Design Tools for Emerging Technologies

S. Johnson, Y. Avniel, J. WhiteMassachusetts Institute of Technology

Cambridge, MA

Abstract The rapidly expanding diversity of technologyavailable at the nanoscale is disrupting the existing transistor-centric microelectronics design paradigm, resulting in nearlydecade-long delays between prototype demonstration andtechnology deployment. The key to reducing these innovation-inhibiting delays is to develop algorithmic approaches that canstart with first principles based descriptions of novelnanotechnology and rapidly and reliably synthesizemanufacturable designs. Design tools are evolving this direction,with new extremely efficient yet customizable physicalsimulators, automatic paramterized low-order model extraction,and ever improving algorithms for robust optimization-newtechniques that generate manufacturable designs by optimizingboth system performance and robustness to manufacturingvariations. In this paper we give a few examples of the coupling ofsuch approaches, but mostly offer pointers to literature forresearchers interested in contributed to this rapidly growing fieldof coupled simulation and robust optimization.

Keywords-optimization, simulation, nanotechnology, computer-aided design

I. INTRODUCTION (HEADING 1)

S. BoydStanford University

Stanford, CA

micromachining, or MEMS, industry. Failure to addressproduct development cycle delays as technology diversifies atthe nanoscale will effectively halt progress, because the delayswill cascade. Without a previous generation's technology inuseable form, it is unlikely that the next generationtechnological investigation can thoroughly begin.

Figure 2. Carbon nanotube transistor electrostatics[2]

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DNA motion

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Figure 1. Nanofluidic DNA filter [2].

The diversity of technological options being generated byrecent developments in micromachining, nanophotonics,carbon nanotubes, quantum wells, and molecular electronics(Figs. 1-4) is causing a major disruption to a decade's oldmicroelectronics design paradigm; one that is based on a fewtechnological alternatives focused solely on generatingsemiconducting transistors. The major ramification of thisdisruption is an enormous delay between prototype proof ofconcept and deployable technology, as is evidenced by the neardecade-long product development cycle associated with the

It is easy to suggest that one simply apply the very rapidlyimproving optimization software to the problem of optimizingnanoscale devices, but there are three key problems that mustbe addressed. The first problem is that classic optimizationapproaches can often find a theoretical "optimum" that isimpractical, because this optimum is too rapidly degraded bymanufacturing imperfections. To address this problem, we canexploit emerging "Robust optimization" approaches. Robustoptimization refers to a new class of optimization techniques[1] that optimize not only the performance of a system, but alsoits robustness in the face of inevitable deviations of the designparameters. A second problem is to determine how best toconvexify the problem formulations and robustness constraintsso as to insure efficient and reliable optimization. The thirdissue is that for devices in leading-edge technologies, the onlymodel may be a three-dimensional physics-based simulation.The optimization approach will need to be somehow combinedwith simulation without making the problem computationallyintractable. Although these are daunting challenges, we seedeveloping this kind of robust optimization based synthesis ofmanufacturable devices as the most promising approach to

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Figure 2. 3d photonic crystal [4].

B

Figure 4: Quantum-well Q-bit [5].

II. ROBUST OPTIMZATIONThe idea of applying numerical optimization techniques to

engineering design problems has a long history, but recentlythis idea has been augmented by new methods of "robustoptimization," whose importance and general ideas wesummarize below.

In principle, optimization can be applied as soon as atractable forward model, i.e., a simulator, has been developed.In the simplest approach, an optimization routine adjusts thedesign variables, then carries out an analysis (using the forwardmodel), and repeats this process. For a design problem with justa modest number of design variables, and a forward model thatcan be evaluated reasonably quickly, including gradients(perhaps by an adjoint method), this approach can work well.Two traditional pitfalls in applying optimization technology,however, are parameter variations (that occur in manufacturingor operation) and model errors (both intentional andunintentional). In either case, the performance of the system (asbuilt or manufactured) deviates from that predicted by themodel. Sometimes, the design is not too sensitive to thesedeviations, but in other cases the "optimized" design isnotoriously sensitive to the mismatch between the real systemand the model. This results in real performance that is farinferior to that predicted by the model. There are many ways to

deal with this basic problem. One is to limit the number ofdesign variables to a small handful, carefully chosen so thatnone of the possible designs is too sensitive. Another standardtechnique is based on posterior sensitivity analysis, in whichthe design is at least checked for sensitivity to parametervariations or model uncertainty.

