  In two-dimensional crystals that lack symmetry under reflections on the
horizontal plane of the lattice (non-$\sigma_{\rm h}$-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 HfSe$_{2}$, 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-$\sigma_{\rm h}$-symmetric
two-dimensional crystals, even if alternative forms of coupling are considered.

Superconductivity