Modeling of diffusional creep and stress relaxation in copper grains during manufacturing of microelectronic integrated circuits
The finite element technique was developed to study diffusional creep and stress relaxation in Cu grains with several atomic monolayers thick grain boundary region of enhanced diffusivity. The model was motivated by the need to study nanoscale back-end interconnect structures of microelectronic circuits. These structures have the length scale that does not conform to the assumptions of classical dimensional theories of diffusional creep. Both diffusion and elasticity governing equations are considered in the coupled formulation of mass flow and stress analysis. Vacancy concentration field in the grains subjected to external load is coupled to stress field through diffusional creep strains. The formulation has been implemented in the commercially available finite element software package MSC.Marc. We validated the model for the case of stress relaxation in one-dimensional grain array by comparing the finite element simulations to the predictions of classical Nabarro-Herring and Coble theories. The numerical results show good correspondence to analytical predictions, suggesting that this model may be used to predict diffusive stress relaxation in more advanced systems of practical importance, such as Cu interconnects at elevated temperatures. We have used our model to study the effect of grain size on creep rate in a polycrystal under external load. The approach has been applied to study the stress relaxation in a typical Cu-Ta-dielectric structure subjected to thermal loads. To improve the computational efficiency of the diffusional creep modeling, we developed the numerical technique of equivalent viscoplastic finite elements. This approach was found to improve the computational efficiency by reducing the coupled elasticity-mass flow problem to the equivalent mechanical creep analysis. The predictions of the equivalent element viscoplastic model showed good correspondence to the stress relaxation results obtained with coupled elasticity-mass flow FEA approach.
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