Anisotropic Eliashberg theory and the two-band model for the superconducting properties of MgB2

We present a framework for the anisotropic Eliashberg theory for the superconducting properties of MgB2 and reduce the framework to a zeroth-order two-band model by neglecting the variation of the electron-phonon interaction within the sigma and pi sheets, of the Fermi surface. We then derive a second-order-corrected two-band model using a systematic expansion of the theory with respect to the intrasheet variation of the electron-phonon interaction. Our results show that the zeroth-order two-band model reproduces successfully the results of the anisotropic Eliashberg theory with fairly small errors and that, with the second-order corrections, the model coincides with the full theory. The second-order-corrected two-band model reproduces not only the intersheet but also the intrasheet variation of the superconducting energy gap on the Fermi surface.