To model the transport mechanism in polysilicon correctly, an appropriate
diffusion model for polysilicon has to be designed. For general polysilicon
layers it is not possible to account for the exact microstructure of the
grains and grain boundaries. The most popular approach for polysilicon
material is to split up the available dopants into a grain interior
concentration 
and grain boundary concentration 
. To
evaluate the grain boundary concentration it is necessary to use a
simplified model of the microstructure of the polysilicon grain
boundaries. The grain boundaries which are assumed to be two-dimensional
interfaces with a certain area density of free states are expanded to a
volumnar concentration by scaling with the local grain size. Therefore, the
grain boundary concentration reads 
, where 
is the
area dopant density within the grain boundary, r the local grain size. Due
to this homogenization the detailed grain boundary information, which is
practically not available for simulation purposes, vanishes. The remaining
grain interior concentration is than given by subtraction of from
the available total dopant concentration 
(4.3-50).
As the polysilicon layers are excessively doped clustering occurs in the
grain bulks. Therefore, we calculate the active and mobile part of the dopant
species 
with a static clustering model (4.3-51), where
is the solubility limit for the dopant species and
is a fitting parameter related to the cluster size.
Using the static clustering approach implies the assumption that dopants can be delivered fast enough from the grain interior regions into the grain boundaries so that no temporary leak of dopants occurs in the grain boundaries.
With the diffusion mechanisms described in the previous section
we can define the dopant diffusion flux
for the grain boundaries 
and for the grain bulks
as given by (4.3-52) and (4.3-53),
respectively.
The electric field in the grain interior regions is depicted in
(4.3-54). The fast diffusion in the grain boundaries is captured by
the corresponding first term of (4.3-52) in combination with the
extensively high diffusivity 
, where the second term denotes the
dopant flux caused by the grain growth. The prefactor 
is a tensor
which acts as weighting factor for the diffusion flux. This weighting factor
is calculated from a combination of local grain size, grain main axis
orientation and lateral to vertical grain size aspect ratio to model the
anisotropic diffusion behavior of polysilicon.
The segregation kinetics between grain interiors and grain boundaries is
accounted for by the generation/recombination factor 
given by
(4.3-55).
The exchange of dopants is modeled by means of a trapping factor t and an
emission factor e, where 
denotes the maximum number of free
states in the grain boundary. If the grain boundary is not filled with
dopants, active dopants from the grain interior are delivered, whereas
dopants diffuse into the grain interiors if the maximum number of free
states in the grain boundary has been exceeded. This mechanism is also able
to capture temporary grain boundary saturation effects.
A full description of the polysilicon model including the dynamic grain growth behavior (see section polysilicon grain growth) for a given dopant reads:
After implantation of the dopant into the polysilicon material only a small
portion 
resides at grain boundaries, where the major part rests at
interstitial sites within the grains. represents the total area
which is covered by grain boundaries, hence, it depends on the initial grain
size 
and the grain boundary thickness 
by 
. It appears from simulations that is not a critical
parameter, because after a few seconds the grain boundaries are fully
occupied due to their unique energetic properties for hosting dopant atoms
.
