Intrinsic valley Hall transport in atomically thin MoS2

Electrons hopping in two-dimensional honeycomb lattices possess a valley degree of freedom in addition to charge and spin. In the absence of inversion symmetry, these systems were predicted to exhibit opposite Hall effects for electrons from different valleys. Such valley Hall effects have been achieved only by extrinsic means, such as substrate coupling, dual gating, and light illuminating. Here we report the first observation of intrinsic valley Hall transport without any extrinsic symmetry breaking in the non-centrosymmetric monolayer and trilayer MoS2, evidenced by considerable nonlocal resistance that scales cubically with local resistance. Such a hallmark survives even at room temperature with a valley diffusion length at micron scale. By contrast, no valley Hall signal is observed in the centrosymmetric bilayer MoS2. Our work elucidates the topological origin of valley Hall effects and marks a significant step towards the purely electrical control of valley degree of freedom in topological valleytronics.

lectron valley degree of freedom emerges as local extrema in the electronic band structures. Inequivalent valleys, well separated in the Brillouin zone, can be energetically degenerate due to symmetry and serve as novel information carriers controllable via external fields [1][2][3][4][5][6] . A feasible means to manipulate such a valley degree of freedom is through a valley Hall effect (VHE) [5][6][7][8][9] . Analogous to an ordinary Hall effect, in which a transverse charge current is driven by a uniform magnetic field in real space, a transverse valley current in the VHE is produced by valley-contrasting Berry curvatures in momentum space. Upon the application of an external electric field, the curvatures drive carriers from different valleys to traverse in opposite directions. Therefore, the VHE has been a major theme in the study of valleytronics, particularly in those 2D materials featuring K and K' valleys in their hexagonal Brillouin zones [10][11][12][13][14][15][16][17][18][19] .
As Berry curvature is even under spatial inversion (P) and odd under time reversal (T), the VHE cannot survive when both P and T symmetries are present. To achieve VHEs in monolayer and bilayer graphene, an elaborately aligned h-BN substrate 10 and a strong dual gating field 11,12 were respectively utilized to break the P symmetry. To excite VHEs in specific valleys 17,18 , circularly polarized lights [20][21][22] were used for breaking the T symmetry in atomically thin transition-metal dichalcogenides (TMDC). Monolayer TMDCs have direct band gaps of optical frequencies at two inequivalent K-valleys 23,24 , due to the intrinsic P asymmetry in their unit cells depicted in Fig. 1a. Thus, Berry curvatures with opposite signs naturally emerge at the two K-valleys. Moreover, the T and mirror symmetries lock the spin and valley indices of the sub-bands split by the spin-orbit couplings, both of which are flipped under T; the spin conservation suppresses the inter-valley scattering. Therefore, monolayer TMDCs have been deemed an ideal platform for realizing intrinsic VHE without extrinsic symmetry breaking 15,16 .
However, the quantum transport in atomically thin TMDCs has been a long-standing challenge due to the low carrier mobility and the large contact resistance in their field-effect devices prepared by an exfoliation method. Recent breakthroughs in the fabrication of low-temperature ohmic contacts for high-mobility 2D TMDC devices [25][26][27][28] have already facilitated the observation of transport hallmarks of Q-valley electrons 28,29 , K-valley electrons 30,31 , K-valley holes [32][33][34] , and Γ-valley holes 35 . These discoveries have revealed the rich and unique valley physics in the platform of atomically thin TMDCs.
In this work, we design nonlocal, layer-dependent, transport measurements to systematically examine the intrinsic VHEs in ntype 2H-MoS 2 . For the first time, we observe nonlocal resistances that exhibit cubic power-law scaling with the local resistances in the monolayers and trilayers, evidencing intrinsic VHEs. Because of the large intrinsic bandgaps and spin-valley locking of TMDCs, such VHEs can even be observed at room temperature in our monolayer devices. Beyond critical carrier densities (∼4.0 × 10 11 cm −2 for monolayers and trilayers), the cubic scaling turns into linear scaling. Notably, only linear scaling is observed in bilayer MoS 2 , where the P symmetry is restored. Intriguingly, although the monolayer and trilayer feature respectively K-and Q-valleys near their conduction-band edges, they display comparable valence-band Berry curvatures, valley Hall signatures, and micron-sized valley diffusion lengths. Our results not only offer the first experimental evidence for the intrinsic VHE but also help elucidate its topological origin 6 in odd-layer TMDCs and pave the way for realizing room-temperature low-dissipation valleytronics by purely electronic means.

