Electron-phonon interactions and superconductivity in Si, Ge, and Sn

The pseudopotential-total-energy method is used to calculate the phonon frequency, the electron density of states at the Fermi level, and the electron-phonon coupling constant for the group-IV elements in the metallic \ensuremath{\beta}-Sn structure. For these elements, the normal-state behavior is similar to that found in other simple and transition metals; the phonon frequencies, force constants, and electron-phonon matrix elements increase with increasing average electron density. With use of a semiempirical treatment of the electron-phonon coupling calculated for one phonon wave vector, the superconducting transition temperatures at normal and high pressures are examined. The superconducting transition temperature decreases while the magnitude of its pressure coefficient increases in going to heavier elements. This behavior is in good agreement with experiment. For Si and Ge, the superconducting behavior is similar to that of white tin. Because of competition and compensation between the cutoff in the phonon spectrum and the electron-phonon matrix element, the electron-phonon coupling \ensuremath{\lambda}'s are similar for the three elements. Hence, the Debye temperature, which is the prefactor of the McMillan equation, dominates in determining the superconducting critical temperatures.