Drift-Diffusion Modeling

1. Drift-Diffusion Modeling Prepared by Dragica Vasileska Professor Arizona State University

2. Outline of the Lecture • Classification of PDEs • Why Numerical Analysis? • Numerical Solution Sequence • Flow-Chart of Equilibrium Poisson Equation Solver • Discretization of the Continuity Equation • Numerical Solution Techniques for Sparse Matrices • Flow-Chart of 1D Drift-Diffusion Simulator

3. Classification of PDEs

4. Classification of PDEs Different mathematical and physical behaviors: • Elliptic Type • Parabolic Type • Hyperbolic Type System of coupled equations for several variables: • Time : first-derivative (second-derivative for wave equation) • Space: first- and second-derivatives

5. Classification of PDEs (cont.) General form of second-order PDEs ( 2 variables)

6. PDE Model Problems • Hyperbolic (Propagation) • Advection equation (First-order linear) • Wave equation (Second-order linear)

7. PDE Model Problems (cont.) • Parabolic (Time- or space-marching) • Burger’s equation (Second-order nonlinear) • Fourier equation (Second-order linear) (Diffusion / dispersion)

8. PDE Model Problems (cont.) • Elliptic (Diffusion, equilibrium problems) • Laplace/Poisson (second-order linear) • Helmholtz equation

9. Well-Posed Problem Numerically well-posed • Discretization equations • Auxiliary conditions (discretized approximated) • the computational solution exists (existence) • the computational solution is unique (uniqueness) • the computational solution depends continuously on the approximate auxiliary data • the algorithm should be well-posed (stable) also

10. Boundary and InitialConditions R • Initial conditions: starting point for propagation problems • Boundary conditions: specified on domain boundaries to provide the interior solution in computational domain R s n

11. Numerical Methods • Complex geometry • Complex equations (nonlinear, coupled) • Complex initial / boundary conditions • No analytic solutions • Numerical methods needed !!

12. Why Numerical Analysis?

13. Coupling of Transport Equations to Poisson and Band-Structure Solvers D. Vasileska and S.M. Goodnick, Computational Electronics, published by Morgan & Claypool , 2006.

14. Drift-Diffusion Approach Constitutive Equations • Poisson • Continuity Equations • Current Density Equations S. Selberherr: "Analysis and Simulation of Semiconductor Devices“, Springer, 1984.

15. Poisson/Laplace Equation Solution Poisson/Laplace Equation No knowledge of solving of PDEs With knowledge for solving of PDEs Method of images Theoretical Approaches Numerical Methods: finite difference finite elements Poisson Laplace Green’s function method Method of separation of variables (Fourier analysis)

16. Numerical Solution Sequence

17. Numerical Solution Details Governing Equations ICS/BCS System of Algebraic Equations Equation (Matrix) Solver ApproximateSolution Discretization φi (x,y,z,t) p (x,y,z,t) n (x,y,z,t) Continuous Solutions Finite-Difference Finite-Volume Finite-Element Spectral Boundary Element Hybrid Discrete Nodal Values Tridiagonal SOR Gauss-Seidel Krylov Multigrid D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State University, Tempe, AZ.

18. What is next? • MESH • Finite Difference Discretization • Boundary Conditions

19. MESH TYPE The course of action taken in three steps is dictated by the nature of the problembeing solved, the solution region, and the boundary conditions. The mostcommonlyused grid patterns for two-dimensional problems are Common grid patterns: (a) rectangular grid, (b) skew grid, (c) triangular grid,(d) circular grid.

20. Mesh Size

21. Example for Meshing

22. 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, andcentraldifferences.

23. Finite Difference Schemes We can approximate derivative of f(x), slopeor 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,

24. 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.

25. Finite Difference Schemes The approach used above in obtaining finite difference approximations is rather intuitive. A more general approach is using Taylor’s series. According to the wellknown expansion, and Upon adding these expansions, where O(x)4 is the error introduced by truncating the series. We say that this erroris of the order (x)4 or simply O(x)4. Therefore, O(x)4represents terms thatare not greater than (x)4. Assuming that these terms are negligible,

26. Finite Difference Schemes Subtracting from We obtain and neglecting terms of theorder (x)3 yields This shows that the leading errors of the order (x)2. Similarly, the forward and backward difference formula have truncation errors of O(x).

27. Poisson Equation

28. Poisson Equation Linearization • The 1D Poisson equation is of the form:

29. Φ => Φ + d

30. Renormalized Form LD=sqrt(qni/εVT)

31. Finite Difference Representation Equilibrium: Non-Equilibrium: n calculated using PM coupling and p still calculated as in equilibrium case (quasi-equilibrium approximation)

32. Criterion for Convergence There are several criteria for the convergence of the iterative procedure when solving the Poisson equation, but the simplest one is that nowhere on the mesh the absolute value of the potential update is larger than 1E-5 V. This criterion has shown to be sufficient for all device simulations that have been performed within the Computational Electronics community.

33. Boundary Conditions

34. 1D Discretization

35. 2D Discretization

36. 2D Discretization (cont’d)

37. Flow-Chart of Equilibrium Poisson Equation Solver

39. Discretization of the Continuity Equation

40. Sharfetter-Gummel Discretization Scheme

41. (a) Linearized Scheme • Within the linearized scheme, one has that • This scheme can lead to substantial errors in regions of high electric fields and highly doped devices.

42. (b) Sharfetter-Gummel Scheme

43. Bernouli Function Implementation Others as Well …

44. Numerical Solution Techniques for Sparse Matrices

45. Numerical Methods Objective: Speed, Accuracy at minimum cost • Numerical Accuracy(error analysis) • Numerical Stability(stability analysis) • Numerical Efficiency (minimize cost) • Validation (model/prototype data, field data, analytic solution, theory, asymptotic solution) • Reliability and Flexibility(reduce preparation and debugging time) • Flow Visualization(graphics and animations)

46. Solution Methods

47. LU Decomposition (cont’d)

48. Iterative Methods • Iterative (or relaxation) methods start with a first approximation which is successively improved by the repeated application (i.e. the “iteration”) of the same algorithm, until a sufficient accuracy is obtained. • In this way, the original approximation is “relaxed” toward the exact solution which is numerically more stable. • Iterative methods are used most often for large sparse system of equations, and always when a good approximation of the solution is known. • Error analysis and convergence rate are two crucial aspects of the theory of iterative methods.

49. Error Equation

50. Jacobi, Gauss-Seidel Methods