Page 1
arXiv:1105.5483v1 [cond-mat.mtrl-sci] 27 May 2011
Observation of Dirac Holes and Electrons in a Topological Insulator
A. A. Taskin, Zhi Ren, Satoshi Sasaki, Kouji Segawa, and Yoichi Ando
Institute of Scientific and Industrial Research, Osaka University, Ibaraki, Osaka 567-0047, Japan
We show that in the new topological-insulator compound Bi1.5Sb0.5Te1.7Se1.3 one can achieve a
surfaced-dominated transport where the surface channel contributes up to 70% of the total conduc-
tance. Furthermore, it was found that in this material the transport properties sharply reflect the
time dependence of the surface chemical potential, presenting a sign change in the Hall coefficient
with time. We demonstrate that such an evolution makes us observe both Dirac holes and electrons
on the surface, which allows us to reconstruct the surface band dispersion across the Dirac point.
PACS numbers: 73.25.+i, 71.18.+y, 73.20.At, 72.20.My
The three-dimensional (3D) topological insulator (TI)
hosts a metallic surface state that emerges due to a non-
trivial Z2topology of the bulk valence band [1, 2]. This
peculiar surface state offers a new playground for study-
ing the physics of quasiparticles with unusual dispersions,
such as Dirac or Majorana fermions [3, 4]. However, most
of the known TI materials are poor insulators in their
bulk, hindering transport studies of the expected novel
surface properties [3, 4]. Last year, we discovered that
the TI material Bi2Te2Se (BTS) presents a high resis-
tivity exceeding 1 Ωcm [5], which made it possible to
clarify both the surface and bulk transport channels in
detail. Also, we found that in our BTS sample the surface
channel accounts for ∼6% of the total conductance. For
this BTS compounds, Xiong et al. recently reported a
higher resistivity in the range of 5–6 Ωcm, together with
pronounced surface quantum oscillations which possibly
signify fractional-filling states [6]. Since the source of the
residual bulk carriers in BTS is the acceptor states [5],
reducing the number of anti-site defects workingas accep-
tors in this promising material is an important challenge
for the advancement of the TI research.
In this work, we tried to optimize the composition of
BTS by reducing the Te/Se ratio and introducing some
Sb into Bi positions [7], while keeping the ordering of
the chalcogen layers as in BTS (Fig.
X-ray powder diffraction patterns shown in Fig. 1(b)
demonstrate that the chalcogen ordering is still present
in Bi1.5Sb0.5Te1.7Se1.3(BSTS), and we focus on this com-
pound in this Letter. We found that in BSTS one can
achieve an even larger surface contribution (up to 70%)
than in BTS. We also found that the surface state of
BSTS changes with time, and, intriguingly, we observed
quantum oscillations from Dirac holes, the negative en-
ergy state of the Dirac fermions, as well as those from
Dirac electrons in the same sample at different time
points. We show that this time evolution can be used to
reconstruct the surface band structure across the Dirac
point, providing a unique way to determine the disper-
sion relation of the surface state.
1(a), [8]). The
Single crystals of BSTS were grown by melting high
purity (6N) elements of Bi, Sb, Te, and Se with a mo-
FIG.
Bi1.5Sb0.5Te1.7Se1.3 (BSTS). (b) Comparison of the X-ray
powder diffraction patterns of BSTS, Bi2Te2Se, and Bi2Te3.
Dashed lines indicate the positions of the peaks characteristic
of the ordering of Se and Te (Te/Se) layers. (c) Temperature
dependence of ρxx measured repeatedly in time in a cleaved
30-µm-thick BSTS sample. (d) Temperature dependence of
the low-field RH, presenting a sign change with time. Dashed
line represents the Arrhenius-law fitting to the data above 150
K, which is also shown in the inset.
1:(Color online) (a) Basic structureunit of
lar ratio of 1.5:0.5:1.7:1.3 at 850◦C for 48 h in evacuated
quartz tubes, followed by cooling to room temperature
over one week.For transport measurements, crystals
were cut along the principal axes, and cleaved just before
the measurements. Ohmic contacts were prepared by us-
ing room-temperature cured silver paste. The resistivity
ρxxand the Hall resistivity ρyxwere measured simulta-
neously by a standard six-probe method [9] by sweeping
the magnetic field B between ±14 T at 0.3 T/min while
stabilizing the temperature T to within ±5 mK.
