132
J. Liu et al.
n = n 0 + A
cos(qx) + cos(qy)
+ B cos(qx) cos(qy)
(6.1)
F[n] =
dr
n
2
[r + (∇
2
+ 1)
2
(∇
2
+ Q
2
)
2
]n +
n
4
4
(6.2)
F
[n] = F[n] +
drM(r)
n(r, t) − n s (r(v, t), t)
(6.3)
∂n
∂t
= ∇
2 δ F
[n]
δn
(6.4)
The above equations and parameters are referred to Chap. 5, Sects. 5.2.6.1 and 5.3.2.
The remainder of this chapter is organized as following. First, an example demonstrating the salient features of the model and the protrusion process is given. Then the
protrusion behavior under different external applied mechanical loading, as well as
effects from grain structure, temperature and geometry are discussed. Finally, general
perspectives on TSV protrusion and an outlook for future work are presented.
6.2 Model Setting and an Example of TSV Protrusion
Figure 6.1 illustrates the model configuration of a TSV structure with loading applied.
Note that a trapezoidal TSV is chosen because of the tapered nature of a drilled via [6].
The initial atomic-scale microstructure is generated by placing crystal nuclei inside
the TSV and then initiating solidification. The microstructure is further equilibrated
after solidification until it remains unchanged [3]. External layers of solid phase on
the left and right sides are manually added to surround the TSVs for application of
strains, which can be controlled by the boundary condition of the PFC model, i.e. the
“penalty term" in Eq. 6.3. The orientations of the grains in the external layers are set
according to their neighbouring grains in the TSV, as highlighted in Fig. 6.1. During
the loading stage, the atoms in these external layers are motivated to move, simulating the application of loading to the TSV. As for the external layer on top of the TSV,
it consists of a single grain with a horizontal orientation, providing with a cover of
solid-phase. The parameters used in the PFC model in this section are set as follows:
(r, n 0 , A, B, q, |v|) = (−1, 0.59, −0.31, −0.14, 1.0, 1.0 × 10
−4
). The parameter v
defines the speed and direction of the motion of atoms in the external layers. Such a
model setting has limitations that in real TSV structures, the cover layer is TiN and the
TSV is surrounded by a diffusion barrier (e.g. Ta, TaN), a dielectric layer (e.g. SiO 2 )
and Si. Recent progress on extension of the PFC models to multi-component and
multi-phase systems offers the opportunity to study such heterogeneous interfaces
in TSVs. The model parameter r = −1 fixes the system temperature to be approximately 700
◦ C, as discussed in [3]: the parameter r is systematically varied and the
corresponding glide velocities of dislocations are recorded, then the temperature is
determined by the relationship between the gliding velocity and temperature, i.e.
u = u 0 e
−G/k B T , as shown in Fig. 6.2. In experimental work, Cu-TSVs are usually
annealed at 400∼450
◦ C, leading to Cu protrusion [2, 7–9]. In our simulation, the
J. Liu et al.
n = n 0 + A
cos(qx) + cos(qy)
+ B cos(qx) cos(qy)
(6.1)
F[n] =
dr
n
2
[r + (∇
2
+ 1)
2
(∇
2
+ Q
2
)
2
]n +
n
4
4
(6.2)
F
[n] = F[n] +
drM(r)
n(r, t) − n s (r(v, t), t)
(6.3)
∂n
∂t
= ∇
2 δ F
[n]
δn
(6.4)
The above equations and parameters are referred to Chap. 5, Sects. 5.2.6.1 and 5.3.2.
The remainder of this chapter is organized as following. First, an example demonstrating the salient features of the model and the protrusion process is given. Then the
protrusion behavior under different external applied mechanical loading, as well as
effects from grain structure, temperature and geometry are discussed. Finally, general
perspectives on TSV protrusion and an outlook for future work are presented.
6.2 Model Setting and an Example of TSV Protrusion
Figure 6.1 illustrates the model configuration of a TSV structure with loading applied.
Note that a trapezoidal TSV is chosen because of the tapered nature of a drilled via [6].
The initial atomic-scale microstructure is generated by placing crystal nuclei inside
the TSV and then initiating solidification. The microstructure is further equilibrated
after solidification until it remains unchanged [3]. External layers of solid phase on
the left and right sides are manually added to surround the TSVs for application of
strains, which can be controlled by the boundary condition of the PFC model, i.e. the
“penalty term" in Eq. 6.3. The orientations of the grains in the external layers are set
according to their neighbouring grains in the TSV, as highlighted in Fig. 6.1. During
the loading stage, the atoms in these external layers are motivated to move, simulating the application of loading to the TSV. As for the external layer on top of the TSV,
it consists of a single grain with a horizontal orientation, providing with a cover of
solid-phase. The parameters used in the PFC model in this section are set as follows:
(r, n 0 , A, B, q, |v|) = (−1, 0.59, −0.31, −0.14, 1.0, 1.0 × 10
−4
). The parameter v
defines the speed and direction of the motion of atoms in the external layers. Such a
model setting has limitations that in real TSV structures, the cover layer is TiN and the
TSV is surrounded by a diffusion barrier (e.g. Ta, TaN), a dielectric layer (e.g. SiO 2 )
and Si. Recent progress on extension of the PFC models to multi-component and
multi-phase systems offers the opportunity to study such heterogeneous interfaces
in TSVs. The model parameter r = −1 fixes the system temperature to be approximately 700
◦ C, as discussed in [3]: the parameter r is systematically varied and the
corresponding glide velocities of dislocations are recorded, then the temperature is
determined by the relationship between the gliding velocity and temperature, i.e.
u = u 0 e
−G/k B T , as shown in Fig. 6.2. In experimental work, Cu-TSVs are usually
annealed at 400∼450
◦ C, leading to Cu protrusion [2, 7–9]. In our simulation, the