have to be used. However, these implants
are hard to control regarding accuracy and reproducibility. In addition
clustering and precipitation effects come along with the prediction of the
final active doping profile [Pac95]. The transient activation model is
a first approach to simulate the activation process of phosphorus dopants
after a sub-amorphizing channeling implant into bare silicon.
Previous work to model the diffusion process of phosphorus in silicon incorporating point defect interaction has been presented by various authors. Yoshida et al. [Yos74] presented an extensive series of experimental data for the vacancy mechanism of phosphorus during oxidation of silicon. Fair and Tsai [Fai77] extended the vacancy mechanism by including doubly negatively charged vacancies to model the ``Kink'' and ``Tail'' behavior of high concentration phosphorus. A rigorous description of phosphorus diffusion was obtained by Dunham [Dun92] including also pair/pair interactions, but several assumptions were taken to simplify the model. All the above mentioned models are dealing with high concentration phosphorus and point defects injected from surface oxidation.
We present a new model for activation of low concentration phosphorus after a previous dopant implant into bare silicon. The excess point defects are generated by lattice damage of dopant implant or by previous silicon implants.
We use a point defect assisted activation mechanism to determine the active profile from the chemical doping profile during the annealing process, where vacancy and interstitial interaction plays a crucial role. Present models of point defect impurity diffusion in silicon favor the pairing of substitutional dopants with point defects [Dun92][Bac92]. These models are based on fully activated initial dopant profiles including enhancement due to point defects. The model we present describes the activation process as well as the diffusion process. We start up with an unactivated dopant species assuming pairing of dopants at interstitial sites with interstitial point defects (I). Due to recombination these dopant/defect pairs are split into substitutional dopants and recombining point defects (I-V).
The basic rate equations are given from (4.3-28) to (4.3-33), taking into account the actual state of the species, where i means interstitial, and s substitutional at the lattice position:
I, V, and A represent interstitial, vacancy, and dopant species and AI, and AV the paired dopants. (4.3-28) is known as kick-out reaction, (4.3-29) describes the corresponding vacancy mechanism, where (4.3-30) is the Frank-Turnbull mechanism and (4.3-31) the opposite vacancy reaction. Finally, (4.3-32) represents the pair/pair recombination and (4.3-33) the bulk recombination. Note, that there are three states possible for the dopant (interstitial, substitutional, and paired) during the diffusion process. If we consider the above reactions to set up a diffusion model, a full description of coupled diffusion reads:
refers to the according generation/recombination reaction and
refers to the diffusion flux of the species. Only paired dopants
and the point defects are able to diffuse, the substitutional and
interstitial dopant species are immobile and can vary by rate
reactions. Under intrinsic doping conditions, we can neglect pair/pair
recombination, because this effect occurs only at high concentrations. From
oxidation and nitridation experiments it is known that phosphorus diffuses
via interstitial mechanism under intrinsic conditions
[Fah83]. Consequently, all terms with dopant/vacancy pairs can be
eliminated. There is also negligible field enhancement at intrinsic
conditions, which leads to a pure diffusive flux given by (4.3-40),
where 
is the charge dependent diffusivity of the diffusing species
I, V, AI and modeled as given by (4.3-3). With the above
assumptions the system of diffusion equations can be simplified for
intrinsic phosphorus to:
We assume that all ionization processes are near equilibrium, because the
electronic interactions are fast compared to the atomic diffusion processes,
and as having intrinsic conditions we do not need to distinguish between
different charge states of the dopants. Therefore, the forward and backward
reaction rates of the different charge levels can be summed up to the
neutral reaction rates. This simplifies the structure of the net pairing
rates to one reaction rate 
and one equilibrium reaction constant
for each generation/recombination term to extract, see for instance
(4.3-47) to (4.3-49).
