Introduction

Quantum transport in Dirac electron systems has been attracting much attention for the half-integer quantum Hall effect (QHE), as typically observed in graphene1,2. A single Dirac fermion under a magnetic field is known to show the quantized Hall effect with the Hall conductance σxy=(n+1/2)e2/h with n being an integer, e the elemental charge and h the Planck constant. This 1/2 is the characteristic of the Dirac fermion compared with the usual massive electrons. In graphene with such a Dirac electron, however, there is fourfold degeneracy due to the spin and valley degrees of freedom, and hence the quantized Hall conductance shows up experimentally as σxy=4(n+1/2)e2/h. The recently discovered topological insulator (TI) possesses metallic Dirac states on the edge or surface of an insulating bulk3,4,5,6. With the application of a magnetic field (B), the unique features of Dirac bands may be exemplified via the formation of Landau levels (LLs). The QHE is the hallmark of dissipationless topological quantum transport originating from one-dimensional chiral edge modes driven by cyclotron motion of two-dimensional (2D) electrons. Unlike the case of graphene, the degeneracy is completely lifted in the spin-polarized Dirac state of 2D and three-dimensional (3D) TIs. The Hall conductance σxy of 3D TI is expected to be given by the sum of the two contributions from the top and bottom surfaces and hence σxy=(n+n’+1)e2/h with both n and n’ being integers. When the two contributions are equivalent, that is, n=n’, only the odd integer QHE is expected. For such 3D TI films, the top and bottom surfaces support surface states with opposite spin-momentum locked modes when the top and bottom surfaces are regarded as two independent systems. Such a helicity degree of freedom in real space can be viewed as the pseudo-spin variable and is hence expected to yield a new quantum state via tuning of surface magnetism and/or Fermi level (EF) that is applicable to quantum computation functions7,8,9. Although intensive research has been carried out for bulk crystals, thin films and field-effect devices10,11,12,13,14,15,16,17, parasitic bulk conduction and/or disorder in the devices continues to hamper efforts to resolve quantum transport characteristics of the Dirac states on chalcogenide 3D TIs surfaces. The most venerable example of the QHE with least bulk conduction has been achieved in a 70 nm strained HgTe film18. Compared with the HgTe system, 3D-TIs of Bi-chalcogenides such as Bi2Se2Te and (Bi1−xSbx)2Te3 have a good potential for exploring the Dirac surface states with wide controllability of transport parameters (resistivity, carrier type and density) and band parameters (energy gap, position of Dirac point and Fermi velocity) by changing the compositions19,20.

In the following, we report on the QHE in field-effect transistors based on 3D TI thin films of (Bi1−xSbx)2Te3 (x=0.84 and 0.88). With electrostatic gate-tuning of the Fermi level in the bulk band gap under magnetic fields, quantized Hall plateaus (σxy=±e2/h) at the filling factor ν=±1 are resolved, pointing to the formation of chiral edge modes at the top/bottom surface Dirac states. In addition, the emergence of a σxy=0 state around the charge neutral point (CNP) reflects a pseudo-spin Hall insulator state when the location of Fermi level is between the non-degenerate top and bottom surface Dirac points.

Results

Transport properties with electrostatic gate-tuning

We grew 3D TI thin films of (Bi1−xSbx)2Te3 (x=0.84 and 0.88; both 8 nm thick) using molecular beam epitaxy (MBE)19 and insulating InP (111) substrates. The EF of as-grown film was tuned near to the bulk band edge by precisely controlling the Bi/Sb composition ratio19,20. Films were then fabricated into photolithography-defined gated Hall-bar devices to allow electrostatic tuning of EF. A cross-sectional schematic of the device structure and the top-view image are shown in Fig. 1a,b, respectively. The device consists of a Hall bar defined by Ar ion-milling, and an atomic-layer-deposited AlOx insulator isolated Ti/Au top gate with electron-beam-evaporated Ti/Au electrodes (see Methods). The magnetotransport measurements were carried out in a dilution refrigerator by low-frequency (3 Hz) lock-in technique with a low excitation current of 1 nA to suppress heating (see Supplementary Fig. 1 and Supplementary Note 1).

