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Quantum Capacitance in Topological Insulators
Faxian Xiu,1* Nicholas Meyer,1 Xufeng Kou,2 Liang He,2 Murong Lang,2 Yong Wang,3
Xinxin Yu,2 Alexei V. Fedorov,4 Jin Zou,5 and Kang L. Wang2*
1
Department of Electrical and Computer Engineering, Iowa State University, Ames, USA
Department of Electrical Engineering, University of California, Los Angeles, California
90095, USA 3State Key Laboratory of Silicon Materials and Center for Electron
Microscopy, Department of Materials Science and Engineering, Zhejiang University,
Hangzhou, 310027, China 4Advanced Light Source Division, Lawrence Berkeley
National Laboratory, 1 Cyclotron Road, Berkeley, California, 94720, USA 5Materials
Engineering and Centre for Microcopy and Microanalysis, The University of Queensland,
Brisbane QLD 4072, Australia *To whom correspondence should be addressed. E-mail:
[email protected], [email protected]
2
Supplementary Information
1. Figure S1| A profile of terrace height
2. Figure S2| Transport measurements for a MBE-grown Bi2Se3 thin film: surface
properties
3. Figure S3| STM Measurements
4. Figure S4| A regular MOS capacitor without Bi2Se3
5. Figure S5| Theoretical calculations of quantum capacitance
6. Figure S6| Bulk properties through transport measurements
7. Figure S7| High temperature capacitance measurements
8. Figure S8| Capacitance simulations for a 2-DEG.
9. Figure S9| Notes on Fe/Al contacts
10. Figure S10| Magnetic-field-dependent capacitance measurements
11. Figure S11| Ambipolar field effect
12. More details on photolithography process
13. Theory for frequency responses of majority carriers
14. Simplified band diagrams for the MOS capacitor and notes on SdH oscillations
1
1. Figure S1| A profile of terrace height
b
a
2.5
Height (nm)
2.0
100 nm
Figure S1. (a) An AFM
image of Bi2Se3 (b) A
depth profile showing the
height of terrace along the
green line in (a). The
individual
terrace
is
estimated to be 1 nm in
height.
1.5
1.0
0.5
0.0
0
150 300 450
Distance (nm)
2. Figure S2| Transport measurements for a MBE-grown Bi2Se3 thin film:
surface properties
Figure S2| a, Longitudinal resistance Rxx versus magnetic field. Solid arrows indicate
integer Landau levels from the valley values; open arrows give the peak values. b, Rxx
versus 1/B at different temperatures. c, Landau level index plot: 1/B versus n. The open
2
(closed) circles are the Landau level of the maxima (minima) of Rxx. d, Normalized
conductivity amplitude xx(T)/xx(0) as a function of temperature (at B=7.6 T). The
effective mass is deduced to be ~0.07 me. e, Dingle plot of ln[(R/R0)Bsinh(] versus
1/B. The plot is used to determine carrier lifetime (τ) and mean free path (ℓ. Some
parameters
can
be
calculated
 F  k F / mcyc  3.6 105 m / s
follows,
  1.3  1012 s
   F  4.6  10 7 m
,
EF  mcyc F2  53meV
,
,
as
and
the
surface
,
mobility
  e / mcyc  13,200cm2 / V  s . The estimated Fermi level of 53 meV suggests the
surface carriers are electrons and the Fermi level is located inside the bulk band gap,
which is closed to the extracted Fermi levels from the quantum capacitance (61.5-79.4
meV, Fig. 4h inset in the main text). Note that the sample shown in Fig. S2 was grown on
a semi-insulating CdS substrate using the same growth condition as those grown on Si
substrates (for the quantum capacitance measurements). There are other excellent reports
on transport measurements which show methods of extracting important parameters1-19.
ARPES20-31 and STM32-36 data can also be used to verify these parameters.
2. Figure S3| STM Measurements
200
STM
100
measurements
150
(acknowledge
50
This paper
Analytis etc. [21]
Lang etc. [28]
100
Professor Naichang Yeh from Caltech,
50
Physics). The Dirac point is about ~ 70 
0
-50
0
(c) 0.8
0.7
dI/dV (a.u.)
250
E - ED (meV)
Figure S3| Spatially averaged density
(b)
150 measurements from
of states (dI/dV)
300
-50
20 meV below the Fermi surface, in
-0.8
0.0
0.8
K//(Å-1reasonable
)
1
2
3
0.5
Dirac
point
0.4
0.3
-400 -300 -200 -100 0
E (meV)
-100
0
0.6
4
12
-2
agreement
ns (10with
cm ) our
100 200
estimation from the SdH oscillations
(~53 meV) and the quantum capacitance measurements (61.5-79.4meV).
3
Fig. 4
3. Figure S4| A regular MOS capacitor
without Bi2Se3
Figure S4| A capacitor device without
Bi2Se3. It shows a p-type characteristic
originating from the Si substrate. At negative
it saturates to the capacitance of COX, from
Capacitance (pF)
Saturate to COX
7.2
6.8
bias,
6.4
which the effective dielectric constant can be
-20
-10
0
10
20
Gate voltage (V)
extracted (  r  7.8 ).
4. Figure S5| Theoretical calculations of quantum capacitance
2
Cq (F/cm )
0.8
0.6
0.4
0.2
-0.4
-0.2
0.0
0.2
0.4
Va (V)
For the TI surfaces, the E-k relation can be described as E (k )  vF | k | and the
g
| E | , thus the quantum capacitance of the
density of states is gTI ( E ) 
2 ( vF ) 2
topological surface states can be described as the following37,
[ q( n  p )]
g
CQ 
q
Va
2 ( vF ) 2


