Understanding why graphene and all 2D materials are stable remains in some ways unsolved. In our recently paper published in Nature Physics we bring together novel ideas into this long-standing puzzle by combining symmetry arguments with anharmonicity.
Many of the applications of graphene rely on its uneven stiffness and high thermal conductivity, and yet its mechanical properties are far from understood. The problem is that standard theories for vibrational properties, such as the harmonic approximation, predict that the flexural vibrational mode has a quadratic dispersion, which yields the unphysical result that in-plane acoustic waves cannot propagate sound. The quadratic dispersion has thus been questioned, and it has been argued that anharmonic phonon–phonon interactions linearize it.
Even though this correction can explain why sound propagates, it also implies that the bending rigidity of graphene diverges and becomes dependent on the sample size — a rather anomalous result. And these problems are not limited to graphene: they are present in all strictly two-dimensional systems, and raise big question marks over our understanding of their mechanical properties.
It is symmetry that imposes a quadratic dispersion for the flexural modes in the harmonic approximation: the mode energies are calculated from second derivatives of a potential that is invariant upon rotation, leading to quadratic behaviour in strictly two-dimensional materials. As we have now demonstrated, anharmonic phonons also come from second derivatives of the full free energy of the system, which incorporates phonon–phonon interactions and is also rotationally invariant — implying a quadratic dispersion for the flexural modes, too. We verified this hypothesis by calculating the phonon frequencies of graphene from the second derivatives of the anharmonic free energy both within atomistic simulations and within the membrane models that are widely used in the community working on the physics of flexural modes.
Our results suggest that the anharmonic phonons obtained from the derivatives of the full free energy are indeed quadratic, as imposed by symmetry, despite the presence of phonon–phonon interactions. This proves true in both the atomistic simulations and the membrane model. Remarkably, even if in the harmonic case a quadratic dispersion implies non-propagating sound, within our anharmonic theory this is no longer true because in-plane phonons can propagate sound even if the flexural modes have a quadratic dispersion. One of the main implications of these results is that the bending rigidity is non-divergent at any temperature, overturning the conventional wisdom for two-dimensional membranes that assumed a linearization of the flexural modes and a consequent sample-size-dependent bending stiffness. Another consequence of our results is that the amplitude of the ripples in graphene are not suppressed by anharmonicity as previously believed, but increase with the sample size.
Our results offer a consistent, fresh perspective on the mechanical and thermal properties of graphene and indeed all strictly two-dimensional materials.
This work represents a huge effort from many people specially Unai Aseginolaza and Josu Diego, who have worked brilliantly in this problem.
Reference: Nature Physics (2024)