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Got a Litho Question? Ask the Experts Q When calculating an aerial image or
resist profile with PROLITH, when do I need to use vector simulations instead of scalar?
A Scalar simulations ignore vector effects by assuming that the electric fields of the various directions of light summing to form the image are completely overlapped (pointing in the same direction), so that the vector sum of the electric fields is equal to the scalar sum. This is always true when the various beams of light are traveling in the same direction. However, as the angle between the rays of light increases, the electric field directions will move away from a perfect overlap. This reduced overlap gives less interference between the rays, thus less modulation and a lower contrast of the image. These larger angles of light become more prevalent when imaging small patterns in a high numerical aperture lens and/or when using off-axis illumination. There is a significant difference between forming an image in air versus an image in resist. As light moves from air into the resist, refraction causes a decrease in the angle of light according to Snell's law. The smaller angles of light in resist mean that vector effects as described above are reduced. So when is vector simulation needed? There is no easy answer to this question, since it depends on many parameters and on the desired accuracy of the simulations. As an example, a scalar simulation of 90 nm lines and spaces imaged with a 0.85NA 193 nm lens using quadrupole illumination predicted a depth of focus that was 40 percent larger than that predicted using vector simulations. If this kind of difference is important, vector simulation are definitely required in this case.
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Chris A. Mack
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How should I interpret residuals from a model fit to my lithographic data?
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Residuals are the difference between the actual experimental data and the predictions of that data made by a best-fit model to the data. When fitting data with a model (using Archer Analyzer for overlay data or ProDATA for focus-exposure matrix data, for example), several factors may contribute to a less than perfect fit of the model to the data. The model, with its limited number of terms and the simplified physical assumptions that went into its derivation, may not adequately describe the actual physical behavior. This type of error usually produces systematic deviations between the model and the data. Experimental noise, caused either by the metrology or by the experimental conditions themselves, can produce either random or systematic differences between model and data. Examining the residuals can give useful information about the nature of the errors in the data and/or the model. For purely random errors in the data the mean of the resulting residuals should be very close to zero. Thus, any significant departure of the mean of the residuals from zero is an indication of systematic errors. By plotting the residuals in the same way as the original data is plotted, observed patterns in the residuals are usually a good indication of a systematic error. For random errors the residual graph should look like a scatter plot. And of course, simply looking at the range or standard deviation of the residuals is a good indicator of how well the model is describing the data.
Do you have a lithography question? Just e-mail lithocolumn@kla-tencor.com and have your questions answered by Chris Mack or another of our experts. Summer 2003
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