Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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: 10.8.2007 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: 10.8.2007 Fast Optimal Design of Semiconductor Devices Equadiff 07, TU Wien Standard Approach Eliminate Poisson and continuity equations, implicit relation Unconstrained optimization in C 10.8.2007 Fast Optimal Design of Semiconductor Devices Equadiff 07, TU Wien Sensitivities for Standard Approach Use chain rule Solve coupled linearized model 10.8.2007 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 Equadiff 07, TU Wien Optimality Condition Karush-Kuhn-Tucker system for solutions of optimal design problem 10.8.2007 Fast Optimal Design of Semiconductor Devices Equadiff 07, TU Wien Gummel Iteration Analogue of Gummel iteration for optimal design problem Note: Last equation is easy to solve 10.8.2007 Fast Optimal Design of Semiconductor Devices Equadiff 07, TU Wien Stabilizing Functional Examples 10.8.2007 Fast Optimal Design of Semiconductor Devices Equadiff 07, TU Wien 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 Equadiff 07, TU Wien 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 Equadiff 07, TU Wien 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 10.8.2007 Fast Optimal Design of Semiconductor Devices Equadiff 07, TU Wien Numerical Result: MESFET Optimized Doping Profile (Almost piecewise constant initial doping profile) 10.8.2007 Fast Optimal Design of Semiconductor Devices Equadiff 07, TU Wien Numerical Result: MESFET Optimized Potential V 10.8.2007 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