Topological insulators are a novel class of quantum matter with a gapped insulating bulk, yet gapless spin-helical Dirac fermion conducting surface states. Here, we report local and non-local electrical and magneto transport measurements in dual-gated BiSbTeSe2 thin film topological insulator devices, with conduction dominated by the spatially separated top and bottom surfaces, each hosting a single species of Dirac fermions with independent gate control over the carrier type and density. We observe many intriguing quantum transport phenomena in such a fully tunable two-species topological Dirac gas, including a zero-magnetic-field minimum conductivity close to twice the conductance quantum at the double Dirac point, a series of ambipolar two-component half-integer Dirac quantum Hall states and an electron-hole total filling factor zero state (with a zero-Hall plateau), exhibiting dissipationless (chiral) and dissipative (non-chiral) edge conduction, respectively. Such a system paves the way to explore rich physics, ranging from topological magnetoelectric effects to exciton condensation.
A three-dimensional (3D) topological insulator (TI) is characterized by an insulating bulk band gap and gapless conducting topological surface states (TSS) of spin-helical massless two-dimensional (2D) Dirac fermions1,2. Such surface states are topologically non-trivial and protected by time-reversal symmetry, thus immune to back scattering. The potential novel physics offered by this system, such as topological magnetoelectric (TME) effects3,4, Majorana fermions5 and effective magnetic monopoles6, has drawn intense interest. One of the most iconic transport signatures for 2D Dirac electronic systems is the half-integer quantum Hall effect (QHE) in a perpendicular magnetic field (B), as first observed in graphene7,8 and later also studied in HgTe9,10. The Landau levels (LLs) of 2D Dirac fermions have energies EN=sgn(N)vF(2eBħ|N|)1/2, where sgn is the sign function, N is the LL index (positive for electrons and negative for holes), vF is the Fermi velocity, e is the elemental charge and ħ is the Plank’s constant h divided by 2π. The zeroth LL at E0=0 is equally shared between electrons and holes, giving rise to the half-integer shift in the quantized Hall conductivity σxy=g(N+1/2)e2/h, where g is the number of degenerate species of Dirac fermions (for example, g=4 for graphene, and g=1 for TSS with a single Dirac cone). This 1/2 can also be related to the Berry-phase due to the spin or pseudospin locking to the momentum of Dirac fermions7,8,9,10.
In most commonly studied TI materials such as Bi2Se3, Bi2Te3 and other Bi/Sb-based chalcogenides, it is often challenging to observe characteristic TSS transport (particularly QHE) due to bulk conduction caused by unintentional impurity doping. Only very recently has well-developed QHE arising from TSS been observed in exfoliated flakes from BiSbTeSe2 (BSTS) single crystals11 and molecular beam epitaxy grown (Bi1−xSbx)2Te3 or Bi2Se3 thin films12,13. In this work, we fabricate dual-gated14,15,16 TI devices from exfoliated BSTS thin flakes with undetectable bulk carrier density and conduction at low temperature11. Such a dual-gating structure is also promising for exploring exciton condensation proposed for TIs17 and topological quantum phase transitions induced by displacement electric field18.
In our dual-gated BSTS devices, the independent, ambipolar gating of parallel-conducting top and bottom surfaces realize two independently controlled species of 2D Dirac fermions, allowing us to investigate such interesting transport phenomena as the minimum conductivity of TSS at Dirac point (DP), and two-species (two-component) Dirac fermion QHE of electron+electron, electron+hole and hole+hole types, involving various combinations of top and bottom surface half-integer filling factors νt and νb, respectively. When (νt, νb)=(−1/2, 1/2) or (1/2, −1/2), there’s an intriguing ν=0 state characterized by zero-Hall plateau and a large longitudinal resistance peak11,12, attributed to the formation of dissipative and non-chiral edge states. We also perform non-local transport measurements and compare them with the normal local measurements in our dual-gated 3D TI devices in the quantum Hall (QH) regime to probe the nature of edge-state transport for both standard QH states and the novel ν=0 dissipative QH-like state. We further demonstrate that the dissipative edge states at ν=0 have temperature-independent conductance, revealing that the transport in such a quasi-one-dimensional (1D) dissipative metallic edge channel could evade standard localization.