E
( f p  f n )

gq kT
E  qVa
ln[2(1  cosh F
)]
2
2 ( vF )
kT
2
4
Va
dE
where g=gsgv is the degeneracy factor considering both spin and energy band
degeneracies for TI surface states, fp and fn are probability of holes and electrons at given
energy followed by the Fermi distribution rule, and Va is the voltage dropped on the TI
surface. From the quantum capacitance measurements, the Fermi level is located 61.5 to
79.4 meV above the Dirac point (Fig. 4h in the main text). This corresponds to the
minimum of the quantum capacitance when Va ~ 0.07 V. The resulting quantum
capacitance of an ideal topological insulator is similar to that of graphene38. Note that the
simulation is simplified to only consider the surface contribution.
6. Figure S6| Bulk properties through transport measurements
(b)
Exp
Fitted
(c) 20
40
200
20
100
0
-20
Exp
Fitted
Surface
Impurity Band
0
-100
T = 220 K
-40
T = 20 K
-20
-4
0
4
B (T)
8
-8
-4
0
4
B (T)
8
-8
-4
8
2
Surface States
Impurity Band
Bulk
 (cm /Vs)
14
4
10
4
2
10
13
10
12
10
Surface States
Impurity Band
Bulk
2
3
10
  T -2.0
4
11
10
0
4
B (T)
(e)
15
10
-2
T = 40 K
-200
-8
n2D (cm )
0
-10
-60
(d)
Exp
Fitted
Bulk
Surface
Impurity Band
10
Gxy (mS)
60
Gxy (S)
Gxy (mS)
(a)
2
2
4
6 8
2
10 T (K)
4
6 8
2
2
100
4
6 8
2
10 T (K)
4
6 8
2
100
Fig. 2
Figure S6| Hall conductance of a ~10 nm Bi2Se3 film, from which three conducting
channels were extracted including the bulk, the impurity band and the surface states.
Magnetic field dependent conductivity Gxy at temperatures of (a) 220 K, (b) 20 K and (c)
40 K. Open circles are experimental data and solid lines are fitted results. d, 2D carrier
5
density versus temperature. The solid yellow line represents the Fermi-Dirac distribution
of the bulk carrier density equivalent to 2D. e, Mobility versus temperature for the three
channels. The dashed line represents the power law dependence of   T-2, which
suggests the mechanism of phonon scattering in these temperature ranges.
Figure S6d illustrates the temperature dependence of the two-dimensional carrier
density of three channels. The bulk carrier densities (green triangles) decrease
exponentially as carriers are frozen to the impurity band. Since the carriers in the bulk
conduction band are electrons, their temperature dependent density can be described as a


classic Fermi-Dirac distribution of nb  n0 / e Ea / k BT  1 (solid yellow line in Fig. S6d),
where n0 is the bulk carrier density of ~ 4  1015 cm-2, and Ea of 20 meV is the energy gap
between the impurity band and the bottom of the bulk conduction band. The carrier
densities of the impurity band (blue squares) and surface states (red circles) remain
approximately constant at 1.3  1013 cm-2 and 4  1011 cm-2, respectively. Figure S6e
shows the corresponding mobility at different temperatures. The mobility of bulk
electrons (green triangles) exhibits a power law   T-2 at high temperatures, consistent
a
b
impurity band reaches a constant value of ~
380 cm2/Vs.
7.
Figure
S7|
High
Capacitance (a.u.)
temperatures (T < 20 K), the mobility of the
9T
100 KHz
7T
40 KHz
5T
3T
0T
temperature
Capacitance (a.u.)
with the dominant phonon scattering. At low
10 KHz
4K Hz
1K Hz
400 Hz
100 Hz
T=60 K
capacitance measurements
4
c
8
T=60 K
12
Gate Voltage (V)
4
d
8
12
Gate Voltage (V)
100 KHz
9T
Figure S7| (a, b), Magnetic field- and
f=100
measurements
KHz.
at
(c,
T=75
d),
K.
The
same
Quantum
5T
3T
0T
Capacitance (a.u.)
and
Capacitance (a.u.)
frequency-dependent capacitance at T=60 K
4K Hz
7T
1K Hz
400 Hz
100 Hz
T=75 K
4
6
8
T=75 K
12
Gate Voltage (V)
4
8
12
Gate Voltage (V)
oscillations completely disappear at 75 K. It is speculated that at high temperatures, the
bulk carriers could be activated to reach a high enough concentration that the scattering
between bulk and surfaces is no longer negligible. As a result, the mobility of the surface
carriers may be degraded, in which case the capacitance oscillations are not as strong as
those observed at low temperatures.
8. Figure S8| Capacitance simulations for a 2-DEG.
10
2
C2DEG (F/cm )
8
6
4
2
0
-0.4
-0.2
0.0
Va (V)
0.2
0.4
Figure S8| Simulated quantum capacitance for a 2-DEG. The E-k relation of the 2-DEG
2 2
m
k
can be described by E (k ) 
and the DOS is g 2 DEG ( E )  2 N ( E ) , where N ( E ) is

