## Main

In crystalline solids, a topological current can be induced by the Berry phase of the electronic wavefunction6. Examples include the quantum Hall current in a magnetic field, and the spin Hall current arising from spin–orbit coupling7. Such topological transport is robust against impurities and defects in materials—a feature that is much sought after in potential electronic applications. In such applications, the ability to switch and to continuously tune the topological transport is crucial. The topological current is in principle dictated by the crystal symmetry, which is difficult to change in bulk materials. Bilayer graphene (BLG), however, offers new opportunities in which inversion symmetry can be controllably broken by an external electric field in the perpendicular direction.

The topological current controlled by the inversion symmetry breaking is associated with carriers’ valley pseudospin, which characterizes the twofold degenerate band edges located at the corners of the hexagonal Brillouin zone. The topological Hall current, odd under time reversal but even under inversion, is strictly zero in pristine mono- and bilayer graphene, which respect both symmetries. When the inversion symmetry is broken, however, time-reversal symmetry requires the Hall currents to have opposite signs and equal magnitudes in the two valleys (that is, a valley Hall effect), as recently demonstrated in monolayer graphene4. Microscopically, inversion symmetry breaking opens a bandgap at the charge neutral point (CNP), and produces valley-contrasted Berry curvatures2. The Berry curvatures act similarly to a momentum-space magnetic field that causes the Hall effect at finite doping6. In BLG, the inversion symmetry (and the Berry curvature) is for the first time controlled by gate electric field. Valley Hall current is therefore expected to be fully tunable by external gates, bringing topological transport in line with modern electronic technology.

In this study, we demonstrated tunable topological valley Hall transport in gate-biased BLG with nonlocal measurement in a Hall bar configuration (Fig. 1a and c inset), which has been used to detect nonlocal transport in other spin/pseudospin systems4,8,9,10,11. The charge current injected at one end of the Hall bar induced a pure valley Hall current in the transverse direction and, because of the inverse valley Hall effect, converted to a charge imbalance at the other end of the sample; we therefore detected a nonlocal response as a voltage drop. We demonstrated that the giant, tunable, nonlocal response can be induced by the perpendicular gate electric field, but is absent in the gapless BLG in which inversion symmetry is present. Such gate tunability indicates the essential role of inversion symmetry breaking in nonlocal transport, and provides unambiguous evidence that our nonlocal signal was a result of the valley Hall effect. The nonlocal transport persists up to room temperature and over long distances (up to 10 μm). Our results represent major progress in the quest for a robust, tunable valley pseudospin system among various alternatives3,4,5,12,13, and indicate the possibility of using the nonlocal topological transport in practical applications under ambient conditions.

The structure of our dual-gate graphene field-effect transistors (FETs) is shown in Fig. 1a, b. We constructed the device by sequentially transferring BLG and hexagonal boron nitride (hBN) flakes onto an hBN substrate supported on a SiO2/Si wafer (SiO2 thickness = 300 nm). Before the deposition of the top hBN flake, BLG was etched into a well-defined Hall bar geometry (Fig. 1a, broken line) for easy characterization in both local and nonlocal configurations. The whole stack was etched again after the deposition of the top hBN to expose the graphene edge for making electrical contacts with metallic leads14 (Cr/Au, 8 nm and 70 nm, respectively). The final device had BLG sandwiched between the top and bottom gates (Fig. 1b), and the hBN gate dielectric ensured excellent sample quality (see Methods).

