5 Concentration Correction and Material Conservation
As a result of the numerical error in the FDM, the sum of the
concentrations drift from unity and some of their values
may be negative. There accumulated effect of the dose-stepping
method can also result in a significant loss or gain of
material.
IMPETUS II employs a correction method and a
conservation method. The main philosophy of these methods is
that their alteration of the material distribution arrived at
by the finite difference method should be just sufficient to
bring them within the bounds of the natural laws,
transgressions in the natural laws are resolved as locally as possible
and the alteration must not create oscillations in the solution.
The numerical approximations inherent in the finite difference
method have the consequence that
solution concentrations after each dose step will generally
sum to something other than unity at all grid points and, at some grid points,
the concentration will take a negative value. The most straightforward
method of correcting the solution is as follows:
for i=1,2,...,n do
[^(q)]ij ¬ max([^(q)]ij,0)
for i=1,2,...,n do
[^(q)]ij ¬ [([^(q)]ij)/(si=1n [^(q)]ij)]
where the original qij for i=1,2,...,n, j=1,2,...,N are the
approximations to the concentrations resulting from the FDM.
This is the method employed in the original IMPETUS. The main
criticism of this method is that material is effectively added and removed
arbitrarily, any tendency to conserve material inherent in the finite
difference approximation (that is when summing all positive and
negative conventrations) to the mathematical model is lost.
If it is assumed that there is a tendency of the FDM to conserve material
in [0,xR] then the correction method must be such that the
the total i-material is almost unaltered.
The following method was developed to correct the [^(q)]ij
and without altering the total material. In the method, the correction
starts at j=1 and progresses to j=N. At any stage j,
gi is used to represent the 'gain' in the i-material as a result
of the correction method.
for i=1,2,..,n
gi ¬ 0
for j=1,2,..,N
for i=1,2,..,n
[^(q)]*ij ¬ [^(q)]ij
for i=1,2,..,n
[^(q)]ij ¬ max([^(qij)]-gi,0)
for i=1,2,..,n
[^(q)]ij ¬ [([^(qij)]-gi)/(Si=1n [ [^(qij)]-gi])]
for i=1,2,..,n
gi ¬ gi + [^(q)]ij - [^(q)]*ij
On completion of the method the [^(qij)] values have clearly
been corrected. The gi will generally be non-zero on completion of
the method so that the correction method does not exactly conserve
material. However, the gi is simply an estimate of the material
gain(loss) and the estimated gain(loss) of any material in this method will
generally be minute in comparison to the that in the earlier correction
method. This latter method is employed in the IMPETUS II code.
At any dose the conservation law requires that the material in the
structure, and the sputtered material are equal in quantity
to the material of the original structure and the total bombarding
ions. More precisely, the law requires that the material in any region
of the material at
any dose f2 is equal to the material at an earlier
dose f1, plus the material that has flowed into
the region over that dose interval minus the material that has
flowed out.
The determination of the quantity of material requires integrations
of the yield-dose history and the concentration depth then
numerical integration methods need to be employed and hence we
must be mindful that only an approximation to material quantities
can be made. Hence an exact conservation of material cannot be
carried out. Generally, there is a marginal gain or loss in materials
as a result of the computational solution.
The flow fi(x,f) of i-particles towards the surface
at depth x and dose f is given by
fi(x,f) = (q(x,f)+ u(f)) qi(x,f) (x,f) Î [0,¥) ×[0,¥) .
(14)
The flow of particles through the surface is precisely the yield, so
that yi(f)=fi(0,f).
Conservation of material requires that in any subregion [x1, x2] Í [0,¥) and over any
dose interval [fA,fB] Í [0,¥) the i-material
present in [x1, x2] at dose fB must equal the i-material
present at dose fB with any net flow of i-material
that has flowed into the region.
That is the following equation must hold:
ó õ
x2
x1
qi(x,fB) dx =
ó õ
x2
x1
qi(x,fA) dx +
ó õ
fB
fA
(fi(xA,f) -fi(xB,f)) df
(15)
As discussed earlier, we cannot necessarily expect
the above equation to hold after the application of the FDM
In IMPETUS II it is assumed that the linear diffusion methods of the
previous section effects a negligible gain or loss of material
and the materials are accounted for only in the domain of the FDM, after
the completion of every 2 d dose step.
Let [x1, x2] Í [0,xR] then the gain in i-material
gi over the dose step f* ® f*+ 2 d
is approximated as follows:
gi » \sf simp[
ó õ
x2
x1
qi(x,phi*+d) dx ] -\sf simp[
ó õ
x2
x1
qi(x,f*) dx ] - 2 d( fi(xA,f) -fi(xB,f))
where simp represents a Simpson rule approximation.
Conservation of i-material can now be effected by a proportional decrease
(increase) to correct the gain(loss)
in the discrete qi(x,f*+d) values.
The technique for conserving material could theoretically be applied
across any subrange of [0, xR]. However, if it is applied over every
region [(j-1)D, jD] then local numerical errors can be
magnified. Equally, if it is applied over a wide range then the
redistributive effects of the technique tends to shift material from
part of the region to another in practice. The method finally adopted
in IMPETUS II
was that of evenly dividing the domain of the FDM into its subranges
[x1,x2] and, if there are interfaces within the
region, a division of the region is placed there. This latter
point ensures that the redistributive effects of the method
are minimised.
The conservation technique is independently applied in each sub-range.
In IMPETUS II the conservation/correction method is applied after every
d dose step. Note that it is difficult to satisfy all
the physical constraints simultaneously.
Note also that in the conservation method,
the total i-material in the region (for each i) is not precisely obtained
and hence it is infeasible to attempt to conserve material exactly.
In the IMPETUS II code the FDM solution is processed by first
applying the
correction, then the conservation method. Unfortunatley, the
conservation method may have the effect of returning concentrations
that sum to a value greater than one, so the correction method is
applied a second time.
After this process it is clear that the concentrations are non-negative
and sum to unity. However, although the one application of the
conservation method was applied, the second correction
method will generally imply some gain (or loss) of material (although it is
generally much smaller than the original gain (loss)).
In order to apply the
conservation principle rigourously, any measured gain(or loss) in
in each material after each d dose step is recorded and
an effort is made to remove (replace) the material at the following dose
level.
A good indication that materials are conserved in IMPETUS II
is given in yield-depth curves of figure 6 resulting from the
original material structure of figure 5. Each of the three layers
contains equal volume concentrations of two materials and
this is reflected in the results.
Further tests on IMPETUS II each consisting of 100 angstrom
layer with sharp interfaces, the first interface at a depth of
200 angstroms were generally found to conserve material
to within 1% in both material systems tested. For thin
layers and sharp interfaces close to the surface,
as considered in sections 2 and 3, the conservation rule is
still more difficult to enforce and a greater proportional
loss of material may be observed in these cases.