Quantum confinement in a quasi-2D geometry changes the electronic properties of a material with respect to the bulk, leading to the formation of subbands. For superconducting systems, such changes of dimensionality imply a modification of the critical properties.
In a quantum-confined superconductor, Cooper pairing is of a multi-band nature because all subbands are coupled. Furthermore, the pairing matrix elements depend on the shape of the confinement potential. We have developed numerical and analytical tools for solving the problem of a BCS superconductor in a square potential well at any electron density , including self-consistency in the chemical potential . The quasi-2D critical temperature shows oscillations as a function of density and confinement length, called shape resonances. Depending on the confinement strength, Tc can either increase with decreasing film thickness, before reaching a maximum and eventually dropping to zero, or it can be lower than the bulk for all thicknesses.
We have explored the possibility that the dome-shaped superconductivity at the LaAlO3/SrTiO3 interface and in bulk SrTiO3 share a common origin, with the interface dome resulting from a shape resonance due to quantum confinement applied to the bulk . To this purpose, we consider a two-band model with pairing strength reproducing the bulk Tc of SrTiO3. We apply our method to solve this model in a quasi-2D configuration, with confinement length corresponding to the experimental observations at the LaAlO3/SrTiO3 interface.
We indeed find that the resulting Tc is of the same order of magnitude as in bulk SrTiO3, and that it forms a dome as a function of density. We show that two different mechanisms explain the suppression of Tc on the underdoped and overdoped sides of the quasi-2D dome.
 D. Valentinis, D. van der Marel and C. Berthod, Phys. Rev. B 94, 054515 (2016)
 D. Valentinis, D. van der Marel and C. Berthod, Phys. Rev B 94, 024511 (2016)
 D. Valentinis, S. Gariglio, A. Fête, J.-M. Triscone, C. Berthod and D. van der Marel, arXiv:1611.07763 (2016)