With cognizance of a point response it is evident, that we get the dopant
distribution in a more-dimensional space by convolution of this distribution
function, i.e. by adding up the point responses to the final
concentration 
. For multilayered, nonplanar structures such as in
Figure 2.3-3, we take a vertical profile 
at a lateral position 
from Section 2.2.4, and obtain the
concentration profile from a lateral convolution (2.3-7).
First, we must consider the beam tilt angle 
. Many implantation
machines tilt the wafer by about 
to reduce channeling effects
[Gil88], larger tilt angles may be used to obtain advantageous
electrical characteristics [Hor88]. We rotate the complete geometry and
also the grid by 
as shown in Figure 2.3-4 using the
transformation matrix 
(2.3-8).
Then, we cut the geometry in thin vertical slices parallel to the direction
of the ion beam (Figure 2.3-4). We use an even number of slices,
in order to promote numerically symmetrical calculations for geometrically
symmetric structures. For each slice 
we determine the quantities
required for the calculation of a one-dimensional vertical distribution
function [Sch91]. These quantities are the scaling factor 
,
the shift 
for the vertical distribution function in layer 
, and
the interface positions 
.
The vertical coordinate of the entry point of the ions 
makes the
calculation independent of the actual origin of the coordinate system. The
interface positions simplify the determination of the layer index
for a given vertical coordinate 
(Figure 2.3-5). We
calculate 
and pretending to deal with a laterally
infinite structure. Vacuum layers 
in a slice require
special treatment and, therefore, the original equations
(2.2-32) - (2.2-35) are slightly modified.
The distribution function vanishes in vacuum layers and is continued in the
next non-vacuum layer (2.3-13). For a layer , 
points to
the previous non-vacuum layer, i.e. in most cases holds 
.
After having completed this initialization phase we have all parameters
for the vertical distribution function 
at any lateral
position . To get the implanted dopant concentration 
at a certain
grid point 
(in the rotated coordinate system) we have to calculate
the convolution integral (2.3-14).
We adjust the actual integration limits to the maximum lateral standard
deviation 
of the layers (2.3-15).
Contributions to the integral outside the interval (2.3-16) are
neglected. The integration boundary parameter 
is
to be selected according to the desired accuracy.
This integration has to be performed numerically by one of the numerous
quadrature formulae, e.g. Chebyshev, Euler-Maclaurin, Gauß, Maclaurin,
Newton-Cotes, Romberg [Sto83]. We perform the integrations
(2.3-14) and (2.3-11) numerically with a Simpson
integrator, which is based on a Newton-Cotes formula. This Simpson
integrator defines its own grid for the integration and refines the grid
locally if desired. It starts with an initial equidistant grid of 
lines
in the right half and lines in the left half of the integration domain.
The grid in the left and/or right half is refined successively until certain
absolute and relative accuracy conditions are fulfilled.
As mentioned above the distribution functions in the
lateral interval 
are required for the calculation of the concentration
at a point . Usually, the real geometry is given for exactly
the domain where we want to calculate the concentration. Some of the
boundaries do not exist in the physical structure and are only introduced to
limit the simulation domain. These boundaries are labeled artificial. At artificial boundaries we extend the geometry by
, such that the concentration can
be calculated at any grid point in the whole real structure (see
Figure 2.3-6), without loosing any contribution to the
convolution integral (2.3-14). It is necessary to extend the
geometry at all (lateral and vertical) artificial boundaries, because the
wafer might be tilted against the ion beam, and the lateral integration is
always performed perpendicular to the ion beam.
We want to emphasize that this implementation for the calculation of the dopant concentration is completely independent of the grid used for the dopant profiles. Furthermore, any vertical and lateral distribution function may be employed for the different target materials to calculate the profiles in the one-dimensional slices. For a list of available distribution functions and parameter sets see Section 2.4.
The basic assumptions for analytical approaches based on moments of a distribution function restrain their applicability to half spaces. The shown expansions to multilayered and nonplanar structures mostly give acceptable results but suffer from lacking underlying physics. The results can be in a high degree inaccurate for instance if the interfaces between layers are not really perpendicular to the initial ion beam direction, or when steep surface contours have to be handled [Hob89].
