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Academic Press is an imprint of Elsevier 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States 525 B Street, Suite 1800, San Diego, CA 92101-4495, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 125 London Wall, London, EC2Y 5AS, United Kingdom First edition 2016 Copyright © 2016 Elsevier Inc All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein) Notices Knowledge and best practice in this field are constantly changing As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein ISBN: 978-0-12-804796-5 ISSN: 0081-1947 For information on all Academic Press publications visit our website at https://www.elsevier.com/ Publisher: Zoe Kruze Acquisition Editor: Poppy Garraway Editorial Project Manager: Shellie Bryant Production Project Manager: Vignesh Tamil Cover Designer: Maria Ines Cruz Typeset by SPi Global, India CONTRIBUTORS T Dumelow Universidade Estado Rio Grande Norte (UERN), Mossoro´, Brazil H Kachkachi PROMES, CNRS-UPR 8521, Universite de Perpignan Via Domitia, Perpignan, France D.S Schmool Groupe d’Etude de la Matie`re Condensee GEMaC, CNRS (UMR 8635) Universite de Versailles/Saint-Quentin, Universite Paris-Saclay, Versailles, France vii PREFACE It is our great pleasure to present the 67th edition of Solid State Physics The vision statement for this series has not changed since its inception in 1955, and Solid State Physics continues to provide a “mechanism … whereby investigators and students can readily obtain a balanced view of the whole field.” What has changed is the field and its extent As noted in 1955, the knowledge in areas associated with solid state physics has grown enormously, and it is clear that boundaries have gone well beyond what was once, traditionally, understood as solid state Indeed, research on topics in materials physics, applied and basic, now requires expertise across a remarkably wide range of subjects and specialties It is for this reason that there exists an important need for up-to-date, compact reviews of topical areas The intention of these reviews is to provide a history and context for a topic that has matured sufficiently to warrant a guiding overview The topics reviewed in this volume illustrate the great breadth and diversity of modern research into materials and complex systems, while providing the reader with a context common to most physicists trained or working in condensed matter The chapter “Collective Effects in Assemblies of Magnetic Nanoparticles” provides an overview of emergent behavior arising from collections of interacting magnetic particles from the perspective of experiment, and also in terms of modeling and theory The second chapter, “Negative Refraction and Imaging from Natural Crystals with Hyperbolic Dispersion,” describes aspects of material optics with a focus on the fascinating properties of hyperbolic materials whose surprising properties can be found in naturally occurring single-phase materials, as opposed to metamaterials in which these properties are engineered through design The editors and publishers hope that readers will find the introductions and overviews useful and of benefit both as summaries for workers in these fields, and as tutorials and explanations for those just entering ROBERT E CAMLEY AND ROBERT L STAMPS ix CHAPTER ONE Collective Effects in Assemblies of Magnetic Nanaparticles D.S Schmool*,1, H Kachkachi† *Groupe d’Etude de la Matie`re Condensee GEMaC, CNRS (UMR 8635) Universite de Versailles/Saint-Quentin, Universite Paris-Saclay, Versailles, France † PROMES, CNRS-UPR 8521, Universite de Perpignan Via Domitia, Perpignan, France Corresponding author: e-mail address: dschmool@fc.up.pt Contents Introduction Magnetic Nanoparticle Assemblies: Theoretical Aspects 2.1 Model 2.2 Equilibrium Properties : Magnetization and Susceptibility 2.3 Dynamic Properties Experimental Aspects 3.1 Magnetometry 3.2 AC Susceptibility 3.3 Magnetization Dynamics €ssbauer Spectroscopy 3.4 Mo 3.5 Neutron Scattering Experiments Summary References 17 24 25 30 33 47 56 85 90 INTRODUCTION Investigating the properties of ensembles of magnetic nanoparticle is a rich and challenging physics problem, from both the experimental and theoretical points of view Indeed, one encounters the typical difficult situation where intraparticle and interparticle effects meet into a formidable manybody problem with both short-range and long-range interactions The intraparticle effects are related with the intrinsic properties of the nanoparticles, such as the underlying material, size, shape, and energy potential In particular, for small sizes the features of the single-nanoparticle physics are dominated by finite-size and surface effects that drastically affect their