Fig. 1: Sample characteristics and TMF measurement scheme.

a The schematic of monolayer graphene-multilayer WSe₂ heterostructures. b Scanning electron microscope image of the device with a TMF measurement configuration (a scale bar: 2 μm; a distance between the injector a and the collector c: \(L \, \approx \, 4.0\)  μm; a probe width: \(w \, \approx \, 0.3\)  μm). The two semicircles (\({S}_{\pm }\)) illustrate trajectories of the carriers at different spin-orbit-coupled bands \({S}_{\pm }\) shown in c under perpendicular magnetic field \(B\). c The energy dispersion of graphene on WSe₂ derived from the effective Hamiltonian Eq. (1) in the main text using SOC strengths of \(\lambda={\lambda }_{R}=8.9\;{{{{{\rm{meV}}}}}}\;({\lambda }_{{SOC}} \equiv \sqrt{{\lambda }^{2}+{{\lambda }_{R}}^{2}}=12.5\;{{{{{\rm{meV}}}}}})\). The inset shows the simulated local current density map at the focusing peak marked by a down triangle in d. d The corresponding TMF spectra, the conductance \(G\) between the injector and collector as a function of magnetic field, calculated for an effective three-terminal device at 6 different hole densities, n (in 10¹²cm⁻²) = −0.78, −0.93, −1.09, −1.24, −1.40, and −1.56 (from bottom to top) using the tight-binding model. See Supplementary Note 1 for more details. e The carrier density (\(n\)) dependence of the four-probe resistances \({R}_{{xx}}\) of two different samples 1 (blue solid line) and 2 (red broken line) measured at 1.5 K. Both exhibit a sharp resistance peak at zero density, indicating high device quality. The inset shows the non-local Hall resistance \({R}_{{ae},{bf}}\left(\equiv {V}_{{bf}}/{I}_{{ae}}\right)\), exhibiting a large negative signal on the hole side originating from the ballistic transport. f The Landau fan—the \(\log \left({R}_{{xx}}\right)\) as a function of \(n\) and \(B\)—plotted in a color scale (the darker color corresponds to the lower resistance), showing a high-quality quantum Hall effect measured at 1.5 K. On the hole side, the broken-symmetry states begin to appear at ~3 T (indicated by black down-triangles), which indicates higher quality on the hole side, consistent with the large negative \({R}_{{ae},{bf}}\) on the hole side shown in the inset of e.