Download Transport Experiments with Dirac Electrons

Transport Experiments with Dirac
Electrons
Joseph George Checkelsky
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: N. Phuan Ong
June 2010
c Copyright 2010 by Joseph George Checkelsky.
All rights reserved.
Abstract
This thesis presents transport experiments performed on solid state systems in which
the behavior of the charge carriers can be described by the Dirac equation. Unlike
the massive carriers in a typical material, in these systems the carriers behave like
massless fermions with a photon-like dispersion predicted to greatly modify their spin
and charge transport properties. The first system studied is graphene, a crystalline
monolayer of carbon arranged in a hexagonal lattice. The band structure calculated
from the hexagonal lattice has the form of the massless Dirac Hamiltonian. At the
charge neutral Dirac point, we find that application of a magnetic field drives a transition to an insulating state. We also study the thermoelectric properties of graphene
and find that the states near the Dirac point have a unique response compared to
those at higher charge density. The second system is the 3D topological insulator
Bi2 Se3 , where a Dirac-like dispersion for states on the 2D surface of the insulating 3D
crystal arises as a result of the topology of the 3D bands and time reversal symmetry.
To access the transport properties of the 2D states, we suppress the remnant bulk
conduction channel by chemical doping and electrostatic gating. In bulk crystals we
find strong quantum corrections to transport at low temperature when the bulk conduction channel is maximally suppressed. In microscopic crystals we are able better
to isolate the surface conduction channel properties. We identify in-gap conducting
iii
states that have relatively high mobility compared to the bulk and exhibit weak antilocalization, consistent with predictions for protected 2D surface states with strong
spin-orbit coupling.
iv
Acknowledgments
I would like to thank my adviser Phuan Ong for being a deeply generous mentor.
He has been generous with his knowledge of science as well as afforded me countless
opportunities in and out of the laboratory. Practically every step forward in my work
has come as a product of insightful discussions with him. His enthusiasm and attitude
regarding research will serve as a lasting model for me.
I am also particularly indebted to Minhyea Lee who guided me from the moment
I sat down at the microscope. One could not ask for a better teacher nor a more
devoted friend. Lu Li (Fig. 2.2) was also a great teacher and compadre. We survived
many long weeks at the magnet lab because of his patience and perseverance.
I have benefited greatly from the other members of the lab. Shu-Wen Teng,
Yoshi Onose, Wei-Li Lee, Phil Casey, and Yufang Wang have all been supportive
during my time in Princeton. The collaboration with Professor Cava and his group
in chemistry has been vital to my projects. In particular working with Yew San Hor
has been fruitful; Yew San is never short of crystals and career advice. Conversations
with Professor Hasan, Professor Haldane, Professor Petta, Professor Shayegan and
Professor Bernevig have all been extremely important. Support at the National High
Magnetic Field Laboratory in Tallahassee, Florida from Scott Hannahs, Tim Murphy,
and the support staff is also greatly appreciated.
The staff in the physics department has been a great resource. Mike Peloso spent
v
many hours teaching me skills in the student machine shop that have allowed me
to solve many experimental problems. The members of the purchasing department
Claude Champagne, Barbara Grunwerg, and Mary Santay helped me day after day.
Laurel Lerner and now Kim Fawkes have been knowledgeable resources throughout
the years. Jim Kukon and Geoff Gettelfinger were particularly helpful as a flexible
resource for liquid helium.
I have met many (mostly basement dwelling) friends along the way- Qiucen Zhang,
Bart McGuyer, Mike Kolodrubetz, Michael Schroer, Aakash Pushp, Colin Parker,
Lukas Urban, Shashank Misra, and Justin Brown to name a few. Above ground,
Dima Abanin and Mira Parish have been great theoretical colleagues and Ted Laird
helped me make it through the roller coaster that was generals.
Finally I wish to thank my family. They have been endlessly supportive and
understanding, particularly my parents for tolerating the din of vacuum pumps in
the background during our Sunday phone calls. I thank them for this and more.
vi
Contents
Abstract
iii
Acknowledgments
v
Table of Contents
vii
1 Introduction
1
1.1
Dirac materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
3D Topological Insulators . . . . . . . . . . . . . . . . . . . . . . . .
6
1.4
Organization
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Experimental setup
12
14
2.1
Cryostats and Magnets . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.2
Transport Experiments . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3 High field ground state of Graphene at the Dirac Point
23
3.1
Electronic Band Stucture of Graphene in H . . . . . . . . . . . . . .
23
3.2
Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
vii
3.2.1
Quantum Hall effect and Landau Levels at ν = 0 . . . . . . .
28
3.2.2
Sample Quality Dependence . . . . . . . . . . . . . . . . . . .
33
3.2.3
Device Heating . . . . . . . . . . . . . . . . . . . . . . . . . .
39
3.2.4
Measurements with pW dissipation . . . . . . . . . . . . . . .
41
3.3
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
3.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4 Thermopower and Nernst effect in Graphene in a high magnetic
field
4.1
56
. . . . . . . . . . . . . . . . . .
57
4.1.1
Thermoelectric Measurements . . . . . . . . . . . . . . . . . .
57
4.1.2
Girvin and Jonson Theory . . . . . . . . . . . . . . . . . . . .
60
Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
4.2.1
S with H = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
4.2.2
S and Syx in H . . . . . . . . . . . . . . . . . . . . . . . . . .
67
4.3
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
4.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
4.2
Thermoelectric Measurements in 2D
5 Quantum Interference in Macroscopic Crystals of Nonmetallic Bi2 Se3 76
5.1
Transport measurements in 3D Topological Insulators . . . . . . . . .
77
5.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
5.2.1
Nernst effect in Cax Bi2−x Se3 : Tuning the Chemical Potential .
81
5.2.2
Nernst effect in Cax Bi2−x Se3 : Low T behavior . . . . . . . . .
85
5.2.3
Transport properties of non-metallic Bi2 Se3
. . . . . . . . . .
89
5.2.4
Angular Dependence of non-metallic crystals . . . . . . . . . .
95
viii
5.3
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
5.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6 Evidence for metallic surface states in voltage-tuned crystals of Bi2 Se3 107
6.1
Gate voltage chemical potential control in Bi2 Se3 . . . . . . . . . . . 108
6.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.3
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7 Concluding Remarks
126
Appendix
129
A Brief Review of Quantum Corrections to Transport
129
Bibliography
135
ix
Chapter 1
Introduction
1.1
Dirac materials
Dirac electrons are distinct from electrons found in normal metals because of their
energy-momentum dispersion relation. Depicted in Fig. 1.1a, electrons in a typical
metal such as Cu behave like massive particles so that their energy has a quadratic
dependence on momentum, obeying the Schrodinger equation. However, in a Dirac
material this dependence is linear (Dirac-like), as in Fig. 1.1b . In other words,
the charge carriers behave as zero mass particles like photons or neutrinos, but (in
the case of the prototypical Dirac material graphene) with a velocity ∼ 0.3% of the
speed of light. Unlike other zero mass particles, however, Dirac electrons carry a
charge. Compared to electrons in Cu, these particles are more strongly influenced
by magnetic fields and have stronger Coulomb (charge) interactions. Despite these
exotic features, a surprising number of materials, some quite familiar to solid state
experiments, are proposed to be host to these states. The electrons in graphene,
1
1.1. Dirac materials
2
Figure 1.1: (a) Electrons in ordinary materials have a quadratic dependence of energy
E as a function of wavevector k and obey the Schroedinger equation. In the absence
of spin-orbit coupling, the bands are spin degenerate as indicated by the arrows.
(b) Dirac electrons have a linear dispersion analogous to photons with a modified
velocity. In graphene and topological insulators, the bands are pseudo-spin or spin
polarized due to symmetries of the crystal. (c) In the 2D quantum spin Hall state,
the bulk of the crystal is insulating and currents are carried by counter-propagating,
spin-polarized edge modes.
bismuth, Bi1−x Sbx , Bi2 Se3 , gray tin (α Sn), and quasiparticles (broken cooper pairs)
in d-wave superconductors are all examples. We refer to these as Dirac materials.
One example of the unique nature of Dirac electron systems is the ubiquity of
quantum spin Hall (QSH) physics. The 2D QSH effect is depicted in Fig. 1.1c.
Analogous to the integer quantum Hall effect, the bulk of the system is insulating
and currents are carried by edge modes which are protected from scattering. Unlike
the quantum Hall case, however, edge modes come in counter propagating pairs.
Furthermore, these modes are spin polarized. This is an exciting state of matter
because it presents a source of highly robust, spin-filtered states potentially useful for
spintronic or quantum computing applications.
Proposals for the QSH state arise in a number of Dirac materials. As discussed
below, strong spin-orbit coupling leads to the realization of this state in 2D HgTe
3
1.2. Graphene
quantum wells and 3D topological insulators [1, 2]. In graphene, the pseudo-spin
associated with the underlying lattice predicts the same spin polarized counter propagating modes for the Landau levels near the Dirac point [3]. Whether or not the
QSH state is realized in a given Dirac system is an experimentally important question.
If the QSH state is absent, it is also important to understand the nature of the true
ground state. We introduce the basic electronic properties of two candidate Dirac
materials below.
1.2
Graphene
Graphene is a single layer of carbon atoms arranged in a hexagonal lattice. Since it
was first realized experimentally in 2005 [4], a vibrant subfield of condensed matter
physics has developed to probe its experimental, theoretical, and industrially-useful
properties (for a review, see [5, 6]). The hexagonal lattice of graphene is shown in
Fig. 1.2a. In terms of crystal structure, the carbon atoms form a triangular lattice
with two atoms forming the basis at rA and rB in the unit cell (shown in gray). The
primitive lattice vectors are
a1 =
!
a, 0
, a2 =
√ !
a 3
,
a
2 2
(1.1)
where a is the carbon-carbon bond length ∼ 1.42 Å. Of each carbon’s 6 electrons, 2 are
tightly bound in 1s2 orbitals and the remaining 4 occupy the 2s and 2px,y,z orbitals.
Hybridization of the in-plane orbitals gives the crystal structure great strength in
plane, and leaves the remaining pz orbital to weakly bind adjacent layers (in the
case of stacks of graphene forming graphite) or to roam in the plane (for single layer
4
1.2. Graphene
Figure 1.2: (a) Crystal structure of graphene. There are two atoms in the basis
which decorate a triangular lattice. Basis vectors and bond vectors are labeled. The
two interpenetrating triangular lattices are shown colored in red and blue. (b) Band
structure of graphene using the tight binding calculation in the text. The energy
goes to zero at two distinct momenta K and K’. Near these points (shown in (c))
the dispersion is linear.
graphene).
Such a configuration of orbitals lends itself to a calculation of the electronic band
structure in the tight binding approximation. We take as the atomic orbital the pz
orbital on each atom and consider hopping between atoms with transfer integral t.
Considering nearest neighbor hopping, examination of Fig. 1.2a shows that electrons always hop from sublattice A to B or vice versa (colored red and blue). The
corresponding bond vectors for hopping bi are
!
√ !
−a 3
b1 = a, 0 , b2 =
,
a , b3 =
2 2
√ !
−a − 3
,
a
2
2
(1.2)
A natural choice of basis to perform the tight binding calculation is amplitude of the
wavefunction ψ per sublattice. Since each hopping is between sublattices, if we set the
on-site (atomic) energy ǫ = 0, the 2 × 2 Hamiltonian H will be purely off-diagonal.
5
1.2. Graphene
Assuming the states are Bloch-like, the off diagonal term HAB is
HAB = −t
3
X
eik·bi
"i=1
= −t e
ikx a
−ikx a/2
+ 2e
cos
√
3ky a
2
!#
(1.3)
∗
and HBA = HAB
. Solving for the energy eigenvalues for H by use of the secular
equation yields the spectrum shown in Fig. 1.2b. A particularly interesting feature is
that at 6 points, the energy E crosses through zero at a sharp point (a Dirac point).
In reality, there are only 2 such distinct points labeled K and K’, the others being
related by reciprocal lattice vectors. Expanding Eq. 1.3 in δk around one of these
√
points K = (0, 4π/3 3a), we have
#
"
2π √3a
2π iδkx a cos
sin
−
δky
HAB = −t 1 + iaδkx + 2 1 −
2
3
2
3
i
3 h
=
at − iδkx + δky
(1.4)
2
Then H is

0
3 
H = at 
2
+iδk + δk
x
y



−iδkx + δky 
 0 p− 
 = v

0
p+ 0
(1.5)
where in the last step we have explicitly written H in the form of the massless Dirac
Hamiltonian with Fermi velocity v = 3at/2 and p± /~ = ±δkx +δky [7]. This dispersion
is shown in Fig. 1.2c; the energy E for charge carriers near this point is
1/2
E(k) = ±~v δkx2 + δky2
(1.6)
with corresponding eigenfunctions


1  1  ik·r
ψ± (k) = √ 
e
2 ±eiφ
(1.7)
1.3. 3D Topological Insulators
6
This electronic structure is at the heart of the interest in graphene and more generally in Dirac materials. Eq. 1.6 shows that (low-energy) charge carriers in graphene
have a light-like dispersion relation, yet are Fermionic and carry charge. These massless Dirac fermions have behavior quite distinct from massive electrons, as becomes
particularly clear with application of H in Chapter 3.
1.3
3D Topological Insulators
3D TIs are bulk bulk insulating crystals with 2D conducting surfaces. After the
prediction [1, 2, 8] and realization [9] of analogous states in 2D, there has been a flurry
of activity in confirming the existence of the 3D TIs [10, 11, 12, 13, 14, 15, 16, 17].
To understand the unique nature of the 2D states, it is useful to view the Angle
Resolved Photoemission Spectroscopy (ARPES) results that have largely founded
the experimental side of this field [13, 14, 15, 16, 17]. Shown in Fig. 1.3a-c are
ARPES results for the proposed TI Bi2 Se3 , while Fig. 1.3d shows a schematic band
structure based on theoretical expectations.
TIs are predicted to have an energy gap between the bulk conduction and valence
band within which a 2D surface band resides. The surface band itself is spin-polarized,
has a Dirac dispersion, and (as described below) is protected from being destroyed
by most kinds of impurities. These elements depicted in the cartoon in Fig. 1.3d are
all reproduced in ARPES. For example, Fig. 1.3c shows that there is a 300 meV gap
between the bulk valence and conduction band, with the 2D linear dispersion band
in the gap. The band is continuous, showing no signs of a disorder induced gap. Spin
polarization experiments are shown in Fig. 1.3b, demonstrating that the spin s has
1.3. 3D Topological Insulators
7
Figure 1.3: ARPES results (a-c) for topological insulator Bi2 Se3 from [15] and
schematic band structure (d). (a) View along z direction of Dirac cone of surface
states. The spin direction is noted with arrows. (b) Spin polarization measurement
of electrons. The polarizations are antiparallel on opposite sides of the Dirac cone.
(c) Bulk band gap and in-gap surface state Dirac cone. The band gap is 300 meV. (d)
Schematic of prediction of band structure for 3D topological insulators. There is a
bulk band gap with an odd number of Dirac cones of spin-polarized, linear dispersion
states inside.
1.3. 3D Topological Insulators
8
a non-zero polarization and is opposite on each side of the Dirac cone. The nature of
this spin polarization is depicted in Fig. 1.3a: s is locked perpendicular to the wave
vector k. It is this combination of linear dispersion, spin polarization, and protection
that drives interest in these materials.
Before outlining the concepts of topological band theory underlying the connection
between the 3D band structure and these novel 2D states, some intuition for the roots
of these properties may be found by considering electrons on the 2D surface of a 3D
crystal. Fig. 1.4 shows a schematic TI, focusing on the electrons on the top surface
of the crystal. Inversion symmetry is broken at the surface of the crystal so that
the electron experiences a trapping electric field Etrap in ẑ that arises as a gradient
of the surface potential [18]. In the reference frame of the electron moving with
wavevector k, this Etrap is felt as an effective magnetic field B ∼ k × Etrap . Then, it
will be energetically favorable for s to point in the plane and perpendicular to k to
minimize the energy s · B. This so-called Rashba contribution to the Hamiltonian is
well known in systems with large spin-orbit coupling [19]. One can see immediately
that any charge current in the plane would then be accompanied by a spin current.
However, the sense of the spin polarization is exactly the opposite on the bottom
surface (Etrap is in the opposite direction), so one needs to isolate the current on
either crystal face to take advantage of the spin texture.
This simple picture can also help us understand in what sense the surface bands
are protected. Fig. 1.4a shows a schematic of the surface band structure with the
spin configuration discussed above. In these materials there are no magnetic orderings, magnetic fields applied, or related phenomena, so viewing the behavior of the
electrons should look the same if the arrow of time is reversed. More mathematically,
1.3. 3D Topological Insulators
9
Figure 1.4: (a) Simplified band structure of surface states demonstrating action of the
time reversal operator T . (b) Electrons on the top surface of a topological insulator
have their spin s pointed perpendicular to their wavevector k as determined by the
effective B. (c) A band structure with an even number of Dirac cones is not protected
by Kramers theorem from the opening of a gap.
application of the time reversal operator T should leave the states in the system unchanged. If we consider the surface electrons, that means for each electron moving
with a particular k, s, and energy E, there must be an electron moving at the same E
with −k and −s. In this way, time reversal symmetry is preserved. This preservation
is known as Kramers theorem [20]. This is demonstrated schematically in Fig. 1.4a.
The texture of spins from the effective B obeys this symmetry. It is interesting to
note that if inversion symmetry was preserved E(k, s) = E(−k, s), which together
with time reversal symmetry would imply E(k, s) = E(k, −s). Thus, in the bulk of
the materials where inversion symmetry is preserved the bands are spin degenerate.
For the simplest case here, we have two lines of states defining E(k) (Fig. 1.4a).
In principle, all odd powers in k are allowed. Consider the T operation again on the
cone of states in Fig. 1.4a, but now acting at k = 0 (the Dirac point) instead of
1.3. 3D Topological Insulators
10
a finite k as before. For time reversal symmetry to be preserved, there must exist
2 states (with opposite s) at this point. While the cones can be deformed in this
parameter space, the cones can never be split apart and still retain 2 states at k = 0
at the same E. In this sense, the continuity of the states is a topological property and
the surface band is topologically protected from the opening of a gap. If instead we
were to have 2 cones as in Fig. 1.4c, this argument would not be true- that structure
belongs to a different topological class. There, Kramers theorem can be satisfied even
with the opening of a gap (there are still an even number of states at each E).
Originally formulated in a model of graphene due to Haldane [21], these ideas
evolved with the theoretical prediction by Kane and Mele that graphene would be
host to the 2D analog of these materials- a 2D insulating bulk wrapped with counterpropagating, spin polarized edge states [1]. This is an alternate realization of the
QSH state proposed for graphene discussed in chapters 3 and 4, though in that case
Landau level quantization is involved. However, since graphene is made of carbon
and thus has small spin-orbit coupling, experimentally realizing the zero H QSH
state proved difficult. It was realized soon after that HgTe based quantum wells
would be a more promising candidate for realizing this state [8], and experimental
verification soon followed [9]. This verification came from the transport experiment
shown in Fig. 1.5. Different samples corresponding to topologically trivial insulators
and QSH insulators were prepared. The trivial class was found to be true insulators
when the chemical potential was tuned via a gate potential in to the bulk gap (black
curve in Fig. 1.5), while those predicted to have conducting edge states showed
conductance σxx = 2e2 /h. Importantly, this σxx did not depend on the geometry
of the sample; the current was shown to be 1D. Further experiments with non-local
1.3. 3D Topological Insulators
11
transport measurements further demonstrated the existence of conducting edge modes
[22]. The theoretical generalization to 3D materials followed soon after [10, 12, 23].
Figure 1.5: Proof of 2D topological insulator in HgTe system from [9]. Device I is a
trivial insulator and shows a diverging resistance in the bulk band gap. Devices III
and IV are topological insulators and show conductance G = 2e2 /h consistent with
edge mode transport. Device II is a non-trivial insulator but too large for the edge
modes to persist unscattered.
The theoretical framework developed in these works is the concept of topological
band theory. In 2D, states can be characterized by a quantity known as the Chern
number nC , which may be calculated as a integral over the bulk band structure of
a given material [24]. Materials with different nC belong to topologically distinct
classes. For example, the vacuum has nC = 0, while the quantum Hall insulator has
nC = 1. One important concept in topological band theory is that when materials
of different nC are brought in to contact, a bound state is formed at the interface.
An example familiar to many is the bound state trapped between the quantum Hall
insulator and vacuum- the edge state (skipping orbit) shown in Fig. 1.6a. As was
1.4. Organization
12
Figure 1.6: (a) Depiction of state trapped between two insulators with different Chern
number. The skipping orbit is trapped between the quantum Hall insulator and
vacuum. (b) A Majorana bound state is trapped between superconductivity and
magnetism induced gapped states in a topological insulator.
shown by Thouless and co-workers [25], one can understand edge states and the
quantum Hall effect that arises as σxy = nC e2 /h. The next step forward was to
consider topological classes of materials that, unlike the quantum Hall case with a
large magnetic field H, exist in the presence of time reversal symmetry. This new
topological class characterized by the so-called Z2 invariant was shown by tricks of
symmetry to be straightforward to calculate [11], and soon particular material systems
based on Bi alloys were proposed. Experimental verification led by Hasan, Cava, and
coworkers by ARPES followed in Bi1−x Sbx [13, 14], followed by the discovery of
additional realizations in Bi2 Se3 [16, 15] and Bi2 Te3 [17, 26]. Proposals for realizing
exotic states of matter such as Majorana fermions (Fig. 1.6b) have fueled intense
interest in these materials [27].
