Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons [1 ed.] 9781614700166, 9781613249376

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Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

LASERS AND ELECTRO-OPTICS RESEARCH AND TECHNOLOGY

OPTICAL LATTICES

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STRUCTURES, ATOMS AND SOLITONS

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LASERS AND ELECTRO-OPTICS RESEARCH AND TECHNOLOGY

OPTICAL LATTICES STRUCTURES, ATOMS AND SOLITONS

BENJAMIN J. FUENTES

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

EDITOR

Nova Science Publishers, Inc. New York

Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

Copyright © 2012 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers‟ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data Optical lattices : structures, atoms, and solitons / [edited by] Benjamin J. Fuentes. p. cm. Includes bibliographical references and index. 1. Optical lattices. 2. Optoelectronics. I. ISBN:  (eBook) Fuentes, Benjamin J. TA1750.O547 2011 621.36--dc23 2011018630

Published by Nova Science Publishers, Inc. †New York Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

CONTENTS Preface Chapter 1

Optical Lattices Prepared by Laser Treatment on Polymers O. Lyutakov, J. Siegel, V. Hnatowicz and V. Švorčík

Chapter 2

Photoluminescence of ZnO Thin Films and Nano Powders Doped with Mono, Di and Trivalent Cations B. S. Acharya

Chapter 3

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vii

Chapter 5

Chapter 6

Giant Spatial Dispersion in the Region of Plasmon-Phonon Interaction in One-Dimensional- Incommensurate Crystal the Higher Silicide of Manganese (HSM) S. V. Ordin and W. N. Wang

1

59

101

Peculiarities of Magnetooptical Properties in Crystals with Mixed Valence Centers Lubov Falkovskaya and ValentinMitrofanov

131

The Interplay of Optical Lattices with Localized Nonlinearity: One-Dimensional Solitons Nir Dror and Boris A. Malomed

161

A New Multilayer Structure Based Refractometric Optical Sensing Element Anirudh Banerjee

197

Chapter 7

Matter Wave Dark Solitons in Optical Superlattices Aranya B. Bhattacherjee and Monika Pietzyk

203

Chapter 8

Optical Superlattices: Where Photons Behave Like Electrons M. Ghulinyan, Z. Gaburro, L. Pavesi, C. J. Oton, N. Capuj, R. Sapienza, C. Toninelli, P. Costantino and D. S. Wiersma

213

Index

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PREFACE An optical lattice is formed by the interference of counter-propagating laser beams, creating a spatially periodic polarization pattern. The resulting periodic potential may trap neutral atoms via the Stark shift. Atoms are cooled and congregate in the locations of potential minima. The resulting arrangement of trapped atoms resembles a crystal lattice. This book presents current research from across the globe in the study of optical lattices, including optical lattices prepared by laser treatment on polymers, peculiarities of magnetooptical properties in crystals with mixed valence centers; the interplay of optical lattices with localized nonlinearity and matter wave dark solitons in optical superlattices. Chapter 1 - Interference of coherent laser beams leads to appearance of periodical regions of different energy density. This periodical pattern (so-called optical lattice) is an interesting object for fundamental physics, since atoms trapped by periodical potential are suitable objects for studying propagation of spin waves, properties of super-solids or behaviour of Cooper pairs. At the other hand, the interference of the laser beams has also practical applications in production of optical gratings, meta-materials, and substrates for assembling of particles or cells. Laser light can also be used to change morphology or structure of thin polymer films. Experimental arrangement may consist of one or two laser beams irradiating the polymer surface. Diffraction pattern are formed either by interference of two incoming beams or by interference of primary and scattered laser beam. Formation of new structures on the polymer surface arises from periodical space distribution of the energy density replicating the light interference pattern. Another, simpler way of polymer modification is irradiation with focused laser beam by which a temperature gradient may be created in the irradiated spot. The temperature and surface tension gradients lead to redistribution of polymer material. In this way sub-micron structures can be prepared on the polymer surface. Depending on the experimental conditions different mechanisms such as ablation of materials, changes of polymers chemistry, photo-migration and photo-orientation processes, flow under temperature and tension gradients may govern the polymer modification by the laser light. In view of many factors taking part in the modification process a satisfactory explanation of the structure formation on polymers by laser irradiation has not been found yet. Microelectronic and optoelectronic technologies ever tend to a higher integrity and density of components. One of the prerequisites of this effort is preparation of well-defined structures on various substrates. Nano-patterning of polymers by the laser beams is novel and useful technique for controlled creation of parallel sub-micrometer patterns on polymer substrates which can be employed for the production of polymer based semiconductors and insulators. Besides

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viii

Benjamin J. Fuentes

common methods such as electron or ultraviolet laser lithography which are nowadays frequently used for the preparation of sub-micrometer patterns on vast range of substrates, the polarized light of excimer laser is an outstanding tool allowing mask-less, photoresist-less formation of periodic structures on polymers with periodicity of the order of the light wavelength. Chapter 2 - Zinc Oxide, a unique material, exhibiting semiconducting, piezoelectric, photonic and anti bactericidal properties has been studied extensively for its structural, optical, electrical and light emitting characteristics by various techniques. Due to formation of nanostructures like rod, tube, needle, comb, flower spring and helixes etc. optical properties have not been determined unambiguously. Nano ZnO films have been studied for its application as nano generator. The material has been recognized as a promising photonic material in uv-vis region also for its lasing action. In this system oxygen vacancies, impurities and Zn interstitials have been observed to play key role in deciding the photo, thermo and stimulated luminescence phenomena taking place in the system. The present review intends to clarify certain aspects of these structural and optical properties taking into account the various impurities like Ag, Al, Cu, Ce, Gd and Eu. Different deposition processes like spray pyrolysis, vacuum deposition and low temperature RF plasma and sonication have been studied and presented. On the basis of available literature on photoluminescence and other luminescence the data have been discussed and tentative model has been proposed. Chapter 3 - If you are faced with unusual (not described by the standard models) physical properties, that, naturally, you begin with checkout of the delivered experiment and searching not considered both directed by experiment, and in modeling of factors. Therefore detection of "anomalies" in optical properties of a thermoelectric crystal a HSM has caused a large cycle of investigations of the various factors influencing optical and kinetic properties of crystals. As originally "anomaly" has been observed in reflexion spectra a HSM checkout of agency on an observable spectral singularity of a state of investigated surfaces of crystals, first of all, has been led. Chapter 4 - This chapter studies the characteristics of the Faraday effect and magnetic circular dichroism in crystals containing iron-group ions of different valences. The cause of such ions appearance can be nonisovalent substitutions or vacancies in the anion and cation sublattices. Quite often, mixed-valence complexes have features characteristic of the JahnTeller ion. Genealogically this is related to the fact that usually one of two states of a single ion with configurations dn or dn1 is orbitally degenerate. Local low-symmetry fields produced by the source of the excess charge, remove this degeneracy. Nevertheless, the ground state of the whole complex of mixed valence may be degenerate due to the effects of the excess charge transfer between 3d-ions of the complex. In this chapter the authors consider just this case, when along with the traditional properties of the Jahn-Teller centers features appear caused by the redistribution (reorientation) of the excess charge between the ions of the complex, and the degeneracy is lifted due to external perturbations or cooperative interactions. Removal of the orbital degeneracy of the cluster leads to a nonuniform distribution of excess charge q between the magnetic ions and the appearance of a relatively large dipole moment of the cluster, proportional to qR, where R is the distance between the source of the excess charge and the nearest 3d-ions in the cluster. The presence of mixed valence clusters in the crystal may lead to an anomalous increase of the antisymmetric components of the dynamic dielectric tensor and, consequently, of the Faraday effect and the magnetic circular dichroism. If a set of energy levels of such clusters with localized on them

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Preface

ix

excess charge contain degenerate levels with unfrozen orbital angular momentum, their contribution to the magneto-optical effects can be significant. The influence of the mixed valence centers on the complex Faraday effect of crystals with spinel, garnet and perovskite structures is considered. It is shown that the electric dipole transitions in such clusters lead to a significant magneto-optical activity in both infrared and visible ranges of spectrum. The contribution of mixed valence clusters in a magneto-optical properties of crystals becomes 17 19 3 quite noticeable even at relatively low concentrations (  10  10 cm ). The intensity of the corresponding transitions is comparable to or higher than of the allowed electro-dipole crystal field single ion transitions. Chapter 5 - Standard models of periodically modulated nonlinear media, such as photonic crystals and Bose-Einstein condensates (BECs) trapped in optical lattices (OLs), are often described by the nonlinear Schroedinger/Gross-Pitaevskii equations with periodic potentials. The authors consider a model including such a periodic potential and the attractive or repulsive nonlinearity concentrated at a single point or at a set of two points, which are represented by delta-functions. For the attractive or repulsive nonlinearity, the model gives rise to ordinary solitons or gap solitons (GSs). These localized modes reside, respectively, in the semi-infinite gap, or finite bandgaps of the system‟s linear spectrum. The solitons are pinned to the delta-functions. Realizations of these models are relevant to optics and BEC. The authors demonstrate that the single nonlinear delta-function supports families of stable ordinary solitons and GSs in the cases of the self-attractive and repulsive nonlinearity, respectively. The authors also show that the delta-function can support stable GSs in the first finite bandgap in the case of the self-attraction (it is usually assumed that the attractive nonlinearity gives rise to unstable GSs in finite bandgaps). The stability of the GSs in the second finite bandgap is investigated too. In addition to the numerical analysis, analytical approximations are developed for the solitons in the semi-infinite gap and two lowest finite bandgaps. In the model with the symmetric pair of delta-functions, the authors investigate the effect of the spontaneous symmetry breaking of the pinned solitons. Chapter 6 - A new multilayer structure based refractometric optical sensing element is suggested for sensing very small refractive index changes of a medium. This new multilayer structure exhibits isolated narrow transmission peaks in the output spectrum in stop band wavelength regions and a slight change in refractive index of material layers induces large transmission peak shifts in the output spectrum. This new multilayer based refractometric sensing element is not only remarkably smaller but is also very sensitive to refractive index changes. Chapter 7 - In this chapter, the authors study the behaviour of matter-wave band gap spectrum and eigenstates as the periodicity of the optical superlattice is increased. The authors show that the band gap (between the two lowest bands) which opens up in a doubly periodic superlattice decreases as the periodicity increases further. This is interpreted as a decrease in the Periels-Nabarro barrier which the dark soliton experiences as it goes from one well to the next. For higher periodicity the mobility of the dark soliton is restored. Chapter 8 - The authors report on optical analogues of well-known electronic phenomena such as Bloch oscillations and electrical Zener breakdown. The authors describe and detail the experimental observation of Bloch oscillations and resonant Zener tunneling of light waves in static and time-resolved transmission measurements performed on optical superlattices. Optical superlattices are formed by one-dimensional photonic structures (coupled

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Benjamin J. Fuentes

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microcavities) of high optical quality and are specifically designed to represent a tilted photonic crystal band. In the tilted bands condition, the miniband of degenerate cavity modes turns into an optical Wannier-Stark ladder (WSL). This allows an ultrashort light pulse to bounce between the tilted photonic band edges and hence to perform Bloch oscillations, the period of which is defined by the frequency separation of the WSL states. When the superlattice is designed such that two minibands are formed within the stop band, at a critical value of the tilt of photonic bands the two WSLs couple within the superlattice structure. This results in a formation of a resonant tunneling channel in the minigap region, where the light transmission boosts from 0.3% to over 43%. The latter case describes the resonant Zener tunneling of light waves.

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In: Optical Lattices: Structures, Atoms and Solitons Editor: Benjamin J. Fuentes

ISBN: 978-1-61324-937-6 © 2012 Nova Science Publishers, Inc.

Chapter 1

OPTICAL LATTICES PREPARED BY LASER TREATMENT ON POLYMERS O. Lyutakov1, J. Siegel1, V. Hnatowicz2 and V. Švorčík1* 1

Department of Solid State Engineering, Institute of Chemical Technology, 166 28 Prague, Czech Republic 2 Nuclear Physics Institute of Academy of Sciences of the Czech Republic, Rez near Prague, Czech Republic

ABSTRACT

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Interference of coherent laser beams leads to appearance of periodical regions of different energy density. This periodical pattern (so-called optical lattice) is an interesting object for fundamental physics, since atoms trapped by periodical potential are suitable objects for studying propagation of spin waves, properties of super-solids or behaviour of Cooper pairs. At the other hand, the interference of the laser beams has also practical applications in production of optical gratings, meta-materials, and substrates for assembling of particles or cells. Laser light can also be used to change morphology or structure of thin polymer films. Experimental arrangement may consist of one or two laser beams irradiating the polymer surface. Diffraction pattern are formed either by interference of two incoming beams or by interference of primary and scattered laser beam. Formation of new structures on the polymer surface arises from periodical space distribution of the energy density replicating the light interference pattern. Another, simpler way of polymer modification is irradiation with focused laser beam by which a temperature gradient may be created in the irradiated spot. The temperature and surface tension gradients lead to redistribution of polymer material. In this way sub-micron structures can be prepared on the polymer surface. Depending on the experimental conditions different mechanisms such as ablation of materials, changes of polymers chemistry, photo-migration and photo-orientation processes, flow, under temperature and tension gradients, may govern the polymer modification by the laser light. In view of many factors taking part in the modification process a satisfactory explanation of the structure formation on polymers by laser irradiation has not been found yet. *

E-mail address: [email protected]

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2

O. Lyutakov, J. Siegel, V. Hnatowicz and V. Švorčík Microelectronic and optoelectronic technologies ever tend to a higher integrity and density of components. One of the prerequisites of this effort is preparation of welldefined structures on various substrates. Nano-patterning of polymers by the laser beams is novel and useful technique for controlled creation of parallel sub-micrometer patterns on polymer substrates which can be employed for the production of polymer based semiconductors and insulators. Besides common methods such as electron or ultraviolet laser lithography which are nowadays frequently used for the preparation of submicrometer patterns on vast range of substrates, the polarized light of excimer laser is an outstanding tool allowing mask-less, photoresist-less formation of periodic structures on polymers with periodicity of the order of the light wavelength.

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1. INTRODUCTION Laser micro- and nano-nanofabrication of various passive and active structures and devices in polymer solid media have attracted a lot of research efforts in recent years. There are several optical processes and techniques that may be utilized for producing structures on a flat surface or in the polymer bulk: surface relief grating, localized photochemical reactions or surface ablation, interference lithography, holography, laser scanning and so on. These techniques utilize the change of polymer structure, chemistry or surface relief under photon absorption. Interference patterns of laser light are used to prepare a periodical structure in a relatively simple way. The process can be performed on different polymers, including those commercially available or specially synthesized in laboratory. In the following some of these perspective techniques are described in more details. Optics has traditionally dealt with dielectric materials. Here, the incident electric-field vector of the light excites microscopic electric dipoles that re-emit electromagnetic waves just like a dipole radio antenna. Other dipoles are excited by this re-emission and so on. This successive excitation and re-emission clearly modifies the phase velocity of light and determines the optical properties of the material, in particular its electric permittivity ε (or dielectric function). One of the “little” revolutions in optics that the concept of artificial effective materials (“meta-materials”) has brought about is the fact that, similarly, magnetic dipoles can be excited by the magnetic-field component of the light, which can be cast into the effective magnetic permeability μ of the material [1-8] (or, alternatively, into spatial dispersion). These metamaterials, man made structures which exhibit extraordinary physical properties e. g. negative differential resistance [9-11] or negative refractive index [3,4], are typically binary composites of conventional materials (a matrix with inclusions of a given shape, arranged in a periodic structure). Since the times of Maxwell, Lord Rayleigh, and Maxwell Garnet up to today, many authors have contributed to the calculation of the bulk macroscopic response in terms of the dielectric properties of its constituents [12-15]. The interplay of ε and μ has given rise to interesting new aspects of electromagnetism that have been reviewed several times [16-20]. In general, both ε and μ are tensors, i.e., the dipole direction is not necessarily identical to the exciting vector direction. A further set of new possibilities originates from the so-called “cross terms”. This means that, in general, magnetic dipoles can not only be excited by the magnetic field but also by the electric field. Similarly, electric dipoles can not only be excited by the electric field but also by the magnetic field [21].

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Mutual orientation of the dipole vectors and exciting field plays an important role regarding to resulting interaction. Whether the dipoles vectors are oriented perpendicular to the exciting fields the resulting interaction is called “bianisotropy“. On the contrary, if the dipole vectors are oriented parallel to the exciting fields, another special case arises, namely chirality. Bianisotropy and chirality are very well-established parts of electromagnetism and a bulk of corresponding theoretical literature does exist [22-39]. Recent technologies allow the manufacture of ordered composite materials with periodic structures, thus giving arrise unique optical properties. For instance, high-resolution electron beam lithography and its interferometric counterpart have been used in order to make particular designs of nano-structured composites, producing various shapes with nanometric sizes [40,41]. Moreover, ion milling techniques are capable of producing high quality air hole periodic and nonperiodic two-dimensional (2D) arrays, where the holes can have different geometrical shapes [42,43]. On the other hand besides common method nowadays widely used for material nano-structuring (see above), new promissing method are currently developed using polarized beam of excimer laser lights [44-46]. Therefore, it is possible to build devices with novel macroscopic optical properties [47]. For example, a negative refractive index has been predicted and observed [48] for a periodic composite structure of a dielectric matrix with noble metal inclusions of trapezoidal shape [49]. Nano-structured metallic films are having an important development as well. On one hand, the existence of surface plasmon-polariton (SPP) modes, excited on the metal–air interface, yields several related phenomena such as an enhancement of optical transmission through sub-wavelength holes [50-53]. Besides the single coupling to SPP modes, double resonant conditions [54], and waveguide modes [55] seem to play an important role in the optical enhancement for metallic gratings with very narrow slits and for compound gratings [56]. On the other hand, a very strong polarization dependence in the optical response of periodic arrays of oriented subwavelength holes on metal hosts [42,43,57] and single rectangular inclusion within a perfect conductor [58] have been recently reported. These studies did not rely on SPP excitation as a mechanism to explain their optical results. Although discovered only 10 years ago, negative refractive index materials (NIMs) have been the target of intense study, drawing researchers from physics, engineering, materials science, optics, and chemistry. These artificial “metamaterials” are fascinating because, as it was shown above, they allow the design of substances with optical properties that simply do not occur in nature [16,59-61]. Such materials make possible a wide range of new applications as varied as cloaking devices and ultrahigh-resolution imaging systems. The variety of possible applications would be even greater if such materials could be engineered to work at optical wavelengths. Since the first demonstration [2] of an artificial negative refractive index materials (NIMs) in 2000, metamaterials have exhibited a broad range of properties and potential applications: nearly zero reflectance; nanometer-scale light sources and focusing; miniaturization of devices, such as antennas and waveguides; and novel devices for medical imaging, especially magnetic resonance imaging. For example, metamaterials may lead to the development of a flat superlens [47] that operates in the visible spectrum, which would offer superior resolution over conventional technology and provide image resolutions much smaller than one wavelength of light. Subsequent theory and experiments [3,4,62-68] confirmed the reality of negative refraction. The development of NIMs at microwave frequencies [24,62,63] has progressed to the point where scientists and engineers are now vigorously

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O. Lyutakov, J. Siegel, V. Hnatowicz and V. Švorčík

pursuing microwave applications. In contrast, research on NIMs that operate at higher frequencies [64-68] is at an early stage, with issues of material fabrication and characterization still being sorted out. As it was already discussed, metamaterials are typically formed by combination of dielectrics and metals forming binary composites. Comprehensive rewiev about basic structures with negative refractive index was published in [19]. Frequently used structure that exhibits negative refractive index is split-ring resonator (SRR). This structure exhibits a band of negative µ values even though it is made of non-magnetic materials. Creation of so-called double SRR allows operation at longer wawelength (in orders of cm). A negative μ at 10 GHz requires SRR dimensions of the order of 1 mm. To obtain negative ε, one needs to arrange long and thin wires in a simple cubic lattice. Negative ε at gigahertz frequencies might be obtained with wires few tens of micrometers in diameter and spaced several millimeters apart. By using an array of SRRs and thin wires in alternating layers, several groups [2-4,62,63] showed negative refractive index at giga hertz frequencies. Although the choice of the metal constituting the structure is not critical in the microwave regime, it is crucial in the optical and the visible regime because the metamaterial losses are dominated by metal losses. Silver exhibits the lowest losses at optical frequencies, and indeed, going from gold [64-66] to silver drastically reduced the losses at similar frequencies [67]. Furthermore, the use of silver has enabled the preparation of first negative-index metamaterials at the red end of the visible spectrum [68] (wavelength 780 nm). Another group has also reported a negative refractive index [41,69], but this has been questioned recently [70]. Only 6 years after their first demonstration, negative-index metamaterials have been brought from microwave frequencies toward the visible regime. However, for applications to come within reach, there are still many difficulties to be overcome. With emerging techniques such as micro-contact printing, nano-embossing, holographic lithography, and quantum tailoring of large molecules, it seems likely that these technical challenges can be successfully met. In what follows we shall briefly discuss the common optical processes for polymer patterning, then the attention was focused on nonlinear optical processes and materials for polymer optical modification and the new technique for materials patterning by laser light. At the end, the examples of application of prepared structures will also be given.

2. LITHOGRAPHY The traditional polymer patterning can include the optical lithography and direct laser writing. Tendencies to improve optical resolution of these techniques and the quality of prepared structures have leaded to appearance of immersion lithography, deep ultraviolet lithography, and near field optical lithography [71-73].

2.1. Traditional Photolithography Traditional photolithography is a common process which utilizes the changing of polymers properties under light illumination. Thin layers of polymer bead photoresist are

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usually prepared by spin-coating technique and then illuminated by UV light through a mask. Depending on the polymer, the illuminated parts become insoluble or more soluble. At the next step the parts of photoresist layers are selectively removed by wet processes [74] (see Figure 1).

Figure 1. Standard photolithography process. Step A – resist deposition, B – illumination through mask, C – development by dissolving of unexposed resist areas.

Using these techniques two-dimensional features can be formed by optical irradiation of a photoresist, followed by chemical development of the exposed photoresist. The smallest feature size that can be achieved (i.e. the resolution) is limited by diffraction of light by the photomask and by feature broadening during the chemical development process. Hence, the best resolution obtainable using common UV photolithography is on the order of 1µm [75].

2.2. Direct Laser Writing Another conventional technique is direct laser writing. This process involves multiphoton absorption [76]. Similar to standard photolithography, structuring is accomplished by illuminating negative-tone or positive-tone photoresists by light of a certain wavelength [77]. In comparison with photolithography, the fundamental difference is absence of complex optical systems or photomasks. Materials that are used for direct laser writing are usually transparent at the laser operating wavelength and multiphoton absorption is utilized to induce a dramatic change in the solubility of the resist. [78]. By scanning and properly modulating

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O. Lyutakov, J. Siegel, V. Hnatowicz and V. Švorčík

the laser beam and its power, a chemical change (usually polymerization), occurring at the focal spot of the laser beam, may be induced at required positions at the substrate surface (Figure 2).

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Figure 2. Principle of direct laser writing – the light affects the material only within the focus (e.g. by two photon absorption).

Direct laser writing can be performed with continuous or femtosecond lasers. A continuous-wave laser is easier to direct, and it is less expensive than the femtosecond lasers used in the past [79]. This and the fact that commercially available photoresists was demonstrated to be useful in direct laser writing [80], creates a situation in which this technique for a variety of applications becomes attractive. Instead of traditional lithography direct laser writing allow the preparation of three dimensional structures [80]. In direct laser writing by multiphoton polymerization, a photoresist is illuminated by laser light at a frequency below the single-photon polymerization threshold of the resist. When this laser light is tightly focused inside the photoresist, the light intensity inside a small volume around the focus may exceed the threshold for initiating multiphoton polymerization. The size and shape of these so-called voxels depend on the microscope objective, and the exposure threshold for multiphoton processes of the photoresist. The voxel sizes down to 120 nm for illumination at a wavelength of 780 nm as was reported [81]. An arbitrary threedimensional periodic or non-periodic pattern can be prepared by controlling the laser power and position of focal spot. This method was used for rapid prototyping of structures with fine features (Figure 3) [82]. Additionally, this technique can be used for creation of photonic crystal. However, from the point of view of practical applications of direct laser writing there are two sufficient limitations. First, process is relatively slow and exposition of great polymers area needs sufficient time interval. Second, direct laser writing is very sensitive to mechanical vibration which can prevent the preparation of structures on long distances. Main disadvantage of both, photolithography and direct laser writing methods can be attributed to restricted class of materials, which can be used, necessity of wet steps, and limitation in spatial resolution. There are several ways to improve photolithography resolution which are now commercially used (immersion lithography, EUV lithography) or relatively new near field optical lithography.

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Figure 3. Microfabrication and nanofabrication at subdiffraction-limit resolution by direct laser writing with two-photon absorption mechanism, scale bars 2 µm [82].

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2.3. Immersion Lithography The key benefit of immersion lithography is the possibility of constructing projection optics with numerical apertures (NAs)>1 by introducing an immersion fluid with a refractive index larger than one. Currently preferred immersion medium is water which has low cost, low toxicity, and a refractive index of 1.44. It was demonstrated that 45 nm patterns can be achieved by using a chemically amplified resist [71]. However, there are several key problems which must be addressed before immersion lithography gains widespread acceptance. These problems relate to the use of an immersion fluid and resist materials for immersion lithography. Immersion fluid can‟t contain any microor nano-size bubbles or contamination, which can scatter the incoming radiation and affect the imaging process. The development of resists materials for immersion lithography will have to deal with the obvious issues related to the exposure of the imaging stack to aqueous or other immersion media. The effects like penetration of the immersion media into the imaging stack, immersion media uptake on photoresist chemistry, the leaching of photoresist components into the immersion media must be excluded [83].

2.4. EUV (Extreme Ultraviolet) Lithography Another approach to improve lithography is application extreme ultraviolet wavelength – so called EUV lithography [72]. In EUV lithography photons with a wavelength of 13.4 nm are used to pattern a photoresists. The primary events induced by absorption of these highenergy photons occur within the inner shells of atoms. As a result, EUV absorption depends

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not on molecular composition but on atomic composition. Given that EUV includes a significant reduction in wavelength compared to current lithography wavelengths, one would expect significantly better resolution. The resolution down to 13.4 nm could be expected [84]. However, the resolution is ultimately determined by the interaction volume in the image recording medium, i.e. a photoresist. When EUV photon is absorbed, photoelectrons and secondary electrons are generated by ionization. These secondary electrons have energies of a few to tens of eV and travel tens of nanometers inside photoresist, before initiating the desired chemical reaction. A contributing factor for this rather large distance is the fact that polymers have significant amounts of free volume. In a recent actual EUV print test, it was found that 20 nm spaces could not be resolved, even though the optical resolution and the photoresist composition were not the limiting factor [85,86].

2.5. Near Field Optical Lithography

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Near field optical lithography, based on a scanning near-field optical microscopy configuration, uses the field-enhancement effect, a so-called lightning-rod effect appearing at the extremity of a metallic tip when illuminated with an incident light polarized along the tip axis (see Figure 4) [73] (see Figure 4) [87].

Figure 4. Field enhancement apertureless method of near field optical lithography [87].

Figure 5. SEM and AFM images of lines prepared by Near Field Optical Lithography technique [89].

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The local enhancement of the electromagnetic field straight below the tip's take place and can cause a photoisomerization reaction. Obtained polymer patterning depends on the a few experimental parameters such as the polarization state, the illumination mode and the tip's geometry. This approach presents an alternative and elegant method of improving the lithography resolution. The advantage of method compared with the other kinds of lithography is to 'be an optics-based technique' without any wavelength resolution limit. Indeed, in the case of near-field illumination, the spatial confinement of the near-field lightmatter interaction is not limited by the light wavelength but rather by both the probe size and the probe-to-matter distance. A large variety of materials (involving numerous photo-induced physical/chemical effects) was used in this method [88]. Additionally, various experiments was performed using several kinds of optical near-field illumination, including fibre tip with aperture, field enhancement apertureless method, near-field mask illumination and so on. This method (see Figure 5, [89]) allows achieving the resolution below 20 nm [87].

3. NONLINEAR OPTICAL PROCESSES IN POLYMERS Application of laser beam or light interference of coherent light opens a simple way for polymer patterning. Several physical or chemical processes can be initiated by the laser beam. Besides of traditional, like photopolymerization, polymer ablation or changing of polymer chemistry there are several other ones, recently discovered and dependent on specific properties of polymer.

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3.1. Processes under Excimer Laser Modification When pulsed UV-laser source becomes commercially available, several pulsed UV-laser applications for surface modification have been explored in many fields. Major processes occurring during polymer modification by excimer laser can be subdivided to: material transport under ablation threshold, ablation and polymer surface chemistry modification [9092]. When polymer surfaces are exposed to a polarized pulsed laser to more than several hundred pulses with a sub-threshold fluence for ablation, ripple structures appear if the photons are absorbed with a high absorption coefficient in the surface layer only several hundred angstroms thick [90]. Possible explanation of observed phenomena consists in the materials transport under light interference pattern. The incident laser beam can interact with the reflected or scattered by surface inhomogeneities light and form the light interference pattern. Then the polymer flow occurs due to effects of temperature or electrical gradients. This method can be applied to the most of conventional polymers. The resulting periodicity is limited by the applied laser wavelength and by the complex response of the material to the laser illumination. This method is suitable for generating structures hundreds nanometers in size. Appearance of ripple structures depends on the polymer chemistry and conformation, surface tension, and stability of material to UV light [91]. When polymer films are exposed to pulsed laser beams at a proper wavelength with beam energy above the ablation threshold energy, photoablation takes place [92]. In a low-fluence

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range an incubation period exists, i.e. after absorbing the laser energy the polymer starts to swell, and subsequently ablated materials are ejected. In a high-fluence range a shock wave front is formed and ablated material is ejected instantaneously with high speed. Usually the ablation of polymers includes depolymerisation of materials and evaporation of lowmolecular weight products. Several theoretical and computer simulation have been published addressing this phenomenon [93,94]. Repeated exposure of polymer surfaces to laser pulses gives rise to special morphological changes of polymer surfaces. Another example of polymer modification by pulsed laser exposure is the surface chemistry modification [95]. For this purpose pulse fluences of the UV laser should be higher than those used for the periodic ripple formation [90]. By exposing polymer surfaces to pulsed UV lasers, chemical properties of polymer surfaces can be improved to more desirable ones. The pulse period could be longer than that used for the ripple formation. This is a useful tool for improving the adhesion of several agents on the polymer surface. In [96] improving of the surface chemistry of polytetrafluoroethylene (PTFE) by this method was described. PTFE is well-known to be a difficult material to be coated with any material with good adhesion because of its very stable chemical structure. A highly caustic treatment is used to coat PTFE with metal. Irradiation of PTFE films by ArF excimer laser in trimethyl borane vapour changes the surface to oleophilic, from its original oleophobic nature, and the modified PTFE surface will accept printing ink. The ArF excimer laser irradiation dissociates both the C-F bonds of PTFE and trimethyl borane and as a result the fluorine atoms are replaced by methyl groups. A similar laser-induced photochemical reaction occurs after exposition of PTFE films to 193 nm ArF excimer laser light in a hydrazine vapour [97]. This process was used for PTFE electroless plating with nickel. The nickel layer adheres strongly to the PTFE surface and it cannot be peeled off using a conventional tape adhesion test. Not only ripple structure can be prepared by polymer modification with pulsed laser beam. Laser ablation of polymers with a stencil mask can generate varieties of surface images on polymer surfaces, which can be used in practice for via-hole fabrication in the electronic packaging industry. In [90] after first stage of modification by laser ablation the sample was rotated by 90° and the previously generated structures were over-illuminated by an additional laser pulses. Another way of generating dot patterns is the use of double beams with their polarization planes orthogonal to each other. Transformation of linear pattern into dot pattern with increasing number of laser pulses in polymer modification by single beam method, with fluence below the threshold value was observed [90]. S. Ibrahim and D. A. Higgins reported [98] a dependence of resulting structure shape on the initial film thickness. It was found, that below a critical thickness of the spin-coated films the line-shaped structures transformed into droplets. This droplet formation was explained by the laser-induced melting across the whole film thickness and subsequent de-wetting on the substrate. The thickness of the layer melted by the laser illumination was computed by a heat-conduction model. Very good agreement with the critical thickness for spin-coated films was found. Patterning of polymers in this way is useful for increasing roughness of the too-smooth surfaces, and for liquid crystal molecular alignment [99]. Additionally, polymer pattern was successfully transferred to silicon surfaces by a series of reactive ion etching [100]. Excimer UV-laser irradiation is effective in enhancement of hydrophilicity, wettability, and gas permeability. Pulsed laser beams can also be used for surface modification of metal deposition on polymer surfaces like PTFE, ABS, and others, which were exposed in air, reactive gases, or aqueous solutions depending on their final uses [101,102].

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However, by pulsed laser modification it is impossible to prepare polymer grating with good optical quality. This limits sufficiently the application of described technology in optics and optoelectronics.

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3.2. Two-Photon Absorption Non-resonant two-photon absorption (TPA) can be defined as the simultaneous absorption of two photons, via a virtual state [103]. The theory of the simultaneous absorption of two photons was developed by Goeppert-Mayer in 1931, and experimentally verified later, after construction of pulsed laser [104]. In the TPA process, molecules exposed to high intensity light can undergo simultaneous absorption of two photons mediated by a so-called „virtual state‟. The combined energy of the two photons enables to access a stable excited state of the molecule (see Figure 6). As light passes through a molecule, the virtual state may form, persisting for a very short duration (of the order of a few femtoseconds) and TPA can occur if a second photon arrives before decay of this virtual state [103]. Two-photon absorption thus involves the concerted interaction of both photons that combine their energies to produce an electronic excitation analogous to that conventionally caused by a single photon of a correspondingly shorter wavelength. Unlike single-photon absorption, whose probability is linearly proportional to the incident intensity, the TPA process depends on both a spatial and temporal overlap of the incident photons and takes on a quadratic (non-linear) dependence on the incident intensity. Two-photon transitions can be described by two different mechanisms. For non-polar molecules with a low-lying, strongly absorbing state near the virtual level, only excited states that are forbidden by single photon selection dipole rules can be populated via two photon absorption. In contrast, strong TPA can occur in polar molecules by a mechanism in which a large change in dipole moment occurs upon excitation of the ground to an excited state. In this case, both the ground and excited states can participate in the formation of the virtual state, enhancing TPA [105].

Figure 6. Scheme of two photon absorption process ΔE = 2 hν.

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There are several processes which can be initiated by TPA and which are reported in the literature as useful instrument for polymer modification. Most of these processes are related to two-photon polymerization, crosslinking or material ablation [106,107]. In practice, TPA can occur only in the place with “great” light intensity. Required spatial distribution of light can be achieved by focusing the light, by light interference or by intersection of laser light beams. It must be noted, that TPA open a wide way for locally polymer modification with the possibility to provide these modification with 3D resolution [108]. Multiphoton absorption has the similar nature with TPA but occurs via several virtual states [105]. As can be expected, multiphoton absorption depends more strongly on the light intensity than TPA. Electronic excitation of the polymer by multiphoton absorption results in its depolymerisation and evaporation from the substrate surface via either photochemical or photothermal processes. That‟s way multiphoton absorption processes can be used for polymer dry etching.

3.3. Photorefractive Effect

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The laser beam, when transmitted through materials, can cause a change in the refractive index of the material. This refractive index inhomogeneity distorts the wavefront of the transmitted laser beam and is described as laser damage because it prevented the use of some materials in optical applications. Later it was found that the materials could return to their initial, homogeneous state if they are subjected to homogeneous illumination or heated to a certain temperature. This particular light-induced change in refractive index is now known as the photorefractive effect [109].

Figure 7. Scheme of photorefractive effect, A – light illumination, B – charge generation, C – charge migration and D – resulting periodic structure. Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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The photorefractive effect has been firstly studied in a variety of inorganic electro-optic crystals such as LiNb03 and BaTi03, ceramics, and semiconductors, including GaAs and InP [109]. In the polymers the photorefractivity is a combined effect of a linear electrooptic response and photoconductivity simultaneously present in a material [110]. It arises from a light-induced generation and migration of charges in the polymer, which give rise to internal space charge fields, and change the refractive index of material. Photorefractivity can be conditionally separated in the next steps: 1 – generation of charge in a photorefractive medium by a spatially modulated light intensity, 2 - migration of charge through drift and/or diffusion processes, 3 – trapping of charge, 4 - appearance an internal electric field which, in turn, modulates the material's refractive index via the linear electro-optic effect [111]. The interaction of two coherent light beams inside a photorefractive material produces a spatially modulated light intensity and cause refractive index grating appearance [112] (see Figure 7).

3.4. Surface Relief Grating

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Very interesting and promising results were reported for surface modification of azocontaining polymers [113]. Dramatic modification of the free surface of a thin azo functionalized polymer films, exposed to an interference pattern of appropriately polarized light beams was demonstrated. The sinusoidally modulated surface structures on the azobenzene functionalized polymer films, known as surface relief grating (SRG), appears due to large-scale polymer chain migration [114] (see Figure 8, [115]). The formation of efficient SRG at a temperature substantially below the glass transition temperature presents new insight into photophysics of polymer surfaces. Essential condition of SRG formation is azobenzene group presence. Azobenzene groups are known to exist in two isomeric states, a thermodynamically stable trans- and a metastable cis- [114] (see Figure 8, [115].

Figure 8. Scheme depicting the surface of a grating with spacing Λ produced by laser inscription; light intensity maxima in the interference surface pattern correspond to surface relief minima on the polymer film [115]. Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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When irradiated with light of appropriate wavelength, the azobenzene chromophore undergoes a reversible trans-cis-trans photoisomerization processes (see Figure 9). Absorption of the appropriate wave-length light by the azo dye molecules elevates them to an electronically excited state. A non-radiative decay from the excited state puts the molecules back to the ground state either in the cis- or the trans- state. The metastable cis state goes to the trans- state either by a spontaneous thermal back reaction or a reverse cis-trans photoisomerization cycle. Additionally, the formation of SRG on azo polymer films is found to be strongly dependent on the polarization of the writing beams [116]. For example, under the intensity recording condition, the two s-polarized writing beams produce the largest light intensity variation. In the direct photo-fabrication of large amplitude holographic SRGs, the free surface of the azobenzene chromophore functionalized polymer thin films is dramatically modified when irradiated with polarized interfering light beam patterns. Several mechanisms have been proposed to explain the origin of the driving force responsible for surface relief production, due to the inherent shape mechanisms, pressure gradients resulting from isomerisation and the supposition of interactions between the azo dipoles and the electric field of the light interference pattern. Models involving thermal effects appear to be the most readily proposed, but suffer the drawback that such mechanisms would be sensitive to the intensity of the light interference pattern, but not its polarization, counter to observations [117]. Models of asymmetric diffusion suggest that individual molecules (chromophores or polymer chains) undergo transient motion upon photo-excitation and thermal re-conversion. Illumination by interfered laser beam creates a concentration gradient of chromophores in different states, depending on the intensity of light in separated polymer region. Molecule has diffused into a dark region, there is a zero probability of it being excited back into a light region, and hence the azo molecules (and their host backbone chains) would be expected to congregate coincident with the interference pattern, and produce a surface relief grating [118]. A mechanism based on pressure gradients inside the polymer film was also proposed. It is known, that chromophores in cis- or trans- state require different volume. Free volume increase required for isomerisation of the bulky azo chromophores leads to a local pressure in areas of high light intensity, proportional to the incident light intensity.

Figure 9. Scheme of azobenzene chromophores trans- and cis- photoisomerisation.

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Upon irradiation, the trans-cis isomerisation of the chromophores requires a change in size and shape of the free volume pocket surrounding each azo group. Insufficient free volume available for this geometrical change results in a pressure exerted on the neighbouring polymer chains, and hence represents a mechanical force in regions of high light intensity. Order-of-magnitude estimates were used to propose that this mechanical pressure is above the yield point in these soft polymers, and hence plastic flow would be expected to result from regions of high light intensity to regions of low light intensity [114]. Mechanisms based on electromagnetic forces naturally include both the intensity and polarization state of the incident radiation. In a mean field model each chromophore is subject to a potential resulting from the dipoles of all other chromophores. Under irradiation, chromophores at any given point will be oriented perpendicular to the light polarization at that point. The mean-field that they generate will tend to align other chromophores in the same direction, and also causes an attractive force between side-by-side chromophores oriented in the same direction. Overall this results in a net force on chromophores in illuminated areas, causing them to order and aggregate. This mechanism predicts a collection of mass in the areas of high light intensity [119]. Another comprehensible theory suggests that the spatial variation of the light (both intensity and polarization) leads to a variation of the material susceptibility at the film surface. The electric field of the incident light then leads to a polarization of the material, whose magnitude is related both to the intensity of the light, and the susceptibility. Forces occur between a polarized material and a light field gradient. Thus, the grating inscription would be related to the spatially-varying material susceptibility, the magnitude of electric field, and the gradient of the electric field. This naturally includes the polarization dependence observed in experiment. This mechanism qualitatively explains the magnitude of SRGs obtained for all combinations of polarization states of writing beams. In this mechanism, azobenzenes are required for two reasons. Firstly, the light-induced orientation of azobenzenes is required to modulate the susceptibility of the material, and second, a photoinduced plasticization is assumed permitting material mobility well below glass transition temperature [120].

4. MATERIALS Laser modification of polymers can be performed on common polymers, like PMMA, PS, PE, PET or on specially synthesized polymers, which contain chromophores, electric conductive parts of molecules and so on. Properties of polymers can be modified in desired way by additional of some low-molecular weight dopants [121].

4.1. Common Polymers Modification of “common” polymers usually includes their ablation or mass redistribution and requires high laser power. The modification by femtosecond laser pulses has been reported for most of traditional polymers, prepared in the form of thin films. Primary laser beam interferes with scattered by surface inhomogeneity waves and produces a

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periodical intensity pattern on the polymer surface, leading to polymer redistribution or ablation [90]. This mechanism can take place in all kind of polymers and can‟t be attributed to specific properties of materials. Surface periodic pattern is produced in this way and depending on the type of polymers, laser wavelength or intensity different parameters of produced surface structures can be achieved. The quality of the prepared structures depends on the polymer used and its inherent properties [122]. Generally it is difficult to produce “fine” periodic structure with “great” amplitude.

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4.2. Photoresist Another wide class of polymers suitable for laser light modification is photoresponsive materials developed for photolithography process [123]. Usually this materials contain some chemical group (for example – epoxy group), which can undergo crosslinking under photoillumination. There are several commercially available photoresist, which are used not only for UV lithography processes, but for modification by laser light too. These polymers or oligomers become a common material for the fabrication of complex electronic and optic structures because of their excellent mechanical durability, thermal stability, and dielectric properties, combined with easy processability. They can be easily spun from solution to form a uniform glassy transparent film on various substrates. Their chemical structure allows fast thermal and light-initiated crosslinking, resulting in a rigid network polymer with excellent chemical stability. Various devices, such as microcantilevers and tribological coatings, photonic crystals, light waveguides, microgears, microcoils and pumps, and microchannels have recently been fabricated from commercially available photoresist [124,125]. However, the recent application of advanced optical microfabrication methods such as holographic or interference lithography for the creation of complex 2D and 3D microstructures on these materials introduce some questions of the actual distribution of crosslinking density and, thus, the corresponding spatial distribution of mechanical, optical and thermal properties within these structures. Especially this question is relevant in the case of preparation structures features with size below 100 nm. Inhomogeneous crosslinking or polymer shrinkage during exposure processes will lead to sufficient deterioration of the applied technology resolution. Some way to resolve this problem is development the new specific photoresists or technologies which allow to developed small size features without polymer shrinkage or increasing of surface roughness [126].

4.3. Chromophore Some of optical processing of polymers, including preparation of surface relief grating or application of photoeffect is strongly related to specific class of materials – chromophore containing polymers. The use of these materials opens a wide way for fully utilization of one of the most important properties of polymers – ability to modify of polymer properties by modification of their chemical or physical structure [127]. It should be noted, that polymers are very flexible materials comparing with e.g. semiconductors. Optical chromophores can be introduced into polymers not only by trivial doping, but can be covalently or non-covalently linked to polymer main chain or back chain. Chromophores can aggregate into crystal forms,

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can be connected to polymer chain through the different spacers or can be attributed to individual parts of macromolecule in the case of copolymers and so on [128-130]. Additionally, properties of polymer matrix can sufficiently influence the chromophore behaviour – polymers weight, chemical structure, conformation, crystallinity, crosslinking allow altering the chromophore distribution and behaviour [131,132]. Another interesting field is the construction of optical chromophores. Chromophores behaviour, stability, and photoresponce can be affected by their chemical structure. Sufficient progress in chromophore design was achieved during the last years [133]. Recent works on preparation new chromophores have focused their attention on the easiness of chromophores poling, their optical response and stability of prepared molecules. It is well known, that the response of materials to external electric field are given by the materials polarizability (linear polarization of material) and hyperpolarizabilities (nonlinear polarization of second and third orders) [134]. To exhibit second-order nonlinear polarization the medium should not posse a center of symmetry, i.e. it should be noncentrosymmetric. All media and molecules exhibit third-order response. Basically all forms of matter exhibit nonlinear phenomena. But to be useful in a device, the material must exhibit a high degree of nonlinearity [134]. The nonlinear optical and photorefractive properties of molecular materials are largely controlled by the polarizability and hyperpolarizability of electrooptical chromophores and their poling behaviour in external electromagnetic fields [135]. There are two approaches to maximizing the nonlinearities in polymers. First, by synthetic design of molecules with large dipole moment and nonlinear coefficients. Second, by maximizing the applied poling field strength (orientation of chromophores by electric field) or by employing steric forces, which ordered the chromophores in the same direction [133]. Nonlinearity is the qualifying property for functional behaviour. Delocalized electrons are an obvious source of nonlinearity and materials based on π-conjugated molecules or polymers combine promising properties with the flexibility of organic synthesis Inserting electron-donor (D) and acceptor (A) groups into the π -conjugated backbone lowers excitation energies and increases relevant transition dipole moments, leading to improved NLO responses at the molecular level. Dipolar, D- π - A, quadrupolar, D- π -A- π -D or A- π -D- π A, and multipolar molecules was devised as optimized structures for various applications, and reliable structure–properties relationships was obtained at the molecular level (Figure 10). The molecular parameters, which are open to synthetic modifications, are (a) the relative electron affinities of the donor (D) and acceptor (A) groups in the dipole and (b) the length and nature of the connecting system [136].

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After incorporation into polymer matrixes, chromophores must be poled with an electric field (see Figure 11) to achieve a noncentrosymmetric dipole alignment. After poling their remarkably high microscopic nonlinearities must be efficiently translated into macroscopic nonlinearities To obtain device-quality materials, polymers with chromophores must exhibit good processibility, which can be achieved by incorporation of chromophores into polymer matrices; high macroscopic nonlinearity, which requires an efficient dipole alignment induced by an electric field (good poling efficiency); high temporal stability of dipole orientation, which can be achieved by attaching chromophores to high-Tg polymers or cross-linked organic networks and low optical loss in the case of „optical“ polymers. Optimization of any individual above-mentioned property is not difficult; however, simultaneous realization of all requirements to achieve device-quality materials is not a trivial task. For example, an attempt to gain a high temporal stability of oriented chromophores by incorporation into high-Tg polymers matrix often results in unsatisfactory processibility, low poling efficiency, and high optical loss. During the poling, the decomposition of chromophores, especially those with high molecular hyperpolarizabilities, may occur. A great number of works have been performed and published, with the aim to achieve a good balance of processibility, nonlinearity, temporal stability, and optical loss of chromophore-doped polymers.

Figure 11. Schematic representation of chromophore poling.

4.4. Azo-Chromophore One of the most promising candidates for photorefractive application seems to be azodye-polymer [137]. Azobenzene systems possess advantages for high optical nonlinearities due to photoinduced trans–cis isomerisation, molecular reorientation and nonlinear absorption. Azobenzenes exist in two isomeric forms, the E (trans-) and Z (cis-) form. Switching between these forms can be governed by illumination at different wavelengths. In principle, azobenzenes can function as molecular switches by applying light of different wavelengths to obtain varying amounts of cis and trans isomers, where the excess of one of the two isomers may be detected by a change in UV absorption spectra [137]. Photoswitching properties make azobenzene system suitable for variable kinds of applications. For example, holograms of very high diffraction efficiency and exceptional stability can be written (and erased) by laser illumination in this material [138]. Azobenzene polymers are interesting because they combine the properties of anisotropy with photoresponsive behaviour that give

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rise to applications in areas such as NLO materials, LC displays, information storage devices and etc [139].

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4.5. Porphyrine In our works we used another interesting type of chromophores – porphyrins and their derivatives. Porphyrins and their derivatives are known to be interesting photonic materials in view of their large third-order optical nonlinearity and optical limiting properties. Porphyrin is a hetero- and macrocyclic compound derived from four interconnected, coplanar pyrrolelike subunits. The porphyrin ring in the common dianonic form has 26 π electrons and forms a highly macrocylic structure with unique properties in stability, optical spectroscopy, nonlinear optical properties, etc. Porphyrins are aromatic compounds. That is, they obey Hückel's rule for aromaticity, possessing 4n+2 π electrons (n=4 for the shortest cyclic path) that are delocalized over the macrocycle. The macrocycles, therefore, are highly-conjugated systems [140]. Additionally, the porphyrins molecules can be easily modified by connection with ligands or by forming the complex structure with several metals. Chemical modifications to fathom and harness the extraordinary structural and electronic properties of this compound were been of great interests in the literature. [140]. The electronic absorption spectrum of a typical porphyrin consists of a strong transition to the second excited state (S0→ S2) at about 400 nm (the Soret band) and a weak transition to the first excited state at about 550 nm (the Q band). Internal conversion from S2 to S1 is rapid so fluorescence is only detected from S1. The Soret and the Q bands both arise from π–π* transitions and can be explained by considering the four frontier orbitals (HOMO and LUMO orbitals).Explanation of absorption spectrum of porphyrins was firstly proposed by [141] in the 1960s. According to this theory, the absorption bands in porphyrin systems arise from transitions between two HOMOs and two LUMOs, and it is the identities of the metal center and the substituent‟s on the ring that affect the relative energies of these transitions. Mixing splits these two states in energy, creating a higher energy state with greater oscillator strength, giving rise to the Soret band, and a lower energy \ state with less oscillator strength, giving rise to the Q-bands [142].

5. SURFACE RELIEF GRATING One of the most interesting and promising way of the single step modification of polymer by light interference pattern is related to azo-containing polymers. Exposing the azo polymer film to the interference pattern lead to migration of the main chain of the polymer and results in creation of sinusoidally periodic pattern, so-called surface relief grating (SRG) [113] (see Figs. 11, 12). The uniqueness of the single step erasable photofabrication of surface relief structures on azo polymer films is mainly due to the photo-isomerisation and photoanisotropic behaviour of the azobenzene group. Several mechanisms were proposed to describe SRG formation (see above). It must be noted, that azo group play a key role in the SRG formation. Some experiments were performed with another chromophores, with the same properties – light absorption, polarizability, cis-trans isomerisation and very weak SRG formation or it

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absence was found [115]. Azobenzene groups are known to exist in two isomeric states, a thermodynamically stable trans- and a metastable cis-. When irradiated with light of appropriate wavelength, the azobenzene chromophore undergoes a reversible trans-cis-trans photoisomerization process. Absorption of the appropriate wave-length light by the azo dye molecules elevates them to an electronically excited state. A non-radiative decay from the excited state puts the molecules back to the ground state either in the cis or the trans- state. The metastable cis state goes to the trans- state either by a spontaneous thermal back reaction or a reverse cis-trans photoisomerization cycle.

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Figure 12. AFM image of a typical surface grating [115].

Typical experimental set-up for SRG formation represents a laser beams, arising from one source, splittered and intersected at the polymer surface [116]. Some time, the probe laser beam with another wavelength and small intensity not affecting chromophore behaviour are used for control SRG formation by observation of diffraction pattern [115]. The grating spacing was usually controlled by changing the angle between the two writing beams and was found to be consistent with the theoretically calculated spacing for the interference pattern. It is clear that the interfering polarized laser beams produce the surface relief patterns. Under the optimum conditions, surface modulation depth greater than 500 nm could be produced [115]. Azobenzene containing polymer systems can be classified as doped and functionalized systems. In the azo dye doped polymer system, also known as the guest–host system, the azo are co-dissolved in appropriate solvents and coated onto glass slides to form films. In functionalized polymer systems, the azobenzene chromophore is covalently linked to the polymer backbone. Some of the recent research on the azo dye doped polymer systems is aimed at comparing the diffraction effciency, birefringence and the formation of SRG with the functionalized systems with the aim to understand the fundamental process. Consequently, the surface relief features are very weak as compared to azo functionalized glassy polymer films and the azo dye doped polymer films are not favourable for forming SRGs [143]. The versatility of covalently linked azo chromophores comes from the fact that this group can selectively be attached to the side chain, main chain or chain ends of a wide class of polymer systems. Appropriate substitution of the azobenzene chromophores may be used to

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tune the chromatic features from UV through the visible spectrum. In addition, the possibility of achieving efficient, optically induced effects even with low incident light power is advantageous. A large variety of azo functionalized amorphous polymers with different chemical structures, molecular weights (MW) and glass transition temperatures (Tg) have since been extensively investigated [144]. The photo-fabrication of surface relief structures on these polymer films is expected to have many interesting applications in photonics and the emerging area of nanotechnology.

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6. INTERFERENCE LITHOGRAPHY Interference lithography can be described as recording the optical interference pattern in the suitable media. Interference lithography was first proposed and implemented by Berger et al. to fabricate two-dimensional 2D hexagonal patterns in a photosensitive polymer, which subsequently served as an etch mask for transfer to a high-index silicon substrate [145]. This approach was extended by Shoji et al. by introducing up to two additional laser beams to create low index contrast in 3D structures with face- or body-centered cubic like symmetry [146]. This technique produces defect free, nanometer-scale structures over large substrate areas in a single step fabrication. Suitable media for recording can be the monomers syrup or photorefractive polymers. Physical base of the interference lithography will be attributed to polymerization process or to photorefractive phenomenon. This process will lead to formation of a surface profile of materials or refractive index grating [147] (see Figure 13, [148]). The optical interference lithography method offers a unique opportunity for efficient large-scale micro-fabrication of 3D structures [149]. By employing the 3D interference pattern of four or more laser beams to expose a photopolymerizable material such as a photoresist or polymerizable resin, nearly perfect LRO can be maintained over different length scales. At the same time, this technique allows straightforward control over the periodic properties of prepared structures through the laser beam parameters. The photo-resist material undergoes a chemical alteration when the total light intensity due to the interference pattern, at local position is maintained over a time interval such that the “exposure” exceeds a specified threshold. For a negative photo-resist, the “underexposed” region can be selectively removed using a developer substance which leaves the “overexposed” regions intact.

Figure 13. Simple example of interference lithography [148].

For a positive photoresist, overexposed regions are removed, leaving the underexposed regions intact [147] (see Figure 14, [150]). Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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Figure 14. SEM images of 3D polymer structures produced with 50.9° prism by interference lithography. A - the top view of the sample. B - tilled view with top surface and its cross section [150].

Recently several experimental set ups for interference lithography have been proposed. In a simpler way multiple beams forming the interference pattern can be obtained by two independent optical elements and steps: Splitting the laser output into two beams either by a dielectric beam splitter or a grating, then superposing them at the exposure area by another specially designed prism [151]. The more widely used technique is the holographic lithography based on the interference of four laser beams. Other proposed configurations include the use of multiple exposures and phase shifts between exposures and a hybrid five beam configuration with three beams forming a two-dimensional pattern and two other beams forming a one-dimensional pattern in the out of plane direction [152]. However, these configurations require careful manipulation of the relative phases between beams and between exposures. In contrast, in a four beam, single exposure configuration, the relative phases between laser beams does not change the shape of the interference pattern. In theoretical reports, four beam holographic lithography configurations were used to create structures which emulate particular cases of triply periodic minimal surfaces which was shown to produce large 3D structure [147]. As will be mentioned below, in a typical case, for generation of periodic microstructure the interference of four noncoplanar laser beams in a photoresist film typically tens nanometers thick is used. The intensity distribution in the interference pattern has three dimensional translational symmetry and its primitive reciprocal lattice vectors are equal to the differences between the wavevectors of the beams. The four laser beam wavevectors determine the translational symmetry and lattice constant of the interference pattern. Eight parameters remain, describing the intensities and polarization vectors of the four beams that are further required to define the intensity distribution within a unit cell (that is, the basis of the interference pattern). These parameters allow considerable freedom in determining the distribution of dielectric material within the unit cell. Several structures, very closed to the calculated optical intensity pattern of four laser beam interference were reported in works [147,153]. Additionally, it is of considerable interest to incorporate some additional features into prepared periodic structure. Optical interference lithography offers a unique opportunity for the inclusion of such features. After the photoresist was exposed to interference patterns, localized exposure technique such as direct laser writing by two-photon absorption can be used to pattern the photoresist further with desired features. In such a scheme, features can be

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introduced into the bulk photo-resist before the removing of unexposed part (in the case of negative photoresist). In the latter situation, complications can arise due to scattering of the “writing” laser beam at the numerous air-dielectric interfaces. These possibilities suggest that optical interference lithography together with direct laser writing can provide a powerful platform for the eventual creation of different optical structures and devices [153]. In the case of application of pulsed laser for interference lithography the important parameter is the number and duration of the laser pulses which must be short compared to the timescales of physical and chemical processes induced by the exposure. In such a case the interference pattern will not be perturbed by photoinduced changes in the refractive index of the precursor. The short exposure also eases constraints on the mechanical stability of the optical components. The threshold exposure for the production of insoluble polymer is determined by the optical fluence, by the number of laser pulses, polymer density, absorption cross-section and quantum efficiency of initiator molecules, and by the chain-length and branching ratio in the polymerization. Attenuation of the laser fluency by absorption in the photoresist therefore limits the thickness of the structures that can be made dimensionally homogeneous by this technique. If required, the attenuation can be reduced by offsetting a smaller initiator concentration with a higher exposure dose [154]. Interference lithography can also be performed by application laser light with different wavelength. Using of laser operating at infrared wavelength is more suitable because of the full utilization of some physical process, like two photon absorption. At the other hand, application of infrared wavelength sufficiently restricts the periodicity of prepared structure. UV light can produce structure with much lower periodicity [146,155]. Unfortunately, to obtain good reproducibility of interference patterns using a pulsed laser is still a challenge. Low penetrability of UV light may limit the thickness of the patterns, thus hampering the broader application of the multibeam interference method on a variety of photosensitive materials. An alternative lithographic route based on interference of three or four beams of visible light in continuous wave mode and laser-initiated cationic polymerization of epoxy has been reported. The laser offers stable beam output and the resist was sensitized by visible light, which is more flexible and applicable to a wide range of photosensitive materials. In the visible region, the transmission increases significantly and the interference of visible light will provide a more even exposure throughout the thick films and better control of the interference pattern. For careful controlling the photochemistry the addition of dopants to eliminate the background exposure caused by noncoplanar polarization of the multiple beams can be used. Then, the polymerization can be controlled in a two-step reaction (exposure and postexposure bake) without perturbing the interference patterns. The creation of series of 2D and 3D defect free porous structures with periods of 0.9-8 µm in an area larger than 4 mm was described in the work [151]. The main disadvantage of fabrication strategies that rely on interference of multiple independent beams comes from alignment complexity and inaccuracies due to differences in the optical path length and angles among the interfering beams as well as vibration instabilities in the optical setup. Simple method was proposed to avoid these drawbacks in interference lithography technology which includes application of diffraction gratings or specific refracting prism deposited directly on the top of resist layer [150,156]. In the work [156] an approach for recording 3D periodic structures in a photosensitive polymer using a single diffraction element mask was described (see Figure 15, [156]). The mask had a central opening surrounded by three diffraction gratings oriented 120° relative to

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one another such that the three first order diffracted beams and the nondiffracted laser beam give a 3D spatial light intensity pattern. Structures patterned in the polymer using 1.0 and 0.56 µm grating periods had hexagonal symmetry with micron- to submicron-periodicity over large substrate area. The structure with feature sizes as small as 50 nm was prepared by this technique. In other words, the single diffraction element mask was demonstrated to create a four-beam interference pattern recorded in a photosensitive polymer This arrangement exhibits improved alignment and stability of the optical setup for obtaining 3D periodic structures. Another approach for easy fabrication of two-dimensional 2D hexagonal and threedimensional 3D structure in a photosensitive polymer is to apply a simple single refracting prism [150]. This method enables the simultaneous splitting and recombining of an incoming laser beam using the same optical element. Thus, antivibration equipment and complicated optical alignment system to adjust the angles between the interfering beams are not required, leading to a very simple optical setup. Large-scale 2D hexagonal and 3D structure with specific symmetry structure was produced in this way. In the context of mass production, this method is much more practical and robust, and is able to produce periodic uniform structure over larger area than application of two independent-element setups. In addition, this method can be easily extended to generate more complex nanostructures, such as diamond lattice, by designing the refracting prism properly. The main progress in interference lithography research will be attributed to the preparation of 3D photonic crystal. Common technique, which are used in the semiconductor technology for the production of photonic crystals use a tightly focused scanning laser beam to initiate chemical vapour deposition or two-photon photopolymerization. It is obvious, that two-photon absorption at near-infra red on photosensitive materials is a promising lithographic technique to construct defect-free 3D structures, but the serial pinpoint writing in a two-photon process may not be suitable for massive production.

