Nature of orbital and spin Rashba coupling in the surface bands of ${\mathrm{SrTiO}}_3$ and ${\mathrm{KTaO}}_3$

Tight-binding models for the recently observed surface electronic bands of ${\mathrm{SrTiO}}_3$ and ${\mathrm{KTaO}}_3$ are analyzed with a view to bringing out the relevance of momentum-space chiral angular momentum structures of both orbital and spin characters. The orbital and the accompanying spin angular momentum structures reveal complex linear and cubic dependencies in the momentum $\mathbf{k}$ (linear and cubic Rashba effects, respectively) in a band-specific manner. Analytical expressions for the cubic orbital and spin Rashba effects are derived by way of a unitary transformation technique we developed, and compared to numerical calculations. The cubic Rashba effect appears as in-plane modulations due to the $C_{4v}$ symmetry of the perovskite structure.

Figure 1

(a)–(c) OAM averages for the three bands obtained from numerical diagonalization of Eq. (1). Parameters are for STO (Table 1) with ${\lambda }_{\mathrm{so}}=0$. Magnitude of OAM is represented on a false-color scale in units of $\hslash$. Directions and magnitudes of OAM vectors are consistent with Eq. (5).

Figure 2

(a), (d) $t_{2g}$-band structures in STO and KTO obtained from respective tight-binding models. For each subband pair, outer (inner) band is denoted as solid (dotted) line. Energy on the $y$ axis is in units of eV. (b), (e) Fermi surface at the energy $E_{\mathrm{F}}$ indicated in (a), (d) and corresponding OAM average vectors in STO and KTO. OAM averages of outer (inner) bands are represented as red (blue) arrows imposed on the Fermi surface contour. Directions of OAM vectors in outer and inner Rashba-split states are the same in STO [7], whereas those in KTO are opposite. (c), (f) Spin vectors on the Fermi surface in STO and KTO. Spin direction of outer (inner) band in STO is antiparallel (parallel) to that of OAM, whereas it is always antiparallel to OAM in KTO. Magnitude of OAM (spin) is represented on a false-color scale in units of $\hslash$ ($\hslash /2$).

Figure 3

Band-specific (a)–(c) OAM vectors and (d)–(f) SAM vectors in KTO obtained from the first-principles calculation. In each figure, upper (lower) plot depicts the OAM or SAM structure of outer (inner) band. Each subband pair reveals opposite OAM and SAM direction for outer and inner bands. In addition to the rotating pattern around the $\Gamma$ point in all the bands, a significant number of linear-polarized OAM and SAM vectors are present in the $E^{\left(m\right)}$ and $E^{\left(u\right)}$ bands. Magnitude of OAM (SAM) is represented on a false-color scale in units of $\hslash$ ($\hslash /2$).