Solid State Physics, Volume 67 ISSN 0081-1947 http://dx.doi.org/10.1016/bs.ssp.2016.08.001 # 2016 Elsevier Inc All rights reserved D.S Schmool and H Kachkachi magnetic properties, both in equilibrium and out of equilibrium On the other hand, assembled nanoparticles into 1D, 2D, or 3D arrays, organized or not, reveal interesting and challenging issues related with their interactions among themselves and with their hosting medium, a matrix or a substrate The ensuing collective effects show up through novel features in various measurements, such as ferromagnetic resonance (FMR), AC susceptibility and M€ ossbauer spectroscopy, to cite a few Now, for assemblies of small particles ($3–10 nm) one has to deal with the interplay between surface effects and interparticle interactions whose study requires tremendous efforts In addition, during a few decades one had to struggle with at least two distributions, namely that of the particles size and the anisotropy (effective) easy axes Today, the situation has improved owing to the huge progress in the production of nearly monodisperse assemblies in well-organized patterns This is one of the reasons for which more theoretical works have appeared recently focusing on such newly devised systems Needless to say that, already at equilibrium, no exact analytical treatment of any kind is ever possible even in the one-spin approximation (OSP), i.e., ignoring the internal structure of the particles and thereby surface effects Only numerical approaches such as the Monte Carlo technique can alleviate this frustration Indeed, applications of this technique to the case of Ising dipoles can be found in reference [1] The same technique has been used in reference [2] to study hysteretic properties of monodisperse assemblies of nanoparticles with the more realistic Heisenberg spin model, within the OSP approximation where each particle carries a net magnetic moment In reference [3], the Landau–Lifshitz thermodynamic perturbation theory [4] is used to tackle the case of weakly dipolar-interacting monodisperse assemblies of magnetic moments with uniformly or randomly distributed anisotropy axes The authors studied the influence of dipolar interactions (DI) on the susceptibility and specific heat of the assembly Today, the literature thrives with theoretical works on the effect of DI on the magnetic properties of assemblies of nanoparticles, most of which make use of numerical techniques [2, 5–25], because the main interest is for dense assemblies for which experimental measurements are relatively easier to perform and the applications more plausible However, it is important to first build a fair understanding of the underlying physics This can only be done upon studying model systems that are simple enough for performing analytical developments and still rich enough to capture the main qualitative features of the targeted systems Analytical expressions come very handy in that they allow us to figure out what are the main relevant physical parameters and how the Collective Effects in Assemblies of Magnetic Nanaparticles physical observables of interest behave as the former are varied and the various contributions to the energy compete which other A brief account of our contribution will be given in the following section The magnetic properties of magnetic nanoparticles can be rather difficult to measure, as we saw in the earlier chapter on single particle measurements, where very specialized methods and adaptations are required [26] To overcome some of the problems with the weak experimental signals, many measurements are made on assemblies of nanoparticles and elements This means that the results obtained are generally an average over the sample and assembly and must also be interpreted taking into account the magnetic interactions between the particles There have been extensive studies using many techniques In the following, we aim to give a brief overview of selected studies and techniques and will not be an exhaustive review In particular, we focus on well-known experimental techniques, which have been applied to the study of nanoparticle systems Standard techniques, such as magnetometry and AC susceptibility, have been applied to the study of magnetic nanoparticle systems Measurements can be made under the usual conditions since the material quantity is not an issue, as stated previously Where these techniques have shown to be of importance is in the study of the superparamagnetic (SPM) behavior observed in magnetic nanoparticle assemblies This arises due to the thermal instability introduced when the magnetic anisotropy, which usually defines the orientation of the magnetization of the magnetic particle, is insufficient to maintain its normal