Thermal oxidation of
silicon is one of the most important steps in the fabrication of highly
integrated electronic circuits, and is mainly used for efficient
isolation of adjacent devices from each other.
If a surface of silicon has contact with an oxidizing atmosphere, the
chemical reaction of the oxidant (oxygen or steam) with silicon results
in silicon dioxide. This reaction consumes silicon, and the newly
formed
silicon dioxide has more than twice the volume of the original silicon.
If a silicon dioxide domain is already existing, the oxidants diffuse
through the oxide domain and react at the interface of oxide and
silicon to form new oxide so that the dioxide domain is penetrated.
Thermal oxidation is a complex process where the three subprocesses
oxidant diffusion, chemical reaction, and volume increase occur
simultaneously. The volume increase is the main source of mechanical
stress and strain, and these cause displacement.
From the mathematical point of view, the problem can be described by a
coupled system of partial differential equations, one for the diffusion
of the oxidant through the oxide, the second for the conversion of
silicon into silicon dioxide at the interface, and a third for the
mechanical problem of the silicon-silicon dioxide-body, which can be
modeled as an elastic, viscoelastic, or viscous body.
For a realistic and accurate oxidation simulation, the three
subproblems
should be coupled. However, most oxidation models decouple them into a
sequence of quasi-stationary steps. Our model takes into account that
the diffusion of oxidants, the chemical reaction, and the volume
increase occur simultaneously in a so-called reactive layer. This
reactive layer has a spatial finite width, in contrast to the sharp
interface between silicon and silicon dioxide in the conventional
formulation. The oxidation process is numerically described by a
coupled system of equations for reaction, diffusion, and displacement.
In order to solve the numerical formulation of the oxidation process,
the finite element scheme is applied.
During the last year a viscoelastic mechanical model has been designed
and implemented which is more realistic than the elastic one, because
both silicon oxide and silicon nitride have a viscoelastic behavior.
The current work deals with the so-called stress-dependent oxidation,
because the oxidant diffusivity and the intensity of the chemical
reaction are significantly influenced by the local stress values in the
material. Because of this fact the resulting geometry of the silicon
dioxide strongly depends on the stress distribution.
|