A Phenomenological Model for Self-rippling Energy of Free Graphene Monolayer and Its Application

Abstract

Several candidate phenomenological expressions are studied for self-rippling energy that drives ripple formation of free single-layer graphene sheets. One phenomenological expression is admittedbased on the stability criterion of periodic ripple mode, while all others are rejected because they cannot admit stable periodic ripple mode. The admitted phenomenological expression contains two terms: one quadratic term which acts like a compressive force and has a destabilizing effect, and another fourth-order term which acts like a nonlinear elastic foundation and has a stabilizing effect. The two associated coefficients depend on specific mechanism of self-rippling and can be determined based on observed wavelength and amplitude of ripple mode. Based on the admitted expression, the effect of an applied force on ripple formation is studied.The present model predicts that the rippling can be controlled or even suppressed with an applied tensile force, or collapsed into narrow wrinkles (of deformed wavelengths down to around 2 nm) under an applied compressive force, and the estimated minimum tensile strain to suppress rippling is in remarkable agreement with some known data. Our results show that self-rippling energy dominates ripple formation of sufficiently long free graphene ribbons, although it cannot drive self-rippling of sufficiently short free graphene ribbons. Consequently, a critical length is estimated so that self-rippling occurs only when the length of free single-layer graphene ribbons is much longer than the critical length. The estimated critical length is reasonably consistent with the known fact that self-rippling cannot occur in shorter free graphene sheets (say, of length below 20 nm). Finally, the effect of a rigid substrate on ripple formation is studied, and the results show that rippling can be suppressed and the graphene monolayer can be ultra-flat in the appearance of a rigid substrate due to the van der Waals interaction between the substrate and graphene. For varying distance between graphene and substrate, there exists a critical distance below which rippling in graphene monolayer cannot exist due to the strong van der Waals interaction. However, when the distance between substrate and graphene is at least few times larger than the critical distance. the effect of substrate can be ignored and the graphene monolayer can be treated approximately as a free-standing graphene monolayer.