Magnetism-induced massive Dirac spect
ra and topological defects in the
surface state of Cr-doped Bi
2
Se
3
-bilayer topological insulators
C.-C. Chen
1,2
, M. L. Teague
1,2
,
L. He
3
, X. Kou
3
, M. Lang
3
, W. Fan
1
, N. Woodward
1
, K.-L. Wang
3
and N.-C. Yeh
1,2,4*
1
Department of Physics, California Institu
te of Technology, Pasadena, CA 91125, USA
2
Institute of Quantum Matter and Information, Calif
ornia Institute of Technology, Pasadena, CA 91125,
USA
3
Department of Electrical Engineering, University of California, Los Angeles, CA 90095, USA
4
Kavli Nanoscience Institute, California Ins
titute of Technology, Pasadena, CA 91125, USA
*
E-mail:
ncyeh@caltech.edu
Supplementary Information
The supplementary information includes the following figures and related information:
Supplementary Figure 1: SMOKE measurement of
Cr-doped monolayer and bilayer topological
insulators
Supplementary Figure 2: Correlation between
the surface topography and
the gap maps of the
Cr-doped TI bilayer samples
Supplementary Figure 3: Temperature dependent
2D and 3D contributions to the proximity-
induced surface gap in Cr-doped TI bilayer systems
1.
Surface Magneto-Optic Kerr Effect (SMOKE)
measurements of Cr-doped monolayer
and bilayer topological insulators
Details of the experimental setup and data an
alysis of the SMOKE measurements have been
described in
Nano Lett.
13
, 4587 (2013). Briefly, the experimental
setup used elliptical polarized
reflected light to probe the magnetic properties of
materials. The SMOKE
system was integrated
with a temperature-controlled cryostat to
achieve cryogenic temperatures down to 2.8 K.
The SMOKE measurements involve spatially av
eraged signals over la
rge areas (10’s ~ 100’s
of
m
2
), which are about
five to six orders of magnitude larger
than the typical areas investigated
by our local-probe STM studies. Because of th
e large-area averaged effect, finite SMOKE
signals can only arise from non-vanishing net
ferromagnetic signals over the areas studied.
Therefore, for randomly oriented magnetic domai
ns plus inhomogeneity
within each magnetic
domains as seen by our local-probe STM experiments at
T
<
T
c
2D
, the SMOKE signal would
essentially average to zero for
T
c
3D
<
T
<
T
c
2D
. On the other hand, when
T < T
c
3D
, clear
manifestation of the Kerr angle
K
and the anomalous Hall resist
ance have been verified, and
these results have also been discussed in detail in
Nano Lett.
13
, 4587 (2013). Here we include
experimental results taken on a similar bilayer system (with
T
c
3D
~ 25 K and the
magnetoresistance and anomalous Hall resist
ance results all consistent with the Bi
2
Se
3
/Bi
2-
2x
Cr
2x
Se
3
bilayer system studied
in this work) of
d
1
-thickness of (Bi
0.5
Sb
0.5
)
2
Te
3
(
d
1
= 0, 3, 6-QL)
on top of 6-QL Cr
0.08
(Bi
0.57
Sb
0.39
)
2
Te
3
, as illustrated below in Supplementary Figure 1.
Supplementary Figure 1| Temperature-depe
ndent SMOKE measurement of Cr-doped
monolayer and bilayer topological insulators:
(a-c) Out-of-plane magnetizations (reflected in
H
c
) measured by a polar-mode SMOKE setup. All
of the modulation-doped samples have the
same 6-QL Cr
0.16
(Bi
0.54
Sb
0.38
)
2
Te
3
bottom
d
2
-layer, with different top
d
1
-layer thicknesses (with
d
1
= 0, 3 and 6-QL). The largest magnetization is
produced in the 3-QL TI/6-QL Cr-doped TI
structure, which is consistent with the transport measurements and has been explained by a
model based on RKKY interaction mediated by surf
ace Dirac fermions, as discussed in detail in
Nano Lett.
13
, 4587 (2013).
2.
Correlation between the surface topography
and the gap maps of the Cr-doped TI
bilayer samples
The spatially inhomogeneous surface gap found
in Cr-doped TI bilayer structures with
d
1
≤
3-QL
has been attributed to three factors, incl
uding inhomogeneous Cr distributions, unaligned
magnetic moments of Cr, and height variations
in the MBE-grown TI bilayer samples as the
result of terrace-like structures.
