Fermi Surface Topology and the Upper Critical Field in Two-Band Superconductors: Application to MgB₂

Recent measurements of the anisotropy of the upper critical field B_c2 on MgB₂ single crystals have shown a puzzling strong temperature dependence. Here, we present a calculation of the upper critical field based on a detailed modeling of band structure calculations that takes into account both the unusual Fermi surface topology and the two gap nature of the superconducting order parameter. Our results show that the strong temperature dependence of the B_c2 anisotropy can be understood as an interplay of the dominating gap on the σ band, which possesses a small c-axis component of the Fermi velocity, with the induced superconductivity on the π-band possessing a large c-axis component of the Fermi velocity. We provide analytic formulas for the anisotropy ratio at T=0 and T=T_c and quantitatively predict the distortion of the vortex lattice based on our calculations.