Rippling transition from electron-induced condensation of curvature field in graphene

A quantum field theory approach is applied to investigate the dynamics of flexural phonons in a metallic membrane like graphene, looking for the effects deriving from the strong interaction between the electronic excitations and elastic deformations. Relying on a self-consistent screening approximation to the phonon self-energy, we show that the theory has a critical point characterized by the vanishing of the effective bending rigidity of the membrane at low momentum. We also check that the instability in the sector of flexural phonons takes place without the development of an in-plane static distortion, which is avoided due to the significant reduction of the electron-phonon couplings for in-plane phonons at large momenta. Furthermore, we analyze the scaling properties of the many-body theory to identify the order parameter that opens up at the point of the transition. We find that the vanishing of the effective bending rigidity and the onset of a nonzero expectation value of the mean curvature are concurrent manifestations of the critical behavior. The results presented here imply that, even in the absence of tension, the theory has a critical point at which the flat geometry becomes an unstable configuration of the metallic membrane, with a condensation of the mean curvature field that may well reproduce the smooth distribution of ripples observed in free-standing graphene.

Figure 1

Diagrammatic equations defining the self-consistent screening approximation. The thick (thin) curly line represents the dressed (free) flexural phonon propagator and the thick (thin) dashed line represents the dressed (undressed) flexural phonon interaction.

Figure 2

(a) Plot of the effective bending rigidity for $2\pi T=5$ meV and values of the deformation potential $g=0,28.0,28.6,28.68,28.72$ eV (curves from top to bottom). (b) Plot of the real and imaginary parts of the effective bending rigidity for $2\pi T=10$ meV and values of the deformation potential $g=28.7,28.74,28.8$ eV (curves from top to bottom in both sides of the figure). $\Lambda$ stands for the cutoff in the momentum integrals, which is of the order of the inverse of the lattice spacing.

Figure 3

Plot of the real part of the effective bending rigidity for (a) $2\pi T=10$ meV and (b) $2\pi T=20$ meV. The curves correspond in both figures to a frequency $\omega =0.25$ eV and values of the deformation potential $g=28.0,28.7,28.84,28.87$ eV (from top to bottom).

Figure 4

Diagramatic equation defining the ladder approximation to the vertex with insertion of the operator ${\nabla }^{2}h{\nabla }^{2}h$ (zig-zag line). The curly line represents the flexural phonon propagator and the dashed line corresponds to the flexural phonon interaction.

Figure 5

Plot of the absolute value of the coefficients $c_1^{\left(n\right)}$ in the expansion of $c_1\left(\tilde{G}\right)$ as a power series of the effective coupling $\tilde{G}$.

Figure 6

Plot of the electron-phonon couplings obtained from the on-site deformation potential in the case of (a) in-plane acoustic longitudinal phonons and (b) out-of-plane acoustic phonons (for phonon momenta $\mathbf{q}={\mathbf{q}}^{\prime }$). The dashed lines represent the respective linear and quadratic behaviors extrapolated from the long-wavelength limit.