Analog Integr Circ Sig Process (2006) 48:49–56 DOI 10.1007/s10470-006-8121-x
Optimization of lithographic masks using genetic algorithms ¨ U. Triltsch · A. Phataralaoha · S. Buttgenbach · D. Straube · H.-J. Franke
Received: 30 June 2005 / Revised: 30 November 2005 / Accepted: 22 December 2005 C Springer Science + Business Media, LLC 2006
Abstract Due to its cost effectiveness and reliability, wetchemical etching of silicon is still one of the key technologies for producing bulk-silicon microstructures. In this paper we present an approach for the design of advanced mask sets for anisotropic, wet-chemical etching of silicon. The optimization method of genetic algorithms is used to derive suitable masks for cases where geometrically calculated compensation structures fail. The underlying etch simulation is described as well as the optimization algorithm itself. Design tasks of current research projects are used as examples to illustrate the advantage of using the presented tool.
structures, which are additionally added to the mask structures. There have been several approaches to use geometrical algorithms to calculate such compensation structures [1, 2]. These approaches work well for several cases, but whenever the design space is limited or undercutting and etching from top and bottom-side is required, these approaches fail. This paper reports the latest developments of using genetic algorithms for generating suitable mask layouts in a given design space. After a detailed description of the approach, the design optimization of a tactile micro sensor and the optimization of an accelerometer are used to illustrate the usefulness of this method.
Keywords Genetic algorithms . Lithography . Anisotropic etching . Bulk-micro-machining . Tactile sensor 2. Optimization of masks 1. Introduction Wet-chemical etching of silicon is still one of the key technologies for producing bulk-silicon microstructures. The main advantages are its cost effectiveness and fast process times. However, the anisotropic behavior of the silicon crystal does not only limit the variety of achievable geometries, it also makes the design of suitable lithographic masks more demanding, compared to dry-etching processes. Usually the anisotropic behavior is compensated by especially calculated U. Triltsch () · A. Phataralaoha · S. B¨uttgenbach Institute for Microtechnology (IMT), Technical University of Braunschweig, Germany e-mail:
[email protected] D. Straube · H.-J. Franke Institute for Engineering Design (IK), Technical University of Braunschweig, Germany
The idea of using genetic algorithms (GA) for calculating lithographic masks was described in detail in earlier publications [3–5]. These approaches have now further been enhanced to speed up convergence and to make configuration of the GA easier. The algorithm was implemented to the software module OMAGA (Optimization of MAsks using Genetic Algorithms), which is forming one part of the T-CAD environment for MEMS developed at the TU Braunschweig within the collaborative Research Center (Sonderforschungsbereich 516) titled, ‘Design and Fabrication of Active Microsystems.’ Genetic Algorithms are very well suited to be used in this context, since a large amount of parameters have to be optimized and the optimization problem has to search a multi-dimensional, non-linear design space. An outline of the algorithm is depicted in Fig. 1. Inside the main iteration an individual, represented by a mask layout, is manipulated in the manner of a DNA string evolving over generations. The
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Fig. 1 Outline of the optimization algorithm
behavior of the algorithm is highly dependent on its configuration. Different strategies for recombination and selection will be described in more detail in Section 2.2. A similar approach has been reported by Antonsson and co-workers [6]. They treat the mask layouts as 2D simple polygons which are described by the edge lengths and the edge directional angles. The number of polygon sides is constant throughout the synthesis procedure. The advantage of this method is its low time consumption. The approach presented in this paper is based on a 2D bitmap description of mask layouts. This leads to a higher overall time consumption of the optimization run. However, the important advantage of the 2D bitmap encoding is its high flexibility. Together with the powerful anisotropic etch simulator SUZANA the presented tool is capable of treating nearly arbitrary geometries. Even designs including etch stop layers or concepts depending on simultaneous etching from top and bottom side can be optimized.
