# American Institute of Mathematical Sciences

February  2017, 10(1): 119-139. doi: 10.3934/dcdss.2017007

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

 1 SISSA, Via Bonomea 265, 34136 Trieste, Italy 2 University of Sussex, Department of Mathematics, Pevensey 2 Building, Falmer Campus, Brighton BN1 9QH, United Kingdom 3 University of Würzburg, Institute of Mathematics, Emil-Fischer-Straße 40, 97074 Würzburg, Germany

Received  January 2015 Revised  July 2015 Published  December 2016

Fund Project: This work was partially supported by the DFG grant SCHL 1706/2-1. The research of G.L. was supported by the ERC grant No. 290888.

In the context of nanowire heterostructures we perform a discrete to continuum limit of the corresponding free energy by means of Γ-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.

Citation: Giuliano Lazzaroni, Mariapia Palombaro, Anja Schlömerkemper. Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 119-139. doi: 10.3934/dcdss.2017007
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##### References:
In the face-centred cubic lattice the nearest-neighbour structure of the atoms provides a subdivision of the space into tetrahedra (A) and octahedra (B). Figure (C) shows a quarter of an octahedron in the same unit cell. Grey dots denote points lying on the hidden facets
The hexagonal close-packed lattice is associated with a tessellation of tetrahedra and octahedra as the ones in the figure. Only some of the bonds and some of the polyhedra of the pretriangulation are displayed
The body-centred cubic lattice is associated with a tessellation of irregular tetrahedra as the one in the figure
Cubic cell in the diamond lattice $\mathcal{L}^{\rm{D}}$. Atoms from the sublattice $\mathcal{L}^{\rm{D}}_1$ are represented in black/grey, while white atoms are from the sublattice $\mathcal{L}^{\rm{D}}_2$. Nearest-neighbour bonds are displayed by solid thick lines. Moreover, the picture shows a tetrahedron from the Delaunay pretriangulation of $\mathcal{L}^{\rm{D}}_1$: its edges (solid and dashed thin lines) correspond to next-to-nearest neighbours in $\mathcal{L}^{\rm{D}}$. A white atom lies at the barycentre of the tetrahedron, which is further divided into four irregular tetrahedra by the bonds between the barycentre and each vertex
Bonds and triangulation in a honeycomb lattice. The lattice is given by $\mathcal{L}^✡:=\mathcal{L}^✡_1 \cup \mathcal{L}^✡_2$, where $\mathcal{L}^✡_i:={\rm{u}}_i^✡ + \xi_1{\rm{v}}^✡_1+\xi_2{\rm{v}}^✡_2 \colon \ \xi_1,\xi_2 \in\mathbb{Z}\}$, ${\rm{v}}^✡_1:=(1,0)$, ${\rm{v}}^✡_2:=(\frac12,\frac{\sqrt3}2)$, ${\rm{u}}^✡_1:=(0,0)$, ${\rm{u}}^✡_2:=(0,\frac{\sqrt3}3)$. This results into two interpenetrating sublattices $\mathcal{L}^✡_1$ and $\mathcal{L}^✡_2$, both being hexagonal (i.e., equilateral triangular). Atoms from $\mathcal{L}^✡_1$ and $\mathcal{L}^✡_2$ are displayed in different colors in the picture, respectively in black and in white. In the left part of the figure we indicate nearest neighbour (solid) and next-to-nearest neighbour bonds (dashed lines). The right part of the figure shows a possible triangulation, that is the natural triangulation of $\mathcal{L}^✡_1$ enriched by considering the nearest-neighbour bonds between atoms $x\in\mathcal{L}^✡_1$ and $y\in\mathcal{L}^✡_2$. This corresponds to ignoring the bonds between atoms of $\mathcal{L}^✡_2$, cf. Section 2.3
By cutting a cubic lattice along certain transverse planes, one finds two-dimensional hexagonal Bravais lattices. (A) face-centred; (B) body-centred
Dislocations in a honeycomb-type lattice. The bonds at the interface are chosen in the following way: First one considers only black atoms and finds a Delaunay pretriangulation, which is then refined to a triangulation (dashed lines); the same is done for white atoms (dotted lines). The dashed and dotted lines thus obtained give the bonds between next-to-nearest neighbours. Finally, each white (resp. black) atom lying inside a triangle formed by three black (resp. white) atoms is connected to the vertices of that triangle by nearest-neighbour bonds (solid lines)
The tetrahedron $\mathcal S$ and its image $F(\mathcal S)$
The octahedron $\mathcal O$
(A) The image of $\mathcal{O}$ through a piece-wise affine map $u$ such that $l_i=1$ for each $i\neq 3$. (B) The projection of $u(\mathcal{O})$ on the plane $p$, where $O=\Pi(Q_1)=\Pi(Q_4)$
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