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Elastic Green’s function method for multiscale modeling of point defects and extended defects in solids
Multi-scale model – discrete lattice structure near a point defect and continuum model near a free surface or interface in the same formalism.
Vinod TewaryNIST, Boulder
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Motivation
Properties of thin films and semi-infinite solids sensitive to concentration of vacancies near a free surface or interface between film and substrate
Examples
Solid-state lighting devices using III-V semiconductors
Diffusion near grain boundaries in Copper interconnects
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Measurable quantity- strain at or near a free surface
Strains are caused by point defects
Strain – a macroscopic quantity defined in the continuum model.
Continuum model applicable to extended defects such as free surfaces and interfaces
Lattice model very difficult near an extended defect- too much disorder; probably not really needed.
Lattice distortion due to point defects sensitive to discrete lattice structure near the defect- continuum model not valid
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Need for a multi-scale model – discrete lattice structure near a point defect and continuum model near a free surface or interface in the same formalism.
Lattice statics Green’s function for point defects.
Lattice statics GF reduces to continuum model GF asymptotically
Continuum GF for extended defects
Objective: To calculate lattice distortion, strains, relaxation energy, change in phonon spectra due to defects, and elastic interaction between the defects
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Point defects in a Crystal Lattice
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0 1 2
0’ 1’ 2’
0” 1” 2”
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l, l’ - lattice sites (monatomic Bravais)
l, l’d matrix – force constants
Obtained by the first and second derivatives of the interatomic potential
F(l) – Force on atom at l
Obtained by the first derivative of the interatomic potential
[ l, l’]ijV(x)/xi xj,
[F(l)]i = – V(x)/xi.
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Born- von Karman model Pair potential Cyclic boundary conditions Supercell
u(l) - displacement of the atom at l
W = - F(l) u(l) +
(1/2) l, l’u(l) u(l’)
Static displacement
W/ u(l) = 0
u(l) = l’G* l, l’F(l’)
u = G* FLattice-statics Green’s function
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G* = [3N x matrices
Relaxation energy: W = -(1/2) FG*F
Static interaction energy between "a" and "b"
Wa,b = Wa+b - Wa - Wb
Thermodynamic interaction energy- Free energy.
Force Const for perfect lattice
,
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G* = G + G G*,
where
G = [
u = G* F.
u = (G + G G*) F
u = G F*
F* = F + u. Kanzaki Force
G(l,l’) has translation symmetry
G(l) = (1/N) qG(q) exp[ ιq.l]
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G(q) = [q
Solution of the Dyson equation
F and nonvanishing only in defect space- matrices of finite dimensions
g* = g + g g*
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g* = (I - g g
u = g* F.
Calculate u in defect space
Calculate Kanzaki force in def space
u = G F*
G reduces to cont GF for large l
Replace sum by an integral
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Treat l and q as continuous variables
Write x for position vector of site l
Gx(/2)3Gc (q) exp (ιq.x) dq,
Gc(q) = Limq0 G(q) = Limq0 [q
= [q
ij (q) = cikjl qk ql,
u(x) = l’Gc x-l’F*(l’)
Gc(x-l’) can be calculated in terms of the derivatives of the continuum Green’s function.
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i3(x) = ci3jk ejk(x) = 0 (x3=0)
ejk = uj(x)/ xk
Efficient methods for calculating GF and derivatives for anisotropic solids are available
Mindlin solution for isotropic solids
4ur/f = -rh/R3 + r/[(+)(R-h)R]
4uz/f = (R2 + h2)/R3 + /[(+)R]
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MSGF method can model a large crystallite at the atomistic level without excessive CPU effort
In the same formalism, include the extended defects using the standard techniques of the continuum model.
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