The valley Nernst effect in WSe2

The Hall effect can be extended by inducing a temperature gradient in lieu of electric field that is known as the Nernst (-Ettingshausen) effect. The recently discovered spin Nernst effect in heavy metals continues to enrich the picture of Nernst effect-related phenomena. However, the collection would not be complete without mentioning the valley degree of freedom benchmarked by the valley Hall effect. Here we show the experimental evidence of its missing counterpart, the valley Nernst effect. Using millimeter-sized WSe\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{2}$$\end{document}2 mono-multi-layers and the ferromagnetic resonance-spin pumping technique, we are able to apply a temperature gradient by off-centering the sample in the radio frequency cavity and address a single valley through spin-valley coupling. The combination of a temperature gradient and the valley polarization leads to the valley Nernst effect in WSe\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{2}$$\end{document}2 that we detect electrically at room temperature. The valley Nernst coefficient is in good agreement with the predicted value.

S emiconducting transition metal dichalcogenides (TMDCs) have gained a great attention because of their fascinating electronic and optical properties when downing the thickness to one monolayer 1,2 . Many of the TMDCs have since been shown to exhibit a strong excitonic feature, showing strong and spectrally narrow photoluminescence and absorption making them very good candidates for optoelectronic devices 3,4 . The combination of a bandgap (1-2 eV) and an atomically thin conducting channel also makes TMDCs ideal materials for next generation field effect transitors in microelectronics 5 . Finally, in the monolayer form, due to the breaking of inversion symmetry and strong spin-orbit coupling, the carriers in TMDCs carry a valley index noted K þ or K À . This valley degree of freedom refers to local extrema in the band structure of TMDC monolayers at the K points of the two-dimensional Brillouin zone and it can be exploited to develop novel valleytronic devices [6][7][8] . In MX 2 (M = Mo, W and X = S, Se) monolayers, the broken inversion symmetry and strong spin-orbit coupling originating from the d-orbitals of the transition metal atom align the electron spins near opposite valleys K þ and K À to opposite out-of-plane directions thus preserving time reversal symmetry. The broken spatial inversion symmetry also generates large Berry curvatures near the K valleys, the Berry curvatures being opposite at opposite valleys 9 . This effect gives rise to unconventional electronic properties such as the valley Hall effect 6,10 . An interesting growing field of investigation consists in coupling the valley physics with thermoelectrics. Several spin/valley-dependent thermoelectric effects have already been predicted theoretically in monolayer TMDCs such as the spin Nernst effect (thermoelectric counterpart of the spin Hall effect [11][12][13] ) and the valley Nernst effect (VNE) 14,15 . Hence, the experimental demonstration of those effects could give rise to a new field called valley caloritronics (by analogy with spin caloritronics 16 ) with the advantage that the coupled valley-spin degree of freedom is more robust than the spin with respect to external magnetic fields due to the large intrinsic spin-orbit coupling. One main obstacle to the observation of the VNE comes from the absence of large area TMDC monolayers which is a prerequisite to apply macroscopic temperature gradients. Here we use the van der Waals epitaxy to grow high-quality WSe 2 mono-and multilayer on epitaxial graphene on SiC over 1 cm 217,18 . By this, we are able to apply temperature gradients in WSe 2 on mm scale. However, in order to detect electrically the VNE, one needs to lift the K þ -K À valley degeneracy, i.e., breaking time reversal symmetry and creating a population imbalance between the two valleys. To do so, we use the ferromagnetic resonance-spin pumping (FMR-SP) technique [19][20][21][22] by growing in-situ the Al/NiFe/Al magnetic stack on top of the WSe 2 layer. By suitably adjusting the position of the sample into the cavity, at the FMR of NiFe (Ni 81 Fe 19 ), we are able to: (i) generate a temperature gradient along the sample and (ii) address a single valley by spin pumping when the magnetic field is applied perpendicular to the film. Eventually, we detect the VNE which is in quantitative agreement with theoretical predictions.