Figure 1: Gating of topological insulator (Bi0.16Sb0.84)2Te3 thin film.
figure 1

(a,b) Cross-sectional schematic and top-view photograph of a Hall-bar device. Broken line in b indicates the position for a. Scale bar, 300 μm. (c) Top gate voltage VG dependence of longitudinal conductivity σxx at various temperatures. (d,e) Effective gate voltage (VG−VCNP) dependence of longitudinal and transverse resistance (Rxx and Ryx) and inverse of Hall coefficient 1/RH under magnetic field B=3 T at temperature T=40 mK. The VG for the charge neutral point (CNP), VCNP, is defined as the gate voltage where Ryx is crossing zero.

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First, device operation was examined at B=0 T. Figure 1c shows the electric field effect controlled conductivity σxx of the x=0.84 film as a function of top gate voltage VG with changing temperature. The minimum conductivity of roughly 2e2/h is observed at VG=−1.7 V. On both sides of this minimum, σxx shows a linear but asymmetric increase with increasing or decreasing VG. The weak temperature dependence of longitudinal conductivity σxx for a wide range of VG is characteristic of the gapless nature of Dirac states under finite disorder21. Thus, we ascribe the conductivity of this TI film below 1 K to the Dirac surface states with a small contribution from bulk conduction. To verify the ambipolar nature of the device, the longitudinal and transverse resistance Rxx and Ryx as a function of VG at B=3 T were measured (Fig. 1d). As expected, this results in a sign change of Ryx at a certain VG, which we hereafter define as the gate voltage corresponding to EF being located at the charge neutral point (CNP), VCNP. This point coincides closely with the VG at which Rxx reaches a maximum. At the VCNP, the Hall effects from the top and bottom surface states appear to cancel, resulting in Ryx∼0, although electron-rich and hole-rich puddles are thought to still exist, as in the case of graphene22. To capture the essence of the observed phenomena, we hereafter take the working hypothesis that EF shifts equally on top and bottom states, which in turn retain their difference in energy position. The inverse of Hall coefficient 1/RH is shown in Fig. 1e, which would be proportional to 2D charge carrier density n2D=1/(eRH), in the simplest case. Asymmetric behaviour of 1/RH between positive and negative VG regions with respect to VCNP is in accord with the asymmetric σxx behaviour at zero magnetic field as shown in Fig. 1c. The more efficient increase in σxx with electron accumulation (positive VG) is often observed in ambipolar TI transistors13,14,15,16,17 and may be related to the difference in vF (Fermi velocity) above and below the Dirac point19. In addition, the proximity of the Dirac point to the valence band edge needs to be taken into account for the present thin film system; in most of the negative region of VG−VCNP, EF must go inside the valence band in which doped holes appears to be fully localized at low temperatures below 1 K, as argued later.

Observation of quantum Hall states

With applying higher magnetic field of B=14 T, clear signatures of QHE are revealed in the temperature dependence of Rxx and Ryx, as shown in Fig. 2a–d for the x=0.84 film and in Fig. 2e–h for the x=0.88 film. We first focus on the results of the x=0.84 film. Rxx increases steeply with lowering the temperature for VG corresponding to the CNP, yet decreases rapidly toward zero on both sides at higher and lower VG. Concomitantly to the decrease in Rxx, Ryx reaches the values of quantum resistance ±h/e2=±25.8 kΩ and forms plateaus in VG at T=40 mK. This correspondence between the rapidly declining Rxx value and the quantized Ryx plateaus at ±h/e2 are distinct evidence for QHE at Landau filling factor ν=±1, respectively, as schematically shown in Fig. 2i (magnetic field dependence is shown in Supplementary Fig. 2 and discussed in Supplementary Note 2). The LL splitting energy of Dirac dispersion (En) is given by , where n is the LL index.

