IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 20, NO. 10, MAY 15, 2008
761
Fuzzy Learning of Talbot Effect Guides Optimal Mask Design for Proximity Field Nanopatterning Lithography Mehmet F. Su, Mahmoud M. Reda Taha, Christos G. Christodoulou, and Ihab El-Kady
Abstract—Processing methods used in photonics and nanotechnology possess many limitations restricting their application areas such as high cost, inability to produce fine details, problems with scalability, and long processing time. Proximity field nanopatterning is a lithography method which surpasses these limitations. By using interference patterns produced by a two-dimensional phase mask, the technique is able to generate a submicron detailed exposure on a millimeter-size slab of light sensitive photopolymer, which is then developed like a photographic plate to reveal three-dimensional interference patterns from the phase mask. While it is possible to use simulations to obtain the interference patterns produced by a phase mask, realizing the mask dimensions necessary for producing a desired interference pattern is analytically challenging due to the intricacies of light interactions involved in producing the final interference pattern. An alternative method is to iteratively optimize the phase mask until the interference patterns obtained converge to the desired pattern. However, depending on the optimization technique used, one either risks a significant probability of failure or requires a prohibitive number of iterations. We argue that an optimization technique that is to take advantage of the physics of the problem using machine learning methods (here fuzzy learning) can lead to competent mask design. This technique is described in this letter. Index Terms—Finite-difference time-domain (FDTD) methods, nanotechnology, numerical analysis, optimization methods, photolithography.
I. INTRODUCTION
P
ROXIMITY field nanopatterning (PnP) lithography [1], [2] is a novel production technique with a great potential to address some problems ailing techniques used in nanotechnology research and applications, such as high cost, extensive calibration requirements, vulnerability to external factors, low yield, and insufficient resolution. PnP crosses the Talbot effect [3], which describes 3-D self-imaging characteristics in interference Manuscript received November 26, 2007; revised January 18, 2008. Research described in this work was suported by Sandia National Laboratories (SNL). Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000. M. F. Su and C. G. Christodoulou are with the Electrical and Computer Engineering Departments, University of New Mexico, Albuquerque, NM 87131 USA (e-mail:
[email protected];
[email protected]). M. M. Reda Taha is with the Department of Civil Engineering, University of New Mexico, Albuquerque, NM 87131 USA (e-mail:
[email protected]). I. El-Kady is with the Department of Photonics Microsystems Technologies, Sandia National Laboratories, Albuquerque, NM 87185 USA, and also with the Electrical and Computer Engineering Department, University of New Mexico, Albuquerque, NM 87131 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/LPT.2008.919511
patterns produced by 2-D phase masks, with chemical properties of certain polymers known as photoresists. In particular, light exposure at certain frequencies cause a change in photoresist solubility in certain liquids known as developers. In PnP, interference patterns produced by a carefully designed phase mask are imprinted on a photoresist slab over which the phase mask is placed. Then the photoresist is developed using the developer much like a photographic plate and baked to harden it prior to use. While it is possible to predict patterns produced by a given mask design via computer simulations of the PnP environment, the mask design necessary to obtain a desired interference pattern cannot be computed directly. An integrated method, described in a previous article [4], addresses this issue by combining a finite-difference time-domain (FDTD) modeling [5] of the PnP exposure process, image pattern recognition techniques comparing the simulation results to desired interference pattern, and numerical optimization to predict new PnP mask parameters. Starting with an initial guess of the variables describing a PnP mask with simple cubic patterned air holes (mask periodicity, hole diameter, and mask height), an initial interference pattern can be simulated. The resulting interference pattern is filtered to reveal the photoresist burn image expected from an experiment using the specific phase mask. Next, fuzzy similarity [4], [6] of the burn image to the 3-D desired image for the PnP end product obtained in [2] is quantified. Based on similarity analysis, improvements to mask variables are projected via systematic optimization. Two possible classes of optimization techniques can be used to project improvements and manage the design process: First, derivative-based optimization techniques that use local gradients of the similarity function to lead to the nearest feasible solution; second, derivative-free optimization techniques maintaining a set of diverse candidates and attempt to iteratively improve the whole set to obtain as many feasible solutions as possible [7], [8]. Whereas the former achieve convergence much faster than the latter, this advantage is offset by the higher risk of convergence failure of the former. The method described in [4] utilized a derivative based optimization technique to achieve 90% or better similarity in 69% of 100 test cases bootstrapped with random initial mask parameters ( nm nm, nm nm, nm nm) using results from 42 simulations ( : mask periodicity; : mask hole diameter; mask height). While the speed of convergence is exceptionally well compared to a derivative-free technique such as genetic algorithm (GA), the success rate remains marginal. The success rate of any derivative-based optimization technique depends on successful initial guesses for the design variables. We
