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
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.