TJ
T.M.L. Janssen
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1
Optimization in the Photolithography Bay
Scheduling and the Traveling Salesman Problem
In a semiconductor factory, integrated circuits (or chips) are constructed on top of slabs of silicon, called wafers. The construction of these wafers is complicated and many different processing steps are needed to gradually building the chip layer by layer. Of these steps, photolithography uses the most expensive equipment. Therefore, the photolithography equipment is often the bottleneck of the factory. Photolithography is used to transfer the geometric pattern of a chip on a wafer. First a light-sensitive photoresist is put on the wafer. ThenUV light is sent through a photomask on the photoresist. The exposed parts of the photoresist will chemically react, creating the pattern. After the exposure, chemical reactions and metal depositions make a layer of circuits on the wafer. In this thesis, we try to increase the production of the semiconductor factory by reducing the time needed for the photolithography. In the first part, we look at themachine level. The time to process a wafer on a lithography steppermachine is determined by different elements of the process (Chapter 2). It turns out that the blade movement required in the exposure step has a significant impact total time required to process a wafer. The blade movement in turn depends on the order in which the different images are processed. Hence we want to find an ordering of the images, such that the blade movement is minimized. This problem turns out to be equivalent to the a priori traveling salesmen problem in the scenario model. The practical problem instances found are solved a limited amount of time using an integer linear programming solver and the average blade movement is reduced by approximately 20%, which reduces the average exposure time 1.6%.
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In a semiconductor factory, integrated circuits (or chips) are constructed on top of slabs of silicon, called wafers. The construction of these wafers is complicated and many different processing steps are needed to gradually building the chip layer by layer. Of these steps, photolithography uses the most expensive equipment. Therefore, the photolithography equipment is often the bottleneck of the factory. Photolithography is used to transfer the geometric pattern of a chip on a wafer. First a light-sensitive photoresist is put on the wafer. ThenUV light is sent through a photomask on the photoresist. The exposed parts of the photoresist will chemically react, creating the pattern. After the exposure, chemical reactions and metal depositions make a layer of circuits on the wafer. In this thesis, we try to increase the production of the semiconductor factory by reducing the time needed for the photolithography. In the first part, we look at themachine level. The time to process a wafer on a lithography steppermachine is determined by different elements of the process (Chapter 2). It turns out that the blade movement required in the exposure step has a significant impact total time required to process a wafer. The blade movement in turn depends on the order in which the different images are processed. Hence we want to find an ordering of the images, such that the blade movement is minimized. This problem turns out to be equivalent to the a priori traveling salesmen problem in the scenario model. The practical problem instances found are solved a limited amount of time using an integer linear programming solver and the average blade movement is reduced by approximately 20%, which reduces the average exposure time 1.6%.
In this paper, we consider the a priori traveling salesman problem in the scenario model. In this problem, we are given a list of subsets of the vertices, called scenarios, along with a probability for each scenario. Given a tour on all vertices, the resulting tour for a given scenario is obtained by restricting the solution to the vertices of the scenario. The goal is to find a tour on all vertices that minimizes the expected length of the resulting restricted tour. We show that this problem is already NP-hard and APX-hard when all scenarios have size four. On the positive side, we show that there exists a constant-factor approximation algorithm in three restricted cases: if the number of scenarios is fixed, if the number of missing vertices per scenario is bounded by a constant, and if the scenarios are nested. Finally, we discuss an elegant relation with an a priori minimum spanning tree problem.
...
In this paper, we consider the a priori traveling salesman problem in the scenario model. In this problem, we are given a list of subsets of the vertices, called scenarios, along with a probability for each scenario. Given a tour on all vertices, the resulting tour for a given scenario is obtained by restricting the solution to the vertices of the scenario. The goal is to find a tour on all vertices that minimizes the expected length of the resulting restricted tour. We show that this problem is already NP-hard and APX-hard when all scenarios have size four. On the positive side, we show that there exists a constant-factor approximation algorithm in three restricted cases: if the number of scenarios is fixed, if the number of missing vertices per scenario is bounded by a constant, and if the scenarios are nested. Finally, we discuss an elegant relation with an a priori minimum spanning tree problem.
Abstract
(2018)
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Neda Hesam Mahmoudi Nezhad, Mohamad Ghaffarian Niasar, Ali Mohammadi-Gheidari, Teun Janssen, Kees Hagen, Pieter Kruit
To optimize the design of a system of electrostatic lenses can be quite challenging. Especially when many lens electrodes are involved, the number of design parameters, such as electrode thickness, radius, gaps between electrodes and voltage, increases rapidly. Therefore, it would be really helpful when optimization routines can be used. There have been some attempts to develop optimization programs, such as Szilagy et al. [1] and Adriaanse et al. [2], but they used to be not very accurate. In the meantime, computers have become much more powerful, making it attractive to revisit the problem. In this work we apply a Genetic Algorithm [3] for the optimization, and MATLAB was chosen for coding.
...
To optimize the design of a system of electrostatic lenses can be quite challenging. Especially when many lens electrodes are involved, the number of design parameters, such as electrode thickness, radius, gaps between electrodes and voltage, increases rapidly. Therefore, it would be really helpful when optimization routines can be used. There have been some attempts to develop optimization programs, such as Szilagy et al. [1] and Adriaanse et al. [2], but they used to be not very accurate. In the meantime, computers have become much more powerful, making it attractive to revisit the problem. In this work we apply a Genetic Algorithm [3] for the optimization, and MATLAB was chosen for coding.