Results
Devices for nonlocal measurements. The structure of a monolayer MoS 2 field-effect transistor is sketched in Fig. 1b. Its bright-field cross-sectional scanning transmission electron microscopy (STEM) image in Fig. 1c clearly shows the layered BN-MoS 2 -BN structure without any impurities in the interfaces down to the atomic scale. The device fabrication process includes a dry transfer step followed by a reactive ion etching step 27,28,35 (see Methods and Supplementary Fig. 1 for details). A low contact barrier formed on the n-type MoS 2 is evidenced by the I-V curves, contact resistances ( Supplementary Fig. 2), and the field-effect mobilities μ varied from 500-4000 cm 2 V −1 s −1 for monolayers, 4000-23000 cm 2 V −1 s −1 for bilayers, and 10000-25000 cm 2 V −1 s −1 for trilayers at T = 2 K (Supplementary Figs. 3 and 4). The impurity-free STEM images and the high mobilities coincide well with the low residue carrier densities (n * = 4 × 10 10 cm −2 , see Supplementary Fig. 5).
As for the electronic measurement, an inverse VHE is exploited to detect a valley current, as sketched in Fig. 1d. An applied current I 12 through probes 1 and 2 induces charge imbalance in a remote region, as measured by the voltage drop V 34 between probes 3 and 4 ( Supplementary Fig. 6). The nonlocal resistance R NL = V 34 /I 12 mediated by the valley Hall current was predicted 36 to present cubic power-law dependence on the local resistance R L = V 24 /I 13 .
Nonlocal transport in monolayer MoS 2 . Nonlocal resistance R NL in an n-type monolayer MoS 2 (sample B of length L = 6 μm and width W = 1.5 μm illustrated in the inset of Fig. 1e), measured as a function of gate voltage V g at varied temperatures, is shown in Fig. 1e. A giant R NL is observed in the range of V g ∼ −15 to −25 V that amounts to the electron density n ∼ 10 10 to 10 11 cm −2 . In particular, the observed R NL ∼ 10 6 Ω exceeds the classical ohmic contribution R CL ¼ R L W πL e ÀπL=W $ 10 4 Ω by two orders of magnitude in the range of V g ∼ −15 to −18 V at 2 K and V g ∼ −22 to −25 V at 300 K. Another unexpected feature of R NL is its V g dependence. In sharp contrast to the classical contribution R CL , which decreases gradually with increasing V g , the observed R NL drops by at least one order of magnitude within an increase of several volts in V g . Both the pronounced nonlocal signal and its unusual sensitivity to V g suggest that the observed R NL has a physical origin different from the classical ohmic contribution R CL .
The temperature dependence of R L and R NL uncovers the mesoscopic mechanism of both the local and nonlocal transport. The conduction can be separated into three regimes: the thermal activation (TA) at 250 K > T > 130 K, the nearest-neighbor hopping (NNH) at 130 K > T > 60 K, and the variable-range hopping (VRH) below 60 K (sample A of L = 3.6 μm and W = 1.5 μm, see Fig. 2f and Supplementary Fig. 7a and 7b). These transport regimes are consistent with previous studies 37,38 . Since pronounced nonlocal signals are observed in all three transport regimes, there appears no clear connection between the transport regimes and the onset of strong nonlocal signals. Interestingly, the characteristic temperatures of both NNH and VRH for R NL are much larger than those for R L in the range of V g ∼ −60 to −58 V (Supplementary Fig. 7d and 7e). This indicates a higher energy barrier in the nonlocal transport and an anomalous origin of the nonlocal signal.
