Thermal ripples in bilayer graphene

TL;DR

This paper investigates thermal fluctuations in free-standing bilayer graphene, revealing how interlayer interactions and wave-vector dependent bending rigidity influence membrane behavior across different length scales and temperatures.

Contribution

It develops a nonlinear elastic model for bilayer graphene, deriving parameters from first-principles and analyzing fluctuation regimes using self-consistent screening approximation.

Findings

Identification of a crossover in bending rigidity behavior at different wavelengths.

Demonstration of a transition from harmonic to strong-coupling fluctuation regimes.

Analysis of temperature-dependent interplay between different fluctuation regimes.

Abstract

We study thermal fluctuations of a free-standing bilayer graphene subject to vanishing external tension. Within a phenomenological theory, the system is described as a stack of two continuum crystalline membranes, characterized by finite elastic moduli and a nonzero bending rigidity. A nonlinear rotationally-invariant model guided by elasticity theory is developed to describe interlayer interactions. After neglection of in-plane phonon nonlinearities and anharmonic interactions involving interlayer shear and compression modes, an effective theory for soft flexural fluctuations of the bilayer is constructed. The resulting model, neglecting anisotropic interactions, has the same form of a well-known effective theory for out-of-plane fluctuations in a single-layer membrane, but with a strongly wave-vector dependent bare bending rigidity. Focusing on AB-stacked bilayer graphene, parameters governing interlayer interactions in the theory are derived by first-principles calculations. Statistical-mechanical properties of interacting flexural fluctuations are then calculated by a numerical iterative solution of field-theory integral equations within the self-consistent screening approximation (SCSA). The bare bending rigidity in the considered model exhibits a crossover between a long-wavelength regime governed by in-plane elastic stress and a short wavelength region controlled by monolayer curvature stiffness. Interactions between flexural fluctuations drive a further crossover between a harmonic and a strong-coupling regime, characterized by anomalous scale invariance. The overlap and interplay between these two crossover behaviors is analyzed at varying temperatures.