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 $Rp$ defined for the one-dimensional case by

$$Rp = \int\limits_{-\infty}^{\infty} x\cdot f(x) \cdot \;dx ,$$ (3.3)

while the higher order moments $m_i$ are defined by

$$m_i = \int\limits_{-\infty}^{\infty} (x-Rp)^i\cdot f(x) \cdot \;dx.$$ (3.4)

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

$$\sigma = \sqrt{m_2}$$ (3.5)

$$\gamma = \frac{m_3}{\sigma ^3}$$ (3.6)

$$\beta = \frac{m_4}{\sigma ^4}$$ (3.7)

$\gamma$ and $\beta$ 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 $Rp$ and the straggling $\sigma$.

$$p(x) = \frac{1}{\sigma\cdot\sqrt{2\cdot\pi}}\cdot \exp\left( -\frac{(x-Rp)^2}{2\cdot \sigma ^2}\right)$$ (3.8)

$$\gamma = 0$$ (3.9)

$$\beta = 3$$ (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].

$$p(x) = \LARGE\left\{ \begin{array}{rl} \sqrt{\frac{2}{(b+c)^2\cdo... ...{2\cdot c^2} \right) & \text{\normalsize for $x \geq a$}\\ \end{array} \right.$$ (3.11)

$$Rp = a+\sqrt{\frac{2}{\pi}}\cdot(c-b)$$ (3.12)

$$\sigma = \sqrt{(b^2-b\cdot c+c^2)-\frac{2}{\pi}\cdot (c-b)^2}$$ (3.13)

$$\gamma = \frac{\sqrt{\frac{2}{\pi}}\cdot (c-b)\cdot \left( \left(... ...t (b^2 + c^2) + \left(3-\frac{8}{\pi}\right) \cdot b \cdot c\right)}{\sigma ^3}$$ (3.14)

There is no explicit formula to calculate the parameters $a$, $b$ and $c$ from the moments $Rp$, $\sigma$ and $\gamma$. 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 $\gamma$ is restricted to [71]

$$\vert\gamma \vert < \frac{4-\pi}{\pi-2}\cdot \sqrt{\frac{2}{\pi-2}} \cong 0,99527 .$$ (3.15)

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

$$\frac{df(y)}{dy} = \frac{(y-a)}{b_0+b_1\cdot y+b_2\cdot y^2}\cdot f(y), \quad y = x - Rp.$$ (3.16)

Depending on the values of the parameters $a$, $b_0$, $b_1$ and $b_2$ different types of solutions are generated, mainly determined by the roots of

$$g(y) = b_0+b_1\cdot y+b_2\cdot y^2 .$$ (3.17)

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

$$a = -\frac{\gamma\cdot \sigma \cdot (\beta+3)}{A}$$ (3.18)

$$b_0 = -\frac{\gamma^2\cdot (4\cdot \beta -3\cdot \gamma^2)}{A}$$ (3.19)

$$b_1 = a$$ (3.20)

$$b_2 = -\frac{2\cdot \beta -3\cdot \gamma^2 - 6}{A}$$ (3.21)

$$A = 10\cdot \beta - 12\cdot \gamma^2 -18$$ (3.22)

The Pearson functions can be distinguished by different values of $\gamma$ and $\beta$.

$$\left. \begin{array}{rcccl} &&\gamma &\neq& 0\\ 1+\gamma^2 &<& \beta &<& 3+1.5\cdot \gamma^2 \end{array}\hspace*{1.4cm}\right\}$$ (3.23)

$$\left. \begin{array}{rcccl} &&\gamma &=& 0\\ &&\beta &<& 3 \end{array} \hspace*{3.1cm}\right\}$$ (3.24)

$$\left. \begin{array}{rcccl} &&\gamma &\neq& 0\\ &&\beta &=& 3+1.5\cdot \gamma^2 \end{array} \hspace*{1.4cm}\right\}$$ (3.25)

$$\left. \begin{array}{rcccl} 0 &<& \gamma &<& 32\\ & & \beta &>& ... ... 6\cdot \sqrt{(\gamma^2+4)^3}}{32-\gamma^2} \end{array} \hspace*{0.2cm}\right\}$$ (3.26)

$$\left. \begin{array}{rcccl} 0 &<& \gamma &<& 32\\ & & \beta &>& ... ... 6\cdot \sqrt{(\gamma^2+4)^3}}{32-\gamma^2} \end{array} \hspace*{0.2cm}\right\}$$ (3.27)

$$\left. \begin{array}{rcccl} & & \gamma &\neq& 0\\ 3+1.5\cdot \ga... ... 6\cdot \sqrt{(\gamma^2+4)^3}}{32-\gamma^2} \end{array} \hspace*{0.2cm}\right\}$$ (3.28)

$$\left. \begin{array}{rcccl} &&\gamma &=& 0\\ &&\beta &>& 3 \end{array} \hspace*{3.2cm}\right\}$$ (3.29)

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

$$\begin{split}\ln(f(y)) = & \frac{1}{2\cdot b_2}\cdot \ln\vert... ...b_1+\sqrt{b_1^2-4\cdot b_0\cdot b_2}} \right\vert . \end{split}$$ (3.30)

The Pearson type V function has a solution of

$$\ln(f(y)) = \frac{1}{2\cdot b_2}\cdot \ln\vert b_0+b_1\cdot y+b_2\cdot y^2\vert+ \frac{\frac{b_1}{b_2}+2\cdot a}{2\cdot b_2\cdot y+b_1}$$ (3.31)

