Surface Reconstruction From Microscopic Images in Optical Lithography

This paper presents a method to reconstruct 3D surfaces of silicon wafers from 2D images of printed circuits taken with a scanning electron microscope. Our reconstruction method combines the physical model of the optical acquisition system with prior knowledge about the shapes of the patterns in the circuit; the result is a shape-from-shading technique with a shape prior. The reconstruction of the surface is formulated as an optimization problem with an objective functional that combines a data-fidelity term on the microscopic image with two prior terms on the surface. The data term models the acquisition system through the irradiance equation characteristic of the microscope; the first prior is a smoothness penalty on the reconstructed surface, and the second prior constrains the shape of the surface to agree with the expected shape of the pattern in the circuit. In order to account for the variability of the manufacturing process, this second prior includes a deformation field that allows a nonlinear elastic deformation between t0he expected pattern and the reconstructed surface. As a result, the minimization problem has two unknowns, and the reconstruction method provides two outputs: 1) a reconstructed surface and 2) a deformation field. The reconstructed surface is derived from the shading observed in the image and the prior knowledge about the pattern in the circuit, while the deformation field produces a mapping between the expected shape and the reconstructed surface that provides a measure of deviation between the circuit design models and the real manufacturing process.

Surface reconstruction from microscopic images in optical lithography

Virginia Estellers Casas, Maria Gabrani, Jean-Philippe Thiran

We propose a shape-from-shading method to reconstruct surfaces of silicon wafers from images of printed circuits taken with scanning electron microscope. Our method combines the physical model of the optical acquisition system with prior knowledge about the shapes of the patterns in the circuit. The reconstruction of the surface is formulated as an optimization problem with a combined criterion based on the irradiance equation and a shape prior that constrains the shape of the surface to agree with the expected shape of the pattern. To account for the variability of the manufacturing process, the model allows a non-linear elastic deformation between the expected patterns and the reconstructed surface. Our method provides two outputs: a reconstructed surface and a deformation field. The reconstructed surface is derived from the shading observed in the images and the prior knowledge about circuit patterns, which results in a shape-from-shading technique stable and robust to noise. The deformation field produces a mapping between the expected shape and the reconstructed surface, which provides a measure of deviation between the models and the real manufacturing process.

Institute of Electrical and Electronics Engineers2014

Geometric Variational Models for Inverse Problems in Imaging

Virginia Estellers Casas

This dissertation develops geometric variational models for different inverse problems in imaging that are ill-posed, designing at the same time efficient numerical algorithms to compute their solutions. Variational methods solve inverse problems by the following two steps: formulation of a variational model as a minimization problem, and design of a minimization algorithm to solve it. This dissertation is organized in the same manner. It first formulates minimization problems associated with geometric models for different inverse problems in imaging, and it then designs efficient minimization algorithms to compute their solutions. The minimization problem summarizes both the data available from the measurements and the prior knowledge about the solution in its objective functional; this naturally leads to the combination of a measurement or data term and a prior term. Geometry can play a role in any of these terms, depending on the properties of the data acquisition system or the object being imaged. In this context, each chapter of this dissertation formulates a variational model that includes geometry in a different manner in the objective functional, depending on the inverse problem at hand. In the context of compressed sensing, the first chapter exploits the geometric properties of images to include an alignment term in the sparsity prior of compressed sensing; this additional prior term aligns the normal vectors of the level curves of the image with the reconstructed signal, and it improves the quality of reconstruction. A two-step recovery method is designed for that purpose: first, it estimates the normal vectors to the level curves of the image; second, it reconstructs an image matching the compressed sensing measurements, the geometric alignment of normals, and the sparsity constraint of compressed sensing. The proposed method is extended to non-local operators in graphs for the recovery of textures. The harmonic active contours of Chapter 2 make use of differential geometry to interpret the segmentation of an image as a minimal surface manifold. In this case, geometry is exploited in both the measurement term, by coupling the different image channels in a robust edge detector, and in the prior term, by imposing smoothness in the segmentation. The proposed technique generalizes existing active contours to higher dimensional spaces and non-flat images; in the plane, it improves the segmentation of images with inhomogeneities and weak edges. Shape-from-shading is investigated in Chapter 3 for the reconstruction of a silicon wafer from images of printed circuits taken with a scanning electron microscope. In this case, geometry plays a role in the image acquisition system, that is, in the measurement term of the objective functional. The prior term involves a smoothness constraint on the surface and a shape prior on the expected pattern in the circuit. The proposed reconstruction method also estimates a deformation field between the ideal pattern design and the reconstructed surface, substituting the model of shape variability necessary in shape priors with an elastic deformation field that quantifies deviations in the manufacturing process. Finally, the techniques used for the design of efficient numerical algorithms are explained with an example problem based on the level set method. To this purpose, Chapter 4 develops an efficient algorithm for the level set method when the level set function is constrained to remain a signed distance function. The distance function is preserved by the introduction of an explicit constraint in the minimization problem, the minimization algorithm is efficient by the adequate use of variable-splitting and augmented Lagrangian techniques. These techniques introduce additional variables, constraints, and Lagrange multipliers in the original minimization problem, and they decompose it into sub-optimization problems that are simple and can be efficiently solved. As a result, the proposed algorithm is five to six times faster than the original algorithm for the level set method.