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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 20, NO. 10, MAY 15, 2008
suggest a method to overcome the low competency of the results associated with random assignment of the initial mask design variables when using gradient-based optimization. In this communication, we argue that the design process can be significantly enhanced if the initial guess of the design variables are estimated through machine learning of the physics underlying the PnP process. II. IMPROVEMENT OF INITIAL PARAMETER ESTIMATION The Talbot effect defines the distances of self-image interfer, ence planes produced by a periodic structure to be where is the periodicity and is exposure wavelength [3]. While the Talbot effect does not readily yield a method to identify the mask variables for a desired PnP end product, it indicates which mask variable should have the most pronounced effect on interference pattern formation. Indeed, simulations performed using a constant wavelength light source indicate that mask periodicity has the most profound effect on the PnP end product. These conclusions suggest that the overall success rate and performance of the PnP mask optimization method could be significantly improved by focusing on prediction of the optimal PnP mask periodicity. Therefore, a multistage optimization technique for efficient mask design was suggested in [4]. is first idenIn this process, the optimal mask periodicity tified followed by determining the optimal hole diameter and mask height [4]. A drawback in using gradient-based optimization is its need to a considerable number of objective function (similarity) evaluations for finite-difference approximations to enable computing the local gradient. Besides, gradient-based optimization frequently oscillates around the minimum point without convergence due to rapidly improving objective function values in the vicinity of the correct mask periodicity value. Fortunately, efficient methods addressing this problem are available. One optimization method is known as golden section search [8]. Given an interval containing a single minimum of an objective function, golden section search converges to the minimum by evaluating the given objective function twice and reducing the interval by about 30% in successive iterations. The process is terminated after the interval is smaller than a threshold. Specifically, given an objective function and an interval in iteration , is evaluated at the following two points inside the interval [8]:
(1) . where is the golden section ratio, The interval is then reduced using the following rules: if otherwise
(2)
The golden section search assumes the given interval contains a single minimum. When provided with such intervals of mask periodicities in the 100 test cases described above, the golden section search method converged to the correct periodicity value after eight simulations only (on average), proving the effectiveness of the method. The remaining challenge is that
Fig. 1. Structural features (a : structure periodicity, d : blob diameter) of the PnP end product as functions of mask periodicity (a ).
such intervals must be deduced automatically if the golden section search is expected to determine PnP mask periodicities. Structural features of a PnP end product (including structure peand blob diameter are complicated functions of riodicity the mask periodicity (Fig. 1). One way to identify viable mask periodicity intervals for golden section search is through machine learning of the complex relationship that relates the desired structural features to the mask periodicity . III. FUZZY LEARNING OF TALBOT EFFECT Given the imprecision in realizing the similarity of the electric field intensity patterns and the desired structure and the limited significance of accuracy in evaluating the golden section search interval boundaries, we argue that a fuzzy inference system can be used to learn the inverse Talbot effect and provide a mask periodicity interval when fed with the structural features of the desired structures. We name this method the fuzzy golden section (FGS) method. The structural features of a simulation are determined from the black/white images obtained from application of an experiment-calibrated threshold, iteratively determined [4], to the FDTD generated electric field intensity patterns [Fig. 2(a)]. This enabled converting all gradients to black/white patterns depicting where the photoresist would be retained after development [Fig. 2(b)]. The desired structural features are determined from CAD generated reference images comprised of the desired black/white photoresist patterns. The structural features obtained for a spanning set of mask periodicities are used to train a fuzzy inference system using the fuzzy-learning-from-examples (FLFE) technique [9]. Given a representative training set, FLFE constructs fuzzy membership functions and establishes a fuzzy knowledge rule-base that realizes the inverse relation of the Talbot effect. IV. RESULTS AND CONCLUSION To test the effectiveness of the proposed methods, we used the same design problem used in [4], retaining all FDTD simulation parameters (28.3 nm/grid; 800-nm excitation plane wave; 5000 time steps) and constraints for optimization. Quantification of similarity between 2-D sectional images of the 3-D electric field intensity patterns simulated by FDTD and 2-D sections of the 3D-CAD generated patterns was performed using image pattern
SU et al.: FUZZY LEARNING OF TALBOT EFFECT GUIDES OPTIMAL MASK DESIGN
Fig. 2. (a) A 2-D slice of electric field intensity patterns. (b) White portions are retained after emulated SU-8 photoresist development through application of an experiment-calibrated intensity threshold.
