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Next: 2.1.3 Evaluation of Statistical Up: 2.1 Statistical Moments and Previous: 2.1.1 Central Moments and

2.1.2 Probability Weighted Moments and their Distribution Functions

 

The estimated characteristic parameters of distributions fitted by central moments are often markedly less accurate than those obtained by other estimation procedures. We present an alternative approach, where we introduce the so-called probability weighted moments (PWMs) in combination with the L-moments to specify statistical distributions [Hos89]. To our knowledge, we used this approach the first time in semiconductor simulation engineering. These L-moments are analogous to conventional central moments, but can be estimated by linear combinations of order statistics. L-moments are able to characterize a wider range of distribution functions and are more robust against outliers in the given data set than central moments. These L-moments can be defined in terms of probability weighted moments tex2html_wrap_inline4865 by a linear combination. These probability weighted moments can be defined in terms of the cumulative distribution function F(x) as given in (2.1-15) after [Gre79], where p,r, and s are positive integers. x(F) denotes the quantile or inverse cumulative distribution function of the random variable x.

  equation456

The quantities tex2html_wrap_inline4865 may now be used to describe and characterize probability distributions. If we express the probability weighted moments in terms of the density function f(x) an alternative definition is given by (2.1-16).

  equation462

Note, that the moments tex2html_wrap_inline4883 are the noncentral conventional moments (see 2.1-3). We shall instead use quantities tex2html_wrap_inline4885 where the random variable x enters linearly, and in particular the moments tex2html_wrap_inline4889 given in (2.1-17), which we shall also refer to as PWMs.

  equation472

Although probability weighted moments are useful to characterize a distribution, they have no particularly meaning. It is useful to define some functions of PWMs, which can be seen as descriptive parameters of location, scale and shape of a probability function, the L-moments. The linear combination between the L-moments tex2html_wrap_inline4891 and the PWMs tex2html_wrap_inline4889 are given in (2.1-18) to (2.1-21) for the first four moments.

     eqnarray480

We can describe a distribution by L-moments even when some of the conventional moments of the distribution do not exist. Therefore, only the mean value of the distribution must exist. The characterization of distributions by L-moments offers new possibilities such as the incorporation of up to now not considered density functions. These are the generalized gamma, generalized logistic, generalized pareto or the four-parameter kappa distribution function. It should be noted at this point that conventional density functions like the Gaussian or Pearson can not be defined over PWMs or L-moments, because the quantile function x(F) is not explicitly defined for these function.

To show the applicability of L-moments for ion implantation profiling, we introduce the four-parameter kappa distribution function to fit a dopant profile. All distribution functions F(x), f(x), and the quantile function x(F) are existing for the four-parameter kappa distribution and are given in (2.1-22),(2.1-23), and (2.1-24), respectively.

    eqnarray489

The four-parameter kappa distribution is a combination of the generalized logistic, the generalized extreme-value and the generalized Pareto distribution, where tex2html_wrap_inline4769 is a location parameter, tex2html_wrap_inline4771 a scaling parameter and h,k shape parameters. The estimation of these characteristic parameters based on L-moments requires a Newton-Raphson iteration method, because no explicit solution of the probability weighted moments for the kappa parameters is possible [Hos91].

     figure503
Figure 2.1-1: The location parameter tex2html_wrap_inline4769 of the four-parameter kappa distribution functions for different dopants, implantation targets and implantation energies.
Figure 2.1-2: The scale parameter tex2html_wrap_inline4771 of the four-parameter kappa distribution functions for different dopants, implantation targets and implantation energies.

In Figure 2.1-1 the location parameter tex2html_wrap_inline4769 of the four-parameter kappa density function is shown for the most frequently used dopants and for silicon and oxide implantation targets. The parameter tex2html_wrap_inline4769 acts similar to the well-known projected range of conventional distributions. When the implantation energy is increased the dopants penetrate deeper into the substrate which leads to higher values for tex2html_wrap_inline4769 . The relationship between tex2html_wrap_inline4769 and the implantation energy is about linear for phosphorus, arsenic and antimony, only boron exhibits nonlinear behavior. The spread of the kappa density function is controlled by the scale parameter tex2html_wrap_inline4771 . Figure 2.1-2 shows the scale parameter tex2html_wrap_inline4771 with respect to dopants, implantation targets and implantation energy. The higher the scale parameter tex2html_wrap_inline4771 the broader the density function. By using implantation energies above 100keV for boron implantations the scale parameter tends to saturate. The other dopants show nearly linear dependence to variation of the implantation energy.

Figure 2.1-3 shows the comparison of distributions obtained from central moments (Gauß, Pearson) and L-moments (Kappa). For non-symmetric profiles the four-parameter kappa distribution gives better results than the conventional distributions.

   figure515
Figure 2.1-3: Comparison of several analytical implantation profiles, based on central or L-moments for a Boron implant at 40keV energy, tex2html_wrap_inline4935 dose, and tex2html_wrap_inline4937 tilt angle. The corresponding result of an amorphous Monte Carlo simulation is also shown for reference purpose.


next up previous contents
Next: 2.1.3 Evaluation of Statistical Up: 2.1 Statistical Moments and Previous: 2.1.1 Central Moments and

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