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## 3.1.1 One-Dimensional Point Response Functions

Generally a point response function can be characterized by a set of parameters which can be expressed by the moments of the function. The first moment is the projected range defined for the one-dimensional case by

 (3.3)

while the higher order moments are defined by

 (3.4)

Instead of using the higher moments directly, the point response functions are characterized by modified moments expressed in terms of the straggling which is the square root of the second order moment.

 (3.5)

 (3.6)

 (3.7)

and are called the skewness and the kurtosis, respectively.

The most prominent one-dimensional point response functions are:

Gaussian function:
It is characterized by the projected range and the straggling .

 (3.8)

 (3.9)

 (3.10)

Joined half Gaussian function [27]:
It is a sum of two Gaussian functions which join at a modal projected range. One additional parameter is necessary to characterize this function [71].

 (3.11)

 (3.12)

 (3.13)

 (3.14)

There is no explicit formula to calculate the parameters , and from the moments , and . Therefore iterative numerical solutions of (3.12) - (3.14) have to be performed. Since the joined half Gaussian function contains only three parameters the kurtosis is a function of the other moments. Worth mentioning is that the value of the skewness is restricted to [71]

 (3.15)

Pearson Functions:
These functions are defined as the solution of the differential equation

 (3.16)

Depending on the values of the parameters , , and different types of solutions are generated, mainly determined by the roots of

 (3.17)

The Pearson parameters can be expressed in terms of the moments , , and which allows a convenient characterization of the different types of Pearson functions [69] [71].

 (3.18)

 (3.19)

 (3.20)

 (3.21)

 (3.22)

The Pearson functions can be distinguished by different values of and .

 Type I (3.23)

 Type II (3.24)

 Type III (3.25)

 Type IV (3.26)

 Type V (3.27)

 Type VI (3.28)

 Type VII (3.29)

The solutions of Pearson type I, type III and type VI functions are [38]

 (3.30)

The Pearson type V function has a solution of

 (3.31)

Finally the Pearson type II, type IV and type VII functions are solved by

 (3.32)

Only bell-shaped solutions are suitable for the definition of a point response function for which reason only the Pearson type II, IV and VII functions can be applied.

Functions with an exponential tail [82]:
These are modeled as a sum of one of the above functions and an exponential tail function.

 (3.33)

is one of the above functions, while is the same function combined with an exponential tail function .

 (3.34)

 (3.35)

and determine the shape of the tail, and is the starting position of the tail function which is set to

 (3.36)

is the location where the main function has its peak value.

 (3.37)

is determined by the condition that has to be continuous at , while ensures a normalized point response function (3.1).

 (3.38)

Three additional parameters (, and ) are introduced by this distribution function which are mainly used to model the channeling tail of implantations into crystalline materials.

Dual Pearson functions [62]:
This is a superposition of two Pearson function, similar to functions with an exponential tail. Instead of two parameters five additional parameters are introduced.

 (3.39)

and are independent Pearson functions with an individual set of parameters , , , , while is a proportionality coefficient.

Several parameter sets for various implantation conditions and ions have been published using one of the above analytical functions. Tab. 3.1 summarizes some of these publications.

Figure 3.6: References for parameters for analytical ion implantation.
Targets Ions species Implantation conditions Function type Ref.
 amorphous silicon silicon dioxide silicon nitride
 boron phosphorus arsenic antimony
 Energy: 25 keV - 300 keV
Pearson [82]
 (100) silicon
 antimony
 Energy: 30 keV - 180 keV Dose: cm - cm Tilt: 7 , Rotation: 0
Joined half Gaussian with exponential tail [82]
 (100) silicon
 boron BF arsenic phosphorus antimony indium
 Energy (boron): 10 keV - 160 keV Dose (boron): cm - cm Energy (BF): 5 keV - 60 keV Dose (BF): cm - cm Energy (phosphorus): 30 keV - 180 keV Dose (phosphorus): cm - cm Energy (arsenic): 20 keV - 160 keV Dose (arsenic): cm - cm Energy (antimony): 10 keV - 180 keV Dose (antimony): cm - cm Energy (indium): 30 keV - 180 keV Dose (indium): cm - cm Tilt: 7 , Rotation: 0
Pearson IV with exponential tail [84]
 (100) silicon with 10 nm - 120 nm SiO
 arsenic
 Tilt 0 and 7
Joined half Gaussian with exponential tail [83]
 (111) silicon silicon dioxide silicon nitride
 boron arsenic
 Energy (boron): 30 keV - 210 keV Energy (arsenic): 30 keV - 400 keV Dose: cm Tilt: 7
Pearson [42]
 (111) silicon (100) silicon
 boron
 Energy: 30 keV - 150 keV Dose: cm - cm Tilt: 7
Pearson [70]
 (100) gallium-arsenide (100) gallium-arsenide with 50 nm nitride
 silicon
 Energy: 50 keV - 300 keV Dose: cm - cm Tilt: 7 , Rotation: 45
Pearson [86]
 (100) silicon
 boron phosphorus
 Energy (boron): 1 MeV - 7.1 MeV Dose (boron): cm - cm Energy (phosphorus): 1 MeV - 5 MeV Dose (phosphorus): cm - cm Tilt: 7
Pearson [30]

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A. Hoessiger: Simulation of Ion Implantation for ULSI Technology