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Fast Optimal Design of
Semiconductor Devices
Martin Burger
Institute for Computational and Applied Mathematics
European Institute for Molecular Imaging (EIMI)
Center for Nonlinear Science (CeNoS)
Westfälische Wilhelms-Universität Münster
joint work with Rene Pinnau, Michael Hinze
Introduction
 Models for Semiconductor Devices (Poisson
+ Kinetic)
 Optimal Design Tasks in Semiconductor
Devices
 Standard approach, sensitivities, difficulties
 One shot approach, advantages, globally
convergent Gummel iterations
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
Microelectronic System Design
 Modern microelectronics is full of
advanced design problems, which one
could / should tackle as optimization tasks
 The design of modern microelectronic
systems involves a variety of scales (nano
to macro) - and of mathematical models
 In this talk we consider a typical
microscale problem
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
Design of Semiconductor Devices
Typical microscale problem:
Design the device doping profile to optimize
the device characteristics (current-voltage
curves)
E.g.: maximize on-state current keeping
small off-state current
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
Mathematical Models
Model Structure: Poisson equation for
potential V, coupled to continuity
equations for (a vector) u
in W (subset of Rd)
Q(u) is the charge generated by u
Doping Profile C(x) enters as source term
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
Mathematical Models
Model Structure: Continuity equations K can
represent kinetic / quantum model, e.g.
 Drift-diffusion, energy transport, 6th order
 Quantum drift diffusion, Schrödinger, …
 Boltzmann statistics
 Hydrodynamic models
 ….
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
Drift-diffusion
Bipolar Drift Diffusion Model:
Vector u consists of electron density n and
hole density p
Scaled charge:
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Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
Device Characteristics
 Outflow current on a contact G (part of the
boundary)
 Optimal design: minimize a functional
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
Optimization Problem
 Example: locally maximize outflow current
around given state (with doping C*)
 Design functional:
 Stabilization functional:
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Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
Standard Approach
 Eliminate Poisson and continuity
equations, implicit relation
 Unconstrained optimization in C
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Fast Optimal Design of Semiconductor Devices
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Sensitivities for Standard Approach
Use chain rule
Solve coupled linearized model
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Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
Sensitivities for Standard Approach
Adjoint method
Solve coupled system
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
Standard Approach
Used for drift-diffusion model by Hinze-Pinnau
02, 03, Stockinger et. al 98, Plasun et. al. 98
Problem 1: implicit relation well-defined only
close to equilibrium (possible non-uniqueness)
Problem 2: existence and computation of
deriva-tives of objective functional with
respect to C (non-wellposedness of linearized
model)
Problem 3: numerical computations and effort
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
New Approach
Alternative to overcome difficulties:
Use
as the new design variable instead of doping
W corresponds to a scaled total charge
New objective:
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
New Constraints
Poisson + continuity equations
Note: triangular structure of the equations
Doping profile eliminated, can be determined
a-posteriori
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
New Approach
Used for drift-diffusion model mb-Pinnau 04
Energy transport Holst 07
Advantage 1: implicit relation between W and I
well-defined everywhere (triangular structure)
Advantage 2: existence and computation of
derivatives of objective functional with respect
to W (global wellposedness and simple
structure of linearized model)
Advantage 3: numerical computations, effort
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
New Approach
Advantage 4: Global convergence of
Gummel iteration for the design problem !
10.8.2007
Fast Optimal Design of Semiconductor Devices
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Optimality Condition
Karush-Kuhn-Tucker system for solutions of
optimal design problem
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Fast Optimal Design of Semiconductor Devices
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Gummel Iteration
Analogue of Gummel iteration for optimal
design problem
Note: Last equation is easy to solve
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Fast Optimal Design of Semiconductor Devices
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Stabilizing Functional
Examples
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Fast Optimal Design of Semiconductor Devices
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Gummel Iteration
This Gummel iteration is a descent method
for the reduced problem
Global convergence to solution of optimal
design problem can be obtained with
standard line-search
Total computational effort compareable to
two device simulations !
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
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
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
Numerical Result: p-n Diode
Optimized potential and CV-characteristic of
the diode, e =10-3
10.8.2007
Fast Optimal Design of Semiconductor Devices
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Numerical Result: p-n Diode
Optimized electron and hole density in the
diode, e =10-3
10.8.2007
Fast Optimal Design of Semiconductor Devices
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Numerical Result: p-n Diode
Objective functional, F, and S during the
iteration, e =10-2,10-3
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
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
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
Numerical Result: MESFET
Finite element mesh: 15434 triangular
elements
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Fast Optimal Design of Semiconductor Devices
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Numerical Result: MESFET
Optimized Doping Profile
(Almost piecewise constant initial doping
profile)
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Fast Optimal Design of Semiconductor Devices
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Numerical Result: MESFET
Optimized Potential V
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Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
Numerical Result: MESFET
Evolution of Objective, F, and S
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
Efficiency
Comparison to previous optimizations:
- Black-box, gradients by FD (Strasser et.
al.): 62 design parameters, >4000 solves of
drift-diffusion
- Semi-Black-box, gradients by adjoint
method (Hinze, Pinnau): > 100 design
parameters, > 200 drift-diffusion solves
- New one-shot approach, arbitrary design
parameters (here > 15000), < 3 drift-diffusion
solves
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
Next Step
 On-State / Off-State Design:
Maximize drive current by keeping leakage
currents small
 On-state treated similar as above, off-state
via linearization around equilibrium
Similar treatment possible, globally
convergent Gummel iteration
 Similar tasks for Ion Channels
mb-Engl-Eisenberg, SIAP 07
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien
Download and Contact
Papers and talks at
www.math.uni-muenster.de/u/burger
Email
[email protected]
10.8.2007
Fast Optimal Design of Semiconductor Devices
Equadiff 07, TU Wien