Second-order charge and spin transport in LaO/STO system in the presence of multiple Rashba spin orbit couplings

Fig. 1 shows a comparison between the second-order charge current flowing perpendicular to the applied electric field calculated using the Schliemann-Loss and relaxation-time approximation approaches, in which the relaxation time was taken as the mean value of $\tau^{\parallel}_{\mu}$ over the Fermi surfaces of both bands. The RTA produces qualitatively similar results to the Schliemann-Loss approach but underestimates the magnitudes of the second-order response.

Fig. 2 shows the first- and second-order responses for the spin accumulations perpendicular to the magnetization direction and the spin currents with spin polarizations parallel to the magnetization directions flowing parallel and perpendicular to the applied electric field. The results at $\phi_{\mathrm{m}}=0,\pi$ and at $\phi_{\mathrm{m}}=\pm pi/2$ are consistent with Tables I and II in the main manuscript, respectively. Interestingly, a finite second-order spin accumulation perpendicular to the magnetization directions at intermediate magnetization angles between the $\pm x$ and $\pm y$ directions emerges in Fig. 2b.

Fig. 3 shows the variation of $\delta J^{(2)}_{y}$ with $\alpha$, $\beta_{3}$, and $\eta_{3}$. For the parameter ranges in this studyt, $\alpha_{3}$ does not have a substantial effect on the sign of $\delta J^{(2)}_{y}$, as indicated by the fact that the dotted black lines denoting the points where $\delta J^{(2)}_{y}=0$ are almost vertical. Fig. 3a and b show that the sign of $\delta J^{(2)}_{y}$ can be switched by varying $\eta_{3}$, and Fig. 3c and d show that the sign of $\delta J^{(2)}_{y}$ can be switched by varying $\beta_{3}$.