The equations in PROMIS allow the treatment of a very wide range
of diffusion problems. For simple diffusion models at sufficiently low
concentrations, that are concentrations smaller than the intrinsic carrier
concentration [Fai81], the equations for the different quantities are
practically independent, i.e. decoupled. For many sophisticated models, for
instance, when taking into account the Frenkel-pair mechanism for point
defects, the equations are strongly coupled. Since we have to treat all
cases, decoupled iteration schemes are inappropriate for our purposes and we
have to use a Newton-like method (3.5-2) for the solution of
.
At the 
iteration 
is the Fréchet
derivative (Jacobian) of 
and 
is the
residual vector 
which is frequently called the right-hand-side. We start at an initial guess 
. This initial
guess is obtained from the solution of previous time steps by linear
extrapolation or, in case of the first time step, it is taken from the
initial condition.
Given this initial guess , we solve the linear equation
(3.5-2) for 
and compute a new solution
from (3.5-3).
The classical Newton method (
) suffers from a phenomenon called
overshoot: the update 
is frequently overestimated
- even by many orders of magnitude. To control this overshoot, a damping
factor 
, usually called Deuflhard-damping factor, is
introduced [Deu74], [Ban81]. We choose the value for 
from
the decreasing sequence (3.5-4). The largest value of
is taken which satisfies the condition of sufficient decrease
(3.5-5), where 
denotes the machine epsilon.
We do not allow arbitrarily small values for . If falls short of
, where 
is the maximum number of Deuflhard-damping steps, we
accept either a full Newton update without regard to the decrease condition
or terminate - depending on switches set by the user.
An additional nonlinear damping of the variables is used in
PROMIS. When a solution 
exceeds physically reasonable
values 
, 
(e.g. negative dopant concentrations), its value
is clipped to 
for exponential quantities, such as
concentrations (cf. Section 3.6.2) or to 
for linear quantities, such as the electrostatic
potential, respectively. Although this approach is mathematically hazardous,
because the direction of the correction vector is altered, it has proven quite
satisfactory in practical applications.
Alternative selections of were proposed by Bank and Rose [Ban80],
[Ban81] and Coughran et al. [Cou83] which are successfully applied
to semiconductor equations, e.g. in [Fic81], [Bür90],
[Hei91b]. For typical process simulation problems a damping 
was necessary only in rare cases, e.g. in some configurations during
pair diffusion and oxygen precipitation calculations.
When we have found our improved solution , we calculate
again the Fréchet derivative 
and continue
the Newton update (3.5-2), (3.5-3) until
the residual of the nonlinear system is sufficiently small
(3.5-6). Sufficiently small means smaller than a desired Newton
accuracy 
.
For the Newton method we need the Fréchet derivative 
of
equations. Furthermore, the termination criterion (3.5-6)
claims for an appropriate scaling of the equations 
, and we have
to solve a large sparse linear system (3.5-2). These aspects
are addressed in Section 3.5.2, Section 3.5.3 and
Section 3.5.4.
For the structure of the implemented modified Newton algorithm see Program 3.5-1.