1.4
Organization
We describe the general experimental apparatus in Chapter 2. Further details relevant
to each experiment are given at the beginning the appropriate chapter. Chapter
3 describes the zero energy state of graphene in high magnetic field. Chapter 4
1.4. Organization
13
describes the thermoelectric response of the Dirac states in graphene. Chapter 5
presents a materials study of topological insulator Bi2 Se3 , focusing on low temperature
transport. Chapter 6 focuses on probing the Dirac states in Bi2 Se3 in a field effect
transistor geometry. Finally, we summarize our results in chapter 7. A brief overview
of quantum corrects to transport is given in the appendix.
Chapter 2
Experimental setup
Experiments presented in this thesis are all transport measurements. Electric, thermoelectric, and thermal transport properties were all performed. These measurements
are done as a function temperature T , magnetic field H, and electron density n varied by an electrostatic gate or chemical composition. Here we introduce the basic
machinery used to vary the parameters T and H. Then we present an overview of
measurement techniques.
2.1
Cryostats and Magnets
A liquid helium (LHe) cooled superconducting 14 T magnet is the main environment for our experiments (Oxford Instruments). A schematic of the configuration
for measurements for T down to 4 K is shown in Fig. 2.1a. A magnet submerged in
LHe is enclosed within a vacuum isolated dewar. A power supply (PS-120, Oxford
Instruments) supplies current to the magnet. Inside of the bore of the magnet, we
place a stainless steel cryostat consisting of an outer vacuum jacket, inner vacuum
14
2.1. Cryostats and Magnets
15
Figure 2.1: (a) Schematic of magnet and He-4 cryostat. The insert (probe) is shielded
from a LHe bath by several layers of vacuum. The bath itself is shielded from air by a
vacuum jacket. T between 4.2 K and 400 K can be achieved varying the amount of He
exchange gas in the vacuum jackets and heating power of the probe. (b) Schematic of
He-3 cryostat used for measurements down to 0.3 K. The sample is placed in vacuum
in physical contact with the He-3 pot where liquid He-3 is condensed and cooled. (c)
View of He-4 probe sample space. The sample is contained within a vacuum chamber
made by a taper seal. Electrical connections come from the top of the probe and
terminate in contact pads. T is monitored with a cernox thermometer. A resistive
heater is used to warm the sample.
2.1. Cryostats and Magnets
16
jacket, and space to put an experimental probe. All three of these spaces are kept at
low pressure by a diffusion pump (Edwards Diffstak). The sample is mounted on a
home-made experimental probe on a Cu cold finger shown in Fig. 2.1c. Experimental
wiring (phosphor bronze wires for AC measurements, Cu wires for DC measurements)
comes down a long stainless steel tube that connects to a mating copper taper seal
and cap. The top of the probe has a multipin connector which allows connection to
measurement equipment. The sample is connected with Au wires to the contact pads
and the chamber sealed with vacuum grease. The probe is pumped down to 10−6
Torr with a turbomolecular pump before loading in the cryostat.
T is monitored by a cernox resistor (Lakeshore). Temperature control is accomplished by PID control with a Lakeshore 340 temperature controller. Care is taken to
minimize the power dissipated in the cernox to avoid unintentional heating, typically
using an excitation current of 10 µA. The heating power is supplied by a thin film
25 Ω resistor (Minco flexible heater) mounted on the cold finger, while the cooling
power comes from the LHe bath. The amount of cooling power can be greatly varied
by introducing different amounts of He gas in to the vacuum chambers. In this way,
T can be controlled within 10 mK over the range 10 K to 400 K with the sample in
vacuum. In order to go down to the base T of 4.2 K, He gas is added to the sample
space. However, this can be problematic for thermal transport measurements as the
He gas presents an alternate path for heat current to flow.
For measurements below 4 K, we use a closed cycle He-3 cryostat (Oxford Heliox)
in place of the 3 chambered He-4 cryostat. A schematic of the He-3 setup is shown
in Fig. 2.1b. He-3 is contained at the top of the probe in a large cylinder. When the
probe is inserted in to the LHe bath, the He-3 is cryopumped down to the Sorb where
2.1. Cryostats and Magnets
17
Figure 2.2: Experimental setup in the National High Magnetic Field Lab in Cell 5.
Labmate Lu Li can be seen loading the He-3 probe in the the cryostat. The cryostat
itself is not in direct contact with the magnet to minimize vibrational noise. The
maximum H in this magnet is 31.2 T.
2.2. Transport Experiments
18
it is absorbed by charcoal. By heating the charcoal to 45 K, we can force He-3 to flow
down toward the He-3 pot, on the outside of which the sample is mounted. Cooling
the He-3 via the 1 K pot (LHe from the bath which is cooled to ∼1.5 K via lowering
of the vapor pressure with a mechanical pump) condenses the He-3 and allows it to
collect in the He-3 pot. Once He-3 liquid is collected, the sorb is cooled to 4 K and
begins to slowly reabsorb the He-3 gas evaporating from the He-3 pot. This lowers
the vapor pressure of the He-3 and cools the sample down to the base T of 0.3 K.
After running this condensing cycle for 1 hour, the hold time is more than 10 hours.
Measurements are also performed at the National High Magnetic Field Laboratory
in Tallahassee, Florida. A picture of the cryostat and magnet is shown in Fig. 2.2.
For this cryostat (He-3 system B in Cell 5), the maximum field is 31.2 T. A long probe
with the sample mounted on a 16-pin dip header is inserted in to a cryostat with liquid
He-3 inside. The principle of operation is the same as detailed for Fig. 2.1b, but here
the sample is immersed in the liquid rather than isolated in vacuum. The largest
difference between the experimental setup is that the magnet in Fig. 2.2 is a water
cooled, copper magnet instead of a superconducting magnet. The cooling water used
to carry heat away from the magnet causes significant mechanical vibration in the
system and causes measurements to be noisier.
2.2
Transport Experiments
Transport experiments on macroscopic crystals are performed in a configuration
similar to Fig. 2.3. We typically measure in-plane (defined as the xy plane with
i, j = (x, y)). For experiments involving thermal gradients, the crystal is mounted
19
2.2. Transport Experiments
vertically using silver epoxy (Epotek H20E) on a sapphire substrate. A ceramic heater
(Film Microelectronics) is mounted to the top of the crystal which has a resistance
1 kΩ, independent of H and T . Gold wires (0.001 inch, California Fine Wire) are
attached to the sample using conducting Ag paint (Dupont) to measure electric potentials Vij and inject electrical current I. These wires connect to contact pads on
the measurement probe. The sapphire substrate is attached the cold finger in Fig.
2.1b with thermal joint compound (Type 120, EG&G Wakefield Engineering). Care
is taken to isolate the wires thermally from the cold bath but suspending them with
nylon pillars. The principle behind this design is to force the heat generated by Joule
heating of the resistor to flow down the sample to the sapphire, which is in good
thermal contact with a Cu cold finger. In this way, we know the heat heating power
put in to the sample.
Temperature gradients ∂i T are measured by thermocouples (type K, Omega Engineering) for T greater than 5 K and with calibrated miniature RuO2 temperature
sensors (Lakeshore) for T below 5 K down to 0.3 K. We can then measure the resistivity tensor ρij , thermoelectric tensor Sij , and thermal conductivity tensor κij
through
Ei = ρij Jjc
(2.1)
Ei = Sij ∂j T
(2.2)
Jiq = κij ∂j T
(2.3)
where J c is the input charge current, J q is the input heat current and Ei is the electric
field determined from the voltage measurement. In principle it is possible to prepare
2.2. Transport Experiments
20
a sample to measure all of these quantities, though not simultaneously. In practice,
due to complexity in mounting, only one of Eq. 2.1 to 2.3 is measured in a single
experiment. For an example of a purely electrical measurement, see Fig. 5.22. This
does not require the sample to be mounted vertically, since no J q are involved.
Figure 2.3: Typical bulk crystal (Bi2 Se3 ) prepared for transport measurements. The
crystal (shiny surface) is standing vertically with a heater mounted on top of the
crystal. Gold wires and thermocouples suspended by nylon pillars are attached to
the sample. The pillars and crystal are epoxied to a sapphire substrate.
We can measure the same quantities in microscopic samples, except for κ since
the sample is not mounted vertically. Fig. 2.4 shows a graphene device patterned for
measurement of ρij and Sij . Crystals are cleaved by mechanical exfoliation on to a ntype Si substrate with 300 nm of thermally grown SiO2 . All electrodes are patterned
with ebeam lithography followed by thermal evaporation of 5 nm Cr and 30 nm of
Au. A heater is patterned near the sample; putting mA order current through the
heater generates a mK scale gradient across the sample, which is enough to detect
thermoelectric voltages. The details of the Sij measurement are given in Chapter 4.
2.2. Transport Experiments
21
Figure 2.4: (a) Graphene microdevice prepared for thermoelectric and electric measurement. A heater is patterned on the right that generates a temperature gradient
across the crystal. The gradient can be monitored by measuring the resistance of the
patches of Au trace which are calibrated as thermometers (labeled therm). The remaining leads are used to measure Vxx and Vxy either due to ∇T or an applied current
bias between the therm leads. The graphene crystal is outlined with the dashed white
line. (b) Field effect transistor geometry used to induce carriers in thin crystals. A
voltage is applied between the Si back gate and a Au electrode in contact with the
crystal. This generates an electric field pointing vertically though the dielectric which
attracts carriers.
22
2.2. Transport Experiments
The largest difference experimentally between the two situations is electrostatic
discharge problems. Microdevices of the type shown in Fig. 2.4 can be easily destroyed
by a discharge of static across the measurement leads or large currents flowing on the
ground plane. When the device is not being actively measured, all leads are grounded
together using a shorting rotary switch (Electroswitch 04-302). Care is taken to
have all instruments share the same ground as well as to ground the operator before
touching the experimental apparatus.
Another difference is the ability to use the electric field effect (EFE) to vary the
density of electrons n in a very thin sample. The setup is shown in Fig. 2.4b. Applying
a gate voltage VG changes the chemical potential of the crystal by bringing carriers
in from the Au leads. For graphene devices, we find that the number of carriers nG
induced in the sample follows the expectation for a parallel plate capacitor geometry,
nG =
CVG
e
(2.4)
where C is the capacitance per unit area of the gate dielectric 1.14 ×10−4 F / m2 .
AC voltages are monitored with Stanford Research (SR) 830 lock-in amplifiers.
Using SR560 preamplifiers we can reduce the noise floor below 10 nV. Alternatively,
a Keithley 6221 current source can be interfaced with a Keithley 2182A to average
positive and negative DC bias measurements (Delta mode). This removes effects of
thermal drift and DC offsets. When pure DC measurements are required, Keithley
2001 voltmeters with a Keithley 1801 pre-amplifiers are used. Effects of DC drift
are minimized by using all Cu clamped connections with solder free joints. This is
particularly important for those connections coming from low T to atmosphere. With
these precautions, DC noise levels of 20 nV are possible. Data are collected via GPIB
488.2 connections using a National Instruments data acquisition card.
Chapter 3
High field ground state of
Graphene at the Dirac Point
This chapter focuses narrowly on one basic question: What is the electronic ground
state of graphene in a magnetic field H at the Dirac point? The reason for asking
this particular question is that in terms of electron density ne and H, this is where
interaction effects between electrons should be strongest. We start with a simple
introduction to Landau level quantization in graphene, present experimental results
demonstrating a unique, insulating ground state [28, 29], and discuss implications for
broken symmetry in the ground state.
3.1
Electronic Band Stucture of Graphene in H
We begin by introducing H in to the electronic structure calculation from Eq. 1.5.
Application of a magnetic induction µ0 H = ∇ × A modifies H by k = −i∇ →
−i∇ + eA, where A is the vector potential. Again we work in the vicinity of valley
23
3.1. Electronic Band Stucture of Graphene in H
24
K. Following Ando [30], we define the following operators
(−ikx + ky )ℓB
√
(3.1)
2
p
where ℓB is the magnetic length defined as ℓB ≡ h/eB. Rewriting H with Eq. 3.1,


√
2v  0 a
H=
(3.2)


ℓB a † 0
a≡
(ikx + ky )ℓB
√
2
a† ≡
It follows that a and a† obey the commutation relation a, a† = 1, and that we
may use Eq. 3.1 as lowering and raising operators to construct the Landau level
wavefunctions ψn by
n
a†
ψn = √ ψ0
n!
(3.3)
where aψ0 = 0. The action of a and a† together is aa† ψn = nψ. Using this raising
and lowering structure, we can see that the structure of the energy level spacing is
p
v
En = sign(n) 2 |n|
ℓB
(3.4)
with the wavefunction given by


E
 n − 1 
ψ(n) =  E 
n
(3.5)
There are several interesting aspects of Eq. 3.4. The energy spectrum of the Landau
levels in particular is distinct from the massive electron case using the Schrodinger
equation. Whereas the energy level spacing in the latter is that of the simple harmonic
oscillator with E ∝ n + 1/2, here the energy level spacing is not uniform in n.
Moreover, E0 = 0, meaning that there is a state at zero energy for n = 0. Considering
3.1. Electronic Band Stucture of Graphene in H
the wave function in Eq. 3.5, for n = 0 we have


 0 
ψ(0) =  E
0
25
(3.6)
meaning that there is no amplitude for the wavefunction on the A sublattice. As
Figure 3.1: Schematic representation of electron amplitude (orange) for a single valley
at the n = 0 Landau Level. Both valleys are polarized on opposite sublattices.
depicted in Fig. 3.1, this would imply that the carriers for valley K are localized on
one sublattice only. However, valley K’ contributes exactly the opposite carriers so
there is no charge-density wave when valley symmetry is preserved. Nonetheless, this
is an indication that interaction effects may produce unique ground states near the
Dirac point at high H. One view of this sub-lattice polarization is that for the n = 0
state the valley and sublattice degeneracy are equivalent. This is only true at n = 0
[31].
3.1. Electronic Band Stucture of Graphene in H
26
Figure 3.2: Breaking the four-fold degeneracy
√ of the n = 0 Landau Level. Two
scenarios are the exchange energy Eex ∼ B causes the physical spin (a) or the
valley spin (b) to split first.
More generally, graphene has a four-fold degeneracy of 2 spins and 2 valleys at
each Landau level. Upon increasing the Coulomb interaction in a high H, one expects
this degeneracy to be lifted. A question related to the charge density wave picture
above is whether the spin, valley, or some combination of the two is most sensitive
to being lifted in intense H. Figure 3.2 shows two possible scenarios for lifting the
degeneracy to produce a broken symmetry ground state.
A large amount of theoretical work has been devoted to this issue [32, 31, 33,
34, 35, 36]. Depending on the relative role of long-range and short-range Coulomb
interactions, a variety of broken symmetry ground states at n = 0 are predicted
to be driven by exchange energy. In one scenario, the exchange energy leads to
ferromagnetic polarization of the physical spins [3, 37]. This produces a spin gap
in the bulk without affecting the valley degeneracy, as shown in Fig 3.2a. Near the
edge of the sample, the residual valley degeneracy is lifted by the edge potential (see
Fig. 3.3). In a simple picture, because at n = 0 the valley degeneracy and sublattice
degeneracy are equivalent, it may not be surprising that the boundary condition at
3.1. Electronic Band Stucture of Graphene in H
27
Figure 3.3: In the physical spin splitting scenario, the edge potential breaks the
symmetry between the two valleys. According to the boundary condition at the edge,
this causes the two edge modes to counter propagate (shown in (a)). This leads to
the counterpropogating modes to carry opposite spin polarization, as depicted in (b).
the edge can distinguish between K and K’. An important consequence of this edge
splitting at n = 0 is the existence of spin-filtered counter propagating edge modes.
These result in a residual conductance of 2e2 /h regardless of the magnitude of the
spin gap in the bulk, and are a realization of the quantum spin Hall effect [38]. If
however, exchange polarizes the valley spin first (shown in Fig 3.2b) the CPE modes
are not present [39, 40, 41]. Yet another possibility is some spontaneous ordering of
the spin described by the valley spin / real spin space [42].
28
3.2. Experimental Results
3.2
3.2.1
Experimental Results
Quantum Hall effect and Landau Levels at ν = 0
Measurements of graphene in large H at temperature T = 0.3 K are shown in Fig.
3.4. The Hall conductivity σxy exhibits quantum Hall plateaus corresponding to the
filling ν relation
σxy
e2
1
4e2 n+
=ν =
h
h
2
(3.7)
where n is the Landau level index, in agreement with the original experimental findings [43, 44]. The factor of 4 is due to the spin and sublattice degeneracy, while the
half-integer shift is a result of the non-zero Berry’s phase of the electron wavefunction [44, 45]. The Landau level at filling n = 0 stands out. Whereas the longitudinal
resistance Rxx is nearly unchanged for levels n = ±1 for H = 8 T to 14 T, the peak in
resistance for n = 0 dramatically increases. Accordingly, in σxy an additional plateau
at σxy = 0 appears. This is more evident in Fig. 3.5b.
We define R0 to be the resistance at charge neutrality. It is evident from Fig. 3.5a
that R0 increases as T decreases. At 14 T, the n = 0 Landau level is split into 2 subbands even at T = 100 K (Fig. 3.5b). It is also instructive to view the T dependence
of σxx , shown in Fig. 3.6a. As a function of decreasing T , it is evident that the longitudinal conductivity σxx crosses over from metallic (σxx increasing with decreasing
T ) to non-metallic (σxx decreasing with decreasing T ) as H is increased. There is
also an obvious saturation at low T ; we address this further below. One noteworthy
measure is that the separatrix between metallic and non-metallic conductivity occurs
at σxx ∼ 4e2 /πh.
29
3.2. Experimental Results
-40
-20
100
0
20
T = 0.3 K
40
(a)
K7
80
14 T
R
xx
(k
)
60
11
40
8
20
14 T
11
8
0
8
(b)
11
8
14 T
2
(e /h)
4
xy
0
-4
-8
-40
-20
0
20
40
V ' (V)
g
Figure 3.4: (a) Longitudinal resistance Rxx of device K7 at T = 0.3 K in the quantum
Hall regime. The effect of increasing H on the n = ±1 Landau level is negligible,
while the n = 0 level responds with a sharp increase in Rxx . The inset shows a
false-color micrograph of the sample. The scale bar is 5 µm. (b) Hall conductivity
σxy shows quantum Hall plateaus following the filling structure in Eq. 3.7. A plateau
at σxy = 0 develops at the highest fields.
30
3.2. Experimental Results
-10
-5
200
0
5
(a)
10
H = 14 T
0.3 K
1
150
)
2
5
R
xx
(k
100
10
50
20
40,100
0
3
4
(b)
2
3
xy
xy
2
0
xx
(e /h)
2
xx
2
(e /h)
1
(0.3 K)
-1
100
1
40
-2
20
10
0
-10
0.3-2 K
5
-5
0
V
g
5
-3
10
(V)
Figure 3.5: (a) Resistance as a function of gate voltage VG near the Dirac point in
the quantum Hall regime. Decreasing temperature causes the resistance to grow dramatically. (b) Longitudinal conductivity σxx in the same range as (a). Here, splitting
of the central Landau level can be seen up to 100 K. At the lowest temperature 0.3
K, Hall conductivity σxy is also shown. The plateau at σxy = 0 is pronounced.
31
3.2. Experimental Results
T (K)
1
10
100
2.0
6 T
7
(e /h)
1.5
8
2
9
10
0
xx
1.0
11
12
0.5
13
(a)
14
0.0
200
0.3 K
(b)
2
120
0
(k
)
160
R
5
80
20
40
40
0
5
10
15
H (T)
0
Figure 3.6: (a) Temperature dependence of σxx at the Dirac point. At low fields,
metallic behavior is observed below 40 K. However, upon increasing H above 10
T, a non-metallic temperature dependence is observed. Below ∼ 2 K, saturation is
seen. (b) Dirac point resistance R0 over the measured field range at selected T . The
divergent nature of the R0 (H) profile gets progressively stronger as T is lowered.
3.2. Experimental Results
32
Viewing R0 (H), we see that as T decreases, R0 becomes an increasingly divergent
function of H. At 0.3 K and 14 T, R0 reaches a value of nearly 200 kΩ for this
device. As we describe below, this behavior of R0 is characteristic for high quality
(high mobility µe ) crystals.
Figure 3.7: The central Landau level is split with increasing of H from 6 T to 14 T.
σxx at charge neutrality is pushed below 0.15 e2 /h at the highest field 14 T.
