3.1 Analytical Method

The analytical method is a purely empirical method. The distribution of the implanted particles is calculated by applying an analytical distribution function (point response function) $p(\vec{x})$ to the simulation domain.

The point response function describes the distribution resulting from an implantation through a single point (ideally focused ion beam) of the surface of the simulation domain. Depending on the dimension of the simulation it is either a one-dimensional, two-dimensional or three-dimensional function fulfilling the condition that the integral over space is one.

$$\int\limits_\mathcal{V} p(\vec{x}) \cdot \;d\mathcal{V} = 1$$ (3.1)

The impurity concentration $C(\vec{x})$ at position $\vec{x} = (x,y,z)$ is calculated summing up the contributions of all point response functions at all surface points of the simulation domain and multiplying by the implantation dose.

Figure 3.1: Schematic figure of the calculation of the impurity concentration by a convolution of the point response functions.

$$C(\vec{x}) = D\cdot \iint\limits_{\mathcal{A}} p(\vec{x}-\vec{x_s}) \cdot \; d\mathcal{A}$$ (3.2)

$\vec{x_s}$ is a point at the surface of the simulation domain, and the integration is performed over the whole surface $\mathcal{A}$. $D$ denotes the implantation dose. Due to the fact that point response functions are always defined in a coordinate system where the $z$ axis is parallel to the ion beam the convolution in (3.2) is also performed in a coordinate system aligned with the ion beam.