Magnetic phases in periodically rippled graphene

We study the effects that ripples induce on the electrical and magnetic properties of graphene. The variation of the interatomic distance created by the ripples translates in a modulation of the hopping parameter between carbon atoms. A tight binding Hamiltonian including a Hubbard interaction term is solved self consistently for ripples with different amplitudes and periods. We find that, for values of the Hubbard interaction $U$ above a critical value $U_C$, the system displays a superposition of local ferromagnetic and antiferromagnetic ordered states. Nonetheless the global ferromagnetic order parameter is zero. The $U_C$ depends only on the product of the period and hopping amplitude modulation. When the Hubbard interaction is close to the critical value of the antiferromagnetic transition in pristine graphene, the antiferromagnetic order parameter becomes much larger than the ferromagnetic one, being the ground state similar to that of flat graphene.