First principles study of Al and C-doped MgB$_2$: evolution of two gaps and critical temperature

We have studied the electron-phonon and superconducting properties of the Mg$_{1-x}$Al$_x$B$_2$ and MgB$_{2(1-x)}$C$_{2x}$ alloys within the framework of density functional perturbation theory, using a mixed-basis pseudopotential method and the virtual crystal approximation (VCA) for modeling the alloys. For both systems, the Eliashberg spectral function ($\alpha^{2}F(\omega)$) and the electron-phonon coupling parameter ($\lambda$) have been calculated in the two band model ($\sigma$,$\pi$) for several concentrations until $x(\mathrm{Al})=0.55$ and $x(\mathrm{C})=0.175$. Using the calculated $\alpha^{2}_{ij}F(\omega)$ and a diagonal expression for the Coulomb pseudopotential matrix, $\mu^{*}$, we solved numerically the Eliashberg gap equations in the two band model without interband scattering. We reproduce the experimental decreasing behavior of $\Delta_{\sigma}(x)$, $\Delta_{\pi}(x)$, and $T_{c}(x)$ for both alloy systems. The role of the interband scattering in the observed behavior of the superconducting gaps and $T_c$ in the Al- and C-MgB$_2$ alloys is discussed. This research was supported by Consejo Nacional de Ciencia y Tecnolog\'ia (Conacyt) under Grant No. 43830-F.