon energy hν
= 29 eV, electrons at the Fermi energy (EB =0 eV) have a kz that corresponds to
the L point in the third bulk BZ. By adjusting ϑ such that the in-plane momentum
kx is fixed at approximately -0.8 Å−1 , at a photon energy hν = 20 eV, electrons at a
binding energy of -2 eV have a kz near 0.3 XL.
There is a clear kz dependence of the dispersion of measured bands A, B and C,
pointing to their bulk nature. The bulk origin of bands A, B and C is confirmed
by their good agreement with tight binding calculations (bands 3, 4 and 5 in Figures 3.10(g) and (i)), which include a strong spin-orbit coupling constant of 1.5 eV
derived from bismuth [47]. The bands labeled 3 to 6 are derived from 6p-orbitals and
their dispersion is thus strongly influenced by spin-orbit coupling. The fact that there
is a close match of the bulk band dispersion between the data and calculations further
confirms the presence of strong spin-orbit coupling. The slight differences between
the experimentally measured band energies and the calculated band energies at ky
3.3. Experimental results
45
kz
(a)
(c)
(b)
(d)
T
X
L
L
ky
kx
X
(e)
kz = L
E B(eV)
(f)
(g)
kz = L
kz = 0.3XL
(h)
6
00
(i)
0
0.0
EB(eV)
-1-1
-2-2
-0.5
-3-3
-1.0
-1
5
C
B
4
-2
A
3
-3
-4-4
-0.15
-0.15
0.0 0.15
0.15
deg
-0.4
C
5
B
4
-4
-0.2
-0.2
0.0
0.0
deg
0.2
0.2
0.4
-0.2
0.0
0.2
-1
ky(Å )
-0.4
-0.2
-0.2
0.0
0.0
0.2
0.2
0.4
-0.2
0.0
0.2
deg
Figure 3.10: (a), Energy distribution curves (EDCs) along a k-space cut given by
the upper yellow line shown in schematic (c) which goes through the bulk L point
in the 3rd BZ (hν = 29 eV). The corresponding ARPES intensity in the vicinity of
L is shown in (e). (b), EDCs along the lower yellow line of (c) which goes through
the point a fraction 0.3 of the k-distance from X to L (hν = 20 eV). (This cut was
taken at a kx value equal in magnitude but opposite in sign to that in (a). (f,h) The
ARPES second derivative images (SDI) of the raw data shown in (a) and (b) to reveal
the band dispersions. The flat band of intensity at EF is an artifact of taking SDI.
(g,i) Tight binding band calculations of bismuth including spin-orbit coupling, using
Liu and Allen model [47], along the corresponding experimental cut directions shown
in (f) and (h). Band 3 drops below −5 eV at the 0.3 XL point. The inter-band gap
between bands 5 and 6 is barely visible on the scale of (g). The circled curves mark
the surface state dispersion, which is present at all measured photon energies (no kz
dispersion). (d) Tight binding valence band (5) dispersion of bismuth in the ky -kz
momentum plane showing linearity along both directions.
3.3. Experimental results
46
= 0 Å−1 shown in Figure 3.10(f-i) are due to the fact that the ARPES data were
taken in constant ϑ mode. This means that electrons detected at different binding
energies will have slightly different values of kz , whereas the presented tight binding
calculations show all bands at a single kz . We checked that the magnitude of these
band energy differences is indeed accounted for by this explanation. Even though the
La and Ls bands in Bi0.9 Sb0.1 are inverted relative to those of pure Bi, calculations
show that near EF , apart from an insulating gap, they are “mirror” bands in terms
of k dispersion (see bands 5 and 6 in Figure 3.10(g)). Therefore an overall close
match to calculations, which also predict a linear dispersion along the kz cut near EF
(Figure 3.10(d)), provides strong support that the dispersion of band C, near EF , is
in fact linear along kz .
Focusing on the Λ-shaped valence band at L, the EDCs (Figure 3.10(a)) show a
single peak out to ky ≈ ±0.15 Å−1 demonstrating that it is composed of a single band
feature. Outside this range however, an additional feature develops on the low binding
energy side of the main peak in the EDCs, which shows up as two well separated
bands in the SDI image (Figure 3.10(f)) and signals a splitting of the band into bulk
representative and surface representative components (Figure 3.10(a),(f)). Unlike
the main peak that disperses strongly with incident photon energy, this shoulderlike feature is present and retains the same Λ-shaped dispersion near this k-region
(open circles in Figures 3.10(g) and (i)) for all photon energies used, supporting its
2D surface character. This behaviour is quite unlike bulk band C, which attains the
Λ-shaped dispersion only near 29 eV (Figure 3.9).
Having established the existence of an energy gap in the bulk state of Bi0.9 Sb0.1
near L (Figures 3.8 and 3.9) and observed linearly dispersive bulk bands uniquely
consistent with strong spin-orbit coupling model calculations [50, 51, 52, 47], we now
discuss the topological character of its surface states, which are found to be gapless
3.3. Experimental results
47
(a)
(b)
G
M
SS of Bi1-x Sbx
-0.2
-0.2
0.0
0.0
0.2
0.2
0.4
0.4
0.6
0.6
0.8
0.8
1.0
0.0
2
M1
-0.1
0.6 0.8 1.0
1.2
k x (Å-1 )
0.6
0.8
1.0
-k x (Å-1 )
(d)
Topological Hall insulator
1
M2
0.0
1.2
1.0
k x (Å-1 )
(c)
0.1
0.04
-0.04
0.0
0.0
EB(eV)
-1
kDeg
y (Å )
0.2
0.2
-0.2
-0.2
EB(eV)
k y (Å-1 )
high
M’
low
G
M1
3 4,5
(e)
0.0
G
M2
M
-0.1
0.0
0.2
0.4
0.6
-1
G
- k X (Å )
0.8
M
1.0
-0.2
-0.1
0.0
EB(eV)
Figure 3.11: (a) Spin-integrated ARPES intensity map of the SS of Bi0.91 Sb0.09 at EF .
(b) High resolution ARPES intensity map of the SS at EF that enclose the M̄1 and
M̄2 points. Corresponding band dispersion second derivative images (SDI) 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. (c) The surface band dispersion SDI of
Bi0.9 Sb0.1 along Γ̄-M̄. The shaded white area shows the projection of the bulk bands
based on ARPES data, as well as a rigid shift of the tight binding bands to sketch
the unoccupied bands above the Fermi level. The Fermi crossings of the surface state
are denoted by yellow circles, with the band near −kx ≈ 0.5 Å−1 counted twice owing
to double degeneracy. The red lines are guides to the eye. The EDCs along Γ̄-M̄ are
shown to the right (d). (e) Schematic of the surface Fermi surface observed in our
experiments, which is consistent with a ν0 = 1 topology.
3.3. Experimental results
48
(Figure 3.9). On the (111) surface of Bi0.9 Sb0.1 , the four TRIM are located at Γ̄
and three M̄ points that are rotated by 60◦ relative to one another. Owing to the
three-fold rotational symmetry of the bulk crystal (A7 crystal structure) and the timereversal symmetry of the two-dimensional surface states, the surface state dispersion
around these three M̄ points are equivalent, and we henceforth refer to them as a
single point, M̄. This is also experimentally verified in Figure 3.11(b). The surface
state Fermi surface of Bi0.9 Sb0.1 (111) was obtained by collecting ARPES intensity in
a narrow energy window about EF as a function of kx and ky , which is displayed in
Figure 3.11(a). It shows a hexagonal Fermi surface enclosing Γ, a petal shaped Fermi
surface along the Γ-M̄ line that does not enclose any TRIM, and a dumbbell shaped
Fermi surface that encloses M̄. Due to the narrowness of the Fermi surface near M̄,
it is necessary to study its band dispersion below EF to deduce the topology.