recognition method by [6]. Structural features obtained for 14 different mask periodicities ranging from 406 to 686 nm were used to train the fuzzy inference system once. Using the structural features extracted from the 3D-CAD patterns, the fuzzy knowledge rule-base was used to identify the interval boundaries of mask periodicity for the golden section search. The optimal mask periodicity was then found using the golden section search representing Stage 1 of the optimization process. In Stage 2, the optimal hole diameter and mask height were found using a gradient-based optimization technique. This stage of the optimization was bootstrapped with random initial values. Here we compare in Fig. 3 the ability of the fuzzy aided optimization method, described in Section III and denoted FGS, and the two-stage optimization using the gradient descent method described in [4] [denoted classical gradient descent (CGD)]. It is observed that the FGS method showed a much higher success rate compared with the CGD method. The FGS method always required a lower number of simulations to find the optimal solution in 100 test cases compared with the CGD method. Upon further testing, 90% or higher similarity was obtained in 99% of 100 test cases for the FGS method compared with 69% in the GCD method. In these tests, the two optimization stages using the FGS method required results from 16 simulations. If the fuzzy learning overhead is considered, a single PnP mask optimization requires 30 simulations to be completed, which means 29% improvement in convergence in comparison with the 42 simulations for the CGD method [4]. It is important to note that subsequent designs using the same mask lattice geometry make reuse of the already trained infer-
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Fig. 3. (a) Comparison between number of simulations required for reaching optimal solution for the FGS method versus the ffCGD method showing the ability of the FGS method to provide fast optimization in all 100 test cases. (b) 69% success rate for CGD method versus (c) 99% success rate for FGS method.
ence system. Hence, the number of average simulations is reduced to 16 for FGS method and the speed of convergence is improved by 62% compared with CGD. It is worth noting that a similar design using GA required 1000 simulations (50 generations of size 20). The proposed method thus has 33-fold and 62-fold faster convergences than classical GA with 99% success rate. By considering the underlying physics, we were able to develop a method to efficiently design PnP masks for a desired end product. REFERENCES [1] S. Jeon et al., “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” in Proc. Nat. Acad. Sci. USA (PNAS), 2004, vol. 101, pp. 12428–112428. [2] S. Jeon et al., “Molded transparent photopolymers and phase shift optics for fabricating three dimensional nanostructures,” Opt. Express, vol. 15, no. 10, pp. 6358–16358, 2007. [3] M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt., vol. 43, no. 10, pp. 2139–2164, Oct. 1, 1996. [4] M. F. Su et al., “A novel integrated method realizing iteratively optimized modeling for proximity field patterning nanolithography,” Photon. Nanostructures, Fundamentals Appl., DOI: 10.1016/j.photonics.2007.11.002. [5] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagation, vol. AP-14, no. 3, pp. 302–307, May 1966. [6] G. Tolt and I. Kalaykov, “Measures based on fuzzy similarity for stereo matching of color images,” Soft Comp., vol. 10, p. 1117, 2006. [7] A. Ravindran et al., Engineering Optimization, 2nd ed. New York: Wiley, 2006. [8] J. S. R. Jang et al., Neuro-Fuzzy and Soft Computing: A Computational Approach to Learning and Machine Intelligent. Englewood Cliffs, NJ: Prentice-Hall, 1997. [9] T. J. Ross, Fuzzy Logic with Engineering Applications. New York: Wiley, 2004.
