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Inverse Problems for Electrodiffusion
Martin Burger
Johannes Kepler University LinzSFB Numerical-Symbolic-Geometric Scientific ComputingRadon Institute for Computational & Applied Mathematics
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Collaborations
Heinz Engl, Marie-Therese Wolfram (Linz)
Peter Markowich (Vienna)
Rene Pinnau (Kaiserslautern)
Michael Hinze (Dresden)
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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
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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
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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
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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
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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
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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
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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
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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
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Mathematical Model Stationary Drift Diffusion Model:
PDE system for potential V, electron density n and hole density p
in (subset of R2) Doping Profile C(x) enters as source term
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Boundary ConditionsBoundary of homogeneous Neumann boundary conditions on N (insulated parts) and
on Dirichlet boundary D (Ohmic contacts)
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Device Characteristics Measured on a contact 0 part of D : Outflow current density
Capacitance
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Scaled Drift-Diffusion SystemAfter (exponential) transform to Slotboom
variables (u=e-V n, p = eV p) and scaling:
Similar transforms and scaling for boundary
conditions
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Scaled Drift-Diffusion SystemSimilar transforms and scaling for boundary
Conditions
Essential (possibly small) parameters
- Debye length - Injection Parameter - Applied Voltage U
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Scaled Drift-Diffusion System
Inverse Problem for full model ( scale = 1)
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Optimization ProblemTake current measurements on a contact 0 in the following
Least-Squares Optimization: minimize
for N large
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Optimization Problem
Due to ill-posedness, we need to regularize, e.g.
C0 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
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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
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SensitiviesPrimal equations
with N different boundary conditions
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SensitiviesBoundary conditions on contact 0
homogeneous boundary conditions else
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Sensitivies
Optimality condition (H1 - regularization)
with homogeneous boundary conditions for C - C0
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Numerical Solution
Structure of KKT-System
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Close to Equilibrium
For small applied voltages one can use linearization of DD system around U=0Equilibrium potential V0 satisfies
Boundary conditions for V0 with U = 0→ one-to-one relation between C and V0
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Linearized DD System Linearized DD system around equilibrium(first order expansion in for U = )
Dirichlet boundary condition V1 = - u1 = v1 = depends only on V0:
Identify V0 (smoother !) instead of C
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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
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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 !
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Numerical TestsTest for a P-N Diode
Real Doping Profile Initial Guess
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Numerical TestsReconstructions with first source
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Numerical TestsReconstructions with second source
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The P-N DiodeSimplest device geometry, two Ohmic contacts, single p-n junction
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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
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Instationary ProblemSimilar to problem with many measurements, but correlation between the problems (different time-steps)
More data (time-dependent functions)
BFGS for optimization problem (Wolfram 2005)
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Unipolar Diode
Time-dependent reconstruction, 10% data noise
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Unipolar Diode N+NN+
Current Measured Capacitance Measured
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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
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Optimal Design Increase of currents at different voltages, reference state C0
Maximize „drive current“ at drive voltage U
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Numerical Result: p-n DiodeBallistic pn-diode, working point U=0.259V
Desired current amplification 50%, I* = 1.5 I0
Optimized doping profile, =10-2,10-3
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Numerical Result: p-n Diode
Optimized potential and CV-characteristic of the diode, =10-3
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Numerical Result: p-n Diode
Optimized electron and hole density in the diode, =10-3
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Numerical Result: p-n Diode
Objective functional, F, and S during the iteration, =10-2,10-3
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Numerical Result: MESFETMetal-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
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Numerical Result: MESFETFinite element mesh: 15434 triangular elements
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Numerical Result: MESFETOptimized Doping Profile(Almost piecewise constant initial doping profile)
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Numerical Result: MESFETOptimized Potential V
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Numerical Result: MESFETEvolution of Objective, F, and S