3. Discretization

IN ORDER to solve the system of partial differential equations, it is necessary to discretize them on an appropriate grid. In general, the solution must be calculated by means of numerical methods. The domain in which the solution is sought is decomposed in a large number of subdomains, in which the solution can be approximated by functions with a given structure. In that way one obtains a fairly large system of, in general nonlinear, algebraic equations. Due to the discretization it is impossible to obtain an exact solution of the analytically formulated problem. There are several techniques to discretize the equations. Among them are the finite difference method, the box integration method, and the finite element method. The former two will be discussed in more detail.

Subsections

• 3.1 Grid Generation
• 3.2 Finite Difference Method
  • 3.2.1 One-Dimensional TAYLOR Expansion
  • 3.2.2 One-Dimensional POISSON's Equation
  • 3.2.3 Final Remarks
  
  
• 3.3 Box Integration Method
  • 3.3.1 POISSON's Equation
  • 3.3.2 Six Moments Transport Model
    • 3.3.2.1 Continuity and Balance Equations
    • 3.3.2.2 Flux Equations
    • 3.3.2.3 Growth Function
  • 3.3.3 Final Remarks

M. Gritsch: Numerical Modeling of Silicon-on-Insulator MOSFETs PDF