Periodic forcing of graphene as geometric ripples on its surface

Abstract

We explore the possibility of using modulated high frequency periodic driving of mono-layer graphene to create effects of curved geometry. The low energy continuum limit of graphene is modeled using Dirac equation in (2+1) dimensions. We suggest that the modifications to the Dirac equation when written in a curved background space can also be induced by a suitable driving scheme. The time dependent system yields, in the approximate limit of high frequency pulsing, an effective time independent Hamiltonian that governs the time evolution, except for an initial and a final kick. We use a specific form of 4-phase pulsed forcing with suitably tuned choice of modulating operators to mimic the effects of weak metric perturbations and thereby effectively induce mild wrinkles on the surface. The strength of the perturbation is found to be directly related to ω -1 the time period of the driving field at the leading order. We apply the method to engineer some specific ‘nearly flat’ metrics and we find that the imprint of curvilinear geometry modifies the band structure significantly. The emergence of band gap at the Dirac point is crucial in this regard. We suggest that this method shall be useful in studying the response of various properties of such materials to non-trivial geometry without requiring any actual physical deformations.