We report a theoretical study on the phase transition between superconducting states with and without spontaneous surface current. The phase transition takes place due to the formation of surface Andreev bound states in unconventional superconductors. Based on the quasiclassical theory of superconductivity, we examine the influence of atomic-scale surface roughness on the surface phase transition temperature $T_s$. To describe the surface effect, the boundary condition for the quasiclassical Green's function is parametrized in terms of specularity (the specular reflection probability in the normal state at the Fermi level). This boundary condition allows systematic study of the surface effect ranging from the specular limit to the diffuse limit. We show that diffuse quasiparticle scattering at a rough surface causes substantial reduction of $T_s$ in the $d$-wave pairing state of high-$T_c$ cuprate superconductors. We also consider a $p$-wave pairing state in which Andreev bound states similar to those in the $d$-wave state are generated. In contrast to the $d$-wave case, $T_s$ in the $p$-wave state is insensitive to the specularity. This is because the Andreev bound states in the $p$-wave superconductor are robust against diffuse scattering, as implied from symmetry consideration for odd-frequency Cooper pairs induced at the surface; the $p$-wave state has odd-frequency pairs with $s$-wave symmetry, while the $d$-wave state does not.

• Received 16 May 2018

DOI:https://doi.org/10.1103/PhysRevB.98.134516