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Accuracy-Optimized Function Approximation Method for Optical Metrology

IP.com Disclosure Number: IPCOM000022148D
Publication Date: 2004-Feb-27

Publishing Venue

The IP.com Prior Art Database

Abstract

An accurate function approximation method is disclosed, which can be used to speed up computations for applications such as optical metrology.

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Accuracy-Optimized Function Approximation Method for Optical Metrology

Kenneth C. Johnson

Background

In optical metrology applications such as scatterometry, a measurement sample's optical signal characteristic (e.g., its reflectance spectrum) is computed from a theoretical model, and parameters of the model (e.g. film thickness, linewidth, etc.) are iteratively adjusted to minimize the discrepancy between theoretical and measured signal characteristics. The simulation parameter set that gives the best fit between measurement and theory represents a best estimate of the sample's physical structure.

Typically, the theoretical model involves very complex and time-consuming operations that can significantly reduce measurement throughput, so it is advantageous to use function approximation methods in the theoretical model to speed up the computations. Once a good estimate of the measurement result is obtained using the function approximation method, the result may optionally be refined using an exact theoretical model. Parallel-processing computer hardware may be used to speed up all phases of the measurement process, including database preparation, computation of the approximating function, and theoretical computations. (One effective way to parallelize the computations is to have different processors do computations for different wavelengths or incidence angles.)

Function approximation methods can be applied to the measurement of any kind of structure whose optical characteristics can be modeled. This includes multilayer film structures, periodic structures such as line/space arrays and contact hole arrays, and aperiodic structures such as isolated lines. In addition, such methods can be applied to structures that cannot be reliably modeled (due to either complexity or insufficient information about the structure's physical composition), but whose optical properties can nevertheless be empirically characterized over a range of processing conditions.

One type of function approximation is polynomial fitting, as described in Ref's. 1, 2. Several possible functional forms can be used (e.g., linear, quadratic, cubic, etc.). In addition, Ref.1 describes a "range splitting" method in which separate functional approximations are used in different parameter subranges.

Another function approximation method, interpolation, is described in Ref's. 3 and 4. This method is similar to the Ref. 1 polynomial (typically cubic) approximation method with range splitting. However, whereas Ref. 1 suggests that "one or two partitions per parameter should be sufficient in most cases," the interpolation method typically uses a large number of subrange partitions for each parameter. Furthermore, the interpolation function is constructed to match the theoretical function value at the subrange limits. (When there are multiple measurement parameters the subranges are rectangular grid cells in a multidimensional parameter space, and the interpolation

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