To investigate how much the height variati
on may influence the surface gap distribution,
we directly compare the topographic map and th
e gap map of a Cr-doped
bilayer sample, (3+6)-
10%, as shown in Supplementary Figure 2. We cal
culate the mean of
the correlation function
C
between the two maps by the following relation:
,
2
2
,,
mn
mn
mn
mn
mn
mn
mn
AABB
C
AA
BB
.
Here the subscripts
m
,
n
refer to the pixel indices of the “
A
” and “
B
” maps that respectively
correspond to the surface topography and the su
rface gap taken on the same
sample and in the
same area, whereas
A
(
B
) denotes the mean value of the entire surface topography (surface gap)
map. We found that the maps show
n in Supplementary Figure 2 yield
C
0.08, which
corresponds to weak correlation. In general,
C
= 1 implies perfect correlation,
1 implies
perfect anti-correlation, a
nd 0 implies no correlation.
Supplementary Figure 2| Comparison of the su
rface topography and the gap map of the
same area in a bilayer sample (3+6)-10% at H = 0:
(a) Surface topography over a (24×24) nm
2
area. (b) The gap map of the same
area as in (a).The correlation
C
calculated using the relation
specified above for the two maps is found
to be 0.08, suggesting weak correlation.
a b
3.
Temperature dependent 2D and 3D contributio
ns to the proximity-induced surface gap
in Cr-doped TI bilayer systems
We assume that the proximity-induced surface gap is given by
= (
J
eff
M
) where
M
is the surface
magnetization that increases monotonically below
T
c
2D
, and
J
eff
is the ferromagnetic coupling
constant with both contributi
ons from the 3D components (
J
dipole
+
J
bulk
)
J
3D
and the 2D
component
J
Dirac
J
2D
associated with surface Dirac fe
rmion-mediated RKKY interaction. As
discussed in the manuscript and further elaborat
ed in the end of this section, we expect
J
3D
to
dominate at low
T
and
J
2D
to dominate at elevated temperatures. Moreover, the long Fermi
wavelength of surface Dirac fermions could result
in much enhanced RKKY
interaction in the
bilayer system so that
J
2D
>
J
3D
.
Based on the aforementioned assumption, the surface gap
(
T
<
T
c
2D
) is expected to
exhibit a non-monotonic
T
-dependence, which is consistent
with our experimental findings
depicted in figure 4:
(
T
) first increases with decreasing
T
for
T < T
c
2D
, and then exhibits a
“dip” at
T
x
where
T
c
2D
>
T
x
>
T
c
3D
, and
T
x
is a dimensional crossover temperature below which
J
Dirac
becomes negligible. While we do not have sufficient information to model
(
T
)
quantitatively, we demonstrate in Supplementary Figure 3 how the qualitative
T
-dependence of
J
eff
and
M
proposed in Subsection 4.1 of th
e manuscript yields non-monotonic
-vs.-
T
behavior
similar to the empirical results in figure 4.
In order to qualitatively emulate the aforementioned
T
-dependence of
J
2D
,
J
3D
,
J
eff
and
M
,
we employ the following functional forms
eff
TJTMT
;
2D
2D
2D
c0
c
c
1;
0;
MT T
M
TT
MT T
eff
2D
3D
JT JT JT
;
3/2
2D
0
c
2D
2D
exp
T
JT J
T
;
2
0
3D
0
3D
3D
c
exp
T
JT J J
T
.
Using the above functional forms and the parameters:
00
02D3D
0
1,
1,
0.315,
0.1,
MJ J
J
we find that for
T <
T
x
, the
T
-dependence of
J
eff
is largely determined by that of
J
3D
, whereas for
T
x
<
T
<
T
c
2D
, the
T
-dependence of
J
eff
is largely determined by that of
J
2D
. Further, the surface
gap also exhibits a non-monotonic
T
-dependence with a minimum at
T
x
.
Although the functional forms given
above can produce a non-monotonic
T
-dependence
of
that is in reasonable agreement with empirical
results, we caution that one should not over-
interpret the physical significance
of the specific functional fo
rms given above because of too
many parameters that can only be determined
from knowing the microscopic details. Further
theoretical studies will be necessary to full
y understand the temper
ature dependence of the
proximity-induced surface gap.
Supplementary Figure 3| Temp
erature (T) dependent 2D and 3D contributions to the
proximity-induced surface gap in th
e Cr-doped TI bilayer system.
