Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires

TL;DR

This paper rigorously derives continuum models for heterogeneous nanowires from discrete lattice energies, demonstrating how dislocations form as nanowire radius increases, using geometric rigidity and Gamma-convergence techniques.

Contribution

It provides a detailed discrete geometric rigidity result for 3D lattices and performs a simultaneous continuum limit and dimension reduction for nanowire models.

Findings

Dislocations occur in large-radius nanowires.

The model captures the transition from defect-free to dislocated structures.

Rigidity results apply to various 3D lattice types.

Abstract

In the context of nanowire heterostructures we perform a discrete to continuum limit of the corresponding free energy by means of $\Gamma$-convergence techniques. Nearest neighbours are identified by employing the notions of Voronoi diagrams and Delaunay triangulations. The scaling of the nanowire is done in such a way that we perform not only a continuum limit but a dimension reduction simultaneously. The main part of the proof is a discrete geometric rigidity result that we announced in an earlier work and show here in detail for a variety of three-dimensional lattices. We perform the passage from discrete to continuum twice: once for a system that compensates a lattice mismatch between two parts of the heterogeneous nanowire without defects and once for a system that creates dislocations. It turns out that we can verify the experimentally observed fact that the nanowires show dislocations when the radius of the specimen is large.