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7.1 Summary

I have presented new and powerful mesh refinement techniques for three-dimensional unstructured tetrahedral meshes. Based on tetrahedral bisection three sophisticated refinement strategies, namely the layer refinement, gradient refinement, and the Hessian refinement method have been derived and investigated. Since anisotropy is more and more a must for any modern three-dimensional discretization scheme, a general technique for introducing a metric tensor based refinement algorithm which allows anisotropy to be taken into account was developed and presented in several simulations.

Achieving a reasonable accuracy is one of the main goals in numerical analysis, since TCAD tools are supposed to be predictive and the results should be reliable. To tackle this issue an error estimator based on the idea of well-known gradient recovery estimators for finite elements combined with an attribute driven refinement mechanism was developed. This combination allows to introduce refinement steps during a simulation on demand and increases the overall accuracy.

With the introduction of copper based lines in semiconductor interconnect stacks electromigration plays a more important role than ever. The formation of voids and the movement of these traps is difficult to model. A lot of investigation related to this topic has been carried out at the Institute for Microlectronics in two spatial dimensions. The extension to three-dimensional unstructured tetrahedral meshes has been developed. Also a new mechanism with a combined mesh refinement and mesh coarsement strategy was invented and tested on a typical interconnect structure. This refinement/coarsement mechanism follows the moving void on its path through the interconnect line and guarantees a good spatial resolution of the interface between metal and the void.

Transistor dimensions reached a point at which first-order assumptions about physical effects broke down. So far device simulation was primarily based on the classical drift-diffusion or the hydrodynamic equations. These equations represent macroscopic balance equations between electron and hole concentrations versus the corresponding current densities. In the sub $ 0.1  \MR{\mu m}$ regime these classical device equations are no longer predictive. One way to overcome this problem is to use full band Monte Carlo simulations. For such simulations a numerical representation of the band structure in the unit cell of the reciprocal lattice, the so-called first Brillouin zone, is used to capture the dependence of the carrier energy on the wave vector. A sophisticated unstructured meshing approach of the first Brillouin zone was developed and compared to a structured approach. It was shown that the unstructured approach gains more accurate solutions with fewer elements compared to the structured approach, which has an important impact on a significantly reduced memory consumption.


next up previous contents
Next: 7.2 Outlook Up: 7. Summary and Outlook Previous: 7. Summary and Outlook

Wilfried Wessner: Mesh Refinement Techniques for TCAD Tools