Transport properties at zero and low magnetic field
Qualitatively, similar data are measured in multiple samples, while results from a typical sample A (channel length L=9.4 μm, width W=4.0 μm, with ∼100 nm-thick BSTS and 40 nm-thick h-BN as top-gate dielectric, see schematic in Fig. 1a) are presented below unless otherwise noted. The h-BN as a substrate or gate dielectric is known to preserve good electronic properties for graphene, resulting from the atomic flatness and relatively low density of impurities in h-BN19. The carrier densities of the top and bottom surface of the BSTS flake are tuned by top-gate voltage Vtg and back-gate voltage Vbg, respectively.
Figure 1b,c shows the double-gated electric field effect measured at T=0.3 K. The longitudinal resistivity ρxx (=Rxx × W/L, with Rxx being longitudinal resistance) at magnetic field B=0 T (Fig. 1b) and Hall resistivity ρxy (=Rxy, Hall resistance) at B=1 T (Fig. 1c) are plotted in colour scale as functions of both top and bottom gate voltages (Vtg and Vbg). The extracted field effect and Hall mobilities are typically several thousands of cm2 V−1 s−1. A minimum carrier density n* ∼9 × 1010 cm−2 per surface can be extracted from the maximum Hall coefficient (absolute value) ∼3.5 kΩ T−1 (when both surfaces are slightly n-type or p-type) measured in Fig. 1c. A set of exemplary Vtg-sweeps with Vbg=3 V is shown in Fig. 1c inset. By adjusting Vtg (or Vbg), the device can be gated through a Rxx peak, identified as the charge-neutrality DP of the top (or bottom) surface, marked by the blue (or red) dashed lines in Fig. 1b. Gating through the DP, the carriers in the corresponding surface change from hole-like to electron-like (that is, ambipolar), as evidenced by Hall measurements (Fig. 1c). The slight deviation of the two lines from being perfectly vertical and horizontal arises from the weak capacitive coupling between the top (bottom) surface and the back (top) gate16. The crossing of these two lines corresponds to the double DP (both surfaces tuned to DP), where ρxx (σxx=1/ρxx) reaches a global maximum (minimum). Within the gate voltage range used, the carriers predominantly come from the TSS and we observe relatively good particle-hole symmetry in the transport properties (for example, the symmetrical appearance of ρxx on both sides of DP in each surface in Fig. 1b and the similar absolute values of the positive and negative maximum Hall coefficient in Fig. 1c).
We have studied six dual-gated BSTS devices with different thicknesses (t) and aspect ratios (L/W). These devices are measured at low temperatures (T<2 K) and the results are repeatable after multiple thermal cycles. When both surfaces are tuned to DP, the minimum 2D conductivity σmin at B=0 T exhibits relatively constant value (3.8±0.1)e2/h for all the devices measured (with the uncertainty representing 90% confidence interval), whose thicknesses range from ∼50 to ∼200 nm and L/W range from 1.3 to 3.5 (Fig. 1d). Our observation indicates that the conductivity at the DP for each major surface (top or bottom) is ∼2e2/h (one unit of conductance quantum), within the range of values (2∼5 e2/h) reported by Kim et al.14 on thin flakes of Bi2Se3 (∼10 nm). The better consistency over multiple samples in our dual-gated BSTS devices may be attributed to the more insulating bulk (whose conduction is immeasurably small at low temperature) and uniformity of the exfoliated BSTS flakes, which are sandwiched between SiO2 and h-BN to achieve better device stability. The minimum conductivity at DP has also been discussed in graphene with considerable interest20,21,22,23,24,25.
The experiments in graphene revealed that the minimum conductivity is strongly affected by carrier-density inhomogeneities (puddles) induced by disorder on or near graphene24,25, such as the absorbates or charged impurities in the substrates. In 3D TIs, one source of impurities likely relevant to the observed quasi-universal minimum conductivity in our dual-gated BSTS devices could be bulk defects (located near surface)26,27, such as those revealed in scanning tunnelling microscopy studies28.
