Theory of a two-dimensional superconductor with broken inversion symmetry

Abstract

A detailed theory of a phase diagram of a two-dimensional surface superconductor in a parallel magnetic field is presented. A spin-orbital interaction of the Rashba type is known to produce at a high magnetic field $h$ (and in the absence of impurities) an inhomogeneous superconductive phase similar to the Larkin-Ovchinnikov-Fulde-Ferrel (LOFF) (Zh. Eksp. Teor. Fiz. 47, 1136 (1964) [Sov. Phys. JETP. 20, 762 (1965)]; Phys. Rev. 135, A550 (1964)) state with an order parameter $\ensuremath{\Delta}(\mathbf{r})\ensuremath{\propto}\mathrm{cos}(\mathbf{Q}\mathbf{r})$. We consider the case of a strong Rashba interaction with the spin-orbital splitting $\ensuremath{\alpha}m{v}_{F}$ (where $\ensuremath{\alpha}$ is the Rashba velocity) much larger than the superconductive gap $\ensuremath{\Delta}$, and show that at low temperatures $T\ensuremath{\leqslant}0.4{T}_{c0}$ the LOFF-type state is separated from the usual homogeneous state by a first-order phase transition line. At higher temperatures, a different inhomogeneous ``helical'' state with $\ensuremath{\Delta}(\mathbf{r})\ensuremath{\propto}\mathrm{exp}(i\mathbf{Q}\mathbf{r})$ intervenes between the uniform BCS state and the LOFF-like state at $g{\ensuremath{\mu}}_{B}h\ensuremath{\approx}1.5{T}_{c0}$. The modulation vector $Q$ in both phases is of the order of $g{\ensuremath{\mu}}_{B}h∕{v}_{F}$. The superfluid density ${n}_{s}^{yy}$ vanishes in the region around the second-order transition line between the BCS state and the helical state. Nonmagnetic impurities suppress both inhomogeneous states and eliminate them completely at ${T}_{c0}\ensuremath{\tau}\ensuremath{\leqslant}0.11$. However, once an account is made of the next-order term over the small parameter $\ensuremath{\alpha}∕{v}_{F}⪡1$, a relatively long wave helical modulation with $Q\ensuremath{\sim}g{\ensuremath{\mu}}_{B}h\ensuremath{\alpha}∕{v}_{F}^{2}$ is found to develop from the BCS state. This ground state carries zero current in the thermodynamic limit; however, under the cyclic boundary conditions a kind of ``spin-orbital Little-Parks oscillations'' [Phys. Rev. Lett. 9, 9 (1962)] is predicted. The long-wave helical modulation is stable with respect to disorder. In addition, we show that vortex defects with a continuous core may exist near the phase boundary between the helical and the LOFF-like states. In particular, in the LOFF-like state these defects may carry a half-integer flux.