Research ArticlePHYSICS

Direct observation of valley-coupled topological current in MoS2

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Science Advances  19 Apr 2019:
Vol. 5, no. 4, eaau6478
DOI: 10.1126/sciadv.aau6478


The valley degree of freedom of electrons in two-dimensional transition metal dichalcogenides has been extensively studied by theory (14), optical (59), and optoelectronic (1013) experiments. However, generation and detection of pure valley current without relying on optical selection have not yet been demonstrated in these materials. Here, we report that valley current can be electrically induced and detected through the valley Hall effect and inverse valley Hall effect, respectively, in monolayer molybdenum disulfide. We compare temperature and channel length dependence of nonlocal electrical signals in monolayer and multilayer samples to distinguish the valley Hall effect from classical ohmic contributions. Notably, valley transport is observed over a distance of 4 μm in monolayer samples at room temperature. Our findings will enable a new generation of electronic devices using the valley degree of freedom, which can be used for future novel valleytronic applications.


Electronic devices exploring carrier transport with spin and valley degree of freedom (DOF) have emerged as promising candidates for next-generation information storage and transport, because pure spin and valley currents do not accompany energy dissipation associated with Joule heating. The ability to electrically generate and detect these pure spin and valley currents in these devices is of particular importance. Over the past decade, driven by the emergence of the spin-orbit coupling engineering, tremendous experimental progress has been made to efficiently generate spin currents by electric currents. On the other hand, electrical control of the valley DOF has just started to attract interest in the past few years, initiated by theoretical studies of valleytronics in two-dimensional honeycomb lattice systems, such as gapped graphene and transition metal dichalcogenides (TMDs) (14, 14, 15), revealing the interplay of their unique band structures and topologies. Experimentally, topological valley transport has been observed in graphene systems when a superlattice structure or perpendicular electric field is used to break the inversion symmetry of this zero-bandgap semiconductor (1618).

In contrast, monolayer TMDs, such as molybdenum disulfide (MoS2), is a direct bandgap semiconductor. Electronic transport in these materials is dominated by the inequivalent K and K′ valleys of the Brillouin zone located at band edges. Because of the inherent absence of inversion symmetry in monolayer TMDs, carriers in these two valleys have nonzero Berry curvature (Ω) without needing the assistance of external mechanisms to break the symmetry as in graphene systems. K and K′ valleys are related by time-reversal symmetry, which forces the Berry curvature to flip its sign, i.e., Ω(K) = − Ω(K′), and allows optical selection through optical pumping of valley polarization (57). Ω acts as a pseudo-magnetic field in the momentum space and results in an anomalous transverse velocity in the presence of an electric field, i.e., ν=eE × Ω(k). Consequently, carriers from K and K′ valleys develop opposite ν, providing a route to electrically generate pure valley currents transverse to the applied electric field. This so-called valley Hall effect (VHE) has been used by Mak et al. (10) in monolayer MoS2 devices to measure valley polarization created by circularly polarized light and has successfully generated polarization in gated bilayer MoS2 that was then visualized by Kerr rotation microscopy (11). It is important to note that this unique VHE phenomenon would not appear in thick multilayer MoS2 devices, because inversion symmetry is completely protected in samples with even layers and starts to recover in thick odd-layer samples (19). In addition, carrier transport in these indirect bandgap multilayer samples does not involve K and K′ valleys, which can result in very small or even zero Berry curvature. In our experiments, we use multilayer MoS2 devices as direct comparison or control samples to monolayer devices. Figure 1A illustrates the VHE occurring in the left vertical electrode of a monolayer MoS2 Hall bar device. Analogous to the spin current, this valley current consists of carriers of opposite (valley) polarization moving along opposite directions, resulting in charge-neutral valley current along the x axis. Onsager reciprocity (20) then ensures the reciprocal effect, a phenomenon defined as the inverse VHE (iVHE) that converts a nonzero valley current into a transverse electric field, and finally develops charge accumulation across the right vertical electrode of the Hall bar in Fig. 1A. Here, we demonstrate electrical generation and detection of valley current in monolayer MoS2 by combining VHE and iVHE in the above-described nonlocal Hall bar device geometry. We observe large nonlocal signals at distances more than 4 μm away from the charge current path and a unique temperature dependence that is consistent with valley transport physics.

Fig. 1 Valley-coupled topological current.

(A) Schematic of valley-coupled topological current due to VHE and iVHE in monolayer MoS2 and the device geometry (bottom), where W1 = 1 μm, W = W2 = 2 μm, L1 = 4.5 μm, and L = 0.5 μm. (B) Schematics of two measurement setups: type I and type II. (C) Patterned MoS2 flake (green) and lithographically defined metal electrodes (yellow).


