A maximum likelihood approach to the inverse problem of scatterometry

Abstract

Scatterometry is frequently used as a non-imaging indirect optical method to reconstruct the critical dimensions (CD) of periodic nanostructures. A particular promising direction is EUV scatterometry with wavelengths in the range of 13 - 14 nm. The conventional approach to determine CDs is the minimization of a least squares function (LSQ). In this paper, we introduce an alternative method based on the maximum likelihood estimation (MLE) that determines the statistical error model parameters directly from measurement data. By using simulation data, we show that the MLE method is able to correct the systematic errors present in LSQ results and improves the accuracy of scatterometry. In a second step, the MLE approach is applied to measurement data from both extreme ultraviolet (EUV) and deep ultraviolet (DUV) scatterometry. Using MLE removes the systematic disagreement of EUV with other methods such as scanning electron microscopy and gives consistent results for DUV.

Scatterometry reference standards to improve tool matching and traceability in lithographical nanomanufacturing

“…However, to achieve even significantly smaller measurement uncertainty values for the final standard calibration a combined data analysis of all measurements including AFM, SEM and polarimetry data will be applied. The procedures and methods for this hybrid metrology approach based on Bayes algorithm have already been developed and tested [17,18]. With this approach we expect to reach measurement uncertainties (k=2) of below 1 nm for the CD and 0.3° for the side wall angles.…”

“…We have characterized the structure quality on these samples using both high resolution microscopy such as Atomic Force Microscopy (AFM) and SEM, and different scatterometry methods such as optical, EUV as well as X-Ray scatterometry and spectroscopic ellipsometry. For the calibration a combined analysis of DUV and EUV scatterometry, spectroscopic ellipsometry or Mueller polarimetry and GISAXS measurement data supported by AFM and CD-SEM results will be applied using Bayes algorithms [16][17][18].…”

Comparison of CD measurements of an EUV photomask by EUV scatterometry and CD-AFM

“…In this measurement comparison with AFM, we took data for eight grating structures with the nominal line geometries as given in Table 1. The details for the MLS and the absorber layer structure are summarized in Table 2 To solve the inverse problem, we have applied the maximum likelihood estimation 31 . Along with the three geometrical parameters for the line shape, three parameters for the MLS (κ, h sc , h fc , see Table 2), one for the roughness parameter σ LR , and two parameters for the statistical measurement uncertainty were estimated.…”

“…Figure 8 shows a comparison of absorber line width (CD) at ½ structure height as measured by scatterometry and AFM. For scatterometry, the measurement uncertainty was derived within the maximum likelihood estimation of the parameters 31 . It represents the combination of statistical measurement errors and errors induced by model deficiencies.…”

Novel mathematical and statistical approaches to uncertainty evaluation: introducing a new EMRP research project

“…In scatterometry, inverse methods are used to reconstruct grating geometries of photo masks from measured diffraction efficiencies (see Figure 1) [10,11]. The efficiency is a measure of the power of the diffracted light in relation to that of the incident light beam.…”