inverse problems for electrodiffusion

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Inverse Problems for Electrodiffusion

Martin Burger

Johannes Kepler University LinzSFB Numerical-Symbolic-Geometric Scientific ComputingRadon Institute for Computational & Applied Mathematics

Inverse Problems for PNP-Systems

Chicago, January 2005 2

Collaborations

Heinz Engl, Marie-Therese Wolfram (Linz)

Peter Markowich (Vienna)

Rene Pinnau (Kaiserslautern)

Michael Hinze (Dresden)



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



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 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 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



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



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



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



Semiconductor Devices MOSFET / MESFET



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



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



Boundary ConditionsBoundary of homogeneous Neumann boundary conditions on N (insulated parts) and

on Dirichlet boundary D (Ohmic contacts)



Device Characteristics Measured on a contact 0 part of D : Outflow current density

Capacitance



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



Scaled Drift-Diffusion SystemSimilar transforms and scaling for boundary

Conditions

Essential (possibly small) parameters

- Debye length - Injection Parameter - Applied Voltage U



Scaled Drift-Diffusion System

Inverse Problem for full model ( scale = 1)



Optimization ProblemTake current measurements on a contact 0 in the following

Least-Squares Optimization: minimize

for N large



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



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



Sensitivies

Define Lagrangian



SensitiviesPrimal equations

with N different boundary conditions



SensitiviesDual equations



SensitiviesBoundary conditions on contact 0

homogeneous boundary conditions else



Sensitivies

Optimality condition (H1 - regularization)

with homogeneous boundary conditions for C - C0



Numerical Solution

Structure of KKT-System



Numerical Solution3 x 3 Subsystems

with



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



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



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



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 !



Numerical TestsTest for a P-N Diode

Real Doping Profile Initial Guess



Numerical TestsDifferent Voltage Sources



Numerical TestsReconstructions with first source



Numerical TestsReconstructions with second source



The P-N DiodeSimplest device geometry, two Ohmic contacts, single p-n junction



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



Results for C0 = 1020m-3



Results for C0 = 1021m-3



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)



Unipolar Diode

Time-dependent reconstruction, 10% data noise



Unipolar Diode N+NN+

Current Measured Capacitance Measured



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



Optimal Design Increase of currents at different voltages, reference state C0

Maximize „drive current“ at drive voltage U



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



Numerical Result: p-n Diode

Optimized potential and CV-characteristic of the diode, =10-3




Optimized electron and hole density in the diode, =10-3




Objective functional, F, and S during the iteration, =10-2,10-3



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



Numerical Result: MESFETFinite element mesh: 15434 triangular elements



Numerical Result: MESFETOptimized Doping Profile(Almost piecewise constant initial doping profile)



Numerical Result: MESFETOptimized Potential V



Numerical Result: MESFETEvolution of Objective, F, and S



Download and Contact

Papers and Talks:

www.indmath.uni-linz.ac.at/people/burger

e-mail: [email protected]

inverse problems for electrodiffusion

Documents

data inverse problems

identification problems

design of parameters

advanced design problems

stabilityinverse problems

problems illposedness

simulation model

model simulation