Spin-orbit coupling (SOC) for $d$-electron gas can be substantially enriched compared with the $sp$-electron gas due to the delicate ordering of the multiple $d$ subbands. Here, we demonstrate the nontrivial Rashba SOC effect at ${\mathrm{SrTiO}}_3$-based interfaces (${\mathrm{LaAlO}}_3\text{/}{\mathrm{SrTiO}}_3$ and ${\mathrm{LaVO}}_3\text{/}{\mathrm{SrTiO}}_3$) directly related to the Ti $3d$ subband ordering via magnetotransport characterizations. Unusual $k$-cubic Rashba SOC contributed from the $d_{xz/yz}$ states is revealed. More strikingly, when a gate voltage is swept to tune the band filling, the SOC strength initially increases and then decreases to form a dome feature, accompanied by an apparent single- to two-carrier transition. These two concomitant effects strongly indicate that the SOC behavior is largely determined by the Ti $3d$ subbands regardless of the overlayer boundary conditions, with the SOC strength peaked at the $d_{xy}-d_{xz/yz}$ crossings due to the band hybridization effect as predicted. The present findings offer new insights into exploration of oxide-based quantum phases and spintronic devices.

• Received 30 January 2015
• Revised 19 May 2015

DOI:https://doi.org/10.1103/PhysRevB.92.075309