of 7
Scanning Tunnelling Spectr
oscopic Studies of Dirac Fermions in
Graphene and Topol
ogical Insulators
N.-C. Yeh
1
, M. L. Teague
1
, R. T.-P. Wu
1
, H. Chu
2
, D. A. Boyd
2
, M. W. Bockrath
3
, L. He
4
, F.-X. Xiu
5
and K.-L. Wang
4
1
Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA
2
Department of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA
3
Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA
4
Department of Electrical Engineering, University of California, Los Angeles, CA 90095, USA
5
Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011, USA
Abstract.
We report novel properties derived from scanning tunnelling spectroscopic (STS) studies of Dirac
fermions in graphene and the surface state (SS) of a strong topological insulator (STI), Bi
2
Se
3
. For mono-layer
graphene grown on Cu by chemical vapour deposition (C
VD), strain-induced scalar and gauge potentials are
manifested by the charging effects and the tunnelling conductance peaks at quantized energies, respectively.
Additionally, spontaneous time-reversal symmetry breaking is evidenced by the alternating anti-localization
and localization spectra associated with the zero-mode of
two sublattices while global time-reversal symmetry
is preserved under the presence of pseudo-magnetic fields. For Bi
2
Se
3
epitaxial films grown on Si(111) by
molecular beam epitaxy (MBE), spatially localized unit
ary impurity resonances with sensitive dependence on
the energy difference between the Fermi level and the Dirac point are observed for samples thicker than 6
quintuple layers (QL). These findings are characteristic of the SS of a STI and are direct manifestation of
strong topological protection against impurities. For samp
les thinner than 6-QL, STS studies reveal the open-
up of an energy gap in the SS due to overlaps of wave functions between the surface and interface layers.
Additionally, spin-preserving quasiparticle interference wave-vectors are observed, which are consistent with
the Rashba-like spin-orbit splitting.
1 Introduction – Dirac materials
The Dirac materials are a new class of matter that
exploits the mapping of electronic band structures and an
embedded spin or pseudo-spin degree of freedom onto
the relativistic Dirac equati
on [1-6]. These materials,
including graphene [1] and the surface state (SS) of three-
dimensional (3D) strong topol
ogical insulators (STI) [2-
6], have emerged as a new paradigm in condensed matter
for investigating the topological phases of massless and
massive Dirac fermions. Among the novel characteristics
of massless Dirac materials, a common phenomenon is
the topological protection of Dirac fermions against
backscattering as the result of prohibition of inverting
spin [2] or pseudo-spin [1] u
nder time reversal operation
due to the large spin-orbit or inter-valley interaction.
In this work, we report studies of massless Dirac
fermions in two types of Dirac materials: the mono-layer
graphene and the SS of a 3D-STI, Bi
2
Se
3
, by means of
scanning tunnelling microscopy and spectroscopy
(STM/STS). For mono-layer CVD-grown graphene on
Cu, we find spectroscopic evidences for strain-induced
scalar and gauge potentials [7]. The gauge potentials
result in both integer and fractional quantum Hall (IQH
and FQH) states, and the finding of FQH states is
attributed to strong short-range Coulomb interaction
mediated by the underlying Cu substrate. Moreover,
alternating anti-localization and localization spectra
associated with the zero-mode of the two sublattices are
observed as the result of local spontaneous time-reversal
symmetry breaking. For MBE-grown Bi
2
Se
3
epitaxial
films on Si(111) in the 3D limit, spatially localized
unitary impurity resonances with sensitive dependence on
the energy difference between the Fermi level (
E
F
) and
the Dirac point (
E
D
) are demonstrated [8]. These findings
are consistent with theoretical predictions for impurities
in the SS of a STI [9,10]. In contrast, STS studies of
samples in the 2D limit reveal the opening of a SS energy
gap and the appearance of spin-preserving quasiparticle
interference (QPI) wave-vectors.
