Origin of the pressure-dependent T$_c$ valley in superconducting simple cubic phosphorus

Motivated by recent experiments, we investigate the pressure-dependent electronic structure and electron-phonon (\emph{e-ph}) coupling for simple cubic phosphorus by performing first-principle 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 \emph{e-ph} coupling. Below the first Lifshitz transition, the phonon hardening and shrinking of the $\gamma$ Fermi surface with $s$ orbital character results in a decreased T$_c$ with increasing pressure. After the second Lifshitz transition, the appearance of $\delta$ Fermi surfaces with $3d$ orbital character generate strong \emph{e-ph} inter-band couplings in $\alpha\delta$ and $\beta\delta$ 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 besides \emph{e-ph} coupling, electronic correlations along with plasmonic contributions may be relevant for simple cubic phosphorous. 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.