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

$$\begin{split}\ln(f(y)) =& \frac{1}{2\cdot b_2}\cdot \ln\vert ... ...t y+b_1}{\sqrt{4\cdot b_0\cdot b_2-b_1^2}}\right) . \end{split}$$ (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.

$$p(x) = r\cdot p_a(x) + (1-r)\cdot p_b(x)$$ (3.33)

$p_a(x)$ is one of the above functions, while $p_b(x)$ is the same function combined with an exponential tail function $p_T(x)$.

$$p_b(x) = \left\{ \begin{array}{rl} k\cdot p_a(x) & \text{for $x <... ...kappa\cdot (k\cdot p_a(x)+p_T(x)) & \text{for $x \geq x_T$} \end{array} \right.$$ (3.34)

$$p_T(x) = k\cdot p_a(x_p)\cdot \exp\left[-\left( \frac{x-x_p}{L}\right) ^\alpha \right]$$ (3.35)

$L$ and $\alpha$ determine the shape of the tail, and $x_T$ is the starting position of the tail function which is set to

$$x_T = \left\{ \begin{array}{rl} Rp + \sigma & \text{if $p_a(x)$\ ... ... c & \text{if $p_a(x)$\ is a joined half Gaussian function} \end{array} \right.$$ (3.36)

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

$$x_p = \left\{ \begin{array}{rl} Rp & \text{if $p_a(x)$\ is a Gaus... ... a & \text{if $p_a(x)$\ is a joined half Gaussian function} \end{array} \right.$$ (3.37)

$\kappa$ is determined by the condition that $p_b(x)$ has to be continuous at $x_T$, while $k$ ensures a normalized point response function (3.1).

$$k\cdot p_a(x_T) = \kappa\cdot (k\cdot p_a(x_T)+p_T(x_T))$$ (3.38)

Three additional parameters ($r$, $\alpha$ and $L$) 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.

$$p(x) = r\cdot p_a(x) + (1-r)\cdot p_b(x)$$ (3.39)

$p_a$ and $p_b$ are independent Pearson functions with an individual set of parameters $a$, $b_0$, $b_1$, $b_2$, while $r$ 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: $4{\cdot}10^{12}$cm$^{-2}$ - $1{\cdot}10^{15}$cm$^{-2}$ Tilt: 7 $\,^\circ\;$, Rotation: 0 $\,^\circ\;$	Joined half Gaussian with exponential tail	[82]
(100) silicon	boron BF$_2$ arsenic phosphorus antimony indium	Energy (boron): 10 keV - 160 keV Dose (boron): $5{\cdot}10^{12}$cm$^{-2}$ - $5{\cdot}10^{15}$cm$^{-2}$ Energy (BF$_2$): 5 keV - 60 keV Dose (BF$_2$): $5{\cdot}10^{12}$cm$^{-2}$ - $5{\cdot}10^{15}$cm$^{-2}$ Energy (phosphorus): 30 keV - 180 keV Dose (phosphorus): $1{\cdot}10^{13}$cm$^{-2}$ - $1{\cdot}10^{16}$cm$^{-2}$ Energy (arsenic): 20 keV - 160 keV Dose (arsenic): $5{\cdot}10^{12}$cm$^{-2}$ - $1{\cdot}10^{15}$cm$^{-2}$ Energy (antimony): 10 keV - 180 keV Dose (antimony): $4{\cdot}10^{12}$cm$^{-2}$ - $1{\cdot}10^{15}$cm$^{-2}$ Energy (indium): 30 keV - 180 keV Dose (indium): $4{\cdot}10^{12}$cm$^{-2}$ - $1{\cdot}10^{15}$cm$^{-2}$ Tilt: 7 $\,^\circ\;$, Rotation: 0 $\,^\circ\;$	Pearson IV with exponential tail	[84]
(100) silicon with 10 nm - 120 nm SiO$_2$	arsenic	Tilt 0 $\,^\circ\;$and 7 $\,^\circ\;$	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: $1{\cdot}10^{15}$cm$^{-2}$ Tilt: 7 $\,^\circ\;$	Pearson	[42]
(111) silicon (100) silicon	boron	Energy: 30 keV - 150 keV Dose: $1{\cdot}10^{13}$cm$^{-2}$ - $1{\cdot}10^{16}$cm$^{-2}$ Tilt: 7 $\,^\circ\;$	Pearson	[70]
(100) gallium-arsenide (100) gallium-arsenide with 50 nm nitride	silicon	Energy: 50 keV - 300 keV Dose: $3.5{\cdot}10^{12}$cm$^{-2}$ - $3.5{\cdot}10^{13}$cm$^{-2}$ Tilt: 7 $\,^\circ\;$, Rotation: 45 $\,^\circ\;$	Pearson	[86]
(100) silicon	boron phosphorus	Energy (boron): 1 MeV - 7.1 MeV Dose (boron): $1{\cdot}10^{13}$cm$^{-2}$ - $1{\cdot}10^{16}$cm$^{-2}$ Energy (phosphorus): 1 MeV - 5 MeV Dose (phosphorus): $1{\cdot}10^{14}$cm$^{-2}$ - $6{\cdot}10^{15}$cm$^{-2}$ Tilt: 7 $\,^\circ\;$	Pearson	[30]