2 Evidence of strain-induced gauge and
scalar potentials and spontaneous time-
reversal symmetry breaking in graphene
EPJ Web of Conferences
DOI: 10.1051
/
C

Owned by the authors, published by EDP Sciences, 2012
, 0
epjconf 201
223 00021
/
23
2012)
0021 (
T
h
i
s
i
s
a
n
O
p
e
n
A
c
c
e
s
s
a
r
t
i
c
l
e
d
i
s
t
r
i
b
u
t
e
d
u
n
d
e
r
t
h
e
t
e
r
m
s
o
f
t
h
e
C
r
e
a
t
i
v
e
C
o
m
m
o
n
s
A
t
t
r
i
b
u
t
i
o
n
-
N
o
n
c
o
m
m
e
r
c
i
a
l
L
i
c
e
n
s
e
3
.
0
,
which
permits
unrestricted use, distribution, and reproduction in any noncommercial medium, provided
the original work is properly cited.
Article available at
http://www.epj-conferences.org
or
http://dx.doi.org/10.1051/epjconf/20122300021
EPJ Web of Conferences

ln
ln
22
xx
yy
xx
yy
xy
xy
uu
uu
ta
uu
aa

 





A

2
sgn
2
FS
n
EnevBn



22
33
F
EaEv
N
The electronic properties of graphene are known to be
highly susceptible to the surrounding environment and
disorder. In general, there are two primary effects
associated with disorder on the electronic properties of
graphene [1]. The first effect is a local change in the
single site energy that leads to an effective shift in the
chemical potential and thus scattering effects for Dirac
fermions [1]. The second type of disorder effect arises
from changes in the distance or angles between the
p
z
orbitals [1]. In this case, the hopping energies between
different lattice sites are modified, leading to the addition
of a new term in the appearance of a vector (gauge)
A
and scalar potentials
to the original Dirac Hamiltonian
[1]. The addition of a disorder-induced vector potential
A
in the Hamiltonian leads to the occurrence of an
effective magnetic field
B
S
= (
c
/
ev
F
)
×
A
for the Dirac
cone at K and an opposite field
B
S
= (
c
/
ev
F
)
×
A
*
for
the inequivalent Dirac cone at K
, where
v
F
denotes the
Fermi velocity of the Dirac fe
rmions [11]. Therefore, the
global time-reversal symmetry is preserved under the
addition of such a gauge potential. Similarly, the presence
of a spatially varying scalar potential can lead to local
charging effects known as self-doping [1].
2.1 Pseudo-magnetic fields, charge interactions,
and integer & fractional quantum Hall states
2.1.1 Strain-induced pseudo-magnetic fields
One mechanism for inducing pseudo-magnetic fields and
charging effects is strain [7,11-13]. Specifically, the
strain-induced gauge potential
A
= (
A
x
,
A
y
) may be
related to the 2D strain field
u
ij
(
x,y
) by the following
relation (with
x
-axis along the zigzag direction) [1,7]:
(1)
where the variable
t
denotes the nearest hopping constant,
a
represents the nearest carbon-carbon distance,
is a
constant ranging from 2 to 3 [1], and the strain field
u
ij
(
x,y
) is related to a 2D displacement field
u
= (
u
x
,
u
y
) by
the definitions
u
xx
(
u
x
/
x
),
u
xy
(
u
x
/
y
) and
u
yy
(
u
y
/
y
). For non-trivial strain that leads to an effective
magnetic field
B
S
= (
c
/
ev
F
)
×
A
, the local density of
states (LDOS) of Dirac fermions is modified so that
LDOS peaks at quantized energies (
E
n
)
(2)
appear in addition to the linear density of states
N
(
E
) per
unit cell of unstrained graphene, where
(3)
In Eq. (2)
n
denotes either integers or fractional numbers.
Hence, strain-induced pseudo-magnetic fields
B
S
may be
inferred from the quantized tunneling conductance peaks
in the STS studies of strained graphene by identifying the
slope of the (
E
n
E
D
) vs. |
n
|
1/2
plot, where
E
D
is the Dirac
energy [11].
In this work, we performed STM/STS studies of
graphene grown by chemical vapor deposition (CVD) on
Cu [7]. Details for the CVD growth process and
STM/STS measurements have been given elsewhere [7].