Another basic method is to add a simple regularization termto the objective that penalizes sensitive designs. The extra termpenalizes designs that are sensitive to variation in theparameter, and thus avoids fragile, nonrobust designs. Thisbasic regularization method is readily extended to far morecomplex situations, with many parameters, and sophisticatedmodels for the expected variations, such as multivariablestochastic models or ellipsoidal bounding models.Unfortunately, this approach often leads to very difficultoptimization problems, with nondifferentiable objectives.A relatively recent approach is robust optimization, whose

development was begun around 1995 by Nemirovski, Ben-Tal,El Ghaoui, and others. In robust optimization, the parametervariations and model mismatch are explicitly taken intoaccount, using models for the variation that are stochastic, or,more often, unknown but bounded. Thus, for example, a set ofmodel parameters might be known to be in a given uncertaintyellipsoid or a given uncertainty polyhedron. The constraints arerequired to hold for all possible values of the parameters, andthe objective for the robust optimization problem is taken to bethe maximum value of the objective, over all possible values ofthe parameters. This results in minimax optimization problems,which can be very difficult to solve. The surprising new resultsare that for certain specific classes of problems, includingclassic least-squares, linear programming, and quadraticprogramming, such problems can be solved exactly efficiently,using new algorithms for more general convex optimization.To give one specific example, the problem of solving a robustlinear program, with the constraint that the coefficients lie ingiven ellipsoids, reduces to a so-called second order coneprogram (SOCP), and can be solved exactly, and veryefficiently. (A sparse SOCP with 105 variables and 106constraints can be solved in minutes on a desktop PC.) Adecade ago, no one would have considered such a problemtractable, let alone routinely solvable.

Robust optimization is described in Ben-Tal andNemirovski [1], as well as in Boyd and Vandenberghe [6]. Theresults so far are for convex problems, and mostly basic ones.But there is enough known that the methods can be useful inother problems. For example, a standard sequential quadraticprogramming (SQP) method, for general nonlinear problems,can be adapted to handle robustness by replacing the QP with arobust QP.

III. COUPLING OPTIMIZATION TO SIMULATIONAs mentioned above, robust optimization will be most

useful for nanotechnology exploration if the optimization canbe combined with a detailed simulation model of the device.There are three approaches for combining optimizationalgorithms with simulation: combining Hessian-freeoptimization algorithms with efficient adjoint-based gradientcomputation; simultaneously optimizing and simulating the

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model by onfly implicitly constructing the Hessian; andgenerating parameterized reduced-order models of the devicefrom simulation and then using the parameterized reduced-order model in the optimization. Below we describe the lattertwo strategies as there have been recent promising results.

A. Simultaneous Optimization and simulation using anImplicit Hession

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3000

2500

2000

1500O

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Figure 5. Optimization time vs.explicit Hessians [7].

# parameters for implicit and

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When combining an interior point based optimizationtechnique with a fast solver or finite-element based devicesimulator, it is possible to avoid the expensive process ofextracting the Hessian from the simulator for use in theoptimizer. For example, it was recently shown for a problem inbiomolecule electrostatic optimization that combining a fast 3-D electrostatic solver with a primal-dual optimizationalgorithm, effectively implicitly computing the Hessian,reduced the optimization time by orders of magnitude over theuse of an explicit Hessian [7].

There are a number of advantages to combining thesimulation and the optimization when using robustoptimization methods. There are the obvious computationalefficiencies, as demonstrated above in the biomoleculeoptimization problem. In addition, the detailed simulatordiscretization equations are available in the optimizationalgorithm, and these discretization equations typically containthe smoothest relation between design parameters andequations. This smooth relation can then be fit with first ordervariations, yielding the kind of explicit relation betweenconstraints and design parameter variations needed to takeadvantage of recent developments in robust optimization.

