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Discrete spin systems on random lattices at the bulk scaling
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 |
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.
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. |
[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. |
[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.
|
[4] |
S. Fanzon, M. Palombaro and M. Ponsiglione, A variational model for dislocations at semi-coherent interfaces, Preprint arXiv: 1512. 03795 (2015). |
[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. |
[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. |
[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. |
[8] |
G. Grosso and G. Pastori Parravicini, Solid State Physics, 2nd edition, Academic Press, Elsevier, Oxford, 2014.
![]() |
[9] |
P. M. Gruber and J. M. Willis, eds. , Handbook of Convex Geometry, Volume A. , Elsevier Science Publishers B. V. , North-Holland, Amsterdam, 1993. |
[10] |
K. L. Kavanagh,
Misfit dislocations in nanowire heterostructures, Semicond. Sci. Technol., 25 (2010), 024006.
|
[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. |
[12] |
G. Lazzaroni, M. Palombaro and A. Schlömerkemper,
Dislocations in nanowire heterostructures: From discrete to continuum, Proc. Appl. Math. Mech., 13 (2013), 541-544.
|
[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. |
[14] |
B. Schmidt,
A derivation of continuum nonlinear plate theory from atomistic models, Multiscale Model. Simul., 5 (2006), 664-694.
doi: 10.1137/050646251. |
[15] |
V. Schmidt, J. V. Wittemann and U. Gösele,
Growth, thermodynamics, and electrical properties of silicon nanowires, Chem. Rev., 110 (2010), 361-388.
|
[16] |
F. Theil,
A proof of crystallization in two dimensions, Comm. Math. Phys., 262 (2006), 209-236.
doi: 10.1007/s00220-005-1458-7. |
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. |
[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. |
[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.
|
[4] |
S. Fanzon, M. Palombaro and M. Ponsiglione, A variational model for dislocations at semi-coherent interfaces, Preprint arXiv: 1512. 03795 (2015). |
[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. |
[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. |
[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. |
[8] |
G. Grosso and G. Pastori Parravicini, Solid State Physics, 2nd edition, Academic Press, Elsevier, Oxford, 2014.
![]() |
[9] |
P. M. Gruber and J. M. Willis, eds. , Handbook of Convex Geometry, Volume A. , Elsevier Science Publishers B. V. , North-Holland, Amsterdam, 1993. |
[10] |
K. L. Kavanagh,
Misfit dislocations in nanowire heterostructures, Semicond. Sci. Technol., 25 (2010), 024006.
|
[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. |
[12] |
G. Lazzaroni, M. Palombaro and A. Schlömerkemper,
Dislocations in nanowire heterostructures: From discrete to continuum, Proc. Appl. Math. Mech., 13 (2013), 541-544.
|
[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. |
[14] |
B. Schmidt,
A derivation of continuum nonlinear plate theory from atomistic models, Multiscale Model. Simul., 5 (2006), 664-694.
doi: 10.1137/050646251. |
[15] |
V. Schmidt, J. V. Wittemann and U. Gösele,
Growth, thermodynamics, and electrical properties of silicon nanowires, Chem. Rev., 110 (2010), 361-388.
|
[16] |
F. Theil,
A proof of crystallization in two dimensions, Comm. Math. Phys., 262 (2006), 209-236.
doi: 10.1007/s00220-005-1458-7. |









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