CoordenaÃÃo de AperfeiÃoamento de NÃvel Superior / Este trabalho à dedicado aos estudos da preparaÃÃo e caracterizaÃÃo das ferritas de nÃquel e cobalto. Estas ferritas foram sintetizadas pelo processo de sol-gel em suspensÃo de Ãgua de coco liofilizada (ACP) fabricada no CearÃ. Este mÃtodo de sÃntese permitiu obter de forma eficiente nanopartÃculas de ferritas. As amostras foram calcinadas durante 4 horas a 400, 600, 800, 1000 e 1200 ÂC. Um estudo preliminar da ferrita de cobalto variando a concentraÃÃo da Ãgua de coco acima da sua concentraÃÃo micelar crÃtica (8,3 mmol/litro) mostrou como isso pode influenciar na qualidade e na homogeneizaÃÃo das nanopartÃculas. A difraÃÃo de raios-X (XRD), a espectroscopia Raman e a Microscopia EletrÃnica de Varredura (SEM) foram aplicadas para estudar a dependÃncia do tamanho da nanopartÃcula de ferrita de nÃquel com a temperatura do tratamento tÃrmico, e a correlaÃÃo de suas propriedades elÃtricas e os resultados de espectroscopia MÃssbauer com as suas caracterÃsticas morfolÃgicas. A cristalinidade das ferritas aumenta com a temperatura do tratamento tÃrmico. Medidas da permissividade complexa realizadas em cavidades ressonantes a 5,9 e 9,0 GHz mostraram que seus valores da parte real decrescem com o aumento da temperatura do tratamento tÃrmico (3,6 - 2,4), enquanto os da parte imaginÃria permanecem baixos para as temperaturas mais altas do tratamento tÃrmico (cerca de 10-3). Medidas de impedÃncia complexa a frequÃncias mais baixas (75 KHz a 30 MHz), mostraram que ela aumenta com o aumento da temperatura do tratamento tÃrmico. Os resultados de espectroscopia MÃssbauer mostraram que as ferritas apresentam uma estrutura espinÃlio inverso com os Ãons Ni2+ ocupando os sÃtios B e os Ãons Fe3+ distribuÃdos igualmente nos sÃtios A e B. Para temperaturas de tratamento tÃrmico abaixo de 800ÂC favoreceram a presenÃa nos grÃos de uma fase mais condutora (Grain Boundary), observada tanto no comportamento das propriedades elÃtricas quanto nas respostas da espectroscopia MÃssbauer. / The present work is dedicated to the studies of the preparation and characterization of the nickel ferrites and cobalt. These ferrites were synthesized by the sun-gel process in suspension of water of coconut lyophilized (ACP) manufactured in CearÂa. This synthesis method allowed to obtain in way efficient ferrite nanoparticles. The samples were treated for 4 hours at 400, 600, 800, 1000 and 1200 C. A preliminary study of the cobalt ferrite varying the concentration of the coconut water above your critical micelle concentration (8,3 mmol/l) it showed as that can influence in the quality of the nanoparticles. The Xray difraction (XRD), the Raman spectroscopy and Scanning Eletron Microscopy (SEM) were applied to study the dependence of the size of the nanoparticle of nickel ferrite with the temperature of the thermal treatment, and the correlation of your electrical properties and the results of MÂossbauer spectroscopy with your morphologic characteristics. The cristalinity of the ferrites increases with the temperature of the thermal treatment. Measurements of the complex permitivity carried out in cavity resonators at 5,9 and 9,0 GHz have showed that your values of the real part decrease with the increase of the temperature of the thermal treatment (3,6 - 2,4), while the imaginary part stay low for more elevated temperatures of the thermal treatment (about 10−3). Measurements of complex impedance to lower frequencies (75 KHz to 30 MHz), have showed that it increases with the increase of the temperature of the thermal treatment. The results of MÂossbauer spectroscopy showed that the ferrites presents a structure inverse spinel with the ions Ni2+ occupying the site B and the ions Fe3+ equally distributed in the sites A and and B. To temperatures of thermal treatment below 800 C, favored the presence in the grains of a phase Grain Boundary (greater condutivity) observed so much in the behavior of the electric properties as in the answers of the MÂossbauer spectroscopy.

