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Mermin-Wagner theorem, flexural modes, and degraded carrier mobility in 2D crystals with broken horizontal mirror ($σ_{\rm h}$) symmetry

We show that the electron mobility in ideal, free-standing two-dimensional `buckled' crystals with broken horizontal mirror ($σ_{\rm h}$) 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 $σ_{\rm h}$ symmetry and by the diverging number of long-wavelength ZA phonons, consistent with the Mermin-Wagner theorem. Non-$σ_{\rm h}$-symmetric, `gapped' 2D crystals (such as semiconducting transition-metal dichalcogenides with a tetragonal crystal structure) are affected less severely by the broken $σ_{\rm h}$ 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.