Suspended graphene exhibits ripples of size ranging from 50 to 100 Å and height $\sim 10$ Å, however, their origin remains undetermined. Previous theoretical works have proposed that rippling in graphene might be generated by the coupling between the bending modes and the density of electrons. These theoretical studies proposed that, in the thermodynamic limit, a membrane of single layer graphene becomes unstable for large enough electron-phonon coupling, which signals a phase transition from a flat phase to a rippled one. Here, we find the stable configuration of a suspended monolayer of graphene at $T=0$ by minimizing the average energy of a membrane where the Dirac electrons of graphene couple to elastic classical deformation fields. We find that the electron-phonon coupling controls a transition from a stable flat configuration to a stable rippled phase. We propose a scaling procedure that allows us to effectively reach larger system sizes. We find that the critical value of the coupling $g_c$ rapidly decays as the system increases its size, in agreement with the experimental observation of an unavoidable stable rippled state for suspended graphene membranes. This decay turns out to be controlled by a power law with a critical exponent $\sim 1/2$. Consistent arguments based on bifurcation theory indicate that the phase transition is discontinuous at large scaling parameter $k$, that the jump in the order parameter decreases as $k^{-1/2}$, and that the phase transition becomes continuous at $k=\infty$, with the order parameter scaling as ${(g-g_{c,\infty })}^{1/4}$.

• Received 3 December 2019
• Revised 1 June 2020
• Accepted 2 June 2020

DOI:https://doi.org/10.1103/PhysRevB.101.235428