5 Phase-Field-Crystal Model: A Tool for Probing Atoms in TSV
125
(a)
(b)
Fig. 5.10 Polycrystalline structure in a rectangle TSV: (a) triangle lattice and (b) square lattice
5.3.2 Square Lattice
The two-mode PFC model was first developed by Wu [12, 38]. This model can
describe the FCC lattice, e.g. Cu and Al, and 2D square lattice. The square lattice in
2D has an order parameter:
n = n 0 +
n=1,2
A n e
iK n ·r
(5.42)
= n 0 + 2A 1 [cos(qx) + cos(qy)] + 4A 2 cos(qx) cos(qy)
(5.43)
where A 1 and A 2 are the amplitudes of two sets of density waves with wave vectors
K 1 = =1, 0 and K 2 = =1, 1, respectively. q = 2π/a is a parameter related to the
lattice constant a. The corresponding free energy functional is:
F[n] =
dr
n
2
[r + (∇
2
+ 1)
2
(∇
2
+ Q
2
)
2
]n +
n
4
4
(5.44)
125
(a)
(b)
Fig. 5.10 Polycrystalline structure in a rectangle TSV: (a) triangle lattice and (b) square lattice
5.3.2 Square Lattice
The two-mode PFC model was first developed by Wu [12, 38]. This model can
describe the FCC lattice, e.g. Cu and Al, and 2D square lattice. The square lattice in
2D has an order parameter:
n = n 0 +
n=1,2
A n e
iK n ·r
(5.42)
= n 0 + 2A 1 [cos(qx) + cos(qy)] + 4A 2 cos(qx) cos(qy)
(5.43)
where A 1 and A 2 are the amplitudes of two sets of density waves with wave vectors
K 1 = =1, 0 and K 2 = =1, 1, respectively. q = 2π/a is a parameter related to the
lattice constant a. The corresponding free energy functional is:
F[n] =
dr
n
2
[r + (∇
2
+ 1)
2
(∇
2
+ Q
2
)
2
]n +
n
4
4
(5.44)