Figure 15. Schematic representation of the diffraction element used to produce the 3D periodic structure in a photosensitive polymer by interference lithography [156]. Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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Structures are produced layer-by-layer, so these techniques are relatively slow; they are currently limited to micrometer length scales. Semiconductor microfabrication, which has the required resolution, is an expensive process at the limit of current technology and has not produced structures more than a few unit cells deep. By contrast, the optical properties of microstructures made by interference lithography may be optimized by controlling the form of the interference pattern. This fabrication method has the high spatial resolution required to produce photonic crystals for the visible spectrum, as well as creating the connected air and dielectric networks that are important for the opening of a full bandgap. The process is cheap, rapid and potentially scalable for large-scale production. It was also shown that these polymeric structures may be used as templates for the construction of photonic crystals with higher refractive index contrast [157]. Probably, it is a promising technology that should allow optical photonic crystals to be realized.

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7. HOLOGRAPHY Holography is a technique that, allows recording the light interference pattern in the photosensitive media, like the interference lithography. However, this technique is very often used to registration of light, scattered from an object [158]. When two coherent light beams overlap, one containing information about an arbitrary image or object (“object beam”) and the other being a plane wave (“reference beam”), a characteristic interference pattern evolves (see Figure 16). By overlapping these beams in an appropriate medium, the material‟s absorption and/or index of refraction are altered nonuniformly as a result of the nonuniform illumination. A holographic “grating” (hologram) is formed which can later diffract light. This hologram contains intensity and phase information about the object. The information stored in the material can then be read out by a plane-wave incident from the original direction of the reference beam. The virtual image of the object evolves as a diffracted signal in the original direction of the object [159]. Common practice of holography is the formation of relief and index patterns in polymer films. Holography process can be based on photopolymerization, or photorefractive effect. Under polymerization insoluble polymers is created from monomers syrup. This process was interested as a new fabrication technique in multi-dimensional structure formation. In photopolymerizable media diffractive patterns were fabricated as a result of light initiating photopolymerization at the brighter regions of the interference, by local and non local polymerization [160]. Since the non-local polymerization followed by diffusion of monomers can be optically triggered and chemically amplified, patterns can be obtained with high homogeneity over larger area to afford simple pattering process. Sub-100 nm periodicities can be easily formed by this way; indeed for blue reflection gratings these periodicities are approached. Additionally, these structures can be formed in a matter of seconds and over large areas. This combination of capabilities is hard to find in most other techniques due to their sequential and directionality properties [161]. Very often in this technique the mixture of different monomers, some time doped by nanoparticles or liquid crystal are used [162,163]. Firstly main attention has been concentrated on one-monomer systems [164].

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Figure 16. Typical holography scheme.

Interference of two or more coherent laser beams within a photo-reactive monomer syrup results in a periodic intensity distribution that initiates a self-similar periodic polymerization process. A higher intensity results in a higher local free-radical concentration, which in turn results in a locally faster polymerization rate. This rate anisotropy causes a spatial distribution of high molecular weight polymer to develop, which changes over time. The intensity of the two writing beams can be used to modulate the temporal evolution of the anisotropy. Geometrical parameters of the periodic structure can be controlled by the interference angle of the two beams [161]. The direction of the grating planes can be at any angle to the surface (orthogonal like lithographic structures or parallel in the case of reflection gratings). The aspect ratio can be large allowing true Bragg gratings to be formed [165]. After polymerization unpolymerised part was selectively removed. The formation of periodic polymer regions separated by voids, generated first by illumination and then by wet chemical processing techniques, allows achieving large periodic refractive index differences through the bulk of a film (see Figure 17, [166]). Latter a mixtures of monomers or monomer and neutral component, with different refractive indexes n1/n2 or absorption coefficients a1/a2 have been studied. In these systems the hologram recording involves a spatial transfer of several molecular components. The distinguishing features of these multicomponent systems are: the mixture consists of at least two components – monomers, that photopolymerize independently, with substantially different rates; the refractive indices or absorption coefficients for the components are different; and spatial mass transfer takes place during the spatially inhomogeneous polymerization process when the mixture is exposed to light fringes [167]. Different systems of monomers were studied with the aim to achieve better resolution and better properties of obtained structure [167-169]. The influences on the storage mechanism by other additives, such as cross-linking agents and plasticizers were also investigated [169]. The introduction of the cross-linking agent usually stabilizes the formed

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structures against thermal and environmental relaxations probably through a matrix densification mechanism. The introduction of plasticizers has a significant effect on the spatial modulation of the refractive index in the recorded grating patterns [168].

Figure 17. SEM images of 3D structures generated by holographic lithography. A - Polymer structure generated by exposure of a 10-µm film of photoresist to the interference pattern, scale bar 10µm. B – top image 1 µm, C – lateral image, scale bar 1 µm [166].

Additionally this technique allows direct controlling of the spatial distribution of liquid crystal or particles in soft matter in two- and three-dimensions [170]. Phase separation between polymers and liquid crystal take place due to mass transfer processes under inhomogeneous light illumination. The propensity of phase separation in mixtures of high molecular weight polymers and low molar mass liquid crystals in sufficient concentration combined with the spatial and temporal anisotropy of holographic polymerization results in the periodic phase separation of nanoscale-sized LC domains. These small domains are spatially separated by polymer regions on a periodicity defined by the holography process. This periodic sub-micrometer Bragg gratings exhibit high diffraction efficiencies that can be modulated using applied external electric fields. The liquid crystal molecules can be replaced by various sized particles [171]. Similar anisotropic physical processes driving the segregation and subsequent phase separation of the liquid crystals induce anisotropic movement and sequestering of the particles. Examples of preparation ordered polymer nanocomposites with luminescent nanocrystals, doped by rearearth ions, was recently demonstrated. These materials have specific properties and wide application range and can substitute corresponding bulk materials, for example in light emitting devices, sensors, micro-lasers and amplifiers.

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Holographic recording in cast polymer layers occurs in real-time and in-situ, meaning that no wet-chemical or postthermal/photochemical processing are required. Additional and very important features can be achieved by the application of photorefractive polymer for holography record. [172]. In these materials the light interference pattern generates the free charge, their trapping and localization. If those materials are exposed to second external treatment (temperature or light illumination) the localization of charge is suppressed and materials are suitable for next hologram recording. At the other words, uniform illumination smears out the charge distribution and erases the hologram, that is the materials are suitable for dynamic (i.e. reversible) and real-time holography. This phenomenon seems to be very important for optical data storage application [173]. However, holography technique has some drawback. Firstly, the exposure sensitivities are still one to three orders of magnitude lower than that of bleached silver halide emulsions, traditionally used for holography record; secondly, polymerization may in some cases induce layer shrinkage; and thirdly, their pre-exposure, as well as post-exposure shelf-lives are relatively short. These problems can be resolved by application of holography in the bulk of photorefractive polymers. The formation of microstructures in new photopolymer and their applications is now intensively investigated [174]. In summary, mutual diffusion of the components or local shrinkage of the materials during polymerization could cause mass transfer. Both the refractive index and absorption coefficient can be modulated in this way. This means that the light fingerscan produce phase and amplitude holograms in photopolymers.

8. SCANNING BY ONE LASER BEAM

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8.1. Nanopattering via Light Absorption of Doped Polymer In our experiments the possibility of surface patterning of arbitrary polymer films was proposed and experimentally verified. The main approach of our experiments consists in doping of thin films of polymers with optical chromophores – particularly porphyrin molecules and scanning of doped materials by focused laser beam. Doped polymers films were shown to respond mechanically on laser scanning, the response depending on light wavelength. Addition of simultaneously continual mechanical movement results in formation of regular grating pattern. Particular experimental arrangement consists in surface writing, which was performed by scanning of 0.25 mW laser emitting at 405 nm wavelength. The laser beam was focused to a spot with a diameter of approx. 0.5 µm. The laser beam was linearly polarized. Polymer films were exposed to periodical laser scanning under ambient conditions. Scanning was performed line by line. Mechanical movement was applied in the direction parallel to scanned lines. The pattern formation was monitored by confocal microscope either by evaluation of reflected laser light or in usual optical regime (see Figure 18). Doping of polymers was performed with the aim to increase the absorption properties of materials at the certain wavelength. Doping was performed either in the thin surface layers or in the polymer bulk. In the case of surface doping two techniques was applied. Two processes of the surface layer doping were used. First one consists of deep coating process, i.e. polymer

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film was immersed in the solution of chromophore in suitable solvent for several seconds, and then dried at the ambient conditions. The second one consists of vacuum evaporation of chromophore molecules onto the polymer surface. Both techniques exhibit specific advantages and restrictions. Deep coating is simpler, but leads to surface damages. Vacuum evaporation of chromophores doesn‟t change the surface morphology but chemical destruction of chromophores during the doping process can occurs. Bulk doping of polymers was performed in a simple way i.e. by mixing polymers and chromophores solutions and preparation, by spin-coating method, of the composite films on a suitable substrate. This method can be applied only for polymers with good solubility. Our experiments were performed on different polymers, including poly(methyl methacrylate) (PMMA) with different molecular weight, polystyrene, Su-8 and so on. The fluid temperature and structure of polymers seems to be important parameters for patterning process. Another parameter, affecting the patterning is the initial thickness of the polymer films, especially in the case of doped polymers or polymers containing covalently linked chromophores molecules. After preparation and doping the polymer films was exposed to periodic dotty irradiation using focused laser beam (see above) scanned over an defined area. The laser energy was absorbed by chromophore molecules and in this way local heating of polymer film was induced. The initially featureless surface of polymers undergoes surface distortion at the boundaries of the exposed area. Addition of concurrent mechanical movement leads to formation of regular, periodic pattern on the polymer surface. The periodicity of the structure (optical grating) can be changed in the range from tenths to tens of micrometers by selecting suitable scanning rate of the laser beam, its intensity and sample speed (photo in enclosure) (see Figure 19, [175]).

Figure 18. Principle of structure creation by single laser beam. Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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Figure 19. Various stages of the surface patterning as a function of the laser intensity and the velocity of the mechanical movement. Laser scanning was applied across area at the center of the image. Part (a) corresponds to laser intensity of 0.03 mW, (b) 0.12 mW, (c) 0.12 mW and simultaneous mechanical movement, and (d) 0.25 mW [175].

The benefit of the proposed approach is the possibility to prepare microstructures with optical quality in an extremely simple way. No additional etching is necessary. The procedure can be used in integrated photonics, namely for efficient coupling of optical radiation into optical waveguide through an optical grating. It must be noted, that proposed process requires relative small laser intensity. Creation of the linear structures is created if the laser intensity exceeds some threshold value. The threshold depends mainly on the dopant concentration, its absorption coefficient and the properties of the polymer matrix. For PMMA films, doped with 0.1% mesotetraphenylporphine 0.1 mW laser power, which scans polymer area 50×50 microns, was found to be enough for structure creation. Irradiation with smaller intensity does not lead to surface patterning, regardless of the time of irradiation. At the other hand, when the laser intensity is too high, polymer degradation occurs and a structure with poorly expressed periodicity is formed (see Figure 19, [175]). Surface morphology of the samples prepared in this way was examined by atomic force microscopy. It confirms that the simple sinusoidal pattern is obtained. The resulting profile consists of periodic array of lines on which the polymer mass was pulled up and above initial film surface (see Figure 20, [175]). Pattern parameters were found to be a function of the experimental conditions and polymers properties. The periodicity primary depends on the velocity of the sample movement. The grating constant of several tenths to several tens of

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micrometers was achieved. Simple linear dependence between the velocity of the sample movement and pattern periodicity was observed in the experiments. In this experiment pattern periodicity down to about 300 nm was obtained which was limited by achievable velocity of sample movement. It was also shown that the pattern form depends on the porphyrin concentration and the time of immersion. With the similar procedure a single lattice of a pillar array with predetermined geometrical parameters, location, and orientation can be obtained. Additionally it was found, that the pattern amplitude depends mostly on the concentration of dopant and the laser intensity. The grating width of the pattern components from several nanometers to one micrometer was obtained by changing the patterning procedure.

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Figure 20. AFM image of polymer grating prepared by single beam laser scanning [175].

Temperature treatment under polymer glass transition temperature of the prepared structures for several hours does not lead to surface smoothing and leaves the grating structure unchanged. Considering this thermal behaviour and the laser intensity threshold it can be concluded that the polymer molten state is a prerequisite of the pattern formation. Possible explanation of the pattern formation is in the combination of the laser scanning and polymer flow induced by temperature and surface tension gradients (so-called Marangoni effect). First, the polymer surface is fused and then periodical heating of polymer surface introduce the surface tension gradient slipped in the direction of laser scanning. The gradient in the surface tension causes the polymer flow away from region of low surface tension. It is obvious that this phenomenon must be more pronounced in the direction of "quick" laser scanning, when the temperature gradient has not enough time to be relaxed. On the boundary of the exposed area the surface tension gradient reaches the greatest value. The material flow stops after untouched polymer outside of the exposed area is achieved. As a result the polymer surface is locally pushed up. Additional mechanic movement of the sample results in the drift of the exposed area boundary and in the grating formation. Not only linear pattern can be prepared in this way. After first stage of the modification by laser scanning the sample can be turned by 90° and the previously generated linear structure can be over-scanned in an additional modification step. As a result, the initial linear pattern is changed into the system of dots, with good quadratic symmetry (see Figure 21).

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Figure 21. SEM image of a structure prepared by two step procedure. After first stage of modification by laser scanning the sample was rotated by 90 ° and the previously generated structures were overscanned by an additional modification step.

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8.2. Single Laser Beam Nanopattering of Polymer This chapter will give an overwiev of an unconventional method for nano-structuring of polymeric material and for lithography-less preparation of metal nano-wires. Such nanostructured composite material may find its application especially in optics like a metamaterials with negative refractive index or like difraction grids. Several studies have shown that the illumination of polymers by polarized UV laser beam induces self-organized ripple structure formation within a narrow fluence range well below the ablation threshold [176-180]. The properties of these periodic structures have been frequently studied since the first report about them [181]. The smallest features on polymers were produced by our group by 157 nm F2 laser irradiation of polyethyleneterephthalate [44]. The period of the ripples depends on the laser wavelength and on the angle of incidence of the laser beam, and their direction is related to the direction of the laser beam polarization [182]. The spacing of the structures can be described by the relation: n–sin 

eq. (1)

where  is the lateral periodicity of the structures,  is the wavelength of the excitation laser light, n is the effective refractive index of the material and  is the angle of incidence of the laser beam [182]. These structures develop on the original material surface having a small roughness, as a result of treatment with one laser beam with an uniform intensity distribution. It is known that the interference between the incoming and the surface scattered waves plays an important role in the structure formation [183]. The interference causes an inhomogeneous intensity distribution, which together with a feedback mechanism results in the enhancement of the

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modulation depth [178]. However, the mechanisms involved are complex and, especially for polymers, there is no conclusive explanation of the structure formation process, yet. Nevertheless, different processes have been reported as responsible for ripple formation [44,184-187] such as thermal and non-thermal scissoring of polymer chains, amorphisation of crystalline domains, local surface melting and photo-oxidation and material transport.

8.2.1. Threshold Fluence and Periodic Structure Formation As was already mentioned above, many studies have shown [176-180] that the illumination of polymers by polarized UV laser beam induces self-organized ripple structure formation within a narrow fluence range well below the ablation threshold. In these processes certain fluence threshold always exists below which the periodic ripple structure is not created on the polymer surface. For the irradiation of the PET samples, the light from a KrF excimer laser (see Figure 22, Lambda Physik Compex, wavelength 248 nm, pulse duration of 20 to 40 ns) was used in combination with a Glan-Thompson polarizer prism (see Figure 23, [45]). After the light throughpass laser beam was divided into the two mutually perpendicularly polarized beams () inclined by 110°. For the modification only  beam was used, beam  pointed out of the sample. For homogeneous illumination of the samples, only the central part of the beam profile defined by an aperture with a length of 10 mm and a width of 0.5 mm was used. The samples were mounted onto a translation stage and scanned in a fix position. The PET surfaces were irradiated with 6000 pulses per area at different laser fluences between 2.0 and 8.4 mJ.cm-2 and different angles of incidence of the laser beam (0°, 22.5° and 45°). The experiments were performed in ambient air atmosphere. Figure 3 displays AFM images of pristine PET and PET irradiated at different laser fluences and under the incidence angle of 0°. The sample irradiated with 3.4 mJ.cm-2 fluence exhibits a rougher surface than the pristine PET. There is a noticeable nodulation, although no ripple formation is visible. At higher laser fluences periodic ripple structures have developed over the irradiated area. At the laser fluence of 6.6 mJ.cm-2, a regular and uniform coverage of the PET surface with ripples is reached. Similar development of the surface morphology has been observed during F2 excimer laser irradiation of PET foil in ref. [44]. In this case the ripple threshold was found at 3.8 mJ.cm-2, whereas the most regular and uniform filling of irradiated area with ripples was achieved during irradiation at the fluence of 4.2 mJ.cm-2. The measured period of the structures was about 130 nm. As it has been already mentioned, applied fluences in the case of KrF laser (248 nm) as well as in the case of F2 laser (157 nm) irradiation were well below the ablation thresholds of PET which are 29.6 mJ.cm-2 [44] and 40 mJ.cm-2 [188], respectively.

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Figure 23. Glan-Taylor polarizing prism with -polarized beam after the throughpass [45].

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Further experiments performed on the laser irradiated PET proved, that the height and the roughness of the ripples depend strongly on the applied laser fluence. This fact is illustrated in Figure 25 which shows the height and the roughness of the ripples as a function of the applied laser fluence. The dependence follows from AFM measurements shown in Figure 24. There is a good correlation between height and surface roughness of the ripples over whole laser fluence range examined (see Figure 25). Both parameters reach the maximum value at the fluence of 6.6 mJ.cm-2, which corresponds to the ripple height of about 90 nm. From the technological point of view this is the most promising structure.

Figure 24. AFM images of the PET irradiated at different KrF laser fluences; the numbers in the inset refer to the laser fluences in mJ.cm-2 employed for irradiation of the PET foils, while pristine stand for un-irradiated pristine PET [46].

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Figure 25. Dependence of the ripple height (○) and roughness (□) on the KrF laser fluence employed for the PET irradiation [46].

Since the periodicity of the formed nano-structures (ripples) could be altered by changing the irradiation parameters (according to eq. (1)), other experiments proving the validity of this formula were done. Figure 26 shows AFM images of the PET irradiated under different incidence angles of the laser beam. For larger angles of incidence, the spacing between two neighbouring ripples is wider. For the incidence angle of 0° and 22.5°, the observed spacing of the ripples is in good agreement with the value calculated by eq. (1) with an effective index of refraction n  1.2. The agreement for the incidence angle of 45° is less convincing. The discrepancy may be due to changes of the polymer refractive index induced by the UV laser irradiation as reported earlier in [189]. Present experiment confirmed the possibility of tailoring the width of ripples by changing the incidence angle of the laser beam.

8.2.2. Metal Coating and Nanowire Formation For successful construction of optically active metamaterial, it is essential to perform a selective metal coating of the previously formed periodic structures. There are many conventional ways how to reallize this technological step, however, only a few of them are relatively simple and inexppensive [45,46,190]. Metal deposition on polymers can be carried out by sputtering, vacuum evaporation, and also by various electrochemical procedures [191193]. The thicknesses of the prepared layers are from few to hundreds of nanometers. Here, an effective process of selective metallisation by sputtering method will be presented. In Figure 27A, we compare the morphology of F2 (laser fluence of 4.4 mJ.cm-2) and KrF (6.6 mJ.cm-2) irradiated samples.

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Figure 26. AFM images of the PET irradiated at a KrF laser fluence of 6.6 mJ.cm-2 under the different incidence angle of the laser beam (0°, 22.5° and 45°). The numbers in the insets in the upper left corner refer to the angle of incidence of the laser beam and in the insets in the upper right corner to the ripple period in nm [46].

The ripples formed under KrF laser exposure have larger width and height in comparison to that formed by the F2 laser. The periodicity of ripples, , was about 208 nm in the case of KrF laser irradiation, while in the case of F2 laser irradiation we obtained periodicity of about 140 nm. The heights of the ripple structure (top-bottom) were about 100 nm and about 15 nm for the KrF laser and the F2 laser irradiation respectively. The sputtering of gold layers onto the ripple structures was accomplished and the surface morphology was examined by AFM. For easier comparison of the growth mechanism of the gold layer, the gold-to-ripple height ratio was kept constant. F2 irradiated samples were sputtered with 50 nm thick Au layer, whereas KrF irradiated samples were covered with 200 nm thick Au layer. Figure 27B shows morphology of sputtered gold layers on the laserinduced ripple structures for both laser light wavelengths.

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Figure 27. AFM images of PET samples: (A) irradiated by KrF (6.6 mJ.cm-2) and F2 (4.4 mJ.cm-2) lasers and (B) irradiated and than sputtered with 200 nm (KrF) and 50 nm (F2) Au layer. Ra is the average surface roughness in nm [46].

For the both wavelengths, the ripple structure of the PET surface is transferred into the relief of the gold layer. The roughness values Ra were nearly identical for the surface with and without gold coating in both cases. However, for the F2 laser irradiated samples it seems that the gold coated surface has more pronounced granular structure than the uncoated ripple structure. Here one has to keep in mind that the AFM images show a convolution of the surface morphology with the geometry of the AFM tip. Therefore, small features and features with a high aspect ratio may not be feasible in the AFM images. Supplementary to the AFM analysis, the samples were cut by FIB and the FIB cuts were investigated with the SEM (see Figure 28). The gold sputtered onto the KrF laser induced ripples is deposited in the form of “nano-wires”, which grow on the ridges of the ripples. The FIB cuts reveal that the gold layer is not continuous and there are gaps between the individual wires. Additionally, grains are visible along the wires, but the FIB cut images suggest that these grains are interconnected. The morphology of the gold layer deposited on the F2 laserinduced ripples is distinctly different. The gold is also deposited in the valleys of the ripple structure (see Figure 28) and the gold coating is continuous. The Au coating partially smoothes out the initial ripple profile. In general the Au layers show cracks which however are not directly related to the ripple pattern.

8.2.3. Chemical Composition of Nano-Structured PET The chemical composition of the polymer surface can affect the growth morphology of the gold layers on polymer surfaces [194]. Therefore, we analyzed KrF and F2 irradiated PET Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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with the ARXPS method [186]. By this method it was shown that the oxygen content in first ten atomic layers of pristine PET is depth dependent [186]. This phenomenon is explained by the reorientation of the polar groups in the surface layers leading to lower oxygen content in near surface layers in comparison with theoretical value for pristine PET (29 at %) [195]. The dependence of the oxygen concentration (at %) derived by ARXPS on the angle of incidence of the XPS primary beam is shown in Figure 29 for pristine PET and PET irradiated with KrF and F2 lasers. For pristine PET the oxygen content decreases from 25.5 at % for an incidence angle of 0° to 19.5 at % at an incidence angle of 80°. For F2 laser irradiation, the oxygen content is about 18 at % and is slightly decreasing to about 16 at %, for KrF laser the O content increases from 28.5 to 33 at % when going from the surface to the bulk.

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Figure 28. SEM images of FIB cuts of lasers irradiated and Au sputtered samples: (A) PET irradiated by KrF laser (6.6 mJ.cm-2) and sputtered with 200 nm of Au layer, (B) PET irradiated by F2 laser (4.4 mJ.cm-2) and sputtered with 50 nm of Au layer [45].

Figure 29. Dependence of the oxygen concentration (at %) on the angle of incidence of the XPS primary beam for pristine PET (PET) and KrF (PET/KrF) and F2 lasers (PET/F2) irradiated PET [45].

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The ARXPS analysis of the irradiated PET shown that the ridges and the valleys of the F2 laser-induced ripples have nearly similar chemical composition, at least regarding the oxygen content. The oxygen content in F2 irradiated PET is lower than in pristine PET, while in the case of KrF laser irradiation the oxygen content is higher and towards the surface the oxygen content increases. It may therefore be concluded that in case of KrF the ridges of the ripple structure contain more oxygen groups than the areas at the valley, which are shielded at higher angles of incidence of the primary X-ray beam. The oxygen on the ridges is even higher than that in the pristine PET. This could be due to a thermal degradation (connected with oxidation) of the polymer at the ripple ridges during the irradiation process. The fact that the F2 laser treated surface shows a lower oxygen content than that of pristine PET may be connected with a reorientation of the polar groups toward the polymer bulk [186,195] or by photochemical scission of the polymer chains in combination with preferential release of CO and CO2 groups (Norrish type II reaction [196]). Figure 30 shows detail of XPS spectra with the carbon C1s peaks for the incidence angles 0º and 80º. The main peak in Figure 30A at 284.7 eV can be associated with PET „bulk“ carbon. Figure 30B shows C1s spectrum for an incidence angle of 80º. The main peak position (a) at 290.1 eV is shifted with respect to a C1s peak in Figure 30A, since the XPS set up does not allow full surface charge compensation at these high angles. Here the primary Xray beam is nearly parallel to the examined surface, thus the information originate from the very top of the ripples. While for F2 laser irradiation mainly one carbon peak is visible in Figs. 30A and 30B, related to "bulk" carbon, the spectra for KrF irradiated samples show pronounced side peaks. The feature (b) in Figure 30B has a similar position as "conductive" carbon originating from conjugated carbon bonds. It may be formed by degradation of the polymer especially at the ridges of the KrF-laser ripple structures. KrF laser irradiation of PET foils can induce the formation of nano-ripple structures with a periodicity of about 200 nm and a height of about 100 nm. Gold sputtering onto these PET surfaces leads to the formation of gold “nano-wires” at the ridges of the ripple structures. The gold layers seem to be non-continuous, with gaps between neighbouring wires. The gold layer morphology is distinctly different for gold sputtered onto F2-laser induced ripples, which are narrower and considerably shallower than those induced by the KrF laser irradiation. Here the layers are continuous showing only irregularly distributed cracks. From the XPS analysis it is possible to conclude that in the case of KrF irradiation, the exposure leads to degradation (oxidation) especially at the ridges of the ripple structures with an increased occurrence of oxygen containing groups. The altered chemical composition probably results in improved metal deposition on the tops of the structures rather than in the valleys between. Together with shadowing effects, this may be the reason for the formation of gold wires with a height considerably higher then the ripple structure itself.

8.2.4. Angle Dependent Irradiation and Sputtered vs. Evaporated Coatings In previous paragraph it was shown that by the combination of excimer laser nanofabrication of PET and Au coating by sputtering technique it is possible, at certain experimental conditions, to prepare gold nanowires. This procedure is very effective, since it does not require rather complex conventional techniques utilizing e.g. lithograpy and mask processing. This paragraph will demonstrate that modification of the PET surface with linearly polarized light from pulsed KrF laser has a significant effect on the properties of subsequently deposited gold nanolayers and that the choice of the gold deposition technique

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is crucial for the quality of the gold coatings. Here, especially the difference between the sputtering and evaporation deposition techniques will be discussed.

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Figure 30. Carbon C1s XPS spectrum of PET irradiated by KrF and F2 lasers (incidence angle of primary X-ray beam 0º (A) and 80º (B)). Peak (a) corresponds to PET “bulk“ carbon, peaks (b) represents an additional carbon compound [45].

Figure 31 shows the AFM topography of laser-induced ripples on PET foils and the ripples subsequently coated by gold layers prepared by sputtering (sputt.) and by evaporation (evap.). The ripples on the PET surface were prepared by KrF laser irradiation under incidence angles of 0° and 22.5°, respectively. The wider ripples obtained with under the angle of incidence of 22.5° lead also to wider structures in the gold coatings. The evaporated and sputtered layers have almost the same topography. The main difference is in the roughness of the nano-structures (see Figure 32), which is slightly lower in the case of evaporated layers. It should be mentioned that the AFM images represent a convolution of the surface morphology with the geometry of the AFM tip. The dependence of the surface roughness on the laser fluence for PET without coating and for gold coated PET is shown in Figure 32. There are two distinct intervals of the roughness values depending on the laser fluence. The first interval, up to the fluence of 4.0 mJ.cm-2, is characterized by a small increase of the surface roughness (compared to pristine PET), which correlates with small changes in the surface topography in Figure 24 for the fluence of 3.4 mJ.cm-2 at which no ripple structure is created. At fluence above a threshold of about 4.2 mJ.cm-2, periodic ripple structures are formed (second range) and the surface roughness increases rapidly between the two “intervals”. In the first interval, gold sputtering causes only a slight increase of the surface roughness, and after gold evaporation the roughness remains practically unchanged for all fluences in this interval. In the second interval, gold deposition results in a decrease of the surface roughness (in comparison to samples only exposed to laser radiation without gold coating) for both

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sputtering and evaporation techniques. The roughness of the evaporated layer seems to show less variations for different fluences compared to the roughness of the sputtered layers.

Figure 31. AFM images of ripples at KrF laser irradiated PET with a fluence of 6.6 mJ.cm-2 under the incidence angles of (left) 0° and (right) 22.5°. The AFM images show ripples without coating (Laser) and ripples subsequently coated with 200 nm thick gold layer by either sputtering (Laser/sputt.) or evaporation (Laser/evap.) [46].

FIB cuts of the laser irradiated and gold coated PET samples were investigated by SEM (see Figure 32). After sputtering, the gold is deposited in the form of “nano-wires”, which grow mainly at the ridges of the ripples. The FIB cuts reveal that there could be gaps between the individual wires and that the metal layer may be discontinuous. The width of the gold nano-wires directly correlates with the width of the ripples formed before the gold deposition. Additionally, a granularity is visible along the wires, but the FIB cut images suggest that the grains may be interconnected. The morphology of the gold layers deposited by evaporation is distinctly different. The gold is deposited in the valleys of the ripple structure too. Comparison with Figure 31 reveals that there is a difference between AFM and FIB-SEM analysis. i.e., the gaps between the sputtered gold nano-wires are not observed on the AFM

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images. We attribute this difference to the fact, that small objects with high aspect ratio may not be observed in the AFM images due to the above mentioned convolution of the surface topology with the AFM tip geometry.

Figure 32. Dependence of the surface roughness on the KrF laser fluence for irradiated PET without

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coating (○) and for irradiated and gold coated PET (sputtering (□) and evaporation (△)) [46].

Figure 33. FIB-SEM images of the gold coatings on PET samples irradiated by a KrF laser under incidence angles 0° and 22.5° (fluence 6.6 mJ.cm-2). The gold deposition was performed either by sputtering (Sputt.) and evaporation (Evap.) [46].

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The nano-wire structure of the sputtered gold layer can be observed also after the gold deposition onto PET samples irradiated by the laser under an angle of incidence of 45°. Again, the gold evaporation leads to a continuous coverage copying nano-structured polymer surface. The reason for the different observed gold morphologies after sputtering (nano-wires) and evaporation (homogeneous gold coverage) is still unclear. The different particle energies in both processes are one possible reason. For sputtering, the particle energy may be considerably higher because of sample charging effect¸ while the evaporated materials should be slower (i.e., colder) and closer to thermodynamical equilibrium. The electrical charge of the sputtered particles can have direct influence on layer formation too, while the evaporated material should be mainly neutral. Other reasons may be the different deposition rates, which were a factor of two lower for sputtering than for evaporation, and possible differences of the substrate and gold layer temperature during the deposition by the two different techniques. The above described results lead to conclusion that the growth mechanism of gold nanostructures depends strongly on the deposition technique. Figure 34 shows the proposed growth mechanism in the coating of nano-structured PET by sputtering technique. Separate Au atoms nucleate preferentially on the top of the ripple ridges. On the other hand, during evaporation (see Figure 35) no preferential nucleation is observed and thus the gold layer is copying nano-structured PET surface forming homogeneous Au coverage.

9. APPLICATION

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Structures prepared on polymers surface or in bulk were extensively used in photonics, electronics, optoelectronics, medicine and in some other fields.

Figure 34. Scheme of the growth mechanism of the gold layer on nano-structured PET surface during sputtering process.

Figure 35. Gold growth mechanism on nano-structured PET surface during evaporation process.

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9.1. Coupling of Light Applications in photonics include waveguide couplers, diffraction grating, sensors and so on. It is well known, that the light can be introduced in the waveguide through prism or grating [197]. Diffraction gratings were extensively used as input and output couplers in integrated optics. However, these gratings were traditionally made using multi-step photolithographic techniques [198]. The recently developed single step photofabrication process of SRG has offered a quick formation of grating couplers with desired periods. Utilization of this one-step fabrication of surface gratings as slab waveguide couplers was demonstrated [199]. However application of grating prepared by SRG technique seems to be promising in laboratory conditions but can not be perspective for mass production. Efficiency of light coupling strongly depends on the surface profile of coupled grating. Ideal possibility is grating with right angle [197]. Grating with sinusoidal form can not efficiently couple the light. That‟s why application of interference lithography technique seems to be more perspective for coupling gratings mass fabrication.

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9.2. Optical Filters Polymer periodic structures become more and more important in the area of integration optics. Transmission capacity of optical communication systems has been drastically increased by the advent of the wavelength division multiplexing WDM technique where multiple lightwaves with different wavelengths are multiplexed and transmitted through a single optical fibre. Tuneable wavelength filters with a narrow bandwidth and a wide tuning range are useful devices for the purpose of selecting one wavelength carrying desired information among the WDM signals. Integrated optical tuneable wavelength filters have been investigated with different operating principles in various substrates such as grating assisted codirectional couplers and multiperiod Bragg reflectors in semiconductors, multistage asymmetric Mach–Zehnder interferometers in lithium niobate, and polymers, etc [200]. Polymeric optical waveguide devices have been widely investigated because of the simple and low-cost fabrication. Compared to other filters, the Bragg grating filter provides a narrow bandwidth, a low crosstalk, and a flat-top passband. In the work [201,202], was demonstrated a tuneable wavelength filter with a Bragg grating on a low-loss polymer waveguide. To control the peak wavelength of the Bragg reflector, an electrical power is applied to the device electrode so that the effective index of the polymer waveguide is changed by the joule heat. Then the Bragg reflection wavelength is shifted to the shorter wavelength proportional to the effective index decrease or the heating temperature rise. Because the polymer has a low thermal conductivity and a high thermo-optic effect, the electrical power consumption would be low [200].

9.3. Organic Distributed Feedback Lasers (DFB) Incorporation of different chromophores into polymers seems to be perspective for preparation optically active materials [203,204]. Simultaneously, these chromophores will be responsible for polymer patterning by interference lithography technique. It means that by

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practically one step optically active material can be prepared and patterned. For DFB laser applications, the use of a photoreactive material that changes its refractive index under UV irradiation is advantageous [205]. In this case, the patterning process can comprise only one step (the irradiation) and no necessary subsequent development steps are needed. All wet chemical steps such as etching and removal of resists can be avoided. In a similar fashion, photoreactive polymers that produce relief gratings upon UV irradiation are useful for DFB applications. Examples of photoinduced relief formation in organic polymers are UV polymerizable formulations containing acrylates, photopatterning of polymers bearing pendant azobenzene units, and the formation of DFB gratings in dye-doped poly(methyl methacrylate) and polycarbonate by interference illumination [206].

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9.4. Liquid Crystal Alignment Another interesting application of patterned polymers is a possibility to achieve alignment of liquid crystal. A liquid crystal display (LCD) is one of the typical examples of molecular electronics, and the alignment of liquid crystals has been of keen interest for industrial applications [207]. Interference lithography opens the way for unique simultaneous patterning of polymers and alignment of liquid crystal. A submicron, spatially periodic, structure consisting of a sequence of oriented layers of a nematic liquid crystal (NLC), separated by isotropic polymeric walls, was obtained experimentally in [170]. This was accomplished by polymerization induced by the interference pattern of UV laser radiation in a NLC-containing prepolymer mixture. When the initial mixture is irradiated with a spatially periodic intensity distribution (an interference pattern) a periodically nonuniform degree of polymerization is induced and the mass transfer starts long before phase separation occurs (and it could be completed before the isotropic-liquid–nematic (I–N) phase transition starts). Thus, it is possible to obtain quite uniform regions of nematic and polymeric isotropic phases, provided that during photoinduced polymerization phase separation and a transition are prevented by means of an additional external action. Then no random processes occur, and quite regular modulation of the nematic concentration within a period of the grating is obtained. Finally it was established that such a structure occurs when phase separation and nematic ordering are prevented during the polymerization process. These structures are the diffraction elements, whose efficiency is much greater than that of a standard grating of polymer-dispersed liquid crystals which is obtained in the same initial mixture [208].

9.5. Optical Data Storage Interference lithography technique can be used for three dimensional controllable materials modification. This possibility will be fully exploited in the preparation of photonic band gap materials and optical data storages [209,210]. The need for data storage is explosively increasing, triggered by the development of multimedia and electronic communication networks which is expected to cross 1020 bits/storage media.. Due to this astronomical increase in the requirement of data storage, intense research activity is on going to find alternate methods and storage media for such large amounts of data. Recently, the focus has shifted from the two-dimensional storages to three-dimensional ones [211]. Several

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approaches to 3D optical data storage, such as, holographic recording with photorefractive media, hole burning, and photon echo, are currently being investigated. The use of twophoton processes for optical data storage was first introduced by Rentzepis and subsequently by Webb [212,213]. Since then there have been several other reports also proposing the use of two-photon processes for optical data storage. The advantages of two-photon based memory systems are volume storage with high data storage densities of the order of 1012 bits/cm, fast read/write times, random access, and low cost storage media. As the two-photon excitation has a quadratic dependence on the pump intensity, the excitation and subsequent photoreaction related to the writing process occurs only in the near vicinity of the focal point. An excellent resolution during the writing process is possible due to this property of twophoton induced processes. The basic components of a two-photon memory fabrication are, a medium which exhibits a change in its optical properties (absorbance, fluorescence, refractive index, etc.) after two-photon absorption, appropriate read and write beams, and a mechanism to precisely access any volume element in the medium [165]. At higher light intensity the two photons induced photochemical process shifts the optical properties of the written region. The read back can be subsequently done either in a two-photon excitation mode using a beam with longer wavelength or in a single-photon excitation mode using a laser with shorter wavelength. The data storage is in the form of stacked compact disk arrangement, i.e., in multiple layers with storage capacity exceeding terabits per cubic centimeter. Although the method can be used for both digital and analog storage, in this case we use a digital bitmap for storage which demonstrates gray scale control [211].

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9.6. Photonic Crystal Photonic crystal (PC) is an artificial periodic dielectric structure that controls the behaviour of photons just as semiconductor crystal control electrons [214]. In these synthetic crystals, the artificial periodic dielectric structure of the PC creates a photonic band gap, much as the crystal potential of a semiconductor produces an electronic band gap. With partial or complete photonic band gaps they give rise to a number of unique optical properties such as optical confinement and the superprism effect, which may be exploited in the future to fabricate high-density optical integrated circuits. PC also exhibits a variety of new physical phenomena, including the suppression or enhancement of spontaneous emission, lowthreshold lasing, and quantum information processing. Complete gaps between bands correspond to forbidden wavelengths, a feature of particular utility for passive photonic devices such as filters, waveguides, and resonant cavities. Structures with partial gaps i.e., in restricted dimensions may also have application in the emerging field of PC-based optical elements [214]. Experimental studies of three dimensional PC have been addressed to the methods for fabricating complex 3D micro/ nanostructures. Various techniques, such as electron-beam lithography, self-assembly, multiphoton polymerization, and interference lithography have been proposed and demonstrated with different levels of success [215]. The optical properties of PC materials scale with their lattice constants. PC destined for use in optical telecommunications circuits must have submicron lattice constants. For this reason interference lithography or holography seems to be very promising techniques for PC preparation [152]. In traditional realization the polymer films are exposed to marginally four

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laser beams, which create interference pattern inside the polymer bulk. Then the unexposed polymer materials are removed by wet process and appeared voids are filled by suitable semiconducting materials with great refractive index. Application of holography or interference lithography technique allows precise controlling of the periodicity and symmetry of prepared photonic band gap materials by changing the wavelength, angle of intersection and the number of laser beams. In the literature the preparation of photonic crystal with the several types of symmetry of cell units by interference lithography has been reported [153]. The main disadvantage of the interference lithography for photonic crystal fabrication is vibration instabilities in the optical setup. For overcoming this problem several experimental set ups, based on diffraction mask or prism has been proposed [156]. In the work [150] an approach to fabricate 3D periodic structures by using a single diffraction mask was described. Masks with a 1.0 and 0.56 µm grating period was used for production micron- and submicron-scale 3D features in photosensitive polymers with good definition and uniformity over large substrate areas.

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9.7. Sensors Application Polymers can play a very important role in sensors fabrication and application. If wide spectrum of light propagates through waveguide with diffraction grating certain wavelengths can be reflected at the grating. Reflected wavelength depends both on the grating geometry and material constant. Changing of the material constant (for example by absorption of external molecules) or grating geometry (external pressure, temperature) will change the reflected wavelength [216]. In literature several sensors prepared by SRG or interference lithography techniques on planar channel or fibre waveguides have been reported [216,217]. Fiber Bragg gratings have been receiving increasing applications in optical sensing. Their Bragg wavelength shift is proportional to the temperature or strain experienced by the gratings. Polymer fibre gratings are many times more sensitive to these influences than silica fibre gratings. Prepared polymer structures can be used in medicine as e.g. biosensors. By using the suitable polymers or by covering polymer structure with suitable materials it is possible to achieve appropriate interaction with some molecules (for example DNA) [218]. Biosensors can be used as analytical tools for the elucidation of molecular interactions. Evanescent optical sensing techniques that detect adlayer changes close to or at surfaces have found increasing application in the realtime analysis of molecular binding events. Novel optic techniques can register refractive index changes or measure the thickness of formed adlayers. These methods are of particular interest because the labelling of bioanalytes is not required. Instrumentation based on surface plasmon resonance or on wave guiding techniques is available, enabling the detection of molecular adlayers on sensor surfaces under carefully controlled and reproducible conditions.

9.8. Optical Metamaterials Optics has traditionally dealt with dielectric materials. Here, the incident electric-field vector of the light excites microscopic electric dipoles that re-emit electromagnetic waves.

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Other dipoles are excited by this re-emission and so on. This successive excitation and reemission clearly modifies the phase velocity of light and determines the optical properties of the material. Later it was concepted in optics that the magnetic dipoles can be excited by the magnetic-field component of the light. The interplay of permittivity (response to electric field) and permeability (response to the magnetic field) has given rise to interesting new aspects of electromagnetism and optics [219]. One of the most popular and perspective application is the preparation of optical metamaterials. Metamaterials are composite structures engineered to have properties that may not be found in nature [220]. In general, properties of metamaterials can be rather attributed to their structure than to their composition where small features create effective macroscopic behaviour. In optical terms metamaterial means materials with negative refractive index. In traditional case optical metamaterial represents an ordered array of artificially constructed electro-magnetic dipoles. Specifically, each dipole is a small LC circuit, which exhibit resonance behaviour in the electromagnetic field with certain frequency. Collective response of this circuit on the external electromagnetic field gives the materials unusual properties e.g. negative refractive index [219]. To affects the electromagnetic field metamaterials must have structural features smaller than the wavelength of light. Patterning of polymers is very useful and powerful tools for preparation of optical metamaterials. Patterned polymer films can be used as substrates for subsequent selective deposition of conductive metallic structure [21]. Several technique and process have been demonstrated for preparation of optical metamaterials on the polymer patterned substrate. Patterning of polymers was achieved by direct laser or electron beam writing. However, these techniques are relatively slow and application of interference lithography, holography, SRG can be expected for preparation of polymer substrates for metamaterials deposition in the near future. Attention to the optical metamaterials preparation and investigation is evoked because of a possibility to create in this way superlenses which can have a spatial resolution below the light wavelength. Another interesting application is optical invisibility. Beside these, optical metamaterials will find its application in sensor construction, smart solar power management, high-frequency battlefield communication and so on [219,220].

9.9. Application in Biotechnology During the past decade, a miniaturization trends has been going on in construction of devices for biological applications. An integrated analysis system for sample handling for biological characterization has recently been developed. Microstructures are of great importance for the coming biotechnology revolution. There is tremendous increase in both academic and corporate research on micromachined laboratories on a chips or on microsize analysis systems [221]. These devices will find applications in areas such as genomic and proteomic studies, which will require extensive parallelism to allow many small simultaneous experiments the integration of multiple experiments on a single carrier requires a miniature format. To minimize the chance of cross contamination when handling biological samples, a single-use device is preferred. Therefore, these devices should be disposable and thus be produced using inexpensive materials and patterning techniques. Polymers might be a good option. One of the more important advantages of polymers is that these materials can also

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deliver active functions. Polymer surfaces can be chemically modified in a variety of ways, and this property is important in microstructures, which have a high surface-to-volume ratio. For example, surface-bound processes may be used to alter biomolecular function. Because thin polymer films may easily be prepared by spin coating, they can be integrated into functional systems. This capability may be important in the development of inexpensive, disposable chemical detectors for genomics and proteomics. Crucial to the development of biologically integrated devices is the ability to organize multiple biomolecules on surfaces with resolutions from the micron to the nanometer scale. Several techniques have been examined for creating micron-level two-dimensional arrays of biomolecule on surfaces, including the use of interference lithography, photochemistry and surface relief grating [222,223]. While each of these methods may be useful for some applications, each has its inherent limitations, particularly in the areas of multiple protein binding, non-specific binding, and in the ability to immobilize proteins while retaining their maximum activity.

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9.10. Modification of Mechanical Behaviour Three dimensional patterning of polymers can significantly improve their mechanical properties. Combination of hard (e.g., calcite) and soft (e.g., proteins) components, inspired by natural structures, may lead to creation of materials with extraordinary mechanical properties. Holographic interference lithography was used to create a 3D polymer microframe with sub-micrometer periodicity, low density, and 200 nm feature size. These structures exhibited unusual deformational characteristics with ultimate strains reaching ca. 300%, much higher than the strains attainable in bulk polymers films [224]. In the work [225] elastical properties on both micro and macro levels was studied. In order to investigate the actual material properties of the complex photopatterned materials fabricated by interference lithography, attention was focused on the elucidation of the spatial distribution of elastic and plastic properties of a relatively simple 2D microstructure. Using atomic force microscopy, high-resolution nanomechanical studies of a thin polymer film was conducted. A two-dimensional polymer (SU-8) structure with six-fold symmetry was fabricated via interference lithography. Nonuniform spatial distribution in the elastic modulus, with a higher elastic modulus obtained for nodes (brightest regions in the laser interference pattern) and a lower elastic modulus for beams (darkest regions in the laser interference pattern) of the photopatterned films was found. It was suggested that such a nonuniformity and unusual plastic behaviour can be related to the variable material properties “imprinted” by the interference pattern. At other words, a spatial distribution of the local elastic modulus can be directly related to the symmetry of the light-intensity distribution within the original interference pattern in the photoresist. The extremely plastic behaviour of the films in the course of their fracturing was related to the essentially composite nature of the 2D perforated films with a high crosslink density obtained at the nodes and low crosslink density at the beams (darkest regions in the laser interference pattern). It was found, that the perforated SU-8 films are capable of highly plastic behaviour with significant local deformation of individual cells and largescale deformation of the whole net. High shearing and bending was observed for large film regions, which is completely

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uncharacteristic of glassy polymeric materials. It can be speculated, that precise control of the nonuniform internal elastic properties of polymer microstructures makes these materials highly deformable and opens potential paths for photopatterned polymeric materials with efficient energy absorption on a sub-micrometer scale [225].

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CONCLUSION In this chapter the application of laser light for polymer patterning was discussed in details. Electronics and photonics technologies need new effective methods for polymer modifications. Periodical structures on the polymers surface or in polymer bulk, created by different patterning techniques, may be used for preparation of photonic crystal, sensors, metamaterias ad so on. Additionally, patterned polymers can serve as substrate for next deposition of organic, metal or semiconducting materials. Utilization of laser light seems to be perspective technique for the polymer patterning. The laser light patterning comprises several techniques, based on different mechanisms and principles which were described in this chapter. Traditional methods of polymer patterning by lithography or direct laser writing were discussed to necessary details too. Also some relatively new methods, which may improve pattern resolution, like deep UV lithography were briefly described and attention was given to the application of interference pattern created by intersection of two or more coherent beams. Generally the application of laser beams for polymer patterning seems to be very perspective because its simplicity and ability to produce well defined and controllable structures. Main disadvantages of these techniques are that they can be applied to limited set of polymers and problems with preparation of structures ordered on longer distances. The polymer patterning with laser beam was illustrated on author‟s experiments performed on PET films. Modification of the PET surface with linearly polarized light from pulsed KrF laser has a significant effect on the properties of subsequently deposited gold nanolayers and the choice of the deposition technique is crucial owing to the quality of prepared coatings. Subsequent deposition of 200 nm thick gold layer is causing a decrease of the surface roughness. While by evaporation a continuous metal coverage is formed, copying nanostructured polymer surface, in the case of sputtering a nanowire-like structure of the gold coating can be observed. It was shown that the width of the nano-wires can be tailored by the width of the ripples formed by preceding laser irradiation. The promising techniques for creation of nanopatterned, regular gold structures (nano-wires) were reviewed. In principle, those techniques could be employed for the creation of metal-polymer composites with interesting electrical, mechanical, and optical properties, which could find novel applications in micro- and nanotechnology or like meta-materials. At the end of the chapter some applications of patterned polymers are described. Periodical polymer structures can play a great role in the photonic and electronic elements. In photonics, for example, they can be useful as coupling elements, wavelength filters, mirrors and so on. Temperature, pressure and bio-sensors based on polymers seem to be very promising to.

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By the next deposition of metal or semiconducting materials on patterned polymer surface or in polymer bulk new classes of materials – photonic crystals or metamaterials can be prepared. Additionally, patterning of polymer can sufficiently effect and improve their mechanical behaviour.

ACKNOWLEDGMENTS The work was supported by the GA CR under the projects 106/09/0125, 108/10/1106, 108/11/P840, 108/11/P337 and Ministry of Education of the CR under Research program No. LC 06041 and GAAS CR under the projects KAN400480701 and KAN200100801. The authors thank Ms. O.Kesselová for the technical assistance.

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[151] Moon, J.H.; Ford, J.; Yang, S. Polym. Adv. Technol. 2006, 17, 83. [152] Escuti, M.J.; Crawford, G.P. Opt. Engin. 2004, 43, 1973. [153] Chan, T.Y.M.; Toader, O.; John, S. Phys. Rev. E 2005, 71, 046605. [154] Gombert, A.; Blasi, B.; Buhler, C.; Nitz, P.; Mick, J.; Hossfeld, W.; Niggemann, M. Opt. Engin. 2004, 43, 2525. [155] Wu, L.J.; Tong, W.Y.Y.; Zhong, Y.C.; Wong, K.S.; Hua, J.L.; Haussler, M.; Lam, J.W.Y.; Tang, B.Z. Appl. Phys. Lett. 2006, 89, 191109. [156] Divliansky, I.; Mayera, T.S.; Holliday, K.S.; Crespi, V.H. Appl. Phys. Lett. 2003, 82, 1667. [157] Vogelaar, L.; Nijdam, W.; van Wolferen, H.A.G.M.; de Ridder, R.M.; Segerink, F.B.; Flück, E.; Kuipers, L.; van Hulst, N.F. Adv. Mater. 2001, 13, 1551. [158] Hariharan, P. Optical holography: Principles, Techniques and Application, Camridge University Press, 1996. [159] Ackermann, G.K.; Eichler, J. Holography. A practical Approach, Wiley, 2007. [160] Kirkpatrick, S.M.; Baur, J.W.; Clark, C.M.; Denny, L.R.; Tomlin, D.W.; Reinhardt, D.W.; Kannan, R.; Stone, M.O. Appl. Phys. A - Mater. Sci. Process. 1999, 69, 461. [161] Meerholz, K. Angew. Chem. 1997, 36, 945. [162] Wiederrecht, G.P. Ann. Rev. Mater. Res. 2001, 31, 139. [163] Crawford, G.P.; Eakin, J.N.; Radcliffe, M.D.; Callan-Jones, A.; Pelcovits, R.A. J. Appl. Phys. 2005, 98, 123102. [164] Kardinahl, T.; Franke, H. Appl. Phys. A - Mater. Sci. Process. 1995, 61, 23. [165] Gu, C.; Xu, Y.; Liu, Y.S.; Pan, J.J.; Zhou, F.Q.; He, H. Opt. Mater. 2003, 23, 219. [166] Campbell, M.; Sharp, D.N.; Harrison, M.T.; Denning, R.G.; Turberfield, A.J. Nature 2000, 404, 53. [167] Lawrence, J.R.; O'Neill, F.T.; Sheridan, J.T. Optic 2001, 112, 449. [168] Weiss, V.; Millul, E. Appl. Surf. Sci. 1996, 106, 293. [169] Neipp, C.; Gallego, S.; Ortuno, M.; Marquez, A.; Belendez, A.; Pascual, I. Opt. Commun. 2003, 224, 27. [170] Kim, M.H.; Kim, J.D.; Fukuda, T.; Matsuda, H. Liq. Cryst. 2000, 27, 1633. [171] Juhl, A.T.; Busbee, J.D.; Koval, J.J.; Natarajan, L.V.; Tondiglia, V.P.; Vaia, R.A.; Bunning, T.J.; Braun, P.V. ACS NANO 2010, 4, 5953. [172] Li, Q.; Wang, P. Appl. Phys. Lett. 2010, 96, 111109. [173] Matharu, A.S.; Jeeva, S.; Ramanujam, P.S. Chem. Soc. Rev. 2007, 36, 1868. [174] Blanche, P.A.; Bablumian, A.; Voorakaranam, R.; Christenson, C.; Lin, W.; Gu, T.; Flores, D.; Wang, P.; Hsieh, W.Y. Kathaperumal, M.; Rachwal, B.; Siddiqui, O; Thomas, J.; Norwood, R.A.; Yamamoto, M.; Peyghambarian, N. Nature 2010, 468, 80. [175] Lyutakov, O.; Hüttel, I.; Siegel, J.; Švorčík, V. Appl. Phys. Lett. 2009, 95, 173103. [176] Arenholz, E.; Heitz, J.; Himmelbauer, M.; Bäuerle, D. Proc. SPIE 1996, 2777, 90. [177] Bolle, M.; Lazare, S.; Le Blanc, M.; Wilmes, A. Appl. Phys. Lett. 1992, 60, 674. [178] Csete, M.; Bor, Z. Appl. Surf. Sci. 1998, 133, 5. [179] Lazare, S.; Benet, P. J. Appl. Phys. 1993, 74, 4953. [180] Csete, M.; Eberle, R.; Pietralla, M.; Marti, O.; Bor, Z. Appl. Surf. Sci. 2003, 208-209, 474. [181] Birnbaum, M. J. Appl. Phys. 1965, 36, 3688. [182] Bäuerle, D. Laser Processing and Chemistry, Springer-Verlag, Berlin-Heidelberg-New York, 2000.

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[183] Sipe, J.E.; Young, J.F.; Preston, J.S.; van Driel, H.M. Phys. Rev. B 1983, 27, 1141. [184] Bolle, M.; Lazare, S. J. Appl. Phys. 1993, 73, 3516. [185] Geil, P.H. Europhys. Conf. Abstr. 1988, 12D, 22. [186] Kotál, V.; Švorčík, V.; Slepička, P.; Bláhová, O.; Šutta, P.; Hnatowicz, V. Plasma Proc. Polym. 2007, 4, 69. [187] Švorčík, V.; Hubáček, T.; Slepička, P.; Siegel, J.; Kolská, Z.; Bláhová, O.; Macková, A.; Hnatowicz, V. Carbon 2009, 47, 1770. [188] Heitz, J.; Arenholz, E.; Bäuerle, D.; Sauerbrey, R.; Phillips, H.M. Appl. Phys. A 1994, 59, 289. [189] Dunn, D.S.; Ouderkirk, A.J. AIP Conf. Proc. 1989, 191, 375. [190] Toma, A.; Chiappe, D.; Massabó, D.; Boragno, C.; Mongeot, F.B. Appl. Phys. Lett. 2008, 93, 163104. [191] Švorčík, V.; Zehentner, J.; Rybka, V.; Slepička, P.; Hanatowicz, V. Appl. Phys. A 2002, 75, 541. [192] Efimenko, K.; Rybka, V.; Švorčík,V.; Hnatowicz, V. Appl. Phys. A 1999, 68, 479. [193] Gustafson, G.; Cao, Y.; Colaneri, N.; Heeger, A.J. Nature 1992, 357, 477. [194] Švorčík, V.; Siegel, J.; Slepička, P.; Kotál, V.; Špirková, M. Surf. Interf. Anal. 2007, 39, 79. [195] Kim, K.S.; Ryu, C.M.; Park, C.S.; Sur, G.S.; Park, C.E. Polymer 2003, 44, 6287. [196] Wu, G.; Paz, M.D.; Chiussi, S.; Serra, J.; Gonzalez, P.; Wang, Y.J.; Leon, B. J. Mater. Sci. Mater. Med. 2009, 20, 597. [197] Hunsperger, R.G. Integrated Optic: Theory and technology, Springer, 2002. [198] Waldhausl, R.; Schnabel, B.; Dannberg, P.; Kley, E.B.; Brauer, A.; Karthe, W. Appl. Opt. 1997, 36, 9383. [199] Bang, C.U.; Shishido, A.; Ikeda, T. Macromol. Rapid Commun. 2007, 28, 1040. [200] Murphy, E. Integrated Optical Circuits and Components: Design and Applications, CRC Press, 1999. [201] Toyoda, S.; Kaneko, A.; Ooba, N.; Hikita, M.; Yamada, H.; Kurihara, T.; Okamoto, K.; Imamura, S. IEEE Photon. Techn. Lett. 1999, 11, 1141. [202] Nam, S.H.; Kang, J.W.; Kim, J.J. Opt. Commun. 2006, 266, 332. [203] He, G.S.; Yuan, L.X.; Cheng, N.; Bhawalkar, J.D.; Prasad, P.N.; Brott, L.L.; Clarson, S.J.; Reinhardt, B.A. J. Opt. Soc. Amer. B - Opt. Phys. 1997, 14, 1079. [204] Ju, K.; Lim, J.S.; Lee, C.; Choi, D.H.; Kim, D.W. Mol. Cryst. Liquid Cryst. 2008, 491, 152. [205] Kavc, T.; Langer, G.; Kern, W.; Kranzelbinder, G.; Toussaere, E.; Turnbull, G.A.; Samuel, I.D.W.; Iskra, K.F.; Neger, T.; Pogantsch, A. Chem. Mater. 2002, 14, 4178. [206] Kranzelbinder, G.; Toussaere, E.; Josse, D.; Zyss, J. Synth. Met. 2001, 121, 1617. [207] Yeh, P.; Gu, C. Optics of Liquid Crystal Display, Wiley, 1999. [208] Yamamoto, T.; Hasegawa, M.; Kanazawa, A.; Shiono, T.; Ikeda, T. J. Phys. Chem. 1999, 103, 9873. [209] Lu, C.; Lipson, R.H. Laser. Photon. Rev. 2010, 4, 568. [210] Buckley, G.S.; Roland, C.M. Polym. Eng. Sci. 1997, 37, 138. [211] Pudavar, H.E.; Joshi, M.P.; Prasad, P.N.; Reinhardt, B.A. Appl. Phys. Lett. 1999, 74, 1338. [212] Dvornikov, A.S.; Taylor, C.M.; Liang, Y.C.; Rentzepis, P.M. J. Photochem. Photobiol. A. Chem. 1998, 112, 39.

