Geometric Modeling of a Line of Alternating Disclinations: Application to Grain Boundaries in Graphene

TL;DR

This paper presents a conformal geometric model for graphene grain boundaries using (2+1)-dimensional gravity, analytically describing defect structures and their geometric effects, aligning with experimental observations.

Contribution

It introduces a novel geometric model linking topological defect theory with atomistic features of graphene through analytical solutions of Einstein equations.

Findings

Curvature is localized near defect lines.

Geometry becomes asymptotically flat at large distances.

Model captures experimental 5-7 grain boundary features.

Abstract

We develop a conformal geometric model for grain boundaries in graphene based on a periodic line of alternating disclinations. Within the framework of (2+1)-dimensional gravity, we solve a reduced form of the Einstein equations to determine the conformal factor, from which the induced metric, scalar curvature, and holonomy are obtained analytically. Each pentagon-heptagon pair is modeled as a disclination dipole, forming a continuous distribution that captures the geometric signature of experimentally observed 5-7 grain boundaries. We show that the curvature is localized near the defect line and that the geometry becomes asymptotically flat, with trivial holonomy at large distances. This construction provides a tractable and physically consistent realization of the Katanaev-Volovich framework, connecting topological defect theory with atomistic features of graphene.