Quantum spin hall effect in LaAlO$_3$/SrTiO$_3$ nanostructures

LaAlO$_3$/SrTiO$_3$ heterostructures are known to exhibit strong spin-orbit coupling. We investigate local and non-local transport behavior of nanoscale Hall crosses created by conductive AFM lithography. The four-terminal resistance of these structures is consistently found to be $\sim$$h/e^2$, independent of the length of the channel. We also observe large (1-10 k$\Omega$) non-local resistances and zero-field Hall resistance that are attributed to quantum spin Hall phase with a spin-orbit derived pseudo-magnetic fields $B_{eff}\sim 15$ T. The pure spin current is blocked by Cooper pairs that form below Tc~200 mK, leading to a collapse of the non-local and zero-field Hall resistances.