Bending rigidities and universality of flexural modes in 2D crystals

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.