Probing Intrinsic Elastic Properties of Multilayer Graphene -- a New Mechanical Constant

TL;DR

This work addresses the challenge of extracting intrinsic in-plane elasticity and phonon sensitivity (Grüneisen parameter) in multilayer graphene by combining in situ bulge tests with a new Bulge Test of Multilayer vdW Materials (BTM) model that accounts for interlayer sliding. It reveals that measured properties vary with layer number due to interlayer shear, while first-principles benchmarks show thickness-independent intrinsic values; a corrective framework yields an intrinsic elastic Grüneisen parameter and introduces the elastic Grüneisen modulus $E_\gamma = E\gamma^{-3/2}$, which remains robust against interlayer effects. The results demonstrate a 2D-to-3D crossover around $N_{\rm lay} \approx 10$, propose corrections to extract intrinsic properties, and show that $E_\gamma$ aligns with DFT across thicknesses. The methodology and the intrinsic modulus concept are argued to generalize to other vdW layered materials, offering a practical route to characterize intrinsic mechanics in experiments with unavoidable interlayer sliding.

Abstract

We present measurements on in-plane Young's modulus and the Grüneisen parameter of multilayer graphene with varying number of layers, obtained through {\it in situ} bulge tests. Accurate determination of their elastic parameters poses a significant experimental challenge due to the substantial differences in mechanical behavior between intra- and inter-layers. To address this, we develop a novel theoretical model with first-principles calculations to investigate thickness-dependent incomplete strain transfer between the layers. Our findings show that the experimentally measured elastic constants, which deviate from computed intrinsic values, fail to fully capture ideal mechanical couplings between layers. As a solution, we propose a new mechanical modulus that integrates the Grüneisen parameter and in-plane Young's modulus, providing a more reliable representation of their mechanical properties, independent of unavoidable interlayer effects.

Figures (4)

• Figure 1: (a) Graphene with various numbers of layers ($N_\text{lay}$) on perforated SiO2/Si substrate with different numbers of samples ($N_S$). (b) Optical transmittance as a function of $N_\text{lay}$. For $N_\text{layer}\le 7$, Raman spectroscopy is used to measure $N_\text{lay}$ and the transmittance for $N_\text{lay}> 7$. (c) In situ AFM-bulge test data and schematics (inset) and (d) Raman spectra around G peak (red dots) with varying pressures (P).
• Figure 2: (a) Schematic diagram for bulge test with perfect clamping condition or without any sliding. $R$ is the radius of bulge hole and $h$ is the bulge height. (b) Similar diagram with interlayer sliding. $h_0$ is the same as $h$ in (a) while $h_1$ is the additional height pulled in from the supported region. (c) Cross-sectional view of the distribution of interlayer stresses in the supported region under an applied pressure of $P$ in case of $N_\text{lay}=5$. The pairwise arrows pointing opposite directions shown in-between neighboring layers represent interlayer shear stress resisting external pressure. For each layer, the force arising from the difference in shear stresses above and below act in positive radial direction and is balanced by the gradient of intralayer stress upto $r=r_{\rm max}$.
• Figure 3: (a) Values of $E$ from experiments, DFT calculations, and the BTM model as a function of $N_\text{lay}$. Dashed lines are the theoretical boundaries for $E_\text{ap}$ of a single layer on SiO$_2$ and on graphene. Red solid circles represent the average of measured values for each $N_{\rm lay}$. (b) $h$ as a function of $P$ for a trilayer graphene. Inset represents $\sigma$ vs. $\epsilon$ curve to obtain Young's modulus, $E$. (c) $k^\text{eff}$ as a function of $P$ for a single-layer graphene with $\tau = \tau_{\rm gg}$ and $\tau_{\rm gs}$. The hatched area is our experimental pressure range. (d) Nonlinear behavior of $h^3$ with respect to $P$ depending on $\tau$ for a single-layer graphene. Dashed line is for a perfect clamping case.
• Figure 4: (a) Computed $\ln\omega_G$ as a function of $\epsilon$ from DFT calculation compared with measured Raman $G$ peak for graphene for $N_\text{lay}=3$. There are six modes from DFT for each $\epsilon$. Green lines show linear fitting curves. (b) Theoretical $\gamma$ from DFT calculations for varying $N_\text{lay}$. (c) Measured Grüneisen parameter of $\gamma$ for varying $N_\text{lay}$ (blue solid rectangles). Green solid circles represent $\tilde{\gamma}$ corrected from measured $\gamma$. (d) Measured elastic Grüneisen modulus ($E_\gamma$) with varying $N_\text{lay}$ (blue crosses) and the constant red dashed line from DFT.