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

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

Giuliano Lazzaroni 1, , Mariapia Palombaro 2, and  Anja Schlömerkemper 3,

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.

Keywords: Nonlinear elasticity, discrete to continuum, dimension reduction, Rod theory, geometric rigidity, non-interpenetration, Gamma-convergence, crystals, dislocations, heterostructures.
Mathematics Subject Classification: Primary: 74B20, 70G75, 49J45; Secondary: 74K10, 74N05.
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:
[1]

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. Google Scholar

[2]

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.  Google Scholar

[3]

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.   Google Scholar

[4]

S. Fanzon, M. Palombaro and M. Ponsiglione, A variational model for dislocations at semi-coherent interfaces, Preprint arXiv: 1512. 03795 (2015). Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8] G. Grosso and G. Pastori Parravicini, Solid State Physics, 2nd edition, Academic Press, Elsevier, Oxford, 2014.   Google Scholar
[9]

P. M. Gruber and J. M. Willis, eds. , Handbook of Convex Geometry, Volume A. , Elsevier Science Publishers B. V. , North-Holland, Amsterdam, 1993. Google Scholar

[10]

K. L. Kavanagh, Misfit dislocations in nanowire heterostructures, Semicond. Sci. Technol., 25 (2010), 024006.   Google Scholar

[11]

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.  Google Scholar

[12]

G. Lazzaroni, M. Palombaro and A. Schlömerkemper, Dislocations in nanowire heterostructures: From discrete to continuum, Proc. Appl. Math. Mech., 13 (2013), 541-544.   Google Scholar

[13]

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.  Google Scholar

[14]

B. Schmidt, A derivation of continuum nonlinear plate theory from atomistic models, Multiscale Model. Simul., 5 (2006), 664-694.  doi: 10.1137/050646251.  Google Scholar

[15]

V. Schmidt, J. V. Wittemann and U. Gösele, Growth, thermodynamics, and electrical properties of silicon nanowires, Chem. Rev., 110 (2010), 361-388.   Google Scholar

[16]

F. Theil, A proof of crystallization in two dimensions, Comm. Math. Phys., 262 (2006), 209-236.  doi: 10.1007/s00220-005-1458-7.  Google Scholar

show all references

References:
[1]

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. Google Scholar

[2]

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.  Google Scholar

[3]

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.   Google Scholar

[4]

S. Fanzon, M. Palombaro and M. Ponsiglione, A variational model for dislocations at semi-coherent interfaces, Preprint arXiv: 1512. 03795 (2015). Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8] G. Grosso and G. Pastori Parravicini, Solid State Physics, 2nd edition, Academic Press, Elsevier, Oxford, 2014.   Google Scholar
[9]

P. M. Gruber and J. M. Willis, eds. , Handbook of Convex Geometry, Volume A. , Elsevier Science Publishers B. V. , North-Holland, Amsterdam, 1993. Google Scholar

[10]

K. L. Kavanagh, Misfit dislocations in nanowire heterostructures, Semicond. Sci. Technol., 25 (2010), 024006.   Google Scholar

[11]

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.  Google Scholar

[12]

G. Lazzaroni, M. Palombaro and A. Schlömerkemper, Dislocations in nanowire heterostructures: From discrete to continuum, Proc. Appl. Math. Mech., 13 (2013), 541-544.   Google Scholar

[13]

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.  Google Scholar

[14]

B. Schmidt, A derivation of continuum nonlinear plate theory from atomistic models, Multiscale Model. Simul., 5 (2006), 664-694.  doi: 10.1137/050646251.  Google Scholar

[15]

V. Schmidt, J. V. Wittemann and U. Gösele, Growth, thermodynamics, and electrical properties of silicon nanowires, Chem. Rev., 110 (2010), 361-388.   Google Scholar

[16]

F. Theil, A proof of crystallization in two dimensions, Comm. Math. Phys., 262 (2006), 209-236.  doi: 10.1007/s00220-005-1458-7.  Google Scholar

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
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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
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Figure 3.  The body-centred cubic lattice is associated with a tessellation of irregular tetrahedra as the one in the figure
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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
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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
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Figure 6.  By cutting a cubic lattice along certain transverse planes, one finds two-dimensional hexagonal Bravais lattices. (A) face-centred; (B) body-centred
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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)
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Figure 8.  The tetrahedron $\mathcal S$ and its image $F(\mathcal S)$
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Figure 9.  The octahedron $\mathcal O$
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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 $O=\Pi(Q_1)=\Pi(Q_4)$
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