Handbook of Magnetic Materials, Volume 18

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1 3

VOLUME EIGHTEEN

5 7

HANDBOOK OF MAGNETIC MATERIALS

9 11 13 15 17 19 21 23 25

Editor

K.H.J. Buschow University of Amsterdam, Van der Waals-Zeeman Institute, Amsterdam, The Netherlands

27 29 31 33 35 37 39 41 43 44

Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sidney Tokyo North-Holland is an imprint of Elsevier

North-Holland is an imprint of Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands Linacre House, Jordan Hill, Oxford OX2 8DP, UK First edition 2009 Copyright r 2009 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://www.elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher 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. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-08-054814-2 ISSN: 1567-2719 For information on all North-Holland publications visit our website at elsevierdirect.com Printed and bound in Great Britain 09 10 11 12 13 10 9 8 7 6 5 4 3 2 1

PREFACE

TO

VOLUME 18

The Handbook series Magnetic Materials is a continuation of the Handbook series Ferromagnetic Materials. When Peter Wohlfarth started the latter series, his original aim was to combine new developments in magnetism with the achievements of earlier compilations of monographs, producing a worthy successor to Bozorth’s classical and monumental book Ferromagnetism. This is the main reason that Ferromagnetic Materials was initially chosen as title for the Handbook series, although the latter aimed at giving a more complete cross-section of magnetism than Bozorth’s book. In the last few decades magnetism has seen an enormous expansion into a variety of different areas of research, comprising the magnetism of several classes of novel materials that share with truly ferromagnetic materials only the presence of magnetic moments. For this reason, the Editor and Publisher of this Handbook series have carefully reconsidered the title of the Handbook series and changed it into Magnetic Materials. It is with much pleasure that I can introduce to you now Volume 18 of this Handbook series. In Chapter 1 of Volume 18 of this Handbook, a review of the filled skutterudites is given. The skutterudites are derived from a class of wellknown compounds characterized by the chemical formula TX3 where the transition metal T ¼ Co, Rh, Ir and the pnictogen X ¼ P, As, Sb. These materials have attracted much interest during the past three decades due to the wide variety of their electronic and magnetic phenomena, including their promising thermoelectric properties. The filled skutterudites, described in Chapter 1, are structurally closely related to the normal skutterudites and have the chemical formula MT4X12. The ‘‘filler’’ atom M comprises the elements La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, U, and Th, the transition metal T mainly including the elements Fe, Ru, and Os. The interest in the filled skutterudites originates from the fact that they display an immense variety of correlated electron phenomena. These phenomena include spin fluctuations, itinerant electron ferromagnetism, local moment ferromagnetism, antiferromagnetism, heavy-fermion behavior, non-Fermi-liquid behavior, Kondo behavior and also conventional BCS superconductivity, unconventional superconductivity, and even hybridization gap semiconducting behavior. Needless to say that the large reservoir of structurally similar compounds that can be obtained by varying the components M, T, and X has provided an almost ideal testing ground for many theoretical models. Many of these compounds do not form easily and in order to obtain single-phase materials considerable experimental v

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Preface to Volume 18

expertise is required. For this reason the authors of Chapter 1 have included a section dealing with the synthesis of the filled skutterudites. Chapter 2 of this Handbook Volume deals with spin dynamics in nanometric magnetic systems. It is emphasized that ferromagnetic resonance (FMR) is a powerful experimental tool in the study of the magnetic properties of ferromagnetic materials and can be applied to the entire range of materials, from bulk ferromagnetic and ferrimagnetic crystals to magnetic thin films and multilayers. More recently, it has also been used to characterize nanogranular and nanoparticle systems. FMR is an excellent method for the evaluation of magnetic properties in low-dimensional structures or nanometric systems indeed. The FMR technique provides a simple method for measuring the effective magnetic field inherent in a given spin system. In this chapter the author first discusses the basic elements of FMR theory, in particular those that are relevant for a general understanding when this technique is applied to nanometric systems. An outline is given of the main quantities that are necessary to deal with nanometric systems, by considering the various components which make up the free energy of the magnetic systems, and hence its effective field. Specific cases of surface anisotropy and interparticle interactions are discussed in detail since they are of particular importance in nanometric systems. The author stresses how mainly in the last decade a number of novel developments have taken place in experimental techniques that have been used to measure magnetization dynamics. These developments include: pump-probe or femtosecond spectroscopy, pulse generation, scanning probe (cantilever) FMR, network analyzer FMR, bolometric detection of FMR, and high-frequency electrical measurements of magnetodynamics. Many of these techniques are easily adaptable to nanometric systems or were acrually developed for the purpose of measuring magnetization dynamics in nanometric structures. These techniques are mainly based on the delivery of the high-frequency excitation signal to the sample, usually through a microstripline upon which the sample is located. Much of these advances are a direct consequence of the need to study ever smaller sized magnetic structures and their temporal response, with particular emphasis on ultrafast dynamics as involved in modern applications in the fields of telecommunication and data storage. A discussion of some of these techniques is included in this chapter. Hand in hand with the development of novel techniques, very extensive numerical calculations have been performed on nanometric systems and for obvious reasons the author has outlined some of the main approaches that have been used. For instance, there has been an enormous research effort based on the dynamical effects of spin current on the magnetization state in nanostructured elements and multilayers. Here the effect of transfer of spin angular momentum, and spin and charge accumulation effects play an important role. Because of the importance of spin torque transfer effects in


vii

data storage systems, the author has included in his chapter a discussion of some of the main details of these effects from both the experimental and the theoretical points of view. In the last chapter of this volume of the Handbook a review is given of the various types of magnetic sensors nowadays available, including a discussion of the materials and principles that have been used for their construction. Sensors play an ever important role in our daily life, in the personal as well in the industrial sphere. Of all sensors, magnetic sensors take the most prominent position. The term magnetic sensor is actually used in at least two different ways. The most common type of magnetic sensor refers to cases where one wishes to probe magnetic fields of various origins, from the Earth’s magnetic field to the stray fields produced by bits of magnetically stored information. In these cases the materials of which the sensor is composed need not necessarily be magnetic. By contrast, the magnetic sensors of the second type include sensors that use magnetic materials or magnetic principles. They may be exploited for measuring either magnetic or nonmagnetic quantities. Usually the first meaning of the term magnetic sensor is used without explicitly specifying that it serves for sensing or measuring a magnetic field. By far most of the produced magnetic sensors are devices based on the Hall effect. These are semiconductor sensors which are cheap and can be made of small size, but their resolution and stability is very limited. If higher accuracy is desired, soft magnetic materials are generally employed, either as yoke, as field concentrator, or as functional element. The fast development of ferromagnetic magnetoresistors used in modern magnetic reading heads has led to the appearance of devices based on similar principles also on the sensor market. Although such sensors based on magnetic thin films or multilayers are much more sensitive than Hall sensors, their performance is still limited. Sensors with cores, yokes, or field concentrators made of bulk magnetic material are more sensitive and stable than thin-film sensors. As outlined by the author in this chapter, the most critical parameter for magnetic sensors is hardly the sensitivity, because amplification is relatively inexpensive. Nonlinearity and temperature dependence of sensitivity is equally important but it can often be suppressed by a feedback. Noise matters, but usually the most serious problems of sensors containing magnetic material are remanence, cross-field sensitivity, and temperature stability of offset. All these items are addressed by the author in his chapter, including the general trend of miniaturization and integration of electronic elements. A specific comparison is made between traditional miniature fluxgates using wire cores based on the longitudinal fluxgate effect and sensors using the transverse fluxgate effect or sensors based on the giant magnetoimpedance effect. Also an overview of magnetic sensors for mechanical quantities is presented, with special emphasis given on torque sensors.

viii


Volume 18 of the Handbook on the Properties of Magnetic Materials, as the preceding volumes, has a dual purpose. As a textbook it is intended to be of assistance to those who wish to be introduced to a given topic in the field of magnetism without the need to read the vast amount of literature published. As a work of reference it is intended for scientists active in magnetism research. To this dual purpose, Volume 18 of the Handbook is composed of topical review articles written by leading authorities. In each of these articles an extensive description is given in graphical as well as in tabular form, much emphasis being placed on the discussion of the experimental material in the framework of physics, chemistry, and material science. The task to provide the readership with novel trends, and achievements in magnetism would have been extremely difficult without the professionalism of the North Holland Physics Division of Elsevier Science B.V. K.H.J. Buschow Van der Waals-Zeeman Institute, University of Amsterdam.

CONTRIBUTORS Yuji Aoki Department of Physics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan K.H.J. Buschow University of Amsterdam, Van der Waals-Zeeman Institute, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands Hisatomo Harima Department of Physics, Kobe University, Kobe 657-8501, Japan Pavel Ripka Czech Technical University, Department of Measurement, Technicka 2, 166 27 Prague 6, Czech Republic Hideyuki Sato Department of Physics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan David Schmool Departamento de Fı´sica and IFIMUP 2 IN, Faculdade de Cieˆncias, Universidade do Porto, Rua Campo Alegre 687, 4169-007 Porto, Portugal Hitoshi Sugawara Faculty of Integrated Arts and Science, University of Tokushima, Tokushima 7708502, Japan Karel Za´ve˘ta Institute of Physics, Academy of Sciences of the Czech Republic, v. v. i., Na Slovance 2, 182 21 Prague, Czech Republic

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CHAPTER ONE

Magnetic Properties of Filled Skutterudites H. Sato1,, H. Sugawara2, Y. Aoki1 and H. Harima3 Contents 1. Introduction 2. Synthesis of Filled Skutterudites 2.1. Synthesis under ambient pressure 2.2. Synthesis under high pressures 3. Background that Realizes Unique Behaviors in the Filled Skutterudites 3.1. Characteristics of the band structure and strong c-f hybridization 3.2. The interaction among multipoles via a main conduction band 3.3. Crystalline electric field effect 4. Filled Skutterudite with Rare Earth or Actinide Elements 4.1. La-based filled skutterudites 4.2. Ce-based filled skutterudites 4.3. Pr-based filled skutterudites 4.4. Nd-based filled skutterudites 4.5. Sm-based filled skutterudites 4.6. Eu-based filled skutterudites 4.7. Gd-based filled skutterudites 4.8. Tb-, Dy-, Ho-, Er-, and Tm-based filled skutterudites 4.9. Yb-based filled skutterudites 4.10.Filled skutterudites with actinoid ions 5. Other Filled Skutterudites 5.1. AT4Sb12 (A ¼ alkaline and alkaline earth; T ¼ Fe, Ru, Os)

1 3 3 5 7 7 10 12 14 14 18 25 50 53 63 65 65 68 69 71 71

Corresponding author. Tel.: + 81 42 677 2507

E-mail address: [email protected] 1 2 3

Department of Physics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan Faculty of Integrated Arts and Science, University of Tokushima, Tokushima 770-8502, Japan Department of Physics, Kobe University, Kobe 657-8501, Japan

Handbook of Magnetic Materials, Volume 18 ISSN 1567-2719, DOI 10.1016/S1567-2719(09)01801-0

r 2009 Elsevier B.V. All rights reserved.

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H. Sato et al.

5.2. Ge cage-forming filled skutterudite APt4Ge12 (A ¼ Sr, Ba, La, Ce, Pr, Nd, Eu, U, Th) 5.3. YT4P12 (T ¼ Fe, Ru, Os) and I0.9Rh4Sb12 6. Summary Acknowledgments References Appendix A

74 78 78 79 79 92

1. Introduction The word ‘‘skutterudites’’ was first used by Haidinger (1845) as the name of a new mineral with chemical formula (Co, Ni, Fe)As3 found in a cobalt mining at ‘‘Skutterud’’, Modum, Norway. Nowadays, ‘‘skutterudite’’ is used as a name for a series of cubic compounds with chemical formula TX3 (T ¼ Co, Ni, Rh, etc.; X ¼ pnictogen). The crystal structure shown in Fig. 1.1a was first determined by Oftedal (1927). The main subject of this section are the filled skutterudites AT4X12 which were synthesized by Jeitschko and Braun (1977) for the first time in the course to search for new ternary metal-P compounds. In the conventional cubic unit

Figure 1.1 (a) Crystal structure of binary skutterudite made of eight TX3 formulae. In a simple cubic sublattice of the transition metal ions (large circle), nearly square rings of pnictogen (small circle) occupy some of the center voids. The pnictogen rings result from the pnictogen--pnictogen chemical bonding. The large dotted circle shows voids of the pnictogen ring. On filling the voids with the ion A, the filled skutterudite structure AT4X12 is obtained. (b) Another type of drawing where A is located at the center of a distorted icosahedron cage formed by 12 pnictogen atoms X (small sphere). The transition metal ions T (large sphere) are located between the cages, forming a simple cubic sublattice.

Magnetic Properties of Filled Skutterudites

3

cell made of eight cubes shown in Fig. 1.1a, there are two large empty spaces around dotted circles surrounded by 12 pnictogens called hereafter as a cage, and the filled skutterudite is the filled up version of the cage by an element A (dotted circles in Fig. 1.1a) that could be rare earth, actinide, alkaline, or alkaline earth elements. This system has been intensively investigated from two viewpoints: as scientific target materials to investigate their novel attractive features and as a potential candidate for thermoelectric material of the next generation. For the latter viewpoint, a nice review article by Sales (2003) has already been reported. Therefore, in this article, we are going to focus on the former subject, putting emphasis on the electronic structures and magnetic properties related with the 4f- and d-electrons. For the detailed explanation of the thermoelectric characteristics, the crystal chemistry, and the more intense discussion on the crystal stability, please refer to the review work of Sales. The crystal structure of the filled skutterudite AT4X12 is shown in Fig. 1.1b. It belongs to the space group Im3¯ (T5h , x204) and the atomic positions of A, transition metal T, and pnictogen X ions are (0, 0, 0), (1/4, 1/4, 1/4), and (0, u, v), respectively. The values of the parameters u and v depend on the combination of A, T, and X, at around u ¼ 0.33520.360 and v ¼ 0.1420.16 (Kaiser and Jeitschko, 1999; Sales, 2003; Uher, 2001). The distorted icosahedron cage is formed by 12 pnictogen atoms, and a rare earth ion is located at the center of the cage. The transition metal ions T are located between the cages, forming a simple cubic sublattice. The size of the icosahedron cage formed by the pnictogen ions increases with increasing ionic radius of pnictogen as P-As-Sb. The lattice constant shown in Fig. 1.2 reflects this feature. The similarities between the filled skutterudite structure and the perovskite structures (for example, SrTiO3) have been pointed out by Jeitschko and Braun (1977) (refer to Fig. 1.2 in Sales, 2003). From such a viewpoint, the characteristic feature for the filled skutterudite structure is the tilting of the octahedra, resulting in the formation of the enlarged voids, in which rare earth ions are accommodated.

2. Synthesis of Filled Skutterudites The preparation of filled skutterudite samples is not so easy a task, especially when one needs high-quality single crystals with a large enough size for particular experiments. That is because this family of compounds itself is usually incongruent, and constituent elements include both high melting temperature materials and high vapor pressure ones. In addition, some elements are toxic. The most widely used way to grow single crystals is the flux method under ambient pressure (Braun and Jeitschko, 1980a, 1980b; Jeitschko and Braun 1977). The method has been successfully applied to

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H. Sato et al.

