Fast Optimal Design of Semiconductor Devices Martin Burger Institute for Computational and Applied...

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

10.8.2007 Fast Optimal Design of Semiconductor Devices Equadiff 07, TU Wien

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


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


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


Mathematical Models

Model Structure: Poisson equation for potential V, coupled to continuity equations for (a vector) u

in (subset of Rd)

Q(u) is the charge generated by u

Doping Profile C(x) enters as source term


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 ….


Drift-diffusion

Bipolar Drift Diffusion Model:

Vector u consists of electron density n and hole density p

Scaled charge:


Device Characteristics Outflow current on a contact (part of the boundary)

Optimal design: minimize a functional


Optimization Problem

Example: locally maximize outflow current around given state (with doping C*)

Design functional:

Stabilization functional:


Standard Approach

Eliminate Poisson and continuity equations, implicit relation

Unconstrained optimization in C


Sensitivities for Standard Approach

Use chain rule

Solve coupled linearized model


Sensitivities for Standard ApproachAdjoint method

Solve coupled system


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


New Approach

Alternative to overcome difficulties:Use

as the new design variable instead of dopingW corresponds to a scaled total charge

New objective:


New Constraints

Poisson + continuity equations

Note: triangular structure of the equationsDoping profile eliminated, can be determined a-posteriori


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


New Approach

Advantage 4: Global convergence of Gummel iteration for the design problem !


Optimality Condition

Karush-Kuhn-Tucker system for solutions of optimal design problem


Gummel Iteration

Analogue of Gummel iteration for optimal design problem

Note: Last equation is easy to solve


Stabilizing Functional

Examples


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 !


Numerical Result: p-n Diode

Ballistic pn-diode, working point U=0.259V

Desired current amplification 50%, I* = 1.5 I0

Optimized doping profile, =10-2,10-3



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: 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



Finite element mesh: 15434 triangular elements



Optimized Doping Profile(Almost piecewise constant initial doping profile)



Optimized Potential V


Numerical Result: MESFETEvolution of Objective, F, and S


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


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 Channelsmb-Engl-Eisenberg, SIAP 07


Papers and talks at

www.math.uni-muenster.de/u/burger

Email

[email protected]

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