EPFL2013

Advanced analytical electron microscopy applied to Solid Oxide Cell materials and their degradation

Stéphane François Jean Poitel

Electron microscopy offers a wide range of techniques to study materials. Environmental Scanning Electron Microscopy (ESEM), in particular, allows the characterisation of samples at high-temperature using a laser heating stage, and under various gaseous atmospheres leading to direct observation of the sample surface morphology dynamics in conditions that are close to those in real operating devices (i.e. at high temperature in various gases). Solid Oxide Fuel Cells (SOFC) operating around 800°C in both oxidising and reducing gases, with proven co-generation efficiencies of 90%, offer very promising solutions in the ongoing energy transition. However, production costs and materials degradation in those demanding operating conditions remain significant barriers to their large-scale deployment. Studying SOFC materials' degradation through a combination of ESEM with other advanced microscopy techniques fulfilled the double aim of demonstrating the power of an ESEM and improving the understanding of SOFC materials' degradation during operation. Two materials from the SOFC stack were selected for degradation case studies at around 800°C in oxidising/reducing gases: coated ferritic steel interconnect and Ni-YSZ anode. The objectives were to explore the parameters' influence (temperature, heating ramp, pressure, gas exposure...), set the boundaries and define the guidelines of the ESEM technique while demonstrating its capabilities. The two different materials sets were selected for their relevance both for the SOFC field and for their different specific characteristics allowing a wide range of experiments to be explored. A specific method for the study of the oxidation of coated steel was developed combining ESEM, Focused Ion Beam (FIB) and Scanning Transmission Electron Microscopy (STEM). Post-test segmentation of the in-situ observed images demonstrated the possibility to obtain quantitative data. FIB sample preparation from the exact site of ESEM observation allowed to correlate the underlying microstructure with the in-situ surface observation. Applying this method, oxidation mechanisms of several coatings were characterised and specific behaviours observed and explained such as the appearance of oxide ridges and the behaviour of a cerium barrier layer in case of cobalt-cerium coated steel. The heating ramp speed and the oxidation temperature were demonstrated to be crucial parameters to obtain the desired microstructure. The nitriding pretreatment of interconnect was investigated too and shows promising results for improving the steel oxidation resistance. Another combination of techniques was developed for the study of the Ni-YSZ anode by complementing the ESEM observation with continuous Mass Spectrometer (MS) acquisition. The first reduction of the NiO-YsZ anode was analysed under gas flows of hydrogen and hydrocarbons. Redox cycling effect on the anode microstructure was investigated and the resulting observations were in line with models from the literature for the reduction and re-oxidation kinetics. Finally, the demonstration of an active solid oxide cell in the microscope operating at more than 700°C under mixed gas paves the way to further possible studies approaching the device's observation in real operating conditions. Limitation of the ESEM techniques are discussed and recommendations for further improvements and perspectives given.