Abstract
Electrons hopping in twodimensional 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 noncentrosymmetric monolayer and trilayer MoS_{2}, 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 MoS_{2}. 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.
Introduction
Electron 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 valleycontrasting 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 hBN 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 transitionmetal dichalcogenides (TMDC). Monolayer TMDCs have direct band gaps of optical frequencies at two inequivalent Kvalleys^{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 Kvalleys. Moreover, the T and mirror symmetries lock the spin and valley indices of the subbands split by the spinorbit couplings, both of which are flipped under T; the spin conservation suppresses the intervalley 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 longstanding challenge due to the low carrier mobility and the large contact resistance in their fieldeffect devices prepared by an exfoliation method. Recent breakthroughs in the fabrication of lowtemperature ohmic contacts for highmobility 2D TMDC devices^{25,26,27,28} have already facilitated the observation of transport hallmarks of Qvalley electrons^{28,29}, Kvalley electrons^{30,31}, Kvalley holes^{32,33,34}, and Γvalley 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, layerdependent, transport measurements to systematically examine the intrinsic VHEs in ntype 2HMoS_{2}. For the first time, we observe nonlocal resistances that exhibit cubic powerlaw scaling with the local resistances in the monolayers and trilayers, evidencing intrinsic VHEs. Because of the large intrinsic bandgaps and spinvalley 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 Qvalleys near their conductionband edges, they display comparable valenceband Berry curvatures, valley Hall signatures, and micronsized 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 oddlayer TMDCs and pave the way for realizing roomtemperature lowdissipation valleytronics by purely electronic means.
Results
Devices for nonlocal measurements
The structure of a monolayer MoS_{2} fieldeffect transistor is sketched in Fig. 1b. Its brightfield crosssectional scanning transmission electron microscopy (STEM) image in Fig. 1c clearly shows the layered BNMoS_{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 ntype MoS_{2} is evidenced by the IV curves, contact resistances (Supplementary Fig. 2), and the fieldeffect 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 impurityfree 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 powerlaw dependence on the local resistance R_{L} = V_{24}/I_{13}.
Nonlocal transport in monolayer MoS_{2}
Nonlocal resistance R_{NL} in an ntype 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 \(R_{{\mathrm{CL}}} = R_{\mathrm{L}}\frac{W}{{{\mathrm{\pi }}L}}{\mathrm{e}}^{  {\mathrm{\pi }}L/W} \sim 10^4\,\Omega\) 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 nearestneighbor hopping (NNH) at 130 K > T > 60 K, and the variablerange 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 \(R_{{\mathrm{NL}}} \propto R_{\mathrm{L}}^3\). Indeed, a diffusive model has predicted such powerlaw 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 \(\sigma _{{\mathrm{xy}}}^{\mathrm{V}}\) (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 intervalley scattering length), and σ_{XX} and R_{L} have the simple relation of \(\sigma _{{\mathrm{xx}}} = \frac{L}{{R_{\mathrm{L}}W}}\). We will focus on such a VHEbased 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 intervalley scattering by the smear of the lowest conduction subband 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 intervalley scattering and edge roughness, l_{V} ∼ 0.36 μm if we assume \(\sigma _{{\mathrm{xy}}}^{\mathrm{V}}\sim 1e^2/h\). In the limit of strong intervalley scattering and edge roughness, l_{V} ∼ 0.43 μm if we assume \(\sigma _{{\mathrm{xy}}}^{\mathrm{V}}\sim 0.1e^2/h\). 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 meanfree 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 valenceband contribution to \(\sigma _{{\mathrm{xy}}}^{\mathrm{V}}\) 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 valenceband contribution to \(\sigma _{{\mathrm{xy}}}^{\mathrm{V}}\) is dominant. In light of this analysis, the gatinginduced 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 oddlayer (evenlayer) 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 plots of R_{L} and R_{NL} in Fig. 3g exhibit clear changes in slop from 1 to 3 near V_{g} = −18.4 V, further confirming the observation of the nonlocal signal of VHE in trilayer MoS_{2}. To illustrate the temperature dependence, Fig. 3h plots the scaling relation between R_{L} and R_{NL} at different temperatures for the case of V_{g} = −20 V. Again, there is a clear change in slop from 1 to 3 near 30 K. Moreover, the valley diffusion length can be extracted based on Fig. 3h and Eq. (1). We obtain l_{V} ∼ 0.5 μm and ∼1 μm, respectively, for the aforementioned two limits \(\sigma _{{\mathrm{xy}}}^{\mathrm{V}}\sim 1e^2/h\;{\mathrm{and}}\sim \hskip 2pt 0.1\;e^2/h\). 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 oddlayer TMDCs, as discussed below.