<p>U ovom radu prikazani su rezultati ispitivanja strukturnih i magnetnih osobina čistih cink- ferita, ZnFe₂O₄ i cink-ferita sa dodatkom indijuma Zn_1-xIn_xFe₂O₄ i itrijuma, ZnY_xFe_2-xO₄, gde je 0 &le; x &le; 0,6. Prahovi na bazi cink-ferita su sintetisani koristeći nisko temperaturnu metodu sinteze iz tečne faze &ndash; metodu koprecipitacije. Osnovni cilj ove doktorske disertacije je bio da se utvrdi veza između uslova sinteze, uticaja različitih katjona, strukture i osobina čistih cink-feritnih nanočestica, kao i cink-ferita sa dodatkom indijuma i itrijuma. Radi praćenja uticaja veličine čestica dobijenog praha na strukturu i osobine ovih materijala, sintetisani čist cink-ferit je kalcinisan na različitim temperaturama. Posebno je razmatran i uticaj dodatka različitih jona na distribuciju i preraspodelu katjona u spinelnoj strukturi. Pored toga ispitivan je uticaj tako pripremljenih prahova na njihove magnetne osobine. Rentgenostrukturna i TEM analize potvrdili su da ispitivani uzorci spadaju u klasu nanomaterijala spinelne strukture. Analiza Raman i M&ouml;ssbauer spektara je ukazala na moguću raspodelu katjona između tetraedarskih i oktaedarskih mesta, te formiranje delimično inverznog spinela. Ispitivanja magnetnih osobina su pokazala da histerezisne petlje ne pokazuju saturaciju u prisustvu jakog magnetnog polja, &scaron;to je potvrdilo superparamagnetnu i jednodomensku prirodu čestica. Pokazano je da pored uticaja veličine čestica, dodatak različitih katjona (u ovom slučaju itrijum i indijum) ima veliki uticaj na uređenje strukture, a posledično i na magnetno pona&scaron;anje ispitivanih nanočestičnih sistema.</p> / <p> This thesis presents the results of the investigation of the structural and magnetic properties of nanostructured zinc ferrites, ZnFe2O4 and zinc ferrites supstituted with different amount of indium and yttrium, Zn1-xInxFe2O4 and ZnYxFe2-xO4 (0 &le; x &le; 0,6). Powders based on zinc ferrites were synthesised by a low temperature wet-chemical method &ndash; coprecipitation. The main purpose of this thesis was to establish the relationship between the synthesis, dopants, structure and properties of zinc ferrite based materials. Nanoparticles of ZnFe2O4 were calcined at different temperatures in order to elucidate the influence of the particle size on the magnetic properties of the obtained nanoparticles. In addition, we have investigated the effect of dopant addition on cation distribution in spinel structure, in order to modify the magnetic properties and to obtain the magnetic ceramics with improved properties compared to the bulk-counterparts. The results of X-ray and TEM analyses confirmed the nanosized nature and spinel type structure of the investigated samples. Raman and M&ouml;ssbauer spectroscopy studies implied on the possible cation distribution between the tetrahedral and octahedral sites and formation of the partially inversed spinel. The study of the magnetic properties showed that hysteresis loops do not saturate even in the presence of high magnetic fields, which confirmed the superparamagnetic and single domain nature of the samples. These observations imply that, besides the particle size, doping (e.g. yttrium and indium) causes significant structural rearrangements which in turn induce changes in magnetic behavior of the investigated nanoparticulate systems.</p>

Nuclear magnetic resonance is a method, which provides information about the magnitude of hyperfine field present on izotope's nuclei with non-zero spin. The hyperfine field is sensitive to the local ordering of atoms surrounding the nuclei. The hyperfine field changes can be observed in response to changes in surroundings caused by substitution. The NMR spektra of 57Fe nuclei in the hexagonal ferrites with magnetoplumbit structure which contain cation substitution of trivalent cations Nd, Pr in crystallographic position of divalent cations Sr are measured and interpreted in this work. All experiment are measured at 4.2 K and we investigate the influence of the cation substituion on hyperfine field in thr measured oxides.

