3.3.5.5 De-channeling

Due to the fact that the point defects disturb the regular order of the atoms in crystalline materials they influence the ion trajectories especially if an ion moves within a channel. The probability that a particle is removed from the channel (de-channeling) is increased by the presence of point defects. Therefore it is very important to consider the material damage during the calculation of a particle trajectory because the average range of a particle especially in the channeling region is significantly reduced by the accumulating damage.

Point defects are treated in the simulation by randomly neglecting lattice atoms and by randomly inserting interstitial atoms as collision partners for a moving particle (Sec. 4.4.1). The probability $P_{\alpha}$ that the collision partner is a point defect is determined by the vacancy and the interstitial concentration. A linear model just considering the amorphization concentration $\rho_{\alpha}$ as a saturation level has turned out to be sufficient to adequately model the de-channeling effect.

$$P_{\alpha} = \Large\left\{\begin{array}{cl} \frac{\rho_V + \rho_I... ... 1} & \text{\normalsize for $\rho_V + \rho_I > \rho_\alpha$} \end{array}\right.$$ (3.174)

$\rho_m$ is the atomic density of the material which is $5{\cdot}10^{22}$cm$^{-2}$ in case of silicon.

When inserting interstitial atoms in the simulation care has to be taken to conserve the local material density $\rho$ determined by the material density $\rho_m$, the vacancy concentration $\rho_I$ the interstitial concentration $\rho_V$.

$$\rho = {\rho_m} + \rho_I - \rho_V$$ (3.175)

In addition reasonable positions for the interstitial atoms must be found. Mainly three methods are proposed for placing the interstitials.

Figure 3.15: Schematic figure of the position of tetrahedral interstitial sites in the diamond lattice. The black spheres denote silicon atoms while the patterned spheres represent the tetrahedral interstitial positions.

Figure 3.16: Schematic picture of crystalline silicon with some interstitials (yellow spheres) placed at tetrahedral interstitial positions. The view in the left figure is parallel to the $<$100$>$ direction while it is parallel to the $<$110$>$ direction in the right figure.

First they can be placed at tetrahedral interstitial sites. This is a reasonable assumption because it is one of the two stable interstitial positions in a diamond lattice ([66]). At these positions the interstitials form regular tetrahedra with lattice atoms as indicated in Fig. 3.3.5. But as stated by Hobler et al. [37] de-channeling from the $<$100$>$ channel is underestimated because the tetrahedral interstitial sites are in the $<$100$>$ atom rows while they are in the middle of the $<$110$>$ channel as shown in Fig. 3.16. The probability for a collision of particle with an interstitial placed at a tetrahedral interstitial sites is therefore significantly lower if the particle moves within an $<$100$>$ channel than if it moves within an $<$110$>$ channel.

To overcome this problem the actual position of an interstitial can be randomly smeared around the tetrahedral interstitial position as proposed by Posselt [64].

Alternatively an interstitial can be placed completely random within a certain area around the current position of the incident particle (Sec. 4.4.1). If this done only care has to be taken that the material density is preserved (3.175). Additionally it is not necessary to individually model the influence of vacancies and interstitials on a particle trajectory.