The atomic mixing model is a set of partial differential equations.
The current qi(x,f) and q(x,f) and diffusion terms
Di(x,f) are material-dependent functions; their values at x
are dependent on the material distribution in the region [0,x].
The equations are thus highly non-linear in the surface region
and hence progress toward
a solution cannot be achieved through analytic methods.
It is natural to look to computational methods of
solution. The most obvious techniques that could be employed
are the finite element, finite difference or related methods.
In IMPETUS II a finite difference method is employed. However,
it is only necessary to apply such methods over a truncated
range of x and to solve a linearised version of the PDEs
over the remanining region.
At depth in the material structure the current of atoms and the
diffusion are small and hence the
PDEs that govern the atomic mixing model in this region can
be simplified there without significant loss of accuracy.
Substituting the expressions for the currents (2)
and (3) into (1) immediately shows that the governing equations
are of parabolic type and are highly non-linear. They may be classed
as being convection-diffusion equations with the u [(¶qi)/(¶x)] term expressing the convection and
the [(¶qi)/(¶x)] being the diffusion.
The magnitude of the currents are related to the diffusivity function
which is in turn related to the nuclear energy deposition function.
From the form of the equations (4) it is clear that the
nuclear energy function and hence the diffusivities decay with
increasing x. Assuming that the currents are small at depths
greater than xH, the equations (1) can be replaced by
the following approximations:
¶qi(x,f)
¶f
» u(f)
¶qi(x,f)
¶x
(x,f) Î [xH, ¥) ×[0,¥)
(26)
for i=1,2,...,n.
The main feature of the equation corresponding to (26) is
that it is hyperbolic. However, its solution is trivial, it is simply
a translation of q at speed u. The equation represents the
moving coordinate system inherent in the model. The approximation
represents the translation of the distribution toward the
surface with negligible mixing.
The governing PDE is thus apparently parabolic,
it is strongly parabolic in the lower surface
region where the diffusivity is high, but at depths x ³ xH
it is almost completely hyperbolic. A further conclusion from this
is that there is a transition from the parabolic equation
to the equation that is predominantly hyperbolic in the region [0,xH].
The approximations (26) can be regarded as a linearisation
of the original PDEs (1). If a linearised term for the
diffusion is included then the original model can be replaced by
the following approximations:
¶qi(x,f)
¶f
» u(f)
¶qi(x,f)
¶x
+Di(x,f)
¶2 qi(x,f)
¶x2
(x,f) Î [xP, ¥) ×[0,¥)
(27)
for i=1,2,...,n. In effect the linearisation (27) models the
minimal diffusion and translation due to surface recession in the
region [xP,¥).
Let the point xR demarcate the depth of the mixing window
and let it be fixed throughout the computational experiment.
The particular value of xR is not crucial, it is set so
that it is safe to assume that there is minimal mixing beyond it.
In IMPETUS II the PDEs of the atomic mixing model (1) are
generally solved using the FDM in a region [0,xP]
where 0 £ xP £ xR. The value of xP is allowed to
vary with f and it depends on the deposition
functions and nature of the
material distribution in [0,xR].
The first principle in choosing xP is so that the energy and
range functions at the particular dose take negligible
values in the range [xP,¥]. However, this principle
is over-ridden if there are insufficiently diffuse interfaces
or thin layers in the domain, in which case xP is chosen to
be the largest value for which the cocentration functions
are sufficiently smooth in [0,xP]. The reason for this is that
it is felt that an exact solution to the linearised equation
will give a more accurate and more physically realistic
result than the numerical solution to the true model in
the neighbourhood of a sharp interface.
In the region [xP,¥), methods based on approximation
(27) are employed. Sharp interfaces are smoothed to
error functions through the solution of (27). In the
region [xR,¥) this is termed linear pre-diffusion
(PRE-D). The primary atoms may range into the region
[xP,xR] and hence a modified method, termed linear post-diffusion
(POST-D) is applied there. Figure 6 illustrates the strategy
employed in advancing the solution through each dose step using
the finite difference method.
Figure 6. Illustration of overall computational strategy.
Other special techniques
are employed in IMPETUS II to control the stepsize
and the grid size to give the method limited adaptive capabilities.
The special case of steady state solutions (that
is when the original material structure contains
wide homogeneous layers) benefit from special
treatment. Details on the methods employed in IMPETUS II
for carrying out these tasks are given in reference [6].