For the rest of the paper, we focus on the transport phenomena in the QH regime under a high magnetic field B perpendicular to the top and bottom surfaces. Figure 2a,c shows in colour scales the longitudinal conductivity σxx (=ρxx/(ρxx2+ρxy2)) and Hall conductivity σxy (=ρxy/(ρxx2+ρxy2)) for Sample A as functions of Vtg and Vbg at B=18 T and T=0.3 K. The colour plots in Fig. 2a,c divide the (Vtg, Vbg) plane into a series of approximate parallelograms, centred around well-developed or developing QH states with vanishing or minimal σxx (Fig. 2b) and quantized σxy in integer units of e2/h (Fig. 2d). These QH parallelograms are bounded by approximately (but slightly tilted) vertical and horizontal lines, which represent the top and bottom surface LLs, respectively. By increasing (decreasing) either Vtg or Vbg to fill (exhaust) one LL on the top or bottom surface, σxy increases (decreases) by e2/h, taking consecutive quantized values of νe2/h, where integer ν=νt+νb=Nt+Nb+1. The Nt(b) is the corresponding top (bottom) surface LL integer index that can be adjusted by top (back) gate to be of either Dirac electrons or holes. In Fig. 2d, different fixed Vbg values (from −17 to 40 V) set νb around consecutive half integers −3/2, −1/2, 1/2, 3/2 and 5/2 (such that the bottom surface contributes σxyb=νbe2/h to the total σxy), explaining the vertical shift of e2/h at QH plateaux of consecutive Vtg-sweeps.
It is also notable that in Fig. 2, there are a few states with zero-quantized Hall conductivity (σxy=0, manifesting as white regions in Fig. 2c, separating the electron-dominated regions in red and the hole-dominated regions in blue) and non-zero σxx minimum, marked by equal and opposite half-integer values of νt and νb thus total ν=0, for example (νt, νb)=(−1/2, 1/2), (1/2, −1/2) and (3/2, −3/2). These states with total ν=0, exhibiting zero-Hall plateaux (see also Fig. 2d), have non-zero σxx minimum (Fig. 2a,b) but very large Rxx maximum (see next, Fig. 3).
Non-local transport at ν=0 states
To further characterize the observed QH and ν=0 states, we have performed non-local transport measurements of Rnl (=Vnl/I, I is the current and Vnl is the non-local voltage, see the schematic measurement setup in the inset of Fig. 3b) as functions of Vtg and Vbg at B=18 T and T=0.3 K and compared the results with the standard (local) measurements of the longitudinal resistance Rxx (Fig. 3a). It is intriguing that unlike other QH states typified by a zero or minimum in Rxx, the states with ν=νt+νb=0 (labelled by (νt, νb) in Fig. 3a with νt=−νb =±1/2 or ±3/2) are accompanied by a Rxx maximum. The best-developed ν=0 states are those at (νt, νb)=(−1/2, 1/2) or (1/2, −1/2), where Rxx reaches ∼220 kΩ (ρxx ∼100 kΩ), exceeding the resistance quantum (h/e2=∼25.8 kΩ) by an order of magnitude. The non-local Rnl also becomes very large (∼100 kΩ) and the similar order of magnitude as Rxx at these two ν=0 states, while negligibly small at other (νt, νb) QH states (see Fig. 3b and also the representative cuts in Fig. 3c).
The simultaneously large local and non-local resistance at ν=0 states in the QH regime has been reported in other 2D electron-hole systems29,30 and understood in a picture of dissipative edge channels. We emphasize that the pronounced Rnl signal cannot be explained from Rxx by a classical Ohmic non-local resistance from the stray current connecting the remote leads. Such a contribution (=∼ρxxe−πL/W) would decay exponentially with L/W (=2.4 in our case), and be three orders of magnitude smaller than the local Rxx (which is the case at B=0 T, Supplementary Fig. 1). As another comparison, the middle panel of Fig. 3c shows the cuts in Fig. 3a,b at Vbg=3 V, crossing the double-DP (also zeroth LL) of both top and bottom surfaces at (νt, νb)=(0, 0), where we observe a relatively large peak in Rxx but significantly smaller Rnl. Such a result is consistent with the ‘extended’ state transport (at the center of zeroth LL) as the current flows through the bulk of the 2D surface.
From the colour plots in Figs 2 and 3, the parallelogram centred around (νt, νb)=(−1/2, 1/2) state is enclosed by boundaries representing Nt=0 and −1, Nb=0 and 1 LLs. Similarly, the (νt, νb)=(1/2, −1/2) state is bound by Nt=0 and 1, Nb=0 and −1 LLs. We conclude that such a ν=0 state can exist when the potential difference V between top and bottom surfaces (equivalently the energy separation between top and bottom surface DPs) is in the range of 0<|V|<2E0−1 (≅2 × 50 meV at B=18 T, where E0−1 is the 0−1 LL separation of TSS Dirac fermions11). The large energy scale of E0−1 can help make the ν=0 and ν=±1 QH states observable at significantly elevated temperatures as demonstrated below.
