Ditza Auerbach, ' Celso Grebogi, '1 Edward Ott, ' and James A. Yorke ' Laboratory for Plasma Research, University of Maryland, College Park, Maryland 20742 '2~institute for Physical Sciences and Technology and Department ofMathematics, University of Maryland, College Park, Maryland 20742 t 1Department of Physics and Department of Electrical Engineering, University of Maryland, College Park, Maryland 20742 (Received 29 June 1992)

Photomask degradation via haze defect formation is an increasing troublesome yield problem in the semiconductor fab. Wafer inspection is often utilized to detect haze defects due to the fact that it can be a bi-product of process control wafer inspection; furthermore, the detection of the haze on the wafer is effectively enhanced due to the multitude of distinct fields being scanned. In this paper, we demonstrate a novel application for enhancing the wafer inspection tool's sensitivity to haze defects even further. In particular, we present results of bright field wafer inspection using the on several photo layers suffering from haze defects. One way in which the enhanced sensitivity can be achieved in inspection tools is by using a double scan of the wafer: one regular scan with the normal recipe and another high sensitivity scan from which only the repeater defects are extracted (the non-repeater defects consist largely of noise which is difficult to filter). Our solution essentially combines the double scan into a single high sensitivity scan whose processing is carried out along two parallel routes (see Fig. 1). Along one route, potential defects follow the standard recipe thresholds to produce a defect map at the nominal sensitivity. Along the alternate route, potential defects are used to extract only field repeater defects which are identified using an optimal repeater algorithm that eliminates "false repeaters". At the end of the scan, the two defect maps are merged into one with optical scan images available for all the merged defects. It is important to note, that there is no throughput hit; in addition, the repeater sensitivity is increased relative to a double scan, due to a novel runtime algorithm implementation whose memory requirements are minimized, thus enabling to search a much larger number of potential defects for repeaters. We evaluated the new application on photo wafers which consisted of both random and haze defects. The evaluation procedure involved scanning with three different recipe types: Standard Inspection: Nominal recipe with a low false alarm rate was used to scan the wafer and repeaters were extracted from the final defect map. Haze Monitoring Application: Recipe sensitivity was enhanced and run on a single field column from which on repeating defects were extracted. Enhanced Repeater Extractor: Defect processing included the two parallel routes: a nominal recipe for the random defects and the new high sensitive repeater extractor algorithm. The results showed that the new application (recipe #3) had the highest capture rate on haze defects and detected new repeater defects not found in the first two recipes. In addition, the recipe was much simpler to setup since repeaters are filtered separately from random defects. We expect that in the future, with the advent of mask-less lithography and EUV lithography, the monitoring of field and die repeating defects on the wafer will become a necessity for process control in the semiconductor fab.

In studying their systems, physical scientists write differential equations derived from fundamental laws. These equations are then used to understand, analyze, predict, and control the system's behavior, provided one is able to determine the solutions. As the role of nonlinearity grows in importance for the study of physics, solutions often cannot be obtained in closed form, and numerical solutions must be relied on. Computers are now an integral part of the physicist's modus operandi . A basic question always present when obtaining numerical solutions is to what extent they are valid. This question is especially meaningful when dealing with chaotic dynamics, since local sensitivity to small errors is the hallmark of a chaotic system. Floating-point calculations commonly used to approximate solutions of differential equations or compute discrete maps produce pseudo-trajectories , which differ from true trajectories by new, small errors at each computational step. Despite the sensitive dependence on initial conditions, the methods of shadowing have shown that for chaotic systems that are hyperbolic or nearly hyperbolic, locally sensitive trajectories are often globally insensitive , in that there exist true trajectories with adjusted initial conditions, called shadowing trajectories, very close to long computer-generated pseudo-trajectories. A dynamical system is hyperbolic if phase space can be spanned locally by a fixed number of independent stable and unstable directions which are consistent under the operation of the dynamics. In the absence of hyperbolic structure, much less is known about the validity of long computer simulations. Recently it was shown that trajectories of a chaotic system with a fluctuating number of positive finite-time Lyapunov exponents fail to have long shadowing trajectories. In other words, they are globally sensitive to small errors. Such hyperchaotic system has two positive Lyapunov exponents, although finite-time approximations of the smaller of the two fluctuate about zero, due to visits of the trajectory to regions of the attractor with a varying number of stable and unstable directions. The destruction of hyperbolicity caused by this phenomenon leads to global sensitivity — only relatively short pseudo-trajectories will be approximately matched by true system trajectories. Our discussion of the global sensitivity of trajectories for these non-hyperbolic systems is limited in this review to the comparison between physical models and computer simulations, but the same questions arise whenever comparing the time behavior of two systems evolving under similar, but slightly different dynamical rules. For example, a natural system and its theoretical model differ by modeling errors. In the presence of fluctuating Lyapunov exponents, global sensitivity may lead to trajectory mismatch, in particular when long times are considered. The result is that no trajectory of the theoretical model matches, even approximately, the true system outcome over long time spans.