Finally, an interesting way of viewing the H dependence of the charge transport
is the profile of σxx across the lowest Landau level with increasing H. As shown in
Fig 3.7, H breaks the degeneracy at n = 0, though it is not evident if it is physical
or valley spin being separated.
33
3.2. Experimental Results
3.2.2
Sample Quality Dependence
An important question regarding the acute dependence of the Dirac point resistance
R0 to H is if it is the intrinsic behavior of the electronic structure of graphene or
an impurity effect. To answer this, several devices of varying quality were measured.
While there are many metrics to characterize devices, the two most appropriate here
are µe and the offset gate voltage V0 . While from section 1.2 one would expect the
chemical potential to be at the Dirac point in devices to occur in undoped graphene,
in practice one finds that in fabrication charge impurities are trapped in devices that
require a (usually positive) voltage (V0 ) to be applied to the back gate to reach charge
neutrality. This trapped negative charge markedly effects the quality of devices. Fig.
3.8b shows a variety of devices with different V0 . For each device, the width of the
resistive peak characterizes the electron mobility through G = neµe , as VG = ne/C
where C is the capacitance of the 300 nm SiO2 dielectric (1.14 × 10−4 F / m2 ). The
devices with smaller V0 clearly have narrower peaks and thus higher µe . Table 3.1
collects these parameters for several representative devices.
units
K5
K7
K8
K18
K22
K29
V0 R0 (14 T) µe
V
kΩ
1/T
3
80
0.3
1
190
1.3
12
15
0.6
20
7.5
0.9
-0.6
> 280
2.5
22.5
7
0.2
Table 3.1: Sample parameters. V0 is the gate voltage required to reach charge neutrality. R0 (14 T) is the resistance measured at charge neutrality at 14 T and 0.3 K.
µe is the electron mobility at H = 0.
3.2. Experimental Results
34
Figure 3.8: (a) Annealing of sample K32. The offset gate voltage V0 can be moved
from greater than 20 V to below 4 V, but the mobility of the sample does not improve
significantly. (b) R(VG ) for several devices measured as-prepared. Those devices with
naturally lower V0 have significantly narrower profiles.
3.2. Experimental Results
35
The relevance to electron transport near charge neutrality becomes evident upon
plotting the Rxx profile of devices with different V0 in the quantum Hall regime. Fig.
3.9a shows the result for devices K7, K5, K8, and K29. As reported in Table 3.1,
lower V0 and higher µe correlates with large R0 . This can be seen in both the VG
dependence and detailed H dependence in Fig. 3.9a and Fig. 3.9b, respectively. One
method known in the literature for decreasing V0 is annealing. While there are many
different methods ranging from baking devices in a Ar/He atmosphere [46] to driving
a large current through the device to anneal through Joule heating [47], we find that
while annealing can reduce V0 , it does not substantially improve the performance of
the device compared to as prepared devices with lower V0 . This is demonstrated in
Fig. 3.8a, as progressively annealing a device moves V0 down without substantially
narrowing the resistive profile.
In extremely intense H > 25 T, however, we can see that increasing sample quality
through annealing marginally improves devices. Fig. 3.10 shows a device measured
to H = 33 T before and after annealing in a He atmosphere at 250◦ C for 1 hour.
Annealing pushes the measured R0 to larger than MΩ values at 33 T after the anneal,
whereas before there was not an obvious divergence. This can also be seen in the T
dependence of R0 for the same device in Fig. 3.11. Before annealing, Rxx is only
weakly dependent on T and H, though it does exhibit a crossover from metallic to
non-metallic behavior. After annealing, the device increases in resistance much more
dramatically at 33 T upon decreasing T .
However, while annealing this device allows us to see that increasing mobility
favors the high resistance state, it does not produce a device of high quality to produce
the state in H < 20 T.
36
3.2. Experimental Results
200
200
(a)
(b)
T = 0.3 K
H = 14 T
(k
)
R
K22
0
100
K5
R
100
T = 0.3 K
xx
(k
)
K7
K7
K8
K18
K29
0
-10
0
10
V
g
(V)
20
5
10
15
0
H (T)
0
Figure 3.9: (a) Comparison of resistance profiles of the n = 0 Landau level for 4
samples of varying quality. As the offset gate voltage V0 becomes progressively larger,
the peak in resistance becomes lower. All measurements are performed at 14 T and
0.3 K. (b) Dirac point resistance R0 as a function of H for three different samples.
Samples of higher mobility and lower V0 diverge at systematically lower H. See Table
3.1 for parameters. For sample K7, the solid curve is a trace taken with fixed Vg while
the points represent data obtained from taking the peak resistance from a sweep of
VG at fixed H.
3.2. Experimental Results
37
Figure 3.10: (a) Resistance near the n = 0 Landau level in high H for device K60
before annealing. The peak does not reach above 105 Ω even at 33 T. (b) After
annealing, V0 has been lowered by ∼ 20 V, and a MΩ peak is seen in resistance.
3.2. Experimental Results
38
Figure 3.11: (a) Crossover from metallic to non-metallic behavior in the relatively low
quality device K60 before annealing. A field of 20 T is required to see the crossover.
(b) After annealing, the crossover occurs below 10 T. The anneal was performed in a
He atmosphere at 250◦ C for 1 hour.
3.2. Experimental Results
3.2.3
39
Device Heating
A clear question to ask is how large R0 becomes in large H. In Fig. 3.9b, for example,
R0 (H) is only shown to 200 kΩ at ∼ 10 T. The reason for the truncated measurement
is that when the device begins to have a divergent resistance, Joule heating becomes
a serious problem. Several spurious measurements are shown in Fig. 3.12 measured
with the constant current bias indicated.
Figure 3.12: (a) Spurious measurement of divergence peak with a fixed current bias of
10 nA in sample K52. Upon increasing H, the peak appears to invert. An experiment
that avoids Joule heating shows monotonically increasing resistance profile with H
(see Fig. 3.18). (b) Heating of devices K22 and K23 beginning at H near the divergent
resistance. Even relatively low current excitation of 1 nA causes a clear break in slope
followed by unreliable measurements.
More directly, Fig. 3.13 shows that upon increasing the current bias above nA
40
3.2. Experimental Results
excitation, the resistive peak is suppressed. This shows that power dissipation as low
as pW can be problematic for measuring the high resistance state. While this seems
incredibly low, being a single layer of atoms only loosely coupled to the heat bath,
it is in fact not unreasonable for such a low excitation to cause thermal runaway
problems.
200
T = 300 mK
150
15 nA
H = 14 T
2 nA
100
R
xx
(k
)
0.6 nA
50
0
-3
0
3
V (V)
G
Figure 3.13: Resistance near the charge neutral point for device K7 measured at 14
T and 0.3 K. The applied current bias is changed between 15 nA, 2 nA, and 0.6 nA.
Clearly the sample heating becomes problematic between 2 nA and 15 nA as seen by
the suppression of the peak. This is consistent with heating the sample to several K.
To measure the high resistance state carefully, then, we have to adopt a technique
that allow for ultra-low dissipation in the device. The measurement circuit we use
is shown in Fig. 3.14. Instead of applying a constant current, we instead apply a
3.2. Experimental Results
41
constant voltage to the circuit. Then, by measuring the current in the circuit at a
low AC frequency as well as the voltage drop across the device, we can do phase
sensitive measurements to ∼ 40 MΩ with dissipation dropping down to the aW level.
As we still use a four-probe measurement scheme, the input-impedance of our voltage
preamplifier’s limits the resistance we can measure below 100 MΩ. Because we will
be interested in the detailed functional form of R0 (H), we prefer this method to a
2-probe scheme which could resolve R in to the GΩ regime.
3.2.4
Measurements with pW dissipation
Adopting the measurement scheme in Fig. 3.14 and going to extremely high H, we
can show that R0 is a divergent function of H. Fig. 3.15 shows that while at higher T
R0 increases only mildly with H, at 0.3 K the value diverges to our measurement limit.
As we show below, such a sharp increase of R0 cannot not be well characterized with
a power law fit in H. Instead, we will have to invoke an exponential dependence. The
impact on σxx and σxy is dramatic. Shown in a separate device in similar conditions,
in Fig. 3.16 it can be seen that σxx is suppressed to a wide plateau at zero in H > 28
T. Similarly, the σxy zero filling plateau observed in lower fields is extended to a wide
range (Fig. 3.17).
It is instructive to look at the detailed H dependence of the resistance divergence.
Fig 3.18b shows a color plot of Rxx as a function of H and VG at 0.3 K. The high
resistance state is confined to the lowest densities. In fact, the divergence is confined
to below the filling ν = ±1, as indicated by the dashed lines.
42
3.2. Experimental Results
Voltage divider
W
100 k
p
-filter
Lock-in A
low-pass
filter
pre-amplifiers
p
Lock-in V2
-filters
Ammeter
Lock-in V1
W
1 k
Sample J18
sample
Figure 3.14: Schematic of the low-dissipation voltage-regulated circuit. Lock-in amplifier A produces a regulated voltage at 3 Hz that is reduced to an amplitude of 40
µV by a 100:1 voltage divider. The signal goes through a π-filter, a buffer resistor
of 100 kΩ, and a low-pass filter before entering the dewar. The ac current passing
through the graphene sample is measured by a picoammeter (DL 1121), whose output
is phase detected by lock-in A. The longitudinal voltage Vxx and Hall voltage Vxy are
phase detected by lock-ins V1 and V2, respectively, after transmission through a bank
of filters and high-impedance preamplifiers. As shown, all wires entering the dewar
are buffered by 1 kΩ nichrome thin-film resistors. The inset shows sample J18.
3.2. Experimental Results
43
Figure 3.15: Resistance at the Dirac point for sample K52 measured to 31.2 T. Below
5 K, the resistance increases from ∼ 10 kΩ to more than 40 MΩ (our measurement
limit). The profile at the lowest T is divergent. The inset shows the low R behavior.
44
3.2. Experimental Results
2.5
20
25
J18b
2.0
28
15
T = 1.6 K
31.2
1.5
1.0
xx
(e
2
/ h)
10 T
0.5
0.0
0
5
10
V
G
15
(V)
Figure 3.16: σxx for sample J18 for field up to 31.2 T. At 10 T, the n = 0 degeneracy
is intact, but it is progressively broken with increasing H. At the highest H, breaking of the remaining degeneracy can also be seen between the split subbands. The
conductivity at 31.2 T at and around the Dirac point is nearly zero.
45
3.2. Experimental Results
6
J18b
T = 1.6 K
4
xy
(e
2
/ h)
10 T
15
2
20
25
28
0
31.2
-2
-4
-6
-10
-5
0
5
V
G
10
15
20
25
(V)
Figure 3.17: Quantum Hall plateaus for sample J18 to 31.2 T. At the highest fields
the σxy = 0 plateau extends over more than 5 V.
46
3.2. Experimental Results
30
10
10
B (T)
28
10
26
10
24
10
22
7
W
6
5
4
3
2
10
(b)
20
-10
<10
-5
0
11
n 2D (10
5
-2
cm
10
)
Figure 3.18: (a) Detailed field dependence of the resistance profile in device K52
increasing H from 10 T to 31 T. (b) Color plot of measurements taken from (a).
The high resistance state is confined below the lowest filling ν = ±1 (marked with
dashed line). The 2D density is calculated assuming n2D = CVG /e, where C is the
capacitance per unit area of the 300 nm SiO2 gate dielectric 1.14 × 10−4 F / m2 .
47
3.3. Discussion
3.3
Discussion
We now attempt to characterize the divergence in resistance seen at the Dirac point.
As mentioned in 3.1.3, the functional form of R0 (H) in Fig. 3.15 cannot be satisfactorily fit by a power law in H. We find instead that empirically we can capture the
sharp divergence in R0 with the form
√
R0 = Ae2b
1−h
(3.8)
where h ≡ H/HC , HC being a fit parameter determining the critical field at which the
device becomes an electrical insulator. A and b are also fit parameters. The quality
√
of this fit is best shown by plotting R0 on a logarithmic scale against 1/ 1 − h, which
should yield a straight line. For device K52 measured to high H, we find that over 3
decades in R the fit is extremely good with A = 440, b = 1.54, and HC = 29.1 T.
We gain further confidence in this fit by comparing the results of several samples.
Those devices taken to high H > 30 T are shown in Fig. 3.20. Each shows the
characteristic divergence (see Fig. 3.20a), and upon casting the data in the form of
Eq. 3.8 we find that all the data collapse on to a universal curve with the same A
and b, but different HC . The fitting results are collected in Table 3.2. Another fact
clear from Table 3.2 is that low HC correlates well with low V0 but poorly with the
size of the device characterized by the voltage lead separation L.
The physical inspiration for Eq. 3.8 is the Berezinskii Kosterlitz Thouless (BKT)
transition [48, 49]. In the context of 2D superfluids, this is the notion that fluctuations in the phase of the order parameter determine the nature of the transition.
Topological defects (vortices) occur in the phase angle to disrupt long-range order.
48
3.3. Discussion
Figure 3.19: Fit of R0 (H) from sample K52 (see Fig. 3.15) to the form in Eq. 3.8.
The fit shows excellent agreement over more than 3 decades in resistance.
units
K22
K7
K52
J18
J24
V0
V
-0.6
1
3
20
24
L
HC
µm
T
3
11 ± 1
1.5
18
3.5
29
2.75
32
2
36
Table 3.2: Sample parameters and fit results. V0 is the offset gate voltage. L is the
distance between the voltage leads. HC is the critical magnetic field determined by
the fit of Eq. 3.8.
49
3.3. Discussion
108
T = 0.3 K
)
107
106
0
(
K52
5
10
J24
J18
104
103
(a)
0
10
20
0
108
30
H (T)
0
(
)
107
106
105
104
103
(b)
1
2
3
4
-1/2
(1 - H/H )
c
Figure 3.20: (a) Resistance divergence in three samples J18, J24, and K52 measured
to high H. All the sample reach our measurement limit. All measurements are
performed at 0.3 K. (b) Casting the data in (a) to the form of Eq. 3.8. With
our measurement error, all the data collapse on to a single curve. The only scaling
parameter in the fit to change for the 3 devices is the critical field HC .
50
3.3. Discussion
A simple energetics argument gives the BKT transition temperature TBKT , which is
lower than the mean field transition temperature. The energy of a single vortex Ev is
Ev = η ln
R
a
(3.9)
where η is, for example, in the case of a 2D superfluid, the superfluid density, R is
the system size, and a is the size of a vortex core. The entropy of a vortex can be
calculated from the number of possible possible states Ω as
S = kB ln Ω = kB ln
R2
a2
(3.10)
where kB is Boltzmann’s constant, and we estimate the number of positions of the
vortex to by the ratio of the system size πR2 and the vortex size πa2 . The free energy
F contribution of a vortex is then
F = E − T S = η ln
R
R
− 2T kB ln
a
a
(3.11)
so that it becomes energetically favorable to have vortices below TBKT = η/2kB .
Below TBKT , the vortices bind together as vortex-antivortex pairs and there is long
range order in the system.
For 2D superconductors, the BKT transition is only valid when R (and a) is
smaller than the 2D magnetic screening length. In this regime, the energy of a vortex
follows Eq. 3.9. Experimental evidence for BKT physics in this regime is very strong.
Fig. 3.21 shows two independent verifications of the BKT theory [50]. In 3.21a,
a non-linear IV curve is measured as a result of pulling apart the vortex-antivortex
pairs. In 3.21b, the resistance going through the superconducting transition is plotted
√
√
on a log scale against 1/ T − TC ∝ 1/ 1 − t, defining t ≡ TC /T . This relation is
51
3.3. Discussion
obeyed over more than 4 decades. In this regime, the resistance scales as the BKT
correlation length ξ which is has an exponential dependence on T
√
ξ = Aeb/
1−t
(3.12)
It is this result we draw an analogy with in Fig. 3.19, replacing the reduced temperature t with reduced magnetic field h. Note that b is not a universal number, but
should be of order unity, as observed here.
Figure 3.21: Experimental evidence for KT transition in thin film superconductor
amorphous indium / indium oxide from [50]. (a) Non-linear IV curves show evidence
for unbinding vortex-antivortex pairs with increasing bias. The different curves are
taken at different T from 1.94 K (a) to 1.46 K (m). (b) The resistance cast in an
analogous form to Eq. 3.8 for T instead of H for the resistance of the film passing
through the KT transition. The fit is in good agreement over more than 4 decades in
R.
While in the 2D superconducting case the state destroyed by vortex-antivortex
unbinding is the ordered zero resistance state, here we hypothesis that above HC
graphene is in an ordered, insulating state. As H is decreased below HC (analogous
3.3. Discussion
52
to increasing T across TBKT ), the ordered state at large H is destroyed by the spontaneous appearance of defects which increase exponentially in density. The fall of R0
below H0 then reflects the current carried by these vortices as ξ 2 . As a prediction we
note that in this regime there should be analogous non-linear IV behavior as seen in
the 2D superconducting case in Fig. 3.21a.
There are several alternative scenarios. One obvious implication of the insulating
state is that the proposed CPEs with conductance 2e2 /h appear to be absent. However, these modes are not protected against magnetic impurities, so it can be argued
that localization along the edge of the sample is occurring. This could be driven
with increasing H as the edge modes are pushed closer to the boundary. However, as
shown in Table 3.2, there is no obvious correlation between sample size and HC . Presumably longer samples would be exponentially more susceptible to this localization.
Moreover, there is an obvious correlation between high sample quality (as determined
by low V0 ) and low HC . One would expect samples with higher impurities levels to
be more easily localized, but we see the opposite trend. Higher quality samples have
progressively lower HC which we take to imply the insulating state is a property of
the pure system.
A subtle point is that the BKT transition is relevant for 2D systems, but this
relativistic system should be 2+1 dimensional. A possible solution to this scenario
has been proposed by Nomura, Ryu, and Lee [42]. Instead of breaking the spin or
valley symmetry with increasing H, they propose that the transition at ν = 0 is due
to a spontaneous ordering of the pseudospin in the physical/valley spin plane. This,
in analogy to the 2D XY ferromagnet model, would allow for a 2D BKT transition.
They also predict a decreasing HC for higher mobility crystals.
53
3.3. Discussion
Figure 3.22: Representation of U(1) order proposed in [42] (a) The Kekule bonddensity-wave order with two defects marked by a filled circle. These defects cause a
change in the sense of the bond ordering, denoted by the different colors. The defects
are charged. (b) Representation of the bond ordering in terms of a U(1) phase. The
charge defects cause vortices and anti-vortices in the phase. Both (a) and (b) taken
from [42].
The ground state wavefunction produced by this ordering breaks the symmetry
between A and B sublattice. The wavefunction is given by
1 †
√
ψ0 =
aKms + eiφ a†K ′ ms |0i
2
m,s=↑,↓
Y
(3.13)
where the creation operator is creating an electron on the mth n = 0 Landau level
with spin s and in valley K or K’ and φ is the angle in the spin-valley plane. According
to [42], this represents a bond density wave (see Fig 3.22a). It has been shown that
such a density wave, known as a Kekule ordering, can cause insulating behavior on
the hexagonal lattice [51]. Charged impurities are responsible for randomizing φ, as
can been seen in Fig. 3.22b where the direction of the Kekule order is depicted. One
therefore expects the defects to carry charge with a conductivity proportional to the
number of vortices nv times their mobility µv . As nv is proportional to ξ −2 , we thus
expect ρ ∼ ξ 2 , as observed.
3.3. Discussion
54
Figure 3.23: Temperature dependence of the Dirac point resistance in device J18 at
selected H approaching HC = 32 T. At 20 T, the increase in R0 is relatively weakly T
dependent. At the highest H, the profile is progressively more well fit to an activated
form R ∼ e∆/T . The dashed lines show an activated curve with ∆ = 14 K for the
highest H.
3.4. Summary
55
Finally, we return to the question of T saturation of the R0 below 2 K. Fig 3.23
shows the variation of R0 in device J18 as a function of T for H approaching HC = 32
T. At relative low H, the curves are clearly saturating at low T . However, at higher
H the curves begin to fit better an activated form R ∼ e∆/T , where ∆ = 14 K at
31.2 T (the dashed line shows the activated form). This implies that the saturation
behavior disappears in the insulating state above the critical field HC and grows as
H drops far below HC .
3.4
Summary
The ground state of graphene at the Dirac point in high H is an insulator. The
scenario that describes the phase transition in H most accurately attributes the
broken symmetry to a spontaneous ordering of the valley-spin pseudospin. Very recently, measurements on significantly higher mobility devices made with free-standing
graphene have been reported [52, 53]. In these devices with V0 < 0.5 V and mobilities
higher than 100,000 cm2 / Vs, the field at which insulating behavior is as low as 5
T. This confirms that the transition to an insulating state is an intrinsic property of
graphene.
Chapter 4
Thermopower and Nernst effect in
Graphene in a high magnetic field
The thermoelectric properties of graphene have been investigated far less than the
electric properties. Here, we measure the thermopower S and Nernst signal Syx in
graphene to high magnetic field H [54]. Of particular interest is the response in the
quantum Hall regime. We compare our results to a theory due to Girvin and Jonson
(GJ) for edge state contributions to thermopower [55, 56]. It has been successful in describing systems with quadratic dispersion, such as GaAs heterostructures [57, 58, 59].
