Quantum Oscillations and Hall Anomaly of Surface States in the Topological Insulator Bi2Te3

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Science  13 Aug 2010:
Vol. 329, Issue 5993, pp. 821-824
DOI: 10.1126/science.1189792

Carrier Mobility in Topological Insulators

In addition to an energy gap, which is a characteristic of all band insulators, the electronic structure of the recently discovered three-dimensional topological insulators Bi2Te3 and Bi2Se3 contains a surface state with a Dirac-like dispersion. This state is predicted to be associated with high carrier mobility. However, the transport properties of the surface state are obscured by the bulk material and challenging to measure. Qu et al. (p. 821, published online 29 July) produced crystals of Bi2Te3 with the Fermi energy lying in the bulk gap and detected quantum oscillations whose magnetic field dependence reveals that they come from a two-dimensional Fermi surface. An anomaly in the Hall conductance originating from the surface state was also observed. The two measurements independently yield mutually consistent high electron mobilities.


Topological insulators are insulating materials that display massless, Dirac-like surface states in which the electrons have only one spin degree of freedom on each surface. These states have been imaged by photoemission, but little information on their transport parameters, for example, mobility, is available. We report the observation of Shubnikov–de Haas oscillations arising from the surface states in nonmetallic crystals of Bi2Te3. In addition, we uncovered a Hall anomaly in weak fields, which enables the surface current to be seen directly. Both experiments yield a surface mobility (9000 to 10,000 centimeter2 per volt-second) that is substantially higher than in the bulk. The Fermi velocity of 4 × 105 meters per second obtained from these transport experiments agrees with angle-resolved photoemission experiments.

Recently, the existence of a new class of bulk insulators called topological insulators has been predicted (16). In a topological insulator, the bulk energy gap is traversed by surface states in which the spin of the electron is locked perpendicular to its momentum by strong spin-orbit interaction. On each surface, the electrons have only one spin degree of freedom (fixed helicity). The spin-resolved nature of the surface states has been confirmed in angle-resolved photoemission spectroscopy (ARPES) experiments on BiSb (7), Sb (8), Bi2Se3 (9), and Bi2Te3 (10). The spin locking in BiSb has also been studied by scanning tunneling microscopy (11). However, it has been a challenge to resolve the conductance of the surface states from the dominant bulk contribution (12, 13). The lack of transport information, especially the mobility, is a serious hindrance. Moreover, detection of the surface currents is a crucial first step in the investigation of phenomena, such as the Majorana fermion (5) and unconventional electrodynamics (6), in topological insulators.

We report dual evidence for surface state conduction in Bi2Te3 from Shubnikov–de Haas (SdH) oscillations and from the weak-field Hall effect. As in the case of the selenide Bi2Se3 (12), as-grown crystals of Bi2Te3 usually display a metallic resistivity ρ versus temperature T (in Bi2Te3, the Fermi energy EF lies in the valence band VB). By selective cleaving of many crystals from a boule of Bi2Te3 grown with a weak compositional gradient (14), we have obtained nonmetallic crystals (Fig. 1A). In the nonmetallic samples Q1, Q2, and Q3, ρ rises when T is lowered below 150 K and saturates to values 4 to 5 mohm cm at 4 K or ~50 times larger than the value in the metallic sample N1. The surface state dispersion obtained by ARPES (10) is sketched in Fig. 1B, with EF in our samples indicated. Even in the most resistive sample Q3, the bulk conductance is ~300 times larger than the surface term (see below). Nonetheless, the latter may be resolved by detecting the SdH oscillations in the resistivity and the Hall resistivity ρyx at low T. Quantum oscillations are well resolved in the raw trace of the Hall conductivity σxy = ρyx/(ρxx2 + ρyx2) measured at T = 0.3 K (Fig. 1, C and D). In metals, SdH oscillations correspond to successive emptying of Landau levels (LL) as the magnetic field H is increased. The LL index n is related to the extremal cross section SF of the Fermi surface (FS) by Embedded Image(1)where γ = 0 or ½, e is the electron charge, h is Planck’s constant (ħ = h/2π), and B is the magnetic flux density.