We found that large and non-uniform strain was induced
by the expansion of graphene and the contraction of the
copper substrate upon cooling from the CVD-growth
temperature (~ 1000°C) to low temperatures [7], which
resulted in significant lattice distortions at atomic scales,
as exemplified in figure 1(a). In addition, local variations
in the Dirac energy
E
D
are also present, as shown in
figure 1(b) for the map of Dirac energies over the same
area as shown in figure 1(a).
Fig. 1.
Topographic and conductance studies of strained CVD-
grown graphene on Cu:
(a)
Atomically resolved topography of
graphene, showing strongly distorted lattices over a (3
3) nm
2
area.
(b)
Variations in the Dirac energy
E
D
over the same area
as shown in (a).
2.1.2 Manifestation of two zero modes due to local
spontaneous time-reversal symmetry breaking
While the strain-induced pseudo-magnetic fields do not
break global time-reversal symmetry, the opposite gauge
fields associated with the two pseudo-spins (valleys) K
and K
in reciprocal space in fact give rise to a peculiar
zero mode, which is a condensate where Dirac fermions
become completely delocalized over the sample and yet
remain alternately localized and anti-localized in the
pseudo-spin projection in the real space [14]. More
specifically, the presence of a pseudo-magnetic field due
to a repulsive second-nearest-neighbor interaction, such
as that provided by a triaxial strain in a graphene nano-
bubbles [13], can provide a net flux of a non-Abelian
gauge field that preserves the time reversal symmetry and
breaks the chiral symmetry of the free Dirac Hamiltonian.
This non-Abelian gauge field is shown theoretically [14]
to serve as a catalyst for local spontaneous breaking of
the time-reversal symmetry in graphene while preserving
the chiral symmetry.
Empirically, the local spontaneous time-reversal
symmetry breaking may be manifested by the alternating
presence and absence of the tunneling conductance peak
at
n
= 0 for two inequivalent sublattice sites in graphene.
Indeed, our spatially resolved STS studies of the CVD-
grown graphene on Cu revealed tunneling spectra with
alternating presence and absence of the zero mode, as
exemplified in figures 2(a) – (b
). This finding is the first
00021-p.2
Eurasia Pacific Summer School and Conference on Strongly Correlated Electrons
00
,
xx
yy
xy
u
u

 
u

††
,,
ij
ij
ii
ij
i
tababUnn
 



H
direct evidence of spontaneous time-reversal symmetry
breaking due to strain-induced gauge fields in graphene.
Fig. 2.
Manifestation of strain-induced pseudo-magnetic fields
in the tunnelling conductance (
dI
/
dV
) versus
(
E
E
D
) spectra of
CVD-grown graphene on Cu at
T
= 77 K:
(a)
Left panel
,
representative point spectra at A-sites after subtraction of the
free Dirac fermion spectral background [7], showing distinct
conductance peaks at integer and fractional quantized energies
E
n
as defined in Eq. (2), where
E
n
is proportional to
|
n
|
1/2
.
Peaks are clearly visible for
n
= 0,
1,
2,
3. Additional humps
at fractional values
n
=
5/3,
2/3 and
1/3 are also identifiable.
Right panel
, atomically resolved topography of the graphene
showing the locations (represented by the white dots 1, 2, 3)
where the zero mode is present.
(b)
Left panel
, representative
point spectra at B-sites after subtraction of the free Dirac
fermion spectral background, showing distinct conductance
peaks at
n
=
1/3,
1,
2,
3, with absence of zero mode at
n
=
0. Right panel
, atomically resolved topography of the graphene
showing the locations (represented by the white dots 1, 2, 3)
where the zero mode is absent.