Voltages applied on top and bottom gates (Vt and Vb) enabled us to independently control the gap opening and carrier doping in the BLG. The bandgap Eg is determined by the average of top- and bottom-gate-induced electrical displacement fields, D = (Dt + Db)/2, which breaks the inversion symmetry of the BLG (ref. 15). The carrier doping n can be tuned by the difference of the two displacement fields, n = ɛ0(DtDb), where ɛ0 is the vacuum permittivity. In our experiment, the drain electrode was grounded, and the displacement fields were related to top- and bottom-gate voltages by Dt = −ɛt(VtVt0)/dt and Db = ɛb(VbVb0)/db, where ɛ and d are the dielectric constant and thickness of the dielectric layers, respectively, and Vt0 and Vb0 are effective offset voltages caused by environment-induced carrier doping. The resistance of our sample measured in a standard four-terminal set-up (referred to as local resistance RL) exhibited the typical behaviour of BLG: the CNP manifested as a peak in RL as the carrier density was varied in a pristine sample (Fig. 1c, blue), and the peak value increased substantially as a bandgap was opened by a finite field D (Fig. 2a–d, blue). RL plotted as a function of both Vt and Vb shows the effect of gate bias more clearly in Fig. 2e. Along the line defined by the (Vt, Vb) pairs at the CNP, the bandgap is fully tuned by D, whereas the sample remained charge neutral. The sample exhibited a net charge doping perpendicular to the line in the (Vt, Vb) plane, with the bandgap remaining constant.

A pronounced nonlocal response appeared as the bandgap opened in BLG at low temperatures. We detected the response by sending a current I through the local leads, and sensing the nonlocal voltage VNL at the far end of the sample (see Fig. 1c inset for the measurement set-up). The nonlocal resistance RNL, defined as RNL = VNL/I, is negligible in a pristine sample with zero gap opening (Fig. 1c, orange). A peak in RNL, however, appeared at a threshold displacement field of D = 0.28 V nm−1, and increased rapidly to the order of a few hundred Ω at large D (Fig. 2a–d, data obtained at temperature T = 70 K). The two-dimensional plot of RNL as a function of Vt and Vb shows the general behaviour of the nonlocal response at T = 70 K (Fig. 2f). RNL generally peaks at the CNP, but two features distinguish the behaviour of RNL from that of the local resistance RL: the RNL peak is generally sharper than the RL peak, and RNL drops to zero outside the peak whereas RL maintains an order of approximately 100 kΩ at finite doping. Both features indicate the distinct origins of RNL and RL, and are general behaviour of nonlocal transport observed in other graphene pseudospin and spin systems4,8,10. In the present study, we noted that the peak positions of RNL and RL were not always aligned with each other; we attribute the misalignment to inhomogeneities that were present in our samples (Supplementary Section IV). We emphasize that the observed RNL was not from the stray charge current that contributes an ohmic nonlocal resistance8. Such ohmic contribution decreases exponentially with the sample length-to-width ratio L/w, and is up to two orders of magnitude lower than the observed RNL in the device shown in Fig. 2a–d (L = 5 μm and w = 1.5 μm; the ohmic contribution to RNL is represented by the broken line). A pronounced nonlocal signal was observed in devices with an even higher L/w ratio of up to 6.7 (Fig. 3e).

The temperature dependence of RNL and RL revealed crucial information on the microscopic mechanism of both the local and nonlocal transport. Three distinct transport regimes were observed at large fields (Fig. 3a): thermal activation at high temperature; nearest-neighbour hopping at intermediate temperature; and variable-range hopping at low temperature. These transport regimes were consistent with previous studies16,17 (Supplementary Section V). In particular, the high-temperature activation behaviour enabled us to extract the BLG bandgap Eg as a function of D, which agreed well with theoretical calculations and previous measurements15,16 (Fig. 3a inset and Supplementary Section V).

Pronounced nonlocal signal is observed in both thermal activation and hopping regimes (Fig. 3b). We found that RNL also followed an exponential activated behaviour in the thermal activation regime at high temperatures (Fig. 3b inset), although with an exponent higher than Eg/2kBT (Supplementary Fig. 5d). The connection between RNL and RL became obvious when lnRNL was plotted against lnRL (Fig. 3c). Data sets for different D were all in straight lines with the same slope α = 2.77 ± 0.02 in the thermal activation regime; hence, a simple relation RNL RLα can be established. We noted that a diffusive nonlocal topological transport model indeed predicted a power-law relation RNL RL3σxy2${\text{e}}^{\text{-}L/{\lambda }_{\text{v}}}$/λv (ref. 18), where σxy is the valley Hall conductivity, and λv is the valley diffusion length. Our observation agreed reasonably well with this predicted scaling between RNL and RL, and the deviation of α from 3 may indicate the complicated role of σxy and/or λv in RNL. α varied among samples, probably as a result of differences in sample disorder (Supplementary Fig. 6). Intriguingly, our data show that the simple relation RNL RLα persists beyond the thermal activation regime and into the hopping regime until the curves eventually level off (Fig. 3c). The same power law also describes the scaling relation between RNL and RL (with an α close to 3) at finite carrier densities (Supplementary Section X).