Title: Preparation and characterization of substituted Y ferrites in the form of ceramics and thin films Author: Dorota Pulmannová Department: Department of Inorganic Chemistry, Faculty of Science, Charles University in Prague Supervisor: RNDr. Daniel Nižňanský, Ph.D. Consultant: Ing. Josef Buršík, CSc. Abstract: In this work we describe a preparation and characterization of a hexagonal ferrite series with composition BaSrCoZnXFe11O22 where X=Fe, Al, Ga, In and Sc. We have prepared these ferrites in the powder and ceramic form using the citrate synthesis and in the thin film form using the chemical solution deposition method. Using the powder neutron diffraction we have found that the sample containing only Fe has collinear magnetic structure that belongs to the C2/m or C2'/m' group. Magnetic structure of the samples substituted with In and Sc is similar, but the magnetic moments of the 18hVI site atoms are not aligned parallely with the other moments. Magnetic structure of Ga-substituted sample is different, it is modulated with a propagation vector k ≈ (0, 0, 3/4). Propagation vector of the Al-substituted ferrite is k ≈ (0, 0, 3/2). Substituting elements show strong preferences for the cation sites. Al and Ga prefer the 3bVI site, Zn prefers the tetrahedral 6cIV and In and Sc prefer the 6cVI site. Room...

In this investigation, the cation distribution in ferrites of spinel-type structure is spectroscopically investigated with respect to the observed magnetic behavior. The ferrite thin films were fabricated by pulsed laser deposition and consequently annealed at different temperatures as well as atmospheres. Structural properties were determined with various methods and the crystalline quality was examined. The dielectric function line-shape was parametrized based on empirical evidence and was found to be dominated by electronic transitions between d orbitals of Fe2+ cations as well as transitions from O 2p to 3d and 4s orbitals of iron and zinc cations. The strongest magneto-optical response was observed for transitions involving cations, which correspond to lattice disorder and inversion within the normal spinel structure. With the decrease in the substrate temperature during fabrication, a decrease in the magnetic response was observed. The diminishing ferrimagnetic order was directly correlated to the decrease in strength of the transitions, involving Fe3+ on tetrahedral lattice sites. After thermal treatment in argon atmosphere and at a temperature below the deposition temperature, the increase in the magnetic response was explained through the facilitation of oxygen vacancies. With the increase in treatment temperature, a decrease in ferrimagnetic order was related to the recrystallization of the disordered spinel structure toward a more stable normal configuration, evident in the dielectric function spectra. The cationic configuration distribution in the surface as well as the bulk region, as a function of Zn concentration, was determined from approximation of the XPS and the dielectric function spectra, respectively. The difference in the cation configuration distribution, in films of predominantly inverse configuration, was related to the weak magnetic response, as opposed to films of predominantly normal spinel configuration. Our results demonstrate that a defect-rich surface region could serve as a possible explanation for the ferrimagnetic order in a nominally non-magnetic normal spinel ZnFe2O4. In combination with structural property determination, the net magnetic behavior is explained through the local cationic disorder, determined from the parametrization of the dielectric function spectra in a wide spectral range.:1 Introduction 2 Theoretical background and fundamental considerations 2.1 Spinel ferrite crystal structure 2.2 Crystal field stabilization energy 2.3 Band structure description 2.4 Verwey transition 2.5 Magnetic exchange interactions 3 Sample preparation and modification 3.1 Macroscopic spinel film formation 3.2 Pulsed laser deposition 3.3 Thermal treatment 3.4 Sample overview 4 Methods and general properties 4.1 Structure characterization techniques 4.1.1 X-ray diffraction 4.1.2 X-ray reflectivity 4.1.3 Energy dispersive X-ray spectroscopy 4.1.4 Focused ion beam and scanning electron microscopy 4.1.5 Raman spectroscopy 4.2 Surface properties 4.2.1 Atomic force microscopy 