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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: (Coloronline) (a)Basic structureunitof

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-

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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∗.

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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-

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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).

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[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.