In this work, we present a sound mathematical result that may correct the pixelization bias in the calculation of the line edge roughness (LER) even when the pixel size of SEM images is much larger than its rms value. This result can be useful in the LER metrology of current and future lithographic patterns where the desired rms value of LER is in the deep sub-nm range and smaller than the used pixel sizes. We computationally justify the predictions of the mathematical result and then demonstrate its application in synthesized SEM images where we can separate the impact of noise and focus on the pixelization effects in LER measurements. Furthermore, this result emphasizes the power of mathematical and computational methods to enhance the metrological output of SEM images in deep nanometer scale.
In this paper, we focus on the across-edges pixelization effects on LER measurement accuracy. The pixelization across edges rounds the detected edge position according to pixel size and is expected to impact the accuracy of LER measurements (rms, correlation length, PSD curve). Given the similar size of targeted LER values in IRDS and of the pixel size used in SEM measurements, these effects are expected to gain an increasing interest and therefore to be worth studying. Our investigation is utilized with the aid of synthesized images (characterized by complete control of roughness and image parameters), with the final aim being the exploration of the link between the selected pixel size of the SEM image and the observed deviation of the measured rms (rms_m) from the true rms value (rms_t). More specifically, the goal is to find the role of the ratio of pixel size to rms_t in the measurement of rms and how we can predict it.
We propose a methodology for the mathematical and quantitative characterization of the deviation of rough line/space patterns from their ideal smooth shape to identify defects related to line mass and shape. The methodology is applied in real AFM images of line/space patterns while a modelling framework is elaborated for the generation of rough line/space patterns with controlled top and sidewall roughness and size variations. We also explore the consequences of the proposed methodology in the metrology of LER and its relation to 3D patterns.
NIL patterns frequently suffer from the presence of defects such as missing lines or dots which degrade their properties and functionality. Due to their low density and nanosize, the measurement of their fraction is challenging nanometrology trade-off between resolution and measurement range. In this paper, we focus on the use of range-limited SEM images and explore the benefits of a computational modeling approach to simulate the measurement process and estimate the statistics and accuracy of the measurement of missing lines in patterns. The main questions we address have to do with the choice of the parameters available in the measurement process such as the number of acquired images, their magnification defining the lines included in images and the position (overlapped or not) at line pattern. The missing lines can have both uncorrelated and correlated positions in pattern. In the case of positive correlations, the defects are aggregated whereas in the opposite case of negative correlations they are arranged in periodic-like positions. We found that for uncorrelated defects, the critical parameter is the total number of lines included in the measurement process while the image position do not have any impact on the measurement accuracy. On the contrary, when correlations in defect positions are considered, the number of images and the number of lines per image differentiate their effects on the accuracy of the result while the arrangement of images along pattern also plays a crucial role in the measurement process.
The aim of this work is to clarify the quantitative relationship between the Edge Placement Error (EPE) and Line Edge Roughness (LER) in a rough pattern. Using a computational modelling approach to isolate this relationship, we show that despite the dominant role of Rms(LER), EPE is also affected by the correlation length especially at long length of interests. Similarly, we demonstrate and quantify the positive correlation of the edge correlation coefficient with Image Placement Error. The ultimate concern is to get design-rule determinations based on EPE definitions and characterization which is informed by modern manufacturing and metrology aspects of LER.
KEYWORDS: Line edge roughness, Scanning electron microscopy, Denoising, Signal to noise ratio, Image processing, Data modeling, Image denoising, Edge detection, Interference (communication), Image filtering
Deep Learning (DL) techniques based on Denoising Convolutional Neural Networks (DeCNN) are applied in the denoising of SEM images of line patterns to contribute to noise-reduced (unbiased) LER nanometrology. The models of DeCNN are trained in a sufficiently large set of synthesized SEM images with controlled Gaussian and Poisson noise level. Due to the image-based nature of the DL approach, it can be combined sequentially with the state of the art PSD-based method especially for highly noisy images where the use of the PSD-based method alone fails. The results for test synthesized images show the high predicting capability of the DL assisted method for the commonly used LER parameters and functions (Rms, ξ, α, PSD) of the true (zero-noise) values revealing its potential for future use toward an unbiased LER metrology.
KEYWORDS: Data modeling, Line edge roughness, Denoising, Matrices, Scanning electron microscopy, Edge detection, Machine learning, Image processing, Detection and tracking algorithms, Algorithm development
In this paper we propose the implementation of a machine learning technique based on Hidden Markov Models (HMMs) to provide denoising of line edges and unbiased LER measurement. HMMs are widely used with great success in speech recognition and image processing for denoising and filtering noise. Here HMMs are used for similar purposes, working with the observed (noisy) edge data, acquired through SEM imaging and an edge detection algorithm in an effort to retrieve a parent edge signal that is statistically close to the real one.
