IMPETUS II is a Fortran code that has been developed to
simulate the
atomic mixing and particle emission that takes place when a material
is bombarded with energetic particles, as in secondary ion
mass spectrometry (SIMS) or secondary neural mass spectrometry
(SNMS). In the code,
the physical process is taken to be governed by the
model of references [1-3]. The purpose of this paper along with the
companion paper [4], is to describe and demonstrate the computational methods
employed in the code. Ref [5] is a user manual for the resulting software.
To begin, a brief outline of the atomic mixing model
is given. Let there be n distinct materials in the material system
(the materials of the structure with the bombarding species).
The governing equations for the concentration of each
i-material is as follows:
¶qi(x,f)
¶f
= (q(x,f)+u(f))
¶qi(x,f)
¶x
+
¶q(x,f)
¶x
qi(x,f) -
¶qi(x,f)
¶x
(1)
for i=1,2,..,n, 0 £ x < ¥, 0 £ f < ¥,
where x is the depth below the receding surface and f is the
dose.
At any dose f the material distribution is determined by the
functions qi(x,f) for i=1,2, ..,n,
the fraction by volume of i-atoms at
depth x. The functions qi satisfy the constraints
Si=1n qi(x,f) = 1, qi(x,f) ³ 0 for all x, f.
The surface recession speed is denoted by u(f). If X0(f)
denotes the depth of erosion at dose f then X0(f) = ò0f u(f¢) d f¢.
The relationship between the fixed and moving coordinate system is such that
X=X0(f) + x.
The scaled current of i-atoms crossing
the point at depth x below the surface in the direction of
increasing depth is denoted qi(x,f) and the collective current
by q(x,f) where
q(x,f)=Si=1n qi(x,f).
The qi are given by the following equation
qi(x,f) = -
¶
¶x
{ Di(x,f) qi(x,f) } ,
(2)
for i=2,3,..,n and
q1(x,f) = -
¶
¶x
{D1(x,f) q1(x,f) }+ t1
ó õ
¥
x
fR(x¢,f) dx¢
(3)
in the domain (x,f) Î [0,¥) ×[0,¥), where fR(x,f) is the probability density functions (with respect to
depth x) for the range of the individual primary particle in the material
structure at dose f, Di(x,f) is the diffusivity of the
i-atoms at depth x and dose f in the material structure and
t1 is the effective volume of a primary particle.
The model relates
the nuclear and ionisation energy functions and the
range deposition functions of the primary particles
to to composition of the material. The sputter
yield and surface recession speed are determined by the
surface concentrations and the experimental conditions.
The given initial material structure and the
conservation of material condition at the surface layer
determines the solution. A detailed description of the
methods employed to compute the energy and range functions,
the sputter yield and the recession speed are given in
reference [4].
The partial differential equations given above
form the true atomic mixing model. The
finite difference method (described in [4])
provides us with a consistent
method for their solution and hence this method should
be applied as widely as possible.
However, the FDM is generally only
applied in a section of the domain [0,xP] since it would be computationally
expensive to apply it over the full depth and the material
may contain sharp interfaces (dicontinuities) which would lead
to oscillations in the progressive solutions. Furthermore, the nature
of the problem is such that most of the action is concentrated within the
surface region; the equations may be simplified at depth without
significant loss of accuracy. Modification of the solution
returned by the FDM is also necessary to ensure that the
concentrations are non-negative and sum to one throughout the
domain and materials are conserved.
IMPETUS II is a general purpose code for simulating the
SIMS or SNMS profiles of a range of material structures within
a set of material systems and under a range of conditions.
Facilities have therefore been included in the code to ensure that
IMPETUS is able to respond effectively to the form
of the particular form of the material structure in hand.
For example, if the material structure contains thin layers (say less than
20 angstroms) or there is a sharp interface in the domain
of the FDM then the computational model must adapt by decreasing
the mesh size to ensure a satisfactory solution. The methods
for adapting the finite difference mesh to the problem in hand
are described in section 2.
If the diffusion term is linearised 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
(4)
for i=1,2,...,n. The approximation (4) tends to provide
a satisfactory approximation to (1) at greater depths where
it models the
minimal diffusion and translation due to surface recession.
A computational model based on (4) is used in IMPETUS II
in the remaining region [xp,¥) and this is described in section
3.
If the material
structure contains wide homogeneous layers (that is of at least of the
order of a micron wide) then the solution tends to a steady state.
In IMPETUS II a method for carrying out the transition to
the steady state is employed and this is described in section 4.
The atomic mixing process must obey the natural laws that materials
must be conserved and the concentrations of each material must be
non-negative and sum to unity. The underlying mathematical
model does satisfy these natural laws
(see reference [6]). However, as we have noted earlier,
the finite different method is prone to numerical error. In
practice the computed
solutions transgress these natural laws.
Moreover, if the problem was left unmanaged, the breach in the physical
laws could increase from dose step to dose step.
Hence in IMPETUS II the solution is examined at
every dose step
and the concentrations must also be corrected so that they are non-negative
and they sum to unity and a material conservation method is enforced. Details
on these techniques are given in section 5.
In the
final section of this paper an algorithm outlining the framework for
the method selection within the code is given.