Experimental Evidence of Cubic Rashba Effect in an Inversion-Symmetric Oxide

We present evidence of cubic Rashba spin splitting in a quasi-two-dimensional electron gas formed at a surface of (001) ${\mathrm{SrTiO}}_3$ single crystal from the weak localization or antilocalization (WAL) analysis of the low-temperature magnetoresistance. Our WAL data were well fitted by the model assuming $m_j=\pm 3/2$ for the spin-split pair, in which $2\pi$ rotation of the electron wave vector ${\mathbf{k}}_{\parallel }$ in the $k_x\mathrm{\text{−}}{k}_y$ plane accompanies $6\pi$ rotation of the spin quantization axis. This finding pertains to the $p$ symmetry of the $t_{2g}$ electronic band derived from $d$ electrons in ${\mathrm{SrTiO}}_3$, which provides insights into the surface electronic state of (001) ${\mathrm{SrTiO}}_3$.

Figure 1

(a) Photograph of a ${\mathrm{SrTiO}}_3$ FET. Carriers are accumulated beneath the Hall-bar shaped gate electrode (Au). The Parylene gate insulator is homogeneously deposited within the view of this photograph and cannot be seen. (b) Sheet resistivity as a function of $E_G$ at 2 K. (c) Hall carrier density vs $E_G$ at 2 K showing the monotonic increase of $n_s$ with $E_G$. (d) The Hall resistance -$R_{xy}$ vs $B$ for STO-$A$ at 2 K.

Figure 2

(a) Magnetoresistance data (open circles) of STO-$B$ at $E_G=4.63\mathrm{MV}/\mathrm{cm}$ and the fit to $B^2$ background (dashed line). (b) Conductance correction $\Delta \sigma$ in unit of $e^2/\pi h$ ($e$ is the unit charge and $h$ is the Planck’s constant) due to weak antilocalization derived by subtracting the $B^2$ background. (c) Theoretical fits given by the ${\mathbf{\Omega }}_1$-only and the ${\mathbf{\Omega }}_3$-only conditions (dashed and solid curves, respectively) within the ILP theory [32] to the experimental data $\Delta \sigma$ (open circles) of STO-$B$ at $E_G=4.38\mathrm{MV}/\mathrm{cm}$. Simplified diagrams of ${\mathbf{\Omega }}_1$ and ${\mathbf{\Omega }}_3$ are also shown. The ${\mathbf{\Omega }}_3$ model perfectly reproduces $\Delta \sigma$.

Figure 3

Experimental $\Delta \sigma$ (closed symbols) and theoretical fit by the ${\mathbf{\Omega }}_3$-only condition in the ILP theory (solid curves) for (a) STO-$A$, (b) -$B$, and (c) -$C$, respectively. (d) Spin-precession length vs sheet carrier density $n_s$. The inset shows the phase coherence length vs $n_s$. (e) Spin-precession time (open symbols) and momentum scattering time (closed symbols) as a function of $n_s$. The inset shows ${\tau }_{\mathrm{SO}}$ vs ${\tau }_{\mathrm{tr}}$ for nearly constant $n_s$ ($n_1\sim 3.5\times {10}^{12}$ and $n_2\sim 4.1\times {10}^{12}{\mathrm{cm}}^{-2}$), ruling out the Elliot-Yafet mechanism for both conditions (see text).

Figure 4

(a) Schematic spin configuration for the cubic-Rashba effect [Eq. (2)]. The spin quantization axis rotates $3\theta$ as $\mathbf{k}$ rotates $\theta$ around the Fermi circles, in contrast to the linear Rashba effect. (b) Energy diagram of the conduction band (CB) bottom of gated-${\mathrm{SrTiO}}_3$ as a function of an in-plane wave vector ($k_{\parallel }$). The band with a small effective mass forms two spin-split bands (light and dark blue curves) with energies $E={\hslash }^2{k}^2/2m^{*}\pm {\alpha }_3{k}^3$. The approximate position of the Fermi level $E_F$ is shown by a dashed horizontal line. (c) Deduced values for the coefficient ${\alpha }_3$ for the cubic-Rashba effect as a function of $n_s$. The inset shows spin splitting energy $\Delta$ as a function of $n_s$. $2\Delta$ corresponds to the difference in the energy of the two spin-split bands at $E_F$.