Download Photoemission Stuides of a New Topological Insulator Class: Experimental Discovery of the Bi 2 X 3 Topological Insulator Class

Photoemission studies of a new
topological insulator class:
experimental discovery of the Bi2X3
topological insulator class
YuQi Xia
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Physics
Adviser: Zahid Hasan
June 2010
c Copyright by YuQi Xia, 2010.
°
All Rights Reserved
Abstract
Topological insulators are materials with a bulk band gap, which carry conducting
surface states that are protected against disorder. In three dimensions, the insulators
carry 2D Dirac fermions on their surfaces. The opening of a magnetic surface gap can
exhibit a topological magnetoelectric effect, and support Majorana fermions which
can be manipulated for quantum computation. Previous spin and angle-resolved photoemission studies have shown that Bi1−x Sbx alloy belongs to this class of materials,
with a characteristic number ν0 = 1. Some materials challenges with Bi1−x Sbx alloy however are the significant degree of bulk disorder and a small band gap. Both
problems make gating difficult for the manipulation and control of the charge carriers.
While ordinary materials such as superconductors and liquid crystals can be described by an order parameter, topological insulators are not associated with a local
order parameter resulting from a spontaneous broken symmetry. Rather, they manifest a topological order which requires a direct probe of how their energy bands are
connected. Measurement techniques designed to detect a particular order parameter
are therefore insufficient to identify the topological character of a material. Alternatively, one can look for properties analogous to the quantum Hall effect as a signature
of a topologically ordered system. However, using transport probes to isolate the surface states of the topological insulator requires a pristine bulk with minimal charge
carrier density. While advances have been made recently in this direction, a good
candidate for such measurements has been elusive.
In this thesis, we describe a systematic study of a new topological insulator class
with a large band gap and a single surface state Fermi surface. Using synchrochonbased angle-resolved photoemission spectroscopy (ARPES), we measured the topological character of these materials by observing the dispersion of their metallic electronic states confined to the surface. Additionally, we confirmed the unusual spin
texture of these surface states using spin-sensitive ARPES. In Chapter 1, we first
iii
give a brief summary of the theoretical developments leading to the proposal of the
topological insulator. In Chapter 2, a description of the experimental techniques of
spin and angle-resolved PES is provided. Chapter 3 presents experimental data for
three members of the topological class: Bi2 Se3 ,Bi2 Te3 and Sb2 Te3 . In each of the
discussions, a comparison with the respective theoretical surface state calculations is
presented. Finally, in Chapter 4, we present several techniques for manipulating the
metallic surface states of the topological insulators.
iv
Acknowledgements
I would like to thank my labmate David Hsieh, who has been an invaluable source
of guidance and encouragement throughout my graduate career. It was he who first
introduced me to the idea of topological insulators, in the wee hours of an ALS
beamtime run in 2007. Our postdoc Dong Qian had taught me most of what I know
about performing experiments. I had tried hard in my own work to emulate his
uncanny intuition and poise that make a good experimentalist. I thank Lewis Wray
for his curiosity and discussions during many of our experiments together. I would
also like to thank my adviser Zahid Hasan, who has been generously supportive of
my academic interests, and taught me how to be a better writer.
Our experiments would not have gone smoothly without the assistance of all the
beamline scientists I have worked with - this includes Alexei Fedorov, and Sung Kwan
Mo of the ALS, and Donghui Lu and Rob Moore of the SSRL. They had been terrific
in helping us with problems, answering our phone calls at all hours of the day. I also
thank Hugo Dil and Fabian Meier, who had been wonderful hosts at the SLS where
we performed our spin measurements. Sebastian Janowski had also been extremely
helpful during our month-long stays at the SRC. For my Compton scattering work,
I had the opportunity to work with Itou-san and Sakurai-san of Spring-8. I thank
them for their patience and time to teach me about the technique.
In Princeton I had the opportunity to work with many wonderful scientists. Our
experiments would not have been possible without Professor Robert Cava and Yew
San Hor, who had provided us with all the samples used in this work. I had also
enjoyed the many insightful discussions with members of Professor Ali Yazdani’s
group and Professor Phuan Ong’s group, who had studied the same systems through
their own perspectives. I would also like to thank Professor David Huse, for his
guidance on my short simulation project, as well as our other discussions later on.
For additional theoretical collaboration I thank Hsin Lin of Northeastern University,
v
who had helped us with most of the surface band calculations in this work.
Finally, I would like to thank my parents, my brother and Tzehui. My academic
life would not have been possible without them.
vi
To my parents.
vii
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
1 Theoretical Background
1
1.1
The quantum Hall effect and topological order . . . . . . . . . . . . .
1
1.2
The quantum spin Hall effect and the helical edge states . . . . . . .
3
1.3
The topological Z2 Invariant . . . . . . . . . . . . . . . . . . . . . . .
7
2 Experimental Setup
2.1
2.2
14
Angle-resolved photoemission spectroscopy . . . . . . . . . . . . . . .
14
2.1.1
Technical background . . . . . . . . . . . . . . . . . . . . . . .
14
2.1.2
Instrumental setup . . . . . . . . . . . . . . . . . . . . . . . .
19
2.1.3
Sample preparation . . . . . . . . . . . . . . . . . . . . . . . .
21
Spin-resolved ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.2.1
Technical background . . . . . . . . . . . . . . . . . . . . . . .
22
2.2.2
Rashba Effect . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.2.3
Instrumental setup . . . . . . . . . . . . . . . . . . . . . . . .
28
3 Experimental Discovery of the Bi2 X3 Topological Insulator Class
33
3.1
Bi2 Se3 Topological Insulator . . . . . . . . . . . . . . . . . . . . . . .
33
3.2
Bi2 Te3 Topological Insulator . . . . . . . . . . . . . . . . . . . . . . .
50
3.3
Sb2 Te3 Topological Insulator . . . . . . . . . . . . . . . . . . . . . . .
58
viii
4 Surface State Control and Manipulation
67
4.1
Chemical potential tuning in Bi2 Se3 . . . . . . . . . . . . . . . . . . .
67
4.2
Surface gap opening with magnetic impurities . . . . . . . . . . . . .
80
4.3
Effects of surface engineering in Bi2 Te3 . . . . . . . . . . . . . . . . .
83
5 Conclusions
92
A List of Works Published
102
B List of Works on Preprint
105
ix
Chapter 1
Theoretical Background
1.1
The quantum Hall effect and topological order
The physics of most condensed matter systems can be described by the non-relativistic
Schrodinger equation. However, in systems such as graphene, which contain massless
Dirac fermions, the dynamics is governed by the relativistic Dirac equation. Consequently, many unusual behaviors can occur. For example, a minimum non-zero
quantum conductivity has been observed for graphene at low temperatures [1]. In
normal 2D materials, disorder usually leads to a localization of carriers which induces
an insulating state. The conductivity therefore goes to zero. On the other hand, for
2D Dirac systems there is a minimum mean free path, and the conductivity is at least
e2
.
h
A second unusual behavior is the half-integer quantum hall effect. The energy of a
q
massless relativistic fermion in a magnetic field is given by E = 2e~c2 B(N + 21 ± 21 ).
Therefore, the lowest energy level is given by N = 0, when the pseudospin is negative.
All higher energies are doubly degenerate. In other words, the lowest energy level has
half the degeneracy as all the other levels. Accordingly, the first Hall plateau occurs
at half the normal filling. All subsequent levels have normal filling, so their corresponding plateaus are shifted by integer numbers from the first one. The result is a
1
collection of half integer plateaus. Such characteristics have been observed through
Hall conductivity measurements. Using ARPES, the 2D Dirac fermions in graphite
have also been imaged [2].
While ordinary condensed matter systems are characterized by the symmetry they
break, integer [3] and fractional [4] quantum hall (QH) states, including the one described above, do not have an analogous symmetry that is spontaneously broken.
The QH effect occurs when a strong magnetic field is applied to a two dimensional
electron gas. At low temperatures, if the magnetic field is high enough, the electrons
begin to circulate in a direction determined by the magnetic field, creating a current
propagating along the edge of the sample. Furthermore, the effect is topologically
robust. Specifically, the edge state is robust against impurities inside the insulating
bulk. When encountered with an impurity site, the propagating electrons simply distort their paths to go around it. The transverse magneto-conductance σxy is precisely
quantized, at fractional multiples of
e2
.
h
A topological treatment of the QH state was first proposed by Laughlin [5] using
a gauge invariant argument in an annulus geometry. Halperin [6] later associated the
Landau level filling number n with the localized edge states of the sample. Specifically,
he suggested that n counts the number of physical surface states propagating along
the edges. Furthermore, the edge state is a chiral Fermi liquid which cannot be
localized by disorder inside the bulk. The bulk contributes no current to the QH
state, and the edge conductance could be directly measured by transport probes.
Moving into systems with periodic potentials requires additional topological consideration for the bulk. Thouless, Kohmoto, Nightingale and den Niks (TKNN) [7]
proposed a topological invariant to explain the geometric “Berry’s phase” of a lattice.
The invariant was then used to rewrite the quantized Hall conductance expression
in terms of the bulk states. Specifically, TKNN found that the transverse magnetoconductance σxy is given by the Chern number (Cs ) over the entire Brillouin zone. If
2
the Fermi level is inside the rth energy gap, σxy is given by:
σxy
where
1
Cs =
2πi
Z
r
e2 X
Cs
=
h s=1
(1.1)
−
→
→
−
[∇k × As (kx , ky )]z d2 k
(1.2)
BZ
and
−
→
s
→i
→ |∇k |u−
As = hus−
k
k
(1.3)
Hatsugai[8, 9] then found that this bulk topological invariant is closely related to
the edge state expression. Specifically, the winding number of the edge states n is
equal to the summation of all the Chern numbers characterizing the occupied bulk
states. Without an external magnetic field, time reversal symmetry is preserved in
−
→
the system. The ∇k × As (kx , ky ) term in the above equation however is odd under
time reversal symmetry. Therefore, for a system with inversion symmetry the integral
must vanish, and σxy = 0. If an external magnetic field is introduced to the system,
a topological defect is created, and propagating currents are induced along the edges
of the sample.
1.2
The quantum spin Hall effect and the helical
edge states
The quantum spin Hall state is another state of matter that is topologically distinct
from most ordinary materials as well as the QH state. Unlike the QH state, the spin
Hall state (with currents that are not necessarily quantized) does not require an external magnetic field. Rather, as Murakami, Nagaosa and Zhang [10] suggested, it is
driven by the internal spin-orbit coupling, which can be even stronger at room tem-
3
perature. Accordingly, good candidates to harbor this phenomena are semiconductors
with band structures that are significantly coupled to spin-orbit interactions.
A detailed discussion of the quantization of σxy through spin-orbit coupling was
later offered by Kane and Mele, as well as Bernevig and Zhang [11, 12]. The Lorentz
−
→ →
force, which drives the QH state, attributes a A · −
p term in the Hamiltonian. In
−
→
terms of the symmetric gauge the vector potential A can be written as:
−
→ B
A = (y, −x, 0)
2
(1.4)
which then produces a Hamiltonian term of the form:
HLorentz ∝ B(xpy − ypx )
(1.5)
Therefore, Bernevig et. al. argued, the goal is to look for another force in nature which
produces a similar Hamiltonian. The obvious candidate is the spin-orbit coupling
−
→ →
→
→
force. Its Hamiltonian is of the form (−
p × E) · −
σ , where −
σ is the Pauli spin
−
→
→
−
matrix. Instead of an external B field, an E field is used, which preserves time
−
→
reversal symmetry. If one considers a E field of the form E(x̂ + ŷ), the corresponding
Hamiltonian becomes:
HSO ∝ Eσz (xpy − ypx )
(1.6)
The equation suggests that electrons of opposite spins will experience effective magnetic field which are pointed in opposite directions. Accordingly, each spin would be
associated with its own set of induced Landau levels. The spin-↑ electrons would be
chiral, with a conductance quantized in fractional multiples of
e2
,
h
while the spin-↓
2
electrons would be anti-chiral, with a conductance in fractional multiples of − eh .
In the QH state, the electronic edge states propagate along different directions
on the top and bottom sides of the sample (Figure 1.1(a)). The right and left-
4
B-Field
(a)
(b)
Quantum Hall System
Quantum Spin Hall System
Figure 1.1: (a) A quantum Hall system contains a chiral state propagating along the
edge of the sample, where backscattering is prohibited. (b) In a quantum spin Hall
system the edge states are helical, and backscattering remains forbidden with an odd
number of right or left-moving channels at each part of the sample.
moving chiral edge states are spatially separated from each other. Accordingly, they
are protected from backscattering and therefore dissipation-less when exposed to impurities. In spin Hall states, the channels of propagation are each split into two
(Figure 1.1(b)). The top portion of the sample contain right-moving spin-↑ electrons
as well as left-moving spin-↓ electrons, while the bottom contain left-moving spin-↑
electrons along with right-moving spin-↓ electrons. While there are both left-moving
and right-moving channels in each part of the sample, backscattering by non-magnetic
impurities is still forbidden. The reason is that when a right-moving spin-↑ electron
backscatters from an impurity along a clockwise path, it becomes a left-moving spin-↓
electron, picking up a phase of π in the process. Alternatively, a right-moving spin-↑
electron can also pick up a phase of −π, by traveling along a counterclockwise path.
The two reflected wavefunctions would then interfere destructively with each other,
thereby allowing perfect transmission. However, this scenario only works when there
is an odd number of right or left-moving channels. For example, if there were instead
5
two right-moving and two left-moving channels: a left-moving spin-↑ electron can
then scatter into the right-moving spin-↑ channel without picking up a phase difference, which allows it to interfere destructively. This odd-even effect was pointed out
by Qi and Zhang [13], which ties directly to the Z2 number in the context of topological insulators. One must also note that if the impurity were magnetic, time-reversal
symmetry is broken. As a result, perfect destructive interference is no longer possible
and backscattering would be allowed.
As suggested by Bernevig, Hughes and Zhang [14], good candidates to exhibit
quantum spin hall states are semiconductors with band inversions, such as HgTe.
Hg and Te are both heavy elements, so they carry very strong spin-orbit couplings.
The effect of the spin-orbit interaction is to invert the s and p orbitals of their band
structures. In conventional semiconductors, the s orbital forms the conduction band,
while the p orbital forms the valence band. But in these two elements the p orbital
is pushed above the s orbital, and the two bands are inverted. One can then create
a quantum well comprised of HgTe, sandwiched between two CdTe layers. As CdTe
carries much weaker spin-orbit coupling, the overall spin-orbit strength of the quantum well could be tuned by varying the thickness of the HgTe layer. A confirmation
was provided by Konig et. al. [15], who found a transition in the quantum well
conductance with changing thickness. Below a critical thickness of dc ∼ 6.5nm, the
transport data showed an infinite resistance, indicating an insulating band structure
with no states between the band gap. As d was increased above the critical value,
a plateau at R =
h
2e2
was observed. The quantization was a signature of the helical
edge states, which emerged between the band gap as a result of the band inversion.
However, the “spin” degree of freedom was not probed directly in this measurement.
Because each of the edge states is spin degenerate, transport measurements cannot
determine the spin at each of the propagation channels. Other scattering techniques
to probe this degree of freedom are also difficult, since the edge states are typically
6
very small. While these surface states are usually localized to within a few atomic
layers, the typical spot size of most instruments is much longer, on the order of 10−1
mm.
1.3
The topological Z2 Invariant
Naively, one can then propose a topological TKNN invariant akin to that for the QH
state. One can define characterization numbers n↑ , which represents the number of
spin-up states, and n↓ = −n↑ , which represents the number of spin-down states. The
quantum spin Hall state carries a zero charge current, given by
e2
(n↑
h
+ n↓ ). The spin
current, on the other hand, is non-vanishing, with a conductance given by
e
(n↑ −n↓ ).
4π
However, it was found by Kane and Mele [11] that in fact such a TKNN characteristic
number n↑ − n↓ would break down when σz is not conserved. Perturbations such
as disorder, mirror-symmetry breaking, and band mixing can destroy the spin Hall
conductance quantization. Nevertheless, it was shown that the topological order
of the phase should be preserved [16]. A different characteristic number was then
proposed by Kane et. al. to truly capture the topological character of the quantum
spin Hall state-specifically, the existence of gapless edge states which is robust in the
presence of disorder.
In two dimensions, the difference between a conventional band insulator and a
quantum spin Hall state is represented by a single Z2 number. Materials which are
topologically equivalent to the spin Hall state are described as strong topological
insulators, and carry edge states that are robust against disorder. Such systems carry
values of Z2 = 1, while band insulators carry values of Z2 = 0. In three dimensions,
the number of Z2 invariants increases to four, but only one of which is robust against
disorder. As each invariant takes binary values of 0 or 1, there are a total of sixteen
possible topological classes.
7
In the bulk of a three dimensional material, time-reversal symmetry holds. Explicitly, this means:
−
→
−
→
E( k , ↑) = E(− k , ↓)
(1.7)
Space-inversion symmetry on the other hand requires that
−
→
−
→
E( k , ↑) = E(− k , ↑)
(1.8)
−
→
→
−
E( k , ↑) = E( k , ↓)
(1.9)
Therefore,
and the energy bands are degenerate. One way to lift this degeneracy is by lifting the
space-inversion symmetry requirement. This is allowed on the surface, so the bands
can split by a Rashba-type spin-orbital coupling [17]. The degeneracy however should
be preserved at high symmetry points, since
E(0, ↑) = E(0, ↓).
(1.10)
More generally, these high symmetry points should follow the criteria:
→
−
−Γr = Γr + G
(1.11)
−
→
where G is a reciprocal lattice vector. In two dimensions, there are four such points,
while in three dimensions there are eight. In the latter case, one can then project
−
→
the eight Kramers points onto a plane perpendicular to G . This projection produces
four distinct points Γt in the two dimensional Brillouin zone, with each reduced pair
differ by
−
→
G
.
2
As these are unique points in the Brillouin zone that are time reversal
invariant and satisfy equation 1.11, they are also referred to as the time reversal
invariant momenta (TRIM).
8
The strong topological invariant, the Z2 number, is then given by a binary number
ν0 defined as [18]:
(−1)
ν0
=
8
Y
δr
(1.12)
r=1
where δr is given by:
p
det[S(Γr )]
.
δr =
Pf[S(Γr )]
(1.13)
For 2N occupied states, S is a 2N × 2N unitary matrix, given by [19]:
Smn (k) =< u−k,m |Θ|uk,n >
(1.14)
where Θ is the time reversal operator with Θ2 = −1. In the denominator, the Pfaffian
operator satisfies the property det[S] = Pf[S]2 . If one labels the eight TRIMs as:
→
−
→
−
→
1 −
Γr=(l1 ,l2 ,l3 ) = (l1 b1 + l2 b2 + l3 b3 )
2
(1.15)
the three additional invariants νk are then given by:
Y
(−1)νk =
δr=(l1 ,l2 ,l3 )
(1.16)
lk =1,li6=k =0,1
where the four δr s in the product represent Γr s lying in the same plane. In two
dimensions the four topological invariants collapse into one, given by:
(−1)
ν0
=
4
Y
δr
(1.17)
r=1
Alternatively, one can express the Pfaffians in terms of the parity eigenvalue ξ2n of
the occupied energy bands. For 2N occupied states equation 1.13 then becomes:
δr =
N
Y
n=1
9
ξ2n (Γr )
(1.18)
(a)
(b)
Bulk Cond.
(c)
Bulk Cond.
EF
EF
Bulk Valence
Γa
Bulk Valence
Γb
Γa
Γb
Γa
Γb
Figure 1.2: (a) A band structure corresponding to πa πb = 1, which is topologically equivalent to that of a band insulator. (b) A band structure corresponding to
πa πb = −1, where a “partner switching” behavior is observed. (c) Example of a band
