A phenomenological model for self-rippling energy of free graphene monolayers

摘要

Several candidate phenomenological expressions are studied for self-rippling energy that drives ripple formation of free single-layer graphene sheets. One phenomenological expression is admitted, 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).