Abstract
(2018)
-
Neda Hesam Mahmoudi Nezhad, Mohamad Ghaffarian Niasar, Ali Mohammadi-Gheidari, Teun Janssen, Kees Hagen, Pieter Kruit
In electron lens design, finding the optimum lens system for theapplication at hand, is still quite a challenge. The situation becomes especially more complicated when many lens electrodesare involved, because the number of free parameters of the optimization, such as electrode thickness, radii, gaps between electrodes and voltages, increases rapidly. Therefore, fast optimization routines are needed to tackle the problem. In the past, there have been some attempts to develop such optimization programs. Szilagy et al. [1] and Adriaanse et al. [2], have published someresults in 1989 on rough optimization of electrostatic lenses. However, using the above-mentioned methods, one could not get very accurate results. Now that we have more powerful computers and significantly better software, we revisit the problem. First we applied the so called “SOEM” (Second Order ElectrodeMethod) [2] for a fast (∼0.1sec) calculation of the axial potential. However, the results of the optimization were not accurate enough. To improve the accuracy of the SOEM-based optimization, we integrated a finite element based potential calculation method (using COMSOL). This way the potential calculation and the objective function calculation is more accurate, although the optimization becomes much slower. We propose a new approach that improves on the low speed of optimization while keeping the high accuracy results of the finite element method based potential calculation. This is done by first using a rough optimization based on the SOEM approach, resulting in a few approximately optimized systems. Then, using the parameters of the systems found, new sets of systems were defined using a small range of values around these parameters. Then the more accurate, COMSOL-based optimization was applied to this set of limited systems. We have tested our method on multi electrode systems up to 7 electrodes. We succeeded to very accurately optimize these systems within a few hours, with the electrode radii, gaps and voltages as free parameters, and the focus position as a constraint. [1] M.Szilagi. Yakowitz and M. Duff, Appl. Phys.Lett. 44, pp. 7-9, 1984. [2] J.P. Adriaanse, H.W.G Van der Steen and J.E. Barth, J.Vac. Sci. Technol. B7, pp. 651-666, 1989.
...
In electron lens design, finding the optimum lens system for theapplication at hand, is still quite a challenge. The situation becomes especially more complicated when many lens electrodesare involved, because the number of free parameters of the optimization, such as electrode thickness, radii, gaps between electrodes and voltages, increases rapidly. Therefore, fast optimization routines are needed to tackle the problem. In the past, there have been some attempts to develop such optimization programs. Szilagy et al. [1] and Adriaanse et al. [2], have published someresults in 1989 on rough optimization of electrostatic lenses. However, using the above-mentioned methods, one could not get very accurate results. Now that we have more powerful computers and significantly better software, we revisit the problem. First we applied the so called “SOEM” (Second Order ElectrodeMethod) [2] for a fast (∼0.1sec) calculation of the axial potential. However, the results of the optimization were not accurate enough. To improve the accuracy of the SOEM-based optimization, we integrated a finite element based potential calculation method (using COMSOL). This way the potential calculation and the objective function calculation is more accurate, although the optimization becomes much slower. We propose a new approach that improves on the low speed of optimization while keeping the high accuracy results of the finite element method based potential calculation. This is done by first using a rough optimization based on the SOEM approach, resulting in a few approximately optimized systems. Then, using the parameters of the systems found, new sets of systems were defined using a small range of values around these parameters. Then the more accurate, COMSOL-based optimization was applied to this set of limited systems. We have tested our method on multi electrode systems up to 7 electrodes. We succeeded to very accurately optimize these systems within a few hours, with the electrode radii, gaps and voltages as free parameters, and the focus position as a constraint. [1] M.Szilagi. Yakowitz and M. Duff, Appl. Phys.Lett. 44, pp. 7-9, 1984. [2] J.P. Adriaanse, H.W.G Van der Steen and J.E. Barth, J.Vac. Sci. Technol. B7, pp. 651-666, 1989.
In this paper, we consider the a priori traveling salesman problem (TSP) in the scenario model. In this problem, we are given a list of subsets of the vertices, called scenarios, along with a probability for each scenario. Given a tour on all vertices, the resulting tour for a given scenario is obtained by restricting the solution to the vertices of the scenario. The goal is to find a tour on all vertices that minimizes the expected length of the resulting restricted tour. We show that this problem is already NP-hard and APX-hard when all scenarios have size four. On the positive side, we show that there exists a constant-factor approximation algorithm in three restricted cases: if the number of scenarios is fixed, if the number of missing vertices per scenario is bounded by a constant, and if the scenarios are nested. Finally, we discuss an elegant relation with an a priori minimum spanning tree problem.
...
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In this paper, we consider the a priori traveling salesman problem (TSP) in the scenario model. In this problem, we are given a list of subsets of the vertices, called scenarios, along with a probability for each scenario. Given a tour on all vertices, the resulting tour for a given scenario is obtained by restricting the solution to the vertices of the scenario. The goal is to find a tour on all vertices that minimizes the expected length of the resulting restricted tour. We show that this problem is already NP-hard and APX-hard when all scenarios have size four. On the positive side, we show that there exists a constant-factor approximation algorithm in three restricted cases: if the number of scenarios is fixed, if the number of missing vertices per scenario is bounded by a constant, and if the scenarios are nested. Finally, we discuss an elegant relation with an a priori minimum spanning tree problem.