To determine the origin of the observed R NL , we investigate the scaling relation between R NL and R L as functions of V g at different temperatures for both sample A (Fig. 2) and sample B ( Supplementary Fig. 8). For a fixed V g , both R L and R NL increase when the temperature is lowered. In sample A, two regimes with distinct scaling behaviors become clearly visible in Fig. 2c, d, the logarithmic plot of R L and R NL at different V g . Above 160 K, the slopes of the lnR NL versus lnR L curves are 1, indicating that R NL ∝ R L . Below 160 K, the slopes turn to 3 in the low electron density regime (R L ≈ 10 8 to 10 9 Ω), which amounts to R NL / R 3 L . Indeed, a diffusive model has predicted such power-law relations 36 , in which a cubic scaling holds for a spin or valley Hall effect 36 . As introduced above and calculated later, the massive Dirac band structure of monolayer MoS 2 produces large valley Hall conductivity σ V xy (see below) but much weaker spin Hall conductivity 36 (see Supplementary Note 1), Therefore, it is natural to attribute the observed nonlocal signal to the VHE, and the obtained cubic scaling may be analyzed by the predicted formula 36 where l V is the valley diffusion length (or inter-valley scattering length), and σ XX and R L have the simple relation of σ xx ¼ L R L W . We will focus on such a VHE-based hypothesis now and elaborate more on the exclude of spin Hall effect in Discussion.
The R NL and R L data measured at different temperatures for the case of V g = −60 V are plotted in Fig. 2d. The cubic law is not applicable above 160 K, due to the enhancement of inter-valley scattering by the smear of the lowest conduction sub-band spin splitting (estimated as E s /k B ∼ 169 K, see Supplementary Fig. 9) at high temperatures. Below 160 K, Eq. (1) can be employed to estimate l V . For the case of intermediate inter-valley scattering and edge roughness, l V ∼ 0.36 μm if we assume σ V xy $ 1e 2 =h. In the limit of strong inter-valley scattering and edge roughness, l V ∼ 0.43 μm if we assume σ V xy $ 0:1e 2 =h. These values of l V are comparable to those obtained in graphene systems [10][11][12][13][14]17,18 .
We further investigated the length dependence of the nonlocal valley transport. Apart from sample A (L = 3.6 μm) and sample B (L = 6 μm), two more samples (L = 11 μm and 16 μm) are investigated ( Supplementary Fig. 8). The semilog plot of R NL at n = 4 × 10 11 cm −2 (extracted from the Hall measurement, see Supplementary Fig. 10) versus the sample length yields an estimate of l V ∼ 1 μm (Fig. 2g). This value is very close to W and much larger than the electron mean-free path l m ∼ 20 nm (estimated from the sample mobility μ for the range of n where the cubic scaling appears) and the localization length ξ ∼ 50 nm  (see Supplementary Fig. 11). Nevertheless, these estimates based on the observed nonlocal signals are suggestive of l v in the order of micron. In sample B, the cubic scaling remains even at room temperature, attributed to the dominant valence-band contribution to σ V xy and particularly the large intrinsic bandgap that is impossible for graphene systems.
Nonlocal transport in bilayer and trilayer MoS 2 . For bilayer MoS 2 , the measured R L and R NL as functions of V g at different temperatures are plotted in Fig. 3a, b. As the carrier density increases, R L and R NL decrease in a similar fashion in the temperature range of 5-50 K. This yields a linear scaling behavior between R L and R NL , as analyzed in Fig. 3c, d, and no cubic scaling is detected. We note that extrinsic P symmetry breaking can be introduced into atomically thin bilayers via external gating, as achieved in bilayer graphene 11,12 , and that detecting a nonlocal signal in gated bilayer graphene requires a threshold gating strength 11,12 . In our devices, however, V g is too low to reach the threshold estimated by an recent optical experiment 18 , the estimated potential difference between the top and bottom layers is ∼ 9.2 meV at V g = −60 V. This weak symmetry breaking produces little change in the total Berry curvature as compared with the pristine case ( Supplementary Fig. 12), given the facts that the induced potential is much smaller than the bandgap and that the valence-band contribution to σ V xy is dominant. In light of this analysis, the gating-induced P symmetry breaking is negligible in our bilayer MoS 2 . Therefore, we conclude that the absence of cubic scaling in bilayer MoS 2 indicates the crucial role of strong P symmetry breaking in generating VHE. This is consistent with the theoretical understanding of VHE [5][6][7][8][9] , as aforementioned in Introduction.