4.2.2 X-ray photoelectron spectroscopy 4.3 Dielectric tensor properties 4.4 Spectroscopic ellipsometry 4.5 Magneto-optical Kerr effect 4.6 Magneto-static properties 5 Results and discussion 5.1 Magnetic and optical properties of Fe3O4 thin film and single crystal 5.2 Magneto-optical properties of ZnxFe3−xO4 thin films 5.3 Fabrication temperature dependent ferrimagnetic order 5.4 Thermally induced structural stabilization 5.5 Cation configuration in dependence on the Zn concentration 5.5.1 Structural property determination 5.5.2 Composition characterization 5.5.3 Magneto-static behavior 5.5.4 Section summary and discussion 6 Summary and outlook Bibliography List of article contributions Selbstständigkeitserklärung Acknowledgments

Les moteurs électriques, les transformateurs, les électroaimants et autres appareils électriques nécessitent des matériaux qui sont en mesure de canaliser les lignes de champ magnétique tout en limitant les pertes qu’elles entraînent. Les matériaux utilisés à cet escient sont appelés matériaux magnétiques doux. Un des moyens utilisé pour limiter les pertes dans ces matériaux est d’augmenter leur résistivité. Pour y arriver, on unit des matériaux très résistifs à des matériaux ferromagnétiques. On obtient ainsi des composites magnétiques doux. Depuis plus de 100 ans, le type de composite magnétique doux le plus utilisé consiste en un empilement de tôles de fer laminées séparées par un matériau isolant. Ces matériaux sont très efficaces mais ont leur lot d’inconvénients. Depuis quelques années, une nouvelle technique s’appuyant sur la métallurgie des poudres a été développée. Elle consiste à envelopper des particules de fer d’un matériau isolant et de les compacter. On obtient ainsi un matériau qui peut être très résistif. Ce projet avait pour but de développer un composite magnétique doux à base de poudre métallique dont le matériau isolant serait de la ferrite NiZn nanométrique. Pour y arriver deux techniques ont été étudiées. La première consistait à recouvrir les particules de fer par placage de la ferrite et la seconde consistait à ajouter nanoparticules de ferrite NiZn à la poudre de fer. Les résultats ont permis de constater que les deux techniques pouvaient être utilisées pour le développement de composite magnétiques doux. Plus spécifiquement, on a obtenues des pertes magnétiques de 11,9 W/kg et de 93 W/kg à 60 Hz et 400 Hz respectivement pour les échantillons préparés par placage de la ferrite et de 13,5 W/kg et de 137 W/kg à 60Hz et 400 Hz respectivement pour les échantillons préparés par ajout de nanoparticules. / Electric motors, transformers, electromagnets and many other electric devices require materials that can provide a path for magnetic field lines while minimizing losses that they generate. Materials used for these applications are called soft magnetic materials. One way to minimize losses in such materials is to increase their resistivity. In order to do so, highly resistive materials are coupled with ferromagnetic materials. These are called soft magnetic composites. For more than one hundred years, the most common type of soft magnetic composite was made by stacking sheets of rolled iron separated by a thin layer of insulating materials. These were very simple and efficient but also had their share of drawbacks. During the last decades, a new technique based on powder metallurgy was developed. It consists in coating iron particles with an isolating material prior to compaction. This type of materials can be highly resistive. The objective of this project was to develop a soft magnetic composite using metal powders in which the insulating materials would be nanometric NiZn ferrite. Two different techniques were studied in order to achieve this goal. The first one consists in coating iron powders with NiZn ferrite using ferrite plating and the second one consists adding nanoparticles to iron powder. The results obtained throughout this study showed that these two techniques could certainly be used to develop metal powder based soft magnetic composites. More specifically, magnetic weight losses of 11,9 W/kg and 93 W/kg were obtained at 60 Hz and 400 Hz respectively for components prepared using the ferrite plating technique while losses of 13,5 W/kg and 137 W/kg were obtained at 60 Hz and 400 Hz respectively for components prepared by adding ferrite nanoparticles.

