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Inverse Problems for Electrodiffusion

Inverse Problems for Electrodiffusion. Martin Burger. Johannes Kepler University Linz SFB Numerical-Symbolic-Geometric Scientific Computing Radon Institute for Computational & Applied Mathematics. Collaborations. Heinz Engl, Marie-Therese Wolfram (Linz) Peter Markowich (Vienna)

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Inverse Problems for Electrodiffusion

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

  2. Collaborations • Heinz Engl, Marie-Therese Wolfram (Linz) • Peter Markowich (Vienna) • Rene Pinnau (Kaiserslautern) • Michael Hinze (Dresden) Inverse Problems for PNP-Systems Chicago, January 2005

  3. 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 Inverse Problems for PNP-Systems Chicago, January 2005

  4. 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 Inverse Problems for PNP-Systems Chicago, January 2005

  5. 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 simulationUsed to predict data • Inverse problem used to relate model to data Inverse Problems for PNP-Systems Chicago, January 2005

  6. 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 Inverse Problems for PNP-Systems Chicago, January 2005

  7. Ill-Posed Problems • Ill-posedness is of particular significance since dataare 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 Inverse Problems for PNP-Systems Chicago, January 2005

  8. 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 a controls smallest eigenvalue and yields stability Inverse Problems for PNP-Systems Chicago, January 2005

  9. 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 Inverse Problems for PNP-Systems Chicago, January 2005

  10. Semiconductor Devices • MOSFET / MESFET Inverse Problems for PNP-Systems Chicago, January 2005

  11. Dopant Profiling • Typical inverse problems: • Design the device doping profileto optimize the device characteristics • Identify the device doping profilefrom measurements of the device characteristics • Optimal design used to improve manufacturing, identification used for quality control Inverse Problems for PNP-Systems Chicago, January 2005

  12. Mathematical Model • Stationary Drift Diffusion Model: PDE system for potential V, electron density n and hole density p in W (subset of R2) • Doping Profile C(x)enters as source term Inverse Problems for PNP-Systems Chicago, January 2005

  13. Boundary Conditions Boundary ofW : homogeneous Neumann boundary conditions onGN(insulated parts) and on Dirichlet boundaryGD (Ohmic contacts) Inverse Problems for PNP-Systems Chicago, January 2005

  14. Device Characteristics • Measured on a contactG0 part ofGD: • Outflow current density • Capacitance Inverse Problems for PNP-Systems Chicago, January 2005

  15. Scaled Drift-Diffusion System After (exponential) transform to Slotboom variables (u=e-V n, p = eV p) and scaling: Similar transforms and scaling for boundary conditions Inverse Problems for PNP-Systems Chicago, January 2005

  16. Scaled Drift-Diffusion System • Similar transforms and scaling for boundary • Conditions • Essential (possibly small) parameters • - Debye lengthl • - Injection Parameterd • Applied Voltage U Inverse Problems for PNP-Systems Chicago, January 2005

  17. Scaled Drift-Diffusion System Inverse Problem for full model ( scaled = 1) Inverse Problems for PNP-Systems Chicago, January 2005

  18. Optimization Problem Take current measurements on a contact G0 in the following Least-Squares Optimization: minimize for N large Inverse Problems for PNP-Systems Chicago, January 2005

  19. Optimization Problem • Due to ill-posedness, we need to regularize, e.g. • C0is 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 Inverse Problems for PNP-Systems Chicago, January 2005

  20. 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 Inverse Problems for PNP-Systems Chicago, January 2005

  21. Sensitivies Define Lagrangian Inverse Problems for PNP-Systems Chicago, January 2005

  22. Sensitivies Primal equations with N different boundary conditions Inverse Problems for PNP-Systems Chicago, January 2005

  23. Sensitivies Dual equations Inverse Problems for PNP-Systems Chicago, January 2005

  24. Sensitivies Boundary conditions on contactG0 homogeneous boundary conditions else Inverse Problems for PNP-Systems Chicago, January 2005

