Metastable states and energy flow pathway in square graphene resonators

Nonlinear interaction between flexural modes is critical to heat conductivity and mechanical vibration of two-dimensional materials such as graphene. Much effort has been devoted to understand the underlying mechanism. In this paper, we examine solely the out-of-plane flexural modes and identify their energy flow pathway during thermalization process. The key is the development of a universal scheme that numerically characterizes the strength of nonlinear interactions between normal modes. In particular, for our square graphene system, the modes are grouped into four classes by their distinct symmetries. The couplings are significantly larger within a class than between classes. As a result, the equations for the normal modes in the same class as the initially excited one can be approximated by driven harmonic oscillators, therefore, they get energy almost instantaneously. Because of the hierarchical organization of the mode coupling, the energy distribution among the modes will arrive at a stable profile, where most of the energy is localized on a few modes, leading to the formation of "natural package" and metastable states. The dynamics for modes in other symmetry classes follows a Mathieu type of equation, thus, interclass energy flow, when the initial excitation energy is small, starts typically when there is a mode that lies in the unstable region in the parameter space of Mathieu equation. Due to strong coupling of the modes inside the class, the whole class will get energy and be lifted up by the unstable mode. This characterizes the energy flow pathway of the system. These results bring fundamental understandings to the Fermi-Pasta-Ulam problem in two-dimensional systems with complex potentials, and reveal clearly the physical picture of dynamical interactions between the flexural modes, which will be crucial to the understanding of their abnormal contribution to heat conduction and nonlinear mechanical vibrations.