Our research activities focus on the development of mathematical and computational methods for the metrological characterization of nanostructure morphologies. The motivation for the mathematical and computational support of nanometrology is emanated from the extreme requirements of accuracy on nanoscale, the extensive use of microscopy images and the increased complexity of nanomorphologies. Since plasma technology is used in both well-defined nanopatterns (transfer of resist patterns to substrates) and the stochastic roughening of open (freeform) surfaces, our research activities in mathematical and computational nanometrology can be separated in two parts:

Α. Nanometrology of patterned structures

Line Edge Roughness (LER) concerns the sidewall roughness of line/space patterns and degrades pattern uniformity and dimensional control. For this reason, it is of primary importance in current semiconductor research and industry where the feature dimensions are shrinking down to the sub-20nm regime. During the last two decades, we have pioneered the metrology of LER with the development and application of novel mathematical and computational tools focusing on the spatial and frequency analysis of LER. The latter have been proven critical in the understanding and control of LER origins and degradation effects in modern semiconductor circuits. The proposed tools include the Height-Height Correlation Function (HHCF), Power Spectral Density (PSD), (multi)fractal analysis and the related parameters of the correlation length, roughness exponent and fractal dimension. This analysis has been shaped in the widely accepted three-parameter model for the characterization of LER (rms, correlation length and roughness exponent).

Finally, we are a member of the initiative of SEMI standards and participate in the committee for the redefinition of the protocol LER measurement to homogenize these in semiconductor research and industry.

Our recent contributions in LER metrology consider both accuracy and completeness issues as well as its effects on device performance.

A.1 Accuracy issues

LER is usually measured through the analysis of top-down Scanning Electron Microscope images. However, they suffer from the presence of noise which degrade the accuracy of LER measurements on nanoscale. In order to mitigate noise effects and control their impact, we have developed two strategies.

  • The first is based on the use of Fourier analysis (Power Spectral Density, PSD) to decompose image noise contribution to LER and acquire unbiased results (see Fig.1) and
  • The second is the development of machine/deep learning techniques for the identification of noise effects on the whole image (convolutional neural network) or the edge profile (Hidden Markov Models) and the consequent denoising of images (see Fig.2).

Figure 1

Figure 1. The PSD-based method for noise-free unbiased measurement of the LER rms value. We allow the presence of noise in edge detection to get a flat floor at high frequencies of the PSD diagram. The area under the flat floor at the whole frequency regime can be used to calculate the contribution of SEM noise to the measured (biased) variance (rms2) of LER. The unbiased rms is then the square subtraction of the noise rms from the measured biased rms.

Figure 2. The architecture of our Deep Learning method (SEMD) based on a convolutional neural network trained to predict the estimated noise of the image from a noisy image. The estimated zero noise image can be calculated by subtracting the estimated residual mapping from the observed high noise image.

Additionally, we devised a computational modeling of top-down SEM images to generate synthesized grayscale images with controlled levels of noise and LER characteristics which imitate closely the real SEM images. These images have been used to provide a testbed for the validation of PSD-based method and the training and testset in the convolutional neural network (see Fig.3). Also, we have extended the PSD-based method to the Height-Height Correlation Function to enable the calculation of the noise-free correlation length and roughness exponent.

Figure 3. Examples of (a) Zero Noise synthesized SEM images (1st column), High Noise synthesized SEM images degraded by varying noise levels (2nd column) and Denoised images obtained by the application of SEMD denoising model on High Noise test images (b) Noisy real SEM images degraded by unknown noise levels (1st column) and denoised SEM images obtained by the application of SEMD denoising model on Noisy real SEM images (2nd column).

A.2 Completeness issues

We continue our work to enrich LER metrology with novel methods to meet the metrological challenges posed by recent advances in patterning techniques. In particular,

  • We developed a novel methodology (c-factor correlation function and length) to capture the cross-line correlations between the roughness of edges and lines in a pattern, which are critical in patterns with high densities (small pitch) and those produced by Directed Self-Assembly and Multi-Patterning lithographies (see Fig.4). The methodology has been applied in real patterns of different lithographies with interesting results (see Fig.5).
  • We initiated a concise hierarchical approach for the mathematical characterization of a real pattern with roughness taking into account stochastic deviations at all scales (LER, Line Width Roughness, Line Center Roughness, pitch variations). The key ingredients of this approach is the identification of the two scales (pitch and CD) defining the line patterns and the two directions (across and along line direction) that correlations and deviations can appear (see Fig.6).
  • We applied the multifractal analysis properly adapted in LER characterization to disclose the whole spectrum of fractal behaviors at the peaks and valleys of LER. Using this analysis, we identified the impact of pattern transfer in a double patterning lithography process (see Fig.7).
  • In the modern 3D multi-layered electronic devices, the Edge and Pattern Placement Error (EPE/PPE) in line/space patterns play a critical role since they degrade device performance. The aim has been to clarify the quantitative relationship between the EPE/PPE and pattern roughness including Line Edge Roughness (LER) and edge correlations (c-factor). Using a computational modelling approach, we showed that despite the dominant role of Rms (LER), EPE is also affected by the correlation length especially at long length of interests. Similarly, we demonstrated and quantified the positive correlation of the edge correlation coefficient with Pattern Placement Error (see Fig.8). The ultimate concern has been to get EPE/PPE-aware design rules of modern circuits, which are well-informed by modern manufacturing and metrology aspects of LER.

Figure 4. Schematic diagram of the flowchart of LER analysis in (a) conventional top-down lithography where edge data is transformed to line width level and (b) DSA lithography through the passage from Line Center level. One can notice the analogy between the couples LWR-CDU and LCR-LPE as well as between LER-LWR and LCR-pitch roughness.