2.1.3 Fractional quantum Hall (FQH) states due to
interacting Dirac fermions
To date the theoretical descriptions for strain-induced
gauge potentials [11,12] have assumed non-interacting
Dirac fermions in graphene. Hence, only integer quantum
Hall (IQH) states are expected under pseudo-magnetic
fields. On the other hand, for CVD-grown graphene on
Cu, the 2D electron gas (2DEG) associated with the
surface state of Cu substrates provides an additional
channel for short-range charge interaction through the
electrostatic coupling between the Dirac fermions in
graphene and the 2DEG in Cu. Although long-range
Coulomb interactions of Dirac fermions are screened by
the underlying 2DEG, whether short-range Coulomb
interactions may be neglected depends on the strength of
onsite Coulomb potential
U
relative to the nearest-
neighbor hopping constant
t
. That is, if we consider the
Hamiltonian
(4)
where
()
ii
aa
annihilates (creates) an electron with spin
(
=
,
) on site
i
of the sublattice A,
()
jj
bb
annihilates (creates) an electron with spin
on site
j
of
the sublattice B,
t
2.8 eV [1] is the nearest-neighbor
hopping energy for hopping between different sublattices,
and
n
i
and
n
i
denote the density of electrons with spin
up and spin down at site
i
, respectively. Given the onsite
Coulomb repulsion
U
Cu
~ 3 eV for the surface state of Cu
[15], the onsite Coulomb potential
U
for graphene in Eq.
(4) is approximated by the relation
U
=
U
Cu
(
d
Cu
/
d
)
2
,
where
d
Cu
~ 0.255 nm and
d
= 3
1/2
a
~ 0.246 nm are the
lattice constants of copper and graphene, respectively.
Hence, we find
U ~
3.2 eV, which is larger than the
hopping constant
t
, implying that the Coulomb interaction
among Dirac fermions cannot be neglected. Thus,
fractional quantum Hall (FQH) states become feasible,
with the Laughlin state of a filling factor
=
1/3 and the
related states of
=
2/3,
4/3 and
5/3 being the
dominant contributions. This rationale is consistent with
our experimental findings exemplified in figure 2.
2.2 Strained-induced scalar potential
The strain found in the CVD-grown graphene on copper
is not purely shear but also contains dilation/compression
components, which is theoretically predicted to gives rise
to an effective scalar potential
(
x
,
y
) and therefore a
static charging effect [12,16,17] in addition to the
pseudo-magnetic field. Specifically, the scalar potential
(
x
,
y
) is given by [12,16,17]:
(4)
where
0
~ 3 eV, and
|
u
|
is the dilation/compression
strain. While the charging effect may be largely screened
if the height variation
l
is much smaller than the magnetic
length
l
B
[
0
/(2
B
S
)]
1/2
, where
0
is the flux quantum
[11,12], our finding of
B
S
ranging from 30 to 50 Tesla
corresponds to values of
l
B
ranging from 5.5 nm to 3.5
nm [7], which is comparable to the height variation
l
~ 1
nm over the sample area shown in figure 1(a). Hence, we
expect significant charging effect in the CVD-grown
graphene on Cu, which may be manifested by the spatial
variations in the constant-voltage tunneling conductance.
Fig. 3.
Correlation of the compression/dilation strain and the
charging effect is manifested by the (3×3) nm
2
spatial maps of
(a)
the zero-bias conductance, and
(b)
the compression/dilation
strain, where the strain is estimated by considering the spatial
variations in the height
z
(
x
,
y
), [(
z
/
x
)
2
+(
z
/
y
)
2
].
(a)
(b)
1
2
3
1
2
3
00021-p.3
EPJ Web of Conferences
imp
imp
imp
imp
~5sgn
ln
D
EUUU

As shown in figures 3(a) and 3(b) for the spatial
maps of the tunnelling conductance at zero bias and of
the dilation/compression strain, respectively, there is
clear correlation between the spatial distributions of the
effective strain [(
z
/
x
)
2
+(
z
/
y
)
2
] and the zero-bias
conductance, where
z
(
x
,
y
) denotes the spatial variations
of the height. This apparent correlation thus provides
direct evidence for
the strain-induced charging effect.
