Solitary Waves on Graphene Superlattices

Abstract

This chapter reviews the basic theoretical aspects of the propagation of solitary electromagnetic waves in graphene superlattices, a one atom thick sheet of graphene deposited on a superlattice, made by several periodically alternating layers of SiO\(_2\) and h-BN. The electronic band structure of graphene and the techniques of band gap engineering are briefly presented. The analysis of the electronic properties of graphene superlattices by using both the transfer matrix method and the Kronig–Penny model are summarized. The nonlinear wave equation for the vector potential of the electromagnetic wave field is derived. This graphene superlattice equation (GSLeq) generalizes the sine-Gordon equation (sGeq). Hence, it also has kink and antikink solutions propagating at a constant speed. There is no closed-form expression for their shape. A straightforward asymptotic method is applied in order to analytically approximate its shape. The interactions of kinks and antikinks is studied by using a numerical method, the Strauss–Vazquez, which is a conservative, finite difference scheme. This numerical method is second-order accurate in both space and time, and nonlinearly stable, exactly conserving a discrete energy. Extensive numerical results for the kink–antikink interactions are presented as a function of a asymptotic parameter. For small values of this parameter, the interaction is apparently elastic, without noticeable radiation, being very similar to that expected for the sGeq. For large values of the asymptotic parameter, the inelasticity of the interaction results in the emission of wavepackets of radiation. In summary, the whole set of results suggest that the GSLeq behaves as a nearly integrable perturbation of the sGeq. Consequently, graphene superlattices can be used to study nonlinear wave phenomena with electromagnetic waves in the THz scale.