Distribution function of thermal ripples in h-BN, graphene and MoS2

Deformations induced by external loading in two-dimensional (2D) materials are well described by traditional thin plate and film theories. However, for randomly distributed ripples induced by thermal fluctuations, a systematic theory is still lacking to describe the structural state at finite temperature. In this work, we performed molecular dynamics simulations and statistical mechanics to investigate the distribution of thermal ripples in widely studied 2D materials like h-BN, graphene, and MoS₂. Our results indicate that thermal ripples are different from the external force-induced deformations in 2D materials, and their probability decays exponentially as the ratio of ripple height to diameter (t/D) increases. By simply substituting the energy in Boltzmann distribution with classical bending energy of 2D materials, we found that the partition function is in proportion to the square root of the ratio of bending stiffness to temperature. The derived distribution function is in good agreement with molecular dynamics simulations, indicating that the probability conforms to the law of Boltzmann distribution. Our simulations and theoretical derivation will give a deeper understanding of ripple morphology in 2D materials and advance the development of the state theory of 2D systems at finite temperature.