  25. Sensitivies Optimality condition (H1 - regularization) with homogeneous boundary conditions forC - C0 Inverse Problems for PNP-Systems Chicago, January 2005

  26. Numerical Solution Structure of KKT-System Inverse Problems for PNP-Systems Chicago, January 2005

  27. Numerical Solution 3 x 3 Subsystems with Inverse Problems for PNP-Systems Chicago, January 2005

  28. Close to Equilibrium For small applied voltages one can use linearization of DD system around U=0 Equilibrium potential V0 satisfies Boundary conditions forV0withU = 0 →one-to-one relation betweenCandV0 Inverse Problems for PNP-Systems Chicago, January 2005

  29. Linearized DD System • Linearized DD system around equilibrium(first order expansion in e forU = e F ) • Dirichlet boundary conditionV1 = - u1 = v1 = F depends only onV0: Identify V0(smoother !) instead ofC Inverse Problems for PNP-Systems Chicago, January 2005

  30. 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 Inverse Problems for PNP-Systems Chicago, January 2005

  31. 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 = eV0 • Well-known problem from Impedance Tomography • Caution: The inverse problem is always non-linear, even for the linearized DD model ! Inverse Problems for PNP-Systems Chicago, January 2005

  32. Numerical Tests Test for a P-N Diode Real Doping Profile Initial Guess Inverse Problems for PNP-Systems Chicago, January 2005

  33. Numerical Tests Different Voltage Sources Inverse Problems for PNP-Systems Chicago, January 2005

  34. Numerical Tests Reconstructions with first source Inverse Problems for PNP-Systems Chicago, January 2005

  35. Numerical Tests Reconstructions with second source Inverse Problems for PNP-Systems Chicago, January 2005

  36. The P-N Diode Simplest device geometry, two Ohmic contacts, single p-n junction Inverse Problems for PNP-Systems Chicago, January 2005

  37. Identifying P-N Junctions • Doping profiles look often like a step function, with a single discontinuity curve G(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 Inverse Problems for PNP-Systems Chicago, January 2005

  38. Results for C0 = 1020m-3 Inverse Problems for PNP-Systems Chicago, January 2005

  39. Results for C0 = 1021m-3 Inverse Problems for PNP-Systems Chicago, January 2005

  40. 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) Inverse Problems for PNP-Systems Chicago, January 2005

  41. Unipolar Diode Time-dependent reconstruction, 10% data noise Inverse Problems for PNP-Systems Chicago, January 2005

  42. Unipolar Diode N+NN+ Current Measured Capacitance Measured Inverse Problems for PNP-Systems Chicago, January 2005

  43. 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 Inverse Problems for PNP-Systems Chicago, January 2005

  44. Optimal Design • Increase of currents at different voltages, reference state C0 • Maximize „drive current“ at drive voltage U Inverse Problems for PNP-Systems Chicago, January 2005

  45. Numerical Result: p-n Diode Ballistic pn-diode, working point U=0.259V Desired current amplification 50%, I* = 1.5 I0 Optimized doping profile,e =10-2,10-3 Inverse Problems for PNP-Systems Chicago, January 2005

  46. Numerical Result: p-n Diode Optimized potential and CV-characteristic of the diode,e =10-3 Inverse Problems for PNP-Systems Chicago, January 2005

  47. Numerical Result: p-n Diode Optimized electron and hole density in the diode,e =10-3 Inverse Problems for PNP-Systems Chicago, January 2005

  48. Numerical Result: p-n Diode Objective functional, F, and S during the iteration,e =10-2,10-3 Inverse Problems for PNP-Systems Chicago, January 2005

  49. 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 I0 Inverse Problems for PNP-Systems Chicago, January 2005

  50. Numerical Result: MESFET Finite element mesh: 15434 triangular elements Inverse Problems for PNP-Systems Chicago, January 2005

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