[Paper Review] Stress intrepretation of graphene E-2g and A-1g vibrational modes: theoretical analysis

D. Sfyris, Georgios I. Sfyris+1 more|arXiv (Cornell University)|Jun 12, 2017

Graphene research and applications22 references12 citations

TL;DR

This paper develops a theoretical framework to interpret the stress tensor components linked to graphene's E₂g and A₁g vibrational modes using the Cauchy stress tensor derived from the IKN procedure. It reveals that both axial and shear stress components are inherently coupled to these modes, with shear stress (σ₁₂) becoming dominant at 45° tensile loading, challenging the assumption of axial dominance in strain analysis.

ABSTRACT

We here focus on only one graphene ring and examine to which stress tensor components the E2g and the A1g vibration mode of graphene correspond. These modes are typically related with the G-peak and the D-peak, respectively, and are strongly related to the stress distribution along the specimen. We adopt the theoretical framework of Admal and Tadmor ([1]) for the macroscopic definition of the Cauchy stress tensor and we introduce into this framework the E2g and the A1g as appropriate perturbations. We use these perturbations to the stress tensor expression and evaluate which stress tensor components are related to each vibrational mode. This approach, though qualitative in nature, incorporates all the main physics and reveals that E2g and A1g vibration modes should be related to shear as well as axial stress components when graphene is at rest (i.e. no external applied loading). To bring our framework closer to more concrete results, we evaluate the instantaneous Hardy stress tensor for a pair potential which correspond to the E2g and A1g modes at rest. Our analysis expands to take into account an applied external tensile field. Taking the armchair direction to be along the x-axis, when tension applies along the armchair direction, it is the axial $σ$11 stress component which dominates over $σ$12, $σ$22, which are of smaller order. When tension is along the zig-zag direction, it is the axial $σ$22 stress component that dominates over $σ$12, $σ$11. When tension is at an arbitrary direction between the armchair and the zig-zag direction, all stress components are of the same order and should all be taken into account even at small strains.

Motivation & Objective

• To clarify the connection between graphene's E₂g and A₁g vibrational modes and the components of the Cauchy stress tensor.
• To address the challenge of translating Raman spectroscopic data into macroscopic stress states in 2D materials.
• To investigate how different loading directions affect the relative contributions of axial and shear stress components.
• To extend the IKN-based stress tensor framework to include vibrational mode perturbations in graphene.
• To evaluate the role of the Hardy stress tensor in relating atomic-scale dynamics to continuum stress fields.

Proposed method

• Adopting the IKN procedure for defining the Cauchy stress tensor in many-body systems, the authors generalize it to include E₂g and A₁g vibrational modes as perturbations.
• The stress tensor is decomposed into dynamic and kinetic parts, with non-uniqueness arising from potential energy extensions, which vanish under spatial averaging.
• The framework incorporates the E₂g and A₁g modes as harmonic perturbations to the stress tensor expression at rest and under external tensile fields.
• The Hardy stress tensor is evaluated for a pair potential model to link atomic-scale dynamics to continuum stress components.
• The analysis considers tensile loading along armchair (x-axis), zig-zag (y-axis), and arbitrary angles θ, with θ = 45° as a critical case.
• The method evaluates the relative magnitudes of σ₁₁, σ₂₂, and σ₁₂ as functions of the applied tensile angle θ, using perturbation theory and trigonometric expansions.

Experimental results

Research questions

• RQ1 Which components of the Cauchy stress tensor are associated with the E₂g and A₁g vibrational modes in graphene at rest?
• RQ2 How does the relative magnitude of axial and shear stress components change under uniaxial tensile loading at different orientations?
• RQ3 To what extent can the Hardy stress tensor capture the stress state induced by specific vibrational modes in graphene?
• RQ4 What is the role of shear stress (σ₁₂) in Raman-detectable strain when loading is applied at 45° to the crystallographic axes?
• RQ5 How do the stress tensor components scale with applied strain and loading angle, particularly near θ = 45°?

Key findings

• At rest, both E₂g and A₁g vibrational modes induce not only axial but also shear stress components, indicating intrinsic coupling to shear deformation.
• When tensile loading is applied along the armchair direction (θ = 0°), the axial stress component σ₁₁ dominates over σ₁₂ and σ₂₂.
• When loading is applied along the zig-zag direction (θ = 90°), the axial stress component σ₂₂ dominates over σ₁₂ and σ₁₁.
• At θ = 45°, the shear stress component σ₁₂ reaches its maximum and becomes of the same order as the axial components σ₁₁ and σ₂₂, making it non-negligible even in small-strain regimes.
• For arbitrary tensile loading angles, all stress components (σ₁₁, σ₂₂, σ₁₂) must be considered simultaneously, especially near 45°, due to the equal-order scaling of axial and shear terms.
• The analysis suggests that assuming only axial stress components in Raman strain mapping may lead to inaccuracies, particularly under off-axis loading, and that shear contributions must be included for quantitative accuracy.