Ginzburg-Landau theory of noncentrosymmetric superconductors
The data of temperature dependent superfluid density $n_s(T)$ in Li$_2$Pd$_3$B and Li$_2$Pt$_3$B [Yuan {\it et al.}, \phrl97, 017006 (2006)] show that a sudden change of the slope of $n_s (T)$ occur at slightly lower than the critical temperature. Motivated by this observation, we microscopically derive the Ginzburg-Landau (GL) equations for noncentrosymmetric superconductors with Rashba type spin orbit interaction. Cooper pairing is assumed to occur between electrons only in the same spin split band and pair scattering is allowed to occur between two spin split bands. The GL theory of such a system predicts two transition temperatures, the higher of which is the conventional critical temperature $T_c$ while the lower one $T^*$ corresponds to the cross-over from a mixed singlet-triplet phase at lower temperatures to only spin-singlet or spin-triplet (depending on the sign of the interband scattering potential) phase at higher temperatures. As a consequence, $n_s (T)$ shows a kink at this cross-over temperature. We attribute the temperature at which sudden change of slope occurs in the observed $n_s (T)$ to the temperature $T^*$. This may also be associated with the observed kink in the penetration depth data of CePt$_3$Si. We have also estimated critical field near critical temperature.