structure with “band-switching” behavior.
Accordingly, one can calculate Z2 by first modeling the band structure with a tight
binding model. The parity eigenvalues at the TRIMs can then be extracted, from
which ν0 is calculated using equations 1.12 and 1.18.
One can now return to the question of edge states, and interpret it in the context
of the topological invariant. For a pair of three dimensional TRIMs Γt1 and Γt2 that
project to the same point Γt in the surface Brillouin zone, one can define a surface
fermion parity given by:
πt = δt1 δt2
(1.19)
For each pair of surface TRIMs Γs and Γt , one can then use the product πs πt to
characterize how the surface bands are connected between the two Kramers points.
Specifically, πs πt = −1 indicates that an odd number of surface bands intersect the
Fermi level between Γs and Γt (Figure 1.2(b)). Similarly, πs πt = +1 corresponds
to an even number of surface bands intersecting the Fermi level between Γs and Γt
(Figure 1.2(a)). When πs πt = 1, a pair of bands originating from Γs would again pair
up at Γt to satisfy Kramers Theorem. This kind of band structure is topologically
equivalent to that of a band insulator. The surface bands can be distorted and pushed
10
(a)
(b)
Bloch Insulator
Conduction
Band
(Z2 = +1)
Strong Topological
Insulator
(Z2 = -1)
Valence
Band
EF
Γ
Γ
Γ
Γ
Figure 1.3: (a) A band structure corresponding to ν0 = 0. The surface bands can be
distorted and pushed into the bulk continuum.(b) A band structure corresponding to
ν0 = 1, where the surface state is robust against disorder.
into the bulk continuum (Figure 1.3(a)). On the other hand, for πs πt = −1 the
surface bands undergo a “partner switching” behavior. Each band originating from
Γs would pair up with another band from a different pair before coming together at
Γt . Accordingly, it is impossible to remove all the surface states by distorting the
band structure. In other words, the surface states are robust against weak disorder
(Figure 1.3(b)). This even-odd argument is reminiscent of that in the previous section,
in the context of spin current channels. In fact, for the simplest case of a single surface
state lying between the band gap in momentum space, the real space equivalent is a
pair of spin currents moving in opposite directions.
At the surface TRIMs, the two dimensional surface bands in a three dimensional
strong topological insulator should disperse according to Dirac equation. The surface
Fermi surface divides the entire Brillouin zone into two regions, with one region
11
E
(a)
E
(b)
μ
μ
μ
-k
k
E
(c)
k
-k
Fermi Gas
½ Dirac Gas
(Topological)
2D Fermi Gas
Topo. Ins. Surface
-k
k
Dirac Gas
Graphene
Figure 1.4: The (top) surface band dispersion, (middle) edge current, and (bottom)
surface Fermi surface for a (a) conventional metal, (b) a strong topological insulator
and (c) graphene. While the electrons in a metal follow the Schrodinger equation,
those in a strong topological insulator and graphene follow the Dirac equation. However, while the spin currents are degenerate in graphene, those in a strong topological
insulator are spin-polarized.
12
carrying a fermion parity +1 and one carrying a parity −1 (Figure 1.4(b)). As an
extension of the even-odd argument, a close Fermi contour should include an odd
number of Dirac points. An electron following such a path would pick up a Berry’s
phase of π, so localization by disorder is forbidden [20].
13
Chapter 2
Experimental Setup
2.1
2.1.1
Angle-resolved photoemission spectroscopy
Technical background
Angle-resolved photoemission spectroscopy is a technique based on the photoelectric
effect to study the electronic structure of condensed matter systems. As first observed
by Hertz, when photons of energy hν strike a sample surface, electrons absorb the
energy and escape into the vacuum with certain kinetic energies. A good approximation of the photoemission process can be described by the three-step model [21],
where (Figure 2.1(a)):
• An electron is first excited by the photon inside the bulk.
• The excited electron then travels to the sample surface.
• The electron escapes from the sample into the vacuum.
An electrostatic analyzer then collects the photoelectrons, as a function of kinetic
energy and surface emission angles θ and φ (Figure 2.2(b)). As a monochromatic
beam with high flux is typically required, a synchrotron radiation source is used to
provide the incident photons. Alternatively, one can also use monochromatic sources
14
(a)
(b)
Ekin
Ef
G
1
Ef
3
2
Excitation Transport Escape to
to Surface Vaccum
hν
Evac
EF
Ei
E0
hν
Ei
Sample
k
Φ
V0
-π/2
0
π/2
Vac.
z
Figure 2.1: (a) The three-step model used to describe the photoemission process
inside the sample. (b) The band structure corresponding to a photoelectron with the
free-electron final state assumption, which produces the dispersion on the right.
such as He lamps. However, in that case, both signal intensity and energy tunability
would be significantly reduced.
The wavefunction of the final state after an electron is excited inside the bulk can
be written as:
−
→
k N −1
ΦN
f = Aφf Φf
(2.1)
−
→
−
→
Here, φ k is the wavefunction of a photoelectron with momentum k and A is the
antisymmetric operator. The final state comprises of the N − 1 electron system, in
addition to the excited electron which is now free to move within the bulk. The
N − 1 electron system can then remain at different excited states. Accordingly, the
photoemission intensity is proportional to:
X
i,f
−
→
k 2
|
|Mi,f
X
|pi,r |2 δ(Ekin + ErN −1 − EiN − hν)
(2.2)
r
where,
−1
|ΦiN −1 > |2
|pi,r |2 = | < ΦN
r
15
(2.3)
is the probability that the N − 1 electron system will remain at the excited state r.
−
→
k 2
The |Mi,f
| term is an one-electron matrix element, which depends on the electron
momentum as well as the energy and polarization of the impinging photon.
The excited photoelectron then follows the kinematics below which describes the
−
→
→
momentum −
p = ~ k (Figure 2.2(b)):
kx =
1p
2mEkin sinθcosφ
~
1p
2mEkin sinθsinφ
~
1p
2mEkin cosθ
kz =
~
ky =
(2.4)
(2.5)
(2.6)
From the conservation of energy, the kinetic energy of the photoelectron is then given
by (Figure 2.3):
Ekin = hν − φ − |EB |
(2.7)
where φ is the work function of the sample and EB is the binding energy of the
electrons. For most metals, the work function φ is on the order of 4eV. From the
conservation of momentum, the component of the momentum parallel to the sample
surface is given by:
−
→
1p
2mEkin sinθ
| k k| =
~
(2.8)
−
→
where | k k | can also be written as (kx , ky , 0) as noted above.
However, due to the surface potential experienced by the escaping electron, the
component of the momentum perpendicular to the sample surface is not conserved.
If one assumes that the final state of the photoelectron is free electron-like, its band
structure should then be parabolic as shown in figure 2.1(b). The bands are offset
by a inner potential value of V0 , which captures the energy difference between the
16
(a)
V1
R1
V2
R2
2D Detector
E
Lens
hν
θ
Sample
z
(b)
φ
hν
e
y
θ
x
Figure 2.2: (a) The Scienta analyzer setup used to measure emitted photoelectrons
from the sample surface. The two hemispheres deflect the electrons to produce a
2D image of energy vs. momentum in one shot. (b) The geometry of the detector
relative to the sample surface. The momentum of the electron inside the sample can
be extracted from the measured values of Ekin , θ and φ.
17
sample and the vacuum. Explicitly, this can be written as:
Ef = Ekin + φ
(2.9)
Since
2
~2 (kk2 + k⊥
)
−
→
− |E0 |
Ef ( k ) = Ekin + φ =
2m
(2.10)
one finds,
|k⊥ | =
1p
2m(Ekin cos2 θ + V0 )
~
(2.11)
Accordingly, given V0 , one can extract all the momentum components of the excited
electron inside the bulk from measured values of Ekin , θ and φ. The value of V0 can
be determined experimentally by measuring the periodicity of the E(k⊥ ) dispersion
at normal emission. However, for samples where the dispersion perpendicular to the
sample surface is small, such a measurement can be difficult. Alternatively, one can
provide a theoretical estimate using the muffin-tin zero. Here, the value is an average
of the potential over the atomic interstitial region defined in most full-potential ab.
initio codes. The muffin-tin radius used is typically on the order of 2-3 Bohr. However,
as the radius is not physical and somewhat an approximation, one needs to cross
check with experimental data for consistency. The typical value of V0 in our system
is approximately 10eV.
In addition to the bulk bands, the electronic structures of many semiconductors
also contain bands that reside in the surface. One way a non-bulk state can form
is if the material has different coordination numbers between the surface and bulk
atoms [22]. Such a mismatch could be due to problems such as poor cleaving or poor
sample growth. However, these states are typically considered “surface core shifts”
instead of surface states. To form surface states one must introduce decaying states
→
−
(i.e.,making k imaginary) into the metallic band gap. These states are then trapped
between the surface of the crystal and the surface barrier potential. Depending on
18
Ekin
Spectrum
EF
hν
E
Sample
Evac
N(Ekin)
Φ
EF
EB
E0
hν
Core Level
N(E)
Figure 2.3: A schematic of the energy levels determining the kinematics of the the
photoelectrons. Inside the crystal, the photoelectron energy is referenced with respect
to the binding energy EB and the Fermi level EF . After escaping into the vacuum, a
shift in energy is applied, corresponding to the sample work function φ.
the symmetry of its wave function, the state is either called a Shockley state or a
Tamm state. The 2D surface states should have no dependence on k⊥ . Therefore,
their corresponding ARPES signals should be independent of incident energy. Higher
incident energies should make the signal intensities weaker, since the escape depths
of the photoelectrons are increased. As a final check on the location of the bands, the
surface states should appear at the gap of the bulk bands projection [23].
2.1.2
Instrumental setup
Angle-resolved photoemission experiments discussed in this work were performed at
beamlines 10.0.1 and 12.0.1 of the Advanced Light Source (ALS) at the Lawrence
19
Berkeley National Laboratory (LBNL) in Berkeley, California, beamline 5-4 of the
Stanford Synchrotron Radiation Laboratory in Menlo Park, California, and beamlines
PGM-A and U1-NIM of the Synchrotron Radiation Center in Stoughton, Wisconsin.
Unlike conventional photon sources such as discharge lamps, synchrotron sources
allow for wide spectral ranges with much higher fluxes, producing continuous spectra
that are both intense and highly polarized.
The configuration of the measurement setup is described in figure 2.2(a). A Scienta analyzer is comprised of two hemispheres of radii R1 and R2 , which form a
deflector across a potential difference ∆V = V2 − V1 . Electrons with a narrow kinetic energy window centered at Epass = e∆V /(R1 /R− R2 /R1 ) will pass through the
analyzer onto a two-dimensional position-sensitive detector. The analyzer deflects
electrons with different kinetic energies by different angles. Therefore, an image as a
function of energy is produced along the y axis of the detector, spread over a range
of typically ±0.1Epass . The analyzer entrance slit on the other hand selects the photoelectrons with varying emission angles. Accordingly, a 2D image of energy versus
momentum can be captured with every shot. Depending on the Scienta analyzer
model, the angular range of the entrance slit is either 9◦ , 14◦ or 38◦ . To minimize
sample surface damage from degased atoms inside the chamber, measurements are
typically performed at pressures better than 5 × 10−11 torr.
To determine the sample work function φ, the Fermi level of the detected photoelectrons is found by studying the ARPES spectra of a polycrystalline sample such
as gold. EF is then determined by fitting the resulting energy distribution curve
with a Fermi-Dirac distribution function. The function is convoluted with a Gaussian distribution, with a half-width corresponding to the instrumental resolution.
Taking into account the energy resolutions of the the beamline monochrometer and
the Scienta analyzer, the total energy resolutions of the measurements are typically
around 15meV. Additional fine tuning of the resolution can be done by adjusting the
20
pass energy window Epass . Typically a Epass of either 5eV or 10eV was used in our
measurements.
2.1.3
Sample preparation
Single crystals of Bi2 Se3 were grown by melting stoichiometric mixtures of high purity elemental Bi and Se in a 4 mm inner diameter quartz tube. The sample was
cooled over a period of two days, from 850 to 650 ◦ C, and then annealed at that
temperature for a week. Single crystals of Bi2−x Mnx Te3 were grown by melting stoichiometric mixtures of elemental Bi (99.999%), Te (99.999%), and Mn (99.95%) at
800 ◦ C overnight in a sealed vacuum quartz tube. The crystalline sample was cooled
over a period of 2 days to 550 ◦ C and maintained at the temperature for 5 days.
The same procedure was carried out, with Sb (99.99%) and Te (99.999%) for Sb2 Te3
crystals. In all three cases, single crystals were obtained and could be easily cleaved
from the boule. Powder X-ray diffraction measurements were taken to ensure that
the crystals were single phase. The cleaving surface directions were then determined
by X-ray Laue diffraction, on a Bruker D8 diffractometer using Cu Kα radiation with
λ = 1.54Å. From the resulting patterns, it was found that all the cleaving surfaces
were in the (111) direction.
The sample was then mounted in a geometry as depicted by figure 2.4. The crystals
were first cut into pieces of approximately 2mm×2mm×0.5mm. A small piece was
then mounted onto a copper post, using the commercial epoxy resin Torr seal. For
other samples with poor electrical conductivity, silver epoxy was used instead to
ensure good electrical contact with the sample holder. A ceramic top post was then
attached onto the sample with another layer of epoxy. The entire object was then
covered with silver paint, again to ensure good electrical contact, thereby preventing
sample charging from photon exposure. An additional layer of graphite was placed
over the object, to reduce background signal from the sample holder. After the sample
21
Silver paint
Graphite
Cermic post
Torr Seal
Sample
Silver epoxy/
Torr Seal
Copper post
Figure 2.4: A schematic of the sample geometry mounted inside the ARPES measurement chamber. The ceramic top post was used to cleave the sample surface in
situ, to ensure a clean surface for measurements.
was mounted inside the ARPES measurement chamber, it was cooled and pumped
down for a few hours. The sample was then cleaved in situ at pressures of less than
5 × 10−11 torr, by knocking down the ceramic top-post with a cleaver. The exposed
surface was typically visually shiny, indicating a flat surface. Additional surface
characterization could be performed using low energy electron diffraction (LEED), to
detect surface structural reconstruction and other relaxation effects after cleavage.
2.2
2.2.1
Spin-resolved ARPES
Technical background
The interaction between an incident photon and an electron can be modeled with a
perturbation term:
Hint = −
→ −
e −
A ·→
p
mc
22
(2.12)
−
→
→
where −
p is the momentum of the electron and A is the electromagnetic vector potential. Additional spin-dependent terms in the perturbation become negligible when
the incident beam is linearly polarized. Accordingly, to the first order the photoemission process conserves spin, since the above energy term is spin-independent. On
the other hand, when circularly polarized light is used the effect of spin-polarization
by the photoexcitation process becomes more significant. A polarization of 43% has
been observed for electron beams produced by light from a circularly polarized laser
[24].
One origin of spin-polarized electrons in a solid is the strong correlation due to
electron exchange interactions, which occurs in most magnetic materials. In this case,
the doubly degenerate electronic states in the band structure become split, with each
resulting state carrying a different spin. Additionally, spin-splitting can occur from
spin-orbit interactions. In this case, while the sample itself does not carry a net
magnetization, the excited photoelectrons are spin-polarized. The polarization of an
electron is defined as [25]:
−
→ 2
P = (< Sx >, < Sy >, < Sz >)
~
(2.13)
where the brackets represent expectation values of the spin operators. The polarization has values less than 1, after normalizing by the prefactor. The Rashba Hamiltonian term describing the spin-orbit interaction is given by:
Hso =
−
→ →
µB −
(→
v × E) · −
σ
2
2c
(2.14)
−
→
→
−
Here, µB is the Bohr magneton, −
v is the velocity, →
σ is the spin, and E is the
average electric field seen by the electron. As noted in the previous chapter, spinorbit coupling breaks time reversal symmetry. The Kramers degeneracy is therefore
broken and the electronic states are spin-split.
23
After the electron escapes into the vacuum, its spin can be detected by Mott
electron polarimetry. The electron spin inside the solid can then be extrapolated,
assuming the photoemission process conserves spin. The technique essentially measures the spin by colliding the electron with a target atom. Depending on the spin
of incident electron, the ensuing scattering vector is different. Accordingly, electrons
of varying spins are separated by the scattering event. When a relatively slow moving electron scatters off a nucleus with charge Ze, the scattering is described by the
Rutherford cross section σR :
σR
Ze2
1
=(
)2 4 θ
dΩ
16π²0 Ekin sin ( 2 )
(2.15)
where θ is the scattering angle and Ekin is the kinetic energy of the electron. However,
when Ekin or Z is large, the rest frame of the electron experiences a magnetic field
induced by the electric field of the nucleus:
−
→
→
1→ −
B =− −
v ×E
c
(2.16)
From Coulomb’s law the electric field is given by:
→
−
Ze →
E = 3−
r
r
(2.17)
→
−
→
Ze → →
Ze −
B = 3−
r ×−
v =
L
3
cr
mcr
(2.18)
Accordingly,
−
→
where L is the orbital angular momentum of the electron. The spin carried by the
electron then interacts with the effective magnetic field, adding an additional spinorbit term into the scattering potential. The resulting cross section then becomes:
−
→
σ = σR [1 + S(θ) P · n̂]
24
(2.19)
Here, the scattering unit vector is given by n̂ =
−
→
n
,
→
|−
n|
where:
−
→
−
→
→
−
n = ki× kf
(2.20)
−
→
−
→
−
→
and k i and k f are the incident and final vectors of the electron. P again is the
polarization describing the electron spin, while S(θ) is the Sherman function, which
characterizes the scattering asymmetry for a 100% polarized electron beam.
For a beam of electrons with N↑ spin-up electrons and N↓ spin-down electrons,
the polarization can be rewritten as:
Pz =
N↑ − N↓
N↑ + N↓
(2.21)
where the z-axis is defined along the direction of the spins. After colliding with the
target nuclei, the asymmetry of the scattered electrons is given by:
Az =
IL − IR
IL + IR
(2.22)
where IL is the intensity of the electron signal observed by a left positioned detector,
and IR is that observed by a right positioned detector. From the scattering crosssections one knows:
IL ∝ N↑ [1 + S(θ)] + N↓ [1 − S(θ)]
(2.23)
IR ∝ N↑ [1 − S(θ)] + N↓ [1 + S(θ)]
(2.24)
and
Therefore,
Pz =
Az
S(θ)
(2.25)
Accordingly, the polarization of the incident electrons in each direction can be uniquely
determined by a set of two Mott detectors. Theoretically, the Sherman function at
25
the denominator can take values up to 0.5. However, for much thicker targets with
multiple and inelastic scattering events the values are typically smaller. For the setup
used in this work, Sef f usually takes values of around 0.08. While the function can
be evaluated from Monte-carlo simulations, the complexities involved, including varying sample geometries, could make the calculation extremely difficult. Accordingly,
the Sherman function is usually experimentally calibrated, using samples with known
polarizations.
2.2.2
Rashba Effect
In this section, we elaborate on the spin-splitting effect induced by spin-orbit interactions. To help rewrite the Hamiltonian term Hso discussed in the previous section,
→
−
one can set E = OV , where V is the electric potential at the sample surface [26].
This potential gradient, which is perpendicular to the surface, comes from the sudden
termination of the crystal lattice. Using a Lorentz transformation, in the rest frame
of a fast moving electron at the surface, the corresponding electric field becomes a
magnetic field. The magnetic field then induces a Zeeman splitting of the energy
levels, producing an energy difference between the spin-up and spin-down electronic
states. The energy splitting, which depends on the magnetic field, is a function of
the electron momentum.
In the case where the electrons are free to move inside the surface plane, V = V⊥ :
OV =
dV
nˆz
dz
(2.26)
where the direction normal to the sample surface is labeled as z. The spin-orbit term
then becomes:
→
→
Hso = αRB (nˆz × −
p)·−
σ
26
(2.27)
The complete Hamiltonian is then given by [27]:
H = H0 + Hso
(2.28)
~2 2
O)
H0 = I(EΓ −
2m
(2.29)
where
and
Hso = −αRB (iσy
∂
∂
− iσx )
∂x
∂y
(2.30)
Here I is the 2 × 2 unit matrix, σx , σy are the standard Pauli matrices, and EΓ is the
energy at the zone center. The coupling constant αRB is dependent on the atomic
number Z, as well as the structural contribution from the surface potential gradient.
Solving the Hamiltonian one finds eigenvalues of the form:
and
−
→
−
→
→
−
~ 2 | k |2
+ αRB | k |
E+ ( k ) = EΓ +
2m
(2.31)
−
→
−
→
−
→
~ 2 | k |2
− αRB | k |
E− ( k ) = EΓ +
2m
(2.32)
with eigenvectors:


−i(φ+ π2 )
→
−
1 e
| k , + >= √ 
2
1
and

(2.33)

−i(φ− π2 )
−
→
1 e
| k , − >= √ 
2
1
27




(2.34)
where φ =arctan( kkxy ). Calculating the expectation of the spinors < Si >, one finds:


−sinφ

→
−
→ −
~

S +( k ) = 
cosφ

2


0
and

(2.35)

 sinφ 

→
−
→ −
~
S −( k ) = 
−cosφ


2

0
(2.36)
−
→
In other words, the spin expectation vector is always perpendicular to k and has
no z component in this model. The constant energy surface therefore comprises of
two concentric circles, with one carrying a spin rotating clockwise as one moves along
one circle, and one carrying a spin rotating counterclockwise as one moves along the
other circle. The corresponding energy dispersion comprises of two parabolic Rashba
−
→
→
−
branches offset from each other by a momentum 2 k0 . The splitting 2 k0 is strongly
Z dependent, which was found to be 0.03Å−1 for Bi (Z = 83), 0.024Å−1 for Au
(Z = 79), and 0.0013Å−1 for Ag (Z = 47).
2.2.3
Instrumental setup
Spin-resolved photoemission experiments were conducted at the Complete Photo
Emission Experiment (COPHEE) endstation [28] at the Swiss Light Source in Villigen, Switzerland. Similar to a Scienta ARPES spectrometer, a hemispherical analyzer
is used to produce an energy vs. momentum spectrum prior to spin measurements
(Figure 2.5). After the the electrons exit the analyzer they are accelerated to 40keV,
before scattering off two sets of gold foil targets. Two sets of Mott detectors are
then used to detect the scattered signal, with each set comprised of four silicon diode
detectors aligned in orthogonal directions. A discriminator is used to separate the
28
scattered signals, operating at a level where the dark count rate is less than 1Hz. As
mentioned in the previous section, every pair of detectors can determine the electron
polarization in a particular direction. Accordingly, each polarimeter can map the polarization in two directions, e.g. in the x̃ − z̃ and ỹ − z̃ planes. The two polarimeters
shared a common axis z̃. Data along this direction can be used to calibrate signal
over the two detectors.
An alternative Mott polarimeter design, which is not used in our experiments, has
been in development to eliminate the high voltage required in the COPHEE setup
[27]. A high voltage requires a detector with a large physical size, with complex
electronics which cannot easily integrate with standard components. The alternative
design, called the Rice University type Mott detector, is a retarding potential detector
which slows down the electrons after scattered off the Au foils. The electron detector
can then be set at ground potential, which also improves the overall efficiency of the
Mott detector. However, such a design has the disadvantage of low stability, which
originates from poor focusing of the electrons onto the Au foil, as well as distortion
by the retarding field after the scattering event.
A vector lying along the (θ = 0, φ = 0) direction in the sample coordinate frame
corresponds to a 45◦ rotation about the zb axis in the detector frame. In general,
polarization vectors in the two frames are related by:
−
→
−̃
→
P = T̂ P
(2.37)
where the transformation matrix T̂ is given by:

