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.

中文翻译：

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LaAlO3 / SrTiO3界面处的量子限制对超导临界温度的调制

超导电性在体掺杂中发展 ${\mathrm{钛酸锶}}_3$ 并在 ${\mathrm{拉铝}}_3$/${\mathrm{钛酸锶}}_3$ 与临界温度的圆顶形密度相关性交界 ${Ť}_C$，尽管尺寸和几何形状不同。我们建议${Ť}_C$ 圆顶 ${\mathrm{拉铝}}_3$/${\mathrm{钛酸锶}}_3$ 是由于超导本体的量子约束而引起的形状共振 ${\mathrm{钛酸锶}}_3$。通过比较三维和准二维二维BCS间隙方程的精确解，我们证实了这种解释。这种比较突出了重乐队对${Ť}_C$在两个几何中。散装${\mathrm{钛酸锶}}_3$，我们从拟合值到实验数据中提取配对相互作用的密度依赖性。我们在有限深度的方势阱中应用量子约束，并计算${Ť}_C$在密闭配置中。我们比较计算出来的${Ť}_C$ 进行实验并解释为什么最优 ${Ť}_C$在二维界面和三维散装材料中彼此非常接近。