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[213] Strickler, H.; Webb, W.W. Opt. Lett. 1991, 16, 1780. [214] Joannopoulus, J.D.; Meade, R.D.; Winn, J.N. Photonic Crystal. Molding the Flow of Light, Princeton University Press, 1995. [215] Krauss, F.; De la Rue, R.M. Progress Quant. Electron. 1999, 23, 51. [216] Peng, G.D.; Chu, P.L. Fiber Integrated Opt. 2000, 19, 277. [217] Zhang, Y.; Feng, D.J.; Liu, Z.G.; Guo, Z.Y.; Dong, X.Y.; Chiang, K.S.; Chu, B.C.B. IEEE Photon. Techn. Lett. 2001, 13, 618. [218] Cosnier, S. Electroanalysis 2005, 17, 1701. [219] Eleftheriades, G.V.; Balmain K.G. Negative-refraction metamaterials: fundamental principles and applications, Wiley, 2005. [220] Cai, W.; Shalaev, V. Optical Metamaterials: Fundamentals and Applications, Springer, 2010. [221] Jeon, S.; Park, J.U.; Cirellit, R.; Yang, S.; Heitzman, C.E.; Braun, P.V.; Kenist, P.J.A.; Rogers, J.A. PNAS 2004, 101, 12428. [222] Guo, L.J. J. Phys. D - Appl. Phys. 2004, 37, R123. [223] Lahann, J.; Balcells, M.; Rodon, T.; Lee, J.; Choi, I.S.; Jensen, K.F.; Langer, R. Langmuir 2002, 18, 3632. [224] Zhang, G.; Yan, X.; Hou, X.L.; Lu, G.; Yang, B.; Wu, L.X.; Shen, J.C. Langmuir 2003, 19, 9850. [225] Jang, J.H.; Ullal, C.K.; Choi, T.Y.; Lemieux, M.C.; Tsukruk, V.V.; Thomas, E.L. Adv. Mater. 2006, 18, 2123.

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Chapter 2

PHOTOLUMINESCENCE OF ZnO THIN FILMS AND NANO POWDERS DOPED WITH MONO, DI AND TRIVALENT CATIONS B. S. Acharya † IMMT, Bhubaneswar, India Dept. of Physics,CVRCE, Bhubaneswar, India

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ABSTRACT Zinc Oxide, a unique material, exhibiting semiconducting, piezoelectric, photonic and anti bactericidal properties has been studied extensively for its structural, optical, electrical and light emitting characteristics by various techniques. Due to formation of nanostructures like rod, tube, needle, comb, flower spring and helixes etc. optical properties have not been determined unambiguously. Nano ZnO films have been studied for its application as nano generator.The material has been recognized as a promising photonic material in uv-vis region also for its lasing action. In this system oxygen vacancies, impurities and Zn interstitials have been observed to play key role in deciding the photo, thermo and stimulated luminescence phenomena taking place in the system. The present review intends to clarify certain aspects of these structural and optical properties taking into account the various impurities like Ag, Al, Cu, Ce, Gd and Eu. Different deposition processes like spray pyrolysis, vacuum deposition and low temperature RF plasma and sonication have been studied and presented. On the basis of available literature on photoluminescence and other luminescence the data have been discussed and tentative model has been proposed.

INTRODUCTION ZnO is a promising material for next generation electronic,opto electronic ,sensing and food packaging devices .This is due to its wide band gap (3.37eV) and large exciton binding †

E-mail address: [email protected]

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energy (≈60meV),at 300 K and remarkable surface properties..These characteristic features make it suitable for light emitting devices in the visible range of ultraviolet, blue,green or white light. It has additional advantage over GaN due to availability of high quality single crystals and better electronic parameters. ZnO thin films find application as transparent conducting oxide(TCO) in UV-A and UV-B region. Thus , it is important to investigate the optical properties of ZnO films and powders prepared by various methods. It helps us to optimize the growth parameters aswellas its application for optical devices. For this purpose, the band structure of the semiconductor and transitions processes taking place in this material need to be understood properly before its use in optical or electronic devices.. Several theoretical models have been used to calculate the band structure of different crystal structure of ZnO(1-6). Normally x-ray, uv absorption/reflection and emission spectral techniques are used to measure electronic core levels. There are some other techniques such as photoelectron spectroscopy (PES) and angle-resolved photoelectron spectroscopy (ARPES) technique which have been used recently for characterizing the nano zinc oxides. Although photoluminescence (PL) is an old technique used for identification of defect centres involved in the luminescence process, this has now become a handy tool in the hands of researchers to characterize the nano materials. Light emission by any means other than blackbody radiation is called luminescence. It is a non-equilibrium process that needs a lamp or laser or high energy as excitation source. Depending on the type of excitation sources, luminescence is categorized as photoluminescence (PL) , electroluminescence,(EL), cathodo luminescence (CL) or tribo and the other kinds of luminescence. PL is one of the most widely used experimental methods for study of semiconductors; especially wide-band gap materials [6-12]. PL , again can be classified into two major types: intrinsic and extrinsic luminescence. The intrinsic luminescence can be broadly categorized as band-to-band luminescence, exciton luminescence and cross-luminescence. The recombination of an electron in the conduction band with a hole in the valence band generates band-to-band transition luminescence, which is exhibited in high purity crystals, such as Si, Ge and GaAs. At low temperatures, this luminescence is often regarded as exciton luminescence. The light emission from bright light-emitting diodes and semiconductor lasers is usually due to the band-to-band transition process. Extrinsic luminescence is normally generated by the impurities called activators which are intentionally or unintentionally incorporated. There are two kinds of extrinsic luminescence in semiconductors i. e. localized and delocalized. These defect related luminescence properties in semiconductors are studied by the steady-state PL (SS-PL) spectra, time-resolved PL (TR-PL), PL excitation (PLE) spectra,optically stimulated luminescence(OSL) and/or optically detected magnetic resonance (ODMR), a modification of the PL technique. In low dielectric semiconductors, the shielding of charges is relatively weak and coulomb potential between the negatively charged electrons and the positively charged holes are higher than their combined thermal energies .So,electrons and holes may form weakly bound electron –hole pairs called Wannier excitons(Figure 1)Excitons in the direct band gap semiconductors are highly effective in producing luminescence, in particular, zinc oxide (ZnO) has an exceptionally large exciton binding energy and hence, high luminescence efficiency. To study the exciton dynamics, nanostructures are the appropriate candidates in which the exciton binding energy is enhanced due to spatial confinement of the electron and hole wavefunctions. The large exciton binding energy of ZnO makes it an ideal sample to study the fundamental properties of excitons, without restructuring the material in the

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nanometer regime, which inherently introduces defects and/or heterogeneities. In ZnO the exciton state is intrinsically stable, in a crystalline sample even at room temperature.

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Figure 1. Wannier exciton with Bohr radius aBohr.

Formation of excitons is a two-step process: first, the thermalization and then the cooling of hot electron-hole plasma.This allows the formation of excitons with excess center of mass momentum (K≠0). Second, relaxation to the K=0 state occurs. These two processes can be best probed by a time-resolved inter band luminescence (7) technique which can give fast information as regards the fast formation with slow momentum relaxation or slow formation with fast momentum relaxation.Temperature is a crucial parameter in the formation of excitons. The exciton formation (electron-hole pair) rate increases by a factor of ~2.5 while going from 20-K to 1400K. This illustrates the importance of phonon modes in the exciton formation process. Inelastic carrier-phonon scattering allows the excess excitation energy to be dissipated into the lattice [13]. At higher temperatures, free charge density increases due to thermally mediated exciton dissociation. In Figure 2 (a), the parabolic curve represents the exciton kinetic energy as a function of centre of mass momentum (K). After initial photon absorption (dashed line), a hot exciton, with K≠0, is formed by emission of an optical phonon (dotted line), needed to conserve momentum. Subsequent relaxation to the emissive K=0 state occurs through further emission of optical phonons (large curved arrows). As the exciton falls to the bottom of the well, acoustic (small curved arrows) phonons are emitted. Thus for a hot electron-hole gas, rapid longitudinal optical (LO) phonon emission is the most important energy loss pathway. Slow acoustic phonon emission has been proposed to explain the slow rise of exciton luminescence (~100 ps [13]) within the framework of the “hot exciton cascade” [14]. The rate determining step for reaching the K=0 state (and thus luminescence) requires the loss of the last few quanta of excess energy. However, the slow decay in the free charge response, similar to that of confined systems [15], suggest that the motion of electrons and holes remains uncorrelated for much of the formation process. In this case, a more realistic description involves cooling through the electron hole continuum (Figure 2.b), where the uncorrelated electron and holes have, on average no net centre of mass momentum. Later in the cooling process, correlated motion between the paired electrons and holes (exciton formation) introduces a non-zero centre of mass momentum.

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Figure 2. Exciton formation illustrated in terms of the “hot exciton cascade” [15].

Photo-exciting a direct semiconductor such as ZnO generates hot electrons in the conduction band and hot holes in the valence band. The subsequent decay of this plasma is dependent on its density. When the average distance (r) between excitations is much larger than the Bohr radius (aBohr) the plasma decays to an exciton gas, with bound electron-hole pairs. This process is phonon mediated and therefore, faster for higher temperatures. At high excitation densities, when r  a Bohr , an exciton gas does not form, but the excited plasma decays (through many body radiative recombination) to an electron-hole liquid (EHL) phase. This process is very fast, occurring while the plasma is still hot. In contrast to exciton formation, EHL formation is not phonon mediated and is essentially independent of lattice temperature. The conduction band of ZnO with wurtzite crystal structure is constructed from s-like Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

c

state ( 7 ), which is symmetric in shape. The valance band is a p-like state. It is split into three bands due to crystal field and spin-orbit interaction [16], which dominates the near bandgap intrinsic absorption and emission spectra. The related free-exciton transitions involving an electron from the conduction band and a hole from these three valance bands are named as A, B and C (Figure 3), which correspond to heavy hole, light hole and crystal field split band respectively. The transition energies of the intrinsic excitons were measured with the various techniques, such as the low temperature absorption, reflection, photo-reflectance and PL spectroscopy techniques etc. Several research groups have reported PL spectra of zinc oxide system [17-21]. The emission line at 3.378eV is considered as the A-free excitons and their first excited state transitions. Similar results have also been reported by Teke et al. [26]. Other research groups [22- 25] investigated the excitonic peak energies for high quality ZnO single crystals and found good agreement with the experimental results. However, there is little quantitative difference, which may be caused by different ZnO samples or various experimental conditions. The discrete electronic energy levels in the bandgap of ZnO are generated by the dopants or defects in semiconductor material. It influences the optical absorption and emission processes. The type and band structure of the semiconductor material influences the electronic states of the bound exciton. Neutral or charged donors and acceptors can bind exciton, which results in bound excitons. The neutral shallow donor-bound exciton (DBE) normally dominates in the low-temperature PL spectrum of high-quality ZnO films, due to the presence of donor sources coming from unintentional impurities or other shallow-

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level defects. The acceptor-bound exciton (ABE) is also sometimes observed in some ZnO films which contain substantial concentration of acceptors. The sharp lines in PL spectra generated by the recombination of bound excitons are the signal, which is used to identify different defects or impurities source. Most sharp donor bound and acceptor-bound exciton lines are observed in the range from 3.34eV to 3.37eV in high quality ZnO films.

Figure 3. Band structure and symmetries of hexagonal ZnO [Meyer et al., phys. stat. sol. (b) 241, No. 2, (2004), 231].

The conduction band of ZnO with wurtzite crystal structure is constructed from s-like c

state ( 7 ), which is symmetric in shape. The valance band is a p-like state. It is split into three bands due to crystal field and spin-orbit interaction [16], which dominates the near bandgap intrinsic absorption and emission spectra. The related free-exciton transitions involving an electron from the conduction band and a hole from these three valance bands are named as A, B and C (Figure 3), which correspond to heavy hole, light hole and crystal field split band respectively. The transition energies of the intrinsic excitons were measured with the various techniques, such as the low temperature absorption, reflection, photo-reflectance and PL spectroscopy techniques etc. Several research groups have reported PL spectra of zinc oxide system [17-21]. The emission line at 3.378eV is considered as the A-free excitons and their first excited state transitions. Similar results have also been reported by Teke et al. [26]. Other research groups [22- 25] investigated the excitonic peak energies for high quality ZnO single crystals and found good agreement with the experimental results. However, there is little quantitative difference, which may be caused by different ZnO samples or various

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experimental conditions. The discrete electronic energy levels in the bandgap of ZnO are generated by the dopants or defects in semiconductor material. It influences the optical absorption and emission processes. The type and band structure of the semiconductor material influences the electronic states of the bound exciton. Neutral or charged donors and acceptors can bind exciton, which results in bound excitons. The neutral shallow donor-bound exciton (DBE) normally dominates in the low-temperature PL spectrum of high-quality ZnO films, due to the presence of donor sources coming from unintentional impurities or other shallowlevel defects. The acceptor-bound exciton (ABE) is also sometimes observed in some ZnO films which contain substantial concentration of acceptors. The sharp lines in PL spectra generated by the recombination of bound excitons are the signal, which is used to identify different defects or impurities source. Most sharp donor bound and acceptor-bound exciton lines are observed in the range from 3.34eV to 3.37eV in high quality ZnO films. Two electron satellite (TES) transition process in high quality ZnO occurs can be observed in low temperature PL spectra in the photon energy, range from 3.30eV to 3.34eV. This transition process is generated by the radiative recombination of an exciton bound to a neutral donor, leaving the donor in the excited state. The energy position of the excited state of shallow donor can be roughly estimated from the energy difference between the groundstate neutral-donor bound excitons (DBE) and TES lines. Then the donor binding energy can be calculated because the excited-to-ground levels distance equals to 3/4th of the donor binding energy (ED). Teke et al. [26] found the separations between A-free exciton and the ground-state neutral DBEs as 16.5meV, 15.3meV and 12.1 meV. The binding energy of the DBEs should be proportional to the binding energy of the corresponding donor with a factor 0.3 [29]. Shallow donor-acceptor-pair (DAP) emission exhibiting the main zero-phonon peak at about 3.22eV followed by at least two LO phonon replicas is important processes in the optical properties of ZnO. The LO phonon energy in ZnO is about 72meV, compared to 91meV in GaN, another competitive wide band semiconductor. Temperature dependent PL measurements are used to study the temperature evolution of peaks in PL spectra of ZnO. At low temperature, A and B exciton peaks can be identified clearly, while at high temperature it is hard to identify the peak positions. Then these free exciton peaks start to quench with the increasing of the temperature. The ABE peaks also quench with the increase in the temperature and hence, can't be identified when temperature is higher than 400K. The DBE peaks can be traced at the whole temperature range. The main peaks and its TES with their LO-phonon replicas quench with increasing temperature.The intensity ratio between DBE and free exciton decreases with the increase in temperature. At low temperature, thermal energy is lower than the binding energy. So the bound exciton transition process dominates at low temperature, while at high temperature, the free exciton transition process prevails. PL measurement at low temperature is the most common technique to investigate the point defects in ZnO. The green luminescence in high-quality undoped ZnO dominates the defect-related part of the PL spectrum [30]. In PL spectra of ZnO, there is normally a sharp peak near 3.22eV with at least 2 LO phonon replicas are commonly observed. The emission lines between 3.25eV and 3.4eV comes from the exciton recombination processes. With increasing temperature, the DAP line quenches and gives way to e-A line, due to thermal ionization of the shallow donor [31,32] above 30˚K.

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The origin of the green luminescence (GL at about 2.5eV) in high quality ZnO has been argued for a long time. Dingle‟s paper [33] concluded that the green luminescence is related to the copper impurities in ZnO. However, Van de Pol [34] attributed the GL band to the oxygen vacancy (VO) present in ZnO. Recent studies in GL band with a fine structure is related to the copper impurities, while the structureless GL band may come from the native point defects such as VO or VZn. This green emission in ZnO is attributed to donor–acceptor complexes [35] and antisite oxygen [36, 37]. It is likely that the green emission from ZnO would be due to intrinsic defects rather than extrinsic impurities. Gao and Wang [8] attributed the 500-550nm emission from the ZnO nanorods on the surface states. It is likely that the oxygen vacancies could be one of the most probable surface defects in ZnO crystals. Yellow luminescence (YL) band in some undoped bulk ZnO [26] have been reported after long time irradiation with a He-Cd laser. The GL gives way to the YL band. The excitation intensity with a power of 10-3Wcm-2 saturates the YL band, which means low concentration of the related defects. The temperature dependence of YL intensity exhibits no quenching up to 2000K. In some ZnO a red luminescence (RL) band is observed. The plausible explanation offered is due to competitive process taking place between the holes & acceptors responsible for GL and RL bands. When temperature is higher than 2000K, the GL starts to quench and gives way to the RL bands [26]. Presence of hydrogen was found to have a great influence on the intensity of the green luminescence band in ZnO films prepared at 8000K substrate temperature [38]. At a relatively low temperature (3000K), hydrogen is adsorbed on the surface of ZnO, resulting

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ZnO  H 2  ZnH  OH The defect resulting from the combination of oxygen vacancy and hydrogen reduces the intensity of green emission by five fold when the sample is maintained in hydrogen atmosphere at 8200K. So complexes of the type (VO-H) in ZnO are regarded as luminescence inefficient [39].

EXPERIMENTAL DETAILS Synthesis of silver doped ZnO powders was performed using a sonicator bath operated under continuous and pulsed mode. In this method 0.2M zinc nitrate (AR grade) 0.001M silver nitrate (AR grade) were added to 10 mL distilled water. 2mL of 25% NH3 (GR grade) was added drop by drop till mild precipitation occured and then 10 drops of NH3 was added to make the solution clear. This clear solution was subjected to continuous and pulsed mode sonication in a sonicator model VC-375(Sonic & Material Inc.) operated at 112.5 watt. The frequency of the sonicator was maintained at 20 KHz ± 50Hz. In continuous mode (CS) sonication was carried out for one hour continuously whereas in pulsed mode (PS) took 2.5 hour. The clear solution became silvery white and gradually a solid suspension was formed as the sonication continued. During sonication, the temperature of the bath was maintained around 70oC. The heat generated during this process was responsible for release of ammonia from the solution resulting the dissociation of Zn (NH3)42+ complex. The pH of the solution

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was measured at the beginning and at the end of the experiment. In both the cases the starting point pH was 10.4, but the pH comes down to 9.6 at the end of continuous mode and 8.4 for pulsed mode. The powder formed was washed thoroughly 2 to 3 times with DD water and acetone by centrifugation with 3500 rpm .The resulting powder was first dried at room temperature and then kept overnight at 1100c in air atmosphere . The detail of reaction mechanism has been given: Zn(NO3)2 +AgNO3+NH4OH+H2O------>Zn(OH)2+AgOH+ NH4NO3+ H2O Zn(OH)2+AgOH+NH4NO3+H2O+NH4OH----->[Zn(NH3)4](NO3)2+[Ag(NH3)2]NO3+H2O In Solution pH>10 [Zn(NH3)4](NO3)2 +H2O------> Zn(OH)2 + NH4NO3 +x NH3------>(2-x) NH4OH+H20 [Ag(NH3)2]NO3 +H2O------> AgOH + NH4NO3+x NH3------>(2-x) NH4OH+H2O Zn(OH)2+AgOH------>ZnO+Ag2O+ H2O Similarly other powders of rare earth doped ZnO were prepared following the above procedure. Details of sample preparation by spray pyrolysis technique have been given in our earlier paper.

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RESULTS AND DISCUSSIONS De-convoluted emission spectra of ZnO:Al samples excited at 200nm has been shown in Figure 4.In the spectrum range of 3.5eV to 1.8eV six emission bands have been present at 3.14eV, 2.98eV, 2.79eV, 2.57eV, 2.37eV and 2.18eV. Their relative intensities were calculated and the values have been given in Table-I. Similar results were also observed for emissions by excitation at 233nm wavelength (Table-II) for all compositions of Al in ZnO (The deconvolution of the emission spectra excited at 233nm are not shown). The photoexcitation spectra of ZnO and ZnO:Al thin films were recorded by placing the monochromator at 415nm. Three more excitation bands around 233nm, 265nm and 276nm were observed along with 200nm band for all the thin films studied in this work. The absorption or the transmittance spectra of the films do not show any band. Since the absorption features of semiconductor nanoparticles are different than the bulk, the exciton absorption band does not appear in the spectra taken at room temperature. However, broad transitions and unresolved steps are observed for semiconductor nanoparticles. In the strong confinement region (Radius of nanoparticle (R) < Bohr radius (a)) there is an increased overlapping of electron and hole wave function and this increases with decrease in cluster volume. Since photoluminescence excitation (PLE) is obtained by monitoring the emission at a particular wave length, transitions that overlap in direct absorption are resolved in PLE. The four excitation bands observed for our ZnO system [Figure 5] (Include these figs.) relate to four different states situated at different levels in the band gap and contributing to 415nm (blue) emission. The broad excitation bands are due to inhomogeneous broadening resulting

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out of size fluctuation in nano particle system. Such type of broadening has been observed by several workers reporting on CdS [41] and CdSe systems [42,43]. Undoped and copper doped zinc oxide films were deposited on borosilicate glass substrates from aqueous solution of zinc acetate and cupric acetate using dry laboratory air as the carrier gas. Both pure and doped ZnO films were deposited by spray pyrolysis method according to the methodology described earlier (our ref). Calculated amount of copper acetate solution was added to the zinc acetate solution and thoroughly homogenized by stirrer. The solution was sprayed on to the glass substrate kept at 2500C (±50) and the process of spraying continued for 2 seconds. After a break of one minute, again the spraying process continued for 2 seconds. This is termed as one cycle. For film deposition 5 such cycles of spraying were performed. This resulted in uniform thin films. In this manner, it was made possible to deposit ZnO and ZnO doped with 1.0-5.0 at wt % of copper. The photoluminescence emission spectra were resolved by curve fitting method, assuming the emission bands to be Voigtian. (This assumption is justified because the inhomogeneous broadening present due to size effects also manifests in emission spectra and the assumption of Gaussian peak shape does not result the best fit when least square refinement is made. By this fitting method six emission bands were observed, which are given in Table-I). The band around 3.14eV (~385nm) has been attributed to free exciton emission band in ZnO as argued by others [43,44].

Figure 4. Splitting of Photoluminescence Emission spectrum of undoped (i), 2.5 at wt % Al (ii), 5.0 at wt % Al (iii) and 7.5 at wt % Al doped (iv) spray pyrolytic ZnO thin film excited with 200nm light. Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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Figure 5. Curve fitted PL excitation spectrum of undoped ZnO thin film (i) and 5.0 at wt % Al doped ZnO thin film (ii) prepared by spray pyrolysis method (em=416nm).

Table I. Deconvolution of Photoluminescence spectra of ZnO and ZnO:Al by curvefitting method. All the emission peaks were assumed to be Voigtian. ex=200 nm

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Peak No.

1 2 3 4 5 6

Undoped Centre (eV) 3.14 2.98 2.79 2.57 2.37 2.18

Rel. Int. (%) 100 123 77 53 38 14

2.5 at wt % Al doped Centre Rel. Int. (eV) (%) 3.12 100 2.97 124 2.80 97 2.58 64 2.37 43 2.19 14

5.0 at wt % Al doped Centre Rel. Int. (eV) (%) 3.13 100 2.97 122 2.79 84 2.58 53 2.37 35 2.20 11

7.5 at wt % Al doped Centre Rel. Int. (eV) (%) 3.16 100 2.98 122 2.79 63 2.58 44 2.37 27 2.20 8

As the bound excitons get thermally ionized at room temperature their emission cannot be detected in PL spectra recorded at room temperature [45]. All the emission bands in the emission spectra other than the exciton band were found to be in the visible region. The intensities of all the bands increased for 2.5 at wt % of Al in ZnO, then decreased for higher concentrations of Al. When the relative intensity of each visible band to that of exciton band was determined, the RI of 2.98eV band (violet band) remains constant irrespective of Al concentration in the film, but the RI of the 2.79eV and 2.57eV band (blue band) increased for 2.5 at wt % Al dopant concentration and then decreased for higher concentrations. This indicates the involvement of Al3+ ion in the process of blue emission. This could happen in the following manner. ZnO is a degenerate semiconductor having lot of structural defects. These defects may be Schottky type or Frankel type. The ionic radii of Al3+ and Zn2+ are 0.053nm & 0.074nm respectively [44]. For charge compensation in the system some of these Al3+ ions may occupy interstitial sites or in the thin films these may be present on the surface of the film. Because of the one-dimensional nature of the film, dangling bonds at the surface may form surface states with the Al3+ ions.

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Figure 6. Variation of the FWHM (Full width at haif maximum) of different emission bands in Photoluminescence spectrum verses doping concentration in spray pyrolytic ZnO thin film excited with 200nm light (i) and 233nm light (ii).

Table II. Deconvolution of Photoluminescence spectra of ZnO and ZnO:Al by curve fitting method. All the emission peaks were assumed to be Voigtian. ex=233nm

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Peak No.

1 2 3 4 5 6

Undoped Centre (eV) 3.10 3.00 2.82 2.58 2.39 2.23

Rel. Int. (%) 100 101 56 37 19 23

2.5 at wt % Al doped Centre Rel. Int. (eV) (%) 3.10 100 2.98 131 2.81 56 2.58 42 2.39 21 2.22 12

5.0 at wt % Al doped Centre Rel. (eV) Int. (%) 3.09 100 2.99 111 2.83 49 2.58 33 2.39 14 2.24 11

7.5 at wt % Al doped Centre Rel. Int. (eV) (%) 3.09 100 2.99 97 2.81 30 2.57 24 2.39 10 2.24 10

In the violet region of the emission spectrum two bands were found i.e. 3.14eV & 2.98eV (Table.I & Table-II). This confirms the idea of bound exciton taking part in the process of photoluminescence contrary to the observation made earlier [45]. It may be due to nondeconvolutions of the spectra made in earlier studies. Further, the green emission band (520nm) (2.38eV) in this system has been attributed to deep traps present inside the band gap as reported by earlier workers [46, 47]. An emission band model has been proposed by us earlier (21). It has also been observed that all thin films produce strong excitonic emissions and these are stronger than the trapped emission indicating good surface passivation of the nanoparticles in these films. Such type of observations have been found in CdS nanocrystals [48]. To study the excitation-energy dependence of photoluminescence of ZnO:Al clusters, the PL spectra were recorded by exciting at 200nm, 233nm, 263nm and 275nm band light. The emission spectra were found to be broad in nature featuring some shoulders. When resolved, these spectra were found to be composed of six emission bands. The peak position changed

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with the change in excitation energy (Table-III). Higher energy excitation gives more intense emission bands. It has been argued by Rodrigues et al. [44] that if the size distribution of the crystallites is broad, then large number of particles are excited thereby giving broad spectrum with no distinct features and if the distribution is extremely narrow, the emission peak always occurs at same energy. Only in the case of intermediate crystallite sizes, appropriate excitation energy results in a broad spectrum that contains more than one peak whose relative intensities and peak position vary with energy (h). Our observation of six discrete peaks confirms the above idea of intermediate crystallite size, which has also been confirmed from X-ray study. The luminescence efficiency of the system decreases with the increase in Al3+ ion (size decreases) because more non-radiative channels occur at upper levels of the quantum box. Our observation supports the intrinsic mechanism proposed by Benisty et al. [45] for the poor luminescence properties of quantum box systems. The reduction in PL intensities with increasing Al concentration can be attributed to the decrease of the crystallite size in the deposited films. The reduction in crystallinity favours the non-radiative recombination processes by introducing more recombination centres. Table III A. Deconvolution of Photoluminescence spectra of ZnO and ZnO:Al by curvefitting method. All the emission peaks were assumed to be Voigtian. ex=200 nm

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Peak No.

1 2 3 4 5 6

Undoped Centre (eV) 3.14 2.98 2.79 2.57 2.37 2.18

Rel. Int. (%) 100 123 77 53 38 14

2.5 at wt % Al doped Centre Rel. Int. (eV) (%) 3.12 100 2.97 124 2.80 97 2.58 64 2.37 43 2.19 14

5.0 at wt % Al doped Centre Rel. Int. (eV) (%) 3.13 100 2.97 122 2.79 84 2.58 53 2.37 35 2.20 11

7.5 at wt % Al doped Centre Rel. Int. (eV) (%) 3.16 100 2.98 122 2.79 63 2.58 44 2.37 27 2.20 8

Table III B. Deconvolution of Photoluminescence spectra of ZnO and ZnO:Al by curve fitting method. All the emission peaks were assumed to be Voigtian. ex=233nm. Peak No.

1 2 3 4 5 6

Undoped Centre (eV) 3.10 3.00 2.82 2.58 2.39 2.23

Rel. Int. (%) 100 101 56 37 19 23

2.5 at wt % Al doped Centre Rel. Int. (eV) (%) 3.10 100 2.98 131 2.81 56 2.58 42 2.39 21 2.22 12

5.0 at wt % Al doped Centre Rel. Int. (eV) (%) 3.09 100 2.99 111 2.83 49 2.58 33 2.39 14 2.24 11

7.5 at wt % Al doped Centre Rel. Int. (eV) (%) 3.09 100 2.99 97 2.81 30 2.57 24 2.39 10 2.24 10

Raman spectra recorded for powder and thin films at room temperature have been depicted in Figure 7. For ZnO, a total of 12 phonon modes (one longitudinal acoustic, two Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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transverse acoustic, three longitudinal optical and six transverse optical) exist out of which six modes are Raman active only. Ashkenov et al. [46] reported that the six Raman-active modes E12 (low), A1(TO), E1 (TO), E22 (high), A1(LO) and E1 (LO) correspond to the bands at 102, 379, 410, 439, 574 and 591cm-1 respectively. However, even in the pure ZnO powder all these Raman active modes are not observable. The peak at 582cm-1 appears midway between the A1 (LO) and E1 (LO) modes. Although several strong and weak bands are observed for undoped thin film, the bands change their position except the band at 437cm-1(E2 high) when doped with 5.0 at wt % of Al. In the low frequency range (LFR), the main difference between the powder and thin film samples is the appearance of the peak at 113cm-1 (E1 low mode) in the latter case.There appears also a peak at 332cm-1 in this range for powder as well as thin films which does not represent any of the Raman active modes. It is ascribed to multiple phonon processes. In case of undoped ZnO thin film, the intensity of the A1 (TO) and E2 (high) mode reduced to a great extent, which reduced further in case of 5.0 at wt % Al doped thin film. In case of ZnO thin film, the A1(TO) and E1(TO) merge to result a broad shoulder with peak centre at 404cm-1. In the high frequency range (HFR), the peak at 582cm-1 is redshifted to 567cm-1 along with evolution of two new peaks at 487cm-1 and 542cm-1 in ZnO thin films compared to powder sample. When Al is doped in the film, the peak at 567cm-1 is found to be blue-shifted, whereas that at 487cm-1 red-shifted. For transition metal oxides, the Raman spectra reported in literature vary from one sample to another, depending upon the size, stress and morphology of the crystallites. It acquires a special significance in the light of nano crystalline thin films of ZnO. The lattice strain causes the surface tension to increase when the crystal size is reduced in nanocrystalline thin films compared with bulk materials. The high-frequency Raman shift has usually been suggested to be activated by surface disorder [46] and explained in terms of surface stress [47, 48] or phonon quantum confinement [49, 50], as well as surface chemical passivation. Hwang et al. [51] indicated for CdSe nanodots embedded in different glass matrices that the effect of lattice contraction must be considered to explain the observed differences in the red-shift or blue-shift of phonon energies. To obtain the phonon frequency as a function of the dot radius (Kj) with contribution of lattice contraction, it was assumed that

 K j    L   D K j   C K j ) 

(1)

where L is the LO phonon frequency of the bulk, D(Kj) is the peak shift due to phonon dispersion and C(Kj) is the peak shift due to lattice contraction [52]. The phonon frequency change at a temperature is given by

 ( K j )

L



 D ( K j )

L



C ( K j )

L

 A  BK j 2

where

    2  1 L np     C b d 0 2 A  3 (   )(T  Tg ) and B    2   L     ,

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Figure 7. Non-resonant Raman spectra of bulk 99.9999% pure ZnO powder (a), as-grown undoped ZnO film (b) and 5.0 at wt % Al doped ZnO film (c) by spray pyrolysis method.

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 is the Grüneisen parameter, ‟ and  are the linear thermal expansion coefficients of the substrate (glass) and the nanodot respectively, T and Tg are the testing and the heat-treatment temperatures respectively, L describes the phonon dispersion, np is the nonzero npth root of the equation of tan( np )   np , c is the compressibility the bulk and b is the parameter describing the size-dependent surface tension of the crystal. The value of B in Eqn-2 is given by the difference of the phonon negative dispersion and the size-dependent surface tension. Thus, a positive value of B indicates that the phonon negative dispersion exceeds the size-dependent surface tension and consequently causes the red-shift of phonon frequency. On the contrary, if the size-dependent surface tension is stronger than the phonon negative dispersion, blue-shift of phonon frequency occurs. In case of balance of the two effects, i.e., B=0, the size dependence disappears. At the lower end of the size limit, the (Kj) ∞ diverges in a Kj-2 way. Therefore, this model may fail to reproduce the Raman frequency shifts satisfactorily near the lower end of the size limit. However, the crystallite size in our samples (≈9-15nm) does not fall at the lower limit. So this model can be effectively applied to explain the observed phonon shift. The blue-shift of acoustic phonon energies is suggested to be the consequence of the compressive stress whereas the red-shift caused by phonon confinement can be attributed to negative dispersion. Hence, the confinement of phonons with negative dispersion within nanocrystals is assumed to be dominating the effect of stress with reduction in the crystallite size in the undoped thin films. With Al doping, the effect of compressive stress increases causing the blue-shift in the acoustic phonon energy. The phonon confinement model [49] attributes the red-shift of the asymmetric Raman line to relaxation of the q-vector selection rule for the excitation of the Raman active phonons due to their localization. The relaxation of the momentum conservation rule arises from the finite crystallite size and the diameter distribution of the nanosolid in the films. When the size is decreased, the rule of momentum conservation will be relaxed and the Raman active modes will not be limited at the center of the Brillouin zone [47]. The large surface-to-volume ratio of a nanodot strongly affects the optical properties mainly due to introduction of surface polarization and surface states [52]. Using a phenomenological Gaussian envelope function of phonon amplitudes, Tanaka et al. [55] showed that the size dependence originated from the relaxation of the q=0 selection rule based on the phonon confinement argument with negative phonon dispersion. The weight of Zn atom (65.39amu) is about four times higher than that of Oxygen atom (15.999amu). Hence in the wurtzite structure of ZnO the vibrations from Oxygen atom are supposed to be involved, giving a comparatively intense E2 (high) mode in each spectrum. The LFR (low frequency Raman) peaks in the range 100-200cm-1 are assigned to lattice vibrations and peaks in the range 200-300cm-1 to deformation modes. With the incorporation of Al3+ ion, the deformation modes disappeared. The band at 380cm-1, attributed to A1 (TO) mode reflects the strength of polar lattice bond. The decrease in 380cm-1 band, A1 (TO) mode supports the fact that polar character of the wurtzite structure of ZnO is reduced when Al3+ ions are introduced into this lattice. Our observation of 436cm-1 band was due to non-polar phonon mode (E22 high) in ZnO:Al thin films and the intensity of this non-polar interaction was also reduced with the introduction of Al3+ into ZnO. This may be the result of size induced stress. A small red-shift of the longitudinal-optical-phonon mode frequencies (bands in the range 560-600cm-1) of the

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undoped ZnO film with respect to the bulk material was tentatively assigned to the existence of vacancy point defects within the films [56]. With Al incorporation this band is blue-shifted again indicating the reduction of point defects, thereby improving the crystal quality of the film.

Figure 8. PL Intensity of ZnO thin film on Si n-type substrate deposited by PVD technique. Curve-a is excitation spectra at emission wavelength 416nm and curves-b, c & d are emission spectra at 203nm, 234nm and 275nm excitation respectively.

The PL spectrum of ZnO has been always analyzed in three major regions of uv-visible spectrum: a UV band edge or near-band edge emission peak around 380nm, a green emission around 510nm, and a red emission around 650nm. The optical properties of ZnO, studied using photoluminescence reflect the intrinsic direct band gap, a strongly-bound exciton state and gap states due to point defects [ 57, 58, 59,60]. A strong room temperature near band edge UV photoluminescence peak at ~3.2eV is attributed to exciton state, as the exciton binding energy is of the order of 60meV [61]. Visible emission is ascribed to defect states. A blue–green emission, centered at around 500nm in wavelength, has been explained on the basis of transitions involving self-activated centers formed by a doubly ionized zinc vacancy and an ionized interstitial Zn+ [57], oxygen vacancies [62, 63, 64, 65], donor–acceptor pair recombination involving an impurity acceptor [66], and/or interstitial O [67, 68, 69, 70]. In the present case of ZnO thin film, three excitation bands at 203nm, 234nm and 275nm were found which contribute to the emission line 416nm [Figure 8]. Accordingly, the emission spectra were obtained by exciting the sample with these excitation energies. When the sample was excited with 200nm light, the emission bands were observed at 385nm, 419nm, 481nm and 526nm. In the PL emission spectrum with 200nm excitation, band edge

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emission is dominated by emissions in the violet region. When excitation energy decreases the bands show blue shift. Further, it is interesting to observe that Cu doping in ZnO, leads all the peaks to be blue shifted [Figure 9].

Figure 9. ZnO thin film doped with Cu (5.0 at wt %) on Sip-type substrate deposited by PVD technique. Curve-a represents the excitation spectrum at emission wavelength 416nm, where as curvesb, c and d represent the emission spectra at 200nm, 235nm and 277nm excitation respectively.

In ZnO, since majority donors are oxygen vacancies (Vo) and zinc interstitials (Zni), the luminescence process in the blue-green region is dominated by defect related recombination processes. The green emission band (2.39eV) originates from the transition by the defect level of antisite oxide (OZn). But, antisite oxide (OZn) states are formed in O-rich conditions as they have relatively low formation energy [71] than interstitial oxides (Oi) and zinc vacancies (VZn). Vanheusden et al. [37,72] reported this emission to be arising out of the recombination of a photogenerated hole with a singly charged ionized state of oxygen vacancy. Egelhaaf et al. [73] reported that these luminescence bands arise from radiative transitions between oxygen vacancies (shallow donors) and Zn vacancies (deep acceptors). With the decrease of oxygen vacancies, the zinc vacancies may lead to pronounced emissions in the green-yellow region (2.16eV). In our study, the thin films were grown in vacuum, where the probability of formation of antisite oxygen and oxygen interstitial states is minimized. Rather the formation of oxygen vacancies is quite high. As discussed earlier, the emission band at 526nm (≈2.36eV) in undoped ZnO, may result from the transitions of electrons from the VoZni to the valence band. The emission band at 481nm (≈2.57eV) may be due to the transition between a shallow donor

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like Zni and a deep acceptor like VZn, where as the emission at 418nm (≈2.97eV) may result due to transition between the conduction band and the zinc vacancy (VZn). The increase in intensity of the bands 382nm (3.25eV) and the violet band (2.97eV) compared to other bands in the PL spectrum of 5.0 at wt % of Cu doped film can lead to the conclusion that Cu doping favours the interband transitions (conduction band to valence band) in PVD grown ZnO films. It also increases the concentration of zinc vacancies. However, the decrease in the PL intensity of the 5.0 at wt % Cu doped ZnO film compared to undoped ZnO film may be due to concentration quenching of luminescence in PVD grown CZO samples as argued by several workers. The possibility of introduction of non-radiative recombination channels with Cu doping may be another reason for the reduction in the PL intensity. A fundamental understanding of the thermal as well as electrical properties in terms of low and high-field carrier transports requires precise knowledge of the vibration modes of the single crystal. In the wurtzite ZnO case, the number of atoms per unit cell is 4 and there are total of 12 phonon modes, namely, one longitudinal-acoustic (LA), two transverse-acoustic (TA), three longitudinal-optical (LO), and six transverse-optical (TO) branches. The A1 and E1 branches are both Raman and infrared active, the two non-polar E2 branches are Raman active only, and the B1 branches are inactive. The A1 and E1 modes each split into LO and TO components with different frequencies due to the macroscopic electric fields associated with the LO phonons. Because the electrostatic forces dominate the anisotropy in the short-range forces, the TO-LO splitting is larger than the A1-E1 splitting. For the lattice vibrations with A1 and E1 symmetries, the atoms move parallel and perpendicular to the c-axis, respectively. The low-frequency E2 mode is associated with the vibration of the heavy Zn sublattice, while the high-frequency E2 mode involves only the oxygen atoms. In the case of highly oriented ZnO films, if the incident light is exactly normal to the surface, only A1(LO) and E2 modes are observed, and the other modes are forbidden according to the Raman selection rules. The complete set of phonon mode frequencies was measured using Raman spectroscopy for both bulk and thin films [Figure 10 & 11]. The peaks were assigned according to Ashkenov et al. [74, 75, 76] and given in Table-IV. All the peaks found in bulk powder were found to be red-shifted in the thin film of Zn. In addition, some modes were found in the low frequency range of the Raman spectrum of the Zn film, which correspond to presence of defects as described earlier. With Cu doping, the intensity of all these peaks were enhanced with blue shift in their positions. The presence of E2 (high mode), originating from the vibrations of oxygen atom indicates adsorbtion of oxygen from atmosphere. When annealed the peaks in the undoped ZnO film were shifted towards low wavenumbers compared to those in the undoped Zn film (as-deposited) with a reduction in the intensity. The intensity was further reduced with Cu doping. Also the peaks were blue shifted with Cu doping. The size-dependent low frequency Raman (LFR) modes (acoustic modes) for different nanosolids are governed by following equation. The LFR frequency depends linearly on the inverse of R.

 ( R)   ( R)   ()  

A' , R

where R=D/2, D being the crystallite size.

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Figure 10. Raman spectra of as-deposited Zn film by PVD technique. (a) & (b) stand for undoped & 5.0 at wt % Cu doped respectively.

When the R approaches infinity, the LFR peaks disappear, which implies that the LFR modes and their blue-shifts originate from vibration of the individual nanoparticle as a whole and provide information about the strength of the interparticle interaction. The mechanical coupling between the nanoclusters can be assumed to play a key role in the LFR [77] modes. The mechanism for LFR mode enhancement is analogous to the case of surface-plasma enhanced Raman scattering from molecules adsorbed on rough surfaces. Hence the reduction in the intensity of the Raman peaks in the spectra of the annealed thin films as observed in our case can be related to the improvement in the crystallite size, inter-particle interaction and the smoothness of the surface.

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Figure 11. Raman spectra of ZnO films annealed in air at 3500C for 2 hours. (a) & (b) stand for undoped and 5.0 at.wt % Cu doped respectively.

To realize the device applications, an important issue is to prepare / deposit both highquality p-type or/and n-type ZnO films. However, like most wide band-gap semiconductors, ZnO has the “asymmetric doping” limitation [78-82], i.e., it can be an easily doped to give high-quality n-type [79, 83] material film, but it is difficult to dope for p-type film. The difficulty of making p-type ZnO has been attributed to low energy native point defects such as O vacancies (Vo) and Zn interstitials (Zni) [84, 85], or to unintentional, but ever-present, hydrogen [86]. The band gap of ZnO can be tuned via divalent substitution on the cation site. For example- Cd substitution leads to a reduction in the band gap to ~3.0eV, where as Mg doping on the Zn site can increase the same up to ~4.0eV, still maintaining the wurtzite structure [87].

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Table IV. Assignment of Raman Peaks in Zn and ZnO films grown by PVD Sample description

Phonon modes observed in cm-1

ZnO (99.9999%) pure powder

147, 209, 332, 378, 407, 437, 582

Asdeposited

115, 172, 206, 269, 323, 362, 433, 471, 509, 552 111, 170, 206, 267, 320, 362,

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Annealed in air at 3500C for 2hrs.

Zn film Zn:Cu (5.0 at wt %) ZnO ZnO:Cu (5.0 at wt %)

Phonon modes assigned according to literature [74, 75 & 76] 106-115: E2 low 323-334: multiple phonon processes 362-378: A1(TO] 407: E1(TO) 423-439: E2 high 540-582: A1LO & E1LO

106, 160, 197, 257, 313, 423, 451, 540 115, 167, 206, 269, 323, 361, 431

The optical properties of ZnO studied using photoluminescence, photoconductivity and absorption reflect the intrinsic band gap, a strongly bound exciton state and gap states due to point defects. In addition, visible emission is also observed due to defect states. The blue green emission has been explained within the context of transitions involving self activated centers formed by a doubly ionized zinc vacancy and an ionized interstitial Zn+ [88], oxygen vacancies [89, 90], donor acceptor pair recombination involving impurity acceptor [91] and/or interstitial O [92]. In the foregoing discussions Cu doped ZnO (CZO) thin films deposited over glass substrate by spray pyrolysis and physical vapor deposition method will be presented. It is well known that Cu can exist in either +1 or +2 valence state depending on its chemical environment. This is due to its affinity to capture electrons. There have been very few reports in the literature regarding Cu-doping in ZnO system. Mollwo et al. [57] reported the existence of an acceptor level located 190meV below the conduction band edge. In ZnO, p-type dopants introduce deep acceptor levels. So the possibility of Cu as a p-type dopant can not be avoided. But reports on Cu-doping relating to polycrystalline ZnO are silent about this aspect. If Cu is to act as an acceptor when substituted at a Zn site, it has to take a valency of +1, which according to Fons et al. [94] depends on the Cu concentration and growth temperature. The role of Cu in the ZnO is different from other dopants such as Mn, Co and Bi, Pr, etc. [95– 100]. There are some reports on the Cu doping in the ZnO, but the effects of Cu on the grain and grain boundary of ZnO still remain unclear. Thus it is important to understand the behaviour of Cu in ZnO nanocrystalline thin films by confirming various deposition process. Since the crystal structure of a system plays an important role in governing its structural, electrical and optical properties, it is essential to look into this aspect by x-ray diffraction, a universal tool for structure determination. Thus x-ray diffraction pattern obtained from a Philips X‟Pert MPD diffractometer for thin films of ZnO and ZnO:Cu has been shown in Figure 12. Undoped ZnO thin film shows seven reflections with enhanced intensities compared to the ZnO:Cu thin film. The humpy nature of the diffraction patterns in the region of 2θ ranging from 160 - 300 shows the cryptocrystalline nature of the spray pyrolysis film.

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The d-values obtained from x-ray diffraction were matched with the standard pattern values of ICDD-PDF-card no.1436 and it was found to have hexagonal wurtzite structure.

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Figure 12. X-ray diffraction spectra of undoped (curve-a) and 1.0, 2.0, 3.0, 4.0 & 5.0 at wt% Cu doped (curve-b, c, d, e & f respectively) ZnO thin film prepared by spray pyrolysis.

Doping with 1.0 at wt % of copper reduces the intensities of all the reflections. However, with higher concentrations of doping of Cu, the intensities increase to some extent. It can be seen that incorporation of Cu does not show any new peak in the pattern, signifying the absence of any CuO or Cu2O phase. The strain caused by the incorporation of Cu into the lattice increases with the increase in dopant concentration. This is further reflected in their dvalues which shifted towards higher side. Reduction of crystallite size with the increase in copper concentration in the film leads to reduction in crystalline order thereby showing the short range order to prevail in these films. The density was calculated from the diffraction data with hexagonal structure and was found to decrease with the increase in dopant concentration. The decrease in density may be due to unequal increase in mass and volume of the deposited mass which can be seen from their lattice constants values (a, c) in the Table-V. In a recent study on thin films of zinc oxide prepared by reactive sputtering, decrease in X-ray density has been attributed the induced stress in the film [101]. The thickness of the films was determined from the x-ray fluorescence method (ref). The thickness determined from XRF study agrees well with the thickness obtained by weight difference method. In spray dried films of ZnO:Cu the thickness of the films increase with the increase in copper concentration when other conditions like distance of the substrate, number of spraying cycles and the time of deposition etc. are kept constant (Table-V). The densities calculated from x-ray diffraction data indicate a decrease in the density with the increase in Cu concentration. Thus the volume and consequently the thickness increases as in a twodimensional thin film system area of deposition remains the same.

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Table V. Variation of density, crystallite size and band gap of ZnO film with Cu doping Conc. of Cu

Thickness (±5nm)

Lattice constants in nm

Lattice

Crystl.

Band gap

(gm/cm )

strain

size

(eV)

± 0.002

(η)

(nm)

± 0.02

3

a

c

c/a

(at wt %)

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Density

0

152

0.3255

0.5218

1.603

5.732

3.25

15

3.20

1.0

158

0.3252

0.5198

1.598

5.677

3.34

14

3.05

2.0

175

0.3251

0.5211

1.603

5.665

3.79

12

3.13

3.0

202

0.3248

0.5228

1.610

5.655

3.84

12

3.16

4.0

213

0.3245

0.5215

1.607

5.659

3.88

12

3.11

5.0

280

0.3252

0.5204

1.600

5.665

4.04

12

3.11

During deposition, the local temperature of the substrate and oxygen concentration in the locality influence quality of thin films. Since copper has strong affinity towards oxygen compared to zinc, increasing the copper concentration in the solution used for spray drying influences the volatility and stoichiometry of the vapour phase, before the product is deposited on glass substrate. In the present study air has been used as the carrier gas during the whole process of deposition and hence could affect the growth kinetics of the films. Such type of results has been reported for thermal and other types of deposition processes [102]. Optical transmittance spectra for all the films were recorded in the wavelength range of 350 to 800nm and then the absorption co-efficients were calculated using Lamberts equation. These have been shown in Figure 13. Compared to the undoped ZnO thin film, the doped films show higher absorbance in the visible region when doped with 1.0-4.0 at wt % of Cu. It decreases with further increase in the copper concentration above 4.0 at wt % [Figure 13 (ii)]. The band gap was calculated for each composition and is shown verses dopant concentration in Figure 13 (iii). The undoped thin film was found to have a direct band gap of 3.20(±0.02)eV. The band gap decreased linearly when doped with various concentrations (at wt %) of Cu. As one looks into Table-VI and Table-VII, it is seen that the band edge emission i.e. 3.14eV band shifts towards lower energy when the concentration of Cu goes up in ZnO film. The band-shift is of the order of 20-30meV. This amount of red-shift occurred in case of 2.98eV band also. But, the position of the green bands (2.57eV, 2.37eV and 2.18eV) remained unchanged irrespective of the dopant concentration and also the excitation energy. The relative intensity (RI) of each emission band was calculated by taking ratio of its intensity to that of the exciton band (3.14eV) in the emission spectrum for a particular sample. The RI of all the peaks except 2.98eV went on increasing with the increase in dopant concentration. The RI of the band at 2.98eV decreased for 1.0 at wt % Cu doping, but then increased for higher doping concentrations. This behaviour was not observed for emission bands obtained at lower excitation energy i.e. 5.32eV (233nm). The RI of the 3.00eV band showed a decrease for 1.0 at wt % Cu doping, whereas rest of the bands the same. No significant change in the RI of all peaks beyond were observed for films containing Cu concentration above 3.0 at wt %.

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Figure 13. Transmittance (%) (i) of Cu doped ZnO thin film prepared by spray pyrolysis. Curves-a, b, c, d, e & f stand for undoped, 1.0, 2.0, 3.0, 4.0 & 5.0 at wt % Cu doped respectively. The absorption coefficients (ii) and the band gap (iii) calculated from the transmittance spectra.

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Figure 14. Splitting of Photoluminescence Emission spectrum of undoped (i), 1.0 at wt % (ii), 2.0 at wt % (iii), 3.0 at wt % (iv), 4.0 at wt % (v) and 5.0 at wt % (vi) Cu doped spray pyrolytic ZnO thin film excited with 200nm light.

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Table VI. Deconvolution of Photoluminescence spectra of spray pyrolytic ZnO and ZnO:Cu by curvefitting method. All the emission peaks were assumed to be Voigtian. ex=200nm Peak No.

1 2 3 4 5 6

undoped Cent. (eV) 3.14 2.98 2.79 2.57 2.37 2.18

R. I. (%) 100 123 76 53 38 14

1.0 at wt % Cu doped Cent. R.I. (eV) (%) 3.12 100 2.97 53 2.79 113 2.57 72 2.39 48 2.25 34

2.0 at wt % Cu doped Cent. R. I. (eV) (%) 3.09 100 2.95 99 2.78 119 2.57 90 2.37 66 2.19 24

3.0 at wt % Cu doped Cent. R. I. (eV) (% ) 3.09 100 2.93 138 2.77 115 2.58 125 2.37 82 2.20 34

4.0 at wt % Cu doped Cent. R. I. (eV) (%) 3.06 100 2.91 118 2.74 120 2.57 123 2.37 109 2.18 34

5.0 at wt % Cu doped Cent. R.I. (eV) (%) 3.06 100 2.94 110 2.77 168 2.58 171 2.37 137 2.20 53

Table VII. Deconvolution of Photoluminescence spectra of spray pyrolytic ZnO and ZnO:Cu by curvefitting method. All the emission peaks were assumed to be Voigtian. ex=233nm Peak No.

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1 2 3 4 5 6

undoped Cent. (eV) 3.10 3.00 2.82 2.58 2.39 2.23

R. I. (%) 100 101 56 37 19 23

1.0 at wt % Cu doped Cent. R. I. (eV) (%) 3.06 100 2.95 69 2.74 70 2.57 62 2.41 36 2.22 49

2.0 at wt doped Cent. (eV) 3.05 2.92 2.76 2.58 2.39 2.21

% Cu R. I. (%) 100 82 46 46 33 12

3.0 at wt % Cu doped Cent. R. I. (eV) (% ) 3.05 100 2.94 107 2.77 67 2.58 60 2.39 45 2.21 20

4.0 at wt % Cu doped Cent. R. I. (eV) (%) 3.04 100 2.93 96 2.76 63 2.57 71 2.39 47 2.23 28

Figure 15. Curve fitted PL excitation spectrum of undoped ZnO thin film (i) and 5.0 at wt % Cu doped ZnO thin film (ii) prepared by spray pyrolysis method (em=416nm).

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Figure 16. Variation of FWHM (Full width at half maximum) of the emission peaks in the PL spectra with Cu concentration in the ZnO film [(1) 3.14eV, (2) 2.98eV, (3) 2.79eV, (4) 2.57eV, (5) 2.37eV and (6) 2.18eV].

The FWHM (full width at half maximum) of the 3.14eV band was observed to be maximum at 1.0 at wt % of copper concentration and then decreased till 3.0 at wt % of Cu concentration [Figure 16 (i) & (ii) line-1]. This was true for all the bands except the band at 2.98eV, which follows a reverse trend. When the dopant concentration increased beyond 3.0 at wt %, the bands did not follow any general trend. When the excitation energy was reduced to 5.32eV (233nm), all bands except the bands in the green region behaved in a similar way. Photoluminescence (PL) spectra of both undoped ZnO and ZnO:Cu films are really complex in nature. Although the excitation and emission spectra depict same nature of spectral features, the PL emission peak intensities are different for ZnO and CZO films. Most of the research articles on the luminescence in ZnO single crystal or nanopowders report two prominent bands centering around 380nm (UV visible) and 520nm (green region) [103]. The band around 380nm (3.25eV) has been attributed to radiative recombination of exciton. Since the band gap of bulk ZnO is 3.37eV, this emission band has been assigned to band edge emission. PL emission spectra of our samples at room temperature (≈270C), several bands have been observed (Table-VI & VII) and we assign 3.14eV emission band to exciton. The FWHM of 3.14eV emission observed increases with decrease in crystallite size and then saturates at higher doping limit (Figure 16). This shows that the electronic density of states in ZnO are dependent on crystallite size which in turn is dependent on impurity concentration in ZnO thin films. Such type of observations has been made by Dijken et al. [67]. It is well known that excitonic feature dominate the emission and absorption spectra of direct band gap solids and wide band gap semiconductors of II–VI compounds are no exception to this. In III-V compound semiconductors the role of two dimensional electron hole confinement enhances the binding energy and oscillator strength [68]. In multiple quantum well structures observed in GaAs/Ga, Al or As system gives rise to exciton related absorption even at room temperature. In a similar situation for polar II-VI compound semi conductor, stronger coupling with longitudinal optical phonons (LO), give rise to excitonic feature in room temperature absorption and emission spectra. The flatness of the absorption spectra in the UV near visible range for ZnO thin films doped with copper confirms this

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(Figure 13). The life-time broadening [full width at half maximum] induced by LO Phonon has been found to be temperature dependent [106] and is given by

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(T )  inh  phT 

LO exp( LO K  T )  1

(4)

where Γinh is the temperature dependent term that denotes the inhomogeneous line width due to exciton-exciton, exciton-carrier interaction and the scattering by defects, impurities and size fluctuation. The second term ΓphT is due to acoustic phonon scattering. Third term is due to the LO phonon scattering. ΓLO is the exciton–LO phonon coupling strength and ħωLO is the LO phonon energy. At high temperatures, the LO phonon Fröhlich scattering dominates and at low temperature the scattering is mainly ruled by acoustic phonons. In case of ZnO quantum wells, it has been observed that ΓLO is larger compared to the ΓLO of the bulk. For IIVI quantum wells [103], this large value of ΓLO suggests the exciton-LO phonon Fröhlich interaction to be dominant at room temperature and high temperature absorption and emission processes. Thus the variation in FWHM in ZnO:Cu doped films can be explained on the basis of inhomogeneous exciton–exciton broadening due to size variation and exciton–exciton LO phonon Fröhlich interaction. The other main emission band in the green region appears in the range 500-530nm (2.35– 2.50eV) and has been studied by various workers for both ZnO and ZnS doped with copper. There has been divergent view about this band as some researchers attributed this to Cuimpurity ion [107] whereas some others attribute this to oxygen vacancies [108, 109]. Van Dijken et al. [67] have studied the exciton luminescence in nanocrystalline ZnO particles and have attributed the green emission band to be due to transition of photo-generated electron from the conduction band to deeply trapped hole. However, Liu et al. [110] have correlated this green emission to interstitial Zn and oxygen ions. We have observed a comparatively low intensity green emission band at 2.36eV in pure ZnO films as well as Cu doped ZnO films. Van de walle [111] have calculated the formation energies and electronic structure of ZnO using the first principle, plane wave pseudo-potential technique together with super-cell approach. In this theory concentration of a defect in crystal depends upon its formation energy (Ef) in the following form

C  N site exp  E f K BT 

(5)

where, Nsite is the concentration of the defect sites in the crystal. Low formation energy implies a high equilibrium concentration of the defect, whereas high formation energy means that defects are unlikely to form. The relative intensity for the band at 2.36eV and 2.57eV increases with increase in dopant concentration. This 2.57eV band can be attributed to Cu ion as the increase in normalized intensity is substantial up to 3.0 at wt % of copper and after which luminescence is quenched. The band around 2.37eV may be attributed to deep traps created by oxygen vacancies as argued by others. This is supported by the fact that with the increase in dopant concentration lattice distortion causes more no. of oxygen vacancies to occupy deep trap states. In a pure sample of ZnO, only oxygen vacancies contribute to these two bands,

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whereas in the doped sample both copper and oxygen may take part in the process of luminescence. In a recent study on the origin of „p‟ type conductivity in ZnO doped with K+, Park et al.(109) have observed that due to large atomic size of (K) potassium, formation of Vo around K is possible and positively charged complex (KZn+Vo) defect centers can be formed under oxygen rich conditions [112].Recently, Merz et.al.(114) have shown that local variations in surface potential and band bending in surface passivated ZnO influences the surface and optical properties of the thin films.In ZnO:Cu film, deposited in ambient pressure at 2700C, Cu atoms replace Zn atoms in the substitutional site and oxygen vacancies near Cu atoms were also formed. Further, the increase in lattice strain (as observed from XRD data) around the impurity atom leads to the formation of other compensating defects such as vacancies around the impurity centre. So as the concentration of copper increases more oxygen vacancies are created around Cu atom thus influencing the green emission. In the beginning, when no impurities are present in ZnO, oxygen vacancies govern the emission process, but addition of impurities like Cu, leads to vacancy formation near Cu atoms as well as presence of Cu at interstitial and surface sites, can lead to their participation in the emission process is making it a competitive one.The gradual reduction in the intensity of the visible emission with Cu concentration can be attributed to the non-radiative recombination of photogenerated electron in the conduction band getting trapped at the surface and then recombining non-radiatively with the surface trapped hole. Possibility of concentration quenching of luminescence can not be ruled out also. In ZnO with wurtzite crystal structure there are two possible interstitial sites i.e. one is tetrahedrally coordinated and another is octahedrally coordinated. It has been concluded from the study of Van der Walle [111] and Kohan [113] that two most common defects in ZnO, zinc vacancies and oxygen vacancies can occur depending on the partial pressure of zinc or oxygen. In particular oxygen vacancies (Vo) have lower formation energy than zinc interstitial and hence should be more abundant in zinc rich conditions. In O-rich conditions Zn vacancies should dominate. From the electronic structure data of Kohan [113], Zn vacancy is expected to have doubly negative charge in n-type ZnO where the formation is more favourable. The transition level between -1 and -2 charge states of VZn occurs at 0.8eV above valence band. Thus we may expect transition with emission energy ≈2.40eV if we take the band gap of our ZnO thin film to be 3.20eV (as observed from absorption measurement). Accordingly the broad green luminescence band observed for spray pyrolysis deposited films can be assigned to VZn. Our observation of this band at 2.37eV and its relative intensity decreases with the decrease in excitation energy. The broad green emission band when decomposed gives another band around 2.58eV and we attribute this to same VZn as the behaviour of this band is similar to 2.37eV band (Figure 15). Many workers have attributed this green band emission to be due to Vo centers. However, Van der walle [111] has pointed out that even in Zn rich conditions where Vo, VZni and Zno defects could be formed. Actually Vo defects are not formed as the oxygen vacancy is deep donor and lies at an energy of 2.7eV above valence band. Thus it is clear that oxygen vacancies cannot supply electrons to the conduction band and hence cannot be a source of n-type conductivity. Raman spectra recorded for powder, undoped and doped (with 5.0 at wt % Cu) thin films at room temperature have been depicted in Figure 17. It can be seen that the bands differ in their position and intensity when doped with 5.0 at wt % of Cu. The six Raman-active modes E12 (low), A1 (TO), E1 (TO), E22 (high), A1 (LO) and E1 (LO) are reported by Ashkenov et al.

