Collaborator: W. Dreyer
(FG 7),
F. Duderstadt
(FG 7),
J. Sprekels
(FG 1),
B. Wagner
(FG 1)
Cooperation with: S. Brodie, C.M. Brown (Heriot-Watt University Edinburgh,
UK), P. Colli, G. Gilardi (Università di Pavia, Italy), S. Eichler
(Freiberger Compound Materials GmbH), T. Hauck (Motorola, München),
W.H. Müller (Technische Universität Berlin), B. Niethammer
(Universität Bonn), E. Radkevich (Lomonosov University, Moscow,
Russia)
Supported by: BMBF: ``Mathematische Modellierung und Simulation der
Entstehung, des Wachstums und der Auflösung von
Arsenausscheidungen
in einkristallinem Galliumarsenid''
(Mathematical modeling and simulation of the formation, growth and
dissolution of arsenic precipitation in single crystal gallium
arsenide)
Description:
There are two models for different aspects of phase
transitions that were studied during the last
report period.
(1) The standard model, which was designed by Dreyer/Müller, is able to
describe morphological changes of binary
alloys on the 
scale. In particular, a study of the coupling of
mechanical stresses,
diffusion and interface motion is of special interest for practical applications. An
important difference between the Dreyer/Müller model and other similar models
concerns the higher gradient part, which models surface
energy of the interface boundary. That part is
given by second derivatives of the concentration as 
. The anisotropy and the concentration dependence of the matrix aij(c) leads to quite new sharp interface limits that have not been
studied before. While the numerical exploitation of the consequences
was already done in former years, B. Wagner started a serious
mathematical treatment of the problem. Moreover, the microscopic model to calculate
the explicit structure of the matrix aij(c) that was used up to
now has been changed during the last report period. The atomic
interactions were formerly described by central forces, however, their
capability to describe properly atomic interaction in anisotropic
crystals of, say, fcc or tetragonal lattice symmetry has become
apparent. For this reason Dreyer/Müller started a rederivation of all
atomic contributions to the free energy containing in particular the
gradient matrix. This rederivation relies on microscopic embedded
force potentials, where noncentral
forces are modeled by the introduction of more than two body forces.
Regarding the applications of the model there are two aspects that
were considered. The data basis for tin/lead alloys has been further
extended and a collaboration with Bosch has been started to find out
whether the Dreyer/Müller model can be described similarly to the
formation of shear bands, the appearance of localized phase
transitions that were observed in the Bosch labs and which might lead
to the initiation of cracks. On the other hand, the collection of data
for the intermetallic silver/tin is still of high importance, because
due to environmental reasons silver/tin will probably substitute the
tin/lead system as solder joint material
in microelectronic devices in the very near future.
(2) The other model that was developed and extensively studied is
designed to describe a process in semi-insulating gallium arsenide
(GaAs), where a complex coupling between eigenstrain fields, due to
dislocations and misfit, and diffusion, which is initiated by a heat
treatment of GaAs, leads to the appearance of liquid arsenic
droplets. Their inhomogeneous
distribution on the mm scale means a serious problem regarding the
use of GaAs as wafer material for optoelectronic devices.
The original mathematical model, which was developed at WIAS, contains
a coupled system of partial differential equations for the variables
strain, (As) concentration and a time- and space-dependent distribution
indicating the size of the droplets. It has turned out during the last
report period that a description of semi-insulating GaAs with only these
variables is insufficient for its realistic thermodynamic
description. There are further constituents which appear in
semi-insulating GaAs. These are oxygen (O), silicon (Si), boron (B) and
carbon (C) in very small quantities, but nevertheless these trace
elements induce very important phenomena. Moreover, these elements
as well as vacancies and the arsenic may carry charges and there are
chemical reactions among these constituents
and with electrons and holes, which must also be considered.
Several conferences with the wafer manufacturer FCM and with other
experimenters have led to an extended description to simulate the
evolution of arsenic droplets in GaAs. There are now 16 constituents included
in the extended model that is designed as follows: Roughly speaking,
there are two different time scales. Mechanical and chemical equilibrium is
reached so much faster than diffusional and interfacial equilibrium that
the mechanics and the chemistry are described by quasistatic equations whose
time dependence is given by the evolution of diffusion and of interface
motion. This leads to an elliptic second-order system to describe the
strains and stresses and to 16 nonlinear algebraic equations to describe the
chemistry. There is only a single diffusing constituent, which has turned
out to be the uncharged arsenic interstitial, and there is thus (as before)
a single diffusion equation. However, that equation is now of a quite
complicated structure, because it couples to the solution of the chemical
algebraic system and (as before) to the mechanical system.
Due to these new complexities, the evolution of droplets is currently
described as a free boundary value problem for a single droplet. If
all results of this simplification are fully exploited, the complete
Becker/Döring model will be coupled again
to the described model. To this end a multiscale
transition is needed and will be carried
out in collaboration with B. Niethammer (Bonn), who is the
leading expert in Becker/Döring and related LSW models.
References:
- C.M. BROWN, W. DREYER, W.H. MÜLLER,
Discrete Fourier transforms and their application to stress-strain
problems in composite mechanics: A convergence study, Proc. Roy.
Soc. London Ser. A, 458 (2002), pp. 1-21.
- E. BONETTI, P. COLLI, W. DREYER, G. GILARDI, G. SCHIMPERNA, J. SPREKELS, On a
model for phase separation in binary alloys driven by mechanical effects,
Phys. D, 165 (2002), pp. 48-65.
- W. DREYER, W.H. MÜLLER, Toward
quantitative modeling of morphology changes of solids with phase field
theories: Atomistic arguments for the determination of higher gradient
coefficients, in: Bioceramics 15. Proceedings of the 15th
International Symposium on Ceramics in Medicine, Sydney, Australia,
December 4-8, 2002, B.B. Nissan, D. Sher,
W. Walsh, eds., vol. 240-242 of Key Engineering Materials, Trans Tech
Publications Ltd, Ütikon-Zürich, 2003, pp. 901-914.
- W. DREYER, F. DUDERSTADT, Modeling of the
appearance of arsenide droplets in gallium arsenide, to appear as
WIAS Preprint.
- W. DREYER, On jump conditions of moving
interfaces in crystals, to appear as WIAS Preprint.
LaTeX typesetting by I. Bremer
5/16/2003