Linear diffusion methods are employed in IMPETUS II to
provide an efficient computational method for modelling the
effect of the minimal energy deposition beyond the
surface region and to smooth the interfaces in the material before
they are passed into the domain of the finite difference method. In the situation
in which there is a sharp interface within [0,xR], the
value of xp is restricted so that the sharp interface
lies outside the domain of the FDM, [0,xP]. Hence the emphasis
is on maintaining the qualitative properties of the solution in
such cases.
The methods of linear diffusion
are based on the approximation (4). In this section an
integral equation method is employed to obtain an
efficient computational solution to the equation corresponding to
(4) with
sharp interfaces in the original material structure.
3.1 General Solution of the Linear Diffusion Model
In the fixed coordinate system the surface recession translation
term (u [(¶qi)/(¶x)] or (4)) is subsumed into
the straightforward diffusion equation:
¶Qi(X,f)
¶f
» Di(X-X0(f),f)
¶2 Qi(X,f)
¶X2
(X,f) Î [xR+X0(f), ¥) ×[0,¥).
If we consider solutions about a single point X*, then the equation may be
linearised and simplified further:
¶Qi(X,f)
¶f
» Di*(f)
¶2 Qi(X,f)
¶X2
(5)
where Di*(f) = Di(X*-X0(f),f),
X » X*, f Î [0,¥).
The following equation, based on approximation (5), is used
to model the pre-diffusion on the original material distribution
¶
^
Q
i
(X,f)
¶f
= Di*(f)
¶2
^
Q
i
(X,f)
¶X2
(6)
with [^(Q)]i(X,0) = Gi(X) describing the
initial material distribution.
The Greens function for the equation (6) is as follows
G(X,f) =
1
2
Ö
pI(f)
exp{
-(X-X*)2
4 I(f)
}
(7)
where I*(f) represent the total diffusion experienced at the point
X* after dose f;
I*(f)=
ó õ
f
0
D*(f¢) d f¢.
(8)
It is easy to show that G(X,f) is a solution to equation (6).
Note that ò- ¥¥ G(X,f) dX = 1.
Given that G(X,f) is of significant value only when |X-X*|
is small then the following Green's function solution
is an acceptable approximation
^
Q
i
(X,f) »
ó õ
¥
-¥
G(X-Y,f)
-
Qi
(Y) dY .
(9)
The smoothing due to the pre-diffusion significantly effects the
material distribution only near the interfaces in Gi(X).
Let X* be an interface and let it be assumed that the initial
concentrations have the following form for X » X*:
At any dose f the linear diffusion methods require a value for the
total energy deposited at each changing point (interface) X*.
In reference [4] the method of obtaining a value for the
numclear energy function
FE inside the mixing window was explained and
the equations that relate a technique
for obtaining the diffusivities
from the value of FE and the known material concentrations were given.
The value of the diffusivities beyond the mixing window are required
to model the smoothing of the interfaces. This is explained in
detail in the next subsection.
The total diffusion at a point X* and dose f is
obtainable from a numerical integration based of the formula (8).
In IMPETUS II this is carried out by computing
the diffusivities of each species at each interface and at each dose and
applying trapezium integration rule to compute the integral
I*.
Having computed the I* the solution concentrations can
be obtained from (10).
3.3 The Evaluation of the Diffusivities outside the Mixing Window
The term D*(f) refers to the rate of diffusion
at the fixed point X* at dose f. For the solution method outlined in the
previous subsections, its instantaneous values are required at each sharp
interface X* up until the interface enters the domain of
the finite difference method, the integration over the dose
quantifies the total diffusion at X*. The diffusivities are
directly related to the nuclear energy deposition function.
The original reasoning behind the choice of a value for xR
is to ensure that the energy retained there is only a small
fraction (circa 1 %) of the energy of bombardment. Hence it is considered
sufficient to only obtain reasonably rough estimates to the energy functions
beyond xR.
In IMPETUS II, approximations to the energy functions in [xR,¥)
are obtained by the following method. Let it be assumed that the energy
functions are known at the point xA ( ³ xR) and that the xAi that
satisfy Ei(xAi) = E(xA) for i=1,2, ..,n are also known.
An approximation to the energy functions at a point xB ( > xA) is
now required.
First an approximation to the energy retained at xB is obtained
by the following formula:
^
E
(xB,f) ¬ E(xA,f) -
n å
i=1
ó õ
[`(q)]ABi (xB-xA)
xA
(FEi(x) + FIi(x)) dx
(11)
where the [`(q)]ABi represents the average concentration of
i-material for i=1,2, ..,n. The xBi for i=1,2,..,n
are then chosen so that Ei(xBi)=[^E](xB,f) using the
bisection method. The nuclear energy deposition at xB may then be
approximated as follows:
The linear pre-diffusion method is applied at points outside the
mixing window. The technique described in the previous subsection
are used to compute the initial estimate of qi(x). However,
in general, the materials at the interfaces diffuse at different
rates. The outcome of this is that the concentrations do
not necessarily sum to one.
(In the true model, the unequal diffusions of the materials
is compensated for in the collective current; materials
are shifted to ensure that the concentrations sum to one.)
In the method employed in IMPETUS II the values of the
concentrations are then re-scaled; the concentration
of each is divided by the sum of the concentrations obtained
from the method of approximation (10), so
that the resulting values sum to one.
In IMPETUS II, the linear post-diffusion method is employed in
preference to the FDM when there are sharp interfaces in the
mixing window. The method is similar to the pre-diffusion method
of the previous subsection. It differs in that the method
must also model the deposition of the primary particles.
The primary particle distribution at dose f+ d is
estimated to be the concentration at dose f plus the
deposion of primary particles during the dose d:
q1(x,f+d) » q1(x,f) + tdfR(x,f)
(12)
The initial estimates of the concentrations are again obtained from the
approximation (10) but in this case the concentrations
of the non-primary species are
re-scaled (divided by 1-q1) so that S2n qi = 1 - q1 where
the qi where the q1 has been obtained from (12).
The effectiveness of the linear diffusion methods employed in IMPETUS
II is demonstrated through using the code to compute the SIMS
profiles for material structures with a sharp interface close to
the surface. The test structures were each of the form illustrated
in figure 3, they consist of a 100 angstrom block of
Germanium at depths of depths of 40, 60, 80 and 100 angstroms in
Silicon. The bombarment energy is 5 keV and the angle of incidence
is 45 degrees to the normal.
The resulting yield-depth curves are given in figure 4.
Figure 3. Material consisting of a 100 angstrom layer near the surface.
Figure 4. Yield-depth curve for Germanium arising from the structure of figure 3.
