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

TL;DR

This paper explains the dome-shaped dependence of the superconducting critical temperature at the LaAlO$_3$/SrTiO$_3$ interface as a quantum confinement shape resonance, supported by theoretical modeling and comparison with experimental data.

Contribution

It introduces a quantum confinement model to explain the $T_c$ dome at the LaAlO$_3$/SrTiO$_3$ interface, highlighting the role of heavy bands and shape resonance effects.

Findings

The $T_c$ dome is explained by quantum confinement effects.

Comparison of 3D and 2D models highlights the importance of heavy bands.

Calculated $T_c$ matches experimental transport data.

Abstract

Superconductivity develops in bulk doped SrTiO$_3$ and at the LaAlO$_3$/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 LaAlO$_3$/SrTiO$_3$ is a shape resonance due to quantum confinement of superconducting bulk 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 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.