Figure 2: Observation of the quantum Hall effect.
figure 2

(a,b,e,f), Effective gate voltage VG−VCNP dependence of Rxx and Ryx at various temperatures of T=40–700 mK with application of magnetic field B=14 T for the x=0.84 (a,b) and x=0.88 (e,f) films of (Bi1−xSbx)2Te3. (c–e,h), Effective gate voltage VG−VCNP dependence of σxx and σxy at various temperatures of T=40–700 mK with the application of magnetic field B=14 T for the x=0.84 and x=0.88 films of (Bi1−xSbx)2Te3 as deduced from the corresponding Rxx and Ryx data. Triangles in d and h show the dips of σxx. (i) Schematics of the Landau levels (LLs) of the surface state of (Bi1−xSbx)2Te3 (x=0.8–0.9) thin film in case of the degenerate top and bottom surface state. At a high field, for example, 14 T, the n=−1 LL of the surface state locates below the top of the valence band. When the Fermi energy (EF) is tuned between the LLs, the quantum Hall state with index ν emerges. (j) Schematics of the LLs of the top and bottom surface states in case the n=0 (Dirac point) energy is different between the two surfaces.

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Using the data of Rxx and Ryx, σxy and σxx as functions of VG−VCNP for the x=0.84 film are plotted in Fig. 2c,d. Again, the Hall plateaus at σxy=±e2/h as well as minima of σxx approaching zero (black triangles) are observed and are indicative of the QHE with ν=±1. In this plot, however, two additional features are to be noted. The first is an unexpectedly wide σxy plateau and thermally activated behaviour of σxx for the ν=+1 (σxy=e2/h) state in the corresponding VG−VCNP (negative) region. As already noted in the VG asymmetric change of σxx (Fig. 1c) in the negative region of VG−VCNP, that is, hole-doping, EF readily reaches the valence band top. The energy position of Dirac point of the x=0.84 film lies by at most 30 meV above the valence band top, while the LL splitting between n=0 and n=−1 levels amounts to 70 meV at 14 T, according to resonant tunneling spectroscopy on similarly grown thin films of (Bi1−xSbx)2Te3 (ref. 19). From the consideration of the Fermi velocity of Dirac cone, the VG−VCNP value at which the EF reaches the valence band top is estimated to be around −1.5 V or even a smaller absolute value. Therefore, in the VG−VCNP region where the quantum Hall plateau or its precursor is observed, the EF locating between n=0 and n=−1 LLs of the surface state is close to or already buried in the valence band, as schematically shown in Fig. 2i. Although the doped but localized holes in the valence band may hardly contribute to transport, that is, σxy(bulk)≪σxy(surface), the relative EF shift with negatively sweeping VG−VCNP becomes much slower as compared with the positive sweep case owing to the dominant density of states of the valence band. This explains a wider plateau region for ν=+1 in the hole-doping side, contrary to the normal behaviour of electron accumulation side, ν=−1.

The second notable feature in Fig. 2c is the emergence of the ν=0 state around VG=VCNP, as seen in the step of σxy and (finite) minimum in σxx as functions of VG−VCNP. This state is more clearly resolved in the x=0.88 film, as shown in Fig. 2e–h, on which we focus hereafter. In a similar manner to the x=0.84 film, the x=0.88 film also shows with lowering temperature divergent behaviour of Rxx around VG=VCNP, while approaching zero around VG−VCNP=−1.5 V. The similar divergent (at VG=VCNP) and vanishing (at VG−VCNP=−1.5 V) behaviours of Rxx are observed also with increasing B at 40 mK (see Supplementary Fig. 3 and Supplementary Note 3). The Ryx reaches 25.8 kΩ around VG−VCNP=−1.5 V forming the ν=+1 quantum Hall state, whereas in the electron-doping regime, Ryx reaches −20 kΩ, short of the quantized value. The failure to form the fully quantized ν=−1 state is perhaps related to the disorder of the surface Dirac state, which is induced by the compositional/structural disorder of the as-grown film and cannot be overcome by gate tuning.

