Delineating the role of ripples on the thermal expansion of 2D honeycomb materials: graphene, 2D h-BN and monolayer (ML)-MoS 2 .

We delineated the role of thermally excited ripples on the thermal expansion properties of 2D honeycomb materials (free-standing graphene, 2D h-BN, and ML-MoS2 ), by explicitly carrying out three-dimensional (3D) and two-dimensional (2D) molecular dynamics simulations. In 3D simulations, the in-plane lattice parameter (a-lattice) of graphene and 2D h-BN shows thermal contraction over a wide range of temperatures and exhibits a strong system size dependence. The 2D simulations of the very same system show a reverse trend, where the a-lattice expands in the whole computed temperature range. In contrast to graphene and 2D h-BN, the a-lattice of ML-MoS2 shows thermal expansion in both 2D and 3D simulations and their system size dependence is marginal. By analyzing the phonon dispersion at 300 K, we found that the discrepancy between 2D and 3D simulations of graphene and 2D h-BN is due to the absence of out-of-plane bending modes (ZA) in 2D simulations, which is responsible for the thermal contraction of the a-lattice at low temperature. Meanwhile, all the phonon modes are present in the 2D phonon dispersion of ML-MoS2 , which indicates that the origin of the ZA mode is not purely due to the out-of-plane movement of atoms and also its effect on thermal expansion is not significant as found in graphene and 2D h-BN.