[Paper Review] Anomalous Hooke's law in disordered graphene

I. V. Gornyi, V. Yu. Kachorovskii+1 more|arXiv (Cornell University)|Mar 1, 2016

Graphene research and applications2 references3 citations

TL;DR

This paper demonstrates that Hooke's law breaks down anomalously in both clean and disordered graphene due to strong thermal fluctuations and anharmonic coupling, leading to a nonlinear strain-tension relationship ξ ∝ σ^(η/(2−η)) in clean graphene and ξ ∝ σ^(η/(8−η)) in strongly disordered graphene, with linear elasticity only emerging at high tensions σ ≫ σ*. The anomalous scaling arises from renormalized bending rigidity κq ∝ q^−η and disorder-induced static ripples.

ABSTRACT

The discovery of graphene, a single monolayer of graphite, has closed the discussion on stability of 2D crystals. Although thermal fluctuations of such crystals tend to destroy the long-range order in the system, the crystal can be stabilized by strong anharmonicity effects. This competition is the central issue of the crumpling transition, i.e., a transition between flat and crumpled phases. We show that anharmonicity-controlled fluctuations of a graphene membrane around equilibrium flat phase lead to unusual elastic properties. In particular, we demonstrate that stretching $ξ$ of a flake of graphene is a nonlinear function of the applied tension at small tension: ${ξ\proptoσ^{η/(2-η)}}$ and ${ξ\proptoσ^{η/(8-η)}}$ for clean and strongly disordered graphene, respectively. Conventional linear Hooke's law, ${ξ\proptoσ}$ is realized at sufficiently large tensions: ${σ\ggσ_*},$ where $σ_*$ depends both on temperature and on the disorder strength.

Motivation & Objective

• To investigate the breakdown of linear Hooke's law in free-standing graphene under small applied tension.
• To understand how thermal fluctuations and anharmonic coupling between in-plane and out-of-plane phonons affect elasticity.
• To examine the role of quenched disorder in modifying elastic response and stabilizing anomalous scaling.
• To derive the scaling exponent α in the nonlinear strain-tension relation ξ ∝ σ^α for both clean and disordered graphene.
• To reconcile theoretical predictions with recent experimental and numerical findings on graphene's nonlinear elasticity.

Proposed method

• Formulated a 2D membrane model with energy functional including bending rigidity κ, in-plane elasticity, and curvature coupling.
• Used functional renormalization group (RG) techniques to derive the flow of κ and disorder strength b in the presence of thermal fluctuations.
• Calculated dynamic and static correlation functions H_q^d and H_q^s for out-of-plane displacements, accounting for tension σ.
• Derived the leading-order corrections to H_q^d and H_q^s via differentiation with respect to σ, revealing anomalous scaling.
• Determined the effective stiffness k_eff(σ) from the strain response, showing power-law behavior in σ with non-trivial exponents.
• Compared theoretical scaling with numerical simulations from Ref. katsnelson16, adjusting for finite-size effects to match the thermodynamic limit.

Experimental results

Research questions

• RQ1 How does Hooke's law fail in graphene at infinitesimally small tensions due to thermal fluctuations?
• RQ2 What is the scaling exponent α in the nonlinear strain-tension relation ξ ∝ σ^α for clean graphene?
• RQ3 How does quenched disorder alter the elastic response and modify the scaling exponent in disordered graphene?
• RQ4 What is the role of renormalized bending rigidity κq ∝ q^−η in determining the anomalous elasticity?
• RQ5 How do the theoretical predictions compare with numerical simulations and experimental data on graphene's nonlinear elasticity?

Key findings

• In clean graphene, strain ξ scales as σ^(η/(2−η)) at small tensions, with η ≈ 0.6–0.7, indicating strong deviation from linear Hooke's law.
• In strongly disordered graphene, the scaling exponent becomes η/(8−η), reflecting dominance of static disorder-induced ripples over dynamic fluctuations.
• Linear Hooke's law is recovered only at high tensions σ ≫ σ*, where σ* depends on temperature and disorder strength.
• The effective stiffness k_eff(σ) vanishes in the thermodynamic limit at zero strain, confirming the absence of finite-size residual stiffness.
• Theoretical scaling exponents agree quantitatively with numerical simulations after removing finite-size cutoffs.
• The anomalous elasticity is attributed to the interplay between fluctuating ripples and disorder, with static ripples dominating in disordered graphene.