Fig. 1

(A) The resistivity profiles ρ versus temperature T of samples Q1, Q2, Q3, and N1. (B) Sketch of the surface state dispersion near the Γ point [traced from (10)]. The Fermi energies of samples Q1, Q2, Q3, and N1 are indicated by short lines (CB, conduction band). Curves of the Hall conductivity σxy versus the magnetic field H in sample Q1 (C) and in samples Q2 and Q3 (D) showing well-resolved SdH oscillations. μ0 is the permeability.

For a two-dimensional (2D) FS, the peak positions depend only on the field component H = Hcosθ along the axis c normal to the cleavage plane (θ is the tilt angle between H and c). To test this hypothesis, we tracked how the SdH extrema shift with θ, in fields up to 30 T [supporting online material (SOM) section S2]. We compare the derivative dρxx/dH in nonmetallic Q1 with that in the metallic sample N1 (Fig. 2, A and B, respectively). In sample Q1, the period of the oscillations depends only on H: The magnetic field corresponding to the n = 3 minimum varies with θ as 1/cosθ up to 65° (Fig. 1C). For 65° < θ < 90°, the oscillations are not resolved. By contrast, the oscillations in N1 survive up to θ = 90° (Fig. 1B). As θ → 90°, SF in N1 deviates strongly from the 1/cosθ trend, to saturate at a value ~1.5× larger than that at θ = 0. From the comparison, we conclude that the quantum oscillations in the nonmetallic sample arise from a 2D FS (SOM section S2).

Fig. 2

The resistivity derivative dρxx/dH versus 1/H = 1/(Hcosθ) in (A) sample Q1 and (B) sample N1 at selected tilt angles θ. In (A), the minima lie on the vertical dashed lines consistent with a 2D FS, whereas in (B), the minima shift systematically with θ. (C) The field position of the n = 3 LL peak for sample Q1 (red circles) varies with θ as 1/cosθ (black curve), consistent with a 2D FS. (D) The period SF for sample N1 (blue circles) increases by 35% as θ → 90°, deviating strongly from 1/cosθ (black curve), consistent with a 3D FS.

To extract more specific information on the surface states, we next analyzed how the SdH amplitudes vary with T in samples Q2 and Q3 (measured with H||c). As shown in Fig. 3A, the oscillation amplitudes in dρxx/dH decrease rapidly as T is raised from 0.3 to 20 K. Although measurements were not carried out at intermediate θ in Q2 and Q3, we verified that the SdH peaks were absent at θ = 90°. In the index plot (Fig. 3B), we confirmed that the (inverse) peak fields 1/B fall on a straight line versus the integers n (Eq. 1). For Q2 and Q3, the slopes yield SF = 33.3 and 28.6 T, with Fermi wave vector kF = 0.032 and 0.030 Å−1, respectively (Table 1). In Q1, the shallower slope yields kF = 0.036 Å−1. Extrapolation of the high-field SdH peaks in Q1 is consistent with 0 < γ < ½ (figs. S3 and S4). To find the corresponding EF, we next determine the Fermi velocity vF.

Fig. 3

(A) Derivative dρyx/dH versus 1/H in sample Q3 measured at temperatures T between 0.3 and 20 K. (B) ΔG (conductance obtained after subtracting a smooth background based on curves measured above 20 K) for sample Q2 at selected T over the same interval. (C) LL index plot 1/B versus n for samples Q1, Q2, and Q3. Our results are consistent with 0 < γ < ½ (fig. S3). Values of SF and kF derived are reported in Table 1. (D) T dependence of the normalized conductivity amplitude Δσxx(T)/Δσxx(0) at 0.3 K in samples Q2 (with H = 12 T) and Q3 (7.8 T). The curves are best fits to λ(T)/sinhλ(T) (Eq. 2). Dingle plots of log[(ΔR/R0)Bsinhλ] versus 1/H, used to determine τ and vFτ for (E) sample Q2 and (F) sample Q3.

