Distribution functions in nonequilibrium theory of superconductivity and Andreev spectroscopy in unconventional superconductors

We develop a theoretical formulation of nonequilibrium superconducting phenomena, including singlet and triplet pairing, which is especially well suited for spatially inhomogeneous problems. We start from the general Keldysh-Nambu-Gor’kov Green’s functions in the quasiclassical approximation and represent them in terms of $2\times 2$ spin-matrix coherence functions and distribution functions for particle-type and hole-type excitations. The resulting transport equations for the distribution functions may be interpreted as a generalization to the superconducting state of Landau’s transport equation for the normal Fermi liquid of conduction electrons. The equations are well suited for numerical simulations of dynamical phenomena. Using our formulation we solve an open problem in quasiclassical theory of superconductivity, the derivation of an explicit representation of Zaitsev’s nonlinear boundary conditions (A. V. Zaitsev, Zh. Éksp. Teor. Fiz. 86, 1742 (1984) [Sov. Phys. JETP 59, 1015 (1984)]; A. L. Shelankov, Fiz. Tverd. Tela (Leningrad) 26, 1615 (1984) [Sov. Phys. Solid State 26, 981 (1984)]) at surfaces and interfaces. These boundary conditions include nonequilibrium phenomena and spin singlet and triplet unconventional pairing. We eliminate spurious solutions as well as numerical stability problems present in the original formulation. Finally, we formulate the Andreev scattering problem at interfaces in terms of the introduced distribution functions and present a theoretical analysis for the study of time reversal symmetry breaking states in unconventional superconductors via Andreev spectroscopy experiments at normal-metal–superconductor interfaces with finite transmission. We include impurity scattering self-consistently.