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# 5. Modeling of Oxidation

Thermal oxidation of silicon is one of the most important steps in fabrication of highly integrated electronic circuits, being mainly used for efficient insulation of adjacent devices. Those parts of silicon which shall not be oxidized are masked by a layer structure of silicon nitride that is not effected by an oxidation. Exposed parts of already existing silicon oxide are penetrated by the oxidant (oxygen diffusion), which reacts finally at the interface of silicon and silicon dioxide to form new dioxide. This chemical reaction consumes silicon, and due to the increased volume of the newly generated dioxide the old dioxide layer is lifted up.

From the mathematical point of view the problem can be described by a coupled system of partial differential equations:

• diffusion equation describing the penetration of the oxidant through the existing silicon dioxide
• chemical reaction describing the transformation of material due to the storage of oxygen molecules into the silicon layer reacting to silicon dioxide
• displacement of the oxide layer usually modeled as an elastic, visco-elastic or viscous flow. Due to the unknown motion of the interface between silicon and silicon dioxide this leads to a free boundary problem.

During the last decades several approaches for simulation of two-dimensional local oxidation of silicon have been published. They are all based on the fundamental work of Deal and Grove [Dea65] who described the phenomenon in one dimension (see also Chapter 2). The common feature of all approaches is that they decouple the diffusion and displacement problem into a sequence of quasi-stationary time steps. The calculated results of the diffusion are used to extract the necessary conditions for the free boundary problem. The displacement computation yields the geometry of the newly generated silicon dioxide range for the next time step.

All published approaches can be classified essentially into three groups:

• Mapping the silicon dioxide domain onto a simple numerical domain (usually a rectangle) is the first approach. Although conceptionally easy it is limited to comparably `simple' geometries like bird's beak structures [Lor85].
• The second approach uses a boundary element approach for diffusion and displacement [Mat83] which suffers from its restriction to essentially linear problems whereas there is much physical evidence that oxygen diffusion and chemical reaction depends on mechanical stresses in silicon dioxide rendering the overall problem nonlinear even in its quasi stationary steps.
• The most general approach models the domain of computation by finite elements [Pon85]. The main disadvantage of this method is the necessity to update the finite element mesh after each time step due to the newly produced silicon dioxide. It turned out to be a quite nasty task to preserve the grid quality during oxidation. Even `tricky' procedures are often not able to manage this problem in an acceptable manner, especially in three dimensions. Therefore most published finite element approaches remesh the silicon dioxide domain after each time step, which is very costly and a source of latent errors especially if the physical model (e.g. visco-elasticity) makes it necessary to interpolate tensors like stresses or strains to a new finite element mesh.

Moreover, none of the published approaches seems to be able to treat problems where not only the geometry but also the topology of the silicon dioxide range changes.

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