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[38] at 102, 379, 410, 439, 574 and 591cm-1 respectively. However, all these bands are not observed even in pure ZnO powder [Figure 17.a]. When compared with Raman spectrum of the pure zinc oxide powder, the band at 332cm1 observed to shift to 320cm-1 in ZnO film. The band at 378cm-1 merges with the band at 407cm-1 to give a broad peak. A new peak at 284cm-1 is found in thin film. The peak at 582cm-1 splits into two peaks (542cm-1 and 567cm-1). With Cu doping, all the peaks are blue shifted except the peak at 437cm-1 which is red shifted. The peak at 332cm-1 does not represent any of the Raman active modes described for ZnO system. It is assigned to multiple phonon processes. The low frequency modes arise from the vibration of the entire nano solid interacting with the host matrix. For a free standing nano solid, these modes correspond to inter cluster interaction. The optical mode is the relative motion of the individual atoms in a complex unit cell that contains more than one atom.The 432cm-1 band represents the nonpolar phonon mode (E22 high) and the reduction in its intensity with the introduction of Cu into ZnO, may be due to size induced stress. A blue-shift of the longitudinal-optical-phonon mode frequencies (bands in the range 560-600cm-1) for Cu doped ZnO film can be tentatively assigned to the formation of more defect centres. An increase in the intensity of the peak at 572cm-1 can be ascribed to the increased concentration of vacancies as described in the earlier section. The evolution of the low frequency modes are related to the decrease in size and increase in the roughness of the surface due to surface passivation. As reported by earlier studies [39], the acoustic modes (AM) could be related to defect induced modes. When Cu concentration is very low, moderate quantity of Cu atoms could be considered to exist as interstitials that shared the oxygen with Zn atoms and hence decrease the defects of O vacancies (Vo) or O interstitials (Oi), but when Cu concentration is as high as above solubility limit i.e. 5.0 at. wt. %, excess Cu atoms can energetically favor to for metallic copper clusters or Cu2O and hence induce more defects in the ZnO lattice. This can lead to the enhancement of the acoustic modes of ZnO. A possible physical mechanism of the AMs is that the impurity centers break the translational symmetry of the crystal, thereby relaxing the conservation of wave vector and hence, can lead to scattering by phonons in the host materials that have wave vectors far from the zone center [40]. The behaviour of powders synthesized by sonication process is quite different from the thin films deposited by PVD or spray pyrolysis. The excitation and the emission spectra have been given in Tables.IX,X&XI. Nanoparticles of noble metals show interesting properties when the shape changes from sphere to rod or cuboids. The relationship between shape and size is important when the size is reduced and surface properties become prominent. In this respect recent studies on gold nano particles have been reported Pol et.al [115]. Nano silver particles in solution and composites have also been studied by Zao etal(116) and Li etal [117] but when it is doped in wide band semi-conductor like ZnO, the detailed structural and optical properties are not available. X-ray diffraction is a standard method to see the effect of concentration on the structure of the semi-conductor. In this investigation, it was observed that irrespective of method of synthesis such as continuous or pulsed mode ultrasonication, the structure remains the same, i.e. Wurtzite. But the crystallite size increases to 55 nm and 47 nm in ZnO: Ag powder compared to pure ZnO of continuous and pulsed mode powder respectively. In an earlier investigation by us [118] we have observed smaller crystallite size like 24 and 20 nm for CS and PS mode samples of ZnO. Thus, the increase in crystallite size can be attributed to incorporation of Ag ion into ZnO. It was interesting to observe that ZnO powder doped with Ag synthesized by CS mode the amount of Ag to be 0.179 wt%, where

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the amount of Ag present in pulsed mode powder was, i.e. 0.385 wt% although the starting concentration of Ag for both (CS and PS mode) solution remain the same i.e. 0.84 wt%.

Figure 17. Non-resonant Raman spectra of bulk ZnO powder (a), as-grown undoped ZnO film (b) and 5.0 at wt % Cu doped ZnO film prepared by spray pyrolysis.

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B. S. Acharya Table VIII. PL emission and excitation spectra of Cu doped ZnO prepared by sonication

ZnO-Cu (continuous) ZnO-Cu (pulsed) ZnO-Cu (continuous) ZnO-Cu (pulsed) ZnO-Cu (continuous) ZnO-Cu (pulsed)

Emission spectra in eV (nm) at ex 293 nm 3.11 --2.95 2.77 2.64 2.54 (398) (420) (446) (469) (486) 3.13 3.0 2.92 2.79 2.65 2.56 (395) (412) (424) (444) (466) (483) Emission spectra in eV (nm) at ex 354 nm 2.37 3.15 2.99 2.78 2.62 (352) (393) (413) (444) ---(471)

2.43 (509)

----

3.14 2.99 2.85 2.69 ---2.28 (395) (414) (433) (459) (541) Excitation spectra in eV (nm) at emission wavelength 560nm -2.71 -3.003 3.059 3.125 (457) (412) (405) (396) 2.65 2.75 2.81 3.00 -3.14 (467) (450) (441) (413) (394)

2.21 (560)

2.04 (605)

---

----

3.163 (392) 3.24 (382)

Table IX. Excitation and Emission bands observed after deconvolution of PL spectra in ZnO:Ag powders prepared by sonication Sample

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ZnOAg(continuous) ZnO-Ag(pulsed)

Emission spectra in eV (nm) at ex 293 nm 2.39 (517) ----

2.56 (483) 2.56 (483)

2.65 (466) 2.65 (468)

2.74 (452) 2.79 (444)

2.84 (436) ---

2.97 (416) 2.93 (400)

3.13 (395) ---

Emission spectra in eV (nm) at ex 354 nm ZnOAg(continuous)

2.09 (590)

2.29 (541)

2.67 (464)

2.99 (413)

3.17 (391)

---

---

ZnO-Ag(pulsed)

2.03 (608)

2.23 (554)

2.73 (452)

3.02 (409)

3.15 (393)

----

----

(3.58) 346 3.54 (350)

(3.89) 318 3.92 (315)

Excitation spectra in eV (nm) at emission wavelength 560nm ZnOAg(continuous) ZnO-Ag(pulsed)

(2.69) 460 2.64 (468)

--2.84 (435)

(3.05) 406 3.01 (411)

(3.21) 385 3.16 (392)

(3.39) 364 3.36 (368)

Thus, pulsed mode sonication is an effective way of incorporating more Ag ions into ZnO system. The size reduction due to pulsed sonication can be explained in the line of argument putforth by Suslik et.al.[119] In the sonochemical reaction process there are three regions [119]

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(i)

gas phase within the collapsing cavity where elevated temperature and pressure are produced. (ii) thin liquid layer surrounding the collapsing cavity i.e. interfacial region, where the temperature is lower than the cavity but still high enough for sonochemical reaction. (iii) Bulk solution at ambient temperature, where reactions take place between solute molecules and reducing radicals. Table X. Showing emission spectra of europium and gadolinium doped ZnO synthesized through sonication Spectral band positions obtained by deconvolution of emission spectra, assuming the peaks to be Gaussian. Excitation wavelength =300nm Sample Gadolinium 5% Gadolinium 10%

Crystallite size in nm 46.0 57.0

Emission spectra in eV and (nm) 3.7 (335) 3.59) (345)

3.41 (363) 3.09 (400)

3.02 (409) 2.96 (417)

2.83 (437) 2.83 (437)

2.64 (469) 2.65 (466)

2.43 (509) 2.60 (476)

2.38 (519)

Spectral band positions obtained by deconvolution of emission spectra, assuming the peaks to be Gaussian. Excitation wavelength =350nm

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Sample Gadolinium 5% Gadolinium 10%

Crystallite size in nm 46.0 57.0

Emission spectra in eV and (nm) 3.13 (395) 3.15 (393)

2.98 (416) 3.00 (412)

2.80 (442) 2.78 (445)

2.64 (468) 2.7 (458)

2.45 (506) 2.55 (485)

2.2 (563) 2.20 (561)

Table XI. Spectral band positions obtained by deconvolution of emission spectra, assuming the peaks to be Gaussian. Excitation wavelength =300nm Sample prepared by sonication Sample

Europium 10%

Crystallite size in nm 57

Emission spectra in eV and (nm) 3.75 (330)

3.51 (352)

3.107 (399)

2.93 (423)

2.74 (452)

2.58 (480)

2.41 (515)

Spectral band positions obtained by deconvolution of emission spectra, assuming the peaks to be Gaussian. Excitation wavelength =350nm Sample

Europium 10%

Crystallite size in nm 57

Emission spectra in eV and (nm) 3.15 (393)

3.01 (411)

2.76 (449)

2.59 (477)

2.43 (509)

2.25 (550)

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2.01 (614)

92

B. S. Acharya

Table XII. Emission spectra of ZnO deposited over ALN substrate by spray pyrolysis method

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Excitation wavelength (nm) 293 354

Emission wavelength in eV(nm) 2.58(480) 2.2(563)

2.65(466) 2.38(521)

2.81(450) 2.6(477)

2.74(441) 3.18(390)

2.93(421) ---

3.16(391) ---

Low vapour pressure of AgI complex eliminates the possibilities of sonochemical reduction taking place in gas phase region. The acceleration in the rate of sonochemical reduction of AgI in ammonical solution may be due to volatile nature of ammonical solution when the bath temperature is ≈70oC. The reducing radical thus produced in the cavitation accelerates the reduction of Ag I to Ag0 [120]. In the pulsed sonication process more number of Ag0 are dispersed into ZnO matrix and these metallic particles sit on the surface of ZnO nanostructure making the surface area more and showing plasmonic resonance peak in the UV-Vis spectra. The SEM and TEM results support this idea (Figures have not been shown here)(121).The intermittent stoppage of sonication brings down the temperature and the gas phase with in the collapsing cavity stops abruptly. Thus reduction of AgI in local high concentration region act as very effective protecting agent to keep the freshly prepared reduced particles in solution.[122] X-ray diffraction results show compression of „c‟ axis in pulsed mode sample, whereas the „a‟ axis remains the same. Thus the c/a ratio was found to be less in PS mode sample compared to continuous mode powder. It is really intriguing to observe compressed c and a axes compared to ZnO synthesized by sonication process. Since the ionic radius of Ag+ is 1.22 Å compared to Zn (0.72 Å) it is expected that silver occupies interstitial position which in turn can lead to expansion of lattice parameters. But this does not happen. So we presume that some of the Ag+ ion replace Zn++ ion and rests go to surface of the ZnO. It was further observed that the surface area of PS powder was twice than that of CS mode powder .The particle size analysis by laser PSA indicates two types of particles to be present in the PS mode powder and their distribution is not Gaussian. When these powders were seen under scanning electron transmission microscope, the result supports our presumption. One can see hexagonal platelets and rods stacked to form star like structure (in CS mode powder). In the pulsed mode powder porous and star-like structures break to give aggregates (121). Similarly, TEM observation gives nail shaped rod having porous structure and maximum length of 1000 nm and thickness  100 nm for CS mode powder. These nail shaped rods bunch up to make star-like bundle having the looks of a spindle (Figure 9). In case of PS mode powder hexagonal structured base for smaller rods are observed and these rods are randomly oriented. The selected area electron diffraction shows the crystallite size to be smaller and the silver nano particles sitting over ZnO rods and making them smaller (121). Thus,to know the characteristics of these structure we proceeded to observe the photoluminescence ,optical absorption and FTIR spectra on these powders. The FTIR spectra recorded from 400 cm-1 to 4000 cm-1 show several hydroxyl ion bands, but the metal oxygen (M-O) bonds for ZnO can be seen at 455 cm-1 and 557 cm-1 in pulsed mode sample. These two bands are also seen as shoulders in continuous mode sample. In an earlier investigation Behera etal. [123] have observed bands at 434 cm-1 and 558 cm-1 in the Raman spectra of A ZnO. Thus the bands around 455 cm-1 can be attributed to E22 high mode of phonon vibration and 558 cm-1 to A1

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(LO) mode. Similarly, presence of H ions has also been observed in Ag doped ZnO and a new band around 1109 cm-1 appears prominently for pulsed mode sample. Recently, Xu et. al. [124] have observed a band around 1082 cm-1 in Ag doped ZnO deposited over glass surface by RF magnetron sputtering. They have attributed this to A1 (2LO) mode of ZnO which was induced by ordered distribution of Znin defects in ZnO lattice. In our system doped with Ag, we have observed a small band around 1050 cm-1 for CS mode and a medium intense band around 1109 cm-1 for pulsed mode powder. These bands can be attributed to A1 (2LO) mode of ZnO. The UV-visible spectra recorded for ZnO powder doped with Ag dispersed in ethanol show one prominent band at 373 nm for CS mode powder and this shifts to 370nm for PS mode powder. This may be attributed to absorption of Ag+ ion in ZnO. In pulsed mode powder the band shifts to low wave length side (blue shift) and there is increase in absorbance on higher wavelength side of the spectra. The blue shift in the PS mode powder can be due to smaller crystallite size resulting quantum confinement in the system. The absorbance in PS mode sample has decreased although analysis of the powder shows more Ag concentration in this sample. This may be attributed to dispersion of more Ag atoms on the surface of the ZnO whose surface area has been observed to increase for PS mode sample . Accordingly , increase in absorbance on high wavelength side can be due to surface scattering by Ag atoms. This has beeen corroborated with our results on SEM and TEM investigation (121). Photoluminescence in ZnO is a complex process due to its inherent stoichiometric deficiency in oxygen content. Recently Kurbanov et.al [125] have studied the spectral behaviour of 3.31eV emission band in ZnO single crystal .Doping with Ag through sonication process makes it more complex when one tries to look into the role of Ag ion. We tried to find out the excitation and emission spectra of the system. The excitation spectra were found to have six or seven bands depending upon the mode of sample preparation . Continuous mode powder with 0.179% Ag shows strongest excitation band at 3.05 eV (406 nm) where as ZnO:Ag pulsed mode powder (0.385% Ag) show 3.16 eV (392 nm) band to be the strongest. A new band at 2.84 eV (435 nm) appears in pulsed mode powder. The deconvoluted PL spectra band positions have been given in Table-IX. Considerable difference in band position as well as their numbers can be observed from the table. Some salient features of these emission spectra can be enumerated as follows: a.

Excitation with 293 nm light leads to absence of two prominent bands at 3.13 eV (395 nm) and 2.39 eV (483 nm) in pulsed mode ZnO:Ag sample. b. Excitation with 354 nm light gives rise to similar band positions in both the samples except that bands have shifted considerably towards low energy (red shift) end of the spectra. However, in both the sample emission related to exciton (3.13eV) remains the same. Thus, from these observations one is led to believe that different centres are responsible for these emission bands and excess Ag ions quench the exciton emission band in pulsed mode powder when excited with 293 nm band light, but the same is not true for 354 nm excitation. This can be explained in the light of presence of Ag ion at 0.23 eV below the conduction band as advocated by Xu et.al [124] . Excitation with 293 nm light leads to two competitive processes in the system. The electrons and the excitons are excited simultaneously. Since Ag atom has already been excited, it has high propensity for absorbing the exciton. Thus,the exciton emission is quenched. However, during 354 nm excitation (E354

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B. S. Acharya

< E293) the exciton emission becomes dominant. Further, PL emission due to Ag atoms also takes place which gives rise to 2.03 eV (608 nm) emission. So, the 2.03 eV emissions can be related to impurity atom like Ag which is selectively observed for 354 nm excitation. Other emission bands can be explained in the light of Zn and oxygen vacancies present in the system as shown in our previous investigation [6]. Peak at 413m and 409nm may be due to radiative defect Zni and VZn related to the interface trap existing at the grain boundaries between silver nano particle and ZnO particles and emitted from the radiative transition Liu et.al [126] and Kang et.al [127] Thus from these above observations and discussion we are led to these following conclusions tentatively. Sonication is an effective method for incorporation of Ag ions /atoms into nano ZnO powders. Pulsed sonication increases the surface area of nano ZnO with more amount of Ag in the ZnO surface , thus making it a possible candidate for photo catalytic use. b. Inclusion of silver in ZnO through sonication promotes higher crystallite size. These silver atoms increase the scattering and are observed to be on the surface of nanorods. c. Evolution of nano structure of ZnO: Ag having Wurtzite structure takes place when more Ag atoms are added to the lattice (PS mode). d. Silver atoms participate in the photoluminescence process in a selective way thus giving rise to luminescence at 608 nm and quenching excitonic luminescence in PS mode sample

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a.

Similar observations in the case of rare earth doped ions have been made and can lead to an important conclusion that rare earth ions in ZnO nano powders do not exhibit enhanced luminescence,rather they decrease the luminescence efficiency of the system. A cursory look at the table-XII,one can see that AlN substrate has profound influence on the emission spectra of the oxide. Excitation with two different wavelength of light leads to two different set of emission bands except the band responsible for exciton. Silicon substrates like and coated with ZnO also behaves in a similar way. Thus we can generalize our conclusions as under.

GENERAL CONCLUSIONS Although it has been possible to collect PL,SEM,TEM and diffuse reflectance(DRS) on powders of many monovalent,divalent & trivalent ions like:B with,N,Li,Au,La &Pr etc. doped in ZnO , thin films deposited by several methods ,in this short review data on few samples have been presented.However,some general conclusions emerging from these observations can be enumerated as follows. a.

Room temperature PLspectra of ZnO doped with monovalent divalent and trivalent ions differ considerably for both powders and thin films

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b. Passivation of ZnO thin film surface takes place due to impurity doping in the lattice.This results in unsaturated dangling bonds on the surface of the paricles/films leading to variation in Raman spectra aswell as PL spectra. c. Sonication has been found to be an effective way for incorporation of rare earth ions in ZnO powders which cannot be done by conventional techniques.But the se ions donot produce their characteristic emission bands with large intensities.Thus energy transfer from ZnO to rare earth ions is not quite efficient in producing strong luminescence in this wide band gap semiconductor. d. Sonication has been found to be produce larger crystallite size unlike spray pyrolysis or physical vapour deposition which reduce the size to strong confinemet region.Agglomeration of crystallites takes place thus leading to various shapes and sizes of nano particles.

ACKNOWLEDGMENT The author would like to thank D.Behera and others who have made this work possible.

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[96] D Hahn and R Nink, J. Phys. Condensed Mater. 3 (1965), 311. [97] E C Lee, Y S Kim, Y G Jin and K J Chang, Physica B 308-310 (2001), 912. [98] C Bundesmann, N Ashkenov, M Schubert, D Spemann, T Butz, E M Kaidashev, M Lorentz and M Grundmann, Appld. Phys. Lett. 83(10) (2003), 1974. [99] F X Liu, J L Yang, and T P Zhao, Phys. Rev. B 55 (1997), 8847. [100] M Talati and P K Jha, Phys E-Low-D Systems & Nanostructures 28 (2005), 171. [101] A E Jimenez-Gonzalez, J. Solid State Chem. 128 (1997), 176. [102] S Major, A Banerjee and K L Chopra, J. Mater. Res. 1 (1986), 300. [103] W E Carlos, E R Glaser and D C Look, Physica B 308-310 (2001), 976 [104] K Minegishi, Y Koiwai and K Kikuchi, Jpn. J Appl. Phys. 36 (1997), L1453. [105] D C Look, D C Reynolds, C W Litton, R L Jones, D B Eason and G Cantwell, Appl. Phys. Lett. 81 (2002), 1830. [106] K K Kim, H S Kim, D K Hwang, J H Lim and S J Park, Appl. Phys. Lett. 83 (2003), 63. [107] Y R Ryu, T S Lee and H W White, Appl Phys. Lett. 83 (2003), 87. [108] F Toumisto, K Saarinen and D C Look, phys. Rev. Lett. 55 (1985), 132. [109] C H Park, S B Zhang and Su-Huai Wei, Phys. Rev. B 66 (2002), 073202. [110] M Liu, A H Kitai and P Mascher, J. Luminescence 54(1) (1992), 35. [111] C G Van de Walle, Physica B, 308-310 (2001), 899. C G Van de Walle, Phys. Rev. Lett. 85 (2000), 1012 [112] S B Zhang, S –H Wei and A Zunger, Phys. Rev. Lett. 84 (2000), 1232 [113] A F Kohan, G ceder, D Morgan and C G Van de Walle, Phys. Rev. B 61 (2000), 15019 [114] T.A.Merz,D.R.Doutt,T.Bolton,Y.Dong and L.J.Brillson, Surface Science 605(2011)L20 [115] V.G. Pol, J.M. Calderón-Moreno, A. Gedanken Chem. Mater., 15 (2003), 1111 [116] X.D. Gao, X.M. Li, W.D. Yu, Thin Solid Films 484 (2005) 160. [117] F.Li, X.Liu, Q.Qin, J.Wu, Z.Li, X.Huang, Cryst.Res.Techno.44(11)(2009) 1249-1254. [118] S.S. Kurbanov, T.W. Kang, J. Lum. 130(5) (2010) 767 . [119] K.S. Suslick, S.B.Cha, A.A.Cichowlas, M.W.Grinstat T., Nature (1991) 353414. K.S. Suslick, M. Fang, T. Hyeon, J. Am. Chem. Soc. 118 (1996)11960. K.S. Suslick, G.J. Price, Ann. Rev. Mater. Sc. 29 (1999) 295 . [120] K.Okitu, H.Bandow, Y.Maeda, Y.Nagata, Sonochemical preparation of ultrafine palladium particle, Chem.Mater.8(1996)315-317. [121] D.Sahu, B.S.Acharya & A.K.Panda. Ultrasonics Sonochemistry. 18 (2011) 601 [122] N.Arul Dhas, A.Gedanken,Sonochemical preparation and properties of nanostructured palladium metallic clustes, J.MaterialsChem8(2)(1998)445. [123] D.Behera, B.S.Acharya, IONICS,16( 2010)543-548. [124] Q. Xu, X.Zhu, F.Zhang, L. Yang, W. Jiang , X. Zhou, Study on Raman line at 1080.2 cm−1 in ZnO thin films prepared under high RF power , Vacuum(2010) . [125] X. Jia, H. Fan, F. Zhang , L.Qin, Ultrasonic Sonochem . 17(2) (2009) 284-287 . [126] K. Liu, B. Yang, H. Yan, Z. Fu, M. Wen, Y. Chen , J.Lum. 129(9) (2009) 969 . [127] H.S. Kang, J.S. Kang , J.W. Kim, J. Appl. Phys.95(2004) 1246.

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In: Optical Lattices: Structures, Atoms and Solitons Editor: Benjamin J. Fuentes

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Chapter 3

GIANT SPATIAL DISPERSION IN THE REGION OF PLASMON-PHONON INTERACTION IN ONE-DIMENSIONAL- INCOMMENSURATE CRYSTAL THE HIGHER SILICIDE OF MANGANESE (HSM) S. V. Ordin‡ and W. N. Wang§ A.F.Ioffe‟s Physico-Technical Institute of the Russian Academy of Sciences, Russia Bathh University Great Britain Academy of Sciences, UK

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INTRODUCTION If you face with unusual (not described by the standard models) physical properties, that, naturally, you begin with checkout of the delivered experiment and searching not considered both directed by experiment, and in modeling of factors. Therefore detection of "anomalies" in optical properties of a thermoelectric crystal a HSM has caused a large cycle of investigations of the various factors influencing optical and kinetic properties of crystals. As originally "anomaly" has been observed in reflexion spectra a HSM checkout of agency on an observable spectral singularity of a state of investigated surfaces of crystals, first of all, has been led. The surface investigations have revealed a lot of interesting, characteristic for the given crystal, but have shown that the observable dependence of reflectivity of the ORDINARY WAVE on orientation of a wave vector of radiation concerning a symmetry axis C of a crystal a HSM is determined, preferentially, by bulk properties of the given crystal. Volume character of absence of the contribution of the free carrying agents at orientation of a wave vector along an axis C was confirmed also with transmittance measurings at the given orientation. The obtained result contradicted both to the standard principles of crystal optics,

‡ §

E-mail address: [email protected] E-mail address: [email protected]

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and the standard description of a deflection from classical crystal optics - the space variance described as the allowance on parameter infinitesimally of Landau (order of smallness). Since the author did not consider possible to hide the given contradictions the obtained experimental effects of investigations the HSM and the key deductions a long time did not admit and were not accepted for printing. The conducted complex investigation a HSM has shown that "anomality" of symmetry of physical properties a HSM is exhibited not only in optics, but also in kinetics, and in acoustics. Thus, it has been proved that symmetry "anomality" is put in a crystal structure of an is one-dimensional- incommensurate crystal a HSM and, according to a Curie principle, is exhibited in its physical properties. It was shown that not considered parametre was in the three-dimensional symmetry groups giving the strict description only at strict transmitting invariance of a crystal lattice. Also it was shown that used calculation of allowances at the expense of a space variance is not applicable for the modulated crystals which parameters of modulation make object boundary between optical and radio ranges. For an explanation of the detect unusual physical properties the HSM was required not only use already before the developed theories of incommensurability and spatial dispersion. Qualitative reconsideration of their basic representations which have been in addition experimentally checked on extreme anisotropic commensurate crystals, boron nitride and graphite was required. Therefore the obtained results touch not only a HSM, and the whole class-room of the incommensurate and modulated crystals. It has define the plain-text form the given book. Each paragraph gives short introduction in basic representations in the viewed area, added by their improvements-corrections and the qualitative advances obtained on the basis of investigation a HSM, as modelling is one-dimensional-disproportionate object. As demonstration of validity injected is qualitatively new representations finishing results of experimental researches of perfect crystals a HSM and some quantitative calculations of observable effects within the limits of the improved representations are presented. At that as obtainment-cultivation of perfect crystals a HSM became possible in the course reconsiderations of thermodynamics of an incommensurate phase and a Bridgman method modification, and setting of finishing experiments, and their quantitative description reconsiderations of properties of incommensurate medium and conditions of an origin of the effects related to spatial dispersion of light became possible in the course. And last as the author in the Open Letter «the Taboo on progress in science», scored now the physicist strongly fragmenting. As consequence, even the adjacent(contiguous) fields of physics contain many mutually exclusive concepts and representations in basic models. Therefore there is a flow of scientific publications containing as interior contradictions (with invariants of physics), and exterior contradictions with publications on the subject. The author has tried to obtain peakly common, consistent pattern of the detect effect. Therefore it was specially repelled not from contradictory, and frequently, primely erratic original publications on the given subjects and the given crystal, and from the basic generalized models consistently described in fundamental monographers. It has found the reflexion both directed by experiments, and at conducting of the analysis and calculations, and in the reduced list of the used publications.

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SPATIAL DISPERSION The electric field E and magnetic field B give rise to polarization of medium which is characterized accordingly by a vector of electric P and magnetic M polarization. Resultant fields, a electric induction vector

D = E + 4 P and a magnetic-field vector

H  B  4 M write down in the form of an algebraic sum of an outside field and the polarization directed by it [1]. In the elementary case normally used in crystal optics of anisotropic medium, the linear response of medium to external actions is considered:

Pi   ij E j M i   ij H j where coefficients independent of external fields

ij -a dielectric susceptibility tensor, ij - a

magnetic susceptibility tensor. At that, naturally, resultant fields are related to external fields also through the linear coefficients independent of external fields

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Di   ij E j Bi  ij H j permittivity tensor  ij  ij  4ij and permeability tensor ij  ij  4ij . At the given, elementary approach the dependence of resultant fields on a time and space neighborhood of a viewed local point of application of an outside margin is initially eliminated from viewing. After lockout at the moment of a time t0 , an electric field the E resultant electric field D disappears not at once. The response to exterior action is prolonged some finite period t which is define by atomic and electronic resonances of medium.

Necessity is related to it to consider a time dielectric dispersion  ij  t  which, through time Fourier transform, is related to its frequency dispersion

 ij   measured in experiments:

Di     ij    E j  

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S. V. Ordin and W. N. Wang On the other hand, switching on an electric field in E a local point of space r0 results in

polarization of medium in some neighborhood of the given point r0  r . Hence it is necessary to consider in addition a spatial dispersion of a permittivity  ij  r  which is analogous, but through space Fourier transform, it is related to its dependence on light wave vector k :  ij  k  . As it is impossible to view, in principle, vibrations in a time for an

infinitesimal period, and it is impossible to view an advance of waves on an infinitesimal segment. Restriction of viewing of optical properties by the registration of only time dispersion gives the incomplete, partial description working, naturally, only in special cases. For crystals the expansion of time vibrations and space waves on normal, independent (immiscible) modes along main crystallographic directions which are one-to-one correspondence to an external action along corresponding crystallographic directions is admissible (to a first approximation) [2]. At that expression for resultant along the extracted crystallographic direction of a field becomes simpler for a time-frequency dispersion:

Di     ii    Ei   also that is especially essential, for a space dispersion

Di  xi    ii  xi   Ei  xi  Di  ki    ii  ki   Ei  ki 

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As the wave vector k of probing radiation is very small, in crystal optics figure its equal to null and neglect a space dispersion. Are at that restricted to only frequency dependence of a permittivity along main crystallographic directions  ii   . In this case the crystal optical

anisotropy is completely set by orientation of an electric vector of radiation E . The key moment, falling out at the given simplified approach, that the dependence of a

permittivity on a wave vector  ii  ki  for various crystallographic directions, cannot be degenerated and at aspiration k to null is. It follows from anisotropy of area of the directed polarization and in a static electric field. Moreover, from the law of a dispersion of excitations in anisotropic medium follows that polarization not completely describes an optical anisotropy. Atomic and electronic resonances- defined excitations are spread in medium. At that, their frequencies depend not only on polarization, but also from orientation and quantity of a wave vector q . Hence, for an ordinary wave, contributions defined by resonance frequencies to a permittivity depend not only on polarization, but also from orientation and quantity of a wave vector of light k too. Viewing of the factors not considered at the partial description we will lead on an uniaxial crystal instance. The law of a dispersion of optical phonons in an uniaxial crystal, for example, in rhombohedral a boron nitride [3], qualitatively looks like, figured on Figure 1.

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At electric vector orientation normally to symmetry axes of a crystal E , the dispersion law has a set of the allowed states for any orientations of a wave vector of a phonon q . I.e., as is shown in Figure 1, it represents a surface. Laws of a dispersion of radiation for wave vector orientation in parallel k and normally k  to symmetry axes, are diagrammatically figured by dotted lines, and their crosses with the given surface of phonon states for corresponding wave vectors q and q are marked out by light circles. These crosses, according to law of conservation energy and an impulse, give frequencies of raised phonons. At that, as polarizations of the phonons excited along main crystallographic directions is define by polarizations of a transversal electromagnetic wave, transversal phonons are raised. Surfaces of the allowed states essential change occurs far from the Brillouin zone centre:

q

 c and q

 a , and at small wave vectors it is inappreciable. Therefore interaction

of radiation by the dataful transversal optical phonons, corresponding in an uniaxial crystal to an ordinary wave, originate on frequency  and weakly depends on wave vector orientation

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in commensurate crystals.

Figure 1. A diagrammatic representation of dispersion laws of transversal phonons in a Brillouin zone of an uniaxial crystal and their cross with the law of a dispersion of light: and c a , periods of translation of a crystal lattice lengthways and normally to symmetry axes, accordingly.

At electric vector orientation in parallel crystal symmetry axes E , from transversality of a probing electromagnetic wave and a law of conservation of momentum, we have single allowed orientation of a wave vector of optical phonons - q it is normal to a symmetry axis.

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At that, the dispersion law of the given phonons is degenerated in figured on Figure 1 for E in a curve in a plane  , q  . Cross in this plane of the a phonon dispersion law with wave vectors q with the dispersion law of radiation with a wave vector k  gives frequency of the transversal optical phonons  corresponding to an extraordinary wave. That these resonance frequencies correspond to transversal phonons, requires introduction additional, an upper index: T and  T . In polar crystals of frequency of transversal phonons T and  T define a low-frequency edge of the characteristic peak of lattice reflexion for ordinary and an extraordinary wave, accordingly, as for example, in nitride a boron. Frequencies of high-frequency edges of peaks of lattice reflexion traditionally connect with corresponding longitudinal phonons: L and  L But a wave vectors of longitudinal phonons their polarizations are parallel, i.e. are perpendicular to wave vectors of exciting them of transversal photons. Therefore, strictly speaking, frequencies L and  L correspond simply drifted on frequency T and  T :

                 L 2 

T 2 

L 2

T 2

2

p

2

p

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where plasma frequency  p and

 p corresponds to local plasma oscillations of ions along a

direction of polarization of corresponding transversal phonons. These plasma additives to frequency define breadth of peak of lattice reflexion which corresponds to region of the forbidden states for electromagnetic waves in a crystal. Thus, the spatial dispersion in a broad sense is exhibited, in principle even in the elementary anisotropic crystals. In narrow sense with a spatial dispersion connect occurrence of in the region of the forbidden states for an ordinary wave of the allowed states at some orientations of its wave vector and, accordingly, dips in peak reflexions [4]. These effects normally observe not in low-frequency lattice reflexion, and the high-frequency peaks of reflexion defined by electronic local plasma oscillations. They well are described as the firstorder correction and the second-order correction on a parametre infinitesimally of Landau a  , where  - the wave-length of light related to the registration of inhomogeneity of a crystal on atomic gauge [5]. Basic representations of crystal optics of commensurate crystals [6] have been experimentally checked on extreme anisotropic commensurate crystals [7], lattice reflexion in the region on highly ordered crystals of boron nitride [8, 9] and in the region of plasma reflexion on highly ordered crystals of graphite [10, 11]. These experiments have shown that observable changes in spectral dependences of reflectivity for an ordinary wave at rotational displacement of its wave vector for weak decaying lattice and plasma oscillations inappreciable (can be connected to reflecting surface change of state). I.e., to a first approximation at small wave vectors the region of the states

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prohibited for light in a crystal is concluded between figured on Figure 1 a surface of states of phonons and drifted upwards on frequency:

T  L a surface cross with which the law of

a dispersion of radiation is scored by black points. At that it is possible to neglect a feeble space dispersion of plasmons, but it is necessary to consider restriction of region of their admissible wave vectors, connected to fracture of plasmons at the expense of their signal attenuation at lengths of waves there is less than electron free length. Qualitatively given pattern persists and for electricity-conductive anisotropic crystals, of the type of graphite, with that only odds that frequency of a transversal phonon and T  0 , accordingly    p  . L

And so, presented on 1 Figure the volume pattern of a space dispersion of excitations in an anisotropic crystal confirms the deductions of crystal optics obtained on the basis of normally used plane patterns of a dispersion. Moreover, the volume pattern, to a first approximation, is built on the basis of plane patterns. But extensionality reflects the basic bond of a time and spatial dispersion and allows to see the additional effects originating in weed the periodic and modulated crystals. So, for example, doubling of a translation period along a symmetry axis C results in decrease in a 2 of time of a Brillouin zone in a direction of wave vectors q and to an origin are additional optical branches of phonons (Figure 2) As it is figured on Figure 2, for an ordinary wave, there is a convolution of the initial surfaces of the allowed states of phonons along a wave vector q , both startingly optical

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surface, and startingly acoustic surface.

Figure 2. A diagrammatic representation of the allowed states of phonons in the contracted Brillouin zone.

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At that excitation by light of additional phonon frequencies was possibly for any orientations of a wave vector of light: light circles mark out cross points of laws of a light dispersion with surfaces of the allowed states of phonons. Doubling of a translation period along a symmetry axis C , as seen from Figure 1, for the phonons polarized along an axis C , to any additional phonon branches does not result and, hence, phonon frequencies are completely define by minimum unit cell for an extraordinary wave. The complete setting of electrodynamics problems demands the registration of boundary conditions. In particular, in dielectrics – surface waves [12] and additional boundary conditions [13]. In electro conducting mediums boundary effect can result in a permittivity sign reversal at infinitesimal k  0 at the expense of accumulation of charges on a surface and, thereby, treatments in 0 static conductivity [14]. However, as it will be shown below, in an investigated one-dimensional incommensurate crystal the HSM, the excitations polarized normally to of an axis of incommensurability C , for wave vectors q and q , are in essence various in crystal volume. Therefore difference of responses of a crystal the HSM for an ordinary wave at wave vectors of light k and k  also, in principle, is define by crystal volume. Thus, the discovered giant spatial dispersion is define, in principle, by volume incommensurability. Therefore at the expense of the surface polaritons and additional waves we will not carry out detailing of developing process of a spatial dispersion.

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INCOMMENSURABILITY Incommensurability is exhibited in the presence of two periods of translation in one direction [15]. In crystals incommensurability arises in the presence of independent (or loosely coupled) sublattices in crystal lattice. If a ratio of periods of independent sublattices to equally rational number, strictly speaking, we have a commensurate state with a period of repeater of much more interatomic spacing. If this ratio to count irrational the total period of translation tends to infinity. Initiation the one-dimensional incommensurability can show on an instance of a two-dimensional regular grid in which we will lead the cross-section which is not coinciding with principal axes (Figure 3) As seen from Figure 3, lowering of dimensionality is lower number of independent periods of translation can to hide the transmitting invariance proper in object and accordingly, its complete symmetry. Only use of a totality of independent periods of fundamental translations allows to find the true symmetry group of a crystal [16]. If in a crystal there are two sublattices having independent sublattices periods along one symmetry axis, that, as consequence, for a complete description of properties of a crystal is necessary to consider all four independent periods of translation. Thus, according to a Curie principle [17], the symmetry group of a three-dimensional crystal with the one-dimensional incommensurability is four-dimensional. Cross-sections of a four-dimensional pattern of physical properties of an is one-dimensional-incommensurate crystal will correctly reflect experimentally observable effects for each specific orientation of a crystal relatively probing action.

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Figure 3."Irrational" cross-section of a two-dimensional regular grid through arbitrarily given point (a) and quasiregular vibrations of displacement of the proximal points of a grid concerning the secant line (b).

Some singularities of selection rules for the infrared vibration spectra of incommensurate crystals are viewed in [18]. In real crystals it is impossible to state about strict correspondence of a ratio of number of periods of sublattices N M to an irrational number, i.e. it is impossible to count the total period of repeater as infinite. Very large commensurate periods of translation also are not thermodynamically stable states since, on the one hand, the minimum of potential corresponding to them is less than energy of phonons, and on the other hand, amplitudes of the resonance oscillations corresponding to very large period of translation appear one order with interatomic spacing and by that, set uncertainty of the given total period of translation [19, 20]. According to thermodynamic conditions by the linear, purely interference viewing of static waves of displacement of atoms, there is simply modulated crystal structure which period of modulation is define by a thermodynamic potential minimum at a spinodal decomposition of solid solutions [21]. But the linear viewing, well describing decay of solid solutions, do not correspond to a real state of crystals in which there are two loosely coupled sublattices with various periods of translation. If interaction of sublattices is not negligiblely small thermodynamic instability of large periods of translation results in that the proximal is implemented to it corresponding to the given temperature and the given composition thermodynamically stable period of translation. For the quantitative description of incommensurability in crystals the parameter of incommensurability   cN cM  N M is inducted [22]. Interaction between sublattices results in nonlinear effects which give a non-monotone dependence of displacement of atoms of sublattices from each other along a common symmetry axis for two sublattices. Dissection of a crystal into the commensurate sections divided by walls, solitons at that follows. Not the monotonicity of displacement of atoms results in composition change at boundaries of commensurate sections, i.e. walls are concentration solitons. Thus, in strict correspondence with a symmetry conservation law at

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phase change from highly symmetric state in an incommensurate state, there is a dissection of a crystal into domains. Initiation the regular grid of solitons and their period well are characterized in the onedimensional case by a nonlinear sine-Gordon equation.

 2   sin   0 x 2

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where M  - number of periods of one of sublattices in the proximal commensurate state M , increased by a phase  characterizing a displacement of atoms of softer sublattice of a rather strictly incommensurate standing of atoms in this sublattice, - the  parameter considering balance of binding energy of sublattices and warmth, a setting thermodynamic minimum of potential for implementable displacement of atoms of sublattices from each other. The known solution of a sine-Gordon equation is presented on figure 4. Within the limits of given soliton theories specificity of thermodynamics of an incommensurate phase, as consequence of phase change on non-Livshits star which boundary is set not only temperature, but also a prehistory of the sample is described. In particular, as is shown in Figure 5, at the fixed composition the temperature of transition from the isotropic phase in incommensurate on cooling and at a heating, in principle, does not coincide.

Figure 4. A diagrammatic representation of transition from purely incommensurate state to dissection of a grid into the commensurate regular segments divided by soliton walls, at a prevalence of binding energy of sublattices over thermal energy.

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The strict description of optical and kinetic properties incommensurate, in principle, represents nonlinear medium the difficult nonlinear problem. However its qualitative simplification in frames or colour symmetry, or, as is shown in Figure 3, the extension of dimensionality of object that is equivalent is possible. The given simplification is admissible in the event that is admissible not only presented on Figure 4 of the description of static incommensurability of a crystal lattice, but also the extension of the description of dynamic effects (Figure 1, 2), excitations in incommensurate medium: phonons, plasmons, polaritons, to a four-dimensional Brillouin zone. Actually it is the Curie principle extension on four-dimensional symmetry groups according to which extreme possible anisotropy of a crystal is restricted by anisotropy of a crystal lattice. At the elementary, one-parametric actions the four-dimensional tensor of response of such uniaxial incommensurate crystal can be degenerated to three-dimensional or even to the two-dimensional. For example, the tensor of static electro conductivity and for normal three-dimensional crystals and for incommensurate crystals has a 2 independent components. At multiparameter actions as on an instance, at Hall effect measuring, degeneration of a four-dimensional tensor of response, in principle, does not occur. At that, there are the additional nondegenerate cross components characterising various response of an incommensurate crystal on the equivalent actions for commensurate crystals.

Figure 5.The Diagrammatic representation of the phase diagramme of an incommensurate state.

MODEL OF A CRYSTAL STRUCTURE THE HSM Investigating alloys of manganese with silicon E.N.Nikitin's with employees [23, 24] have discover that in the region of compositions MnSi1,71-1,75 corresponding approximately to formula Mn4Si7, forms quasi-homogeneous compound - the higher silicide of manganese

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(HSM) with perspective thermoelectric properties. Numerous X-ray diffraction studies have shown that in the denoted area of compositions there is a set of structures: Mn4Si7, Mn11Si19, Mn15Si26, Mn26Si45, Mn27Si47, described by formulas of the type of MnNSiM [25]. All structures exhibited tetragonal symmetry. At that, the translation period a in a plane to a perpendicular tetragonal axis C , for all structures was close: a 5,53 Å. Observable translation periods of structures along an axis C changed in the interval, approximately, 17÷118 Å. Crystal lattices of all structures have common motif and, on literary data, symmetry the HSM is characterised by the space group of symmetry with inversion-rotational axis of the 4order - D2d. However, even if to assume that one of atom of structure occupies a 2 crystallographic not equivalent positions, even the elementary structure Mn4Si7 does not answer any of 230 Fedorov space groups [26]. The carried out analysis has shown [27] that the complex change in relative position of atoms at transition from one structure to another unambiguously is characterized within the limits of model 2 of sublattices, loosely coupled with each other along an axis C (Figure 6). The translation period СMn of strictly ordered rigid sublattice Mn (Figure 6а) weakly depends on composition of in the region of homogeneity a HSM and amounts to approximately 4,37 Å. The translation period of СSi of weakly ranked and having a smaller constant rigidity Si, sublattices Si (Figure 6b), traces composition change (and also temperatures and pressures) according to a relation: M*СSi = N*СMn, in the interval 2,4952,533 Å. At that formed by tetrahedrons from atoms Si the spiral can be twisted and uncoiled slightly about an axis C .

Figure 6. Model of a crystal structure the HSM.

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X-ray diffraction studies have confirmed existence, besides earlier detect, of some other commensurate superstructures forecast within the limits of given model [28, 29]. Key for a crystal lattice the HSM is that despite lacking strictly certain crystallographic atomic arrangement Si concerning sublattice Mn, the translation period СSi well is observed [30], though a halfwidth of a X-ray reflex corresponding to it 10 times more than for СMn. On the one hand, it explains unusual chemical formulas a HSM to which there does not correspond any Fedorov group. On the other hand, allows to explain the coherent heterogenization observed the HSM as consequence of phase change on non-Livshitsstar [31], i.e. initiation a phase characterized in parameter of an incommensurability of translation periods of Mn and Si sublattices  = СMn/СSi – N/M. Thermodynamic instability of large translation periods) is exhibited that very large periods of superstructures in crystals the HSM are not observed. At that, as well as follows from a symmetry conservation law dissection of a crystal a HSM on the domains, accompanied by composition change at their boundaries is observed [32] (Figure 7). Within the limits of the one-dimensional model 2 of weakly bound sublattices symmetry reasons about solid-state phase change in the HSM prove to be true and concretized by microscopic calculations of atomic position Si [33]. The rigid sublattice Mn sets allocation (sinusoidal - in model of Fraenkel-Kontorova) potential Vo along an axis C which translation period), is generally incommensurate to a translation period of the weak sublattice Si. As a result of interaction of these sublattices arises displacement of atoms Si along an axis C , described by thermodynamically stable regular net of solitons.

Figure 7. A scanning electron microscopy highly ordered HSM crystal in a contrast mode in mass of atoms (a, b) and optical polarizing microscopy weakly ordered HSM crystal (c) with X-ray the microanalysis (d).

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S. V. Ordin and W. N. Wang

In difference from an initiation in volume of a crystal of a dislocation of Fraenkel: a soliton and an antisoliton simultaneously, existence of the regular net only solitons (or only antisolitons) is rigidly related to change in composition of a crystal. Solitons view normally as dynamic objects, stable in a time at the expense of a cancellation of effects of an anharmonicity and a dispersion. However if for allocation of atoms of each sublattice along an axis C to inject the complex amplitude which phase is define by its translation period we will obtain two "frozen" (not dependent on a time) crystalline waves. In the linear approach interaction superposition of 2 crystalline waves will give a normal interference. At the registration of anharmonic effects displacement of atoms Si concerning minimums Vo is characterized by the sin-equation of Gordon not dependent on a time, where where =1/2 (VoM/)* M2, VoM = (/)*(Vo/Si* СSi2). At with = 22/16 the soliton grid is a system ground state. At large Vo and small Si and  (с) the weak grid is completely fine-adjusted under rigid and becomes precisely commensurable. A temperature dependence of a size of a soliton: 1/ (Т-Тс)2, at Тс 800оС, agrees well to periods of coherent heterogenization in the samples the HSM quenched from different temperatures. However within the limits of this approximate calculation to completely commensurate structure there corresponds an one-domain crystal. In real crystals the size almost completely commensurate domains is terminating and agrees well to results of estimations on the basis of described above an interference of crystalline waves. The size of soliton L can be found from a following relation between wave vectors soliton grids q*, commensurate qC and incommensurate qI phases: q* = /L =qC -qI, гдеqC =/М*СMn, а qI = /N*СSi. For the presented on Figure 7 highly ordered crystal the HSM following parametres are obtained: СMn = 4,3665Å, СSi = 2,5129Å. Considering that the silicic sublattice is stretched, we will obtain the peak period of solitons for Mn27Si47: L = 6,6. That agrees well to a period of the strata enriched by manganese (Figure 6а, b). Beatings with a period 0,25 concerning superstructure Mn4Si7 are also observed by means of a scanning electronic microscope, but on an etched surface. The electronic appearing through microscopy has shown that in the domain crystal lattice modulation (Figure 8), reflecting an interference of "crystalline" waves of two weakly interacting sublattices is also observed. The cited X-ray data and electronic photographers are obtained on highly ordered crystal the HSM, obtainment - growing which became possible thanks to understanding, both common regularities of an incommensurate state, and specificity of formation the HSM. Crystals the HSM forming on peretectic reaction at temperature 1145 C, grew up in a temperature gradient near temperature of peretectic. At that, as has shown data analysis of an optical polarizable and electronic scanning microscopy, there is a stochastic regime of formation of the irregular net of solitons which provoke phase precipitates of a monosilicide. The stochastic regime arose that naturally in an incommensurate phase jumped transition and received weakly ordered crystals. On cooling a melt on peretectic reaction the isotropic solid phase the HSM from which naturally forms first, at temperature nearby 1000 C there is a transition in an incommensurate phase [34]. Considering that velocities of solid-phase transitions 10 times less velocities of transition from a liquid phase to a solid phase, formation highly ordered incommensurate crystal the HSM it was carried out their temperature-compensated isotropic state in an additional temperature gradient with small velocity of its passage. As it was shown above, are obtained

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highly ordered incommensurate crystals the HSM which microscopic structure is quantitatively determined by parameters of its crystal lattice.

Figure 8. An appearing through electron microscopy the HSM.

OPTICAL ANISOTROPY THE HSM The HSM proves, at the unicomponent actions, as an uniaxial crystal with obviously expressed anisotropy of the kinetic coefficients [35]. In the interval temperatures 100-1100оК conductance   in a plane of a normal to axis C is isotropic and exceeds parallel axes C

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conductance approximately

 in 10 times. At temperatures below 800 C as and   has 

metallic character. Under the representations of crystal optics of uniaxial crystals [36], measurings of reflexion the HSM were carried out in polarized light at two orientations of an electric vector E: lengthways and normally to principal axis of symmetry C and on two faces of a crystal analogously oriented concerning an axis C (Figure 9). At reflectivity measuring conditions of excitation of normal modes have been implemented. It was revealed that an ordinary wave reflexion spectrum: E  C within the limits of representations of crystal optics independent of orientation of a wave vector of light k , in crystals the HSM monotonically changes at its rotational displacement concerning an axis C . Between states k C (Figure 9a) and k  C (Figure 9b) primely quantitative is observed not, but qualitative difference of reflexion spectra. The given difference of spectra of an ordinary wave begins in a short-range infrared gamut (about 3000 sm-1). At magnification of a wave-length of light:  3 the difference increases that is opposite to the effects of spatial dispersion observed in a visible and ultraviolet gamut, a characterized parameter infinitesimally of Landau a  . To fathom the nature of observable effect spectroscopic investigations the HSM have been led in maximum wide spectral gamut: from an ultraviolet gamut to a radio-frequency range. The basic prominent features of reflexion spectra and a transmission and their bond with various excitations in a crystal the HSM have been analyzed.

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116

Figure 9. The characteristic reflexion spectra and uptakes the HSM at indoor temperature: a –for the isotropic plane, a normal to axis C , b and c –for a plane of a parallel axis a reflexion spectrum from the isotropic plane.

C with superimposition of

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At the analysis of excitations in the capacity of the starting basic backgrounds the adiabatic approach was used: an opportunity of partitioning of frequencies of electronic vibrations, atomic vibrations and rotational vibrations [37]. However, in noт-crystalline materials [38] and in so complex crystals as the HSM as it will be shown below, prime partitioning of frequencies of excitations into three groups of strongly spaced delivered frequencies is often broken. Therefore interpretation of spectra of a crystal the HSM has demanded improvement and the extension of standardly used models of resonances in a solid state. Presented, around 40000 sm-1 on Figure 9, ultraviolet peak of reflexion corresponds to interbandtransitions [39]. As seen from Figure 9, it is isotropic, weakly depends not only on a direction of a wave vector of light, but also from polarization. The characteristic dip of a reflectivity of in the region of visible light (about 17000-3500 cm-1) is formed by a reflexion low-frequency edge on the scored interband transitions and a high-frequency edge of reflexion on local plasma oscillations. Localization of electrons at atoms normally is considered [40]. But in the given spectral singularity in crystals the HSM is viewed correlation with anisotropy of conductance on a constant current: the high-frequency edge of reflexion on the localised carrying agents depends on polarization and inversely proportional static conductance. Therefore the observable polarizable dependence of the given edge of reflexion can be connected to a degree of localization of carrying agents in some elementary volume formed by regularly located potential wells which in different crystallographic directions have excellent periods. Longitudinal and transversal frequency of the space oscillator arising at that are define by width of layers with the localised carrying agents and their period. The anomalous spectroscopic dependence of a reflectivity from k is exhibited at lengths of waves  3 , mainly, as total absence at k  C plasmareflexion on the free carriers. The Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

characteristic absorption on the free carriers, corresponding to conductance the HSM of normally to axis C , at orientation k  C (Figure 9, Theory ) in absorption spectra (Figure 9,

 Exp. ) also is not observed. The uptake low-frequency edge  Exp. well is characterised by the lattice absorption. The high-frequency absorption edge originally has been interpreted as Opt .

direct interband transition with breadth of a band-gap Eg

 0.67eV [41].

However the analysis of high-frequency reflexion spectra has shown that interaction of sublattices the HSM essentially affects electronic states and is exhibited as in an overflowing(overflow) of electrons from split-off valence s-zone Si in d-zone Mn, and in s-d resonance scattering [42]. Splitting of bands arising at that near to a Fermi level on E (Konn-singularity) is a necessary condition for an origin in the HSM of the regular potential barriers for the free carrying agents and an observable coherent heterogenization (Figure 7, 8). In the region of resonance splitting of bands, at d  E  dk

0 arises Van Hove

singularities: to a maximum of probability of transitions between the splitted bands at

h  E* . It explains both a maximum of reflexion around a 1 eV , and a high-frequency

absorption edge  Exp. . At that the frequency dependence of an absorption constant same as

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S. V. Ordin and W. N. Wang

for direct transitions 

2

 

h  Eg . But in a dependence of an absorption edge on

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temperature the basic bond with E* phonons is exhibited (Figure 9a, Figure 10).

Figure 10. Transmission spectra of the HSM plates perpendicular to an axis C a thickness 15 microns (curve 1 and 2) and 8.5 microns (curve 3): curve 1s and 3s correspond to room temperature of the sample, a 2 curve – 77К.

At k C plasma reflexion on the free carrying carriers misses practically to a radiofrequency range (Figure 9a). At that at frequencies the  500cm1 continuous characteristic(reference) set of the lattice oscillators is observed. At k  C and E  C characteristic for   plasma reflexion is observed (Figure 9b). At that at the expense of the additive contribution to a permittivity plasma and lattice vibrations the high-frequency part of the characteristic set of the lattice oscillators also is exhibited. On the one hand it, in the complete correspondence with volume convolution of a Brillouin zone (Figure 2), denotes existence over periodicity along an axis C . On the other hand, the observable characteristic spectrum of lattice reflexion for samples of different composition from area of homogeneity the HSM which assigned different superperiods, is qualitatively similar for all samples. What for an ordinary wave, defining a super-set of lattice vibrations, normal to an axis is C over periodicity along an axis C , direct neutron investigations of lattice vibrations of crystals the HSM have directly confirmed [43, 44, 45].

