Phase diagram of a surface superconductor in parallel magnetic field

Detailed theory of phase diagram of clean 2D surface superconductor in a parallel magnetic field is presented. Regular spin-orbital interaction of the Rashba type is known to produce inhomogeneous superconductive state similar to the Larkin-Ovchinnikov-Fulde-Ferrel (LOFF) state with $\Delta({\bf r})\propto \cos({\bf Qr})$athighmagneticfields,with$Q \sim g\mu_B h/v_F$. We consider the case of relatively strong Rashba interaction and show that at low temperatures $T\le 0.4T_{c0}$ the LOFF-type state is separated from the usual homogeneous state by the first-order phase transition line. At higher temperatures new “helical” state with $\mathrm{\Delta }(\mathbf{r})\propto \exp (i\mathbf{Q}\mathbf{r})$ intervene between uniform BCS state and LOFF-like state. One component of superfluid density tensor $n_s$ vanishes on the second-order transition line between BCS state and helical state. Nonmagnetic impurities suppresses both inhomogeneous states, and eliminate them completely at $T_{c0}\tau \le 0.11$.