cosθcos
√ φ+sinφ

2

−cosθsin
T̂ = 
√ φ+cosφ

2

sin
√ θ
−cosθcos
√ φ+sinφ
2
cosθsin
√φ+cosφ
2
−sin
√ θ
2
2
29

−sinθcosφ

sinθsinφ 


cosθ
(2.38)
Lens
hν
θ
Analyzer
φ
Sample
Deflector
Polarimeter 1
Py Pz
Polarimeter 2
B1
B2
Px Pz
L1
R2
x
y
y
R1
z
F1
x
L2
F2
Au Foil
z
Figure 2.5: A schematic of the experimental setup for spin-resolved photoemission
measurements. A set of hemispherical analyzer is used to produce an energy vs.
momentum image, after which two sets of polarimeters detect the spin polarizations
of the electrons. In the sample reference frame, the electron spin angles are given by
(θ, φ).
30
→
−
A typical tilt angle of θ ∼ 2◦ corresponds to a correction of ∼1% for P . According,
a θ = 0 approximation is usually used in the transformation.
The efficiency of the polarimeter is given by:
²=
N 2
S
N0 ef f
(2.39)
Due to the fact that only a small amount of the incident electrons (N0 ) are scattered
into the detectors, the value of ² generally ranges from 10−3 to 10−4 . The standard
deviation of the measured polarization is given by:
−̃
→
∆P
1
√
=
−̃
→
²N0
P
(2.40)
Accordingly, long scans on the order of a few hours are typically required to place the
error bar within ±0.01, with N ∼ 5 × 105 electrons.
A typical problem in polarization measurements is the existence of significant
background signal [27]. Despite the fact that in non-magnetic systems the background
should be unpolarized, it nevertheless can affect the line shape and magnitude of the
measured polarization. Specifically, with a stronger background the polarization is
reduced. The effect is due to the definition of asymmetry A. While the contribution
from the background is canceled out in the numerator of A, the denominator is
increased with the additional contribution. The origins of the background include
secondary, inelastic and quasi-elastic scattering events, unpolarized bulk bands, and
sample and beam quality. The background can additionally be dependent on emission
angle, and the kinetic energies of the photo-electrons.
Therefore, due to the numerous complications from the background signal as well
as resolution broadening, additional analysis is required to extract the spin polarization from the raw data. A “two-step fitting routine” suggested by Meier et. al.
[29] is typically used. In the first step, the measured intensity profile is fitted by a
31
summation of Lorentzian distributions at the peak locations:
Itot =
n
X
Ii + B
(2.41)
i=1
where B is the background. For each of the peaks, a set of spin angles (θi , φi )
assumption is then used, giving:
−
→
P i = (Pxi , Pyi , Pzi ) = ci (cosθi cosφi , cosθi sinφi , sinθi )
(2.42)
where ci gives the magnitude of the polarization vector. Inverting equation 2.21 one
then finds,
1
Iαi;↑,↓ = I i (1 ± Pαi )
6
(2.43)
where α runs along x̃, ỹ, z̃ and +(−) is used for ↑(↓) electrons. Summing all the peaks
together one finds,
Iα↑,↓ =
n
X
Iαi;↑,↓ +
i=1
B
6
(2.44)
The polarization is then given by:
Pα =
Iα↑ − Iα↓
Iα↑ + Iα↓
(2.45)
Finally, the spin angles (θi , φi ) and magnitude ci assumptions are varied, to fit the
above expression to measured data. The result gives a set of spin directions, for each
of the peaks in the polarization profile.
32
Chapter 3
Experimental Discovery of the
Bi2X3 Topological Insulator Class
3.1
Bi2Se3 Topological Insulator
We begin by presenting the experimental discovery of a single Dirac cone topological
insulator, before any prediction from theoretical calculations. Recent experiments
and theories have suggested that strong spin-orbit coupling effects in certain band
insulators can give rise to a new phase of quantum matter, the so-called topological
insulator, which exhibits novel properties [20, 30, 31, 15, 32, 33, 18]. Such systems
feature two-dimensional surface states whose electrodynamic properties are described
not by the conventional Maxwell equations but rather by an attached axion field,
originally proposed to describe interacting quarks [34, 35, 36, 37, 38, 39, 40, 41].
It has been proposed that a topological insulator [30] with a single Dirac cone interfaced with a superconductor can form the most elementary unit for performing
fault-tolerant quantum computation [40]. Here we present an angle-resolved photoemission spectroscopy study that reveals the first observation of such a topological
state of matter featuring a single surface Dirac cone realized in the naturally occur-
33
ring Bi2 Se3 class of materials. Our results, supported by our theoretical calculations,
demonstrate that undoped Bi2 Se3 can serve as the parent matrix compound for the
long-sought topological device for quantum computation. Our study further suggests
that the undoped compound reached via n-to-p doping should show novel transport
phenomena even at room temperature.
It has been experimentally shown that spin-orbit coupling can lead to new phases
of quantum matter with highly nontrivial collective quantum effects [15, 32, 33].
Two such phases are the quantum spin Hall insulator [15] and the strong topological
insulator [32, 33, 18] both realized in the vicinity of a Dirac point but yet quite
distinct from graphene [42]. The strong-topological-insulator phase contains surface
states (SSs) with novel electromagnetic properties [18, 34, 35, 36, 37, 38, 39, 40, 41].
It is currently believed that the Bi1−x Sbx insulating alloys realize the only known
topological-insulator phase in the vicinity of a three-dimensional Dirac point [32],
which can in principle be used to study topological electromagnetic and interface
superconducting properties [34, 35, 36, 40]. However, a particular challenge for the
topological-insulator Bi1−x Sbx system is that the bulk gap is small and the material
contains alloying disorder, which makes it difficult to gate for the manipulation and
control of charge carriers to realize a device. The topological insulator Bi1−x Sbx
features five surface bands, of which only one carries the topological quantum number
[33]. Therefore, there is an extensive world-wide search for topological phases in
stoichiometric materials with no alloying disorder, with a larger gap and with fewer
yet still odd-numbered SSs that may work as a matrix material to observe a variety
of topological quantum phenomena.
The topological-insulator character of Bi1−x Sbx [32, 33] led us to investigate the
alternative Bi-based compounds Bi2 X3 (X=Se, Te). The undoped Bi2 Se3 is a semiconductor that belongs to the class of thermoelectric materials Bi2 X3 with a rhombohedral crystal structure (space group D5 3d (R3̄m); [43, 44] (Figure 3.1). The unit
34
Se/Te1
Se/Te2
Bi
y
Quintuple Layer
z
x
y
x
(b)
(a)
Figure 3.1: (a) The crystal structure of the Bi-based compounds Bi2 X3 (X=Se, Te).
(b) The unit cell contains five atoms, with quintuple layers ordered in the Se(1)-BiSe(2)-Bi-Se(1) sequence.
cell contains five atoms, with quintuple layers ordered in the Se(1)-Bi-Se(2)-Bi-Se(1)
sequence. Electrical measurements have reported that, although the bulk of the material is a moderately large-gap semiconductor, its charge transport properties can
vary significantly depending on the sample preparation conditions [45], with a strong
tendency to be n-type [46, 47] owing to atomic vacancies or excess selenium. An intrinsic bandgap of approximately 0.35eV is typically measured in experiments [48, 49],
whereas theoretical calculations estimate the gap to be in the range of 0.24-0.3eV
[46, 50].
It has been shown that spin-orbit coupling can lead to topological effects in materials that determine their spin Hall transport behaviors [15, 32, 33, 18]. Topological
quantum properties are directly probed from the nature of the electronic states on
the surface by studying the way surface bands connect the material’s bulk valence
and conduction bands in momentum space [32, 33, 18]. The surface electron behavior
35
is intimately tied to the number of bulk band inversions that exist in the band structure of a material [18]. The origin of topological Z2 order in Bi1−x Sbx is bulk-band
inversions at three equivalent L-points [32, 18] whereas in Bi2 Se3 only one band is
expected to be inverted, making it similar to the case in the two-dimensional quantum
spin Hall insulator phase. Therefore, a much simpler surface spectrum is naturally
expected in Bi2 Se3 . All previous experimental studies of Bi2 Se3 have focused on the
material’s bulk properties; nothing is known about its SSs. It is this key experimental
information that we provide here that, for the first time, enables us to determine its
topological quantum class.
The bulk crystal symmetry of Bi2 Se3 fixes a hexagonal Brillouin zone (BZ) for
its (111) surface (Figure 3.2(a)) on which M̄ and Γ̄ are the time-reversal invariant
momenta (TRIMs) or the surface Kramers points. We carried out high-momentumresolution angle-resolved photoemission spectroscopy (ARPES) measurements on the
(111) plane of naturally grown Bi2 Se3 . The electronic spectral weight distributions
observed near the Γ̄ point are presented in Figure 3.4(a)-(c). Within a narrow bindingenergy window, a clear V-shaped band pair is observed to approach the Fermi level
(EF ). Its dispersion or intensity had no measurable time dependence within the
duration of the experiment. The ‘V’ bands cross EF at 0.09 Å−1 along Γ̄-M̄ and at
0.10 Å−1 along Γ̄-K̄, and have nearly equal band velocities, approximately 5 × 105
ms−1 , along the two directions. A continuum-like manifold of states - a filled U-shaped
feature - is observed inside the V-shaped band pair. All of these experimentally
observed features can be identified, to first order, by a direct one-to-one comparison
with the LDA band calculations. Figure 3.2(c) shows the theoretically calculated
(111)-surface electronic structure of bulk Bi2 Se3 along the K̄ − Γ̄− M̄ k-space cut. The
calculated band structure with and without SOC are overlaid together for comparison.
The bulk band projection continuum on the (111) surface is represented by the shaded
areas, blue with SOC and green without SOC. In the bulk, time-reversal symmetry
36
kz
(a)
kx
(111)
K
(b)
Γ
M2
M
ky
ky
Γ
F
Z
M1
Γ
M3
kx
L
Theoretical Calculations
(c)
K
Bulk
no soc
soc
Surface
no soc
soc
Figure 3.2: (a) A schematic diagram of the full bulk three-dimensional BZ of Bi2 Se3
and the two-dimensional BZ of the projected (111) surface. (b) The surface Fermi
surface (FS) of the two-dimensional SSs along the K̄ − Γ̄ − M̄ momentum-space cut
is a single ring centered at Γ̄ if the chemical potential is inside the bulk bandgap.
The band responsible for this ring is singly degenerate in theory. The TRIMs on the
(111) surface BZ are located at Γ̄ and the three M̄ points. The TRIMs are marked
by the red dots. In the presence of strong spinorbit coupling (SOC), the surface
band crosses the Fermi level only once between two TRIMs, namely Γ̄ and M̄ ; this
ensures the existence of a π Berry phase on the surface. (c) The corresponding local
density approximation (LDA) band structure. Bulk band projections are represented
by the shaded areas. The band-structure topology calculated in the presence of SOC
is presented in blue and that without SOC is in green. No pure surface band is
observed to lie within the insulating gap in the absence of SOC (black lines) in the
theoretical calculation. One pure gapless surface band is observed between Γ̄ and M̄
when SOC is included (red dotted lines)
37
demands E(~k, ↑) = E(−~k, ↓) whereas space inversion symmetry demands E(~k, ↑) =
E(−~k, ↑). Therefore, all the bulk bands are doubly degenerate. However, because
space inversion symmetry is broken at the terminated surface in the experiment, SSs
are generally spin-split on the surface by spin-orbit interactions except at particular
high-symmetry points-the Kramers points on the surface BZ. In our calculations,
the SSs (red dotted lines) are doubly degenerate only at Γ̄ (Figure 3.2(c)). This is
generally true for all known spin-orbit-coupled material surfaces such as gold [51,
52] or Bi1−x Sbx [32]. In Bi2 Se3 , the SSs emerge from the bulk continuum, cross
each other at Γ̄, pass through the Fermi level (EF ) and eventually merge with the
bulk conduction-band continuum, ensuring that at least one continuous band-thread
traverses the bulk bandgap between a pair of Kramers points. Our calculated result
shows that no surface band crosses the Fermi level if SOC is not included in the
calculation, and only with the inclusion of the realistic values of SOC (based on atomic
Bi) does the calculated spectrum show singly degenerate gapless surface bands that
are guaranteed to cross the Fermi level. The calculated band topology with realistic
SOC leads to a single ring-like surface FS, which is singly degenerate so long as the
chemical potential is inside the bulk bandgap. This topology is consistent with the
Z2 = −1 class in the Fu-Kane-Mele classification scheme [18].
The calculated valence band dispersion is in good correspondence while making
a comparison with the experimental data taken in both the Γ̄-M̄ and Γ̄-K̄ directions
(Figure 3.3). The theoretical band calculations were performed with the LAPW
method in slab geometry using the WIEN2K package [53]. The generalized gradient approximation of Perdew, Burke and Ernzerhof [54] was used to describe the
exchange-correlation potential. SOC was included as a second variational step using
scalar-relativistic eigenfunctions as basis after the initial calculation was converged to
self-consistency. The surface was simulated by placing a slab of 12 quintuple layers
in vacuum. A grid of 21×21×1 points was used in the calculations, equivalent to 48
38
ARPES
Γ
M
ARPES
Γ
K
M
Γ
Γ
0
EB (eV)
-1
0.35
-2
-3
0.0
0.0
(a)
0.4
0.0
0.8
-1
ky (Å )
0.4
0.0
0.8
(b)
0.4
0.8
0.0
0.4
0.8
-1
kx (Å )
Figure 3.3: The band second-derivative images and first-principles calculation results
along Γ̄-M̄ (a) and Γ̄-K̄ directions (b) are presented. The color of the calculated
bands represents the fraction of electronic charge residing in the surface layers. A
rigid shift of EF is included to match the lowest energy excitations in the ARPES
data with calculations, a consequence of the system being somewhat electron-doped.
The strongest signals are observed from the surface states.
39
k points in the irreducible BZ and 300 k points in the first BZ. To calculate the kz
p
of the ARPES measurements (kz = (1/~) 2m(Ekin cos2 θ + V0 )), an inner potential
V0 of approximately 11.7eV was used, given by a fit on the ARPES data at normal
emission.
The first and twelfth quintuple blocks are taken as the surfaces of the slab. The
fractions of electron charge residing in the atomic spheres of these two surface blocks
are presented by the color of the bands. Bright lines indicate bands located inside
the bulk sample. These bands should overlap with a bulk energy band at a particular
kz value, after projected onto the 2D surface BZ. The calculation agrees with data
after shifting the EF of the calculated result to 200meV above the bottom of the
lowest conduction band. Doping the semiconductor matrix with electrons allows one
to study the connectivity of the surface and bulk states at all energies between the
valence and conduction bands, thereby determining the unique and specific class of
topological order of the parent materials [18].
In the measured spectra, the strongest quasiparticle signals are typically observed
near normal electron emission, at around 1.5-2.0eV. This band corresponds to a pure
surface band lying outside the bulk projection of states. Strong quasiparticle signals
near Γ̄ are of surface origin, based on their direct correspondence with the band calculation. Our energy dependent study of the bands suggests insignificant kz dependence
of these states, confirming the 2D (surface) nature of the bands. Similar observation
has been previously reported [47], where the measured Γ − Z dispersion of bands
with binding energies up to 3eV is on the order of the instrumental resolution of
100meV, while the calculated values for the corresponding valence bands are at least
twice the measured values.energy excitations in the ARPES data with calculations,
a consequence of the system being somewhat electron-doped. The strongest signals
are observed from the surface states.
We would like to note that the calculated bandwidth is smaller than the measured
40
Low
High
EF
0.1
M
Γ
M
K
Γ
K
SS
SS
EB (eV)
Intensity (arb. units)
0.0
-0.1
-0.2
-0.3
-0.4
-0.15
0.00
0.15
-0.15
-1
(a)
ky (Å )
0.00
0.15
-0.12
0.00
0.12
-1
(b)
kx (Å )
(c)
-1
ky (Å )
Figure 3.4: High-resolution ARPES measurements of surface electronic band dispersion on Bi2 Se3 (111). Electron dispersion data measured with an incident photon
energy of 22eV near the Γ̄-point along the Γ̄ − M̄ (a) and Γ̄ − K̄ (b) momentum-space
cuts. (c) The momentum distribution curves corresponding to (a) suggest that two
surface bands converge into a single Dirac point at Γ̄. The V-shaped pure SS band
pair observed in (a-c) is nearly isotropic in the momentum plane, forming a Dirac
cone in the energy-kx − ky space (where kx and ky are in the Γ̄ − K̄ and Γ̄ − M̄ directions, respectively). The U-shaped broad continuum feature inside the V-shaped
SS corresponds roughly to the bottom of the conduction band.
values. A better agreement requires a renormalization of the theoretical results by a
self-energy given by:
−
→
→
−
→
−
Σ( k , E) = Σ0 ( k , E) + iΣ”( k , E)
(3.1)
A similar adjustment of ab. initio results to match experimental band data was
used for graphite. It is believed that the underestimation of quasiparticle bandwidths
by LDA is due to its failure to account for many body effects in describing the
−
→
excited final states. Approximating Σ0 ( k , E) by a linear function stretches the band
dispersion [55]. For Bi2 Se3 , an approximately 30% stretch is required to match the
lowest energy excitations to theory.
41
A global agreement between the experimental band structure (Figure 3.4(a-c)) and
our theoretical calculation (Figure 3.2(c)) is obtained by considering a rigid shift of the
chemical potential by about 200 meV with respect to our calculated band structure
(Figure 3.2(c)) of the formula compound Bi2 Se3 . The experimental sign of this rigid
shift (the raised chemical potential) corresponds to an electron doping of the Bi2 Se3
insulating formula matrix. This is consistent with the fact that naturally grown Bi2 Se3
semiconductor used in our experiment is n-type, as independently confirmed by our
transport measurements. The natural doping of this material, in fact, comes as an
advantage in determining the topological class of the corresponding undoped insulator
matrix, because we would like to image the SSs not only below the Fermi level but
also above it, to examine the way surface bands connect to the bulk conduction band
across the gap.
A unique determination of the surface band topology of purely insulating Bi1−x Sbx
[32, 33] was clarified only on doping with a foreign element, Te. In our experimental
data on Bi2 Se3 , we observe a V-shaped pure SS band to be dispersing towards EF ,
which is in good agreement with our calculations. More remarkably, the experimental
band velocities are also close to our calculated values. By comparison with calculations combined with a general set of arguments presented above, this V-shaped band
is singly degenerate. Inside this ‘V’ band, an electron-pocket-like U-shaped continuum is observed to be present near the Fermi level. This filled U-shaped broad feature
is in close correspondence to the bottom part of the calculated conduction band continuum (Figure 3.2(c)). Considering the n-type character of the naturally occurring
Bi2 Se3 and by correspondence to our band calculation, we assign the broad feature
to correspond roughly to the bottom of the conduction band.
To systematically investigate the nature of all the band features imaged in our
data, we have carried out a detailed photon-energy dependence study, of which selected data sets are presented in figure 3.5(a) and (b). A modulation of incident
42
High
Low
EB (eV)
19 eV
21 eV
0.0
31 eV
-0.2
-0.4
-0.6
(a)
-0.2 0.0 0.2 -0.2 0.0 0.2 -0.2 0.0 0.2
-1
ky (Å )
hν
(eV)
31
28
25
(b)
3.2
kz (Å -1)
22
21
20
19
18
17
16
15
M
Γ
M
Γ
F
