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Crystalline sheets (e.g., graphene and transition metal dichalcogenides)liberated from a substrate are a paradigm for materials at criticality becauseflexural phonons can fluctuate into the third dimension. Although studies ofstatic critical behaviors (e.g., the scale-dependent elastic constants) areplentiful, investigations of dynamics remain limited. Here, we use moleculardynamics to study the time dependence of the midpoint (the heightcenter-of-mass) of doubly clamped nanoribbons, as prototypical grapheneresonators, under a wide range of temperature and strain conditions. Bytreating the ribbon midpoint as a Brownian particle confined to a nonlinearpotential (which assumes a double-well shape beyond the buckling transition),we formulate an effective theory describing the ribbon's tunneling rate acrossthe two wells and its oscillations inside a given well. We find that, fornanoribbbons compressed above the Euler buckling point and thermalized above atemperature at which the non-linear effects due to thermal fluctuations becomesignificant, the exponential term (the ratio between energy barrier andtemperature) depends only on the geometry, but not the temperature, unlike theusual Arrhenius behavior. Moreover, we find that the natural oscillation timefor small strain shows a non-trivial scaling $\tau_{\rm o}\simL_0^{\,z}T^{-\eta/4}$, with $L_0$ being the ribbon length, $z=2-\eta/2$ beingthe dynamic critical exponent, $\eta=0.8$ being the scaling exponent describingscale-dependent elastic constants, and $T$ being the temperature. These unusualscale- and temperature-dependent dynamics thus exhibit dynamic criticality andcould be exploited in the development of graphene-based nanoactuators.

Vibrations and tunneling of strained nanoribbons at finite temperature