saddle points whenever concave-convex shapes are mixed. Fr¨uhauf [1] tries to overcome these problems by performing two-dimensional calculations in a first step and calculating three-dimensional sidewalls from the generated line structures. However, simulations for complex undercut geometries and processes involving etching from top and bottom side are not possible. To overcome such drawbacks and to allow simulation of arbitrary shapes based on the generally used (100)-, (110)and (111)-faces, the etch simulator SUZANA is used to assess the quality of the individual masks. This simulator was developed at the Institute for Microtechnology and performs the underlying structuring process using a cellular automaton model [7]. Cellular automata may be considered as discrete systems containing large numbers of simple identical components with local interactions. They consist of a lattice of “cells,” each with a finite number of possible states. The cells evolve synchronously in discrete time steps according to identical transition rules. These rules can be regarded as a function, whose arguments are the states of the cell under consideration and the neighboring cells. Its value is the new state of the considered cell. In the present case the silicon crystal structure was chosen as the lattice of the cellular automaton, meaning each silicon atom corresponds to a cell. However, in order to reduce the large amount of data that has to be handled, it is necessary to define macro cells consisting of a cluster of atoms (see Fig. 2). Of course, the resolution is lowered depending on the number of atoms in a macro cell. 2.2. Configuration of the genetic algorithm Each chromosome is formed as a two-dimensional bitmap mask. Based on 2D bitmap descriptions of the mask layouts, OMAGA offers high flexibility and is capable of treating nearly any mask geometry. According to the discrete
2.1. Assessment of individuals The crucial point of any optimization algorithm is the assessment of the quality of a derived solution. In this case the grade of a lithographic pattern has to be evaluated. This is done by performing an etch simulation with a defined set of parameters. Two different types of etch simulators have emerged during the last few years. One is using the geometrical model based on the construction of Wulff-Jaccodine, the other is an atomistically modeled etch process. The delimiting problem of the Wulff-Jaccodine algorithm is the occurrence of Springer
Fig. 2 The cellular automaton model uses macro cells which represent a block of silicon crystal cells
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structure of the underlying etch simulation the solution is gained at a certain problem resolution. To speed up the optimization a low start resolution can be chosen and the model can be refined after a certain number of generations or after reaching a predefined quality of the mask. A configurable amount of individuals is used to form the initial population (see Fig. 1), which can be regarded as the genetic pool. Such a population represents one generation in the optimization process, whilst the designer specifies the actual number of generations. If the desired silicon structure needs synchronous etching of both wafer sides, an individual is made up of a top-side and a bottom-side mask, both undergoing optimization. Having configured an initial population, further parameters have to be defined to determine the algorithms behavior. The selection-pressure determines the rate at which a certain genotype is selected and kept in the population. A selection pressure of 1.2 has been proven to work for most configurations. The user has to choose a selection criterion. The algorithm can be forced to take a certain number of best individuals to the next generation. This has the disadvantage that a genotype, which represents a local optimum, may spread out in the population, thus not leading to a global optimum. To avoid this, another method is to take a number of best individuals to the new generation after a stochastic mutation has changed them. The new individuals of the next generation are formed by a recombination of the best genotypes of the parent generation. As a recombination method a uniform block crossover was chosen. Stochastically chosen rectangular regions determine the crossover points (Fig. 3). This strategy, developed by Steffensen [5], has the disadvantage of eventually destroying locally found optima. To speed up convergence the latest approach considers regions. The optimization area can be divided into four or six sectors. The fitness of each individual is calculated for any such sector separately and sectors with higher fitness lead to blocks of greater height and width. Additionally mutation is used to enrich the genetic pool with new information. A mutation probability has to be set
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and a certain mutation operator can be chosen. Currently the uniform block mutation method is used as it has shown the best performance. This operator takes the same sample action as the crossover method: stochastically chosen rectangular areas are selected and the chromosomes are inverted in these areas. A reference structure can be modeled in a 3D-modeler like r . Afterwards the resulting volume model has to SolidWorks be converted into the discrete lattice model that corresponds to the output of the underlying etch simulation. This makes a geometrical comparison between the reference and the actual result much easier. Finally the assessment method must be configured. The quality of the result is highly dependent on the evaluation criteria. Therefore a weighting system is introduced giving the designer the opportunity to define regions of the mask layout of greater or lesser importance to the resulting structure by specifying appropriate weighting factors. As creation of new individuals is significantly dependant on the performance of their parents, fitness evaluation plays an important role in the GA procedure. In the underlying etch simulation the transformation process from 2D layout to 3D shape makes no algebraic model available to derive an explicit fitness function. To find a solution, a strategy was chosen that compares the similarity between the reference shape and the currently achieved simulation results. The approach has to be computationally fast and at the same time should be able to handle critical parts of the structure, e.g. underetched regions, that may cause a 2D algorithm to fail. Cross correlation is used in the field of image processing with great success for template matching. The device topology is currently compared by normalized cross correlation, which is motivated by the measure of the squared Euclidean distance between the surface of the modified cellular lattice and reference structure related to the wafer surface. The method is independent of the feature size and calculates the cross correlation coefficient ks as a measure of fitness (where F and T are the mean values of f(x, y)) and t(x, y) respectively).