31
eV
Z
2.8
L
22
21
2.4 19
Γ
15
-0.8
-0.4 0.0
EB (eV) (c)
0.0 0.8
-1
ky (Å )
Figure 3.5: (a)The energy dispersion data along the Γ̄-M̄ cut, measured with the
photon energy of 21 eV (corresponding to 0.3 k-space length along Γ-Z k kz ), 19 eV
(Γ) and 31 eV (-0.4 k-space length along Γ-Z of the bulk three-dimensional Γ BZ)
are shown. (b) The energy distribution curves obtained from the normal-emission
spectra measured using 15-31 eV photon energies reveal two dispersive bulk bands
below -0.3 eV (blue dotted lines). The Dirac band intensity is strongly modulated by
the photon energy changes due to the matrix-element effects (which is also observed in
BiSb; [32]). (c) A k-space map of locations in the bulk three-dimensional BZ scanned
by the detector at different photon energies over a theta (θ) range of ±30◦ . This map
(kz , ky , Ephoton ) was used to explore the kz dependence of the observed bands.
43
photon energy enables us to probe the kz dependence of the bands sampled in an
ARPES study (Figure 3.5(c)), allowing for a way to distinguish surface from bulk
contributions to a particular photoemission signal [32]. Our photon-energy study did
not indicate a strong kz dispersion of the lowestlying energy bands on the ‘U’, although the full continuum does have some dispersion (Figure 3.5). Some variation of
the quasiparticle intensity near EF is, however, observed owing to the variation of the
electron-photon matrix element. In light of the kz -dependence study (Figure 3.5(b)),
if the features above -0.15 eV were purely due to the bulk, we would expect to observe dispersion as kz moved away from the Γ-point. The lack of strong dispersion
yet close one-to-one correspondence to the calculated bulk band structure suggests
that the inner electron pocket continuum features are probably a mixture of surfaceprojected conduction-band states, which also include some band-bending effects near
the surface and the full continuum of bulk conduction-band states sampled from a
few layers beneath the surface. Similar behavior is also observed in the ARPES study
of other semiconductors [22]. In our kz -dependent study of the bands (Figure 3.5(b))
we also observe two bands dispersing in kz that have energies below -0.3eV (blue dotted bands), reflecting the bulk valence bands, in addition to two other non-dispersive
features associated with the two sides of the pure SS Dirac bands. The red curve is
measured right at the Γ-point, which suggests that the Dirac point lies inside the bulk
bandgap. Taking the bottom of the ‘U’ band as the bulk conduction-band minimum,
we estimate that a bandgap of about 0.3 eV is realized in the bulk of the undoped
material. Our ARPES estimated bandgap is in good agreement with the value deduced from bulk physical measurements [49] and from other calculations that report
the bulk band structure [46, 50]. This suggests that the magnitude of band bending
near the surface is not larger than 0.05 eV. We note that in purely insulating Bi2 Se3
the Fermi level should lie deep inside the bandgap and only pure surface bands will
contribute to surface conduction. Therefore, in determining the topological character
44
of the insulating Bi2 Se3 matrix the ‘U’ feature is not relevant.
We therefore focus on the pure SS part. The complete surface FS map is presented in figure 3.6. Figure 3.6(a) presents electron distribution data over the entire
two-dimensional (111) surface BZ. All the observed features are centered around Γ̄.
None of the three TRIMs located at M̄ are enclosed by any FS, in contrast to what
is observed in Bi1−x Sbx [32]. The detailed spectral behavior around Γ̄ is shown in
figure 3.6(b), which was obtained with high momentum resolution. A ring-like feature formed by the outer ‘V’ pure SS band (a horizontal cross-section of the upper
Dirac cone in figure 3.4 surrounds the conduction-band continuum centered at Γ̄.
This ring is singly degenerate from its one-to-one correspondence to band calculation (Figure 3.7). An electron encircling the surface FS that encloses a TRIM or a
Kramers point obtains a geometrical quantum phase (Berry phase) of π mod 2π in
its wavefunction [18]. Therefore, if the chemical potential (Fermi level) lies inside the
bandgap, as it should in purely insulating Bi2 Se3 , its surface must carry a global π
mod 2π Berry phase. In most spin-orbit materials, such as gold (Au[111]), it is known
that the surface FS consists of two spin-orbit-split rings generated by two singly degenerate parabolic (not Dirac-like) bands that are shifted in momentum space from
each other, with both enclosing the Γ̄-point [51, 52]. The resulting FS topology leads
to a 2π or 0 Berry phase because the phases from the two rings add or cancel. This
makes gold-like SSs topologically trivial despite their spin-orbit origin.
To further confirm the π surface Berry’s phase on Bi2 Se3 , spin-polarized ARPES
was performed on the system (Figure 3.8). A spin-resolved momentum distribution
curve was taken along the M̄ − Γ̄ − M̄ direction, at a binding energy of -140meV
below the Fermi level. Figure 3.8(d) shows the corresponding polarization in the
Mott detector reference frame, also labeled as x̃, ỹ, and z̃ in the first chapter. The
polarization in the y direction clearly shows that the surface bands are spin-polarized.
Furthermore, Py is significantly larger than that along the x and z directions. Accord-
45
Bi2Se3
(a)
0.2 (b)
ky (Å-1)
1
SS
M
M
Γ
-1
-1
0
kx (Å-1)
1
-1
ky (Å )
0
Γ
0.0
Gold
Bi2Se3
-0.2
Γ
K
-0.2
0.0
-1
kx (Å )
M
0.2
(c)
Bulk Cond.
0
EB (eV)
EF
SS
-0.2
SS
Bi2Se3
Gold
-0.4
Bulk Valence
M
Γ
(d)
Bulk Valence
M
M
-0.1
M
Γ
(e)
(f)
0.0 0.1
k (Å -1 )
Figure 3.6: (a) The observed surface FS of Bi2 Se3 consists of a small electron pocket
around the center of the BZ, Γ̄. (b)High-momentum-resolution data around Γ̄ reveal
a single ring formed by the pure SS V-shaped Dirac band. For the naturally occurring
Bi2 Se3 , the spectral intensity in the middle of the ring is due to the presence of the
’U’ feature, which roughly images the bottom of the conduction-band continuum (see
the text). The observed topology of the pure surface FS of Bi2 Se3 is different from
that of most other spin-orbit materials such as gold (Au(111)). (c) The Au(111)
surface FS features two rings (each non-degenerate) surrounding the Γ̄ point. An
electron encircling the gold FS carries a Berry phase of zero, characteristic of a trivial
band insulator or metal, and can be classified by Z2 = +1 [18]. The single surface
FS observed in Bi2 Se3 is topologically distinct from that of gold. The single nondegenerate surface FS enclosing a Kramers point (Γ̄) constitutes the key signature
of a topological-insulator phase characterized by Z2 = −1. (d), (e) Schematic SS
topologies in gold and Bi2 Se3 for direct comparison.
46
νi={1,000}
Bi2Se3
M
Γ
M
K
Γ
K
Bulk Cond.
EF
EB (eV)
0.0
Bi2Se3
0.4
-0.2 0.0-1 0.2-0.2 0.0 0.2
M
-1
(a)
ky (Å )
kx (Å )
(c)
0.2
ky
M
E
kx
0.1
k y (Å -1 )
Bulk Valence
Γ
0.0
-0.1
-0.2
-0.2 -0.1 0.0 0.1 0.2
(b)
-1
kx (Å )
(d)
Figure 3.7: The (a) surface dispersion underlying the (b) Fermi surface with a chiral
left-handed spin texture. (c), (d) Schematic of the surface and bulk band topology.
47
Intensity (arb. units)
3 (a)
3
Iy¯
Iy­
2
(b)
Iz
(c)
1
2
0
3
1
2
1
0
0
Spin polarization
-0.1
(d)
Ix
0.0
-1
k (Å )
-0.1
0.1
0.0
0.1
-1
k (Å )
0.5
Px
0.0
Pz
Py
-0.5
-0.1
0.0
0.1
-1
k (Å )
Figure 3.8: (a) Spin-polarized momentum distribution curve along the M̄ − Γ̄ − M̄
direction at a binding energy of -140meV, measured along the y direction. The data
shows a clear splitting of the two surface bands across Γ̄. Similar momentum distribution curves measured in the (b) x and (c) z directions suggest slight polarizations
as well. (d) The corresponding polarization curves however indicate that the spins
are mostly inside the (111) plane, lying tangential to the surface Fermi surface.
48
ingly, the spins are pointed largely inside the (111) plane and tangential to the Fermi
surface. The spin-resolved momentum distribution curves for the spin components
parallel (Iy↑ ) and antiparallel (Iy↑ ) to the y direction are shown in figure 3.8(a). The
peaks of the two curves are consistent with the surface bands being spin-polarized at
the two sides of the Brillouin zone center. While along the x and z direction one also
finds slight polarizations in the spin-polarized momentum distribution curves, one
must remember that their respective polarization is at least a factor of two smaller
than that along y. The measured polarization in these directions could be due to
instabilities of the Mott detectors and variability of the incident photon beam.
Our theoretical calculation supported by our experimental data suggests that in
insulating Bi2 Se3 there exists a singly degenerate surface FS which encloses only one
Kramers point on the surface Brillouin zone. This provides evidence that insulating
Bi2 Se3 belongs to the Z2 = −1 topological class in the Fu-Kane-Mele topological
classification scheme for band insulators. On the basis of our ARPES data we suggest
that it should be possible to obtain the fully undoped compound by chemically holedoping the naturally occurring Bi2 Se3 , thereby shifting the chemical potential to lie
inside the bulk bandgap. The surface transport of Bi2 Se3 prepared as such would
therefore be dominated by topological effects as it possesses only one Dirac fermion
that carries the non-trivial Z2 index. The existence of a large bulk bandgap (0.3 eV)
within which the observed Z2 Dirac fermion state lies suggests the realistic possibility
for the observation of topological effects even at room temperature in this material
class. Because of the simplest possible topological surface spectrum realized in Bi2 Se3 ,
it can be considered as the ‘hydrogen atom’ of strong topological insulators. Its
simplest topological surface spectrum would make it possible to observe and study
many exotic quantum phenomena predicted in topological field theories, such as the
Majorana fermions [40], magnetic monopole image [35, 36] or topological exciton
condensates [41], by transport probes.
49
3.2
Bi2Te3 Topological Insulator
In this section, we investigate the bulk and surface dispersions of Bi2 Te3 [58]. Interestingly, the surface Fermi surface is very similar to that in Bi2 Se3 . Calculated band
structure along the K̄-Γ̄-M̄ direction shows that spin-orbit coupling induces a single
metallic surface band near the zone center (Figure 3.9(a)). The resulting surface
Fermi surface is a single non-degenerate ring, centered at Γ̄ (Figure 3.9(e)).
To determine the topological character of the insulator, a systematic study of
its band structure was then performed. The high energy valence bands measured
by ARPES is well described by the calculations, after taking into account spin-orbit
coupling (Figure 3.9(b)). As seen in Bi2 Se3 , a band structure strongly dependent
on spin-orbit coupling could suggest a nontrivial topological invariant. Data taken
near the Fermi level show surface states that are metallic, characterized by a single
Dirac cone crossing EF (Figure 3.9(c)). The density of states at the Fermi level forms
a single ring around the surface Brillouin zone center. As suggested by theory, the
underlying surface band is spin polarized. Accordingly, the Fermi surface feature
carries a Berry’s phase of π, and the electronic structure belongs to the Z2 = −1
topological class.
Theoretical calculations on the system suggest that Bi2 Te3 is an indirect gap
insulator in the bulk. The valence band maximum (VBM) lies in the Γ − Z − L
plane in the three-dimensional Brillouin zone, conventionally labeled as b. The bulk
conduction band minimum (CBM) above EF is slightly displaced, lying at the d
point in the Γ − Z − L plane. As found by Youn et. al. and Mishra et. al. [50, 59],
six copies of such points exist in the 3D Brillouin zone, forming indirect band gaps
inside the bulk. To confirm that the sample studied is indeed a bulk insulator, a
series of ARPES scans at various kz values cutting through the VBM and CBM
were taken. The presented data correspond to scans measured with hν = 31, 35
and 38eV incident photons (Figure 3.10(b-d)). No kz dispersion of the Dirac bands
50
Bi2-x MnxTe 3 (111)
(a)
0.6
(b)
0
K M
0
0.4
G
-0.2
E B (eV)
(eV)
EEBB(eV)
0.2
0
-0.2
-0.4
-1
(e)
-2
0.1
0.1
-0.6
0.0
0.4
-1
k (Å )
Single Spin-Dirac cone
-1
-0.4
k y (Å )
-0.8
0.0
0.0
-0.1
-0.1
Intensity (arb. units)
0.0
E B (eV)
-0.1
-0.2
-0.3
-0.4
-0.2
(c)
kx
-0.1
0.0
0.1
-1
k x (Å )
-0.05 0
0.05
-1
G
k x (Å )
-0.3 -0.2 -0.1 0.0
0.2
(d)
E B (eV)
Figure 3.9: (a) Calculated band structure along the K̄ − Γ̄ − M̄ direction in the (111)
surface Brillouin zone. Bulk band projections are represented by the shaded region.
The band structure results with SOC are presented in blue and that without SOC
in green. Without SOC (black lines), no pure surface states are observed. On the
other hand, one set of surface states are observed with SOC (red lines). (b) Second
derivative image of valence band spectrum along Γ̄− M̄ . (c) The surface states imaged
1 hour after cleavage. The dashed lines are guides to the eye. (d) The corresponding
energy distribution curves of the data. (e) The surface Fermi surface measured finds
a single ring located around the zone center.
51
kz
(a)
a
d
3
2
z
b
L
1
G
kx
ky
(b) hn = 31eV
EB (eV)
0.0
1
(c) hn = 35eV
2 Bulk gap
(d) hn = 38eV
3
-0.2
-0.4
-0.6
d
b
-0.8
0.2
-1
0.0
0.2
kx (Å )
-1
kx (Å )
0.0
0.2
-1
0.0
kx (Å )
Figure 3.10: (a) The three dimensional Brillouin zone of Bi2 Te3 . (b-d) ARPES scans
along cuts 1, 2, 3 labeled in (a), taken with hν = 31, 35 and 38eV incident photon
energies one hour after cleavage.
52
E B (eV)
Bi 2-x Mn x Te 3 (x = 0)
0
(a) 8min
low
high
(g) k = 0
-2
(c) 40min
(b) 20min
-1.5
(x = 0.05)
low
E B (eV)
(d) 15min
high
(e) 4hours
E B (eV)
-0.2
-1
(f) 9hours
0.0
0
0.0
0.0
-0.1
-0.1
-0.1
-0.2
-0.2
-0.2
-0.3
-0.3
-0.3
-0.5
-0.2
-0.1
-0.10 -0.05
0
0.00
0.05
0.1
0.10
-0.1
0
0.1
-1
k (Å )
-0.10 -0.05
0.00
0.05
D
0.10
0
-0.1
-0.10 -0.05
0
0.00
0.05
0.1
0.10
(x = 0.05)
15 mins
4 hrs
9 hrs
Int. (arb. units)
Figure 3.11: ARPES scans of Bi2 Te3 along the Γ̄ − M̄ direction taken at 30eV (a) 8
minutes (b) 20 minutes and (c) 40 minutes after cleavage. Similar ARPES scans of
Bi1.95 Mn0.05 Te3 along the Γ̄ − M̄ direction taken at 30eV (d) 15 minutes (e) 4 hours
minutes and (f) 9 hours after cleavage. (g) The corresponding energy distribution
curves showing the surface state time evolution after cleavage.
is observed with variations in hν, confirming that their density of states lie mostly
in the surface layers. However, near kx = 0.27Å−1 , a hole-like band is found to
strongly disperse with kz . The maximum of this band rises up to -150meV below
EF , with hν = 35eV incident photon energy. Using the free electron approximation,
this corresponds to (0.27,0,0.27)Å−1 in the 3D Brillouin zone, a close agreement with
calculations. The CBM at the d = (0.17, 0, 0.27)Å−1 corresponds to an incident
photon energy of hν = 38eV. However, scans taken at this energy did not produce
any signal from the CBM, indicating that EF lies inside the bulk band gap. Tunneling
[60] and optical [61] measurements have found indirect band gaps of > 150meV in
the system, in agreement with our measurements.
As reported previously by Noh et. al. [62], the surface states in Bi2 Te3 showed
significant time dependence shortly after cleavage. In this study, the binding energy
53
of the surface Dirac node was found to be EB ∼ −100meV at 8 minutes after cleavage
(Figure 3.11(a)), which increased to EB ∼ −130meV at 40 minutes (Figure 3.11(c)).
Such effect can be due to the breaking of Te(1)-Te(2) bonds inside the quintuple layers,
thereby inducing an electric field near the surface. The relatively long time scale of the
band bending process suggests that the surface charge accumulation could be due to
a slow relaxation of the lattice. The origin of this delay can be significant site defects
[60, 63], which create local lattice or charge density instabilities. To study the effect
of site defects on the equilibrium time scale, Bi atoms were systematically substituted
with Mn in the crystal lattice. The relaxation time scale was found to lengthen up
to an order of magnitude, as additional defects were introduced. Specifically, for
Bi1.96 Mn0.05 Te3 , the valence band edges were shifted by 100meV over a period of 15
hours, approaching the equilibrium position (Figure 3.11(g)).
In the last part of this section, we explore the effect of a higher order perturbation
on the spin texture of the Fermi surface. Recently, theorists have predicted that [64]
significant modulations of the surface states can occur if one includes a unconventional
hexagonal warping term in the surface band Hamiltonian analogous to the cubic
Dresslhaus spin-orbit orbit term for a bulk rhombohedral system:
Hw =
λ 3
3
(k + k−
)σz
2 +
(3.2)
Specifically, the surface Dirac cone would be hexagonally deformed, deviating from
the linear geometry observed in Bi2 Se3 . Additionally, as one moves 2π around the
Kramer’s point, electrons lying on this deformed cone would carry spins that are
twisted out-of-plane.
As the strength of the warping term increases with distance from Dirac node, we
have doped Bi2 Te3 in the bulk with excess electrons to raise its chemical potential.
Figure 3.12(a) and (b) show the corresponding ARPES spectra in the Γ − M and
54
Figure 3.12: High-resolution ARPES measurements of surface electronic band dispersion on Bi2 Te3+x along (a) Γ − M and (b) Γ − K. Evolution of the Fermi surface (FS)
features as the chemical potential is rigidly lowered to (d) -20meV, (e) -60meV, (f)
-120meV and (g) -230meV. (h)-(k) The corresponding constant energy contours from
first principle calculations. (c) A schematic of the FS evolution in the surface BZ.
55
Γ − K directions. Compared to undoped Bi2 Te3 the Dirac node now lies at a higher
binding energy, at 0.3eV. Additionally, an electron pocket is observed near the zone
center. Figures 3.12(d)-(g) present the photoelectron emission intensity from the
sample at several different binding energies integrated over a finite energy window.
Near the Fermi level, the constant energy contour is snow-flake like. As the binding
energy is increased the vertices gradually retreat, and the contour becomes a hexagon
with vertices in the Γ − M direction. Further lowering of the binding energy finds a
circular feature, which is similar to that observed in Bi2 Se3 .
The Fermi surface topology is described fairly well by first principles calculations,
as seen in figures 3.12(h)-(k). Fu has found [64] that the constant energy contour
evolution arises naturally from the Hw perturbation. The hexagonal warping term is
strongest along the Γ − K direction and weakest along the Γ − M direction. This
behavior is in good agreement with the ARPES data, as the contour extends the
furthest along Γ − M .
Another consequence of the warping term is the addition of an out-of-plane component for the surface electron spins. Figure 3.13(a) presents the calculated σz spin
profile for Bi2 Te3+x . In the horizontal plane, a projection shows the Pz polarization
intensity in the surface BZ. Since σz is odd with respect to inversion symmetry, Hw
vanishes along the Γ − M plane. Accordingly, Pz should vanish in the Γ − M direction
and reaches its maximum along Γ−K. In figure 3.13(b) we present the z-polarization
data measured with spin-resolved ARPES at various points on the surface FS. In the
same figure the calculated Pz is presented, as a function angle θ away from the Γ − M
direction. The data is in good agreement with the calculation, displaying a
2π
3
peri-
odicity.
Additionally, as suggested by the evolution of the constant energy contours, Pz
should also be dependent on binding energy. Specifically, the polarization should increase as one moves further away from the Dirac node, as the energy contour deviates
56
Figure 3.13: (a) Calculated σz spin texture for Bi2 Te3 as a function of binding energy
(vertical) and surface momentum (horizontal). Measured z-polarization as a function
of (b) angle θ away from Γ − M and (c) binding energy. The calculated polarization
