Non-Commutative Geometry and Measurements of Polarized Two Photon Coincidence Counts

Employing Maxwell's equations as the field theory of the photon, quantum mechanical operators for spin, chirality, helicity, velocity, momentum, energy and position are derived. The photon ``Zitterbewegung'' along helical paths is explored. The resulting non-commutative geometry of photon position and the quantum version of the Pythagorean theorem is discussed. The distance between two photons in a polarized beam of given helicity is shown to have a discrete spectrum. Such a spectrum should become manifest in measurements of two photon coincidence counts. The proposed experiment is briefly described.Comment: Latex, 13 pages, 3 figure

On the Green's Function of the almost-Mathieu Operator

The square tight-binding model in a magnetic field leads to the almost-Mathieu operator which, for rational fields, reduces to a $q\times q$ matrix depending on the components $\mu$, $\nu$ of the wave vector in the magnetic Brillouinzone. We calculate the corresponding Green's function without explicit knowledge of eigenvalues and eigenfunctions and obtain analytical expressions for the diagonal and the first off-diagonal elements; the results which are consistent with the zero magnetic field case can be used to calculate several quantities of physical interest (e. g. the density of states over the entire spectrum, impurity levels in a magnetic field).Comment: 9 pages, 3 figures corrected some minor errors and typo

Monopole and Berry Phase in Momentum Space in Noncommutative Quantum Mechanics

To build genuine generators of the rotations group in noncommutative quantum mechanics, we show that it is necessary to extend the noncommutative parameter $\theta$ to a field operator, which one proves to be only momentum dependent. We find consequently that this field must be obligatorily a dual Dirac monopole in momentum space. Recent experiments in the context of the anomalous Hall effect provide for a monopole in the crystal momentum space. We suggest a connection between the noncommutative field and the Berry curvature in momentum space which is at the origine of the anomalous Hall effect.Comment: 4 page

Role of phason-defects on the conductance of a 1-d quasicrystal

We have studied the influence of a particular kind of phason-defect on the Landauer resistance of a Fibonacci chain. Depending on parameters, we sometimes find the resistance to decrease upon introduction of defect or temperature, a behavior that also appears in real quasicrystalline materials. We demonstrate essential differences between a standard tight-binding model and a full continuous model. In the continuous case, we study the conductance in relation to the underlying chaotic map and its invariant. Close to conducting points, where the invariant vanishes, and in the majority of cases studied, the resistance is found to decrease upon introduction of a defect. Subtle interference effects between a sudden phason-change in the structure and the phase of the wavefunction are also found, and these give rise to resistive behaviors that produce exceedingly simple and regular patterns.Comment: 12 pages, special macros jnl.tex,reforder.tex, eqnorder.tex. arXiv admin note: original tex thoroughly broken, figures missing. Modified so that tex compiles, original renamed .tex.orig in source

From Feynman Proof of Maxwell Equations to Noncommutative Quantum Mechanics

In 1990, Dyson published a proof due to Feynman of the Maxwell equations assuming only the commutation relations between position and velocity. With this minimal assumption, Feynman never supposed the existence of Hamiltonian or Lagrangian formalism. In the present communication, we review the study of a relativistic particle using ``Feynman brackets.'' We show that Poincar\'e's magnetic angular momentum and Dirac magnetic monopole are the consequences of the structure of the Lorentz Lie algebra defined by the Feynman's brackets. Then, we extend these ideas to the dual momentum space by considering noncommutative quantum mechanics. In this context, we show that the noncommutativity of the coordinates is responsible for a new effect called the spin Hall effect. We also show its relation with the Berry phase notion. As a practical application, we found an unusual spin-orbit contribution of a nonrelativistic particle that could be experimentally tested. Another practical application is the Berry phase effect on the propagation of light in inhomogeneous media.Comment: Presented at the 3rd Feynman Festival (Collage Park, Maryland, U.S.A., August 2006

The Noncommutative Harmonic Oscillator based in Simplectic Representation of Galilei Group

