2 Analytical Solution of Korhonen's Equation

Korhonen's equation can be analytically solved for different cases of initial and boundary conditions [51,7,80,83], however from the practical point of view, the most important solution is for the situation where the vacancy flux is blocked at the both ends of a finite line, i.e.

$$\quad J_V(0,t)=J_V(-l,0)=0,$$ (232)

on the segment $[-l,0]$, assuming the initial vacancy concentration to be $C_V(x,0)=C_V^{eq}$. In this case, the solution of (4.21) is given in [83] as,

$$\nu(\xi,\zeta)=\frac{C_V(x,t)}{C_V^{eq}}=A_0-\sum_{n=1}^{\infty}A_n (-B_n\zeta+\alpha\xi/2),$$ (233)

where,

$$A_0=\frac{C_V(x,\infty)}{C_V^{eq}}=\frac{\alpha \text{exp}(\alpha\xi)}{1-\text{exp}(-\alpha)},$$ (234)

is the steady-state solution and,

$$A_n=\frac{16n\pi[1-(-1)^n\text{exp}(\alpha/2)]}{(\alpha^2+4n^2\pi^2)^2}\Bigl(\text{sin}(n\pi\xi)+\frac{2n\pi}{\alpha}\text{cos}(n\pi\xi)\Bigr),$$ (235)

$$B_n=n^2\pi^2+\alpha^2/4.$$ (236)

For the sake of simplicity, in the equations (4.23)-(4.26) we used the substitutions,

$$\xi=\frac{x}{l},\quad\zeta=\frac{D_Vt}{l^2},\quad\alpha=\frac{Z^*eEl}{kT}.$$ (237)

Figure 4.2: Build-up of vacancies at the blocking boundaries for $\zeta =0.01$ (dashed line), $\zeta =0.05$ (points), $\zeta =0.2$ (full line).

Figure 4.3: Build-up of hydrostatic pressure at the blocking boundaries for $\zeta =0.01$ (dashed line), $\zeta =0.05$ (points), $\zeta =0.2$ (full line).

For $\alpha=4$ in Figure 4.2 and Figure 4.3 the distribution of the normalized vacancy concentration $\nu$ and the reduced hydrostatic pressure,

$$\eta=\frac{\Omega\sigma}{kT},$$ (238)

are presented for different electromigration stressing times. The stress build-up continues until the electromigration driving force is counter-balanced by intrinsic stress in the line. That is the case in Figure 4.3 for $\zeta =0.2$ where the stress profile is a straight line. The condition for the stress-electromigration equilibrium is,

$$\frac{\Omega}{kT}\frac{\partial\sigma}{\partial x}= \frac{Z^*e\rho J}{kT} .$$ (239)

Equation (4.29) is discussed in more detail in [48]. If for the given operating conditions the stress-electromigration equilibrium is reached before a critical stress threshold is build, no damage is produced and the interconnect is virtually ``immortal'' [48,49,47].