Abstract
Topological insulators are a novel class of quantum matter with a gapped insulating bulk, yet gapless spinhelical Dirac fermion conducting surface states. Here, we report local and nonlocal electrical and magneto transport measurements in dualgated BiSbTeSe_{2} 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 twospecies topological Dirac gas, including a zeromagneticfield minimum conductivity close to twice the conductance quantum at the double Dirac point, a series of ambipolar twocomponent halfinteger Dirac quantum Hall states and an electronhole total filling factor zero state (with a zeroHall plateau), exhibiting dissipationless (chiral) and dissipative (nonchiral) edge conduction, respectively. Such a system paves the way to explore rich physics, ranging from topological magnetoelectric effects to exciton condensation.
Introduction
A threedimensional (3D) topological insulator (TI) is characterized by an insulating bulk band gap and gapless conducting topological surface states (TSS) of spinhelical massless twodimensional (2D) Dirac fermions^{1,2}. Such surface states are topologically nontrivial and protected by timereversal symmetry, thus immune to back scattering. The potential novel physics offered by this system, such as topological magnetoelectric (TME) effects^{3,4}, Majorana fermions^{5} and effective magnetic monopoles^{6}, has drawn intense interest. One of the most iconic transport signatures for 2D Dirac electronic systems is the halfinteger quantum Hall effect (QHE) in a perpendicular magnetic field (B), as first observed in graphene^{7,8} and later also studied in HgTe^{9,10}. The Landau levels (LLs) of 2D Dirac fermions have energies E_{N}=sgn(N)v_{F}(2eBħN)^{1/2}, where sgn is the sign function, N is the LL index (positive for electrons and negative for holes), v_{F} is the Fermi velocity, e is the elemental charge and ħ is the Plank’s constant h divided by 2π. The zeroth LL at E_{0}=0 is equally shared between electrons and holes, giving rise to the halfinteger shift in the quantized Hall conductivity σ_{xy}=g(N+1/2)e^{2}/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 Berryphase due to the spin or pseudospin locking to the momentum of Dirac fermions^{7,8,9,10}.
In most commonly studied TI materials such as Bi_{2}Se_{3}, Bi_{2}Te_{3} and other Bi/Sbbased 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 welldeveloped QHE arising from TSS been observed in exfoliated flakes from BiSbTeSe_{2} (BSTS) single crystals^{11} and molecular beam epitaxy grown (Bi_{1−x}Sb_{x})_{2}Te_{3} or Bi_{2}Se_{3} thin films^{12,13}. In this work, we fabricate dualgated^{14,15,16} TI devices from exfoliated BSTS thin flakes with undetectable bulk carrier density and conduction at low temperature^{11}. Such a dualgating structure is also promising for exploring exciton condensation proposed for TIs^{17} and topological quantum phase transitions induced by displacement electric field^{18}.
In our dualgated BSTS devices, the independent, ambipolar gating of parallelconducting 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 twospecies (twocomponent) Dirac fermion QHE of electron+electron, electron+hole and hole+hole types, involving various combinations of top and bottom surface halfinteger filling factors ν_{t} and ν_{b}, respectively. When (ν_{t}, ν_{b})=(−1/2, 1/2) or (1/2, −1/2), there’s an intriguing ν=0 state characterized by zeroHall plateau and a large longitudinal resistance peak^{11,12}, attributed to the formation of dissipative and nonchiral edge states. We also perform nonlocal transport measurements and compare them with the normal local measurements in our dualgated 3D TI devices in the quantum Hall (QH) regime to probe the nature of edgestate transport for both standard QH states and the novel ν=0 dissipative QHlike state. We further demonstrate that the dissipative edge states at ν=0 have temperatureindependent conductance, revealing that the transport in such a quasionedimensional (1D) dissipative metallic edge channel could evade standard localization.
Results
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 nmthick BSTS and 40 nmthick hBN as topgate dielectric, see schematic in Fig. 1a) are presented below unless otherwise noted. The hBN 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 hBN^{19}. The carrier densities of the top and bottom surface of the BSTS flake are tuned by topgate voltage V_{tg} and backgate voltage V_{bg}, respectively.