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That the characteristic set of the lattice oscillators observed in the HSM qualitatively same for samples with various super-periods denotes that static incommensurability results in an origin also to dynamic incommensurability. On loosely coupled sublattices along an axis C are spread weakly interacting phonons. In the region of homogeneity a HSM change in a period of translation of a sublattice of manganese negligiblely is not enough, and change in a period of translation of a sublattice of silicon inappreciably. Therefore it is qualitative laws of a phonons dispersion each of sublattices from the sample to the sample do not change. At that for both sublattices a super-period it is define by a period of concentration solitons along an axis C . For each of sublattices in the domain formed by solitons there is a set of standing transverse waves with which light interaction, in principle, is is possible. So the period of concentration solitons is much more than periods of translations of sublattices, light interaction was possibly on almost as much as large lengths of waves. But phonons of the sublattices which do not have interior dipoles, in itself are not an IR - active. However in a plane, a perpendicular axis C atoms of sublattices have the strong ioncovalent bond defining both resonance frequency of a high-frequency lattice oscillator

T  474cm1 and its dipole moment which sets width of this oscillator: L  512cm1 . Hence, excitation by light was possibly for pairs standing, along an axis C , waves of different sublattices. At that for each independent subsystem of phonons the given excitation is parametric. Parametric excitation generally is characterized by the equation of Hill:

d 2x  p t  x  0 . dt 2

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In that specific case, at the harmonic excitation – the formula of Mathieu:

 d 2x 2 2   1  cos  t    x 0, 0 2 dt 2  0  which gives a series of resonances: 1 20 , 0 ,3 2 0 , 20 ,.... , where 0 - the basic resonance, sharply weakening with a growth of frequency [46]. Parametric interaction normally is not considered in crystaloptics. However it essentially is exhibited in spectra of lattice vibrations of anisotropic crystals [47]. Moreover, parametric interaction of normal modes determines initiation the one-dimensional phase change in an anisotropic crystal [48]. The characteristic frequency division for parametric resonance of a high-frequency basic lattice oscillator 1 2 T  237cm1 in a series of the lattice oscillators a HSM is observed in an ordinary wave (Figure 9a, 9b). Thus, the phonon spectrum of a crystal the HSM has two independent (to a first approximation) wave vectors: for a sublattice of manganese q

Mn

and for a silicon sublattice

q Si . I.e. according to a four-dimensional symmetry group of a crystal the HSM we have a Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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S. V. Ordin and W. N. Wang

four-dimensional phonon spectrum. But similarly to, as in commensurate anisotropic crystals parametric bond of the normal modes corresponding to different crystallographic directions is exhibited, in an incommensurate crystal parametric bond of the normal modes corresponding to two independent wave vectors of sublattices is exhibited. At that specificity of the one-dimensional incommensurability is exhibited, in addition, that as q

Mn

, and q

Si

determine interaction of a crystal lattice with an ordinary wave that

shows the contribution of lattice vibrations at k  C (Figure 9b). As it was shown above, convolution of a volume Brillouin zone along an axis C does not affect a phonons dispersion line the polarized along an axis C . As consequence, for an extraordinary wave: E C (Figure 9c), a spectrum of lattice reflexion the HSM contains the only one powerful oscillator which frequency

T  427cm1 is a little below frequency T

for an ordinary wave. The high-frequency lattice oscillator of an extraordinary wave resides on the edge of plasma reflexion around 200 sm-1 which agrees will qualitatively be agreed to quantity of conductance  along an axis C . The reasons of an origin concerning small, the second order of smallness of oscillations for an extraordinary wave on the edge of plasma reflexion and on the lattice peak on

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frequency of an orthogonal mode T  474cm1 was possibly are related to additional developing process of parametric interaction, but was possibly and with a margin error the polarizable measurings related to a degree of orientational ordering of a crystal the HSM. Since these rather small oscillations of in essence observable spectroscopic pattern do not change, in a given work we will not analyze them. The key moment following from the fact of recording in an extraordinary wave of a powerful lattice oscillator is that, just as rigidity and the dipole moment of the interatomic bonds defining local bonds of atoms of sublattices along an axis C are great. Hence, to observable effect of a slide of sublattices as whole, from each other along an axis C brings inappreciable decrease of rigidity of local bond of atoms of two sublattices along an axis C :

T  T . On higher frequency it appears energies of orthogonal phonons [49] enough for displacement of separate atoms of sublattice Si along an axis C . Thus, the phonon component is an one-dimensionally-incommensurate phase defining at an origin in the HSM. And so, practically all observed in reflexion spectra and transmissions the HSM effects qualitatively and, in many respects, agree quantitatively well to one-dimensionallyincommensurate model of its crystal lattice. As is shown in Figure 9b, the observable giant spatial dispersion is define by a dependence of a spectrum of plasma reflexion of an ordinary wave on orientation of a wave vector of light. It not only does not contradict the given model of a crystal lattice, but is a direct consequence of incommensurability along an axis C .

WAVEGUIDE MODEL OF A CRYSTAL THE HSM Within the limits of model of homogeneous anisotropic plasma [50, 51] anisotropy a reflexion plasmon-PHONON in a crystal the HSM cannot be completely described. Whereas

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121

  follows that thickness of skin layer at E  C , it is

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much less than:   0.7  . Anisotropy of an edge of reflexion of local plasma oscillations the HSM (Figure 9с) denotes that the part of the carriers of current localized at E C , at is E  C free. Localization of carriers of current occurs, first of all, in walls of the concentration solitons which thickness in high-ordered HSM crystals do not exceed a 300 A (Figure 7b). From a large transparency of the HSM crystal at k C follows, just as essentially smaller skin layer corresponding to higher conductivity of concentration solitons, appears more thickness of conducting layers. As consequence, at k C the separate conducting layer is not capable to form the complete plasma reflexion and uptake. At kС in an IR-range of a spectrum  there is more than period of modulation of a crystal lattice and plasma in the HSM crystal. Therefore for reflexion from a plane of a parallel axis C (Figure 9b, 9c) application of model of effective medium based on electrostatics, i.e. on a condition k=0, in principle, is admissible. Thus, for a complete description of optical properties of plasma the HSM at k C it is impossible to start with homogeneous environment model. The total reflectivity of crystals the HSM from a plane of a perpendicular axis C forms of a set of reflexions of separate layers which are dephased both at the expense of their space separation, and at the expense of a phase displacement at boundary metal-dielectric. Calculations of optical properties of the modulated media, similar to the crystal a HSM, were carried out by the interference methods for diffraction gratings [52] and a method of Bloch waves for photon crystals [53]. Calculation of a multipass interference in the HSM crystal us also has been carried out. However this calculation has demanded, on the one hand introduction of some presumable parameters, on the other hand, the obtained unwieldy analytical forms can be qualitatively obtained within the limits of more simple model reflecting adjective of optical and electrodynamic properties of the HSM crystal in an IRrange. For a qualitative explanation of singularities of symmetry of optical properties the HSM arising at the expense of the one-dimensional modulation of plasma along a symmetry axis, it is possible to use the waveguide model with distributed parameters (Figure 11, at the left). In waveguide models presented on Figure 11 highly to conducting layers the distributed inductance L , and intermediate, by the dielectric – a distributed capacitance C is compared. The given modulated medium is the uniaxial. But in difference from homogeneous uniaxial medium exhibits, naturally, a dependence of properties not only on polarization, but also on a direction of propagation of a wave actived in it. Interleaving of inductivity and capacity in the given model results in an asymmetry secondary member, is analogous to colour, a case in point to the elementary, black-and-white. Lengthways highly symmetric directions taking into account polarisation of a wave the s

model has a 4 precise analytical forms for input impedance Z input (s=1,2,3,4), two of which (s=3,4) are equivalent:

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122

S. V. Ordin and W. N. Wang s Z input 

2 Z1s Zs 1  1  2 1s Z2

,

s

(2)

s

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where Z 1 - impedance either L , or C , and – Z 2 on the contrary, impedance either C , or L , accordingly.

Figure 11. The waveguide model of medium with modulated along an axis С electroconductivity and a 3 its equivalent circuits corresponding to its normal modes, each of which has the frequency dependence of impedance. s

Three nonequivalent Z input correspond to extending of normal modes on three nonequivalent chains figured on Figure 10 and allow to calculate precisely, in principle, a 3 spectroscopic reflectivity dependences r by formula:

r

Z vakk  Z input Z vak  Z input

,

where Z vak  377 - impedance of vacuum. Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

(3)

Giant Spatial Dispersion in the Region of Plasmon-Phonon Interaction …

123

It is possible to inject in addition the resistances ensuring signal attenuation (the elementary complication which is not changing a view of common formulas – is switching on serial and-or parallel C and-or L resistances). But for a qualitative explanation of the key differences between nonequivalent impedances it is enough to view singularities and s

asymptotics of purely reactive impedance Z input . 1, 2 Z input there correspond orientations kС, i.e. to reflexion from a plane  С.

1 1 2 LC * . In low-frequency region the impedance has the inductive character ( (2   1 )L ), but * At ЕС Z input (Figure 10, on the right: 1) has a pole near to frequency f1 

1

has a finite value ( 1 L ) on zero frequency. In high-frequency – the capacitive character ( *

1 Ñ ) also is tended to 0 on infinity. 2 2 At the ЕС Z input  i L , i.e. normal inductive (Figure 10, on the right: 2). 1 3 3( 4 )

Degenerate impedances Z input correspond to orientation kС, i.e. to reflexion from a plane С (Figure 10, on the right: 3). At that for normal incidence of light: EС, has a pole near to frequency f 3 

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*

2 . In low-frequency region the impedance has normal inductive LC

2    3* character ( L ), and in high-frequency – the capacitive (  ), but is tended to a C  finite value (  * ) on infinity. 3C The giant spatial dispersion of a normal wave is a consequence of nonequivalence of 2 3 chains (2 and 3) and impedances corresponding to them: Z input and Z input .

At kС it is applicable also as it is mentioned above the model of effective medium [54] 1, 2

giving qualitatively concordant results with impedances Z input . 2

At ЕС ( Z input ) also we will obtain purely plasma response:

2p  1    (1  2 ) and  2    , 

(4)

where  p - the plasma frequency defined by medial electro conductivity. 1

At ЕС ( Z input ) in a layered medium also there is a resonance responsible for occurrence of cross vibration defined by the free carrying agents on frequency

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S. V. Ordin and W. N. Wang

 T* 

d1  p , d1 and d2 – a thickness highly and was low conducting layer, d1  d 2

accordingly. On the same frequency the longitudinal plasmon should be observed within the limits of this model at EС that corresponds to purely inductive impedance. Thus the high-frequency lattice oscillator observed at ЕС (Figure 9с) was possiblly corresponds to vibrations of the plasma localized transversely a highly conducting layer. At Е С on the same effective frequency the longitudinal plasmon is exhibited. The free current carriers in a crystal a HSM are holes. Hence, the given oscillator corresponds to vibrations of negatively charged sublattice of silicon concerning the localized positive charge of the free current carriers. The opportunity of dual interpretation of the given oscillator in addition underlines conditionality of the adiabatic approach of partitioning of frequencies in the complex crystals.

Analysis of Conditions of an Initiation of the Spatial Dispersion Used above model not only allow, proceeding from reflexion spectra, to define various parameters of the modulated medium, but also bring an attention to the question on completeness of the phenomenological description of the spatial dispersion. The traditional description of the spatial dispersion in crystals is based on the primary assumption of homogeneity of a field at a wave-length of light  of much more interatomic spacing a [55]. Therefore initially contribution of the spatial dispersion was considered as

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the small correction on a parameter infinitesimally of Landau a

 ij ( , k )   ij ( ,0)   ijl * kl  ijlm * kl * km



1:

,

where   a   1 ,   (a  )  1 . For plasma the field exponentially changes on a depth of penetration, defined thickness of skin layer  which can be much less wave-lengths of light  . I.e. the field can be viewed as homogeneous on gauges much less  . On the other hand, in the modulated structures homogeneity of a material define by gauge of modulation which L , naturally, is much more a than interatomic spacing a . Therefore the parametrecharacterising a spatial dispersion, L  is much more unities and the spatial dispersion cannot be considered in the form of a series, as the correction on smallness parameter. At that, in principle, the dependence of a permittivity on a wave vector direction even is possible k at its aspiration to null for an ordinary wave: 2

 ii  ki k 0   jj  k j k 0 i

j

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Besides, in an one-dimensionally-incommensurate crystal the quasi-momentum conservation law k, in principle, cannot be observed at orientation along a incommensurability axis. At that the contribution to a permittivity is possible for all frequencies of phonons, without dependence from their wave vector q . For plasma oscillations as it is shown at calculation of impedances, disturbance of synchronism of vibrations of layers along a incommensurability axis can expel completely the contribution of plasma reflexion to a permittivity.

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FOUR-DIMENSIONAL PERMITTIVITY OF THE MODULATED MEDIA The standard computational method of the modulated crystals on the basis of Bloch functions [53] does not give a complete description of optical properties of a crystal a HSM. It is related by that at multiparameter action some equivalent, for three-dimensional crystals, experimental conditions become nonequivalent for crystals with an incommensurate symmetry axis, i.e. with pinch for an incommensurate crystal of dimensionality of a tensor of response. The electromagnetic wave, in principle, cannot be considered as the unicomponent action since contains an electric and magnetic component. But except for response of magnetic crystals, response as the dielectric, and conducting commensurate crystals can be well described as response to a alternating electric field within the limits of the frequency dependence of a permittivity of a crystal with the small correction on a spatial dispersion. However, the image of branches of various polarizations in 3-dimensional k-space is a projection onto it the allowed states from 6-dimensional pulse-field (pulse-co-ordinate) space. For homogeneous anisotropic medium wave vector orientation can be neglected. At that, a permittivity a tensor of the second order a connecting 2 vector: Dl = lmEm. The tensor of the second order can be given to main reference axes: Dm = mmEm and completely is characterised by a 3 independent components. For an uniaxial crystal of a 2 of its component are degenerate:

 DN    N     DN    0 D   0  L 

0

N 0

  EN      *  EN   L   E L  0 0

(8)

On it anisotropy of an uniaxial crystal can be mapped in the form of a 3 of twodimensional cross-sections 2s from which are equivalent. In case of regularly nonuniform medium with a period of inhomogeneities of the order of a light penetration depth it is necessary to consider orientation not only E, but also k. At that a permittivity a tensor of rank 4 a connecting 2 of tensor of rank 2: Dlr = lrmjEmj. A tensor of rank 4 too it is possible to result in principal axes (modulations): Dmj = mjmjEmj. But, since a tensor of an electromagnetic wave is antidiagonal:

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S. V. Ordin and W. N. Wang

 0  Emj=  E N( N )  E ( L)  N

E N( N ) 0 E N( L )

E L( N )   E L( N )  0 

(9)

Therefore to use this tensor difficult. It is necessary to analyses and compute each of 81 its components. Symmetry properties of the is one-dimensional-modulated medium it is possible to describe completely, having injected a four-dimensional vector of a field. In this case the permittivity a four-dimensional tensor of rank 2 also is resulted in to the diagonal view from a 4 independent components:

 E N( L )   ( L)  E  Ej =  (NN )  Dj =jjEjгдеjj = E  N   E (N )   L 

  N( L )   0  0   0 

0



( L) N

0 0

0 0

 N( N ) 0

      L( N )  0 0 0

(10)

Anisotropy of the uniaxial modulated crystal can be mapped in the form of a 4 of twodimensional cross-sections 2s from which are equivalent, in the complete correspondence with the offered waveguide model.

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REFERENCES [1] [2] [3]

[4] [5]

[6] [7] [8]

Turov E.A., Electrodynamics constitutive-material equations, Moscow,“Science”, Main edition of the physical and mathematical literature, 1983, 160 pp. H. Poulet, J.-P.Matheu, Vibration spectra and symmetry of crystals. "World", Moscow, 1973, 437 pp. S.V. Ordin, [B.N. Sharupin], J. Semiconductors (FTP), 32 (9), 924-932, 1998, Normal Lattice Oscillations and Crystalline Structure of Non-Isotropic Modifications of a Boron Nitride. L.D. Landau,E.M. Lifshits,Electrodynamics of continuous mediums, Moscow, “Science”, Main edition of the physical and mathematical literature, 1982, p.491-508. V. M Agranovich,M. D.Galanin, Transfer of energy of electronic excitation in condensed mediums, Moscow, “Science”, Main edition of the physical and mathematical literature, 1978, 383 pp. G.N.Zhizhin, B.N.Mavrin, V.F.Shabanov, Vibration spectra of crystals, Moscow, "Science", 1984, 232с. F.Bassani, J. Pastori-Parravichini, Electronic states and optical transitions in solids Moscow,“Science”, 1983, 392 pp. S.V.Ordin, B.N. Sharupin, IR Investigation Of Boron Nitride Structure Ordering, Abstract MRS95 - L, Symposium AAA at the 1995 Fall Meeting Symposium Title: Gallium Nitride and Related Materials.

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Giant Spatial Dispersion in the Region of Plasmon-Phonon Interaction … [9]

[10]

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[16]

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S.V.Ordin, B.N.Sharupin, M.I.Fedorov, V.I.Rumjantsev, E.V.Tupitsina, A.S.Osmakov, Normal Lattice Oscillations and Crystalline Structure of Anisotropic Modifications of a Boron Nitride, Reports of an open All-Russia symposium (with participation of scientists from the countries CIS): Amorphous and microcrystalline semiconductors, St. Petersburg, Russia, july, 1998, Publishing House of the Russian Academy of Science, 1998, p. 154. S.V.Ordin, M.I.Fedorov, V.I.Rumjantsev, E.V.Tupitsina, O.O.Taranovsky., Anisotropyof plasma oscillationsinpyroliticgraphites, Proceedings of IX Inter state Seminar: Thermoelectricsandtheirapplications, November, 1998, St.-Petersburg, RAS, pp. 114-122. S.V. Ordin,A.S. Osmakov,V.I. Rumyantsev,E.V. Tupitsina,A.I., Shelyh, Features of an electronic spectrum and a microstructure of graphite, IX National conference on growth of crystals, theses of reports, Moscow, Crystallography Institute of the Russian Academy of Science, 2000, p.534. N.L. Dimitruk,V. G. Litovchenko, V.L. Strizhevsky, Surface polaritons in semiconductors and dielectrics, Kiev, “NaukovaDumka”, 1989, 375 pp. S.I.Pekar, Crystal optic and additional light waves, Kiev, “NaukovaDumka”, 1982, 295 pp. Ordin S.V., Okamoto E, Fokin A.F., Dimensional plasma effects in porous SiC with metal, Proceedings of IX Interstate Seminar: Thermoelectrics and their applications, November, 2004, St.-Petersburg, RAS, pp. 154-162. Group-theoretical methods in the physics, Proceedings of the international seminars under the editorship of V. A. Koptsika, a v.1-3, Moscow, “Science”, Main edition of the physical and mathematical literature, 1986, 1500 pp. O. V. Kovalev, Nonreducible and induced representations and presentations of a Fedorov group, Moscow, “Science”, Main edition of the physical and mathematical literature, 1986, 367 pp. I.S.Zheludev, Physics of crystals and symmetry, Moscow, “Science”, 1987, 188 pp. Th. Rasing,P. Wyder,A.Janner, T. Janssen, Sol. State Commun.,41, 715, (1982). M.Born, H.Kun, the Dynamic theory of crystal lattices, Moscow, “Foreign literature”, 1958, 488 pp. A.M.Kosevich, Bases of a mechanics of crystal lattices, Moscow, “Science”, Main edition of the physical and mathematical literature, 1972, 280 pp. A.G. Hachaturjan, Theory of phase transitions and structure of solid solutions, Moscow, “Science”, 1974, 384 pp. H. Butger, Principles of the dynamic theory of a lattice, Moscow, “World”, 1986, 390 pp. E.N.Nikitin,J. Sov. Solid State Phys. (FTT), 1959,т.1, с.340. E.N. Nikitin,V. G.Bazanov,V. I. Tarasov, Thermoelectric properties of silicides of transition metals. The Invention № 18631с a priority from 5/12/1960. P. V. Geld, F.A.Sidorenko,Silicide of transition metals of the fourth period, Moscow, "Metallurgy", 1971, 584 pp. R.Noks, A.Gold, Symmetry in a solid, Moscow, “Science”, Main edition of the physical and mathematical literature, 1970, 424 pp.

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[27] S.V.Ordin,V.K. Zaytsev, K.A.Rahimov,A.E. Engalychev,J. Sov. Phiys. Solid State(FTT), 23(2), 1981, pp. 621-623, Peculiarities of a Crystalline Structure and Thermopower of Higher Manganese Silicide, [28] S.V. Ordin, V.K.Zaitsev, M.P. Shcheglov, Superstructure Ordering of Higher Manganese Silicide Single Crystal in Incommensurate Region, Proceedings of V Interstate Seminar: Thermoelectrics and their applications, November, 1996, St.Petersburg, Publishing House RAS, 1997, p. 106-108, p.213-214. [29] S.V.Ordin V.K.Zaitsev,M.P. Shcheglov,M.I.Fedorov, Superstructure orderingand thermoelectric properties of higher manganese silicide based materials, Proc. XVII th Int. Conf. Thermoelectrics(ICT ’ 98),1998, pp.296-299. [30] S.V. Ordin,M.P.Scheglov, Incommensurability in Higher Silicide of Manganese, Proceedings of VII Interstate Seminar: Thermoelectrics and their applications, November, 2000, St.-Petersburg, Publishing House of the Russian Academy of Science, 2000, p. 203-208, p.436-437. [31] J.A. Izjumov, Neutron diffraction on long-period structures, Moscow, “EnergoAtom”, 1987,200 pp. [32] S.V.Ordin V.K. Zaytsev, A.E. Engalychev,V.A.Solovjov,Electronically microscopic studies of materials on the basis of the higher silicide of manganese, Proceedings an all-Union seminar: Materials for thermoelectric converters, Leningrad, Publishing House of the Russian Academy of Science 1987, pp.99-100. [33] S.V. Ordin,A.E. Engalychev,A.V. Fokin,Incommensurability and solid-phase transitions in the higher silicide of manganese: weakly ordered crystals, Proceedings of IX Interstate Seminar: Thermoelectrics and their applications, November, 2004, St.Petersburg, RAS, pp.21-25. [34] S.V. Ordin,A.E.Engalychev, Incommensurability and solid-phase transitions in the higher silicide of manganese: highly orderedcrystals, Proceedings of IX Interstate Seminar: Thermoelectrics and their applications, November, 2004, St.-Petersburg, RAS, pp. 26-42. [35] S.V. Ordin etc, J. Sov. Phiys. Solid State, 23 (2), 1981, pp. 621-623, Pecularities of a Crystalline Structure and Thermopower of Higher Manganise Silicide. [36] Ju.I. Sirotkin, M.P.Shaskolsky, Crystal Physics Basis, Moscow, “Science”, Main edition of the physical and mathematical literature, 1975, 680 pp. [37] M.V.Volkenshtejn, L.A.Gribov, M.A.Eljashevich, B.I.Stepanov, Vibrations of molecules, Moscow, “Science”, Main edition of the physical and mathematical literature, 1972, 699 pp. [38] N.F.Mott, E.A.Davis, Electron processes in non-crystalline materials, Clarendon Press, Oxford, 1979, 750 pp. [39] S.V.Ordin, A.I.Shelyh, Electronic Reflection and Energy Band Structure in Higher Silicide of Manganese, Proceedings of VIII Interstate Seminar: Thermoelectrics and their applications, November, 2002, St.-Petersburg, RAS, pp. 66-71, 439. [40] J.M.Ziman, Principles of the theory of solids, Cambridge, University Press, 1972, 450 pp. [41] S.V. Ordin, V.K. Zaitsev, V.I. Tarasov and M.I. Fedorov, Optical Properties of Higher Silicide Manganese, J. Sov. Phys. Sol. State (FTT) 21, 1979, p.1454. [42] W.A.Harrison, Electronic Structure and the Properties of Solids, W.H.Freeman and Company, San Francisco, Moscow, “World”, 1983, 790 pp.

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[43] M.A.Krivoglaz, the Diffuse scattering of a X-ray and neutrons on fluktuatinginhomogeneities in imperfect crystals, Kiev, “NaukovaDumka”, 1984, 287 pp. [44] R.Currat, J.Etrillard, S.B.Vakhrushev, S.V.Ordin, L.Bourgeois, P.Bourges, Collective Dynamics of an Incommensurate MnSi Composite, Proc. of International Conference on diffusing neutrons, Paris, France, 2004, p. 211-212. [45] L. Bourgeois, P. Bourges, R. Currat, J. Etrillard, S.B. Vakhrushev, S.V. Ordin, Structures and Phase Transitions, in book Laboratoire Leon Brillouin's report 2004, p.10. [46] Academician L.I. Mandelshtam, Lectures on a vibration theory, Moscow, “Science”, 1972, 470 pp. [47] Ordin S.V., Sokolov I.A, Zjuzin A.J., Parametrical Interaction of Normal Oscillations in anisotropic crystals, Physico-Technical institute of A.F.Ioffe of the Russian Academy of Sciences, St.-Petersburg, Russia, Proceedings of X interstate seminar: Thermoelectrics and their application, on November, 14-15th 2006, pp.144-149. [48] S.V. Ordin, A.S. Osmakov, V.I. Rumyantzev, E.V. Tupitsina, and A.I. Shelykh, J. Surface, X-ray, synchrotron and neutron researches (Moscow) 5, 108, 2003. [49] Ch. Kittel, Thermal Physic, John Willey and Sons, Inc., New York, Moscow, “Science”, Main edition of the physical and mathematical literature, 1977, 366 pp. [50] M.C.Steele, B.Vural, Wave Interaction in Solid State Plasmas, McGraw-Hill Book Company, New York, Moscow, “Atom”, 1973, 248 pp. [51] V.V.Nikolsky, T.I.Nikolsky, Electrodynamics and wave propagation, Moscow, “Science”, Main edition of the physical and mathematical literature, 1989, 543 pp. [52] V.P.Shestopalov, A.A.Kirilenko, S.A.Masalov, J.K.Sirenko, Kiev, Resonancewave dispersion, “NaukovaDumka”, 1986, 232 pp. [53] A.Yariv, P.Yen, Optical waves in crystals, Moscow, ”World”, 1987, 616 pp. [54] Max Born, Emil Volf, Principles of Optics, Moscow, “Science”, Main Publishing house of the Physical and mathematical Literature, 1970, 855 pp., Second (revised) Edition, Pergamon press, 1964. [55] V. M. Agranovich, V. L. Ginzburg, Crystal optic taking into account the spatial dispersion, Moscow, “Science”, Main edition of the physical and mathematical literature, 1980, 365 pp.

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In: Optical Lattices: Structures, Atoms and Solitons Editor: Benjamin J. Fuentes

ISBN: 978-1-61324-937-6 © 2012 Nova Science Publishers, Inc.

Chapter 4

PECULIARITIES OF MAGNETOOPTICAL PROPERTIES IN CRYSTALS WITH MIXED VALENCE CENTERS Lubov Falkovskaya1** and Valentin Mitrofanov2 1

Institute of Metal Physics, Ural Branch of Russian Academy of Sciences, Ekaterinburg, Russia 2 Institute of Metallurgy, Ural Branch of Russian Academy of Sciences, Ekaterinburg, Russia

ABSTRACT

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This chapter studies the characteristics of the Faraday effect and magnetic circular dichroism in crystals containing iron-group ions of different valences. The cause of such ions appearance can be nonisovalent substitutions or vacancies in the anion and cation sublattices. Quite often, mixed-valence complexes have features characteristic of the Jahn-Teller ion. Genealogically this is related to the fact that usually one of two states of a single ion with configurations dn or dn1 is orbitally degenerate. Local low-symmetry fields produced by the source of the excess charge, remove this degeneracy. Nevertheless, the ground state of the whole complex of mixed valence may be degenerate due to the effects of the excess charge transfer between 3d-ions of the complex. In this publication we consider just this case, when along with the traditional properties of the Jahn-Teller centers features appear caused by the redistribution (reorientation) of the excess charge between the ions of the complex, and the degeneracy is lifted due to external perturbations or cooperative interactions. Removal of the orbital degeneracy of the cluster leads to a nonuniform distribution of excess charge q between the magnetic ions and the appearance of a relatively large dipole moment of the cluster, proportional to qR, where R is the distance between the source of the excess charge and the nearest 3d-ions in the cluster. The presence of mixed valence clusters in the crystal may lead to an anomalous increase of the antisymmetric components of the dynamic dielectric tensor and, consequently, of the Faraday effect and the magnetic circular dichroism. If a set of energy levels of such clusters with localized on them excess charge contain degenerate levels with unfrozen orbital angular momentum, their contribution to the magneto-optical effects can be significant **

E-mail address: [email protected], [email protected]

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132

Lubov Falkovskaya and Valentin Mitrofanov The influence of the mixed valence centers on the complex Faraday effect of crystals with spinel, garnet and perovskite structures is considered. It is shown that the electric dipole transitions in such clusters lead to a significant magneto-optical activity in both infrared and visible ranges of spectrum. The contribution of mixed valence clusters in a magneto-optical properties of crystals becomes quite noticeable even at relatively low 17

19

3

concentrations (  10  10 cm ). The intensity of the corresponding transitions is comparable to or higher than of the allowed electro-dipole crystal field single ion transitions.

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1. INTRODUCTION Among the inorganic materials, transition-metal oxides have acquired special attention owing to the vast diversity of structures and a wide spectrum of exotic properties (such as superconductivity, colossal magneto-resistance, multiferroism, metal-insulator transition, magneto-optics, spin-dependent transport) with numerous potential technological applications. These materials belong to strongly correlated systems, and in most cases their behavior cannot be explained within the framework of one-electron band theory [1]. An important problem of the physics of these strongly correlated systems concerns the nature of their electronic structure and energy spectrum. Optic and magneto-optic techniques are well suitable for studying them. In addition, magneto-optical (MO) spectroscopy seems to be the favorable method for monitoring the distribution of magnetic ions within transition-metal oxides (for instance spinel type ferrites MexFe3-xO4 [2], CoFe2O4 [3]), where it is possible to correlate between the electronic transition‟s intensity and the concentration of the metal ions directly in the corresponding crystal site. The present chapter is devoted to investigation of peculiarities of the Faraday effect (FE) and magnetic circular dichroism (MCD) in transition metal oxides containing iron-group ions of different valences. The cause of such ions appearance can be non-isovalent substitutions or vacancies in the anion and cation sublattices. In such cases the important role is played by intervalence charge transfer (IVCT) or intervalence site transfer (IVST) transitions in which an electron, through optical excitation, is transferred from one cation to a neighboring one [4]. Compounds containing an element in two different oxidation states, mixed valence compounds, often show intense absorption and magneto-optical activity in the visible region which can be attributed in particular to IVCT transitions. The distinction between IVCT, IVST transitions and crystal field (CF) transitions is that the two first transitions involve two cations while the third one is a single-ion transition [2]. Possible candidates include YIG:Ca [5], where the presence of Fe4+ ions along with Fe3+ ions in tetrahedral positions takes place due to the charge compensation of the diamagnetic Ca2+, Co3O4 [6], manganites where the charge transport is believed to be due to Mn3+–Mn4+ polaron hopping similar to the Fe2+–Fe3+ hopping in Fe3O4 and of partially Mg2+ or Al3+ substituted Fe3O4 [7]. It should also be noted that when the spinel ferrites present a large surface/volume ratio they can be oxidized into mixed-valence defect ferrites [8]. These original spinels have magnetic, magneto-optical and optical properties making them interesting materials for high density recording media. In this chapter we suppose that near the defects of crystal lattice (nonisovalent substitutions or vacancies) the magnetic clusters that is the mixed valence (MV) centers

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Peculiarities of Magnetooptical Properties in Crystals …

133

containing 3d-ions of one and the same element with different valences are formed. At small impurity concentrations the MV centers do not interact with each other and the double exchange takes place mainly on MV clusters. Quite often, mixed-valence (MV) complexes have features characteristic of the JahnTeller (JT) ion. Genealogically this is related to the fact that usually one of two states of a single ion with configurations dn or dn1 is orbitally degenerate. Local low-symmetry fields produced by the source of the excess charge, remove this degeneracy. Nevertheless, the ground state of the whole complex of MV may be degenerate due to the effects of the excess charge transfer between 3d-ions of the complex. In this case when along with the traditional properties of the JT centers features appear caused by the redistribution (reorientation) of the excess charge between the ions of the complex, and the degeneracy is lifted due to external perturbations or cooperative interactions. Removal of the orbital degeneracy of the cluster leads to a nonuniform distribution of excess charge q between the magnetic ions and the appearance of a relatively large dipole moment of the cluster, proportional to qR, where R is the distance between the source of the excess charge and the nearest 3d-ions in the cluster. The presence of MV clusters in the crystal may lead to an anomalous increase of the antisymmetric components of the dynamic dielectric tensor and, consequently, the FE and the MCD. If a set of energy levels of such clusters with localized on them excess charge contain degenerate levels with unfrozen orbital angular momentum, their contribution to the magnetooptical effects can be significant. It is shown that the electric dipole transitions in such clusters lead to an appreciable magneto-optical activity in both infrared and visible ranges of spectrum. In this the magnetooptical properties of crystals depend strongly on the Hund's-rule coupling on a single 3d-ion

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n

m

with electron configuration t 2 g e g , the symmetry of the cluster, the relative magnitudes of the transfer integrals of the excess charge between the ions of the cluster and the lowsymmetry fields produced by defects. The comparison of experimental data with the results of theoretical calculations of magnetooptical activity of crystals containing non-isovalent substitutions in spinels, YIG and manganites was carried out.

2. MIXED VALENCE CENTERS Let us consider one of the most typical irregularity in crystals with spinel structure represented by the trigonal clusters involving three 3d-ions with the electronic configurations

3d n  3d n  3d n 1 (n = 2…9). Since the sites occupied by the 3d-ions are equivalent the excessive charge of the 3d n 1 ion (hole) can be delocalized among the cations. This trigonal irregularity has four equivalent dispositions related to the four C3 axes in a cubic lattice. Although each cluster has trigonal symmetry and therefore is optically anisotropic, this anisotropy remains hidden and the crystal is expected to possess cubic symmetry. In such orientationally degenerate clusters exhibiting delocalization of the excess charge between 3dcations at least one of the metal ions turns out to be orbitally degenerate in the ground or excited configuration and therefore possesses an unquenched orbital angular momentum. A trigonal cluster in the spinel structure consisting of two magnetic ions with 3d n configuration that are supposed to have an orbitally non-degenerate ground state and one

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Lubov Falkovskaya and Valentin Mitrofanov

magnetic ion with configuration 3d n 1 and orbitally degenerate ground term is presented on Figure 1. Such a cluster type irregularity in a cubic lattice can appear due to the presence of the cation vacancy or non-isovalent cation substitution in the octahedral positions. For sake of definiteness hereunder the case of a trigonal cluster with the typical electronic configuration

3d 3  3d 3  3d 2 (for example, Cr3+- Cr3+- Cr4+) will be considered. The ground state of the ions with the configuration 3d 3 t 23g in a cubic crystal field (Cr3+) is the orbital singlet 4A2, while the ground state of 3d 2

  t  ions is the orbital triplet ( T -term). The model of the 2 2g

3

1

system includes two main physical factors, namely, the influence of the low symmetry crystal field on the orbitally degenerate ions and the double exchange that appears due to the delocalization of the extra hole within the cluster. Consequently, we shall represent the Hamiltonian of this kind of cluster irregularity by:

H  H0   H i  H res .

(1)

i

In Eq. (1) H0 is the zeroth order Hamiltonian that includes cubic crystal fields acting on the isolated centers, Hi is the Hamiltonian of low-symmetry crystal fields that act on the constituent Cr ion residing in the site i ( i =1,2,3), Hres is the Hamiltonian describing the

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resonant interaction (double exchange) in the three center irregularity. This interaction is responsible for the hopping of the “extra” hole between the Cr ions.

Figure 1. Scheme of the trigonal MV cluster near cation vacancy in spinel lattice. V-vacancy, C-cation, А-anion, h-excess charge.

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Peculiarities of Magnetooptical Properties in Crystals …

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The orbital triplets are split by the joint action of the trigonal and rhombic crystal fields. Therefore the Hamiltonians Hi involving both contributions can be represented as:

1 3 H1  (l(x1)l(y1) T2  l(x1)lz( 1) T2  l(y1)lz( 1) T2 )  h0 (  E  E ), 2 2 1 3 H2  (l(x2 )l(y2 ) T2  l(x2 )lz( 2 ) T2  l(y2 )lz( 2 ) T2 )  h0 (  E  E ), (2) 2 2 H3   (l(x3)l(y3) T2  l(x3)lz( 3) T2  l(y3)lz( 3) T2 )  h0  E . Here  is the parameter of the trigonal field that is represented in the Cartesian frames,

l(xi ) , l(yi ) , lz( i ) are the direction cosines of the local trigonal axis directed from ion with number    i towards the opposite oxygen ion ( l1  1 1 1, l2  1 1 1, l3  1 1 1 ),  are the orbital operators determined in basis of orbital 3T1-triplet functions, h0 is the parameter describing the low-symmetry (rhombic) component of the crystal field produced by the cation vacancy. Let‟s suppose that the trigonal field gives the main contribution to splitting of 3T1-term. At this trigonal doublet turns out to be the lowest state. So the orbital operators E , E in the ground trigonal basis have the following form:

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 0 1

0

 E    ,  1 0

i

 E    .   i 0

(3)

The resonant interaction in an impurity cluster resulting in the hole delocalization can be described by the following Hubbard type Hamiltonian:

H res 

mm  c m c m   (t  1

1

mm1  ,

 h.c.) ,

(4)

 where c m and c m are the creation and annihilation operators of t 2 g -hole on the center

m in an orbital state  and spin projection is  . For the sake of simplicity we assume that the Hund coupling (ferromagnetic intraatomic exchange interaction) is much larger than the mm

hopping integrals t  1 that seems to be a realistic assumption. 2

 

The ion with configuration 3d is in the orbital triplet state 3 T1 t 22g that is split by the trigonal field into an orbital doublet and a singlet. For the sake of brevity let us confine ourselves by trigonal doublet states of these ions. In the absence of the transfer processes the “excessive” t 2 g -hole of the d

2

ion occupies six-fold degenerate state (two-fold on-site

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136

Lubov Falkovskaya and Valentin Mitrofanov

degeneracy and three equivalent positions). When the transfer processes are involved the degeneracy is removed. One can see that the hole transfer occurs through the spin sites and therefore the rate of the jumps is spin-dependent in accordance with the general concept of the double exchange [9, 10]. The double exchange is known to produce a strong ferromagnetic effect [10] (although it is not always valid for the so called frustrated systems) and therefore stabilizes the levels corresponding to the maximum value of the total spin projection Ms (parallel orientation of the spins). To take advantage of the symmetry properties the six stationary states of the system entire can be classified according to the irreducible representations Γ of the cluster symmetry group C3v. The result is the following:

  A1  A2  2E .

(5)

The cluster wave functions can be presented in the form [11]: 3

    (C( ),k k (  )  C( ),k k ( )) . k 1

In Eq. (6) the functions

(6)

k () are represented in terms of fully antisymmetric products

of the wave functions of three ions for which the excess t2g-hole is located on site k (k = 1-3),

 

while the corresponding cation with configuration d 2 t 22g

is in one of trigonal states

E(   ).

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The energies of the system are found as [11]:

EA1  h0  b1  2b2 ,

EA2   h0  b1  2b2 ,

EE   b1 / 2  {( h0  b2 )2  ( 1.5b1 )2 } 1 / 2 , b1  [t1  2t 2  t 3 ] / 3 ,

(7)

b2  [t1  t 2  2t 3 ] / 3 ,

where b1 and b2 are the transfer integrals for a t2g-hole, 12 13 23 12 13 23 t xy , xy  t xz , xz  t yz , yz  t 1 , t yz , yz  t xy , xy t xz , xz  t 2 ,

(8)

12 13 23 t yz , xz  t xy , yz  t xz , xy  t 3 .

The scheme of the energy levels, Eq. (7), of the cluster states is presented on Figure 2 for the case when b1  b2 .

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Peculiarities of Magnetooptical Properties in Crystals …

Figure 2. The energy levels of the system with t2 g -hole as functions of the parameter

137

b1 .

3. THE SPECTRUMS OF ABSORPTION AND COMPLEX FARADAY EFFECT IN THE INFRARED REGION OF ENERGIES

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Let us investigate the contribution of MV clusters described above to the optical and magnetooptical properties of crystals in the infrared region of energies [12-14]. The situation when the two-fold degenerate state E turns out to be the lowest one presents the essential interest. According to (7) it realizes at h0 , b2  0 and b1 /b2  1 /2 . The impurity clusters lead to the essential renormalization of the symmetric (diagonal) growth of the antisymmetric (non-diagonal)

 () and anomalous

 a () components of dielectric permeability

tensor 4

 ( )     4    Px ,rs Px ,rs   , s r 1 4



(9)



 a ()  2i   Px ,rs Py ,rs     Py ,rs Px ,rs   . s r 1

(10)

Here   is the sum of electronic and lattice contributions to the dielectric permeability at the values of frequencies much larger than the resonance frequencies in the MV compleх, the symbols s and r number the impurity clusters and related trigonal axes correspondingly (there are four C3 axes in a cubic lattice), ......  stands for the Fourier-transform of the twotime Green function,

z

axis in the coordinate system

x, y , z

is parallel to the crystal

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138

Lubov Falkovskaya and Valentin Mitrofanov The dipole momentum operator P of MV center can be represented in the following form 3

P  q  R k ˆ k ,

(11)

k 1

where q is the excess charge of the hole, Rk is the radius vector of ion with number k in a triangular system, operator ˆ k describing the inhomogeneous distribution of hole charge density on triad ions has the following matrix elements













1 1  k 2 2  C(1)1 ,k C(2 ) 2 ,k  C(1)1 ,k C(2 ) .2 ,k .

(12)

The diagonal elements of the operator P matrix determine the value of the dipole momentum in the given state, while the non-diagonal ones are related to the electric dipole transitions between states of different types. If absorption and FR are determined by renormalization of dielectric permeability, the absorption coefficient () and complex specific rotation Ф(  ) of electromagnetic wave polarization plane have the form:

 ( )  2k0 Im  ( ) /[Re  ( )   ( )]

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Ф() 

k0 2

1 /2

/2 ,

(13)

() ()[(1   a () / ())1 / 2  (1   a () / ())1 / 2 ] .

(14)

where k0   , с is the light velocity,  ( ) is the symmetric (diagonal) component of

c

magnetic permeability tensor.Let‟s investigate here the contributions to

 () and  a ()

connected with the removing of degeneracy on the cluster ground state. The main mechanisms determining the splitting of the E-term are spin-orbit interaction ( Hso ) and random crystal fields ( Hrand ) . The Hamiltonian describing the energy of an isolated MV cluster associated with the trigonal axis r may be presented in the following form: (r ) H r  H so  H rand  V    V    V( r )  , (r ) H so 





3  mx( r ) m(yr ) nz  mx( r ) mz( r ) n y  mz( r ) m(yr ) nx  2





 D (2nz 2  nx2  n 2y )   3(nx2  n 2y )  , H rand  h    h   , 1 1

0

1 0 1

1  0 i   . 0

    ,     ,    2  0 1 2 1 0 2i

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(15)

Peculiarities of Magnetooptical Properties in Crystals …

139

In Eqs. (15)  and D are the parameters of the first and second orders of spin-orbit interaction for the cluster, m(r ) is a unit vector directed along the trigonal axis of the cluster, the coordinate system x, y, z is connected with the C4 axes of a cubic crystal, nx , ny , nz are the direction cosines of magnetization.    The orbital operators   ,   ,   are determined in the space of functions of the triad doublet levels EE , h and h are the components of random crystal field acting on the MV 

cluster. The normal distribution with the dispersion  for these random fields will be used [15]:

g( h , h )   2 exp{( h2  h2 ) / 2 }. The absorption coefficient

(16)

() has the most simple form for samples magnetized

along the crystallographic [001] axes

 ( )   ( )k0 p 2CJT  (  i ) 2[  i   /(2 3)]exp{[4(  i ) 2   2 / 3]/  2 },

(17)

i

 ()  32 2 2 /[Re  ()   () ]1/2 , C JT is the volume concentration of MV centers, (z) is teta-function, i  E( i )  E(E ), i  A1 , A2 , E . The parameter p depends upon the type of transition E   : p   (E  )qR / 3 ,  (E  ) is the Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

where

reduction factor for the corresponding transitions: 2

 (E  A1 )  1  (b2  h0 ) /(( h0  b2 )2  (1.5b1 )2 )1/2 2

 (E  A2 )  1  (b2  h0 ) /(( h0  b2 )2  (1.5b1 )2 )1/2 2

 (E  E )  9 / 4b12 /(( h0  b2 )2  (1.5b1 )2 ) . The expression for the Faraday rotation () at the arbitrary M direction can be presented in the form 4

( )  9 [ ( ) / ( )]1/2 k0 p 2C JT  C 2 (n, r ) ( / Er ) /[ri2   2 ] r 1 i

C(n, r )  m(xr )m(yr )nz  m(xr )m(zr )ny  m(yr )m(zr )nx , ri  i  Er /2,

Er  [32C 2 (n, r )  h2  h2 ]1/2 ,

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av

,

(18)

140 where symbol ...

Lubov Falkovskaya and Valentin Mitrofanov av

means averaging with respect to the random fields, Er is splitting

energy of the ground E  -term by random fields and spin-orbit interaction. The typical frequency dependencies for the absorption coefficient and the Faraday rotation angle are presented on Figutres 3-5. The form of absorption lines in the presence of random crystal fields is shown on Figure 3. In this case the values i , which are proportional to the transfer integrals, b1, 2 turn out to be threshold for the frequency dependence of absorption coefficient.

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Figure. 3. Spectral distribution of absorption lines random crystal fields (M

[001]),

E  i

from MV complexes in the presence of

w   /i ,  /i  0.1 ,  /i  0.1.

Figure 4. The influence of spin-orbit interaction on the form of lines in the absence of random crystal fields (M

[111]),

E  i

in the spectrum ()

w   /i ,  /i  0.22 ,  /i  0.

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Peculiarities of Magnetooptical Properties in Crystals … Just these same frequencies

141

i determine the resonance frequencies in the dependence

() . It should be noted that these transitions are characterized by the fine structure of lines (Figure 4) connected with the spin-orbit interaction. Correspondingly the spectrum should depend upon the magnetization direction. The characteristic peculiarity of () spectrum is the dependence of Faraday effect

sign (Figure 5) upon the type of the transition: for the transitions E  A1 , A2 and

E  E , E these signs turn out to be opposite (the transition E  E connects the split states of the ground E -term).

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Figure 5. The frequency dependence of the impurity contribution to the Faraday rotation due to the transitions

E  E , E , A1 and A2

 /h0  0.087.

(M [001]) , w   /h0 , b1  b2  0.3h0 ,

4. FARADAY EFFECT AND MAGNETIC CIRCULAR DICHROISM IN THE VISIBLE PART OF SPECTRUM 4.1. Spinel Type Crystals Let us now examine the contribution of trigonal impurity clusters to the magnetooptical properties of spinel in the visible range of spectrum [16]. The case of a trigonal cluster with the typical electronic configuration 3d5  3d5  3d4 (for example, Mn2+- Mn2+- Mn3+) will





be considered here. The ground state of the ions with the configuration 3d 5 t23g e g2 in a cubic 2+

6

crystal field (Mn ) is the orbital singlet A1, while the ground state of the high spin





3d 4 t23g eg ions is the orbital doublet (5E-term). The Hamiltonian of the cluster has again the form (1)-(2). The explicit form of the orbital operators

 E depends upon the state of ion

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142

Lubov Falkovskaya and Valentin Mitrofanov





with configuration 3d 4 . For the orbital doublet ( 3d 4 t 23g eg configuration) they are given by Pauli matrices:

1

0 

 0 1

E    , E    .  0  1  1 0 The

  3z

standard 2

basis



r ,  3 x  y 2

2

of

2



(19)

E E and 4 2 2 is used. For ion with configuration 3d (t2 g e g ) the the

cubic

E

doublet

orbital operators are determined on the trigonal basis functions E and E analogously to the

 

case of ion with configuration 3d 2 t 22 g (see Eq. (3)). Again the six-fold degenerate states of the cluster with e g -or t2 g - holes being transferred among three complex ions can be expanded into irreducible representations (5) of











C3v group. The corresponding wave functions of the d 5 t23g eg2  d 5 t23g eg2  d 4 t23g eg



system can be presented in the following form:

(gr ) 

3

(C(),kk (E )  C(),kk (E )) ,

k 1

where the functions

(20)

k (E ) are the products of the wave functions of the constituent ions

with the excess eg-hole instantly localized on the site k (k=1, 2, 3). It is explicitly indicated





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that the cation in configuration d 4 t23g e g gives rise to a two-fold orbitally degenerate cubic term 5E. The symbol    ,  numerates the functions of the cubic E-basis. ) The expressions for energies E( gr  of the ground state of the system are found as [17]:

E(Agr1 )  h0  t4  3(t5  t4 ) / 2 , E(Agr2 )   h0  t 4  (t5  t4 ) / 2 ,



) EE( gr  (t4  t5 ) / 4  h0  (t5  t4 ) / 22  9(t4  t5 )2 / 16 



1 /2

,

(21)

t312z 2  r 2 , 3 z 2  r 2  t313y 2  r 2 , 3 y 2  r 2  t323x 2  r 2 , 3 x 2  r 2  t4 , tx122  y 2 , x 2  y 2  tz132  x 2 , z 2  x 2  tz232  y 2 , z 2  y 2  t5 , where the energies are counted off the ground ferromagnetic state of the crystal without regard for the spin-orbit interaction. The energy levels

) ) EE( gr , EE( gr belong to the repeated  

E–representations (Eq. (5)). It is reasonable to assume that in the case of 90 - superexchange

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Peculiarities of Magnetooptical Properties in Crystals …

143

the main role plays the t 5 pathway while t 4 contribution is weak ( t 4  t5 ). A scheme of the energy levels, Eq. (21), obtained at t4  0.1t5 is presented in Figure 6.

Figure 6. The energy levels of mixed valence irregularity with the delocalized

e g -hole as functions of

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the transfer integral t 5 .





The excited state of the 3d 4 ion is the orbital triplet state 5T2 t22g e g2 that is split by the trigonal field into an orbital doublet and a singlet. The energy pattern of the MV cluster in the excited configuration











d 5 t23g eg2  d 5 t23g eg2  d 4 t22g eg2



is determined by the transfer

integral between the t2g-orbitals (in contrast to the case of eg-hole in the ground state). The cluster wave functions in the excited state can be presented in the form [11]

(ex ) 

3

(C(),kk ( )  C(),kk ( )) .

(22)

k 1

Here the functions

k () are analogous to those in Eq.(6).

The energies of the system in excited state are found as [11]

E(Aex1 )   cub  h0  b1  2b2 ,

E(Aex2 )   cub  h0  b1  2b2 ,

) EE( ex   cub  b1 / 2  {( h0  b2 )2  ( 1.5b1 )2 } 1 /2 ,  Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

(23)

144

Lubov Falkovskaya and Valentin Mitrofanov

where cub is the value of 3d states splitting by cubic crystal field (cub=10Dq), b1 and b2 are the transfer integrals for a t2g-hole, determined in Eq.(7). The scheme of the energy levels, Eq. (23), of the excited cluster states is presented on Figure 7 (all the energies are counted off the value  cub / h0 and it is assumed by way of example that b1  b2 ).

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Figure 7. The energy levels of the system in excited state as functions of the parameter b1 .

We have thus the two sets of the energy levels corresponding to the ground and excited configurations of the ion being in orbitally degenerate state separated by the energy  cub . Let us consider the electric dipole transitions in the visible range of frequencies that is ) (ex ) transitions between the levels of ground  (gr  and excited   groups. It is evident that

because of the orthogonality of the functions of e g and t2 g states the direct matrix elements of the operator

 k between functions from different sets, Eqs. (20) and (22), are equal to

zero and as a result the corresponding matrix elements of the dipole momentum operator, Eq. (11), vanish. The contribution of MV clusters to the complex tensor of dielectric permeability  ij () takes place in the region of the mentioned frequencies only due to the mixing of the wave functions of excited, Eq. (22), and ground, Eq. (20), states by the trigonal field or/and by the double exchange interaction. The functions of the same symmetry are mixed and the mixing coefficients are proportional to the ratio

 (   /  cub , t312z 2  r 2 , xy /  cub )

~ (gr )  (gr )  (ex ) .

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Peculiarities of Magnetooptical Properties in Crystals …

145

As a result the matrix elements of the dipole momentum operator between the functions

(gr )

(ex)

and  may be expressed in terms of the matrix elements relating only to functions (ex)

of excited states  as follows:

~ (gr ) P (ex )   (ex ) P (ex ) .

(25)

In order to evaluate the dielectric permeability tensor it is necessary to determine the (ex)

(ex)

from the set (23), the proper wave functions  of the sublevels and the matrix elements of the dipole momentum operator within these functions. The main mechanisms determining the splitting of the E-term are spin-orbit interaction ( Hso ) and random crystal fields ( Hrand ) (see Eqs. (15)-(16)). splitting of the doublet levels EE



The matrices of the dipole momentum operators in the space of the functions ) ) ( ex) ( ex) ( ex) A( ex , A( ex2 ) , E( ex  ,   .  ,   , 1 E E E 







obtained after splitting the doublet levels have the

following form:

   Px( r )  m(xr ) (U( r )  3U( r ) ),    Py( r )  m(yr ) (U( r )  3U( r ) ) ,   Pz( r )  2 m(zr )U( r ) ,   3qR 6 .

(26)

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In Eq. (26) R is the distance between the source of the excess charge and 3d-ions of MV

 (r)  (r)

irregularity. The matrices of the operators U , U are given in [16]. Finally, the expression for the complex specific rotation Ф of the light wave polarization plane can be presented as

Ф 

V  (r )

2

 ( ) 2 2 k0     exp( E / T )  ( ) Z s r 1 3r  ,  4

2  ( 2  E    2 )  2i 2   2 (E   2    2 )2  4 2  2

Y ( ) ,



(27)



V( r )  6 mx( r ) m(yr ) nz  mx( r ) mz( r ) n y  mz( r ) m(yr ) nx ,



r  V2  V2  (V( r ) )2



1/2

.

Here Z means the statistical sum including all levels, index  numbers the levels belonging to the ground group, while  numerates the excited levels, E  E  E ,

 is the half-width of the individual line. The coefficient Y ( ) takes the values Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

146

Lubov Falkovskaya and Valentin Mitrofanov

(1   ),  2 or (1   2 ) , depending on the pair of levels (  , ) for which the electric dipole transition

in

  2(b2  h0 )

the

visible

range

of

frequencies

takes

place,

4(b2  h0 )2  9b12 .

The real part of the complex specific rotation Ф describes the Faraday effect appearing in the rotation of polarization plane of linearly polarized wave having passed through a magnetic crystal. The imaginary part of the complex specific rotation Ф is related to the magnetic circular dichroism, arising from the difference in absorption coefficients for right- and left circular polarized light. The results of numerical calculations of FR and MCD spectrums are presented in Figs. 811. It was taken into account that in the case of the 900 superexchange the inequality

t 4  t5 is well, so it was considered that t4  0, t 5  t . In order to reduce the number of parameters the following relation was chosen between the transfer integrals in the excited configuration: b2  1 8 b1 . The dimensionless parameters for frequency of electromagnetic radiation and energetic characteristics of spectrum are scaled to the spin-orbit coupling parameter  . The following set of parameters was used for all numerical simulations: T/ = 0.1, h0/ = 0.25, b2/ = 0.5, t/ = 1.5, cub/ =200, D =0,   , / = 0.001. Let us first consider the peculiarities of the frequency dependencies of FR caused by MV irregularities. At the chosen values of parameters the A1 level turns out to be the lowest one

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in the set of the ground state levels and at low temperature we see the transitions from this ) ) level to the split doublet levels E(ex and E(ex   . It can be seen from Figs. 8-9 that in the case when magnetization is parallel to the trigonal axis [111] the number of peaks is twice in comparison with the case when M is along the tetragonal direction [001]. This may be explained by the fact that for M parallel to the [001]-axis the frequency dependencies of FR and MCD for four types of clusters with different trigonal axes which are present in a cubic crystal coincide while in the case of magnetization parallel to axis [111] the total dispersion curve is the result of superposition of two frequency dependencies, one arising from centers with trigonal axis [111] and the second

   

one related to the centers with trigonal axes 111 , 111 , 111 . In the case of arbitrary direction of magnetization all four types of clusters with different trigonal axes give different contributions to FR and MCD. Figure 10 and Figure 11 show the dependencies of FR and MCD upon the direction of magnetization M when it rotates in the plane (110). The spectra have strongly anisotropic character. Let‟s now turn our attention to the role of random crystal fields in the formation of the electric dipole contribution of MV clusters in the spectra of FR and MCD. It is obvious (see Figs. 8-11) that the random crystal fields decrease the intensity of FR and MCD spectra. This observation is related firstly to the fact that the random crystal fields give a contribution to the splitting of degenerate ground and excited states and respectively they reduce the (r)

magnetooptical activity by the factor V

r . Furthermore, these crystal fields randomly

renormalize the transition frequencies and this leads to the corresponding diffusion of FR and MCD spectra.

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Figure 8. Frequency dependences of impurity contribution to FR (magnetization M is parallel to axis 001 ), /= 0 (1), 0.75 (2).

Figure 9. Frequency dependences of impurity contribution to FR (magnetization M is parallel to axis [111]), / = 0(1), 0.75(2).

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Figure 10. Angular dependences of FR at different values of random crystal fields dispersions, / = 0(1), 0.1(2), 0.2(3).

Figure 11. Dependence of impurity contribution to MCD upon the angle between M in (110) plane and [001] axis, / = 0(1), 0.1(2), 0.2(3). Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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Let us discuss briefly the experimental results of measuring both the optical and MO polar Kerr spectra of pure Fe3O4 and of Mg2+ and Al3+ substituted Fe3O4 [7]. Fe3O4, or magnetite, crystallizes in the inverse spinel structure, with cation distribution (Fe3+)[Fe3+Fe2+]O4. In this structural formula the parentheses denote tetrahedral (A) sites and the square brackets denote the octahedral (B) sites. It was shown that the relative intensity of MO transitions at 0.56 eV and 1.94 eV depends on both [Mg2+] (which substitutes [Fe2+]) and on [Al3+] (which substitutes [Fe3+]). These electric dipole transitions were identified unambiguously as IVCT transitions:

Fe Fe

Fe ) Fe

2 4 2 3 3 2 1 (t 2 g e g ), Fe2 (t 2 g e g )

t 2 g t 2 g

2 4 2 3 3 2 1 (t 2 g e g ), Fe 2 (t 2 g e g

t2 g e g

 at 0.56 eV and ) at 1.94 eV.

3 3 2 2 4 2 1 (t 2 g e g ), Fe2 (t 2 g e g )

3 3 2 2 3 3 1 (t 2 g e g ), Fe2 (t 2 g e g

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In the IVCT model proposed by us the transitions in Fe3-xMexO4 should take place in MV clusters Fe3+-Fe2+- Fe2+ or Fe3+-Fe3+- Fe2+, which have trigonal symmetry and are localized near Mg2+ or Al3+. The dipole moment appears on the cluster due to the difference of ionic radii of Fe2+, 2+ Mg and Al3+, Fe3+. The double exchange, the low-symmetry crystal fields and spin-orbit interactions determine the structure of MV cluster energy spectrum in the ground and excited states. At this the excitation spectrum differs for clusters Fe3+-Fe2+- Fe2+ and Fe3+-Fe3+- Fe2+. The observed paramagnetic line shape [19] for the transitions at 0.56 eV and 1.94 eV can be connected with splitting of the cluster ground state by the spin-orbit interaction or by the difference of the ocscillator strengths for transitions caused by right and left circularlypolarized light when both the ground and the excited states are split.

4.2. Garnet Type Crystals This section deals with crystals having the garnet structure. The opportunity of appearance of MV clusters consisting of iron ions in yttrium-iron garnet (YIG) at nonisovalent substitutions or vacancies in one of sub-lattices is investigated. The elementary cell of YIG is presented on Figure 12. The nearest-neighbor inter-ionic distances for magnetic ions in this lattice are given in the Table below [18]. Table 1. Nearest-neighbor distances between magnetic ions in YIG

1 2 3 4 5

Ion Y3+ Fe3+(a) Fe3+(d)

Inter-ionic distances (A) 4 Fe3+(a) at 3.46 Fe3+(d) at 3.09 (2); 3.79 (4) 6 Fe3+(d) at 3.46 4 Fe3+(a) at 3.46 4 Fe3+(d) at 3.79

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Figure 12. Scheme of the trigonal MV cluster near cation vacancy in garnet lattice. V-vacancy, a – octahedral iron sites, d – tetrahedral iron sites.

The detail investigation of YIG structure shows that in all cases when four iron ions turn out to be the nearest neighbors to the vacancy or the substituted ion they form the tetrahedron which has four edges of one length and two – of the other one (the lines 1, 4 and 5 of the Table) or the tetrahedron with edges of three different lengths (case 2). In each of these variants the symmetry of environment of the cluster consisting of four iron ions, which are involved in the transfer of excess charge is found to be so low that the appearance of orbital degenerate state among its energy levels is impossible. The same reason makes the couple of ions Fe3+(d) near Y3+ ion (case 2) not interesting for our problem because the symmetry group C2 does not have degenerate representations. The favorable in this sense is the configuration of ions in case 3 when Fe3+(a) ion is surrounded by improper octahedron, which has two sides being equilateral triangles (one of them containing Fe3+(d) ions with numbers 1, 2 and 3, is presented on Figure 12). Further investigation showed that the set of states of such trigonal MV cluster contains degenerate (doublet) levels what is the necessary condition of appearance of the impurity contribution to the non-diagonal component of dielectric permeability tensor. So let us consider a trigonal cluster in the garnet structure (Figure 12), which appears due to the presence of the cation vacancy in the octahedral position. Such a cluster consists of three nearest tetrahedral iron ions and the excess charge (hole), which is transferred over them. Detail consideration shows that the tetrahedrons of oxygen ions surrounding Fe3+(d) ions are elongated along one of the cubic crystal axes (for the first ion it‟s elongated along zaxis, for the second ion – along x-axis and for the third one – along y-axis). Thus the local symmetry of triad ions is described by D2 d group while the symmetry of a triad itself is C3 (in contrast to the case of triad in spinel structure where both the local symmetry of triad ions and the symmetry of the triad itself was C3v ). Two ions of the triad have 3d (e t 2 ) 5

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Peculiarities of Magnetooptical Properties in Crystals …

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configuration and an orbitally non-degenerate ground state and one has 3d 4 configuration with orbitally degenerate ground term. This cluster can give contribution to the complex FR in the visible range of spectrum if the electric dipole transitions take place between the energy levels of two groups of such MV clusters: the ground one d 5 ( e 2t23 )  d 5 ( e2t23 )  d 4 ( e 2t22 ) and the excited one d 5 ( e2t23 )  d 5 ( e2t23 )  d 4 ( et23 ) .