Figure 1.2 Lattice parameter of the filled skutterudites for compounds with R ¼ LaBEu. The trend reflects that the size of the X12 icosahedron cage increases as the pnictogen is changed as P-As-Sb.

P- and Sb-based filled skutterudites containing light rare earth elements, although there is a limit on the size of grown samples. However, for the filled skutterudites containing As and/or heavy rare earth elements, the synthesis under high pressures was practically proven to be useful (Shirotani et al., 1996). In the following, the basic part of crystal growth methods is explained, under ambient pressure and under high pressures in separate subsections.

2.1. Synthesis under ambient pressure From the viewpoint of sample quality for various physical property measurements, the flux method is the best way to grow single crystals. For example, single crystals of LaFe4P12 with residual resistivity ratio, RRR


5

( ¼ rRT/r0, where rRT is the room temperature resistivity and r0 the residual resistivity), greater than 1000 and those of CeRu4Sb12 with the size larger than 5 mm 5 mm 5 mm have been obtained (Sugawara et al., 2000, 2002a), in which de Haas2van Alphen effect has been observed. The flux method under ambient pressure is the most convenient way without any special apparatus. Basically, only a high-vacuum pump and a resistance heater furnace are necessary, and that is a laboratory-friendly method for the high-quality single crystal growth. In this section, the method of crystal growth under ambient pressure is briefly described. For a complete review on the general principle and the growth techniques by the flux method, the following review papers are recommended: Canfield and Fisk (1992) and Fisk and Remeika (1989). The method we applied is basically the same as was reported by Jeitschko and Braun (1977). However, we have modified various conditions, in the process to search for the best condition to grow larger and higher quality samples, which are described below. The raw materials, better than 3N (99.9% pure)-rare earth (ingot), 4N-Fe, Ru, Os (powder), 6N-P (chunk), and 5N-Sn (shot), were encapsulated in a quartz tube with a molar ratio R:T:P:Sn ¼ 1:4:20:40. Sometimes, the reaction between the constituent materials and quartz is a serious problem, for example, Os attacks quartz at high temperatures above 9501C. For such a case, Al2O3 can be used as a crucible to avoid direct contact between the raw materials and quartz. The carbon coating on inside wall of quartz tube is another possible solution to avoid the reaction (Bauer et al., 2001c). The ampoule is heated up to 10501C in a furnace, kept at this temperature for 100 hours, cooled down to 6501C at the rate of 11C/hour, and then furnace is cooled down to room temperature. The estimation of minimum Sn ratio needs special care in order to avoid explosion of the ampoule during the heating process. Actually, when we tried crystal growth in the ratio R:T:P:Sn ¼ 1:4:20:20, the ampoule exploded at around 8501C. The excess Sn-flux was dissolved in concentrated hydrochloric acid (HCl), since HCl does not practically attack filled skutterudite. For the RT4Sb12 system, one can use Sb itself as flux. In the Sb self-flux method, the growth in the molar ratio R:T:Sb ¼ 1:4:20 is usually successful. The crystal growth process is basically the same as in the previous report (Braun and Jeitschko, 1980b). The crystal growth of RRu4Sb12 and ROs4Sb12 is easier compared to that of RFe4Sb12. After growing crystals, the excess Sb-flux was dissolved in aqua regia. In this process, the soak interval in aqua regia requires special care, since aqua regia also attacks the sample. Typically, a few hours are enough for removing Sbflux. In the case of RFe4Sb12, single crystal growth is promoted by the troublesome solid-state reaction, since this compound is not stable above B7001C (Morelli and Meisner, 1995), which is far below the melting temperature. Moreover, this compound is easily attacked by aqua regia,

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which makes it difficult to extract grown single crystals from Sb-flux. Single crystals might be mechanically isolated from Sb-flux (Mori et al., 2007). In addition, the quality of RFe4Sb12 crystals is generally worse compared to other Sb-based skutterudites, partly due to the incomplete filling of R sites, typically 70290% (Butch et al., 2005; Ikeno et al., 2008). Better crystals with almost fully occupied R sites can be obtained using the high-pressure synthesis described in Section 2.2. Recently, Ge-based filled skutterudites MPt4Ge12 have been synthesized by Bauer et al. (2007) and Gumeniuk et al. (2008a) in polyscrystalline form, almost simultaneously. The synthesizing process is basically the same as that used for polycrystalline RFe4Sb12 (Danebrock et al., 1996; Morelli and Meisner, 1995; Toda et al., 2008). Fortunately in this case, we can use an arc furnace in the first step to mix the constituent materials because of relatively low vapor pressure of the raw materials. For M ¼ Sr or Ba, excess amount of M in the ratio of M:Pt:Ge ¼ 1.2:4:12 is prepared to compensate the loss of M due to evaporation (Sugawara et al., unpublished data).

2.2. Synthesis under high pressures Single crystals of both P- and Sb-based skutterudites could be grown by the flux method at ambient pressure, at least for light rare earth components. However, the method does not work for the systems containing heavy rare earth elements except for the divalent or intermediate valence Eu and Yb with a larger ionic radius. Therefore, samples for such systems are hardly obtainable even in polycrystalline form. To overcome the difficulty, Shirotani et al. (2003b, 2005a) applied the high-pressure technique to obtain polycrystalline samples of heavy rare earth-filled skutterudites. They have succeeded in synthesizing RFe4P12 for all the rare earth elements by wedge-type cubic anvil pressure cell up to B4 GPa. For As-based system, due to the high vapor pressure of As, the highpressure synthesis has been the best way to prepare samples for experiments, since no suitable flux had been found. Recently, Henkie et al. (2008) reported a clever way to overcome the problem utilizing As-rich Cd as flux and an enhanced pressure technique without any apparatus to reach really high pressure of BGPa. In this technique, elemental components of a skutterudite along with As-rich Cd flux are sealed in a quartz ampoule in the same manner as in ordinary flux method. The ampoule is loaded in a welded end of a steel tube within a furnace. From the other end of the tube, the pressure inside the tube can be controlled to compensate an estimated inner ampoule pressure up to 30 atm at 8501C. By this technique, they have succeeded in growing single crystals of As-based filled skutterudites containing Fe, Rs, Os, and light rare earth elements. For the Sb-based skutterudites, as mentioned above, there remains a serious problem concerning the vacancy of the guest ions; the filling


7

fraction is usually less than 1 and strongly depends on the synthesizing condition. In some cases, the filling fraction changes the magnetic ground state of the system (Ikeno et al., 2007; Tanaka et al., 2007). Furthermore, in recent works on several Ce-based skutterudites, crucial disagreement was found in some of the observed physical properties between two groups; that is, on CeRu4As12, Baumbach et al. (2008) reported a semimetallic single crystal exhibiting non-Fermi-liquid (NFL) behaviors grown at B1.5 MPa, while Sekine et al. (2007) reported a Kondo semiconducting polycrystalline sample synthesized at 4 GPa. In the Ce-based filled skutterudites with strong c-f hybridization, carrier discompensation caused by small amount of Ce vacancies might cause such a drastic change in the electrical properties. On the other hand, in the polycrystalline samples synthesized under high pressures, one cannot rule out the possible grain boundary contribution to the electrical properties. In order to settle the problem, single crystals grown under high pressure are essential. Recently, single crystals of the filled skutterudites have been successfully grown under pressure 425 GPa for the first time (Tatsuoka et al., 2008). The constituent elements R, T, and As in the ratio of 1:4:20230 are put into a cylindrical boron nitride crucible with an inner space of 6.5 mmf 5.5 mm which was surrounded by a cylindrical graphite tube heater. These along with a pair of thermocouples are set into a cylindrical hole in a cube (18 mm3) made of pyrophyllite, and subjected to high pressures using a cubic multianvil-type press shown in Fig. 1.3 (Piglet II, Oh!sawa-system). After being pressurized up to a desired pressure (325 GPa) at room temperature, the crucible is heated to Tmax and kept

Figure 1.3 (a) Cubic anvil cell used for flux growth of filled skutterudite single crystals under high pressure up to 6 GPa. Photograph was taken when a top anvil was absent. (b) Starting materials are put into 6 mm inner diameter BN crucible surrounded by a carbon heater tube within a pyrophyllite cube (18 mm 18 mm 18 mm).

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Figure 1.4 LaFe4As12 single crystals grown by the As self-flux method under 4 GPa. Single crystals were taken out by As distillation.

for some interval Dt (0.523 hours) depending on material, and quickly decreased to a starting temperature Tst, then cooled down slowly to Tsp with a rate of dT/dt (0.326 1C/hour), followed by furnace cooling down to room temperature. The pressure is kept constant until the temperature is close to the room temperature. As an example for RFe4As12 (R ¼ La, Ce, Pr), the parameters, Dt ¼ 1 hour, Tmax ¼ 11001C, Tsta ¼ 9501C, dT/dt ¼ 6 1C/hour, and Tstp ¼ 8001C, lead to high-quality single crystals with a maximum length over 1.5 mm and the RRR over 250 (Fig. 1.4), on which the de Haas-van Alphen (dHvA) oscillations have been successfully observed (Namiki et al., unpublished data).

3. Background that Realizes Unique Behaviors in the Filled Skutterudites 3.1. Characteristics of the band structure and strong c-f hybridization The crystal structure of the filled skutterudite AT4X12 (A ¼ R: rare earths in this section) is shown in Fig. 1.1. The molecular orbitals from the R-f, T4-d, and X12-p electrons are classified in Table 1.1. Thirty-six bands from X12-p hybridize with 20 T4-d bands, resulting in the 36 bonding and 40 nonbonding/antibonding bands, from a simple tight-bonding calculation. Therefore, the unfilled skutterudite compound T4X12 (T ¼ Co, Rh, Ir;

9


Table 1.1 The single-valued irreducible representations and the number of the molecular orbitals in a filled skutterudite RT4X12. Bethe symbols in Th symmetry are used in this article. Otherwise, we present them with the relevant point symmetry, like G5 (Oh). The parity ( + or ) is also dropped sometimes for simplicity. T4-d

X12-p

Mulliken

Bethe

Degeneracy

R-f

ag au eg eu tg tu

Gþ 1 G 1 Gþ 23 G 23 Gþ 4 G 4

1 1 2 2 3 3

0 1 0 0 0 2

1 0 2 0 5 0

2 1 2 1 4 5

7

20

36

Total

X ¼ P, As, Sb) becomes an insulator or a semimetal (Takegahara and Harima, 2003a). The filled skutterudites, due to the lack of one d-electron in T site, should be one-hole system with R3 + , or an insulator or a semimetal with R4 + . The latter case corresponds to Ce- or Th-filled skutterudites (Takegahara and Harima, 2003b). Based on the full-potential linear augmented plane-wave (FLAPW) band-structure calculations for La-filled skutterudites, the conduction bands consist of two or three bands. In all cases, the au band from X12-p crosses the Fermi level, so it is the main conduction band. The ag band, mainly from T4-d electrons, forms a closed hole Fermi surface (FS) around the G point in LaFe4P12 (Sugawara et al., 2000), and the larger contribution from T4-d electrons in heavier pnictogen is expected (Harima and Takegahara, 2003a). Only in the case of LaRu4P12, eu bands appear at the Fermi level with the au band. It is noticed that the au band hybridizes well with G 5 (G 7 in Oh) state of f-electrons, while the ag band never hybridizes with any f-electron orbital at the G point. This situation is typically realized in CeOs4Sb12, as will be shown later. The main conduction band consists of the unique X12-p molecular orbital, with the symmetry au in Th or xyz, as shown in Fig. 1.5. When one applies a simple tight-binding model to this molecular orbital, energy E is obtained, ky a kx a kz a cos E ¼ 0 þ 8t cos cos , 2 2 2

(1)

where t (W0) is the two-center integral (for the X12-p molecular orbitals), e0 the energy origin of the molecular orbital, and a the lattice constant. The above energy function is of the same form as the simple s-band for BCC lattice (Slater and Koster, 1954), though t ¼ (sss) (o0) in the s-band case.

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Figure 1.5 X12-p molecular orbital of the main conduction band with symmetry au or xyz. Black and white color means a sign of p orbital. A p orbital is located at each site of the X12 icosahedron, in which a rare earth ion is located in the center (Harima and Takegahara, 2003b).

When one electron occupies the band, the Fermi level is located at E ¼ 0; then the FS is found to be exactly a cube. In the extended zone scheme, the FS is connected along the edges (the D-axes in the BZ), to form the edgeshared cubes, that is, three-dimensional checkerboard. Because |kx| ¼ p/a, |ky| ¼ p/a, or |kz| ¼ p/a gives E ¼ 0, the FS exhibits perfect nesting with q ¼ (1, 0, 0)(2p/a) (hereafter presented just as q ¼ (1, 0, 0) in this chapter) and there is an infinite peak at E ¼ 0 in the density of states. This nesting property persists even in the real system, as revealed by the band-structure calculation for LaFe4P12 (Sugawara et al., 2000). Experimentally, the superstructure with q ¼ (1, 0, 0) has been observed in PrFe4P12 (Iwasa et al., 2002) and PrRu4P12 (Lee et al., 2001) suggesting the structural phase transition. The detailed features arising from the nesting property will be discussed later. It is this nesting property that is the most unique and important characteristic of the filled skutterudites. The distance between R and X is 3.08 Å for CeFe4P12 and 3.48 Å for CeOs4Sb12, larger than 2.97 Å for CeP and 3.21 Å for CeSb. Therefore, the hybridization in one pair of R and X, that is, two-center integral, is expected to be not too large. However, each rare earth ion has 12 nearest


11

neighbor pnictogens, so the total hybridization could become quite large. Moreover, in the filled skutterudites, each pnictogen belongs to one rare earth ion, and the mixing matrix elements of R-4f and X12-p electrons are independent of k in the BZ, that is, RX12 molecule behaves like one huge ion. Without the spin2orbit coupling, f orbitals are split into one singlet (au) and two triplets (tu), as shown in Table 1.1. au corresponds to G 5 in the double-valued irreducible representations; that is, with the spin2orbit interaction, tu splits into G 5 and G67 . au (G5 ) conduction band hybridizes well with the G5 (G6 or G7 in Oh) of f-electrons. Therefore, in many of CeT4X12, Ce-4f electrons hybridize well with the au band, and they show semiconducting behavior. In CeRu4Sb12 and CeOs4Sb12, which are not regarded as simple semiconductors, a very large hybridization gap can be found between G 5 bands. The band structure calculated for CeOs4Sb12 is shown in Fig. 1.6, where the ag conduction band prevents a real gap opened at the Fermi level. In CeOs4Sb12, orbital-dependent hybridization is observed in the photoemission spectroscopy (PES) experiments (Matsunami et al., 2009).

3.2. The interaction among multipoles via a main conduction band The multipoles arising from 4f-electrons, as shown in Fig. 1.7, are considered to understand the variety of physical properties in the filled

Figure 1.6 Band structure and density of states for CeOs4Sb12. The main conduction band (at the C point G 5 in the double-valued irreducible representations) hybridizes well with the G 5 f-electrons (Harima and Takegahara, 2003b).