The developed HMM method was trained with the assistance of synthesized rough edges, on a wide spectrum of predefined and controlled noise levels and roughness characteristics. This ensures the method adapts on a variety of LER parameters and noise levels. The edges were then used to validate its effectiveness in a broad range of line patterns. Our results so far specify the characteristics of the training data set which are required to make the method effective in the unbiased LER measurement.
Power spectral density (PSD) analysis is playing a more critical role in the understanding of line-edge roughness and linewidth roughness (LWR) in a variety of applications across the industry. It is an essential step to get an unbiased LWR estimate, as well as an extremely useful tool for process and material characterization. However, PSD estimates can be affected by both random and systematic artifacts caused by image acquisition and measurement settings, which could irremediably alter its information content. We report on the impact of various setting parameters (smoothing image processing filters, pixel size, and SEM noise levels) on the PSD estimate. We discuss also the use of a PSD analysis tool in a variety of cases. Looking beyond the basic roughness estimate, we use PSD and autocorrelation analysis to characterize resist blur, as well as low and high frequency roughness contents, applying this technique to guide the EUV material stack selection. Our results clearly indicate that, if properly used, PSD methodology is a very sensitive tool to investigate material and process variations.
Two fundamental challenges of line edge roughness (LER) metrology are to provide complete and accurate measurement of LER. We focus on recent advances concerning both challenges inspired by mathematical and computational methods. Regarding the challenge of completeness: (a) we elaborate on the multifractal analysis of LER, which decomposes the scaling behavior of edge undulations into a spectrum of fractal dimensions similarly to what a power spectral density (PSD) does in the frequency domain. Emphasis is given on the physical meaning of the multifractal spectrum and its sensitivity to pattern transfer and etching; (b) we present metrics and methods for the quantification of cross-line (interfeature) correlations between the roughness of edges belonging to the same and nearby lines. We will apply these metrics to quantify the correlations in a self-aligned quadruple patterning lithography. Regarding the challenge of accuracy, we present a PSD-based method for a noise-reduced (sometimes called unbiased) LER metrology and validate it through the analysis of synthesized SEM images. Furthermore, the method is extended to the use of the height–height correlation functions to deliver noise-reduced estimation of the correlation length and the roughness exponent of LER.
Power spectral density (PSD) analysis is playing more and more a critical role in the understanding of line-edge roughness (LER) and linewidth roughness (LWR) in a variety of applications across the industry. It is an essential step to get an unbiased LWR estimate, as well as an extremely useful tool for process and material characterization. However, PSD estimate can be affected by both random to systematic artifacts caused by image acquisition and measurement settings, which could irremediably alter its information content. In this paper, we report on the impact of various setting parameters (smoothing image processing filters, pixel size, and SEM noise levels) on the PSD estimate. We discuss also the use of PSD analysis tool in a variety of cases. Looking beyond the basic roughness estimate, we use PSD and autocorrelation analysis to characterize resist blur[1], as well as low and high frequency roughness contents and we apply this technique to guide the EUV material stack selection. Our results clearly indicate that, if properly used, PSD methodology is a very sensitive tool to investigate material and process variations
The aim of this paper is to investigate the role of etch transfer in two challenges of LER metrology raised by recent evolutions in lithography: the effects of SEM noise and the cross-line and edge correlations.
The first comes from the ongoing scaling down of linewidths, which dictates SEM imaging with less scanning frames to reduce specimen damage and hence with more noise. During the last decade, it has been shown that image noise can be an important budget of the measured LER while systematically affects and alter the PSD curve of LER at high frequencies. A recent method for unbiased LER measurement is based on the systematic Fourier or correlation analysis to decompose the effects of noise from true LER (Fourier-Correlation filtering method). The success of the method depends on the PSD and HHCF curve. Previous experimental and model works have revealed that etch transfer affects the PSD of LER reducing its high frequency values. In this work, we estimate the noise contribution to the biased LER through PSD flat floor at high frequencies and relate it with the differences between the PSDs of lithography and etched LER. Based on this comparison, we propose an improvement of the PSD/HHCF-based method for noise-free LER measurement to include the missed high frequency real LER.
The second issue is related with the increased density of lithographic patterns and the special characteristics of DSA and MP lithography patterns exhibits. In a previous work, we presented an enlarged LER characterization methodology for such patterns, which includes updated versions of the old metrics along with new metrics defined and developed to capture cross-edge and cross-line correlations. The fundamental concept has been the Line Center Roughness (LCR), the edge c-factor and the line c-factor correlation function and length quantifying the line fluctuations and the extent of cross-edge and cross-line correlations. In this work, we focus on the role of etch steps on cross-edge and line correlation metrics in SAQP data. We find that the spacer etch steps reduce edge correlations while etch steps with pattern transfer increase these. Furthermore, the density doubling and quadrupling increase edge correlations as well as cross-line correlations.
