Experimentally, it is comparatively easy to measure vertical atom
profiles to improve our modeling of implanted depth distributions. Although
several two-dimensional attempts exist [GJ89], [Sub90],
[Sub92], [Cer92], still less progress has been made in measuring
lateral atom profiles. So, simple Gaussian distributions
(2.2-10) have been used almost universally to model the lateral
spread of implanted profiles [Fur72], [Rys83a], [Gil88]. For
symmetry reasons all odd moments of the lateral distribution must vanish
(
, 
).
Monte Carlo simulations [Hob87a], [Hob88b] have shown that Gaussian
distributions are not sufficient for the representation of lateral profiles,
at least the fourth moment, the lateral kurtosis 
has to be
considered
. Hobler et al. [Hob87a] have
proposed a modified Gaussian function for 
and a
Pearson VII for 
. We get the modified Gaussian distribution
function replacing the square in (2.2-10) by a positive exponent
, which has to be determined from the lateral kurtoses . For
the considered range of kurtoses holds
.
The lateral kurtosis for the modified Gaussian function can be
expressed in terms of using the Gamma function 
.
Unfortunately, the inverse of function (2.2-12) cannot be
calculated analytically. Therefore, the approximation
(2.2-13) - (2.2-16) [Hob87a] is used. The other
parameters 
and 
are then obtained easily,
(2.2-17) - (2.2-18). For the derivation of expressions
(2.2-13) - (2.2-18) and a detailed discussion of the
properties of the modified Gaussian distribution see [Hob88a].
For high kurtoses () which appear for instance with damage
profiles, the Pearson VII [Joh70] is prefered to the modified Gaussian.
The parameters are calculated from (2.2-20), (2.2-21)
and (2.2-22) [Hob87a].
For the sake of completeness, we shall present the Pearson II distribution
(2.2-23), which is also unskewed (), and might
be used as an alternative to the modified Gaussian for lateral kurtoses
. Its free parameters are to be calculated from
(2.2-24) - (2.2-26).
