Dimensionality effects on superconductivity in LaAlO$_3$/SrTiO$_3$ nano-structures

We investigate electron transport, especially superconductivity, in LaAlO$_3$/SrTiO$_3$ nanostructures with respect to the dimensionality by creating channels with widths varying from nanometers to micrometers using c-AFM lithography. Superconducting properties such as upper critical magnetic field and critical current are compared as a function of electron doping (gating) and temperature. The superconducting properties among these devices show a marked deviation from naive scaling expectations. We discuss the results in regard to ferroelastic domains and possible edge/boundary mode transport scenarios.