The surface state band dispersion along a path connecting Γ̄ to M̄ is shown in
Figure 3.11(c). Like in pure bismuth, two surface bands emerge from the bulk band
continuum near Γ̄ to form a central electron pocket enclosing Γ̄ and an adjacent
hole lobe [56, 57, 58]. It has been established that these two bands result from the
spin-splitting of a Kramers pair and are thus singly spin degenerate [59, 58]. On
the other hand, the surface band that crosses EF at −kx ≈ 0.5 Å−1 , and forms
the narrow electron pocket around M̄, is clearly doubly degenerate, as far as we can
determine within our experimental resolution. This is indicated by its splitting below
EF between −kx ≈ 0.55 Å−1 and M̄, as well as the fact that this splitting goes
to zero at M̄ in accordance with Kramers theorem. In semimetallic single crystal
bismuth, only a single surface band is observed to form the electron pocket around M̄
[60, 61]. Moreover, this surface state overlaps, hence becomes degenerate with, the
bulk conduction band at L owing to the semimetallic character of Bi. In Bi0.9 Sb0.1
on the other hand, the states near M̄ fall completely inside the bulk energy gap
3.3. Experimental results
49
(a)
x = 0.1
k y (Å-1)
0.06
(b)
2
4,5
0.00
1
M
-0.06
0.6
(c)
0.8
1.0
EB(eV)
0.0
-0.1
-0.2
0.04
1.0
0.8
2
EB(eV)
0.0
-0.04
-0.08
0.6
-0.2
M
-0.1
0.0
Intensity (arb. units)
0.6
(e)
(d)
Intensity (arb. units)
1
1.0
0.8
-1
- k x(Å )
-0.08
(f)
-0.04
0.0
EB(eV)
Figure 3.12: (a) ARPES intensity integrated within ±10 meV of EF originating
solely from the surface state crossings. The image was plotted by stacking along the
negative kx direction a series of scans taken parallel to the ky direction. (b) Outline of
Bi0.9 Sb0.1 surface state ARPES intensity near EF measured in (a). White lines show
scan directions “1” and “2”. (c) Surface band dispersion along direction “1” taken
with hν = 28 eV and the corresponding EDCs (d). The surface Kramers degenerate
point, critical in determining the topological Z2 class of a band insulator, is clearly
seen at M̄, approximately 15 ± 5 meV below EF . (We note that the scans are taken
along the negative kx direction, away from the bulk L point.) (e), Surface band
dispersion along direction “2” taken with hν = 28 eV and the corresponding EDCs
(f). This scan no longer passes through the M̄-point, and the observation of two well
separated bands indicates the absence of Kramers degeneracy as expected.
3.3. Experimental results
50
preserving their purely surface character at M̄. The surface Kramers doublet point
can thus be defined in the bulk insulator (unlike in Bi [59, 56, 57, 58, 60, 61]) and is
experimentally located in Bi0.9 Sb0.1 samples to lie approximately 15 ± 5 meV below
EF at k = M̄ (Figure 3.11(c)). These results show that the surface band crosses
EF five times between Γ̄ and M̄, which is consistent with a ν0 = 1 topology where
π(Γ̄)π(M̄) = −1. The dumbbell shaped Fermi surface segment enclosing M̄ must
therefore be regarded as two Fermi surfaces, which leads to an overall Fermi surface
(Figure 3.11(e)) that encloses the four surface TRIM a total of 7 times.
The imaging of the Kramers point at M̄ is an important demonstration that our
alignment is strictly along the Γ̄-M̄ line. To see the effects of slight misalignment, we
contrast high resolution ARPES measurements taken along the Γ̄-M̄ line with those
that are slightly offset from it. Figure 3.12 shows that with ky offset from the Γ̄-M̄
line by less than 0.02 Å−1 , the Kramers degeneracy is lifted and the top branch of
the Kramers pair crosses EF to form part of the bow-shaped electron distribution
(Figure 3.12(a) and (b)).
In order to confirm that the surface states of insulating Bi1−x Sbx belong to the
ν0 = 1 class, the spin-polarization of the surface bands must be measured. Below we
present results of the spin-resolved ARPES analysis on bulk insulating Bi0.91 Sb0.09 .
For this experiment, we used incident photons in the VUV regime and analyzed
the photoelectron spins using a single Mott polarimeter. By using VUV photons,
spin conserving photoemission processes (where the electric field of light only acts
on the orbital degree of freedom of the electron inside a solid) dominate over spin
non-conserving processes (which arise from coupling to the magnetic field of light)
[38]. Therefore the spin polarization of a photoemitted electron is representative of
its spin polarization inside the crystal.
3.3. Experimental results
(e)
(c)
Bi 0.91 Sb 0.09
l3
l2
l1
r1
EB(eV)
0.0
0.0
-0.1
-0.1
-0.2
-0.2
0.1
0.0
Pz’
Py’
-0.1
-M
-0.2
-0.4
-0.6
Intensity (arb. units)
l4,5
Polarization
0.1
0.1
-0.6
G
-0.4
k x (Å )
Intensity (arb. units)
5
4
r2
r1
l1
l2
l3
l1
l2
r1
l3
l4,5
2
1
-0.6
-0.4
0.0
-0.2
k x (Å-1 )
(f)
0.5% Te doped BiSb
1
1
-M
l4,5
M
G
l3 l2
l1
0.0
0.0
I tot
Py 0
0 Pz
2
-1
-1
1
-1
0
-0.4
3
0.1
0.1
EB = -25 meV
3
-0.6
Spin up
Spin down
4
0.0
(d)
6
5
k x (Å-1 )
-1
(b)
-0.2
-0.2
0.0
k x (Å-1 )
0
Px
1 0
1
Pin plane
EB(eV)
(a)
51
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
-0.4
-0.2
0.0
0.2
0.4
k x (Å-1 )
Figure 3.13: (a) The surface band dispersion ARPES second derivative image (SDI)
along the Γ̄ to -M̄ direction of bulk insulating Bi0.91 Sb0.09 . Dashed white lines are
guides to the eye. The intensity of bands l4, 5 is scaled up for clarity. (b) MDC of the
spin averaged spectrum at EB = -25 meV [green line below EF in (a)] using a photon
energy hν = 22 eV, together with the Lorentzian peaks of the fit. (c) Measured spin
polarization curves (symbols) for the y 0 and z 0 (Mott coordinate) components together
with the fitted lines. (d) The in-plane and out-of-plane spin polarization components
in the sample coordinate frame obtained from the spin polarization fit. The symbols
refer to those in (b). The fitted parameters are consistent with 100% polarized spins.
(e) Spin resolved spectra for the y component based on the fitted spin polarization
curves shown in (c). Spin up (down) refers to the spin direction being approximately
parallel to the +(-)ŷ direction. (f) The surface band dispersion SDI centered about
Γ̄ of (Bi0.925 Sb0.075 )0.995 Te0.005 . Electron doping through Te reveals that bands l2 and
l3 are connected above EF .