(a) Schematic temperature
(
T
) dependence of the ferromagnetic coupling constant
J
eff
that consists of a 3D component
J
3D
that dominates at low
T
, and a 2D component
J
2D
that dominates at hi
gher temperatures. The
specific functional forms for
J
3D
and
J
2D
are given in the text. (b) Schematic
T
-dependence of the
surface magnetization
M
, with the specific functional form sp
ecified in the text. (c) The surface
gap
=
J
eff
M
reveals a non-monotonic
T
-dependence, consistent with
our experimental findings
shown in figure 4 and reproduced in (d) for a (1+6)-5% sample.
Finally, we discuss our
rationale for a dominant
J
2D
at elevated temperatures. First of all,
we note that the wavefunction of Dirac fermions
may be expressed as a linear combination of
orthonormal eigenfunctions of th
e Hamiltonian. While the eigenf
unctions are independent of
temperature, the wavefunction di
stribution among the eigenfunctions
is certainly a function of
temperature. In addition, the RKKY interaction associated with
J
2D
must involve spatially
extended electronic wavefuncti
ons along the c-axis, and the
spatial extension unavoidably
involves multiple atomic layers so that thermally assisted processes (
e.g.
, assisted by phonons or
impurity states in the bulk) sim
ilar to the case of variable-ra
nge-hopping conductiv
ity are likely.
In this context, higher
temperatures could eff
ectively reduce the hopping
barrier and enhance the
RKKY interaction mediated by th
e surface Dirac fermions. More sp
ecifically, we note that at
elevated temperatures the electronic wavefunc
tions of surface Dirac fermions will involve not
only the ground-state eigenfunction
(which is confined to 2D)
but also the excited-state
eigenfunctions (which could ex
tend beyond 2D). Therefore, a
ny interaction matrix element
involving the surface Dirac fermions will also be
temperature dependent. In particular, if the
interaction involves phonon-
assisted processes, more signifi
cant temperature dependence may be
expected.
Empirically, there are two pieces of evidences
for strong temperature dependence in the
electronic contributi
ons of Dirac fermions, which could be
considered as indirectly support for
our conjecture of temperature dependence in
the 2D RKKY interaction mediated by Dirac
fermions. The first piece of evidence was associat
ed with the observation
of quantum anomalous
Hall effect (QAHE) by C-Z Chang
et al.
in
Science
340
, 167 (2013): While theoretical
predictions for the occurrence
of QAHE based on purely topologic
al nature of the magnetic
topological insulators would have only required th
e sample temperature to be sufficiently lower
than the Curie temperature ~ 15 K, empi
rically QAHE could only be observed at
T
~ 30 mK;
slight increase of the temperature to
T
~ 90 mK already caused signi
ficant deviations from the
ideal behavior of QAHE. Although there has b
een no theoretical account for this unexpected
temperature dependence of the
QAHE in topological insulators
, the empirical finding strongly
suggests the relevance of temperature to the electr
onic contributions of Di
rac fermions so that
increasing temperature could cha
nge the 2D characteristics of Di
rac fermions in the ground state
to more 3D like in the excited states, thus di
minishing the QAHE for a strictly 2D system.
The second piece of evidence came from our de
tailed temperature dependent studies of
topological insulator films [L. He
et al.
, Nano Letters
12
, 1486 (2012)], which revealed that the
electrical conductivity of topol
ogical insulators was dominated by the surface conductance only
at sufficiently low temperatures; at interm
ediate temperatures both surface and bulk
contributions were relevant, a
nd then at high temperatures bulk effects dominated. Thus, the
electronic contributions of the
surface Dirac fermions in topologi
cal insulators appear to be
strongly dependent on the temperat
ure, which imply that the 2D
RKKY interaction mediated by
the surface Dirac fermions in our bilayer system
would also be temperature dependent.
In summary, to achieve proximity-induced su
rface ferromagnetism in our bilayer systems
through the 2D RKKY interac
tion mediated by the surface Dirac fermions, the electronic
wavefunctions of the surface Dirac fermions
must extend beyond the surface layer. While the
mechanism(s) that mediate the spread of wave
functions are not clear
yet, possible physical
mechanisms such as phonon and/or
impurity-states assisted hopping
are all strongly temperature
dependent. Empirically it has also been shown th
at the electronic contributions of surface Dirac
fermions to such phenomena as the QAHE and elec
trical conductivity reveal
strong temperature
dependence, suggesting that the
wavefunctions of surface Dirac fe
rmions in the excited states
differ significantly from those in
the ground state. Consequently, it is
feasible to suggest that the
magnetic coupling coefficient
J
2D
that involves the surface Dirac
fermions is also temperature
dependent, with negligible contributions at lo
w temperatures due to strongly surface-confined
electronic wavefunctions in the
ground state and thus diminished
wavefunction overlaps with the
underlying Cr atoms.