3
I
mpurity resonances and effects of
dimensional crossover in the surface
state of topological insulator Bi
2
Se
3
An exciting new development in condensed matter
physics over the last decade is the beautiful realization of
topological field theories [18] in strongly correlated
electronic systems, where topological field theories are
shown to provide a classification of order due to
macroscopic entanglement that is independently of
symmetry breaking [19]. The FQH state is the first
known example of such a quantum state that exhibits no
spontaneous broken symmetry, and its properties depend
only on its topology rather than geometry, which is
topologically distinct from all other quantum states
classified by broken symmetry. Recently, the quantum
spin Hall (QSH) states and the topological insulators (TI)
have emerged as a new class of topological states and
have stimulated intense research activities [2]. As
described in Section 1, one of the novel properties
associated with the TI is the presence of a Dirac spectrum
of chiral low-energy excitations, which is a salient feature
of the Dirac materials. In the case of 3D-STI, an odd
number of massless Dirac cones in their SS are ensured
by the Z
2
topological invariant of the fully gapped bulk
[2–6]. Backscattering of Dirac fermions is suppressed due
to topological protection that preserves the Dirac
dispersion relation for any time-reversal invariant
perturbation [2].
3.1. Spectroscopic evidences for Fermi-energy
dependent impurity resonances
While direct backscattering is prohibited in both the SS
of 3D-STI and in graphene, sharp resonances are not
excluded because Dirac fermions with a finite parallel
momentum may be confined by potential barriers [1]. In
fact, theoretical calculations for Dirac fermions in the
presence of non-interacting impurities have predicted the
occurrence of strong impurity resonances [9,10].
Nonetheless, to date no direct empirical observation of
strong resonances has been demonstrated despite reports
of spectral evidences for quasiparticle interferences (QPI)
associated with impurity or step-edge induced scattering
[20–22] of Dirac fermions.
Here we report direct STS observation of impurity
resonances in a 3D-STI system, Bi
2
Se
3
[8]. The samples
that we investigated are epitaxial Bi
2
Se
3
films grown on
Si(111) by molecular beam epitaxy (MBE). Details of the
MBE film growth process have been described elsewhere
[23]. Transmission electron microscopy (TEM) on these
films exhibited perfect triangular lattice structures, and
ARPES (angle resolved photoemission spectroscopy)
studies revealed a single Dirac cone for the SS [23]. We
performed STS studies [8] on samples of three average
thicknesses: 60, 7 and 3 quintuple layers (QLs), because
the energy difference (
E
F
E
D
) of MBE-grown Bi
2
Se
3
epitaxial films was found to increase with decreasing
thickness [24]. In particular, a crossover of the SS from a
gapless Dirac cone to gapped spectra occurred when the
average thickness became less than 6 QLs, and the energy
gap further increased with decreasing thickness [24, 25].
Generally the spectra in our STS studies were found
consistent throughout a large flat area [8]. Representative
point spectra for the 60-QL and 7-QL samples are given
in the main panels of figures 4(a) and 4(b), and the ranges
of the Dirac energy
E
D
for all areas investigated are
summarized in the insets of figures 4(a) and 4(b), which
correspond to
E
D
= (
78±38) meV and
E
D
= (
100±25)
meV, respectively.
Fig. 4.
Representative tunnelling conductance spectra (main
panel) of MBE-grown Bi
2
Se
3
epitaxial films of
(a)
60-QL and
(b)
7-QL thickness. The insets are histograms of the Dirac
energy
E
D
over extended areas of the 60-QL and 7-QL samples,
showing
E
D
= (
78±38) meV for 60-QL and
E
D
= (
100±25)
meV for 7-QL. We further note that in STS studies the Fermi
energy
E
F
coincides with the zero energy.
Despite the relatively consistent tunneling spectra for
most areas investigated, we note the presence of a few
quantum impurities, as exemplified by the high
conductance spots in figure 5(a) for the constant-bias
conductance map of a 60-QL sample over a (5.0
8.0)
nm
2
area and in figure 5(b) for a 7-QL sample over a (5.1
8.8) nm
2
area. The tunneling spectra on top of the
quantum impurities are exemplified in figure 6(b) for the
60-QL and 7-QL samples.