12

H. Sato et al.

(a)

(b)

(c)

(d)

+ − − + Figure 1.7 Schematic shape of multipoles: (a) charge (electric monopole), (b) spin (magnetic dipole), (c) electric quadrupole, and (d) magnetic octupole. (Reproduced with permission from Kusunose (2007)).

skutterudites. For example, the antiferro-quadrupole ordering is realized in PrOs4Sb12, and a higher multipole ordering is discussed in PrFe4P12 PrRu4P12 and SmRu4P12. It is discussed that such multipoles couple with each other via the main conduction band in the filled skutterudites (Harima, 2008). We consider the wave function of the main conduction band for skutterudites, jxyz(|r|). Originally, it consists of p-electrons on pnictogen site, but the molecular orbital has the symmetry of au in Th or xyz, which corresponds to l ¼ 3. The energy origin e0 of the main conduction band is written as Z

0 ¼

cxyz ðrÞV ðrÞcxyz ðrÞ dV ¼ hxyzjV jxyzi.

(2)

If the energy origins e0 differ from the corner to the center in the BCC lattice, the gap easily opens in the main conduction band, as A0 aB0 ! hxyzjV A jxyziahxyzjV B jxyzi.

(3)

Here A and B mean the corner site and the center site in the BCC lattice. V can be expanded in the local symmetry as V ðrÞ ¼ V 0 ðrÞ þ V 4 ðrÞ þ V c6 ðrÞ þ V t6 ðrÞ.

(4)

The last term is typical in the Th symmetry as discussed in Section 3.1 (Takegahara et al., 2001), but not significant. Simple consideration reveals that V6 is active with the third-order wave function jxyz(|r|). The main conduction band in the filled skutterudites couples with up to V6 term potential. Such higher order components in the potential always exist as the total symmetric representations from the surrounding ions in a crystal. Next, let us investigate possible origins of the difference of the higher order potential DV4 + DV6. V4 and V6 can be obtained not only from the surrounding ions, but also from the anisotropy of the electron charge distribution, including multipoles of the f-electrons. Hexadecapole (24 ¼ 16) and hexacontatetrapole (26 ¼ 64), which are allowed anisotropic

13


charge densities formed by f-electrons in Th symmetry, cause V4 and V6, respectively. Rough estimation of /V4 + V6S could help us to imagine the order of the energy gap. Let rf(r) be the f-electron charge density, then we estimate

rxyz ðrÞ rf ðrÞ rf ðrÞrxyz ðrÞ f , xyz f xyz hri r r

(5)

av

where rxyz(r) is the charge density of the main conduction band, |f S the f-electron wave function, and /rSav the distance between f charge and the pnictogen position. The anisotropic part of the right term shows the crystal field for f-electrons from the main conduction band. This term is usually of the order of 100 K. Therefore, we easily expect a several tenth Kelvin of energy gap D from the antiferro-higher order multipole ordering, up to /DV6S, that is, antiferro-hexacontatetrapole ordering. It should be noticed that even though the ordering is not realized, the higher order multipoles interact with the main conduction band in the filled skutterudites RT4X12.

3.3. Crystalline electric field effect The local point symmetry Th(m3) of the rare earth ion is the key considering the crystalline electric field (CEF) effect. As pointed out by Takegahara et al. (2001), this feature, the difference compared to the ordinary cubic (Oh) symmetry, leads to an extra term in the CEF Hamiltonian HCEF. In the Stevens’ operator equivalent methods (Hutchings, 1964), following the notation given by Lea et al. (1962) (Lea, Leask, and Wolf, LLW), it can be expressed as H CEF ¼ Bc4 O4 þ Bc6 Oc6 þ Bt6 Ot6 c t O4 O6 O þ ð1 jxjÞ þy t 6 , ¼W x Fð4Þ Fð6Þ F ð6Þ

ð6Þ

where O4 ¼ O04 þ 5O44 , Oc6 ¼ O06 21O46 , and Ot6 ¼ O26 O66 . The set of the CEF parameters {Bc4 , Bc6 , and Bt6 } or {W, x (jxj 1), and y} identifies the CEF level scheme of the f-electron states in a particular compound. The last term characteristic of the Th site symmetry disappears in the case for the site symmetry O, Td, and Oh (i.e., y ¼ 0), and the wellknown cubic CEF Hamiltonian of LLW results. In the CEF level scheme, this extra term does not lead to further splitting of the degenerate levels but modifies the eigenfunctions and eigenvalues quantitatively. Therefore, this important term cannot be neglected when magnetic properties are discussed using the CEF Hamiltonian of Eq. (6). Since this term includes the sixth-order tensor operators, multiplets with the total angular momentum JW5/2 will be affected in the Russell-Saunders coupling

14

H. Sato et al.

scheme. For example, in the case of Pr3 + ion with y ¼ 0, the J ¼ 4 ground-state multiplet splits into four sublevels; a singlet G1 (Oh), a nonKramers nonmagnetic doublet G3 (Oh), and two magnetic triplets G4 (Oh) and G5 (Oh). When y6¼0, the G4 (Oh) and G5 (Oh) states mix with each ð2Þ other and result in two states named as Gð1Þ 4 and G4 . The non-Kramers doublet (G3 (Oh) for y ¼ 0) is classified into G23. With the increasing value of ð2Þ |y|, the magnetic (more generally, multipolar) properties of the Gð1Þ 4 and G4 states become closer and, in addition, the energy splitting between the two states becomes larger, as demonstrated in Fig. 1.8. In fact, the remarkable experimental observations, for example, in PrFe4P12, PrRu4P12, and PrOs4Sb12, could not be understood without this feature as discussed below. More naively, when nonzero magnetic field is applied along the z-axis of the crystal with the Th site symmetry, the x- and y-axes are no longer equivalent, in marked contrast to the crystals with the O, Td, and Oh site symmetry. One should keep in mind that such an effect appears in the anisotropy of the physical quantities, especially in tensor-type physical quantities (including Hall coefficients and elastic constants). The energy2momentum relation of

Figure 1.8 An example of |y| dependence of the magnetic moment /JzS and the ð2Þ 3+ energy levels of the Gð1Þ 4 and G4 states for the J ¼ 4 ground-state multiplet of Pr ion for x ¼ 0.6 (data from Takegahara et al., 2001).


15

the conducting electrons also possesses the Th symmetry of the crystal structure. This feature can be easily seen in the shapes of the FS topology of filled skutterudites.

4. Filled Skutterudite with Rare Earth or Actinide Elements 4.1. La-based filled skutterudites All the La-based filled skutterudites except LaFe4As12 (Tatsuoka et al., 2008) and LaFe4Sb12 (Viennois et al., 2005) exhibit superconductivity below a superconducting (SC) transition temperature Ts that varies from 0.74 K for LaOs4Sb12 to 10.3 K for LaRu4As12 depending on T and X as shown in Table A1 (note the references in Table A1). The absence of SC transition in the two Fe-based filled skutterudites is probably correlated with the relatively high electronic density of states near Fermi level D(EF) dominated by Fe 3d-electrons (Tatsuoka et al., 2008; Viennois et al., 2005). According to the band-structure calculation by Harima and Takegahara (2003a) based on a FLAPW method, the 3d-band in these compounds is located nearer to EF compared to 4d- and 5d-bands in the Ru- and Osbased filled skutterudites as shown in Fig. 1.9. The D(EF) increases monotonically as X changes from P to As and Sb. The experimental specific

Figure 1.9 Comparison of the density of electronic states in LaFe4X12 (X: P, As, Sb) (Harima and Takegahara, 2003a).

16

H. Sato et al.

heat coefficients (gex) of 57, 170, and 195 mJ/K2 mol for LaFe4X12 (X ¼ P, As, Sb), respectively, are consistent with the variation in D(EF) (Meisner et al., 1984; Tatsuoka et al., 2008; Viennois et al., 2005). LaFe4P12 is an ordinary Pauli paramagnet with wp ¼ (628) 104 emu/mol and exhibits an SC transition at Ts ¼ 4.6 K (Meisner et al., 1984). The high SC transition temperature of a compound containing such high concentration of Fe is consistent with the minor contribution of the 3dband at EF. Actually, the Mo¨ssbauer measurement indicates almost zero magnetic moment (less than 0.01mB) on Fe (Shenoy et al., 1982). The FS along with the cyclotron effective mass has been determined by dHvA experiments (Sugawara et al., 1998, 2000), and the FS topology is well explained by FLAPW band-structure calculation (Harima, 1998). LaFe4As12 is only one La-based filled skutterudite exhibiting ferromagnetism that is realized by the high D(EF) dominated by Fe 3d-electrons (Tatsuoka et al., 2008). The temperature dependence of w1 m and the field dependence of M at selected temperatures are shown in Fig. 1.10. The large Rhodes-Wohlfarth ratio (the ratio of the effective moment deduced from the Curie constant to the saturation moment per magnetic ion) of 32 indicates that LaFe4As12 is the itinerant electron weak ferromagnet (Moriya, 1985; Rhodes-Wohlfarth, 1963). LaFe4Sb12 is an enhanced Pauli paramagnet close to a ferromagnetic (FM) quantum critical point with a spin-fluctuation temperature Tsf ¼ 50715 K. Viennois et al. (2005) reported NFL behaviors in the basic physical properties; that is, the magnetic susceptibility wm(T)BT 1.35, resistivity r(T)BT1.7 and Gruneisen parameter r(T)BT22/3. In the systematic study of MFe4Sb12 (M ¼ Na, K, Ca, Sr, Ba, La, Yb), Schnelle et al. (2008) pointed out that the g value in LaFe4Sb12 under zero magnetic field is almost twice that of the filled skutterudites with monovalent and divalent guest ions, although D(EF) decreases with increasing valence of guest ions. Furthermore, the spin-fluctuation contribution BT3 ln(T/Tsf) is needed to better fit the temperature dependence of specific heat C(T), while the specific heat data for the other La-based skutterudites could be well fitted to a simple model C(T) ¼ gT + bT3 + dT5 composed of only the electronic and the Debye contributions for To7 K. The g value is suppressed linearly with applied magnetic field, the magnitude of which depends on the filling fraction. These facts indicate extraordinarily strong spin fluctuations in this compound, although the character of spin fluctuation, either FM or antiferromagnetic (AFM), is still controversial. At 1.3 K, the magnetization monotonically increases up to 48 T where it nearly saturates to 0.620.7mB (Yamada et al., 2007). A question arises why only LaFe4As12 exhibits FM among the LaFe4X12 series, if the Stoner exchange parameter is not much different between LaFe4As12 and LaFe4Sb12. Both LaFe4P12 with the lowest D(EF) and LaFe4Sb12 with the highest D(EF) are Pauli paramagnets, although the latter is strongly


17

Figure 1.10 (a) Temperature dependence of the inverse of magnetic susceptibility with an inset showing the magnetization versus temperature plot at 0.01 T near TC. (b) Field dependence of magnetization at selected temperatures with an inset showing expanded view in low fields at 2 K (Tatsuoka et al., 2008).

exchange-enhanced. In a similar case on the itinerant electron ferromagnet Ni3Al and the exchange-enhanced paramagnet Ni3Ga, the stronger critical fluctuation in Ni3Ga was theoretically demonstrated as a factor to destabilize the FM state in Ni3Ga (Aguayo et al., 2004). The larger ratio D(EF; LaFe4Sb12)/D(EF; LaFe4As12) ¼ 1.39 (Harima and Takegahara, 2003a) compared to D(EF; Ni3Ga)/D(EF; Ni3Al) ¼ 1.06 indicates a

18

H. Sato et al.

considerably strong SF effect in LaFe4Sb12. In fact, most of the experimental results on LaFe4X12 (X ¼ P, As, Sb) are consistent with such a scenario. Both LaFe4Sb12 and LaFe4As12 exhibit NFL behaviors, which suggest their closeness to the FM quantum critical point. The resistivities in both LaFe4As12 and LaFe4Sb12 exhibit T 5/3-dependence at low temperatures suggestive of the critical fluctuation in the three-dimensional weak ferromagnets (Moriya, 1985), in contrast to the Fermi-liquid behavior of T 2-dependence in LaFe4P12 (Sato et al., 2000). The other La-filled skutterudites exhibit only weak magnetism as naturally expected; however, the study of their basic properties is indispensable to understand the magnetic properties of rare earth filled skutterudites containing 4f-electrons. Therefore, their basic characteristics are briefly mentioned below. LaRu4P12 is important as a reference compound for PrRu4P12 that exhibits an M2I transition below 62 K (Sekine et al., 1997) whose primary order parameter is believed to be a scalar-type multipole of 4f-electrons. The FS determined by the dHvA effect (Saha et al., 2005) has a good nesting condition with q ¼ (1, 0, 0) predicted by the band-structure calculation based on FLAPW method with the local-density approximation (LDA) (Harima, 2000). The observed cyclotron effective masses of (2.6211.8)m0, about twice enhanced compared to the band mass, are consistent with the high SC transition temperature of this material (Shirotani et al., 1997), suggesting strong electron2phonon interactions. Among LaT4P12 (T ¼ Fe, Ru, Os), only the FS of LaOs4P12 has no suitable nesting condition, which was confirmed by the dHvA experiment and a band-structure calculation (Sugawara et al., 2008a). The electron2phonon mass enhancement evaluated from the ratio of the dHvA cyclotron effective mass to the band mass is 1.922.4 depending on the frequency branches. LaRu4As12 has the highest SC transition temperature Ts ¼ 10.3 K among all the stoichiometric La-based filled skutterudites (Shirotani et al., 1997). LaRu4Sb12 shows an SC transition at Ts ¼ 3.58 K (Uchiumi et al., 1999). LaOs4Sb12 has the lowest, Ts ¼ 0.74 K, among all the stoichiometric La-based filled skutterudites (Sugawara et al., 2005a, 2005b). It should be noted that Ts is enhanced in the Pr-homolog PrOs4Sb12 containing 4f-electrons; that is, Ts ¼ 1.82 K (Bauer et al., 2002c). Nakai et al. (2008) found an evident difference in the nuclear spin-lattice relaxation rate divided by temperature 1/T1T between La and Sb sites. 1/T1T at Sb site is explained by the Korringa mechanism, while that at La site exhibits a broad maximum around 50 K (as shown in Fig. 1.11), suggesting some additional relaxation mechanism at La site. The additional low-lying excitations have been attributed to the rattling motion of the La ions, which is commonly observed in Sb-based filled skutterudites by the macroscopic methods (Goto et al., 2004; Ogita et al., 2007).


19

Figure 1.11 Temperature dependence of 1/T1T at the La and Sb sites under various magnetic fields. The broad maximum is found at around 50 K only at the La site. (Reprinted figure with permission from Nakai, et al. (2008), by the American Physical Society r 2009.)

RRh4X12 is unstable at ambient pressure. However, LaxRh4P12 was synthesized for the first time by forcing La into a binary skutterudite RhP3 at pressure around 4 GPa and temperature 11001C by Shirotani et al. (2005b). The lattice constant was found to increase from 7.996 to 8.0581 Å after synthesis under high pressure. It exhibits an SC transition at around 17 K that is the highest among the filled skutterudites.

