Intrinsic mode-coupling and thermalization in nanomechanical graphene drums

Nanomechanical graphene resonators display strong nonlinear behavior, which leads to coupling between normal modes. This coupling allows for intermodal energy-transfer, which facilitates the redistribution of energy initially localized in a single mode. Further, the mode-coupling intrinsically limits the quality factor of the device. We study the mode-coupling in a circular graphene resonator using molecular dynamics and continuum mechanics. Mimicking a ring-down setup, the fundamental mode is excited with a given energy, and the time-evolution of this energy is computed. At $T>0$, we find a relaxation rate independent of system size and proportional to $T^*/\epsilon_{\rm pre}^2$, where $T^*$ is the effective temperature and $\epsilon_{\rm pre}$ is the pre-strain of the system \footnote{D. Midtvedt, Z. Qi, A. Croy, H. S. Park, A. Isacsson,arXiv:1309.1622}. At low temperatures, the system enters a metastable state where only very few low-frequency modes are excited, the life-time of which increases exponentially with decreasing excitation energy. This is similar to what is seen in the much studied Fermi-Pasta-Ulam (FPU) problem. We make a detailed comparison between the dynamics of a graphene drum and the FPU system, and propose to use graphene drums as test beds for FPU physics.