3.3. Experimental results
52
Figure 3.13(b) shows a spin-averaged momentum distribution curve (MDC) along
the Γ̄ to -M̄ direction taken at EB = -25 meV, which intersects the surface bands
at the k positions shown in Figure 3.13(a). This MDC was obtained by averaging the signal from the four electron detectors surrounding the Mott polarimeter.
A sum of Lorentzian lineshapes I i and a non-polarized background B are fitted to
this MDC, which are used as inputs to the two-step fitting routine developed by
Meier et al. [42] as described in Chapter 2. Because a single Mott polarimeter is
being used, only two spin components are measured. Therefore an additional constraint that the electron spins are fully polarized with magnitude h̄/2 (i.e. |P| = 1)
had to be imposed in the fit. It is known that spin polarization of surface bands
can become less than 1 as a band approaches the bulk band continuum [62]. Because the surface band intersections with the EB = -25 meV line (Figure 3.13(a))
are located far from the bulk continuum, this imposed constraint is realistic. Moreover, common strong spin-orbit coupled materials such as gold and bismuth based
surface alloys have been experimentally shown to exhibit 100% spin polarized surface states [63, 42]. To review the fitting procedure briefly, a spin polarization veci
tor P~M
= (Pxi0 , Pyi0 , Pzi0 ) = (cos ϑi cos ϕi , cos ϑi sin ϕi , sin ϑi ) is assigned to each band,
where ϑi and ϕi are referenced to the primed Mott coordinate frame. A spin-resolved
spectrum is then defined for each peak i using the relation Iαi;↑,↓ = I i (1 ± Pαi )/6,
where α = x0 , y 0 , z 0 , and + and − correspond to the spin direction being parallel (↑) or antiparallel (↓) to α. The full spin-resolved spectrum is then given by
P
Iα↑,↓ = i Iαi;↑,↓ + B/6, from which the spin polarization of each spatial component
can be obtained as Pα = (Iα↑ − Iα↓ )/(Iα↑ + Iα↓ ). This latter expression is a function
of ϑi and ϕi , which are the parameters that are varied to fit the experimental spin
polarization data.
3.3. Experimental results
53
(a)
EB
(b)
EB
spin down
La
EF
G
H
kx
La
EF
Ls
spin up
T
spin down
L
M
Ls
spin up
T
G
H
kx
L
M
Bulk conduction band
Bulk valence band
Figure 3.14: Two possible band connection scenarios in Bi1−x Sbx that are generated
from (a) first principles calculations and (b) tight-binding calculations.
The spin polarization data for the y 0 and z 0 components (i.e. Py0 and Pz0 ) are obtained by taking the difference between the intensities of the left-right (or top-bottom)
electron detectors over their sum, normalized by the Sherman function. These data
are shown as symbols in Figure 3.13(c), and the solid lines are the fits based on the fitting procedure described above. The best fit parameters (Pxi0 , Pyi0 , Pzi0 ) are transformed
into the sample coordinate frame and are displayed in Figure 3.13(d). Although the
measured polarization curves only reaches a magnitude of around ±0.1, which is similarly seen in studies of Bi thin films [59], this is not a true measure of the actual spin
polarization but is rather an artifact of the non-polarized background and overlap of
adjacent peaks with different spin polarization. A true measure of the spin polarization can only be extracted through fitting, which yield polarization values consistent
with 1 (Figure 3.13(d)).
3.3. Experimental results
54
The fitted spin polarization vectors of the surface bands suggest that all spins
are nearly aligned parallel to the in-plane ±ŷ direction along the Γ̄-M̄ cut, and Figure 3.13(e) shows the spin-resolved spectrum calculated from Py . This suggests that
the electric field in the Rashba spin-orbit Hamiltonian (p × E) · σ points predominantly normal to the sample surface. The fact that the fitted spin polarizations are
not strictly along the ±ŷ direction is likely due to slight misalignments of the scan
away from the Γ̄-M̄ cut, which is difficult to eliminate given the very narrow Fermi
surface features. Bands l1 and r1 having nearly opposite spin because time-reversal
symmetry demands spins directions to be reversed on either side of Γ̄. Bands l1 and
l2 also have opposite spin, which shows that they form a Kramers pair. These observations confirm that the surface states of bulk insulating Bi1−x Sbx are representative
of the ν0 = 1 class. Because of a dramatic intrinsic weakening of signal intensity near
crossings l4 and l5, and the small energy and momentum splittings of bands l4 and
l5 lying at the resolution limit of modern spin-resolved ARPES spectrometers, we are
unable to extract the spin polarizations of these two bands. However, whether bands
l4 and l5 are both singly or doubly spin degenerate does not change the fact that an
odd number of spin-polarized Fermi surfaces enclose the surface TRIM.
Recent theoretical work [64] has uncovered a discrepancy between Bi(111) surface
state calculations using the tight-binding parameters given by Liu and Allen [47] and
using first principles methods [65]. In particular, they disagree as to whether bands l1
and l2 cross each other above the Fermi level as shown in Figure 3.14. Our observation
that bands l2 and l3 have the same spin suggests that they originate from the same
band, and thus the band dispersion above the Fermi level should follow that shown
in Figure 3.14(a). A more direct way to show that bands l2 and l3 connect above
EF is to map their energy dispersion above EF . While inverse photoemission [33]
can be used for this purpose, these results can often be difficult to interpret. Instead,
3.3. Experimental results
55
Te-doped BiSb
0.1
(a)
0.1
E B (eV)
Deg
-1
k y (Å )
0.2
0.2
0.0
0.0
1
-0.2
-0.2
-0.2
(b)
0.0
3
2
0.2
0.4
0.6
0.8
1.0
-0.1
-0.1
-0.1
-0.2
-0.2
-0.1
E B (eV)
-1
Deg
-0.2
0.0
0.2
0.1
undoped BiSb
-0.2
0.0
0.2
0.4
0.6
0.8
0.2
1.0
1.2
(g)
-1
Sn-doped BiSb
Deg
-0.2
-0.2
-0.2
0.0
0.0
0.2
0.2
0.4
0.4
0.6
0.6
-1
k x (Å )
0.8
0.8
1.0
1.0
1.2
1.2
(h)
-0.1
-0.1
-0.1
-0.1
-0.2
-0.2
-0.2
-0.1
0.0
0.1
-0.1
0.0
0.1
(j)
-0.1
(k)
0.0
0.0
0.0
-0.1
-0.1
-0.1
-0.1
-0.2
-0.2
-0.1
0.0
0.1
0.0
0.1
0.0
0.1
(l)
0.0
-0.2
0.2
0.1
0.1
3
0.0
(i)
0.0
M
E B (eV)
k y (Å )
-0.2
-0.2
0.1
0.0
K
0.0
0.2
0.0
0.1
0.0
0.0
0.0
0.2
G
-0.2
0.1
2
-0.2
(c)
(f)
0.0
0.1
-0.2
-0.2
(e)
0.0
1.2
0.0
0.0
Sn-doped BiSb
0.1
0.0
-0.2
0.2
0.2
k y (Å )
1
(d)
0.0
-0.2
Te-doped BiSb
undoped BiSb
0.1
-0.1
0.0
0.1
-0.1
-1
k x (Å )
Figure 3.15: The (111) surface state Fermi surfaces of (a) 1% Te doped, (b) undoped
and (c) 1% Sn doped Bi1−x Sbx . (d) to (l) show ARPES dispersion maps along the
constant kx cut directions marked by the white arrows in (a) for all three samples.