In this work we study symplectic unitary representations for the Galilei group. As a consequence the Schr\"odinger equation is derived in phase space. The formalism is based on the non-commutative structure of the star-product, and using the group theory approach as a guide a physical consistent theory in phase space is constructed. The state is described by a quasi-probability amplitude that is in association with the Wigner function. The 3D harmonic oscillator and the noncommutative oscillator are studied in phase space as an application, and the Wigner function associated to both cases are determined.Comment: 7 pages,no figure

Extraction of three-dimensional information on image obtained by top view electron microscopy

L’industrie de la microélectronique est animée par une croissance exponentielle ininterrompue depuis le milieu du XXème siècle. Cette croissance, longtemps soutenue par la réduction de la taille de grille des transistors, est aujourd’hui portée par les innovations sur les formes complexes des transistors de nouvelle génération (Fin-FET, etc…). Afin de contrôler les étapes de conception de ces transistors, l’industrie du semi-conducteur a besoin d’outils de métrologie adaptés à ces nouvelles architectures pour lesquelles les caractéristiques géométriques influent directement sur les performances. Depuis plusieurs décennies, le CD-SEM (Critical Dimension Scanning Electron Microscope) est l’outil de référence pour mesurer la taille des motifs dans un environnement de production. Cependant, le CD-SEM ne permet pas, aujourd’hui, d’obtenir des mesures tridimensionnelles et les équipements de métrologie spécialisés dans les mesures 3D (AFM, FIB-STEM, Scattérométrie) ne sont pas compatibles avec les contraintes de production (temps de mesure, coût, destructivité, etc…).Depuis plusieurs années, des travaux de recherche, à l’instar de cette thèse, ont pour objectif de déterminer une méthode de métrologie tridimensionnelle basée sur l’utilisation du microscope électronique à balayage. L’approche retenue dans le cadre de ces travaux de thèse est la reconstruction géométrique basée sur l’inversion d’un modèle de simulation d’images de microscopie électronique à balayage. Cette approche nécessite l’utilisation d’un modèle de simulation rapide, performant et avec un minimum d’information a priori sur la géométrie. Au cours de cette thèse, nous avons développé deux modèles de simulation d’images SEM : Synthsem2 et Synthsem3. Le premier est un modèle paramétrique, rapide et performant, mais pas suffisamment indépendant de la géométrie pour l’application visée. En revanche, ce modèle s’avère utile pour d’autres applications et a fait l’objet d’un transfert industriel à une entreprise partenaire. Le deuxième modèle développé, Synthsem3, est un modèle totalement indépendant de la géométrie et calibré à partir de données obtenues par simulation Monte-Carlo. Ce modèle a également fait l’objet d’un transfert industriel à une entreprise partenaire.Le modèle Synthsem3 nous a permis d’étudier la sensibilité du signal de microscopie électronique aux variations de paramètres géométriques d’intérêt. Plusieurs conditions d’acquisition (énergie, inclinaison du faisceau) ont été étudiées afin de construire des tables de sensibilité pour chaque paramètre en fonction des caractéristiques géométriques du motif. Il en ressort notamment que l’estimation de la hauteur d’un motif à partir d’un signal SEM formé par un faisceau non incliné, est hautement incertaine, alors que l’utilisation d’un faisceau incliné améliore nettement l’incertitude. Nous avons ensuite procédé à la résolution du problème inverse par la méthode de Gauss-Newton, en utilisant un calcul analytique du gradient du modèle Synthsem3. Nous avons montré la possibilité de reconstruire une géométrie, sans information a priori sur celle-ci, à partir d’une image de microscopie électronique à balayage sans bruit, avec un faisceau non incliné. En présence de bruit dans le signal, la résolution est instable, conformément aux résultats de l’analyse de sensibilité paramétrique. Enfin, nous avons montré que l’inclinaison du faisceau améliore nettement la stabilité de la résolution du problème inverse.Ces travaux sont le point de départ de plusieurs projets à l’étude au sein du laboratoire d'accueil (CEA Leti) et de l’entreprise partenaire à laquelle nous avons transféré la technologie, en vue d’une future commercialisation.The microelectronics industry has been driven by an uninterrupted exponential growth since the mid-twentieth century. This growth, long supported by the reduction of the transistors gate size, is now driven by innovations on the complex shapes of new-generation transistors (Fin-FET, etc ...). In order to control the design stages of these transistors, the semiconductor industry needs metrology tools adapted to these new architectures for which the geometric characteristics directly influence the performances. For several decades, the Critical Dimension Scanning Electron Microscope (CD-SEM) has been the reference tool for measuring pattern size in a production environment. However, the CD-SEM does not allow, today, to obtain three-dimensional measurements and metrology equipment specialized in 3D measurements (AFM, FIB-STEM, Scalerometry) are not compatible with the production constraints (measurement time, cost, destructiveness, etc ...).For several years, research studies, like this thesis, have aimed to determine a three-dimensional metrology method based on the use of the scanning electron microscope. The approach adopted in this thesis is the geometric reconstruction based on the inversion of a simulation model of scanning electron microscopy images. This approach requires the use of a fast simulation model, efficient and with a minimum of prior information on the geometry. During this thesis, we developed two SEM image simulation models: Synthsem2 and Synthsem3. The first is a parametric model, fast and efficient, but not sufficiently independent of the geometry for the intended application. On the other hand, this model is useful for other applications and has been the subject of an industrial transfer to a partner company. The second model developed, Synthsem3, is a model totally independent of geometry and calibrated from data obtained by Monte-Carlo simulations. This model has also been the subject of an industrial transfer to a partner company.The Synthsem3 model allowed us to study the sensitivity of the electron microscopy signal to the variations of geometric parameters of interest. Several acquisition conditions (energy, inclination of the beam) have been studied in order to build sensitivity tables for each parameter according to the geometrical characteristics of the pattern. In particular, the estimation of the height of a pattern from a SEM signal formed by a non-tilted beam is highly uncertain, while the use of a tilted beam greatly improves the uncertainty. We then proceeded to solve the inverse problem by the Gauss-Newton method, using an analytical calculation of the gradient of the Synthsem3 model. We have shown the possibility of reconstructing a geometry, without prior information on it, from a noiseless scanning electron microscopy image, with a non tilted beam. Using a noisy signal, the resolution is unstable, in accordance with the results of the parametric sensitivity analysis. Finally, we have shown that the inclination of the beam clearly improves the stability of the resolution of the inverse problem.This work is the starting point for several projects under study in the host laboratory (CEA Leti) and the partner company to which we transferred the technology for future commercialization

