[Paper Review] Interlayer Configurations in Twisted Bilayers of Folded Graphene

Johannes Rode, Christopher Belke+2 more|arXiv (Cornell University)|Aug 29, 2016

Graphene research and applications3 citations

TL;DR

This study uses atomic force microscopy (AFM) to investigate interlayer configurations in twisted bilayer graphene (TBG) formed by folding monolayer graphene, revealing a strong twist-angle-dependent modulation of interlayer distance up to ~3 Å—significantly exceeding theoretical predictions. Enhanced friction due to superlubricity in incommensurate layers and a curvature-radius scaling with folded edge length, consistent with carbon nanotube models, are key findings.

ABSTRACT

The folding of monolayer graphene leads to new layered systems, termed twisted bilayer graphene (TBG), generally displaying a certain interlayer rotation away from crystallographic alignment. We here present an atomic force microscopy study on folded graphene, revealing unexpectedly large twist angle dependent modulations of ~3 angstrom in interlayer distance. At the TBG surface, we find enhanced friction attributable to superlubricity in between incommensurate layers. At the bended edge, the radius of curvature scales with the folded length, congruent to earlier studies on carbon nanotubes.

Motivation & Objective

• To systematically measure interlayer distance in twisted bilayer graphene (TBG) formed by folding monolayer graphene, addressing the lack of experimental data beyond AB-stacking.
• To investigate the dependence of interlayer distance and friction on twist angle in TBG, particularly in incommensurate configurations.
• To examine the mechanical properties of folded edges, focusing on curvature radius and its scaling with folded edge length.
• To establish a geometric method for determining twist angle θ via angle φ between folded edge and crystallographic direction, leveraging graphene’s sixfold symmetry.
• To compare experimental results with theoretical models of interlayer coupling and carbon nanotube behavior, validating predictions on curvature and stability.

Proposed method

• AFM topography and lateral force microscopy (LFM) were used to measure interlayer distance Δh and friction in folded TBG, with contact mode to minimize mechanical crosstalk.
• Interlayer distance Δh was extracted from height histograms of AFM data, fitted with Gaussian distributions for monolayer (MLG), substrate, and TBG regions.
• Twist angle θ was determined geometrically as twice the angle φ between the folded edge and a crystallographic edge, leveraging graphene’s sixfold symmetry.
• Curvature radius r at the folded edge was measured from height profiles across folded regions and correlated with the length ℓ of the folded edge.
• An exponential fit was applied to r vs. ℓ data to model saturation behavior, compared to theoretical predictions for carbon nanotubes.
• Friction was normalized to the AB-stacked bilayer step ΔV_AB and reported as ΔV/ΔV_AB to compare with single-crystal fewlayer graphene.

Experimental results

Research questions

• RQ1 How does the interlayer distance Δh in twisted bilayer graphene (TBG) vary with twist angle θ, and does it deviate significantly from the theoretical AB-stacking distance of 3.35 Å?
• RQ2 What is the origin of enhanced friction in TBG compared to single-crystal fewlayer graphene, and how is it related to interlayer superlubricity?
• RQ3 How does the radius of curvature r at the folded edge of TBG scale with the length ℓ of the folded edge, and does this behavior align with predictions for carbon nanotubes?
• RQ4 To what extent do experimental measurements of Δh and r in folded TBG support or contradict existing theoretical models of interlayer coupling and curvature stability?
• RQ5 Can the geometric method of determining θ via φ (angle between folded edge and crystallographic direction) be reliably applied to folded graphene flakes with high accuracy?

Key findings

• Interlayer distance Δh in TBG exhibits a pronounced dependence on twist angle θ, varying over a range of ~3 Å, with maximal values significantly exceeding the theoretical AB-stacking distance of 3.35 Å.
• The lower limit of Δh (~3.4 Å) corresponds to AB-stacked bilayer graphene, confirming it as the energetic minimum, while larger distances are observed for intermediate twist angles.
• Enhanced friction in TBG, normalized to AB-stacking, is attributed to superlubricity and increased pliancy in incommensurate layers, contradicting the expected decreasing trend with layer count.
• The radius of curvature r at the folded edge increases with the length ℓ of the folded edge, following an exponential trend that saturates near 6–7 Å, in excellent agreement with theoretical predictions for carbon nanotubes.
• The angular distribution of twist angles shows a strong preference for θ ≈ 0° and θ ≈ 30°, with few intermediate values, mirroring trends observed in CVD-grown TBG.
• The geometric method for determining θ via φ yields an accuracy of ±1.5°, enabling reliable twist angle assignment across a range of folded graphene flakes.