An impedance-based core loss measurement technique for power ferrites, the modeling and analysis of mechanisms of high-frequency losses, and design methodology for optimization for high-frequency magnetics are presented. The high-frequency losses of ferrite materials are characterized employing a large-signal impedance measurement technique. The impedance analyzer controlled through an IEEE-488 interface, measures the impedance of the inductor under test under large signal excitation via a power amplifier. The core loss is a form of a parallel resistance is derived from measured impedance characteristics. A wideband impedance probe, enables core loss characterization up to 100 MHz. A comprehensive analysis of all major loss mechanisms in ferrites is presented. A new form of residual losses due to a magnetoelectric effect is postulated to account for losses at high frequencies. Two models of losses in ferrites are proposed, one with emphasis on analysis of loss mechanisms, and the other with an emphasis on the design of high-frequency magnetic components. Both models include the important effect of static bias field, which is the case in many power electronics applications. Magnetic losses due to magnetostriction are measured. Dependence of magnetoelastic resonances on the magnetic bias. core material, core shape and size is studied. The influence of diffusion after-effect on core loss under time-varying bias field is investigated. Thermal stability of high-frequency magnetics is studied. A verification of one- and two- dimensional models of winding losses for solid and litz wire is performed. The optimum design method for high-frequency power transformers and inductors is proposed. / PhD

L'objectif de ce travail est d'obtenir un outil numérique permettant de simuler et d'optimiser le procédé de mise en forme par compression et frittage de poudres métalliques. La démarche proposée repose sur une étude expérimentale du comportement des matériaux, une modélisation phénoménologique et l'utilisation du modèle dans un code éléments finis. Une étape de validation des calculs à partir de pièces fabriquées en milieu industriel complète la démarche. La première partie de ce manuscrit est consacrée à l'étude de poudres métalliques (mélanges fer-graphite et carbure de tungstène-cobalt). Une vaste étude expérimentale a plus particulièrement été entreprise sur le mélange fer-graphite. Les conditions expérimentales à adopter pour obtenir le comportement intrinsèque du matériau ont été étudiées. Une loi élastoplastique isotrope à une variable d'écrouissage a été ensuite retenue pour simuler la compression d'une pièce de référence complexe. La confrontation entre résultats numériques et expérimentaux montre quelques écarts qui sont discutés. Les déformations au cours du frittage sont étudiées sur les mélanges de poudres de carbure de tungstène et de cobalt à partir d'essais de dilatométrie. Les résultats expérimentaux ont permis d'identifier les paramètres d'une loi phénoménologique simple. La seconde partie de ce travail est consacrée à l'étude de pâtes d'hexaferrite de strontium destinées à la fabrication d'aimants permanents. Le comportement du matériau a été étudié sous diverses sollicitations mécaniques et thermiques. Nous avons montré que la simulation de la compression d'un segment ferrite à partir d'une approche monophasique était mal adaptée parce qu'elle n'était pas suffisamment représentative des phénomènes observés. Enfin les retraits anisotropes au cours du frittage ont été calculés à partir d'un modèle thermoélastique. L'agrément obtenu entre les résultats numériques et expérimentaux valide les calculs proposés.

Die vorliegende Arbeit befasst sich mit magnetischen Tunnelkontakten (magnetic tunnel junctions, MTJs) auf Basis des Oxids Zinkferrit (ZnxFe3-xO4). Dabei soll das Potential dieses Materials durch die Demonstration des Tunnelmagnetowiderstandes (tunnel magnetoresistance, TMR) in zinkferritbasierten Tunnelkontakten gezeigt werden. Dazu wurde ein Probendesign für MTJs auf Basis der „pseudo spin valve“-Geometrie entwickelt. Die Basis für dieseStrukturen ist ein Dünnfilmstapel aus MgO (Substrat) / TiN / ZnxFe3-xO4 / MgO / Co. Dieser ist mittels gepulster Laserabscheidung (pulsed laser deposition, PLD) hergestellt. Im Rahmen dieser Arbeit wurden die strukturellen, elektrischen und magnetischen Eigenschaften der Dünnfilme untersucht. Des weiteren wurden die fertig prozessierten MTJ-Bauelemente an einem im Rahmen dieser Arbeit entwickeltem und aufgebautem TMR-Messplatz vermessen. Dabei ist es gelungen einen TMR-Effekt von 0.5% in ZnxFe3-xO4-basierten MTJs nachzuweisen. Das erste Kapitel der Arbeit gibt eine Einführung in die spintronischen Effekte Riesenmagnetowiderstand (giant magnetoresistance, GMR) und Tunnelmagnetowiderstand (TMR). Deren technologische Anwendungen sowie die grundlegenden physikalischen Effekte und Modelle werden diskutiert. Das zweite Kapitel gibt eine Übersicht über die Materialklasse der spinellartigen Ferrite. Der Fokus liegt auf den Materialien Magnetit (Fe3O4) sowie Zinkferrit (ZnxFe3-xO4). Die physikalischen Modelle zur Beschreibung der strukturellen, magnetischen und elektrischen Eigenschaften dieser Materialien werden dargelegt sowie ein Literaturüberblick über experimentelle und theoretische Arbeiten gegeben. Im dritten Kapitel werden die im Rahmen dieser Arbeit verwendeten Probenpräparations- und Charakterisierungsmethoden vorgestellt und technische Details sowie physikalische Grundlagen erläutert. Die Entwicklung eines neuen Probendesigns zum Nachweis des TMR-Effekts in ZnxFe3-xO4-basierten MTJs ist Gegenstand des vierten Kapitels. Die Entwicklung des Probenaufbaus sowie die daraus resultierende Probenprozessierung werden beschrieben. Die beiden letzten Kapitel befassen sich mit der strukturellen, elektrischen und magnetischen Charakterisierung der mittels PLD abgeschiedenen Dünnfilme sowie der Tunnelkontaktstrukturen.

