  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 STM tip explains recent
experiments. In particular, we observe a reversible behavior for small values
of the STM current and an irreversible transition from flat rippled membrane to
rigid buckled membrane when the current surpasses a critical value. This work
opens the possibility to test mechanical properties of graphene under different
temperature and electrostatic conditions.
