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 ${T}_{c}$, followed by an eventual decrease for higher pressures. We demonstrate that this ${T}_{c}$ 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 $\ensuremath{\gamma}$ Fermi surface with $s$-orbital character results in a decreased ${T}_{c}$ with increasing pressure. After the second Lifshitz transition, the appearance of $\ensuremath{\delta}$ Fermi surfaces with $3d$-orbital character generate strong e-ph interband couplings in $\ensuremath{\alpha}\ensuremath{\delta}$ and $\ensuremath{\beta}\ensuremath{\delta}$ channels, and hence lead to an increase of ${T}_{c}$. For higher pressures, the phonon hardening finally dominates, and ${T}_{c}$ decreases again. Our study reveals that the intriguing ${T}_{c}$ valley discovered in experiment can be attributed to Lifshitz transitions, while the plateau of ${T}_{c}$ 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.