combine with 
electrons to form a cluster. These
clusters are immobile and therefore the diffusivity in the high
concentration region is vanishing. The total dopant concentration 
is divided into a mobile active 
and a immobile clustered 
part. Once the clusters are formed, they are not preserved during the
diffusion process. There is an exchange between the active and clustered
species during the transient diffusion process. In the case of thermal
equilibrium this exchange process is referred to as static clustering,
and as dynamic clustering otherwise. Based on rate equation
(3.1-33) and the law of mass action, the exchange rate of dopants due
to dynamic clustering 
can be expressed by (4.3-12), where
and 
are the clustering and declustering rate,
respectively.
The net active concentration 
is given by (4.3-13), where
the charge state for the clustered species reads 
. The
according electron concentration n and the electric field are given in
(4.3-14) and (4.3-15), respecively.
The dynamic clustering diffusion system for the immobile ( 
)
and the mobile dopants is then given by (4.3-16) and (4.3-17),
respectively, where the diffusion flux of the active dopants is given by a
field enhancement model.
If the cluster formation takes place under thermal equilibrium ( 
), we can define the equilibrium clustering rate 
with a mass
action law to
where the carrier concentration n is determined by (4.3-13).
Therefore, the total dopant concentration can be expressed as given
by (4.3-19).
After scaling of (4.3-19) to the solubility limit of the dopant
and further approximations using an artificial clustering factor
, we get the static clustering relation for the dopant as
depicted in (4.3-20).
Finally, the static clustering diffusion system for the dopant 
reads
to (4.3-21), where the clustering kinetics is considered by the
active concentration used as diffusing species instead of the total ones.
The electric field is calculated as previously shown in (4.3-15),
where is substituted by 
. As the clustering
reaction is in equilibrium for the static clustering model the solubility
limit is fixed during the whole diffusion process. On the contrary, the
dynamic clustering model allows temporary variations of the effective
solubility limit. The static clustering model benefits by its easy
implementation and the fact that it needs no additional cluster equation to
solve. Within PROMIS-NT the static clustering model is implemented by
an additional diffusion current model, which automatically corrects the net
doping according to clustering conditions.
