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We consider a simple spin-membrane model for rippling in graphene. The modelexhibits transitions from a flat but rippled membrane to a buckled one. At hightemperature the transition is second order but it is first order at lowtemperature for appropriate strength of the spin-spin coupling. Driving thesystem across the first order phase transition in nonequilibrium conditionsthat mimic interaction of the graphene membrane with a STM tip explains recentexperiments. In particular, we observe a reversible behavior for small valuesof the STM current and an irreversible transition from flat rippled membrane torigid buckled membrane when the current surpasses a critical value. This workopens the possibility to test mechanical properties of graphene under differenttemperature and electrostatic conditions.

STM driven transition from rippled to buckled graphene in a spin-membrane model