Figure 1b,c shows the doublegated electric field effect measured at T=0.3 K. The longitudinal resistivity ρ_{xx} (=R_{xx} × W/L, with R_{xx} being longitudinal resistance) at magnetic field B=0 T (Fig. 1b) and Hall resistivity ρ_{xy} (=R_{xy}, Hall resistance) at B=1 T (Fig. 1c) are plotted in colour scale as functions of both top and bottom gate voltages (V_{tg} and V_{bg}). The extracted field effect and Hall mobilities are typically several thousands of cm^{2} V^{−1} s^{−1}. A minimum carrier density n^{*} ∼9 × 10^{10} cm^{−2} per surface can be extracted from the maximum Hall coefficient (absolute value) ∼3.5 kΩ T^{−1} (when both surfaces are slightly ntype or ptype) measured in Fig. 1c. A set of exemplary V_{tg}sweeps with V_{bg}=3 V is shown in Fig. 1c inset. By adjusting V_{tg} (or V_{bg}), the device can be gated through a R_{xx} peak, identified as the chargeneutrality 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 holelike to electronlike (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) gate^{16}. The crossing of these two lines corresponds to the double DP (both surfaces tuned to DP), where ρ_{xx} (σ_{xx}=1/ρ_{xx}) reaches a global maximum (minimum). Within the gate voltage range used, the carriers predominantly come from the TSS and we observe relatively good particlehole symmetry in the transport properties (for example, the symmetrical appearance of ρ_{xx} 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 dualgated 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 σ_{min} at B=0 T exhibits relatively constant value (3.8±0.1)e^{2}/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 ∼2e^{2}/h (one unit of conductance quantum), within the range of values (2∼5 e^{2}/h) reported by Kim et al.^{14} on thin flakes of Bi_{2}Se_{3} (∼10 nm). The better consistency over multiple samples in our dualgated 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 SiO_{2} and hBN to achieve better device stability. The minimum conductivity at DP has also been discussed in graphene with considerable interest^{20,21,22,23,24,25}.
The experiments in graphene revealed that the minimum conductivity is strongly affected by carrierdensity inhomogeneities (puddles) induced by disorder on or near graphene^{24,25}, such as the absorbates or charged impurities in the substrates. In 3D TIs, one source of impurities likely relevant to the observed quasiuniversal minimum conductivity in our dualgated BSTS devices could be bulk defects (located near surface)^{26,27}, such as those revealed in scanning tunnelling microscopy studies^{28}.
Twocomponent QHE
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 σ_{xx} (=ρ_{xx}/(ρ_{xx}^{2}+ρ_{xy}^{2})) and Hall conductivity σ_{xy} (=ρ_{xy}/(ρ_{xx}^{2}+ρ_{xy}^{2})) for Sample A as functions of V_{tg} and V_{bg} at B=18 T and T=0.3 K. The colour plots in Fig. 2a,c divide the (V_{tg}, V_{bg}) plane into a series of approximate parallelograms, centred around welldeveloped or developing QH states with vanishing or minimal σ_{xx} (Fig. 2b) and quantized σ_{xy} in integer units of e^{2}/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 V_{tg} or V_{bg} to fill (exhaust) one LL on the top or bottom surface, σ_{xy} increases (decreases) by e^{2}/h, taking consecutive quantized values of νe^{2}/h, where integer ν=ν_{t}+ν_{b}=N_{t}+N_{b}+1. The N_{t(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 V_{bg} values (from −17 to 40 V) set ν_{b} around consecutive half integers −3/2, −1/2, 1/2, 3/2 and 5/2 (such that the bottom surface contributes σ_{xy}^{b}=ν_{b}e^{2}/h to the total σ_{xy}), explaining the vertical shift of e^{2}/h at QH plateaux of consecutive V_{tg}sweeps.
It is also notable that in Fig. 2, there are a few states with zeroquantized Hall conductivity (σ_{xy}=0, manifesting as white regions in Fig. 2c, separating the electrondominated regions in red and the holedominated regions in blue) and nonzero σ_{xx} minimum, marked by equal and opposite halfinteger values of ν_{t} and ν_{b} thus total ν=0, for example (ν_{t}, ν_{b})=(−1/2, 1/2), (1/2, −1/2) and (3/2, −3/2). These states with total ν=0, exhibiting zeroHall plateaux (see also Fig. 2d), have nonzero σ_{xx} minimum (Fig. 2a,b) but very large R_{xx} maximum (see next, Fig. 3).
Nonlocal transport at ν=0 states
To further characterize the observed QH and ν=0 states, we have performed nonlocal transport measurements of R_{nl} (=V_{nl}/I, I is the current and V_{nl} is the nonlocal voltage, see the schematic measurement setup in the inset of Fig. 3b) as functions of V_{tg} and V_{bg} at B=18 T and T=0.3 K and compared the results with the standard (local) measurements of the longitudinal resistance R_{xx} (Fig. 3a). It is intriguing that unlike other QH states typified by a zero or minimum in R_{xx}, the states with ν=ν_{t}+ν_{b}=0 (labelled by (ν_{t}, ν_{b}) in Fig. 3a with ν_{t}=−ν_{b} =±1/2 or ±3/2) are accompanied by a R_{xx} maximum. The bestdeveloped ν=0 states are those at (ν_{t}, ν_{b})=(−1/2, 1/2) or (1/2, −1/2), where R_{xx} reaches ∼220 kΩ (ρ_{xx} ∼100 kΩ), exceeding the resistance quantum (h/e^{2}=∼25.8 kΩ) by an order of magnitude. The nonlocal R_{nl} also becomes very large (∼100 kΩ) and the similar order of magnitude as R_{xx} at these two ν=0 states, while negligibly small at other (ν_{t}, ν_{b}) QH states (see Fig. 3b and also the representative cuts in Fig. 3c).
