The strongly implicit discretization of the basic semiconductor
equations 3.1, 3.2 and 3.3 is often
applied to solve the transient problem in semiconductor devices. The
difference system obtained by the strongly implicit discretization, linearized
at an equilibrium point, is absolutely stable (stable independent of time
step)
. The difference equations are nonlinear
and coupled. Thereby, they ought to be solved simultaneously, using
e.g. Newton's iterative method. This amounts to solving a large system of
equations, instead of solving three much smaller systems as in a sequential
approach. Moreover, when including the trap-dynamics
equations 3.11, the number of space variables becomes very large
in a simultaneous approach.
Therefore, we will apply a sequential algorithm to solve the basic semiconductor equations 3.1, 3.2 and 3.3, including the trap-dynamics equations 3.11. The discretization methods and schemes for solving the corresponding difference equations in a sequential manner have been considerably investigated for the semiconductor-device problems in the literature [373][314][313][283], but usually assuming vanishing generation-recombination. Some of these results which we directly adapted to solve our problem are repeated here. It is worth to note that the presented algorithm has shown absolute stability and high efficiency in all charge-pumping simulations carried out for different MOS devices and a variety of the terminal pulses.
Let us assume that the solution is known for the moment 
: the potential
, carrier concentrations 
and 
and the trap occupancy function
. We consider the solution for the time step 
.
After integrating equation 3.1 in the interval 
and multiplying by 
we arrive at
Since the explicit discretization leads to difference equations which are not
stable for large time steps ([440]), we prefer to discretize both
time-dependent current-continuity equations 3.1
and 3.2 by the strongly implicit method with respect to carrier
concentrations. After replacing 
and 
with their
values for 
, 
and it follows
The discretized Poisson equation 3.3 reads
In the previous equations and henceforward we drop the discretization in the
position space. Let us neglect the generation-recombination terms at present.
M.Sever (his previous family name was Mock) has proven convergence
([440]) of the difference
equations 3.13 - 3.15 in [314]. He
also claimed this scheme is not stable for the time steps larger than the
minimal effective dielectric time
constant in device 
. This
time constant is very small for common parameters, much shorter than the time
intervals involved in the charge pumping.
To overcome the instability of 3.13 - 3.15
M.Sever has proposed in [314][313] an unconditionally stable
difference scheme. It is based on solving the total current continuity equation
instead of the Poisson equation 3.3. In 3.16,
is the displacement
current. Expression 3.16 follows by
replacing 3.1 and 3.2 into 3.3 after
differentiating 3.3 in time. The trap-related terms vanish after
the substitution, because
holds.
This scheme consists of the linear difference
equations 3.13, 3.14 and
which is a nonlinear difference equation related to the continuity equation for the total current [373][313]. An other proposal for this method is presented in [314] (see also [416]). It consists of equation 3.17 and
Equations 3.18 and 3.19 are the same
as 3.13 and 3.14, but 
and
are evaluated using new potential 
instead of , after
solving 3.17 first. Both systems of difference equations are
absolutely stable, linearized at the equilibrium assuming vanishing
generation-recombination [314][313]. For both systems, the solution
does not satisfy the Poisson equation at 
, as it is evident after
replacing the solution 
in 3.15
.
However, based on this discretization method, M.Sever has constructed an
iterative scheme which exhibits very good behavior [314]:
where 
is an iteration counter. The procedure consists of
sequentially solving 3.20, 3.21
and 3.22. At the beginning
, 
and 
are
assumed. After [314], this procedure linearized at the equilibrium
assuming vanishing generation-recombination converges independent of the time
step 
. For 
(one sequence) it is equivalent
to 3.17 - 3.19. For 
the
solution of the procedure 3.20 - 3.22 converges
to the solution of the nonlinear coupled system obtained by the strongly
implicit discretization of the basic semiconductor
equations 3.1, 3.2 and 3.3
([314]), thus it satisfies the Poisson equation. In practice, after
truncating the procedure at a finite 
, the Poisson equation is satisfied
with some residual error.
To reduce the residual of the Poisson equation B.S.Polsky and J.S.Rimshans
developed in [373] (and further applied in [283]) an absolute
stable extension of the difference
scheme 3.13, 3.14 and 3.17. They
solve this system with the Poisson equation 3.15 extended with
a conveniently chosen damping (stabilizing) term. The potential
from 3.17 is applied as a predictor for the Poisson equation to
obtain , which results in a remarkable reduction of the residual
error in the Poisson equation (see footnote 17). However,
according to [373][283] it seems that they did not apply the
outstandingly good iterative
algorithm 3.20 - 3.22.
In the algorithm we employ, the sequential
procedure 3.20, 3.22 and 3.21 is
applied at the beginning of every new time step. After a given accuracy is
achieved, the time-dependent Gummel procedure ([159]) is applied to
obtain the final solution, using the solution
of 3.20, 3.22 and 3.21 as an
initial solution. For the `switch criterion' we chose that the maximal local
correction in the potential
becomes smaller than the
assumed value 
(Figure 1.2 in [191]). The time-dependent
Gummel procedure we apply follows from 3.13, 3.14
and 3.15:
where is the iteration counter. Additional details of the algorithm are
given in [191]. Since the final solution is obtained by
solving 3.25, the Poisson equation is fulfilled with the
desired (controlled) accuracy at , independent of the time step. No
proof for the convergence of the algorithm is available now. From the practical
side, however, this algorithm has always shown absolute and fast convergence
for time steps ranged from very short (
) to very long (
) in many
simulations. It combines the stability of the initial
procedure 3.20, 3.22 and 3.21
with faster convergence of the final
procedure 3.23, 3.24 and 3.25.
For further discussion the difference equation 3.24 is written discretized in both time and position space
where the matrix 
, due to discretization in the position space, depends
either on 
or 
. 
is the unit matrix.
is the net generation rate averaged in time
(the last right-hand-side term in 3.24).
Since high trap densities commonly occur in practice, for both interface
and bulk traps, the term can introduce a
large local perturbation in the carrier continuity equation. Consequently,
this term is evaluated here in advance for 
. In
deriving 3.26 we used
The same idea is applied in the discretization of the Poisson equation 3.25 in time and position space (see Chapter 7 in [416] also)
where 
is the difference form of the Laplace operator 
. The term
at the main diagonal of the
left-hand-side matrix is crucial for a stable and efficient convergence of
the sequential algorithm presented, when high trap densities are assumed in
device.
