Superconductivity of Li, Al and K under pressure

C Franchini$^{\S}$, N. N. Lathiotakis$^{\dag}$, A. Floris$^{\dag\S}$, A. Sanna$^{\S}$, M. A. L. Marques$^{\dag}$, M.~L{\"u}ders$^{\ddag}$, S. Massidda$^{\S}$, E. K. U. Gross$^{\dag}$, A. Continenza$^{*}$.\\ $^{*}$ CASTI - INFM and Dip. Fis., Univ. di L'Aquila, I-67010 Coppito (L'Aquila) Italy; $^{\S}$ SLACS INFM and Dip. Fis., Univ. di Cagliari, I-09042 Monserrato (Ca), Italy; $^{\dag}$ Institut f{\"u}r Theoretische Physik, Freie Universit{\"a}t Berlin, Arnimallee 14, D-14195 Berlin, Germany; $^{\ddag}$ Daresbury Lab., Warrington WA4 4AD, United Kingdom.\\ Extreme pressure strongly affects the superconducting properties of ``simple'' metals, like Li, K and Al. Using the new ab-initio method of density functional theory of the superconducting state, we report investigations on the superconducting properties of dense Li, K and Al. Our results show an unprecedented agreement with experiments, assess the predictive power of the method over a wide range of densities and electron-phonon couplings, and provide predictions for K, where no experiments exist so far. For fcc K we predict a superconducting phase transition at 18 GPa, with a maximum critical temperature of about 2 K at 23 GPa, the pressure where the crossover between the fcc and the K $III$ structure experimentally occurs. We studied the effect of pressure on the electronic and vibrational properties of alkali, showing a progressive phonon softening near the K point of the Brillouin zone and a concomitant enhancement of the electron-phonon coupling constant $\lambda$.