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5.2.3 Compact Model for Lifetime Estimation

In order to derive an expression for the solder bump lifetime, it is important to define a failure condition. While the ultimate failure condition of any interconnect is an increase of its resistance, the physical condition which must be fulfilled for an initialization of the early phase of failure in solder bumps (i.e. void nucleation) is the attainment of a threshold stress.

An analytical solution of Korhonen's model [93] describes the time evolution of the stress build-up due to electromigration in a 1D structure as follows

\[\begin{equation} \sigma(x,t) = \cfrac{2|Z^*|e\rho j \pi}{\Omega_\text{a}}\sqrt{\cfrac{kt}{\pi}}, \end{equation}\] (5.2)

where Z^* is the effective valence, e is the electron charge, ρ is the electrical resistivity, Ω_a is the atomic volume, and k is a constant. The mechanical stress along the 1D interconnect increases until it reaches a threshold value, which is a usual condition for electromigration void nucleation. In the case of a 3D geometry, more vacancies are available in the cross section of the structure and σ is proportional to R². By following equation (5.2) the compact model describing the time needed to reach the threshold stress σ_thr value in a solder bump, which depends on its radius, is obtained as follows

\[\begin{equation} TTF = \cfrac{A\sigma_\text{thr}^2}{j^2 R^4}, \end{equation}\] (5.3)

where the constant A contains several material properties related to the solder bump.

The time evolution of the stress built-up due to electromigration for a total of six applied current densities is shown in (5.14).

Figure 5.14: Time evolution of the maximum tensile stress in the analyzed structure for six applied current densities. The threshold stress σ_thr for void nucleation is shown in the x-axis.

As expected, at the initial stage the maximum stress exhibits linear growth with time. Subsequently, the stress increases in proportion to the square root of time, until it reaches the threshold value for void nucleation [93]. The value of the stress threshold σ_thr is 7.7MPa and is obtained from equation (3.72) [63] by using values of the initial adhesion-free patch R_p of 10nm, the interfacial free energy of tin γ_Sn of 0.0545N/m, and the critical contact angle θ_c of 45°. At higher current densities the threshold stress for void nucleation is obtained in less time than at lower current densities. This implies that under higher current density stress conditions, the electromigration-induced vacancy transport at the bump/UBM interface is more efficient, since more vacancies are available in a cross section of the bump. Lifetimes are extracted from the stress/time curves and are fitted to the compact model for lifetime prediction presented in equation (5.3). The results obtained from simulations are in good agreement with the compact model. Furthermore, similar to Black's original empirical work [12], the failure time in equation (5.3) follows a j_-2 dependence, which indicates that the void nucleation is the dominant mechanism of electromigration failure in the bump. Therefore, the development of a compact model for the prediction of the void nucleation time is beneficial for the estimation of the TTF for solder bump technologies.

Figure 5.15: TTF depends on the current density. The solid line indicates the fit according to the compact model for lifetime prediction presented in equation (5.3).