is overlaid for comparison (blue line).
further from that of an isotropic 2D Dirac fermion. Near the Dirac node, the data
shows a z-polarization within the error of the measurement. In this case, the spins
lie mostly inside the surface plane, rotating by 2π as one goes around the Kramer’s
point. At 0.3eV away from the Dirac point, near EF , an almost 30% Pz is observed
(Figure 3.13(c)). Because the contour is distorted at this energy, a finite out-of-plane
component is required to conserve the π Berry’s phase carried by a surface electron
traveling around Γ.
Therefore, we have experimentally shown that the hexagonal warping term induces
a σz component for the surface electron spins. It was suggested that the perturbation could serve as an additional method to introduce a gap at the surface Dirac
node. Instead of using fields perpendicular to the sample surface, a relatively modest
field parallel to the surface is sufficient. Additionally, enhanced scattering signal is
expected in scanning tunneling measurements, due to the availability of new scatter57
ing channels with the distorted FS. Interesting new phases could also emerge in this
system, such as spin orderings reflective of an exotic spin-density-wave.
3.3
Sb2Te3 Topological Insulator
So far, all reported STIs have been based on the strong spin-orbit element Bi. Here,
we report the first experimental observation of surface states in Sb2 Te3 . Our photoemission data, coupled with first-principles calculations, suggest that the undoped
material belongs to the strong topological class, characterized by Z2 = 1.
Sb2 Te3 , along with Bi2 Se3 and Bi2 Te3 , have been well studied for their excellent
thermoelectric properties. The thermoelectric figure of merit of the system is strongly
tied to its band structure properties, such as the bandgap and effective mass at the
local extremas [57]. Therefore, much theoretical work has been done to comprehensively map its bulk band structure. It has been shown that to accurately describe the
multiple-valley features of the valence and conduction bands, a strong spin-orbit coupling is required [65]. While optical measurements give a band-gap 280meV at room
temperature [66], calculations using full-potential linearized augmented plane-wave
(FPLAPW) with generalizaed gradient approximation (GGA) give a much smaller
value of 30meV [67]. However, an earlier work using the same method but with unrelaxed lattice parameters gives a closer value of 0.278eV [65]. The crystal structure of
Sb2 Te3 is rhombohedral with the space group D53d (R3̄m). The unit cell contains five
atoms, with the quintuple layers ordered in the Te(1)-Sb-Te(2)-Sb-Te(1) fashion. The
Van der Waals couplings between the Te(1) layers are weak while the rest are strong
covalent bonds. Therefore, it is likely that the crystal is cleaved along the Te(1)
plane. The lattice parameters of the hexagonal cell are a = 4.25Å and c = 30.4Å [67].
Spin-orbit coupling brings the band gap from a direct one at Γ to an indirect one,
with the valence band maximum and conduction band minimum lying at non-high
58
symmetry points in the Γ-a-Z plane (Figure 3.16(e)).
We present surface band calculations for the Sb2 Te3 (111) surface. The calculations
were performed with the LAPW method in slab geometry using the WIEN2K package
[53]. GGA of Perdew, Burke, and Ernzerhof [54] was used to describe the exchangecorrelation potential. SOC was included as a second variational step using scalarrelativistic eigenfunctions as basis after the initial calculation was converged to selfconsistency. The surface was simulated by placing a slab of six quintuple layers in
vacuum. A grid of 35 × 35 × 1 points was used in the calculations, equivalent to 120
k-points in the irreducible BZ and 2450 k-points in the first BZ.
Figure 3.14(a) shows the calculated electronic structure of the Bi2 Se3 (111) surface
band along the K̄ − Γ̄ − M̄ direction. The result with and without SOC are overlaid
together for comparison. The bulk band projections are represented by the shaded
areas in blue with SOC and green without. In the bulk, time-reversal symmetry
demands E(~k, ↑) = E(−~k, ↓) while space inversion symmetry requires that E(~k, ↑) =
E(−~k, ↑). Therefore, all the energy bands should be doubly-degenerate. The bulk
bandgap calculated with SOC is 89meV. The disparity with experimental value is
not unusual, as DFT calculations often underestimate the bandgap. In the Γ̄ − M̄
direction, the conduction band minimum (CDM) is at (kx , kz ) = (0.068, 0.127), while
the valence band maximum (VBM) is at (-0.341,0.239). The detailed locations of the
local extremas have been reported elsewere [67, 68] so will not be discussed here. It
has been shown that the absolute CDM and VBM should lie in the Γ − Z − a plane,
therefore these two points should also be the absolute CDM and VBM. Converting
to hexagonal coordinates, they are given by (0.82, 0.78, 0.78) and (0.52, 0.32, 0.32)
respectively. While the position of the CDM is close to that reported by Wang et.al.
[67], the position of the VBM position is slightly off. However, it is very close to the
second highest valence band peak position reported by that work. This might be due
to the extreme flatness of the top of the bulk valence band, where a slight change of
59
Sb2Te3
(a)
(b)
G
Bulk
M
no soc
soc
K
EB (eV)
Surface
no soc
soc
(c)
0.1
-0.3
-0.145
0.145
k (Å -1 )
Momentum
Figure 3.14: (a) LDA calculation result of the 2D surface state dispersion in the
K̄ − Γ̄ − M̄ direction. Bulk band projections are represented by the shaded areas.
The result with SOC is colored in blue and that without is in green. No pure surface
bands are observed in the insulating gap without SOC (black lines). One pure surface
band EF crossing is observed when SOC is included (red lines). (b) The corresponding
surface FS is a single ring centered at Γ̄. The TRIMs in the (111) surface BZ are Γ̄
and the three M̄ points. (c) An enlargement of the surface dispersion with SOC near
Γ̄.
60
the lattice parameter can change the location of the VBM.
On the surface, space inversion symmetry is broken, so the degeneracy of the
surface bands can be broken by spin-orbit interactions (spontaneous Rashba effect).
Nevertheless, Kramers Theorem dictates that spin degeneracy should be preserved
at some high symmetry points of the surface BZ, called TRIMs. In the Sb2 Te3 (111)
surface BZ, these are given by Γ̄ and three M̄ points, located 60◦ away from each
other and in the middle of two Γ̄ points (Figure 3.14(b)). The calculated result
without SOC shows no pure surface bands crossing EF . SOC drastically changes the
band structure of the SS. One finds two singly degenerate surface bands emerging
from the bulk projection which are paired together at the Γ̄ point. The two bands
however do not recombine at M̄ , a “partner-switching” behavior (Figure 3.17(e)) that
is characteristic of strong topological metals [18] . The top surface band forms an
electron pocket at EF , giving a ring-like surface Fermi surface (FS) reminiscent of
that in Bi2 Se3 [56]. The Fermi velocities (vF ) in the Γ̄-M̄ and Γ̄-K̄ directions are
1.21 eV·Å and 1.16 eV·Å respectively. It should be emphasized that while doubly
degenerate quasiparticle states appear as two overlapping bands in the calculated
band structure, these two surface bands each appears only once, and represents one
eigenstate at a given momentum.
The measured valence band dispersion is then overlaid with the calculated bulk
band spectrum for comparison. Figure 3.15(c) presents the second derivative intensity
(SDI) plot of the quasiparticle signal along the Γ̄ − M̄ direction. Both the theoretical
and experimental plots are taken at kz = 0.1984, where the valence bulk band reaches
the absolute maximum. One finds a very good agreement after shifting the EF of the
calculated result to 300meV below the top of the valence band, a consequence of
doping the semiconductor with holes. The strong intensity of the bands, especially
that at around 0.75eV, suggests that the states cannot be purely due to the bulk.
Rather, they are surface states that are resonant with the bulk bands. At the lowest
61
High
M
EB (eV)
0.0
Low
M
Γ
K
Γ
K
-0.2
-0.4
(a)
(b)
-0.4
0.0
0.4 -0.4
0.0
0.4
k x (Å -1 )
k y (Å -1 )
M
K
EB (eV)
0
-1
(c)
-2
0.0
0.4
(d)
0.0
0.4
k (Å -1 )
Figure 3.15: High momentum scans near EF along (a) Γ̄− M̄ and (b) Γ̄− K̄ directions
show a pair of “/\“ bands that are resonant with the lower part of the calculated
surface Dirac bands. Second-derivative image and first-principles calculation result
of the bulk electron bands along (c) Γ̄ − M̄ and (d) Γ̄ − K̄ at kz =-0.77Γ − Z. A
rigid 300meV upshift of EF is included to match the lowest energy excitations in the
ARPES data with calculations, a reflection that the system is hole-doped.
62
binding energy, one finds a pair of band dispersing towards the Fermi level. While
one might identify them as the lower portion of the spin-polarized surface Dirac
bands, the bands are located inside the calculated bulk projection. Furthermore, these
bands strongly agree with the calculated dispersion of the highest valence band. The
observation suggests that this band pair are not pure surface states, but are resonant
states which are only partly due to the bottom of the surface Dirac band. High
momentum resolution data near EF along the Γ̄−M̄ and Γ̄−K̄ directions are presented
in figures 3.15(a) and (b). The EF crossings are at 0.06Å−1 in both directions, in
agreement with the calculated bulk valence band crossings. The extracted Fermi
velocities are approximately 3eV·Å and 2eV·Å in the Γ̄ − M̄ and Γ̄ − K̄ directions
respectively. In the Γ̄− K̄ direction (Figure 3.15(b)), one finds additional quasiparticle
intensities at normal emission between the band pair. This weak feature is due to the
bulk and provides further evidence that the bands near EF are lying inside the bulk
continuum.
To confirm the surface nature of the observed bands, a incident photon energy
dependence study was performed on the Γ̄ − K̄ cut. By changing the energy of the
incident photon, one moves to different kz value in the 3D bulk BZ (Figure 3.16(f)).
The inner potential used is approximately 9.5eV, given by the muffin-tin zero of
the calculation. The value is obtained by averaging the atomic potential over the
interstitial region, using a muffin-tin radius of 2.5 bohr. Moving the photon energy at
normal emission from 17eV, corresponding to -0.2Γ − Z, to 27eV or 0.4Γ − Z of the
next BZ (Figure 3.16(f)), one finds little kz dispersion of the “/\“ band pair around
Γ̄. A strong variation of the quasiparticle intensity however is observed near 0.3Å−1
and at Γ̄. At 18eV (Figure 3.16(a)), the weak feature between the “/\” bands is easily
observable, reminiscent of that in figure 3.15(b). This feature then gradually weakens
with changing photon energy, and eventually disappears above 24eV (Figure 3.16(c)).
The strong kz dependence suggests that this feature is bulk-like. The “/\“ bands,
63
High
(a) 18 eV
Low
(d) 26 eV
(c) 24 eV
(b) 22 eV
EB (eV)
0.0
-0.2
-0.4
-0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4
-1
kx (Å )
kz
3.0
K
M
kx
Γ
L
kz (Å-1)
ky
M
L
Γ
M
27 eV
2.7
2.4
Γ
F
2.1 17 eV
-1.0 -0.5
a
Γ
Z
(e)
Z
0.0 0.5
F
1.0
-1
(f)
kx (Å )
Figure 3.16: High angular resolution surface band dispersion near the Γ̄ point along
Γ̄-M̄ direction at (a) 18eV, (b) 22eV, (c) 24eV and (d) 26eV. (e) Red lines show the
approximate location of some of these cuts in the bulk 3D BZ. (f) Using the free
electron final state approximation, one can map the incident photon energy to the kz
momentum position. Our energy dependence study, which went from 17eV to 24eV,
corresponds to -0.2Γ − Z to 0.4Γ − Z of the next BZ in k-space. Solid arcs indicate
points in the bulk 3D BZ seen by the detector over a θ range of ±30◦ .
64
which are immediately located around this bulk feature, are therefore lying inside the
bulk continuum. They should be resonant with the bulk band, even though little kz
dependence is observed.
The surface Fermi surface is then presented in figure 3.17(a). One finds a ringlike Γ̄ feature due to the “/\” band, in addition to 6 petals at approximately 0.3Å−1 .
While the Fermi surface topology is reminiscent of that observed in Bi1−x Sbx and pure
Sb [32, 33], it is different in that all the features are spin-degenerate. The underlying
bands near EF are all resonant states located in the bulk continuum (Figure 3.17(c)),
where space-inversion symmetry holds. Nevertheless, due to the strong agreement
between our experimental data and theoretical calculation below the valence bulk
band maximum, we can use our surface band calculation to speculate on the topological character of undoped Sb2 Te3 . The calculation suggests that by moving EF
upwards, perhaps via chemical doping, the six petal RS states disappear, and the Γ̄
ring feature leaves the bulk-continuum to become purely surface-like. The 0.3Å−1 RS,
which is similar to that observed near the M̄ point in pure Sb (Figure 3.17(c)), could
be topologically removed and does not contribute to the Z2 character of the system
[33]. By removing these features, the remaining spin-polarized ring is analogous to
that observed in Bi2 Se3 . The resulting surface FS only encloses the Γ̄ point an odd
number of times, giving it a ν0 = 1.
In conclusion, we have calculated the band structure of Sb2 Te3 (111) surface and
found that SOC induces a single non-degenerate band crossing EF . The measured
ARPES data on the p-type sample agree with the calculation result below the valence
band maximum. The surface FS consists of resonant state features reminiscent of that
in Bi1−x Sbx . Based on the calculation, we speculate that the undoped sample carries
a Fermi surface topology similar to Bi2 Se3 , and belongs to the strong topological class
which supports a nontrivial Berry’s phase protected from disorder.
65
G
(a)
M
(b)
M
0.3
-1
ky (Å )
G
M
0.0
RS
Sb2Te3
-0.3
0.0
0.4
M
0.8
kx (Å-1)
(c)
Sb2Te3
SS
EF
(d)
Sb
EF
SS
RS
RS
RS
M
Γ
Γ
M
Γ
M
(e)
Figure 3.17: (a)The surface FS shows resonant states comprised of a ring around
Γ̄ surrounded by six additional petals at 0.3Å−1 . (b) A schematic of the surface
FS. (c) The underlying band dispersion is similar to that in (d) Sb(111), which also
contains several RS states. (e) A “partner-switching“ behavior is observed in the
surface band calculation, a critical characteristics in generating an odd number of EF
crossings. Therefore, by shifting EF downwards via chemical doping, one can achieve
a spin-polarized FS carrying a Berry’s phase of π.
66
Chapter 4
Surface State Control and
Manipulation
4.1
Chemical potential tuning in Bi2Se3
As a new phase of quantum matter, topological insulator with unique two dimensional edge states is a higher dimensional realization of the quantum spin Hall phase
[18, 20, 31]. Recent theory has suggested that a device comprised of a conventional
superconductor on top of a STI can support fault-tolerant schemes for quantum computation applications [40]. While the first example of a STI was recently discovered
in Bi1−x Sbx [32], the system contains alloying disorder with complex surface states.
The search for a better device candidate results in the large band-gap semiconductor
Bi2 Se3 , with a simple ring-like Fermi surface on the (111) surface [56]. While most
naturally grown Bi2 Se3 semiconductor is of n-type, significant progress has been made
recently to reduce the bulk carrier density [63]. Another key challenge of STI device
applications is controlling the two dimensional edge states. While the density of state
at the Fermi level (DOS(EF )) in a conventional quantum Hall system (QHE in 2DEG)
can be tuned via an external magnetic field, this method cannot be applied to the
67
STIs since a magnetic field breaks time reversal symmetry and the topological protection is thus lost. Furthermore, chemical techniques such as bulk doping to reach
the Dirac point are undesirable as they introduce extra carriers which either turn
the bulk into a semi-metal or an alloy, or increase the residual conductivity beyond
the surface conduction limit. None of these conditions are desirable for a functional
topological insulator. Here, we demonstrate a suite of surface doping methods which
can tune the carrier density on the surfaces of the topological insulator system Bi2 Se3
in a time-reversal invariant way. In studies of the two dimensional Dirac system
graphene, various alkali metal and gas molecules have been used to successfully tune
the carrier density [69, 70]. In this chapter, we present angle-resolved photoemission
spectroscopy which demonstrates the capability of engineering the surface state of
Bi2 Se3 with NO2 molecular absorption [71]. We show that the chemical potential can
be systematically tuned, thereby shifting the surface Dirac point onto the Fermi level.
Moreover, we report that photon stimulation can also control the surface carrier density. The resulting surface state is robust at room temperature, giving a graphene-like
Dirac cone with spin-polarized topological order on the two dimensional surface state.
The molecular doping of the Bi2 Se3 (111) surface is first confirmed by comparing
the measured valence band dispersion before and after exposure to NO2 gas. Additional spectral features are observed after doping, which have been attributed to
NO2 absorption. NO2 molecular doping of the Bi2 Se3 cleaved surface was achieved
via controlled exposures to NO2 gas (Matheson, 99.5%). The absorption effects were
studied under static flow mode: exposing the sample to the gas for a certain time then
taking data after the chamber was pumped down to the base pressure. The sample
temperature was maintained at 10K during exposure to NO2 . Figure 4.1 presents the
experimental valence bands with binding energy above 12eV before and after exposure to NO2 . On the clean samples (Figure 4.1(a)), dispersive Bi2 Se3 valence bands
are clearly observed at binding energy from 0 to 5eV, in addition to weak features
68
0.1
0
-0.1
(a) Before NO2 absorption
0.1
0
-0.1
Intensity (arb Units)
k (A-1)
Intensity (arb Units)
k (A-1)
Bi2-dCadSe3
(b) After NO2 absorption
No2
-10
-5
Binding Energy (eV)
0
Figure 4.1: (a) Before NO2 adsorption, the angle-resolved and integrated valence band
spectra are sharp and dispersive. (b) After exposure to NO2 gas, two non-dispersive
features appear at binding energies of 4eV and 7.5eV. The overall signal from the
Bi2 Se3 sample is weakened with No2 adsorption.
69
at binding energy of 9 11eV. After exposure to NO2 , two additional strong features
appear at binding energy of 4eV and 7.5 eV (Figure 4.1(b)). These new features are
non-dispersive. Accordingly, they could be attributed to photoemitted electrons from
randomly adsorbed molecules. With the additional layers of molecules on the sample
surface, the signal from the underlying Bi2 Se3 layer is weakened.
The surface band dispersions near Γ̄ with different NO2 dosages are shown in
figure 4.2. One finds two significant changes in the photoemission spectra. Firstly,
due to an increasing surface roughness, the measured spectra after NO2 exposure are
not as sharp as that from a clean surface. The total emission intensity is reduced,
as the photoelectron escape depth is increased. Secondly, after NO2 absorption, electrons on the Bi2 Se3 surface migrate to the NO2 molecules. This transfer of electrons
hole-dopes the electronic bands of Bi2 Se3 , thereby shifting the chemical potential
gradually towards the surface Dirac point. After a NO2 exposure of 0.1 Langmuir
(0.1L) (Figure 4.2(c)), the chemical potential is lowered by approximately 150meV.
The conduction bands become fully unoccupied, removing the resonant state feature
near EF . The pure surface band velocity is unchanged after NO2 exposure. However, the surface Dirac point (degenerate Kramers point) is rigidly shifted upwards
to 150meV below EF . Increasing amounts of NO2 doping remove additional electrons
from the Bi2 Se3 surface. Eventually, after an exposure of 2L of NO2 (Figure 4.2(f)),
the surface Dirac point reaches the Fermi level. The resulting surface bands at Γ̄
form a spin-polarized Dirac cone. No further changes of the chemical potential are