A colored scanning electron microscopy image of one of the measured MoS2 Hall bar devices is shown in Fig. 1C. Two types of measurements can be made, as illustrated in Fig. 1B. A conventional four-probe measurement (type II) allows the extraction of sheet resistance and contact resistance, while the nonlocal setup (type I) measures the Hall voltage induced by any carrier distributions due to the VHE or classical ohmic contribution. A back-gate voltage (Vg) is applied to the SiO2/Si substrate to modulate the carrier concentration in the MoS2 channel. Device fabrication is provided in Materials and Methods. Optical and electrical characterizations and measurement details are provided in section S1. Typical n-type MoS2 field-effect transistor behaviors are observed in two-probe measurements of all devices; sheet and contact resistances are extracted from type II measurements for various temperatures ranging from 4 to 300 K (see section S1). Field-effect mobility values of ~10 and ~30 cm2/Vs are typically measured at room temperature for monolayer and multilayer devices, respectively.

The most important spurious signal to be ruled out in our measurements is the ohmic contribution that can result in a van der Pauw–like signal (21) in a typical nonlocal type I measurement. When a DC bias of Vds = 5 V is applied to the left electrode of the Hall bar, a nonlocal Hall voltage (Vnl) measured in the on-state of a monolayer MoS2 device (40 V < Vg < 60 V) can reach ~0.6 V at T = 300 K and increase to ~1.2 V at T = 4 K, as compared to ~10- to 50-mV Vnl readings in the on-state of a multilayer (~9 to 10 layers) device (20 V < Vg < 40 V), as shown in Fig. 2 (A and C). As mentioned above, VHE does not exist in thick multilayer MoS2 because inversion symmetry is either preserved or weakly broken and transport does not occur in K and K′ valleys. Therefore, the detected finite Vnl signals in multilayer devices can only be associated with the ohmic contribution or any other unknown effects. The magnitude of the nonlocal voltage due to the ohmic contribution is expected to be dependent on the sheet resistance (ρsh) of the channel and device geometry: Vohmic=IDCρshWW1eπLW (21), where L is the channel length and W and W1 are the width of the channel and the current electrode, respectively (labeled in Fig. 1A). Using individual IDC and ρsh measured for monolayer and multilayer MoS2, we are able to calculate the ohmic contribution as a function of the back-gate voltage (Vg) for each device, as presented in Fig. 2 (B and D). We notice that the magnitude of the measured Vnl of the multilayer device from Fig. 2C matches the values of the calculated ohmic contribution, while more than one order of magnitude larger Vnl signals are measured in the monolayer MoS2 device with an opposite temperature trend that we will discuss later. This significant magnitude difference in measured nonlocal voltages is also supported by a detailed potential analysis that resembles our experimental setup, as shown in Fig. 3. Using experimentally measured contact resistance and MoS2 sheet resistance obtained from the four-probe measurement, only a fraction of the supply voltage (Vds = 5 V) is actually applied across the injector lead, i.e., Vin = 1.8 V. We then simulate in SPICE a resistor network with 4 × 106 identical resistors uniformly distributed over the Hall bar and observe that when a constant voltage of 1.8 V is applied at the injector, the nonlocal voltage drop across the detector lead in the given geometry due to the ohmic contribution is expected to be ~29 mV. This picture can get more complicated by the gate field–controlled Schottky barrier contacts (22). Nevertheless, we conclude that the magnitude of Vnl due to the ohmic contribution calculated from the resistor network is in good agreement with the experimental measurements in multilayer MoS2 devices. We benchmark the SPICE-based resistor network simulation with the analytical equation in section S2. We also rule out the possibility of magnitude difference coming from the characteristic difference between monolayer and multilayer by evaluating the impact of mobilities on nonlocal signals (see section S1).

Fig. 2 Comparison of nonlocal voltages obtained in monolayer and multilayer MoS2 devices.

(A) Measured nonlocal voltage with respect to global back-gate voltage Vg in monolayer MoS2 using type I setup. Inset: Full range of Vg. Note that data points in the range of Vg < 40 V are not included in analysis because these large device resistances become comparable to the input impedance of the nanovoltmeter. (B) Ohmic contribution calculated from the measured sheet resistance: Vohmic=IDCρshWW1eπLW as a function of Vg, plotted with the same y-axis range as in (A). Inset: Zoom-in data. (C and D) Nonlocal voltage response in a multilayer MoS2 device for the same measurements performed in (A) and (B). Note that the y axis in both plots has a unit of millivolts.

