During the oxidation process a chemical reaction converts silicon into SiO and the nearly sharp Si/SiO-interface moves from the surface into the silicon substrate. The formed SiO has more than twice of the original volume of silicon, which is the main source of stress and displacements in the materials. The oxidation process depends on four parameters: the used oxidant species, the temperature and the pressure in the furnace, and the crystal orientation of the silicon substrate. Since the oxide growth rate is strongly temperature dependent for all species, in practice the oxide growth is mainly controlled only by the temperature.

An enhanced three-dimensional oxidation model was developed which is based on a diffuse interface with a reaction layer. This model takes into account that during oxidation the oxidant diffusion, the chemical reaction, and the volume increase occur simultaneously. The diffuse interface concept avoids the drawbacks of the moving boundary problem, complicated mesh algorithms, and enormous data update. Therefore, the enhanced model enables the simulation of even complex structures with a moderate demand on computer resources. Since SiO and SiN show visco-elastic behavior, besides an elastic also a visco-elastic formulation with a so-called effective shear modulus was introduced for the mechanics.

The effects of thermal oxidation of doped silicon material were described. Because of the built-up Si/SiO-interface segregation leads also to a redistribution of the dopands. For modeling this redistribution the five-stream diffusion model from Dunham was introduced.

In order to solve the mathematical formulation numerically, the finite element method was applied. The finite element discretization with tetrahedrons for the oxidant diffusion, the -dynamics, and the mechanics was explained in detail. The principle of the assembling in order to built up a complete equation system and the handling of Dirichlet boundary conditions and mechanical interfaces was described.

The enhanced oxidation model was implemented in the in-house process simulation tool FEDOS. The architecture and main components of FEDOS were depicted and the simulation procedure for oxidation was explained. The mesh plays a key role for simulation, because the number of finite elements is always a compromise between accuracy and simulation time. This means that an acceptable accuracy should be reached with as small as possible number of elements. The most effective strategy found is to use a static mesh. It was demonstrated that in critical regions, e.g. along the edge of a mask, the mesh should be finer than in the rest of the structure. Due to the diffuse interface concept of the enhanced oxidation model, the procedure for a physical interpretation of the displayed simulation results with a sharp Si/SiO-interface was described. For practical applications of the oxidation model, the simple but effective model calibration with the surface oxidant concentration was shown.

Stress has a significant influence on the oxidant diffusion and the chemical reaction, and so also on the resulting oxide growth rate. A universal stress calculation concept for the simulation of stress dependent oxidation, where the stress in the structure is determined in two steps, was presented. The enhanced oxidation model was applied to simulate two three-dimensional (fabricated) structures. It was demonstrated that only when the stress dependence of the oxidation process is taken into account, the simulation results agree with the real physical behavior.

In copper interconnects stress is an important promoting factor for electromigration, which can lead to void formation and to failure of the interconnect. The procedure for the simulation of thermal stress in a representative interconnect structure was described. First, a electro-thermal simulation was performed in order to obtain the temperature distribution in the interconnect layout due to Joule self-heating. With this temperature distribution the thermal stress can be simulated. The reason for thermal stress are the different thermal expansion coefficients of the respective materials in the adjacent layers. The highest stress values in the interconnect structure were predicted at the bottom of the vias. Therefore, this is the most critical region for electromigration.

The effects of intrinsic stress in deposited thin films were discussed. A negative effect of stress in free-standing MEMS structures was demonstrated with the unwanted deflection of cantilever. For a linear stress gradient the deflection of the cantilever increases quadratically with the length. A number of sources which can generate tensile or compressive stress in the film were described. The whole intrinsic stress comes from microscopic effects like grain dynamics. Macroscopic mechanical formulations for the different intrinsic stress sources, which describe the stress development due to the microscopic effects, were listed. A methodology which can predict a qualitative strain or stress curve over the film thickness was developed. This methodology was applied to determine qualitatively the strain curve for a deposited multilayer SiGe film. The found strain curve was calibrated and applied to simulate the stress and deflection in a fabricated cantilever structure.

Ch. Hollauer: Modeling of Thermal Oxidation and Stress Effects