We find agreement with the model with a correction for Berry’s phase effects for S.
The off-diagonal thermoelectric conductivity αxy has peaks at each Landau level consistent with the GJ model, but is anomalously narrow at the Dirac point. We describe
the method for thermoelectric measurements in graphene, briefly introduce GJ theory, present experimental results and discuss unique features of thermoelectricity at
the Dirac point.
56
57
4.1. Thermoelectric Measurements in 2D
4.1
4.1.1
Thermoelectric Measurements in 2D
Thermoelectric Measurements
In two dimensions (i = (x, y)), the thermoelectric tensor Sij is defined as
Ei = Sij ∂j T
(4.1)
where Ei is the electric field in the i direction, and ∂j (T ) the temperature gradient
in the j direction. The thermoelectric conductivity αij is
Ji = αij ∂j T
(4.2)
where Ji is the electrical current in the i direction. These two quantities are useful
for considering thermoelectric experiments with two different types of boundary conditions. If electric current is allowed to flow in the circuit (for example, an infinite
boundary condition), then it is useful to think in terms of charge currents and Eq. 4.2
is appropriate. If charge currents are not allowed to flow, then Eq. 4.1 is more useful.
In practice, it is more common for calculations of αij to be done with the infinite
boundary condition while experiments usually measure voltages in a circuit with an
open boundary condition and thus measure Sij . As we show below, by additionally
measuring the conductivity tensor σij = Gij , we may calculate αij from measured
quantities.
Our measurements are performed on devices in the form of sample J10 in Fig. 4.1.
The operating principle of the device is borrowed from similar measurements done
on carbon nanotubes by Small, Kim and coworkers [60]. The graphene crystal was
exfoliated onto a thin layer (300 nm) of SiO2 grown on n-doped Si wafer. A strip of
4.1. Thermoelectric Measurements in 2D
58
Figure 4.1: Image of device J10. A heater is patterned near the sample to supply a
temperature gradient ∇T . 4-probe thermometers are patterned on top of the sample
(labeled Therm) that also act as current leads. Signal leads are also patterned to
measure voltages. The scale bar is 3 µm.
gold evaporated near one end of the sample serves as the heat source (labeled “Heater”
in Fig. 4.1). To measure the gradient −∇T , we pattern two 4 probe thermometers.
Their resistances were measured to 4 significant figures between 10 and 300 K. Using
this calibration we can use them as local thermometers with resolution ∼ ±1 mK
above 10 K. They are also used as current leads for measurements of σij . The voltage
signals Vij are detected by the 4 gold lines labeled “Signal Leads”. For a temperature
difference δT ∼10 mK, the uncertainty in the gradient −∇T was ±10%, mostly due
to the uncertainty in the voltage lead spacing (∼2 µm ± 200 nm). To improve the
signal-to-noise ratio, we used an alternating heater current of frequency ω/2π ∼ 3 Hz
to produce an alternating gradient. The thermoelectric response was detected at 2ω,
with a phase shift of −90◦ .
An applied temperature gradient −∇T ||x̂ (with H||ẑ) generates an electric field E
59
4.1. Thermoelectric Measurements in 2D
in the sample with components Ex = Sxx (−∂x T ) and Ey = Syx (−∂x T ). By convention
we define the thermopower S and Nernst signal eN as
S≡
Ex
−∇T
,
eN ≡
Ey
∇T
(4.3)
As referenced above, we can calculate αij from this experimental geometry. In Fig.
P
4.1 with −∇T ||x̂, H||ẑ, the charge current density J is Ji = j [σij Ej + αij (−∂j T )].
Setting J = 0, we solve for E and obtain
Ei = −
X
ρik αkj (−∂j T ) =
k,j
X
Sij (−∂j T ),
(4.4)
j
with ρij = Rij the 2D resistivity tensor. Inverting Eq. 4.4, we may calculate the
tensor αij from measured quantities. We have
αxx = −(σxx Ex + σxy Ey )/|∇T |
αxy = (−σxy Ex + σxx Ey )/|∇T |.
(4.5)
This will be useful in comparing our results with the GJ theory.
Measurement of S and Syx are often more incisive probes of electronic band structure than purely electrical quantities. One way of understanding this is the Mott
relation [61], which relates thermoelectric transport quantities to electric quantities,
2
π 2 kB
T dσij Sij = −
3 |e| dE E=EF
(4.6)
where E is the energy and EF is the Fermi energy. Since Sij is the logarithmic
derivative of Gij with respect to energy, effects of Landau level quantization which
cause changes to the band structure are amplified in the thermoelectric measurement.
60
4.1. Thermoelectric Measurements in 2D
A useful tool is the gate voltage VG , which allows us to directly test Eq. 4.6 by
varying E. We use the chain rule to explicitly include the dependence of VG
2
π 2 kB
T 1 dσij dVG Sij = −
3 |e| σij dVG dE E=EF
(4.7)
and from the density of states and capacitive charging Q = CV ,
dVg
=
dEF
r
√
e 2 VG
πC ~vF
(4.8)
where C is the capacitance per unit area of the 300 nm thick SiO2 gate dielectric
(1.14 × 10−4 F / m2 ) and vF is the Fermi velocity.
4.1.2
Girvin and Jonson Theory
The effect of temperature gradients on edge currents in the quantum Hall state has
been considered by Girvin and Jonson [55, 56]. The geometry of the 2D electron
system is shown in Fig 4.2a. The sample is infinite along ŷ and of length L along x̂.
The field H is applied along ẑ. This boundary condition lends to a calculation of αij ,
as discussed above. In the quantum Hall state, the longitudinal current Ix = 0, and
the Hall current Iy is give by [55]
L
Iy =
2π
Z
dk
X −e ∂ǫkN
N
~L ∂k
nF (ǫkN )
(4.9)
where k is the wave vector, N the Landau index, ǫ the energy, and nF the Fermi
function. Expressing the edge current in this way, it is plain to see the physical origin
of S in the quantum Hall state: the difference in nF due to different T on the two
edges. As shown in Fig. 4.2a, the left and right edge of the sample are at different T
and therefore nF is more smeared out on the hot (left) edge. Thus, an extra current
61
4.1. Thermoelectric Measurements in 2D
density Jy flows along ŷ giving Jy ∆T = αxy . For a finite sample, δIy produces a
quantized Hall potential VH = (h/e2 )δIy whose gradient lies parallel to (−∇T ). The
same reasoning shows that Syx arises due to disorder and should be small. Therefore,
when the chemical potential µ is aligned with the LL energy in the bulk En , Sxx
displays a large peak whereas the Nernst signal Syx is small. The calculation from
GJ is shown in Fig. 4.3.
Just as σxy is quantized in the quantum Hall regime, GJ show that the peak value
of αxy is also quantized. Following the Kubo formulation from above, one would
calculate αxy at each n 6= 0 as,
e
αxy (µ) =
hT
Z
∞
En
∂nF
dǫ (ǫ − µ) −
∂ǫ
,
(4.10)
max
As µ is varied, αxy rises to a narrow peak with peak value αxy
= (kB e/h) ln 2 ≃ 2.32
nA/K, attained when µ equals En in the bulk. This is depicted in Fig. 4.2b. The
width of the peak broadens linearly with T . This implies that for S in a finite sample
we expect the quantized values
S=
kB ln 2
e(n + 1/2)
(4.11)
We would expect there to be many issues regarding the application of this theory
to graphene. First, the Dirac dispersion of graphene leads to a new quantization
condition for σxy ; how does this effect αij and Sij ? Perhaps more interesting is the
question of the n = 0 Landau level, which is not treated by the GJ theory. As
discussed in chapter 3, there is considerable debate to the edge state structure for
n = 0, so measurements of Sij would be informative in that regime.
4.1. Thermoelectric Measurements in 2D
62
Figure 4.2: (a) The effect of −∇T on the edge currents Iy in a system with edge
currents. The red line is the energy of the nth Landau level and the broken line
the chemical potential µ. En increases sharply at the sample edges. For a system
with quantizing magnetic field in +ẑ, Hall currents flow as shown. The result of the
temperature gradient in x̂ is to unbalance the Fermi-Dirac distributions on the two
edges (shown on each side). This results in a thermoelectric response. (b) GJ theory
predicts that at each Landau level crossing with µ there is a peak in the off-diagonal
thermoelectric conductivity αxy that reaches a universal value of ≈ 2.31 nA / K (for
a singly degenerate level).
4.1. Thermoelectric Measurements in 2D
63
Figure 4.3: (a) Calculation for S and Sxy for a sample with moderate disorder from
[56]. Successively higher Landau levels have decreased maxima in S, but Sxy remains
unchanged. The calculation makes no prediction for filling ν = 0. (b) With increased
disorder, Syx grows considerably and Sxx falls far below the quantized value.
64
4.2. Experimental Results
4.2
4.2.1
Experimental Results
S with H = 0
Measurement of S near room temperature with H = 0 are shown in Fig 4.4. Also
200
S
S
Mott
K59
exp
302 K
0.5 x S
S ( V / K)
100
Mott
xx
0
-100
-200
-30
-20
-10
0
V
G
10
- V
0
20
30
(V)
Figure 4.4: Measurement S of device K59 in zero field near room temperature. The
measured Sexp is compared to the prediction from the Mott relation Eq. 4.7. Agreement is best when Smott is divided by 2. The longitudinal conductance σxx is shown
in arbitrary units as a thin dashed line.
shown (in arbitrary unites) is σxx . The gate voltage VG axis has been shifted by
V0 . σxx shows the typical field effect dependence on VG : a minimum at the Dirac
point and a strong increase as either holes (negative VG ) or electrons (positive VG )
are introduced in to the system. The experimentally measured Sexp is also shown. It
changes sign at the Dirac point and is positive for holes and negative for electrons,
reaching a maximum of ∼ 80 µV / K. Comparison to the Mott relation from Eq. 4.7
65
4.2. Experimental Results
is also shown as SM ott . The agreement is poor. Dividing SM ott by 2 yields a more
satisfactory result, but there is still disagreement between theory and experiment. It
has been suggested that this is due to the use of 4 probe instead of 2 probe conductance
in Eq. 4.7 [62], but the issue remains unresolved.
120
280 K
240
200
S ( V / K)
80
40
0
160
80
60
120
-40
-80
J10
H = 0
-10
0
10
V
G
20
30
40
(V)
Figure 4.5: Thermopower S for various T between 10 K and 300 K in device J10. The
peak structure on either side of the Dirac point is steadily scaled down as T drops.
The maximum value attained is ∼ 100 µV / K.
Detailed T dependence of S(VG ) is shown in Fig. 4.5 and Fig. 4.6 for devices
J10 and K59, respectively. As T is lowered, the magnitude of S systematically decreases. Both samples show similar peak structures on either side of the Dirac point.
The maximum magnitude of S is shown against T for both samples in Fig 4.7. The
dependence is roughly linear, as expected from Eq. 4.6. Sample J10 shows a small
66
4.2. Experimental Results
75
80 K
106 K
50
133 K
S ( V / K)
159 K
25
205 K
302 K
0
-25
-50
K59
H = 0
-75
-100
-20
-10
0
10
V
G
20
30
40
(V)
Figure 4.6: Measurement of device K59 over a limited T range. The data largely
reproduce the features in device J10 (Fig. 4.5.) The maximum |S| at 300 K is ∼ 100
µV / K.
67
4.2. Experimental Results
anomaly at low T . This seems to be associated with the appearance of large, reproducible fluctuations in S. An example of this is shown in Fig. 4.8 for sample J10 at
10 K. Five different sweeps of S(VG ) are shown to overlap. One surprising aspect of
these fluctuations is that they are larger than the baseline magnitude of S. Moreover,
the seem too large to be explained by Eq. 4.7.
100
H = 0
60
J10
40
|S
max
| ( V / K)
80
K59
20
0
0
50
100
150
200
250
300
T (K)
Figure 4.7: Maximum value of S measured for devices J10 and K59. Smax (T ) is
approximately linear in T , as expected from the Mott relation. There is a small
anomaly at the lowest T for sample J10.
4.2.2
S and Syx in H
Application of H causes large oscillations to appear in all of the thermoelectric quantities. For device J3, S is shown at H =5, 9, and 14 T. The Dirac point is at ∼ 12
V. Upon tuning away from charge neutrality, we observe a series of peaks in S that
68
4.2. Experimental Results
2
J10
T = 10 K
Sweep 1
S ( V / K)
1
2
3
4
5
0
-1
-2
-10
0
10
V
G
20
30
40
(V)
Figure 4.8: Large fluctuations in S for device J10 at 10 K. The fluctuations are
reproducible; 5 successive scans of VG are shown. The fluctuation grows on the
electron side to overwhelm the baseline S.
4.2. Experimental Results
69
coincide with successive Landau levels crossing µ. At higher H, the spacing in VG
increases as expected for the enhanced degeneracy of each Landau level. As is particularly clear in Fig. 4.9a, the magnitude of S is a decreasing function of the Landau
level index n, similar to that predicted by Eq. 4.11. However, at 20 K and 14 T S at
n = −1 is ∼ 41 µV /K, exceeding the value predicted 39.8 µV / K.
An interesting comparison is between S at 20 K and 50 K (Fig. 4.9c). Shown at
14 T, there is almost no change in the magnitude of the n = 1 peak after the factor
of 2.5 change in T . This implies that at 50 K, S is already in the quantum regime.
Similarly, comparison between S at 9 and 14 T shown very little difference at 20 K; as
for the case of electrical measurements [63], this is further evidence that the Landau
level energy spacings are extremely large even at modest H in graphene.
For each measurement of S, the longitudinal conductance Gxx is also shown for
comparison. A notable trend is that lower n levels seem to have peak values for S
and Gxx at different VG . Also, in the region in which Gxx exhibits a peak at the Dirac
point, S appears to exhibit a faint reversal in sign before going through zero on each
side of the Dirac point. We revisit this feature further below with the calculation of
αij .
The Nernst effect Syx shows similar effects of Landau level quantization, shown
in Fig. 4.10. Here, there is a large peak in Syx at the n = 0 level, followed by a
series of peaks at each successive level. The higher n levels appear to have similar
peaks in Syx and the signal oscillates between positive and negative values. This
is qualitatively consistent with the GJ calculation in Fig. 4.3. As opposed to the
saturating dependence of S in high H, the peak value of Syx at the Dirac point
continues to grow with H. It reaches above 40 µV / K at 14 T and 20 K, and has a
70
4.2. Experimental Results
-20
-10
0
10
20
30
30
(a)
5 T
6
40
J3, T = 20 K
20
10
4
0
2
-10
0
-20
(b)
-1
40
20
4
0
G
xx
2
(e /h)
n=0
-2
2
+1
+2
S ( V / K)
9 T
6
-20
0
6
(c)
14 T
40
50 K
20
4
20 K
0
2
-20
0
-20
-10
0
10
V
G
20
30
40
(V)
Figure 4.9: Pronounced oscillations due to Landau level quantization seen in device
J3 at H = 5 T, 9 T, and 14 T (a, b, and c). The longitudinal conductance Gxx is also
shown for comparison. The Landau level index n is marked in (b). Measurements
are taken at 20 K, except for the curve shown for comparison at 14 T in panel (c).
71
4.2. Experimental Results
-20
6
-10
0
10
20
30
40
(a)
5 T
20
10
4
0
2
J3, T = 20 K
0
10
G
0
2
yx
4
S
20
( V / K)
30
(b)
9 T
xx
2
(e /h)
6
-10
-10
0
(c)
6
14 T
40
30
20
4
10
0
2
-10
0
-20
-10
0
10
V
G
20
30
40
-20
(V)
Figure 4.10: Nernst signal Syx as a function of VG for H = 5 T, 9 T, and 14 T (a, b,
and c). The dominant feature is the peak at the n = 0 Landau level. It grows with
increasing H and has a positive value. Additional oscillations of similar size are seen
at higher n.
72
4.3. Discussion
sign (positive) consistent with the vortex-Nernst effect in superconductors [64].
4.3
Discussion
The magnitude of the quantized peak in S given by Eq. 4.11 is smaller than that
measured in Fig. 4.9, as discussed above. One explanation for this is that the
derivation of Eq. 4.11 does not include the effects of the Berry’s phase shift that
is responsible for the unique quantum Hall structure (Eq. 3.7) in graphene [44].
Taking this shift in to account, we instead would expect
S=
kB
ln 2
en
(4.12)
This would predict a peak value of 59.6 µV / K, which is significantly higher than the
measured value of 41 µV / K. However, it seems likely that samples with less disorder
broadening of Landau levels would have an enhanced S (as well as a suppressed
Syx ) consistent with the GJ calculation as shown in Fig. 4.3. Further experiments
with devices of higher quality will be required to resolve this issue. If one were
able to reliably measure this quantization, in principle it represents a method for
thermometry calibrated only by fundamental constants.
To analyze the results in more detail, we calculate the αxx and αxy by Eq. 4.5.
The results are shown for H = 9 T and 14 T in Fig. 4.11a and b, respectively. Shown
as a thin green line at the top of each panel is the quantum value predicted by GJ with
the appropriate degeneracy (4) for graphene 4kB e/h ln 2. The peaks reach within 20
% of this value. It should be noted that this may be due to the large uncertainty in
the calibration of ∇T simply due to the geometric uncertainty in the device leads.
4.3. Discussion
73
Figure 4.11: The diagonal and off diagonal elements of αij calculated from measured
Sij and σij at 9 T and 14 T (a and b). αxx has a repeating positive/ negative profile
while αxx exhibits a series of peaks at each Landau level where the value reaches near
the predicted maximum of (4kB e/h) ln 2 (shown as a dashed line).
74
4.3. Discussion
Between the peak values, αxy falls to zero. Within error it appears to be always
non-negative. αxx shows a dispersive profile changing from positve to negative values
as the chemical potential crosses the center of each Landau level.
One surprising feature is that even for n = 0, αxy appears to be reaching near
the GJ model’s prediction. However, the edge current model cannot accommodate
the structure of the n = 0 level, so it is unclear why it appears to obey the same
quantization. One noticeable feature of αxy at the n = 0 level is that it is substantially
narrower in VG than the higher n levels. A useful metric for comparison is the area
An (T ) for each n from Eq. 4.10,
An (T ) =
Z
dn2D αxy ∼ Nn
2
4c0 kB
eT
h
(4.13)
where Nn is the maximum value of the density of states of the Landau level and c0 is
a constant ≈3.29. Note that we have performed the integral with respect to the 2D
carrier density n2D which is given by CV /e. At a given T , the area under the peak
of αxy is directly proportional at Nn .
If we naively extend the validity of Eq. 4.10 and 4.13 to the n = 0 Landau level,
we would be forced to conclude that Nn is smaller by roughly a factor of 4 compared
to the higher n. It is difficult to reconcile this with the basic notion of conservation
of states, and presents a puzzle for interpretation at the Dirac point. Whereas in S
we noticed what appeared to be a reversal in sign on either side of the Dirac point,
in αxy states appear to be missing in the same region. While the reversal in sign of S
may hint at the presence of counter-propagating edge modes at n = 0 [3], it is unclear
how these modes could describe the anomalous behavior of αxy . Similarly, the source
of the large Syx at the Dirac point remains unexplained.
4.4. Summary
4.4
75
Summary
Our results reveal that, at 9 T, the thermoelectric response in graphene already falls
in the quantum regime at 50 K. Measurement of S indicate that Berry’s phase effects
modify the quantization relation. The inferred off-diagonal term αxy is a series of
peaks independent of n, B and T . The peak values come close to the predicted value
of (4kB e/h) ln 2. The most surprising feature of the thermoelectric response is that at
n = 0 the peak in αxy is significantly narrower than for higher n, but also approaches
the GJ quantized peak value.
Chapter 5
Quantum Interference in
Macroscopic Crystals of
Nonmetallic Bi2Se3
3D topological insulators (TIs) are a new class of insulators consisting of a 3D insulating bulk wrapped with novel 2D conducting surface states. The surfaces states are
host to spin-polarized Dirac electrons with potential use for a broad range of applications [65]. However, a serious materials drawback is that the bulk is a relatively
poor insulator. Without achieving a significantly improved insulating state, the bulk
of the crystal dominates electrical transport and many of the exciting proposals for
potential uses of TIs are unachievable. Here, we show how through chemical doping of
Ca in to the large band gap TI Bi2 Se3 we can suppress conduction of the bulk bands.
We uncover a unique conduction channel characterized by large quantum corrections
to electrical transport at low temperature T [66]. We present a brief introduction to
76
5.1. Transport measurements in 3D Topological Insulators
77
transport in 3D TIs, show experimental results for doping and low T transport, and
finally discuss implications for the surface bands in these materials.
5.1
Transport measurements in 3D Topological Insulators
There are several specific expectations for the transport properties of 3D topological
insulators.
1. There should be conducting, in-gap states as seen in the 2D case in Fig. 1.5.
2. In-gap states should not experience 2kF scattering (backscattering) as that
would require a spin-flip and break time reversal symmetry. This implies that
they would have relatively high mobility.