orientation In fact the energy barrier is defined as the product of the particles magnetic anisotropy constant K and its volume V Once the thermal energy is of the same order of magnitude as KV, the magnetization becomes unstable, switching spontaneously between the energy minima of the system As a result, the magnetic measurement, which has a characteristic measurement time, will sample the magnetic state as being (super)paramagnetic A combination of measurements as a function of temperature and applied field allows the system to be defined in terms of its energy barrier and the blocking temperature TB, where the magnetization is stable over the measurement time Indeed, for AC susceptibility measurements, a frequency dependence is also important Indeed the average switching time between magnetic easy axes is characterized as an attempt frequency For measurements made with lower characteristic measurement time, such as M€ ossbauer spectroscopy and FMR, corresponding values of the blocking temperature will be much higher due to the Arrhenius behavior associated with superparamagnetism D.S Schmool and H Kachkachi Ferromagnetic resonance is a very sensitive method for measuring the magnetic properties of materials via the precessional magnetization dynamics defined by the systems magnetic free energy The precessional motion of the magnetization is in general strongly influenced by magnetic anisotropies and exchange effects in solids This is often regarded as the internal effective magnetic field experienced by the local magnetic spins of the system This can thus be separated into the various contributions to the local magnetic field, via, magnetocrystalline anisotropy, shape anisotropy, exchange interactions, etc In magnetic nanosystems [26–29], this can be adapted to include surface anisotropy effects as well as magnetic DI between particles This will produce shifts in the resonance fields and can significantly affect the linewidth of resonance absorption lines Once again, measurements as a function of sample temperature can provide further information regarding the magnetic behavior of nanoparticle assemblies as they move through different magnetic regimes Nuclear techniques provide another form of probe for the local magnetic order in solids When applied to magnetic nanoparticle systems, information on the magnetic modifications at a magnetic surface can be established as can the effects of interparticle interactions One such technique is M€ ossbauer spectroscopy, and this has been applied to many Fe-based nanoparticle systems Temperature-dependent measurements provide a sensitive probe of magnetic and SPM effects in these low-dimensional systems It has been seen to be particularly useful for the study of magnetic structures at the surface of nanoparticles M€ ossbauer spectroscopy has also been extensively used to identify the oxide species which frequently form of metallic Fe and Fe oxide nanoparticles Neutron scattering is another nuclear technique which has been broadly used as a research tool for investigating nanoparticles and magnetic nanoparticle assemblies This for the most part concerns the scattering at low angles from the incident neutron beam Such small-angle neutron scattering (SANS) has become a well-established technique in the study of solids and biological samples Here we consider how it can be applied to provide information regarding the size and distribution of nanoparticles in an ensemble Indeed, information regarding the size and shape of samples can be inferred from scattered intensity distributions Using polarized neutrons allows magnetic information to be gleaned, which, as in the case of M€ ossbauer spectroscopy, provides information on the surface of the magnetic particle and with care can be used to establish the spin distribution or surface anisotropy of magnetic nanoparticles Interparticle interactions Collective Effects in Assemblies of Magnetic Nanaparticles will also affect the magnetic scattering and thus SANS can also provide information of magnetic interactions between the particles, where studies are frequently performed as a function of particle concentration Application of a magnetic field to the sample is also used, where in systems of magnetic nanoparticles dispersed in a solvent, or ferrofluid, the interaction between the magnetic moments of the particles produces a spatial ordering of the assembly Core–shell models of magnetic nanoparticles can also be established using a combination of SANS and polarized SANS measurements, with and without applied magnetic fields In the following, we focus on some theoretical aspects related to the treatment of assemblies of magnetic nanoparticles This will discuss the energy considerations for an