This key conclusion can be immediately tested in thicker MoS 2 samples. Given that P symmetry is broken (respected) in pristine odd-layer (even-layer) MoS 2 , one might wonder whether the intrinsic VHE and its cubic scaling could be detected in trilayer MoS 2 . Figure 3e, f display our R L and R NL data measured in trilayer MoS 2 as functions of V g at different temperatures. Evidently, the measured R NL rapidly decreases as V g increases in the narrow range of −20 V < V g < −18.4 V, which is reminiscent of the behavior of R NL in our monolayer devices in the low density regime. Similar to the monolayer case, the logarithmic -55 -50 -60

T (K)
In (R NL )   2  10  20  30  50  70  80  100  130  160  200  250  c Scaling relations between ln R L and ln R NL at V g ranging from −50 V to −60 V. When the electron density is relatively high, i.e., R L and R NL are small, R NL is linearly proportional to R L . When the electron density is relatively low, a crossover from linear to cubic scaling is observed. The critical density nc = 4 × 10 11 cm −2 , with the gate voltage V g = −57 V. d Crossover phenomenon by considering classical diffusion (R NL ∝ R L ) and valley Hall transport (R NL ∝ R L 3 ). The experimental data (solid circles, V g = −60 V) clearly show two different regimes which are fitted by two linear curves (orange dashed line with slope 1 and blue dashed line with slope 3). The critical temperature is around 160 K~200 K, as marked by the blue arrow. e R NL plotted as a function of V g at low temperatures. The ohmic contribution, calculated according to R L and device geometry, is deducted from the measured R NL at different temperatures. f 1/R NL (orange circles) and 1/R L (blue circles) in log scale plotted as functions of 1/T at V g = −60 V. Three distinct transport regimes were observed: the thermal activation (TA) transport, nearest neighbor hopping (NNH) transport, and the variable range hopping (VRH) transport. g Semilog plot of R NL as a function of L at n = 2 × 10 11 1). We obtain l V ∼ 0.5 μm and ∼1 μm, respectively, for the aforementioned two limits σ V xy $ 1e 2 =h and $0:1 e 2 =h. Both the observed amplitude of nonlocal signal and the estimated valley diffusion length in the trilayer MoS 2 devices are comparable to those in the monolayer case. In addition to the crucial role of P symmetry breaking, significantly, these observations are suggestive of a universal physical origin of VHEs in odd-layer TMDCs, as discussed below.
Layer-dependent Berry curvatures. To better understand the thickness dependent observations, we calculate the electronic band structures and Berry curvatures 15,16 for monolayer, bilayer, and trilayer MoS 2 . The band structures in Fig. 4a, c are indeed thickness dependent. In particular, the conduction-band minima lie at the K-valleys for the monolayer, whereas they shift to the Qvalleys for the bilayer and trilayer. Given the low electron densities in our samples (∼4 × 10 11 cm −2 in monolayers and trilayers, ∼1 × 10 12 cm −2 in bilayers), the Fermi levels only cross the lowest conduction sub-bands, as indicated by the green lines in Fig. 4a, c. As bilayer MoS 2 has a restored P symmetry that is intrinsically broken in odd-layer MoS 2 , the sub-bands are spin degenerate in the bilayer yet spin split in the monolayer and trilayer. With these band structures, we further compute the Berry curvatures that drive the VHEs. Berry curvature vanishes if both P and T are present. As plotted in Fig. 4d, f, our calculations reveal that the curvatures are indeed trivial in the bilayer yet substantial in the monolayer and trilayer. This explains the reason why no cubic scaling is observed in bilayer MoS 2 and highlights the role of P symmetry breaking in producing VHEs.