3. A magnetic field H should open a gap at the Dirac point due to breaking of timereversal symmetry. The states away from the Dirac point should be quantized
in to Landau levels and exhibit 2D Shubnikov de-Haas oscillations.
4. As in graphene, Berry’s phase effects should modify the quantum Hall structure,
but here the degeneracy (for a single surface) would be 1/4 of graphene (see
Eq. 3.7),
σxy =
e2 1
n+ )
h
2
(5.1)
5. From the two dimensionality and large spin-orbit coupling, we would expect to
see weak anti-localization [67].
5.1. Transport measurements in 3D Topological Insulators
78
Before any of these surface properties can be measured by transport, however, we
must experimentally address the remnant bulk contributions to transport. In Fig.
Figure 5.1: (a) ARPES measurement for as-grown Bi2 Se3 from [16]. The surface states
reside in the bulk band gap, but the Fermi energy is pinned to the bulk conduction
band. (b) Electron carrier density of Bi2 Se3 crystals grown with increasing excess of
Se from [68]. Increased Se suppresses the size of the bulk conduction band pocket.
5.1a, ARPES results for crystals of Bi2 Se3 are shown [16]. While the 2D surface band
with linear dispersion is evident in the bulk band gap, it is also apparent that the
Fermi level EF is located in the conduction band. While a band structure calculation
would predict the chemical potential would be in the bulk band gap, it has been found
that over time Se escapes from crystals and in the process donates electrons. One
proof of this hypothesis is shown in Fig. 5.1b. Crystals in these experiments were
grown large amounts of excess Se [68]; crystals with more Se tend to have lower carrier
densities n, presumably due to suppression of Se vacancies [69]. We can still measure
crystals from Fig. 5.1a, though, and address whether or not we see signatures of 2D
transport.
5.1. Transport measurements in 3D Topological Insulators
79
Basic transport experiments characterizing these crystals are shown in Fig. 5.2.
The resistivity profile ρ(T ) is that of a metal with a relatively low mobility 500-5000
Figure 5.2: (a) Electrical transport in heavily n-doped Bi2 Se3 . The resistivity profile is
metallic, though the low temperature ρ is that of a poor metal. (b) Shubnikov de-Haas
oscillations in n-doped Bi2 Se3 . The period of the oscillation in 1/H is only weakly
dependent on angle θ. This indicates a bulk conduction pocket that is approximately
spherical.
cm2 / Vs (depending on the particular crystal). The Hall density nH (not shown) for
these crystals is ∼ 1019 e− / cm3 , demonstrating that EF is deep in the conduction
band. Measuring the extremal Fermi surface size Ak is possible through Shubnikov
de-Haas measurements [70], as shown in Fig. 5.2b. Here, the period of the oscillation
in 1/H is proportional to Ak
2πe 1 −1
Ak =
∆
(5.2)
~
H
p
The resulting Fermi wavevector kF = Ak /π is roughly 0.07 Å−1 , consistent with
the conduction band measured by ARPES in Fig. 5.1a. When the field is tilted,
the observed Ak changes only by ∼ 30%, consistent with a slightly ellipitcal Fermi
5.1. Transport measurements in 3D Topological Insulators
80
surface as expected for the bulk conduction band [71]. Thus, we conclude that these
transport properties are coming from the 3D bands. To measure the 2D states, we
will have to move the chemical potential down to the gap.
81
5.2. Results
5.2
5.2.1
Results
Nernst effect in Cax Bi2−x Se3 : Tuning the Chemical Potential
Following the discovery that Ca can be used to donate holes to Bi2 Se3 by substituting
for Bi [72], we have used light amounts (. 0.5%) of Ca to attempt to move the
chemical potential in to the bulk band gap. As our metric for progress, we will
monitor Ak via the Nernst effect eN . As described in Chapter 4, thermoelectric
quantities such as eN are more sensative to changes in the electronic band structure
due to Landau level quantization than purely electrical quantities.
5
7.5 K - 40 K
0
60
-10
80
100
-15
125
-20
150
e
N
( V / K)
-5
-25
175
200
-30
225
250
-35
n
H
19
~ 10
-
275
3
e / cm
300
-40
0
5
10
H (T)
0
Figure 5.3: Measurement of Nernst effect eN for heavily doped Bi2 Se3 crystal M1.
The signal crosses over from negative to positive with decreasing T and shows Landau
level quantization below 40 K.
82
5.2. Results
Typical results for a crystal (sample M1) with relatively high nH (∼ 1019 e−
cm−3 ) are shown in Fig. 5.3. At high T , eN is large and negative. As T decreases, eN
changes sign and begins to exhibit Landau level oscillations periodic in 1/H. From
the temperature dependence of the oscillation magnitude, we estimate the effective
mass to be ∼ 0.1me .
10
15
20
30
40
8
50
60
4
80
2
100
e
N
( V / K)
6
0
150
-2
200
250 K
18
-4
n ~ 10
0
5
-
3
e / cm
10
H (T)
0
Figure 5.4: Measurement of Nernst effect eN for lightly n-doped Bi2 Se3 crystal M4.
A crossover from negative to positive eN is seen, but the negative contribution is
suppressed relative to more heavily doped crystals. Landau level quantization is
apparent at 20 K and below.
Crystals with smaller densities show a similar pattern, but with the negative eN
signal suppressed. Fig. 5.4 shows eN at selected T for crystal M4, with nH ∼ 1018
e− cm−3 . The crossover to positive eN occurs near T = 100 K. Landau quantization
is still seen, but the period is significantly longer. This corresponds to a smaller
conduction band pocket.
5.2. Results
83
Figure 5.5: (a) Fan diagram constructed from Nernst measurements on crystals with
a range of Ca dopings. The larger slopes correspond to crystals with smaller extremal
Fermi surfaces. The sign of the Landau level is determined by the Hall effect and
thermopower. (b) Comparison of Nernst measurement for relatively high and low
carrier densities at 15 K.
84
5.2. Results
Samples M1 and M4 are compared in Fig. 5.5b for T = 15 K. To make comparisons
between samples more clear, we construct the index plot in Fig. 5.5a. We record the
H at which eN shows a maximum, and plot the successive peak values assuming each
corresponds successive integer Landau levels. The steeper the slope in Fig. 5.5a, the
smaller Ak . Note that we choose the Landau levels to be hole-like or electron like
(positive or negative) from measurements of thermopower S and the Hall coefficient
p
RH . The smallest Ak found corresponds to Fermi wavevector kF = A/π ≈ 0.046
Å−1 . Given only this Ak , it is ambiguous if this corresponds to the bulk or surface
Fermi surface. However, given that these crystals exhibit metallic ρ(T ) profiles and
the lack of significant angular dependence of Ak in Fig. 5.2, it seems much more
likely that these are bulk properties.
To further test this idea, we can compare kF determined from these measurements
to that determined from RH . We assume that the Fermi surface is perfectly spherical
so that the density derived from Shubnikov de-Haas oscillations nSdH is
nSdH = 2
(4/3)πkF3
kF3
3 = 2
3π
2π
(5.3)
The result is shown in Fig. 5.6. The scaling between the two is satisfactory, though
there is disagreement by an overall factor of roughly 2. This underestimate of the
density is likely do to the non-zero eccentricity of the Fermi surface [69]. Also plotted
in Fig. 5.6 is the quantity kF ℓ which characterizes the mobility of the crystals. The
act of Ca doping is to push the crystals to the right in Fig. 5.6 (toward more hole
doping). We note that while kF ℓ is monotonically reduced at higher dopings, it is not
dramatically suppressed even from the highest doping. There are also a few crystals
grown near the lowest doping that had significantly higher kF ℓ, which we attribute
5.2. Results
85
to successful attempts to suppress Se vacancies.
Figure 5.6: Comparison of electron densities calculated from Shubnikov de-Haas oscillations and the Hall effect. The two scale closely, differing by an overall factor of
roughly 2. Also shown is the parameter kf ℓ which reflects the mobility of electrons in
the system. Even highly doped crystals (those with large hole dopings) do not have
a significantly suppressed value.
5.2.2
Nernst effect in Cax Bi2−x Se3 : Low T behavior
Before moving on to results for crystals with the chemical potential near the gap, we
briefly detail results of eN and S at low T . Fig. 5.7 shows the result of a measurement of sample M3 down to 0.5 K. The Landau level quantization continues to be
pronounced, but a new feature that appears is a near H independent eN at low T .
This is brought out more clearly by confining the data below 2 T so that the effect
of Landau level quantization does not obscure the signal. Casting the signal as eN /T
in Fig. 5.8 to remove the T dependence inherent in eN from the Mott Formula (Eq.
86
5.2. Results
10
8.7 K
Bi Se
2
3
e
N
( V / K)
8
5.5 K
4.3 K
6
2.2 K
4
2
1.7 K
0.5 K
0
0
5
10
0
H (T)
Figure 5.7: Nernst effect eN for Bi2 Se3 crystal M3 (low n-type doping) at low T . The
most striking feature is the H-independence that appears at the lowest T .
87
5.2. Results
4.6), we see that as T decreases eN /T appears to approach a saturation value of 0.8
µV / K2 at 2 T.
1.0
Bi Se
2
3
1.5
0.9
0.5
2.6
0.8
3.3
5.5
2
e / T ( V / K )
4.3
0.6
6.4
N
0.4
9.4
10.8
12.5
14.8
16.4
18.5 K
8
0.2
0.0
0.0
0.4
0.8
1.2
1.6
2.0
H (T)
0
Figure 5.8: Measurements of Nernst effect eN in crystal M3 divided by T . At low T ,
the traces appear to fall on a universal curve within the uncertainty of the measurement.
For completeness, we also show the result of the thermopower S to compare to
eN . Fig 5.9 shows the result for sample M4 over a large T range. The values of S
at zero T are large at high T : above 170 µ V / K at 250 K. Note that since this is
an electron-like crystal S is negative. The H dependence of S changes from slightly
increasing the magnitude at high T, to suppressing S below 150 K. Starting at 60 K,
a sharp low H dependence develops which resembles a large peak in S for zero H. At
the lowest T Landau level quantization can be observed.
Fig. 5.10 shows S(H) for sample M3 for T below 10 K. The low H peak in S is
88
5.2. Results
200
250
200
-1 x S ( V / K)
160
150
120
100
80
60
80
50
40
30
40
20
15 K
0
-10
-5
0
5
10
H (T)
0
Figure 5.9: Thermopower S for crystal M4. Note that the scale has by multiplied
by -1 since this electron-like crystal has a negative S. The H dependence changes as
function of T , eventually developing a peak at low H for the lowest T .
89
5.2. Results
more pronounced, but then becomes difficult to resolve at the lowest T . Similar to
eN , S is largely H independent below 2 K except for 1/H oscillations from Landau
quantization.
60
-1 x S ( V / K)
8.7 K
40
5.5
4.3
20
2.2
1.7
1
0.5
0
-10
-5
0
0
5
10
H (T)
Figure 5.10: Low T measurement of S in crystal M3. The peak in S(H) is the
dominant feature with Landau level quantization also evident. At low T , S becomes
H independent.
5.2.3
Transport properties of non-metallic Bi2 Se3
Having found the Ca doping (x = xmin ≈ 0.25%) which resulted in the minumum Ak ,
a large number of crystals were tested with nominally the same doping to find crystals
with the chemical potential in the bulk band gap. Among those crystals that showed
measurable Landau level quantization (samples M1-M11), all had metallic resistivity
profiles. However, by focusing on crystals at xmin , several crystals were found with
5.2. Results
90
non-metallic profiles (denoted G3-G8). These profiles are shown in Fig. 5.11a.
Figure 5.11: (a) Resistivity versus temperature profiles comparing non-metallic and
metallic crystals (metallic crystals are multiplied by 20 for clarity). Non-metallic
crystals appear to all begin to rise in ρ near 150 K. (b) Fan diagram for metallic
samples. Crystals with non-metallic profiles were found by growing with a Ca doping
near that which produced the smallest conduction band crystals M11 and M3-M5.
The resistivity at low T for G3-G8 is large, reaching above 70 mΩ cm at 0.3 K
for sample G5. This is more than 150 times larger than the most resistive metallic
crystals. However, up to 31 T (discussed further below in Fig. 5.17), Landau level
quantization was not observed. Therefore, we cannot place this crystals on the index
plot in Fig. 5.11b. However, when cross correlated with the same batch of crystals
in ARPES measurements, we find that xmin is the same doping at which ARPES
91
5.2. Results
measures the chemical potential to be in the bulk gap [15].
Figure 5.12: (a) Resistivity ρ profile for crystal G4. ρ rises below 150 K and has a
sharp upturn below 5 K. This profile is much weaker than would be expected from a
simple activated form (shown in red). (b) Conductance per square as a function of
ln T for crystal G4. The ln T dependence persists up to 2 K with a slope of roughly
200e2 /h.
Focusing on the low T portion of ρ(T ), we see that the rise occurs starting near
150 K. This is inconsistent with the predicted 300 meV gap. Furthermore, the shape
of the increase does not fit an exponential; a comparison of ρ(T ) to an activated
form is shown in Fig. 5.12a. This implies that if the chemical potential is indeed in
the bulk gap, there is another conductance channel contributing significantly. The
conductance G at 0.3 K for sample G4 (the sample which we focus on here) is roughly
8000 e2 /h. It seems unlikely that we can ascribe this to a simple surface conduction
mode, which we would expect in the ballistic limit to be given by
G = Nchan
e2
e2
= wkF
h
h
(5.4)
where Nchan is the number of conducting channels which we estimate as the width of
92
5.2. Results
the sample w divided by kF . As the crystals are mm in lateral size and µm thick, G
is more likely given in the diffusive limit
G = kF ℓ
e2
h
(5.5)
where ℓ is the mean free path. In terms of surface conduction, one sign that is more
encouraging is the extremely low T (less than 5 K) ρ(T ). Here, we see a sharp
pickup of ρ. When plotted as G(ln T ) in Fig. 5.12b, we see that the T dependence
is logarithmic. This is a signature of weak localization [67]. We will discuss this in
detail below. We find the coefficient of this ln T to be approximately 200 e2 /h.
20 K
cm)
1.5
150
15
(m
100
1.0
100
10
cm)
200
50
20
2.0
250 K
(m
25
250
50
200
0.5
5
20
150
(a)
G3
H||c
M5
10
-10
(b)
H||c
0
0.0
-10
-5
0
0
5
H (T)
-5
0
0
5
10
H (T)
Figure 5.13: (a) Magnetoresistance (MR) for non-metallic crystal G3. The profile
changes from a positive MR at high T to negative at low T . At 50 K and below, a
sharp low H dip is apparent. (b) MR for metallic crystal M5. The MR is always
positive and develops Shubnikov de-Haas oscillations at 20 K.
It is useful to further compare the non-metallic and metallic crystals. Fig. 5.13
shows the H dependence for crystal G3 and M5 at selected T . The metallic crystal in
93
5.2. Results
Fig. 5.13 has a roughly quadratic H dependence consistent with Lorentz force magnetoresistance (MR). As T is lowered, the magnitude of ρ decreases and a Shubnikov
de-Haas pattern develops at 20 K. In the non-metallic crystal, ρ(H) is also initially
quadratic and positive, but as T drops and ρ rises it develops in to negative MR at
high H. Moreover, the dominant feature that appears is a sharp low H cusp with
positive MR, seen clearly at 20 K. No Shubnikov de-Haas signatures are seen.
H (T)
H (T)
0
0
4
8
0.1
12
0
1
32
30
1.15
0.85
0.5
28
28
0.3 K
4
26
(m
cm)
(b)
1.15
30
(m
0.85
0.3 K
8
2.5
26
1.8
4 K
24
2.5
1.8
0.5
C (B,T)
cm)
(a)
0.5
24
(c)
1.8 K
0.0
0.85 K
0.3 K
-0.5
0
1
2
3
4
5
0
6
7
8
9
10
H (T)
Figure 5.14: (a) Magnetoresistance for crystal G4 between 8 K and 0.3 K. There is
a strong low H anomaly and obvious fluctuations in ρ at the lowest T . (b) The MR
is linearized over a large range in H on a log scale. As T rises, the range becomes
smaller. (c) Autocorrelation C at selected T . There is a large oscillation seen over
the entire H range.
Going to lower T and focusing on non-metallic crystal G4, we see that the low H
anomaly begins to dominate the ρ(H) profile as T drops below 4 K. This is shown in
Fig. 5.14a. When cast on a log scale (Fig. 5.14b), we see that over a large field range
the MR fits to a log H trend (shown as a dashed line). This is further evidence for
5.2. Results
94
weak (anti) localization, discussed below. The slope dG/d ln H is 140 e2 /h.
Figure 5.15: Fluctuation in conductance δG obtained by subtracting a smooth background from G(H). Shown in geometries with H out of plane (a) and in-plane
perpendicular to the current (b) and parallel to the current (c). Two sweeps for each
geometry at 0.3 K are shown. The fluctuations fades away as T approaches 4 K.
Another feature evidence in Fig. 5.14 at T below 4 K is that the trace appears to
become noisy, fluctuating with magnitude 0.5%ρ. However, we can distinguish this
signal from random noise by repeatedly sweeping H. To see the signal more clearly, we
remove a smooth background from the traces in Fig. 5.14 and display the fluctuation
is as a change in conductance δG. Shown in Fig. 5.15, for each configuration of H, we
see reproducible fluctuations in δG that die out as T rises to 4 K. Each configuration
has two traces taken at 0.3 K to demonstrate that the features are reproduced in fine
detail. It is instructive to view the autocorrelation C(B, T ) defined as
95
5.2. Results
C(B, T ) =
hδG(B ′ , T )δG(B ′ + B, T )iB ′
,
hδG2 i
(5.6)
which measures how peaks in the signal at a particular B = µ0 H are correlated.
Computation of C(B, T ) for these traces at selected T is shown in Fig. 5.14c. The
most obvious feature in C is the periodic structure. There appears to be more than
one frequency- a fast 1 T period signal and an obvious beating of this signal. Upon
observing these features, understanding their origin and relevance to the surface states
was the prime goal of experiments.
5.2.4
Angular Dependence of non-metallic crystals
Before quantitatively analyzing the results, one straightforward approach to addressing the contribution to magnetotransport from the bulk and surface electrons is to
tilt the angle of H in experiments. The results are shown in Fig. 5.16. Focusing on
the low H cusp in Fig. 5.16a, we note that for all configurations of H angle θ the peak
is seen. However, it is reduced by a factor of ∼ 3 in terms of dG/d ln H. Similarly,
the fluctuation, characterized by the rms value over the range 4 to 10 T, has similar
dependence on θ. Shown in Fig. 5.16b, the rms δG has a cosine dependence with
an θ independent offset of roughly 30%. Fig. 5.16c shows that C has a persistent
oscillation for all geometries, but there is no obvious evolution of this periodicity with
θ.
Doing these experiments to high H = 31 T, we find that the patterns remain the
same. Sample G8 was measured at 0.35 K in the same geometry; the result is shown
in Fig. 5.17. Notably there are still no Shubnikov de-Haas oscillations seen.
To bring out the fluctuation signal, we plot dR/dH for each θ in Fig. 5.18. The
96
5.2. Results
0
0.32
0
H (T)
4
(deg)
8
12
0
20
40
60
80
(a)
T = 2.2 K
4
(b)
o
13
G (
0.28
o
33
67
2
37
74
0.26
2
20
=85
z
55
61
47
H
rms [ G] (e
-1
)
/ h)
=4
0.30
y
42
0
C (B, )
0.5
o
o
=4
o
47
(c)
85
0.0
-0.5
0
1
2
3
4
0
5
6
7
H (T)
Figure 5.16: (a) Angular dependence of G(H) at 2.2 K. Both the low H anomaly and
conductance fluctuation are strongest with θ near 0◦ , but persist for all angles. (b)
rms amplitude of the fluctuation as a function of θ. The fit is a cosine with an overall
offset. The offset makes up ∼ 30% of the overall amplitude. (c) Autocorrelation C
at selected θ. The oscillation persists for all θ.
97
5.2. Results
50
40
0
31
o
47
(m
cm)
59
30
75
89
0
0
0
0
0
20
Ca
Bi
0.0025
Se
2-x
c
3
H
T = 0.35 K
10
a
G8
0
-30
-20
-10
0
10
20
30
H (T)
0
Figure 5.17: Angular dependence of magnetoresistance taken to 31 T in crystal G8.
The low H anomaly is seen for all geometries. No Shubnikov de-Haas oscillations are
observed.
98
5.2. Results
fluctuations are strongest when H is perpendicular to the sample plane, but they
are evident for all θ. As the fluctuations are roughly the same size over this large H
range, this makes it unlikely that the signal is coming from Landau level quantization,
which would be small for low H and grow exponentially.
c
H
Ca
Bi
0.0025
a
1.0
Se
2-x
3
T = 0.35 K
/ T)
G8
0
o
15
o
dR / dH (
31
0.5
o
47
o
59
o
75
o
0.0
89
0
0
10
20
30
H (T)
0
Figure 5.18: Derivative of R(H) at 0.35 K up to 31 T. The fluctuation in resistance
remains relatively unchanged over the full H range and persists for all θ.