The widely tunable bandgap in BLG provides another crucial benefit, which is the room-temperature operation of our BLG nonlocal FET. Figure 3d shows the RNL observed up to room temperature in a BLG biased at D = −1.23 V nm−1 (corresponding to Eg = 135 meV). A wide bandgap alone does not guarantee high-temperature nonlocal transport because the nonlocal signal, along with RL, decreases exponentially with temperature in the thermal activation regime. However, RNL does not decrease as rapidly with increasing temperature in the hopping regime. When the onset of the hopping regime occurs at high temperatures, room-temperature operation consequently remains possible. The key, therefore, is to find samples in which the onset of the hopping regime occurs at near room temperature, as we have demonstrated in the samples shown in Fig. 3d and Supplementary Fig. 7.

We now turn to the length dependence of the nonlocal valley transport. Here we note the analogy between the valley transport in biased BLG and the spin transport in which spin-flip scattering causes the spin diffusion current to decay exponentially. The valley current in BLG decreases through intervalley scattering, which requires a large momentum transfer (for example, by atomic-scale disorders). Such disorders are found to be extremely rare in cleaved BLG crystals19,20, implying a large valley diffusion length λv (refs 21, 22). In the present study, an appreciable nonlocal signal was observed in samples up to 10 μm long. Figure 3e shows the length-dependent RNL, measured on a single device under a field of D = −0.47 V nm−1. From a line fit to the semilog plot of the RNL peak value as a function of sample length (Fig. 3e inset), we obtained an order of magnitude estimation λv 1 μm, and sample inhomogeneity (manifested as a shift of RNL peaks in Fig. 3e) prevents a more precise estimation. Such a large length scale agrees reasonably well with recent studies on intervalley scattering21,22, and is also consistent with the areal density of atomic defects (12.05 μm−2) found in Kish graphite23, which is the same type of specimen used in this study.

The observation of a giant nonlocal response in the middle of the energy gap in insulating BLG was unexpected. Although midgap helical edge states in 2D quantum spin Hall systems can support long-range nonlocal conduction24,25,26, such spin helical edge states did not exist in our study because BLG is topologically trivial. Midgap valley helical modes may still exist at topological domain walls or edges of BLG in certain circumstances27,28,29, and may potentially lead to nonlocal conduction. The nonlocal transport through edge states and bulk states, however, exhibit markedly different length dependences; for a given sample width, bulk conduction depends on the active length of the sample, whereas the edge state conduction depends only on the length of the edge8. We took advantage of this difference to fabricate the device shown in Fig. 4 inset: two Hall bars (left and right with shared current injection leads 2 and 5) have the same active sample length but substantially different edge lengths. The fact that a comparable nonlocal response was observed on both Hall bars unambiguously demonstrates the bulk origin of our valley transport. A robust nonlocal response with the same sign and order of magnitude was observed in all of the samples that we fabricated on hBN (6 in total). Helical modes at the domain walls, which differ according to sample if they exist at all, are unlikely to be the origin of the observed nonlocal transport.

The question then arises as to what physical mechanism leads to the nonlocal valley transport that we observed. The valley Hall effect originally proposed in graphene requires finite doping2, and is therefore not applicable in our insulating BLG. To this end, we note that nonlocal valley transport in an insulator without helical edge states is an area still open to theoretical investigation30, and theories on the anomalous Hall effect in insulating ferromagnetic materials may provide important clues31,32,33. Our results call for a continued effort, both experimental and theoretical, to address this outstanding problem.