Microscopic theory of superconductivity in $\text{MgB}_2$-type systems in a magnetic field: A neighborhood of $H_{\text{c}2}$

We calculate the upper critical magnetic field $H_{\mathrm{c} 2}$ in the framework of a microscopic superconductivity theory with two energy bands of different dimensions on the Fermi surface with the cavity topology typical of the compound $\mathrm{MgB}_2$ taken into account (an anisotropic system). We assume an external magnetic field parallel to the crystallographic $z$ axis. We obtain analytic formulas in the low-temperature range $(T/T_{\mathrm{c}}\ll1)$ and also near the critical temperature $\bigl((T-T_{\mathrm{c}})/T_{\mathrm{c}}\ll1\bigr)$. We compare the temperature dependence of $H_{\mathrm{c} 2}$ for a two-band anisotropic system with that of $H_{\mathrm{c} 2}^0$ corresponding to a two-band isotropic system (with Fermi-surface cavities of the same topology). We determine the role of the band-structure anisotropy, the positive curvature of the upper critical field near the critical temperature, and the important role of the ratio $v_1/v_2$ of the velocities on the Fermi surface in determining $H_{\mathrm{c} 2}$. We also obtain the values of the parameters $\Delta_1$ and $\Delta_2$ along the line of the critical magnetic field.