99
5.3. Discussion
5.3
Discussion
We now attempt to understand the transport signatures of the non-metallic crystals.
Upon first viewing δG in Fig. 5.15, the ln H MR in Fig. 5.14b, and the ln T ρ(T )
in Fig. 5.12b, one may be reminded of several phenomena from 2D and mesoscopic
physics. We briefly review the relevant phenomena in the appendix. Some prototypical measurements for 2D mesoscopic systems which bear resemblance to our results
are shown in Fig. A.1.
Samples with non-metallic ρ(T ) profiles invariably show the type of low H cusp
in Fig. 5.19. As T is lowered, the peak in G(H) gets sharper. As shown in Fig.
5.14, the H dependence is well described by ln H between 0.02 T and 2 T. Given that
H suppresses G, it would on these grounds seem to point to weak anti-localization
in a 2D system as described in the previous section. However the coefficient of the
ln H (as well as the ln T behavior) measured here is hundreds of times larger than
predicted by Eq. A.1. We refer to the coefficient A defined as ∆G = Ae2 /h ln H, and
collect it for several samples at 0.3 K in Table 5.1.
units
G3
G4
G5
G6
G7
G8
ρ
mΩcm
30
15
76
18
16
25
c
µm
50
50
80
50
25
10
G
rms δG
e2 /h
e2 /h
8000
5.9
1760
0.8
7000
1
4800
0.9
1050
0.6
A
nH
1018 cm−3
0.7
178
5
35
1
135
7
63
8
20
5
Table 5.1: Sample parameters. ρ is the resistivity at 0.3 K (except for G4, which was
only cooled to 4 K). c is the crystal thickness. G is the conductance per square at 0.3
K. rms δG is reported at 0.3 K. A is the coefficient of the ln H term in ρ(H). nH is
the density inferred from the Hall coefficient RH = 1/ne.
100
5.3. Discussion
One possibility is that the crystal is broken up to hundreds of layers such that many
faces are acting in parallel. However, we have also seen that weak anti-localization
should only depend on H perpendicular to the 2D system, but the H rotation experiment showed that the signal persisted for all geometries (Fig. 5.16 and Fig. 5.17).
This would imply both an orbital and spin contribution to weak anti-localization.
0.44
40 K
G4
H||c
0.40
0.36
20
G (
-1
)
30
10
5
0.32
1.8
1.1
0.28
-2
0.3
-1
0
1
2
H (T)
0
Figure 5.19: Low field magnetoconductance below 2 T for crystal G4. The low H
anomaly grows systematically with decreasing T .
Turning to the fluctuations seen in the conductance, we present the T dependence
of δG, the autocorrelation C, and the Fourier transform F in Fig. 5.20. Comparing
these results to theory of conductance fluctuations rules out a conventional source
(see appendix). In Fig. 5.20a, the signal is seen to persist to H = 12 T, ruling out
AAS oscillations. Fig. 5.20b shows that C oscillates instead of decaying as a power
law, which rules out UCF. The Fourier transform in Fig. 5.20c shows a broad peak
5.3. Discussion
101
corresponding to an AB ring size dAB ≈ 60 nm. This would imply that the crystal
surface is tiled with a period array of features that produce AB patterns on the scale
of 60 nm. However, we can similarly rule this possibility out as follows, unless LT is
extremely large.
Fig. 5.21 shows the F for three tilt angles θ, bending H away from normal to
the surface. While the amplitude of the Fourier components falls, there is no obvious
shift in the peak frequency f . One would expect f to change as the area of the ring
normal to H changed. A stronger argument is that of size scales. Fig. 5.22 shows a
typical crystal above a ruler with mm rulings. The separation of the voltage contacts
is ∼ 1 mm, so that the fluctuations would be suppressed by 1000 for UCF (according
to Eq. A.3) and 108 for a tiling of 60 nm AB rings (suppressed as N 1/2 ).
Despite this quantitative disagreement, the qualitative similarities between the
results for these non-metallic crystals and 2D / mesoscopic transport are striking.
The ln H behavior and conductance fluctuation are seen in all non-metallic samples.
Results for several samples are shown in Table 5.1 and Fig. 5.23. One additional
observation is that the T dependence of the rms δG is faster than the power law
decay in T expected for UCF (shown in Fig. 5.23b).
There are several possible explanations for these effects. The possibility of the
crystal cracking in to many 2D systems seems to be able to explain the large ln H
and ln T dependence, but has difficulty explaining the conductance fluctuation. Alternatively, examination of the ARPES data for these materials as in Fig. 5.1a seems
to indicate that there is a large amount of bulk Fermi surface remnant in the bulk
band gap. One possibility based on this observation is that the signal arises from the
hybridization of the bulk and surface electrons. This may explain why the effects are
102
5.3. Discussion
4
6
8
10
(a)
0.3 K
10
12
0.85
G (e
2
/ h)
2
0
1.8
-10
C(B)
0.5
(b)
0.0
-0.5
2
4
6
8
10
12
H (T)
0
4
F(
B
) (a.u.)
(c)
0
0
2
4
B
-1
6
8
(T )
Figure 5.20: (a) Fluctuations in conductance at selected T obtained by subtracting
a smooth background from G(H). The amplitude of the signal reaches 10 e2 /h. (b)
Autocorrelation at the same T shows periodic oscillations containing more than one
frequency. (c) Fourier spectrum of the fluctuation shows a broad peak near 1 T−1 .
5.3. Discussion
103
Figure 5.21: Fourier spectrum of conductance fluctuation in crystal G4. The amplitude of the spectra are decreased as H is bent away from the surface normal (larger
angle), but the frequency does not dramatically shift.
5.3. Discussion
104
Figure 5.22: Photograph of crystal G3. Au wires are attached to the sample with Ag
paint, mounted on a sapphire substrate. A ruler with mm rulings is shown.
105
5.3. Discussion
0.3 K
(b)
(a)
G4
Hc
G4 (
0.25
|| )
6
5
G6
G (
0.20
Hb
G7
G4 (
)
3
H| a
0.15
0.10
||
2
-1
)
4
G4 (
2
| )
G6
G7
G5
/ h)
H || c
rms ( G) (e
0.30
1
G5
Ha
G5 (
-2
0
H (T)
0
2
0.0
|| )
0.5
1.0
1.5
0
2.0
T (K)
Figure 5.23: (a) Magnetoconductance for crystals G4-G7 at 0.3 K. The low H anomaly
is present in all. (b) Comparison of T dependence of the conductance fluctuation
in several crystals. The signal decays quickly as T increases for each crystal and
measurement geometry.
5.4. Summary
106
so large in terms of G, but does not readily explain all observations. Finally, in regard
to the large conductance fluctuations, it has been recently found that in graphene topgated nanostructures Fabry-Perot resonances of the conduction electrons can generate
reproducible fluctuations up to δG ∼ 10 e2 /h [73]. We hypothesize that step heights
at the crystal surface may act as similar resonator where H can modulate the electron
wavevector k through Lorentz force bending. This could in turn modulate scattering
at step edges by changing the incident angle of the electrons. Scanning tunneling
microscopy experiments would be a sharp test of this hypothesis.
5.4
Summary
We have shown that Ca doping can smoothly change Bi2 Se3 from an n-type to ptype conductor as monitored by nH and Ak . For intermediate dopings with high
ρ, low T transport qualitatively resembles both 2D and mesoscopic transport, but
quantitatively is inconsistent with existing theories. We propose these properties
may be due to a novel hybridization of bulk and surface states or resonant scattering
of surface electrons off of step edges on the crystal surface.
Chapter 6
Evidence for metallic surface states
in voltage-tuned crystals of Bi2Se3
To discern the transport properties of surface states in Topological Insulators (TIs),
it would be ideal to compare crystals dominated by bulk and surface currents. One
method for doing this in-situ is by manipulating the chemical potential µ in a crystal
via an electrostatic gate. In this chapter, we present an experiment in which we
tune the chemical potential µ of thin crystals of Bi2 Se3 from n-type to p-type parts
of the Fermi surface [74]. We show that there are high mobility, conducting in-gap
states that undergo weak anti-localization and are host to conductance fluctuations.
We estimate the surface state conductance σss to be approximately 6e2 /h, and show
that the conductance channel from the surface acts in parallel to the bulk when the
chemical potential is tuned in to the bulk conduction band.
107
6.1. Gate voltage chemical potential control in Bi2 Se3
6.1
108
Gate voltage chemical potential control in Bi2Se3
The crystal structure for Bi2 Se3 is shown in Fig. 6.1d [75]. The basic unit in the
structure is a 5 layer sandwich of Bi and Se layers, referred to as the quintuple layer.
The unit cell is comprised of three of these quintuple layers. The binding between
the quintuple layers is weak compared to the intralayer bonding, so these crystals can
be easily cleaved. Crystals are cleaved on to heavily doped Si substrates with 300
nm of thermally grown SiO2 (see Fig. 6.1c). It was found that crystals of lateral size
up to 10 µm with thickness between 5 nm and 50 nm could be cleaved. Fig. 6.1a
shows an Atomic Force Microscope (AFM) image of such a crystal. The crystal is
approximately 9 nm thick, corresponding to 9 quintuple layers. It has been predicted
[76] and shown experimentally by ARPES [77] that if the top and bottom surface
in Bi2 Se3 are closer than ∼ 4 quintuple layers, the surface states band structure is
distorted and is eventually destroyed. Therefore, we limit our study to crystals thicker
than 5 nm.
One important observation that aided in finding thin crystals is that in an optical
microscope crystals thinner than ∼ 20 nm show a color that darkens progressively
toward the color of the SiO2 layer as the crystal becomes thinner. This is similar
to that reported in graphene [4]. By cross correlating optical images with AFM
measurements, we can create an approximate color map for layer thickness, useful for
estimation within ±2 nm. This is shown in Fig. 6.2.
Our measurement geometry is shown in Fig. 6.1b. Electrodes are patterned by
ebeam lithography followed by deposition of 10 nm of Cr and 100 nm of Au. It is
found empirically that the cleaving and lithographic process pushes µ of the crystal
further in the the conduction band (crystals as-grown are n-type). Therefore, we
6.1. Gate voltage chemical potential control in Bi2 Se3
109
Figure 6.1: (a) AFM image of cleaved Bi2 Se3 crystal. It is approximately 9 nm thick,
3 µm long and 1.5 µm wide. It is suitable for making contacts with ebeam lithography.
(b) Similar crystal with Cr/Au contacts deposited in to measure resistance and Hall
effect. (c) Schematic of field effect geometry used in these experiments. (d) Crystal
structure of Bi2 Se3 from [75]. The unit cell is composed of 3 quintuple layers consisting
of Se:Bi:Se:Bi:Se stacks.
Figure 6.2: Optical image of cleaved Bi2 Se3 crystal showing changing color on different
plateaus with approximate thickness labeled in nm, measured separately by AFM.
6.1. Gate voltage chemical potential control in Bi2 Se3
110
work primarily with initially hole-doped crystals with relatively large Ca content
(0.5% substiting for Bi). This is roughly twice the amount found to minimize the
bulk band contribution to transport for macroscopic crystals in Chapter 5.
Figure 6.3: Measurements of few layer graphene using field effect transistor geometry
from [4]. (a) The resistance R is tuned by the gate voltage VG and goes through
a peak at charge neutrality. (b) The Hall coefficient changes from electron-like to
hole-like at ∼ 40 V. The associated band structure and electron filling is shown.
The technique of cleaving single crystals in the field effect geometry has been
employed successfully for measurement of graphene, as in Chapters 3 and 4. A more
appropriate comparison for these relatively thick crystals is the original experiments
done on few layer graphene [4]. Measurements of few layer graphene in this device
geometry are shown in Fig. 6.3, adapted from [4]. The schematic band structure is
111
6.1. Gate voltage chemical potential control in Bi2 Se3
shown in Fig. 6.3b. By applying positive (negative) gate voltage VG , electrons (holes)
are added to the sample. Accordingly, the hall constant RH can be changed between
electron-like to hole-like, and at the charge neutral point VN a maximum in R is seen
(Fig. 6.3a).
0
800
-50
0 T
-100
)
600
(
R
400
14 T
-200
(
xx
yx
R
-150
)
200
-250
T = 0.3 K
-300
0
-100
-50
0
V
G
50
100
(V)
Figure 6.4: Field effect experiment on a Bi2 Se3 device with thickness greater than 50
nm. The longitudinal resistance Rxx and Hall resistance Ryx can only be marginally
changed. From Ryx we estimate -300 V would be required to reach charge neutrality.
This is approximately a factor of 2 too large to realize with the current gate dielectric.
Similar measurements on early devices of Bi2 Se3 with thickness larger than 50
nm are shown in Fig. 6.4. This device is electron doped when VG = 0. Both the
Hall resistance Ryx and longitudinal resistance Rxx can be manipulated with VG ,
but no change in carrier type is observed. Comparing to Fig. 6.3, we appear to be
exploring only the heavily electron doped side (right side) of the band structure (in
112
6.2. Results
the conduction band in Fig. 1.3). From the curvature of Ryx ,
Ryx =
H
H
=
ne
C(VN − VG )
(6.1)
where we assume the induced charge is given by n = CVG /e with C = 1.14 F / m2 for
300 nm of SiO2 . A fit to the curve in Fig. 6.4 yields VN . From this we estimate that
-300 V would be required to reach charge neutrality. Ideally, the breakdown electric
field Eb for SiO2 is 1 V / nm, but in practice after fabrication is closer to 0.5 - 0.6 V
/ nm for 300 nm thick dielectric layers. Therefore, we are unable to reach VN in this
device. By cleaving thinner crystals, however, we are able to bring the magnitude of
VN down as low as -80 V.
6.2
Results
Table 6.1 shows parameters for five different devices. Devices of thickness t ≤ 20 nm
exhibit similar characteristics. For these devices an applied gate voltage can change
the carrier type from electron-like to hole-like at VN listed in Table 6.1. Each exhibits
a maximum in the resistance per square Rmax of 2 − 3 kΩ. For thicker crystals such as
devices S4 and S5, we reach the threshold for breakdown of the SiO2 gate dielectric
(typically -180 V to -200 V) before we are able to empty out the conduction band
electrons from the system. A comparison of the R(VG ) profiles for samples S3, S4,
and S5 is shown in Fig 6.5. S3 exhibits a peak in resistance for temperatures T below
80 K followed by a change in carrier type (see Figure 6.6a) upon further decreasing
VG . The progressively thicker crystals show systematically less response to VG . S4,
approximately 30 nm thick, can be tuned by the back gate, but there are too many
113
6.2. Results
carriers to empty the conduction band below the breakdown voltage of the insulating
SiO2 layer. S5 is thicker than 40 nm and can only be slightly tuned by the back gate.
units
S1
S2
S3
S4
S5
t
VN
µCB
Rmax
2
nm
V
cm /Vs kΩ
10
-80
330
2.4
6
-90
830
2.3
20
-170
720
2.8
30
< -175
910
1.9
> 40 < -175
1630
0.3
Table 6.1: Sample parameters. t is the sample thickness estimated from an optical
image of the cleaved crystal (see Fig 6.5). VN is the gate voltage required to reach
charge neutrality at 5 K. µCB is the Hall mobility in the conduction band, calculated
at zero gate voltage. Rmax is the maximum resistance recorded over the measured
gating range.
Figure 6.6 shows detailed transport data for device S3. The Hall resistance of
the device changes sign at VG ∼ −170 V, though this is at the extreme limit of our
measurement range. The R(T ) profiles for selected gate voltages are shown in Figure
6.6b. For VG = 0 the R(T ) profile appears as a poor metal below 160 K. Decreasing VG
brings non-monotonic R(T ) profiles as µ is brought near the conduction band edge.
Finally, upon emptying the conduction band (as determined by Ryx ) a re-entrant
metallic profile with a relatively large residual resistivity ratio appears.
We now focus on the thinnest devices S1 and S2 measured at He-4 and He-3
temperatures, respectively. A schematic of the band structure under electrostatic
gating is shown in the inset of Fig. 6.7. As VG is set to larger negative values, we
expect that µ in the crystal will be pulled toward the band gap. Given a crystal that
has µ close enough to the gap at VG = 0 and is thinner than the depletion width, we
expect that at some VG we can move µ uniformly in to the bulk gap. In Fig. 6.7a,
114
6.2. Results
3.0
3.0
3.0
2.5
2.5
S3
40
2.5
H = 0
80
2.0
2.0
2.0
5 K
S4
120
H = 0
)
R (k
1.5
(c)
(b)
(a)
5 K
1.5
160
50
1.5
100
1.0
1.0
1.0
150
S5
200
0.5
0.5
0.0
0.5
0.0
-150
-100
V
G
-50
(V)
0
-150
H = 0
5 K
0.0
-100
-50
V
G
(V)
0
-100
-50
V
G
0
(V)
Figure 6.5: Resistance R as a function of back gate voltage VG for three different
samples (see Table 6.1.) (a) S3 of thickness t = 20 nm at various T . (b) S4 with
t = 30 nm at various T . (c) S5 with thickness larger than 40 nm at 5 K.
115
6.2. Results
-175
(b)
0
2.4
-165
-165
2.0
0
-175
1.6
-150
yx
1.2
-100
0
R
yx
)
(
)
R
-150
-50
-200
-50
-100 V
4 T
(c)
-150
-75
V
-300
0
0.8
1 T
-100
-200
R (k
-100
(
)
(a)
G
2
3
0.4
S3
T = 10 K
(V)
1
0 V
S3
0
H = 0
4
0
H (T)
0
50
0.0
100
150
T (K)
Figure 6.6: Results for S3. (a) Hall resistance Ryx as a function of magnetic field H
for different VG . The overall slope changes from negative to positive for VG ∼ −170
V. The inset shows Ryx at various gate voltages at 1 T and 4 T. (b) Resistance as a
function of temperature T at H = 0 at various VG .
we see that upon applying ∼ −90 V at 10 K, the resistance per square R(VG ) reaches
a maximum and then begins to fall. Combined with the change in Hall sign (shown
below) we take this as evidence that we are accessing surface state conduction. This
becomes more apparent on closer inspection of the R(T ) profile obtained by taking
constant T cuts to R(VG ), show in Fig. 6.7b.
For VG = +25 V, µ is in the conduction band. The R(T ) profile is that of a bad
metal with a small residual resistivity ratio (RRR). In fact, qualitatively this curve
resembles metallic bulk crystals of Bi2 Se3 as in Fig. 5.2a. Both curves have relatively
low RRR and exhibit a shallow minimum near 40 K. Taking a T cut at -50 V, we
6.2. Results
116
Figure 6.7: (a) Resistance R at H = 0 as a function of VG for selected T . The
traces disperse strongly as VG is tuned beyond the maximum in R. The inset shows
band bending driven by the gate potential as well as the depletion length a.(b) Cuts
at selected T show significantly different R(T ) profiles at different VG . Inelastic
scattering appears to become more important at large negative VG . The inset shows
an image of device S1. The scale bar is 4 µm.
117
6.2. Results
find a non-monotonic T dependence that resembles the profiles observed in the nonmetallic crystals G3-G7 in Fig. 5.11a. This implies we are near the conduction band
edge. Finally, a cut at -100 V reveals a large RRR conduction channel that fits a T 2.5
power law up to 100 K. We have no analog for this profile in bulk crystals. Moreover,
the RRR is improved by more than 50% from the bulk channel at VG = +25 V.
Measurements of Ryx at 5 T are shown in Fig. 6.8a. At the lowest T , we find
that Ryx changes sign at ∼ −80 V. As T increases, Ryx rapidly drops to electron-like
values even at the most negative VG . This is consistent with the constant rise in
R(VG ) observed in Fig. 6.7a. The shape of Ryx now qualitatively resembles that of
few layer graphene in Fig. 6.3. It is noticeable, however, that the hole side of Ryx is
suppressed compared to the electron side.
This is made more clear when we plot the Hall conductance Gxy and the Hall
angle tan θH . Both are large on the electron side, but fall near zero and remain attain
only small positive values on the hole side. The strong suppression of the Hall signal
in the gap does not imply a sharply reduced mobility, since the T dependence of G
shows that the mobility is in fact significantly enhanced over that in the bulk.
Under asymmetric gating (as here), Gxy has different values between the top and
bottom surfaces. Hence the Hall currents are unbalanced and the Hall resistance RH
should be calculated in a multiband model,
RH (VG ) =
nt µ2t + nb µ2b
e(nt µt + nb µb )2
(6.2)
where nt,b and µt,b refer to the top and bottom surface state density and mobility,
respectively, which all may depend on VG . The multiband Hall response would yield
a suppressed RH , but it seems unlikely that the character of the two bands could be
6.2. Results
118
Figure 6.8: (a) Hall resistance Ryx at 5 T. At low T we see a change in sign from
electron-like to hole-like response. At elevated T , the entire gating spectrum is
electron-like. (b) Hall conductance Gxy at selected T . A noticeable feature is that
on the hole side, Gxy is positive but extremely small. (c) Hall angle tan θH shows a
similar pattern to Gxy but the fall on the hole side resembles a rounded step function.
