The structure of diffusion equations (3.1-1), (3.1-2) and boundary conditions (3.1-3) which can be treated by PROMIS is rather general.
are the concentrations (dependent variables) and 
is the
number of quantities (equations). 
denotes the electrostatic
potentials and is one of the dependent variables . In the flux
definition (3.1-2) the first term (
) corresponds to
diffusion fluxes. Note, that the flux 
of quantity 
may depend on
the gradients of all concentrations (
).
For instance, dopant fluxes may be driven by gradients in the point defect
concentrations. The second term (
) provides for field
induced fluxes (drift). For almost all applications the local charge
neutrality approximation is sufficient [Hu72], [Shr80]. In this
case the effect of the electric field can be entirely included in the first
term of (3.1-2) and affects only the values of 
(cf. Section 3.2.1).
The third term 
accounts for convective fluxes. The coefficient
is of dimension velocity. Mostly, has the form
, with 
a driving force and 
a mobility. The
fourth term 
is just a utility term without a specific physical
meaning.
The coefficients for the time derivatives 
will most often
be the unity matrix. The last term in the continuity equation
(3.1-1) 
denotes the recombination term.
In the boundary conditions (3.1-3), 
denotes the flux of the 
-th quantity perpendicular to the surface,
flowing into the simulation area. Thus, any known type of boundary
conditions, such as Neumann, Dirichlet, Cauchy for elliptic and parabolic
systems of PDEs can be treated.
All coefficients 
may be functions of temperature,
time and the spatial coordinates. All coefficients but 
may
be functions of the dependent variables . Nevertheless, certain
restrictions on the coefficients must be considered. If the matrix of
is positive semi-definit, the are negative, and
, the system described by
(3.1-1) - (3.1-2) is well-posed ([Mit80],
[Vem81]).
This set of equations allows the treatment of a wide variety of diffusion phenomena including point defect assisted diffusion under oxidizing conditions [Hu83], [Gil89], [Dun89], [Fah90], [Lev90], clustering effects and precipitation [Fai73], [Gue82], [Sol90], point defect and pair formation kinetics [Hu85], [Fai88], [Dür89], [Kim89], [Fai89], [Mor89], [Mul91], [Hu92b], grain boundary diffusion in polysilicon [Jon88], [Tak89], [Ozo90], [Pfi90], [Lau90], [Jon91], [Orl92], and the impact of stress effects on diffusion [Orl90], [Orl91a].
