Superconductivity Induced by Flexural Modes in Non-σ_h-Symmetric Dirac-Like Two-Dimensional Materials: A Theoretical Study for Silicene and Germanene

In two-dimensional crystals that lack symmetry under reflections on the horizontal plane of the lattice (non-σ_h-symmetric), electrons can couple to flexural modes (ZA phonons) at first order. We show that in materials of this type that also exhibit a Dirac-like electron dispersion, the strong coupling can result in electron pairing mediated by these phonons, as long as the flexural modes are not damped or suppressed by additional interactions with a supporting substrate or gate insulator. We consider several models: The weak-coupling limit, which is applicable only in the case of gapped and parabolic materials, like stanene and HfSe₂, thanks to the weak coupling; the full gap-equation, solved using the constant-gap approximation and considering statically screened interactions; its extensions to energy-dependent gap and to dynamic screening. We argue that in the case of silicene and germanene superconductivity mediated by this process can exhibit a critical temperature of a few degrees K, or even a few tens of degrees K when accounting for the effect of a high-dielectric- constant environment. We conclude that the electron/flexural-modes coupling should be included in studies of possible superconductivity in non-σ_h-symmetric two-dimensional crystals, even if alternative forms of coupling are considered.