4.2. Ce-based filled skutterudites The strong hybridization between conduction and f-electrons (c-f hybridization) was realized by the large coordination number; 12 X and 8 T ions surrounding R are believed to play an essential role in the exotic characteristics of skutterudites. As a key feature to reflect such strong c-f hybridization in the Ce-based filled skutterudite, there exists an evident correlation between the energy gap Eg estimated from the temperature dependence of electrical resistivity r(T) (Figs. 1.12a21.12c) and lattice constant as shown in Fig. 1.13. The smaller lattice constant-filled skutterudites such as CeRu4P12 and CeFe4P12 are semiconductors with relatively large Eg (Bauer et al., 2001b, 2001c; Grandjean et al., 1984; Meisner et al., 1985; Morelli and Meisner, 1995; Shirotani et al., 1999), while CeRu4Sb12 with a larger lattice constant is a semimetal exhibiting NFL behaviors at low temperatures. Even in the system showing metallic r(T), such as CeFe4Sb12 and CeRu4Sb12, some properties exhibit a signature of hybridization gap, which might be underlying common

20

H. Sato et al.

(b)

(a)

1 ρ (mΩ cm)

CeRu4P12

ρ (mΩcm)

CeOs4P12

ρ/ρ300K

104

10

CeOs4As12

1

102

CeRu4As12 CeFe4P12

0.1

Tx

CeOs4Sb12 (0 T)

(c)

~ T1/2

CeFe4Sb12 CeOs4Sb12

(14 T)

CeFe4As12 CeRu4Sb12

100 1

10 T (K)

100

0.1 1

10

100 T (K)

1

10 T (K)

100

Figure 1.12 Temperature dependence of electrical resistivity q(T) in (a) CeT4P12 (Shirotani et al., 1999) (b) CeT4As12 (reproduced with permission from Maple et al., 2008), and (c) CeT4Sb12 (Sato et al., 2008). Tx of CeOs4Sb12 in (c) corresponds to an undefined ordering temperature.

Figure 1.13 Energy gap determined from q(T) versus lattice constant in CeT4X12. (Reproduced with permission from Sugawara et al. (2005b), by the American Physical Society r 2009.)

features in these materials. Table 1.2 shows the comparison of the hybridization gap in Ce-based filled skutterudites determined by various measurements for all the combinations of R, T, and X. This strong c-f hybridization realized by the unique crystal structure of the filled skutterudites also plays a key role in the unusual behaviors of other rare earth homologs such as Pr- and Sm-filled skutterudite compounds.

Compound

q(T)

Opt.

INS

NMR, NQR

ELS

Grandjean et al. (1984), Magishi et al. (2006), Matsunami et al. (2008a), Meisner et al. (1985), Sato et al. (2000) Magishi et al. (2006), Matsunami et al. (2008b), Shirotani et al. (1999) Matsunami et al. (2008b), Shirotani et al. (1999) Grandjean et al. (1984), Maple et al. (2008) Maple et al. (2008), Matsunami et al. (2008b), Sekine et al. (2007) Matsunami et al. (2008b), Sekine et al. (2008a) Grandjean et al. (1984), Magishi et al. (2007b), Matsunami et al. (2008b), Mori et al. (2007), Viennois et al. (2007) Adroja et al. (2003), Dordevic et al. (2001), Matsunami et al. (2008b), Takeda et al. (2000a, 2000b), Yogi et al. (2008) 0.0004 Adroja et al. (2007), Bauer et al. (2001c), Matsunami et al. (2003, 2008b, 2009), Nakanishi et al. (2007), Yogi et al. (2005)

CeFe4P12

0.13

0.23

0.1

CeRu4P12

0.086

0.2

0.078

CeOs4P12 CeFe4As12 CeRu4As12

0.034 0, 0.01 0, 0.0043

0.3

CeOs4As12 CexFe4Sb12

0.0047 0

CeRu4Sb12 CeOs4Sb12

0.1 0.05 B0.003

0.04

0.03

0

0.047

0.03

0.03

0.0009

0.03

0.05, 0.027

0.025

References


Table 1.2 Comparison of the hybridization gap Eg (eV) in Ce-filled skutterudite compounds determined by various measurements: electrical resistivity r(T), optical experiment (optical conductivity, PES, and XANES) Opt., inelastic neutron scattering INS, nuclear magnetic resonance NMR, nuclear quadrupole resonance NQR, and elastic experiment ELS.

21

22

H. Sato et al.

CeFe4P12 is a semiconductor with an energy gap EgB0.13 eV (B1500 K) (Grandjean et al., 1984; Meisner et al., 1985; Sato et al., 2000). The electrical resistivity r increases with decreasing temperature without any phase transition down to 0.35 K (Meisner, 1981), although the complex temperature dependence is unexplainable of a simple band semiconductor. The lattice constant of CeFe4P12 is significantly smaller than that expected from the trivalent lanthanide contraction (Jeitschko and Braun, 1977), which suggests that the valence of Ce ion deviates from Ce3 + to Ce4 + . The magnetic susceptibility wm is small (2.6 104 emu/ mol) at room temperature, which is roughly half of that in nonmagnetic LaFe4P12 (Meisner et al., 1985), and almost constant down to B50 K, also suggesting the presence of Ce4 + . The result of Mo¨ssbauer measurement indicates zero magnetic moment on Fe (Grandjean et al., 1984). From the P-NMR measurements, the Knight shift is small and is almost independent of temperature, which is consistent with temperature dependence of wm (Magishi et al., 2006). However, the X-ray absorption near-edge structure (XANES) spectroscopy (Xue et al., 1994) suggests that the valence of Ce is primarily trivalent, although a small peak at high energy B5732 eV expected for Ce4 + was observed besides the main peak for Ce3 + at B5722 eV. Recently, Matsunami et al. (2008a) reported the Ce 3d corelevel PES and X-ray absorption spectroscopy (XAS) experiments, in combination with single-impurity Anderson model calculations, indicating mixed valence in CeFe4P12 due to strong c-f hybridization. The bandstructure calculations by using an LDA have also predicted a c-f hybridization gap formation (Khenata et al., 2007; Nordstrom and Singh, 1996). The semiconducting state is very sensitive to La-substitution; La-substitution by 1% (x ¼ 0.01) destroys the energy gap, suggesting that the coherence among the Ce ions plays an important role in the semiconducting behavior (Sugawara et al., 2006). CeRu4P12 is a semiconductor with an energy gap EgB0.086 eV (B1000 K) (Shirotani et al., 1999). The magnetic susceptibility at room temperature is small (B5 105 emu/mol) and almost constant down to B50 K (Shirotani et al., 1999), which is consistent with the P-NMR measurements (Fujiwara et al., 2000; Magishi et al., 2006). The magnetization curve at 2 K shows a saturation tendency to a value of 0.15mB/Ce at 5 T much less than 2.54mB/Ce expected for Ce3 + free-ion. This fact suggests the existence of Ce4 + ions in CeRu4P12 consistent with the smaller lattice constant expected from Ce3 + ( Jeitschko and Braun, 1977). On the other hand, XANES spectroscopy (Lee et al., 1999) suggests that the valence of Ce is primarily trivalent. Overall features of the physical properties are similar to those of CeFe4P12 except the smaller La-substitution effect compared to CeFe4P12 (Shirotani et al., 1999; Sugawara et al., 2006). CeOs4P12 is a semiconductor with an energy gap EgB0.034 eV (B400 K) (Shirotani et al., 1999). The lattice constant is smaller than that


23

expected from trivalent lanthanide contraction as in CeRu4P12 (Jeitschko and Braun 1977). The magnetic properties are only mentioned to be similar to those of CeRu4P12 (Shirotani et al. 1999). The electronic ground state in CeFe4As12 is still controversial whether semiconducting (Grandjean et al., 1984) or semimetallic (Maple et al., 2008). In a polycrystalline sample, the r shows a semiconducting behavior with a small energy gap EgB0.01 eV (B100 K) (Grandjean et al., 1984). Recently, single crystalline samples of filled skutterudite arsenides were grown by mineralization in a Cd-As flux (Henkie et al., 2008). The r decreases with decreasing temperature down to B150 K, below which it slightly increases as in a semimetal or a semiconductor with a significantly small energy gap (Maple et al., 2008). Polycrystalline CeRu4As12 synthesized by the high-pressure technique (Sekine et al., 2007) was reported to be a semiconductor with a small energy gap EgB0.0043 eV (B50 K). The magnetic susceptibility wm is small (B1.3 103 emu/mol) at room temperature, which is similar to those in CeT4P12 (T ¼ Fe, Ru, Os) (Meisner et al., 1985; Shirotani et al., 1999); however, the sample exhibits a notable broad maximum in wm at around 270 K. In contrast, a single crystalline sample grown by Henkie et al. (2008) is metallic and indicates NFL behaviors of r(T)BT 1.4 below 3.5 K (Maple et al., 2008). The specific heat divided by temperature C(T)/T can be described by either a weak power law C(T)/TBT 0.97 or a logarithmic function C(T)/TBlnT. The T-dependence of wm further supports the apparent NFL behavior. For 1.9 KoTo10 K, wm(T) is described by either a power law wm(T)BT 0.49 or a logarithmic function wm(T)Bln(T) (Maple et al., 2008). In the case of CeOs4As12, both the polycrystalline sample synthesized under B4 GPa (Sekine et al., 2008a) and the single crystal grown under several MPa (Maple et al., 2008) were reported to be semiconductors, although the energy gap (EgB0.0047 and 0.0056 eV, respectively) is slightly different. This is in contrast with CeFe4As12 and CeRu4As12 which exhibit entirely different electronic states at low temperature depending on the synthesizing condition of the samples. The magnetic susceptibility wm at room temperature is small (B1.4 103 emu/mol) and only weakly depends on temperature down to B50 K, similar to CeFe4As12 and CeRu4As12. CeFe4Sb12 is metallic (Grandjean et al., 1984) and exhibits moderately heavy-fermion (HF) behaviors at low temperatures (Gajewski et al., 1998; Morelli and Meisner, 1995). The Mo¨ssbauer measurement indicates absence of magnetic moment on Fe (Long et al., 1999). The temperature dependence of magnetic susceptibility wm(T) is reminiscent of intermediate valence Ce-based compounds (Danebrock et al., 1996; Mori et al., 2007; Viennois et al., 2004); wm(T) increases with decreasing temperature and shows a broad maximum between 115 and 150 K, depending on the

24

H. Sato et al.

sample. The wmB3.5 103 emu/mol at room temperature is almost the same order of magnitude as that of valence fluctuation compound CeSn3. Magnetization increases almost linearly with increasing magnetic field and is only 0.06mB/Ce at 7 T. From the 121,123Sb-NQR measurements, the nuclear spin-lattice relaxation time T1 shows an activated temperature dependence 1/T1pexp(D/kBT) with an energy gap D/kB ¼ 190 K above 50 K, while 1/T1 is proportional to temperature below 30 K, which was explained by a pseudo-gap model suggested to be induced by the hybridization between Ce 4f and conduction electrons (Magishi et al., 2007b). Optical (Matsunami et al., 2008b) and inelastic neutron scattering (INS) measurements (Viennois et al., 2007) suggest the opening of hybridization gap or spin gap (B40 meV) at low temperatures. Numerous reports on the transport and thermoelectric properties on both pure and substitution systems such as Ce1xLaxFe4Sb12 and Ce(Fe1xCox)4Sb12 have been reported, since they were thought to be promising candidates for the thermoelectric material (Sales, 2003). The filling fraction of Ce site is evaluated to be 70290% in CexFe4Sb12 by EPMA measurements, which is a common feature to the RxFe4Sb12 system (Bauer et al., 2002a; Butch et al., 2005). CeRu4Sb12 was reported to be a metal that exhibits NFL behavior at low temperatures under zero magnetic fields (Bauer et al., 2001b; Takeda and Ishikawa, 2000a, 2000b); temperature dependence of r follows rpT n (n ¼ 121.6), and the temperature dependences of C and wm are described by a logarithmic or a power law dependence at low temperatures. Under the magnetic fields, the NFL behaviors are suppressed and the Fermi-liquid behaviors recover (Abe et al., 2002; Takeda and Ishikawa, 2000c). In fact, the dHvA and SdH oscillations from a small hole FS were clearly observed above 6 T (Abe et al., 2002; Sugawara et al., 2002a), which evidences a semimetallic ground state. Temperature dependences of r and wm at high temperatures are similar to that of CeFe4Sb12. Optical (Dordevic et al., 2001; Kanai et al., 2002), INS (Adroja et al., 2003), and 121,123Sb-NQR (Yogi et al., 2008) measurements suggest the opening of hybridization gap or spin gap (B30 meV) at low temperatures. CeOs4Sb12 was first reported to be a semiconductor with a small gap EgB0.0009 eV (B10 K) estimated from the temperature dependence of electrical resistivity r(T) (Bauer et al., 2001c). A little later, the temperature dependence r(T) at low temperatures was recognized to be not that simple. With decreasing temperature from RT, r(T) first decreases in a similar manner as in CeFe4Sb12 and CeRu4Sb12, and it starts to increase after showing a minimum at around 50 K following r(T)p(T)1/2 below B30 K (Sugawara et al., 2005b). A large residual density of states (gB0.2 J/K2 mol) was observed in the specific heat measurement as shown in Fig. 1.14 (Namiki et al., 2003b), unexpected for ordinary semiconductors. With increasing magnetic field, the resistivity at low temperatures is

25


6 0.6

μ0H (T)

CeOs4Sb12

0T

C/T (J/K2mol)

2T 0.4

4 ordered state 2

4T 0

0

1

2

T (K)

0.2

phonon (LaOs4Sb12) 0

0

2

4 T (K)

Figure 1.14 Temperature dependence of the specific heat in CeOs4Sb12 under 0, 2, and 4 T (Namiki et al., 2003b).

gradually suppressed and becomes metallic at high fields as shown in Fig. 1.12c (Sugawara et al., 2005b), while the band-structure calculation (Harima and Takegahara, 2003b) predicts a semimetallic ground state. As another novel feature, it exhibits a phase transition at B0.9 K in zero magnetic field as shown in Fig. 1.14 (Namiki et al., 2003b). The H2T phase diagram shown in Fig. 1.15 mimics that of CeB6 (Fujita et al., 1980) which indicates an antiferro-quadrupole (AFQ) ordering; however, the electronic part of entropy released below the transition temperature (B0.05R ln2 at 0 T) is too small to be attributed to such localized f-electron scenario. Moreover, the CEF excitation expected for the localized felectrons in Ce3 + has not been observed in the INS experiment (Yang et al., 2005). On the other hand, AFM reflections characterized by the wave vector q ¼ (1, 0, 0) with a tiny ordered moment of B0.07mB/Ce was observed in zero magnetic field (Iwasa et al., 2008). The reflections are rapidly suppressed by magnetic fields and only FM correlation is observed above 1 T. At this stage, the origins of the low-temperature, high-resistive state and the unusual ordered phase are still unclear. However, such a drastic effect of magnetic field suggests a magnetic origin, most probably due to the 4f-electrons. The optical (Matsunami et al., 2003, 2009), INS (Adroja et al., 2007), 121,123Sb-NQR (Yogi et al., 2005), and elastic

26

H. Sato et al.

Figure 1.15 H2T phase diagram determined by various experiments in CeOs4Sb12. (Reproduced with permission from Sugawara et al. (2005b), by the American Physical Society r 2009.)