(28)

The cubic T2 term of the ground configuration d 4 ( e2t22 ) is split by the local field of D2d symmetry into E and B2 states. For the sake of brevity we confined ourselves only by doublet states. Then the six-fold degenerate (the product of multiplicities of orientational and orbital degeneracy) states of a MV center with maximum projection of the total spin may be classified according to the irreducible representations of C3 group

  2 A  2E .

(29)

The cubic E term of the excited configuration d 4 ( et23 ) is split by the local field of D2d symmetry into singlets A1 and B1, so in this case we again appear at set (29) of irreducible representations of C3 group for the six-fold degenerate states of the triad. The cluster wave functions in the ground state d 5 ( e 2t23 )  d 5 ( e2t23 )  d 4 ( e 2t22 ) have the

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following form ) ( gr  

3

(C(),kk ( )  C(),kk ( )) .

(30)

k 1

Here the functions

k () are built according the same rules as the function k () in

(6). The expressions for energies

 ( gr ) of the ground state of the system are found as

 A( gr )  (  p2  p3  h0 )  (( p2  p3 )2  ( p1  p4 )2 ) , 1 2

)  E( gr  {( p2  p3  2h0 )  (( p2  p3 )2  4( p12  p1 p4  p42 ) }, 

(31)

p1  t x121 z1 , x2 z2 , p2  t x121 z1 , x2 y 2 , p3  t y121 z 1 , x2 z2 , p4  t y121 z1 , x2 y 2 . Here h0 is the parameter of low symmetry crystal field produced by the cation vacancy; the functions dxi yi , dxi zi , dyi zi are determined in the local coordinate system of the corresponding triad ion:

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xy xy , y1  , z1  z , 2 2 yz yz x2  x , y2  , z2  , 2 2 x1 

x3 

(32)

xz xz , y 3  y , z3  , 2 2

where the coordinate system x, y, z is connected with the C 4 axes of a cubic crystal. 2 3

2 3

3

For the excited state d5 (e t 2 ) - d5 (e t 2 ) - d4 (et 2 ) we have the following cluster wave functions ) ( ex  

3

(C(A ,k)k ( A1 )  C(B ,)kk (B1 )) , 1

1

(33)

k 1

k ( A1 ) and k ( B1 ) are the products of the wave functions of the

where the functions

constituent ions with the excess e-hole instantly localized on the site k (k=1, 2, 3). The energies of the excited state terms are the following

 A( ex )  cub  ( r1  r4 )  (( r1  r4 )2  ( r2  r3 )2 ) , 1 2

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)  E( ex  cub  {(r1  r4 )  (( r1  r4 )2  4( r22  r2r3  r32 ) }, 

(34)

r1  t121 2 , r2  t1212 , r3  t121 2 , r4  t1212 . Here functions

i and  i are determined in the local coordinate system of the

corresponding triad ion 1 2

1  [ 3z12  r 2 ],  1 

3 2 1 3 2 2 [ x1  y12 ] ; 2  [ 3x22  r 2 ],  2  [ y2  z2 ] . 2 2 2

(35)

Again as in the case of spinel structure because of the orthogonality of e and t2 functions

the matrix elements of the operator ˆ k (Eq.(12)) between functions of the ground and excited

states are equal to zero. The electric dipole transitions in the visible range of spectrum can take place only due to the mixing of the wave functions of these states. In contrast to spinel the crystal field of D2d symmetry in tetrahedral positions does not mix these functions. They are mixed only by the double exchange interaction. The consideration analogous to those for spinel (Eqs. (15),(24)-(25)) leads to the following expressions for matrices of the dipole momentum operators in the space of functions

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) ( gr ) ( gr ) ( gr ) A( gr ) , A( gr ) , E( gr  ,   .  ,   E E E 







 ˆ ( r )  ( 1  3 ) ˆ ( r ) ], Px( r )  m(xr ) [( 1  3  )   (r) ( r ) (r) ( r )  ˆ  ( 3   ) ˆ ], Py  my  [( 3   )   (r) ˆ ( r )  ( 3  2 ) ˆ ( r ) ], Pz  m(zr ) [ ( 3  2  )  

(36)

  3qa / 24.  (r)  (r)

Here a is the lattice constant,   exp(2i /3) , the matrices of the operators X , X



 (r)

were obtained analogously to those of the operators U( r ) , U . After some calculations the expression for the FR has again the form of Eq. (27) where the coefficient  should be substituted by  and the coefficients Y ( ) must be replaced 2 2 by Y ( xs , xd ) which takes the values [(1  xs xd )  ((1  xs )(1  xd ) ] depending on the

pair of energy levels (  , ) between which the transition takes place. The coefficients xs and xd have the following form

xs 

p3  p2 (( p2  p3 )2  ( p1  p4 )2 )

, xd 

p2  p3 (( p2  p3 )2  4( p12  p1p4  p42 )

. (37)

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The shape of the lines for EF and MCD depends upon the fact which of the levels of group (31) turns out to be the lowest one. a) Let‟s begin from the case when singlet level A of the cluster is the ground one and the electric dipole transitions take place to the split doublet levels. The following set of the parameters was used at numerical simulations in this case:

T /  cub  0.001, h0 /  cub  0.01,  /  cub  0.017, D  0,    ,  /  cub  0.13, p1 /  cub  0.17, p2 /  cub  0.2, p3 /  cub  0.1, p4 /  cub  0.17, r1 /  cub  0.1, r2 /  cub  0.1, r3 /  cub  0.05, r4 /  cub  0.2 . One can see (Figs. 13 and 14) that the Faraday rotation curves are peaked at the transition wavelength while the MCD shows a dispersive behavior. So the transitions are found to have diamagnetic character [19] typical for the case when the electric dipole transition takes place between an orbital singlet ground state and an excited doublet state which is split in our case by spin-orbit coupling and random crystal fields. b) If the doublet state E from the set of energies (31) turns out to be the lowest one (this case takes place if one changes the sign of only one of the transfer integrals, that is

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p2 /  cub  0.2 ) we obtain the so-called paramagnetic transition [19] when the Faraday rotation curves show a dispersive character and MCD has a bell-shaped behavior (see Figs. 15, 16).

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Figure 13. Frequency dependence of diamagnetic impurity contribution to FR in YIG (magnetization M is parallel to axis [111]).

Figure 14. Frequency dependence of diamagnetic impurity contribution to MCD in YIG (magnetization M is parallel to axis [111]). Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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Figure 15. Frequency dependence of paramagnetic impurity contribution to FR in YIG (magnetization M is parallel to axis [001]),  /   0(1),0.75(2).

Figure 16. Frequency dependence of paramagnetic impurity contribution to MCD in YIG (magnetization M is parallel to axis [001]),

 /  0(1),0.75(2).

The theoretical value of the Faraday rotation is of the order of hundreds deg/cm which agrees with experimentally obtained [20] values in YIG doped with Co2+ ions substituted in octahedral sites. Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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4.3. Manganites Orthorhombic manganites RMnO3 are typical compounds with the 3d-electron orbitally ordered ground state where 3d 2 2 and 3d 2 2 type orbitals for eg-electrons are 3x r

3 y r

3+

alternately ordered on Mn sites in the ab plane and are stacked parallel along the c axis below the orbital ordering temperature Tc 750 K. Such an orbital ordering is stabilized by the cooperative Jahn–Teller effect that lifts the orbital degeneracy of the crystal ground state 5E term for the octahedral t23g eg electronic configuration of the Mn3+ ion. Systems exhibiting different compositions La1-xMexMnO3 (Me=Ca, Sr) have been studied where La1-xCaxMnO3 is probably the best-known object. As a result, Mn4+ appears and then Mn4+-O2--Mn3+ interact through shared oxygen in the perovskite structure giving rise to double exchange interaction [21]. Transitions in octahedral complexes of manganese ions with different valences and the d–d transitions in the manganese ions are responsible for the MO activity of both doped and oxygen-nonstoichiometric manganites. Let us discuss briefly some peculiarities of absorption and MO spectrums of doped manganites in particular of (La0.5Pr0.5)0.7Ca0.3MnO3 films [22], which are brought about by IVCT transitions. The data on temperature dependence of the absorption bands at 0.14 and 0.4 eV indicate that they are connected with transitions in the MV centers. А broad peak in the optical conductivity spectra of Nd0.7Sr0.3MnO3 near 1.2 eV at room temperature is to be interpreted as a charge transfer transition from a Mn3+ eg-level to the unoccupied Mn4+ eglevels on an adjacent site [23]. The appearance of the middle infrared bands can be understood in the framework of our model. At non-isovalent substitution (La3+Sr2+) the simplest MV cluster has the tetragonal symmetry in the ab plane involving four Mn-ions with the electronic configurations Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

3d4  3d4  3d4  3d3 . Since the sites occupied by the Mn-ions are equivalent the excessive charge of the Mn4+-ion can be delocalized among the cations. Thus the ground state of MV cluster is fourfold degenerate. The double exchange lifts the degeneracy and the energies of the system are found as:

E( gr ) ( A1 )  b ,

E( gr ) ( A2 )  b ,

E( gr ) (E)  0 ,

(38)

where A1, A2, E are the irreducible representations of C4v group, b  b 2 2 2 2 is the 3x r ,3y r transfer integral. One more set of low-lying states of MV cluster appears at the excitation of Mn3+ ion ( 3d 2 2 ( 3d 2 2 )  3d 2 2 ( 3d 2 2 )) . In this case we have the following energies 3 x r

3 x r

3 y r

3 x r

of the system

E(ex)( A1 )   JT  b ,

E(ex)( A2 )   JT  b ,

E(ex)(E)   JT ,

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Peculiarities of Magnetooptical Properties in Crystals …

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where  JT is the splitting energy of Eg - term of Mn3+ ion due to the cooperative JT effect. The energies of the corresponding transitions are equal to

E1  E(ex ) ( A1 )  E ( gr ) ( A1 )   JT , E2  E(ex ) ( A1 )  E ( gr ) ( A1 )   JT  2b ,

(40)

E3  E(ex ) ( E )  E ( gr ) ( A1 )   JT  b . One can suppose that the absorption bands at 0.14 and 0.4 eV [22] may be connected with IVCT transitions A1( gr )  E( ex) and A1( gr )  A1( ex) respectively. The appropriate

values of parameters  JT  0.14 eV and b  0.13 eV are quite reasonable. Then a broad peak in the optical conductivity spectra of Nd0.7Sr0.3MnO3 near 1.2 eV [23] can not be interpreted as a charge transfer transition from a Mn3+ eg-level to the unoccupied Mn4+ eglevels on an adjacent site since the parameters  JT and b are overestimated. Most probably that the absorption band near 1.2 eV can be assigned to the crystal field

5

Eg 5 T2 g

transition in Mn3+ ion. On the whole, we can also expect that the doped manganites must exhibit a considerable magneto-optical activity in the region of 2–4 eV [22, 23-25], which may be related in particular

to

the

IVCT

5

Eg [ Mn3  ]4 A2 [ Mn4  ]4 T1 g ( 4T2 g )[ Mn4  ]5 Eg [ Mn3  ]

transitions and investigated in the framework of our model.

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CONCLUSION We have shown that the anomalous values of antisymmetric component of dielectric permittivity tensor are found both in the infrared and visible ranges of frequencies in magnetic crystals with 3d mixed- valence ions if the orbital degeneracy presents in the ground or excited state of MV complex. This result was obtained for crystals with spinel, garnet and perovskite structures where the spectral characteristics of FE and MCD were investigated. The corresponding effects should be rather anisotropic and this anisotropy of FE and MCD spectrums decreases with the increase of the varience of the random crystal field distribution together with the effects themselves. It is important that these effects become appreciable at 17

rather small values of MV center concentrations CJT  10

 1019 cm-3. The contribution

of MV centers into complex FE is larger in the infra-red region because in the visible region it is reduced proportionally to the square of ratio of the double exchange interaction parameter to the splitting by cubic crystal field. It should be noted that the oscillator strengths of transitions associated with charge transfer 2p  3d are of the order f  10-1, those for twocenter d-d CT transitions, including, IVCT and IVST, amounts to f  10-5. Oscillator strengths of optical transitions in our model are of the order f  10-2. 10-3 The magnetooptical properties of crystals with MV irregularities are determined to a considerable extent by the imperfectness of the samples under investigation, violations in the distribution of cations among sublattices and changes of valence states of 3d-ions. It should

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be noted that the contribution of MV clusters to the magnetooptical activity in the visible part of spectrum is formed substantially by imposing of the big enough number of elementary transitions. Thus, as against isolated 3d-ions in a crystal matrix both paramagnetic, and diamagnetic contributions to the FE and MCD take place.

ACKNOWLEDGMENTS The authors are grateful to Anatolii Fishman and Boris Tsukerblat for their interest in the work and valuable discussions. Financial support from the Programs DPS RAS N 09-T-21013 and UD-SD RAS N 09-C-2-1016 is gratefully acknowledged.

REFERENCES [1] [2] [3] [4] [5] [6]

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[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

Moskvin, A.;Pisarev, R. Low Temp Phys. 2010, vol. 36, pp 489-510. Fontijn, W.F.J.; van der Zaag, P.J.; Feiner, L.F.; Metselaar, R.; Devillers, M.A.C. J Appl. Phys. 1999, vol. 85, pp 5100-5105. Tiroch, E.; Shemer, G.; Markovich. G. Chem Mater. 2006, vol.18, pp 465-470. Allen, G.C.; Hush, N.S. In Progress in Inorganic Chemistry; Cotton, F.A.; Ed.; Interscience, N.-Y. 1967. Simsa, J.W.Z.; Simsova, J.;Zemek, J.;Wigen, P.E.;Pardavi-Norvath, M. J. E. Physique. 1988, vol.49, pp C8-275- C8-276. Martens, D.; Peeters, W.L.; van Noort, H.M.; Erman, M. J. Phys. Chem. Solids. 1985, vol.46, pp 411-416. Fontijn, W.F.J.; van der Zaag, P.J.; Devillers, M.A.C.; Brabers, V.A.M.; Metselaar, R. Phys. Rev. 1997, vol.B 56, pp 5432-5442. Tailhades, P.; Bonningue, C.; Rousset, A.; Bouet, L.; Pasquet, I.; Lebrun, S. JMMM. 1999, vol.193, pp 148-151. Zener, C. Phys.Rev. 1951, vol. 81, pp 440-444. Anderson, P.W.; Hasegawa, H. Phys Rev. 1955, vol. 100, pp 675-681. Ivanov, M.; Mitrofanov, V.; Falkovskaya, L.; Fishman, A. Fiz Tverd Tela. 1993, vol.35, pp 2025-2036. Ivanov, M.; Mitrofanov, V.; Falkovskaya, L.; Fishman, A.;Tsukerblat, B. Fiz Tverd T ela. 1996, vol. 36, pp 3628-3641. Mitrofanov, V.; Falkovskaya , L.;.Fishman, A. Fiz Tverd Tela. 1997, vol.39, pp 953955. Falkovskaya, L.; Fishman, A.; Ivanov, M.; Mitrofanov, V.;Tsukerblat, B. Zeit fur Phys. Chem. 1997,vol. 201, pp S.231-242. Ivanov, M.; Mitrofanov, V.; Fishman, A. Phys Stat Sol. 1984, vol.B 121, pp 547-559. Falkovskaya, L.; Fishman, A.; Mitrofanov, V.; Tsukerblat, B. Phys. Let. A. 2010, vol. 374, pp 3067-3075. Mitrofanov, V.; Fishman, A. Fiz Tverd Tela. 1990, vol. 32, pp 2598-2605. Geller, S.; Gilleo, M.A. J. Phys. Chem. Solids. 1957, vol. 3, pp 30-36.

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[19] Wittekoek, S.; Popma, T.J.A.; Robertson, J.M.; Bougers, P.F. Phys. Rev. 1975, vol.B12, pp 2777-2788. [20] Daval, J.; Ferrand, B.; Milani, E.; Paroli, P. IEEE Trans Magn. 1987. vol. Mag-23, pp 3488-3490. [21] Zener, C. Phys. Rev. 1951, vol. 82, pp 403-405. [22] Loshkareva, N.; Sukhorukov, Yu,; Gan‟shina, E.; Mostovshchikova, E.; Kumaritova, R.; Moskvin, A.; Panov, Yu.; Gorbenko, O.; Kaul‟, R. JETP. 2001, vol. 92, pp 462-473. [23] Kaplan, S.G.; Quijada, M.; Drew, H.D.; Tanner, D.B.; Xiong, G.C.; Ramesh, R.; Kwon, C.; Venkatesan, T. Phys. Rev. Lett. 1995, vol.77, pp 2081-2084. [24] Sukhorukov, Yu.; Moskvin, A.; Loshkareva, N.; Smolyak, I.; Arkhipov, V.; Mukovskii, Ya.; Shmatok, A. Tech. Phys. 2001, vol. 46, pp 778-781. [25] Gan‟shina, E.; Granovskii, A.; Vinogradov, A.;Kumaritova, R.; Grenet, J.-C.; Cairo, R.; Revkolevschi, A.; Dhalenne, G.; Berton, J. Crystallography Reports. 2003, vol.48, pp 473-476.

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In: Optical Lattices: Structures, Atoms and Solitons ISBN: 978-1-61324-937-6 c 2012 Nova Science Publishers, Inc. Editor: Benjamin J. Fuentes

Chapter 5

T HE I NTERPLAY OF O PTICAL L ATTICES WITH L OCALIZED N ONLINEARITY: O NE -D IMENSIONAL S OLITONS Nir Dror and Boris A. Malomed Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel

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Abstract Standard models of periodically modulated nonlinear media, such as photonic crystals and Bose-Einstein condensates (BECs) trapped in optical lattices (OLs), are often described by the nonlinear Schroedinger/Gross-Pitaevskii equations with periodic potentials. We consider a model including such a periodic potential and the attractive or repulsive nonlinearity concentrated at a single point or at a set of two points, which are represented by delta-functions. For the attractive or repulsive nonlinearity, the model gives rise to ordinary solitons or gap solitons (GSs). These localized modes reside, respectively, in the semi-infinite gap, or finite bandgaps of the system’s linear spectrum. The solitons are pinned to the delta-functions. Realizations of these models are relevant to optics and BECs. We demonstrate that the single nonlinear deltafunction supports families of stable ordinary solitons and GSs in the cases of the selfattractive and repulsive nonlinearity, respectively. We also show that the delta-function can support stable GSs in the first finite bandgap in the case of the self-attraction. The stability of the GSs in the second finite bandgap is investigated too. In addition to the numerical analysis, analytical approximations are developed for the solitons in the semi-infinite gap and two lowest finite bandgaps. In the model with the symmetric pair of delta-functions, we investigate the effect of the spontaneous symmetry breaking of the pinned solitons.

1.

Introduction

In the course of the last decade, a great deal of interest has been drawn to theoretical and experimental studies of the nonlinear dynamics in systems with periodic potentials. Physical realizations of this topic are well known in nonlinear optics [1, 2] and Bose-Einstein condensates (BECs). In optical media, the periodic (lattice) potentials may be created as

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Nir Dror and Boris A. Malomed

permanent or virtual ones (in the latter case, these are photonic lattices induced in photorefractive crystals [2, 3]). In BECs, similar potentials can be induced in the form of optical lattices (OLs), i.e., interference patterns formed by laser beams shone through the condensate. The OLs make it possible to study a great variety of dynamical effects in BECs [4, 5]. Similar periodic potentials may also be imposed by magnetic lattices [6]. It is commonly known that, in the free space, stable one-dimensional (1D) solitons exist in optical waveguides and BECs with attractive cubic nonlinearity, while in 2D and 3D geometry the solitons are unstable to the collapse [7]. On the other hand, it has been predicted that 2D [8, 9, 10, 11] and 3D [8, 11] solitons can be stabilized by dint of the corresponding OLs. Moreover, low-dimensional OLs, i.e., quasi-1D and quasi-2D lattices in 2D [11] and 3D [11, 12] space, respectively, also provide for the stabilization of fully localized multidimensional solitons. A related prediction is the existence of stable 2D [13, 14] and 3D [15] solitons in models with radial OLs. In BECs with repulsive interactions between atoms, i.e., repulsive intrinsic nonlinearity, solitons cannot exist in free space, but gap solitons (GSs) may be supported by the OL. The principle behind the formation of the GS is that the periodic potential can invert the sign of the effective mass of collective excitations, which may then balance the repulsive nonlinearity. In the 1D case, several species of GSs are known, including fundamental solitons and their two- and three-peak bound complexes, in the first and second bandgaps [16, 17], and subfundamental solitons in the second gap (the latter means a twisted soliton squeezed into a single cell of the potential lattice, whose norm is smaller than the norm of the fundamental soliton existing at the same value of the chemical potential) [18, 19, 20]. The origin of the GS families may be traced back to bifurcations generating them from Bloch waves at the edges of the bandgaps [17, 21]. Multidimensional GSs are represented by fundamental solitons [22, 23, 24, 25] and gap-type vortices [23, 24, 26, 27, 28]. In particular, the simplest gap vortices are composed of four density peaks, and fall into two different categories: densely packed squares (alias off-site-centered vortices), in which the center is positioned around a local maximum of the OL potential [23, 26, 28], and rhombic (on-site-centered) configurations, featuring a nearly empty lattice cell in the middle [22, 23, 24, 25]. While GS families in 2D are also generated by bifurcations from the respective Bloch waves [21, 29], there are gap-vortex families which do not originate from such bifurcations [30]. In the experiment, an effectively one-dimensional GS, composed of a few hundred atoms, was created in the condensate of 87 Rb [31]. In another experiment, which involved a stronger OL, broad confined states were created [32]. They were interpreted as modes intermediate between the GSs and extended nonlinear Bloch waves [33]. In a different framework, solitons can be supported by a spatial modulation of the local nonlinearity (a particular case of the “nonlinearity management” [34]). In terms of solidstate physics, an effective potential structure induced by means of this method is called a pseudopotential [35]. Realizations of such structures are possible in optics and BECs, where they are expected to give rise to many new effects; see the recent review [36]. In optical media, nonlinearity-modulation profiles can be induced by means of nonuniform distributions of resonant dopants, similar to those in resonantly-absorbing Bragg reflectors; see the original works [37] and the review [38]. Such dopant-density patterns can be created by means of available technology [39]. For matter waves in BECs, similar nonlinear

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profiles can be created via the spatial modulation of as (x), the local value of the s-wave scattering length. Such modulations can be induced by means of the Feshbach resonance, controlled by a nonuniform dc magnetic field [40] or resonant optical field [41, 42]. It was also predicted that the Feshbach resonance may be controlled by dc electric fields [43]. Note that the Feshbach resonance allows one to create a pseudopotential corresponding to a sign-changing function as (x) (actually, this is possible in the condensate of 7 Li [44]), which implies the spatial alternation between the attractive and repulsive signs of the nonlinearity. In the context of BECs, many works dealt with solitons and related dynamical states in the framework of 1D pseudopotential structures [45]. Nonlinear structures based on the spatial modulation of the Kerr coefficient were also considered in optics, giving rise to similar states [46]. Solitons supported by 2D pseudopotentials were theoretically studied too, with the conclusion that it is much more difficult to stabilize them than using linear OL potentials [47]. The results accumulated in the studies of solitons in nonlinear lattices are summarized in a recent review [36]. A specific example of such nonlinear pseudopotential settings is a model where the nonlinearity is concentrated at a single point, which is represented by a delta-function. A prototypical model of this type was introduced in Ref. [48]. It may represent a planar linear waveguide with a narrow nonlinear stripe embedded into it (for the case of the secondharmonic-generating quadratic nonlinearity, spatial solitons in a similar setting were considered in Ref. [49]). In terms of the BECs, localized nonlinearity may be induced through the Feshbach resonance imposed by a focused laser beam. Extending the studies in this direction, a model of a double-well pseudopotential, based on a symmetric set of two deltafunctions, or their regularized counterparts, was introduced, with the purpose of studying the spontaneous symmetry breaking (SSB) of localized modes [50]. In particular, it is possible to find full analytical solutions for symmetric, asymmetric and antisymmetric states in the model with two ideal delta-functions. In this model, where the SSB bifurcation is of the subcritical type, the symmetric solutions are stable up to the bifurcation point. Beyond this point, the symmetric states and the emerging asymmetric states are unstable, as well as all the antisymmetric ones. Symmetry breaking in a circular nonlinear lattice, with a smooth spatial modulation of as, was studied, in the framework of both the Gross-Pitaevskii equation (GPE) and the many-body quantum system, in Ref. [51]. A natural extension of the settings outlined above is a model integrating the linear OL potential and the nonlinearity concentrated at one or two points, represented by the respective delta-functions or their regularized versions. These systems are the subject of the present work. In particular, we demonstrate that, while cusp-shaped solitons pinned to the delta-function multiplying the attractive cubic nonlinearity turn out to be unstable, the linear periodic potential readily stabilizes them. Another issue of obvious interest is whether the strongly localized repulsive nonlinearity may support GSs in finite bandgaps of the OLinduced spectrum (we demonstrate that this is indeed possible). The rest of the paper is organized as follows. The model is formulated in section 2.. For the system including the single delta-function and OL potential, analytical approximations, based on the perturbation theory, are developed for the pinned modes in the first and second finite bandgaps, as well as in the semi-infinite gap, in section 3.. A comparison with numerical findings is performed too. Detailed results of the numerical analysis for the existence and, most important, stability of the pinned modes in the semi-infinite,

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first and second gaps are reported in section 4.. Both the attractive and repulsive signs of the delta-functional nonlinearity are considered, for different positions of the delta-function with respect to the underlying lattice. In section 5., an analysis is reported for symmetric, antisymmetric and asymmetric modes supported by a pair of the delta-functions, positioned symmetrically with respect to a maximum or minimum of the OL potential. The paper is concluded in section 6..

2.

The Model

The model featuring the cubic nonlinearity represented by the delta-function was introduced in Ref. [48], in the context of tunneling of interacting particles through a junction, 1 iψt + ψxx − σδ(x)|ψ|2 ψ = 0, 2

(1)

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where ψ is the mean-field wave function in the BEC, or the local amplitude of the guided electromagnetic field in the context of optics (in the latter case, time t is replaced by the propagation distance), with σ = +1 and −1 corresponding to the repulsive and attractive nonlinearity, respectively. Obviously, Eq. (1) amounts to the simple linear equation valid at x < 0 and x > 0, iψt + (1/2)ψxx = 0, which is supplemented by the derivative-jump condition at x = 0, produced by the integration of Eq. (1) over an infinitely small vicinity of x = 0: ψx (x = +0) − ψx (x = −0) = 2σ |ψ(x = 0)|2 ψ(x = 0). (2) Stationary states are looked for in the ordinary form, ψ(x,t) = e−iµt φ(x), where µ is the chemical potential in BEC (−µ is the propagation constant in optics), and the real function φ obeys the equation µφ + (1/2)φ00 − σδ(x)φ3 = 0. (3) An exact soliton solution to Eq. (3) with σ = −1 is obvious: φ0 (x) = Ae−

√ −2µ|x|

,

A2 =

The norm of this solution, N=

Z +∞ −∞

p −2µ.

|ψ(x)|2 dx,

(4)

(5)

is independent of µ, taking a constant value, N = 1, hence the formal application of the Vakhitov-Kolokolov (VK) criterion, dN/dµ < 0 [7, 52], predicts neutral stability. In fact, numerical simulations of the evolution of pinned solitons (4) with small perturbations added to them demonstrate a strong instability (not displayed here in detail): the soliton either decays or suffers the collapse (formation of a singularity), if the exact norm of the perturbed soliton is, respectively, N < 1 or N > 1. These features, including the degeneracy of the norm and the instability leading to the decay or collapse, resemble those known for the 2D Townes solitons [7] in the free 2D space with the cubic attractive nonlinearity, or in the 1D space with the quintic nonlinearity [53]. Moreover, it is easy to find a family of exact

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The Interplay of Optical Lattices with Localized Nonlinearity

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analytical solutions to Eq. (1), with σ = −1, which explicitly describe the approach to the collapse at t → 0−: ( " #) r x0 (|x| − ix0 )2 ψ (x,t) = − exp i , (6) t 2t where x0 is an arbitrary real positive constant, the solution being valid at t < 0. Decaying solutions, at t > 0, are described by the same solution (6), with −x0 replaced by x0 > 0, the decay taking place at t → +∞. The norm of solution (6) is exactly N = 1 (irrespective of the value of x0 ), i.e., the same as that of stationary solution (4). In this work we introduce a natural extension of Eq. (1), adding to it the periodic OL potential, with the objective to stabilize the solitons: 1 iψt + ψxx + ε cos(2x)ψ − σδ(x − ξ)|ψ|2 ψ = 0. 2

(7)

Here, the OL potential, whose period is normalized to be π, is V (x) = −ε cos(2x). In its first period, 0 ≤ x < π, the minimum and maximum of the potential are located, respectively, at x = 0 and x = π/2 (and vice versa for ε < 0), while the δ-function is set at x = ξ, which does not necessarily coincide with the minimum or maximum of the potential. A modification of the model, which is considered in section 5., deals with the nonlinearity represented by a pair of δ-functions, placed symmetrically with respect to the minimum or maximum of the periodic potential:

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1 iψt + ψxx + ε cos(2x)ψ − σ [δ(x − ξ) + δ(x + ξ)] |ψ|2 ψ = 0 2

(8)

(recall that the same double-delta nonlinearity, but without the OL potential, and solely with the attractive sign of the nonlinearity, σ = −1, was introduced in Ref. [50]). In this case, the asymmetry measure of stationary modes is defined as θ=

R +∞ 0

|ψ(x)|2 dx − R +∞ −∞

R0

2 −∞ |ψ(x)| dx 2

|ψ(x)| dx

≡

N+ − N− . N

(9)

Dynamical invariants of Eqs. (7) and (8) are norm N, given by Eq. (5), and the Hamiltonian (written here for the latter equation), H=

Z +∞  1 −∞

 1 |ψx |2 − ε cos (2x)|ψ(x)|2 dx + ∑ |ψ (x = ±ξ)|4 . 2 2 +,−

(10)

N is proportional to the number of atoms trapped in the BEC, or the total power of the trapped beam in optics. Control parameters of the models are ε and ξ, along with N. As said above, in models with periodic potentials solitons may exist in bandgaps of the spectrum of the linearized version of the equation – either in the semi-infinite gap or finite ones, separated by Bloch bands [4, 16, 17]. For the models under the consideration, the spectrum is the same as in the classical Mathieu equation (it is actually displayed by means of shaded areas in Fig. 1 below).

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3.

Nir Dror and Boris A. Malomed

Perturbation Analysis of the Single-Delta Model

Equation (7), as well as the corresponding linear Mathieu equation, can be treated by means of the perturbation theory if the OL strength, ε, is a small parameter. Here, we consider the case when the delta-function is placed symmetrically with respect to the periodic potential, at ξ = 0, the objective being to construct approximate analytical solutions for GSs (gap solitons) supported by the repulsive (σ = +1) or attractive (σ = −1) point-wise nonlinearity, in the first and second finite bandgaps, as well as for ordinary solitons in the the semi-infinite gap. The respective form of the stationary equation is   µφ + (1/2)φ00 + ε cos(2x) − σδ(x)φ2 φ = 0, (11) cf. Eq. (3).

3.1. 3.1.1.

The first finite bandgap Solutions of the linear equation

In the case of small |ε|, the first finite bandgap occupies a narrow interval of values of µ around µ = 1/2. Accordingly, we set µ ≡ 1/2 + ν, with |ν|  1.

(12)

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Then, approximate solutions corresponding to the GS in the first finite bandgap can be sought for as φ(x) = e−λ1 |x| [A cos(x) + B sin (|x|)] , (13) where λ1 is assumed to be a small positive coefficient (subscript 1 indicates that λ1 pertains to the first finite bandgap). The substitution of ansatz (13) into Eq. (11), off point x = 0 (which corresponds to the linear equation), and the subsequent analysis following the usual perturbation theory for the Mathieu equation (in other words, the asymptotic analysis of the linear parametric resonance [54]), yield the following equations for amplitudes A and B:  
1 2 ν + λ1 + ε A − λ1 B = 0, 2  
1 2 ν + λ1 − ε B + λ1 A = 0. (14) 2

The resolvability condition for the linear homogeneous system (14) is  2 1 ε2 ν + λ21 − + λ21 = 0. 2 4

In the first approximation, λ21 /2 may be neglected in the parentheses, which yields r ε2 λ1 = − ν2 , 4

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(15)

(16)

The Interplay of Optical Lattices with Localized Nonlinearity

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hence the solution exists for |ν| < |ε|/2.

(17)

In fact, Eq. (17) is the prediction provided by the perturbation theory for the width of the first finite bandgap, see Fig. 1(a). Further, it follows from Eqs. (14) that, in the first approximation, s B = sgn(ε)

ε/2 + ν A, ε/2 − ν

(18)

where condition (17) is taken into consideration, to identify the correct sign. 3.1.2.

The nonlinear part of the solution

For the stationary solutions, condition (2) takes the form of   dφ ∆ |x=0 = 2σφ3 |x=0 , dx

(19)

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where ∆ stands for the jump of the derivative at x = 0. The substitution of Eqs. (13), (16), and (18) into Eq. (19) yields s ! r ε2 ε/2 + ν − − ν2 − sgn (ε) A = σA3 . (20) 4 ε/2 − ν p 2 2 On the left-hand side of Eq. much smaller than the following p (20), term ε /4 − ν is p 2 2 term [in the general case, ε /4 − ν is small, while (ε/2 + ν) / (ε/2 − ν) is not; in special cases, near the upper edge pof the bandgap for ε > 0, or the lower one for p ε < 0, with |ε/2 + ν|  |ε/2 − ν|, the term (ε/2 + ν) / (ε/2 − ν) becomes small, but ε2 /4 − ν2 is then still smaller]. Thus, Eq. (20) yields, for nontrivial solutions (A2 6= 0): s ε/2 + ν 2 A = σ sgn(ε) . (21) ε/2 − ν This solution exists provided that sgn(ε) = σ, and it does not exist in the opposite case. In other words, it exists if the repulsive delta-function, with σ = +1, is set at the local minimum of the OL potential, or the attractive delta-function is placed at the local maximum. Note that, although the result was obtained by means of the perturbation theory, the amplitude given by Eq. (21) is not small (which does not invalidate the perturbative treatment). In the first approximation, the calculation of the norm of the weakly localized soliton solution based on the above formulas yields
N1 ≈ (2λ1 )−1 A2 + B2 ≡

|ε|

2 (ε/2 − ν)2

.

(22)

In the case of the attraction (then, ε is negative, as shown above), relation (22) yields dN1 dN1 ε ≡ =− < 0, dµ dν (ε/2 − ν)3

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Nir Dror and Boris A. Malomed

hence, according to the VK criterion, the so predicted family of the GSs might be stable. Nevertheless, it is actually unstable, as shown below. As for the repulsive nonlinearity, the VK criterion is irrelevant for it (being sometimes replaced by an “anti-VK” condition [55]). The calculation of Hamiltonian (10) in the present approximation yields H≈

|ε|

4 (ε/2 − ν)2

(24)

[in this approximation, H is dominated by the gradient term in Eq. (10)]. The fact that this expression is always positive demonstrates that the GS cannot realize a ground state (it cannot correspond to an absolute minimum of the energy, which must be either negative or zero). Nevertheless, this does not mean that these solitons cannot be stable against small perturbations (actually, they may be metastable states, in comparison with the ground state).

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3.1.3.

Comparison with numerical results

The first finite bandgap, as predicted by the perturbation theory [Eq. (17)], is shown in Fig. 1(a), together with the numerically found borders of the first bandgap. In accordance with the above prediction, the soliton in the model with the repulsive or attractive nonlinearity exists only for ε > 0 and ε < 0, respectively (this conclusion is true for all values of ε and µ in the first bandgap, not only for small ε). As concerns the stability, the GSs supported by the repulsive delta-function placed at the minimum of the OL potential (σ = +1, ε > 0) are stable, while all the solitons generated in the finite bandgap by the attractive nonlinearity (σ = −1, ε < 0) are unstable [contrary to the formal prediction of the VK criterion, see Eq. (23)]. A detailed stability analysis is reported in the next section. Typical profiles of the soliton solutions are shown in Figs. 2 and 3.1.3. If ε is not too large, the approximation quite accurately predicts the shape of the soliton. In particular, the analytical and numerical results agree very well for µ taken near the middle of the bandgap. Close to the upper edge (for ε > 0) or lower edge (for ε < 0), the analytical shape of the soliton is less accurate, as seen in Figs. 2(c),(f)and 3(a),(d). In fact, this inaccuracy originates from the use of approximation (21) instead of the more general Eq. (20). Figure 4 shows the norm of the solitons versus µ, for both negative and positive ε, which correspond to the attractive and repulsive nonlinearities, respectively. As expected, the results become more accurate as |ε| diminishes, while µ takes values farther from the upper or lower edges of the bandgap, for ε > 0 and ε < 0, respectively.

3.2.

The second finite bandgap

The perturbation theory for the Mathieu equation may also uncover the second finite bandgap, when µ = 2 + ν, with |ν|  1, (25) cf. Eq. (12). To this end, an approximate solution to the linear equation is sought as φ(x) = e−λ2 |x| [A0 + A2 cos (4x) + B2 sin(4|x|) + A1 cos (2x) + B1 sin(2|x|)] ,

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(26)

6

6

4

4

2

2

0

µ

µ

The Interplay of Optical Lattices with Localized Nonlinearity

169

Gap II

0

Gap I −2

−2

−4

−4

−6 0

2

4

6

8

−6 0

2

4

ε

6

8

ε

(a)

(b)

Figure 1. Borders of the first (a) and second (b) finite bandgaps (labeled as Gap I and Gap II, respectively), as predicted by the perturbation approximation, see Eqs. (17) and (32), in comparison with the numerically constructed bandgap structure, in which Bloch bands are shaded. The semi-infinite gap is represented by the white areas at the bottom of the panels. cf. ansatz (13) adopted in the first finite bandgap. Substituting ansatz (26) into Eq. (11), off point x = 0, and neglecting terms ∼ λ2 , we find, in the zeroth-order approximation,

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ε A0 = − A1 , 4

A2 =

ε A1 , 12

B2 =

ε B1 . 12

(27)

Next, the homogeneous system of equations for A1 and B1 takes the following form, cf. Eqs. (14) derived in the first finite bandgap:   1 2 1 νA1 + λ2 A1 + ε A0 + A2 − 2λ2 B1 = 0, (28) 2 2 1 1 νB1 + λ22 B1 + εB2 + 2λ2 A1 = 0, (29) 2 2 or, on substituting expressions (27),   5ε2 λ22 ν− A1 − 2λ2 B1 = 0, + 24 2   ε2 λ22 ν+ + B1 + 2λ2 A1 = 0. 24 2

(30)

As in the case of Eqs. (14), terms λ22 /2 in the parentheses may be neglected in the lowest approximation, which yields r 5 4 1 2 1 ε + ε ν − ν2 . (31) λ2 = 2 576 6

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1.5

µ=0.3

µ=0.5

0.4

1

0.2

0.5

0

0

−0.2

−0.5

−0.4

−20

0

20

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Figure 2. Comparison between the analytical approximation (dashed lines) and the numerically found profiles (solid lines) for solitons in the first finite bandgap of the repulsive model, for ε = 0.5 (a)-(c) and ε = 1 (d)-(f). The results are shown for selected values of µ (indicated in each panel), close to the center of the gap or near each of its edges. Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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As follows from this expression, the perturbation theory predicts the following form of the second finite bandgap, see Fig. 1(b): −1/24 ≤ ν˜ ≡ ν/ε2 ≤ 5/24,

(32)

cf. Eq. (17) for the first finite bandgap. Then, in the lowest approximation, the relation between B1 and A1 is [cf. Eq. (18)]
5ε2 /24 − ν B1 = − A1 . (33) 2λ2 Further, the jump condition at point x = 0 keeps the form of Eq. (19), and, in the lowest approximation, the jump of the first derivative is dominated by term B1 sin (2|x|) in ansatz (26). With regard to relation (33), this leads to the following prediction for the solution supported by the single delta-function in the second finite bandgap:   σ 5 2 2 A1 = − ε −ν . (34) λ2 24

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As follows from Eqs. (34), inside the second finite bandgap (32) this solution exists for σ = −1, i.e., solely for the attractive nonlinearity. In the first approximation, the norm of the soliton is " # ˜ 2 2 (5/24) − ν˜ ((5/24) − ν) N2 ≈ 2 1+ , (35) ˜ ε (5/576)+ (ν˜ /6) − ν˜ 2 (5/576)+ (ν/6) − ν˜ 2 with ν˜ defined as per Eq. (32). It follows from the plot of the norm versus µ, which is shown in Fig. 5 for ε = ±1, that the corresponding GS family satisfies the VK criterion, dN/dµ < 0. Nevertheless, as in the case of the solitons in the first finite bandgap, which was considered above, the GSs in the second bandgap, supported by the attractive nonlinearity, turn out to be unstable at all values of ε. In the general case, when the delta-function is placed asymmetrically with regard to the OL potential, GSs may exist in the second bandgap in the case of the repulsive nonlinearity, and may be stable in that case. A detailed analysis of this case is presented in the next section. Examples of numerically found profiles of the solitons in the second finite bandgap, together with their analytically predicted counterparts [see Eqs. (33) and (34)], are displayed in Fig. 6. The results shown in both Figs. 6 and 5 demonstrate that the perturbative approximation is more accurate for smaller and positive values of ε, when the attractive delta-function is set at a local minimum of the OL potential.

3.3.

The semi-infinite gap

For the sake of completeness of the analysis, we also present results of the application of the perturbation theory (for small |ε|) to solitons supported by the attractive delta-function (σ = −1) in the semi-infinite gap, i.e., at µ < 0. In the zeroth approximation (ε = 0), the Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

The Interplay of Optical Lattices with Localized Nonlinearity 500

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soliton is given by solution (4). Then, it is easy to find the first-order correction to it in the following form: r   ε 5 1 µ φ1 (x) = φ0 (x) − + cos (2x) + − sin(2|x|) , (36) 1 − 2µ 6 2 2 where φ0 (x) is expression (4) [to the first order in ε, solution φ0 (x) + φ1 (x) satisfies both the linear part of Eq. (11) at x 6= 0, and the jump condition (19) at x = 0]. To the same order, the norm of this solution is ε (5 + 2µ) N = 1− , (37) 3 (1 − 2µ)2

which demonstrates that the weak OL potential lifts the degeneracy of the soliton family (4). Note that expression (37) satisfies the VK stability criterion, dN 2ε (11 + 2µ) =− < 0, dµ 3 (1 − 2µ)3

(38)

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for µ > −5.5 in the case of ε > 0, which corresponds to the attractive delta-function set at a local minimum of the OL potential, see Eq. (11); the change of the sign of dN/dµ at µ < −5.5 does not really matter, as dependence (37) is virtually flat in that region. Numerical results demonstrate that the entire soliton family is stable in the case of ε > 0, obeying the VK criterion everywhere in the semi-infinite gap (for a further discussion, see the next section). Figure 7 presents the respective comparison between the analytical approximation and the numerical results. As expected, the prediction is more accurate for smaller ε, closely approximating even complex soliton profiles near the edge of the gap; see panel (e) in Fig. 7. On the other hand, the solitons centered at the local maximum of the OL potential at ε < 0 are definitely unstable (not shown here in detail).

4.

Numerical Results for the Model with the Single Delta-Function

In the case when the local nonlinearity is represented by the single delta-function in Eq. (7), we have found soliton modes in the semi-infinite and two lowest finite gaps, and their stability was investigated by means of numerical methods. The stationary solutions were constructed applying the Newton-Raphson method to the respective nonlinear boundaryvalue problem. The stability was then examined by considering a perturbed solution to Eq. (7), in the form of ∗ φ(x,t) = φs (x) + ge−iλt + f ∗ eiλ t , (39) where φs (x) is the stationary solution, functions g and f are eigenmodes of the infinitesimal perturbation, and λ is the corresponding eigenvalue, which may be complex in the general case. When substituting expression (39) into Eq. (7) and linearizing, one arrives at the following eigenvalue problem:      Lˆ σδ(x − ξ) (φs (x))2 g g =λ , (40) 2 ˆ f f −σδ(x − ξ) (φs(x)) −L

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with Lˆ ≡ −µ − (1/2)d 2/dx2 − ε cos(2x) + 2σδ(x − ξ) (φs (x))2 . This problem can be easily solved using a simple finite-difference scheme, the solution being stable if all the eigenvalues are real. The so predicted stability or instability was verified by means of direct simulations of the evolution of initially perturbed modes. For this purpose, the standard pseudospectral split-step method and the Crank-Nicolson finite-difference algorithm were used. In the course of the analysis, the delta-function was set at different positions within half a period of the OL, 0 < ξ < π/2. In the numerical calculations based on the discretization with stepsize ∆x, the discrete counterpart of the δ-function was defined so that it took a nonzero value, δ˜ = 1/(2∆x), at the single point, x = ξ.

4.1.

Solitons in the semi-infinite gap

Numerical analysis of the solutions in the semi-infinite gap reveals a single soliton family, which exists only in the case of the attractive nonlinearity (σ = −1). A natural result of the stability analysis is that these solitons are stable if the attractive delta-function is located at or near the minimum of the potential. An example for ε = 5 and µ = −4 is displayed in Fig. 8, where shift ξ of the delta-function from the potential minimum, x = 0, takes values within one-half of the spatial period of the potential (recall the period is π). In the semiinfinite gap, the soliton’s stability complies with the VK criterion. In particular, the soliton family is completely stable for ξ = 0. On the other hand, the family is (quite naturally) completely unstable for x = π/2, when the delta-function is set at the point of the potential maximum.

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4.2.

Solitons in the first finite bandgap

The numerical analysis of solutions in the first bandgap demonstrates that GSs cannot exist simultaneously for the attractive and repulsive nonlinearities. If the delta-function is set close to a maximum of the potential (ξ = π/2, for ε > 0), it can support a soliton only with the attractive sign of the nonlinearity. Shifting the position of the attractive deltafunction toward an adjacent minimum of the potential, the soliton ceases to exist at a critical (threshold) value of the coordinate, ξ = ξthr . At the same point, a new soliton appears in the repulsive case, and exists at ξ < ξthr . Figure 9(a) describes the norm of the GS versus the delta-function’s position within one-half of the OL period, for ε = 5 and µ = −1, which is close to the middle of the first finite bandgap. These results agree with the prediction of the perturbation theory obtained for x = 0, according to which the soliton exists only for σ = sgn(ε); see Eq. (21) (recall that ε > 0 and ξ = π/2 are equivalent to ε < 0 and ξ = 0). Close to the existence threshold, the amplitude of the soliton diverges, while its width remains approximately constant. Representative examples of such GSs are displayed in Fig. 10. In particular, Figs. 10(e)-(f)demonstrate a stable soliton (supported by the repulsive nonlinearity) featuring an especially high amplitude, achieved at x = 0.282. The threshold value, ξthr , varies with ε and µ, as shown in Fig. 11. Specifically, for small values of ε and/or small values of µ (close to the lower edge of the first bandgap), the soliton-existence region expands in the case of the repulsive nonlinearity. The stability analysis demonstrates that, in the case corresponding to Fig. 9(a), the GSs in the first bandgap, supported by the repulsive delta-function, are always stable. On

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Figure 7. Panels (a) and (d) display the comparison between the numerical and analytically estimated norms (the solid and dashed curves, respectively) for the solitons in the semiinfinite gap, for ε = 0.1 and 0.5, respectively. Examples of the soliton profiles, taken close to the edge of the gap and deeper inside, are displayed in panels (b),(c) and (e),(f), for the cases corresponding to (a) and (d), respectively. In (b) and (c), the analytically predicted profiles are very close to their numerical counterparts, making the dashed and solid lines virtually indistinguishable.

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Figure 8. (Color online) The norm versus the position of the attractive delta function, for the soliton family in the semi-infinite gap, at ε = 5 and µ = −4. Here and in similar figures shown below, the dashed line depicts the underlying periodic potential (rescaled and shifted upward for the clarity of the picture), while stable and unstable solitons correspond to continuous and dotted lines, respectively.

the other hand, in the case of the attractive nonlinearity, local unstable eigenmodes exist, with large (Im {λ} > 1) purely imaginary eigenvalues, making all the solitons unstable. This finding is not surprising because, as stressed above, in the cases of the repulsion and attraction, the GSs tend to be located, respectively, close to a local minimum or maximum of the periodic potential. The stability analysis was also carried out for other values of ε and µ. The solitons are always stable under the repulsive nonlinearity, while the stability may change in the case of the attraction. Examples are presented in Fig. 12(b)-NvsMuF FGD eltax11, f orε = 5 and ξ = 0.15 (repulsion) and ξ = 0.4, 0.7, 1.1 (attraction). In the latter case, it is seen that, for µ close to the upper edge of the first bandgap, there are small stability areas for the solitons (conspicuous for ξ = 0.4, and extremely small for larger ξ). For fixed µ = 0.7 and varying position ξ of the attractive delta-function, a small stability region is found too, as shown in Fig. 9(b). Although it is small, the existence of the stability area for the GSs in the case of the attraction is a remarkable fact, as it is often assumed that all solitons are unstable in finite band gaps if the nonlinearity is attractive (see, however, Ref. [17], where stable GSs were found in the case of attraction). Apart from the common “localized” instability discussed above (indicated in the stability diagrams by dotted lines), there is additional weaker instability which is often referred to as “oscillatory” (see Refs. [16, 17] for more details). This instability is characterized by complex eigenvalues, λ, and delocalized oscillatory eigenfunctions. Techniques that can be used for the detection of such instabilities are described in Ref. [17]. Regions corresponding to unstable solitons of this type can be clearly seen in Fig. 9(b), distinguished in this and other figures by a dashed-dotted marking. The oscillatory instability can also be found in the case presented in Fig. 12(e), within a

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Figure 9. (Color online) The norm versus the position of the nonlinearity-modulating deltafunction, for solitons in the first finite bandgap, with ε = 5 and µ = −1 (a) or µ = 0.7 (b), and both signs of the nonlinearity. The dashed-dotted vertical lines represent the respective thresholds, ξthr = 0.287 (a) and ξthr = 0.035 (b). As before, continuous lines refer to stable solitons, while dotted and dashed-dotted ones correspond to the strong localized and weak oscillating instabilities, respectively. Shapes of the solitons corresponding to marked points in (a) are displayed in Fig. 10. The evolution of the solitons marked in (b) is displayed in Fig. 13. In panel (b), only the soliton branch corresponding to the attractive nonlinearity appears, as the region of ξ for the repulsive sign is too small, while the corresponding solitons have a very large norm.

tiny area between the non-oscillating unstable solitons and the region of stable solitons, the latter one being very small by itself. Figure 13 illustrates the stability and the simulated development of the solitons’ instabilities, for the representative cases marked in Fig. 9(b). It is relevant to stress that the stability investigation was carried out repeatedly, varying the size of the spatial domain and the number of grid points. In doing so, we have checked that the stable solitons indeed exist, not being simply a case of weak instability. Looking at Fig. 12, it is easy to conclude that, unlike the semi-infinite gap, the stability of the GSs in the first finite bandgap does not obey the VK criterion (cf. Ref. [55], where the same conclusion was reached for models with combined linear and nonlinear periodic potentials). Another noteworthy fact is that, when the delta-function is placed anywhere except maxima or minima of the potential (x = 0, π/2), the GS family does not completely fill the first bandgap. For instance, for ξ = 0.7, the lower cutoff (existence border) for the soliton in the model with attraction is µthr ≈ −2.5, while the lower border of the bandgap is at µ = −2.894.

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The Interplay of Optical Lattices with Localized Nonlinearity 8 7

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Figure 11. (a) The existence threshold, ξthr , for the solitons in the first finite bandgap, as a function of ε, for µ = −1. (b) The same, but as a function of µ for fixed ε = 5.

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4.3.

Soliton solutions in the second finite bandgap

The numerical investigation was also performed for GSs in the second finite bandgap. For the repulsive nonlinearity, a single branch of solitons exists in one-half of the potential period in the plane of (N, ξ). On the other hand, in the case of the attractive nonlinearity, there are two different branches, one located around the minimum of the potential and the other one – around its maximum. A typical example is shown in Fig. 14(a), for ε = 5 and µ = 2, close to the middle of the second bandgap. In this bandgap, there are two thresholds in the region of 0 < ξ < π/2. Similar to what was seen in the first bandgap, the soliton’s amplitude diverges at the threshold, while the soliton’s width remains finite. With the attractive delta-function, GSs are always unstable in the second bandgap (featuring the localized instability), in contrast to the situation in the first bandgap, where a small stability region was found for the case of attraction; see Figs. 9 and 12. On the other hand, in the case of the repulsive delta-function, the stability changes with the variation of parameters ξ, ε and µ. In particular, in the situation displayed in Fig. 14(a) both localized and oscillatory instabilities can be found, and two stability regions are present. If the repulsive delta-function is set at ξ = 0.5, and ε = 5, in which case the second bandgap is 1.05 < µ < 3.724, stable solitons are found only at µ close to the lower edge of the gap; see Fig. 15. For instance, at µ = 1.4 the GSs are stable at almost all values of ξ, except for a narrow interval where the localized instability occurs, as seen in Fig. 14(b). Similar results were found at other values of ε. Similar to the situation in the first bandgap, the GS families cover the entire second bandgap only for ξ = 0 and π/2. For different values of ξ, there are regions of µ in which no solitons are present; see Fig. 15.

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Figure 12. (Color online) The norm of the gap solitons versus µ, in the first finite bandgap, for ε = 5, with the repulsive delta-function placed at ξ = 0 (a) or ξ = 0.15 (b), and the attractive delta-function placed at ξ = 0.4 (c), ξ = 0.7 (d), ξ = 1.1 (e), and ξ = π/2 (f). As above, the stable and unstable solitons correspond to continuous and dotted lines, respectively. A soliton subject to an oscillating instability can be found in the case (e), in an extremely narrow region (barely visible on the scale of this figure) between the stable and locally unstable sections. Blue vertical stripes on both sides indicate Bloch bands between which the first bandgap is sandwiched. The dashed-dotted vertical lines represent thresholds at which the solitons’ norm diverges: µthr = −0.141, −1.572, −2.5, −2.858 for (b), (c), (d) and (e), respectively.

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Figure 13. The evolution of solitons supported by the single attractive delta-function, for the case shown in Fig. 9(b). Panel (a) displays an example of a stable soliton, obtained for ξ = 0.4. Unstable solitons, subject to the oscillatory instability or the strong localized instability, are exhibited for ξ = 0.9 (b) and ξ = 1.3 (c), respectively.

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Figure 14. (Color online) The same as in Fig. 9, but in the second finite bandgap, for µ = 2 (a) and µ = 1.4 (b). Note that, in very narrow regions of ξ, such as the left margin in (b), the norm of the (unstable) solitons, supported by the attractive instability, is especially large, therefore it cannot be displayed on the scale of this figure. The two thresholds (as explained in the text) are represented, as usual, by dashed-dotted vertical lines.

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Figure 15. (Color online) The norm versus µ, in the second finite bandgap for ε = 5, with the repulsive delta function set at ξ = 0.5. The norm diverges at the threshold µ = 3.307, marked by the vertical dashed-dotted curve.

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5.

Numerical Results for the Model with the Two Symmetric Delta-Functions

Equation (8) presents a natural extension of the model, which includes two delta-functions symmetrically positioned around the potential maxima or minima. We consider here the existence and stability conditions for solitons in this model, in the semi-infinite and the first finite gaps. In each case, two settings were explored, with either a potential maximum or minimum located exactly at the midpoint between the two delta-functions. The stability analysis was carried out using the method outlined in Sec. 4., with the nonlinearity coefficient δ(x − ξ) in Eq.(40) replaced by δ(x − ξ) + δ(x + ξ).

5.1.

Solitons in the semi-infinite gap

As in the models with the single delta-function, solitons in the semi-infinite gap exist only for the attractive nonlinearity. First, we examined the changes that the solitons undergo with the increase of distance ξ of each delta-function from the potential minimum located between them, which corresponds to ε > 0 in Eq. (8). For small values of ξ, a region of stable symmetric solitons always exists. Increasing ξ, we reach a bifurcation point, after which the symmetric solutions lose their stability (against non-oscillatory perturbations) and a new asymmetric branch emerges, which may be partially stable. An example is shown in Fig. 16(a), for ε = 5 and µ = −4. In this case, there is a tiny region of stable asymmetric solitons, abutting on the bifurcation point, which is too small to be visible in the figure. Closer to the edge of the semi-infinite gap, the bifurcation occurs at higher values of ξ. For example, ξbif (ε = 2, µ = −1) = 0.777, ξbif (ε = 5, µ = −3) = 0.935, ξbif (ε = 6, µ = −4) = 0.683, and ξbif (ε = 8, µ = −6) = 0.491, and in all these cases, the asymmetric modes are completely unstable. On the other hand,

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the bifurcation occurs at smaller values of ξ for sets of (ε, µ) taken deeper inside gap, i.e., when ε is smaller and/or µ is more negative, the bifurcation giving rise to a conspicuous stability region for asymmetric states in such cases. Typical examples are presented in Fig. 17(a)-NvsDeltax2DeltaS IGe ps2mum4, f or(ε, µ) = (5, −6) and (2, −4). Going still deeper into the semi-infinite gap, the value of ξ at the bifurcation point keeps decreasing. The corresponding domain of stable asymmetric solutions may not necessarily emerge exactly at the bifurcation point, but slightly later. An example is displayed in Fig. 17(c) for (ε, µ) = (2, −6). Bifurcation diagrams in the (µ, θ) plane, displayed in Fig. 17(d) for ξ = 0.4 and ε = 2 and 5, demonstrate the SSB of the supercritical type. For instance, at ε = 2, stable branches of the asymmetric modes emerge at the bifurcation point, where the symmetric branch loses its stability. On the other hand, for ε = 5, the asymmetric branches are not immediately stable after the bifurcation point, as the VK criterion is satisfied for them only when further decreasing µ. At large values of ξ, the asymmetric soliton gradually transforms into a fundamental one, pinned to either of the two delta-functions. As concerns the symmetric modes, with the increase of the distance between the delta-functions (2ξ), they transform into two-soliton bound states that never regain the stability they had prior to the bifurcation. Specifically, near the OL minimum points, where the fundamental soliton is stable (see Fig. 16(a)), eigenvalues accounting for the instability of the bound state decrease as its two constituents are pulled farther apart with the increase of ξ (but still, never fall exactly to zero). As in the two-delta-functions model without the lattice (ε = 0) [50], antisymmetric states exist too and are unstable at small values of ξ. As ξ increases, they develop into antisymmetric two-soliton bound states, for which unstable eigenmodes could not be found (featuring instead zero eigenvalues) exactly in the region where the stable fundamental soliton is supported by the single delta-function. Direct simulations have shown that, while such bound states are stable against symmetric disturbances, asymmetric perturbations break them into mutually incoherent fundamental solitons. Figure 18 presents several examples of the solitons of all the aforementioned types, in the case corresponding to Fig. 16(a). Similar analysis was carried out for two attractive delta-functions placed on both sides of a local maximum of the periodic potential, which implies ε < 0 in Eq. (8). Figures 16(b) and 19 present a typical example of the results in the (ξ, N) plane, for ε = −5 and µ = −4. In this case, both the symmetric and asymmetric solitons are unstable at small values of ξ. In other aspects, the results resemble those reported above for the case of two delta-functions placed symmetrically around a potential minimum. It is relevant to compare these results with those reported in Ref. [50] for two attractive delta-functions in the absence of the periodic potential (ε = 0). In that case, exact analytical solutions are available for the pinned states of all the types—symmetric, asymmetric, and antisymmetric. The symmetric states are stable before the symmetry-breaking bifurcation, and unstable after it, terminating at final N. The bifurcation is of an “extreme subcritical” type, with branches of the asymmetric states going backward and never turning forward, hence they are completely unstable. In fact, these results, obtained with the ideal δ-functions, are degenerate. The numerical analysis with regularized delta-functions lifts the degeneracy, demonstrating that the branches of the asymmetric modes eventually turn forward, stabilizing themselves. Simultaneously, the family of the symmetric modes (un-

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Figure 16. (Color online) Branches of symmetric and asymmetric soliton modes in the semi-infinite gap of the model with two delta-functions, for µ = −4 and ε = 5 (a) or −5 (b), which correspond, respectively, to the local minimum or maximum of the periodic potential located between the delta-functions. As before, the continuous and dotted lines represent stable and unstable portions of the soliton families. Circles correspond to representative examples of solitons that are shown in panels (a)-(d) of Figs. 18 and 19 .

