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 (${\sigma }_{\mathrm{h}}$) 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 ${\sigma }_{\mathrm{h}}$ symmetry and by the diverging number of long-wavelength ZA phonons, consistent with the Mermin-Wagner theorem. Non-${\sigma }_{\mathrm{h}}$-symmetric, “gapped” 2D crystals (such as semiconducting transition-metal dichalcogenides with a tetragonal crystal structure) are affected less severely by the broken ${\sigma }_{\mathrm{h}}$ 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.

Figure 1

Cartoon illustration of the scattering process of an electron absorbing a flexural mode. The upper half of the figure illustrates an incoming electron and a phonon before scattering, the bottom half shows the scattered electron after it has absorbed the phonon (wavy line). Whereas in materials with horizontal mirror (${\sigma }_{\mathrm{h}}$) symmetry, such as graphene or semiconducting TMDs in the hexagonal phase, this scattering process is prohibited by symmetry, in materials without ${\sigma }_{\mathrm{h}}$ symmetry such as silicene or semiconducting TMDs in the tetragonal phase, scattering with flexural modes is allowed. This scattering can be exceedingly strong since the out-of-plane ionic displacement diverges as $1/Q$ at a small phonon momentum $Q [\delta R_{\mathrm{ZA}}\left(Q\right)\propto 1/Q\propto \lambda \to \infty$, instead of the “usual” in-plane behavior $\propto 1/Q^{1/2}]$ and the number of occupied phonon modes, governed by the Bose-Einstein distribution, diverges as well $[N_{\mathrm{ZA}}\left(Q\right)\propto 1/Q^2\to \infty ]$.

Figure 2

Calculated electron momentum relaxation rates in silicene at 300 K. The momentum relaxation rate due to scattering with ZA phonons has been obtained assuming “stiffened” ZA phonons (${\omega }_Q^{\left(\mathrm{ZA}\right)}\sim Q^{\alpha }$, with $\alpha =1$ and 3/2) for wavelengths longer than the cutoff ${\lambda }_0=1 \mu \mathrm{m}$ (essentially without any cutoff, yielding results largely independent of $\alpha$) and 0.1 nm, as indicated. The large value of the relaxation rate obviously indicates a failure of perturbation theory resulting from the diverging thermal population of ZA phonons.

Figure 3

ZA-limited electron mobility as a function of the wavelength cutoff ${\lambda }_0$ calculated for three values of the Fermi energy $E_{\mathrm{F}}$ and for two different stiffened phonon dispersions (${\omega }_Q^{\left(\mathrm{ZA}\right)}\sim Q^{\alpha }$ with $\alpha =3/2$ and 1) for wavelengths longer than the cutoff ${\lambda }_0$. The upper curve labeled “hard-cutoff” is, instead, the electron mobility limited by in-plane (TA and LA) and flexural (ZA) acoustic modes and assuming that ZA phonons with wavelengths longer than ${\lambda }_0$ are fully damped, i.e., that electrons do not scatter with long-wavelength ZA phonons.

Figure 4

As in Fig. 3, but for germanene.

Figure 5

As in Fig. 3, but for the “gapped and parabolic” ${\mathrm{HfSe}}_2$.

Figure 6

As in Fig. 5, but for iodine-functionalized stanene, “iodostannanane.”

Figure 7

Cutoff wavelength ${\lambda }_0$ as a function of the maximum allowed out-of-plane displacement $\delta R$, expressed in units of the lattice constant $a_0$.