(Nakanishi et al., 2007) measurements suggest the opening of hybridization gap or spin gap (B30 meV) at low temperatures. Recent 121Sb-NMR measurements under magnetic fields revealed that AFM spin fluctuations reflect the proximity to a quantum critical point at H ¼ 0 T, and suggested that the high-field ordered phase above 1 T is caused by the freezing of multipole degree of freedom of 4f-eletrons (Yogi et al., 2009).

4.3. Pr-based filled skutterudites 4.3.1. General trend of the Pr-based filled skutterudites The Pr-based filled skutterudites are one of the most intensively investigated series among the filled skutterudites since some of them exhibit unusual strongly correlated electron behaviors and multipole orderings. In the crystal structure, the lattice of Pr ions (with the localized f 2-configuration) interacting with conduction electrons is formed, providing an interesting system, which has not been well understood experimentally and theoretically, in contrast with the Ce f1-configuration case. For example, a theoretical idea of quadrupole Kondo effect has been proposed (Cox and Zawadowski, 1998). In order to search for such behaviors experimentally, Pr-based compounds with the non-Kramers doublet CEF ground state, including PrInAg2 (Yatskar et al., 1996), have been investigated. In Pr-based filled skutterudites studied so far, nonKramers doublet CEF ground states have not been realized as shown in Table 1.3. Nevertheless, it has been found that unique low-energy singlet2triplet CEF level schemes are formed in many cases, and HF states

Compound

CEF-GS

D1st (K)

1st

TTR

c (mJ/K2 mol)

leff (lB/f.u.) References

PrFe4P12

G1?

B16 (OS)

Gð1Þ 4 ?

Tmp ¼ 6.5 K

120022700

3.62

PrRu4P12

G1

68 (RT)

Gð1Þ 4

TMI ¼ 62 K

o60

3.84

PrOs4P12

G1

48

Gð2Þ 4

No (W50 mK) 56.5

3.63

PrFe4As12 PrRu4As12

G5-like? G1

2 B30

2 Gð1Þ 4

TC ¼ 18 K Ts ¼ 2.4 K

3.98 3.30

PrOs4As12

Gð2Þ 4

4.5

G1

TN ¼ 2.3 K 502200 (T1 ¼ 2.2 K) (TWTN), B1000 (To2 K)

340 B95

2.77 (o20 K)

Aoki et al. (2002a), a model proposed in Kuramoto and Kiss (2008), Sato et al. (2000), Torikachvili et al. (1987) Iwasa et al. (2005b), Matsuhira et al. (2002a), Sekine et al. (1997) Matsuhira et al. (2005b), Sekine et al. (1997), Sugawara et al. (2009), Yuhasz et al. (2007) Sayles et al. (2008) Namiki et al. (2007), Shirotani et al. (1997) Yuhasz et al. (2006)


Table 1.3 Reported fundamental features of Pr-based filled skutterudites. CEF ground state (CEF-GS), excitation energy to the first excited state D1st, the type of the first excited state (1st), the transition temperature to the ordered state TTR (Tmp: multipolar; TMI: insulating, M2I transition; TN: antiferromagnetic; Ts: superconducting; TC: ferromagnetic ordering), electronic specific heat coefficient g, and effective magnetic moment meff are listed.

27

28

Table 1.3. (Continued ) Compound

CEF-GS

D1st (K)

1st

TTR

c (mJ/K2 mol)

leff (lB/f.u.) References

PrFe4Sb12 PrxFe4Sb12 (xo1) PrRu4Sb12

G1 G5-like

B22 B28

G5 {4(2)} G1

B300 B1000

4.5 4.224.3

G1

73

Gð2Þ 4

No TC ¼ 4.124.6 K Ts ¼ 1.3 K

59

3.58

PrOs4Sb12

G1

B7

Gð2Þ 4

Ts ¼ 1.85 K

3102750

2.97

Tanaka et al. (2007) Bauer et al. (2002a), Butch et al. (2005) Adroja et al. (2005), Takeda and Ishikawa (2000a) Aoki et al. (2002b), Bauer et al. (2002c), Goremychkin et al. (2004), Kuwahara et al. (2005), Vollmer et al. (2003)

Ferromagnetic, antiferromagnetic, or ferrimagnetic.

H. Sato et al.


29

Figure 1.16 Temperature dependence of electrical resistivity for several Pr-based filled skutterudites (Sugawara et al., 2008a). For superconductors, see Fig. 1.24.

and unprecedented types of multipolar orderings appear (see Fig. 1.16 for the corresponding anomalies in the electrical resistivity). This section places focus on the four most intensively investigated compounds, PrFe4P12, PrRu4P12, PrOs4As12 and PrOs4Sb12 (Maple et al., 2007; Sato et al., 2007a), and comparison with other Pr-based filled skutterudites will be made later. While these compounds were first synthesized by Jeitschko and Braun (1977, 1980), detailed physical property measurements started in 1990s.

4.3.2. Heavy-fermion and multipolar ordered states in PrFe4P12 PrFe4P12 is the first known Pr-based Kondo compound exhibiting almost all the Kondo characteristics, although a few Pr-based compounds exhibiting some HF signatures had been reported. First systematic study of the PrFe4P12 has been done by Torikachvili et al. (1987) where PrFe4P12 was reported to be an antiferromagnet with Neel temperature of 6.5 K, based on both a peak in wm(T) and metamagnetic behaviors in M(H) as shown in Fig. 1.17 (Aoki et al., 2002a). However, various experimental results, such as nuclear specific heat (Aoki et al., 2002a), neutron scattering (Hao et al., 2003; Keller et al., 2001), X-ray diffraction (Iwasa et al., 2003), and muon-spin relaxation

30

H. Sato et al.

0.4 2.0 2.5 K

PrFe4P12 4.2 K

H // [100] M (μB/Pr)

χ (emu/mol)

0.3 1.0

7.0 K

0.2

0.0 0

1

0.1

2 3 μ0H (T)

4

5

PrFe4P12 H // [100] 0.1 T 0.0 0

100

200

300

T (K)

Figure 1.17 Temperature dependence of magnetic susceptibility vm in PrFe4P12. Inset shows the field dependence of magnetization at selected temperatures. (Reproduced with permission from Aoki et al. (2002a), by the American Physical Society r 2009.)

(mSR) (Aoki et al., unpublished data) indicate the nonmagnetic origin of the transition at Tmp ¼ 6.5 K, probably caused by multipole degrees of freedom without magnetic character. The extraordinary behaviors found in the transport properties such as r(T) and S(T) evidencing the Kondo-like effect, have revived the researches on this material (Sato et al., 2000). Figure 1.18a shows the temperature dependence of electrical resistivity under selected magnetic fields. In zero field, r at RT (B300 mO cm) is larger than that of LaFe4P12, and it increases as ln(T) from 150 K down to TP ¼ 13 K with decreasing temperature, where r exhibits a faint maximum indicating development of coherent Kondo state. With further decreasing temperature, r exhibits a sudden increase at Tmp ¼ 6.5 K, reflecting the phase transition, and r(T) does not follow the simple Fermi-liquid behavior in the ordered phase. Figure 1.18b shows the temperature dependence of thermoelectric power S(T) in PrFe4P12 compared with the reference compound LaFe4P12. The large absolute value above Tmp along with the sensitive field dependence in the inset also strongly suggests the dominance of Kondo effect in this material (Yamada et al., 2004). Pourret et al. (2006) reported remarkable features in the thermal transport properties: the drastical increase of the lattice thermal conductivity and the anomalously large values of the Seebeck and Nernst coefficients in the ordered state, reminiscent of the hidden ordered state in URu2Si2. With decreasing temperature across Tso, the Hall coefficient (RH) also shows a jump from 4 109 to + 9 107 m3/C (Sato et al., 2000), indicating the

31


(a)

1200 μ0H

PrFe4P12

H // [100]

=0T 1

ρ (μΩcm)

800

2

4

3

6 8 ~ ln(T )

10 400

14

~T 2 0

1

10 T (K)

100

(b) LaFe4P12 0 50

-50

S (μV/K)

S (μV/K)

PrFe4P12 0 7.0 K -50 H//[100]

4.2 K -100

-100

μ0H(T)

2.5 K -150 0

PrFe4P12

1

2

3

4

5

-150 0

100

200

300

T (K)

Figure 1.18 (a) Temperature dependence of resistivity under selected magnetic fields in PrFe4P12 and (b) the comparison of temperature dependence of thermoelectric power between PrFe4P12 and LaFe4P12 (Aoki et al., 2005b, Sato et al., 2003).

32

H. Sato et al.

Figure 1.19 (a) Phase diagram for PrFe4P12. mp.O is a nonmagnetic ordered phase thought to be a scalar-type multipole order. For only narrow magnetic field direction near [1 1 1], Tayama et al. (2004) reported a different ordered phase H.P. whose origin has not yet settled. (b) Field dependence of the specific heat coefficient c and the square root of the T2 coefficient A in the field-induced heavy-fermion state (Aoki et al., 2005b).

disappearance of a large nested FS predicted by the band-structure calculation (Harima and Takegahara, 2002). Applied magnetic field reduces Tmp monotonically and reaches 0 K at Hc, depending on the field direction as shown in Fig. 1.19a. Above Hc, r(T) follows r0 + AT2 with a large A-coefficient quite well, reflecting the heavy Fermi-liquid state. For the three principal field directions, the coefficient A and the specific heat coefficient g follow Kadowaki2Woods relation well as shown in Fig. 1.19b (Aoki et al., 2005b). The g value in the ordered phase plotted in the figure also indicates the reduced carrier number in the ordered phase. More directly, the FS reconstruction across Hc has been investigated by dHvA experiment (Sugawara et al., 2002b). Figure 1.20a compares the dHvA frequency between PrFe4P12 and the reference compound LaFe4P12 (Sugawara et al., 2000). It should be noted that t-branch is observed only in the ordered phase, while all other branches are detected only in the fieldinduced HF state above Hc. From this figure, the change in FS across Hc is evident, consistent with the bulk properties. Note that the FS volume of t-branch is only 0.15% of the Brillouin zone size. It should also be noted that the dHvA frequencies are remarkably different between PrFe4P12 and LaFe4P12, in contrast with the fair agreement between PrRu4Sb12 and LaRu4Sb12 (Matsuda et al., 2002a). This fact also indicates the stronger c-f hybridization in PrFe4P12 compared to those in other Pr-filled skutterudites. Figure 1.20b shows the field dependence of cyclotron effective mass m for selected field directions, which contains a good deal of important information. (1) The detected maximum mass is 81m0 for o-branch, which directly evidences the strong mass enhancement in this material. Note that

33


(a)

PrFe4P12

LaFe4P12

(110)

(010)

a′ a

Frequency (T)

b

c

103

ω

a′′

100

b′

80

ψ

e

ω

60 χ

40

ψ

20

χ

τ 0

f i

g

(b)

PrFe4P12 θ = 25 deq.

mc* (m0)

104

0

5

10 μ0 H(T)

15

20

h τ 102 -90 [100]

-60 -30 0 30 60 [101] [001] [111] Field Direction (Degrees)

90 [110]

Figure 1.20 (a) Angular dependence of dHvA frequencies in PrFe4P12 in comparison with those in the reference compound LaFe4P12 and (b) field dependence of the cyclotron effective mass in PrFe4P12. Note that only the s-branch is detected in the low-field ordered phase. (Reproduced with permission from Sugawara et al. (2000), by the American Physical Soceity r 2009.)

the signal for o-branch is hardly detectable in lower fields where the heavier effective mass is expected. (2) In the field-induced HF state, m for all the branches decreases with increasing field, which suggests that the magnetic Kondo mechanism dominates the mass enhancement in PrFe4P12. This is consistent with the 31P-NMR experiment by Ishida et al. (2005) where they concluded that the magnetic Kondo mechanism plays a role in PrFe4P12, since the temperature dependence of 1/T1 is similar to that reported for Ce- and Yb-based HF compounds. (3) The m ¼ 10 m0 for t-branch is unusually heavy, if the very small size of FS described above is taken into account. Combining this fact with the very small field dependence of the mass, some novel mass enhancement mechanism is inferred in the low-field ordered state. The discussion on the order parameter in the low-temperature nonmagnetic phase has almost been settled after many twists and turns. The AFQ order scenario was proposed (Iwasa et al., 2003), soon after the first AFM order scenario had been ruled out. The AFQ scenario has been

34

H. Sato et al.

accepted for long time, because many of the experimental results are reasonably explained. However, Kikuchi et al. (2007b) reported a new result on the angular dependence of NMR frequencies that contradicts with the AFQ scenario; that is, the local symmetry at Pr site preserves across the phase transition. Stimulated by this experiment, several models compatible with the experimental result have been theoretically proposed, based on the orbital degrees of freedom. These scalar-type-order (SO) scenarios (Kiss and Kuramoto, 2006; Sakai et al., 2007) have succeeded in explaining all the features contradicting with the AFQ scenario, such as the absence of field-induced staggered moment perpendicular to the field, the isotropic magnetic susceptibility (Aoki et al., 2002a, Sato et al., 2007b), the field-direction dependence of transition temperature (Sato et al., 2008), and even the anisotropic response of wm against uniaxial pressure (Kuramoto and Kiss, 2008; Matsuda et al., 2002b). The CEF level scheme is a remaining subject that should be clarified. However, the high-intensity quasi-elastic scattering due to strong c-f hybridization prevents the successful observation of neutron inelastic scattering peaks accompanying the CEF-level excitation (Iwasa et al., 2003) above Tmp, although sharp peaks were detected at 1.4 and B3.0 mV in the ordered phase. Recently, more elaborate study on the INS experiment in wide ranges of temperatures and magnetic fields has been done; however, no decisive progress in determining the CEF level scheme has been made (Park et al., 2008). Based on the various indirect experimental evidences on PrFe4P12, Kuramoto and coworkers (Kuramoto et al., 2006; Kuramoto and Kiss, 2008; Otsuki et al., 2005) proposed a theoretical CEF model, where the ground state is the G1 singlet with the low-lying Gð1Þ 4 triplet and possibly the G23 doublet excited states. They further suspect that these six levels are almost degenerate in the high-temperature phase of PrFe4P12. As temperature decreases below Tmp, one of the Pr sublattices has the G1 ground state, while the other Pr takes the doublet ground state, based on the neutron scattering of PrFe4P12 where at least two inelastic transitions are visible in the ordered phase (Iwasa et al., 2003). 4.3.3. Metal2insulator transition in PrRu4P12 PrRu4P12 was reported to exhibit a metal2insulator (M2I) transition at TMI ¼ 60 K from the electrical resistivity measurement (Sekine et al., 1997). Magnetic susceptibility shows no distinct anomaly at TMI as shown in Fig. 1.21, which indicates nonmagnetic origin of the transition. A l-shaped second-order-type anomaly at TMI ¼ 62.3 K in the specific heat under zero magnetic field does not change under magnetic field up to 12 T (Sekine et al., 2000a). L2-edge XANES spectra indicate that Pr exists in trivalent state over a wide temperature range, 20 KoTo300 K (Lee et al., 1999). The Raman scattering experiments suggest that the M2I transition

