[Paper Review] Shear-strain-induced Spatially Varying Super-lattice Structures on Graphite studied by STM

S. Choudhary, Anjan K. Gupta|arXiv (Cornell University)|Apr 27, 2007

Graphene research and applications21 references4 citations

TL;DR

This study uses STM to observe shear-strain-induced spatially varying moiré super-lattices on HOPG, transitioning from a 6 nm hexagonal lattice to a 13 nm square lattice and then to 1D fringes. The authors attribute this to in-plane shear strain causing a spatially varying rotation of the top graphite layer, and propose a k-space Fourier analysis model that successfully simulates both the 2D super-lattice and 1D fringes by relating them to the spatial dependence of the rotation angle.

ABSTRACT

We report on the Scanning Tunneling Microscope (STM) observation of linear fringes together with spatially varying super-lattice structures on (0001) graphite (HOPG) surface. The structure, present in a region of a layer bounded by two straight carbon fibers, varies from a hexagonal lattice of 6nm periodicity to nearly a square lattice of 13nm periodicity. It then changes into a one-dimensional (1-D) fringe-like pattern before relaxing into a pattern-free region. We attribute this surface structure to a shear strain giving rise to a spatially varying rotation of the affected graphite layer relative to the bulk substrate. We propose a simple method to understand these moire patterns by looking at the fixed and rotated lattices in the Fourier transformed k-space. Using this approach we can reproduce the spatially varying 2-D lattice as well as the 1-D fringes by simulation. The 1-D fringes are found to result from a particular spatial dependence of the rotation angle.

Motivation & Objective

• To investigate the origin of spatially varying super-lattice structures and 1D fringes observed on the HOPG (0001) surface using STM.
• To determine whether these patterns arise from shear strain-induced lattice rotation in the top graphite layer.
• To develop a simple analytical model based on Fourier space analysis to understand and simulate the observed moiré patterns.
• To explain the emergence of 1D fringes as a consequence of a specific spatial dependence of the rotation angle.
• To connect the observed surface structures with strain-induced distortions in the graphite layer's stacking and periodicity.

Proposed method

• Conducted high-resolution STM imaging in constant current mode under ambient conditions on freshly cleaved HOPG samples.
• Used a home-built STM with PtIr tip and analyzed both filtered and unfiltered data for quantitative analysis.
• Proposed a k-space Fourier transformation model to analyze the interference between a fixed lattice and a rotated lattice.
• Modeled the moiré pattern as a function of the rotation angle θ, with the super-lattice periodicity given by D = d / (2 sin(θ/2)).
• Derived conditions under which 1D fringes emerge by analyzing the phase functions f₁, f₂, f₃ and their spatial dependence.
• Simulated the observed patterns by assuming a spatially varying rotation angle θ(x,y) = C / (kx), which reproduces both the 2D super-lattice and 1D fringes.

Experimental results

Research questions

• RQ1 What causes the spatially varying super-lattice structure observed on the HOPG surface, transitioning from 6 nm to 13 nm periodicity?
• RQ2 How do 1D fringe patterns emerge in connection with a 2D moiré super-lattice on graphite?
• RQ3 Can a simple analytical model based on Fourier space analysis explain the evolution of moiré patterns under shear strain?
• RQ4 What spatial dependence of the rotation angle θ(x,y) leads to the formation of 1D fringes in the observed STM images?
• RQ5 Why do the 1D fringes connect smoothly to the bright spot rows from cos(f₂) and cos(f₃), but not from cos(f₁)?

Key findings

• The observed spatially varying super-lattice structure on HOPG results from in-plane shear strain inducing a spatially varying rotation of the top graphite layer relative to the bulk.
• The transition from a 6 nm hexagonal lattice to a 13 nm square lattice is explained by a rotation angle increasing from ~1.9° to ~3.5°, consistent with moiré theory.
• The 1D fringe pattern arises from a specific spatial dependence of the rotation angle, θ(x,y) ∝ 1/x, which causes the interference fringes to align linearly.
• The simulation using the k-space Fourier model successfully reproduces both the 2D super-lattice and the 1D fringes by varying only the direction of the wavevector in reciprocal space.
• The 1D fringes are found to originate from the interference of cos(f₂) and cos(f₃), while cos(f₁) remains constant in the fringe region, explaining their absence in the observed pattern.
• The model predicts that any function of the form θ(x,y) ∝ 1/(x + γy) can produce 1D fringes, with the observed case being a special limit of this general solution.