The simultaneously large local and nonlocal resistance at ν=0 states in the QH regime has been reported in other 2D electronhole systems^{29,30} and understood in a picture of dissipative edge channels. We emphasize that the pronounced R_{nl} signal cannot be explained from R_{xx} by a classical Ohmic nonlocal resistance from the stray current connecting the remote leads. Such a contribution (=∼ρ_{xx}e^{−πL/W}) would decay exponentially with L/W (=2.4 in our case), and be three orders of magnitude smaller than the local R_{xx} (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 V_{bg}=3 V, crossing the doubleDP (also zeroth LL) of both top and bottom surfaces at (ν_{t}, ν_{b})=(0, 0), where we observe a relatively large peak in R_{xx} but significantly smaller R_{nl}. 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 (ν_{t}, ν_{b})=(−1/2, 1/2) state is enclosed by boundaries representing N_{t}=0 and −1, N_{b}=0 and 1 LLs. Similarly, the (ν_{t}, ν_{b})=(1/2, −1/2) state is bound by N_{t}=0 and 1, N_{b}=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<2E_{0−1} (≅2 × 50 meV at B=18 T, where E_{0−1} is the 0−1 LL separation of TSS Dirac fermions^{11}). The large energy scale of E_{0−1} 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 V_{bg} to set ν_{b} near 1/2 (dashed lines) or −1/2 (solid lines), and the peaks in local R_{xx} and nonlocal R_{nl} corresponds to the (ν_{t}, ν_{b})=(−1/2, 1/2) or (1/2, −1/2), respectively (Fig. 4a,b, detailed raw data are shown in Supplementary Fig. 2). The R_{xx} peaks (>∼150 kΩ) are seen to be more robust up to the highest temperature (T=50 K) measured, while R_{nl} 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 Tdependence of σ_{xx} and σ_{xy} at (ν_{t}, ν_{b})=(−1/2, 1/2), (1/2, −1/2), (1/2, 1/2) and (−1/2, −1/2) in Fig. 4e,f. The σ_{xy} maintains good quantization at νe^{2}/h (ν=0, ±1) up to T=50 K, while σ_{xx} increases with T (the gatedependent σ_{xx} and σ_{xy} traces at different temperatures are shown in Supplementary Fig. 3). The σ_{xx} for ν=±1 states is found to show thermally activated behaviour at high temperatures^{11}, where the finite σ_{xx} is attributed to the thermally excited 2D surface or 3D bulk carriers. Such carriers can shunt the edgestate transport and suppress the nonlocal R_{nl} response at high T (ref. 29). We also note that the σ_{xx} versus T curves for ν=0 and ν=±1 states follow the similar trend and have approximately constant separation. We find the averaged separation Δσ_{xx}=1/2 × (σ_{xx}(−1/2, 1/2)+σ_{xx}(1/2, −1/2)−σ_{xx}(1/2, 1/2)−σ_{xx}(−1/2, −1/2)) to be largely Tindependent with a value of (0.27±0.01)e^{2}/h, which we attribute to the conductivity of the quasi1D dissipative edge channel.
Discussions
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 inplane field and can be viewed as a quasi1D domain boundary that separates the top and bottom surfaces with B pointing outward and inward, respectively, thus can support QH edge states^{31}. 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 σ_{xy}=νe^{2}/h=(ν_{t}+ν_{b})e^{2}/h, restricted to integer multiples of e^{2}/h. When the two surfaces have opposite carrier types but one of the them dominates, welldefined QH states with ν=ν_{t}+ν_{b} may still be observed, such as the (−1/2, 3/2) state with σ_{xy}=(−1/2+3/2)e^{2}/h=e^{2}/h and vanishing ρ_{xx}. Previous studies in InAs/(AlSb)/GaSb heterostructurebased electronhole systems^{32,33} also revealed QH effect with R_{xy}=h/(νe^{2})=h/(ν_{e}−ν_{h})e^{2} (ν_{e} and ν_{h} are electron and holefilling factors, both are positive integers) and vanishing R_{xx} when the AlSb barrier (separating electron and hole gases) is sufficiently thin to enable electronhole 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 halfinteger 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 E_{f} 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 l_{B}=(ħ/eB)^{1/2}≅6 nm at B=18 T); however, it has been suggested that even in the presence of wellquantized LLs, a standard TI Hall measurement would exhibit deviations from perfectly quantized values due to conduction through the side surfaces^{31,34,35,36}. 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 nonchiral edge modes^{37}, possibly explaining the good quantization in σ_{xy} and vanishing ρ_{xx} (also R_{nl}) observed in our experiments.
