1. 1 Inverse Problems for Electrodiffusion Martin Burger Johannes Kepler University Linz SFB Numerical-Symbolic-Geometric Scientific Computing Radon Institute for Computational & Applied Mathematics

2. Inverse Problems for PNP-Systems Chicago, January 20052 Collaborations Heinz Engl, Marie-Therese Wolfram (Linz) Peter Markowich (Vienna) Rene Pinnau (Kaiserslautern) Michael Hinze (Dresden)

3. Inverse Problems for PNP-Systems Chicago, January 20053 Identification For most systems there are some parameters that cannot be determined directly (Parameter to be understood very general, could also be functions or even the system geometry appearing in the model) These parameters have to be determined by indirect measurements Measurements and parameters related by simulation model. Fitting model to data leads to mathematical optimization problem

4. Inverse Problems for PNP-Systems Chicago, January 20054 Optimal Design Modern engineering and increasingly biology is full of advanced design problems, which one could / should tackle as optimization tasks Ad-hoc optimization based on insight into the system becomes more and more difficult with increasing system complexity and decreasing feature size Alternative approach by numerical simulation and mathematical optimization techniques

5. Inverse Problems for PNP-Systems Chicago, January 20055 Inverse Problems Such optimal design and identification problems are usually called inverse problems (reverse engineering, inverse modeling, …) Forward problem: given the design variables / parameters, perform a model simulation Used to predict data Inverse problem used to relate model to data

6. Inverse Problems for PNP-Systems Chicago, January 20056 Inverse Problems Solving inverse problems means to look for the cause of some effect Optimal design: look for cause of desired effect Identification: look for the cause of observed effect Reversing the causality leads to ill-posedness: two different causes can lead to almost the same effect. Leads to difficulties in inverse problems

7. Inverse Problems for PNP-Systems Chicago, January 20057 Ill-Posed Problems Ill-posedness is of particular significance since data are not exact (measurement and model errors) Ill-posedness can have different consequences: - Non-existence of solutions - Non-uniqueness of solutions - Unstable dependence on data To compute stable approximations of the solution, regularization methods have to be used

8. Inverse Problems for PNP-Systems Chicago, January 20058 Regularization Basic idea of regularization: replacement of ill- posed problem by parameter-dependent family of well-posed problems Example: linear equation replaced by (Tikhonov regularization) Regularization parameter  controls smallest eigenvalue and yields stability

9. Inverse Problems for PNP-Systems Chicago, January 20059 Inverse Problems for PNP-Systems Identification or Design of parameters in coupled systems of Poisson and Nernst-Planck equations, describing transport and diffusion of charged particles Parameters are usually related to a permanent charge density Classical application: semiconductor dopant profiling

10. Inverse Problems for PNP-Systems Chicago, January 200510 Semiconductor Devices MOSFET / MESFET

11. Inverse Problems for PNP-Systems Chicago, January 200511 Dopant Profiling Typical inverse problems: - Design the device doping profile to optimize the device characteristics - Identify the device doping profile from measurements of the device characteristics Optimal design used to improve manufacturing, identification used for quality control

12. Inverse Problems for PNP-Systems Chicago, January 200512 Mathematical Model Stationary Drift Diffusion Model: PDE system for potential V, electron density n and hole density p in  (subset of R 2 ) Doping Profile C(x) enters as source term

13. Inverse Problems for PNP-Systems Chicago, January 200513 Boundary Conditions Boundary of  homogeneous Neumann boundary conditions on  N (insulated parts) and on Dirichlet boundary  D (Ohmic contacts )

14. Inverse Problems for PNP-Systems Chicago, January 200514 Device Characteristics Measured on a contact  0 part of  D : Outflow current density Capacitance

15. Inverse Problems for PNP-Systems Chicago, January 200515 Scaled Drift-Diffusion System After (exponential) transform to Slotboom variables (u=e -V n, p = e V p) and scaling: Similar transforms and scaling for boundary conditions

16. Inverse Problems for PNP-Systems Chicago, January 200516 Scaled Drift-Diffusion System Similar transforms and scaling for boundary Conditions Essential (possibly small) parameters - Debye length - Injection Parameter  - Applied Voltage U

17. Inverse Problems for PNP-Systems Chicago, January 200517 Scaled Drift-Diffusion System Inverse Problem for full model ( scale  = 1)

18. Inverse Problems for PNP-Systems Chicago, January 200518 Optimization Problem Take current measurements on a contact  0 in the following Least-Squares Optimization: minimize for N large

19. Inverse Problems for PNP-Systems Chicago, January 200519 Optimization Problem Due to ill-posedness, we need to regularize, e.g. C 0 is a given prior (a lot is known about C) Problem is of large scale, evaluation of F involves N solves of the nonlinear PNP systems

20. Inverse Problems for PNP-Systems Chicago, January 200520 Numerical Solution If N is large, we obtain a huge optimality system of 2(K+1)N+1 equations (6N+1 for DD) Direct discretization is challenging with respect to memory consumption and computational effort If we do gradient method, we can solve 3 x 3 subsystems, but the overall convergence is slow

21. Inverse Problems for PNP-Systems Chicago, January 200521 Sensitivies Define Lagrangian

22. Inverse Problems for PNP-Systems Chicago, January 200522 Sensitivies Primal equations with N different boundary conditions

23. Inverse Problems for PNP-Systems Chicago, January 200523 Sensitivies Dual equations

24. Inverse Problems for PNP-Systems Chicago, January 200524 Sensitivies Boundary conditions on contact  0 homogeneous boundary conditions else

