# A variational problem modelling behavior of unorthodox silicon crystals

J. Hannon; M. Marcus; Victor J. Mizel

ESAIM: Control, Optimisation and Calculus of Variations (2003)

- Volume: 9, page 145-149
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topHannon, J., Marcus, M., and Mizel, Victor J.. "A variational problem modelling behavior of unorthodox silicon crystals." ESAIM: Control, Optimisation and Calculus of Variations 9 (2003): 145-149. <http://eudml.org/doc/245034>.

@article{Hannon2003,

abstract = {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.},

author = {Hannon, J., Marcus, M., Mizel, Victor J.},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Landau/de Gennes order parameter; parametric problem},

language = {eng},

pages = {145-149},

publisher = {EDP-Sciences},

title = {A variational problem modelling behavior of unorthodox silicon crystals},

url = {http://eudml.org/doc/245034},

volume = {9},

year = {2003},

}

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

PB - EDP-Sciences

VL - 9

SP - 145

EP - 149

AB - 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.

LA - eng

KW - Landau/de Gennes order parameter; parametric problem

UR - http://eudml.org/doc/245034

ER -

## References

top- [1] B. Dacorogna and C.E. Pfister, Wulff theorem and best constant in Sobolev inequality. J. Math. Pures Appl. 71 (1992) 97-118. Zbl0676.46031MR1170247
- [2] I. Fonseca, The Wulff theorem revisited. Proc. Roy. Soc. Lond. A 432 (1991) 125-145. Zbl0725.49017MR1116536
- [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.

## Citations in EuDML Documents

top- Alexander J. Zaslavski, On a variational problem arising in crystallography
- Victor J. Mizel, Alexander J. Zaslavski, Anisotropic functions: a genericity result with crystallographic implications
- Victor J. Mizel, Alexander J. Zaslavski, Anisotropic functions : a genericity result with crystallographic implications

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.