Microscopic calculation of coherence lengths and magnetic penetration depth in multiband superconductors

We present an extended Ginzburg-Landau (GL) method for calculating the superconducting coherence length and magnetic penetration depth at temperatures well below the transition temperature $T_{\mathrm c}$. In contrast to conventional GL theory, which expands the free energy in both the order parameters and their gradients, our method applies a perturbative expansion only to the covariant-gradient terms, while retaining the full dependence on the superconducting order parameters. The coefficients of these terms are determined from finite differences of microscopic free energies evaluated at small imposed pair momenta. The method applies to both single-band and multiband superconductors and therefore provides a framework for incorporating more realistic electronic structures. For the models examined here, the extended GL method agrees well with real-space Bogoliubov-de Gennes (BdG) calculations over a wide temperature range, while requiring substantially less computational effort.