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Motivated by recent experiments, we investigate the pressure-dependentelectronic structure and electron-phonon (\emph{e-ph}) coupling for simplecubic phosphorus by performing first-principle calculations within the fullpotential linearized augmented plane wave method. As a function of increasingpressure, our calculations show a valley feature in T$_c$, followed by aneventual decrease for higher pressures. We demonstrate that this T$_c$ valleyat low pressures is due to two nearby Lifshitz transitions, as we analyze theband-resolved contributions to the \emph{e-ph} coupling. Below the firstLifshitz transition, the phonon hardening and shrinking of the $\gamma$ Fermisurface with $s$ orbital character results in a decreased T$_c$ with increasingpressure. 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 hencelead to an increase of T$_c$. For higher pressures, the phonon hardeningfinally dominates, and T$_c$ decreases again. Our study reveals that theintriguing T$_c$} valley discovered in experiment can be attributed to Lifshitztransitions, while the plateau of T$_c$ detected at intermediate pressuresappears to be beyond the scope of our analysis. This strongly suggests thatbesides \emph{e-ph} coupling, electronic correlations along with plasmoniccontributions may be relevant for simple cubic phosphorous. Our findings hintat the notion that increasing pressure can shift the low-energy orbital weighttowards $d$ character, and as such even trigger an enhanced importance oforbital-selective electronic correlations despite an increase of the overallbandwidth.

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