A benchmark study on SEM aberration retrieval

Abstract

This report focuses on the process of aberration coefficient retrieval for measurements of a
Scanning Electron Microscope (SEM). The main goal is the verification of a modified aberration coefficient retrieval method described in an article by Uno et al. (2005) [1]. This method
first extracts spot profiles, represented by Point Spread Functions (PSFs), from a through-focus
series of SEM images using Fourier analysis. The size and shape of a PSF characterize the
performance of a SEM. Afterward, it determines characteristics from these PSFs to obtain
aberration coefficients.
The process consisted of three parts: probe shape extraction, PSF analysis, and aberration coefficient calculation, which included a consistency test. The original method was changed to
include clarification for the use of complex conjugates of functions in the Fourier domain, the
proper statistical definition of standard deviation as characteristic of the width of a PSF, the
curvature definition used in the method, and contour tracing around a PSF to exclude points
outside the contour. These adaptations were made to clarify the methods described in the reference article.
This modified method was tested against generated PSFs that used the wave-optical description
of five different aberrations. By increasing the value of these individual simulated aberration
coefficients C
′
n with factors s ∈ [1, S], S ∈ N, the consistency of this retrieval method could be
determined by calculating the ratio Q between simulated and retrieved aberration coefficients
as a function of the factor s.
The spot shape extraction method was implemented and assessed successfully, as the shapes of
the generated PSFs were successfully retrieved, and the use of complex conjugates of functions
in the Fourier domain resulted in rotated spot shapes.
Contour tracing around the retrieved PSFs had mixed results. For all aberrations, it presented
a good measure of the spot shape. The implementation of an algorithm that excluded points
pj outside the contour was not accurate for one test aberration: two-fold astigmatism A2. The
contour tracing algorithm excludes important parts of the shapes of the PSFs. From research
into the characteristics asymmetry µ, width σ, and curvature ρ it could be inferred that the first
two characteristics µ and σ represent the mean and standard deviation of discrete probability
mass functions. Curvature ρ has no such direct statistical equivalent, it was concluded however
that this characteristic presented a good measure for the effect of third-order spherical aberration in a PSF.
Results could not indicate that this wave-optical description of aberrations is consistent with
all of the coefficients retrieved using the modified method of aberration coefficient retrieval
as the ratio Q remained consistent for defocus Cdf , three-fold astigmatism A2 and third-order
spherical aberration Cs as a function of the factors s.
Continued research into this method of aberration retrieval is recommended, specifically in the
definitions of the characteristic curvature ρ, more advanced contour tracing, and the digitizations of aberration coefficients. More assessment and modification of this method of aberration
coefficient retrieval can allow for a more comprehensive understanding of the method, which
will eventually contribute to an automatic aberration corrector that could speed up SEM measurements significantly.