ensemble of ferromagnetic nanoparticles, where the individual particle energy is considered as well as the additional energy contribution which arises from interparticle (dipolar) interactions This then allows the equilibrium state of the system to be evaluated and the magnetization and susceptibility properties to be obtained These considerations are followed by a general discussion of dynamic magnetic properties and the AC susceptibility response of an assembly of weakly interacting ferromagnetic nanoparticles Section aims to provide a brief overview of experimental studies on magnetic nanoparticle assemblies For each of the methods discussed, we will give a short general introduction to the method, where appropriate We will cover both static and dynamic measurement techniques MAGNETIC NANOPARTICLE ASSEMBLIES: THEORETICAL ASPECTS We have recently provided simple expressions for the magnetization and susceptibility, both in equilibrium and out of equilibrium, which take account of temperature, applied field, intrinsic properties, as well as (weak) DI [11, 12, 21, 22, 30–35] However, this has been done at the price of a few simplifying assumptions, either related with the particles themselves or with the embedding assembly In particular, the study of the effect of DI, which is based on perturbation theory, applies only to a dilute assembly with an interparticle separation thrice the mean diameter of the particles In some cases, we only considered monodisperse assemblies with oriented anisotropy axes For the calculation of the particle’s relaxation time, we only consider weak fields, small core and surface anisotropies A brief account of these works is D.S Schmool and H Kachkachi given in the following sections For the study of interplay between surfacedominated intrinsic properties and DI-dominated collective behavior, we model a many-spin nanoparticle according to the effective-one-spin problem (EOSP) proposed and studied in Refs [34–39] The EOSP model is a better approximation than the OSP model in that it accounts for the intrinsic properties of the nanoparticle, such as the underlying lattice, size, and energy parameters (exchange and anisotropy), via an effective energy potential In the simplest case, the latter contains a quadratic and a quartic contributions in the components of the particle’s net magnetic moment These two contributions should not be confused with the core and surface anisotropy contributions In fact, the effective model is a result of a competition between several contributions to the energy, namely the spin–spin exchange interaction inside the nanoparticle, the on-site anisotropy attributed to the spins in the core and on the surface The outcome of the various competitive effects is an effective model for the net magnetic moment m of the nanoparticle with a potential energy that contains terms with increasing order in its components mα, α ¼ x, y, z The coefficients of these terms are functions of the atomic physical parameters, such as the constant of the on-site anisotropies and exchange coupling, together with those pertaining to the underlying crystal structure In the following section, we will give a brief account of these theoretical developments, related to the intrinsic, as well as collective features of the nanoparticles We will also discuss an excerpt of the main results they lead to, for the magnetization and susceptibility 2.1 Model We will illustrate our theoretical developments in the simplest situation of a monodisperse assembly and oriented anisotropy More general situations of polydisperse assemblies, with both oriented and random anisotropy, can be found in the cited works, e.g., in Ref [31] We commence with a monodisperse assembly of N ferromagnetic nanoparticles carrying each a magnetic moment mi ¼ mi si , i ¼ 1,…, N of magnitude m and direction si, with jsij ¼ Each magnetic moment has a uniaxial easy axis e aligned along the same z-direction The energy of a magnetic moment mi interacting with the whole assembly, and with a (uniform) magnetic field H ¼ Heh, reads (after multiplying by À β À1/kBT) ð0Þ E i ¼ E i + E DI i , (1) Collective Effects in Assemblies of Magnetic Nanaparticles ð0Þ where the first contribution E i is the energy of the free (noninteracting) nanocluster at site i, comprising the Zeeman energy and the anisotropy contribution, i.e 0ị E i ẳ xi si Á eh + Aðsi Þ, (2) where Aðsi Þ is a function that depends on the anisotropy model and is given by > OSP < σ i ðsi Á ei Þ , ! Aðsi Þ ¼ (3) ζ > EOSP: : σ i ðsi Á ei Þ À s4i, x + s4i, y + s4i, z , The second term in Eq (1) represents the DI between nanoclusters, which can be written as X E DI si Á Dij Á sj i ¼ξ (4) j
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