Effects of Al and C doping on the electronic structure and phonon renormalization in ${\text{MgB}}_2$

We have studied the structural, electronic, and lattice dynamical properties of ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ and ${\text{MgB}}_{2\left(1-y\right)}{\text{C}}_{2y}$ alloys within the framework of density-functional theory using the self-consistent virtual-crystal approximation (VCA). The structural properties, electronic band structure, and full phonon dispersion have been analyzed for the $0\le x\left(\text{Al}\right)\le 1$ and $0\le y\left(\text{C}\right)\le 0.3$ ranges of concentrations. We found that both dopants reduce the number of holes in the $\sigma$ bands until they are completely filled at different doping concentrations. These concentrations correlate with the experimentally observed loss of superconductivity in these alloys. The largest influence of doping on the phonon dispersion is found for the branches connected to the $E_{2g}$ and $B_{1g}$ modes. For both alloys a sudden increase in the $E_{2g}$-mode frequency is observed in a well-defined region of Al and C concentrations which correlates with the topological changes in the $\sigma$-Fermi surfaces.

Figure 1

Evolution of the calculated $a$ and $c$ lattice parameters for the ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ and ${\text{MgB}}_{2\left(1-y\right)}{\text{C}}_{2y}$ alloys, and comparison with experimental data (Refs. 14, 17, 19, 20, 21, 22, 23).

Figure 2

Evolution of the $c/a$ ratio for the ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ and ${\text{MgB}}_{2\left(1-y\right)}{\text{C}}_{2y}$ alloys.

Figure 3

Electronic band structure of ${\text{MgB}}_2$ indicating the points where electronic topological transitions can be expected with electron doping.

Figure 4

Evolution of the energies corresponding to the $\sigma$, $\pi$, and (predominantly) Mg bands as a function of $x$ for ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ and $y$ for ${\text{MgB}}_{2\left(1-y\right)}{\text{C}}_{2y}$ alloys.

Figure 5

Evolution of $N_i\left(E_F\right)$ ($i=\sigma$, $\pi$, Mg, and total) as a function of $x$ for ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ and $y$ for ${\text{MgB}}_{2\left(1-y\right)}{\text{C}}_{2y}$ alloys. The calculated values for ${\text{MgB}}_2$ are $N_{\sigma }=0.148$ states ${\text{eV}}^{-1}$/spin and $N_{\pi }=0.200$ states ${\text{eV}}^{-1}$/spin.

Figure 6

Calculated and experimental (Refs. 82, 83) phonon dispersion curves for ${\text{MgB}}_2$.

Figure 7

Phonon dispersion of ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ as a function of $x$. Branches strongly affected by doping are in indicated by gray color.

Figure 8

Phonon dispersion of ${\text{MgB}}_{2\left(1-y\right)}{\text{C}}_{2y}$ as a function of $y$. Branches strongly affected by doping are in indicated by gray color.

Figure 9

Evolution of the $E_{2g}$- and $B_{1g}$-phonon mode frequencies at $\Gamma$ as a function of $x$ for ${\text{Mg}}_{1-x}{\text{Al}}_x{\text{B}}_2$ and $y$ for ${\text{MgB}}_{2\left(1-y\right)}{\text{C}}_{2y}$ alloys.