Temperature dependence of the ν=0 and ±1 states
We have studied the temperature (T) dependence of the QHE and ν=0 states from 0.3 K to 50 K at B=18 T (Fig. 4). At each temperature, the bottom surface density is tuned by Vbg to set νb near 1/2 (dashed lines) or −1/2 (solid lines), and the peaks in local Rxx and non-local Rnl corresponds to the (νt, νb)=(−1/2, 1/2) or (1/2, −1/2), respectively (Fig. 4a,b, detailed raw data are shown in Supplementary Fig. 2). The Rxx peaks (>∼150 kΩ) are seen to be more robust up to the highest temperature (T=50 K) measured, while Rnl peaks decrease rapidly (approximately linearly in T, shown in Fig. 4c) with increasing T and is nearly suppressed above 50 K. We also show the T-dependence of σxx and σxy at (νt, νb)=(−1/2, 1/2), (1/2, −1/2), (1/2, 1/2) and (−1/2, −1/2) in Fig. 4e,f. The σxy maintains good quantization at νe2/h (ν=0, ±1) up to T=50 K, while σxx increases with T (the gate-dependent σxx and σxy traces at different temperatures are shown in Supplementary Fig. 3). The σxx for ν=±1 states is found to show thermally activated behaviour at high temperatures11, where the finite σxx is attributed to the thermally excited 2D surface or 3D bulk carriers. Such carriers can shunt the edge-state transport and suppress the non-local Rnl response at high T (ref. 29). We also note that the σxx versus T curves for ν=0 and ν=±1 states follow the similar trend and have approximately constant separation. We find the averaged separation Δσxx=1/2 × (σxx(−1/2, 1/2)+σxx(1/2, −1/2)−σxx(1/2, 1/2)−σxx(−1/2, −1/2)) to be largely T-independent with a value of (0.27±0.01)e2/h, which we attribute to the conductivity of the quasi-1D dissipative edge channel.
In our measurement setup, the contacts connect to the top, bottom and side surfaces, all of which are probed simultaneously. The side surface only experiences an in-plane field and can be viewed as a quasi-1D domain boundary that separates the top and bottom surfaces with B pointing outward and inward, respectively, thus can support QH edge states31. When the top and bottom surfaces are doped to the same carrier type (either n or p), the corresponding QH edge states (on the side surface) would have the same chirality and give the observed total σxy=νe2/h=(νt+νb)e2/h, restricted to integer multiples of e2/h. When the two surfaces have opposite carrier types but one of the them dominates, well-defined QH states with ν=νt+νb may still be observed, such as the (−1/2, 3/2) state with σxy=(−1/2+3/2)e2/h=e2/h and vanishing ρxx. Previous studies in InAs/(AlSb)/GaSb heterostructure-based electron-hole systems32,33 also revealed QH effect with Rxy=h/(νe2)=h/(νe−νh)e2 (νe and νh are electron and hole-filling factors, both are positive integers) and vanishing Rxx when the AlSb barrier (separating electron and hole gases) is sufficiently thin to enable electron-hole hybridization. Despite the phenomenological similarities, our QH system is distinctive in the sense that the spatially separated electrons and holes residing on the top and bottom surfaces have half-integer filling factors, and the hybridization only happens at the side surface.
We show the schematic energy spectrum when the two surfaces are degenerate with V=0 (refs 34, 35, 36) in Fig. 4g, which depicts the Fermi energy Ef inside the 0−1 LL gap and corresponds to the (1/2, 1/2) QH state. For a relatively thick sample such as ours (∼100 nm>magnetic length lB=(ħ/eB)1/2≅6 nm at B=18 T); however, it has been suggested that even in the presence of well-quantized LLs, a standard TI Hall measurement would exhibit deviations from perfectly quantized values due to conduction through the side surfaces31,34,35,36. On the other hand, it has also been suggested that when net chiral modes exist (Fig. 4g,i show one such net chiral mode), the QH effect may be restored by the local equilibrium between non-chiral edge modes37, possibly explaining the good quantization in σxy and vanishing ρxx (also Rnl) observed in our experiments.
