Modeling Intercalated Group-4-Metal Nitride Halide Superconductivity with Interlayer Coulomb Coupling

Behavior consistent with Coulomb-mediated high-T$_C$ superconductivity is shown to be present in the intercalated group-4-metal nitride halides A$_x$(S)$_y$MNX, where the MNX host (M = Ti, Zr, Hf; X = Cl, Br) is partially intercalated with cations A$_x$ and optionally molecular species (S)$_y$ in the van der Waals gap between the halide X layers, expanding the basal-plane spacing d. The optimal transition temperature is modeled by T$_{C0}$ ${\propto}$ {\zeta}$^{-1}$({\sigma}/$A$)$^{1/2}$, where the participating fractional charge per area per formula unit {\sigma}/$A$ and the distance {\zeta}, given by the transverse A$_x$-X separation ({\zeta} < d), govern the interlayer Coulomb coupling. From experiment results for {\beta}-form compounds based on Zr and Hf, in which concentrations x of A$_x$ are varied, it is shown that {\sigma} = {\gamma}[v(x$_{opt}$ $-$ x$_0$)], where x$_{opt}$ is the optimal doping, x$_0$ is the onset of superconducting behavior, v is the A$_x$ charge state, and {\gamma} = 1/8 is a factor determined by the model. Observations of T$_C$ < T$_{C0}$ in the comparatively more disordered {\alpha}-A$_x$(S)$_y$TiNX compounds are modeled as pair-breaking by remote Coulomb scattering from the A$_x$ cations, which attenuates exponentially with increasing {\zeta}. The T$_{C0}$ values calculated for nine A$_x$(S)$_y$MNCl compounds, shown to be optimal, agree with the measured T$_C$ to within experimental error. The model for T$_{C0}$ is also found to be consistent with the absence of high-T$_C$ characteristics for A$_x$MNX compounds in which a spatially separated intercalation layer is not formed.