Cooper Pair Writing at the LaAlO$_{3}$/SrTiO$_{3}$ Interface*

Superconducting semiconductors offer unique ways to exert electrostatic control over macroscopic quantum phases. The recently demonstrated nanoscale control over conductivity at the LaAlO$_{3}$/SrTiO$_{3}$ interface raises the question of whether nanoscale control over superconducting phases can be realized. Here we report low-temperature magnetotransport experiments on structures defined with nanoscale precision at the LaAlO$_{3}$/SrTiO$_{3}$ interface. A quantum phase transition is observed that is associated with the formation of Cooper pairs, but a finite resistance is observed at the lowest temperatures. Higher mobility interfaces exhibit larger Ginsburg-Landau coherence lengths, a stronger suppression of pairing by magnetic field as well as Shubnikov-de Haas oscillations. Cooper pair localization, spin-orbit coupling, and finite-size effects may factor into an explanation for some of the unusual properties observed.