Fermi Surface Topology and the Upper Critical Field in Two-Band Superconductors: Application to ${\mathrm{M}\mathrm{g}\mathrm{B}}_2$

Recent measurements of the anisotropy of the upper critical field $B_{c2}$ on ${\mathrm{M}\mathrm{g}\mathrm{B}}_2$ 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 $\sigma$ band, which possesses a small $c$-axis component of the Fermi velocity, with the induced superconductivity on the $\pi$-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.

Figure 1

Fermi surface topology used for the calculation of $B_{c2}$ in this work. The $\pi$ band is modeled by a half-torus, the $\sigma$ band by a distorted cylinder.

Figure 2

Temperature dependence of the anisotropy ratio $\Gamma =B_{c2}^{ab}/B_{c2}^c$ for the two-band model described in the text and different interband coupling strengths $\eta$. Solid circles are experimental results taken from Lyard et al. [20].

Figure 3

Anisotropy ratio $\Gamma =B_{c2}^{ab}/B_{c2}^c$ as a function of interband coupling strength $\eta$ for $T=0$ (dashed line) and $T=T_c$ (solid line). The dotted line shows the approximation for $T=0$ given in Eq. (7). At $T_c$ the anisotropy ratio is much more sensitive to small interband coupling strengths $\eta$.

Figure 4

Vortex lattice structure for magnetic field in c-axis direction (left panel) and in ab-plane direction (right panel) at zero temperature near $B_{c2}$ calculated from the two-band model described in the text. For comparison, the area of the unit cell has been kept fixed.