Change of parity of the order parameter in high-temperature superconductors due to impurities

The dependence of the critical temperature on impurity concentration is calculated for a model of a tetragonal high-$T_c$ superconductor described in a previous paper [A. A. Abrikosov, Phys. Rev. B 52, R15 738 (1995)] under conditions when the odd ($d$ symmetry) order parameter is favorable for a clean substance. The critical temperature decreases rapidly with increasing impurity concentration according to the general pair-braking formula and tends to zero at a critical concentration. Since, however, the critical temperature for the phase with an even order parameter is only weakly influenced by impurities, it eventually exceeds the $T_c$ of the odd phase, and this means a phase transition to the even phase. The $T_c$ curve is expected to have a kink and a long tail.