There are several noteworthy spectral characteristics
for the quantum impurities. First, sharp conductance
peaks associated with the presence of impurities,
imp
,
generally occur at energies near the Dirac point
E
D
, which
are consistent with the unitary limit of impurity scattering
so that the impurity potential
U
imp
diverges following the
relation [17]
. (5)
Second, the conductance peaks are strongly localized in
space, within a radius of ~ 0.2 nm around each impurity,
implying rapid recovery of the SS of Dirac fermions
away from the impurities. Third, the intensity of the
00021-p.4
Eurasia Pacific Summer School and Conference on Strongly Correlated Electrons
impurity resonances decreases rapidly with the increasing
energy difference |
E
F
E
D
|.
The rapidly diminishing spectral intensity of impurity
resonances with increasing |
E
F
E
D
| is consistent with the
theory for non-interacting point impurities embedded in a
system of massless Dirac fermions [10]. Following the
theoretical analysis in reference [10] that utilized the
Keldysh Green function formalism, we showed in
reference [8] and reproduced in
figure 6(a) the theoretical
dependence of the intensity of impurity resonances on (
E
F
E
D
), which is in reasonable agreement with our
empirical findings [8].
Fig. 5.
Representative atomically resolved constant energy (
E
)
conductance maps of MBE-grown Bi
2
Se
3
epitaxial films of
(a)
60-QL taken at
E
=
30 meV and
(b)
7-QL taken at
E
=
50
meV. The high conductance spots are associated with quantum
impurities embedded in the atomically ordered Bi
2
Se
3
host, and
the intensity of the impurity resonance is found to be highly
localized so that the generic spectrum for the SS of the 3D-STI
recovers within ~ 0.2 nm [8]. The point spectrum at the location
indicated by the white circle in (a) and that at the location
indicated by the black circle in (b) are given in figure 6(b),
showing sharp resonant peaks near the respective Dirac energies
E
D
, implying that the impurity po
tential is in the unitary limit
[9].
Fig. 6.
(a)
Theoretical predictions for the dependence of the
intensity of impur
ity resonances on
(/)
E
, where
denotes a
cutoff energy and
()
F
D
E
EE

[8, 10].
(b)
Representative
point spectra above the quantum impurities in the 60-QL and 7-
QL samples, showing strong resonances at energies very close
to
E
D
. The spectrum for a point impu
rity of the 60-QL sample
was taken at the location indicated by a white circle in figure
5(a), and that for a point impurity
of the 7-QL sample was taken
at the location denoted by a black circle in figure 5(b).
3.2 Effects of dimensional crossover
While the dispersion relation of the SS of 3D-STI has
been confirmed to be a gapless Dirac cone, for 3D-STI in
the thin-film limit the coupling between opposite
topological surface states in space is theoretically
expected to open up an energy gap [25]. Therefore, the
dispersion relation will convert from a Dirac cone into a
gapped Dirac hyperbola [25]. The thin film may further
break the top–bottom symmetry due to the asymmetric
chemical potentials between the surface and the interface
of the thin film with a substrate, leading to a Rashba-like
coupling and further energy splitting in the momentum
space [25]. These theoretical predictions have been
verified by ARPES studies of MBE-grown Bi
2
Se
3
films
on SiC [24], which demonstrated that the surface spectra
open a visible energy gap when the thickness is below 6
QLs. The energy gap was found to increase with
decreasing thickness in thin films of 2–6 QL.
Fig. 7.
Evidence for quasiparticle interferences (QPI) in the 5-
QL Bi
2
Se
3
sample from Fourier transformed (FT) tunneling
conductance maps that reveal energy (
E
) dependent QPI wave-
vectors (
q
) with preservation of spin texture, where the
q
-value
for each FT-conductance map is obtained by the wave-vector
from the center of the zone to the one of the 6 points on the
hexagon:
(a)
E
=
300 meV,
(b)
E
=

meV, and
(c)
E
=

meV. The energy dependence of the QPI wave-vectors
q
(
E
)
determined from all FT conductance maps is summarized in
(d)
,
showing decreasing |
q
| values as
E
approaches
E
D
.