Results
Growth of WSe 2 and characterization. The growth of monoand multilayer of WSe 2 is performed under ultrahigh vacuum by molecular beam epitaxy (MBE) in the van der Waals regime on epitaxial graphene on SiC. The graphene layer was grown by chemical vapor deposition on the Si face of SiC. The VNE is expected to be very large and dominant at the top of the valence band at the K þ and K À points of the 2D Brillouin zone. Hence WSe 2 layers are p doped by introducing 0.1% of Nb in order to shift the Fermi level to the top of the valence band 23 . In Fig. 1b, ex-situ grazing incidence x-ray diffraction shows the single crystalline character of the WSe 2 layers and the measured lattice parameter which is the one of bulk WSe 2 . More characterization results on the WSe 2 layers are given in the Supplementary Note 1. We demonstrate that the Nb incorporation does not affect the growth and crystallinity of the WSe 2 layers but it makes them conducting at room temperature and provides electronic states for carrier diffusion and the detection of the VNE (see Supplementary Note 4). WSe 2 films are finally capped with a thin %10-nm-thick amorphous Se layer to allow air-transfer to another deposition chamber to grow the Al(7 nm)/NiFe(20 nm)/ Al(5 nm) stack after the thermal desorption of the amorphous Se. Figure 1c shows the cross-sectional scanning transmission electron microscopy (STEM) micrographs with high-angle annular dark-field imaging (HAADF) technique of the stacks made of monolayer WSe 2 ( Fig. 1c (top left) and 1c (bottom left)) and multilayer WSe 2 ( Fig. 1c (top right) and 1c (bottom right)) used for the VNE experiments. The interfaces (graphene/WSe 2 ) and (WSe 2 /metallic stack) with high crystalline quality are atomically resolved and abrupt. The number of WSe 2 layers can be unambiguously determined based on the STEM images, and is in agreement with the estimation done by the combination of atomic force microscopy measurements and quartz crystal microbalance. We clearly note that the uppermost layer WSe 2 is degraded after Al deposition on top which is likely due to an intermixing of Al and Se. For that reason, in order to preserve the desired number of layers, especially in the case of one monolayer WSe 2 , we have systematically deposited at least 30% of matter (Se, W, Nb) more. Figure 2 illustrates the temperature-gradient induced voltages that are at play under the application of an external magnetic field coupled with spin injection via FMR-spin pumping. We also include rectification effects (RE) due to the anisotropic magnetoresistance and anomalous Hall effect in NiFe. They do not contribute to the measured voltage when the magnetic field and the magnetization are perpendicular to the film (i.e., θ H = 90 and 270 ) 20,24 . It is shown that the VNE might be detected in the configuration where the magnetic field is perpendicular to the sample surface. We discuss below how we can electrically detect the VNE thanks to the FMR-SP technique and disqualify the spurious effects based on robust arguments.
FMR-spin pumping experiments. In order to study the VNE, the samples are cut into small 0.4 2.4 mm 2 pieces, electrically contacted with Al wires using a mechanical bonding machine and finally introduced into the 9 GHz electron paramagnetic resonance cavity (Fig. 3a). The radio frequency (RF) power is set to 200 mW. In FMR-SP experiments, thermoelectric effects like the ordinary Seebeck effect (OSE) are considered as parasitic effects and are usually removed from electrical signals 25 . Here, we purposely position the sample away from the center of the cavity in order to generate a temperature gradient along the sample and study the VNE as shown in Fig. 3a. By shifting up (resp. down) the sample by Δy, microwaves are more absorbed by the NiFe layer at the bottom (resp. top) of the sample with respect to the top (resp. bottom) leading to a temperature gradient along Àŷ (resp. þŷ). We measure simultaneously the FMR signal of the NiFe layer (Fig. 3b, c) and the voltage V = V l + V t as a function of the angle θ H from 0 to 360 . The DC magnetic field H static has a fixed direction along þx. V l (resp. V t ) corresponds to the longitudinal (resp. transverse) component of the voltage. When rotating the sample, we find that the magnetization of the soft NiFe layer is parallel to the applied field at θ H = 90 and 270 at the ferromagnetic resonance by using a simple macrospin model and by minimizing the system energy 20 . We confirmed this prediction by out-of-plane anisotropic magnetoresistance measurements (see Supplementary Note 3).   Observation of the valley Nernst effect. We performed the measurements at room temperature for four different samples: 2 samples with one monolayer of WSe 2 (samples A and B), a reference sample without WSe 2 (sample C) and one sample with WSe 2 multilayers (sample D). We first focus on sample A and detect a symmetric voltage in Fig. 4a demonstrating the large Seebeck effect in WSe 2 . The rotating sample disturbs the RF cavity settings and modifies the Q factor (proportional to the RF field h rf ). Here, the Q factor and the Seebeck effect are minimum in the parallel configuration. The Seebeck voltage (V s l ) varies between 10 and 35 μV with a value of 22.6 μV in the perpendicular geometry. If we assume that, at best, 1/10 of the Nb atoms are activated at room temperature, the expected doping level is 10 11 cm À2 giving a Seebeck coefficient S = 500 μV K À126,27 . The temperature difference between the two electrical contacts (ΔT) then ranges between 20 and 70 mK depending on the sample orientation. In the perpendicular configuration, the temperature difference is~40 mK. V a ¼ V a l þ V a t is reported in Fig. 4c as a function of the applied field for two different angles θ H = 0 (parallel configuration) and θ H = 90 (perpendicular configuration), μ 0 H r being the resonance field. In Fig. 4b, we show the full angular dependence of V a for μ 0 H ¼ μ 0 H r . We clearly see that the signal is maximum in the perpendicular configuration.