Fig. 3 Electric potential mapping from a SPICE-based resistor network simulation.

(A) SPICE simulation of a resistor grid with ~4 × 106 uniform resistors, where each resistor corresponds to ~3-nm channel length, with (x = 1500, y = 1400) points. Vds values applied at the two ends of the injector are V1 = 1.8 V and V2 = 0 V, respectively. Values greater than 0.94 V and less than 0.86 V are denoted with the same colors to resolve the nonlocal voltage distribution. (B) Voltage profiles along the y direction for four different positions denoted by arrows (1 to 4) in (A). Nonlocal voltage difference under open circuit condition is calculated to be ~29 mV.

In addition to the magnitude of Vnl, its temperature dependence provides further evidence in support of the VHE being responsible for the nonlocal carrier transport in monolayer MoS2. Figure 4A shows increasing Vnl with decreasing temperature down to 50 K in monolayer MoS2, while a completely opposite trend is observed for the multilayer in Fig. 4B. Note that, because a voltage source (Vds) is used in our measurements (instead of a constant current source), the temperature dependence of Vohmic due to the sheet resistance (ρsh) is expected to be cancelled out. However, finite contact resistance (Rc) needs to be considered in all MoS2 devices, which prevents ρsh to be eliminated in the evaluation of the ohmic contribution. It is expected that Vohmic=Vds(2RC+ρshL1W1)ρshWW1eπLW. Different temperature dependences of Rc and ρsh are observed in four-probe measurements, as presented in section S1. The increasing Vnl with increasing temperature observed in Fig. 4B for the multilayer MoS2 device (dots) can be fitted by the modified Vohmic equation, considering the contribution from the contact resistance (lines). On the other hand, the increasing Vnl with decreasing temperature down to 50 K for the monolayer MoS2 device is expected for enhanced intervalley scattering length (λ) at low temperatures (9, 23, 24), confirming that valley transport is responsible for the observed large signals. More detailed analyses of nonlocal signals and intervalley scattering length will be discussed in the following section.

Fig. 4 Temperature dependence and extraction of intervalley scattering length.

(A) Measured Vnl as a function of temperature at different Vg for monolayer MoS2. (B) Temperature dependence of multilayer MoS2 at different Vg (dots) and the calculated trends (lines) using the modified ohmic equation, Vohmic=Vds(2RC+ρshL1W1)ρshWW1eπLW, with the consideration of the contact resistance contribution (see section S1). Note that the trends of Vnl with respect to temperature in (A) and (B) are completely opposite. (C) Device geometry and corresponding valley-circuit model that define the geometric parameters in Eq. 1. Details are given in section S3. (D) Temperature dependence of Rnlnorm (normalized to the maximum point) measured at Vg = 58 V [orange dots in (A)]. The empirical fittings use λ(T)=5.5T0.470.16 (dashed blue line) and λ=15T0.73 at T > 100 K (green line). Inset: Calculated temperature dependence of valley Hall angle, θ. (E) λ (T) extracted from Rnl and the power-law dependence described in (D). Inset: Theoretically calculated intervalley scattering length (solid line) and λT0.6 to guide the eye (dashed line).

This increasing Vnl with decreasing temperature trend stops at T ~ 50 K and reaches its maximum value. This unique maximum point results from two extreme limits of λ approaching either zero or infinity. While smaller Vnl is expected with increasing temperature due to a shorter λ, large λ at temperatures lower than 50 K can also lead to reduced nonlocal resistances, Rnl = Vnl/IDC. This transition can actually be analogized to the well-studied quenched Hall effect (25, 26), where the Hall voltage vanishes when the carriers’ longitudinal velocity is much higher than the transverse velocity. We suggest that the observed nonmonotonic temperature dependence of Vnl for monolayer MoS2 is an outcome of the monotonically increasing λ with decreasing temperature. We highlight that this temperature dependence of λ is consistent with the recent observation of increased intervalley scattering rate at higher temperatures in TMDs, which is attributed to phonon-activated intervalley relaxation (27).