6.2. Results
119
so different to accommodate such a strong suppression as in Fig. 6.8. For example, it
would require |nt − nb | ≫ |n(0 V)| to produce the observed asymmetry in Ryx (VG ).
Unless µb on the hole side was significantly higher than for the electron µt [78], this
would be difficult to reconcile with the change in sign of RH . Addition of a third
channel from the bulk would involve a significantly lower mobility band and fails
to capture the observed response. An interesting possibility is if the 2 surfaces are
electrically connected by the side walls, there may be an additional contribution or
modification to Eq. 6.2. To test this scenario, one may add a top gate to tune the 2
surfaces independently, or deposit a magnetic film to decouple the 2 surfaces.
Plotting G(VG ) at 5 K in Fig. 6.9 brings out some surprising features. First, we can
see fluctuations in G across the entire VG range. Subtracting a smooth background,
we plot the fluctuation δG in Fig. 6.9b. Here, three separate measurements are
shown. The signal is reproducible and of approximately uniform size for all VG . The
T dependence of the rms δG is shown in Fig. 6.9c. It fits well to a T −1/2 power law.
Application of H uniformly shifts the G(VG ) down by approximately a factor of 2 at
14 T. Surprisingly, the same suppression is seen in δG.
Traces of Rxx (H) are shown for several VG at T = 5 K. Both a low H anomaly
and fluctuations become more obvious as VG approaches VN . We recognize these
features to be qualitatively similar to the quantum corrections to conductance seen
in macroscopic Bi2 Se3 in Chapter 5. To study this features in more detail, we cool
a second sample S2 with similar VN to 0.3 K. Fig. 6.11a and Fig. 6.11a show the
magnetoconductance of device S2 at VG = 0 and VG = VN , respectively. The low
H peak is now prominent, and the fluctuations are more pronounced. Additionally,
there are longer H scale oscillations developing.
6.2. Results
120
Figure 6.9: (a) Conductance G for device S1 at T = 5 K. There is a broad minimum
near charge neutrality. Application of H = 14 T appears to uniformly shift G(H)
down by a factor of 2. (b) Fluctuation in conductance δG obtained by subtracting a
smooth background. Three separate measurements are shown. The fluctuations stay
of the same order across the entire gating spectrum. (c) The rms size of δG decreases
as T increases and closely follows a T −0.5 power law. The rms δG is suppressed by a
factor of 2 at 14 T at 5 K.
121
6.2. Results
5
T = 5 K
S1
4
3
-50 V
R
xx
(k
)
-75 V
2
-25 V
0 V
1
0
0
4
8
12
H (T)
0
Figure 6.10: Magnetoresistance at selected VG approaching VN = −80 V for T = 5 K.
The magnitude of the MR at both low H and high H grows as electrons are drained
from the system. Fluctuations in R(H) are also observed.
122
6.3. Discussion
6.3
Discussion
We first analyze the results to determine the basic properties transport properties of
the surface states. The metallic profiles of R(T ) for in-gap states in Fig. 6.7b is strong
evidence for conducting surface states. As the conduction band has a low RRR, it is
unlikely that impurity driven in-gap states would have such a high mobility.
From Fig. 6.9a, we can estimate σss assuming that the bulk contribution is minimized. Then, considering 2 surfaces, an upper bound for σxx is approximately 6e2 /h.
We can also estimate the mobility µss using σss = neµss
µe =
σss
4πσss
=
ne
ekF2
(6.3)
Taking the Fermi wavevector kF from the ARPES result for the surface states in
Fig. 5.1a as kF = 0.05 Å−1 we find µss ≈ 700 cm2 /Vs. This is roughly twice
the conduction band mobility µCB found from the Hall effect (see Table 6.1). This
together with the high RRR of the in-gap conductance seen in Fig. 6.7b imply the
in-gap states have a relatively high mobility. We therefore associate the relatively
large MR with the surface states, which given the rigid shift of σ(VG ) with H implies
that the surface states act as a parallel conductance channel even when µ overlaps
with the bulk conduction band. This is further implied by the uniformity of the
conductance fluctuations in Fig. 6.9b across the gating spectrum.
We now turn our attention to the low H dependence of the magnetoconductance
at 0.3 K. We associate the sharp peaks in Fig. 6.11a and 6.11b with weak antilocalization. We fit the low field magnetoconductance to the standard theory for
weak anti-localization
1 H0
e2
H0
∆Gxx (H) = A
ln
−ψ
+
h
H
2
H
(6.4)
6.3. Discussion
123
Figure 6.11: (a) Magnetoconductance of device S2 at VG = 0 V and T =0.3 K. The
low H peak is prominent and fluctuations overwhelm the background G(H). (b)
Magnetoconductance at VN . The low H peak grows and further oscillations in G(H)
are observed. (c) Fit of low H profile to theory for weak anti-localization (see text).
The fit is satisfactory considering the large conductance fluctuations. (d) Results of
fitting to Eq. 6.4. The parameter A peaks at charge neutrality near the theoretical
value 1/π for 2 conducting surfaces. A peak is seen concurrently in the dephasing
field H0 .
6.3. Discussion
124
where A is a number predicted from theory, H0 is the dephasing magnetic field, and ψ
is the digamma function. Positive A implies field-suppression of antilocalization in a
2D (two-dimensional) system. For the present system with relatively low mobility and
strong spin-orbit coupling, theory predicts A = 1/2π [79], naively implying for the
present system a contribution of 2A = 1/π for two conducting surfaces. Confining
the fit to H below 0.4 T, two examples are shown in Figure 6.11c at VG = 0 V
and -100 V. The fit is satisfactory considering the large conductance fluctuations in
the system. Fitting results across the entire gate voltage range are shown in Figure
6.11d. At VN the coefficient A peaks at 0.38, which is within 20% of the theoretical
value considering 2 contributing surfaces. It should be noted that our uncertainty in
geometric calculation of Ω per square is of roughly this order. We also find a peak
in the dephasing field H0 at VN . We may explain this by the increased Coulomb
interaction due to the weakining of screening when the density of electrons becomes
low. These two facts together with the charge neutrality measured by the Hall effect
strongly imply that both surfaces have µ within the bulk band gap at VN . It is
noteworthy that even with µ in the conduction band the signature of weak antilocalization persists.
Finally, we return to the suppression of conductance fluctuations with H. While
AAS oscillations are suppressed on the field scale of weak localization (see the appendix), here the rms δG appears to be suppressed with H over much larger scale.
Fig. 6.12 shows a plot of δG(H). It is clear that the amplitude of δG is decreasing
as H increases. We hypothesize that the surface state contribution to conductance is
disproportionately affected by H compared to the remnant bulk channel. Thus the
lessening of δG may reflect a conductance more heavily carried by bulk states.
125
6.4. Summary
T = 5 K
0.1
V
G
= -75 V
2
G (e /h)
0.2
0.0
-0.1
-0.2
4
6
8
10
H (T)
0
Figure 6.12: Fluctuation in conductance δG at VG = −75 V and T = 5 K. The
amplitude of fluctuations is suppressed with increasing H.
6.4
Summary
We have seen that by electrostatic gating we can isolate transport properties of the
surface states in the 3D topological insulator Bi2 Se3 . We are able to tune the crystals
to charge neutrality as determined by the Hall effect to focus on the contributions of
the surface states to electrical transport. We estimate the surface state conductance
to be σss ≈ 6e2 /h, with a mobility enhanced over the bulk as indicated by an increased
residual resistivity ratio and estimates of the electron density from ARPES. At low
T , we find the H dependence of the conductivity is consistent with field suppression
of weak anti-localization for two conducting surfaces at the charge neutral point. We
also measure fluctuations in the conductance δG that persist when µ is moved in to
the conduction band. This, along with the H dependence of σxx , indicates that the
surface acts as a parallel conductance channel with the bulk.
Chapter 7
Concluding Remarks
We have studied two systems with disparate crystal structure but similar electronic
structure described by the Dirac equation. The 2D Dirac (pseudo-)spin polarized
states in graphene arise directly from a tight-binding calculation of the band structure,
whereas for the 3D topological insulator Bi2 Se3 they arise as a result of time reversal
symmetry and the topology of the 3D bands. To conclude, we compare results for
the two systems and suggest directions for future work.
One basic conclusion from these experiments was that for H = 0, for all their
differences, graphene and the 2D surface of Bi2 Se3 both have a conductance of a few
times e2 /h. An important parallel in these systems is the prediction of protected
transport in a quantum spin Hall (QSH) state. These states arise in the two systems
for different reasons. In graphene, symmetry breaking at the lowest Landau level in
the quantum Hall regime was predicted to produce the 2D QSH state with counterpropogating edge modes protected from scattering [3]. In Bi2 Se3 , strong spin-orbit
coupling and time reversal symmetry were shown to produce a 3D QSH state that has
126
127
been observed in ARPES [14]. Our experiments show that the effect of H on these
two proposed QSH states was contrary to expectations. Whereas the graphene QSH
state had been proposed to be protected from scattering, we found that in high H
the system became an electrical insulator. Moreover, as the insulating state was more
robust in cleaner samples, measurements indicate that the CPE modes are absent.
For Bi2 Se3 , application of H breaks time reversal symmetry and should therefore destroy the QSH state. However, for H up to 14 T we found no evidence of this even
for states at the Dirac point.
Future directions for graphene are to further probe the hypothesis of a KT-like
transition to the insulating state. Higher quality samples that can be measured without the need for extremely intense H > 14 T would be ideal. Measurements of
the angular dependence for H of the transition, IV curves near HC , and detailed T
dependence above HC would all be sharp tests of the KT hypothesis. Extending thermoelectric measurements to the insulating regime would also be informative, though
perhaps difficult due to the large fluctuation signal observed at low T .
For the study of the 3D topological insulator Bi2 Se3 , the main challenge to overcome was isolating the properties of the 2D surfaces. We addressed this issue from a
materials perspective, attempting to improve the quality of the insulating bulk state
with chemical doping and electrostatic gating. Electrostatic gate control of µ proved
to be a powerful tool for isolating the surface band contribution to electronic transport. Provided crystals that are sufficiently thin and uniform, we showed one may
overcome the persistent bulk electrons that have thus far hampered experiments in
3D topological insulators. The usefulness of the technique is that it allows in-situ
control of µ to compare bulk and in-gap states. Next, one can imagine replacing the
128
Au electrodes here with magnetic or superconducting materials to probe a variety
of exotic states that have been proposed [27, 80]. With the characterization of surface state conduction here, realizing topologically non-trivial supercurrents and the
physics of spin transport to test the QSH texture in these materials becomes the next
step.
Appendix A
Brief Review of Quantum
Corrections to Transport
Localization
The theoretical framework for localization in electron systems dates back to the original prediction by Anderson that an electron wavefunction could be trapped by disorder [85]. When experiments on 2D systems began to emerge, a ubiquitous feature was
a ln T correction to the resistance at extremely low T . Typical data for a Au-Pd thin
film is shown in Fig. A.1a below 20 K [81]. Two competing theories were presented
to explain this effect. Anderson and coworkers showed that non-interacting electrons
would undergo a ln T localization due to constructive interference of time-reversed
paths [86], while Altshuler and coworkers showed that Coulomb interaction between
electrons predicted the same behavior [87].
One prediction that arose from this debate was that in the same regime in which
the conductance of an electron system obeyed a ln T dependence, the contribution
129
130
Figure A.1: (a) Measurement of resistance of Au-Pd thin film from [81]. A strong log T
correction appears below 5 K. (b) Magnetoconductance of a Au film from [82]. As T is
decreased, the profiles fit to a log H form over increasing range in H. (c) Conductance
fluctuation in thin Bi film from [83]. Two traces are shown offset to demonstrate
reproducibility. The measurement is performed below 1 K. (d) Comparison of in
plane (circles) and out of plane (crosses) magnetoresistance for a thin Cu film from
[84]. The MR is flat for the former but large for the latter.
131
from the non-interacting theory should be suppressed in H with a ln H dependence
[88]. This effect was later observed; typical experimental data are shown for a Au
film in Fig. A.1b [82].
Figure A.2: Real space time reversed paths leading to weak localization from [67]. An
electron can explore a closed path 0 → 1 → 2 by elastic scattering off of impurities
in the diffusive regime. By symmetry, the time reversed path 0 → 1′ → 2′ must also
be allowed. The two paths constructively interfere at the origin, causing the electron
to be localized.
A real space picture of the mechanism underlying the non-interacting electron
picture is shown in Fig. A.2, due to Bergmann [67]. At low T in the diffusive regime,
the time between inelastic scattering events τin exceeds that for elastic scattering
events τel . Therefore, an electron may scatter several times and retain its wavefunction’s phase. It is possible for an electron to explore a closed path, as shown by the
process 0 → 1 → 2 etc. in Fig. A.2. By time reversal symmetry, the opposite path
(0 → 1′ → 2 etc. ) must also be possible with the same likelihood. When these
132
two paths meet at the origin, they will constructively interfere since they have have
covered the same path. This constructive interference leads to an increased likelihood of the electron to stay in the same position- it is said to be localized. As T goes
down and the number of the possible paths grows, the localization grows stronger and
the ln T correction dominates the ρ(T ) profile. The prediction of ln H suppression
can then be qualitatively understood as the dephasing of the electrons going through
these closed paths.
The field was considerably enriched with the prediction that spin-orbit coupling
would lead to destructive rather than constructive interference [79, 89]. This is referred to as weak anti-localization. The Au film result in Fig. A.1b is actually
field suppression of weak anti-localization, as σ drops in H. This theory was confirmed in experiments that measured a crossover from weak localization to weak
anti-localization with the deposition of Au on to a Mg film [67].
Important experimental facts about these effects are that:
1. The ln H and ln T dependence of ρ are results of calculations in 2D.
2. These quantum corrections to transport arise only at low T .
3. The field suppression of localization for a single 2D system has the form
∆σ ≈
e2
ln H
2πh
(A.1)
with factors of order unity.
4. The ln H dependence is only for H perpendicular to the plane since the effect is
based on orbital motion. Experimental data consistent with this for a Cu film
is shown in Fig. A.1d [84].
133
5. Both single particle and interaction effects make a contribution to the ln T
dependence of σxx at low T , but the contributions may be separated with the
application of H.
Conductance Fluctuations
Figure A.3: (a) AB oscillations arise from modulation of the constructive interference
of an electron exploring separate paths about a magnetic flux Φ. The signal that
emerges is periodic in h/e. (b) AAS oscillations arise from modulation of the interference of time reversed paths about an enclosed Φ. Here the signal is periodic in
h/2e. (c) UCF is generated by modulation of interference of multiple electron paths
in a phase coherent region. The signal is aperiodic.
In mesoscopic systems 2D systems, several additional corrections to conductance
arise from interference of electron paths. Generally, the length scale for these effects
is set by the shorter of the phase coherence length Lφ and the thermal length LT
Lφ =
p
Dτin
,
LT =
r
D~
kB T
(A.2)
where D is the diffusion coefficient. Physically LT corresponds to the dephasing of
electron paths of conduction electrons from opposite edges of the thermally broadened
Fermi function. In a typical system at 1 K, LT ∼ 1 µm. Three important phenomena
at these scales are shown in Fig. A.3.
134
The Aharonov Bohm (AB) effect is depicted in Fig. A.3a [90]. In the context
of solid state experiments, we consider an electron which can traverse both paths
around a region which encloses a magnetic flux Φ . The electron explores both
paths, so that Φ modulates the constructive interference when the two paths meet
[91]. This fluctuation is seen, for example, in the geometry of a nanoscale Au ring
in Fig. A.4 [92]. The field scale of the modulation is h/e. As T grows higher,
there is an exponential suppression of this effect as LT becomes shorter than the ring
circumference [93]. Periodic arrays of rings can be used to see AB oscillations over a
larger length scale, but the signal is suppressed by N 1/2 , where N is the number of
rings [94]. The h/e modulation persists to arbitrarily high H.
Figure A.4: (a) Oscillations of resistance of 784 nm Au ring (shown in inset) measured
at 10 mK from [92]. (b) The Fourier spectrum has peaks corresponding to both the
AB and AAS frequencies.
135
A closely related effect is the Altshuler, Aronov, Spivak (AAS) effect shown in
Fig. A.3b [95]. Here, instead of an electron exploring each arm of the ring, electrons
with time reverse paths have their weak localization modulated by Φ, resulting in a
h/2e modulation [96]. The signal only persists to the field scale of weak localization,
typically ∼ 0.1 T. Multiply connected rings have been used to generate this effect
in arrays as large as 106 [97]. These oscillations are also seen in addition to the AB
oscillations in the the Au ring geometry in Fig. A.4. This is evident from the two
separate peaks in the Fourier spectrum corresponding to periods h/eA and h/2eA for
ring size A for the AB effect and AAS effect, respectively.
Finally, in a phase coherent patch electron paths diffuse to form a complex array of
overlapping areas. Application of H in a system of this type leads to modulation of G,
but it is aperiodic as depicted in Fig. A.3c [98, 99, 100, 101]. Typical experimental
data for a thin Bi film is shown in Fig. A.1c [83]. The autocorrelation of these
aperiodic fluctuations decays as a power law H d−4 , where d is the dimension of the
system. The size of the modulation δG for a coherent patch saturates to e2 /h at low T ,
which is a universal number for any system where the charge carriers are electrons. For
this reason, these fluctuations are referred to as Universal Conductance Fluctuations
(UCF). In a region of a sample of size L larger than LT , one considers the system to be
broken up in to many coherent areas that must be classically averaged. Accordingly,
the size of the signal is suppressed as
δG =
e2 LT 2−d/2
h L
(A.3)
The signal should persist to arbitrarily high H; as a function of increasing T the
signal decays as a weak power law T 1/4 or T 1/2 depending on the the relative size of
LT and Lφ [100].
Bibliography
[1] C. L. Kane and E. J. Mele, “Quantum Spin Hall Effect in Graphene,” Physical
Review Letters, vol. 95, no. 226801 10.1103/PhysRevLett.95.226801, 2005.
[2] B. A. Bernevig and S. C. Zhang, “Quantum Spin Hall Effect,” Physical Review
Letters, vol. 96, no. 106802 10.1103/PhysRevLett.96.106802, 2006.
[3] D. A. Abanin, P. A. Lee, and L. S. Levitov, “Spin-Filtered Edge States and
Quantum Hall Effect in Graphene,” Physical Review Letters, vol. 96, no. 176803
10.1103/PhysRevLett.96.176803, 2006.
[4] K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V.
Morozov, and A. K. Geim, “Two-Dimensional Atomic Crystals,” Proceedings
of the National Academy of Sciences of the United States of America, vol. 102,
no. 10.1073/pnas.0502848102, p. 10451, 2005.
[5] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K.
Geim, “The Electronic Properties of Graphene,” Reviews of Modern Physics,
vol. 81, no. 10.1103/RevModPhys.81.109, p. 109, 2009.
[6] A. K. Geim, “Graphene:
Status and Prospects,” Science, vol. 324,
no. 10.1126/science.1158877, p. 1530, 2009.
136
BIBLIOGRAPHY
137
[7] P. A. M. Dirac, “The Quantum Theory of the Electron,” Proceedings of the
Royal Society of London Series a-Containing Papers of a Mathematical and
Physical Character, vol. 117, p. 610, 1928.
[8] B. A. Bernevig, T. L. Hughes, and S. C. Zhang, “Quantum Spin Hall Effect
and Topological Phase Transition in Hgte Quantum Wells,” Science, vol. 314,
no. 10.1126/science.1133734, p. 1757, 2006.
[9] M. Konig, S. Wiedmann, C. Brune, A. Roth, H. Buhmann, L. W. Molenkamp,
X. L. Qi, and S. C. Zhang, “Quantum Spin Hall Insulator State in Hgte Quantum Wells,” Science, vol. 318, no. 10.1126/science.1148047, p. 766, 2007.
[10] L. Fu, C. L. Kane, and E. J. Mele, “Topological Insulators in Three
Dimensions,” Physical Review Letters, vol. 98, no. 106803 10.1103/PhysRevLett.98.106803, 2007.
[11] L. Fu and C. L. Kane, “Topological Insulators with Inversion Symmetry,” Physical Review B, vol. 76, no. 045302 10.1103/PhysRevB.76.045302, 2007.
[12] J. E. Moore and L. Balents, “Topological Invariants of Time-Reversal-Invariant
Band Structures,” Physical Review B, vol. 75, no. 121306 10.1103/PhysRevB.75.121306, 2007.
[13] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A
Topological Dirac Insulator in a Quantum Spin Hall Phase,” Nature, vol. 452,
no. 10.1038/nature06843, p. 970, 2008.
[14] D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier,
G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation
BIBLIOGRAPHY
138
of Unconventional Quantum Spin Textures in Topological Insulators,” Science,
vol. 323, no. 10.1126/science.1167733, p. 919, 2009.
[15] D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder,
L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov, H. Lin, A. Bansil,
D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A Tunable Topological Insulator in the Spin Helical Dirac Transport Regime,” Nature, vol. 460,
no. 10.1038/nature08234, p. 1101, 2009.
