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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.
Next: 5.1 Mechanical Models
Up: PhD - Mustafa Radi
Previous: 4.8.4 Three-Dimensional Temperature Distribution
Mustafa Radi
1998-12-11