Phonon softening and superconductivity in tellurium under pressure

The phonon dispersion and the electron-phonon interaction for the $\beta$-Po and the bcc high pressure phases of tellurium are computed with density-functional perturbation theory. Our calculations reproduce and explain the experimentally observed pressure dependence of the superconducting critical temperature (T$_{\rm c}$) and confirm the connection between the jump in T$_{\rm c}$ and the structural phase transition. The phonon contribution to the free energy is shown to be responsible for the difference in the structural transition pressure observed in low and room temperature experiments.