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] sub(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 [sigma] sub(h) symmetry and by the diverging number of long-wavelength ZA phonons, consistent with the Mermin-Wagner theorem. Non-[sigma] sub(h)-symmetric, "gapped" 2D crystals (such as semiconducting transition-metal dichalcogenides with a tetragonal crystal structure) are affected less severely by the broken [sigma] sub(h) 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.