A natural consequence of the Rashba-like coupling
and the gapped Dirac hyperbola in the thin-film limit of a
3D-STI is that the resulting energy dispersion relation
would involve four split Fermi surfaces with chiral spin
textures for surface-state spin-up, interface-state spin-up,
surface-state spin-down, an
d interface-state spin-down
[25]. Therefore, quasiparticles interferences (QPI) due to
elastic impurity scattering are expected only between
Fermi surfaces with the same spin texture, similar to the
findings from STS studies of the QPI in Bi
x
Sb
1
x
[20].
However, we note that Bi
x
Sb
1
x
is a 3D-STI with much
more complicated Fermi surfaces and therefore multiple
spin-preserved QPI wave-vec
tors are allowed [20]. In
contrast, for the four split Fermi surfaces in Bi
2
Se
3
thin
00021-p.5
EPJ Web of Conferences
films, only one value for the QPI wave-vector
q
is
expected for a given quasiparticle energy due to the
preservation of spin textures from spin-up to spin-up
states and from spin-down to spin-down states. Further,
the |
q
|-values should be smaller than the reciprocal lattice
constants, and should decrease as
E
E
D
.
Our preliminary STS studies of a 5-QL Bi
2
Se
3
thin
film on Si (111) revealed that the Fourier-transformed
(FT) conductance maps at various constant energies
indeed exhibited a single QPI |
q
|-value for a given
energy, as exemplified in figures 7(a) – 7(c). Further, the
magnitude of QPI wave-vectors indeed decreases with
quasiparticle energy
E
approaching the Dirac point
E
D
, as
shown in figure 7(d). These results are qualitatively
consistent with preservation of spin textures in QPI for
3D-STI in the thin-film limit. However, detailed analysis
of our STS data is not in quantitative agreement with the
fitting parameters derived from ARPES studies of Bi
2
Se
3
films on SiC [24]. Possible reasons for the occurrence of
discrepancies may include different substrates used,
different (
E
F
E
D
) values for each
nominal sample
thickness [8, 24], and variations in the physical properties
from microscopic to macroscopic experiments. Further
systematic STS studies of Bi
2
Se
3
films with different
thicknesses and su
bstrates will be necessary to resolve the
quantitative discrepancies.
4 Conclusion
We report in this work scanning tunnelling spectroscopic
(STS) studies of Dirac fermions in graphene and the
surface state (SS) of a three-dimensional (3D) strong
topological insulator (STI), Bi
2
Se
3
. In the case of CVD-
grown graphene on Cu, we find that strain-induced scalar
and gauge potentials are manifested by the charging
effects and the tunnelling conductance peaks at quantized
energies, respectively. The observation of both integer
and fractional quantum Hall (IQH/FQH) states due to
strain-induced pseudo-magnetic fields may be attributed
to significant short-range Coulomb interactions of Dirac
fermions in graphene mediated by the underlying Cu
substrate, so that the resulting Coulomb interaction
U
~
3.2 eV is larger than the nearest-neighbour hopping
energy
t
~ 2.8 eV in graphene. Further, spontaneous time-
reversal symmetry breaking is evidenced by the spatially
alternating presence and absence of the pseudo-magnetic
field-induced zero mode in the tunnelling conductance
spectra while global time-reversal symmetry is preserved.
For MBE-grown Bi
2
Se
3
epitaxial films on Si(111),
unitary impurity resonances with sensitive dependence on
the energy difference between the Fermi level and the
Dirac point are observed for samples thicker than 6
quintuple layers (QL). These resonances are spatially
localized within an area of radius ~ 0.2 nm, implying
strong topological protection against impurities. For
samples thinner than 6-QL, STS studies reveal the open-
up of an energy gap in the SS due to overlaps of wave
functions between the surface and interface layers and
Rashba-like spin-orbit splitting. Additionally, spin-
preserving quasiparticle interference (QPI) wave-vectors
are observed as a function of quasiparticle energy, which
are qualitatively consistent with the dimensional
crossover behaviour of 3D-STI.
Acknowledgement
The research on graphene at Caltech was supported by
the National Science Foundation, and the research on
topological insulators at Caltech and UCLA was
supported by the Center on Functional Engineered Nano
Architectonics (FENA).
