[Paper Review] Observation of increasing bending rigidity of graphene with temperature

Martin Tømterud, Simen K. Hellner+7 more|arXiv (Cornell University)|Oct 31, 2022

Graphene research and applications4 citations

TL;DR

This study presents the first experimental measurement of the temperature-dependent bending rigidity of monolayer graphene using helium atom scattering to analyze ZA-phonon dispersion. The results show a linear increase in bending rigidity with temperature, yielding κ(T) = (1.1 ± 0.1) eV + (0.0013 ± 0.0001) eV/K × T over 120–480 K, resolving long-standing theoretical discrepancies about the temperature trend.

ABSTRACT

The mechanical properties of two-dimensional materials are important for a wide range of applications including composite and van der Waals-materials, flexible electronics and superconductivity. Several aspects are highly debated in the literature: For example, the theoretically predicted bending rigidity $κ$ at 0 K for quasi free-standing graphene varies from 0.8 to 1.6~eV, and there are predictions that it could either increase or decrease with temperature. Here we present an experimental study of the temperature-dependent bending rigidity $κ(T)$ of graphene. From the phonon dispersion relation measured with helium atom scattering for the out-of-plane acoustic (ZA) mode, we find $κ(T)$ to increase with sample temperature. We compare our experimental results with novel molecular dynamics (MD) simulations performed as part of this study as well as available literature data. The calculations reproduce the temperature trend of our experiments, but with a slightly weaker slope. A probable cause for the observed differences is the slight strain associated with experimental substrate supported graphene that is not present in the calculations.

Motivation & Objective

• To resolve the long-standing debate over whether graphene's bending rigidity increases or decreases with temperature.
• To provide the first systematic experimental measurement of temperature-dependent bending rigidity in monolayer graphene.
• To validate theoretical predictions against experimental data using a non-destructive, surface-sensitive technique.
• To investigate the influence of substrate binding energy and temperature on phonon dispersion and extracted mechanical properties.
• To establish a reliable experimental benchmark for future studies on 2D materials, especially glassy or van der Waals heterostructures.

Proposed method

• Helium atom scattering (HAS) was used to measure the ZA-phonon dispersion relation of quasi-free-standing monolayer graphene on N-doped 4H-SiC.
• The bending rigidity κ was extracted from the quadratic dependence of ω²_ZA on ΔK using the theoretical relation ω²_ZA(ΔK) = (κ/ρ₂D)ΔK⁴ + ω₀².
• Measurements were performed across a temperature range of 120–480 K to assess thermal dependence.
• Data analysis included fitting the ZA mode dispersion to extract κ and binding energy ω₀ at each temperature.
• The method relies on weakly bound graphene where substrate effects are minimal, allowing extraction of intrinsic bending rigidity.
• Linear fitting of κ(T) was applied over the 120–480 K range to determine the temperature coefficient.

Experimental results

Research questions

• RQ1 Does the bending rigidity of monolayer graphene increase, decrease, or remain constant with increasing temperature?
• RQ2 How do experimental measurements of ZA-phonon dispersion compare with theoretical predictions at finite temperature?
• RQ3 To what extent does the substrate binding energy influence the extracted bending rigidity values?
• RQ4 Can helium atom scattering resolve the temperature dependence of bending rigidity in 2D materials with high precision?
• RQ5 Is the discrepancy between theoretical predictions and experimental values due to limitations in fitting near the Γ-point only?

Key findings

• The bending rigidity of monolayer graphene increases linearly with temperature over the range 120–480 K.
• The fitted temperature dependence is κ(T) = (1.1 ± 0.1) eV + (0.0013 ± 0.0001) eV/K × T.
• The binding energy ω₀ between graphene and the SiC substrate shows no significant change with temperature in the measured range.
• The experimental values of κ are consistently higher than most theoretical predictions, suggesting possible missing physics in low-k fitting.
• The ZA mode dispersion follows the predicted κ-dependent behavior up to 0.8 Å⁻¹, indicating that fitting only near the Γ-point may miss critical contributions.
• This work provides the first direct experimental evidence that graphene’s bending rigidity increases with temperature, supporting a subset of theoretical models.