Transcript Level Set Methods For Inverse Obstacle Problems

Inverse Problems in
Semiconductor Devices
Martin Burger
Johannes Kepler Universität Linz
Outline
Introduction: Drift-Diffusion Model
Inverse Dopant Profiling
Sensitivities
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Joint work with
Heinz Engl, RICAM
Peter Markowich, Universität Wien & RICAM
Antonio Leitao, Florianopolis & RICAM
Paola Pietra, Pavia
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Inverse Dopant Profiling
Identify the device doping profile from
measurements of the device
characteristics
Device characteristics:
Current-Voltage map
Voltage-Capacitance map
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Inverse Dopant Profiling
Device characteristics are obtained by
applying different voltage patterns (spacetime) on some contact
Measurements:
Outflow Current on Contacts
Capacitance = variation of charge with
respect to voltage modulation
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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 and
on Dirichlet boundary GD (Ohmic Contacts)
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Device Characteristics
Measured on a contact G0 on GD :
Outflow current density
Capacitance
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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
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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
following
on a contact G0 in the
Least-Squares Optimization: minimize
for N large
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Optimization Problem
This least squares problem is ill-posed
Consider Tikhonov-regularized version
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 drift-diffusion system
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Sensitivies
Define Lagrangian
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Sensitivies
Primal equations,
with different boundary conditions
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Sensitivies
Dual equations
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Sensitivies
Boundary conditions on contact G0
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
If N is large, we obtain a huge optimality system of
6N+1 equations
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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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=0
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 linearized
potential 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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Identifiability
Natural question: do the data determine the doping
profile uniquely ?
For a quasi 1D device (ballistic diode), the doping
profile cannot be determined, information content of
current data corresponds to one real number (slope of
the I-V curve)
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Identifiability
For a unipolar 2D device (MESFET, MOSFET),
voltage-current data around equilibrium suffice only
when currents ar measured on the whole boundary
(B-Engl-Markowich-Pietra 01) – not realistic !
For a unipolar 3D device, voltage-current data around
equilibrium determine the doping profile uniquely
under reasonable conditions
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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
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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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Model Reduction 1
Typically small Debye length l
Consider limit l → 0 (zero space charge)
Equilibrium potential equation becomes algebraic
relation between V0 and C
- V0 is piecewise constant
- identify junction in V0 or a = exp(V0 )
Continuity equations
div ( a  u1 ) = div ( a-1  v1 ) = 0
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Identifiability
Since we only want to identify the junction G, we
need less measurements
For a unipolar diode with zero space charge, the
junction is locally unique if we only measure the
current for a single applied voltage (N=1)
Computational effort reduced to scalar elliptic
equation
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Model Reduction 2
If, in addition to zero space charge, there is also low
injection (d small), the model can be reduced further
(cf. Schmeiser 91)
In the P-region, the function u satisfies
Du=0
Current is determined by  u only
Inverse boundary value problem in the P-region,
overposed boundary values on contact 2 (u = 0 on G,
u = 1 on contact 2, current flux = normal derivative of
u measured)
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Identifiability
For a P-N Diode, junction is determined uniquely by
a single current measurement (B-Engl-MarkowichPietra 01)
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Numerical Results
For zero space charge and low injection,
computational effort reduces to inverse free boundary
problem for Laplace equation
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Results for C0 = 1020m-3
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Results for C0 = 1021m-3
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