Using Jacobi Method for Solving Full Semiconductor Equations

Question

I am looking to perform modeling of a MOSFET device and have therefore been researching computational methods for how to do so. Quite often when solving the Poisson equation, the reader is pointed to using the Jacobi method in which the second order differential equation is converted into an algebraic equation involving nearby grid points. This equation is then solved by specifying the boundary conditions and iterating over the grid points until some convergence criteria is reached.

When solving the full semiconductor equations however, meaning both the Poisson equation and the Drift-Diffusion equations along with continuity requirements, I never see the Jacobi method referenced. Instead it is usually suggested that Gummel method or Newton method are used. Is there a reason for this, or perhaps a reason the Jacobi method cannot be used in this instance?

Answer 1

The Jacobi method solves problems in which the matrix is strictly diagonal dominant, which means that for every row, the diagonal term is greater-than or equal to the sum of the remaining terms: $$\vert a_{ii}\vert\geq\sum_{j\neq i}\vert a_{ij}\vert\,\forall i$$

When you start to add terms to your Poisson equation, the matrix you end up with is (usually) no longer strictly diagonal dominant and the Jacobi method cannot be used. Hence, other suggestions appearing.