In Fig. 4e, we repeated the same measurements on sample B with the contacts at different positions and found the same behavior but with different signal amplitudes. Moreover, in sample B, V a is negative at θ H = 90 since we reversed the polarity of the contacts (Vþ and VÀ) with respect to sample A as shown in the insets of Fig. 4b, e. As a comparison, V s l in the reference sample C exhibits very low values in the whole angular range demonstrating that the Seebeck effect is negligible in the Al/ NiFe/Al/graphene/SiC stack (see Supplementary Note 2). Furthermore, the full angular dependence of the measured voltage in sample C (Fig. 5a) shows two things: (i) zero voltage in the perpendicular configuration, indicating that the ordinary Nernst effect in the full stack as well as the ANE in NiFe are negligible and not detectable. We also note that the ANE in NiFe (sample A) can be estimated as follows: V ANE % α ANE ΔT W=L % 0.1 nV with α ANE in NiFe being of the order of 9 nV K À1 (ref. 28 ), ΔT = 70 mK and W % 0.2 mm, L % 1.3 mm being the transverse and longitudinal distances between the electrical contacts. The ANE voltage is indeed negligibly small compared with the measured V a in the perpendicular configuration; (ii) a cosine dependence corresponding well to the ISHE. We do not discuss here about the origin of this ISHE signal which probably comes from the self-conversion into the NiFe layer.
Analysis of the valley Nernst effect in WSe 2 . We then interpret the angular dependence of V a ¼ V a l þ V a t in samples A and B (Fig. 4b, e) as the combination of the ISHE proportional to the inplane component of the spin s == and the VNE proportional to the out-of-plane component of the spin s ? . We can express the experimental data as: Here we do not consider the ordinary Nernst effect in WSe 2 since it is expected to be at least 3 orders of magnitude smaller than the VNE coefficient as discussed in ref. 15 . Moreover, if there exists a longitudinal temperature gradient as described in Fig. 2d, the contribution of the ANE and SSE in NiFe for the   in-plane component can also be ignored as demonstrated in NiFe with FMR-SP experiment 29 . We find (V 0 ISHE = 6.4 μV G À2 , V 0 VNE = 18.9 μV G À2 ) for sample A and (V 0 where R is the WSe 2 transverse resistance between the two electrical contacts. This resistance is measured independently by transferring the same WSe 2 film grown on mica onto a SiO 2 /Si substrate pre-patterned with a transfer length measurement (TLM) arrangement (see Supplementary Note 4). We find the VNE coefficient α VNE % 38.9 pA K À1 for sample A and % 38.8 pA K À1 for sample B. If we define α 0 ¼ ek B =2h where e is the electron charge, h the Planck constant and k B the Boltzmann constant, we obtain α VNE % 0:02α 0 for one monolayer of WSe 2 .