Freshly cleaved thin samples were used for studying
the surface transport in BSTS. As shown in Fig. 1(c),
ρxx in BSTS sharply increases upon lowering tempera-
ture from 300 K, which is characteristic of an insulator,
but it saturates below ∼100 K due to the metallic sur-
Page 2
2
??
??
?
??
?
???
?
?
?
??
?
??
?
???
?
?
?
??
?
?
?
??
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
FIG. 2: (Color online) (a,b) dρxx/dB vs 1/B⊥ measured in
tilted magnetic fields in the p state (5 h) and the n state (730
h). All curves are shifted for clarity. Insets show the Fourier
transforms of the data at θ=0◦. (c,d) Temperature depen-
dences of SdH amplitudes for θ=0◦shown in (a) and (b) and
their theoretical fittings. Insets show ∆Rxx vs. 1/B after
subtracting a smooth background. (e) Landau-level fan dia-
gram obtained from the oscillations in ∆Rxx measured at T
= 1.5 K and θ = 0◦; minima and maxima in ∆Rxxcorrespond
to nosc and nosc+1
2, respectively.
face transport as well as the bulk impurity-band (IB)
transport [5]. This behavior is essentially the same as in
BTS. What is peculiar in BSTS is that ρxx(T) increases
slowly with time and, furthermore, the low-temperature
Hall coefficient RH changes sign with time in thin sam-
ples. As an example, Figs. 1(c) and (d) show ρxx(T)
and RH(T) data of a 30-µm-thick BSTS sample, mea-
sured repeatedly in about one month. In contrast to the
relatively small change in ρxx [Fig. 1(c)], RH at low-
temperature exhibits rather drastic evolution [Fig. 1(d)]
from positive to negative, whereas RH at high temper-
ature is essentially unchanged with time. This suggests
that the source of the time dependence resides in the
surface channel. In passing, RH(T) above 150 K is pos-
itive and demonstrates an activated behavior [shown by
the dashed line in Fig. 1(d)], signifying the thermal ex-
citation of holes into the valence band with an effective
activation energy of about 60 meV. This is similar to
what we observed in BTS [5].
To understand the nature of the time evolution, the
Shubnikov-de Haas (SdH) oscillations were measured in
the aforementioned 30-µm-thick sample twice, 5 h after
cleavage (called p state) and 725 h later (called n state),
between which RH changed sign (the sample was kept
at 300 K in air). In BSTS, the oscillations do not fade
out even after long exposure to ambient atmosphere, as
opposed to other TI materials like Sb-doped Bi2Se3[10].
The SdH oscillations were clearly observed in ρxx(B) in
our samples, but they were not clear in ρyx(B), so the
following SdH analysis is restricted to ρxx(B). Figures
2(a) and (b) show dρxx/dB for both the p and n states,
plotted as a function of 1/B⊥(≡ 1/B cosθ), where θ is
the angle between B and the C3 axis. Several equidis-
tant maxima and minima are easily recognized, and im-
portantly, the positions of maxima and minima depend
solely on B⊥, which signifies a 2D character of the ob-
served oscillations. Insets show the Fourier transform
of the oscillations taken at θ=0◦. Two frequencies are
clearly seen in the p state, but the second one (60 T) is
probably a harmonic of the primary frequency F = 30 T.
On the other hand, only the primary F = 50 T is seen in
the n state. The averaged Fermi wave number kF is ob-
tained by using the Onsager relation F = (?c/2πe)πk2
resulting in kF = 3.0×106cm−1and 3.9×106cm−1for
the p and n states, respectively. This corresponds to the
surface hole (electron) concentration of 7.2×1011cm−2
(1.2×1012cm−2) for a spin-filtered surface state. Fitting
the standard Lifshitz-Kosevich theory [11] to the tem-
perature dependences of the SdH amplitudes [Figs. 2(c)
and (d)] gives the cyclotron mass mc of (0.10±0.01)me
for holes and (0.075±0.003)mefor electrons (me is the
free electron mass). Also, from the B-dependence of the
SdH amplitudes [insets of Figs. 2(c) and (d)] one can
obtain the scattering time τ of 5.8×10−14s (4.2×10−14
s) for holes (electrons) through the Dingle analysis.
