Analysis of rippling in incommensurate one-dimensional coupled chains

Graphene and other recently developed two-dimensional materials exhibit exceptionally strong in-plane stiffness. Relaxation of few-layer structures, either freestanding or on slightly mismatched substrates, occurs mostly through out-of-plane bending and the creation of large-scale ripples. In this work, we present a novel double chain model, where we allow relaxation to occur by bending of the incommensurate coupled system of chains. As we will see, this model can be seen as a new application of the well-known Frenkel-Kontorova model for a one-dimensional atomic chain lying in a periodic potential. We focus in particular on modeling and analyzing ripples occurring in ground state configurations, as well as their numerical simulation.