First-principles calculation of the superconducting transition inMgB2within the anisotropic Eliashberg formalism

We present a study of the superconducting transition in ${\mathrm{MgB}}_2$ using the ab initio pseudopotential density-functional method, a fully anisotropic Eliashberg equation, and a conventional estimate for ${\mu }^{*}.$ Our study shows that the anisotropic Eliashberg equation, constructed with ab initio calculated momentum-dependent electron-phonon interaction and anharmonic phonon frequencies, yields an average electron-phonon coupling constant $\lambda =0.61,$ a transition temperature $T_c=39\mathrm{}\hspace{0.5em}\mathrm{K},$ and a boron isotope-effect exponent ${\alpha }_{\mathrm{B}}=\mathrm{0.32.}\mathrm{}$ The calculated values for $T_c,$ $\lambda ,$ and ${\alpha }_{\mathrm{B}}$ are in excellent agreement with transport, specific-heat, and isotope-effect measurements, respectively. The individual values of the electron-phonon coupling $\lambda (\overrightarrow{k},k^{\prime })$ on the various pieces of the Fermi surface, however, vary from 0.1 to 2.5. The observed $T_c$ is a result of both the raising effect of anisotropy in the electron-phonon couplings and the lowering effect of anharmonicity in the relevant phonon modes.