Model of ripples in graphene

We propose a model of ripples in suspended graphene sheets based on plate equations that are made discrete with the periodicity of the honeycomb lattice and then periodized. In addition, the equation for the displacements with respect to the planar configuration contains a double-well site potential, a nonlinear friction, and a multiplicative white-noise term satisfying the fluctuation-dissipation theorem. The nonlinear friction terms agree with those proposed by Eichler et al. [Nature Nanotech. 6, 339 (2011)] to explain their experiments with a graphene resonator. The site double-well potential indicates that the carbon atoms at each lattice point have equal probability to move upward or downward off plane. For the considered parameter values, the relaxation time due to friction is much larger than the periods of membrane vibrations and the noise is quite small. Then ripples with no preferred orientation appear as long-lived metastable states for any temperature. Numerical solutions confirm this picture.

Figure 1

Neighbors of a given atom $A$ in sublattice 1 (dark blue). The primitive vectors are $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ that connect atom $B$ to three different next-nearest neighbors in its same sublattice.

Figure 2

Ripples formed in a suspended $80\times 80$ honeycomb graphene sheet (with 12 800 atoms). (a) Density plot of the off-plane displacement as a function of in-plane coordinates at $t=1$. (b) The same at $t=50$. (c) The same at $t=100$. (d) Average density plot over all times from $t=1$ to 100 with unit time step. Parameter values are as in the second row of Table .

Figure 3

Ripples in the central part of a $160\times 160$ graphene lattice (with 51 200 atoms).

Figure 4

Same as in Fig. 1 for a $200\times 200$ honeycomb lattice with $w_0=1$ nm (180 000 atoms). Times are (a) $t=10$, (b) $t=250$, and (c) $t=510$. (d) Average density plot from $t=410$ to 510, using a $t=10$ time step. Parameters are as in Table except for the rescaled $\gamma =0.11838$, $\eta =1.64621$, $\varphi =1.25598\times {10}^{-4}$.