B. Parameterized Reduced Order Models (PROM)Recent developments in techniques for generating

geometrically parameterized reduced order models (PROMs)from detailed physical simulation [8] are another alternative tocoupling robust optimization to simulation. For example, wehave used this technique to develop improved opticalinspection methods for the semiconductor industry [9].Currently, optical scattering measurements are made from adevice under inspection and the geometry is determined byexpensive three-dimensional electromagnetic simulation ofgeometrically parameterized candidate structures. In order toaccelerate the approach, we developed a projection-basedmethod for automatically extracting a PROM from theelectromagnetic simulation, and then used the PROM with anoptimization algorithm to determine the parameter values. Ascan be seen from Figure 7, a simple rectangular parallelepipedexample, we were able to accurately predict structure.

Figure 6. E. Coli Inhibitor Chorismate mutase [7].

When using the now-pervasive Primal-Dual interior pointmethods for convex optimization, the computation of eachoptimization step involves solving a large linear system thatincludes a Hessian associated with cost function and theconstraints. For large problems, this linear system can besolved iteratively, typically using a preconditioned Krylovsubspace method. Fast integral equation methods forelectromagnetic analysis (useful for photonic device analysis)or finite-element based multiphysics simulators (used for avariety of nanotechnology device analyses) also form largelinear systems of equations that are typically solved iteratively.

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Figure 7. Optimization on 5(order model predicts shape [9].

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Although the above preliminary results demonstrate a proofof concept, there are a number of challenges to extending thePROM strategy to more realistic robust optimization problems.The first issue to consider is that the model is to be used for

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optimization, and therefore the parameterized reductionstrategy should not just focus on the traditional issues offidelity and reduced model computational complexity, but mustalso consider the issue of maintaining convexity so that themodel is easily optimized. A second critical difficulty is therapid increase in reduced model cost with number of designparameters. To address this problem, one can pursue a strategyin which a kind of singular value decomposition is used toreparameterize the design to a much smaller set of moremutually independent, and more critical, design parameters.Then, the reparameterized design can be optimized, and thephysical problem then inferred. Such reparameterization islikely to improve the conditioning of the subsequentoptimization, but the exact approach to reparameterization canalso ease the robustness constraints just by selectingreparameterizations that are less sensitive to process variations.

IV. CONCLUSIONSIn this brief paper, the authors attempted to expose the readerto some to the issues associated with developing products withemerging nanotechnology, and provide some insight in to theauthors view ofhow design tools much evolve to meet thechallenge of reducing "time-to-market" for products based onthis emerging technology. We hope to encourage moreresearchers to join to exciting field.

The authors would like to acknowledge Homer Reid,Junghoon Lee, Almir Mutapcic, Jay Rockway, Jay Bardhan,Michael Altman and Jongyoon Han for many valuablesuggestions and for someofthe applications refered to in thispaper.

REFERENCES

[1] A. Ben-Tal and A. Nemirovski, Lectures on Modern ConvexOptimization (SIAM, 2001).

[2] H. Reid and J. White, Manuscipt under development.[3] J. Han and H. G. Craighead, "Separation of Long DNA Molecules in a

Microfabricated Entropic Trap Array," Science 288, 1026 (2000).[4] S. G. Johnson and J. D. Joannopoulos, "Three-dimensionally periodic

dielectric layered structure with omnidirectional photonic band gap,"Appl. Phys. Lett. 77, 3490 (2000).

[5] R. Caflisch,

[6] Stephen Boyd and Lieven Vandenberge, Convex Optimization(Cambridge Univ., 2004).

[7] J. Bardhan et al., "Fast methods for biomolecule charge optimization,"Intl. Conf on Modeling and Simulation of Microsystems (San Juan,April 2002).

[8] L. Daniel et al., "A multiparameter moment matching model reductionapproach for generating geometrically parameterized interconnectperformance models," IEEE Trans. Computer-Aided Design Integ.Circuits Syst. 23 (5), 678-693 (2004).

[9] J. Lee et al., "Accelerated optical topography inspection usingparameterized model order reduction," to appear in Microwave TheoryTech. Symp. (June 2005).

ACKNOWLEDGMENTS

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