Topological phase-diagram of time-periodically rippled zigzag graphene nanoribbons

The topological properties of electronic edge states in time-periodically driven spatially-periodic corrugated zigzag graphene nanoribbons are studied. An effective one-dimensionalHamiltonian is used to describe the electronic properties of graphene and the time-dependence is studiedwithin the Floquet formalism. Then the quasienergy spectrumof the evolution operator is obtained using analytical and numeric calculations, both in excellent agreement. Depending on the external parameters of the time-driving, two different kinds (types I and II) of touching band points are found, which have aDirac-like nature at both zero and p  quasienergy. These touching band points are able to host topologically protected edge states for afinite size system. The topological nature of such edge states was confirmed by an explicit evaluation of the Berry phase in the neighborhood of type I touching band points and by obtaining thewinding number of the effectiveHamiltonian for type II touching band points. Additionally, the topological phase diagram in terms of the driving parameters of the systemwas built.