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Figure 17. (Color online) (a)-(c) Branches of the symmetric and asymmetric states in the (ξ, N) plane near the bifurcation point, in the semi-infinite gap of the model with two deltafunctions, for (a) ε = 5, µ = −6, (b) ε = 2, µ = −4, and (c) ε = 2, µ = −6. (d) The bifurcation diagrams in the (µ, θ) plane, for ξ = 0.4 and ε = 2 and 5.

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Figure 18. Typical profiles of modes generated by two attractive delta-functions in the semiinfinite gap. Panels (a)-(d) correspond to the points marked by circles in Fig. 16(a). The arrows indicate the position of the delta-functions. Stable symmetric (a) and asymmetric (b)-(c) solitons are demonstrated, as well as an unstable symmetric bound state (d). In addition, panels (e) and (f) show, respectively, an unstable antisymmetric soliton, and an antisymmetric bound state which is unstable with respect to asymmetric perturbations.

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2 2.5 ξ=0.1 2 1.5 1

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Figure 19. Examples of solitons corresponding to the points marked by circles in Fig. 16(b). Panels (a) and (b) demonstrate unstable symmetric and asymmetric solitons. An example for a stable asymmetric soliton is displayed in (c). Panel (d) shows an unstable symmetric bound state. In addition, examples of an antisymmetric soliton, that features the strong local instability, and antisymmetric bound state, which is unstable against asymmetric perturbations, are shown in panels (e) and (f), respectively.

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stable past the bifurcation point) extends to N → ∞. Antisymmetric states are completely unstable in that model (they become stable if the regularized delta-functions are made broad enough). A comparison with the present findings suggests that the addition of the periodic potential also lifts the degeneracy of the system, even if the δ-functions are kept in the ideal form. In fact, this is similar to how the inclusion of the weak periodic potential lifts the degeneracy of the soliton family (4) supported by the single attractive δ-function and stabilizes the family; see Eqs. (36) and (37).

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5.2.

Solitons in the first finite bandgap

Similar to the case of the single delta-function, GSs in the first finite bandgap can be found for both the attractive and repulsive nonlinearities. We again start by considering the pair of delta-functions placed around a minimum of the potential. For the case of two attractive delta-functions, antisymmetric solitons exist in the region of 0 < ξ < π − ξthr , where ξthr is the same threshold as in the model with the single delta-function. Unlike what was observed in the semi-infinite gap, in the first finite bandgap the asymmetric solitons bifurcate from the antisymmetric branch; see an example in Fig. 20(c) for ε = 5 and ξ = 1. In particular, this setting features a closed bifurcation loop, where both the direct and the reverse bifurcations are of the supercritical type. In the case presented in Fig. 20(c), there is a small region of stable antisymmetric solitons, obtained for high values of µ, near the upper edge of the gap. Decreasing µ, the soliton is destabilized by an oscillatory instability, both the antisymmetric and asymmetric branches being strongly unstable past the bifurcation point. A typical example, plotted in the (ξ, N) plane for ε = 5 and µ = −1, is displayed in Figs. 20(a)-(b),where a small section of the antisymmetric branch is stable. The stability analysis, carried out for different values of ε and µ, indicates a somewhat larger stability region for larger ε, and for µ taken closer to the upper edge of the bandgap. The stability region gradually disappears at smaller values of ε, or close to the lower edge of the bandgap. A weak oscillatory instability also occurs in the present case. In particular, in the situation corresponding to Fig. 20, a very small section (too small to be visible in Fig. 20(a)) of the corresponding weakly unstable antisymmetric solitons exists, starting from the edge of the stable region and extending to slightly smaller values of ξ. (2) In this setting, an additional threshold, ξthr , was found (which is not related to that in the single-delta-function model, ξthr ; see Fig. 11). The new threshold serves as the upper boundary for always-stable symmetric solitons generated by a pair of closely placed repulsive delta-functions, as shown in Fig. 20(b). Simultaneously, the same threshold is a (2) lower border for a branch of unstable symmetric states that exists at ξthr < ξ < π − ξthr , in the case of the attractive nonlinearity. Typical examples of the soliton profiles of the symmetric, antisymmetric and asymmetric types are displayed in Fig. 21. Further, Fig. 22 (2) shows ξthr as a function of ε and µ (within the first finite bandgap). All the symmetric GSs are stable in the case of repulsion, and unstable under attraction. Modes found at ξ > π − ξthr may be considered as bound states of two fundamental GSs. For both signs of the nonlinearity, symmetric and antisymmetric bound states are found precisely where their fundamental counterparts exist. This can be seen in Fig. 20(a), comparing the location of the bound-state branches with those obtained in the model with the single delta-function, cf. Fig. 9(a). It is not surprising that both symmetric and anti-

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symmetric bound states are strongly unstable close to the potential maximum, in the case of the attractive nonlinearity. For the repulsive nonlinearity, the symmetric bound states experience weak local instability in the regions near the minimum of the OL. On the other hand, their antisymmetric counterparts may appear to be stable in terms of the eigenvalues, and in direct simulations with respect to symmetric perturbations, but they split into their fundamental constituents when asymmetric disturbances are imposed (similar to the above-mentioned antisymmetric bound states in the semi-infinite gap). When the two delta-functions are placed around a maximum of the periodic potential, (2) the picture is somewhat simpler. In this case, the secondary threshold, ξthr , does not exist, while, for the attractive nonlinearity, all types of the soliton families—symmetric, asymmetric and antisymmetric ones—exist in the region of 0 < ξ < π/2 − ξthr (not shown here). Also, in contrast to the previous setting, the asymmetric solitons bifurcate from the symmetric branch (not from the antisymmetric one), all the solitons being unstable in the case of the attractive nonlinearity. Results for the bound states are qualitatively similar to those reported for the minimum-centered configuration, with the difference that, in the present case, the weak localized instability occurs for the antisymmetric solutions, while the symmetric bound states are the ones that are unstable only against asymmetric perturbations.

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6.

Conclusions

In this work, we have introduced two settings that combine attractive or repulsive nonlinearity, concentrated in one or two points, and the linear periodic potential. For the model with the single delta-function, we have found stable solutions in the semi-infinite and two lowest finite gaps. In particular, in the case of the attractive nonlinearity (σ = −1), the degenerate family of exact soliton solutions exists in the absence of the periodic potential, being fully unstable. Even a weak potential lifts the degeneracy and stabilizes the entire family in the semi-infinite gap, provided that the delta-function is set in a finite region around a local minimum of the periodic potential. The stability of this soliton family agrees with the VK criterion. In the first finite bandgap, GSs have been found for both attractive and repulsive nonlinearities, although they do not coexist: if the delta-function is placed in a finite area around a local minimum of the periodic potential, the GS exists only under the repulsion, and in the remainder of the period of the potential, GS can be supported solely by attractive nonlinearity. In the first bandgap, all the GSs are stable in the case of the repulsion, while the soliton pinned to the attractive delta-function is stable only in a small region, if any. In the second bandgap, two soliton branches, centered around the attractive deltafunction set at either the minimum or maximum of the periodic potential, were found. While none of them is stable, stability regions were produced for the soliton branch supported by the repulsive delta-function. It exists exactly for those values of shift ξ of the δ-function relative to the underlying potential at which both aforementioned branches are absent in the case of the attraction. In addition to the numerical results, analytical predictions, based on the perturbation theory for the Mathieu equation, were presented for the case in which the δ-function is po-

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Figure 20. (Color online) (a)-(b) The soliton’s norm versus half the distance between the two delta-functions, ξ, in the first finite bandgap, for ε = 5 and µ = −1. In panel (a), the asymmetric branch bifurcates from the antisymmetric one, in the case of the attractive nonlinearity. At larger values of ξ, the soliton branches correspond to unstable antisymmetric bound states of two solitons, for either repulsive or attractive nonlinearity. (b) Branches of symmetric states are shown for both the repulsive and attractive nonlinearities. Not shown in (b) are the unstable symmetric bound states, for larger ξ, as this part of the diagram is virtually identical to the one in panel (a), for the antisymmetric bound states. The dasheddotted vertical lines represent the existence thresholds, π ± ξthr = 2.853, 3.430 (periodic (2) with period π) and ξthr = 0.588 [only in (b)]. Circles indicate solitons whose profiles are displayed in Fig. 21. (c) The bifurcation diagram for the antisymmetric and asymmetric modes, in the (µ, θ) plane, at ε = 5 and ξ = 1.

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sitioned exactly at the maximum or minimum of the periodic potential. This approximation was developed for the solitons in the semi-infinite and two lowest finite gaps, showing a good accuracy in comparison with the numerical findings, in all the cases. In the model with two separated local nonlinearities, represented by the symmetric pair of the delta-functions, the numerical analysis was carried out for the semi-infinite and the first finite gaps. Families of symmetric, antisymmetric and asymmetric solitons were found, with respect to the symmetric set of the two delta-functions. In the semi-infinite gap, with the delta-functions set symmetrically around a minimum of the potential, the symmetric solitons are stable up to the symmetry-breaking bifurcation point, from which asymmetric branches emerge, that turn out to be partially stable. Antisymmetric modes were found too in the semi-infinite gap, being completely unstable. With the two delta-functions placed symmetrically around a point of the potential maximum, all the soliton families obtained at small separations between the delta-functions, as well as all the two-soliton bound states, are unstable. In the first finite bandgap, asymmetric solitons bifurcate from the antisymmetric branch, if the potential minimum is set between the attractive delta-functions. This antisymmetric

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Figure 22. (a) The secondary threshold, ξthr , for the pair of two closely set delta-functions in the first finite band gap, versus ε, for µ = −1. (b) The same as a function of µ, for ε = 5. branch has a very short stability segment, while the asymmetric GSs are completely unstable. Under the repulsion, there is a stable branch of symmetric modes centered around the minimum of the potential. At a certain threshold value of the separation 2ξ between the delta-functions (which is different from the threshold found in the single-delta-function setting), the symmetric mode switches from repulsive to attractive nonlinearity, simultaneously loosing its stability. Finally, in the configuration with the local maximum of the periodic potential fixed at the midpoint, no stable solitons were found for sufficiently small separations 2ξ. For both configurations, with the midpoint coinciding with either the maximum or minimum of the underlying potential, all the families of two-soliton bound states are unstable against various perturbation modes. This work may be naturally extended in other directions. In particular, it may be interesting to consider the nonlinearity represented by a periodic array of delta-functions. A challenging issue is to analyze similar models in two dimensions.

References [1] Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press: San Diego, 2003). [2] F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008); Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Progress in Optics 52, 63 (2009).

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[3] N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002); J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003). [4] O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179 (2006). [5] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, Adv. Phys. 56, 243 (2007). [6] S. Ghanbari, T. D. Kieu, A. Sidorov, and P. Hannaford, J. Phys. B: At. Mol. Opt. Phys. 39, 847 (2006). [7] L. Berg´e, Phys. Rep. 303, 259 (1998). [8] B. B. Baizakov, B. A. Malomed, and M. Salerno, Europhys. Lett. 63, 642 (2003). [9] N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen and M. Segev, Phys. Rev. Lett. 91, 213906 (2003). [10] J. Yang and Z. H. Musslimani, Opt. Lett. 28, 2094 (2003); Z. H. Musslimani and J. Yang, J. Opt. Soc. Am. B 21, 973 (2004). [11] B. B. Baizakov, B. A. Malomed and M. Salerno, Phys. Rev. A 70, 053613 (2004).

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[12] D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan and L. Torner, Phys. Rev. E 70, 055603(R) (2004). [13] Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004); 94, 043902 (2005); Y. V. Kartashov, R. Carretero-Gonz´alez, B. A. Malomed, V. A. Vysloukh, and L. Torner, Opt. Express 13, 10703 (2005). [14] B. B. Baizakov, B. A. Malomed, and M. Salerno, Phys. Rev. E 74, 066615 (2006). [15] D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, Phys. Rev. Lett. 95, 023902 (2005). [16] F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, Phys. Rev. A 64, 043606 (2001); I. Carusotto, D. Embriaco, and G. C. La Rocca, ibid. 65, 053611 (2002); P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage and Y. S. Kivshar Phys. Rev. A 67, 013602 (2003). [17] D. E. Pelinovsky, A. A. Sukhorukov and Y. S. Kivshar, Phys. Rev. E 70, 036618 (2004). [18] N. K. Efremidis and D. N. Christodoulides, Phys. Rev. A 67, 063608 (2003). [19] T. Mayteevarunyoo and B. A. Malomed, Phys. Rev. A 74, 033616 (2006); ibid. 80, 013827 (2009). [20] J. Cuevas, B.A. Malomed, P. G. Kevrekidis, and D. J. Frantzeskakis, Phys. Rev. A 79, 053608 (2009). Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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[21] M. Skorobogatiy and J. Yang, Fundamentals of Photonic Crystal Guiding (Cambridge University Press, Cambridge, England, 2009). [22] B. B. Baizakov, V. V. Konotop, and M. Salerno, J. Phys. B 35, 5105 (2002); P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Phys. Rev. A 67, 013602 (2003). [23] E. A. Ostrovskaya and Y. S. Kivshar, Opt. Exp. 12, 19 (2004); Phys. Rev. Lett. 93, 160405 (2004). [24] H. Sakaguchi and B.A. Malomed, J. Phys. B 37, 2225 (2004). [25] A. Gubeskys, B.A. Malomed, and I. M. Merhasin, Phys. Rev. A 73, 023607 (2006). [26] E. A. Ostrovskaya, T. J. Alexander, and Y. S. Kivshar, Phys. Rev. 74, 023605 (2006). [27] A. Gubeskys and B. A. Malomed, Phys. Rev. A 76, 043623 (2007). [28] T. Mayteevarunyoo, B. A. Malomed, B. B. Baizakov, and M. Salerno, Physica D, 238, 1439 (2008); J. Wang and J. Yang, Phys. Rev. A 77, 033834 (2008). [29] Z. Shi and J. Yang, Phys. Rev. E 75, 056602 (2007). [30] J. Wang and J. Yang, Phys. Rev. A 77, 033834 (2008). [31] B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K.-P. Marzlin, and M. K. Oberthaler, Phys. Rev. Lett. 92, 230401 (2004).

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[32] Th. Anker, M. Albiez, R. Gati, S. Hunsmann, B. Eiermann, A. Trombettoni, and M. K. Oberthaler, Phys. Rev. Lett. 94, 020403 (2005). [33] T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, Phys. Rev. Lett. 96, 040401 (2006). [34] B. A. Malomed, Soliton Management in Nonlinear Systems (Springer: New York, 2006). [35] W. A. Harrison, Pseudopotentials in the Theory of Metals (Benjamin: New York, 1966). [36] Y. V. Kartashov, B. A. Malomed, and L. Torner, Solitons in nonlinear lattices, Rev. Mod. Phys., in press. [37] B. I. Mantsyzov and R. N. Kuz’min, Sov. Phys. JETP 64, 37 (1986); B. I. Mantsyzov, Phys. Rev. A 51, 4939 (1995); A. Kozhekin and G. Kurizki, Phys. Rev. Lett. 74, 5020 (1995); T. Opatrny, B. A. Malomed, and G. Kurizki, Phys. Rev. E 60, 6137 (1999); A. Y. Sivachenko, M. E. Raikh, and Z. V. Vardeny, Phys. Rev. A 64, 013809 (2001); J. Cheng and J. Y. Zhou, Phys. Rev. E 66, 036606 (2002). [38] G. Kurizki, A. E. Kozhekin, T. Opatrny, and B. A. Malomed, in Progress in Optics 42, 93 (E. Wolf, editor: North Holland, Amsterdam, 2001). Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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[39] K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007). [40] S. Inouye, M. R. Andrews, J. Stenger, H. J. Miesner, D. M. Stamper-Kurn, and W. Ketterle, Nature (London) 392, 151 (1998); Ph. Courteille, R. S. Freeland, D. J. Heinzen, F. A. van Abeelen, and B. J. Verhaar, Phys. Rev. Lett. 81, 69 (1998); J. L. Roberts, N. R. Claussen, J. P. Burke, C. H. Greene, E. A. Cornell, and C. E. Wieman, ibid. 81, 5109 (1998). [41] P. O. Fedichev, Yu. Kagan, G. V. Shlyapnikov, and J. T. M. Walraven, Phys. Rev. Lett. 77, 2913 (1996). [42] M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, Phys. Rev. Lett. 93, 123001 (2004). [43] M. Marinescu and L. You, Phys. Rev. Lett. 81, 4596 (1998).

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[44] K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, Nature 417, 150 (2002); L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, Science 296, 1290 (2002). [45] H. Sakaguchi and B. A. Malomed, Phys. Rev. E 72, 046610 (2005); J. Garnier and F. K. Abdullaev, Phys. Rev. A 74, 013604 (2006); D. L. Machacek, E. A. Foreman, Q. E. Hoq, P. G. Kevrekidis, A. Saxena, D. J. Frantzeskakis, and A. R. Bishop, Phys. Rev. E 74, 036602 (2006); M. A. Porter, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, Physica D 229, 104 (2007); J. Belmonte-Beitia, V. M. P´erez-Garc´ıa, V. Vekslerchik, and P. J. Torres, Phys. Rev. Lett. 98, 064102 (2007); F. Abdullaev, A. Abdumalikov, and R. Galimzyanov, Phys. Lett. A 367, 149 (2007); G. Dong and B. Hu, Phys. Rev. A 75, 013625 (2007); D. A. Zezyulin, G. L. Alfimov, V. V. Konotop, and V. M. P´erez-Garc´ıa, ibid. 76, 013621 (2007); Z. Rapti, P. G. Kevrekidis, V. V. Konotop, and C. K. R. T. Jones, J. Phys. A: Math. Theor. 40, 14151 (2007); H. A. Cruz, V. A. Brazhnyi, and V. V. Konotop, J. Phys. B: At. Mol. Opt. Phys. 41, 035304 (2008); L. C. Qian, M. L. Wall, S. L. Zhang, Z. W. Zhou, and H. Pu, Phys. Rev. A 77, 013611 (2008); F. K. Abdullaev, A. Gammal, M. Salerno, and L. Tomio, ibid. 77, 023615 (2008); A. S. Rodrigues, P. G. Kevrekidis, M. A. Porter, D. J. Frantzeskakis, P. Schmelcher, and A. R. Bishop, ibid. 78, 013611 (2008); Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, Opt. Lett. 34, 3625 (2009). [46] R. Y. Hao, R. C. Yang, L. Li, and G. S. Zhou, Opt. Commun. 281, 1256 (2008). [47] H. Sakaguchi and B. A. Malomed, Phys. Rev. E 73, 026601 (2006); G. Fibich, Y. Sivan, and M. I. Weinstein, Physica D 217, 31 (2006); G. Dong, B. Hu, and W. Lu, Phys. Rev. A 74, 063601 (2006); R. Y. Hao and G. S. Zhou, Chin. Opt. Lett. 6, 211 (2008); Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, Opt. Lett. 34, 770 (2009); N. V. Hung, P. Zi´n, M. Trippenbach, and B. A. Malomed, Twodimensional solitons in media with stripe-shaped nonlinearity modulation, Phys. Rev. E 82, 046602 (2010). [48] B.A. Malomed and M. Ya. Azbel, Phys. Rev. B 47, 10402 (1993). Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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[49] A. A. Sukhorukov, Y. S. Kivshar, and O. Bang, Phys. Rev. E 60, R41 (1999). [50] T. Mayteevarunyoo, B. A. Malomed, and G. Dong, Phys. Rev. A 78, 053601 (2008). [51] L. C. Qian, M. L. Wall, S. Zhang, Z. Zhou, and H. Pu, Phys. Rev. A 77, 013611 (2008). [52] M. Vakhitov and A. Kolokolov, Radiophys. Quantum. Electron. 16, 783 (1973). [53] F. K. Abdullaev and M. Salerno, Phys. Rev. A 72, 033617 (2005). [54] L. D. Landau and E. M. Lifshitz, Mechanics (Nauka Publishers: Moscow, 1973).

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[55] H. Sakaguchi and B. A. Malomed, Phys. Rev. A 81, 013624 (2010).

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In: Optical Lattices: Structures, Atoms and Solitons Editor: Benjamin J. Fuentes

ISBN: 978-1-61324-937-6 © 2012 Nova Science Publishers, Inc.

Chapter 6

A NEW MULTILAYER STRUCTURE BASED REFRACTOMETRIC OPTICAL SENSING ELEMENT Anirudh Banerjee1 Advance Research Centre for Optical Communications, Department of Electronics and Communication, Amity School of Engineering and Technology, Lucknow-226010, Amity University, Uttar Pradesh, India

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ABSTRACT A new multilayer structure based refractometric optical sensing element is suggested for sensing very small refractive index changes of a medium. This new multilayer structure exhibits isolated narrow transmission peaks in the output spectrum in stop band wavelength regions and a slight change in refractive index of material layers induces large transmission peak shifts in the output spectrum. This new multilayer based refractometric sensing element is not only remarkably smaller but is also very sensitive to refractive index changes.

1. INTRODUCTION In past years, multilayer or photonic band gap structures [1-3] attracted lot of attention of researchers due to their enormous applications in optical communications, optical electronics and optical instrumentation. These multilayer structures are formed by using two or more materials. These multilayered structures lead to formation of stop bands, in which propagation of electromagnetic waves of certain wavelengths are prohibited. However, these bands or ranges depend upon a number of parameters such as refractive indices of materials, filling fraction and angle of incidence. If all other parameters are kept constant, then any change in refractive index of a material will change the stop bands and ranges of transmission. By monitoring transmission or reflection band change or shift, a slight change in refractive index 1

E-mail address: [email protected]

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Anirudh Banerjee

of a structural material can be detected. Based on this principle, recently Hopman et al. [4] suggested a quasi-one dimensional (1D) photonic crystal based compact building block for refractometric optical sensing. The suggested one dimensional (1D) photonic band gap structure [4] was 76m thick and it produced a wavelength band shift of 0.8nm for a refractive index change of 0.05. In this chapter, a new multilayer structure is suggested as a refractometric optical sensing element. In this new sixteen layer structure, layers with low refractive indices (denoted by L) and layers with high refractive indices (denoted by H) are arranged in the sequence LLLHHLHHLLLHHLHH as shown in Figure 1. This new multilayer structure exhibits isolated narrow transmission peak in the reflectance or stop band region of its transmission spectrum and any change in structural parameters such as refractive indices of materials, filling fraction and angle of incidence etc. induce a large transmission peak shift in the output spectrum. If all other parameters are kept constant, then any change in refractive index of a material will induce transmission peak shift. Therefore, by monitoring transmission peak shift, even a slight change in refractive index of a structural material can be detected. With proper arrangements, this kind of multilayer structure can be used in material adulteration sensors and/or refractometer systems, for detection of adulteration and determination of refractive index of gases or liquids or suitable materials.

2. THEORY

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In Figure 1, the proposed new multilayer structure is shown. Layers with low refractive indices (denoted by L) and layers with high refractive indices (denoted by H) are arranged in the LLLHHLHHLLLHHLHH sequence.

Figure 1. Schematic of the proposed new multilayer structure.

Using the matrix formulation method [5] to calculate the transmittance through this new multilayer structure, the transfer matrices for the layers L and H with thicknesses a and b, respectively, are given below as

 cos  L ML     iq L sin  L

 i sin  L  qL   cos  L 

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(1)

199

A New Multilayer Structure based Refractometric Optical Sensing Element and

MH

 cos  H    iq H sin  H 2

 i sin  H qH cos  H

    2

n b cos H are the layer phase thicknesses.  H n L and n H are refractive indices of layers L and H respectively.

where,  L 



nL a cos L and  H 

(2)

 L and  H are angles of refraction in layers L and H respectively. Parameters q L and q H are given by

q L  nL cos L and q H  nH cos H for TE polarization and

qL  cos L nL and q H  cos H nH for TM polarization. One can obtain the total transmission matrix for this random sequence multilayer as

M  M LM LM LM H M H M LM H M H M LM LM LM H M H M LM H M H

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m12  m   11  m21 m22 

(3)

For normal incidence of light on the structure in surrounding medium of air, the transmission coefficient of this multilayer can be written as t

2 m11  m12   m21  m22 

(4)

and the transmissivity for this structure can be written in terms of transmission coefficient as Tt

2

(5)

3. RESULTS AND DISCUSSIONS Figure 2, shows the transmission spectrum at normal incidence of light for this new multilayer structure with low index layer refractive index nL  1.35 and nL  1.39 . It is assumed that the refractive index of layers may change due to adsorption or adulteration or doping of a material. The refractive index of high index layer was kept constant at nH  4.35

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Anirudh Banerjee

and the thicknesses of low indices layers and high indices layers were taken as a  202nm and b  40nm respectively. It is clearly evident from Figure 2 that the isolated transmission peak exhibited in reflectance or stop band regions by new multilayer structure shift toward higher wavelengths with increase in low index layer‟s refractive index (as happens in case of doping or adulteration).

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Figure 2. Transmission spectra for the proposed multilayer structure with change in refractive index of L layers.

Figure 3. Transmission spectra for the proposed multilayer structure with change in refractive index of L layers. Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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The center wavelength of transmission peak was at 1.538m which shifted to 1.5775m on changing refractive index of low index layer from n L  1.35 to n L  1.39 . Next, for different variations in the values of refractive indices of materials in this multilayer structure, it was observed that transmission peak shifts sufficiently to indicate variation in refractive index of any layer. Figure 3, shows the transmission peak shift by this multilayer for another refractive index and thickness combination with nL  1.45 , nH  4.35 , a  200nm and

b  50nm . It is clear from Figure 3, that initially the center wavelength of transmission peak was at 1.694m which shifted to 1.734m on changing refractive index of low index layer from nL  1.45 to nL  1.50 . These transmission peak shifts are very easily detectable with the help of existing optoelectronic devices.

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CONCLUSIONS A new multilayer structure is suggested as refractometric optical sensing element. It was observed that slight change in refractive index of material layers of the structure causes a sufficiently large transmission peak shift, which can be very easily detected by monitoring transmission spectrum, with the help of existing optoelectronic devices. The narrow transmission peaks with steep side edges increase the detection sensitivity. A much higher sensitivity or transmission peak shift per unit refractive index change [4] can be obtained at a much smaller size [4] with this proposed new multilayer structure. This kind of sensing element finds potential applications in fluid or material adulteration sensors or refractometer systems with suitable arrangements. The applications of this new multilayer structures is not limited to refractive index sensing, but it can be further extended to temperature sensing by infiltrating the low index layer with liquid crystals, in which, refractive index changes can be thermally induced.

ACKNOWLEDGMENTS I am grateful to Sri. Aseem Chauhan, Maj. Gen. K. K. Ohri, Prof. S. T. H. Abidi, Prof. N. Ram and Brig. U.K. Chopra of Amity University, U.P., Lucknow Campus for their constant encouragement and support during this research work.

REFERENCES [1] [2] [3]

Yablonovitch, E. Phys. Rev. Lett. 1987, vol. 58, 2059-2062. John, S. Phys. Rev. Lett. 1987, vol. 58, 2486–2489. Banerjee, A. In Photonic Crystals: Fabrication, Band Structure and Applications; Laine, V. E.; Nova Science Publishers: Hauppauge, NY, 2010. ; pp 135-147.

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202 [4]

Hopman, W. C.; L., Pottier, P.; Yudistira, D.; Lith, J. V.; Lambeck, P. V.; Rue, R. M. D. L.; Driessen, A. ; Hoekstra, H. J. W. M.; Ridder, R. M. D. IEEE J. Sel. Top. Quantum Electron. 2005, vol. 11, 11-16. Born, M.; Wolf, E. Principles of Optics; Cambridge University Press: London, U.K., 1980; pp 66-67.

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[5]

Anirudh Banerjee

Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

In: Optical Lattices: Structures, Atoms and Solitons ISBN: 978-1-61324-937-6 c 2012 Nova Science Publishers, Inc. Editor: Benjamin J. Fuentes

Chapter 7

M ATTER WAVE D ARK S OLITONS IN O PTICAL S UPERLATTICES Aranya B. Bhattacherjee1 and Monika Pietzyk2 1 Department of Physics, ARSD College (South Campus), University of Delhi, India 2 National Quantum Information Centre of Gda´nsk, Sopot, Poland

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Abstract In this work, we study the behaviour of matter-wave band gap spectrum and eigenstates as the periodicity of the optical superlattice is increased. We show that the band gap (between the two lowest bands) which opens up in a doubly periodic superlattice decreases as the periodicity increases further. This is interpreted as a decrease in the Periels-Nabarro barrier which the dark soliton experiences as it goes from one well to the next. For higher periodicity the mobility of the dark soliton is restored.

1.

Introduction

When a gas of ultracold atoms is loaded into an optical lattice,its properties are modified strongly [1]. Ultracold bosons trapped in such periodic potentials have been widely used recently as a model system for the study of some fundamental concepts of quantum physics like Josephson effects[2], squeezed states[3], Landau-Zener tunneling and Bloch oscillations[4] and superfluid-Mott insulator transitions[5]. One of the many advantages of a macroscopic quantum periodic system such as a BEC in an optical lattice, is that the effective periodic potential created by a standing light wave can be easily and precisely manipulated by changing the intensities, polarizations, frequencies or geometric arrangement of the interfering laser beams. For example, the depths of the periodic potential wells induced by an optical lattice can be controlled by tuning the intensities of the laser beams. Using superposition of optical lattices with different periods [6], it is now possible to generate more sophisticated periodic potentials characterized by a richer spatial modulation, the so-called optical superlattices. An important and exciting application of optical superlattice is quantum computation [7]. Theoretical interest in optical superlattices started only recently. Examples include work on fractional filling Mott

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Aranya B. Bhattacherjee and Monika Pietzyk

insulator domains [8], dark[9] and gap[10] solitons, the Mott-Peirels transitions[11], nonmean field effects[12] and phase diagrams of BEC in two-colour superlattices[13]. Porter et al.[14] have shown that optical superlattices can manipulate and control solitons in BEC. The analogue of the optical branch in solid-state physics has also been predicted in an optical superlattice [15]. Rousseau et al.[16] have considered the effect of a secondary lattice on a one-dimensional hard core of bosons (strongly correlated regimes). A detailed theoretical study of the Bloch and Bogoliubov spectrum of a BEC in a one-dimensional optical superlattice has been done by Bhattacherjee [17]. In an interesting work [18], we show that due to the secondary lattice, there is a decrease in the superfluid fraction and the number fluctuation. The dynamic structure factor which can be measured by Bragg spectroscopy is also suppressed due to addition of the secondary lattice. The visibility of the interference pattern (the quasi-momentum distribution) of the Mott insulator is found to decrease due to the presence of the secondary lattice. In a very recent experiment [19], it was observed that the center-of-mass motion of a BEC is blocked in a quasi-periodic lattice. This was interpreted as a result of an increase in the effective mass in an superlattice [20]. Most remarkably, periodicity of the optical lattice potential leads to the effective dispersion of the BEC wavepackets being a function of the band structure. In the majority of condensates currently created experimentally, the interatomic interaction is repulsive. This corresponds to an effectively defocusing nonlinearity of the matter-wave which can support dark solitons-localized dips on the condensate density background with a phase gradient across the localized regions. Similar to other types of solitons, they can remain dynamically stable due to the balancing effects of nonlinearity and the (positive) dispersion. Dark solitons have been created experimentally in repulsive condensates by using phase imprinting technique to apply a sharp phase gradient to a condensate cloud in a magnetic trap [21, 22]. In the case of BECs loaded into optical lattices, i.e. with the possibility for dispersion management, dark solitons can be supported in both repulsive (for positive effective dispersion) and attractive condensates (for negative effective dispersion). Moreover, dark lattice solitons are expected to be easier to create experimentally than bright gap solitons as they are not confined to the spectral gaps and a phase imprinting technique can be applied to a nonlinear Bloch-wave background within a spectral band. The theory of dark solitons has been developed extensively for many types of periodic systems such as discrete atomic chains and waveguide arrays [23, 24, 25, 26, 27]. Applying the concepts of discrete dynamical systems to the physics of the Bose-Einstein condensates in optical lattices, Abdullaev et al. [28] studied dark and bright solitons on non-zero backgrounds in a vertical lattice by employing a discrete mean-field model derived in the tight-binding approximation, i.e. considering a single isolated band of the Bloch-wave spectrum. In contrast, Yulin and Skryabin [29] used a single-gap continuous coupled-mode model in order to examine the stability and existence of out-of-gap dark and bright solitons. A more general analysis based on the continuous Gross-Pitaevskii equation with a periodic potential was presented by Alfimov et al. [30] who showed that for a repulsive condensate in an optical lattice, dark solitons can exist as stationary localized solutions with nonvanishing asymptotics. Alfimov et al. [30] as well as Konotop and Salerno [31] also found numerically stable dark solitons for periodic quasi-one-dimensional BEC systems. The weak spectral instability of the dark solitons in the combined optical lattice and a strong harmonic potential, both in the discrete and continuous mean-field models, has been

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Matter Wave Dark Solitons in Optical Superlattices

205

established in [32]. In this work, we study the behaviour of matter-wave gap spectrum and the eigenstates as the periodicity of the optical superlattice is increased.We study the structure and mobility properties of dark solitons in superlattices by employing the full continuous mean-field model. We show that the band gap (between the two lowest bands) which opens up in a doubly periodic superlattice decreases as the periodicty increases further. This is interpreted as a decrease in the Periels-Nabarro barrier which the dark soliton experiences as it goes from one well to the next. For higher periodicity the mobility of the dark soliton is restored.

2.

The Model

We consider an elongated cigar shaped BEC in an optical superlattice. The dynamics of the BEC can be described in the mean-field approximation by the Gross-Pitaevskii (GP) equation for the macroscopic condensate wavefunction ψ(x, y, z, t). i~

∂ψ(x, y, z, t) ∂t 

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=

 ~2 2 2 − ∇ + VL(x) + V (x, y, z) + g3D |ψ(x, y, z, t)| ψ(x, y, z, t), (1) 2m

where V (x, y, z) is the time-independent magnetic trapping potential and g3D = 4π~2 as /m is the two-body interaction with m and as as the mass and scattering length of the condensate atoms respectively. We will consider only the case of repulsive interaction. For the cases examined in this paper, we use the parameters set by 87 Rb: m = 1.44x10−25 kg and as = 5.7 nm. We consider an anisotropic parabolic magnetic trapping potential V (x, r) 2 of the form V (x, r) = 21 mω⊥ (x2 + Ω2 r 2 ) + VL(x), where r 2 = y 2 + z 2 , and Ω = ω⊥ /ωx . The light shifted optical lattice potential of the superlattice is described as
VL = V0  sin2 k1 x + (1 − ) sin2 k2 x , (2)

where 0 ≤  ≤ 1. The superlattice potential can be obtained by creating two separate far-detuned quasi-1D single-periodic lattices using lasers of different wavelengths (Fig.1). If the two lattices are orthogonally polarized, when they are superimposed, the resulting dipole trapping potential is proportional to the sum of their individual intensities. With this interpretation, V0 is proportional to the total intensity and ε related to the relative intensities of the two standing light waves. The lattice wavevectors are k1 = 2π/λ1 and k2 = 2π/λ2 , and the larger of the two periods is d = λ1 /2. In this paper we choose λ1 /λ2 = κ = 2, 4, 6. All length scales are made dimensionless with respect to aL = d/π and energy scales made dimensionless with respect to twice the single photon recoil energy EL = ~2 /ma2L . Time is made dimensionless with respect to τL = ~/EL. The condensate is elongated along the x direction (cigar shaped) and this can be achieved experimentally by making the magnetic trap frequency along the x direction very weak compared to the magnetic trap frequencies along the radial direction. The condensate wavefunction can be separated as ψ(x, r, t) = χ(r)φ(x, t), with χ(r) well described by the ground-state of aRtwo-dimensional ∞ radially symmetric quantum harmonic oscillator, with the normalization −∞ |χ|2 dydz = 1. . Further since the magnetic trap along the x direction is weak compared to the optical

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trap, we will ignore it along the x direction. Integrating out the radial coordinates, we obtain the 1D GP equation.   ∂φ(x, t) 1 ∂2 2 i = − + V (x) + g |φ(x, t)| φ(x, t), (3) L 1D ∂t 2 ∂x2 where g1D = 2(as /aL)(ωr /ωL ), ωr is the radial trap frequency. In the next section we will calculate the matter-wave band-gap sprectrum and the corresponding eigenstates for the superlattices κ = 2, 4, 6. 0.8 0.8 0.6 VL HxL

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Figure 1. The structure of the superlattice potential described by Eqn.2 for  = 0.3, V0 = 1 and κ = 2 (top left plot, orange color), κ = 4(top right plot, yellow color), κ = 6(bottom plot, green color).

3.

Matter-wave Band Gap Spectrum and Dark Solitons

Stationary states of Eqn. 3 can be written in the form φ(x, t) = Φ(x)exp(−iµt),

(4)

where µ is the corresponding chemical potential. The steady state wavefunction obeys the time-independent GP equation   1 ∂2 2 − V (x) + µ − g |Φ(x, t)| Φ(x, t) = 0. (5) L 1D 2 ∂x2

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Figure 2. Matter-wave band gap spectrum for the non-interacting condensate in an optical superlattice with κ = 2 and  = 0.3. The shaded areas are Bloch bands where k is real and the unshaded regions between the bands are the gaps where κ is complex. Also shown is the density profile of dark solitons in the big well and the smallest well.

In order to find the matter-wave band-gap spectrum, we will consider g1D = 0 (the case of non-interacting condensate). This condition is true if the number of atoms is small. Eqn. 5 becomes linear in Φ(x) and the condensate wavefunction can be represented as a superposition of Bloch waves Φ(x) = a1 Φ1 (x)eikx + a2 Φ2 (x)e−ikx ,

(6)

where Φ1,2 (x) have the periodicity of the lattice potential, a1,2 are constants, and k is the Floquet exponent. The matter-wave spectrum in the linear case consists of bands in which k is real. The bands are separated by band-gaps in which k is complex. The solutions at the band edges are exactly periodic stationary Bloch states. Figure 1 presents the bandgap diagram on the plane (µ, V0 ) for the Bloch wave solutions of Eqn. 5 in the noninteracting case for a lattice potential described by Eqn.2 for κ = 2. The spectrum is obtained by solving the matrix eigenvalue problem corresponding to g1D = 0 in Eqn.5. Only few lowest bands are shown. Also shown along with the band-gap spectrum is the corresponding density profile of the dark solitons. Note that there are two stable solitons

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m

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Figure 3. Matter-wave band gap spectrum for the non-interacting condensate in an optical superlattice with κ = 4 and  = 0.3. Note that the lowest three band-gaps decreases in comparison to the case κ = 2. The profile of the dark solitons also changes with respect to Fig.1. Noticeable changes in the background is seen.

centred at the largest and the smallest well. The density profile consists of the dark soliton and the background. The corresponding potential is also shown for convenience. The background has a spatial structure of a periodic Bloch wave. The matter-wave band-gap spectrum and the corresponding density profiles of the dark solitons for the case κ = 4, 6 are shown in figures 3 and 4 respectively. Increasing the periodicity of the secondary lattice causes structural changes in the matter-wave band-gap spectrum as well as in the spatial structure of the Bloch states. The gaps which opens up in the κ = 2 case changes as κ increases. In particular the lowest gap (between µ1 (k = 1) and µ2 (k = 1)) decreases as κ increases. Correspondingly in the density profile, we find that the width of the dark soliton profile increases as κ increases. The back ground which is a superposition of Bloch waves shows two distinct peaks (one big and one small) for κ = 2. As κ increases, the smaller peak starts to diminish. This is probably because of the fact for higher periodic superlattice the local periodicity over a few lattice sites is restored and the local influence of the secondary lattice over a few lattice sites is insignificant. The above observations leads

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Matter Wave Dark Solitons in Optical Superlattices m6(k=1) m5(k=1)

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Figure 4. Matter-wave band gap spectrum for the non-interacting condensate in an optical superlattice with κ = 6 and  = 0.3. The lowest three band-gaps decreases further compared to the case κ = 4. This is an indication of restoration of mobility of the dark solitons as discussed in the text.

to a speculation that the local mobility of the dark solitons which is reduced for the doubly periodic optical superlattice (κ = 2) is restored for higher periodic superlattice (κ = 4, 6). To check this speculation, we calculate the Peierls-Nabarro barrier for different κ’s.

4.

Peierls-Nabarro Barrier

The energy difference between a soliton centered at a maximum of the periodic potential and one centered at a neighbouring potential minimum corresponds to the height of an effective potential known as the Peierls-Nabarro (PN) potential. The value of the PN potential can be understood as the minimum energy required to move a localized wavepacket by one lattice site and this gives the measure of the mobility of the wavepacket. We calculate the PN potential by calculating at a fixed value of µ the energy difference between dark solitons centered at the biggest minima and the next nearest minima. A fixed value of µ ensures that the amplitude of the Bloch-wave background is constant, which is the case for a dark soliton

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0.20

DE

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Μ

Figure 5. The Peierls-Nabarro (PN) potential versus the chemical potential for κ = 2, 4, 6. The height of the barrier is decreasing with increasing κ.

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moving across the lattice. We analyse the dark solitons originating in the first band. We use the energy functional: )  Z (  1 ∂Φ(x, t) 2 2 4 E= + VL (x)Φ(x, t) + g1D |Φ(x, t)| dx. (7) 2 ∂x Strictly speaking this difference is the PN barrier experienced by the dark soliton centered in the neighboring wells. The dark states of the biggest well and the nearest well correspond to the minima and maxima of the PN potential respectively. This means that the biggest minima dark states should be stable, and the next nearest minima states unstable with respect to the variations in their position relative to the lattice. The knowledge of the PN barrier potential height is essential in answering the questions about mobility of the lattice soliton and its ability to interact with other localized states. Fig. 5 shows the PN barrier experienced by a dark soliton in the three different superlattices κ = 2, 4, 6 as a function of the chemical potential µ. Clearly we see that the PN barrier experienced by a dark soliton in a superlattice decrease as the periodicity increases. This confirms our speculation that the local mobility increases with κ. For moderate values of µ within the first band, the energy difference ∆E = Enext−nearest−well − Ebiggest−well is positive. Therefore we expect that even in a shallow lattice regime the dark biggest well soliton is effectively pinned by the lattice due to the presence of a large PN barrier. As the periodicity increases, the PN barrier decreases and the lattice is no longer able to pin the soliton. As the periodicity increases, the chemical potential required to release the soliton decreases. The variations in energy difference for different periodicity of the lattice suggests that for a fixed chemical potential, variation in the superlattice parameters controls the mobility and interaction properties of the dark solitons.

5.

Conclusions

We have analysed the band-gap structure of the Floquet-Bloch matter waves in optical superlattice structures in the framework of the Gross-Pitaevskii equation. We have shown Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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that band-gaps that initially appear in a double periodic lattice decreases as the periodicity is increased further. This is interpreted as an increase in the soliton mobility and this was confirmed by calculating the Peierls-Nabarro potential barrier which was found to decrease with increasing periodicity of the optical lattice. We have demonstrated that the mobility of the dark solitons can be effectively controlled by changing the periodicity of the optical superlattice.

References [1] Morsch, O.; Oberthaler, M. Rev. Mod. Phys 2006, 78, 179. [2] Anderson, B. P.; Kasevich, M. Science 1998 282, 1686. [3] Orzel, C. ; Tuchman, A. K.; Fenselau, M. L.; Yasuda, M.; Kasevich, M .A. Science 2001 291 , 2386. [4] Morsch, O.: M¨uller, J. M.; Cristiani, M.; Ciampini, D; Arimondo, E. Phys. Rev. Lett. 2001 87, 140402. [5] Greiner, M.; Mandel, O.; Esslinger, T.; H¨ansch, T. W.; Bloch, I. Nature 2002 415, 39. [6] Peil, S et. al. Phys. Rev. A. 2003 67 051603(R). [7] Sebby-Strabley. J et. al. Phys. Rev A 2006 73, 033605. [8] Buonsante. P; Vezzani, A. Phys. Rev. A 2004 70, 033608.

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[9] Louis, P. J. Y.; Ostrovskaya, E. A; Kivshar, Y. S.; J. Opt. B 2004 6, S309. [10] Louis, P. J. Y.; Ostrovskaya, E. A; Kivshar, Y. S.; Phys. Rev. A 2005 71, 023612. [11] Dmitrieva L. A.; Kuperin, Y. A.; Cond-mat/0311468. [12] Rey, A. M.; Hu, B. L.; Calzetta, E.; Roura, A.; Clark, C. W.; Phys. Rev. A 2004 69, 033610 . [13] Roth, R.; Burnett, K,; Phys. Rev. A 2003 68, 023604. [14] Porter, M. A.; Kevrekidis, P. G.; Carretero-Gonzalez, R.; Frantzeskakis, D. J.; Phys. Letts. A 2006 352, 210. [15] Huang, Chou-Chun.; Wu, Wen-Chin. Phys. Rev. A 2005 72, 065601. [16] Rousseau, V. G.; Arovas, D. P.; Rigol, M.; H´ebert, F.; Batrouni, G. G.; Scalettar, R. T.; Phys. Rev. B 2006 73, 174516. [17] Bhattacherjee, A. J. Phys.B. At. Mol. Opt. Phys. 2007 40, 143. [18] Bhattacherjee, A. Eur. Phys. J. D 2008 46, 499. [19] Lye, J. E. et. al. Phys. Rev. A 2007 75, 061603. Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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[20] Bhattacherjee, A.; Pietzyk, M. Cent. Eur. J. Phys. 2008 6, 26. [21] Burger, S. et. al. Phys. Rev. Lett. 1999 83, 5198. [22] Denschlag, J. et. al. Science 2000 287, 97. [23] Peyrard, M.; Kruskal, M. D.; Physica D 1984 14 88. [24] Kevrekidis, P. G; Weinstein, M. I.; Physica D 2000 142 113. [25] Ablowitz, M. J.; Musslimani, Z. H.; Phys. Rev. E 2003 67 025601. [26] Sukhorukov, A. A.; Kivshar, Y. S.; Phys. Rev. E 2002 65 036609. [27] Feng, J.; Kneub¨uhl, F. K. IEEE J. Quantum Electron. 1993 29 590. [28] Abdullaev, F. Kh.; Baizakov, B.B.; Darmanyan, S. A.; Konotop, V. V.; Salerno, M Phys. Rev. A 2001 64 043606. [29] Yulin, A. V.; Skryabin, D. V. Phys. Rev. A 2003 67 023611. [30] Alfimov, G. L.; Konotop, V. V.; Salerno, M. Europhys. Lett. 2002 58 7. [31] Konotop V. V.; Salerno. M. Phys. Rev. A 2002 65 021602.

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[32] Kevrekidis, P. G.; Carretero-Gonzalez, R.; Theocharis, G.; Frantzeskakis, D. J.; Malomed, B. A. Phys. Rev. A 2003 68, 035602.

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In: Optical Lattices: Structures, Atoms and Solitons ISBN: 978-1-61324-937-6 c 2012 Nova Science Publishers, Inc. Editor: Benjamin J. Fuentes

Chapter 8

O PTICAL S UPERLATTICES : WHERE P HOTONS B EHAVE L IKE E LECTRONS M. Ghulinyan1∗, Z. Gaburro1, L. Pavesi1 , C. J. Oton2 , N. Capuj2 , 3 R. Sapienza , C. Toninelli3 , P. Costantino3 and D. S. Wiersma3† 1 Dipartimento di Fisica, University of Trento, Povo (Trento), Italy 2 Departamento de Fisica Basica, University of La Laguna, Tenerife, Spain 2 European Laboratory for Nonlinear Spectroscopy and INFM-MATIS, Sesto Fiorentino (Florence), Italy

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Abstract We report on optical analogues of well-known electronic phenomena such as Bloch oscillations and electrical Zener breakdown. We describe and detail the experimental observation of Bloch oscillations and resonant Zener tunneling of light waves in static and time-resolved transmission measurements performed on optical superlattices. Optical superlattices are formed by one-dimensional photonic structures (coupled microcavities) of high optical quality and are specifically designed to represent a tilted photonic crystal band. In the tilted bands condition, the miniband of degenerate cavity modes turns into an optical Wannier-Stark ladder (WSL). This allows an ultrashort light pulse to bounce between the tilted photonic band edges and hence to perform Bloch oscillations, the period of which is defined by the frequency separation of the WSL states. When the superlattice is designed such that two minibands are formed within the stop band, at a critical value of the tilt of photonic bands the two WSLs couple within the superlattice structure. This results in a formation of a resonant tunneling channel in the minigap region, where the light transmission boosts from 0.3% to over 43%. The latter case describes the resonant Zener tunneling of light waves.

Keywords: Optical superlattice, Wannier-Stark ladder, Bloch oscillations, Zener tunneling, porous silicon. ∗ E-mail † E-mail

address: [email protected] address: [email protected]

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1.

Ghulinyan et al.

Introduction

Transport phenomena of charge carriers in solids have been studied intensively. The energy spectrum of a quantum particle in a semiconductor crystal is described by extended Bloch modes and consists of allowed and forbidden bands. In the momentum space an electron moves in the allowed energy band running its quasi-impulse in the Brillouin zone. With a given energy, in the real space it travels freely through the infinite crystal. When an external bias is applied on the crystal, the electron which travels from the Brillouin zone center towards the zone edge, is firstly accelerated by the field F. Then, upon reaching the Brillouin zone edge, its velocity continuously reduces down to zero and finally it starts to move in the opposite direction. Thus, in the Brillouin zone the electron performs a periodic motion, which can be described as an interference phenomenon brought up by Bragg reflection from the energy band edge once the particle wavelength reaches the crystal lattice period d. This periodic motion in k-space is accompanied by an oscillatory motion in real space, known as electronic Bloch oscillations [1], and has a characteristic period of TB = h/eFd. The energy spectrum of such a particle in a semiconductor crystal, exposed to an external filed, is described by the so-called Wannier-Stark ladder (WSL) [2]. The WSL consists of a set of equidistant states, which are localized in space and have an energy separation proportional to the inverse of TB . The Bloch oscillations are closely related to another phenomenon, known as electrical breakdown or nonresonant Zener tunneling [3], which becomes effective as the electric field increases. In real space the tilt of the energy bands with increasing field gets stronger, and at high enough electric fields an electron tunnels to the continuum of states of an upper energy band without gaining extra energy from the field (this phenomenon has a wide application in microelectronic devices, such as the Zener and tunnel diodes). With the increasing escape probability of the electron to the other energy band the Bloch oscillations are damped strongly. On the other hand, resonant Zener tunneling is possible at high electric fields between charge carriers in WSLs. The experimental study of electronic Bloch oscillations has been an issue for a long time because of the fact that in a semiconductor crystal an electron wave packet looses its coherence on a time scale, which is much shorter than the oscillation period TB . The invention of electronic superlattices [4] opened up new possibilities to study such interference phenomena, which show dynamics faster than the decoherence time of an electron wave packet. This possibility is brought up by the large translational period of a superlattice structure, which implies a smaller Brillouin zone and, therefore, a shorter oscillation period. Recent experiments cover a series of results such as the observation of WSLs [5], Zener breakdown [6], and Bloch oscillations [7]. Resonant tunneling between the anticrossing Wannier-Stark states of neighboring energy minibands has been considered theoretically [8] and observed in experiments [9]. Various analogies between the transport of electrons and the propagation of light waves in dielectric materials have been established [10]. Electronic crystals have an analogue in the form of photonic crystals. Photonic crystals are artificial materials, in which a periodic variation of dielectric constant on a length scale comparable with the wavelength leads to the formation of bands where the propagation of photons is allowed or forbidden [11]. Since photons are uncharged particles and interact extremely weakly with each other, a light wave

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packet, that propagates through such a system, remains coherent for much longer times than charged particles. This means that dynamic interference effects could be isolated and studied more easily with light than with electrons.

a)

c)

Quantum Wells

Electronic miniband

Optical Cavities

Photonic miniband

l/2 cavities

b)

Mirrors

Potential barriers

} }

Bragg mirrors

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Figure 1. In analogy to the electronic coupling of separate quantum wells in a semiconductor superlattice (a), an optical superlattice can be realized when optical cavities are brought together (b) resulting in the formation of a miniband of extended photonic states. (c) A SEM micrograph of a one-dimensional porous silicon optical superlattice. The existence of the optical counterpart of a WSL has been discussed theoretically [12] and observed experimentally in linearly chirped Moir´e gratings [13]. Different photonic systems have been proposed to observe Bloch oscillations of light waves [14, 15]. Optical superlattices of coupled degenerate cavities have been proposed as a potentially ideal system to observe Bloch oscillations for light (see Fig. 1.1) [16]. In these structures an optical path gradient, ∆δ, parallel to the light propagation direction was suggested to mimic the optical equivalent of an external force (the static electric field in the electron case). In this chapter we report on the observations of optical analogues of electronic Bloch oscillations and electrical Zener breakdown. We will describe and detail the experimental observation of Bloch oscillations [17] and resonant Zener tunneling [18] of light waves both in static and time-resolved transmission measurements performed on optical superlattices. The one-dimensional multilayer structures were made of porous silicon and were specifically designed to present a tilted photonic crystal band in close analogy to the tilted electronic miniband of a biased semiconductor superlattice. A controlled optical path gradient along the growth direction of the structure was used to form an optical Wannier-Stark ladder of equidistant photonic states. The frequency separation of these localized states defines the period of the photon Bloch oscillations. When the superlattice is designed such that two minibands are formed within the stop band, at a critical optical path gradient, we observe resonant coupling between two Wannier-Stark ladders, which leads to delocalization of the optical waves and, hence, resonant Zener tunneling.

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Current density j1 j2

Multilayer stack

Effective refractive index 1 n1 n2 3.5

etching time

low porosity high porosity

Figure 2. Multilayer growth in porous silicon: a modulation of the electrochemical current density in time (left) results in a formation of a stack of differently porous layers (middle), where each layer is described by an effective refractive index ne f f (right).

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2.

Porous Silicon-Based One-Dimensional Systems

The unique optical properties of porous silicon (PS) have attracted the researchers intensively in the last decade [19]. The electrochemical anodization procedure of silicon offers a cheap and fast technology for growing porous structures. Despite the structural inhomogeneities at the nanoscale, PS grown on heavily doped p-type silicon substrate shows straight columnar structures and passive optical properties of a dielectric material [20]. The typical pore sizes of 30-50 nm are more than an order of magnitude smaller than the wavelength of visible light, therefore an effective medium approximation allows one to characterize the porous structure with a single effective refractive index1 ne f f , which is lower than the refractive index of bulk silicon. The porosity of a layer can be modulated by changing the current density during the etching process, which results in a modulation of ne f f (Fig. 1.2). An important property of the electrochemical etching is that silicon dissolution preferably takes place at the pore tips, where the local electric field has the highest value. This means that the already grown layers will not be affected by the changes in the current density, i.e. the pore sizes and the thickness of a layer will not change while new layers are growing. Then it becomes possible to realize one-dimensional multilayer structures, like dielectric Bragg mirrors and Fabry-Per´ot filters, by alternating high and low porosity layers. Usually, the preparation of PS structures of good optical quality, composed by several dozens of layers is a difficult task, because the anodization conditions might drift as the sample thickness increases [21]. These drifts affect the optical parameters of the layers, and one has to consider them when trying to maintain constant optical path in photonic structures. The reflectance spectrum is therefore heavily modified due to the inhomogeneities of the layers. Unfortunately, most 1 There

exists, however, some residual scattering especially at the interface between layers of different pore

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Figure 3. Reflectance (front- and backside) and transmittance spectra of a free-standing 10 coupled microcavities sample centered at 1300 nm: (top layer) naturally grown sample; (bottom layer) the same structure with compensated drift in layer thicknesses. The frontand backside reflectance spectra match in the compensated case, and the sample transmittance increases correspondingly, showing almost all features.

porous samples are not detached from the silicon substrate, limiting the investigation of the spectral properties of such photonic structures. Free-standing PS-based multilayers [21], where both transmission and reflection measurements are possible, give richer information about the optical losses due not only to absorption but also to scattering, and allow to characterize the drifts in layer parameters (Fig. 1.3a). The natural drifts of the layer thickness and porosity can be successfully compensated by changing the etching parameters in a controlled way2 (Fig. 1.3b). Moreover, as we will see in the following sections, we can use the optical path gradients for specific purposes and even enhance them in a controlled manner upon will. 2 The optical thickness depends on both porosity and physical thickness of the layer. The physical thickness can be easily controlled by simply varying the duration of the etch, while the refractive index drift can be enhanced by varying the duration of so-called etch-stops (zero current pulses) combined with the use of a magnetic stirrer. Assuming a linear variation of the layer thickness with depth, one can compensate the evaluated drift by modifying the etching time so that a constant layer optical thickness with depth is attained.

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b) 2

Log(|E|2)

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Figure 4. Light intensity distribution inside the optical superlattice structure. The parameters used in the calculations correspond to samples used in the actual experiment. (a) Flat band situation, ∆δ = 0%, (b) tilted band situation, ∆δ = 14%. The dashed lines are a guide to the eye that indicates the theoretical tilting of the miniband. Above each panel the coupled microcavity structure is schematically shown; the grey scale refers to the refractive index variation along the depth in the sample (the darker the larger n).

3.

Bloch Oscillations of Light

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3.1.

The optical superlattice

In a close analogy to the formation of an electronic superlattice when several semiconductor quantum wells are brought together (Fig. 1.1(a)), an optical superlattice is made by coupling degenerate optical resonators (cavities) within the same photonic structure (Fig. 1.1(b))[21, 22]. Optical coupling between the various cavities yields a degenerate mode repulsion [23] and a formation of a miniband of optical states, which are densely packed around the resonant wavelength. In one dimension, an optical superlattice can be formed by stacking two dielectric layers A and B with different refractive indices and quarter-wave thickness in a way to form identical cavities separated by Bragg mirrors (Fig. 1.1(c)). In particular, we choose the following sequence of layers: BABABABAB (AA)1 BABABABAB (AA)2 . . . (AA)m BABABABAB. This structure is essentially a series of m microcavities (AA)m coupled to each other through the BABABABAB Bragg mirrors. The amount of splitting of cavity resonances is given by the strength of the coupling mirrors3 , i.e. the weaker the mirrors are, the larger splitting occurs. The light intensity distribution inside this structure can be calculated using a transfer matrix (TM) formalism [24]. Figure 1.4(a) shows the appearance of a miniband of extended optical states (bright lines) in the photonic stop band (dark regions) in the conditions of a constant optical path throughout the structure. 3 For

a given index contrast between A and B layers, the reflectance of a Bragg mirror improves with increasing the number of periods. Similarly, for a fixed number of periods, the mirror reflectance grows with an increase in the refractive index contrast Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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In order to obtain optical Bloch oscillations one has to introduce a gradient in the optical thickness of the layers. This will result in a spatial tilting of the miniband (Fig. 1.4(b)) and in the formation of an optical Wannier-Stark ladder. The latter manifests as a series of narrow equidistant transmission peaks with a frequency separation of the peaks that defines the period TB of the Bloch oscillations. The linear change of the optical thickness introduces, to first order, a linear tilt of the miniband [25]. The oscillation period will decrease with increase in the miniband tilting.

