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

2.3.1 Models for inelastic energy loss
Bohr used classical mechanics to describe the interaction of a charged moving
particle with the electrons in the target, (1, 2). The energy loss under these
circumstances is based on the classical calculation of the momentum and energy
transferred to an electron by the ion in a collision. This simple picture ignores the fact
that electrons are bound to atomic nuclei. The ionisation energy required to separate the
electron from the atom has to be accounted for and therefore the scattering event
becomes inelastic. Bohr showed that the rate of energy loss varies with the ion velocity.
A number of approximations were developed to take into account possible
energetic states of an electron in the target and the average population of these states.
The work of Bethe (18) and Bloch (19) produced a quantum mechanical approach to the
inelastic energy loss. The Bethe – Bloch formula only describes the energy loss well at
energies beyond the maximum in the dE/dx curve to near relativistic velocities, i.e. in
the region from 10MeV/amu to 2 GeV/amu (see Figure 2.3 later). In this regime there is
an increase in the inelastic stopping power for a decrease in velocity.
For lower energies, two models that better describe the inelastic energy loss are
that of Firsov (20) and Lindhard (21, 22). Neither model comprehensively takes account
of every interaction taking place, but both theories adequately describe the general
behaviour of the stopping power with regard to the energy dependence and the
magnitude.
Firsov’s expression (20) is based on a simple geometric model of momentum
exchange between the projectile and the target atom during the interpenetration of the
electron clouds surrounding the two colliding atoms. Each binary collision is viewed as
leading to an overlap of their electron orbits and the formation of ‘quasi molecules’ with
electrons from both atoms crossing the instantaneous quasi boundary between the
atoms, which then take on the momentum of the atom to which they temporarily attach.
Electrons transferred either temporarily or permanently from the ion to the atom and
vice versa cause a transfer of momentum. Thus electrons from the moving atom lose
momentum in transferring to the initially stationary target atom and those from the
struck atom gain momentum from the incident particle. This momentum exchange
processes leads to energy loss of the incident particle.
Another way to look at the inelastic energy loss is the model of Lindhard and
Scharff (21, 22) based on a free electron gas. In the gas, positively charged nuclei are
embedded in effectively a plasma. The electrons are no longer considered to be attached
to any one atom. The model assumes that the free electron gas consists of electrons at
22