High-$T_\textrm {C}$ superconductivity in Cs$_3$C$_{60}$ compounds governed by local Cs-C$_{60}$ Coulomb interactions

Unique among alkali-doped $\textit {A}$$_3$C$_{60}$ fullerene compounds, the A15 and fcc forms of Cs$_3$C$_{60}$ exhibit superconducting states varying under hydrostatic pressure with highest transition temperatures at $T_\textrm {C}$$^\textrm {meas}$ = 38.3 and 35.2 K, respectively. Herein it is argued that these two compounds under pressure represent the optimal materials of the $\textit {A}$$_3$C$_{60}$ family, and that the C$_{60}$-associated superconductivity is mediated through Coulombic interactions with charges on the alkalis. A derivation of the interlayer Coulombic pairing model of high-$T_\textrm {C}$ superconductivity employing non-planar geometry is introduced, generalizing the picture of two interacting layers to an interaction between charge reservoirs located on the C$_{60}$ and alkali ions. The optimal transition temperature follows the algebraic expression, $T_\textrm {C0}$ = (12.474 nm$^2$ K)/$\ell$${\zeta}$, where $\ell$ relates to the mean spacing between interacting surface charges on the C$_{60}$ and ${\zeta}$ is the average radial distance between the C$_{60}$ surface and the neighboring Cs ions. Values of $T_\textrm {C0}$ for the measured cation stoichiometries of Cs$_{3-\textrm{x}}$C$_{60}$ with x $\approx$ 0 are found to be 38.19 and 36.88 K for the A15 and fcc forms, respectively, with the dichotomy in transition temperature reflecting the larger ${\zeta}$ and structural disorder in the fcc form. In the A15 form, modeled interacting charges and Coulomb potential e$^2$/${\zeta}$ are shown to agree quantitatively with findings from nuclear-spin relaxation and mid-infrared optical conductivity. In the fcc form, suppression of $T_\textrm {C}$$^\textrm {meas}$ below $T_\textrm {C0}$ is ascribed to native structural disorder. Phononic effects in conjunction with Coulombic pairing are discussed.