k s =
( f (x, y) − F) (t (x, y) − T ) 2 2 x,y ( f (x, y) − F) x,y (t (x, y) − T ) x,y
Due to its two-dimensional behavior, this method lacks accuracy on problems where underetching occurs. Therefore this measure is used in combination with a fast boolean (XOR) spatial comparison of the feature volume to meet the requirements for robustness and time consumption. Fig. 3 Normal (above) and ‘wraparound’ (below) crossover in a 2D layout. Crossover positions (CP1, CP2) are chosen at random and the block for which these positions define opposite corners is selected for crossover
2.3. Optimization cycle For each generation every individual has to be evaluated, which is done by comparison of the intended geometry with Springer
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the resulting geometry after etch simulation. Depending on the assessment of the current generation particular individuals can be removed from the population (death) or recombined and then mutated for building the next generation. Attention must be paid to the eventuality, that the genetic information of the single individuals of one generation becomes too similar. In this case the optimization process would run the risk of remaining in a local optimum, which might not produce a result of sufficient quality. If a determined percentage of similarity to the intended structure has not yet been reached, the optimization tool will use mutation for generating new genetic information, thus generating new individuals and causing the process to search a different area of the available design space. If the desired quality of the results is not reached within the specified number of generations, the optimization process stops and a new run or a change in parameters becomes necessary. However, it has to be kept in mind that it might not be possible to etch the intended structure and that therefore a satisfactory solution cannot be found. The optimization procedure requires a substantial amount of processing time, depending on the number of individuals and generations, the size of the masks and the specified resolution for the etch simulation [3].
3. Examples The following examples show applications of OMAGA. In both cases the layout of geometrical compensation structures failed due to a very limited design space (see Fig. 4 and 5). 3.1. Boss membrane A set of five boss structures should be placed on a 25 µm thick membrane, which forms hinges in combination with the structures. Scheibe [2] and Fr¨uhauf [1] reported on special structures for a similar design, but these approaches were not suitable for the given problem, as residual structures would remain on the membrane area. Due to symmetry effects only
Fig. 4 Boss structures etched with conventionally calculated compensation structures
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Fig. 5 Overview of the application example, showing the weight mask, the desired design, an analytically compensated mask and the simulation result and the optimized mask meeting the given specifications
one quarter of the structure was modeled. The reference structure is depicted in the upper right corner of Fig. 5. The following specifications had to be met:
r Height of boss structures 425 µm r Width (w) of membrane between two boss structures 50 µm
r Keep length (l) as long as possible This limits the width of the mask window (b) and consequently the design space for a suitable compensation structure to approximately 650 µm. The optimization project was set-up with an initial population of four masks, which had been used in a first manual optimization process. Special weight was given to the bottom area of the grooves and the central area of the section. The result (see Fig. 5) shows how this approach leads to solutions that cannot be found by analytical calculations. The compensation structures for two convex corners are connected. This has the effect that two relatively short structures withstand the etchant for a longer time. Only the optimization process can calculate the thickness of the joining part in such a way that it is breaking up just at the right time and the compensation structure is completely removed at
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the end of the etch process. Figure 5 shows the result after three runs with iteration over 5000 generations with 20 to 25 individuals each. At the end the best individual showed a match of 96% in comparison to the reference structure. This can be explained by two effects: the number of generations was too low to generate a structure with higher fit or the wet-etching process cannot generate the given reference structure. In the current setup there is a mix of both cases. The given reference structure cannot easily be formed by the main crystal layers of (100)-silicon substrates. The simulation result shows a typical case. The bottom line matches the reference structure and near the surface there is too much material preserved. The other case is shown in the SEM (Fig. 6). This SEM shows the technological result of a mask layout from an earlier optimization step. Here the corner at the surface of the wafer is matching the reference structure but the bottom line is undercut. This states how important expert knowledge is to prepare the initial set-up of the optimization process and to interpret the results. However, for the given problem the optimization led to the desired geometry and it was possible to derive an advanced design for the desired application. Such a boss-membrane structures is applied for a three dimensional tactile force sensor, where the uniformity of stiffness in all three directions is necessary. A conventional boss-membrane structure has a high ratio of stiffness of 40, regarding lateral versus vertical direction. The usage of the membrane structure as bending structure for a three dimensional tactile sensor are therefore limited. Because of the difference of gain sensitivities in vertical and lateral direction, a high contact force in vertical direction is need in comparison to the contact force in lateral direction. Using a set of five boss structures the ratio of stiffness can be reduced, so that a more uniform contact force can be achieved. As a tactile element a hard metal stylus of 5 mm is attached on the boss in the middle of the structure. The force applied