3.3. Experimental results
56
0.1
(a) Sn-doped BiSb
0.1
E B (eV)
0.0
0.0
-0.1
-0.1
-0.2
-0.2
-0.3
E B (eV)
0.10.0
(b) undoped BiSb
0.0
-0.1
-0.1
-0.2
-0.2
0.1
(c) Te-doped BiSb
E B (eV)
0.10.0
-0.1
0.0
-0.2
-0.1
-0.3
-0.2
0.0
-1
k x (Å )
0.8
Figure 3.16: The ARPES intensity map along the Γ̄-M̄ cut of (a) 1% Te doped, (b)
undoped and (c) 1% Sn doped Bi1−x Sbx together with their corresponding energy
distribution curves.
we mapped the surface band dispersion of Te doped Bi1−x Sbx , where Te acts as an
electron donor [66]. Figure 3.13(f) shows that the hole band formed by crossings l2
and l3 in insulating Bi1−x Sbx has sunk completely below EF with 0.5% Te doping,
and are in fact part of the same band.
More generally, we show that the doping level of the surface Fermi surfaces of
Bi1−x Sbx can be controlled by either hole doping the bulk with Sn or electron doping
it with Te. Doping also changes the Fermi level in the sample bulk causing them
to become metallic (Figure 3.5). Figures 3.15(a)-(c) show the (111) surface state
Fermi surfaces of a 1% Te doped, undoped and 1% Sn doped sample of Bi0.93 Sb0.07 .
The effect of Te doping is clearly to enlarge the sizes of the central hexagonal Fermi
surface and the dumbbell shaped Fermi surface around M̄, while shrinking the size
3.3. Experimental results
57
of the petal shaped Fermi surfaces. Because the former two Fermi surface segments
are electron like while the latter is hole like, the effect of Te doping is to raise the
chemical potential of the surface states. The effect of Sn doping on the other hand is
just the opposite. Figure 3.16 shows that in a range of 1% hole or electron doping,
the surface state dispersion can be tuned from having the electron like bands around
M̄ completely pulled above EF to having the hole like bands pushed completely below
EF respectively. This study shows that small amounts of bulk doping can be used to
finely control the carrier density of the topological surface metals.
3.3.2
Topological surface states in metallic Sb
The ν0 = 1 topology of bulk insulating Bi1−x Sbx is predicted to be inherited from the
bulk wavefunctions of pure Sb (Figure 3.4(d)). Because there is a finite direct energy
gap between the bulk valence and conduction bands at each k-point in Sb, ν0 is well
defined for Sb and its surface states should therefore also exhibit a non-trivial energy
dispersion that is adiabatically connected to that of Bi1−x Sbx . In this section, we will
apply the previous experimental procedures to study metals such Sb.
The low lying electronic states of single crystal Sb(111) were studied first using
incident photon energy modulated ARPES, which we employ to isolate the surface
from bulk-like electronic bands over the entire BZ. Figure 3.17(b) shows momentum
distribution curves of electrons emitted at EF as a function of kx (k Γ̄-M̄) for Sb(111).
The out-of-plane component of the momentum kz was calculated for different incident photon energies (hν) using the free electron final state approximation with an
experimentally determined inner potential of 14.5 eV [68, 69]. There are four peaks
in the MDCs centered about Γ̄ that show no dispersion along kz and have narrow
widths of ∆kx ≈ 0.03 Å−1 . These are attributed to surface states and are similar
to those that appear in Sb(111) thin films [68]. As hν is increased beyond 20 eV, a
3.3. Experimental results
58
kz
(a)
M
K
ky
kx
EB = 0 eV
X
L
G
hn = 26 eV
L
H
U
U
X
T
Intensity (arb. units)
(c)
(b)
(111)
G
3.2
L
k z (Å-1 )
3.0
hn = 26 eV
H
2.8
18 eV
2.6
U
14 eV
-1.0 -0.8 -0.6 -0.4 -0.2
-1
k x (Å )
T
0.0
hn = 14 eV
0.2
-0.4
-0.2
0.0
0.2
0.4
-1
k x (Å )
Figure 3.17: (a) Schematic of the bulk BZ of Sb and its (111) surface BZ. The shaded
region denotes the momentum plane in which the following ARPES spectra were
measured. (b) Select MDCs at EF taken with photon energies from 14 eV to 26 eV
in steps of 2 eV, taken in the T XLU momentum plane. Peak positions in the MDCs
were determined by fitting to Lorentzians (green curves). (d) Experimental 3D bulk
Fermi surface near H (red circles) and 2D surface Fermi surface near Γ̄ (open circles)
projected onto the kx -kz plane, constructed from the peak positions found in (c). The
kz values are determined using calculated constant hν contours (black curves). The
shaded gray region is the theoretical hole Fermi surface calculated in [67].
3.3. Experimental results
(a) hn = 24 eV
59
0 Intensity (arb. units) 20
0.0
0.0
0.0
-0.1
-0.1
-0.2
-0.2
-0.2
-0.3
-0.3
-0.4
-0.4
-0.4
-0.5
-0.5
EB(eV)
(b) hn = 20 eV
0.0
0.0
-0.1
-0.1
-0.2
-0.2
-0.2
-0.3
-0.3
-0.4
-0.4
-0.4
-0.5
-0.5
(c) hn = 18 eV
0.0
-0.1
-0.1
-0.2
-0.2
-0.2
-0.3
-0.3
-0.4
-0.4
-0.4
-0.5
-0.5
0.4
-1
k x (Å )
1
(f) hn = 18 eV
0.0
0.0
0.0
0
(e) hn = 20 eV
0.0
-0.4
(d) hn = 24 eV
G
1.0
0.4
M
-1
k x (Å )
Figure 3.18: ARPES intensity maps of Sb(111) as a function of kx near Γ̄ (a)-(c)
and M̄ (d)-(f) and their corresponding energy distribution curves, taken using hν =
24 eV, 20 eV and 18 eV photons. The intensity scale of (d)-(f) is a factor of about
twenty smaller than that of (a)-(c) due to the intrinsic weakness of the ARPES signal
near M̄.
broad peak appears at kx ≈ -0.2 Å−1 , outside the k range of the surface states near
Γ̄, and eventually splits into two peaks. Such a strong kz dispersion, together with
a broadened linewidth (∆kx ≈ 0.12 Å−1 ), is indicative of bulk band behavior, and
indeed these MDC peaks trace out a Fermi surface (Figure 3.17(c)) that is similar in
shape to the hole pocket calculated for bulk Sb near H [67]. Therefore by choosing
an appropriate photon energy (e.g. ≤ 20 eV), the ARPES spectrum at EF along Γ̄-M̄
will have contributions from only the surface states. The small bulk electron pocket
centered at L is not accessed using the photon energy range we employed.
3.3. Experimental results
60
Having distinguished the bulk from surface contributions to the ARPES signal
at EF , we proceed to map their band dispersions below EF . ARPES spectra along
Γ̄-M̄ taken at three different photon energies are shown in Figure 3.18. Near Γ̄ there
are two rather linearly dispersive electron like bands that meet exactly at Γ̄ at a
binding energy EB ∼ -0.2 eV. This behavior is consistent with a pair of spin-split
surface bands that become degenerate at the TRIM Γ̄ due to Kramers degeneracy.