Discussion
In the following, we assume that Nb doping shifts the Fermi level close to the top of the K þ " and K À # valence bands. We also assume that the intervalley scattering time is longer than the characteristic momentum scattering time in the WSe 2 layers. The VNE thus generates a pure transverse valley/spin current. The VNE coefficient vanishes when K þ " and K À # bands are equally populated by time reversal symmetry. If we assume that FMR-SP in the perpendicular configuration allows to populate only the K þ " valley (100% valley polarization), the maximum expected value for the VNE coefficient is %0.09α 0 at room temperature when the Fermi level is located between the K þ ", K À # valence band edge 14 . Experimentally, we find α VNE % 0:02α 0 which is very close to the predicted maximum value. The experimental value suggests that the Fermi level lies approximately ΔE % 0.11 eV above the top of the valence band 14 . The Fermi energy corresponds to a hole density n h % h , T, k B , _ denote valley degeneracy, effective hole mass, temperature, Boltzmann and reduced Planck constants, respectively), corroborating well with the hole doping range in our WSe 2 . Other features related to the VNE such as rf-power quadratic dependence and spatial-shift induced sign change of the voltage (V a at θ H = 90 ) are given in Supplementary Note 3. Our interpretation for the observation of VNE is also supported by the fact that we detect no voltage in the perpendicular configuration in the reference sample C (Fig. 5a, b). Moreover, we performed the same FMR-SP measurements on the multilayer WSe 2 as shown in Fig. 5c, d. In this case, the perpendicular voltage is much weaker than the one of the monolayer WSe 2 and comparable to the inplane signal. This is also true when we convert the voltages into currents by normalizing with the resistance measured in opencircuit condition: 0.18, 0.00 and 0.06 μAG À2 for samples A, C and D, respectively. In particular, the maximum voltage is no longer located at θ H = 90 . This angular dependence of the voltage is still in qualitative agreement with the observation of the valley Nernst effect. First, when increasing the number of layers, WSe 2 becomes an indirect bandgap semiconductor with the top of the valence band at the Γ point. As a consequence, when increasing the number of layers, we expect a sharp decrease of the valley Nernst effect in the K valleys as observed experimentally. Second, the angular dependence of the transverse voltage V a of the monolayer, which shows a maximum when the spin of holes is pinned to the out-of-plane direction, results from the imbalance of the valley populations. Here, any signal arising from the Rashba spin-orbit coupling (induced by the asymmetric environment of WSe 2 or induced by the proximity effect in graphene 30 ) is neglected. In particular, the recently discussed spin-type Berry curvature 31 is orders of magnitude smaller than the intrinsic Berry curvature at the valence band edges of WSe 2 monolayer. Finally, we note that the full angular dependence of the FMR-SP measurements in the case of the multilayer WSe 2 (Fig. 5c) needs further investigation in order to get insight into its behavior, in particular the appearance of the two maxima with opposite signs. We anticipate that the direct-to-indirect bandgap transition and Ising-Rashba spin-orbit effects 32 associated with the restored inversion symmetry in the multilayer would be appropriate arguments to interpret the full angular dependence.
In conclusion, by applying a macroscopic temperature gradient and by addressing a single K valley, we could demonstrate the large valley Nernst effect in monolayer WSe 2 . For this, we have grown high-quality WSe 2 monolayers on graphene by van der Waals epitaxy over large areas. At the ferromagnetic resonance of the NiFe film in contact with WSe 2 , we could both generate a temperature gradient in WSe 2 at the millimeter scale and a valley polarization by spin pumping. This original technique allowed us to give an estimation of the valley Nernst coefficient that agrees well with the predicted value for one monolayer of WSe 2 . The valley Nernst effect could be used as an alternative way to generate valley currents for valley-related physics.

Methods
Sample growth and characterization. The graphene/SiC substrate is first outgased at 750 C during one hour in the MBE chamber with a base pressure in the low 10 À10 mbar range. It is then kept at 500 C during the co-deposition of W (0.01875 Ås À1 ), Se (10 À6 mbar) and Nb (0.01875 Ås À1 ). W and Nb are evaporated thanks to e-gun evaporators and Se with a Knudsen cell. The Nb flux is pulsed in order to reach the right concentration of 0.1%. At the end of the growth, the whole film is annealed at 750 C during 15 minutes under Se flux to improve the crystal quality 33 . We obtain continuous and uniform WSe 2 films on 5 5 mm 2 graphene/SiC substrate.
X-ray diffraction analysis was performed with a SmartLab Rigaku diffractometer equipped with a Copper Rotating anode beam tube (K α = 1.54 Å) and operated at 45 kV and 200 mA. A parabolic mirror and a parallel in-plane collimator of 0.5 C of resolution was used in the primary optics and a second parallel collimator was used in the secondary side. A K β filter was used for all the measurements.
Raman measurements were done with a Horiba Raman set-up using a laser excitation source of 632 nm with spot size of 0.5 μm. The signal was collected by choosing a 1800 grooves/mm grating.
WSe 2 transfer. We adopted a wet transfer process that is described in details in ref. 34 for the transfer of WSe 2 grown on mica onto SiO 2 /Si substrate.
Electrical contacts. The Ti/Pt contacts, which were pre-patterned on SiO 2 /Si substrate by lithography and etching followed by metal deposition, have thicknesses of 5 nm/10 nm, respectively. See Supplementary Note 4 for the distribution of the contact distance.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.