References
1.
A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S.
Novoselov, A.K. Geim, Rev. Mod. Phys.
81
, 109
(2009).
2.
X.-L. Qi and S.-C. Zhang, Phys. Today
63
, 33
(2010).
3.
L. Fu and C. L. Kane, Phys. Rev. B
76
, 045302
(2007).
4.
H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang and
S.-C. Zhang, Nat. Phys.
5
, 438 (2009).
5.
J. E. Moore and L. Balents, Phys. Rev. B
75
, 121306
(2007).
6.
A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W.
Ludwig, Phys. Rev. B
78
, 195125 (2008).
7.
N.-C. Yeh, M. L. Teague, S. Yeom, B. L. Standley,
R. T.-P. Wu, D. A. Boyd, M.
W. Bockrath, Surf. Sci.
605
, 1649 (2011).
8.
M. L. Teague, H. Chu, F.-X. Xiu, L. He, K.-L. Wang
and N.-C. Yeh, submitted to Europhys. Lett. (2011).
9.
R. R. Biswas and A. V. Balatsky, Phys. Rev. B
81
,
233405 (2010).
10.
K. Saha, I. Paul
and K. Sengupta, Phys. Rev. B
81
,
165446 (2010).
11.
F. Guinea, M.I. Katsnelson, A.K. Geim, Nat. Phys.
6
,
30 (2010).
12.
F. Guinea, M. I. Katsnelson, M. A. H. Vozmediano,
Phys. Rev. B
77
, 075422 (2008).
13.
N. Levy, S. A. Burke, K. L. Meaker, M. Panlasigui,
A. Zettl, F. Guinea, A. H. Castro Neto, M. F.
Crommie, Science
329
, 544 (2010).
14.
I. F. Herbut, Phys. Rev. B
78
, 205433 (2008).
15.
I. V. Solovyev, A. I. Liechtenstein and K. Terakura,
Phys. Rev. Lett.
80
, 5758 (1998).
16.
H. Suzuura and T. Ando, Phys. Rev. B
65
, 235412
(2002).
17.
J. L. Manes, Phys. Rev. B
76
, 045430 (2007).
18.
D. Birmingham, M. Blau, M. Rakowski and G.
Thompson, Phys. Rep.
209
, 129
(1991).
19.
X.-G. Wen, Adv. Phys.
44
, 405 (1995).
20.
P. Roushan, J. Seo, C. V. Parker, Y. S. Hor, D.
Hsieh, D. Qian, A. Richardella, M. Z. Hasan, R. J.
Cava and A. Yazdani, Nature
460
, 1106 (2009).
21.
T. Zhang, P. Cheng, X. Chen, J.-F. Jia, X. Ma, K.
He, L. Wang, H. Zhang, X. Dai, Z. Fang, X. Xie, and
Q.-K. Xue, Phys. Rev. Lett.
103
, 266803 (2009).
22.
Z. Alpichshev, J. G. Analytis, J.-H. Chu, I. R. Fisher,
Y. L. Chen, Z. X. Shen, A. Fang, and A. Kapitulnik,
Phys. Rev. Lett.
104
, 016401 (2010).
00021-p.6
Eurasia Pacific Summer School and Conference on Strongly Correlated Electrons
23.
L. He, F.-X. Xiu, Y. Wang, A. V. Fedorov, G.
Huang, X. Kou, M. Lang, W. P. Beyermann, J. Zou,
and K.-L. Wang, J. Appl. Phys.
109
, 103702 (2011).
24.
Y. Zhang, K. He, C.-Z. Chang, C.-L. Song, L.-L.
Wang, X. Chen, J.-F. Jia, Z. Fang, X. Dai, W.-Y.
Shan, S.-Q. Shen, Q. Niu, X.-L. Qi, S.-C. Zhang, X.-
C. Ma and Q.-K. Xue, Nat. Phys.
6
, 584 (2010).
25.
W. Y. Shan, H.-Z. Lu and S.-Q. Shen, New J. Phys.
12
, 043048 (2010).
00021-p.7