q . D . ![]() |
= ![]() ![]() ![]() ![]() |
(3.17) |
q . D . ![]() |
= ![]() ![]() ![]() ![]() |
(3.18) |
This equation system for n2 and n3 can be rewritten as
![]() |
n1 and n4 are the fixed values of the carrier concentration on the left and the right boundary (Dirichlet boundary condition).
The characteristic polynomial of the system matrix is
The solutions of (3.20) are the eigenvalues of (3.19).
![]() |
= - ![]() ![]() ![]() ![]() ![]() |
(3.21) |
![]() |
= - ![]() |
(3.22) |
According to (3.7) the spectral condition number of (3.19) is
![]() ![]() ![]() ![]() ![]() ![]() |
(3.23) |
and, therefore, for large
the spectral condition of the system matrix
will be poor. Thus, if the internal state of a device results in a large value
of
the solver cannot compute the solution of the linear system with
sufficient accuracy. The result will be an increase of iteration steps for the
Newton scheme, if convergence can be achieved at all.
This problem can be alleviated by transforming the linear system
(3.19) as follows. Adding the second equation to the first
one and scaling the second equation with
![]() ![]() ![]() ![]() ![]() ![]() | |
= ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
(3.24) |
results in the new system
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
(3.25) |
The new system matrix has the eigenvalues
|
(3.26) |
hence, for large values of
the spectral condition number
is nearly independent of
because in the limit for
the factors
and
are 1.
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
(3.27) |
The strong influence of the internal state of the device on the spectral condition of the equation system has been eliminated.