The surface origin of this pair of bands is established by their lack of dependence
on hν (Figure 3.18(a)-(c)). A strongly photon energy dispersive hole like band is
clearly seen on the negative kx side of the surface Kramers pair, which crosses EF for
hν = 24 eV and gives rise to the bulk hole Fermi surface near H (Figure 3.17(c)).
For hν ≤ 20 eV, this band shows clear back folding near EB ≈ -0.2 eV indicating
that it has completely sunk below EF . Interestingly, at photon energies such as 18
eV where the bulk bands are far below EF , there remains a uniform envelope of
weak spectral intensity near EF in the shape of the bulk hole pocket seen with hν =
24 eV photons, which is symmetric about Γ̄. This envelope does not change shape
with hν suggesting that it is of surface origin. Due to its weak intensity relative to
states at higher binding energy, these features cannot be easily seen in the energy
distribution curves in Figure 3.18(a)-(c), but can be clearly observed in the MDCs
shown in Figure 3.17(b) especially on the positive kx side. Centered about the M̄
point, we also observe a crescent shaped envelope of weak intensity that does not
disperse with kz (Figure 3.18(d)-(f)), pointing to its surface origin. Unlike the sharp
surface states near Γ̄, the peaks in the EDCs of the feature near M̄ are much broader
(∆E ∼80 meV) than the spectrometer resolution (15 meV). The origin of this diffuse
ARPES signal is not due to surface structural disorder because if that were the case,
electrons at Γ̄ should be even more severely scattered from defects than those at M̄.
In fact, the occurrence of both sharp and diffuse surface states originates from a k
3.3. Experimental results
61
(a)
EB
n 0 = 1 topology (Sb)
K
La
EF
Ls
H
T
G
(b)
EB
kx
M
G
L
M
n0 = 0 topology (Au-like)
K
EF
Ls
La
H
T
G
kx
M
G
L
M
Figure 3.19: Schematic of the bulk band structure (shaded areas) and surface band
structure (red and blue lines) of Sb near EF for a (a) topologically non-trivial and (b)
topological trivial (gold-like) case, together with their corresponding surface Fermi
surfaces are shown.
dependent coupling to the bulk. As will be discussed in Figure 3.20(b), the spin-split
Kramers pair near Γ̄ lie completely within the gap of the projected bulk bands near
EF attesting to their purely surface character. In contrast, the weak diffuse hole like
band centered near kx = 0.3 Å−1 and electron like band centered near kx = 0.8 Å−1
lie completely within the projected bulk valence and conduction bands respectively.
Therefore they are hybrid states of the bulk and surface states, dubbed “resonance
states” [33], and their ARPES spectra exhibit the expected lifetime broadening due
to an increase in elastic decay channels into the underlying bulk continuum [70].
The topological properties of Sb depend on whether the surface Kramers pair observed near Γ̄ switch partners between Γ̄ and M̄, which can be deduced from the shape
3.3. Experimental results
62
of the surface Fermi surface. In topologically trivial spin-orbit metals such as gold, a
free-electron like surface state is split into two parabolic spin-polarized sub-bands that
are shifted in k-space relative to each other [63], which do not switch partners between
TRIM (Figure 3.19(b)). As a result, two concentric spin-polarized Fermi surfaces are
created, one having an opposite sense of in-plane spin rotation from the other, that
enclose Γ̄. Such a Fermi surface, like the schematic shown in Figure 3.19(b), does not
support a non-zero Berry’s phase because the TRIM are enclosed an even number
of times and their geometrical phases cancel. In a topologically non-trivial metal on
the other hand, a partner switching behavior must occur through the spin up and
spin down bands emerging from Γ̄ separately merging into the bulk valence and conduction bands respectively between Γ̄ and M̄ (Figure 3.19(a)). This way, the surface
bands that formed the outer ring-shaped Fermi surface in gold now instead form petal
shaped Fermi surfaces that do not enclose Γ̄, leaving behind a single Fermi surface
enclosing Γ̄.
Figure 3.20 shows a spin-integrated ARPES intensity spectrum of Sb(111) from
Γ̄ to M̄. The previosuly discussed systematic incident photon energy dependence
study of such spectra, previously unavailable with helium lamp sources [71], made it
possible to identify two V-shaped surface states centered at Γ̄, a bulk state located
near kx = −0.25 Å−1 and resonance states centered about kx = 0.25 Å−1 and M̄.
An examination of the ARPES intensity map of the Sb(111) surface and resonance
states at EF (Figure 3.20(c)) reveals that the central surface FS enclosing Γ̄ is formed
by the inner V-shaped SS only. The outer V-shaped SS on the other hand forms
part of a tear-drop shaped FS that does not enclose Γ̄, unlike the case in gold. This
tear-drop shaped FS is formed partly by the outer V-shaped SS and partly by the
hole-like resonance state. The electron-like resonance state FS enclosing M̄ does not
affect the determination of ν0 because it must be doubly spin degenerate as will be
3.3. Experimental results
63
shown later. Such a FS geometry (Figure 3.20(e)) suggests that the V-shaped SS pair
may undergo a partner switching behavior expected in Figure 3.19(a). This behavior
is most clearly seen in a cut taken along the Γ̄-K̄ direction since the top of the bulk
valence band is well below EF (Figure 3.20(d)) showing only the inner V-shaped
SS crossing EF while the outer V-shaped SS bends back towards the bulk valence
band near kx = 0.1 Å−1 before reaching EF . The additional support for this partner
switching band dispersion behavior comes from tight binding surface calculations on
Sb (Figure 3.20(b)), which closely match with experimental data below EF . Our
observation of a single surface band forming a FS enclosing Γ̄ suggests that pure Sb
is likely described by ν0 = 1, and that its surface may support a π Berry’s phase.
Detailed ARPES intensity maps of Sb(111) along the -K̄−Γ̄−K̄ direction are
shown in Figure 3.21(c), which shows that the inner V-shaped band that was observed along the -M̄−Γ̄−M̄ direction retains its V-shape along the -K̄−Γ̄−K̄ direction
and continues to cross the Fermi level, which is expected since it forms the central
hexagonal Fermi surface. On the other hand, the outer V-shaped band that was
observed along the -M̄−Γ̄−M̄ direction no longer crosses the Fermi level along the
-K̄−Γ̄−K̄ direction, instead folding back below the Fermi level around ky = 0.1 Å−1
and merging with the bulk valence band (shaded regions in Figure 3.21(c)). This
confirms that it is the Σ1(2) band starting from Γ̄ that connects to the bulk valence
(conduction) band, in agreement with the tight binding calculations.
Here we give a detailed explanation of why the surface Fermi contours of Sb(111)
that overlap with the projected bulk Fermi surfaces can be neglected when determining the ν0 class of the material. Although the Fermi surface formed by the surface
resonance near M̄ encloses M̄, we will show that this Fermi surface will only contribute
an even number of enclosures and thus not alter the overall evenness or oddness of
TRIM enclosures. Consider some time reversal symmetric perturbation that lifts the
3.3. Experimental results
(a)
SS
M
(b)
e RS
0.4
0.4
0.4
EB(eV)
EB(eV)
h RS
{
BS
{
G
0.0
0.0
high
low
Sb(111)
0.2
0.2
64
-0.2
-0.2
0.000
EF
S1
-0.2
-0.2
n M = -1
-0.4
-0.4
-0.4
-0.6
-0.6
-0.5
0.0
0.5
-1
1.0
kx (Å )
(c)
G
-1
M
kx (Å )
(d) -K ¬ G ® K
(e)
M
0.0
0.0
0.3
0.2
0.2
EB(eV)
-1
S2
0.2
0.2
-0.4
-0.4
Deg
ky (Å
)
Band structure calculation
e RS
-0.1
-0.1
0.1
0.0
0.0
-0.1
-0.2
-0.2
-0.3
-0.5
-0.5
0.0
0.0
0.5
0.5
-1
kx (Å )
-0.2
-0.2
SS
-0.3
-0.3
G
h RS
M
1.0
1.0
-0.2 -0.1 0.0 0.1 0.2
k y (Å-1)
Sb (111) topology
Figure 3.20: (a) Spin-integrated ARPES spectrum of Sb(111) along the Γ̄-M̄ direction.