Nevertheless, the ν=0 feature is clearly resolved for the x=0.88 film, as shown in the VG dependence of σxy (Fig. 2g) calculated from Rxx and Ryx. In addition to the σxy=e2/h (ν=+1) plateau, a plateau at σxy=0 appears at around VG=VCNP. The plateau broadening occurs via centring at σxy=0.5 e2/h as the isosbestic point with elevating temperature. In accordance with the plateaus in σxy, σxx takes a minima at ν=+1 and 0, as shown in Fig. 2h. Here, we can consider the contribution of both the top and bottom surface Dirac states to this quantization of σxy, as schematically shown in Fig. 2j. At the ν=+1 (ν=−1) state, both the top and bottom surfaces are accumulated by holes (electrons) with EF being located between n=0 and n=−1 (n=+1) LLs, giving rise to the chiral edge channel. In contrast, we assign the ν=0 state to the gapping of the chiral edge channel as the cancellation of the contributions to σxy from the top and bottom surface states with ν=±1/2, when EF locates in between the energy levels of the top and bottom surface Dirac points (n=0 levels), as shown in Fig. 2j. This ν=0 state can hence be viewed as a pseudo-spin Hall insulator, if we consider the top and bottom degree of freedom as the pseudo-spin variable. Such an observation of a zero conductance plateau has been reported also in disordered graphene under very high magnetic field23,24,25,26 and analysed theoretically27, as well as in the 2D TIs, the quantum wells of HgTe28 and InAs/GaSb29. From the analyses shown in the following, we propose here that the major origin for the presence of σxy=0 is more like the energy difference of the top/bottom Dirac points rather than other effects such as electron-hole puddles due to composition inhomogeneity.

Discussion

To further discuss the characteristics of these QH states, we investigate the B dependence of σxy (Fig. 3a,b). The analysis of the plateau width against VG determines the phase diagram as shown in Fig 3c,d. The plateau edges are determined from the second derivative of σxy with respective to VG (see Supplementary Fig. 4 and Supplementary Note 4), while the plateau transition points between ν=0 and ν=±1 are defined here by σxy=±e2/2h. The plateau shrinks with decreasing B for the ν=−1 state of the x=0.84 film (Fig. 3a). However, the ν=+1 state for both films appears to be rather robust with reducing VG (doping more holes), since EF positions already below the top of the valence band, perhaps for VG−VCNP<−1.5 V. On the other hand, the ν=0 plateau is only weakly dependent on B, although the plateau width is wider for x=0.88 than for x=0.84. The observation of ν=0 requires the condition that the Fermi level is located in between the energy levels of the top and bottom surface Dirac points (Fig. 2j). From the Hall data in the relatively high positive VG−VCNP region (electron-doping) shown in Fig. 1e, we can know the relation between the sum of the top and bottom Dirac electron density versus VG−VCNP. Then, with the values of the ν=0 plateau width between the σxy=±0.5 e2/h points (δVG∼0.9 and 1.4 V; see Fig. 3c,d) and the Fermi velocity (vF∼5 × 105 ms−1; ref. 19), we can estimate the energy difference (δEDP) between the Dirac points at the top and bottom surface states to be ∼50 meV and ∼70 meV for the x=0.84 and x=0.88 film, respectively (see Supplementary Note 5). These values should be compared with a much larger band gap energy (∼250 meV). The energy difference δEDP is, however, considerably larger than a Zeeman shift (∼9 meV at 14 T; ref. 19), which rationalizes the above analysis with ignoring the Zeeman shift of the n=0 LL. Although the reason why the two films (x=0.84 and 0.88) show such a difference in δEDP is not clear at the moment, we speculate that the monolayer buffer layer of Sb2Te3 (x=1.0) used for the growth of the x=0.88 film (see Methods) may cause the considerably higher energy position of the Dirac point at the bottom surface. Incidentally, for the region of |VG−VCNP|<δVG/2, electron accumulation at the top surface and hole accumulation at the bottom surface should coexist. This may naturally explain the observed (Fig. 1e and see also Supplementary Fig. 5 and Supplementary Note 6) deviation from the linear relationship between 1/RH and VG−VCNP as well as the extrema of 1/RH observed at around ±δVG/2.

Figure 3: Magnetic field dependence of Hall plateaus.
figure 3

(a,b) σxy at T=40 mK under various B as a function of effective gate voltage VG−VCNP for the x=0.84 (a) and x=0.88 (b) films. Traces for lower magnetic fields are each vertically offset by e2/h. Dotted lines represent the plateau transitions as defined at σxy=±0.5 e2/h. (c,d) Quantized σxy phase diagram for the ν=+1, 0 and −1 (blue, green and red shaded, respectively) states associated with the plateau edges (filled squares) and the plateau transition point (filled circles) defined at σxy=±0.5 e2/h in the plane of magnetic field and VG−VCNP. The plateau edges are determined by the second-order VG derivative of σxy.

