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5. Summary and Outlook

In this thesis I have discussed the numerical concepts, methods and algorithms for solving partial differential equations eligible for describing physical phenomena taking place in modern interconnect and process technology. Special focus was put on the nonlinear PDE systems and the PDE system including the double obstacle problem. These problems represent a challenge for numerical implementation, since their discretization and linearization demands complex procedures in order to ensure physical soundness of the obtained simulation results and overall accuracy of the calculations.

The applied solution methods are based on the finite element method and as such strongly dependent on the mesh characteristics of the simulation regions. Important connections between the mesh and applied numerical schemes are investigated. The conditions for optimal space and time discretization control are derived and demonstrated.

In the second chapter I have introduced the basic concepts of the finite element method. The solving procedure for the general time dependent, nonlinear system of partial differential equations is also presented.

The third chapter offers a physically well founded introduction in the advanced diffusion models used in modern TCAD solutions. This introduction is subsequentially used as basis for the detailed discussion of the numerical handling several important models. The chapter is concluded with two simulation examples which demonstrate physical soundness of the applied models and correctness of the developed numerical schemes.

The full spectrum of modern modeling concepts for electromigration TCAD solutions is discussed in the fourth chapter. The diversity of the background physical phenomena and their mathematical formulations is presented, thus motivating an appropriate numerical method which is subsequentially discussed. The models of electromigration promoting factors such as electrical current density, thermal gradients, and mechanical stress are integrated in the numerical procedure together with material transport equations. The advantage of such an approach for predictive electromigration simulation is illustrated with several simulation examples.

The crucial problem of void evolution is simulated using a diffuse interface model. The governing diffuse interface equation for the order parameter coupled with the Laplace equation for the electrical field is solved using the finite element method. A dynamically adapted grid is maintained by a refinement-coarsening algorithm controlled by position, curvature, and width of the simulated void-metal interface, which distributes the mesh density in such a way that it allows an efficient simulation of evolving voids through large portions of a complex interconnect geometry. Due to high electrical current gradients in the proximity of the interconnect corners and overall asymmetry of the electrical field, voids exhibit a specific faceting which was not observed in the case of straight interconnect geometries. The presented method is well suited for long time prediction of the resistance change due to electromigration during the interconnect life time. The applied diffuse interface model extends readily to incorporate additional physical phenomena such as anisotropy, temperature variations, and bulk and grain boundary diffusion.

In order to include all relevant physics there is an ultimative need for three-dimensional simulation. For the physical phenomena discussed in this thesis the comprehensive and sofisticated models are to a large extent already available. However, numerical realization of these models, applicable for the efficient and accurate three-dimensional simulation, frequently poses a huge problem.

First there is a need for high quality mesh generation capable to handle complex multi-segment three-dimensional geometries with non-planar interfaces and surfaces and efficient mechanisms to control these meshes dynamically in connection with numerical algorithm.

Error estimators should be developed which correspond to the specific structure of PDE equations appearing in the given diffusion models and these estimators should be used to control local mesh adaptation and time stepping.

Each PDE problem handeled here reduces to a system of nonlinear algebraic equations and sometimes to tackle the problem the standard Newton's scheme is instable and inefficient expecially for problems which include complex bulk and interfacial models. In such cases an improved, problem adjusted algorithm, should be introduced and applied. The applicability of high order implicit Runge-Kutta methods should be investigated and appropriate solutions integrated in simulation tools.

Considering the electromigration problem, the main emphasis should be put to extend the numerical algorithm for the diffuse interface model to three-dimensional simulations. The approaches for including the different self-diffusion paths in the overall simulation scheme should be studied. This problem also includes efficient mechanisms for the local mesh adaptation.

In the future efforts on integration of the implemented tools with other tools for interconnect and process TCAD developed at the Institute for Microelectronics should be intensified in order to make available a complete TCAD solution.

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Next: A. Vacancies in Solids Up: Dissertation Hajdin Ceric Previous: 4.6 Simulation Results

H. Ceric: Numerical Techniques in Modern TCAD