3.1.2 Two-Dimensional Point Response Functions

Two-dimensional point response functions $p(x,z)$ are modeled by combining two of the functions presented in Sec. 3.1.1. One vertical function is multiplied with a lateral function, which additionally meets the requirement that its projected range is zero. Furthermore the moments of the lateral function depend on the vertical coordinate $z$. Worth mentioning is that a coordinate system is used where $z$ is parallel to the ion beam direction.

$$p(x,z) = p_v(z)\cdot p_l(x,\sigma_l(z),\gamma_l(z), \beta_l(z))$$ (3.40)

Usually it is also assumed that the lateral function is a symmetric function which requires besides $R_{pl}$ also the lateral skewness $\gamma_l$ to be zero. Therefore mainly Gaussian, Pearson II and Person VII functions are used. Besides a so called modified Gaussian function [33]

$$f(x) = a\cdot \exp(-\vert b\cdot x\vert^p) \;\;with \;\;p \geq 2$$ (3.41)

is sometimes applied for the case that $\beta_l < 3$. There is an explicit relation between the kurtosis and the power $p$

$$\beta_l = \frac{\Gamma\left( \frac{1}{p}\right) \cdot \Gamma\left( \frac{5}{p}\right)}{\Gamma\left( \frac{3}{p}\right)^2},$$ (3.42)

where $\Gamma (x)$ is the Gamma function.

[12], [33], [54], [56] provide calibrated two-dimensional point response functions for the implantation with boron, phosphorus, arsenic and antimony ions into amorphous and crystalline silicon, into silicon dioxide and silicon nitride.