Modulation of the superconducting critical temperature due to quantum confinement at the ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ interface

Superconductivity develops in bulk doped ${\mathrm{SrTiO}}_3$ and at the ${\mathrm{LaAlO}}_3$/${\mathrm{SrTiO}}_3$ interface with a dome-shaped density dependence of the critical temperature $T_c$, despite different dimensionalities and geometries. We propose that the $T_c$ dome of ${\mathrm{LaAlO}}_3$/${\mathrm{SrTiO}}_3$ is a shape resonance due to quantum confinement of superconducting bulk ${\mathrm{SrTiO}}_3$. We substantiate this interpretation by comparing the exact solutions of a three-dimensional and quasi-two-dimensional two-band BCS gap equation. This comparison highlights the role of heavy bands for $T_c$ in both geometries. For bulk ${\mathrm{SrTiO}}_3$, we extract the density dependence of the pairing interaction from the fit to experimental data. We apply quantum confinement in a square potential well of finite depth and calculate $T_c$ in the confined configuration. We compare the calculated $T_c$ to transport experiments and provide an explanation as to why the optimal $T_c$'s are so close to each other in two-dimensional interfaces and the three-dimensional bulk material.

Figure 1

Weak-coupling model for the density-dependent superconductivity of ${\mathrm{SrTiO}}_3$. (a) Parabolic approximation for the bottom of the conduction band; $\mathrm{\Delta }E$ is the band splitting at $k=0$. (b) Data from Ref. [7] (squares with error bars) and calculated $T_c\left(n\right)$ curve (solid line). Thick lines show the range of densities visited at the LAO/STO interface. Inset: Dimensionless interaction strength [35] $\tilde{V}=2V{[m_e/\left(2\pi {\hslash }^2\right)]}^{3/2}\sqrt{\hslash {\omega }_D}$ reproducing the experimental $T_c$ (circles) and interpolation (solid line). Colored arrows indicate remarkable values of $T_c$ in (b) and the corresponding values of the chemical potential in (a).

Figure 2

2DEL critical temperature (squares) and thickness (crossed squares; right scale) measured at the LAO/STO interface as a function of the sheet carrier density. Solid lines show the calculated $T_c$ obtained by confining the model shown in Fig. 1 in a square potential well of width $L$ and various depths $U$ as indicated. A continuous interpolation of the thickness as a function of the density (dotted yellow line) was used. For each couple $(n_{2\mathrm{D}},L)$, the pairing interaction is read from the inset in Fig. 1 at the density $n=n_{2\mathrm{D}}/L$. The thick line shows the 3D $T_c$ at density $n$.

Figure 3

(a) Subband structure in a quantum well of width 6.8 nm and depth 39 meV for band splitting $\mathrm{\Delta }E=12.9$ meV, corresponding to $n_{2\mathrm{D}}=2\times {10}^{13}{\mathrm{cm}}^{-2}$. The bound light (heavy) subbands with energies below $U$ ($U+\mathrm{\Delta }E$) are shown on the left (right). The chemical potential $\mu =20.5$ meV is also indicated. (b) Measured critical temperature (squares) and calculated $T_c$ (solid line) for band splitting $\mathrm{\Delta }{E}_{\mathrm{STO}}(1+\lambda /\alpha L_{\exp })$ and potential $U=e^2{n}_{2\mathrm{D}}\alpha L_{\exp }/\left({\epsilon }_0\epsilon \right)$ with $\alpha =0.65$, $\lambda =30$ nm, and $\epsilon =630$. The dashed line shows $T_c$ calculated with the same $\alpha$, $\lambda =12$ nm, and $\epsilon =377$.

Figure 4

Critical temperature as a function of thickness $L$ and sheet carrier density $n_{2\mathrm{D}}$. Spheres show the experimental data from Fig. 2, where the thickness has been reduced by a factor $\alpha =0.65$. The shaded surface shows the $T_c$ calculated with band splitting $\mathrm{\Delta }{E}_{\mathrm{STO}}(1+\lambda /L)$ in a square well of width $L$ and depth $U=e^2{n}_{2\mathrm{D}}L/\left({\epsilon }_0\epsilon \right)$, with $\lambda =30$ nm and $\epsilon =630$.

Figure 5

Same as Fig. 2, except that (a) the well width is kept fixed to the value $L=11$ nm for all densities; and (b) the well width is varied by $\pm 20\%$ around the measured thickness (shaded area) and an average $T_c$ is calculated.