from the surface field. By deriving the effective Hamiltonian $\mathcal{H}_{eff}$ that acts on each two-fold degenerate band in the weak SOC limit where $\mathcal{H}_\mathrm{hop}+\mathcal{V}_\mathrm{sf}$ is dominant over $\mathcal{H}_\mathrm{SOC}$, we obtain Rashba-type band splitting terms near $\Gamma$ and $\mathrm{X}$ as summarized in Table \[tab:heff\] (see Appendix A for details). Table \[tab:heff\] shows the Rashba coupling to be linear in the ISB hopping $\gamma$ divided by the energy difference between two relevant states $\Delta$.

![\[fig:wanhopping\]Hopping strengths (in eV) between Wannier functions for Ta $d$ states at one site and the neighboring site in $x$ direction. Ta $d$ states and O $p$ states are used for the Wannier function construction. The terms relevant to the ISB $\gamma_1$, $\gamma_2$, $\gamma_3$ are presented. Both direct (horizontal arrows) and indirect (oblique arrows, via O $p$) paths are depicted. Empty arrows indicate terms that would be absent without ISB.](fig3.pdf){width="35.00000%"}

One reason why the $e_g$ orbital contribution is crucial for the Rashba splitting in the $t_{2g}$ bands is that the $t_{2g}$-$e_g$ ISB hoppings $\gamma_{2,3}$ are significantly larger than the intra-$t_{2g}$ hopping $\gamma_1$: $\gamma_1 \approx -0.04$ eV, $\gamma_2 \approx -0.25$ eV, $\gamma_3 \approx 0.30$ eV. This is necessary condition for the effective Hamiltonian of Table \[tab:heff\] to give larger 