Effect of spin-orbit interaction on the vortex dynamics in LaAlO₃/SrTiO₃ interfaces near the superconducting transition

Controlling spin-orbit interaction and its effect on superconductivity has been a long-standing problem in two-dimensional inversion symmetry broken superconductors. An open challenge is to understand the role of various energy scales in shaping the complex phase diagram in these systems. From a combined experimental and theoretical study of resistance fluctuations and its higher order statistics, we propose a phase diagram for the superconducting phase in the magnetic-field--spin orbit interaction energy plane for the quasi-two dimensional electron gas at the interface of LaAlO$_3$/SrTiO$_3$ heterostructures. The relative variance of resistance fluctuations increases by few orders of magnitude below the spin-orbit field B$_{SO}$ and a non-Gaussian component to the fluctuations arises for fields below the upper critical field B$_{C2}$. Theoretical calculations show that the non-Gaussian noise predominantly arises due to percolative nature of the superconducting transition. We quantify the strength and the relative importance of the spin-orbit interaction energy, Zeeman energy and the pairing potential. Our work highlights the important role played by the interplay between these energy scales in framing the fascinating phases seen in two-dimensional inversion-symmetry-broken superconductors.