Physical models for the oxidation process are based on the works of Deal and Grove [Dea65], Chin et al. [Chi83], Kao et al. [Kao88] and Rafferty [Raf91].
The diffusion of the oxidant is governed by (4.1-1), where
is the oxidant concentration and 
is the oxidant's diffusion
coefficient in 
. The boundary conditions follow the Deal-Grove
model (4.1-2), with 
the interface reaction coefficient,
the mass transfer coefficient and 
the oxidant solubility in
.
For process temperatures above 
viscous flow of oxide is commonly
assumed [Kao88], [Umi89]. The slow incompressible (
) flow of oxide is expressed by a creeping flow
equation (4.1-3) in which the viscous force is balanced with
the pressure gradient. There 
denotes the oxide viscosity, 
is
the oxide velocity and 
is the hydrostatic pressure.
The following boundary conditions apply to (4.1-3). At the
interface the velocity is determined from the net volume
expansion 
, i.e. the ratio of consumed 
to produced
, and 
is the number of oxidant molecules in a unit volume of
oxide (4.1-4). At the free oxide surface and at an eventual
nitride/oxide interface pressure equilibrium is assumed.
Stress effects on the oxide growth are usually considered by
stress-dependent surface reaction , oxidant diffusivity , and
viscosity , where 
, , and 
denote the normal stress,
hydrostatic pressure and shear stress, respectively. The volumes 
, 
and 
, despite their physical meaning, and 
are used as
fitting coefficients.
In this model, in the one-dimensional case the motion of the oxide is
stress-free, uniform and perpendicular to the surface. According to
equations (4.1-1), (4.1-2)
and (4.1-4) Deal and Grove ended up with a relation for the oxide
thickness 
(4.1-7).
is referred to as parabolic growth rate, and 
is the linear growth
rate. For these two parameters there exist well established models for
dry and wet oxidation including pressure dependence, chlorine dependence and
dry oxide kinetics [Ho83a]. From 
and we derive the stress-free
parameters 
and 
, assuming the mass transfer in the gas phase
being effective, i.e. 
is negligible compared to 
.
We have implemented the models (4.1-9) and (4.1-10) for
and , respectively. 
and 
are the intrinsic
orientation dependent linear and parabolic rate constants, by 
we
take the oxidant pressure dependence into account, 
describes
the chlorine dependence, and 
accounts for the anomalous fast
growth of thin oxides (
).
Tables 4.1-1 to 4.1-2 summarize the implemented
parameter values. Most of the parameter values are modeled by Arrhenius laws
(
).
These parameter models for and are used on the one hand to calibrate
two-dimensional oxidant flow simulations, and on the other hand they are
the basis for one- and two-dimensional analytical oxide shape calculations.
The one-dimensional oxide growth increment 
is calculated
from the actual oxide thickness , the time increment 
and
the actual parameters and using (4.1-11), a numerically
uncritical solution of (4.1-7).
As two-dimensional analytical model we have implemented (4.1-12)
for the amount of consumed silicon 
to cover the four basic LOCOS
cases shown in Figure 4.1-1. The lateral positions 
and 
are a left and right mask edge, respectively, and the fit
parameter 
describes the ratio of lateral to
vertical oxide growth.