Table 1

Parameters in samples Q1, Q2, Q3, and N1. SF and kF are determined from the SdH period. EF and vF are obtained from the T dependence of the amplitude Δσxx(T)/Δσxx(0). ℓ is found from the Dingle factor eD. The length (L), width (W), and thickness (t) are in μm. The uncertainty in vF is ±10%. Dash entries indicate quantities not measured.

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The T dependence of the amplitude Δσxx of conductivity oscillations is given by Δσxx(T) = Δσxx(0)λ(T)/sinhλ(T). For 2D massless Dirac states, the thermal factor λ(T) is given by (15, 16)Embedded Image(2)where kB is Boltzmann’s constant and mcyc = EF/vF2 is the cyclotron mass. In low H, the degree of orbit bending is measured by the Hall mobility μ, given by μ = eτtr/mcyc = eℓ/ħkF, where τtr and ℓ are the transport lifetime and mean free path, respectively. We may find mcyc by fitting the T dependence of the conductivity amplitudes to Eq. 2 (Fig. 3D). Because kF is known, we calculate vF = 3.7 and 4.2 × 105 m s–1 for Q2 and Q3, respectively [from ARPES (10), vF ~ 4 × 105 m s–1]. Lastly, this yields EF = 94, 84, and 78 meV above the Dirac point in samples Q1, Q2, and Q3, respectively (Fig. 1B).

The lifetime τ of the surface states can be found by estimating the Dingle factor eD, where D = 2π2ΓEF/(ħeBvF2), with Γ = 1/τ the scattering rate. Because Embedded Image (where R is resistance), we may find Γ from the slope in the semilog plot of (ΔR/R0)Bsinhλ(T) versus 1/B (Fig. 3, E and F). We find that vFτ = 217 and 219 nm in Q2 and Q3, respectively (these values are lower bounds for the mean free path ℓ = vFτtr because τtr > τ in principle). The metallicity parameter is then kFℓ > 69 and 66 in samples Q2 and Q3, respectively.

The conductance tensor Gij(B) of the 2D Dirac gas may be calculated in the semiclassical Boltzmann approximation (14). Defining G(B) ≡ Gxx(B), we may express the zero-B conductance as G(0) = (e2/h)kFℓ, whereas the Hall conductance isEmbedded Image(3)(including both spin degrees). With kFℓ ~70, we calculate G(0) ≈ 2.4 × 10−3 ohm−1, compared with the observed total resistance R ≈ 1.4 ohm at 0.3 K in sample Q1. Hence, the 2D states account for ~0.3% of the total conductance observed at 0.3 K. Interestingly, the large value of kFℓ implies a large Hall mobility (μ = 10,200 cm2 V–1 s–1). In weak H, the surface Hall current may dominate the low-mobility bulk Hall current.

We next report the detection of this surface Hall current as a distinct anomaly. In Fig. 4A, we plot the observed Hall conductivity σxy in sample Q3. In addition to the SdH oscillations in high fields (>5 T), we observe a prominent “dispersive” anomaly in weak H corresponding to a high-mobility contribution that is n-type. We express the observed Hall conductivity as the sumEmbedded Image(4)where σbxy > 0 is the bulk, p-type Hall conductivity and Gxy < 0 the surface Hall conductance given in Eq. 3, with t the crystal thickness. As shown in Fig. 1C and D, the sign of ρyx in sample Q1, fixed by the bulk carriers, is n-type, whereas Q2 and Q3 are p-type (fig. S4 and SOM section S4). By using Eq. 3 and a Drude-like expression for σbxy, we have achieved a close fit to the nonmonotonic curve in Fig. 4A (fig. S5).

On subtracting σbxy, we isolate the surface term Gxy, which shows a large dispersive anomaly centered at H = 0, with quantum oscillations in the wings (Fig. 4B). The solid curve is the fit to Eq. 3 (the semiclassical expression does not describe the quantum oscillations). By Eq. 3, Gxy attains the maximum value Gmaxxy = (e2/2h)kFℓ at the peak field Bp = 1/μ (arrows in Fig. 4B). From the fit, we find μ ~ 9000 cm2 V–1 s–1, kFℓ = 94, kF = 0.04 Å−1, and ℓ = 235 nm. Given the non-uniformity of the cross-section and the uncertainties in measuring t and the voltage lead spacing L (both ± 10%), these are in good agreement with the numbers inferred from the SdH period and the Dingle analysis (17).

