5 Phase-Field-Crystal Model: A Tool for Probing Atoms in TSV
121
5.2.6.2 Magnetic Field
To couple the magnetic field in the PFC models, a dimensionless free energy F
that
couples three fields, i.e., the number density n(r), the magnetization vector m(r),
and the magnetic field B, is constructed [32–34]:
F
= F +
drω
W
2
0
2
|∇m|
2
+ (r c − βn
2
)
|m|
2
2
+ γ
|m|
4
4
−
α
2
(m · ∇n)
2
− m · B +
|B|
2
2
(5.34)
Note that the magnetic field is described by an external and a self-induced magnetic
field, i.e., B = B ind + B ext . In addition, the induced magnetic field is related to the
magnetization vector m(r) as: B ind = ∇ × A and ∇
2 A = −∇ × m. The details of
this model can be found in [32].
The governing equation for m is nonconserved dynamics and for n is conserved
dynamics:
∂m i (r, t)
∂t
= −
δF
(n, m)
δm i
∂n(r, t)
∂t
= ∇
2 δF
(n, m)
δn
(5.35)
where i = x, y, z and m i are the components of m.
An example of how the magnetic field induces a phase transition is shown in
Fig. 5.7 [33]. The system is equilibrated to the uniform state after 10
5 time-steps as
shown in Fig. 5.7b. Then an external magnetic field H is applied and it induced the
growth of the square lattice in Fig. 5.7c, d.
5.2.6.3 Electric Field
A PFC model that describes the electromigration in metals was developed by Wang,
Bevan, and Provatas [35]. A new term F EM is introduced for the electromigration
driving force:
F ext = F EM =
drA EM (r)eV (r) =
drA 0
Z
∗
n mf n(r)eV (r)
(5.36)
where A 0 is a coupling constant, Z
∗ is the effective electromigration charge parameter,
is the atomic volume, n mf =
dr
exp[−(r − r
)
2
/(2λ mf )]n(r
) is local average
density, e is the electron charge, and V (r) is the local electric potential.
The governing equations are: