First principles study of the electronic structure and phonon properties for Al and C-doped MgB$_2$

We have studied the structural, electronic and lattice dynamic properties of the superconducting alloys Al and C-doped MgB$_2$ within the framework of density functional perturbation theory, using a mixed-basis pseudopotential method and the virtual crystal approximation (VCA) for modeling the alloy. For both systems the structural parameters were determined on the following ranges, 0$\leq$x$\leq$1 for Mg$_{1-x}$Al$_x$B$_2$ and 0$\leq$x$\leq$0.4 for MgB$_{2(1-x)}$C$_{2x}$, finding a very good agreement between the calculated structural parameters and experimental data. The complete phonon dispersion curves were calculated for selected Al and C-concentrations. The calculated phonon bands for MgB$_2$ using the LDA and GGA approximations are compared in detail with the experimental data available in the literature. The evolution of the full-dispersion curves are analyzed as a function of Al and C-concentration, specially the E$_{2g}$-phonon mode frequency. In agreement with the experimental observed behavior, we find strong renormalization of the E$_{2g}$-mode for both Al and C-doped MgB$_2$. Additionally, we found a strong reduction of the E$_{2g}$-band dispersion with the filling of the $\sigma$-band. This research was supported by CONACYT, M\'{e}xico under Grant No. 43830-F.