Theoretical models of Rashba spin splitting in asymmetric ${\mathrm{SrTiO}}_3$-based heterostructures

Rashba spin splitting in two-dimensional (2D) semiconductor systems is generally calculated in a $\mathbf{k}·\mathbf{p}$ Luttinger-Kohn approach where the spin splitting due to asymmetry emerges naturally from the bulk band structure. In recent years, several new classes of 2D systems have been discovered where electronic correlations are believed to have an important role. In these correlated systems, the effects of asymmetry leading to Rashba splitting have typically been treated phenomenologically. We compare these two approaches for the case of 2D electron systems in ${\mathrm{SrTiO}}_3$-based heterostructures, and find that the two models produce fundamentally different behavior in regions of the Brillouin zone that are particularly relevant for magnetotransport. Our results demonstrate the importance of identifying the correct approach in the quantitative interpretation of experimental data, and are likely to be relevant to a range of 2D systems in correlated materials.

Figure 1

Comparison of the conduction band $d$ states in ${\mathrm{SrTiO}}_3$ with the hole band $p$ states in GaAs (see text). The color red (blue) indicates $m_j=\pm \frac{3}{2}$ ($m_j=\pm \frac{1}{2}$) bands to point out the reversal of the heavy and light mass.

Figure 2

$\mathbf{k}·\mathbf{p}$ model (lines) and DFT calculations [33] (dots) of the band structure of bulk ${\mathrm{SrTiO}}_3$ around $\mathrm{\Gamma }$. Left panel: No SO, Eq. (4). The nondegenerate bands along $\mathrm{\Gamma }-M$ require $N\ne 0$ (see text). Right panel: With SO, Eq. (6).

Figure 3

In-plane dispersion in a triangular well with slope $F=1.0$ meV/Å according to the LK Hamiltonian (left panel) and the $\gamma$ model with $\gamma =5.4$ meV (right panel). The color indicates the $he, so$, or $le$ character of the subbands.

Figure 4

Subbands for (a) $F=0.1$ meV/Å, (b) 0.2 meV/Å, (c) 0.5 meV/Å, and (d) spin splitting $\mathrm{\Delta }$ of the lowest subband for four values of $F$. The solid line is the LK model, and the dashed line the $\gamma$ model with ${\gamma }_{\text{fit}}$. The dotted lines in (a) and (c) correspond to larger values of $\gamma$. In (d) we show the spin splitting of the lowest subband. The spin splitting is very sensitive to the choice of the parameter $\gamma$. In (a) we show the subbands for $\gamma =13$ meV, which is the value that gives the same cubic splitting of the $le$ (see Supplemental Material S4 [39]) and in (c) we show the effect of doubling $\gamma$.