Inverse Problems for Electrodiffusion

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)

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

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

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

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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

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

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

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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

10
Semiconductor Devices

• MOSFET / MESFET

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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 W (subset of R2)
• Doping Profile C(x) enters as source term

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Boundary Conditions
Boundary of W homogeneous Neumann boundary
conditions on GN (insulated parts) and on
Dirichlet boundary GD (Ohmic contacts)
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Device Characteristics

• Measured on a contact G0 part of GD
• Outflow current density
• Capacitance

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Scaled Drift-Diffusion System
After (exponential) transform to Slotboom
variables (ue-V n, p eV p) and
scaling Similar transforms and scaling for
boundary conditions
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Scaled Drift-Diffusion System

• Similar transforms and scaling for boundary
• Conditions
• Essential (possibly small) parameters
• - Debye length l
• - Injection Parameter d
• Applied Voltage U

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Scaled Drift-Diffusion System
Inverse Problem for full model ( scale d 1)
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Optimization Problem
Take current measurements on a contact G0
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(K1)N1 equations (6N1 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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Sensitivies
Define Lagrangian
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Sensitivies
Primal equations with N different boundary
conditions
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Sensitivies
Dual equations
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Sensitivies
Boundary conditions on contact G0 homogeneo
us boundary conditions else
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Sensitivies
Optimality condition (H1 - regularization) wi
th homogeneous boundary conditions for C - C0
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Numerical Solution
Structure of KKT-System
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Numerical Solution
3 x 3 Subsystems with
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Close to Equilibrium
For small applied voltages one can use
linearization of DD system around U0 Equilibrium
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 e for U e F )
• Dirichlet boundary condition V1 - u1 v1 F
  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 Tests

• Test for a P-N Diode
• Real Doping Profile Initial Guess

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Numerical Tests

• Different Voltage Sources

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Numerical Tests

• Reconstructions with first source

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Numerical Tests

• Reconstructions with second source

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The P-N Diode

• Simplest device geometry, two Ohmic contacts,
  single p-n junction

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

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Results for C0 1020m-3
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Results for C0 1021m-3
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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)
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Unipolar Diode
Time-dependent reconstruction, 10 data noise
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Unipolar Diode NNN
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 Diode
Ballistic pn-diode, working point
U0.259V Desired current amplification 50, I
1.5 I0 Optimized doping profile, e
10-2,10-3
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Numerical Result p-n Diode
Optimized potential and
CV-characteristic of the diode, e 10-3
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Numerical Result p-n Diode
Optimized electron and hole density in
the diode, e 10-3
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Numerical Result p-n Diode
Objective functional, F, and S during
the iteration, e 10-2,10-3
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Numerical Result MESFET
Metal-Semiconductor Field-Effect Transistor
(MESFET) Source U0.1670 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 MESFET
Finite element mesh 15434 triangular elements
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Numerical Result MESFET
Optimized Doping Profile (Almost piecewise
constant initial doping profile)
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Numerical Result MESFET
Optimized Potential V
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Numerical Result MESFET
Evolution of Objective, F, and S
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Download and Contact

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• www.indmath.uni-linz.ac.at/people/burger
• e-mail martin.burger_at_jku.at