Tutorial From Semi-Classical to Quantum Transport Modeling：从半经典量子输运建模教程.ppt

Tutorial From Semi-Classical to Quantum Transport Modeling：从半经典量子输运建模教程
Outline of the Lecture
Classification of P of Algebraic Equations
Equation (Matrix) Solver
ApproximateSolution
Continuous Solutions
Finite-Difference
Finite-Volume
Finite-Element
Spectral
Boundary Element
Hybrid
Discrete Nodal Values
Tridiagonal
SOR
Gauss-Seidel
Krylov
Multigrid
φi (x,y,z,t)
p (x,y,z,t)
n (x,y,z,t)
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes,
Arizona State University, Tempe, AZ.
What is next?
MESH
Finite Difference Discretization
Boundary Conditions
MESH TYPE
The course of action taken in three steps is dictated by the nature of the problem being solved, the solution region, and the boundary conditions. The most commonly used grid patterns for two-dimensional problems are
Common grid patterns: (a) rectangular grid, (b) skew grid, (c) triangular grid, (d) circular grid.
Mesh Size
Example for Meshing
Finite Difference Schemes
Before finding the finite difference solutions to specific PDEs, we will look at how
one constructs finite difference approximations from a given differential equation. This essentially involves estimating derivatives numerically. Let’s assume f(x) shown below:
Estimates for the derivative of f (x) at P using forward, backward, and central differences.
Finite Difference Schemes
We can approximate derivative of f(x), slope or the tangent at P by the slope of the arc PB, giving the forward-difference formula,
or the slope of the arc AP, yielding the backward-difference formula,
or the slope of the arc AB, resulting in the central-difference formula,
Finite Difference Schemes
We can also estimate the second derivative of f (x) at P as
or
Any approximation of a derivative in terms of values at a discrete set of points is
called finite difference approximation.
Finite Difference Schemes
The approach used above in obtaining finite difference approx