For the 40 K superconductor ${\mathrm{MgB}}_2,$ we have calculated the electronic and phononic structures and the electron-phonon (e-ph) interaction throughout the Brillouin zone ab initio. In contrast to the isoelectronic graphite, ${\mathrm{MgB}}_2$ has holes in the bonding σ bands, which contribute $42\%$ to the density of states: $N\left(0\right)=0.355\mathrm{}$ $\mathrm{states}/\left({\mathrm{MgB}}_2\right)\left(\mathrm{eV}\right)\left(\mathrm{spin}\right).\mathrm{}$ The total interaction strength, $\lambda =0.87\mathrm{}$ and ${\lambda }_{\mathrm{tr}}=0.60,$ is dominated by the coupling of the σ holes to the bond-stretching optical phonons with wave numbers in a narrow range around $590\hspace{0.5em}{\mathrm{cm}}^{-1}.$ Like the holes, these phonons are quasi-two-dimensional and have wave vectors close to $\Gamma A,$ where their symmetry is E. The π electrons contribute merely 0.25 to λ and to ${\lambda }_{\mathrm{tr}}.$ With Eliashberg theory we evaluate the normal-state resistivity, the density of states in the superconductor, and the B-isotope effect on $T_c$ and ${\Delta }_0,$ and find excellent agreement with experiments, when available. $T_c=40\mathrm{}\hspace{0.5em}\mathrm{K}$ is reproduced with ${\mu }^{*}=0.10\mathrm{}$ and $2{\Delta }_0{/k}_B{T}_c=\mathrm{3.9.}\mathrm{}$ ${\mathrm{MgB}}_2$ thus seems to be an intermediate-coupling e-ph pairing s-wave superconductor.

• Received 20 March 2001

DOI:https://doi.org/10.1103/PhysRevB.64.020501