Fig. 3: Analysis of TMF signal.

a The color-scale map of the average difference \(\langle {\delta B}^{2}\rangle \equiv ( \sum [ {(\Delta {B}_{+}/{B}_{0})}^{2}+{(\Delta {B\_} / {B}_{0})}^{2}]) / N\) as a function of \(\lambda\) and \({\lambda }_{R}\) from sample 1 (\(N\): number of data used). \({\Delta B}_{\pm }\) is the difference between the predicted focusing peak positions from the simulation for certain (\(\lambda\), \({\lambda }_{R}\)) and the real peak positions measured in the experiment for the band \({S}_{\pm }\), whereas \({B}_{0}\) is the half of the maximum splitting observed. Thus, the smaller \(\langle {\delta B}^{2}\rangle\) (darker in the map) indicates a better agreement. A dashed white circle draws the best-fit value. We use a criterion \(\langle {\delta B}^{2}\rangle \le 0.1\) to extract the SOC strengths of \({\lambda }_{{SOC}}=13.9\pm 4.0\) meV (and \(12.0\pm 3.5\) meV for sample 2, see Supplementary Fig. 2b). \({\theta }_{{SOC}}\) in a is defined as \({\cos }^{-1}\left(\lambda /{\lambda }_{{SOC}}\right)\). b–e Comparison of the electron trajectories for the second focusing peak (in the absence of the intervalley scattering) and spin configurations for the two cases when there is only the spin-valley Zeeman term (b, c; \({\theta }_{{SOC}}=0\)) and when only Rashba term exists (d, e; \({\theta }_{{SOC}}=\pi /2\)). The shapes of the resulting focusing peaks for each case are shown in the inset of b and d. Without intervalley scattering, due to the spin conservation, the electron at the edge (state A) will be backscattered to B when \({\theta }_{{SOC}}=0\) (b, c), leading to the splitting in the second peak, whereas when \({\theta }_{{SOC}}=\pi /2\), it will be transferred to state C (d, e). When the intervalley scattering is present, the backscattering from state A to C’ can occur even for the \({\theta }_{{SOC}}=0\) case (c), leading to the suppression of the splitting in the second peak. See the main text for details. f The calculated TMF spectra with varying \({\theta }_{{SOC}}\) at \(n=-1.25\times {10}^{12}\) cm^-2 when the overall SOC strength \({\lambda }_{{SOC}}=10\) meV. One can clearly see that the positions of the first focusing peaks remain the same while the second peak shows multiple peaks near \({\theta }_{{SOC}}=0\).