Article Contents
Article Contents

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

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.

Mathematics Subject Classification: Primary: 74B20, 70G75, 49J45; Secondary: 74K10, 74N05.

 Citation:

• Figure 1.  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

Figure 2.  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

Figure 3.  The body-centred cubic lattice is associated with a tessellation of irregular tetrahedra as the one in the figure

Figure 4.  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

Figure 5.  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

Figure 6.  By cutting a cubic lattice along certain transverse planes, one finds two-dimensional hexagonal Bravais lattices. (A) face-centred; (B) body-centred

Figure 7.  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)

Figure 8.  The tetrahedron $\mathcal S$ and its image $F(\mathcal S)$

Figure 9.  The octahedron $\mathcal O$

Figure 10.  (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 $O=\Pi(Q_1)=\Pi(Q_4)$

•  R. Alicandro, G. Lazzaroni and M. Palombaro, Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours Preprint SISSA 58/2016/MATE. A. Braides , M. Solci  and  E. Vitali , A derivation of linear elastic energies from pair-interaction atomistic systems, Netw. Heterog. Media, 2 (2007) , 551-567.  doi: 10.3934/nhm.2007.2.551. E. Ertekin , P. A. Greaney , D. C. Chrzan  and  T. D. Sands , Equilibrium limits of coherency in strained nanowire heterostructures, J. Appl. Phys., 97 (2005) , 114325. S. Fanzon, M. Palombaro and M. Ponsiglione, A variational model for dislocations at semi-coherent interfaces, Preprint arXiv: 1512. 03795 (2015). L. Flatley  and  F. Theil , Face-centered cubic crystallization of atomistic configurations, Arch. Ration. Mech. Anal., 218 (2015) , 363-416.  doi: 10.1007/s00205-015-0862-1. G. Friesecke , R. D. James  and  S. Müller , A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55 (2002) , 1461-1506.  doi: 10.1002/cpa.10048. G. Friesecke  and  F. Theil , Validity and failure of the Cauchy-Born Hypothesis in a two-dimensional mass-spring lattice, J. Nonlinear Sci., 12 (2002) , 445-478.  doi: 10.1007/s00332-002-0495-z. G. Grosso and  G. Pastori Parravicini,  Solid State Physics, 2nd edition, Academic Press, Elsevier, Oxford, 2014. P. M. Gruber and J. M. Willis, eds. , Handbook of Convex Geometry, Volume A. , Elsevier Science Publishers B. V. , North-Holland, Amsterdam, 1993. K. L. Kavanagh , Misfit dislocations in nanowire heterostructures, Semicond. Sci. Technol., 25 (2010) , 024006. G. Lazzaroni , M. Palombaro  and  A. Schlömerkemper , A discrete to continuum analysis of dislocations in nanowire heterostructures, Commun. Math. Sci., 13 (2015) , 1105-1133.  doi: 10.4310/CMS.2015.v13.n5.a3. G. Lazzaroni , M. Palombaro  and  A. Schlömerkemper , Dislocations in nanowire heterostructures: From discrete to continuum, Proc. Appl. Math. Mech., 13 (2013) , 541-544. S. Müller  and  M. Palombaro , Derivation of a rod theory for biphase materials with dislocations at the interface, Calc. Var. Partial Differential Equations, 48 (2013) , 315-335.  doi: 10.1007/s00526-012-0552-x. B. Schmidt , A derivation of continuum nonlinear plate theory from atomistic models, Multiscale Model. Simul., 5 (2006) , 664-694.  doi: 10.1137/050646251. V. Schmidt , J. V. Wittemann  and  U. Gösele , Growth, thermodynamics, and electrical properties of silicon nanowires, Chem. Rev., 110 (2010) , 361-388. F. Theil , A proof of crystallization in two dimensions, Comm. Math. Phys., 262 (2006) , 209-236.  doi: 10.1007/s00220-005-1458-7.

Figures(10)