Superconductivity under pressure: application of the functional derivative

In this paper, we calculate the superconducting critical temperature as a function of pressure, Tc(P ), using a method based on the functional derivative of the critical temperature with the Eliashberg function, dTc/dA2F. The coulomb electron-electron repulsion parameter, mu*(p) at each pressure is obtained in a consistent way by solving the linearized Migdal-Eliashberg equation. This method requires as the starting input only the knowledge of Tc(P ) at the starting pressure. It applies to superconductors for which the Migdal-Eliashberg equations hold. We study Al, a typical BCS weak coupling superconductor with a low Tc . Our results of Tc(P ) as a function of pressure for Al show an excellent agreement with the calculations of Profeta et al. (Phys. Rev. Lett.96, 047003 (2006)) which agree well with experiment.