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3.2.10 A Statistical Measure for the Sharpness of SEM Images

Nien Fan Zhang

Statistical Engineering Division, ITL

Michael T. Postek

Robert D. Larrabee

Precision Engineering Division, MEL

Fully automated or semi-automated scanning electron microscopes (SEM) are now commonly used in semiconductor production and other forms of manufacturing. Testing and proving that the instrument is performing at a satisfactory level of sharpness is an important aspect of quality control. Industrial users of SEMs would like to have these instruments fuction without human intervention for a long periods of time, and to have some simple criterion (or indication) of when they need attention. At the present time, no self-testing is incorporated into these instruments to verify that the instrument is performing at a satisfactory performance level.

In this paper, a statistical measure, known as the multivariate kurtosis, is proposed as one approach to the measurement of the sharpness of SEM images. The application of Fourier analysis techniques to the SEM images is useful for sharpness measurement. The two-dimensional spectrum density of an image is similar to a probability density. In the theory of probability, kurtosis is a measure of a type of departure of a probablity distribution from the normal shape. The value of kurtosis can be compared with 3 to determine whether the distribution is ''peaked" or ''flat-topped" relative to a normal probability density. Literature shows that the smaller the kurtosis, the flatter the top of the distribution. The results have been extended to the multivariate case.

Based on the computed spatial frequency spectrum of selected SEM images, we observe that when an SEM image is visually sharper than a second image, the higher spatial frequency components of the first image are larger than those of the second. Treating the normalized spectrum as a probability density function, a sharper SEM image corresponds to a spectrum which has a larger shoulder or has a flatter shape. Thus, it can be concluded that the corresponding kurtosis of the sharper image is smaller. In addition to that, the marginal kurtosis can be used to measure the shape of the marginal spectrum. The difference between the marginal kurtoses, which are the kurtoses of the marginal distributions can be used to detect possible instrument vibration.

A series of five micrographs are selected as examples depicting a representative set of experiments to demonstrate the sharpness analysis procedure. The first figure is the graphical measure of sharpness following analysis of these five images. Low numbers for kurtosis indicate a better quality image or higher sharpness. Marginal kurtosis analysis also has been done. The second figure shows that something is wrong with Sample 4 because of the larger relative difference of marginal kurtosis. The results of kurtosis analysis coincide with the ranking of the quality of the samples.

This work was presented by N. F. Zhang at the SPIE's (The International Society for Optical Engineering) 1997 International Symposium on Microlithography and has been published in Proceedings SPIE 1997, vol 2050, p. 375-386.


Figure 13: These figures show the sharpness measures for the five samples

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Date created: 7/20/2001
Last updated: 7/20/2001
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