Measuring strain and rotation fields at the dislocation core in graphene

TL;DR

This study measures strain and rotation fields around dislocations in graphene using high-resolution microscopy, compares them with advanced theoretical models, and provides analytical formulas to better understand and engineer strain effects in 2D materials.

Contribution

The paper introduces high-resolution measurements of dislocation fields in graphene and compares experimental results with advanced discrete and analytical models, highlighting asymmetries and nonlinear effects.

Findings

Experimental fields differ from linear elasticity predictions.

Discrete theories closely match experimental data when considering out-of-plane displacements.

Analytical formulas from hyperstress theory effectively fit experimental data.

Abstract

Strain fields, dislocations and defects may be used to control electronic properties of graphene. By using advanced imaging techniques with high-resolution transmission electron microscopes, we have measured the strain and rotation fields about dislocations in monolayer graphene with single-atom sensitivity. These fields differ qualitatively from those given by conventional linear elasticity. However, atom positions calculated from two dimensional (2D) discrete elasticity and three dimensional discrete periodized F\"oppl-von K\'arm\'an equations (dpFvKEs) yield fields close to experiments when determined by geometric phase analysis. 2D theories produce symmetric fields whereas those from experiments exhibit asymmetries. Numerical solutions of dpFvKEs provide strain and rotation fields of dislocation dipoles and pairs that also exhibit asymmetries and, compared with experiments, may yield information on out-of-plane displacements of atoms. While discrete theories need to be solved numerically, analytical formulas for strains and rotation about dislocations can be obtained from 2D Mindlin's hyperstress theory. These formulas are very useful for fitting experimental data and provide a template to ascertain the importance of nonlinear and nonplanar effects. Measuring the parameters of this theory, we find two characteristic lengths between three and four times the lattice spacings that control dilatation and rotation about a dislocation. At larger distances from the dislocation core, the elastic fields decay to those of conventional elasticity. Our results may be relevant for strain engineering in graphene and other 2D materials of current interest.