4.4.1.2 Collision Partner in Crystalline Materials

Figure 4.5: Schematic figure for the selection of collision partners for simultaneous collisions in crystalline materials.

Figure 4.6: Schematic figure for the selection of collision partners in crystalline materials.

In crystalline materials the cores of the target are placed around crystal lattice positions. The collision partner is determined by looking for a lattice position in the direction of motion, considering the following criteria [35]:

• Lattice sites with a distance smaller than the search distance to the current ion position are selected (small blue circles in Fig. 4.6). The search distance has a default value of three times the lattice constant as suggested in [30].

• Several criteria are applied to remove selected lattice sites from this initial list:
  • The lattice site is located opposite to the direction of motion, because an ion cannot collide with an atom opposite to the direction of motion.
  • The free flight path (projection of the vector from the ion position to the lattice site position on the vector of the direction of motion) is shorter than the minimum free flight path which is set to five percent of the lattice constant.
  • The free flight path is larger than the maximum free flight path which is 95% of the lattice constant.
  • The impact parameter (distance from the vector of the direction of motion) is larger than the maximum impact parameter which has a default value of the size of the lattice constant.
  • The distance between the current ion position and the lattice site is smaller than the minimum distance which is 15% of the lattice constant.
  • Collision partners which have been involved in the previous collision are removed from the list to avoid multiply treating a collision partner. This check is only performed if the scattering angle falls below the minimum scattering angle which has a default value of 30 $\,^\circ\;$.
  This list of criteria and the default values of the parameters used in the criteria are suggested in [30], to optimize the performance of the simulation, to avoid a skipping of collision events, to avoid multiply treating the same collision partner and to avoid placing collision partners unphysically close to each other. Up to now the influence of these effects on the simulation results has not been investigated rigorously. Therefore it is not possible to estimate the error which would arises from neglecting these effects.

• The remaining lattice positions are ordered by the free flight path. From this list all lattice sites are accepted as collision partners whose free flight path do not differ more than the interval for simultaneous collisions from the shortest free flight path in the list. Additionally lattice sites are considered with a flight path difference less than the minimum free flight path from the longest flight path according to previous condition (Fig. 4.5). An interval for simultaneous collisions of 10% of the lattice constant is suggested in [30].

• The actual collision partners are placed randomly in a plane perpendicular to the direction of motion, which contains the lattice position. A Gaussian probability distribution function is assumed. Thereby the thermal vibration of the atoms is considered (Sec. 3.3.4). The vibration amplitude is a user defined parameter (command-line parameter atomVibration), which should be set according to the average wafer temperature during the implantation.

As a result of this procedure more than one collision partner can be selected and therefore several nuclear collision events sometimes have to be calculated simultaneously, while each of these collisions is treated as a binary collision event. Simultaneous collisions are not an approximation to a multi-particle interaction, but are used to avoid the skipping of collision events, especially if an ion moves along the center of a crystalline channel, where at least two lattice sites can be characterized approximately by the same criteria in the selection procedure mentioned above. Without multiple collisions the above selection criteria could result in skipping one of these potential collision partners. The free flight path for a simultaneous collision is the largest free flight path to one of the accepted collision partners.

Footnotes

... angle

The minimum scattering angle can be set by the command-line parameter minScatteringAngle