For the (νt, νb)= (−1/2, 1/2) or (1/2, −1/2) state, the carrier density on the top and bottom surfaces are opposite. Since Ef is within the LL gap on both the surfaces, the finite residual σxx and large Rnl we observed are indicative of dissipative edge transport. We show a schematic energy spectrum38 of this ν=0 state with V slightly smaller than E0−1 in Fig. 4h, where the Fermi level Ef resides between the Nt=−1 and 0 LL of top surface (marked in blue), thus νt=−1/2, and also between the Nb=0 and 1 LL of bottom surface, thus νb=1/2. Overall, such energy spectrum represents a (νt, νb)=(−1/2, 1/2) and ν=0 state. The Ef crosses an even number (only two shown in this illustrative example in Fig. 4h) of counter-propagating edge modes (arising from sub-bands of the quasi-1D side surface). The disorder can cause scattering and local equilibrium between the counter-propagating modes, giving rise to non-chiral dissipative transport (depicted by a series of conducting loops that can hop between adjacent ones in Fig. 4j) on the side surface with a large and finite resistance. While the energy spectrum (Fig. 4h) is expected to have a gap (Δ) near the edge (due to the hybridization between top (marked with blue) and bottom (red) surface zeroth LLs and approximately the finite-size confinement-induced gap ≅hvF/t≅10 meV opened at DP of the side surface38), we did not observe a truly insulating state with vanishing σxx and diverging Rxx (Figs 2a and 4a). This is likely due to the disorder potential (spatial fluctuation of DP28) comparable or larger than Δ and thus smearing out this gap (effectively Ef always crosses the non-chiral edge modes). It would be an interesting question for future studies to clarify whether the weak T-dependence (at T<∼50 K) of the observed conductance (Fig. 4e), similar to the behaviour reported in InAs/GaSb based electron-hole systems39, may indicate an absence of localization40,41,42,43 in such quasi-1D resistive edge channels.
Several recent theories have pointed out that the ν=1/2−1/2=0 state in the TI QH system may bring unique opportunities to realize various novel physics. It has been suggested that both the ν=0 state in TI QHE and an analogous quantum anomalous Hall (QAH) state with zero-Hall-conductance plateau in a magnetic-doped TI around the coercive field can be used as platforms to observe the TME effect44,45, where an electric (magnetic) field induces a co-linear magnetic (electric) polarization with a quantized magnetoelectric polarizability of ±e2/2 h. A zero-Hall-plateau state has been recently observed in the QAH case in ultrathin (few-nm-thick) films of Crx(Bi,Sb)2-xTe3 at low temperature (<1 K) (refs 46, 47). In comparison, our samples have much larger thickness (>∼50 nm, suggested to be preferable for better developed TME effect45,48), and our ν=0 state survives at much higher temperatures (∼50 K). It has also been proposed that excitonic condensation and superfluidity can occur in thin 3D TIs at the ν=0 state in QH regime49 (in addition to the at-zero B field17) induced by spontaneous coherence between strongly-interacting top and bottom surfaces. In future studies, much thinner samples are likely needed to investigate the possibility of such exciton superfluidity.
3D TI single crystals BiSbTeSe2 (BSTS) were grown by the vertical Bridgman technique11. BSTS flakes (typical thickness ∼50–200 nm) are exfoliated (Scotch tape method) onto highly doped Si (p+) substrates (with 300 nm-thick SiO2 coating), and lithographically fabricated into Hall bar-shaped devices with Cr/Au contacts. A thin flake of hexagonal boron nitride (h-BN, typical thickness ∼10–40 nm) is transferred19 on the BSTS flake to serve as a top-gate dielectric and a top-gate metal (Cr/Au) is deposited afterwards. The thickness of BSTS and h-BN flakes are measured by atomic force microscopy.
Transport measurements are performed with the standard lock-in technique using a low-frequency (<20 Hz) excitation current of 20 nA in a helium-4 variable temperature system (with base temperature down to 1.6 K) or a helium-3 system equipped with magnetic fields (B) up to 18 T (down to 0.3 K).
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We thank J. Hu, T. Wu, E. Palm, T. Murphy, A. Suslov, E. Choi and B. Pullum for experimental assistance. We also thank F. de Juan, R. Ilan, N. Nagaosa and W. Ku for helpful discussions. This work is supported by the DARPA MESO program (Grant N66001-11-1-4107). A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-1157490, the State of Florida, and the U.S. Department of Energy.
The authors declare no competing financial interests.
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Xu, Y., Miotkowski, I. & Chen, Y. Quantum transport of two-species Dirac fermions in dual-gated three-dimensional topological insulators. Nat Commun 7, 11434 (2016). https://doi.org/10.1038/ncomms11434
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