Origin of the pressure-dependent Tc valley in superconducting simple cubic phosphorus

Motivated by recent experiments, we investigate the pressure-dependent electronic structure and electron-phonon (e-ph) coupling for simple cubic phosphorus by performing first-principles calculations within the full potential linearized augmented plane-wave method. As a function of increasing pressure, our calculations show a valley feature in Tc, followed by an eventual decrease for higher pressures. We demonstrate that this Tc valley at low pressures is due to two nearby Lifshitz transitions, as we analyze the band-resolved contributions to the e-ph coupling. Below the first Lifshitz transition, the phonon hardening and shrinking of the γ Fermi surface with s-orbital character results in a decreased Tc with increasing pressure. After the second Lifshitz transition, the appearance of δ Fermi surfaces with 3d-orbital character generate strong e-ph interband couplings in αδ and βδ channels, and hence lead to an increase of Tc. For higher pressures, the phonon hardening finally dominates, and Tc decreases again. Our study reveals that the intriguing Tc valley discovered in experiment can be attributed to Lifshitz transitions, while the plateau of Tc detected at intermediate pressures appears to be beyond the scope of our analysis. This strongly suggests that aside from e-ph coupling, electronic correlations along with plasmonic contributions may be relevant for simple cubic phosphorus. Our findings hint at the notion that increasing pressure can shift the low-energy orbital weight towards d character, and as such even trigger an enhanced importance of orbital-selective electronic correlations despite an increase of the overall bandwidth.