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).