35


200

100 PrRu4P12

150

60 100 40

χ-1 ( mol / emu )

χ (10-3 emu / mol )

80

50

20

0

0 0

50

100

150

200

250

300

350

T [K]

Figure 1.21 Magnetic susceptibility vm and inverse of magnetic susceptibility w1 m versus temperature for PrRu4P12 (Reproduced with permission from Sekine et al. (1997), by the American Physical Society r 2009.). Note that no anomaly has been found at TMI in vm, indicative of nonmagnetic origin of the transition.

is related to lattice instability (Sekine et al., 1999). No significant change in the lattice volume has been detected in a thermal expansion measurement, but a slight jump in the thermal expansion coefficient was observed at TMI (Matsuhira et al., 2000). Thus, it was considered that the M2I transition is driven neither by magnetic instability nor by charge instability of Pr ions, but is related to tiny lattice instability. Simply assuming a trivalent Pr ion, PrRu4P12 is an uncompensated metal, so that it never becomes an insulator without forming multiple unit cells below TMI. However, no crystallographic transformation with breaking cubic symmetry (Im3¯ ) or lowering translational symmetry was identified, until the electron diffraction measurement was done (Lee et al., 2001). The origin of the M2I transition was discussed with the nesting property of the FS with q ¼ (1, 0, 0), as shown in Fig. 1.22 (Harima and Takegahara, 2002; Sugawara et al., 2000). Such a nesting property originates in the p-molecular orbital of the X12 cage with symmetry xyz, as mentioned in Section 3.1. A structural phase transition with q ¼ (1, 0, 0) has actually been observed by using electron diffraction technique, although the space group in the low-temperature phase is unclear (Lee et al., 2001). Then it was indicated that this structural phase transition is associated with AFQ ordering (Curnoe et al., 2002a, 2002b). Assuming structural distortions of P12 cages, the LDA + U calculation, where 4f 2-electrons in Pr are treated as localized ones, predicts opening of a gap at the Fermi level (Curnoe et al., 2004; Harima and Takegahara 2002). The optical conductivity spectrum clearly revealed that an energy gap of B10 meV opened below TMI (Matsunami et al., 2005).

36

H. Sato et al.

Figure 1.22 Band structure and Fermi surface in PrRu4P12 (Harima et al., 2002; Harima and Takegahara, 2002; Harima and Takegahara, 2003b; Aoki et al., 2005b).

It was clarified that the M2I transition is associated with the structural phase transition originating in the nesting property of the FS; however, the role of 4f-electrons remains unclear. It is a key to understand the mechanism of M2I transition, because the non-f reference compound LaRu4P12 with a similar FS (Saha et al., 2005) does not show the structural phase transition, but undergoes superconductivity. Synchrotron radiation X-ray diffraction measurements (Lee et al., 2004a, 2004b) and extended X-ray absorption fine structure (EXAFS) experiments (Cao et al., 2005) have revealed that the crystal structure of the lowtemperature phase is Pm3. These experiments suggest that the local point symmetry of Pr sites does not change, but there exist two inequivalent Pr sites below TMI. These facts exclude the possibility of AFQ ordering scenario, because lowering of the local symmetry is necessary in the quadrupole ordering. Moreover, the observed distortions are too small (B0.13% of the lattice constant) to obtain a real gap expected from the band calculations, even though the distortion is compatible with Pm3. The distortion of 0.5% is necessary for the gap formation (Curnoe et al., 2004). The microscopic characteristics of the two inequivalent Pr sites have been confirmed by an INS experiment (Iwasa et al., 2005b). It has been revealed that the well-defined crystal field excitations on two Pr sites show different temperature dependence below TMI, as shown in Fig. 1.23. To understand the M2I transition, Takimoto (2006a, 2006b) has introduced antiferro-hexadecapole ordering scenario, which explains the role of 4f-electrons and the temperature dependence of the crystal field excitations on two Pr sites. Kuramoto and coworkers have discussed a scalar order parameter in the ordered phase, in comparison with the ordering of PrFe4P12 (Kiss and Kuramoto, 2006; Kuramoto et al., 2006). The scalar order parameter means a totally symmetric one into which hexadecapole and higher rank multipoles can be classified. Harima (2008) explained why such higher rank multipoles are relevant to be considered in the filled


37

Figure 1.23 Temperature dependence of CEF level schemes of the two inequivalent Pr1 and Pr2 in the ordered phase in PrRu4P12. (Reprinted with permission from Iwasa et al. (2005a), by the American Physical Society r 2009.)

skutterudites, as mentioned in Section 3.2. Now it is believed that Pr sites split into two inequivalent Pr sites with a singlet ground state and a triplet ground state, with staggered higher electric multipole moments under the cubic symmetry, and the staggered moments bring about the gap in the main conduction band. However, there remains a question why magnetic Pr ions with the triplet ground state do not show long-range ordering.

4.3.4. Heavy-fermion superconductivity and quadrupolar ordering in PrOs4Sb12 The study of unconventional superconductivity in HF systems has been one of the major subjects in solid-state physics for more than two decades (Grewe and Steglich, 1991; Pfleiderer, 2009; Steglich et al., 1979; Stewart, 1984). In these materials, hybridization of conduction electrons with the localized f-electrons of the magnetic ions leads to the formation of quasiparticles with a strongly enhanced effective mass at low temperatures. Below the SC transition temperature Ts, such heavy quasiparticles condense, forming Cooper pairs. The commonly observed large specific heat jump at Ts provides clear evidence for this phenomenon. The attractive interactions that form heavy Cooper pairs are considered to

38

H. Sato et al.

originate mainly from magnetic fluctuations, instead of phonon excitations as in a conventional s-wave superconductor. HF superconductivity in PrOs4Sb12 was first reported by Bauer et al. (2002c). Since then, intensive research work has revealed that the physical properties of this compound are quite different from those of the wellknown Ce- and U-based HF superconductors. In those HF superconductors, f-electrons are itinerant by forming quasiparticles below a coherence temperature T (WTs). In contrast, 4f-electrons in PrOs4Sb12 exhibit well-localized character coexisting with HF superconductivity, suggesting entirely new physics. After the review of the normal state, where the multipolar degrees of freedom of 4f-electrons play a key role, the unusual features of the HF superconductivity will be discussed. Normal state properties: multipole moments of the localized 4felectrons. The electrical resistivity r of PrOs4Sb12, shown in Fig. 1.24, monotonically decreases with decreasing temperature without showing a noticeable lnT-dependence (Bauer et al., 2002c; Sugawara et al., 2005a), unlike those of ordinary Ce- or U-based Kondo compounds. Below 10 K, r shows a roll-off behavior with no signature of T2 temperature dependence characteristic of a Fermi-liquid state above Tsc ¼ 1.85 K (Frederick and Maple, 2003; Sugawara et al., 2003). The roll-off behavior is caused by conduction electron scattering from the thermally populated CEF-split Pr3 + energy levels. The existence of low-lying CEF energy levels emerges more clearly as a broad peak at B3 K both in magnetic susceptibility w(T) (Fig. 1.24) and in specific heat C(T) (Fig. 1.25), indicating a nonmagnetic CEF ground state. The CEF level scheme has been determined by several experimental techniques: specific heat, magnetization, INS, and others (Aoki et al., 2002b; Goremychkin et al., 2004; Kohgi et al., 2003; Kuwahara et al., 2004; Rotundu et al., 2004; Tayama et al., 2003; Tou et al., 2005). The key feature of the CEF level scheme relevant to the low-temperature physics is the combination of a nonmagnetic G1 singlet ground state and a magnetic triplet Gð2Þ 4 excited state with a small energy separation DCEF/kBB8 K. The other two states G23 and Gð1Þ 4 are located at far higher energies (W130 K). The well-localized character of the Pr 4f-electrons is also reflected in the close resemblance of the Hall coefficient and thermoelectric power between PrOs4Sb12 and LaOs4Sb12 (Sugawara et al., 2005a). There appears a field-induced AFQ ordering in the normal state. Specific heat measurements revealed a pronounced l-type anomaly in applied fields as shown in Fig. 1.25, providing clear evidence for the existence of the field-induced ordered phase (FIOP) (Aoki et al., 2002b; Vollmer et al., 2003). The large associated entropy release indicates that the FIOP originates from the 4f-electrons of the Pr ions. The phase transition has been further investigated by several other experimental techniques


39

Figure 1.24 Electrical resistivity q and magnetic susceptibility v as a function of T for PrOs4Sb12 and related materials. The arrows in the q versus T plot indicate the superconducting transitions. The large difference in the T dependences of q and v in the normal state between PrOs4Sb12 and PrRu4Sb12 reflects different values of the CEF energy separation DCEF between the nonmagnetic singlet ground state and the magnetic triplet excited state between the two compounds (Aoki et al., 2007b).

(Maple et al., 2003a, 2003b; Oeschler et al., 2004; Rotundu et al., 2004; Sugawara et al. 2003, 2005a; Tayama et al., 2003; Tenya et al., 2003). Figure 1.26a shows the determined magnetic field versus temperature phase diagram. The FIOP appears for all field directions, and the phase boundary has a noticeable anisotropy with respect to the applied field direction (Rotundu et al., 2004; Tayama et al., 2003). For H// [0 0 1], the FIOP has the largest area of the ordered state in the H versus T plane and the transition temperature has a maximum value of 1.3 K at m0HB9 T.

40

H. Sato et al.

Figure 1.25 Specific heat C versus T of PrOs4Sb12 in several magnetic fields (Aoki et al., 2002b). For H ¼ 0, the jump due to the superconducting transition is indicated by the arrow. The Schottky peak at 3 K is due to the singlet--triplet CEF thermal excitation. The clear k-type peak developing in high fields gives clear thermodynamical evidence for the existence of a field-induced AFQ phase.

Neutron diffraction measurements have clarified the order parameter of the FIOP (Kohgi et al., 2003). For H//[0 0 1], magnetic superlattice reflections characterized by the wave vector q ¼ (1, 0, 0) originating from an AFM moment mAF of Pr ions parallel to the [0 1 0] direction were detected (see Fig. 1.26b). Note that the observed AFM structure demonstrates that the [1 0 0] and [0 1 0] directions are not equivalent in H//[0 0 1]. As discussed in Section 3.3, this feature reflects the absence of four-fold symmetry around the [0 0 1] axis in the Th site symmetry of Pr ions. The small AFM component (mAFE0.025 mB/Pr ion) provides evidence that the FIOP is primarily an AFQ ordered phase (the observed AFM component is induced as a secondary effect). In this model, a singlesite mean-field Hamiltonian can be expressed as: H ¼ H CEF gJ mB J H lJ h J0 i J

X

li hO0i iOi ,

(7)

i

where HCEF, J, and Oi are the CEF Hamiltonian (Takegahara et al., 2001), total angular momentum, and ith quadrupole moment of Pr ion in a sublattice, respectively. Oi includes O02 and O22 (G3-type), and Oxy, Oyz,


41

Figure 1.26 (a) Magnetic field versus temperature phase diagram for PrOs4Sb12. In the superconducting (SC) state, the possible SC phase boundary proposed from the thermal conductivity measurement (broken line) (Izawa et al., 2003) and the anomaly appearing in the lower critical field Hc1(T) (open circle) (Cichorek et al., 2005) are shown. The phase boundaries of the FIOP are constructed mainly from magnetization measurements (Tayama et al., 2003) combined with specific heat (Aoki et al., 2002b; Rotundu et al., 2004) and electrical resistivity (Sugawara et al., 2005a) data. Schottky anomaly due to excitation between two lowest CEF singlets in applied fields (Rotundu et al., 2004) is also shown (open squares). (b) Magnetic structure in the FIOP for H//[0 0 1] determined by neutron scattering measurements (Kohgi et al., 2003) (the thick arrows) depicted schematically. The distorted charge distribution of Pr 4f-electrons in the antiferro-quadrupole ordered state is also shown. (c) Zeeman splitting of the C1 singlet and Gð2Þ 4 triplet CEF levels. The spatial 4f charge distribution for the CEF states in zero field is depicted (Aoki et al., 2007b).

and Ozx (G5-type) terms (Shiina et al., 1997). /JuS and hO0 i i represent the thermal average of these quantities in the other sublattice. This model reasonably explains the phase diagram (Aoki et al., 2007b; Shiina and Aoki, 2004). G5-type AFQ interaction plays a dominant role in the field-induced AFQ phase and the primary order parameter is Oyz for H//[0 0 1]. The Ot6 ¼ O26 O66 term in HCEF, unique to the Th symmetry, is essential to realize the AFM moment parallel to the [0 1 0] direction. For H//[1 1 0], neutron scattering suggests Oxy-type AFQ ordering (Kaneko et al., 2007), which is consistent with the above-mentioned model. The appearance of the AFQ ordering is intimately linked to the lowenergy singlet2triplet CEF levels of Pr ions. In the applied fields, the excited Gð2Þ 4 triplet is split by the Zeeman effect, and the lowest level crosses the singlet G1 ground state at B9 T forming a quasi-doublet ground state as shown in Fig. 1.26c. For H//[0 0 1], the quasi-doublet state has a quite large Oyz-type quadrupole moment, which orders by the G5-type AFQ interaction. In applied pressure, the AFQ ordered phase shifts to lower fields (Tayama et al., 2006).

42

H. Sato et al.

The ordering wave vector q ¼ (1, 0, 0) is just the same as that in PrRu4P12 (Hao et al., 2004; Lee et al., 2001) and PrFe4P12 (Iwasa et al., 2002), where the nesting is considered to be a driving force for the anomalous ordering in those compounds. In contrast, r(T) for PrOs4Sb12 does not show any noticeable increase across the AFQ transition (Maple et al., 2003a, 2003b; Sugawara et al., 2005a), suggesting that such nesting instability does not play a key role in the AFQ ordering. Actually the determined FS does not seem to possess a q ¼ (1, 0, 0) nesting instability (Sugawara et al., 2002c). Quadrupolar excitons in the Pr-ion lattice. INS has revealed that nonmagnetic AFQ interactions between Pr ions actually dominate in PrOs4Sb12 (Kuwahara et al., 2005). Figure 1.27 summarizes the wave vector q and temperature dependences of the low-energy G1 Gð2Þ 4 CEF excitations for q ¼ (z, 0, 0) in zero field. The excitations show a clear dispersion as well as strong temperature dependence. The intensity decreases toward the zone corner q ¼ (1, 0, 0) where the softening is

Figure 1.27 Reduced wave vector dependences of (a) the energy E and (b) the integrated intensity of the excitation peak observed in inelastic neutron scattering measurements. Temperature dependences of (c) E and (d) the width of the peak DE at q ¼ (1, 0, 0). Theoretical curves for the excitons are shown by the dashed lines. (Reproduced with permission from Kuwahara et al. (2005), by the American Physical Society r 2009.)