From the measured values of kF and mc, one obtains
the effective Fermi velocity v∗
m/s and 6.0×105m/s for holes and electrons, respec-
tively. Now we discuss that this difference between the
two v∗
Fbears crucial information regarding the Dirac
cone: If the surface state consists of ideal Dirac fermions,
the Fermi velocity is independent of k and is particle-
hole symmetric. However, the energy dispersions of the
surface states in Bi-based TI materials generally deviate
from the ideal Dirac cone, and it can be described as [12]
F,
F(≡ ?kF/mc) of 3.5×105
E(k) = EDP+ vF?k +
?2
2m∗k2, (1)
where EDP is the energy at the Dirac point (DP), vF is
the Fermi velocity at the DP, and m∗is an effective mass
that accounts for the curvature in E(k). The effective
Fermi velocity v∗
Freflects the local curvature in E(k) and
can be expressed as v∗
The p and n states correspond to the situations where the
Fermi energy EF is below or above the DP, respectively,
and the time evolution of RH is a manifestation of the
time-dependent change of the surface chemical potential.
By using the kF and v∗
Fvalues obtained for the p and
F(k) = (∂E/∂k)/? = vF+ ?k/m∗.
Page 3
3
n states, we can solve simultaneous equations to obtain
vF = 4.6×105m/s and m∗= 0.32me. This mean that
the time evolution of the transport properties allowed us
to do a “spectroscopy” of the surface state to determine
its dispersion, from which we can further estimate the
position of EF to lie 80 meV below (140 meV above) the
DP in the p (n) state. Finally, the mean free path ℓs
= v∗
Fτ is about 20 nm (40 nm) and the surface mobility
µSdH
s
= (eℓs)/(?kF) is about 1.0×103cm2/Vs (9.8×102
cm2/Vs) in the p (n) state.
To infer the Dirac nature of the surface state from the
SdH oscillations [13, 14], Fig. 2(e) shows the Landau-
level (LL) fan diagram, which presents the values of 1/B
at the nosc-th minima in the ρxxoscillations as a function
of nosc[14]. In the case of ideal Dirac particles, the dia-
gram for holes (electrons) intersects the nosc-axis at −0.5
(0.5), reflecting the π Berry phase [13–15]. However, in
recent SdH studies of TIs [5, 6, 10, 16–18] exact π Berry
phase has rarely been reported and this deviation has
been a puzzle. The Zeeman coupling of the spins to the
magnetic field can be a source of this discrepancy [10],
and in addition, the deviation of E(k) from the ideal lin-
ear dispersion also causes the Berry phase to shift from
π [14]. We therefore followed the scheme of Ref. [14]
to see if the LL fan diagram obtained for BSTS can be
understood by considering these additional factors: The
solid lines in Fig. 2(e) show the theoretical LLs for the
non-ideal Dirac fermions [14] with the band parameters
already obtained (vF = 4.6×105m/s and m∗= 0.32me)
and a surface g-factor gs= 20 (which is the sole fitting
parameter). Those lines agree reasonably well with the
experimental data for both the p and n states, supporting
not only the Dirac nature of the observed surface state
but also the conjecture that the holes and electrons reside
on the same Dirac cone.
Now we discuss the mechanism for the time evolution
of the transport properties in BSTS. At low tempera-
ture, the Fermi level is pinned to the IB in the bulk of
the material [5], so the observed development of trans-
port properties most likely comes from a change in the
surface as schematically shown in Figs. 3(a) and (b). To
understand the p state where holes dominate the Hall re-
sponse, one must assume that an upward band-bending
occurs just after cleavage, putting the Fermi level below
the DP and simultaneously creating a hole accumulation
layer (AL) near the surface [Fig. 3(a)]. The air expo-
sure apparently causes n-type doping on the surface as
was reported for Bi2Se3[19], leading to a downward band
bending [Fig. 3(b)].
