3.3.1 Energy Loss Mechanisms

Entering the target an ion starts loosing energy by interacting with the nuclei and the electrons of the target material. On entrance the kinetic ion energy is in the keV range which results in a speed of some hundred m/s. The final point of the ion trajectory is reached when the kinetic energy of the particle reaches the thermal energy, which is of the order of 35 meV. Assuming a constant slow down, the time an ion needs to come to rest (ballistic time) can be estimated by

$$t=\sqrt{\frac{2\cdot M\cdot s^2}{E}}.$$ (3.50)

It is of the order of 10$^{-12}$ s if the ion trajectory length $s$ is about 1 $\mu$m and the implantation energy $E$ is in the hundred keV range. $M$ is the ion mass. For several application the ballistic time will be even shorter.

In the general case the energy loss would have to be calculated by solving the Schrödinger equation with the Hamiltonian operator

$$\begin{split}H &= Hk_{ion} + \sum_{i=1}^N (Hk_{n_i} + W_{ion\... ...1}^n\sum_{j=1,j\neq i}^n W_{e_i\leftrightarrow e_j} \end{split}$$ (3.51)

$$Hk_{ion} = \frac{\vec{p}_{ion}(t)^2}{2\cdot M_{ion}}$$ (3.52)

$$Hk_{n} = \frac{\vec{p}_{n}(t)^2}{2\cdot M_{n}}$$ (3.53)

$$Hk_{e} = \frac{\vec{p}_{e}(t)^2}{2\cdot M_{e}}$$ (3.54)

$$W_{ion\leftrightarrow n} = \frac{Z_{ion}\cdot Z_{n}\cdot e^2}{\vert\vec{r}_{ion}(t) - \vec{r}_n(t)\vert}$$ (3.55)

$$W_{ion\leftrightarrow e} = \frac{Z_{ion}\cdot e^2}{\vert\vec{r}_{ion} - \vec{r}_e\vert}$$ (3.56)

$$W_{n_i\leftrightarrow n_j} = \frac{Z_{n_i}\cdot Z_{n_j}\cdot e^2}{\vert\vec{r}_{n_i}(t) - \vec{r}_{n_j}(t)\vert}$$ (3.57)

$$W_{e_i\leftrightarrow e_j} = \frac{e^2}{\vert\vec{r}_{e_i}(t) - \vec{r}_{e_j}(t)\vert}$$ (3.58)

$$W_{n_i\leftrightarrow e_j} = \frac{Z_{n_i}\cdot e^2}{\vert\vec{r}_{n_i}(t) - \vec{r}_{e_j}(t)\vert}$$ (3.59)

$Hk_{ion}$, $Hk_{n}$, $Hk_{e}$ are the kinetic energies of the ion, the target nuclei and the target electrons, $\vec{p}$ are their moments, and $M$ are their masses. Beside the kinetic energies the interaction potentials $W$ between the ion and the target nuclei ( $ion\leftrightarrow n$) and electrons ( $ion\leftrightarrow e$) are considered. Finally the interactions of the target nuclei and electrons among themselves ( $n_i\leftrightarrow n_j$, $e_i\leftrightarrow e_j$, $n_i\leftrightarrow e_j$) are taken into account. $Z_{ion}$, $Z_{n}$ are the charges of the ion and of the nuclei, and $\vec{r}_{ion}$, $\vec{r}_n$, $\vec{r}_e$ are the coordinates of the ion, the nuclei and the electrons.

Since a rigorous treatment of this problem is not feasible several assumptions have to be made to provide an appropriate theoretical background. First of all is the old assumption made by Bohr [10] to separate the interaction of the ion with the nuclei of the target from the interaction with the electrons of the target. The interaction of the ion with the nuclei (nuclear stopping process) can be treated as an elastic collision process, while the interaction with the electrons can be treated as an inelastic process without any scattering effects (electronic stopping process). The total stopping process can be modeled as as sequence of alternating nuclear and electronic stopping processes.