Theoretical model for Rashba spin-orbit interaction in d electrons

We show that the Rashba spin-orbit interaction in $d$ electron solids, which originates from the broken inversion symmetry at surfaces or interfaces, is strongly dependent on the orbital characters of the bands involved. This is studied by developing a tight-binding model in the presence of a uniform perpendicular electric field and spin-orbit coupling. We argue that for valence electrons, the spin-orbit coupling strength scales only as the square of the atomic number. The electric field distorts the $d$ orbitals through the admixture of $p$ and $f$ states and also introduces intersite overlap parameters. Expressions for Rashba coefficients for the bands are obtained in both weak and strong spin-orbit interaction limits and are shown to be orbital dependent. The results are compared with first-principles calculations for model systems, showing good agreement. Our study demonstrates the orbital-dependent gate control of the Rashba effect for the purposes of oxide electronics.