Usadel

Usadel refers to the Usadel equations, a set of diffusion-like equations used in the quasiclassical theory of superconductivity to describe the behavior of electronic excitations in diffusive metals and superconducting heterostructures. They were developed by Klaus D. Usadel in 1970, derived from the Eilenberger equations in the dirty limit where impurity scattering is strong and the electronic mean free path is short compared with the superconducting coherence length. The Usadel equations govern the spatial and energy dependence of the quasiclassical Green's functions in superconductors and proximity-coupled systems, including the retarded, advanced, and Keldysh components in a unified framework.

Mathematically, they are nonlinear diffusion-type equations for matrix Green's functions in Nambu space, subject to the normalization condition G^2+FF^†=1. The superconducting order parameter Δ, exchange fields, and applied potentials enter through commutator terms, and the diffusion constant D sets the scale for spatial variation. In practice, one solves separate equations in superconducting and normal regions with appropriate interfacial boundary conditions.

Common boundary conditions include the Kuprianov–Lukichev prescriptions and Nazarov's more general boundary conditions, which model SN and SNS junctions and other mesoscopic geometries. Linearized forms near the critical temperature Tc describe weak induced superconducting correlations, while the full nonlinear equations describe strong proximity effects and non-equilibrium transport in diffusive devices.

Usadel theory has become a standard tool for studying the proximity effect, Josephson currents in diffusive metals, superconducting quantum interference devices, and other mesoscopic superconducting phenomena. It complements the Eilenberger equations for ballistic systems and Gor'kov's microscopic formulation within the broader quasiclassical approach to superconductivity.