Theory of a two-dimensional superconductor with broken inversion symmetry

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 Δ(r)∝cos(Qr) . We consider the case of a strong Rashba interaction with the spin-orbital splitting αmv_F (where α is the Rashba velocity) much larger than the superconductive gap Δ , and show that at low temperatures T⩽0.4T_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 Δ(r)∝exp(iQr) intervenes between the uniform BCS state and the LOFF-like state at gμ_Bh≈1.5T_c0 . The modulation vector Q in both phases is of the order of gμ_Bh/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_c0τ⩽0.11 . However, once an account is made of the next-order term over the small parameter α/v_F≪1 , a relatively long wave helical modulation with Qtilde gμ_Bhα/v_F² 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.