Systematic study of the superconducting critical temperature in two- and three-dimensional tight-binding models: A possible scenario for superconducting H3S

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Ever since BCS theory was first formulated it was recognized that a large electronic density of states at the Fermi level was beneficial to enhancing $T_c$. The A15 compounds and the high temperature cuprate materials both have had an enormous amount of effort devoted to studying the possibility that such peaks play an important role in the high critical temperatures existing in these compounds. Here we provide a systematic study of the effect of these peaks on the superconducting transition temperature for a variety of tight-binding models of simple structures, both in two and three dimensions. In three dimensions large enhancements in $T_c$ can occur, due to van Hove singularities that result in divergences in the density of states. Furthermore, even in more realistic structures, where the van Hove singularity disappears, large enhancements in $T_c$ continue due to the presence of `robust' peaks in the densities of states. Such a peak, recently identified in the bcc structure of H$_3$S, is likely the result of such a van Hove singularity. In certain regimes, anomalies in the isotope coefficient are also expected.