The first model including lateral spread of implanted ions was proposed by
Furukawa et al. [Fur72]. It is based on the statistical distribution
function 
for one ion, i.e. the response to a punctiform beam. Such
a point response is shown in Figure 2.3-1 for boron implanted with
an energy of 
.
Without loss of generality 
is the point of entrance of the ion.
Furukawa et al. and later Runge [Run77] used a 2D Gaussian function for
the statistical distribution function which may be written as
(2.3-1). This approach has been refined by Ryssel [Rys83a]
to allow an arbitrary vertical distribution function (2.3-2).
All these models assume that the 2D distribution function is
separable, i.e. it may be written as a product of two independent
distribution functions for vertical and lateral direction, 
and
, respectively (2.3-3).
Separability entails that any correlation between the vertical 
and the
lateral 
coordinate is ignored. The natural way to consider correlation
between the two coordinates is the introduction of mixed moments 
(2.3-4)
An attempt to construct distribution functions from such moments has been accomplished, e.g. by Winterbon [Win86] and Lorenz et al. [Lor89] with moderate success.
A different method to include correlation between the two coordinates is to allow a depth dependent lateral distribution function [Hob87a]. As can be seen from Figure 2.3-2 the moments of the lateral distribution function depend on depth.
The statistical distribution function for one ion may generally be written
as (2.3-5). The lateral distribution function is seen here as
a function of 
with parameters, which depend on the depth 
. For any
depth a normalization condition (2.3-6) must hold.
The lateral moments (standard deviation 
and kurtosis
) are obtained by fitting the results of Monte Carlo
simulations. Fitting formulae and parameters are listed in [Hob87a].
Assuming a Gaussian distribution for 
no higher lateral moments
than the standard deviation 
can be taken into account. As
mentioned in Section 2.2.2, the Pearson VII and the modified
Gaussian distribution functions are well suited for lateral distribution
functions including the lateral kurtosis 
too.
