Bending rigidities and universality of flexural modes in 2D crystals

Abstract

The existence of flexural modes with a quadratic phonon-dispersion is a distinguishing property of two-dimensional materials and has important consequences for their properties. Here, we deduce theoretically within the harmonic approximation the conditions for which orthotropic two-dimensional materials display a flexural mode. Further, we derive formulae for the calculation of the corresponding bending rigidities using the equilibrium structure and the second-order force constants as input. This completes the description of the elasticity of 2D crystals. Our findings are exemplarily validated by ab initio calculations of the phonon dispersions of four representative materials.

General invariance and equilibrium conditions for lattice dynamics in 1D, 2D, and 3D materials

“…This is in agreement with the theoretical model discussed in Ref. 21 and this work for bending waves, where the first-order dispersion vanishes as the bending conditions regardless of the crystal type of LD solids. As expected, such two invariance conditions are found to have negligible influence on the lattice dynamics of bulk materials, even for low-symmetry crystals.…”

“…As revealed by a few recent studies [19][20][21] , the violation of the rotational invariance and vanishing stress condition are expected to have a remarkable effect on the lattice-dynamical properties of 2D materials, which can lead to an unphysical dispersion relation of the out-of-plane ZA branch. The influence of such two invariance conditions on the phonon dispersions of infraredinactive 2D materials is illustrated in Fig.…”

“…20 ). It has been recently argued [19][20][21] that the lack of rotational invariance in the harmonic lattice Hamiltonian with non-vanishing external stress from DFT calculations is responsible for the linear dispersion of the ZA mode; it becomes quadratic after including rotational invariance and vanishing stress condition. Like translational invariance, there are various reasons for the broken rotational invariance in the interatomic force constants (IFCs) obtained from DFT and DFPT; these include incomplete basis sets, insufficient Brillouin zone sampling or numerical stability of algorithms.…”

“…Indeed, Huang had already demonstrated this in the case of bulk ionic crystals 35,36 , in which the long-range Coulomb interactions need a special treatment. However, this condition seems to have been omitted in recent investigations on the rotational invariance of 2D materials [19][20][21] . The detailed implementation of these conditions for correcting the IFCs either from real-space small displacements or reciprocal-space DFPT calculations are then presented.…”

Electronic transport in graphene with out-of-plane disorder

“…Being the first and the most investigated twodimensional (2D) material, graphene continues attracting attention as a platform for exploring novel physics and realizing prospective technological applications [1]. The 2D nature of graphene gives rise to soft flexural modes that result in low-energy out-of-plane disorder otherwise absent in bulk, three-dimensional materials [2][3][4][5]. The interplay between bending upon in-plane compression and the interlayer adhesion results in several distinct types of out-of-plane disorder: ripples, wrinkles and folds (see Refs.…”