Using first-principles calculations for understanding MgB$_{2}$

Electronic structure methods have been the key to understand the peculiar properties of the multi-band superconductor MgB$_2$. After briefly reviewing the current state of our understanding of the material we will discuss some of the more recent observations on Al or C doped MgB$_2$.\\ We will demonstrate, based on first-principles band structure calculations and Eliashberg theory, that the experimentally observed decrease of the critical temperature $T_c$ of Al and C doped MgB$_2$ samples can be understood mainly in terms of a band filling effect due to the electron doping by Al and C. A simple scaling of the electron-phonon coupling constant $\lambda$ by the variation of the density of states (DOS) as function of electron doping is sufficient to capture qualitatively the observed behavior. Using the virtual crystal approximation to account for the electron doping we calculate the change of the DOS due to doping. The expected hardening of the $E_{2g}$ phonon frequency at $\Gamma$ is also well reproduced by our calculations using this approximation. These two contributions together, the change of DOS and phonon frequency, reproduce the observed dependence of $T_c$ on doping concentration astonishingly well not using any free parameter to fit the data.\\ Further, we also explain the long standing open question of the experimental observation of a nearly constant $\pi$ gap as function of doping by a compensation of the effect of band filling and interband scattering. Both effects together generate a nearly constant $\pi$ gap and shift the merging point of both gaps to higher doping concentrations, resolving the discrepancy between experiment and theoretical predictions based on interband scattering only.\\ The author acknowledges many fruitful collaborations and discussions on this exciting topic with O.K. Andersen, I.I. Mazin, A.A. Golubov, O.V. Dolgov, O. Jepsen and R.K. Kremer over the past years.