# 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

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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 (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

top- B. Dacorogna and C.E. Pfister, Wulff theorem and best constant in Sobolev inequality. J. Math. Pures Appl.71 (1992) 97-118. Zbl0676.46031
- I. Fonseca, The Wulff theorem revisited. Proc. Roy. Soc. Lond. A432 (1991) 125-145. Zbl0725.49017
- J. Hannonet al., Step Faceting at the (001) Surface of Boron Doped Silicon. Phys. Rev. Lett.79 (1997) 4226-4229.
- H.C. Jeong and E.D. Williams, Steps on Surfaces: Experiment and Theory. Surface Sci. Reports34 (1999) 175-294.
- W.W. Mullins, Theory of thermal grooving. J. Appl. Phys.28 (1957) 333-339.
- J.E. Taylor, II-Mean curvature and weighted mean curvature. Acta Metall. Mater.40 (1992) 1475-1485.

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