For the (ν_{t}, ν_{b})= (−1/2, 1/2) or (1/2, −1/2) state, the carrier density on the top and bottom surfaces are opposite. Since E_{f} is within the LL gap on both the surfaces, the finite residual σ_{xx} and large R_{nl} we observed are indicative of dissipative edge transport. We show a schematic energy spectrum^{38} of this ν=0 state with V slightly smaller than E_{0−1} in Fig. 4h, where the Fermi level E_{f} resides between the N_{t}=−1 and 0 LL of top surface (marked in blue), thus ν_{t}=−1/2, and also between the N_{b}=0 and 1 LL of bottom surface, thus ν_{b}=1/2. Overall, such energy spectrum represents a (ν_{t}, ν_{b})=(−1/2, 1/2) and ν=0 state. The E_{f} crosses an even number (only two shown in this illustrative example in Fig. 4h) of counterpropagating edge modes (arising from subbands of the quasi1D side surface). The disorder can cause scattering and local equilibrium between the counterpropagating modes, giving rise to nonchiral 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 finitesize confinementinduced gap ≅hv_{F}/t≅10 meV opened at DP of the side surface^{38}), we did not observe a truly insulating state with vanishing σ_{xx} and diverging R_{xx} (Figs 2a and 4a). This is likely due to the disorder potential (spatial fluctuation of DP^{28}) comparable or larger than Δ and thus smearing out this gap (effectively E_{f} always crosses the nonchiral edge modes). It would be an interesting question for future studies to clarify whether the weak Tdependence (at T<∼50 K) of the observed conductance (Fig. 4e), similar to the behaviour reported in InAs/GaSb based electronhole systems^{39}, may indicate an absence of localization^{40,41,42,43} in such quasi1D 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 zeroHallconductance plateau in a magneticdoped TI around the coercive field can be used as platforms to observe the TME effect^{44,45}, where an electric (magnetic) field induces a colinear magnetic (electric) polarization with a quantized magnetoelectric polarizability of ±e^{2}/2 h. A zeroHallplateau state has been recently observed in the QAH case in ultrathin (fewnmthick) films of Cr_{x}(Bi,Sb)_{2x}Te_{3} 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 effect^{45,48}), 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 regime^{49} (in addition to the atzero B field^{17}) induced by spontaneous coherence between stronglyinteracting top and bottom surfaces. In future studies, much thinner samples are likely needed to investigate the possibility of such exciton superfluidity.
Methods
Sample preparation
3D TI single crystals BiSbTeSe_{2} (BSTS) were grown by the vertical Bridgman technique^{11}. BSTS flakes (typical thickness ∼50–200 nm) are exfoliated (Scotch tape method) onto highly doped Si (p+) substrates (with 300 nmthick SiO_{2} coating), and lithographically fabricated into Hall barshaped devices with Cr/Au contacts. A thin flake of hexagonal boron nitride (hBN, typical thickness ∼10–40 nm) is transferred^{19} on the BSTS flake to serve as a topgate dielectric and a topgate metal (Cr/Au) is deposited afterwards. The thickness of BSTS and hBN flakes are measured by atomic force microscopy.
Transport measurement
Transport measurements are performed with the standard lockin technique using a lowfrequency (<20 Hz) excitation current of 20 nA in a helium4 variable temperature system (with base temperature down to 1.6 K) or a helium3 system equipped with magnetic fields (B) up to 18 T (down to 0.3 K).
Additional information
How to cite this article: Xu, Y. et al. Quantum transport of twospecies Dirac fermions in dualgated threedimensional topological insulators. Nat. Commun. 7:11434 doi: 10.1038/ncomms11434 (2016).
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Acknowledgements
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 N660011114107). A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR1157490, the State of Florida, and the U.S. Department of Energy.
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Y.P.C supervised the research. I.M. synthesized the crystals. Y.X. fabricated the devices, performed the transport measurements and analysed the data. Y.X. and Y.P.C wrote the paper.
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Xu, Y., Miotkowski, I. & Chen, Y. Quantum transport of twospecies Dirac fermions in dualgated threedimensional topological insulators. Nat Commun 7, 11434 (2016). https://doi.org/10.1038/ncomms11434
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DOI: https://doi.org/10.1038/ncomms11434
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