2m
the number of contributing bands at a given energy39. From these equations, we can
derive the capacitance for 2-DEG, as follows,

[ p  n ]

1
1

[ q  v ( E )(

)dE ]

E  EC  qVa
EC  E  qVa
Va
Va
1  exp(
) 1  exp(
)
kT
kT


1
1
 q  v( E )
(

)dE

E

E

qV
E
 E  qVa
Va 1  exp(
C
a
) 1  exp( C
)
kT
kT
EC  E  qVa
 exp(
)

1
q
q
E  E  qVa
kT
Where
[
]

sec h 2 [ C
]
Va 1  exp( EC  E  qVa ) kT [1  exp( EC  E  qVa )]2 4kT
kT
kT
kT
C2 DEG  q
Finally we can get:
7
C2 DEG 
C2 DEG 
mq


q
4kT
2
mq
2
Ec

2
 v( E ')sec (
2
2 kT
2
 v( E )[sec (
0
EF  E  qVa
 E  E  qVa
)  sec 2 ( F
)]
2kT
2kT
E ' qVa
)dE
2kT
The simulated capacitance of 2-DEG is shown in Fig. S8. It is noted that the capacitance
from a 2-DEG essentially saturates when the magnitude of Va is sufficiently large,
exhibiting a dramatic difference compared to that from the surface states (Fig. S5).
9. Figure S9| Notes on Fe/Al contacts
In our capacitor device, we used Fe/Al as source/drain contacts. It is noted from
experiments that the capacitance measurements do not show significant difference when
Ti/Au is used for the source/drain contacts.
a
b
9T
5T
3T
0T
Capacitance (a.u.)
Capacitance (a.u.)
7T
100 KHz
40 KHz
10 KHz
4K Hz
1K Hz
400 Hz
100 Hz
T=25 K
4
8
T=25 K
12
4
Gate Voltage (V)
c
8
12
Gate Voltage (V)
d
e
9T
5T
3T
0T
100 KHz
Capacitance (a.u.)
Capacitance (a.u.)
7T
40 KHz
10 KHz
4K Hz
1K Hz
400 Hz
100 Hz
T=25 K
4
8
T=25 K
12
Gate Voltage (V)
Figure S9. Comparison of
quantum
capacitance
on
devices with Fe/Al contacts (a
and b) and Ti/Au contacts (c
and
d).
No
significant
difference
of
quantum
capacitance was observed,
suggesting the minor effect
from the Fe layer. (e) shows a
microscope image of the new
device with Ti/Au contacts.
4
8
12
Gate Voltage (V)
8
40 μm
10. Figure S10| Magnetic-field-dependent capacitance measurements
To study the magnetic-field-dependent capacitance measurements, we have carried
out additional experiments at 25 K using the original device (Fe as electrodes). The DC
voltage during the capacitance measurements was kept constant at +8 V. The excitation
ac frequency was set to be 100 KHz. Then the magnetic field was swept at a rate of 50
Oe/s, from 3 to 9 T, while the capacitance was recorded. Figure S10 shows the quantum
oscillations of capacitance as a function of magnetic field (1/B) at 25 K. It suggests the
formation of Landau levels. Combining Figures 3, 4 and S10, we can calculate the
velocity of the surface carriers using the following equation: p  mcycVF  k F . From
Figure S10, the periodicity of the SdH oscillations yields important parameters: the Fermi
vector kF  0.029 Å-1. Therefore the Fermi velocity can be calculated by VF  k F / mcyc ,
yielding a value of ~3.7×105 m/s (where mcyc =0.091 m0 at Vg=+8 V). This value is in a
good agreement with that from the transport measurements (~3.6×105 m/s).
Vg=+8 V
Capacitance (a.u.)
Figure
R10.
Capacitance
measurements with a fixed DC gate
voltage of +8 V and an ac frequency
of 100 KHz. The magnetic field was
swept from 3 to 9T at 25 K.
25 K
f=100 KHz
0.1
0.2
0.3
1/B (1/T)
9
11. Figure S11| Ambipolar field effect
It is a very challenging task to obtain ambipolar field effect through the capacitance
measurement for the 10 nm-thick sample. The primary reason is the occurrence of the
depletion capacitance when we apply negative voltages. Unlike quantum capacitance in
graphene which has no bulk contributions, topological insulators always have bulk
properties. When a negative gate voltage is applied, the depletion capacitance
dynamically changes in accordance with the magnitude of the DC bias. Figure S11 shows
preliminary results from a 6 nm-thick thin film, which suggests an ambipolar field effect.
2.60
B=0 T
T=25 K
f=100 KHz
Figure S11. Preliminary quantum
capacitance data for ~6 nm
Bi2Se3 thin film.
2