In the above picture, there must be three transport
channels in the p state: surface Dirac holes, bulk IB,
and the surface AL due to the band bending. Hence,
one may wonder if the SdH oscillations observed in the
p state might actually be due to the AL, rather than the
Dirac holes. Fortunately, one can see that this is not the
case, by analyzing the non-linear B dependence of ρyx.
FIG. 3: (Color online) (a,b) Schematic picture of the bulk and
surface states and the surface band bending for the p and n
states, respectively. EC (EV) is the energy of the conduction-
band bottom (valence-band top), and the shaded band depicts
the impurity band. (c) ρyx(B) data measured at 1.5 K in the p
and n states. The dashed (solid) line is the fitting of ρyx(B) in
the n-state (p-state) using the two-band (three-band) model.
(d) Decomposition of the three-band ρyx(B) fitting in the p
state (see text).
In the following, we discuss the analyses of the ρyx(B)
data, starting from the simpler case of the n state.
As in BTS [5], the ρyx(B) curves in the n state of BSTS
can be well fitted with a simple two-band model described
in Ref. [5]. The dashed line in Fig. 3(c) is a result of such
fitting to the 1.5-K data, where the fitting parameters are
the bulk electron density nb= 2.3×1016cm−3, the bulk
mobility µb= 190 cm2/Vs, and the surface mobility µs
= 1250 cm2/Vs (the surface electron density was fixed at
1.2×1012cm−2from the SdH data). These parameters
give the residual bulk conductivity σbof 0.73 Ω−1cm−1,
and the surface contribution to the total conductance
can be estimated as Gs/(Gs+ σbt) ≈ 0.1, where Gs ≈
2.4×10−4Ω−1is the sheet conductance of the surface
and t = 30 µm is the thickness.
In the p state, the AL must also be taken into account,
so we tried a three-band analysis in which we assumed
that the bulk carriers are the same as in the n state.
The solid line in Fig. 3(c) shows a result of the fitting
to the 1.5-K data, where the fitting parameters are the
AL mobility µs′ = 770 cm2/Vs, the AL sheet conduc-
tance Gs′ = 2.2×10−3Ω−1, and the Dirac-hole mobility
µs= 1170 cm2/Vs (the Dirac hole density was fixed at
7.2×1011cm−2from the SdH data). To understand the
relative importance of the three channels, it is instruc-
tive to examine the individual contributions to the total
ρyx: As shown in Fig. 3(d), the putative ρyx(B) due
solely to the surface Dirac holes (curve 1) is strongly
modified when the residual bulk contribution is added
(curve 1+2), but it is still very different from the mea-
Page 4
4
???????????
?
?
?
??
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
FIG. 4: (Color online) Temperature dependences of ρxx in
0 T measured in a BSTS sample before and after reducing
its thickness. Upper inset shows the plot of Rxx(T) for the
same set of data. Lower inset shows the high-temperature
activation behavior in RH(T) for the 8-µm sample, which is
compared with the similar behavior observed in the 30-µm
sample [Fig. 1(d) inset] shown as a dashed line.
sured ρyx(B); only when the third contribution of the AL
is added (curve 1+2+3), the ρyx(B) behavior is satisfac-
torily reproduced.
Based on the above analysis, one can see that it is
impossible to interpret the SdH oscillations in the p state
to originate from the AL: If the SdH oscillations were
due to the AL, the third transport channel must be the
surface Dirac holes; however, the sheet carrier density ns′
of the third channel is estimated as ns′ = Gs′/(eµs′) ≈
1.8×1013cm−2, which is too large for Dirac holes for
which kF ? 5×106cm−1[20] and hence ns′ must be ?
2×1012cm−2. Therefore, one can safely conclude that
the SdH oscillations are due to the Dirac holes.
Lastly, we show that the surface conductance in BSTS
can be reasonably estimated by simply changing the
thickness. Figure 4 shows the temperature dependences
of ρxx of a different sample, measured first in 67-µm
thickness, and later cleaved down to 8-µm thick [21].
