Mermin-Wagner theorem, flexural modes, and degraded carrier mobility in two-dimensional crystals with broken horizontal mirror symmetry

We show that the electron mobility in ideal, free-standing two-dimensional "buckled" crystals with broken horizontal mirror (sigma(h)) symmetry and Dirac-like dispersion (such as silicene and germanene) is dramatically affected by scattering with the acoustic flexural modes (ZA phonons). This is caused both by the broken sh symmetry and by the diverging number of long-wavelength ZA phonons, consistent with the Mermin-Wagner theorem. Non-sh-symmetric, "gapped" 2D crystals (such as semiconducting transition-metal dichalcogenides with a tetragonal crystal structure) are affected less severely by the broken sh symmetry, but equally seriously by the large population of the acoustic flexural modes. We speculate that reasonable long-wavelength cutoffs needed to stabilize the structure (finite sample size, grain size, wrinkles, defects) or the anharmonic coupling between flexural and in-plane acoustic modes (shown to be effective in mirror-symmetric crystals, like free-standing graphene) may not be sufficient to raise the electron mobility to satisfactory values. Additional effects (such as clamping and phonon stiffening by the substrate and/or gate insulator) may be required.