Title: Origin of the pressure-dependent T$_c$ valley in superconducting simple cubic phosphorus,
Abstract: 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.