), the diffusivities of the dopants show some
concentration dependence. The gradient of the ionized dopant causes an
electric field, which operates as additional drift term for the diffusion
flux. The modified dopant diffusion flux is given by (3.1-24),
where 
is the charge state of the dopant, -1 for singly charged
donors and +1 for singly charged acceptors, and 
denotes the
electrostatic potential.
The electrostatic potential is determined by the Poisson equation (3.1-25), where n and p are the electron and hole concentrations, respectively.
The quantity 
represents the total net concentration of dopants,
which is defined for z ionized dopants by:
Under special assumptions, like Boltzmann statistics and charge neutrality
(3.1-27) it is not necessary to solve Poisson's equation
numerically. In this case, the electron and hole concentration can be
expressed analytically by (3.1-28) and (3.1-29), where 
is the built-in potential.
With (3.1-27) to (3.1-29) the built-in potential can then be calculated explicitly.
Due to the fact that electric interactions between charged species are occurring on a much faster time scale than the chemical interactions, the above assumptions are valid under diffusion conditions. Examinations on applying the numerical solution (3.1-25) and the approximation of the Poisson equation (3.1-30) to diffusion profiles can be found in [Jün86].
When the build-in potential is substituted into (3.1-24), the influence of the electric field on the dopant diffusion is modeled consistently. If we think in terms of multiple dopant diffusion, the impact of the electric field is quite different. Whether the electric field is enhancing or retarding the diffusion flux depends on the local doping conditions. For an acceptor diffusing in a p-doped region, enhancement takes place, while there is retardation for the acceptor if the net-doping changes its sign at an adjacent p-n junction.
Besides the field effect at extrinsic doping conditions, we further have to
modify the diffusivities for the dopants to account for concentration
dependence and Fermi level dependence. As we expressed the multiply charged
point defects in terms of the equilibrium concentrations (see 3.1-9),
the intrinsic dopant diffusivity (3.1-23) needs to be extended, to
account for multiple charges states of dopants and point-defects. Let us
assume for the sake of completeness that point defects exists in either
neutral, singly charged or doubly charged states. According to the given
atomic diffusion mechanism, there are several combinations of charged
dopant-defect pairs AX possible. Each charged pair r may also be related
to different diffusivities 
. To calculate the effective dopant
diffusivity 
, we have to sum up all particular diffusivities by
(3.1-31), where 
denotes the intrinsic diffusivity 
of the dopant [Fai81].
The temperature dependence of the diffusivities obeys an Arrhenius law 
. Diffusivities for the most common dopants under
extrinsic conditions are shown in Table 3.1-1.
Table 3.1-1: Arrhenius parameters for the
diffusivities of B, As, P and Sb in silicon after
[Fai81]
The vacant and occupied positions in Table 3.1-1 refer to the mechanism involved during diffusion. While the data for boron and arsenic are reliable, the diffusion behavior of phosphorus is much more complicated than Table 3.1-1 suggests. It depends on the doping level, whether phosphorus diffuses via interstitials or vacancies or both. Some authors also included pair-pair diffusion mechanisms to model experimental data for phosphorus diffusion obtained by [Yos79] more or less accurately [Dun92] [Gha95]. Till now these experimental data could not be reproduced satisfactory.
