Anharmonic renormalization of flexural acoustic modes in graphene and their effect on the mechanical stability of the membrane

The mechanical stability and long-range crystalline order of two dimensional materials has always been under debate [1], however, since the discovery of graphene, the debate is only theoretical. 2D materials are invariant upon any rotation by putting the axis in the plane where the system is. This symmetry obliges the harmonic dispersion of the acoustic out-of-plane modes to be quadratic close to the point gamma in the first Brillouin zone, instead of linear, and this functionality makes the membrane unstable. The instability is translated as diverging atomic displacements as a function of the sample size and finite line width of phonons with very low crystal momentum[2].

These instabilities arise within the harmonic approximation, however, an anharmonic approach may suppress them. In this work we apply an anharmonic method named stochastic self-consistent harmonic approximation[3]. We see that the inclusion of anharmonic effects suppresses the divergence in the atomic displacements and provides a vanishing line width to the phonons with very low crystal momentum.

[1] Mermin, N David, Phys. Rev. B, 176, 250 (1968).
[2] M. Katsnelson et. al, Account of Chemical Research, 46, 97-105 (2012).
[3] I. Errea et. al, Phys. Rev. B, 89, 064302 (2014).