Superconductivity and Lattice Instability in Compressed Lithium from Fermi Surface Hot Spots

Lithium, a simple metal not superconducting above 5mK at ambient pressure, becomes a 20 K superconductor at 50 GPa. This high T$_c$ is shown to arise from critical (formally divergent) electron-phonon coupling to the transverse phonon branch along intersections of Kohn anomaly surfaces with the Fermi surface. First principles linear response calculations of the phonon spectrum and spectral function $\alpha^2 F(\omega)$ reveal (harmonic) instability already at 25 GPa. Our results imply that the fcc phase is anharmonically stabilized in the 25-38 GPa range.