It is puzzling to understand and compare the nonlocal signals of VHEs in monolayer and trilayer MoS 2 . Similar cubic scaling behaviors and their transitions to linear ones above the critical densities or temperatures are observed in both cases. However, the conduction-band Berry curvatures (the difference between the blue and orange curves in Fig. 4d, f) are large in the monolayer Kvalleys yet negligibly small in the trilayer Q-valleys. This implies that the geometric explanation of VHE requiring finite doping 5 should not be the origin 11 , which is further evidenced by the fact that the cubic scaling behaviors weaken rapidly with increasing the electron densities.
On the other hand, these facts appear to be in harmony with the topological VHE 6-9 that arises from the valley Hall conductivity (see Supplementary Note 1). This conductivity amounts to the total valley-contrasting Berry curvature contributed from all occupied states, i.e., all the states below the Fermi level if at zero temperature. In our case, the monolayer and trilayer share almost identical substantial valence-band Berry curvatures (the orange curves in Fig. 4d, f), due to the extremely weak interlayer couplings. By contrast, the conduction-band contributions are different but very minor (the difference between the blue and orange curves in Fig. 4d, f)    (1) can be done for a theoretical hypothesis of spin Hall effect as well, it appears that this is not the case for three reasons. First, the spin Hall conductivities are predicted to be very small for pristine odd-layer TMDCs when the valence bands are fully filled 16 (see Supplementary Note 1). Second, the observed nonlocal resistances have little response to a magnetic field up to 9 T (Supplementary Fig. 13). Third, the spin diffusion length in TMDCs is at the scale of several tens of nanometers 40,41 , which is 1-2 orders smaller than the extracted diffusion lengths based on our experimental data or Eq. (1).
In conclusion, the pronounced nonlocal signals are observed in our MoS 2 samples with length up to 16 μm and at temperature up to 300 K. The valley diffusion lengths are also estimated to be in the order of micron. The low carrier concentration ensures the low possibility of bulk inter-valley scattering and maintains a long valley diffusion length. In addition, the mirror and T symmetries lock the spin and valley indices of the lowest sub-bands, preventing bulk inter-valley scattering via spin conservation. Our observed intrinsic VHEs and their long valley diffusion lengths are promising for realizing room-temperature low-dissipation valleytronics. To better elucidate the outstanding problems of both geometric 5 and topological [6][7][8][9] VHEs, our observations and analyses call for future efforts, particularly complementary experiments in p-type TMDCs (where spin Hall conductivities are predicted to be much larger 16 ) such as the one 42 that we became aware of during the peer review process.

Layer numbers and stacking orders.
To determine the number of layers for a MoS 2 sample, we carried out micro-Raman and photoluminescence measurements before making a device (Supplementary Fig. 14). We also took cross-sectional STEM (JEOL JEM-ARM200F Cs-corrected TEM, operating at 60 kV) images after the electronic measurement. The STEM image can clearly determine the number of MoS 2 layers (Supplementary Fig. 14) and distinguishes the 2H stacking order from other stacking orders such as 1T and 3R (Supplementary Fig. 15).

Selective etching process.
A hard mask is patterned on the heterostructure by the standard e-beam lithography technique using PMMA (step 2 in Supplementary  Fig. 1). The exposed top BN layer and MoS 2 are then etched via reactive ion etching (RIE), forming a Hall bar geometry (steps 3 & 4 in Supplementary Fig. 1). Then a second-round e-beam lithography and RIE is carried out to expose the MoS 2 layer (steps 5 & 6 in Supplementary Fig. 1). The electrodes are then patterned by a thirdround e-beam lithography followed by a standard e-beam evaporation (steps 7 & 8   in Supplementary Fig. 1). To access the conduction band edges of MoS 2 , we choose Titanium as the contact metal, as the work function of Titanium (∼4.3 eV) matches the band-edge energy of MoS 2 (∼4.0-4.4 eV depending on the layer numbers).
Electronic measurement. The I-V curves are measured by Keithley 6430. Other transport measurements are carried out by using: (i) low-frequency lock-in technique (SR 830 with SR550 as the preamplifier and DS 360 as the function generator, or (ii) Keithley 6430 source meter (>10 16 Ω input resistance on voltage measurements). The cryogenic system provides stable temperatures ranging from 1.4 to 300 K. A detailed discussion of the nonlocal measurement is presented in Supplementary Fig. 6.
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