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Figure 4 summarizes the flow of conductivity tensor (σxy, σxx) plotted with the two experimental subparameters (T and VG) at 14 T. With decreasing T, the flow in (σxy, σxx) tends to converge toward either of (σxy, σxx)=(−e2/h, 0), (0, 0) or (e2/h, 0) at high magnetic field (for example, 14 T), which corresponds to ν=−1, 0 and +1 QH state, respectively. Incipient convergence to ν=0 is discerned for the x=0.84 film (Fig. 4a), while the ν=−1 state is not discernible for the x=0.88 film (Fig. 4b). Among these three QH states, the unstable fixed point appears to lie on the line of σxy=±0.5 e2/h (approximately with the critical σxx value of ∼0.5 e2/h) which corresponds to the crossing of EF at the n=0 LL (or Dirac point) of the bottom and top surface state (see Fig. 2j), respectively30.

Figure 4: Flows of σxy and σxx.
figure 4

(a,b) (σxy, σxx) are displayed with the two experimental subparameters (T and VG) for the x=0.84 (a) and x=0.88 (b) films. Each line connecting between points represents the flow behaviour of (σxy, σxx) with lowering temperature from 700 to 40 mK at the specific value of VG at B=14 T. The flows direct from upper to lower with decreasing temperature.

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In conclusion, we have successfully observed the QHE at ν=±1 and 0 in 3D TI thin films of (Bi1−xSbx)2Te3 (x=0.84 and 0.88). Due to a considerable difference of the Dirac point (or n=0 LL) energies of the top and bottom surfaces of the thin film, the ν=0 state observed at σxy=0 is interpreted as a pseudo-spin Hall insulator with the top/bottom degree of freedom as the pseudo-spin. Further studies on non-local transport in mesoscopic structures will open the door to dissipationless topological-edge electronics on the basis of the 3D topological insulators.

Methods

MBE film growth

Thin films of (Bi1−xSbx)2Te3 (x=0.84 and 0.88) were fabricated by MBE on semi-insulating InP (111) substrate. The Bi/Sb composition ratio was calibrated by the beam equivalent pressure of Bi and Sb, namely 8 × 10−7 Pa and 4.2 × 10−6 Pa for x=0.84 and 6 × 10−7 Pa and 4.4 × 10−6 Pa for x=0.88. The Te flux was over-supplied with the Te/(Bi+Sb) ratio kept at ∼20. The substrate temperature was 200 °C and the growth rate was ∼0.2 nm per minute. Fabrication procedures for the x=0.84 and 0.88 films are slightly different at the initial growth on InP surfaces. We grew the 0.84 film with supplying Te and (Bi+Sb) from the initial stage. For the x=0.88 film, we started with supplying Te and Sb for a monolayer growth of Sb2Te3 buffer layer followed by Bi shutter opening. This difference may be an origin of the larger energy gap δEDP of the Dirac points between the top and bottom surfaces (Fig. 2j) for the x=0.88 film. After the epitaxial growth of 8 nm-thick thin films, in situ annealing was performed at 380 °C to improve surface morphology.

Device fabrication

The AlOx capping layer was deposited at room temperature with an atomic layer-deposition system immediately after the discharge of the samples from MBE. This process turned out to be effective to protect the surface from degradation. The device structure was defined by subsequent photolithography and Ar ion-milling processes. Ohmic-contact electrodes and top gate electrode were Ti/Au and deposited with an e-beam evaporator. Here, ion-milling was performed under 45-degree tilt condition on a rotating stage, resulting in the ramped side edge as schematically shown in Fig. 1a. This ensured electrical contact to the top and bottom of the film.

Additional information

How to cite this article: Yoshimi, R. et al, Quantum Hall effect on top and bottom surface states of topological insulator (Bi1−xSbx)2Te3 films. Nat. Commun, 6:6627 doi: 10.1038/ncomms7627 (2015).