Simulation of ripples in single layer graphene sheets and study of their vibrational and elastic properties

We present the results of our theoretical investigation on ripples and elastic properties of single layer graphene sheets in both membrane and ribbon conformations. The formation of ripples in both the systems is simulated and analyzed using two-dimensional vibrating membrane model. We have chosen both square graphene membrane, armchair and zigzag graphene nanoribbons with different sizes. The amplitude of vibrational modes of each system is determined using this model. We observed that the vertical displacement (amplitude of the ripples) reaches a maximum height of about 0.99 nm from the mean plane in both conformations whose lengths are integral multiple of the basic armchair/zigzag units. We have studied the dynamical elastic properties through the calculation of parameters like normalized stiffness, speed parameter, Cauchy number and critical velocity with reference to a new aspect ratio of graphene sheets. We have made correlations between the calculated parameters with the formation of ripples and found that the out-of-plane deformations are spontaneous and significant in square conformation of graphene than the graphene nanoribbons. The vibrational modes obtained for GNRs and membranes are acoustic modes. The results of our study will be very much useful in selecting graphene sheets with suitable conformation and chirality for designing nanoscale devices.

► Present model is consistent with real situation in which the graphene layer is freely suspended.
► Maximum ripple height is ∼1 nm and it is in agreement with the experimental observation.
► The deformations are spontaneous and significant in graphene nanoribbons with high aspect ratio.
► All vibrational modes obtained using the present theoretical model are acoustic modes.
► Dynamical ripples do not affect the propagation of Dirac fermions and crystallinity of graphene.