When the atoms at the cathode side are pushed to the anode side by electrons, voids and extrusion are formed at the cathode side and anode side, respectively. At this time, because atoms aggregate, the compression stress will generate at the anode side. On the other hand, because Sn atoms leave the cathode side, tensile stress generates here. We can use an equation to describe the phenomenon.
dx
where C is the atomic concentration, D is the diffusion resistivity, Z* is the effective charge number of ions, E is the electric field, K is the Boltzmann’s constant, T is the absolute temperature, σ is the maximum stress, Ω is the atomic volume. The first term represents the momentum transferred by electron wind force, when electrons hit the atoms. The second term represents the back stress generated from the gradient. Under the circumstance, the difference of the stress between the cathode side and the anode of the stripe would create a gradient of chemical potential,
Ω
=
∇ dx dσ
µ , and this gradient would have influence on the drift velocity.
So, if stripe length is short enough to cause larger gradient be able to balance the strength made from the electron wind force, electromigration would have no effects on the stripe and no depletion would occur and neither does extrusion. At this time, the length of the stripe is called critical length, and the equation can be written as:
ρ
The critical length is the shortest stripe length, below which no electromigration would have influence on the stripe.
To obtain the critical length, we applied a constant current density to different stripe length. In this study, the electromigration behavior was investigated at different stripe lengths (330 µm, 73 µm, 23 µm) stressed at the current density of 1 × 105 A/cm2 at room temperature. The stripe length is made by etching, so it is difficult to have a regular length. The depletion area are about 820 µm2, 1020 µm2 and 380 µm2 for the stripe lengths of 330 µm, 73 µm, 23 µm, respectively; the width of the depletion area are 42 µm, 69 µm and 66 µm for the stripe lengths of 330 µm, 73 µm, 23 µm, respectively and the stressing time are all 53.5 h. So, the drift velocities are 1 × 10-4, 7.7 × 10-5 and 2.2 × 10-5 µm/s for the stripe lengths of 330µm, 73µm, 23µm, respectively; a shown in figure 4.10.
It is found that the drift velocity decreased as the stripe length is decreased. So, we can have the linear relation between drift velocity and the reciprocal of stripe length as shown in figure 4.11. The critical length was estimated by extrapolating the plot of drift velocity against the reciprocal of stripe length to zero drift velocity.14 The extrapolated value was 18 µm. That means when the stripe length is below 18 µm, no electromigration damage would have influence on the stripe, because when the stripe is shorter than 18 µm, the back stress is high enough to be able to balance the electron wind force.
FIG. 4.10. Plan-view SEM images of the cathode side stressed under the current density of 1 × 105 A/cm2 at room temperature for 53.5 h on different stripe lengths. The drift velocities are (a) 1 × 10-4 µm/s for 330 µm; (b) 7.7 × 10-5 µm/s for 73 µm; (c) 2.2 × 10-5 A/cm2 for 23 µm.
FIG. 4.11. The plot of drift velocity against the reciprocal of the stripe length. The critical length was estimated to be 18 µm.
Chapter 5 Conclusions
Edge displacement method was used in this study to measure the electromigration properties of pure Sn stripes deposited on Ti film. Drift velocities and other important electromigration parameters of pure Sn have been investigated at R.T., 50 ℃, 75 ℃ and 100 ℃ .
We found that when the stressing temperature is constant, faster drift velocities we would obtain under higher current densities. Besides, under the same current density, the higher testing temperature, the faster drift velocity. The threshold current densities were 1.93 × 104, 9.65 × 103, 9.57 × 103 and 7.93 × 103 A/cm2 for R.T., 50 ℃, 75 ℃ and 100 ℃, respectively. The activation energy of 0.32 eV was obtained due to the smaller grain sizes of the Sn film. The products of DZ* were 1.95 × 10-10, 4.84 × 10-10, 1.27 × 10-9 and 1.99 × 10-9 cm2/s for R.T., 50 ℃, 75 ℃ and 100 , ℃ respectively.
In addition, the measured critical length of Sn film was 18 µm at R.T. These results are very important for electromigration in Pb-free solders, because their matrix consists of almost pure Sn.
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