Limits of model-based CD-SEM metrology

International audienceAlthough the critical dimension (CD) is getting smaller following the ITRS roadmap, the scanning electron microscope (CD-SEM) is still the most general purpose tool used for non-destructive metrology in the semiconductor industry. However, we are now dealing with patterns whose dimensions are of the same order of magnitude as the electron interaction volume and therefore, the usual edge-based metrology methods fail. Like scatterometry has extended the resolution of optical imaging metrology through complex modeling of light-matter interaction, some electrons-matter simulation models have been proposed. They could be used to improve accuracy and precision of CD-SEM metrology. However, these model-based approaches also face to fundamental limits mainly due to probe size with respect to the considered structure and noise. This paper analyses these limits assuming the model is perfect and the microscope has no systematic defect. In this simulation study, we have used the model proposed by D. Nyyssonen, assuming to perfectly represent the SEM effects in the image. The feature of interest is limited to isolated trapezoidal lines with various CD, sidewall angles (SWA) and heights. We have carried out the study with several beam energies, tilts and probe sizes. Surprisingly enough, sensitivity analysis shows that with typical noise amplitude, sidewall angle can be determined with a reasonable precision using SEM images. Single tilted beam SEM images can also bring advantage to measure patterns height. Since these precision figures depend on the geometries, we provide useful graphs giving the ultimate precision for various dimensions (CD, height, SWA)

De Haas-van Alphen oscillations and magnetic breakdown : semiclassical calculation of multiband orbits

We use an algebraic method to compute de Haas–van Alphen oscillations in two-dimensional systems in the semiclassical approximation for cases where the Fermi surface lies on more than one sheet of the energy surface. We treat magnetic breakdown by computing the Riemann surface associated with the Bloch energy equation. The topology of this surface, in particular, its fundamental group, is used to classify electronic trajectories in the complexified Brillouin zone. Three examples taken from tight-binding models of quasi-twodimensional organic conductors show how this can be implemented to calculate frequencies and breakdown fields