Communications in Mathematical Sciences

Volume 13 (2015)

Number 5

A discrete to continuum analysis of dislocations in nanowire heterostructures

Pages: 1105 – 1133

DOI: https://dx.doi.org/10.4310/CMS.2015.v13.n5.a3

Authors

Giuliano Lazzaroni (SISSA, Trieste, Italy)

Mariapia Palombaro (Department of Mathematics, University of Sussex, Brighton, United Kingdom)

Anja Schlömerkemper (Department of Mathematics, University of Würzburg, Germany)

Abstract

Epitaxially grown heterogeneous nanowires present dislocations at the interface between the phases if their radius is big. We consider a corresponding variational discrete model with quadratic pairwise atomic interaction energy. By employing the notion of Gamma-convergence and a geometric rigidity estimate, we perform a discrete to continuum limit and a dimension reduction to a one-dimensional system. Moreover, we compare a defect-free model and models with dislocations at the interface and show that the latter are energetically convenient if the thickness of the wire is sufficiently large.

Keywords

nonlinear elasticity, discrete to continuum, dimension reduction, rod theory, geometric rigidity, non-interpenetration, Gamma-convergence, crystals, dislocations, heterostructures

2010 Mathematics Subject Classification

49J45, 70G75, 74B20, 74K10, 74N05

Published 22 April 2015