Unifying phenomenological approach to the London and Ginzburg-Landau theories of superconductivity in the general unsteady case

Abstract

The London equations of superconductivity are derived from a variational principle. The derivation produces an additional scalar equation that determines the superconducting charge density. The analysis is extended to the general dissipative case by means of appropriate kinetic coefficients. After introducing the complex order parameter Ψ a time-dependent Ginzburg-Landau (GL) equation is obtained from the variational principle together with the Josephson phase relation. The real part of the GL equation is found to coincide with the additional scalar equation. Special attention is paid to a correct interpretation of the imaginary part of the GL equation. In the dissipative case the time-dependent GL equation is shown to contain two complex relaxation coefficients. The theory is supplemented by a dissipative version of the Josephson phase relation. In a special case some results given by Schmid and by Gor'kov and Eliashberg are recovered. The analysis is believed to set the scene for a microscopic derivation of a general time-dependent GL equation.