We will now quantitatively analyze this interesting temperature dependence of Rnl for monolayer MoS2 using a self-consistent theoretical model describing the VHE. This model, similar to other theoretical descriptions in the literature (18, 21), assumes a uniform and rectangular geometry without considering the arm lengths (Fig. 4C). Also following (18, 21), we use a circuit model that is equivalent (28, 29) to the standard spin-diffusion equation used in the context of materials with spin Hall effect to describe the VHE by defining the valley Hall angle as θ = σxyxx, where σxx and σxy denote longitudinal and transverse Hall conductivities, respectively. The valley Hall conductivity includes both intrinsic and extrinsic contributions and can be written as σxy=σxyin+σxyex (30). When the Fermi level lies close to the conduction band minima, a condition that is fulfilled by our MoS2 devices (see section S7), σxyin, dominates over σxyex (30). Using σxyin2e2h (16) and measured σxx, we estimate θ ~ 0.4 at T = 50 K for our devices, which is similar to the estimation of Gorbachev et al. (16) (see section S7 for a detailed calculation of θ as a function of temperature). As also noted in (18), when θ is not small (i.e., θ ~ 1), one needs to self-consistently solve Rnl, considering the feedback impact of iVHE that behaves as a load to the generating section (induced by the direct VHE) and the impact of VHE that serves as a load to the detecting section (governed by the iVHE). Our circuit model automatically captures these self-consistencies to arbitrary order when solved in SPICE, but it is possible to derive an analytical equation considering only the iVHE at the generator side and the VHE at the detector side as second-order effects. Furthermore, our model takes the width of the arms explicitly, and we can analytically obtain the following expression for the nonlocal resistance (see section S3 for a detailed derivation)RnlVnlIDC=2ρλWexp[Lλ]sinh[W12λ]sinh[W22λ]θ2(exp[W12λ]W1+2λsinh[W12λ]θ2)(exp[W22λ]W2+2λsinh[W22λ]θ2)(1)where λ is the intervalley scattering length; W1 and W2 are the widths of the generating and detecting arm, respectively; ρsh is the sheet resistance; and W is the width of the channel in Fig. 4C. We combine our VHE model (29) with nonmagnetic circuit models that are also derived from a valley-diffusion equation (without any spin-orbit coupling) to obtain the infinite valley loads on both ends, as well as to obtain the valley diffusion in the middle channel whose length is denoted by L, based on the spin-circuit modeling described in (28). Conversely, the VHE model only considers charge transport in the vertical direction and valley-coupled topological current in the longitudinal direction. It is important to note that Eq. 1 is validated by a self-consistent numerical simulation of the composite valley circuit in SPICE simulations and can be analytically reduced to the expression generally used in the literature (21), if we assume θ2 ≪ 1 and W1,2/λ ≪ 1, yielding Rnl=12(θ2Wσλ)exp(Lλ). It is clear from the complete (Eq. 1) and reduced equation that the two extreme limits of λ naturally lead to an optimal intervalley scattering length to reach the maximum nonlocal resistance value.

This unique behavior enables us to quantitatively extract λ. As suggested by Eq. 1, the temperature dependence of Rnl comes from that of λ and θ. With the calculated θ (T) shown in the inset of Fig. 4D (section S7), we are able to fit the normalized nonlocal resistance, Rnlnormcurve (dashed blue line in Fig. 4D, labeled as “Empirical”) by tuning λ (T). Because different physical mechanisms are responsible for the decreasing Rnlnorm in the low and high temperature regimes, we can separately fit the high temperature trend with a power-law function of λT0.73, which is in line with the temperature dependence of intervalley scattering that will be discussed later. Fitting for T > 75 K regime is shown as the solid green line in Fig. 4D. λ (T) is then quantitatively extracted from Rnlnorm and plotted (blue dots) in Fig. 4E, in a good agreement with the power-law fitting at high temperatures. Furthermore, using the analytical expression in (31) describing both acoustic and optical intervalley phonon scattering together with the field-effect mobility extracted from type II measurements, we are able to analytically derive λ (T) as shown by the solid line in the inset of Fig. 4E (see sections S1 and S8). A power-law fitting of λT0.6 (dashed line) is obtained here, which is consistent with the experimental fitting of λT0.73 at T > 100 K. At low temperatures, the extraction of λ > 1 μm from the experimentally measured nonlocal signals is comparable to other valley Hall systems, as reported in (1618).

In addition to this unique λ extraction through fitting the temperature dependence plot, three distinguished long-channel devices were fabricated on the same monolayer MoS2 flake to allow λ extraction through the channel length dependence (see section S6). Nonlocal signals larger than ~450 mV were observed at room temperature for the device with L = 2 μm and ~120 mV for L = 4 μm, and finally diminish when the channel is 5.5 μm long. λ ~ 600 nm is extracted for valley transport at room temperature, which is consistent with the temperature fitting method within a reasonable range considering sample-to-sample variations. In general, λ is believed to be governed at low temperatures by atom-like defects that provide the necessary momentum required for carriers to scatter between K and K′ valleys in the conduction band. In MoS2, these atom-like defects arise due to molybdenum and sulfur vacancies. Recently, it has been pointed out that owing to the symmetry of atomic defects, only molybdenum vacancies can participate in intervalley scattering (32). Fourier transform scanning tunneling spectroscopy studies also provide further evidence (33, 34). The relatively large λ on the micrometer scale extracted from our devices could be a result of relatively low molybdenum vacancy density in our MoS2 sample.

