Damage formation and annealing studies of low energy ion implants ...

Equations 2.1 and 2.2 show that the force and the potential decrease rapidly with
increasing atomic separation. Collisions leading to deflections occur over a relatively
small range of distances, the closest being about 10 -11 m and obviously the most distant
collisions occur at a distance no greater than half the interatomic spacing.
This situation is not changed significantly when screening functions are
introduced and the entire collision sequences associated with ion implantation over the
energy range of present interest can be described in terms of successive binary
collisions. An additional justification for this approximation is that over the major part
of the highly energetic interaction, the energy transferred in the collisions significantly
exceeds the binding energies of the target atoms in the solid. Hence there is no need to
adjust the mass of the target atoms to compensate for binding effects.
When the distance between the two atoms is of the order of a0, e.g. for 10 – 200
keV H and He in Si, the Coulomb potential is no longer valid, because the positive
charges are likely to be reduced by the presence of inner shell electrons in the region
between the two nuclei. Several analytical and empirical approximations have been
developed to calculate the effects of electron screening which leads to a screened
Coulomb potential, where the Coulomb potential is multiplied by a screening function
Z1Z
2 ⎛ − r ⎞
V()
r = exp⎜
⎟ (2.3)
4πε
0r
⎝ a ⎠
where r is the separation distance and a is the screening radius, or parameter. This
screening parameter is an important concept. It moderates the effect of the nuclear
positive charge on the electrons because the inner electrons shield some of the nuclear
charge. Bohr suggested an empirical exponential form (4). More complicated potentials
were derived based on analytical results, such as those developed by Firsov (5), given in
equation 2.4, Moliere (6), and Lindhard (7), all of which are based on the Thomas –
Fermi free electron gas model (8).
a
a
0
= (2.4)
1 1 2
3 3 3
( Z1
+ Z2
)
For MEIS using 100 – 200 keV H or He ions it is generally accepted that a
screening function as expressed in the Molière potential (6) or a modified form of it is a
good screening function to use (9, 10). The implications for MEIS are discussed further
in chapter 4.
A universal potential (11, 12, 13) developed by Biersack and Ziegler, again
based on treatment of the Thomas – Fermi model (free electron gas) has been shown to
give a good description for intermediate separations (0 < r < a0) which are of vital
17