The work of Neumann and Tewordt is generalized to obtain the first-order correction (in $1-\frac{T}{T_c}$) to the Ginzburg-Landau expression for the free energy of an inhomogeneous super-conductor. From this expression, the generalized Neumann-Tewordt equations for the first-order corrections to the solutions of the Ginzburg-Landau equations are derived. For two important geometries, the normal-superconducting wall and the mixed state of type-II super-conductors, we show that the free energy can be rewritten so that it involves only the solutions of the Ginzburg-Landau equations. We apply this formalism to the calculation of the $N-S$ wall energy, where we calculate ${\sigma }_{\mathrm{NS}}$ as a function of $T$ and $\frac{{\xi }_0}{l}$ for $k=\frac{1}{\sqrt{2}}$, and to the mixed state of type-II superconductors, where we calculate $H_{c1\mathrm{}}$ as a function of $T$, $k$, and $\frac{{\xi }_0}{l}$ for singly and doubly quantized isolated vortices.

• Received 2 June 1971

DOI:https://doi.org/10.1103/PhysRevB.4.3016