Quantum elasticity of graphene: Thermal expansion coefficient and specific heat

Abstract

We explore thermodynamics of a quantum membrane, with a particular application to suspended graphene membrane and with a particular focus on the thermal expansion coefficient. We show that an interplay between quantum and classical anharmonicity-controlled fluctuations leads to unusual elastic properties of the membrane. The effect of quantum fluctuations is governed by the dimensionless coupling constant, $g_0\ll 1$, which vanishes in the classical limit ($\hslash \to 0$) and is equal to $\simeq 0.05$ for graphene. We demonstrate that the thermal expansion coefficient ${\alpha }_T$ of the membrane is negative and remains nearly constant down to extremely low temperatures, $T_0\propto \exp (-2/g_0)$. We also find that ${\alpha }_T$ diverges in the classical limit: ${\alpha }_T\propto -\ln (1/g_0)$ for $g_0\to 0$. For graphene parameters, we estimate the value of the thermal expansion coefficient as ${\alpha }_T\simeq -0.23{\mathrm{eV}}^{-1}$, which applies below the temperature $T_{\mathrm{uv}}\sim g_0{\varkappa }_0\sim 500$\:K (where ${\varkappa }_0\sim 1$\:eV is the bending rigidity) down to $T_0\sim 10^{-14}$\:K. For $T<T_0$, the thermal expansion coefficient slowly (logarithmically) approaches zero with decreasing temperature. This behavior is surprising since typically the thermal expansion coefficient goes to zero as a power-law function. We discuss possible experimental consequences of this anomaly. We also evaluate classical and quantum contributions to the specific heat of the membrane and investigate the behavior of the Grüneisen parameter.

20 pages, 5 figures

References in corpus (17)

• Electric Field Effect in Atomically Thin Carbon Films
• The electronic properties of graphene
• 2D materials and van der Waals heterostructures
• Ripple Texturing of Suspended Graphene Atomic Membranes
• Negative Thermal Expansion Coefficient of Graphene Measured by Raman Spectroscopy
• Finite temperature lattice properties of graphene beyond the quasiharmonic approximation
• Moir{é} patterns as a probe of interplanar interactions: graphene on h-BN
• Crumpling transition and flat phase of polymerized phantom membranes
• Graphene as a Prototype Crystalline Membrane
• Thermal fluctuations of free standing graphene
• Theory of anharmonic phonons in 2D crystals
• Lattice Expansion in Seamless Bi layer Graphene Constrictions at High Bias
• In Search of Microscopic Evidence for Molecular Level Negative Thermal Expansion in Fullerenes
• The flat phase of quantum polymerized membranes
• Non-linear fluctuation effects in dynamics of freely suspended film
• Comment on "Thermodynamics of quantum crystalline membranes"
• Reply to 'Comment on "Thermodynamics of quantum crystalline membranes"'

Cited by in corpus (7)

• Thermal properties of graphene from path-integral simulations
• Scale without conformal invariance in membrane theory
• Phonon-induced exciton weak localization in two-dimensional semiconductors
• Flexural deformations and collapse of bilayer two-dimensional crystals by interlayer exciton
• Emergent continuous symmetry in anisotropic flexible two-dimensional materials
• Disorder-driven transition to tubular phase in anisotropic two-dimensional materials
• Perturbative renormalization and thermodynamics of quantum crystalline membranes