Layerdependent 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 conductionband minima lie at the Kvalleys 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 subbands, as indicated by the green lines in Fig. 4a, c. As bilayer MoS_{2} has a restored P symmetry that is intrinsically broken in oddlayer MoS_{2}, the subbands 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 conductionband 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 Qvalleys. 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^{6,7,8,9} that arises from the valley Hall conductivity (see Supplementary Note 1). This conductivity amounts to the total valleycontrasting 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 valenceband Berry curvatures (the orange curves in Fig. 4d, f), due to the extremely weak interlayer couplings. By contrast, the conductionband contributions are different but very minor (the difference between the blue and orange curves in Fig. 4d, f) because of the low electron densities. Therefore, the valenceband contributions dominate the valley Hall conductivities, leading to similar nonlocal signals of VHEs in monolayer and trilayer. Recently, nearly quantized edge transports have been observed along the designed or selected domain walls in graphene systems^{13,14} and even in artificial crystals^{19}. In our case, the roughness of natural edges can cause edge intervalley scattering^{8} and remove any possible edge state^{39}. This also partly reduces the valenceband contributions^{4} which in principle would result in a quantized valley Hall conductivity (valley Chern number^{6,8}) in the massive Dirac model.
Discussion
Finally, we note that the VHE and spin Hall effects are distinct in TMDCs, in spite of the spinvalley locking. The spinvalley locking is a property at Fermi level only when it lies in the lowest conduction or highest valence subband. Yet, all states below Fermi level contribute to the spin and valley Hall conductivities^{6} (see Supplementary Note 1). Although a similar line of analysis based on Eq. (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 oddlayer 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 intervalley scattering and maintains a long valley diffusion length. In addition, the mirror and T symmetries lock the spin and valley indices of the lowest subbands, preventing bulk intervalley scattering via spin conservation. Our observed intrinsic VHEs and their long valley diffusion lengths are promising for realizing roomtemperature lowdissipation 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 ptype 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.
Methods
Van der Waals structures
MoS_{2} bulk crystals are bought from 2D semiconductors (website: http://www.2dsemiconductors.com/), and the hBN sources (grade A1) are bought from HQ graphene (website: http://www.hqgraphene.com/). To fabricate van der Waals heterostructures, a selected MoS_{2} sample is picked from the SiO_{2}/Si substrate by a thin hBN flake (5–15 nm thick) on PMMA (950 A7, 500 nm) via van der Waals interactions. The hBN/MoS_{2} flake is then transferred onto a fresh thick hBN flake lying on another SiO_{2}/Si substrate, to form a BNMoS_{2}BN heterostructure (step 1 in Supplementary Fig. 1).
Layer numbers and stacking orders
To determine the number of layers for a MoS_{2} sample, we carried out microRaman and photoluminescence measurements before making a device (Supplementary Fig. 14). We also took crosssectional STEM (JEOL JEMARM200F Cscorrected 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 ebeam 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 secondround ebeam 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 ebeam lithography followed by a standard ebeam 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 bandedge 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) lowfrequency lockin 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.
Data availability
The authors declare that the major data supporting the findings of this study are available within the paper and its Supplementary Information. Extra data are available from the authors upon reasonable request.
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Acknowledgements
We acknowledge the financial support from the Research Grants Council of Hong Kong (Project Nos. 16300717, 16302215, 16324216, C703617W, C602616W, and HKUST3/CRF13G), the Croucher Foundation, the Dr. Taichin Lo Foundation, the National Key R&D Program of China (with grant number 2017YFB0701600), US Army Research Office (under grant number W911NF1810416), and UTDallas Research Enhancement Funds. We acknowledge the technical support from Wing Ki Wong and the Raith–HKUST Nanotechnology Laboratory at MCPF. Z.W. acknowledges useful conversations with Danru Qu. F.Z. is grateful to Allan MacDonald, Joe Qiu, and Di Xiao for valuable discussions.
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Z.W. and N.W. conceived and designed the experiments. F.Z. provided the theoretical support. N.W. and F.Z. supervised the work. Z.W. fabricated the devices, performed the measurements, and analyzed the data with the help from M.H., J.L., T.H., L.A., Y.W., S.X., G.L., C.C. and K.T.L. X.C. carried out the STEM characterizations. G.B.L. computed the band structures and Berry curvatures. F.Z. and P.C. analyzed the band structures and Berry curvatures and provided theoretical explanations. Z.W., F.Z., B.T.Z. and N.W. wrote the manuscript with contributions from all authors.
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Wu, Z., Zhou, B.T., Cai, X. et al. Intrinsic valley Hall transport in atomically thin MoS_{2}. Nat Commun 10, 611 (2019). https://doi.org/10.1038/s41467019086299
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