observed with additional exposures.
Therefore, the carrier density on the Bi2 Se3 surface can be controllably tuned via
NO2 absorption. Figure 4.3 shows the Fermi surface (FS) of the topological surface
state with different NO2 dosages. Without any NO2 exposure, the FS comprises
of a nearly rounded ring surrounding a resonant state feature at the center. After
exposing to 0.1L of NO2 , the FS becomes a small circle formed by the surface Dirac
70
Bi2Se3
0
(a) 0 L
(b) 0.01 L
(c) 0.1 L
(d) 0.5 L
(e) 1 L
(f) 2 L
-0.2
Binding Energy (eV)
-0.4
0
-0.2
-0.4
0
-0.2
-0.4
-0.1
0
0.1
-0.1
0
0.1
Momentum k (A-1)
Figure 4.2: (a) High resolution surface band dispersion near the Γ̄ point on the
Bi2 Se3 (111) surface. Two spin-polarized surface bands intersect at 300meV below EF ,
forming a Dirac point. Additional spectral weight from a resonant state is observed
within the “V” shape band. (b-f) With NO2 absorption, the chemical potential is
lowered, thereby removing excess electrons from the Bi2 Se3 conduction bands. The
Dirac point moves towards EF with increasing amounts of NO2 exposure. (c) At
0.1 Langmuir, the conduction band becomes fully unoccupied. (f) The Dirac point
reaches the Fermi level when the NO2 dosage exceeds 2L.
71
NO 2 Exposure (Langmuir)
(a) 0 L
(b) 0.1 L
(c) 2 L
0.4
k y (A -1 )
0.2
0
-0.2
-0.4
-0.2
0
-0.2-0.2
0
0.2 -0.2
0
0.2
k x (A -1 )
(d)
G
M
K
Bi2Se3
Figure 4.3: (a) A high resolution mapping of the pure Bi2 Se3 surface Fermi surface
around Γ̄. A ring with a resonant state at the center is clearly observed. (b) The inner
RS is removed with 0.1L of NO2 , leaving only a small circle formed by the Dirac cone
as the charge carrier density is reduced. (c) A single Dirac point Fermi surface at Γ̄
is obtained with 2L of NO2 gas. (d) A wide angular-range scan finds only one Dirac
point (odd number) between the two TRIMs (Γ̄ and M̄ ) in the extended Brillouin
Zone.
72
cone. A low photon flux beam was used when mapping the FS at 0.1L exposure,
to minimize photon-induced charge transfer (discussed below). Counting the surface
carrier density using Luttinger’s theorem, 0.1L of NO2 removes approximately 0.0066
electrons per unit cell of Bi2 Se3 . A total exposure of more than 2L of NO2 reduces
the FS to a single point within our experimental resolution, with an additional 0.005
electrons per unit cell removed from the surface. A wide angular-range scan of the
FS (Figure 4.3(d)) reveals a single (odd number) surface Dirac point in the extended
Brillouin Zone.
As reported by studies of NO2 absorption on graphene, extensive photon exposure
can remove NO2 molecules from the doped sample surfaces [70]. A very different behavior is observed on the Bi2 Se3 (111) surface. In our experiment, we found that NO2
absorption is very stable against photon exposure. Furthermore, very interestingly,
we observed additional photon-induced charge transfer on the NO2 doped surfaces.
By varying the photon flux, the chemical potential or hole-doping of the surface states
could be easily tuned. We present the photon-induced evolution of the surface band
structure in figure 4.4. To reduce the uncertainties due to photon-related effects,
each spectrum was collected within 1 minute after opening the photon shutter. Figure 4.4(a) presents the surface band dispersion after the sample is first exposed to
0.1L of NO2 . The resulting Dirac point is at about 150meV below EF . With photon
exposure the chemical potential gradually shifts downwards. After a dosage of 2×1013
photons, the Dirac point is shifted to approximately 100meV below EF . Successive
exposures continue to shift the chemical potential. After a total dosage of approximately 6×1014 photons, the chemical potential is stabilized as the Dirac point reaches
the Fermi level. No additional movement of the chemical potential is observed after
closing the photon shutter for at least an hour. The photon-induced effect is observed
for all photon energies used in the experiments (28-55eV). The detailed mechanism
behind this photon-stimulated effect is still under investigation. Nevertheless, we
73
Bi2Se3
0
(a) 1012 photons
(b) 1013 photons
(c) 2x1013 photons
(d) 1x1014 photons
(e) 3x1014 photons
(f) 6x1014 photons
-0.2
Binding Energy (eV)
-0.4
0
-0.2
-0.4
0
-0.2
-0.4
-0.2
-0.1
0
0.1
0.2 -0.2
-0.1
Momentum k (A-1)
0
0.1
0.2
Figure 4.4: (a) After 0.1L exposures of NO2 , the Dirac point is at approximately
150meV below EF . (b-f) Photon exposure induces additional electron transfer from
the Bi2 Se3 surface to the absorbed NO2 molecules, moving the surface chemical potential in a well-controlled manner towards the Dirac point. (c) With a dosage of
2 × 1013 photons, the Dirac point is moved to 100meV below EF . (f) The chemical
potential eventually stabilizes with 6 × 1014 photons, when the Dirac point reaches
the Fermi level.
74
have demonstrated that photon-induced hole doping can be a controllable way to
manipulate the surface bands.
Bi2 Se3 is a semiconductor with a 300meV bulk energy gap, making it a good candidate to exhibit topological effects at room temperature. To check that the surface hole
doping is retained at room temperature, the thermal stability of the doped Bi2 Se3
surface is carefully monitored after different amounts of NO2 exposures. Here, we
present the thermal stability of the Dirac point FS after exposing the sample to 2L of
NO2 . After the Dirac point is moved to the surface chemical potential via NO2 exposure, the sample is slowly warmed to room temperature. During the heating process,
the photon shutter was closed to avoid photon-induced doping effects. Figure 4.5(c)
shows the temperature evolution of the valence bands at 10K, 200K and 300K. One
finds that the intensities of the NO2 quasiparticle peaks do not change going from
the base temperature to room temperature. This observation indicates that within
our detection sensitivity, no NO2 desorption occurred during the heating process.
Accordingly, the hole-doped Bi2 Se3 surface should be robust at room temperature,
giving a stable Dirac point system analogous to graphene. The stability of samples
with other NO2 dosages is demonstrated in figure 4.6. Figure 4.6(a-b) presents the
surface bands spectra with a dosage of 0.5L NO2 at 10K and 300K. The samples were
warmed to room temperature at a rate of about 5 to 10 degrees per minute. The
position of the Dirac point is nearly identical ( 100meV below Fermi level), which
indicates that there is no significant NO2 thermal desorption that can change the
hole doping effect on the sample surface. This is further confirmed by the valence
bands spectra at 10K and 300k (Figure 4.6(c-d)). Few changes on the nondispersive
features at 4eV and 7.5eV are observed from the adsorped NO2 molecules.
Here, we discuss hole doping of Bi2 Se3 by introducing Ca atoms into the sample
bulk instead of the surface. Systematic time-dependent ARPES measurements were
taken for Bi2−δ Caδ Se3 , to study the electronic structure evolution of the system as
75
(a) 300K
Bi2Se3
(c)
-0.2
Intensity (arb. Units)
Binding Energy (eV)
0
-0.4
0
(b) 10k
300K
200K
10K
-0.2
0
-20
-10
Binding Energy (eV)
-0.4
-0.1 0
0.1
Momentum k (A-1)
Figure 4.5: (a) Warming the sample to room temperature from (b) 10K finds no additional change in the hole-doped surface chemical potential. The observation indicates
that the NO2 doped Bi2 Se3 surface is very stable, and the surface band remains a
Dirac cone at room temperature. (c) The NO2 doping is checked by monitoring the
valence band spectra with changing temperature. No obvious changes of the NO2
peak intensities are found during the heating process. Therefore, the desorption of
NO2 molecules at room temperature should be very weak.
76
Binding Energy (eV)
Bi2-dCadSe3
0
(a) 10K
(b) 300K
-0.2
-0.4
-0.1
0
0.1
-0.1
0
Momentum k (A-1)
0.1
(c) 10K
0.1
Momentum k (A-1)
0
-0.1
(d) 300K
0.1
0
-0.1
-10
-5
Binding Energy (eV)
0
Figure 4.6: (a) Surface states of sample after exposure to 0.5L of NO2 at 10K, which
is then slowly warmed to (b) 300K. The corresponding valence bands at (c) 10K and
(d) 300K.
77
Bi2Se3
Ca 0%
Intensity (arb. units)
EB (eV)
0.0
High
Low
Ca 1%
Ca 0.5%
Ca 0.25%
t = 18 hr.
-0.2
-0.4 (a)
-0.1 0.0
(b)
0.1
-0.1 0.0
(d)
(c)
0.1
-0.1 0.0
0.1
-0.1 0.0
0.1
(m)
EF
(e)
0.1
-0.1 0.0
-0.1 0.0
0.1
0.1
-0.1 0.0
0.1
(h)
(g)
(f)
-0.1 0.0
-0.1 0.0
0.1
-0.1 0.0
0.1
EF
(n)
-1
k (Å )
(l)
(k)
(j)
(i)
-1
k (Å )
0.3
25
(o)
(p)
-3
cm )
0.2
Bi0.9 Sb 0.1
20
15
n (10
r (mΩ cm)
20
Bi1.9975 Ca 0.0025 Se3
10
5 Bi1.9933 Sn 0.0067 Te 3
0
0.1
p-type
0.0
n-type
-0.1
-0.2
x = 0 x = 0.04 d = 0 d = 0.0025 d = 0.005
-0.3
x = 0 x = 0.04 d = 0 d = 0.0025 d = 0.005
Doping
Doping
Figure 4.7: Surface band dispersions of Bi2−δ Caδ Se3 within 20 minutes after cleavage
for (a) δ = 0, (b) δ = 0.0025, (c) δ = 0.005 and (d) δ = 0.01. (e-f) The corresponding
momentum distribution curves. Red lines are guides to the eye. (i-l) Schematic of
EF evolution as a function of Ca concentration δ. At long time scales, the surface
band spectra (m) revert to that in the δ = 0 case. (o) Resistivity and (p) Hall density
measurements for Bi2 Se3+x and Bi2−δ Caδ Se3 .
78
Low
15 min
219hours
eV
High
331hours
eV
15 hours
18 hours
0.0
EB (eV)
0.0
-0.5
18 hr.
15 hr.
3 hr.
2 hr.
15 min.
-0.2
-1.0
-0.4
-0.6
-1.5
(a)
(b)
(c)
(d)
-0.2 0.0 0.2 -0.2 0.0 0.2 -0.2 0.0 0.2 -0.2 0.0
(e)
0.2 -0.2
0.0
-2.0
0.2
(f)
Intensity (a. u.)
-1
k (Å )
Figure 4.8: (a) Surface spectrum shortly after cleavage reveals a Dirac point lying
above EF in Bi2−δ Caδ Se3 . (b-e) The spectrum gradually relaxes back to a δ = 0-like
state as time elapses, with the Dirac point moving gradually to higher binding energy.
(f) EDCs through the Γ̄ point suggest a shift of approximately 340meV over this time
interval.
a function of Ca doping. Early time scans taken through the (111) surface Brillouin
zone center are shown in figure 4.7 for several Ca concentrations. The surface band
dispersion of as-grown samples contains a single surface Dirac cone, with the Dirac
node lying 0.3 eV below EF . An additional resonant state is also observed, suggesting
that the Fermi level intersects the electron-like bulk conduction band. With 0.25%
concentration of Ca, EF is significantly lowered, as the Ca atoms donate holes to the
system. Because the bulk CBM lies at a binding energy of approximately -0.1 eV for
δ = 0 (Figure 4.7(a)), a 0.3 eV shift in EF between δ = 0 and δ = 0.0025 suggests
that the CBM lies 0.2eV above EF for δ = 0.0025. Previous tunneling [60] and optical
measurements [49] have found an indirect bandgap of approximately 0.35eV for the
system. Accordingly, EF should lie inside the bulk band gap at this Ca doping. With
increasing Ca concentration EF is lowered even further. At δ = 0.01, EF is moved
below the Dirac node (Figure 4.7(d)) and intersects the hole-like bulk valence band.
However, it was observed that for all samples, EF rises back up to the δ = 0like surface spectrum on a typical timescale of 18 hours (Figure4.7(m)). A similar
79
relaxation of the surface states has been observed in Bi2 Te3 [62]. Figure 4.8 shows
the time evolution of the surface states for δ = 0.5%. Immediately after cleavage
(Figure 4.8(a)), the Dirac point of the surface state lies above the Fermi level. This
spectrum is the most representative of the bulk states, as reflected in transport measurements that show bulk p-type behavior. Over a period of 18 hours, the chemical
potential gradually shifts upward as the Dirac point is moved below EF . Shifts in the
positions of the valence band peaks in the energy distribution curve (EDC) at Γ̄ (Figure 4.8(f)) over an 18 hours time interval reveal a band bending magnitude ∆ ∼340
meV. According, even though bulk Ca doping succeeds in tuning EF between the bulk
valence and conduction bands, it does not change the position of EF relative to the
surface Dirac point at the relaxation equilibrium.
Using high resolution ARPES, we have demonstrated for the first time a method
to controllably manipulate the surface states of a topological insulator. Our technique
produces a room temperature Dirac point at the Fermi level with topological quantum
properties. This work opens many new possibilities for STI device applications, as
well as future investigation of topological order in quantum systems.
4.2
Surface gap opening with magnetic impurities
To determine the nature of the Bi2 Se3 RS states and their connection to the conduction band structure, we have also carried out a systematic electron doping study
of the sample surface. Figure 4.9(b) presents ARPES data obtained at 29eV photon
energy along Γ̄ − M̄ after Fe atoms of less than 1% of a monolayer had been deposited
onto the sample surface. The Fe atoms were deposited using an e-beam heated evaporator at a rate of approximately 0.12Å/min measured by a quartz crystal thickness
monitor. With iron deposition, the material becomes strongly electron doped with
the bottom of the “V” band shifted by more than 200 meV (Figure 4.9). In addition
80
(a)
0.0 1 2 3
EB (eV)
Fe
Bi2Se3
(f)
RS
-0.2
SS
-0.4
-0.6
-0.8
-0.2 -0.1 0.0 0.1 0.2
(b)
-1
Bulk
M
(c)
Γ
M
ky (Å )
EF
Bulk Cond.
EF
RS
RS
Bi2Se3
SS
Bulk Valence
M
Γ
Fe Dep.
SS
(d)
(e)
M
M
Γ
M
Figure 4.9: (a) Iron atoms are deposited on the (111)-surface of Bi2 Se3 . (b) Upon
doping the surface with iron the chemical potential rises significantly. A pair of bands
emerge from the RS feature as the Fermi level is shifted up. (c) The corresponding
energy distribution curves reveal a small gap opening at Γ̄ between the “V” and the
“Λ”-shaped bands. This gap opening signals the breaking of the Kramers degeneracy
due the loss of time reversal symmetry brought about by the local magnetic field. (de) Schematic presentation of the SS bands before and after Fe deposition. The Dirac
point and Kramers degeneracy are intact before Fe deposition whereas a gap opens
after Fe deposition. Fermi level can then be tuned to fall in the gap, making the SS
fully gapped. This case would then be topologically equivalent to spin-orbit coupled
gold. The surface spectrum of Bi2 Se3 is equivalent to Fu-Kane partner switching Z2
topology [18].
81
to a nearly rigid shift in the chemical potential, the outer “V” band also becomes
less dispersive, with a decrease in the band velocity near EF . More interestingly, the
inner RS reveals itself to be the bottom of a pair of parabolic bands shifted from each
other in k-space. These bands are reminiscent of those observed in the SS of Au(111)
[51, 52] and Sb(111) [72] at Γ̄. Our iron-deposition results reveal that the RS feature
crosses EF twice, bringing the total number of crossings between a Γ̄− M̄ pair to three.
The observed odd number of band crossings between a pair of TRIM then guarantees
that the material has an intrinsic Z2 index of -1 independent of our correspondences
drawn between the experimental data and the theoretical band calculation.
It has long been suggested that magnetic impurities on the surface or the bulk
would have a strong effect on the protection of the Dirac (Kramers) point of a Z2
topologically order material. The time reversal breaking imposed by magnetic impurities is likely to induce a gap at the otherwise gapless Dirac point, thereby making
the bands develop some curvature and mass depending on the strength of time reversal breaking perturbation [18]. Since iron is magnetic, its use in doping the surface
in our experiment also naturally provides us with the possibility of observing the system’s response to the breaking of time reversal invariance. In the iron doped samples
we observe a clear bending of the surface state bands away from the Kramers point
(Figure 4.9(c)). Not only do the bands undergo linear to parabolic deformation in the
presence of magnetic impurities, a spectral weight suppression is also evident which
is the sign of a small gap opening at Γ̄ (Figure 4.9(b)). Therefore our data seems
to suggest that the Dirac-like bands near the Kramers point is sensitive to magnetic
impurities on the surface. This observation is in agreement with the Z2 theory of
topological phases [73].
82
4.3
Effects of surface engineering in Bi2Te3
In the previous chapter, various techniques for tuning the surface states of Bi2 Se3
were discussed. For the rest of this work, as a comparison, we will first examine
the effect of NO2 molecular adsorption on the sister compound Bi2 Te3 . A discussion of non-magnetic alkali atom deposition is also presented, as an experimental
control for the work on magnetic Fe impurities. Using angle-resolved photoemission
spectroscopy, we demonstrate the capability of electronic structure engineering of the
Bi2 Te3 surface with alkali atom (potassium, K) deposition and molecular adsorption
without breaking the time reversal or the Z2 invariance. We show that the chemical potential can be systematically tuned to shift the surface Dirac point downwards
in energy. Moreover, we report that the effect of NO2 molecular adsorption can be
further manipulated by photon-assisted stimulation, thereby controlling the surface
carrier density in opposite directions. These methods of tuning carrier densities on
the surface of a topological insulator open up new research avenues for a deeper understanding of topological quantum phenomena in Bi2 Te3 , in connection to its potential
for spintronic and quantum computing applications.
To review, calculated band structure along the K̄-Γ̄-M̄ direction in Bi2 Te3 shows
that spin-orbit coupling induces a single metallic surface band near the zone center
(Figure 4.10(a-b)). The resulting surface Fermi surface is a single non-degenerate
ring centered at Γ̄. Although the experimental results are in qualitative agreement
with calculations, and the chemical potential lying in the gap only intersects the
surface states (Figure 4.10), the observed differences could result from the fact that
all Bi2 Te3 samples have a relaxed cleaved surface [58, 62]. Since Te has charge 2− in
the stoichiometric compound, excess Te atoms were added in an attempt to slightly
hole-dope to move the chemical potential, thereby moving the Dirac point closer to
EF . Because the details of the surface relaxation (thus band bending) in Bi2 Te3 have
no simple correlation with the bulk carrier concentrations of our samples (insulating
83
Bi2Te3
Theoretical Calculation
(a)
E B (eV)
0.4
0.0
0.0
-0.1
Theo.
Dirac
point
G
M
-0.4
K
-0.2
-0.8
(b)
K
G
(d)
M
(e) 1 min. K Dep.
Bi2Te3+δ
(c)
G
(f) 2 min. K Dep.
E B (eV)
0.0
-0.3
Δ
-0.6
(g)
Experiment
-0.1
0.0
0.1
-0.1
0.0
0.1
Momentum k (Å-1)
-0.1
0.0
0.1
0
1x10 4 2x10 4
Intensity (a. u.)
Figure 4.10: (a-b) (111) Surface band (red lines) calculation with spin-orbit coupling
shows a pair of non-degenerate Dirac bands crossing the EF . The bulk band projection
is denoted by the shaded region in orange. The resulting (c) Fermi surface is a single
non-degenerate ring centered at the zone center. (d) ARPES data on Bi2 Te3+δ show
a slight rise of the chemical potential relative to the stoichiometric compound Bi2 Te3
[58]. The surface band structures after depositing potassium (K) for (e) one minute
and (f) two minutes show additional electron-doping of the surface states. The (g)
energy distribution curves at Γ̄ show that relative to the pristine sample (black), the
chemical potential is shifted by 140meV after one minute of (blue) of K deposition.
Little additional change of the EF position is observed upon an additional minute of
K deposition (red).
84
or not) it is not possible to find a unique fingerprint of the bulk insulating state by
just studying the surface Fermi surface shape. We have experimentally determined
that in the presence of relaxation, slightly excess Te atoms added in the bulk can
electron dope the surface (Figure 4.10(d)). These effects taken together may account
for some of the differences between calculated band structure and the experimental
data.
In figure 4.10(e) we present ARPES data taken at 30eV after potassium (K) is deposited onto the sample for approximately one minute. The material becomes strongly
electron doped, with the bottom of the “V” (Dirac band) band rigidly shifted by approximately 140meV. The Fermi velocity of the Dirac band is unchanged with doping.
