Electrically controlled superconducting states at the heterointerface ${\text{SrTiO}}_3/{\text{LaAlO}}_3$

We study the symmetry of Cooper pair in a two-dimensional Hubbard model with the Rashba-type spin-orbit interaction as a minimal model of electron gas generated at a heterointerface of ${\text{SrTiO}}_3/{\text{LaAlO}}_3$. Solving the Éliashberg equation based on the third-order perturbation theory, we find that the gap function consists of the mixing of the spin-singlet $d_{xy}$-wave component and the spin-triplet $\left(p_x\pm i{p}_y\right)$-wave one due to the broken inversion symmetry originating from the Rashba-type spin-orbit interaction. The ratio of the $d$-wave and the $p$-wave component continuously changes with the carrier concentration. We propose that the pairing symmetry is controlled by tuning the gate voltage.

Figure 1

(Color online) The $\mathit{k}$ dependence of the spin-singlet gap function ${\Delta }_{\uparrow \downarrow }\left(\mathit{k},i{\epsilon }_0\right)-{\Delta }_{\downarrow \uparrow }\left(\mathit{k},i{\epsilon }_0\right)$ and the spin-triplet gap function ${\Delta }_{\uparrow \uparrow }\left(\mathit{k},i{\epsilon }_0\right)-{\Delta }_{\downarrow \downarrow }\left(\mathit{k},i{\epsilon }_0\right)$ for $T/t=0.008$, $\lambda /t=0.1$, $U/t=3$, and $n=0.3$, with ${\epsilon }_0=\pi T$. The dotted (green) and solid (black) lines denote the nodal lines and Fermi surfaces, respectively.

Figure 2

(Color online) Eigenvalues of Éliashberg equation against $n$ for $T/t=0.008$ and $U/t=4$.

Figure 3

(Color online) The ratio of the singlet and the triplet components for $T/t=0.008$ and $\lambda /t=0.1$ as a function of $U$.

Figure 4

(Color online) $n\text{−}U$ phase diagram of the symmetry of gap function at $T/t=0.008$. The (red) dashed and (red) dotted lines denote the lines where the corresponding values of $\kappa$ equal to 0 and $\pm 0.4$, respectively. The (black) solid line denotes the boundary between spin-singlet and spin-triplet phase in the absence of the RSOI.