A variational problem modelling behavior of unorthodox silicon crystals
ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 145-149.

Controlling growth at crystalline surfaces requires a detailed and quantitative understanding of the thermodynamic and kinetic parameters governing mass transport. Many of these parameters can be determined by analyzing the isothermal wandering of steps at a vicinal [“step-terrace”] type surface [for a recent review see [4]]. In the case of orthodox crystals one finds that these meanderings develop larger amplitudes as the equilibrium temperature is raised (as is consistent with the statistical mechanical view of the meanderings as arising from atomic interchanges). The classical theory due to Herring, Mullins and others [5], coupled with advances in real-time experimental microscopy techniques, has proven very successful in the applied development of such crystalline materials. However in 1997 a series of experimental observations on vicinal defects of heavily boron-doped Silicon crystals revealed that these crystals were quite unorthodox in the sense that a lowering of the equilibrium temperature led to increased amplitude for the isothermal wanderings of a step edge [3]. In addition, at low temperatures the step profile adopted a periodic saw-tooth structure rather than the straight profile predicted by the classical theories. This article examines a stored free energy model for such crystals involving a (higher order) Landau/de Gennes type ``order parameter" term and provides a proof for the existence of a minimizer.

DOI : 10.1051/cocv:2003004
Classification : 49J45, 49S05
Mots-clés : Landau/de Gennes order parameter, parametric problem 
@article{COCV_2003__9__145_0,
     author = {Hannon, J. and Marcus, M. and Mizel, Victor J.},
     title = {A variational problem modelling behavior of unorthodox silicon crystals},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {145--149},
     publisher = {EDP-Sciences},
     volume = {9},
     year = {2003},
     doi = {10.1051/cocv:2003004},
     mrnumber = {1957095},
     zbl = {1066.49011},
     language = {en},
     url = {https://numdam.org/articles/10.1051/cocv:2003004/}
}
TY  - JOUR
AU  - Hannon, J.
AU  - Marcus, M.
AU  - Mizel, Victor J.
TI  - A variational problem modelling behavior of unorthodox silicon crystals
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2003
SP  - 145
EP  - 149
VL  - 9
PB  - EDP-Sciences
UR  - https://numdam.org/articles/10.1051/cocv:2003004/
DO  - 10.1051/cocv:2003004
LA  - en
ID  - COCV_2003__9__145_0
ER  - 
%0 Journal Article
%A Hannon, J.
%A Marcus, M.
%A Mizel, Victor J.
%T A variational problem modelling behavior of unorthodox silicon crystals
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2003
%P 145-149
%V 9
%I EDP-Sciences
%U https://numdam.org/articles/10.1051/cocv:2003004/
%R 10.1051/cocv:2003004
%G en
%F COCV_2003__9__145_0
Hannon, J.; Marcus, M.; Mizel, Victor J. A variational problem modelling behavior of unorthodox silicon crystals. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 145-149. doi : 10.1051/cocv:2003004. https://numdam.org/articles/10.1051/cocv:2003004/

[1] B. Dacorogna and C.E. Pfister, Wulff theorem and best constant in Sobolev inequality. J. Math. Pures Appl. 71 (1992) 97-118. | MR | Zbl

[2] I. Fonseca, The Wulff theorem revisited. Proc. Roy. Soc. Lond. A 432 (1991) 125-145. | MR | Zbl

[3] J. Hannon et al., Step Faceting at the (001) Surface of Boron Doped Silicon. Phys. Rev. Lett. 79 (1997) 4226-4229.

[4] H.C. Jeong and E.D. Williams, Steps on Surfaces: Experiment and Theory. Surface Sci. Reports 34 (1999) 175-294.

[5] W.W. Mullins, Theory of thermal grooving. J. Appl. Phys. 28 (1957) 333-339.

[6] J.E. Taylor, Ii-Mean curvature and weighted mean curvature. Acta Metall. Mater. 40 (1992) 1475-1485.

  • Zaslavski, Alexander Introduction, Turnpike Phenomenon and Symmetric Optimization Problems, Volume 190 (2022), p. 1 | DOI:10.1007/978-3-030-96973-8_1
  • Zaslavski, Alexander Optimization Problems Arising in Crystallography, Turnpike Phenomenon and Symmetric Optimization Problems, Volume 190 (2022), p. 267 | DOI:10.1007/978-3-030-96973-8_7
  • Rieger, Marc Oliver; Zirnstein, Heinrich-Gregor Parameter Dependence of Multiscale Systems and Nanostructures on Crystals, Mathematics and Mechanics of Solids, Volume 12 (2007) no. 5, p. 513 | DOI:10.1177/1081286506064718
  • Mizel, Victor J.; Zaslavski, Alexander J. Anisotropic functions: a genericity result with crystallographic implications, ESAIM: Control, Optimisation and Calculus of Variations, Volume 10 (2004) no. 4, p. 624 | DOI:10.1051/cocv:2004023

Cité par 4 documents. Sources : Crossref