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3.2.

Sample preparation and optical WSLs

We have grown the optical superlattices by controlled electrochemical etching of heavily doped p-type (100)-oriented silicon. The electrolyte was prepared mixing a 30% volumetric fraction of aqueous HF (48 wt.%) with ethanol. A magnetic stirrer was used to improve electrolyte exchange. The applied current density defined the porosity of the layer. We applied 50 mA/cm2 for the high porosity layer A (porosity 73%, refractive index nA = 1.45) and 7 mA/cm2 for the low porosity layer B (50%, nB = 2.1). The physical thickness of the layers was controlled by adjusting the duration of the etch times. Alternating these currents, we grew a superlattice structure with ten coupled cavities. The superlattice structures were made free-standing by applying an electropolishing current pulse at the end of the growth process [21]. Particular care was taken to control the anodization conditions which usually drift as the total sample thickness increases. Moreover, the natural refractive index drift was compensated by changing the etching times of the layers. The process is known to provide excellent control over the layer properties allowing the growth of Fabry-Per´ot filters with a resonance quality factors up to 3300 [22]. The one dimensional translational symmetry of the superlattice was broken by introducing an optical path gradient in the growth direction of the structure. This was achieved by changing the duration of the etch stop current, which controlled the refractive index and hence the variation in the optical thickness of each layer. We produced samples with gradient values in the range from ∆δ = 2 to 14% (values that were extracted from the best fit parameters of the transmission spectra). As porous silicon samples suffer from lateral inhomogeneities due to doping variations of the silicon wafers, an inhomogeneous widening of the transmission peaks is usually measured when broad probe beams are used. In order to avoid this, some spectra were measured in a high-resolution transmission setup with a very small numerical aperture (NA ∼ 0.0075, leading to a negligible broadening of 0.02 nm at 1550 nm wavelength), where a tunable laser source focused to a 35µm diameter spot was used. Figure 1.5 shows the transmission spectra for different values of gradient ∆δ. The flat miniband situation is reported in the top panel (Fig. 1.5(a)). Optical WSLs are formed with an increase in the gradient over a certain value: the bigger is ∆δ, the larger is the energy separation of the Wannier-Stark states. The states in an optical Wannier-Stark ladder are not extended and have a reduced spatial extent inside the superlattice structure. The transmission resonances become narrower and less intense for the large gradient values (Fig. 1.5(b-d)). This is a clear signature of the strongly inclined miniband, which also means that in order to be transmitted the photons now need to overcome a thicker tunneling barrier.

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Figure 5. High resolution transmission spectra of the optical superlattices with different gradients of the optical thickness of the layers (λ0 = 1.55µm is the central wavelength). The top spectrum corresponds to the non drifted sample (spatially flat miniband), while the others (b), (c), (d) show the occurrence of the optical Wannier-Stark ladder with equidistant resonances: the energy separation of the states increases with the increase of the miniband tilt.

3.3.

Time-resolved transmission measurements

We have performed time-resolved transmission experiments on these samples using an optical gating technique. This technique involves mixing a reference beam together with the transmitted signal in a 0.3mm thick non-linear crystal (β-Barium Borate) to produce a sum frequency signal. The probe beam is obtained from an optical parametric oscillator (Spectraphysics OPAL) pumped by a Ti:sapphire laser at center wavelength 810 nm (pulse duration 130 fs, average power 2.0 W, repetition rate 82 MHz) yielding short pulses tunable from 1300 to 1600 nm wavelength range (average power 100 mW). The reference pulse at 810 nm wavelength is obtained from the residual Ti:sapphire beam (450 mW average power). The sum frequency signal is detected by a photodiode, and a standard lock-in technique is used to suppress noise. A delay line in the reference beam allows to tune the time delay between signal and reference and thus to scan the signal pulse in time. In the top panel

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Figure 6. Time-resolved transmitted signals through the superlattice structures for various values of the optical path gradient ∆δ. The top panel shows the undisturbed probe pulse without sample. An oscillating signal is measured in the photodetector when the optical WSL are excited. The period of the oscillations and the total transmission decrease while increasing ∆δ.

of Fig. 1.6 we plot the system response without sample from which we determine that the temporal resolution of our system is smaller than 250 fs. The apparatus is designed such that the transmission spectrum of the sample can be monitored during the time-resolved measurement by sampling the transmitted light. Time-resolved data indeed show that transmitted signals oscillate with a period TB that decreases as the band tilting increases (Fig. 1.6). It is important to note that the optical WSL in our structures is formed above a threshold gradient value (∼7%). This is confirmed also by the TM-calculations. The reason is that an optical thickness gradient value below 7% is not sufficient to tilt fully the miniband within our sample thickness. One can see how the amplitude of the transmitted intensity decreases as the gradient increases: as a result of an increased tilt of the photonic miniband (see also Fig. 1.4(b)) the oscillating photon wave packet has to tunnel out through a thicker barrier. In Fig. 1.7 the measured periods of the Bloch oscillations are compared to the ones calculated through TM formalism. The

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Figure 7. Measured photonic Bloch oscillation period TB and decay time τB as a function of the gradient ∆δ. The error bars are the standard deviations obtained from various measurements on several positions on the sample and represent therefore the effect of lateral sample inhomogeneities. The solid lines are calculated using the transfer matrix method.

experimental data are in perfect agreement with the theoretical prediction. However, one can measure oscillations even below the critical gradient value of 7%, which in this case are due to the reflection of light from the sample physical boundaries, therefore changing the gradient in this range does not influence the oscillation period. At larger values of gradient the period of Bloch oscillations decreases linearly with the increase in the miniband tilt, as expected. The oscillations decay with a characteristic time τB which saturates at large gradients. This is a sign of the increased confinement of the optical modes in the WSL: as the photonic band tilt gets steeper, tunneling out of the sample becomes more difficult, and the transmission losses decrease accordingly. These intrinsic losses are not the only one present in the structure, as light in porous silicon suffers also from external losses. At large gradient values τB saturates to 1.26 ps, which is caused by scattering on the pores and residual ab−1 −1 sorption losses in the porous silicon sample. One can consider that τB = (τ−1 pBO + τext ) , where τ pBO is the intrinsic decay time of the Bloch oscillation and τext is due to absorption and scattering losses. The solid line in Fig. 1.7 is obtained by assuming τext =1.3 ps that corresponds to an extinction coefficient (total losses, absorption + scattering) of αext =100 cm−1 , in agreement with previously determined loss values [21].

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Figure 8. The calculated intensity distribution of the light inside the double miniband sample plotted as a color scale versus the normalized frequency ω/ω0 , where ω0 is the minigap central frequency, and the depth inside the sample. (a) Flat miniband situation, ∆δ = 0%. Two minibands MB1 and MB2, separated by a minigap region are seen in the calculated (b) and measured transmission spectrum (c).

4.

Resonant Zener Tunneling of Light

4.1.

The double-miniband optical superlattice

The successful demonstration of the optical analogue of electronic Bloch oscillations naturally raises the question whether it is possible to observe the optical analogue of Zener breakdown phenomenon. As a first reasoning towards the study of Zener tunneling of light, one could think of tilting the optical superlattice bands such that the first order (centered at λc) miniband is coupled to the next order one (centered at λc /3). However, this would require very large band tilting, and the realization of such structures, where the optical parameters are controlled and characterized, is impossible in practice. A possible solution is to build an optical superlattice that exhibits two minibands within the stop band. This will allow to use relatively small optical path gradients. For this, one should couple within the same structure two sets of cavities of C and D type, which are centered at different wavelengths λ1 and λ2 , respectively4 . The appropriate sequence of layers looks like the following: BABABAB (CC)1 BABABAB (DD)1 . . . (CC)m BABABAB (DD)m BABABAB. In our studies we have chosen m=6, λ1 = 0.81λc and λ2 = 0.88λc. The samples were grown using the same technique described in the previous section. The refractive indices for this type of structures were determined to be nA = nC = nD = 1.5 and nB = 2.12. We produced samples with controlled gradient values in the range from ∆δ = 0 to 18%. The light intensity distribution inside this structure in the absence of optical path gradient (Fig. 1.8(a)) shows the formation of two flat minibands MB1 and MB2. These appear as two sets of intense lines of extended optical states, stretching through the sample. The two minibands are separated by a photonic minigap (dark region), which shows 4 Note

that C and D layers are also quaterwave-thick, so that (CC) and (DD) form λ1 /2 and λ2 /2 cavities, respectively. Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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Figure 9. (a) Optical WSLs of localized modes are formed in two minibands at ∆δ = 6.7%, which appear as weak narrow peaks in the transmission spectra (graphs in the right). Light transmission drops down to 2%: transfer matrix calculations (b) and the experimental data (c).

negligible transmission of 0.43%. The parameters used in the calculations correspond to those of the samples studied in the actual experiment. The calculated spectra also take into account a loss coefficient of 100 cm−1 (due to absorption and scattering out of the propagation axis) which gives a nearly negligible spectral broadening of the transmission peaks of 0.3 − 0.5 nm. The corresponding calculated transmission spectrum (Fig. 1.8(b)) is compared with the experimental one (Fig. 1.8(c)). The latter is measured with a Varian Cary 5000 spectrophotometer using a broad beam of 1mm in diameter, which explains the appearance of wider and less intense spectral features. The high-resolution transmission setup was utilized in some particular cases, where the observation of narrow peaks with high intensity was essential. An introduction of 6.7% of negative optical path gradient[26] tilts the photonic band structure and results in the formation of optical WSLs in both minibands (Fig. 1.9(a)). Now the spatial confinement of the localized states causes a decrease of the absolute transmission from 50% in the flat band case (delocalized states) down to 2% (Fig. 1.9(b)). In Fig. 1.9(c) we plot the measured transmission spectrum of the superlattice with tilted minibands, where the WSLs can be appreciated. We go on further increasing the optical path gradient and we look at the evolution of photonic miniband picture. At a critical degree of band tilting (∆δ ∼ 10.3% in our case) the WSLs in two minibands couple within the extension of the structure. Coupling induced delocalization of two anticrossing states takes place, which appears as an intense resonant tunneling channel (Fig. 1.10(a)). The resonant Zener tunneling appears as an enhanced transmission peak in both calculated (Fig. 1.10(b)) and measured transmission spectra (Fig. 1.10(c)). This observation, together with the TM-calculations in Figs. 1.8-1.10, nicely demonstrates the physics of resonant Zener tunneling. An efficient transmission channel opens when two internal resonances couple to form a delocalized mode, which has a transmission coefficient much larger than the sum of the transmission coefficients of the two individual

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Optical Superlattices: where Photons Behave Like Electrons Scattering States map

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Figure 10. (a) The two WSLs couple at ∆δ = 10.3% and form a resonant tunneling channel through the sample. Resonant Zener tunneling is predicted by theory (b) and confirmed experimentally (c) as an enhanced transmission peak in the center of the minigap.

resonances (before coupling). Such internal resonances can only couple if the frequency difference between the modes is smaller than their bandwidth. The characteristic property of Zener tunneling is that this frequency difference is tuned by changing the gradient inside the sample and that the internal resonances arise from a double Wannier-Stark ladder. The critical gradient value ∆δZT at which Zener tunneling occurs (in our case 10.3%) depends on the frequency difference between the centers of the minigaps ∆ω = ω1 − ω2 , and the bandwidth of the minibands themselves. At ∆δZT the frequency of the low frequency miniband at the last cavity (high depth) matches the frequency of the high frequency miniband at the first cavity (small depth). In Fig. 1.11 we show some numerical results which reflect the evolution of the optical WSLs towards the eventual coupling: the shrinkage of the minigap is plotted versus the increasing ∆δZT . The calculations are performed for different reflectivities of Bragg mirrors (number of mirror periods) in the optical superlattice. In the limit of very small bandwidth (Bragg mirrors consisting of large number of layers resulting in weak coupling between the cavities) the optical thickness gradient at which Zener tunneling occurs takes its minimum value defined as ∆δZT = ∆ω/ω1 (solid line). At larger bandwidths the situation is more complicated and to obtain ∆δZT one needs to calculate the scattering states map at each gradient value as in Fig. 1.4. We will see that the theoretical value for ∆δZT is indeed 10.3% for our sample. We can observe from the graph in Fig. 1.11 that for the case of strong coupling between the cavities (1.5- and 2.5-period mirror cases) the relative shift of the resonance is negative for small gradients. This is due to the fact that the bandwidth of the miniband decreases slightly when the small gradient only detunes the coupling mirrors but is still not enough to break the translational symmetry of the superlattice. Over a certain value WSLs start to form and the minibands start to expand monotonically with the further increase of titling. Another interesting observation is the behavior of the curve for the strongest coupling case (1.5-periods) in the vicinity of ∆δZT ≈ 18%: the coupling between the two WSL states is so strong that a change in the tilting of the miniband around ∆δZT does not influence the

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Minigap

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Gradient Dd, % Figure 11. The calculated minigap width reduction is plotted versus the increasing gradient of the optical path for various reflectivities of coupling mirrors (number of periods). In the limit of very weak coupling between the cavities (solid line) the optical thickness gradient at which Zener tunneling occurs has a minimum value. The sketch in the top describes the minigap width reduction. relative shift of the resonance. This means that a marked double peak would manifest in the transmission spectrum for all this range of gradients because of enhanced level repulsion between the degenerate states. We decided to examine the Zener tunneling regime in detail. For this we have grown a superlattice structure with lateral (in-plain) variation in ∆δ around 10.3%. To achieve this, the magnetic stirrer was placed close to one edge of the sample during the whole growth process, thus the electrolyte exchange was enhanced differently between the closest (to the stirrer) and farthest points on the sample surface. The latter resulted in a smoothly varying refractive index profile between the antipodal points in each layer of the optical superlattice (see the schematic sketch in Fig. 1.12). Measuring the transmission spectrum at different points over the sample surface (moving the incident beam laterally) allowed us to follow the evolution of miniband coupling around ∆δZT . In this way the system was studied at

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different values of gradient between 6.5% to 10.7%.

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w/w0 Figure 12. (top) A schematic view of the Hi-Res transmission measurement of the superlattice sample with in-plane gradient. (bottom) The comparison between the experimental transmission values of the maximum transmission (line) around the central frequency ω0 as a function of the gradient and the transfer matrix calculations (dots). The inset shows the transmission spectrum of the tilted superlattice around the value of the optical path gradient where the first anti-crossing of the optical Wannier-Stark ladders and hence Zener tunneling occurs. High-resolution transmission measurements are performed for this sample. Fig. 1.12 reports the measured high-resolution transmission spectra taken at different points corresponding to different gradients. In the vicinity of the threshold gradient the transmission spectrum is very sensitive to small changes of the optical path. One can see how the edge states of the two WSLs start to overlap, and the saddle-like curvature transforms gradually into a sharp resonance of 42% transmission. In the inset of Fig. 1.12 the calculated transmission values at ω0 for ∆δ in the range 0 ÷ 25% are compared with the experimental

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data. The correspondence between the experimental and calculated values of the Zener tunneling gradient is very good. In the calculations another maximum, corresponding to the second anticrossing of WSL states, is present at ∼ 21%. The intensity of this peak is much weaker compared with the first one at ∆δZT = 10.3%, which can be understood in terms of decreasing probability of the photon tunneling out of the superlattice as the miniband tilt increases. The analysis of spectra around the Zener tunneling regime shows that, together with the increase of the transmission value, the resonant transmission peak gets wider5 . At resonance its full width half maximum (FWHM) is roughly a factor of two larger than that of the uncoupled WSL peaks at 0.98ω0 and 1.025ω0 . This broader lineshape is the effect of repulsion between the two coupled resonances.

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4.2.

Resonant Zener tunneling in time-domain

When studying narrow spectral features in static transmission measurements, one usually is limited by the setup resolution. Therefore, lineshape differences are often difficult to appreciate even in the high precision spectra. Small variations in the frequency domain result in big fluctuations in the time domain. The Fourier transformation of a single resonance, corresponding to a phase shift of π, results in an asymmetric lineshape with single exponential decay at long times. A double resonance mode lineshape experiences a different phase shift (2π), which affects the shape of the time response, in particular, shifting its maximum towards longer times. Also, the Fourier transformation of a double resonance peak results in a more symmetric pulseshape in the time domain. This feature should allow to distinguish among peaks of the same width, originating from resonances of different multiplicity. We have tested these properties looking at the time response of our optical superlattice sample, performing ultrashort pulse propagation experiment in the Zener tunneling regime. The technique used is the same as that described in detail for the Bloch oscillations case. Figure 1.13 shows three examples of transmitted pulses centered at different wavelengths. A reference pulse, measured in the absence of sample is plotted as dotted line for comparison. When a single Wannier-Stark state is excited (Fig. 1.13(a)), the transmitted signal intensity decays exponentially, which is the characteristic behavior of a localized state. The delay of the pulse, defined as the delay of the center of the mass of the pulse profile, as expected, is not big. When the incident pulse excites two resonances, a complex signal oscillating with a period of ∼ 300 fs, determined by the frequency separation of the excited states, is observed (Fig. 1.13(b)). These are the well known photonic Bloch oscillations. In our specific case, this the oscillations are damped because of the strong coupling of the double resonance with the environment. Finally, Fig. 1.13(c) shows the time response of the peak with enhanced transmission at 1560 nm. The time-resolved profile does not have the typical shape of a single resonance. The observed picture is consistent with a double resonance transport behavior: it is characterized by a rapid decay and a substantial delay of maximum transmission point that amounts to almost 360 fs. The fast decay time is due to the strong coupling of the mode with the sample environment. The delay is caused by the transient 5 The

resonance lifetime, which is inversely proportional to the FWHM, decreases for the coupled states, which provides an additional proof of coupling induced delocalization of WSL states in the Zener tunneling regime. Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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Figure 13. Time-resolved transmitted signals from a double miniband superlattice with optical path gradient in the Zener tunneling regime. The panels (a), (b), and (c) correspond to different probe wavelengths. (a) A excited single resonance shows a characteristic exponential decay whereas (b) damped Bloch oscillations are observed when exciting two Wannier-Stark resonances. In (c) the Zener tunneling peak is excited leading to a strongly delayed but nearly symmetric transmitted pulse. The dotted curves refer to the transmission in the absence of a sample.

time necessary to build up the double resonance of the Zener tunneling mode.

5.

Conclusion

In conclusion we have observed the optical counterparts of electronic Bloch oscillations and Zener tunneling in optical superlattices of porous silicon. A linear variation in the optical constants of the system along the propagation direction allows to form an optical Wannier-Stark ladder and to observe photonic Bloch oscillations resolved in time. Both the oscillation period and the damping time versus the strength of the Wannier-Stark ladder are consistent with predictions from transfer matrix calculations. The observation of resonant Zener tunneling of light waves has been performed via spectral and time-resolved transmission measurements on specifically designed optical superlattices, which exhibit two

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minibands. At a critical gradient of the optical path value, coupling of photonic minibands occurs resulting in delocalization of the optical Wannier-Stark states and, consequently, Zener tunneling of the light waves. The transition from low to high transmission is extremely sensitive to the variation of optical path gradient, making a Zener tunneling light valve a strong possibility. These fascinating parallels between electrons and photons not only show the potential of complex photonic structures to study fundamental problems, but also add new functionalities to silicon.

Acknowledgments The authors would like to thank the financial support by the EC under contract number IST-2-511616 and MIUR through COFIN projects ”Silicon-based photonic crystals: technology, optical properties and theory” and ”Silicon-based photonic crystals for the control of light propagation and emission” and FIRB projects ”Sistemi Miniaturizzati per Elettronica e Fotonica” and ”Nanostrutture molecolari ibride organiche-inorganiche per fotonica”.

References ¨ die Quantenmechanik der Elektronen in Kristallgittern, Z. Phys. 52, [1] F. Bloch, Uber 555-6000 (1928). [2] G.H. Wannier, Possibility of a Zener Effect, Phys. Rev. 100, 1227 (1955).

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[3] C. Zener, A theory of the electrical breakdown of solid dielectrics, Proc. R. Soc. London Ser. A 145, 523-529 (1934). [4] L. Esaki and R. Tsu, Superlattice and negative differential conductivity in semiconductors, IBM J. Res. Dev. 14, 61-65 (1970). [5] E.E. Mendez, F. Agullo-Rueda, and J.M. Hong, Stark localization in GaAs-GaAlAs superlattices under an electric field, Phys. Rev. Lett. 60, 2426-2429 (1988). [6] H. Schneider, H.T. Grahn, K.V. Klitzing, K. Ploog, Resonance-Induced Delocalization of Electrons in GaAs-AlAs Superlattices, Phys. Rev. Lett. 65, 2720-2723 (1990); B. Rosam, D. Meinhold, F. L¨oser, V.G. Lyssenko, S. Glutsch, F. Bechstedt, F. Rossi, K. K¨ohler, and K. Leo, Field-Induced Delocalization and Zener Breakdown in Semiconductor Superlattices, Phys. Rev. Lett. 86, 1307-1310 (2001). [7] T. Dekorsy, P. Leisching, K. K¨ohler, and H. Kurz, Electro-optic detection of Bloch oscillations, Phys. Rev. B 50, 8106-8109 (1994); V.G. Lyssenko, G. Valusis, F. L¨oser, T. Hasche and K. Leo, Direct Measurement of the Spatial Displacement of BlochOscillating Electrons in Semiconductor Superlattices, Phys. Rev. Lett. 79, 301-304 (1997); T. Hartmann, F. Keck, H. J. Korsch and S. Mossmann, Dynamics of Bloch oscillations, New J. Phys. 6, 2-25 (2004). [8] S. Glutsch and F. Bechstedt, Interaction of Wannier-Stark ladders and electrical breakdown in superlattices, Phys. Rev. B 60, 16584-16590 (1999). Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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[9] B. Rosam, K. Leo, M. Gl¨uck, F. Keck, H. J. Korsch, F. Zimmer, K. K¨ohler, Lifetime of Wannier-Stark states in semiconductor superlattices under strong Zener tunneling to above-barrier bands, Phys. Rev. B 68, 125301 (2003). [10] Ping Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, Academic Press, New York, (1995); Wave Scattering in Complex Media, from theory to applications, edited by S.E. Skipetrov and B.A. van Tiggelen, NATO series II, Vol. 107 (Kluwer, Dordrecht, 2003). [11] J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals (Princeton University Press, Princeton, NJ, 1995); Photonic Crystals and Light Localization in the 21st Century, Nato Advanced Research Institute, series C, 563, edited by C.M. Soukoulis (Kluwer, Dordrecht, 2001). [12] G. Monsivais, M. del Castillo-Mussot, and F. Claro, Stark-Ladder Resonances in the Propagation of Electromagnetic Waves, Phys. Rev. Lett. 64, 1433-1436 (1990). [13] C. Martijn de Sterke, J.N. Bright, P. A. Krug, and T. E. Hammon, Observation of an optical Wannier-Stark ladder, Phys. Rev. E 57, 2365-2370 (1998). [14] G. Lenz, I. Talanina, and C. Martijn de Sterke, Bloch Oscillations in an Array of Curved Optical Waveguides, Phys. Rev. Lett. 83, 963-966 (1999); A. Kavokin, G. Malpuech, A. Di Carlo, P. Lugli, F. Rossi, Photonic Bloch oscillations in laterally confined Bragg mirrors, Phys. Rev. B 61, 4413-4416 (2000).

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[15] P.B. Wilkinson, Photonic Bloch oscillations and Wannier-Stark ladders in exponentially chirped Bragg gratings, Phys. Rev. E 65, 56616 (2002). [16] G. Malpuech, A. Kavokin, G. Panzarini, and A. Di Carlo, Theory of photon Bloch oscillations in photonic crystals, Phys. Rev B 63, 035108 (2001). [17] R. Sapienza , P. Costantino, D.S. Wiersma, M. Ghulinyan, C.J. Oton, and L. Pavesi, Optical Analogue of Electronic Bloch Oscillations, Phys. Rev. Lett. 91, 263902 (2003). [18] M. Ghulinyan, C.J. Oton, Z. Gaburro, L. Pavesi, C. Toninelli, D.S. Wiersma, Zener Tunneling of LightWaves in an Optical Superlattice, Phys. Rev. Lett. 94, 127401 (2005). [19] S. Ossicini, L. Pavesi, F. Priolo, Light Emitting Silicon for Microphotonics, Springer Tracts in Modern Physics, Vol. 194 (Springer, Berlin, 2003); Properties of Porous Silicon, edited by L. Canham, (Short Run Press Ltd., London, 1997). [20] C.J. Oton, M. Ghulinyan, Z. Gaburro, P. Bettotti, L. Pavesi, L. Pancheri, S. Gialanella, N.E. Capuj, Scattering rings as a tool for birefringence measurements in porous silicon, J. Appl. Phys. 94, 6334-6340 (2003). [21] M. Ghulinyan, C. J. Oton, Z. Gaburro, P. Bettotti, and L. Pavesi, Porous silicon freestanding coupled microcavities, Appl. Phys. Lett. 82, 1550-1552 (2003). Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook

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[22] M. Ghulinyan, C. J. Oton, G. Bonetti, Z. Gaburro, and L. Pavesi, Free-standing porous silicon single and multiple optical cavities, J. Appl. Phys. 93, 9724-9729 (2003). [23] L. Pavesi, G. Panzarini, and L.C. Andreani, All-porous silicon-coupled microcavities: Experiment versus theory, Phys. Rev. B 58, 15794-15800 (1998). [24] J.B. Pendry, Symmetry and transport of waves in one-dimensional disordered systems, Adv. Phys. 43, 461-542 (1994). [25] Let us consider the flat bandedge energy, E0 ∼ 1/λ0 and the tilted bandedge one, E ∼ 1/λ, where λ = λ0 (1+∆δ), with ∆δ the optical path gradient. Then for the ratio ∆E/E0 (where ∆E = E − E0 ) one can have ∆E/E0 = 1/(1 + ∆δ) − 1, which for small ∆δ is a linear relation.

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[26] For technical reasons, the gradient is defined with reference to the last layer. This causes a small shift of the spectral features to higher frequencies when the gradient is increased. A gradient of 6.7% therefore means that the first layer (depth zero) has an optical thickness that is 6.7% smaller than the optical thickness of the last layer.

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INDEX

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A absorption spectra, 18, 85, 117 acetone, 66 acoustics, 102 adhesion, 10 adlayers, 47 AFM, 8, 20, 31, 33, 34, 35, 36, 37, 40, 41 ambient air, 33 ammonia, 65 amorphous polymers, 21 amplitude, 14, 16, 28, 31, 114, 164, 167, 175, 179, 180, 209, 221 anisotropy, 18, 26, 27, 76, 104, 111, 115, 117, 120, 125, 133, 157 annihilation, 135 anodization, 216, 219 anti bactericidal properties, viii, 59 aqueous solutions, 10 aromatic compounds, 19 aspiration, 104, 124 asymmetry, 121, 165 asymptotics, 123, 204 atmosphere, 33, 65, 66, 76 atomic force, 30, 49 atoms, 1, 7, 10, 43, 76, 87, 88, 93, 94, 109, 110, 112, 113, 114, 117, 119, 120, 162, 165, 203, 205, 207 azobenzene units, 45

B band gap, ix, 25, 45, 46, 47, 59, 60, 62, 63, 66, 69, 74, 78, 79, 81, 82, 85, 87, 95, 161, 166, 167, 168, 169, 170, 171, 172, 173, 175, 177, 178, 179, 180, 181, 182, 183, 188, 189, 190, 191, 192, 197, 203, 205, 207, 208, 209 barriers, 117, 215 base, 21, 92

beams, vii, 1, 2, 9, 10, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25, 26, 33, 46, 47, 49, 50, 162, 203, 219 bending, 49, 87 bias, 214 binding energy, 60, 64, 74, 85, 110 biological samples, 48 biomolecules, 49 biosensors, 47 biotechnology, 48 birefringence, 20, 231 birefringence measurement, 231 blackbody radiation, 60 bonds, 10, 39, 68, 92, 95, 120 Bose-Einstein condensates (BECs), ix, 161 bosons, 203, 204 branching, 23 breakdown, ix, 213, 214, 215, 223 Britain, 101 bulk materials, 27, 71

C Ca2+, 132 Cairo, 159 candidates, 18, 60, 132 carbon, 39, 40 cation, viii, 78, 131, 132, 134, 135, 136, 142, 149, 150, 151 challenges, 4 charge density, 61, 138 charge migration, 12 chemical, 5, 6, 8, 9, 10, 16, 21, 23, 24, 26, 28, 29, 37, 39, 45, 49, 71, 79, 113, 162, 164, 206, 210 chemical properties, 10 chemical stability, 16 chemical structures, 21 chemical vapour deposition, 24 chirality, 3

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234 CIS, 127 clarity, 177 clusters, viii, 69, 88, 131, 132, 133, 137, 141, 144, 146, 149, 151, 158 CO2, 39 coatings, 16, 40, 42, 50 coherent laser beams, vii, 1, 26 color, 206, 223 combined effect, 13 communication, 44, 45, 48 communication systems, 44 compensation, 39, 68, 132 competitive process, 65, 93 complexity, 23 complications, 23 composites, 2, 3, 4, 50, 88 composition, 8, 37, 39, 48, 81, 109, 110, 112, 113, 114, 118 compound semiconductors, 85 compounds, 19, 85, 132, 156 compressibility, 73 compression, 92 computation, 203 conductance, 115, 117, 120, 121 conduction, 10, 60, 62, 63, 76, 79, 86, 87, 93 conductivity, 44, 87, 108, 111, 121, 123, 156, 157, 230 conductor, 3, 85, 88, 96, 230 configuration, 8, 22, 133, 134, 135, 136, 141, 142, 143, 146, 150, 151, 156, 189, 192 confinement, 9, 46, 60, 66, 71, 73, 85, 93, 222, 224 conservation, 73, 88, 105, 109, 113, 125 constituents, 2, 184, 189 construction, 11, 17, 25, 35, 48 consumption, 44 contamination, 7, 48 cooling, 61, 110, 114 cooling process, 61 Cooper pairs, vii, 1 copolymers, 17 copper, 65, 67, 80, 81, 85, 86, 87, 88 correlation, 34, 117 cost, 7, 44, 46 covalent bond, 119 covering, 47 cracks, 37, 39 critical value, x, 213 crystal quality, 74 crystal structure, 60, 62, 63, 79, 87, 102, 109, 112 crystalline, 33, 61, 71, 80, 114, 117, 128 crystallinity, 17, 70 crystallites, 70, 71, 95

Index crystals, vii, viii, ix, 13, 16, 24, 25, 27, 45, 46, 51, 60, 62, 63, 65, 101, 102, 104, 105, 106, 107, 108, 109, 111, 113, 114, 115, 117, 118, 119, 120, 121, 124, 125, 126, 127, 128, 129, 131, 132, 133, 137, 149, 157, 161, 162, 201, 214, 230, 231 cultivation, 102 cycles, 67, 80 Czech Republic, 1

D damages, iv, 29 damping, 229 data analysis, 114 decay, 11, 14, 20, 61, 62, 109, 164, 165, 222, 228, 229 decomposition, 18, 109 deconvolution, 66, 90, 91 defect site, 86 defects, 61, 62, 64, 65, 68, 74, 76, 78, 79, 86, 87, 88, 93, 132, 133 deficiency, 93 deformation, 49, 73 degenerate, viii, x, 68, 125, 131, 133, 134, 135, 137, 142, 144, 146, 150, 151, 156, 184, 189, 213, 215, 218, 226 degradation, 30, 39 Delta, 166, 174, 183, 184 deposition, viii, 5, 10, 24, 35, 39, 40, 41, 42, 43, 48, 50, 51, 59, 67, 79, 80, 81, 95 deposition rate, 43 depth, 20, 33, 38, 124, 125, 217, 218, 223, 225, 232 derivatives, 19 destruction, 29 detectable, 201 detection, viii, 47, 101, 177, 198, 201, 230 dielectric constant, 214 dielectric permeability, 137, 138, 144, 145, 150 dielectric permittivity, 157 dielectrics, 4, 108, 127, 230 diffraction, 5, 18, 20, 23, 24, 27, 44, 45, 47, 79, 80, 88, 92, 112, 113, 121, 128 diffuse reflectance, 94 diffusion, 13, 14, 25, 28, 146 diffusion process, 13 dimensionality, 108, 111, 125 diodes, 60, 214 dipole moments, 17 directionality, 25 discretization, 175 dislocation, 114 disorder, 71 disordered systems, 232

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Index dispersion, 2, 71, 73, 93, 102, 103, 104, 105, 106, 107, 108, 114, 115, 119, 120, 123, 124, 125, 129, 139, 146, 204 displacement, 106, 109, 110, 113, 114, 115, 120, 121 dissociation, 61, 65 distilled water, 65 distribution, vii, viii, 1, 12, 16, 17, 22, 26, 27, 28, 32, 45, 49, 70, 73, 92, 131, 132, 133, 138, 139, 140, 149, 157, 204, 218, 223 diversity, 132 DNA, 47 donors, 62, 64, 75 dopants, 15, 23, 62, 64, 79, 162 doping, 16, 28, 29, 69, 73, 75, 76, 78, 79, 80, 81, 85, 88, 95, 199, 200, 219 DRS, 94 dynamical systems, 204

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E election, 73 electric field, 2, 13, 14, 15, 17, 18, 27, 48, 76, 103, 104, 125, 163, 214, 215, 216, 230 electrical breakdown, 230 electrical properties, 76 electricity, 107 electroluminescence, 60 electrolyte, 219, 226 electromagnetic, 2, 9, 15, 17, 47, 48, 105, 106, 125, 138, 146, 164, 197 electromagnetic fields, 17 electromagnetic waves, 2, 47, 106, 197 electromagnetism, 2, 3, 48 electron, viii, 2, 3, 17, 46, 48, 60, 61, 62, 63, 64, 66, 85, 86, 87, 92, 107, 113, 115, 132, 133, 156, 214, 215 electron beam lithography, 3 electron diffraction, 92 electron microscopy, 113, 115 electronic structure, 86, 87, 132 electrons, 8, 17, 19, 46, 60, 61, 62, 75, 79, 87, 93, 117, 156, 214, 215, 230 elucidation, 47, 49 embossing, 4 emission, 2, 46, 48, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 74, 75, 79, 81, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 230 energetic characteristics, 146 energy, vii, viii, 1, 7, 9, 11, 19, 29, 43, 50, 60, 61, 62, 64, 69, 73, 74, 75, 78, 81, 85, 86, 87, 93, 95, 105, 109, 110, 126, 131, 132, 133, 136, 137, 138, 140, 142, 143, 144, 149, 150, 151, 153, 157, 168, 205, 209, 210, 214, 219, 220, 232

energy density, vii, 1 energy transfer, 95 engineering, 3 England, 194 environment, 79, 121, 150, 228 equilibrium, 43, 60, 86 equipment, 24 etching, 10, 12, 30, 45, 216, 217, 219 ethanol, 93, 219 europium, 91 evaporation, 10, 12, 29, 35, 40, 41, 42, 43, 50 evolution, 26, 48, 64, 71, 88, 164, 175, 178, 182, 224, 225, 226 excitation, 2, 3, 11, 12, 14, 17, 32, 46, 48, 60, 61, 62, 65, 66, 68, 69, 73, 74, 75, 81, 84, 85, 87, 88, 90, 93, 108, 115, 119, 126, 132, 149, 156 exciton, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 74, 79, 81, 85, 86, 93, 94 experimental condition, vii, 1, 30, 39, 62, 64, 125 exposure, 6, 7, 10, 16, 21, 22, 23, 27, 28, 36, 39 extinction, 222

F fabrication, 4, 10, 14, 16, 21, 23, 24, 25, 39, 44, 46, 47, 201 Faraday effect, viii, 131, 132, 141, 146 Fermi level, 117 film thickness, 10 films, vii, viii, 1, 3, 9, 10, 15, 23, 25, 28, 29, 30, 46, 48, 49, 50, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 80, 81, 85, 86, 87, 88, 94, 95, 99, 156 filters, 44, 46, 50, 216, 219 financial, 230 financial support, 230 flatness, 85 flexibility, 17 fluctuations, 228 fluid, 7, 29, 201 fluorescence, 19, 46, 80 fluorine atoms, 10 foils, 34, 39, 40 force, 14, 15, 30, 49, 215 Ford, 56 formation, viii, x, 1, 2, 10, 11, 13, 14, 19, 20, 21, 25, 26, 28, 29, 31, 32, 33, 39, 43, 44, 45, 59, 61, 62, 75, 86, 87, 88, 114, 146, 162, 164, 197, 213, 214, 215, 216, 218, 219, 223, 224 formula, 35, 111, 119, 122, 149 France, 129 free volume, 8, 15 FTIR, 92

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Index

full width half maximum, 228 fundamental physics, vii, 1

G gadolinium, 91 gap solitons (GSs), ix, 161, 162 genomics, 49 geometrical parameters, 31 geometry, 9, 37, 40, 42, 47, 162 glass transition, 13, 15, 21, 31 glass transition temperature, 13, 15, 21, 31 grain boundaries, 94 graphite, 102, 106, 107, 127 gratings, vii, 1, 3, 23, 25, 26, 27, 44, 45, 47, 121, 215, 231 Great Britain, 101 growth, 36, 37, 43, 60, 79, 81, 119, 127, 137, 215, 216, 219, 226 growth mechanism, 36, 43 growth temperature, 79

insulators, vii, 2 integrated circuits, 46 integrated optics, 44 integration, 44, 48, 164 integrity, vii, 2 interface, 3, 94, 216 interference, vii, 1, 2, 9, 12, 13, 14, 16, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 32, 44, 45, 46, 47, 48, 49, 50, 109, 114, 121, 162, 204, 214, 215 invariants, 102, 165 inversion, 112 ionization, 8, 64 ions, viii, 27, 68, 73, 86, 90, 93, 94, 95, 106, 131, 132, 133, 134, 135, 136, 138, 141, 142, 145, 149, 150, 152, 155, 156, 157 iron, viii, 131, 132, 149, 150 iron-group ions, viii, 131, 132 irradiation, vii, 1, 5, 10, 15, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 45, 50, 65 isomers, 18 Israel, 161 issues, 4, 7 Italy, 213

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H Hamiltonian, 134, 135, 138, 141, 168 height, 34, 35, 36, 39, 209, 210 hologram, 25, 26, 28 homogeneity, 25, 112, 118, 119, 124 host, 14, 20, 88 House, 127, 128 hybrid, 22 hydrazine, 10 hydrogen, 65, 78 hydrophilicity, 10 hydroxyl, 92

I illumination, 4, 5, 6, 9, 10, 12, 16, 18, 25, 26, 27, 28, 32, 33, 45 images, 8, 10, 20, 22, 25, 27, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 125 imaging systems, 3 immersion, 4, 6, 7, 31 imprinting, 204 impurities, viii, 59, 60, 62, 64, 65, 86, 87 incubation period, 10 India, 59, 197, 203 information processing, 46 inhomogeneity, 12, 15, 106 initiation, 113, 114, 119

J Jahn-Teller centers, viii, 131 Jahn-Teller ion, viii, 131

K K+, 87 kinetics, 81, 102

L laser ablation, 10 laser light, vii, 1, 2, 3, 4, 6, 10, 12, 16, 23, 28, 32, 36, 50 laser radiation, 40, 45 lasers, 6, 10, 27, 37, 38, 40, 60, 205 lattice parameters, 92 lattices, 112, 127, 149, 162, 163, 194, 205, 215 laws, 105, 108, 119 lead, vii, viii, 1, 3, 16, 19, 21, 30, 31, 40, 43, 49, 75, 76, 87, 88, 92, 94, 104, 108, 131, 132, 133, 137, 197 lens, 213 light, vii, viii, ix, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 21, 23, 24, 25, 26, 27, 28, 32, 33, 36, 39, 44, 46, 47, 48, 49, 50, 59, 60, 62, 63,

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Index 67, 69, 71, 74, 76, 83, 93, 94, 102, 104, 105, 106, 107, 108, 115, 117, 119, 120, 123, 124, 125, 127, 138, 145, 146, 149, 199, 203, 205, 213, 214, 215, 216, 218, 221, 222, 223, 229, 230 light beam, 12, 13, 14, 25 light transmission, x, 213 light-emitting diodes, 60 linear dependence, 31 liquid crystals, 27, 45, 201 liquid phase, 114 lithium, 44 lithography, viii, 2, 3, 4, 6, 7, 8, 9, 16, 21, 22, 23, 24, 25, 27, 32, 44, 45, 46, 47, 48, 49, 50 local mobility, 209, 210 localization, 28, 73, 117, 230 low temperatures, 60 luminescence, viii, 59, 60, 61, 64, 65, 70, 75, 76, 85, 86, 87, 94, 95 luminescence efficiency, 60, 70, 94

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M magnetic circular dichroism, viii, 131, 132, 146 magnetic field, 2, 48, 103, 163 magnetic materials, 4 magnetic resonance, 3, 60 magnetic resonance imaging, 3 magnetization, 137, 139, 141, 146, 147, 154, 155 magnitude, 15, 28, 216 majority, 75, 204 management, 48, 162, 204 manganese, 111, 114, 119, 128, 156 manganites, 132, 133, 156, 157 manipulation, 22 mass, 15, 24, 26, 27, 28, 30, 44, 45, 61, 80, 113, 162, 204, 205, 228 material surface, 32 materials, vii, 1, 2, 3, 4, 6, 7, 9, 10, 12, 16, 17, 18, 19, 21, 23, 24, 27, 28, 43, 44, 45, 46, 47, 48, 49, 50, 51, 54, 60, 71, 88, 117, 128, 132, 197, 201, 214 materials science, 3 matrix, 2, 3, 17, 18, 27, 30, 88, 92, 138, 144, 145, 152, 158, 198, 199, 207, 218, 222, 224, 227, 229 matrixes, 18 matter, vii, ix, 9, 17, 25, 27, 162, 174, 203, 204, 205, 206, 207, 208, 210 measurements, ix, 34, 64, 87, 213, 215, 220, 227, 228, 229 mechanical properties, 49 media, ix, 2, 7, 17, 21, 25, 45, 121, 132, 161, 162, 195 medicine, 43, 47

melting, 10, 33 metal ions, 132, 133 metal oxides, 71, 132 metals, 4, 19, 88, 127 meta-materials, vii, 1, 2, 50 methodology, 67 methyl group, 10 methyl methacrylate, 29, 45 Mg2+, 132, 149 microcavity, 218 microcrystalline, 127 microfabrication, 16, 25 micrometer, vii, 2, 25, 27, 31, 49, 50 microscope, 6, 28, 92, 114 microscopy, 8, 30, 49, 113, 114, 115 microstructure, 22, 49, 127 microstructures, 16, 25, 28, 30, 49, 50 migration, vii, 1, 12, 13, 19 miniaturization, 3, 48 Ministry of Education, 51 mission, 67, 93 mixed valence clusters, viii, 131, 132 mixing, 29, 144, 152, 219, 220 model system, 203 modelling, 102 models, viii, ix, 60, 101, 102, 117, 121, 161, 162, 165, 178, 183, 192, 204 modifications, 17, 19, 50 modulus, 49 molecular reorientation, 18 molecular weight, 10, 15, 21, 26, 27, 29 molecules, 4, 11, 14, 15, 17, 19, 20, 23, 27, 28, 29, 47, 55, 77, 91, 128 momentum, ix, 61, 73, 105, 125, 131, 133, 138, 144, 145, 152, 204, 214 monomers, 21, 25, 26 Moon, 56 morphology, vii, 1, 29, 30, 33, 35, 36, 37, 39, 40, 41, 71 Moscow, 126, 127, 128, 129, 196 multidimensional, 162 multilayered structure, 197 multimedia, 45

N nanocomposites, 27 nanocrystals, 27, 69, 73 nanodots, 71 nanofabrication, 2, 7 nanolayers, 39, 50 nanometer scale, 49 nanometers, 3, 8, 9, 21, 22, 31, 35, 49, 61

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Index

nanoparticles, 25, 66, 69 nanorods, 65, 94 nanostructures, viii, 24, 46, 59, 60, 95, 99 nanotechnology, 21 nanowires, 39 NATO, 231 neglect, 104, 107 Netherlands, 51 neutral, vii, 26, 43, 62, 64, 164 neutrons, 129 next generation, 59 nickel, 10 noble metals, 88 nodes, 49 nonlinear dynamics, 161 nonlinear optics, 161 non-polar, 11, 73, 76 normal distribution, 139 nucleation, 43 null, 104, 124 numerical analysis, ix, 161, 163, 175, 184, 191 numerical aperture, 7, 219

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O oligomers, 16 one dimension, 198, 218, 219 optical activity, ix, 132, 133, 157 optical anisotropy, 104 optical communications, 197 optical gratings, vii, 1 optical lattice, vii, ix, 1, 161, 203, 204, 205, 211 optical lattices (OLs), ix, 161 optical microscopy, 8 optical parameters, 216 optical properties, viii, ix, 2, 3, 19, 25, 46, 48, 50, 59, 60, 64, 73, 74, 79, 87, 88, 101, 104, 121, 125, 132, 133, 216, 230 optical systems, 5 optoelectronics, 11, 43 orbit, 62, 63, 138, 139, 140, 141, 142, 145, 146, 149, 153 organic polymers, 45 orthogonality, 144, 152 oscillation, 214, 219, 222, 229 overlap, 11, 25, 66, 227 oxidation, 33, 39, 132 oxygen, viii, 38, 39, 59, 65, 74, 75, 76, 79, 81, 86, 87, 88, 92, 93, 94, 135, 150, 156

P palladium, 99 parallel, vii, 2, 3, 26, 28, 39, 76, 105, 106, 115, 116, 121, 123, 136, 137, 146, 147, 154, 155, 156, 215 parallelism, 48 passivation, 69, 71, 88 penetrability, 23 periodical pattern, vii, 1 periodical potential, vii, 1 periodicity, viii, ix, 2, 9, 23, 24, 27, 29, 30, 32, 35, 36, 39, 47, 49, 118, 203, 204, 205, 207, 208, 210, 211 permeability, 2, 10, 48, 103, 137, 138, 144, 145, 150 permittivity, 2, 48, 103, 104, 108, 118, 124, 125, 126, 157 perovskite structures, ix, 132, 157 PES, 60 PET, 15, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 50 pH, 65, 66 phase diagram, 111, 204 phase shifts, 22 phase transitions, 114, 127, 128 phonons, 61, 73, 76, 85, 86, 88, 104, 105, 106, 107, 108, 109, 111, 118, 119, 120, 125 photoconductivity, 13, 79 photoelectron spectroscopy, 60 photo-excitation, 14, 66 photographers, 114 photolithography, 4, 5, 6, 16 photoluminescence, viii, 59, 60, 66, 67, 69, 74, 79, 92, 94 photo-migration, vii, 1 photonics, 21, 30, 43, 44, 50 photons, 7, 9, 11, 46, 106, 214, 219, 230 photo-orientation processes, vii, 1 photopolymerization, 9, 24, 25 physical phenomena, 46 physical properties, viii, 2, 101, 102, 108 physical structure, 16 physics, vii, 1, 3, 53, 98, 102, 127, 132, 162, 203, 204, 224 piezoelectric, viii, 59 PL spectrum, 62, 64, 74, 76 plasticization, 15 platelets, 92 platform, 23 PMMA, 15, 29, 30 point defects, 64, 65, 74, 78, 79 Poland, 203 polar, 11, 38, 39, 73, 76, 85, 106, 149 polar groups, 38, 39 polarizability, 17, 19

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Index polarization, vii, 3, 9, 10, 14, 15, 17, 22, 23, 32, 73, 103, 104, 106, 117, 121, 138, 145, 146, 199 polarization planes, 10 poly(methyl methacrylate), 29, 45 polycarbonate, 45 polyethyleneterephthalate, 32 polymer, vii, 1, 2, 4, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 39, 43, 44, 45, 46, 47, 48, 49, 50, 51 polymer chain, 13, 14, 15, 17, 33, 39 polymer composites, 50 polymer density, 23 polymer films, vii, 1, 9, 25, 28, 29, 46, 48, 49 polymer materials, 47 polymer matrix, 17, 18, 30 polymer modification, vii, 1, 9, 10, 12, 50 polymer nanocomposites, 27 polymer properties, 16 polymer structure, 2, 22, 47, 50 polymer systems, 20 polymeric materials, 50 polymerization, 6, 12, 21, 23, 25, 26, 27, 28, 45, 46 polymerization process, 21, 26, 45 polymers, vii, 1, 2, 4, 6, 8, 9, 10, 13, 15, 16, 17, 18, 19, 21, 25, 27, 28, 29, 30, 32, 33, 35, 43, 44, 45, 47, 48, 49, 50, 55 polystyrene, 29 porosity, 216, 217, 219 porphyrins, 19 potassium, 87 precipitation, 65 preparation, vii, 2, 4, 6, 16, 24, 27, 29, 32, 44, 45, 46, 48, 50, 66, 93, 99, 216, 219 pressure gradient, 14 principles, 44, 50, 58, 98, 101 probability, 11, 14, 75, 117, 214, 228 probe, 9, 20, 219, 220, 221, 229 propagation, vii, 1, 121, 129, 164, 197, 214, 215, 224, 228, 229, 230 propagation of spin waves, vii, 1 proteins, 49 proteomics, 49 PTFE, 10 pumps, 16 purity, 60 pyrolysis, viii, 59, 66, 67, 68, 72, 79, 80, 82, 84, 87, 88, 89, 92, 95

Q quanta, 61 quantum confinement, 71, 93 quantum dots, 97

quantum well, 85, 86, 215, 218

R radiation, 7, 15, 30, 40, 45, 60, 101, 104, 105, 106, 107, 146 radicals, 91 radio, 2, 102, 115, 118 radius, 61, 62, 66, 71, 92, 138 Raman spectra, 70, 71, 72, 77, 78, 87, 89, 92, 95 Raman spectroscopy, 76 reaction mechanism, 66 recombination, 60, 62, 63, 64, 70, 74, 75, 76, 79, 85, 87 recombination processes, 64, 70, 75 recommendations, iv red shift, 88, 93 redistribution, vii, viii, 1, 15, 131, 133 reflectance spectra, 217 reflectivity, 101, 106, 115, 117, 121, 122 refractive index, ix, 2, 3, 4, 7, 12, 13, 21, 23, 25, 26, 27, 28, 32, 35, 45, 46, 47, 48, 197, 199, 200, 201, 216, 217, 218, 219, 226 refractive indices, 26, 197, 198, 199, 201, 218, 223 refractometric sensing element, ix, 197 relief, 2, 13, 14, 16, 19, 20, 21, 25, 37, 45, 49 renormalization, 137, 138 repulsion, 177, 188, 189, 192, 218, 226, 228 requirements, 18 resistance, 2, 132 resolution, 3, 4, 5, 6, 7, 8, 9, 12, 16, 25, 26, 46, 48, 49, 50, 219, 220, 221, 224, 227, 228 resonator, 4 restoration, 209 restrictions, 29 restructuring, 60 rings, 231 rods, 92 room temperature, 61, 66, 68, 70, 74, 85, 86, 87, 118, 156 roughness, 10, 16, 32, 34, 35, 37, 40, 42, 50, 88 rules, 11, 76, 109, 151 Russia, 101, 127, 129, 131

S sapphire, 220 scanning electronic microscope, 113, 114 scattering, 23, 61, 77, 86, 88, 93, 94, 117, 129, 163, 205, 216, 217, 222, 224, 225 segregation, 27 selected area electron diffraction, 92

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self-assembly, 46 semiconductor, 24, 46, 60, 62, 64, 66, 68, 95, 214, 215, 218, 231 semiconductor lasers, 60 semiconductors, vii, 2, 13, 16, 44, 60, 78, 85, 127 seminars, 127 sensing, ix, 47, 59, 197, 198, 201 sensitivity, 201 sensors, 27, 44, 47, 50, 198, 201 shape, 2, 3, 6, 10, 14, 15, 22, 62, 63, 67, 88, 149, 153, 168, 228 shock, 10 showing, 39, 80, 92, 191, 217 side chain, 20 signals, 44, 221, 229 signs, 141, 163, 164, 178, 188 silica, 47 silicon, 10, 21, 111, 119, 124, 213, 215, 216, 217, 219, 222, 229, 230, 231, 232 silver, 4, 28, 65, 88, 92, 94 simulation, 10 simulations, 146, 153, 164, 175, 179, 184, 189 Singapore, 97 single crystals, 60, 62, 63 skin, 121, 124 smoothing, 31 smoothness, 77 soft matter, 27 solid phase, 114 solid solutions, 109, 127 solid state, 98, 117 solitons, vii, ix, 109, 110, 113, 114, 119, 121, 161, 162, 163, 164, 165, 166, 168, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 195, 204, 205, 207, 208, 209, 210, 211 solubility, 5, 29, 88 solution, 16, 29, 65, 67, 81, 88, 91, 92, 110, 164, 165, 167, 168, 172, 174, 175, 223 solvents, 20 sorption, 222 Spain, 213 species, 162 spectral techniques, 60 spectroscopy, 19, 60, 62, 63, 76, 132, 204 speculation, 209, 210 spin, vii, 1, 5, 10, 29, 49, 62, 63, 132, 135, 136, 138, 139, 140, 141, 142, 145, 146, 149, 151, 153 spindle, 92 St. Petersburg, 127 stability, ix, 9, 16, 17, 18, 19, 23, 24, 161, 163, 164, 168, 174, 175, 177, 178, 179, 180, 183, 184, 188, 189, 192, 204

stabilization, 162 stable states, 109 standard deviation, 222 stoichiometry, 81 storage, 19, 26, 28, 45 storage media, 45 stress, 71, 73, 80, 88, 178 stretching, 223 structural changes, 208 structural defects, 68 structure, vii, ix, x, 1, 2, 3, 4, 10, 12, 16, 17, 19, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 39, 40, 41, 43, 45, 46, 47, 48, 49, 50, 60, 62, 63, 64, 65, 73, 78, 79, 80, 86, 87, 88, 92, 94, 102, 109, 112, 114, 115, 127, 132, 133, 141, 149, 150, 152, 156, 162, 169, 197, 198, 199, 200, 201, 204, 205, 206, 208, 210, 213, 214, 215, 216, 217, 218, 219, 222, 223, 224, 226 structure formation, vii, 1, 25, 32, 33 structuring, 3, 5, 32 substitutes, 149 substitution, 20, 78, 134, 156, 166, 167 substitutions, viii, 131, 132, 133, 149 substrate, 6, 10, 12, 21, 24, 29, 43, 47, 48, 50, 65, 67, 73, 74, 75, 79, 80, 81, 92, 94, 216, 217 substrates, vii, 1, 2, 16, 44, 48, 67, 94 Sun, 53, 55, 95, 96 superconductivity, 132 superfluid, 203, 204 superimposition, 116 superlattice, ix, x, 203, 204, 205, 206, 207, 208, 209, 210, 211, 213, 214, 215, 218, 219, 221, 223, 224, 225, 226, 227, 228, 229 super-solids, vii, 1 suppression, 46 surface area, 92, 94 surface chemistry, 9, 10 surface layer, 9, 28, 38 surface modification, 9, 10, 13 surface properties, 60, 88 surface structure, 13, 16 surface tension, vii, 1, 9, 31, 71, 73 susceptibility, 15, 103 symmetry, viii, ix, 17, 21, 22, 24, 31, 47, 49, 88, 101, 102, 105, 107, 108, 109, 111, 112, 113, 115, 119, 121, 125, 126, 127, 131, 133, 134, 135, 136, 144, 149, 150, 151, 152, 156, 161, 163, 184, 191, 219, 225 synthesis, 17, 88

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Index

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T techniques, viii, 2, 3, 4, 5, 25, 26, 28, 39, 41, 43, 44, 46, 47, 48, 49, 50, 59, 60, 62, 63, 95, 132 technologies, vii, 2, 3, 16, 50 technology, 3, 11, 16, 23, 24, 25, 50, 57, 162, 216 telecommunications, 46 TEM, 92, 94 temperature, vii, viii, 1, 9, 12, 13, 15, 28, 29, 31, 43, 44, 47, 59, 61, 62, 63, 64, 65, 66, 68, 70, 71, 74, 79, 81, 85, 86, 87, 91, 92, 94, 109, 110, 114, 116, 118, 146, 156, 201 temperature dependence, 65, 114, 156 tension, vii, 1, 9, 31, 71, 73 tension gradients, vii, 1, 31 testing, 73 thermal degradation, 39 thermal energy, 64, 110 thermal expansion, 73 thermal properties, 16 thermal stability, 16 thermalization, 61 thermodynamics, 102, 110 thin films, 15, 28, 60, 66, 67, 68, 69, 70, 71, 73, 75, 76, 77, 79, 80, 81, 85, 87, 88, 94, 99 thin polymer films, vii, 1, 49 topology, 42 toxicity, 7 TPA, 11, 12 transformation, 228 transition metal, 71, 127, 132 transition temperature, 13, 15, 21, 31 translation, 33, 105, 107, 108, 109, 112, 113, 114, 119 transmission, ix, 3, 23, 92, 115, 197, 199, 200, 201, 213, 215, 217, 219, 220, 221, 223, 224, 225, 226, 227, 228, 229, 230 transmittance spectra, 66, 81, 82, 217 transparency, 121 transport, 9, 33, 132, 214, 228, 232 treatment, vii, 10, 28, 31, 32, 73, 167 tunneling, ix, 164, 203, 213, 214, 215, 219, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231

U UK, 101 UV, 5, 6, 9, 10, 16, 18, 21, 23, 32, 33, 35, 45, 50, 60, 74, 85, 92, 93 UV absorption spectra, 18 UV irradiation, 45 UV light, 5, 9, 23

V vacancies, viii, 59, 65, 74, 75, 78, 79, 86, 87, 88, 94, 131, 132, 149 vacuum, viii, 29, 35, 59, 75, 122 valence, vii, viii, 60, 62, 75, 79, 87, 117, 131, 132, 133, 143, 157 valve, 230 vapor, 79 variations, 41, 87, 201, 210, 219, 228 varieties, 10 vector, 2, 47, 73, 88, 101, 103, 104, 105, 106, 107, 108, 115, 117, 120, 124, 125, 126, 138, 139 velocity, 2, 30, 48, 114, 138, 214 vibration, 6, 23, 47, 76, 77, 88, 92, 109, 123, 129

W Wannier-Stark ladder (WSL), x wave propagation, 129 wave vector, 88, 101, 104, 105, 106, 107, 108, 114, 115, 117, 119, 120, 124, 125 wavelengths, 3, 8, 18, 36, 37, 44, 46, 47, 197, 200, 205, 223, 228, 229 wells, 86, 117, 203, 210 wettability, 10 wetting, 10 wide band gap, 59, 85, 95 wires, 4, 32, 37, 39, 41, 43, 50

X XPS, 38, 39, 40 X-ray diffraction (XDR), 80, 87, 88, 92, 112, 113

Y yttrium, 149

Z Zener breakdown, ix, 213, 214, 215 zinc, 60, 62, 63, 65, 67, 74, 75, 76, 79, 80, 81, 87, 88 zinc oxide (ZnO), viii, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 99 ZnO nanorods, 65

Optical Lattices: Structures, Atoms and Solitons : Structures, Atoms and Solitons, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook