First-principles calculation of the superconducting transition in MgB₂ within the anisotropic Eliashberg formalism

We present a study of the superconducting transition in MgB₂ using the ab initio pseudopotential density- functional method, a fully anisotropic Eliashberg equation, and a conventional estimate for μ*. 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 λ=0.61, a transition temperature T_c=39 K, and a boron isotope-effect exponent α_B=0.32. The calculated values for T_c, λ, and α_B are in excellent agreement with transport, specific-heat, and isotope-effect measurements, respectively. The individual values of the electron-phonon coupling λ(k^→,k^→′) 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.