# 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
##### References:

show all references

##### 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 $O=\Pi(Q_1)=\Pi(Q_4)$
 [1] Antonio De Rosa, Domenico Angelo La Manna. A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021059 [2] Raz Kupferman, Cy Maor. The emergence of torsion in the continuum limit of distributed edge-dislocations. Journal of Geometric Mechanics, 2015, 7 (3) : 361-387. doi: 10.3934/jgm.2015.7.361 [3] Annalisa Cesaroni, Valerio Pagliari. Convergence of nonlocal geometric flows to anisotropic mean curvature motion. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021065 [4] Xiaofei Liu, Yong Wang. Weakening convergence conditions of a potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021080 [5] Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065 [6] Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2021, 13 (1) : 1-23. doi: 10.3934/jgm.2020032 [7] Bruno Premoselli. Einstein-Lichnerowicz type singular perturbations of critical nonlinear elliptic equations in dimension 3. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021069 [8] John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti. Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021004 [9] Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185 [10] Emma D'Aniello, Saber Elaydi. The structure of $\omega$-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195 [11] Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475 [12] Jonathan DeWitt. Local Lyapunov spectrum rigidity of nilmanifold automorphisms. Journal of Modern Dynamics, 2021, 17: 65-109. doi: 10.3934/jmd.2021003 [13] Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2725-3737. doi: 10.3934/dcds.2020383 [14] Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021032 [15] Tao Wang. Variational relations for metric mean dimension and rate distortion dimension. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021050 [16] Yuan Gao, Jian-Guo Liu, Tao Luo, Yang Xiang. Revisit of the Peierls-Nabarro model for edge dislocations in Hilbert space. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3177-3207. doi: 10.3934/dcdsb.2020224 [17] Enkhbat Rentsen, N. Tungalag, J. Enkhbayar, O. Battogtokh, L. Enkhtuvshin. Application of survival theory in Mining industry. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 443-448. doi: 10.3934/naco.2020036 [18] Thomas Barthelmé, Andrey Gogolev. Centralizers of partially hyperbolic diffeomorphisms in dimension 3. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021044 [19] Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151 [20] Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311

2019 Impact Factor: 1.233