A Variational Problem Modelling Behavior of Unorthodox Silicon Crystals

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

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

Abstract

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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.

How to cite

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Hannon, J., Marcus, M., and Mizel, Victor J.. "A Variational Problem Modelling Behavior of Unorthodox Silicon Crystals." ESAIM: Control, Optimisation and Calculus of Variations 9 (2010): 145-149. <http://eudml.org/doc/90686>.

@article{Hannon2010,
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.; parametric problem},
language = {eng},
month = {3},
pages = {145-149},
publisher = {EDP Sciences},
title = {A Variational Problem Modelling Behavior of Unorthodox Silicon Crystals},
url = {http://eudml.org/doc/90686},
volume = {9},
year = {2010},
}

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
DA - 2010/3//
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.; parametric problem
UR - http://eudml.org/doc/90686
ER -

References

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  1. B. Dacorogna and C.E. Pfister, Wulff theorem and best constant in Sobolev inequality. J. Math. Pures Appl.71 (1992) 97-118.  Zbl0676.46031
  2. I. Fonseca, The Wulff theorem revisited. Proc. Roy. Soc. Lond. A432 (1991) 125-145.  Zbl0725.49017
  3. J. Hannonet 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. Reports34 (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.  

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