43


observed, which is not explainable as due to magnetic excitations. This behavior provides evidence for the first realization of quadrupolar excitons (propagating quadrupole fluctuations mediated principally by AFQ interactions) (Shiina et al., 2004). Both the energy E(T) and the width DE(T) of the excitation peak at q ¼ (1, 0, 0) decrease with decreasing temperature. Crossing Ts, while no noticeable anomaly appears in E(T) within the experimental accuracy, DE(T) shows a sharp decrease just below Ts indicating enhancement of the exciton’s lifetime in the SC state. The relation between the quadrupolar excitons and the superconductivity should be investigated further. Unconventional superconducting properties in PrOs4Sb12. Many unconventional features of the SC state are reviewed here briefly, focusing on the roles played by Pr 4f-electrons; for further details, refer to Aoki et al. (2007b). The basic SC parameters are listed in Table 1.4, obtained using Ginzburg2Landau theory. The reference compound LaOs4Sb12 with no 4f-electron is a conventional s-wave superconductor Table 1.4 Superconducting parameters of Pr-based filled skutterudites and their isostructural La-based reference compounds. Superconducting transition temperature Ts, specific heat jump at Ts: DC/gnTs, upper critical field Hc2, the slope in Hc2 versus T at around Ts, and CEF excitation energy to the first excited state D1st are listed. Ts (K)

DC/cnTs (T/K)

Hc2 (T)

PrOs4Sb12

1.85

2

2.3

1.9

LaOs4Sb12

0.74

1.18

0.04

0.095

PrRu4Sb12

1.03

1.87

0.2

0.24

LaRu4Sb12

3.58

2.22

0.28

0.12

PrRu4As12

2.33

0.83

0.62

0.37

1.75

0.72

0.08

Material

LaRu4As12 10.3

[dHc2/dT]Ts (T/K)

D1st (K)

8

References

Bauer et al. (2002c), Goremychkin et al. (2004), Kuwahara et al. (2004) Aoki et al. (2005a), Sugawara et al. (2005a) 65 Adroja et al. (2005), Takeda and Ishikawa (2000a) Takeda and Ishikawa (2000a) B30 Namiki et al. (2007), Shirotani et al. (1997) Shirotani et al. (1997)

44

H. Sato et al.

with Ts ¼ 0.74 K, as suggested by a clear Hebel2Slichter peak in 1/T1 of Sb-NQR (Kotegawa et al., 2003). The strongly enhanced Ts for PrOs4Sb12, that is, Ts(PrOs4Sb12)/Ts(LaOs4Sb12) ¼ 2.5, indicates that the Pr 4f-electrons do not cause pair-breaking scattering but play an essential role for the Cooper pairing. Enhanced values of the slope in the upper critical field Hc2 versus T at T ¼ Ts (1.9 T/K for PrOs4Sb12, being 20 times larger than that for LaOs4Sb12) and the size of the specific heat jump at Ts evidences highly enhanced effective masses of quasiparticles that are involved in the Cooper-pair formation. In (Pr1xLax)Os4Sb12, the two completely different SC states are smoothly connected in the Ts versus x phase diagram (Rotundu et al., 2006). This is quite different from the behavior of the majority of HF superconductors, where chemical substitution rapidly suppresses Ts. An Sb-NQR study has revealed gradual development of the unconventional superconductivity with changing x (Yogi et al., 2006), which can be explained by assuming multiband superconductivity (MBSC). In contrast, the SC state in PrRu4Sb12 is a conventional type with Ts ¼ 1.03 K (Takeda and Ishikawa, 2000a; Yogi et al., 2003). This fact suggests that the Pr-ion state with one order of magnitude larger DCEF in PrRu4Sb12 is less effective in realizing unconventional superconductivity. In Pr(Os1xRux)4Sb12, superconductivity also appears for all values of x, although a minimum of Ts exists at xB0.6 (Frederick et al., 2004; Ochiai et al., 2007; Ho et al., 2008). In PrOs4Sb12, the specific heat shows a sample-dependent structure around the SC transition (Aoki et al., 2003a, 2003b; Bauer et al., 2002c; Grube et al., 2006; Maple et al., 2002; Measson et al., 2004; Vollmer et al., 2003). Some samples show clear double jumps as observed in UPt3 (Brison et al., 2000; Joynt and Taillefer, 2002), and from this observation a possibility of multiple SC phases was initially pointed out. However, the structure varies strongly from sample to sample, and the observation of a sharp single jump in a small piece cut out from an as-grown single crystal (Seyfarth et al., 2006) may suggest an inhomogeneous Ts distribution in the samples. Strong-coupling nature of the superconductivity is indicated by the T-dependences of several physical quantities. The estimated sizes of the SC gap are Ds/kBTC ¼ 3.7, 2.7, 2.6, and 2.1 from specific heat (Grube et al., 2006), Sb-NQR 1/T1 (Kotegawa et al., 2003), magnetic penetration depth l in radio frequency (rf ) fields (Chia et al., 2003), and l in transverse-field muon-spin rotation (TF-mSR) (MacLaughlin et al., 2002), respectively, although tunneling spectroscopy (Suderow et al., 2004) yields 1.7 being close to the weak-coupling BCS value of 1.77. Symmetry of superconducting order parameter. The order parameter of an SC state is expressed in terms of the gap function Ds(k), which reflects the symmetries broken in the SC state (Sigrist and Ueda, 1991). In order to


45

determine the SC gap structure (the nodal structure) in PrOs4Sb12, several types of experimental techniques have been applied so far. Oscillating patterns in the magnetic-field-direction dependence of thermal conductivity k(H) indicate possible existence of two distinct SC phases with different symmetries (two- and four-fold) with point nodes (Izawa et al., 2003) (the proposed phase boundary is drawn in Fig. 1.26a). It was argued that a flux-line lattice distortion from an ideal hexagonal lattice observed by small angle neutron scattering measurements supports the existence of nodes that break the cubic symmetry (Huxley et al., 2004). In a flux-flow resistivity study, an enhancement in the pinning force is observed near the proposed phase boundary (Kobayashi et al., 2005); however, no other report on anomalies in other physical quantities has yet been made. In an angle-resolved specific heat study, four-fold oscillations always appear in C(H) without any sign of cubic-symmetry breaking (Sakakibara et al., 2008). The node structure in the SC state, if it exists, can be investigated by measuring the T-dependences of physical quantities in T{Ts by probing thermal quasiparticle excitations. In PrOs4Sb12, however, the existence of the low-energy Pr CEF excitations prevents conclusive interpretations. For example, electronic specific heat Ce(T) is complicated by the Schottky tail extending down to T{Ts. As was inferred from the strong q and Tdependences of the excitonic excitation spectra, it is not easy to subtract such a contribution from the Ce data. The spin-lattice relaxation rate 1/T1 in Sb-NQR study exhibits no Hebel2Slichter peak just below Ts (Kotegawa et al., 2003), attributable to a strong-coupling effect. For 1/3oT/Tso1, an exponential T-dependence is observed in the quasiparticle part of 1/T1, suggestive of no nodes in the gap. The magnetic penetration depth l(T), which is related to the superfluid density, has been measured by different techniques. The l(T) data obtained by TF-mSR (MacLaughlin et al., 2002) shows an exponential T-dependence, indicating a fully opened gap. In contrast, l(T) in the Meissner state for rf fields of 21 MHz shows a T 2-dependence for T/Tso0.3, consistent with the pointnode scenario (Chia et al., 2003). Tunneling spectroscopy measurements indicate that the gap opens over a large part of the FS (Suderow et al., 2004). In spite of all these measurements sensitive to the low-energy quasiparticle excitations, the symmetry of the SC order parameter (cubic or lower than cubic?) and the nodal structure in the energy gap (point nodes or deep minima along the [1 0 0] directions, or fully opened for all the directions?) is not yet settled. Another interesting feature in the SC state is possible MBSC. At temperatures far below Ts, the H-dependence of k shows a sharp increase in low fields accompanied by a plateau behavior around H/Hc2B0.4 (Seyfarth et al., 2005). This behavior has a close similarity with that observed in MgB2, which has been established to be a typical two-band

46

H. Sato et al.

superconductor. An upward curvature in Hc2(T) for TrTs (Measson et al., 2004) and the x-dependent Sb-NQR data in (Pr1xLax)Os4Sb12 (Yogi et al., 2006) can also be understood in the MBSC scenario. The issue of the seemingly contradictory data discussed in this section might be solved by reanalysis taking into account the idea of MBSC. To clarify the spin state, Knight-shift measurements have been made to investigate spin susceptibility ws. The muon Knight shift (Higemoto et al., 2007) for H//[1 0 0] shows no sign of decrease in ws below Ts, indicating a spin-triplet (odd-parity) SC state. Time-reversal symmetry (TRS) is another important factor to characterize an SC state. In a state with broken TRS, the magnetic moments (spin or/and orbital) of Cooper pairs are nonzero and ordered in the k space either ferromagnetically or antiferromagnetically (like a SDW state), and thereby a spontaneous but extremely small internal magnetic fields can appear inside the superconductor. In PrOs4Sb12, zero-field muon-spin relaxation (ZF-mSR) has detected static spontaneous internal fields (122 104 T at T-0), which appeared just below Ts, indicating broken TRS in the SC state (Aoki et al., 2003b). Note that similar behavior was reported in Sr2RuO4 (Luke et al., 1998). Regarding which part of the Cooper pairs (spin and/or orbital) carries a nonzero magnetic moment, a definite conclusion has not yet been reached. In LaOs4Sb12 and PrRu4Sb12, no sign of spontaneous internal fields is observed (Adroja et al., 2005; Aoki et al., 2005a), in good agreement with the widely accepted interpretation that these are conventional superconductors. ZF-mSR study has been extended to (Pr1yLay)Os4Sb12 and Pr(Os1xRux)4Sb12 (Shu et al., 2007), where Ru doping is more efficient than La doping in suppressing the spontaneous fields. Possible mechanisms of superconductivity in PrOs4Sb12. Concerning the mechanism of HF behaviors and the unconventional superconductivity, the rattling motion (or anharmonic vibrations) of Pr ions may be considered as a possible origin. Although several aspects of the anharmonic Pr-ion vibration modes have been detected experimentally (Goto et al., 2004; Iwasa et al., 2006; Kaneko et al., 2006; Ogita et al., 2006), correlation with the superconductivity has not been clarified yet at the moment. The importance of quadrupole degrees of freedom in the well-localized 4f-electrons is reflected in the appearance of the field-induced AFQ ordered phase and the excitonic quadrupole fluctuations. The phase diagram of Fig. 1.26a has a close resemblance to those for the typical HF and cuprate superconductors, where a magnetically ordered phase appears in the vicinity of the SC phase when a certain parameter (pressure, atomic doping, or oxygen content) is varied (Cox and Maple, 1995; Mathur et al., 1998). In such superconductors, it is widely believed that the Cooper pairing is mediated by magnetic fluctuations. Therefore, one can naturally


47

infer from the phase diagram of PrOs4Sb12 that excitonic quantum quadrupole fluctuations of the Pr ions may play an important role in the exotic Cooper pairing, as well as in the heavy quasiparticle formation. 4.3.5. Heavy-fermion multiple ordered state in PrOs4As12 PrOs4As12 shows an AFM transition at TN ¼ 2.3 K, where a sharp cusp appears in the magnetic susceptibility wm(T) (Yuhasz et al., 2006). At high temperatures, a Curie2Weiss behavior shows yp ¼ 29 K and meff ¼ 3.81mB/f.u., which may indicate additional Pauli paramagnetic contribution. A reduced value of meff ¼ 2.77mB/f.u. below 20 K and saturating magnetization of 1.68mB/f.u. at 2 K are attributable to the CEF effect. The localized character of 4f-electrons has been confirmed by INS experiments and the CEF level scheme has been determined to be Gð2Þ 4 , G 1, Gð2Þ et al., 2008). 4 , and G23 at 0, 0.4, 13, and 23 meV, respectively (Chi Therefore, the low-lying well-separated triplet2singlet (Gð2Þ 4 G1 ) governs the low-temperature physics. The electronic specific heat coefficient g is estimated to be 502200 mJ/K2 mol in the normal state and B1 J/K2 mol below TN (Yuhasz et al., 2006). In the paramagnetic state, the electrical resistivity shows a minimum at B15 K suggesting single-ion Kondo scatterings; note that a negative dip appears in the thermoelectric power at 16 K (Wawryk et al., 2008) as a related anomaly. In this temperature range, the field dependences of specific heat and resistivity can be explained by the single-ion Kondo model and the Kondo temperature TK is estimated to be 123.5 K (Maple et al., 2006). The AFM state consists of different phases as shown in Fig. 1.28, where the phase boundaries were determined by magnetization measurements (Ho et al., 2007), specific heat (Maple et al., 2006), and elastic constant (Yanagisawa et al., 2008). The complexity may be relevant to the multipolar degrees of freedom of Pr ions; note that a presence of AFQ-type interactions is inferred from the ultrasound measurement (Yanagisawa et al., 2008). The well-localized 4f-electrons have also been confirmed by dHvA experiments on PrOs4As12 and LaOs4As12 (Ho et al., 2007). The observed cyclotron effective mass (o6m0) decreases with increasing magnetic field, which may be consistent with the aforementioned Kondo behaviors. 4.3.6. Other Pr-based filled skutterudites PrOs4P12 is a metal, which shows no indication of phase transition down to 50 mK (Sugawara et al., 2009). Magnetic susceptibility wm(T) shows a Curie2Weiss behavior with meff ¼ 3.63 mB and yp ¼ 17 K (Sekine et al., 1997; Yuhasz et al., 2007), indicating that magnetic moments of Pr3 + ions have AFM interactions. A broad peak at 12 K in wm(T) suggests nonmagnetic CEF ground state. The electrical resistivity (see Fig. 1.16)

48

H. Sato et al.

RtQu Cu 6

34

JO3 JO4 VO3 VO4

JY JY VY VY VE3 VE4

RO

J"*V+

QR4

CHO

V"*M+

Figure 1.28 H versus T phase diagram of PrOs4As12 determined by temperature and magnetic field dependences of magnetization, specific heat and electrical resistivity (Yuhasz et al., 2006, Maple et al., 2006) The lines are guides to the eye, separating the paramagnetic region from the ordered phases. (Reproduced with permission from Maple et al., 2008.)

