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Discrete power spectrum of line width roughness
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The variation of device characteristics is a challenge to present and future large-scale integrations. One of the origins of the variation is line width roughness (LWR). To facilitate the efforts to cope with LWR, we developed a method to accurately characterize LWR basing on the analysis of power spectral densities (PSDs). Because experimental PSDs are intrinsically discrete, we derive simple analytic formulas of the discrete PSDs by assuming that the autocorrelation function (ACF) exponentially decays with distance. The PSDs calculated by using the formulas agree excellently with experimentally obtained PSDs of photoresist LWR. From the result we find that the photoresist LWR of this study has a standard deviation of 2.5 nm and an exponentially-decaying autocorrelation function with a correlation length of 35 nm. Although the experimental PSDs inevitably contain a component produced by scanning-electron-microscope (SEM)-image noise, the two components of LWR and noise are separately determined by the method of this study. Due to this feature, the resultant variance of LWR is independent of the noise intensity. However, it is still important to reduce the noise component in order to maintain the accuracy of analysis especially in the case of LWR that has an unknown functional form of PSD. This is because even the PSDs that have different functional forms sometimes look alike in the presence of large noise. To reduce the noise effect, it is effective to average the SEM images perpendicularly to fine lines before edge detections. The procedure does not reduce the variance unlike averaging along the lines. The method of this study is applicable not only to LWR but also to other cases as far as the ACF exponentially decays with distance, or equivalently the spectral line shape is Lorentzian. Accordingly, it forms the basis for spectral analysis of most experimental results.
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