In addition, we observe an inner resonant-state (due to surface band bending) feature
that appears to be the bottom of a pair of spin-split parabolic bands shifted from each
other in k-space. These bands are reminiscent of those observed in the SS of Au(111)
and Sb(111) at Γ̄. Assuming that each K atom donates one electron, Luttinger count
theorem suggests that the change of the electron pocket size corresponds to 7% of a K
monolayer deposited onto the sample surface. Additional K doping (Figure 4.10(f))
has little effect on moving the chemical potential. The signal from the Dirac point
however attenuates due to the increasing escape depth of the photoelectrons (Figure 4.10(g)). Nevertheless, no gap is observed as a result of deposition, confirming
that time-reversal symmetry is preserved suggesting the non-magnetic character of
the potassium valence state sticking onto the surface.
We then study the effect of NO2 doping on the Bi2 Te3 (111) surface states which
has shown to be effective in graphene [69, 70]. To check the adsorption or sticking
behavior of the molecular dopants (Figure 4.11(a-d)) we collected valence data with
a wide binding energy range above 15eV prior and after exposure to NO2 gas. On the
clean undoped samples (Figure 4.11(a-b)), dispersive Bi2 Te3 valence bands are clearly
observed at binding energies from 0 to 5eV, with some additional weak features at
85
Chemical Potential Tuning via Surface Doping
Bi2Te3
k (Å -1 ) Intensity k (Å -1 )
(arb. unit)
0.1
(a)
Bi2Te3
E
0.0
(j)
-0.1
(g)
(b) Before NO2 absorption
0.1
(e)
μ
G
(c)
k
0.0
Intensity
(arb. unit)
-0.1
O
Bi2Te3
NO2
(d) After NO2 absorption
-15
-10
O
N O
O
O
O
N
N
-5
(k)
0
Binding Energy (eV)
(e) 0L
(f) 0.15L
(g) 0.3L
(h) 0.6L
(i) 1L
(j) 1.5L
-0.2
Molecular Doping
Binding Energy (eV)
0.0
-0.4
0.0
-0.2
-0.4
-0.1
0.0
0.1
-0.1
0.0
0.1
-0.1
0.0
0.1
-1
Momentum k (Å )
Figure 4.11: Angle-resolved valence band spectra (a) prior to and (c) post exposure
to 1.5 Langmuirs of NO2 . Panels (b) and (d) show the angle integrated spectra for
(a) and (c), respectively. Prior to the NO2 adsorption, the valence band features
are sharp and dispersive with momentum. After exposure to NO2 gas, three nondispersive features appear at binding energies of 4.4eV, 7.5eV and 13eV. (e) High
resolution surface band dispersion data near the Γ̄ point suggest two non-degenerate
surface bands intersecting at 130meV below EF , forming a Dirac point. (f-j) With
NO2 adsorption, the chemical potential is systematically raised, thereby introducing
additional electrons into the Bi2 Te3 surface bands.
86
binding energies of 9-11eV characterizing the Bi2 Te3 matrix. Upon exposure to NO2 ,
three additional intense features appear at binding energies of 4.4eV, 7.5 eV and
13eV (Figure 4.11(c-d)). These new features are non-dispersive and independent of
the choice of incident photon energy, which suggest that they are due to the adsorbed
distributed molecules on the sample surface. Core level signals at similar binding
energies are known to appear upon depositing NO2 on carbon-based materials [70].
These results confirm the sticking or adsorption of NO2 molecules on the Bi2 Te3 (111)
surface.
The surface band dispersions near Γ̄ with varying NO2 dosages are shown in
Figure 4.11(e-j). After equal intervals of data collection time, one finds two significant
changes in the photoemission spectra. Firstly, due to an increasing surface roughness,
the measured spectra after NO2 exposure are not as sharp as that from a clean
surface. The total emission intensity is reduced, as the photoelectron escape depth
is increased. Secondly, after NO2 adsorption, the electron pocket around Γ̄ grows
significantly larger as the chemical potential gradually shifts away from the surface
Dirac point. After a NO2 exposure of 0.15 Langmuir (0.15L) (Figure 4.11(f)), the
chemical potential is raised by approximately 170meV. The surface conduction bands
gain partial occupation, introducing a weak resonant state feature near EF . The pure
surface band velocity is unchanged after NO2 exposure. However, the surface Dirac
point (degenerate Kramers point) is rigidly shifted downwards to 300meV below EF .
Increasing amounts of NO2 doping increase the electron carrier density on the Bi2 Te3
surface. Eventually, after an exposure of 1L of NO2 (Figure 4.11(i)), the surface Dirac
point reaches 350meV below EF . Inside the non-degenerate Dirac cone, a pair of spinsplit parabolic bands from the previously unoccupied surface conduction band are also
observed at around 60meV. Additional exposures deteriorate the sample significantly.
While the position of the Dirac point becomes difficult to locate, judging from the
size of the lowest energy electron pocket the shift of the Fermi level is minimal with
87
further dosages. This observation is in direct contrast with that observed in graphene
[70], where electrons are transferred from the graphene layer to the NO2 molecules.
We then estimate the amount of charge carrier transfer as a result of NO2 deposition by applying the Luttinger count theorem. Relaxing the sample for 1 hour
after cleavage, the charge concentration (Figure 4.11(e)) is approximately 0.00335
electrons per unit cell. After exposure to 1L of NO2 gas (Figure 4.11(i)), the surface
becomes strongly electron doped and approximately 0.039 electrons are transferred
to the surface. The mechanism behind this charge transfer is presently unclear which
makes it difficult to precisely estimate the amount of NO2 molecules that actually
dope the surface. Nevertheless, it is clear that a systematic tuning of the chemical
potential is possible with NO2 doping.
We have further investigated the effect of photo-induced changes on the surface
of Bi2 Te3 . In our experiment, we find that the effect of NO2 adsorption is unstable
against photon exposure. This result suggests that photon exposure could be an additional way with which to tune the electron-doped surface states on Bi2 Te3 (111).
We present the photon-induced evolution of the surface band structure data in Figure 4.12. To reduce the uncertainties due to photon-related effects, each spectrum was
collected within 1 minute after opening the photon shutter. Figure 4.12(a) presents
the surface band dispersion after the sample is first exposed to 0.3L of NO2 . The resulting Dirac point is at about 330meV below EF . With photon exposure the chemical
potential gradually shifts upwards. After a dosage of 1×1015 photons 30eV, the Dirac
point is shifted to approximately 250meV below EF . Successive exposures continue
to shift the chemical potential. After a total dosage of approximately 8×1015 photons
per mm2 , the chemical potential is found to be stabilized as the Dirac point reaches
130meV, the energy location prior to the NO2 exposure. The photon effect remains
after exposure is removed for extended periods. The induced effective EF shift is
observed for a wide range of UV energies (28-55 eV) and is found to be robustly
88
6x1014
Photons
E
μ
NO2
BiTe/Se
G
(m)
k
Photo-assisted manipulation of Dirac Carrier Density
EB (eV)
Bi2Te3
0.0
(a)
0.3L NO2
(b)
1x1015
(c)
3x1015
(d)
5x1015
(e)
7x1015
(f)
8x1015
-0.2
-0.4
-0.1
0.0
0.0
0.1
(g)
0.1L NO2 (h)
-0.15
0.0
-0.1
0.0
0.1 -0.1
1x1013 (i)
0.0
2x1013
0.1 -0.1
(j)
0.0
0.1 -0.1
1x1014 (k)
0.0
0.1 -0.1
3x1014 (l)
0.0
0.1
6x1014
-0.2
-0.4
0.15
-0.15
0.0
0.15 -0.15
0.0
0.15
-0.15
0.0
0.15 -0.15
0.0
0.15 -0.15
0.0
0.15
-1
Momentum k (Å )
Figure 4.12: (a) Upon exposure to 0.3L of NO2 on Bi2 Te3 , the Dirac point is located at
approximately 330meV below EF . (b-f) Photon exposure reverses the electron transfer to the Bi2 Te3 surface, moving the surface chemical potential in a well-controlled
manner towards the Dirac point. With a dosage total of (b) 1×1015 photons, the
Dirac point is moved to 250meV below EF . The Dirac point eventually reaches the
energy location prior to the NO2 exposure with a dosage of 8×1015 photons, which
is at about 130meV below EF . For comparison, the photo-induced manipulation of
Bi2 Se3 surface states exposed with (g) 0.1L of NO2 gas is also presented in (h-l).
With a dosage level of (i)2×1013 photons, the Dirac point is shifted by 50meV to to
100meV below EF . The chemical potential is eventually stabilized at the Dirac point
by the application of 6×1014 photons per mm2 .
89
reproducible.
For comparison with other topological insulators such as Bi2 Se3 , which may have
varying chemically-active surfaces with different relaxation details, we again present
the effect of photon exposure on the Bi2 Se3 (111) surface doped with NO2 molecules
(Figure 4.12(g-l)). We [71] have shown that exposure to NO2 dopants hole-dopes the
Bi2 Se3 surface states, moving the chemical potential to the Dirac point. While this
behavior is opposite to that observed for our Bi2 Te3 (telluride system), the additional
effect of photon exposure is similar for both systems. In figure 4.12(g) we present the
surface band dispersion data taken after a fixed dosage of 0.1L of NO2 . The resulting Dirac point is at a binding energy of approximately 150meV. Similar to Bi2 Te3 ,
photo-induced doping gradually moves the chemical potential downwards. A dosage
of 2×1013 photons at 30eV shifts the Dirac point by 50meV, while a total dosage of
6×1014 photons shifts the Dirac point to the Fermi level (Figure 4.12(l)). While the
effect of photo-induced changes in Bi2 Te3 could be attributed to photo-stimulated
desorption, the detailed mechanism in Bi2 Se3 is still under investigation by us. One
possible candidate is the well-known surface photovoltage effect, which has been observed for various class of semiconductors [74]. Nevertheless, we have demonstrated
that photon exposure can be a controllable mechanism to manipulate the topological spin-polarized surface bands, systematically driving the chemical potential to the
Dirac point via photon flux.
In summary, using high resolution ARPES, we have demonstrated for the first
time a suite of methods to controllably manipulate the topological edge states of
the topological insulator Bi2 Te3 on its (111) surface without breaking time reversal
symmetry. Our systematic data directly showed that both potassium (K) and NO2
electron-dopes the topologically ordered surface over a wide doping-range. Additionally, we showed that the effect of the dosage can be reversed by controlled exposure to
photon-flux. We further showed that photon exposure has a similar effect on Bi2 Se3 ,
90
which can be used to place the chemical potential to the much desired Dirac point.
Our work opens up many new possibilities for future utility of topological insulators
in spintronic and quantum computing research.
91
Chapter 5
Conclusions
This work presents the experimental discovery of the (Bi/Sb)2 X3 class of topological
insulators, using spin and angle-resolved photoemission. The work shows that a
combination of ARPES data and theoretical calculation can be used to identify the
topological insulators, and measure their Z2 topological invariants. Additionally, we
present several methods for tuning the surface states of these systems. The results
suggest that one could use the spin-polarized protected edge states of these quantum
Hall-like systems for spintronics or computation device applications, without high
magnetic fields or cryogenics. Much of the chapters can be found in the publications
[56], [71], and [58].
The next experimental step would be to directly detect the edge states of these
systems using transport measurement techniques, to look for signatures of weak antilocalization via anomalous magneto-optic effects. For detection using Hall measurements, one could introduce surface gaps in bulk systems that have been tuned via
the methods described in chapter 4, or in samples with lower dimensionality such as
nanoribbons [75] or thin films [76]. In device physics, one could construct structures
of topological insulators with superconducting films. Interaction between the helical
surface states and the superconducting states will induce the formation of Majorana
92
fermions [40], which could be manipulated for computation. One could also look for
novel physical phenomena using the topological insulator as a platform, such as magnetic monopoles[34, 35, 36], exciton condensates[41], and topological dislocations[38].
93
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Appendix A
List of Works Published
1. Xia, Y., Qian, D., Hsieh, D., Wray, L., Pal, A., Lin, H., Bansil, A., Grauer, D.,
Hor, Y.S., Cava, R.J. & Hasan, M. Z. Observation of a large-gap topologicalinsulator class with a single Dirac cone on the surface. Nature Phys. 5, 398
(2009).
2. Xia, Y., Qian, D., Wray, L., Chen, G. F., Luo, J. L., Wang, N. L., & Hasan,
M. Z. Fermi surface topology and low-lying quasiparticle dynamics of parent
Fe1+x Te/Se superconductor by orbital-polarization resolved ARPES. Phys. Rev.
Lett. 103, 037002 (2009).
3. Hsieh, D., Xia, Y., Qian, D., Wray, L., Meier, F., Dil, J. H., Osterwalder, J.,
Patthey, Fedorov, A. V., Lin, H., Bansil, A., Grauer, D., Hor, Y. S., Cava,
R. J. & Hasan, M. Z. Observation of time-reversal protected single-Dirac-cone
topological-insulator states in Bi2 Te3 and Sb2 Te3 . Phys. Rev. Lett. 103, 146401
(2009).
4. Hsieh, D., Xia, Y., Qian, D., Wray, L., Dil, J. H., Meier, F., Osterwalder, J.,
Patthey, L., Checkelsky, J. G., Ong, N. P., Fedorov, A. V., Lin, H., Bansil,
A., Grauer, D., Hor, Y. S., Cava, R. J. & Hasan, M. Z. Nature. A tunable
102
topological insulator in the spin helical Dirac transport regime. Nature. 460,
1101 (2009).
5. Hsieh, D., Xia, Y., Wray, L., Qian, D., Pal, A., Dil, J.H., Osterwalder, J.,
Meier, F., Bihlmayer, G., Kane, C.L., Hor, Y.S., Cava, R.J. & Hasan, M.Z.
Observation of unconventional quantum spin textures in topological insulators.
Science. 323, 919 (2009).
6. Hsieh, D., Qian, D., Wray, L., Xia, Y., Hor, Y.S., Cava, R.J. & Hasan, M.Z.
A topological Dirac insulator in a quantum spin Hall phase. Nature. 452, 970
(2008).
7. Xia, Y., Qian, D., Hsieh, D., Wray, L., Hasan, M.Z., Viciu, L. & Cava, R.J.
The observation of gapped quasiparticles near the metal-insulator transition in
Nax CoO2 . J. Phys. Chem. Sol. 69, 2986 (2008).
8. Wray, L., Qian, D., Hsieh, D., Xia, Y., Li, L., Checkelsky, J.G., Pasupathy,
A., Gomes, K.K., Parker, C.V., Fedorov, A.V., Chen, G.F., Luo, J.L., Yazdani,
A., Ong, N.P., Wang, N.L. & Hasan, M.Z. Momentum dependence of superconducting gap, strong-coupling dispersion kink, and tightly bound Cooper pairs
in the high-Tc (Sr,Ba)1−x (K,Na)x Fe2 As2 superconductors. Phys. Rev. B . 78,
184508 (2008).
9. Li, Y.W., Qian, D., Wray, L., Hsieh, D., Xia, Y., Kaga, Y., Sasagawa, T.,
Takagi, H., Markiewicz, R.S., Bansil, A., Eisaki, H., Uchida, S. & Hasan, M.Z.
X-ray imaging of dispersive charge modes in a doped Mott insulator near the
antiferromagnet/superconductor transition. Phys. Rev. B. 78, 073104 (2008).
10. Wray, L., Qian, D., Hsieh, D., Xia, Y., Gog, T., Casa, D., Eisaki, H. & Hasan,
M.Z. Intermediate dimensional character of charge transfer excitation modes in
a two-leg cuprate ladder. J. Phys. Chem. Sol. 69, 3146 (2008).
103
11. Wray, L., Qian, D., Hsieh, D., Xia, Y., Eisaki H. & Hasan, M.Z. Dispersive
collective charge modes in an incommensurately modulated cuprate Mott insulator. Phys. Rev. B. 76, 1 (2007).
12. Qian, D., Hsieh, D., Wray, L., Morosan, E., Wang, N.L., Xia, Y., Cava, R.J. &
Hasan, M.Z. Emergence of Fermi pockets in a new excitonic charge-density-wave
melted superconductor. Phys. Rev. Lett. 98, 117007 (2007).
13. Xia, Y., Qian, D., Hsieh, D., Wray, L., Viciu, L., Cava, R.J. & Hasan, M.Z. Observation of gapped quasiparticles in sodium charge-ordered Na1/2 CoO2 . Physica B. 403, 1007 (2008).
14. Qian, D., Hsieh, D., Wray, L., Xia, Y., Cava, R.J., Morosan, E. & Hasan,
M.Z. Evolution of low-lying states in a doped CDW superconductor Cux TiSe2 .
Physica B. 403, 1002 (2008).
15. Wray, L., Qian, D., Hsieh, D., Xia, Y., Gog, T., Casa, D., Eisaki, H. & Hasan,
M.Z. Dispersive collective charge modes in a spin 1/2 cuprate ladder. Physica
B. 403, 1456 (2008).
104
Appendix B
List of Works on Preprint
1. Lin, H. et. al. Single-Dirac-cone Z2 topological insulator phases in distorted
Li2 AgSb-class and related quantum critical Li-based spin-orbit compounds.
arXiv:1004.0999.
2. Lin, H. et. al. A new platform for topological quantum phenomena : Topological Insulator states in thermoelectric Heusler-related ternary compounds.
arXiv:1003.0155.
3. Hor, Y. S. et. al. The development of ferromagnetism in the doped topological
insulator Bi2−x Mnx Te3 . arXiv:1001.4834.
4. Hsieh, D. et. al. First observation of spin-momentum helical locking in Bi2 Se3
and Bi2 Te3 , demonstration of Topological-Order at 300K and a realization of
topological-transport-regime. arXiv:1001.1590.
5. Hsieh, D. et. al. Direct observation of spin-polarized surface states in the parent
compound of a topological insulator using high-resolution spin-resolved-ARPES
spectroscopy in a Mott-polarimetry mode. arXiv:1001.1574.
6. Wray, L. A. et. al. Observation of intertwined Fermi surface topology, orbital
parity symmetries and electronic interactions in iron arsenide superconductors.
105
arXiv:0912.5089.
7. Wray, L. A. et. al. Observation of unconventional band topology in a superconducting doped topological insulator, Cux -Bi2 Se3 : Topological Superconductor
or non-Abelian superconductor? arXiv:0912.3341.
8. Hsieh, D. et. al. A topological Dirac insulator in a quantum spin Hall phase
(experimental realization of a 3D Topological Insulator). arXiv:0910.2420.
9. Hsieh, D. et. al. Observation of topologically protected Dirac spin-textures
and π Berry’s phase in pure Antimony (Sb) and topological insulator BiSb.
arXiv:0909.5509.
10. Hsieh, D. et. al. Time-reversal-protected single-Dirac-cone topological-insulator
states in Bi2 Te3 and Sb2 Te3 : Topologically Spin-polarized Dirac fermions with
π Berry’s Phase. arXiv:0909.4804.
11. Xia, Y. et. al. Discovery (theoretical prediction and experimental observation)
of a large-gap topological-insulator class with spin-polarized single-Dirac-cone
on the surface. arXiv:0908.3513.
12. Xia, Y. et. al. Topological Control: systematic control of topological insulator
Dirac fermion density on the surface of Bi2 Te3 . arXiv:0907.3089.
13. Xia, Y. et. al. Fermi surface topology and low-lying quasiparticle dynamics
of parent Fe1+x Te/Se Superconductor by orbital-polarization resolved ARPES.
arXiv:0906.5392.
14. Hsieh, D. et. al. First observation of spin-helical Dirac fermions and topological phases in undoped and doped Bi2 Te3 demonstrated by spin-ARPES spectroscopy. arXiv:0904.1260.
106
15. Hor, Y. S. et. al. p-type Bi2 Se3 for topological insulator and low temperature
thermoelectric applications. arXiv:0903.4406.
16. Hsieh, D. et. al. First direct observation of Spin-textures in Topological Insulators : Spin-resolved ARPES as a probe of topological quantum spin Hall effect
and Berry’s phase. arXiv:0902.2617.
17. Hsieh, D. et. al. A topological Dirac insulator in a quantum spin Hall phase :
Experimental observation of first strong topological insulator. arXiv:0902.1356.
18. Xia, Y. et. al. Fermi surface topology and low-lying quasiparticle structure of
magnetically ordered Fe1+x Te. arXiv:0901.1299.
19. Hsieh, D. et. al. Experimental determination of the microscopic origin of magnetism in parent iron pnictides. arXiv:0812.2289.
20. Xia, Y. et. al. Electrons on the surface of Bi2 Se3 form a topologically-ordered
two dimensional gas with a non-trivial Berry’s phase. arXiv:0812.2078.
21. Wray, L. A. et. al. Momentum-dependence of superconducting gap, strongcoupling dispersion Kink, and tightly bound Cooper pairs in the high-Tc
(Sr,Ba)1−x (K,Na)x Fe2 As2 superconductors. arXiv:0812.2061.
22. Wray, L. A. et. al. Tightly-bound Cooper pair, quasiparticle kinks and clues
on the pairing potential in a high Tc FeAs Superconductor. arXiv:0808.2185.
23. Wray, L. A. et. al. Charge collective modes in an incommensurately modulated
cuprate. arXiv:cond-mat/0612207.
24. Qian, D. et. al. Emergence of Fermi pockets in an excitonic CDW melted novel
superconductor. arXiv:cond-mat/0611657.
107