25. Inverse Problems for PNP-Systems Chicago, January 200525 Sensitivies Optimality condition (H 1 - regularization) with homogeneous boundary conditions for C - C 0

26. Inverse Problems for PNP-Systems Chicago, January 200526 Numerical Solution Structure of KKT-System

27. Inverse Problems for PNP-Systems Chicago, January 200527 Numerical Solution 3 x 3 Subsystems with

28. Inverse Problems for PNP-Systems Chicago, January 200528 Close to Equilibrium For small applied voltages one can use linearization of DD system around U=0 Equilibrium potential V 0 satisfies Boundary conditions for V 0 with U = 0 → one-to-one relation between C and V 0

29. Inverse Problems for PNP-Systems Chicago, January 200529 Linearized DD System Linearized DD system around equilibrium(first order expansion in  for U =  ) Dirichlet boundary condition V 1 = - u 1 = v 1 =  depends only on V 0 : Identify V 0 (smoother !) instead of C

30. Inverse Problems for PNP-Systems Chicago, January 200530 Advantages of Linearization Linearization around equilibrium is not strongly coupled (triangular structure) Numerical solution easier around equilibrium Solution is always unique close to equilibrium Without capacitance data, no solution of Poisson equation needed

31. Inverse Problems for PNP-Systems Chicago, January 200531 Advantages of Linearization Under additional unipolarity (v = 0), scalar elliptic equation – the problem of identifying the equilibrium potential can be rewritten as the identification of a diffusion coefficient a = e V 0 Well-known problem from Impedance Tomography Caution: The inverse problem is always non-linear, even for the linearized DD model !

32. Inverse Problems for PNP-Systems Chicago, January 200532 Numerical Tests Test for a P-N Diode Real Doping ProfileInitial Guess

33. Inverse Problems for PNP-Systems Chicago, January 200533 Numerical Tests Different Voltage Sources

34. Inverse Problems for PNP-Systems Chicago, January 200534 Numerical Tests Reconstructions with first source

35. Inverse Problems for PNP-Systems Chicago, January 200535 Numerical Tests Reconstructions with second source

36. Inverse Problems for PNP-Systems Chicago, January 200536 The P-N Diode Simplest device geometry, two Ohmic contacts, single p-n junction

37. Inverse Problems for PNP-Systems Chicago, January 200537 Identifying P-N Junctions Doping profiles look often like a step function, with a single discontinuity curve  (p-n junction) Identification of p-n junction is of major interest in this case Voltage applied on contact 1, device characteristics measured on contact 2

38. Inverse Problems for PNP-Systems Chicago, January 200538 Results for C 0 = 10 20 m -3

39. Inverse Problems for PNP-Systems Chicago, January 200539 Results for C 0 = 10 21 m -3

40. Inverse Problems for PNP-Systems Chicago, January 200540 Instationary Problem Similar to problem with many measurements, but correlation between the problems (different time-steps) More data (time-dependent functions) BFGS for optimization problem (Wolfram 2005)

41. Inverse Problems for PNP-Systems Chicago, January 200541 Unipolar Diode Time-dependent reconstruction, 10% data noise

42. Inverse Problems for PNP-Systems Chicago, January 200542 Unipolar Diode N + NN + Current Measured Capacitance Measured

43. Inverse Problems for PNP-Systems Chicago, January 200543 Optimal Design Similar problems in optimal design Typical goal: maximize / increase current flow over a contact, but keep distance to reference state small Again modeled by minimizing a similar objective functional

44. Inverse Problems for PNP-Systems Chicago, January 200544 Optimal Design Increase of currents at different voltages, reference state C 0 Maximize „drive current“ at drive voltage U

45. Inverse Problems for PNP-Systems Chicago, January 200545 Numerical Result: p-n Diode Ballistic pn-diode, working point U=0.259V Desired current amplification 50%, I* = 1.5 I 0 Optimized doping profile,  =10 -2,10 -3

46. Inverse Problems for PNP-Systems Chicago, January 200546 Numerical Result: p-n Diode Optimized potential and CV-characteristic of the diode,  =10 -3

47. Inverse Problems for PNP-Systems Chicago, January 200547 Numerical Result: p-n Diode Optimized electron and hole density in the diode,  =10 -3

48. Inverse Problems for PNP-Systems Chicago, January 200548 Numerical Result: p-n Diode Objective functional, F, and S during the iteration,  =10 -2,10 -3

49. Inverse Problems for PNP-Systems Chicago, January 200549 Numerical Result: MESFET Metal-Semiconductor Field-Effect Transistor (MESFET) Source: U=0.1670 V, Gate: U = 0.2385 V Drain: U = 0.6670 V Desired current amplification 50%, I* = 1.5 I 0

50. Inverse Problems for PNP-Systems Chicago, January 200550 Numerical Result: MESFET Finite element mesh: 15434 triangular elements

51. Inverse Problems for PNP-Systems Chicago, January 200551 Numerical Result: MESFET Optimized Doping Profile (Almost piecewise constant initial doping profile)

52. Inverse Problems for PNP-Systems Chicago, January 200552 Numerical Result: MESFET Optimized Potential V

53. Inverse Problems for PNP-Systems Chicago, January 200553 Numerical Result: MESFET Evolution of Objective, F, and S

54. Inverse Problems for PNP-Systems Chicago, January 200554 Download and Contact Papers and Talks: www.indmath.uni-linz.ac.at/people/burger e-mail: martin.burger@jku.at