Last, nonlocal signals measured with in-plane magnetic field applied up to 5 T are presented in section S10. As expected, no impact from the magnetic field is observed, indicating the robustness of the valley polarization in monolayer MoS2 and further excluding the possibility that the spin Hall effect is responsible for our measurements. Therefore, these results once again resonate with the mechanism of the VHE (5, 35).

In summary, we report electrically generating and detecting valley-coupled topological current in monolayer MoS2. Our approach provides a unique way to integrate charge, spin, and valley DOFs, which can be useful for emerging device technologies.


Device fabrication

Chemical vapor deposition (CVD)–grown MoS2 films were transferred to 90-nm SiO2 substrates with highly doped Si on the back side serving as a global back gate (Vg). The transfer process includes the following: (i) The sample was spin-coated with polystyrene (PS) followed by immersion in deionized (DI) water, (ii) the PS/MoS2 stack was then detached from the substrate and scooped up by the receiving SiO2 substrate, and (iii) PS was subsequently dissolved by toluene and bathed in acetone and isopropyl alcohol to thoroughly clean it. Standard e-beam lithography using poly(methyl methacrylate) (PMMA) A4 950 resist was used to pattern electric contacts on the CVD MoS2 flakes. Ti/Au (20 nm/80 nm) was deposited in an e-beam evaporator followed by a liftoff process in acetone. The CVD-grown boron nitride (BN) film was transferred from Cu foil onto the devices through a process that involves etching the Cu foil with iron chloride (FeCl3) and immersing it in diluted HCl and DI water alternatingly for a few times before scooping up. This BN layer was inserted to minimize device degradation from PMMA residues after the reactive ion etching (RIE) process. RIE etching mask was defined by e-beam lithography using PMMA A4 950 resist, and BN/MoS2 flakes were etched using Ar/SF6 for 10 s. The final devices were annealed in forming gas (N2/H2) at 300°C for 3 hours, followed by vacuum annealing (~10−8 torr) at 250°C for 4 hours to minimize PMMA residue and threshold voltage shift due to trap charges.


Supplementary material for this article is available at

Section S1. Optical and electrical characterizations

Section S2. Details of resistor network for ohmic contribution

Section S3. Derivation of nonlocal resistance, Rnl

Section S4. Additional nonlocal measurements for multilayer MoS2 devices

Section S5. Additional nonlocal measurements from other monolayer MoS2 devices

Section S6. Channel length and width dependence in monolayer MoS2 device

Section S7. Detailed θ calculation and its temperature trend

Section S8. Detailed λ calculation and its temperature trend

Section S9. Nonlocal internal resistance measurements

Section S10. Applied in-plane magnetic field

Fig. S1. Device layer number confirmations.

Fig. S2. Device electrical characterizations.

Fig. S3. SPICE-based resistor network.

Fig. S4. The lumped valley-circuit model that is used to derive Eq. 1.

Fig. S5. Comparison of analytical equations for Rnl with the full SPICE simulation of the circuit shown in fig. S4.

Fig. S6. Long-channel multilayer MoS2 devices and additional monolayer MoS2 device measurements.

Fig. S7. Length dependence in monolayer MoS2.

Fig. S8. Valley Hall angle and intervalley scattering length.

Fig. S9. Extraction of internal resistance in the nonlocal electrode.

Fig. S10. Vnl measurements with in-plane magnetic field applied.

References (3641)

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Acknowledgments: We would like to acknowledge fruitful discussions with J. Appenzeller and S. Datta. P.U. would like to thank H. Ochoa for useful discussions on intervalley scattering in TMDs. Funding: T.Y.T.H., S.Z., and Z.C. gratefully acknowledge the support of this work by NEW LIMITS, a center in nCORE, a Semiconductor Research Corporation (SRC) program sponsored by NIST through award number 70NANB17H041. P.U. would like to acknowledge the support from the Purdue University Startup Funds. Author contributions: Z.C. conceived and managed the research project. T.Y.T.H. designed and fabricated samples and carried out electrical measurements. T.Y.T.H., K.Y.C., P.U., and Z.C. performed theoretical analysis. S.Z. carried out material preparation and optical characterizations. All authors discussed the results and wrote the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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