KEYWORDS: Scanning electron microscopy, Line edge roughness, Metrology, Edge roughness, Inspection, Process control, Optical lithography, Line width roughness, Error analysis
Recently, a novel method for the calculation of unbiased Line Edge Roughness based on Power Spectral Density analysis has been proposed. In this paper first an alternative method is discussed and investigated, utilizing the Height-Height Correlation Function (HHCF) of edges. The HHCF-based method enables the unbiased determination of the whole triplet of LER parameters including besides rms the correlation length and roughness exponent. The key of both methods is the sensitivity of PSD and HHCF on noise at high frequencies and short distance respectively. Secondly, we elaborate a testbed of synthesized SEM images with controlled LER and noise to justify the effectiveness of the proposed unbiased methods. Our main objective is to find out the boundaries of the method in respect to noise levels and roughness characteristics, for which the method remains reliable, i.e the maximum amount of noise allowed, for which the output results cope with the controllable known inputs. At the same time, we will also set the extremes of roughness parameters for which the methods hold their accuracy.
In this paper, we propose to rethink the issue of LER characterization on the basis of the fundamental concept of symmetries. In LER one can apply two kinds of symmetries: a) the translation symmetry characterized by periodicity and b) the scaling symmetry quantified by the fractal dimension. Up to now, a lot of work has been done on the first symmetry since the Power Spectral Density (PSD), which has been extensively studied recently, is a decomposition of LER signal into periodic edges and quantification of the ‘power’ of each periodicity at the real LER. The aim of this paper is to focus on the second symmetry of scaling invariance. Similarly to PSD, we introduce the multifractal approach in LER analysis which generalizes the scaling analysis of standard (mono)fractal theory and decomposes LER into fractal edges characterized by specific fractal dimensions. The main benefit of multifractal analysis is that it enables the characterization of the multi-scaling contributions of different mechanisms involved in LER formation. In the first part of our work, we present concisely the multifractal theory of line edges and utilize the Box Counting method for its implementation and the extraction of the multifractal spectrum. Special emphasis is given on the explanation of the physical meaning of the obtained multifractal spectrum whose asymmetry quantifies the degree of multifractality. In addition, we propose the distinction between peak-based and valley-based multifractality according to whether the asymmetry of the multifractal spectrum is coming from the sharp line material peaks to space regions or from the cavities of line materis (edge valleys). In the second part, we study systematically the evolution of LER multifractal spectrum during the first successive steps of a multiple (quadruple) patterning lithography technique and find an interesting transition from a peak-based multifractal behavior in the first litho resist LER to a valley-based multifractality caused mainly by the effects of etch pattern transfer steps.
Line Edge Roughness is largely used in the current semiconductor research and industry for the evaluation of materials and processes since it is considered one of the critical factors to degrade device performance. Therefore, its accurate measurement and complete characterization though complicated is highly required. LER measurement is usually based on the analysis of top-down SEM images. As a result, it suffers from the limitations of image-based metrology which among others are the presence of noise and the digital nature of images. Recently, several studies have paid attention on the impact of image noise on LER metrics and the aliasing effects on the Power Spectrum Density curves caused by the discreteness of edge data along line/edge direction. However, image digitization imposes discretization of edge data also in the vertical to edge/line direction. In this paper, we focus on the effects of this aspect of edge data discretization on LER rms value and PSD curve. We explain and analyze the effect using synthesized SEM images and identify the critical role of the ratio of rms to pixel size. We find that for such ratios larger than 0.6, the image digitization has a fixed contribution to the measured rms roughness and PSD which should be removed to mitigate these effects.
Directed self-assembly (DSA) lithography poses challenges in line edge roughness (LER)/line width roughness metrology due to its self-organized and pitch-based nature. To cope with these challenges, a characterization approach with metrics and/or updates of the older ones is required. To this end, we focus on two specific challenges of DSA line patterns: (a) the large correlations between the left and right edges of a line (line wiggling) and (b) the cross-line correlations, i.e., the resemblance of wiggling fluctuations of nearby lines. The first is quantified by the line center roughness whose low-frequency part is related to the local placement errors of device structures. For the second, we introduce the c-factor correlation function, which quantifies the strength of the correlations between lines versus their horizontal distance in pitches. The proposed characterization approach is first illustrated and explained in synthesized scanning electron microscope images with full control of their dimensional and roughness parameters; it is then applied to the analysis of line/space patterns obtained with the Liu–Nealey flow (post-Polymethyl methacrylate removal and pattern transfer), revealing the effects of pattern transfer on roughness and uniformity. Finally, we calculate the c-factor function of various next-generation lithography techniques and show their distinct footprint on the extent of cross-line correlations.
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