By introducing a vector potential and a scalar potential MAXWELL's equations can
often be rewritten in a more practical form. The vector potential
is defined by
|
(2.7) |
which fulfills eqn. (2.2) since
evaluates to zero for
every vector field
. Inserting eqn. (2.7) into eqn. (2.1) gives
|
(2.8) |
Interchanging the order of the time derivative and the curl operator,
|
(2.9) |
and using the associative property of the curl operator,
|
(2.10) |
the argument of the curl operator can be substituted by the gradient of a scalar potential
|
(2.11) |
since
yields zero for every scalar field . The minus
sign on the right hand side of eqn. (2.11) is introduced by convention based on
historical reasons.
In the quasi-stationary case, which holds true for semiconductor devices2.1, the time
derivative of the vector potential can be neglected
|
(2.12) |
POISSON's equation is finally obtained by inserting eqn. (2.12) into
eqn. (2.5)
|
(2.13) |
which is in turn inserted into eqn. (2.4)
|
(2.14) |
In the case of vanishing space charge density POISSON's equation simplifies
to the LAPLACE equation
|
(2.15) |
M. Gritsch: Numerical Modeling of Silicon-on-Insulator MOSFETs PDF