Generally, 
is a function of several physical variables defined on
two given grid points ij (4.1-5), where 
is the electrostatic
potential, 
the net doping, 
the intrinsic carrier density and
C the doping concentration at the grid points i and j, respectively.
We will discuss now several discretizations of diffusion current models starting with the diffusion flux followed after the first Fickian law (4.1-6).
Using the finite-difference approximation the discretized diffusion current at the ij-th grid line becomes to (4.1-7).
The average diffusion coefficient 
is defined as the
projection of the diffusivities at the grid points i and j, 
and
, to the mid-edge point of 
(see 4.1-8).
The current discretization (4.1-7) allows the easy implementation of
a static clustering model. Only the dopant concentrations 
and 
have to be replaced by the mobile species 
and 
, where an
algebraic relation (4.3-20) can be used to calculate the active
species.
To account for the electric field of the charged dopants the diffusion flux
model (4.1-6) has to be extended by a field enhancement term, where
Z denotes the charge state of the dopant. Equation (4.1-9) gives
the diffusion flux for single negatively charged dopants under thermal
equilibrium and by assuming Boltzmann statistics to predict the
electrostatic potential 
, where the electron
concentration is scaled to the intrinsic carrier density 
.
For a general formulation of the field enhancement the electron concentration n has to be replaced by the majority carrier concentration. The discretized diffusion current is then given by (4.1-10).
Using the box integration method offers the possibility of using the Scharfetter-Gummel discretization scheme for the diffusion current [Sch69]. Again, the field enhancement flux reads
The electrostatic potential is scaled to the thermal voltage
and the Einstein relation is used to express
the mobility 
. Assuming a constant diffusion flux along the
discretization interval 
, we obtain a differential equation
for the dopant concentration within the discretization interval
to (4.1-12). The boundary conditions are given to
and 
.
This equation represents an ordinary differential equation for the unknown
, where the electrostatic field is assumed to be constant between
the interval . This implies a linear dependence of the
electrostatic potential within . The electric field is
approximated by the finite difference expression (4.1-13).
By solving the differential equation (4.1-12) for and
re-inserting the solution, the discretized current between the grid points
ij is given by (4.1-14), where B(x) are the the
Bernoulli-coefficients (4.1-15) and 
an auxiliary variable
(4.1-16).