MD simulations, currently the most detailed description of the dynamic evolution of proteins, are based on the repeated solution of a set of differential equations implementing Newton's second law. Many such systems are known to exhibit chaotic behavior, i.e., very small changes in initial conditions are amplified exponentially and lead to vastly different, inherently unpredictable behavior. We have investigated the response of a protein fragment in an explicit solvent environment to very small perturbations of the atomic positions (10(-3)-10(-9) A). Independent of the starting conformation (native-like, compact, extended), perturbed dynamics trajectories deviated rapidly, leading to conformations that differ by approximately 1 A RMSD within 1-2 ps. Furthermore, introducing the perturbation more than 1-2 ps before a significant conformational transition leads to a loss of the transition in the perturbed trajectories. We present evidence that the observed chaotic behavior reflects physical properties of the system rather than numerical instabilities of the calculation and discuss the implications for models of protein folding and the use of MD as a tool to analyze protein folding pathways.

A control scheme is devised for taming the dynamics of convectively unstable extended systems exhibiting chaotic behavior. We show how complex spatiotemporal phenomena can be eliminated in arrays of coupled chaotic elements, in favor of a coherent state in which all elements are synchronized to a prescribed periodic orbit of the uncoupled system. The propagation of the synchronizing front and the resulting steady state properties of the array are analyzed in the presence of noise.

A control scheme for eliminating the chaotic fluctuations observed in coupled arrays of semiconductor lasers driven high above threshold is introduced. Using the model equations, we show that the output field of the array can be stabilized to a steady in-phase state characterized by a narrow far-field optical beam. Only small local perturbations to the ambient drive current are involved in the control procedure. We carry out a linear stability analysis of the desired synchronized state and find that the number of active unstable modes that are controlled scales with the number of elements in the array. Numerical support for the effectiveness of our proposed control technique in both ring arrays and linear arrays is presented.

Chaotic dynamical systems can be organized around an underlying strange set, which is comprised of all the unstable periodic orbits. In this paper, we quantify the complexity of such an organization; this complexity addresses the difficulty of predicting the structure of the strange set from low-order data and is independent of the entropy and the algorithmic complexity. We refer to the new measure as the grammatical complexity. The notion is introduced, discussed, and illustrated in the context of simple dynamical systems. In addition, the grammatical complexity is generalized to include metric properties arising due to the nonuniform distribution of the invariant measure on the strange set.

Invariant sets arising in chaotic dynamics can be organized around their underlying skeleton of unstable periodic orbits. In this paper a scaling function for the eigenvalues of the unstable periodic orbits of strange sets embedded in two dimensions is introduced. For the piecewise-linear Lozi mapping, the scaling function is obtained analytically, as a convergent series in b, the inverse dissipation strength. In general, the scaling function is shown to converge only for uniformly hyperbolic systems, while the inadequacy of this description for other systems is demonstrated and discussed. The converged periodic-orbit scaling function is applied to yield accelerated convergence of the multifractal spectrum of a strange set.

The phase diagram of an Ising model on the hcp lattice with antiferromagnetic nearest-neighbor interactions J in the basal plane and coupling D (ferromagnetic or antiferromagnetic) between planes is analyzed. Previously obtained exact results predict the existence of a Lifshitz tricritical point in its phase diagram. In mean-field theory the model is shown to exhibit a twofold Lifshitz point and a phase diagram with paramagnetic, ferromagnetic, and modulated phases. Monte Carlo simulations that clearly indicate the tricritical nature of the Lifshitz point are reported, and series-expansion techniques are used to identify a commensurate phase in a low-temperature portion of the phase diagram.