Dynamics of thermal rippling of 2D MoSe₂ membranes

Rippling is a ubiquitous structural feature of freestanding monolayer two-dimensional (2D) materials. Phenomenological theory of the thermodynamic behavior of flexible membranes predict a power law for the correlation function of the out-of-plane displacements. We have demonstrated that the ripple structure of monolayer transition-metal dichalcogenides can be reconstructed from a single-frame scanning-transmission electron-microscopy image collected at designated angles. Here we take a sequence of such images of MoSe₂ and deduce the dynamics of rippling by combining the experimental images with pertinent molecular dynamics simulations based on empirical potentials. We first reconstruct the atomic-scale structures of 3D ripple distortions of suspended MoSe₂ membrane from STEM images. Fourier analysis of both the reconstructed images and molecular dynamics snapshots demonstrate that the correlation function H(q) follows the scaling law H(q) ∝ q^-4 fairly well. We not only provide a new tool in atomic structure reconstruction with temporal resolution, but also give the first direct evidence for the classical theory of thermal fluctuations in 2D membranes.