Fig. 4

(A) The Hall conductivity σxy versus H in Q3 measured at 0.3 K, showing quantum oscillations above 5 T and a dispersive anomaly in weak H. (B) Surface Hall conductance Gxy obtained by subtracting the bulk contribution σbxy (Eq. 4). The solid curve is the fit to Gxy in Eq. 3 (details in SOM section S5). The peak field Bp (arrows) gives 1/μ, whereas the maximum value yields (e2/2h)kFℓ. (C) The linear MR measured with H||c at 0.3 and 20 K (ρ ≡ ρxx). (D) Rounding of the nonanalytic cusp at H = 0 in increasing T.

The quantitative agreement with 2D states persuades us that the weak-field Hall anomaly originates from Dirac surface states. Because of the large difference in mobilities and the difference in carrier sign, we are able to resolve directly the surface Hall current as the dispersive anomaly superposed on the bulk Hall current. The curve in Fig. 4B is a snapshot of the Dirac surface current from which μ and kFℓ may be estimated by inspection.

By how much is μ enhanced over the bulk? Unfortunately, in the nonmetallic crystals, it is not straightforward to pin down the hole density p and mobility μb of the bulk states. The bulk Hall signal changes sign abruptly as EF approaches the Dirac point. As shown in fig. S1, samples Q1 and N1 are n-type, whereas Q2 and Q3 are p-type. This suggests competition and compensation between electron and hole bulk bands. A more interesting complication is the unusual magnetoresistance (MR). In samples Q1, Q2, and Q3, the transverse MR measured with H||c displays a highly unusual linear increase versus H [by contrast, metallic crystals do not show the linear MR (fig. S2)]. As shown in Fig. 4C (for sample Q2), the nonanalytic variation Embedded Image extends over 3.5 decades in H (from 30 G to 14 T) at 0.3 K. The expanded view in Fig. 4D shows that, as T is raised to 20 K, the low-H region is rounded. The large linear MR is predominantly caused by coupling of the spin to H. When H is aligned with the current, the longitudinal MR is 50% as large as the transverse MR (fig. S7). This unusual MR, which may arise from a spin-mediated decrease in the bulk carrier density p with H, implies that the bulk states cannot be regarded as conventional impurity bands. However, if we use the bulk value obtained from the fit in Fig. 4Bb ~ 860 cm2 V–1 s–1), we find that the surface mobility is 12 times larger than the bulk value.

By using both the SdH and weak-field Hall anomaly, we have resolved a surface current that is n-type with kF ~ 0.035 Å−1 and a velocity vF ~ 4 × 105 m s–1. The observed mobility and kFℓ are enhanced substantially over the bulk values. These features are consistent with protected surface Dirac states. With the surface current detected, research on topological insulators may now be expanded to include transport experiments.

Supporting Online Material

Materials and Methods

SOM sections S1 to S6

Figs. S1 to S9

References and Notes

  1. Materials and methods are available as supporting material on Science Online.
  2. We may exclude the interpretation of Gxy as a bulk term as follows. If we write Gxy/t as n′eμ2B/[1 + (μB)2] with μ fixed at 10,000 cm2 V–1 s–1, we get a bulk density n′ = 1.4 × 1014 cm−3. This tiny 3D FS pocket would have a Fermi cross section SF′ that is far too small, by a factor of 360, to match the SdH period (the wave vector kF = 1.6 × 10−3 Å−1 is 19 times too small).
  3. The research is supported by NSF under Materials Research Science and Engineering Centers grant DMR-0819860. We acknowledge valuable comments by J. G. Checkelsky, M. Z. Hasan, A. Yazdani, A. Bernevig, and F. D. M. Haldane. High-field measurements were carried out at the National High Magnetic Field Laboratory, Tallahassee, which is supported by NSF cooperative agreement no. DMR-0084173, by the state of Florida, and by the U.S. Department of Energy.
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