Figure 5. Cross-line correlations (quantified by c-factor correlation function) of patterns from different lithographies versus distance (measured in pitches). One can notice the sensitivity of c-factor correlation function to the type of lithography and the propagation of line fluctuations across pattern.

Figure 6. Schematic of a line pattern showing the different scales involved in its hierarchical structuring. First, the two parameters (CD, pitch) defining the horizontal scales of pattern are illustrated when they remain fixed along the whole pattern (a), or exhibit random variations  from line to line (CD and pitch nonuniformity) (b). In the third schematic shown in c) the impact of pattern roughness on local variations of CD and pitch is illustrated along with the two new scales along vertical direction ξLWR (LWR correlation length) and ξLCR (LCR correlation length) defining these variations.

Figure 7. (a) A typical example of the multifractal spectrum of real LER (left edges) along with the physical meaning of the left and right branch of the spectrum and the definitions of δa and δf, (b) Multifractal spectra of the LER of line/space patterns during the first five steps of a Quadruple Patterning Lithography process. Each row shows schematically the pattern and underneath stack (left), a representative top-down SEM image of the pattern (middle) and the multifractal spectrum of the left edges of pattern (right). We can notice the insightful footprint of each process step on the multifractal spectrum (e.g. core etch weakens left branch and strengthen right branch).

Figure 8. The impact of Rms value of LER and edge correlations quantified by Pearson coefficient c-factor (left diagram) on the Pattern Placement Error (middle and right diagrams).

Nanometrology of open (freeform) surfaces

Our aim is to cope with the challenges posed by the rich variety and increased complexity of surface nanomorphologies produced by plasma etching and other surface treatments. Main emphasis has been paid on the quantification of hierarchy and complexity in multiscale structures. To this end, we have developed and implemented methods inspired by (a) Signal analysis (Fourier and Correlation analysis), (b) Stochastic geometry (Point Pattern analysis), (c) Complexity science (multiscale entropies, multifractality) and (d) Machine learning (neural networks, random forests). Fourier and correlation methods are more well-established and widely known. Therefore, we focus on our recent work on stochastic geometry, complexity science and machine learning:

  1. Signal analysis: Plasma etching is largely used in our lab to pattern polymer surfaces with hierarchical structures composed of nanopillars (first hierarchical level) and their bundles (second level) spatially separated. Fourier analysis has been properly applied in the top-down SEM images to quantify the scales and nature of the two level hierarchy of polymer surfaces (see Fig.9). The quantified scales can be linked to the optical properties of surfaces.
  2. Stochastic Geometry: We developed and applied a methodology for the characterization of the degree of order/randomness of the positions of nanofeatures (nanoparticles, nanodots, nanowires) on a substrate taking into account their size based on the generalization of the Nearest Neighbor Index (NNI) used in Point Pattern Analysis to a spectrum of NNI for different feature sizes (see Fig.10). Also, we investigated in detail the impact of image processing steps (denoising, binarization) on NNI in real SEM images with back-scattered and secondary electrons.
  3. Complexity Science: We quantified the complexity of nanostructured surfaces using the concept of statistical/average symmetry. The key idea has been to quantify the complexity of a surface by means of the deviation from the statistical symmetry so that it maximizes between full order and full randomness when the heterogeneity (spatial information) of surfaces enhances. The method has been applied in both synthesized and experimental surfaces and its results have been evaluated in comparison with more conventional approaches (see Fig.11).
  4. Machine Learning: The aim of the machine learning approach here is to develop data-based techniques to predict the link between functionality and structural roughness parameters of complex nanorough surfaces. To this end, we apply conventional and deep neural networks as well as random forests to relate the roughness parameters of a complex scale-limited fractal surface with its active area given that the latter plays a critical role in multiple surface functionalities (catalysis, hydrophobicity etc.). The data-based models were trained in synthesized data generated by inverse Fourier techniques for fractal surfaces. The results are promising and show that the deep neural networks provide the best predictions as compared with the true active area values (see Fig.12). Also, they are able to evaluate the relative importance of roughness parameters with respect to their impact on active surface area.

Figure 9. Fourier spectra (circularly averaged) of top-down SEM images (inset in diagram) depicting PMMA surfaces etched by O2 plasma in our new homemade plasma nanotechnology equipment versus the applied bias voltage (50W, 100W and 200W). One can notice the footprint of the two-level hierarchy of surfaces (1st level: nanopillars, 2nd level: bundles of nanowires) at high and low frequency regimes respectively. The two levels can be quantified by a) the frequency knee defining the scale of the first level, b) the broad maximum (if any) at low frequencies defining the second scale and c) the slopes of the two power laws below and above frequency knee which characterize the fractality of the two level geometries.

Figure 10. A typical spectrum of NNI vs. feature (nanodot) size indicating the variation of nanodot dispersion randomness versus size: It starts at small sizes with complete spatial randomness (0.9<NNI<1.1), then in the middle sizes the dispersion is organized more evenly (NNI>1.1) and finally there appears a strong dip at the aggregate regime before moving to periodic values again at very large sizes.

Figure 11. The proposed complexity measure versus the randomness of the positions and sizes of Gaussian mounds in a rough surface. One can notice the maximum of complexity measure at the most heterogeneous (but no random) surface.

Figure 12. (a) The demonstration of the success of the Deep Neural Network predictions with the true active areas of the test data and (b) the evaluation of the significance of roughness parameters with respect to their impact on active surface area.