CQ (F/cm )
2.56
2.52
2.48
-10 -5
0
5 10
Vg (V)
12. More details on photolithography process
A conventional photo-resist of AZ5214 is used during the photolithography process.
For a standard photolithography recipe, first Bi2Se3 thin films were spin coated with a
thin layer of AZ5214 (4000 rpm) immediately followed by a soft bake at 100°C for 60 s.
Then we use a mask aligner to expose the wafer for 8 s and subsequently develop it in a
dilute solution of AZ400K:H2O (1:5) for 40 s. Depending on the design of photomasks,
sometimes an image reversal photolithography process is used. A critical process for the
fabrication of a capacitor is the deposition of Al2O3 gate dielectric via an atomic layer
deposition technique. A careful surface cleaning by dilute HF is employed for the
removal of oxides from sample surfaces prior to the Al2O3 deposition.
10
13. Theory for frequency responses of majority carriers
When an ac signal is injected into one system, the response of the inside carriers is
limited by the Shockley-Read-Hall (SRH) generation/recombination rate. In our device
structure, we have both high-mobility surface and low-mobility bulk channel in series.
Since our capacitance measurements are conducted under accumulation region, the main
response to external signal comes from the majority carriers (electrons) in both channels.
Therefore, their frequency responses are determined by the different motilities
transferring along each channel40-44.
We can understand this kind of majority response rate by applying the Poisson’s
equation in our n-type material:
 2 ( qV ) 
q2 N D
n
(1 
)
k BT
ND
(S1)
Where qV is the band-bending, ND is the density of ionized donors, and n is majority
electron concentration. When a small signal ac signal is added to the input dc signal, we
can assume n  n0  n and V  V0  V . By substituting these into (S1), we thus obtain
the Poisson form for the ac component:
q 2n
 2 ( qV ) 
k B T
(S2)
Combing this expression with the current continuity equation that   I  q
Poisson equation finally becomes
 2 (V  E Fn ) 
n
, the
t
 n0
q 
(V  E Fn ) 
(V  E Fn )
2
N D n
kT t
If the frequency of the ac component is, the above equation has little
frequency/time-dependence only if
 n0
n
q
1
 1 n0
  
  ( )  0  (
) 
2
kT
N D n or
 D ND
qN D
ND
(S3)
Here 1/D is the transfer rate of the electron in the system. In this scenario, the
electrons can follow up, or equivalently speaking, they cannot be affected by the external
stimulations only when their transfer rates are much faster than the applied signal
frequency. Therefore, since 1/D is proportional to the mobility , the electron response in
those low-mobility bulk mediums can thus be filtered out under high frequency input
conditions, and leaving only the component from high-conduction surface channel.
11
14. Simplified band diagrams for the MOS capacitor and notes on SdH
oscillations
b. Vg < 0
a. Vg > 0


n-type
EF
EC
n-type
EF
EC
Ei
Ei
EV
EV
Figure S14. The MOS capacitor under positive (a) and negative biases (b).
When the Fermi energy moves further away from the Dirac point, the SdH
oscillations become more rapid with decreasing periods. Our experiments, however,
demonstrate an opposite trend. The results suggest that the gate voltage may not have a
linear relationship with the Fermi energy although CQ exhibits a quasi-linear behavior in
the range of 6-12V in Figure 3a. The non-linearity could be in part attributed to the bulk
states, where a certain amount of gate voltage could be shared by the bulk. As the Fermi
level moves upwards, the surface becomes more conductive and the bulk may thus share
a larger portion of the gate voltage. Therefore it is possible that the SdH oscillations
exhibit an opposite tread in such scenarios.
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