Fig. 6 Boss structures processed with optimized compensation structures. The SEM shows an earlier optimization stage then depicted in Fig. 4. The corner of the upper left boss is not fully compensated, yet
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Comparison of stiffness of boss structures Kx Kz (mN/µm) (mN/µm) Kz: Kx
The FEM results Classically compensated mask Optimized mask
0.28 0.30 0.42
1.15 1.72 1.58
4.1 5.7 3.8
to the tactile element tip is transmitted through the stylus to the boss-membrane structure and results in a deformation of the membrane depending on the force direction. Using FEM-analysis with the reference structure the stiffness and its ratio can be determined (Table 1). Figures 7 and 8 show the boss structure under a load in the both direction, whereas only the suspension membrane between the bosses and the frame deform. The FEM results show, that the stiffness is depending on the area of suspension membrane. The mechanical characterization of the boss-membrane structure was performed using an apparatus, which consists
Fig. 7 The FEM result of stress on the boss structure under vertical load
Fig. 8 load
The FEM result of stress on the boss structure under lateral
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of a fine positioning system, a balance for measuring bending forces, data acquisition and control devices. The bossmembrane structure in a holder was mounted on a controlled high precision translation stage. While the micro probe was moved downward and as soon as it contacted the load cell, both the contact force and output signals were simultaneously measured. The measurements were performed for every step of 0.25 µm. The total displacement for the probing process is about 7 µm. Figure 9 shows the complete device with mounted stylus in an application as 3D tactile sensor. In Fig. 10 the stiffness of the boss structure processed with optimized compensation structures for lateral and vertical contact is shown. Compared to the boss structure with the classically compensated mask the boss structure with the optimized mask has a lower stiffness difference between vertical and lateral direction, where Kx and Kz is stiffness in lateral and vertical direction respectively (Table 1). The ratio of stiffness of the boss structure with the optimized mask is quite close to the results from the FEM analysis. This property shows a capability to use the optimized boss structure for three dimensional tactile sensing applications.
Fig. 9 Boss structure with stylus
Fig. 10 Stiffness curve of boss structure with optimized compensated mask for lateral and vertical contacting
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Fig. 11 Initial design overview for the optimisation example
3.2. Design of an accelerometer The design of a three-axis accelerometer will act as an example for concepts including simultaneous etching from top and bottom side. Figure 11 shows a design presented in [9]. The sensitivity of such sensors is highly dependent on the seismic mass that can be created which led to the idea of using bulk-micro machined sensors rather than devices produced by surface micro machining. However, this design still needs a chip area of approximately 144 mm2 . A further reduction is possible by reducing the gap size between the four systems and the masses and the beams. Therefore the manually calculated compensation structures used for the design shown in Fig. 11 cannot longer be used as the design space is further limited from 450 µm down two only 250 µm. To reduce optimization time only a part of the desired structure is considered at once. In this case focus is given to the convex corner of the seismic mass. The project was set-up with an initial population of three mask-sets, which have been used in a first manual design process. Special weight was given to the beam area as its shape is crucial for the behavior of the device and the corner of the seismic mask. Again the result (see Fig. 12) shows that this approach leads to a solution that cannot be found by analytical calculations. The solution was gained after one run with iteration over 5000 generations with 20–25 individuals each. At the end the best individual showed a match of 95% in comparison to the reference structure. However, for the given problem the optimization led to the desired geometry after 5000 generations only. It was possible to derive an advanced design for the desired application, simultaneously deriving a solution for top- and bottom-side masks.
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55 Acknowledgment The Deutsche Forschungsgemeinschaft (DFG) has financially supported this work within a collaborative Research Center (Sonderforschungsbereich 516) titled, ‘Design and Fabrication of Active Microsystems.’
