Extended Ginzburg-Landau formalism: Systematic expansion in small deviation from the critical temperature

Abstract

Based on the Gor'kov formalism for a clean $s$-wave superconductor, we develop an extended version of the single-band Ginzburg-Landau (GL) theory by means of a systematic expansion in the deviation from the critical temperature ${T}_{c}$, i.e., $\ensuremath{\tau}=1\ensuremath{-}T/{T}_{c}$. We calculate different contributions to the order parameter and the magnetic field: the leading contributions ($\ensuremath{\propto}{\ensuremath{\tau}}^{1/2}$ in the order parameter and $\ensuremath{\propto}\ensuremath{\tau}$ in the magnetic field) are controlled by the standard GL theory, while the next-to-leading terms ($\ensuremath{\propto}{\ensuremath{\tau}}^{3/2}$ in the gap and $\ensuremath{\propto}{\ensuremath{\tau}}^{2}$ in the magnetic field) constitute the extended GL (EGL) approach. We derive the free-energy functional for the extended formalism and the corresponding expression for the current density. To illustrate the usefulness of our formalism, we calculate, in a semianalytical form, the temperature-dependent correction to the GL parameter at which the surface energy becomes zero, and analytically, the temperature dependence of the thermodynamic critical field. We demonstrate that the EGL formalism is not just a mathematical extension to the theory: variations of both the gap and the thermodynamic critical field with temperature calculated within the EGL theory are found in very good agreement with the full BCS results down to low temperatures, which dramatically improves the applicability of the formalism compared to its standard predecessor.