Fig. 2: Schematics of the transverse (\({V}_{\mathrm{t}}\)), longitudinal (\({V}_{\mathrm{l}}\)) voltages measured under different configurations and the electrical detection of the VNE.

The external applied magnetic field (a, c) perpendicular and parallel (b, d) to the sample surface. The temperature gradient: in-plane for (a, b) and out-of-plane for (c, d). W and L stand for transverse and longitudinal distances. e Illustration of the VNE with electrical detection in open circuit conditions. f Resume of emerging effects with respect to our experimental set-up in each configuration (a–d): OSE (all the stack), ISHE (NiFe, WSe\({}_{2}\)), (S)VNE (WSe\({}_{2}\)), ANE (NiFe), ONE (all the stack), PNE (NiFe), RE (NiFe) and SSE (NiFe) stand for ordinary Seebeck effect, inverse spin Hall effect, (spin) valley Nernst effect, anomalous Nernst effect, ordinary Nernst effect, planar Nernst effect, rectification effects (due to the anisotropic magnetoresistance and anomalous Hall effect) and spin Seebeck effect, respectively.