Pressure effects on the electronic properties and superconductivity in the $\beta$-pyrochlore oxides: $A$Os$_2$O$_6$ ($A$ = Na, K, Rb, Cs)

We present a first-principles study of the electronic structure and superconducting parameters of the compounds $A$Os$_2$O$_6$ ($A$ = Na, K, Rb, and Cs) and at ambient and applied hydrostatic pressure. We find that the sensitivity of the density of states at the Fermi energy, $E_{\rm F}$, and related electronic properties to the size of the alkali metal atom as well as to applied pressure is driven by a van Hove singularity with energy very close to $E_{\rm F}$. Further, a computation of the superconducting parameters of these materials allows us to show that the observed change of $T_c$, both upon substitution of the alkali metal and under applied hydrostatic pressure, can be well understood within a phonon-mediated pairing scenario. In this regard, we find that the correction to the effective electron mass due to spin fluctuations plays a significant role.