References
Fig. 12 Overview of the application example, showing the weight masks, the desired design and the simulation result gained from the derived optimized mask-set
1. J. Fr¨uhauf, Shape Functional Elements of Bulk-Silicon Microtechnique, Springer-Verlag, Berlin, 2004. 2. C. Scheibe and E. Obermeier, “Compensating corner undercutting in anisotropic etching of (100) silicon for chip separation.” Journal of Micromechanics and Microengineering, vol. 5, pp. 109–111, 1995. 3. S. B¨uttgenbach, U. Hansen, and L. Steffensen, “Computational synthesis of lithographic mask layouts for silicon microcomponents.” In Proc. SPIE, vol. 4407, pp. 126–134, 2001. 4. U. Hansen, L. Steffensen, and S. B¨uttgenbach, “MST-CAD Environment for Anisotropic Etching of Silicon.” In Proc. of the VDE World Microtechnologies Congress (MICRO. tec 2000), vol. 2, pp. 73–76 Hannover, Sept. 2000. 5. L. Steffensen, Entwicklungsumgebung f¨ur den rechnerunterst¨utzten Entwurf von Mikrokomponenten: Shaker, Ph.D. Thesis, Braunschweig, 2000. 6. L. Ma and E.K. Antonsson, “Automated mask-layout and process synthesis for MEMS.” In Proc. Modeling and Simulation of Microsystems, pp. 20–23, San Diego, 2000. 7. O. Than and S. B¨uttgenbach, “Simulation of anisotropic chemical etching of crystalline silicon using a cellular automata model.” Sensors and Actuators A, vol. 45, pp. 85–89, 1994. 8. S. B¨utefisch, A. Schoft, and S. B¨uttgenbach, “3-Axes Monolithic Silicon Low-g Accelerometer.” IEEE Journal of Micromechanical Systems, vol. 9, pp. 551–556, 2000.
4. Summary and perspective A method for deriving suitable mask layouts for arbitrary bulk-silicon structures has been presented. The approach uses a genetic algorithm (GA) to optimize an initial set of masks to meet predefined demands. The underlying etch simulation, based on a cellular automaton model, was described. The set-up of the GA was discussed and selected recombination and mutation methods were presented as well as the possible methods to assign the quality of a derived mask. An example of a current research project was used to illustrate the effectiveness of the presented approach. One critical point of the algorithm is a reliable and fast assessment of the simulation result. Currently there is work ongoing to optimize the algorithms for comparing and scoring single individuals. Furthermore the set-up of suitable project configurations is still complex and needs detailed expert knowledge of the algorithms used and the process itself. Future work will integrate knowledge based, half-automated wizards which make set-up easier for non-experts.
Udo Triltsch was born in Bergisch Gladbach, Germany, in 1976. He received the Dipl.-Ing. degree for Mechanical Engineering from the Technical University of Braunschweig, Germany, in 2002. He is currently working towards his Ph.D. at the Institute for Microtechnology, Braunschweig, Germany. His research interests include: design methodology for MEMS, process simulation and knowledge management. Anurak Phataralaoha was born in Bangkok, Thailand, in 1973. He received the B. Eng. degree for Production Engineering from KMUTT, Thailand in 1995 and Dipl.-Ing. degree for Mechanical Engineering from Technical University of Clausthal, Germany in 2002. He is currently working towards his Ph.D. at the Institute for Microtechnology, Braunschweig, Germany. His research interests include: 3D-tactile sensors, micro machining for silicon, Tribological micro guide.
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¨ Stephanus Buttgenbach obtained the Diploma and Ph.D. degrees in physics from the University of Bonn, Germany, in 1970 and 1973, respectively. From 1974 to 1985, he was with the Institute of Applied Physics of the University of Bonn, working on atomic and laser spectroscopy. In 1983, he was promoted to Professor of Physics. From 1977 to 1985, he was also a Scientific Associate at CERN in Geneva, Switzerland. In 1985, Dr. B¨uttgenbach joined the Hahn-Schickard-Society of Applied Research at Stuttgart as Head of the Department of Microtechnology, where he worked on micromechanics, laser microfabrication, and resonant sensors. From 1988 to 1991, he was the Founding Director of the Institute of Micro and Information Technology of the Hahn-Schickard-Society. In 1991, Dr. B¨uttgenbach became Professor of Microtechnology at the Technical University of Braunschweig. His current research centers on the development and application of micro sensors, micro actuators, and micro systems. Currently, he is Vice President of the Technical University of Braunschweig, where his areas of responsibility are research and technology transfer. Dima Straube was born in Berlin, Germany, in 1977. He received the Dipl.-Ing. degree for Civil Engineering from Technical University of Berlin, Germany, in 2002. He is currently working towards his Ph.D. at the Institute for Engineering Design, Braunschweig. His research interests include: design methodology for MEMS, computer aided design and tolerance management.
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Hans-Joachim Franke was born in Helmstedt, Germany, on February 14, 1944. He received his diploma in mechanical engineering (Dipl.-Ing.) from the Technical University of Braunschweig, Germany, in 1969. From 1969 to 1976 he was research assistant of Prof. Roth at the Institute for Engineering Design. In 1976 he received his Ph.D. degree in mechanical engineering. From 1976 to 1988 he had diverse executive positions at the KSB-AG in Frankenthal, Germany, a company, which produces pumps and valves. Since 1988 he has been the director of the Institute for Engineering Design of the Technical University of Braunschweig. His research interests are in the areas of design methodology, computer aided design and machine elements.
