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We consider a simple spin-membrane model for rippling in graphene. The model exhibits transitions from a flat but rippled membrane to a buckled one. At high temperature the transition is second order, but it is first order at low temperature for appropriate strength of the spin-spin coupling. Driving the system across the first-order phase transition in nonequilibrium conditions that mimic interaction of the graphene membrane with a scanning tunneling microscopy (STM) tip explains recent experiments. In particular, we observe a reversible behavior for small values of the STM current and an irreversible transition from a flat rippled membrane to a rigid buckled membrane when the current surpasses a critical value. This work makes it possible to test the mechanical properties of graphene under different temperature and electrostatic conditions.
Freestanding graphene Crystalline order Atomic-scale