The surface states are denoted by SS, bulk states by BS, and the hole-like resonance
states and electron-like resonance states by h RS and e− RS respectively. (b) Calculated surface state band structure of Sb(111) based on the methods in [47, 64].
The continuum bulk energy bands are represented with pink shaded regions, and the
lines show the discrete bands of a 100 layer slab. The red and blue single bands,
denoted Σ1 and Σ2 , are the surface states bands with spin polarization hP~ i ∝ +ŷ
and hP~ i ∝ −ŷ respectively. (c) ARPES intensity map of Sb(111) at EF in the kx -ky
plane. (d) ARPES spectrum of Sb(111) along the Γ̄-K̄ direction shows that the outer
V-shaped SS band merges with the bulk band. (e) Schematic of the surface FS of
Sb(111) showing the pockets formed by the surface states (unfilled) and the resonant
states (blue and purple).
3.3. Experimental results
65
bulk conduction La band completely above EF so that there is a direct excitation gap
at L. Since this perturbation preserves the energy ordering of the La and Ls states,
it does not change the ν0 class. At the same time, the weakly surface bound electrons
at M̄ can evolve in one of two ways. In one case, this surface band can also be pushed
up in energy by the perturbation such that it remains completely inside the projected
bulk conduction band (Figure 3.21(a)). In this case there is no more density of states
at EF around M̄. Alternatively the surface band can remain below EF so as to form a
pure surface state residing in the projected bulk gap. However by Kramers theorem,
this SS must be doubly spin degenerate at M̄ and its FS must therefore enclose M̄
twice (Figure 3.21(b)). In determining ν0 for semi-metallic Sb(111), one can therefore neglect all segments of the FS that lie within the projected areas of the bulk FS
(Figure 3.20(e)) because they can only contribute an even number of FS enclosures,
which does not change the modulo 2 sum of TRIM enclosures.
As was the case in Bi1−x Sbx , confirmation of a surface π Berry’s phase in Sb rests
critically on a measurement of the relative spin orientations (up or down) of the SS
bands near Γ̄. Spin detection of the photoelectrons was again measured using a single
Mott polarimeter that measures two orthogonal spin components, which are along
the y 0 and z 0 directions of the Mott coordinate frame and lie predominantly in and
out of the sample (111) plane respectively (Figure 3.22(a)). Spin-resolved momentum
distribution curve data sets of the SS bands along the −M̄-Γ̄-M̄ cut at EB = −30 meV
(Figure 3.22(b)) are shown for maximal intensity. Figure 3.22(d) displays both y 0 and
z 0 polarization components along this cut, showing clear evidence that the bands are
spin polarized, with spins pointing largely in the (111) plane. In order to extract the
3D spin polarization vectors from a two component measurement, we carried out the
two-step fitting routine [42] with the spin polarization vectors again constrained to
have length one. Our fitted polarization vectors are displayed in the sample (x, y, z)
3.3. Experimental results
(a)
66
(c)
Scenario 1
Sb
La
EF
Ls
Sb (111)
0.0
0.0
-0.1
-0.1
M
G
(c)
-K ¬ G ® K
-0.2
-0.2
Scenario 2
La
-0.3
-0.3
EF
Ls
G
M
-0.2 -0.1
0.0
0.1
0.2
-1
k y (Å )
Figure 3.21: (a) Schematic of the surface band structure of Sb(111) under a time
reversal symmetric perturbation that lifts the bulk conduction (La ) band above the
Fermi level (EF ). Here the surface bands near M̄ are also lifted completed above EF .
(b) Alternatively the surface band near M̄ can remain below EF in which case it must
be doubly spin degenerate at M̄. (c) ARPES intensity plot of the surface states along
the -K̄−Γ̄−K̄ direction. The shaded green regions denote the theoretical projection
of the bulk valence bands, calculated using the full potential linearized augmented
plane wave method using the local density approximation including the spin-orbit
interaction (method described in [65]). Along this direction, it is clear that the outer
V-shaped surface band that was observed along the -M̄−Γ̄−M̄ now merges with the
bulk valence band.
3.3. Experimental results
67
(a)
40 kV
e beam
accelerating optics
q
sample
(b)
E B (eV)
0.0
l2
z’
z
y’
x’
x
y
¬ -M
Au foil
G
l1
r2
spin || y
spin ||- y
S2
-0.1
S1
l1
-0.2
-0.2
-0.1
0.0
10
0.1
G
r1
0.2
-1
k x (Å )
4
Momentum distribution of spin
(d)
0.4
0.2
0.0
-0.2
Py ’
Pz ’
-0.4
0
-0.1
0.0
0.2
0.1
0.3
k x (Å-1)
(e)
M®
r1
r2
r1
l1
l2
-0.2
Intensity (arb. units)
e
20
EB = -30 meV
Spin polarization
Mott spin detector
hn
Intensity (arb. units)
(c)
-0.2
0.0
0.1
(f)
l2
r1
l1
r2
1
1
2
Py 0
0 Pz
-1
-1
-1
0
-0.2
-0.1
0.0
0.1
0.2
0.2
k x (Å-1)
EB = -30 meV
Iy
Iy
-0.1
0
Px
1 0
1
Pin plane
-1
k x (Å )
Figure 3.22: (a) Experimental geometry of the spin-resolved ARPES study. (b)
Spin-integrated ARPES spectrum of Sb(111) along the −M̄-Γ̄-M̄ direction. The momentum splitting between the band minima is indicated by the black bar and is
approximately 0.03 Å−1 . A schematic of the spin chirality of the central FS based on
the spin-resolved ARPES results is shown on the right. (c) Momentum distribution
curve of the spin averaged spectrum at EB = −30 meV (shown in (b) by white line),
together with the Lorentzian peaks of the fit. (d) Measured spin polarization curves
(symbols) for the detector y 0 and z 0 components together with the fitted lines using
the two-step fitting routine [42]. (e) Spin-resolved spectra for the sample y component based on the fitted spin polarization curves shown in (d). Up (down) triangles
represent a spin direction along the +(-)ŷ direction. (f) The in-plane and out-of-plane
spin polarization components in the sample coordinate frame obtained from the spin
polarization fit. Overall spin-resolved data and the fact that the surface band that
forms the central electron pocket has hP~ i ∝ −ŷ along the +kx direction, as in (e),
suggest a left-handed chirality.
