Reduced-order modeling of thermally induced deformations on reticles for extreme ultraviolet lithography

Bikcora, Can; Weiland, S.; Coene, W.

doi:10.1109/acc.2013.6580705

Cited by 3 publications

(15 citation statements)

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“…The equations for the thermal state and deformation of the short stroke chuck have already been obtained in [6] by using the FEM. In a slightly modified form, this derivation is put in the Appendix for completeness, with the only notable change being that the coefficients of the reticle's CTE curve are now explicit in the notation due to their potential role as parameters to be estimated.…”

Section: B Description Of the Large-scale Thermo-mechanical Modelmentioning

confidence: 99%

“…To surmount this obstacle, model reduction techniques were developed in [6] for the specific case of x 0 = 0, which corresponds to the situation where the whole system is in cold state, at the temperature T ref . However, in a steadily running EUV machine, the initial condition is not necessarily zero at every location in the system as many reticle exchanges occur without waiting for all the adjacent components like the burl layer, clamp, and chuck to cool down.…”

Section: Approximation Of the Large-scale Modelmentioning

confidence: 99%

“…Regarding the modeling procedures, this work builds on and extends our previous research reported in [6], in which we combined the finite element method (FEM) with model reduction techniques to arrive at a reduced-order thermomechanical model for the EUV reticle and its nearby components. Given in state-space form, the state transition was formed by thermal dynamics, whereas the output equations corresponded to the mechanical model that mapped the states to pointwise displacements in order to quantify thermal deformation throughout the analyzed geometry.…”

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confidence: 98%

“…The potential parameters to be estimated are the coefficients of the polynomial for the reticle's CTE, which vary from one reticle to another. By contrast with [6], these parameters are made explicit in the model as required by any estimation scheme. Due to the stochastic nature of the variables, the Bayesian approach to estimation and prediction is chosen.…”

mentioning

confidence: 99%

“…This characteristic of the CTE is among the major differences of the EUV reticles from the DUV reticles where the CTE is taken as constant. The conclusion of [6] was that the proper orthogonal decomposition (POD) method [7] and the discrete empirical interpolation method (DEIM) [8] are the most suitable reduction strategies for linear and nonlinear model reduction, respectively, in the described model. As the basic scenario, the derived loworder model assumed that the illumination of a newly placed unheated reticle begins after its adjacent components (e.g., the clamp, that is, the structure that the reticle is fixed to) have completely cooled down to the reticle's temperature.…”

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confidence: 99%

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Thermal Deformation Prediction in Reticles for Extreme Ultraviolet Lithography Based on a Measurement-Dependent Low-Order Model

Bikcora

Weiland

Coene

2014

IEEE Trans. Semicond. Manufact.

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In extreme ultraviolet lithography, imaging errors due to thermal deformation of reticles are becoming progressively intolerable as the source power increases. Despite this trend, such errors can be mitigated by adjusting the wafer and reticle stages based on a set of predicted deformation-induced displacements. Since this control scheme operates online, an accurate low-order model is necessary. However, finite element modeling of the reticle and its adjacent components leads to a large-scale thermomechanical model that should be simplified. First, parameters of the model's initial thermal condition are reduced to only a few from which numerous initial conditions can be accurately reconstructed. This entails placement of temperature sensors at the corresponding locations, and for this purpose, the discrete empirical interpolation method (DEIM) is utilized. Then, linear and nonlinear model reductions are performed via the proper orthogonal decomposition method and DEIM, respectively. The resultant model is employed in the Kalman filter to estimate the parameters of the reticle's temperature-dependent coefficient of thermal expansion from several displacement measurements and to subsequently predict the displacements that are used for control. By processing the outputs from the simulated large-scale model, this filter is shown to perform successfully, even in the presence of an unexpected initial condition.Index Terms-Discrete empirical interpolation method, extreme ultraviolet lithography, Kalman filter, proper orthogonal decomposition, reticle heating, sensor placement.

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Section: B Description Of the Large-scale Thermo-mechanical Modelmentioning

confidence: 99%

Section: Approximation Of the Large-scale Modelmentioning

confidence: 99%

mentioning

confidence: 98%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Thermal Deformation Prediction in Reticles for Extreme Ultraviolet Lithography Based on a Measurement-Dependent Low-Order Model

Bikcora

Weiland

Coene

2014

IEEE Trans. Semicond. Manufact.

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A Reduced Order Modeling Methodology for Steam Turbine Clearance Control Design

Bıyık

D’Amato

Subramaniyan

et al. 2017

Journal of Engineering for Gas Turbines and Power

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Finite element models (FEMs) are extensively used in the design optimization of utility scale steam turbines. As an example, by simulating multiple startup scenarios of steam power plants, engineers can obtain turbine designs that minimize material utilization, and at the same time, avoid the damaging effects of large thermal stresses or rubs between rotating and stationary parts. Unfortunately, FEMs are computationally expensive and only a limited amount of simulations can be afforded to get the final design. For this reason, numerous model reduction techniques have been developed to reduce the size of the original model without a significant loss of accuracy. When the models are nonlinear, as is the case for steam turbine FEMs, model reduction techniques are relatively scarce and their effectiveness becomes application dependent. Although there is an abundant literature on model reduction for nonlinear systems, many of these techniques become impractical when applied to a realistic industrial problem. This paper focuses on a class of nonlinear FEM characteristic of thermo-elastic problems with large temperature excursions. A brief overview of popular model reduction techniques is presented along with a detailed description of the computational challenges faced when applying them to a realistic problem. The main contribution of this work is a set of modifications to existing methods to increase their computational efficiency. The methodology is demonstrated on a steam turbine model, achieving a model size reduction by four orders of magnitude with only 4% loss of accuracy with respect to the full order FEMs.

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Sparse approximate inverses of Gramians and impulse response matrices of large-scale interconnected systems

Haber,

Verhaegen

2014

Preprint

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In this paper we show that inverses of wellconditioned, finite-time Gramians and impulse response matrices of large-scale interconnected systems described by sparse state-space models, can be approximated by sparse matrices. The approximation methodology established in this paper opens the door to the development of novel methods for distributed estimation, identification and control of large-scale interconnected systems. The novel estimators (controllers) compute local estimates (control actions) simply as linear combinations of inputs and outputs (states) of local subsystems. The size of these local data sets essentially depends on the condition number of the finite-time observability (controllability) Gramian. Furthermore, the developed theory shows that the sparsity patterns of the system matrices of the distributed estimators (controllers) are primarily determined by the sparsity patterns of state-space matrices of large-scale systems. The computational and memory complexity of the approximation algorithms are O(N ), where N is the number of local subsystems of the interconnected system. Consequently, the proposed approximation methodology is computationally feasible for interconnected systems with an extremely large number of local subsystems.

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Reduced-order modeling of thermally induced deformations on reticles for extreme ultraviolet lithography

Cited by 3 publications

References 12 publications

Thermal Deformation Prediction in Reticles for Extreme Ultraviolet Lithography Based on a Measurement-Dependent Low-Order Model

Thermal Deformation Prediction in Reticles for Extreme Ultraviolet Lithography Based on a Measurement-Dependent Low-Order Model

A Reduced Order Modeling Methodology for Steam Turbine Clearance Control Design

Sparse approximate inverses of Gramians and impulse response matrices of large-scale interconnected systems

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