In IMPETUS II, the qi functions obtained at each
dose level via the FDM are monitored so that when they change only by minute
amounts from over a range of dose steps then it is determined that
the solution is tending to a steady state. Steady state solutions
occur when the initial material distribution is homogeneous over a wide
range. Provided that the next interface is also well away from
the mixing window then the FDM is temporarily abandoned in
IMPETUS II and a special
method for handling the steady state is employed.
The most obvious way of dealing with the steady state is to simply
adopt the final FDM solution as the steady state solution and
progress to the next interface with all the functions being
invariant with f. Unfortunately, in order to obtain a satisfactory
solution in this way, the FDM would have to be applied well into
the convergence to steady state.
A more efficient technique is that of expressing the transition to
steady state in an asymptotic form and carrying out a
straightforward extrapolation from the final FDM solution up
until the next interface approaches xR.
Such a method is employed in IMPETUS II.
The extrapolation method consists of two stages, first obtaining
the coefficients that define the extrapolant then using
the extrapolant to
advance the solution.
The exact asymptotic form of the extrapolant for the
qi functions is unknown. In the
IMPETUS II code it is modelled as follows:
qi(x,f) ~ ai(x) +
bi(x)
f- f¢+c
(13)
where f¢( < f) is a reference point. Note that the second
part of the approximation tends to zero as f increases and
ai(x) is the approximation to the steady state solution for qi.
The values of ai(xj) and bi(xj) and c > 0 that are approximately
the best least-squares fit to the qi(xj,f) over the
final set of dose steps of the FDM.
The method employed in IMPETUS II is now outlined. Firstly,
for three values of c the best least-squares values of ai(x1)
and bi(x1) are obtained and the least-squares error for each
is evaluated and, through interpolation of these values and minimisation
of the interpolant, a values of c is selected. For the given value
of c, the values of ai(xj) and bi(xj) that give the
best least-squares fit to the qi(xj,f) are computed.
Once the coefficients ai(xj), bi(xj) and c are obtained, the
solution can be advanced through evaluating (13).
The structure of figure 5 is used to demonstrate the handling
of steady state solutions in IMPETUS II. The material
structure consists of 50 % arsenic and the remaining
50 % made up of Gallium and Aluminium. The layers are
each 1200 angstroms wide. The first layer consists of 0% Aluminium
(50 % Gallium), the second layer has 25 % Aluminium
(25 %) the third layer has 50 % Aluminium (0 %).
The resulting yield-depth curves are given in figure 6.
Figure 5. Material consisting of wide layers.
For this test case the simulation was completed in around 16 minutes on
a 33 MHz 486 PC, approximately half the time it would take if the
FDM was used exclusively.
Figure 6. Yield-depth curve for Aluminium arising from the structure of figure 5.