[16] Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S.
Hor, R. J. Cava, and M. Z. Hasan, “Observation of a Large-Gap TopologicalInsulator Class with a Single Dirac Cone on the Surface,” Nature Physics, vol. 5,
no. 10.1038/nphys1274, p. 398, 2009.
[17] Y. L. Chen, J. G. Analytis, J. H. Chu, Z. K. Liu, S. K. Mo, X. L. Qi, H. J. Zhang,
D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X.
Shen, “Experimental Realization of a Three-Dimensional Topological Insulator,
Bi2 Te3 ,” Science, vol. 325, no. 10.1126/science.1173034, p. 178, 2009.
[18] L. Petersen and P. Hedegard, “A Simple Tight-Binding Model of Spin-Orbit
Splitting of Sp-Derived Surface States,” Surface Science, vol. 459, p. 49, 2000.
[19] Y. A. Bychkov and E. I. Rashba, “Properties of a 2d Electron-Gas with Lifted
Spectral Degeneracy,” JETP Letters, vol. 39, p. 78, 1984.
[20] H. A. Kramers, “Paramagnetic Properties of Crystals of Rare Earths I,”
Proceedings of the Koninklijke Akademie Van Wetenschappen Te Amsterdam,
vol. 35, p. 1272, 1932.
BIBLIOGRAPHY
139
[21] F. D. M. Haldane, “Model for a Quantum Hall-Effect without Landau-Levels Condensed-Matter Realization of the Parity Anomaly,” Physical Review Letters,
vol. 61, p. 2015, 1988.
[22] A. Roth, C. Brune, H. Buhmann, L. W. Molenkamp, J. Maciejko, X. L. Qi, and
S. C. Zhang, “Nonlocal Transport in the Quantum Spin Hall State,” Science,
vol. 325, no. 10.1126/science.1174736, p. 294, 2009.
[23] R. Roy, “Topological Phases and the Quantum Spin Hall Effect in Three Dimensions,” Physical Review B, vol. 79, no. 195322 10.1103/PhysRevB.79.195322,
2009.
[24] S. S. Chern, “Characteristic Classes of Hermitian Manifolds,” Annals of Mathematics, vol. 47, p. 85, 1946.
[25] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Dennijs, “Quantized
Hall Conductance in a Two-Dimensional Periodic Potential,” Physical Review
Letters, vol. 49, p. 405, 1982.
[26] D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder,
L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava,
and M. Z. Hasan, “Observation of Time-Reversal-Protected Single-Dirac-Cone
Topological-Insulator States in Bi2 Te3 and Sb2 Te3 ,” Physical Review Letters,
vol. 103, no. 146401 10.1103/PhysRevLett.103.146401, 2009.
[27] L. Fu and C. L. Kane, “Probing Neutral Majorana Fermion Edge Modes with
Charge Transport,” Physical Review Letters, vol. 102, no. 216403 10.1103/PhysRevLett.102.216403, 2009.
BIBLIOGRAPHY
140
[28] J. G. Checkelsky, L. Li, and N. P. Ong, “Zero-Energy State in Graphene in a
High Magnetic Field,” Physical Review Letters, vol. 100, no. Artn 206801 Doi
10.1103/Physrevlett.100.206801, 2008.
[29] J. G. Checkelsky, L. Li, and N. P. Ong, “Divergent Resistance at the Dirac Point
in Graphene: Evidence for a Transition in a High Magnetic Field,” Physical
Review B, vol. 79, no. Artn 115434 Doi 10.1103/Physrevb.79.115434, 2009.
[30] T. Ando, “Theory of Electronic States and Transport in Carbon Nanotubes,”
Journal of the Physical Society of Japan, vol. 74, no. 10.1143/jpsj.74.777, p. 777,
2005.
[31] J. Alicea and M. P. A. Fisher, “Graphene Integer Quantum Hall Effect in
the Ferromagnetic and Paramagnetic Regimes,” Physical Review B, vol. 74,
no. Artn 075422 Doi 10.1103/Physrevb.74.075422, 2006.
[32] H. A. Fertig and L. Brey, “Luttinger Liquid at the Edge of Undoped Graphene
in a Strong Magnetic Field,” Physical Review Letters, vol. 97, no. 116805
10.1103/PhysRevLett.97.116805, 2006.
[33] M. O. Goerbig, R. Moessner, and B. Doucot, “Electron Interactions in
Graphene in a Strong Magnetic Field,” Physical Review B, vol. 74, no. 161407
10.1103/PhysRevB.74.161407, 2006.
[34] V. P. Gusynin, V. A. Miransky, S. G. Sharapov, and I. A. Shovkovy, “Excitonic
Gap, Phase Transition, and Quantum Hall Effect in Graphene,” Physical Review
B, vol. 74, no. 195429 10.1103/PhysRevB.74.195429, 2006.
BIBLIOGRAPHY
141
[35] K. Yang, S. Das Sarma, and A. H. MacDonald, “Collective Modes and Skyrmion
Excitations in Graphene Su(4) Quantum Hall Ferromagnets,” Physical Review
B, vol. 74, no. 075423 10.1103/PhysRevB.74.075423, 2006.
[36] M. Ezawa, “Intrinsic Zeeman Effect in Graphene,” Journal of the Physical Society of Japan, vol. 76, no. 094701 10.1143/jpsj.76.094701, 2007.
[37] D. A. Abanin, K. S. Novoselov, U. Zeitler, P. A. Lee, A. K. Geim, and L. S.
Levitov, “Dissipative Quantum Hall Effect in Graphene near the Dirac Point,”
Physical Review Letters, vol. 98, no. 196806 10.1103/PhysRevLett.98.196806,
2007.
[38] D. A. Abanin, P. A. Lee, and L. S. Levitov, “Charge and Spin Transport at
the Quantum Hall Edge of Graphene,” Solid State Communications, vol. 143,
no. 10.1016/j.ssc.2007.02.024, p. 77, 2007.
[39] K. Nomura and A. H. MacDonald, “Quantum Hall Ferromagnetism in
Graphene,” Physical Review Letters, vol. 96, no. 256602 10.1103/PhysRevLett.96.256602, 2006.
[40] E. V. Gorbar, V. P. Gusynina, and V. A. Miransky, “Toward a Theory of
the Quantum Hall Effect in Graphene,” Low Temperature Physics, vol. 34,
no. 10.1063/1.2981388, p. 790, 2008.
[41] V. R. Gusynin, V. A. Miransky, S. G. Sharapov, and I. A. Shovkovy, “Edge
States, Mass and Spin Gaps, and Quantum Hall Effect in Graphene,” Physical
Review B, vol. 77, no. 205409 10.1103/PhysRevB.77.205409, 2008.
BIBLIOGRAPHY
142
[42] K. Nomura, S. Ryu, and D. H. Lee, “Field-Induced Kosterlitz-Thouless Transition in the N=0 Landau Level of Graphene,” Physical Review Letters, vol. 103,
no. 216801 10.1103/PhysRevLett.103.216801, 2009.
[43] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V.
Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-Dimensional Gas of Massless Dirac Fermions in Graphene,” Nature, vol. 438, no. 10.1038/nature04233,
p. 197, 2005.
[44] Y. B. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental Observation
of the Quantum Hall Effect and Berry’s Phase in Graphene,” Nature, vol. 438,
no. 10.1038/nature04235, p. 201, 2005.
[45] Y. S. Zheng and T. Ando, “Hall Conductivity of a Two-Dimensional Graphite
System,” Physical Review B, vol. 65, no. 245420 10.1103/PhysRevB.65.245420,
2002.
[46] M. Ishigami, J. H. Chen, W. G. Cullen, M. S. Fuhrer, and E. D.
Williams, “Atomic Structure of Graphene on Si O2 ,” Nano Letters, vol. 7,
no. 10.1021/nl070613a, p. 1643, 2007.
[47] J. Moser, A. Barreiro, and A. Bachtold, “Current-Induced Cleaning of
Graphene,” Applied Physics Letters, vol. 91, no. 10.1063/1.2789673, 2007.
[48] V. L. Berezinskii, “Destruction of Long-Range Order in One-Dimensional and
2-Dimensional Systems Possessing a Continuous Symmetry Group,” Soviet
Physics JETP-USSR, vol. 34, p. 610, 1972.
BIBLIOGRAPHY
143
[49] J. M. Kosterlitz and D. J. Thouless, “Ordering, Metastability and PhaseTransitions in 2 Dimensional Systems,” Journal of Physics C-Solid State
Physics, vol. 6, p. 1181, 1973.
[50] A. F. Hebard and A. T. Fiory, “Critical-Exponent Measurements of a TwoDimensional Superconductor,” Physical Review Letters, vol. 50, p. 1603, 1983.
[51] C. Y. Hou, C. Chamon, and C. Mudry, “Electron Fractionalization in
Two-Dimensional Graphenelike Structures,” Physical Review Letters, vol. 98,
no. 186809 10.1103/PhysRevLett.98.186809, 2007.
[52] X. Du, I. Skachko, F. Duerr, A. Luican, and E. Y. Andrei, “Fractional Quantum Hall Effect and Insulating Phase of Dirac Electrons in Graphene,” Nature,
vol. 462, no. 10.1038/nature08522, p. 192, 2009.
[53] K. I. Bolotin, F. Ghahari, M. D. Shulman, H. L. Stormer, and P. Kim, “Observation of the Fractional Quantum Hall Effect in Graphene,” Nature, vol. 462,
no. 10.1038/nature08582, p. 196, 2009.
[54] J. G. Checkelsky and N. P. Ong, “Thermopower and Nernst Effect in Graphene
in a Magnetic Field,” Physical Review B, vol. 80, no. Artn 081413 Doi
10.1103/Physrevb.80.081413, 2009.
[55] S. M. Girvin and M. Jonson, “Inversion Layer Thermopower in High MagneticField,” Journal of Physics C-Solid State Physics, vol. 15, p. 1147, 1982.
[56] M. Jonson and S. M. Girvin, “Thermoelectric Effect in a Weakly Disordered
Inversion Layer Subject to a Quantizing Magnetic-Field,” Physical Review B,
vol. 29, p. 1939, 1984.
BIBLIOGRAPHY
144
[57] H. Obloh, K. Vonklitzing, K. Ploog, and G. Weimann, “Thermomagnetic Behavior of the Two-Dimensional Electron-Gas in Gallium Arsenide - Aluminum
Gallium Arsenide Heterostructures,” Surface Science, vol. 170, p. 292, 1986.
[58] R. Fletcher, J. C. Maan, K. Ploog, and G. Weimann, “Thermoelectric Properties of Gallium Arsenide - Gallium Aluminum Arsenide Heterojunctions at
High Magnetic-Fields,” Physical Review B, vol. 33, p. 7122, 1986.
[59] R. Fletcher, J. C. Maan, and G. Weimann, “Experimental Results on the HighField Thermopower of a Two-Dimensional Electron-Gas in a Gallium Arsenide
- Gallium Aluminum Arsenide Heterojunction,” Physical Review B, vol. 32,
p. 8477, 1985.
[60] J. P. Small, K. M. Perez, and P. Kim, “Modulation of Thermoelectric Power
of Individual Carbon Nanotubes,” Physical Review Letters, vol. 91, no. 256801
10.1103/PhysRevLett.91.256801, 2003.
[61] N. F. Mott and H. Jones, The Theory of the Properties of Metals and Alloys.
New York: Dover, 1958.
[62] Y. M. Zuev, W. Chang, and P. Kim, “Thermoelectric and Magnetothermoelectric Transport Measurements of Graphene,” Physical Review Letters, vol. 102,
no. 096807 10.1103/PhysRevLett.102.096807, 2009.
[63] K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler,
J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, “Room-Temperature
Quantum Hall Effect in Graphene,” Science, vol. 315, no. 10.1126/science.1137201, p. 1379, 2007.
BIBLIOGRAPHY
145
[64] Y. Wang, L. Li, and N. P. Ong, “Nernst Effect in High-T-C Superconductors,”
Physical Review B, vol. 73, no. 024510 10.1103/PhysRevB.73.024510, 2006.
[65] M. Z. Hasan and C. L. Kane, “Topological Insulators,” arXiv, no. 1002.3895,
2010.
[66] J. G. Checkelsky, Y. S. Hor, M. H. Liu, D. X. Qu, R. J. Cava, and
N. P. Ong, “Quantum Interference in Macroscopic Crystals of Nonmetallic
Bi2 Se3 ,” Physical Review Letters, vol. 103, no. Artn 246601 Doi 10.1103/Physrevlett.103.246601, 2009.
[67] G. Bergmann, “Weak Localization in Thin-Films - a Time-of-Flight Experiment
with Conduction Electrons,” Physics Reports-Review Section of Physics Letters,
vol. 107, p. 1, 1984.
[68] G. R. Hyde, H. A. Beale, I. L. Spain, and J. A. Woollam, “Electronic Properties
of Bi2 Se3 Crystals,” Journal of Physics and Chemistry of Solids, vol. 35, p. 1719,
1974.
[69] H. Kohler and H. Fischer, “Investigation of Conduction-Band Fermi-Surface in
Bi2 Se3 at High Electron Concentrations,” Physica Status Solidi B-Basic Research, vol. 69, p. 349, 1975.
[70] L. Schubnikow and W. J. de Haas, “Magnetic Resistance Increase in Single Crystals of Bismuth at Low Temperatures,” Proceedings of the Koninklijke Akademie
Van Wetenschappen Te Amsterdam, vol. 33, p. 130, 1930.
[71] G. R. Hyde, R. O. Dillon, H. A. Beale, I. L. Spain, J. A. Woollam, and D. J.
BIBLIOGRAPHY
146
Sellmyer, “Shubnikov-De Haas Effects in Bi2 Se3 with High Carrier Concentrations,” Solid State Communications, vol. 13, p. 257, 1973.
[72] Y. S. Hor, A. Richardella, P. Roushan, Y. Xia, J. G. Checkelsky, A. Yazdani,
M. Z. Hasan, N. P. Ong, and R. J. Cava, “P-Type Bi2 Se3 for Topological Insulator and Low-Temperature Thermoelectric Applications,” Physical Review B,
vol. 79, no. 195208 10.1103/PhysRevB.79.195208, 2009.
[73] A. F. Young and P. Kim, “Quantum Interference and Klein Tunnelling in
Graphene Heterojunctions,” Nature Physics, vol. 5, no. 10.1038/nphys1198,
p. 222, 2009.
[74] J. G. Checkelsky, Y. S. Hor, R. J. Cava, and N. P. Ong, “Surface State Conduction Observed in Voltage-Tuned Crystals of the Topological Insulator Bi2 Se3 ,”
arXiv, no. 1003.3883, 2010.
[75] H. J. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S. C. Zhang, “Topological Insulators in Bi2 Se3 , Bi2 Te3 and Sb2 Te3 with a Single Dirac Cone on the
Surface,” Nature Physics, vol. 5, no. 10.1038/nphys1270, p. 438, 2009.
[76] C. X. Liu, H. Zhang, B. H. Yan, X. L. Qi, T. Frauenheim, X. Dai, Z. Fang,
and S. C. Zhang, “Oscillatory Crossover from Two-Dimensional to ThreeDimensional Topological Insulators,” Physical Review B, vol. 81, no. 041307
10.1103/PhysRevB.81.041307, 2010.
[77] Y. Zhang, K. He, C.-Z. Chang, C.-L. Song, L. Wang, X. Chen, J. Jia, Z. Fang,
X. Dai, W.-Y. Shan, S.-Q. Shen, Q. Niu, X. Qi, S.-C. Zhang, X. Ma, and Q.-K.
BIBLIOGRAPHY
147
Xue, “Crossover of Three-Dimensional Topological Insulator of Bi2se3 to the
Two-Dimensional Limit,” arXiv, no. 0911.3706, 2009.
[78] J. H. Chen, C. Jang, S. Adam, M. S. Fuhrer, E. D. Williams, and
M. Ishigami, “Charged-Impurity Scattering in Graphene,” Nature Physics,
vol. 4, no. 10.1038/nphys935, p. 377, 2008.
[79] S. Hikami, A. I. Larkin, and Y. Nagaoka, “Spin-Orbit Interaction and Magnetoresistance in the 2 Dimensional Random System,” Progress of Theoretical
Physics, vol. 63, p. 707, 1980.
[80] L. Fu and C. L. Kane, “Superconducting Proximity Effect and Majorana
Fermions at the Surface of a Topological Insulator,” Physical Review Letters,
vol. 100, no. 096407 10.1103/PhysRevLett.100.096407, 2008.
[81] G. J. Dolan and D. D. Osheroff, “Non-Metallic Conduction in Thin Metal-Films
at Low-Temperatures,” Physical Review Letters, vol. 43, p. 721, 1979.
[82] T. Kawaguti and Y. Fujimori, “Magnetoresistance and Inelastic-Scattering
Time in Thin-Films of Silver and Gold in Weakly Localized Regime,” Journal of the Physical Society of Japan, vol. 52, p. 722, 1983.
[83] N. O. Birge, B. Golding, and W. H. Haemmerle, “Conductance Fluctuations
and 1/F Noise in Bi,” Physical Review B, vol. 42, p. 2735, 1990.
[84] C. Vanhaesendonck, L. Vandendries, Y. Bruynseraede, and G. Deutscher, “Localization and Negative Magnetoresistance in Thin Copper-Films,” Physical
Review B, vol. 25, p. 5090, 1982.
BIBLIOGRAPHY
148
[85] P. W. Anderson, “Absence of Diffusion in Certain Random Lattices,” Physical
Review, vol. 109, p. 1492, 1958.
[86] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan,
“Scaling Theory of Localization - Absence of Quantum Diffusion in 2 Dimensions,” Physical Review Letters, vol. 42, p. 673, 1979.
[87] B. L. Altshuler, A. G. Aronov, and P. A. Lee, “Interaction Effects in Disordered
Fermi Systems in 2 Dimensions,” Physical Review Letters, vol. 44, p. 1288, 1980.
[88] B. L. Altshuler, D. Khmelnitzkii, A. I. Larkin, and P. A. Lee, “Magnetoresistance and Hall-Effect in a Disordered 2-Dimensional Electron-Gas,” Physical
Review B, vol. 22, p. 5142, 1980.
[89] S. Maekawa and H. Fukuyama, “Magnetoresistance in Two-Dimensional
Disordered-Systems - Effects of Zeeman Splitting and Spin-Orbit Scattering,”
Journal of the Physical Society of Japan, vol. 50, p. 2516, 1981.
[90] Y. Aharonov and D. Bohm, “Significance of Electromagnetic Potentials in the
Quantum Theory,” Physical Review, vol. 115, p. 485, 1959.
[91] D. Y. Sharvin and Y. V. Sharvin, “Magnetic-Flux Quantization in a Cylindrical
Film of a Normal Metal,” JETP Letters, vol. 34, p. 272, 1981.
[92] R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, “Observation Of
.Planck’s Constant Divided by Electronic Charge Aharonov-Bohm Oscillations
in Normal-Metal Rings,” Physical Review Letters, vol. 54, p. 2696, 1985.
BIBLIOGRAPHY
149
[93] S. Washburn, C. P. Umbach, R. B. Laibowitz, and R. A. Webb, “TemperatureDependence of the Normal-Metal Aharonov-Bohm Effect,” Physical Review B,
vol. 32, p. 4789, 1985.
[94] C. P. Umbach, C. Vanhaesendonck, R. B. Laibowitz, S. Washburn, and R. A.
Webb, “Direct Observation of Ensemble Averaging of the Aharonov-Bohm Effect in Normal-Metal Loops,” Physical Review Letters, vol. 56, p. 386, 1986.
[95] B. L. Altshuler, A. G. Aronov, and B. Z. Spivak, “The Aaronov-Bohm Effect
in Disordered Conductors,” JETP Letters, vol. 33, p. 94, 1981.
[96] B. L. Altshuler, “Fluctuations in the Extrinsic Conductivity of Disordered Conductors,” JETP Letters, vol. 41, p. 648, 1985.
[97] B. Pannetier, J. Chaussy, R. Rammal, and P. Gandit, “Magnetic-Flux Quantization in the Weak-Localization Regime of a Nonsuperconducting Metal,”
Physical Review Letters, vol. 53, p. 718, 1984.
[98] P. A. Lee, A. D. Stone, and H. Fukuyama, “Universal Conductance Fluctuations
in Metals - Effects of Finite Temperature, Interactions, and Magnetic-Field,”
Physical Review B, vol. 35, p. 1039, 1987.
[99] P. A. Lee and A. D. Stone, “Universal Conductance Fluctuations in Metals,”
Physical Review Letters, vol. 55, p. 1622, 1985.
[100] P. A. Lee and T. V. Ramakrishnan, “Disordered Electronic Systems,” Reviews
of Modern Physics, vol. 57, p. 287, 1985.
BIBLIOGRAPHY
150
[101] A. D. Stone, “Magnetoresistance Fluctuations in Mesoscopic Wires and Rings,”
Physical Review Letters, vol. 54, p. 2692, 1985.