Irreversibility in Response to Forces Acting on Graphene Sheets

The amount of rippling in graphene sheets is related to the interactions with the substrate or with the suspending structure. Here, we report on an irreversibility in the response to forces that act on suspended graphene sheets. This may explain why one always observes a ripple structure on suspended graphene. We show that a compression-relaxation mechanism produces static ripples on graphene sheets and determine a peculiar temperature $T_c$, such that for $T<T_c$ the free-energy of the rippled graphene is smaller than that of roughened graphene. We also show that $T_c$ depends on the structural parameters and increases with increasing sample size.

Figure 1

Log-log plot of $⟨|h(\mathbf{q}){|}^2⟩$ as a function of $\mathbf{q}/q_0$ (log scaled) both in the rough and ripple cases. In the inset, $⟨|h(\mathbf{q}){|}^2⟩$ is shown as a function of $|\mathbf{q}|/q_0$ to clarify a peak around $|\mathbf{q}|\simeq 10q_0$. A graphene sheet incorporates 80 000 atoms at 50 K.

Figure 2

The dependence of the free-energy and internal-energy (entropy) differences as a function of temperature.

Figure 3

Probability distribution function of the total energy $E$ for the rippled and rough states for $T=30$, 50, 70, 85, 100, and 300 K (from top to bottom). The dashed curves are the Gaussian probability distribution function. For clarity, the PDFs were shifted upward.

Figure 4

Transition from rippled state (upper snapshot) to the rough state, due to the increasing of the temperature from 55 to 520 K.