Submitted by Edisangela Bastos (edisangela@ufpa.br) on 2012-12-21T20:53:17Z No. of bitstreams: 2 license_rdf: 23898 bytes, checksum: e363e809996cf46ada20da1accfcd9c7 (MD5) Tese_AnaliseTeoricaNovos.pdf: 4286403 bytes, checksum: 2e6ecba005be9d6281dcd2ba7b390082 (MD5) / Approved for entry into archive by Ana Rosa Silva(arosa@ufpa.br) on 2012-12-28T13:59:17Z (GMT) No. of bitstreams: 2 license_rdf: 23898 bytes, checksum: e363e809996cf46ada20da1accfcd9c7 (MD5) Tese_AnaliseTeoricaNovos.pdf: 4286403 bytes, checksum: 2e6ecba005be9d6281dcd2ba7b390082 (MD5) / Made available in DSpace on 2012-12-28T13:59:17Z (GMT). No. of bitstreams: 2 license_rdf: 23898 bytes, checksum: e363e809996cf46ada20da1accfcd9c7 (MD5) Tese_AnaliseTeoricaNovos.pdf: 4286403 bytes, checksum: 2e6ecba005be9d6281dcd2ba7b390082 (MD5) Previous issue date: 2012 / Este trabalho apresenta um estudo teórico sobre novos circuladores compactos com 3-portas tipos W e Y, baseados em cristais fotônicos bidimensionais. No circulador tipo Y, os guias de onda que o compõem formam ângulos de 120° entre si (com formato assemelhado ao da letra Y). O circulador tipo W é uma modificação do tipo Y, obtido a partir do reposicionamento de uma das portas entre as outras duas com um ângulo de 60° entre os guias de onda (com formato assemelhado ao da letra W). Os parâmetros geométricos dos cristais foram obtidos dos diagramas de bandas proibidas. O circulador de três portas tipo Y, projetado para operar em frequências de micro-ondas, foi investigado com o objetivo de gerar um protótipo inédito, enquanto que o tipo W, para frequências ópticas, foi investigado para demonstrar a possibilidade de desenvolver um circulador mais compacto em comparação com o tipo Y conhecido. O tipo W pode ser também uma alternativa geométrica mais adequada no design de circuitos integrados. Os modelos são bons no sentido em que possuem elevada isolação (maior que -20 dB em ambos os circuladores) e baixa perda de inserção (maior que -0,5 dB no caso do circulador tipo Y). O circulador tipo W apresenta uma largura de banda de operação em torno de 100 GHz para um nível de -20 dB de isolação, centrado no comprimento de onda de 1,5um. As simulações foram feitas utilizando-se o software comercial COMSOL Multiphysics, o qual se baseia no método dos elementos finitos. / This work presents a theoretical study on new compact 3-ports circulators of W- and Y-types, based on two-dimensional photonic crystals. In the Y-type circulator, the waveguides that composes it form angles of 120° between each other (having the shape of a Y). The W-type circulator is a modification of the Y-type, obtained by repositioning one of the ports between the other two with an angle of 60° between them (having the shape of a W). The geometric parameters of crystals were obtained from band-gap diagrams. The three-port circulator of Y-type, designed to operate at microwave frequencies, was investigated with the aim of generating a innovative prototype, while the W-type, for optical frequencies, was investigated to demonstrate the feasibility of developing a more compact circulator compared to the known Y-type. The W-type can also be a more adequate alternative geometry for the design of integrated circuits. The models are reasonable in the sense that they present high isolation (higher than 20 dB in both circulators) and low insertion loss (less than -0.5 dB in the case of Y-type circulator). The W-type circulator has about 100 GHz of operation bandwidth, centered at the wavelength of 1.5 um. The simulations were performed using the commercial software COMSOL Multiphsics, which is based on the finite element method.