Phonon renormalization and anharmonicity in Al-doped MgB$_2$

We have studied the evolution of the $E_{2g}$ phonon mode dynamics in Mg$_{1-x}$Al$_x$B$_2$ as a function of doping using the Frozen Phonon Approximation (FPA). The doping was modeled in the ab-initio Virtual Crystal Approximation (VCA). The results were obtained by means of first-principles total-energy calculations using the full potential Linearized Augmented Plane Wave (LAPW) method and the Generalized Gradient Approximation (GGA) for the exchange-correlation potential. We present results for the evolution of the phonon frequency and anharmonicity of the $E_{2g}$ mode as a function of Al concentration ($x$). From a comparison of the experimental data with the calculated $E_{2g}$ phonon frequency we show that the VCA-FPA reproduces the observed phonon renormalization in the whole range of Al concentrations. More interestingly, we find that the anharmonicity gradually decreases with Al doping and vanishes for $x$(Al)$>$0.5, that behaviour correlates with the evolution of the measured Raman linewidth in Al-doped MgB$_2$. The significance of these results are discussed in the light of the experimentally observed loss of superconductivity in Mg$_{1- x}$Al$_x$B$_2$.This work was supported by Consejo Nacional de Ciencia y Tecnolog{\'\i}a (CONACYT, M\'exico) under Grant. No. 43830-F.