The Stationary Poisson Equation

In semiconductor devices, the potential retardation effects are completely negligible so

1. We can neglect the computation of the magnetic field (at least for the majority of semiconductor devices, but certainly for the Silicon devices)
2. We can adopt the stationary description of the electric potential, i.e. we select and calculate only the Poisson equation among the set of Maxwell's equations.

We, thus, report the Poisson equation

$$\nabla \cdot [ \epsilon ({\bf {x}}) \nabla \phi ({\bf {x}},t) ] = - q [ N_D({\bf {x}}) - N_A({\bf {x}}) - n({\bf {x}},t) + p({\bf {x}},t)]$$ (6.5)

Actually, if we have a two-dimensional regular finite-difference grid, the discretization of the Poisson will give an algebraic system to solve, which is quite complicated to solve, because the boundary conditions are difficult to implement in a generic simulator such as textbfGNU Archimedes and, furthermore, this algebraic system is consuming from the point of view of computer memory (even if we can use well-known methods applied to sparse matrices).
These reasons have influenced the author of GNU Archimedes to adopt a lightly different approach in the simulation of the electrostatic potential. (And for this, I thanks Vittorio Romano for his excellent advices).