3.3. Experimental results
68
coordinate frame (Figure 3.22(f)), from which we derive the spin-resolved momentum
distribution curves for the spin components parallel (Iy↑ ) and anti-parallel (Iy↓ ) to
the y direction as shown in Figure 3.22(e). There is a clear difference in Iy↑ and Iy↓
at each of the four momentum distribution curve peaks indicating that the surface
state bands are spin polarized. Each of the pairs l2/l1 and r1/r2 have opposite spin,
consistent with the behavior of a spin split Kramers pair, and the spin polarization
of these bands are reversed on either side of Γ̄ in accordance with the system being
time reversal symmetric. This measured spin texture of the Sb(111) surface states
together with the connectivity of the surface bands (Figure 3.20), uniquely determines
its belonging to the ν0 = 1 class. Therefore pure Sb can be regarded as the parent
metal of the Bi1−x Sbx topological insulator class. In other words, the topological
order originates from the Sb wavefunctions as predicted by Fu and Kane [22].
In addition to a Z2 topological invariant ν0 , Teo, Fu and Kane predicted in 2008
that spin polarized ARPES measurements can uncover a new type of topological
quantum number nM that provides information about the spin chirality of the surface
Fermi surface [64]. Electronic states in the mirror plane (ky = 0) (Figure 3.23(a))
are eigenstates of the mirror operator M (ŷ) with eigenvalues ±i. M (ŷ) is closely
related to, but not exactly the same as the spin operator Sy . It may be written as
M (ŷ) = P C2 (ŷ): the product of the parity operator P : (x, y, z) → (−x, −y, −z) and
a twofold rotation operator C2 (ŷ): (x, y, z) → (−x, y, −z). For a free spin, P does
not affect the pseudovector spin, and C2 (ŷ) simply rotates the spin. Thus, M (ŷ) =
exp[−iπSy /h̄]. For spin eigenstates Sy = ±h̄/2, this gives M (ŷ) = ∓i. In a crystal
with spin-orbit interaction on the other hand, Sy is no longer a good quantum number,
but M (ŷ) still is.
The energy bands near the Fermi energy in Bi1−x Sbx are derived from states with
even orbital mirror symmetry and satisfy M (ŷ) ∝ −i sgn(hSy i). Unlike the bulk
3.3. Experimental results
69
(a)
(d)
kz
(111)
G
M
Insulating Bi Sb
bulk conduction band
E
K
ky
kx
spin up
L
spin down
EF
L
L
Mirror plane
bulk valence band
M
(b)
(c)
nM = 1
G
bulk conduction band
E
E
E
M
Pure Sb
(e)
n M = -1
G
kx
G
spin up
S1
spin down
S2
EF
EF
EF
S2
S1
bulk valence band
G
kx
G
kx
M
G
kx
M
Figure 3.23: Implications of k-space mirror symmetry on the surface spin states. (a)
3D bulk Brillouin zone and the mirror plane in reciprocal space. (b) Schematic spin
polarized surface state band structure for a mirror Chern number (nM ) of +1 and (c)
-1. Spin up and down mean parallel and anti-parallel to ŷ respectively. The upper
(lower) shaded gray region corresponds to the projected bulk conduction (valence)
band. The hexagons are schematic spin polarized surface Fermi surfaces for different
nM , with yellow lines denoting the mirror planes. (d) Schematic representation of
surface state band structure of insulating Bi1−x Sbx and (e) semi metallic Sb both
showing a nM = −1 topology. Yellow circles indicate where the spin down band
(bold) connects the bulk valence and conduction bands.
3.3. Experimental results
70
states which are doubly spin degenerate, the surface state spin degeneracy is lifted
due to the loss of crystal inversion symmetry at the surface, giving rise to the typical
Dirac like dispersion relations near time reversal invariant momenta (Figure 3.23(b)
and (c)). For surface states in the mirror plane ky = 0 with M (ŷ) = ±i, the spin
split dispersion near kx = 0 has the form E = ±h̄vkx . Assuming no other band
crossings occur, the sign of the velocity v is determined by the topological mirror
Chern number (nM ) describing the bulk band structure. When nM = 1, the situation
in Figure 3.23(b) is realized where it is the spin up (hSy i k ŷ) band that connects
the bulk valence to conduction band going in the positive kx direction (i.e. the spin
up band has a velocity in the positive x direction). For nM = −1 the opposite holds
true (Figure 3.23(c)). Because the central electron-like FS enclosing Γ̄ intersects six
mirror invariant points (Figure 3.23(b) and (c)), the sign of nM distinguishes two
distinct types of handedness for this spin polarized FS.
For both Bi1−x Sbx and Sb, a single surface band with a positive velocity at the
Dirac point, which switches partners at M̄, connects the bulk valence band to the
bulk conduction band, so |nM | = 1. From our spin-resolved ARPES data on both
insulating Bi1−x Sbx and pure Sb, this surface band has hP~ i ∝ −ŷ along the kx
direction, suggesting a left-handed rotation sense for the spins around this central FS
thus nM = −1. Therefore in both Bi1−x Sbx and Sb, the bulk electron wavefunctions
exhibit the anomalous value nM = −1 predicted in [64], which is not realizable in free
electron systems with full rotational symmetry.
There is an intimate physical connection between a 2D quantum spin Hall insulator
and the 2D k-space mirror plane of a 3D strong topological insulator. In the former
case, the occupied energy bands for each spin eigenvalue will be associated with an
ordinary Chern integer n↑,↓ , from which a non-zero spin-Chern number can be defined
ns = (n↑ − n↓ )/2. In the latter case, it is the mirror eigenvalue of the occupied energy
3.3. Experimental results
(a)
l = 1.28eV
l = 1.1eV
(b)
71
(d)
(e)
(f)
(g)
(c)
Figure 3.24: (a) The phase diagram of the Bi1−x Sbx system as a function of a dimerization parameter ∆d and a spin-orbit coupling parameter λ. (b) Surface Fermi surface
near M̄ for and λ = 1.1 eV (c) λ = 1.28 eV. (d),(e) Surface state band dispersion
along the Γ̄-M̄ direction for λ = 1.1 eV and (f),(g) for λ = 1.28 eV. All panels are
taken from [72].
bands that have associated with them Chern integers n+i,−i , from which a non-zero
mirror Chern number can be defined nM = (n+i − n−i )/2.
3.3.3
Evolution of surface state spectrum from Bi to Sb
Having established that the topological invariant ν0 of both insulating Bi1−x Sbx and
metallic Sb is equal to 1, we investigate the topological properties of pure bismuth,
which is predicted to be characterized by ν0 = 0 (Figure 3.4). The topologically
trivial (ν0 = 0) properties of bismuth were deduced from the relative energy ordering
of its La and Ls bands, which were calculated using tight-binding methods [47, 64],
first principles methods [65, 72] as well as more direct numerical methods based on
evaluating Chern invariants [73].
3.3. Experimental results
0.00.0
72
x=0
0.0
-0.1-0.1
E B (eV)
-0.2-0.2
0.0
0.0
-0.1
-0.2
0.0
x = 0.02
0.5
deg
1.0
0.0
-0.1
-0.1
0.5
1.0
0.0
0.5
1.0
0.5
0.5
1.0
1.0
-0.2
0.0
x = 0.04
0.5
deg
1.0
0.0
-0.1
-0.1
-0.2
-0.2
0.0
x = 0.075 deg
-0.1
-0.2
-0.2
0.00.0
x = 0.09
x = 0.065 deg
-0.1
-0.2
0.0
0.0
0.5
0.5
deg
1.0
1.0
G
M
0.0
0.0
-1
G
M
deg
k G M (Å )
Figure 3.25: Surface state band dispersion along the Γ̄-M̄ direction for x=0, x=0.02,
x=0.04, x=0.065, x=0.075 and x=0.09. Each panel includes energy distribution
curves (right) and the corresponding second derivative image plot (left). The energy
distribution curves from kx =0.25Å−1 to kx =1.25Å−1 are scaled up and are shown separately from the energy distribution curves ranging from kx =-0.15Å−1 to kx =0.25Å−1
in order to enhance the weak features near M̄.