The overall behavior of ρxx(T) is similar between the
two curves, but a striking difference lies in their low-
temperature saturation values ρsat
(for 8 µm): it is lower in the thinner sample, imply-
ing a larger relative surface contribution. Note that the
resistance Rxx duly increases upon reducing the thick-
ness, as shown in the upper inset of Fig. 4. Instruc-
tively, the difference in ρxx disappears at high temper-
ature where the bulk conduction dominates. From the
observed difference in ρsat
i
one can estimate the surface
and bulk contributions to the total conductivity by using
ρsat
i
= [(Gs/ti) + σb]−1. We obtain Gs≈ 1.8×10−4Ω−1
and σb≈ 0.1 Ω−1cm−1, and this σbis much smaller than
that in the 30-µm-thick sample. This is probably because
the number of acceptors is smaller in this second sample
[22], as can be inferred in the high-temperature RH be-
1
(for 67 µm) and ρsat
2
havior [lower inset of Fig. 4]. The obtained values of Gs
and σballows us to calculate the fraction of the surface
contribution to the total conductance, Gs/(Gs+σbti), for
the 67- and 8-µm thick samples to be 0.2 and 0.7, respec-
tively. Therefore, when the thickness of a BSTS sample
is reduced to ? 10 µm, one can achieve a bulk TI crystal
where the topological surface transport is dominant over
the bulk transport.
In summary, we show that one can achieve a
surface-dominated transport in the new TI compound
Bi1.5Sb0.5Te1.7Se1.3. The surface band bending and its
time dependence makes it possible to observe the SdH os-
cillations of both Dirac holes and electrons, with which
we could determine the surface band dispersion across
the Dirac point. In addition, by analyzing the non-linear
ρyx(B) curves, we could identify the role of the surface ac-
cumulation layer in the transport properties. Obviously,
this material offers a well-characterized playground for
studying the topological surface state.
This work was supported by JSPS (NEXT Program
and KAKENHI 19674002), MEXT (Innovative Area
“Topological Quantum Phenomena” KAKENHI), and
AFOSR (AOARD 10-4103).
[1] L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007).
[2] J. Moore, L. Balents, Phys. Rev. B 75, 121306(R) (2007).
[3] M.Z. Hasan, C. Kane, Rev. Mod. Phys. 82, 3045 (2010).
[4] X.L. Qi and S.C. Zhang, arXiv:1008.2026v1.
[5] Z. Ren et al. Phys. Rev. B 82, 241306(R) (2010).
[6] J. Xiong et al., arXiv:1101.1315.
[7] Since the anti-site defects occur between Bi and Te, re-
ducing the Te/Se ratio would suppress the anti-site; at
the same time, Se in the outermost layer is prone to va-
cancy creations, which is suppressed by Sb doping.
[8] S. Nakajima, J. Phys. Chem. Solids 24, 479 (1963).
[9] The sample is suspended on gold wires in order to insure
that both surfaces are in the same condition.
[10] J.G. Analytis et al., Nat. Phys. 10, 960-964 (2010).
[11] D. Shoenberg, Magnetic Oscillations in Metals (Cam-
bridge University Press, Cambridge, 1984).
[12] D. Culcer et al., Phys. Rev. B 82, 155457 (2010).
[13] G. Mikitik, Yu. Sharlai, Phys. Rev. Lett. 82, 2147 (1999).
[14] A. A. Taskin and Y. Ando, arXiv:1103.3096.
[15] Y. Zhang et al., Nature (London) 438, 201 (2005).
[16] D.X. Qu et al., Science 329, 821 (2010).
[17] B. Sac´ ep´ e et al., arXiv:1101.2352.
[18] C. Br¨ une et al., Phys. Rev. Lett. 106, 126803 (2011).
[19] J.G. Analytis et al., Phys. Rev. B 81, 205407 (2010).
[20] S.-Y. Xu et al., arXiv:1007.5111; D. Hsieh et al., Nature
(London) 460, 1101 (2009).
[21] The cleaved 8-µm-thick sample showed a gradual increase
in ρxx similar to the 30-µm-thick sample, so we only
used relaxed samples (in which the AL is absent) for the
thickness-dependence study.
[22] Unfortunately the SdH oscillations were not observed in
this sample, probably due to a lower surface mobility.
Download full-text