13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- Metal line 1 Metal line 3 Via Current flowj Metal line 2 Reservoir Reservoir Figure 1. Multilayer interconnection with reservoir structure 2. Simulation method The governing parameter for EM damage is used for constructing the numerical simulation [2]. The parameter is given by formulation of divergence of atomic flux due to EM. The atomic flux vector J is represented by Eq. (1). where N is atomic density, D0 a prefactor, k Boltzmann’s constant, T the absolute temperature, Qgb net activation energy for atomic diffusion, κ the constant relating the change in stress with the change in atomic density under restriction by passivation, ΝT the atomic density under tensile thermal stress σT, N0 the atomic density at a reference condition, Ω the atomic volume, Z * the effective valence and e the electronic charge. ρ is the temperature-dependent resistivity. Symbols j* and ∂N/∂l are the components of the current density vector and atomic density gradient in the direction of J, respectively. In Eq. (1), the back flow of atoms due to the stress gradient and the effect of the stress generated in the metal line on diffusivity are taken into account. Grain boundary diffusion is assumed as dominant diffusion mechanism in the simulation, because wide Cu lines covered with passivation layer were supposed. According to literature [4, 5], in wide Cu interconnects, grain boundaries become preferential EM paths rather than lattice and interface diffusions. Sasagawa et al have introduced a grain texture model for calculating atomic flux divergence [6]. So we used the governing parameter for EM damage based on the model even for Cu lines. Considering atoms going in and out at a unit rectangle, atomic flux divergence in polycrystalline line is formulated as given in Eq. (2). ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ Ω∂ − ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + Ω − − Ω − = ∗ l N N Z e j kT Q N N N kT ND T T gb 0 * 0 0 exp κ ρ σ κ J (1)
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