In the first principles study of Zhang et al. [72], the phase diagram of the Bi1−x Sbx
system is parameterized by a dimerization parameter ∆d of two Bi layers and a spinorbit coupling parameter λ. It is seen in Figure 3.24(a) that at a fixed ∆d, the system
can cross-over from the ν0 = 1 to ν0 = 0 regime by increasing λ. In these two regimes,
the surface states near M̄ show a subtle but qualitative difference. In Sb, the dumbbell
shaped Fermi surface around M̄ does not enclose M̄ (Figure 3.24(b)) whereas in Bi
it does (Figure 3.24(c)). This is because in Sb, the surface bands between Γ̄ and
M̄ switch partners whereas in Bi they do not (Figure 3.24(d) and (e)). In Bi, the
spin-split band pair emerging from the valence band near Γ̄ reconnects again to the
valence band near M̄. This is consistent with the scenario where a band inversion
transition at L takes place near x ∼ 0.04.
Figure 3.25 shows the evolution of the surface band dispersion between Γ̄ and M̄
measured by ARPES. Like the dispersions shown in Figures 3.24(e) and (g), as x
3.3. Experimental results
73
x = 0.03
0.0
M ¬ G ® M
(b)
K ¬ G ® K
(a)
G ¬ M ® G
(c)
0.0
0.0
K ¬ M ® K
(d)
0.0
0.0
-0.1
-0.1
-0.2
-0.2
-0.1
-0.1
-0.1
EB(eV)
-0.2
-0.2
-0.2
x = 0.09
0.0
Low
(e)
High
(f)
0.0
0.0
0.6 0.8 1.0
(g)
-0.1
-0.1
-0.2
-0.2
-0.1
0.0
-0.1
-0.1
-0.1
-0.2
-0.2
-0.2
-0.1 0.0 0.1
-1
k (Å )
-0.1 0.0 0.1
k x (Å-1)
(h)
0.0
0.0
-0.2
0.6 0.8 1.0
k x (Å-1)
-0.1
0.0
0.1
-1
k (Å )
Figure 3.26: High resolution ARPES intensity maps of x=0.03 and x=0.09 along the
(a),(e) K̄-Γ̄-K̄, (b),(f) M̄-Γ̄-M̄, (c),(g) Γ̄-M̄-Γ̄ and (d),(h) Γ̄-K̄-Γ̄ directions. The cut
directions in the surface Brillouin zone are shown in the cartoons above.
is increased from zero, the band close to the Fermi level around M̄ goes from being
relatively flat to acquiring a w-shaped dispersion. At the same time, the Kramers
degeneracy near Γ̄ sinks lower in binding energy consistent with a lowering of the
hole-like band at T . However, exactly how the surface bands connect to the bulk
valence and conduction bands near M̄ could not be directly measured because the
surface bands become very weak in intensity near M̄ and any intensity coming from
the bulk conduction band overlaps the intensity of the surface state bands near M̄.
High resolution ARPES data near M̄ show that the appearance of the w-shaped
feature at higher x comes from a new band that emerges near EF , which was shown in
Figure 3.11 to be its Kramers pair. The presence of such a double band is confirmed
by the two-peak feature in the EDC (Figure 3.27(c)). On the other hand, a one-
3.3. Experimental results
74
peak feature is observed for lower x. By performing a series of such fits for pure
Bi and Bi0.9 Sb0.1 , we extract a dispersion relation between Γ̄ and M̄ that is shown
in Figure 3.27(a) and (b). By superposing the calculated positions of the projected
bulk bands, it is seen that a partner switching behavior is exhibited by both Bi
and Bi0.9 Sb0.1 , which would suggest that Bi0.9 Sb0.1 is also topologically non-trivial.
However, further studies of possible surface state band bending effects [1], which may
alter the energy positions of the surface states relative to the bulk states, are required
to clarify this issue. As of the writing of this thesis, there is no direct evidence for a
topological phase transition at x ∼ 0.04.
3.3. Experimental results
75
(a)
(c)
k x = 0.6 Å
x = 0.0
x = 0.0
Intensity (arb. units)
0.0
EB(eV)
-0.1
-0.2
(b)
x = 0.1
0.0
x = 0.1
-0.1
-0.2
-0.2
0.0
0.2
0.4
0.6
k x (Å-1)
0.8
1.0
1.2
-0.1
0.0
EB(eV)
Figure 3.27: The surface state band dispersion along Γ̄-M̄ obtained by fitting to the
high resolution ARPES data is shown for (a) x=0.03 and (b) x=0.09. The shaded
regions show the projected bulk band structure onto the (111) plane based on a rigid
shift of the tight-binding bands according to the model of Golin [74]. (c) Energy
distributions curves taken at a constant electron emission angle such that electrons
at the Fermi level have kx =0.6Å−1 and ky =0Å−1 are shown for x=0 and x=0.1. The
contrast between the single band and double band behavior is clearly seen.
Chapter 4
Conclusions
The work presented in this thesis constitutes the first experimental evidence of a
strong topological insulator (STI) phase of matter in nature that is realized in bulk
insulating Bi1−x Sbx alloys [75, 76, 77]. Recently, our work has been reproduced by
Nishide et al. [78]. More generally, this work demonstrates that a combination of spinand angle-resolved photoemission spectroscopy is a new tool with which to measure
Z2 topological invariants and identify new exciting topological phases of matter that
do not require a large external magnetic field. Large parts of this work can be found
in two publications [79, 80].
Following our discovery, there have been great advances in terms of new materials
candidates for STIs, as well as new theoretical proposals for novel macroscopic phenomena that are associated with the STI phase. On the materials front, an ARPES
study performed by our group in 2008 [81] showed that the surface states of Bi2 Se3 are
topologically non-trivial and are much simpler than those in Bi1−x Sbx because there
is only one as opposed to five Fermi level crossings. This coincided with theoretical
work by Zhang et al. [82] predicting that Bi2 Se3 , Bi2 Te3 and Sb2 Te3 are all STIs
that are realized without the need for external pressure. However, as-grown Bi2 Se3 is
slightly n-doped in the bulk, which means that a true single surface band STI is not
76
77
realized in this system. Rather, two recent spin-ARPES [83] and ARPES [84] works
have established that Bi2 Te3 may be a true single surface band STI with a large gap
insulating bulk.
On the theoretical front, the surface states of STIs have recently been shown to
give rise to many unusual phenomena including an image magnetic monopole field
[85], axion electrodynamics [86, 87, 88], topological exciton condensation [89] and
non-Abelian quasi-particle statistics when interfaced with an ordinary superconductor
[90]. Verification of these effects currently awaits experiments.
The general field of topological insulators promises to realize new physical phenomena that challenge our fundamental understanding of solid state materials, and
bring forth new physics advances that may be applied towards technologies ranging
from non-energy-dissipative devices to quantum computers. Already the search is
underway for new topological invariants realized in Mott insulators [91] and magnetic
insulators [92], as well as new ways to surface engineer such materials [93].
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