decreases monotonically from B150 mO cm at room temperature and shows a roll-off behavior near 70 K (Yuhasz et al., 2007), which is caused by a low-energy singlet2triplet (G1 Gð2Þ 4 ) CEF excitations. In specific heat C(T), the corresponding Schottky anomaly appears around 13 K and, from this anomaly, the energy separation D(G1 Gð2Þ 4 )B44 K is estimated (Matsuhira et al., 2005b). The electronic specific heat coefficients g of PrOs4P12 and LaOs4P12 are estimated to be 56.5 and 21.6 mJ/K2 mol, respectively, indicating the mass enhancement factor due to 4f-electrons to be 2.6. The cyclotron effective mass mc ¼ 18 m0 obtained from the dHvA effect measurements for a-branch in the 48th main band (Sugawara et al., 2009) is B3.8 times that for LaOs4P12, reflecting mass enhancement associated with f-electron. PrFe4As12 shows an FM transition at TC ¼ 18 K (Henkie et al., 2008; Maple et al., 2008; Sayles et al., 2008). The electrical resistivity shows a metallic behavior with a sharp drop below TC and thermoelectric power exhibits a steep peak (B20 mV/K) at B9 K in the ordered state. A hysteretic behavior in wm(T), a broad peak in C(T), and an anomaly in the elastic constant, all observed near 12 K, suggest a change in magnetic and/ or structural order at around this temperature. In the paramagnetic state,


49

magnetic susceptibility shows a Curie2Weiss behavior with meff ¼ 3.98mB/ f.u., indicating 3d-electron contribution of meff ¼ 1.74mB/Fe4As12. Reduced value of meff ¼ 3.52mB/f.u. below 40 K and saturating magnetization of 2.3mB/f.u. in the FM state suggests a magnetic G5-like triplet ground state. The electronic specific heat coefficient g is 340 mJ/K2 mol in the normal state, indicating moderate mass enhancement. PrRu4As12 is a superconductor with Ts ¼ 2.4 K (Shirotani et al., 1997). The electrical resistivity shows a broad roll-off behavior around 50 K. Thermoelectric power is + 18 mV/K at 300 K (Henkie et al., 2008). The electronic specific heat coefficient g is B95 mJ/K2 mol, which is 1.5 times larger compared with that in LaRu4As12 (Namiki et al., 2007). The specific heat data and a Hebel2Slichter peak in 1/T1 in 75As-NQR studies (Shimizu et al., 2007) suggest a conventional s-wave superconductivity with a weak-coupling constant (2D/kBTC ¼ 3.5). Magnetic susceptibility shows a Curie2Weiss behavior with meff ¼ 3.30mB/f.u. and yp ¼ 11 K (Namiki et al., 2007). It is inferred that the low-energy CEF level scheme consists of singlet2triplet with D(G1 Gð1Þ 4 ) B30 K. PrFe4Sb12 shows different magnetic properties depending on the Pr-site occupancy (or the sample preparation process). For the sample with Pr-site filling fraction x ¼ 0.73 made by arc melting (Bauer et al., 2002a, 2002b) and x ¼ 0.87 grown by Sb-flux method (Butch et al., 2005), magnetic ordering appears at 4.6 and 4.1 K, respectively, exhibiting broad peaks in specific heat. In both samples, a magnetic triplet CEF ground state (G5 (Oh)-like) is inferred. The large value of meff ¼ 4.224.3mB/f.u. estimated from Curie2Weiss behaviors at high temperatures suggests the contribution from the itinerant Fe 3d-electrons (meff ¼ 2.722.8mB/f.u.). A neutron scattering experiment on a sample with x ¼ 0.87 indicates a complex magnetic structure, possibly ferrimagnetic, consisting of Pr and Fe magnetic moments (Butch et al., 2005). The internal fields at Fe sites have been studied by 57Fe Mo¨ssbauer experiments (Reissner et al., 2004). For x ¼ 0.87, magnetic critical exponents obtained by modified Arrott plot analysis differ significantly from ordinary reported magnetic systems. In applied fields of B3 T, NFL behaviors appear in specific heat and resistivity at low temperatures (Bauer et al., 2002b; Butch et al., 2005). In contrast, samples with almost full Pr-site filling fraction synthesized at 4 GPa exhibit no magnetic ordering down to 0.1 K, although meff ¼ 4.5/f.u. estimated at high temperatures is not much different (Tanaka et al., 2007). A broad peak in the magnetic susceptibility and a Schottky peak in the specific heat indicate that the CEF ground state is a singlet G1 with a triplet Gð2Þ 4 first excited state lying at 22 K, which have been confirmed by the INS measurements (Kuwahara et al., unpublished data). A metamagnetic anomaly at 16 T in the magnetization curve is probably due to a level crossing in this CEF level scheme (Yamada et al., 2007). Such drastic change in the CEF level scheme is attributable to the local symmetry lowering

50

H. Sato et al.

caused by the Pr-site deficiency and/or a change in D(EF) due to the Fermi level shift. The latter effect may be reflected in the change in the thermoelectric power and Hall coefficient (Tanaka et al., 2008). PrRu4Sb12 is a Van Vleck paramagnet with a CEF singlet ground state and becomes superconducting below Ts ¼ 1.03 K (Takeda and Ishikawa, 2000a). The electronic specific heat coefficient g ¼ 59 mJ/K2 mol is 60% larger than that of LaRu4Sb12. At Ts, DC/gTs ¼ 1.87, which is slightly larger than the BCS value of 1.43. From a Curie2Weiss behavior above 50 K, meff ¼ 3.58 mB/f.u. and yp ¼ 11 K are obtained. INS shows a G1 singlet ground state with a G5 (Oh)-like triplet lying at 5.6 meV (Adroja et al., 2005). A broad roll-off behavior around 70 K in r(T) (Abe et al., 2002), absence of softening in the elastic constant (Kumagai et al., 2003), and a decrease in the Lorenz number determined by thermal conductivity measurements (Rahimi et al., 2008) are consistent with this CEF level scheme. The dHvA measurements on PrRu4Sb12 and LaRu4Sb12 revealed well-localized nature of 4f-electrons with only minor mass enhancement factor of 1.4 due to 4f-electrons in PrRu4Sb12 (Harima and Takegahara 2002; Matsuda et al., 2002a). PrRu4Sb12 is judged to be a typical weak-coupling s-wave superconductor. A distinct Hebel2Slichter peak appears in Sb-NQR 1/T1 (Yogi et al., 2003), and the penetration depth decreases exponentially (Chia et al., 2004), suggesting 2D(0)/kBTs ¼ 3.123.8. The upper critical field Hc2 is B0.2 T Takeda and Ishikawa (2000a). Thermal conductivity measurements suggest full gap superconductivity with multiband effects (Hill et al., 2008). Detailed comparison with the HF superconductor PrOs4Sb12 is given in the section for PrOs4Sb12. INS and Raman scattering experiments (Iwasa et al., 2007; Ogita et al., 2008) have clarified that the optical modes exhibit softening behavior, which is attributable to the anharmonic motion of the filled Pr ions within the Sb cage. A peak appears in 1/T2 of Sb-NQR at B100 K (Kotegawa et al., 2008), which suggests slowing down of the charge fluctuations of the Sb cage.

4.4. Nd-based filled skutterudites All of the investigated Nd-based filled skutterudites exhibit FM transitions except NdRu4Sb12 whose order parameter has not yet been settled. In all the systems, 4f-electrons are thought to be well localized, although NdOs4Sb12 is reported to exhibit HF behaviors at low temperatures (Ho et al., 2005; Maple et al., 2005). As another common feature to most Nd-based skutterudites, the resistivity increases logarithmically with decreasing temperature from far above and down to BTC. The origin has not yet been clarified, though the strong suppression by magnetic fields indicates magnetic origin (Sato et al., 2000; Torikachvili et al., 1987). Brief description on each material is given below.

51


μ0H = 0 T

1.2

ρ / ρΤ =46Κ

1.5 T 0.8

NdFe4P12 10 T

0.4

I // [100], H // [110] 0 0

10

20

30

40

50

T (K)

Figure 1.29 Temperature dependence of the normalized electrical resistivity under selected magnetic fields (Sato et al., 2003).

NdFe4P12 is an FM metal with TCB2 K (Torikachvili et al., 1987). The resistivity decreases monotonically from B150 mO cm at room temperature and shows a minimum near 30 K. Below 30 K, the resistivity increases with decreasing temperature (Fig. 1.29) up to a peak near TC, below which it sharply decreases down to less than 1 mO cm. The resistivity increase is strongly suppressed by applied magnetic field. At low temperatures below TC, the temperature dependences of the electrical resistivity and the specific heat may be dominated by magnons. For a simple isotropic ferromagnet, one can naively expect the temperature dependences of the electrical resistivity and heat capacity to be T 2 and T 3/2, respectively. However, T 4 (Sato et al., 2000) and T3 (Torikachvili et al., 1987) dependences were experimentally observed below 2 K for r and C, respectively. Both exponents of the temperature dependences can be naturally understood, if the magnon energy dispersion is linear in wave number q rather than the q2-dependence (Sato et al., 2000). The magnetic susceptibility approximately follows Curie2Weiss law above B150 K with an effective moment of 3.53mB, close to the Nd3 + free-ion value of 3.62mB. This is consistent with the de Hass vanAlphen experiment, where the detected FS in NdFe4P12 is close to that in LaFe4P12 (except for the small spin splitting of the dHvA branches due to the FM exchange interaction), indicating well-localized nature of 4f-electrons (Sugawara et al., 2000). The downward deviation of 1/w from Curie2Weiss law below B150 K indicates the presence of CEF splitting. The ordered phase was confirmed to be FM with the ordered moment of 1.6 mB/Nd at 2 K by neutron scattering measurements, in agreement with the bulk saturation magnetization 1.72 mB/Nd at 1.4 K under 5 T (Keller et al., 2001, Torikachvili et al., 1987).

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NdRu4P12 was reported to be a metallic ferromagnet with TC ¼ 1.7 K (Masaki et al., 2008). In the Knight-shift measurements of 31P-NMR and 101 Ru-NQR, Masaki et al. (2008) found that the FM spin correlation coexists with the antiferro-type fluctuation of q 6¼ 0 above TC. As the origin of such dual spin fluctuations, they suggested the instability of localized 4f-electrons stimulated by the FS nesting with q ¼ (1, 0, 0) which is a common feature to RRu4P12 (Harima, 2008). For NdOs4P12, only the crystallographic data have been reported ( Jeitschko and Braun, 1977). NdFe4As12 is a metallic ferromagnet with TCB15 K (Takeda et al., unpublished data). The magnetic susceptibility follows Curie2Weiss law above B150 K, and the effective moment and paramagnetic Curie temperature are estimated as meff ¼ 4.7mB/f.u. and yp ¼ 45 K, respectively. At around 6 K below TC, the specific heat exhibits a clear Schottky peak and the electrical resistivity shows a shoulder-like structure, reflecting an energy level splitting of 4f-electrons. The larger value of meff compared to the Nd3 + free-ion value of 3.62mB, indicates a considerable contribution from the itinerant Fe 3d-electrons, as expected from the itinerant electron ferromagnetism in LaFe4As12. NdRu4As12 is an FM metal with TCD2.3 K and the magnetization of B1.9mB at 2 K under 7 T (Namiki et al. unpublished data). NdOs4As12 is a metal exhibiting an FM-like transition at TCB1.6 K and the magnetization at 2 K under 7 T is B1.8mB (Namiki et al., unpublished data). NdFe4Sb12 was reported to be a ferromagnet by two research groups, however, there was an evident discrepancy between the reported ordering temperatures 13 K (Danebrock et al., 1996) and 16.5 K (Bauer et al., 2002b). Recently, the systematic variation of TC with the filling fraction (x) of the Nd sites has been found as the origin of the discrepancy based on the specific heat and the magnetic measurements on NdxFe4Sb12 single crystals. TC increases linearly from 8.6 to 15 K with increasing x from 0.77 to 0.90 as shown in Fig. 1.30 (Ikeno et al., 2008). The incomplete filling of the rare earth ion sites is now recognized as a general tendency for Sb-based filled skutterudites. Moreover, in Fe-based skutterudites, due to the large 3d-electrons’ contribution to D(EF), the magnetic ground state tends to be stabilized as a combined effect of 3d- and 4f-electrons. NdRu4Sb12 is a metal exhibiting a magnetic transition near 1.3 K. The inverse magnetic susceptibility versus T plot shows a shoulder near 50 K, suggesting crystal field splitting of the same order of magnitude (Takeda and Ishikawa, 2000a). From Curie2Weiss law above 50 K, an effective moment of B3.5 mB and a paramagnetic Curie temperature of around 30 K have been estimated. The magnetic entropy of R ln4 suggests the CEF ground state to be a G68-quartet. Takeda and Ishikawa (2000a) inferred the transition to be AFM based on a peak in AC susceptibility at


53

Figure 1.30 Dependence of M/H versus temperature at 0.01 T on filling fraction x in NdxFe4Sb12 (Ikeno et al., 2008).

1.3 K and the field dependence of magnetization above TC. On the other hand, the field and temperature response of the transport properties point to FM character of the transition (Abe et al., 2002). Note that this material also shows a resistance minimum near 6 K, which is common to Nd-based filled skutterudites (Abe et al., 2002; Sato et al., 2000; Torikachvili et al., 1987). NdOs4Sb12 exhibits a large electronic specific heat coefficient gE520 mJ/K2 mol, suggesting the third HF material in ROs4Sb12 (Ho et al., 2005) after the first Pr-based HF superconductor PrOs4Sb12 and the heaviest Sm-based HF SmOs4Sb12 robust against the magnetic field. The temperature dependence of specific heat at low temperatures is shown in Fig.1.31. As supporting evidence, Imada et al. (2007) reported a Kondolike state ascribable to the hybridization between the Nd 4f and the conduction band states, based on the XAS and SXPES measurements. NdOs4Sb12 exhibits an FM transition at B0.9 K with a saturation magnetization Ms ¼ 1.73mB estimated at 0.4 K. The magnetic susceptibility follows Curie2Weiss law above B100 K with a negative paramagnetic Curie temperature yp ¼ 243 K and an effective moment meffD3.84 mB close to the Nd3 + free-ion value. Ho et al. (2005) proposed two CEF level ð2Þ ð2Þ schemes, (I) G6 (Oh) (0 K), Gð1Þ 8 (Oh) (180 K), G8 (Oh) (420 K) and (II) G8 ð1Þ (Oh) (0 K), G8 (Oh) (220 K), G6 (Oh) (600 K), by fitting the experimental data of the magnetic susceptibility and electrical resistivity. Based on the

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40

C/T (mJ/mol-K2)

1600

C (J/mol-K)

30

1200

20

NdOs4Sb12 800 400 0

100

200

300

400

NdOs4Sb12

T2 (K2)

10

Cel + Clat γ ~ 520 mJ/mol-K2 ΘD ~ 255 K, ΘE ~ 39 K, r ~ 0.48

0

0

5

10 T(K)

15

20

Figure 1.31 Temperature dependence of the specific heat in NdOs4Sb12. The inset shows C/T versus T2 plot at low temperature. (Reprinted figure with permission from Ho et al. (2005), by the American Physical Society r 2009.)

neutron inelastic scattering experiments, Kuwahara et al. (2008) determined the energy splittings from the CEF ground state to the first and the second excited states to be 267 and 350 K, respectively. They proposed two ð1Þ possible CEF level schemes, (III) Gð2Þ 8ð1Þ (Oh) (0 K), G6 (Oh) (267 K),G8 (Oh) ð2Þ (350 K) and (IV) G8 (Oh) (0 K), G8 (Oh) (267 K), G6 (Oh) (350 K), both of which also well reproduce the temperature dependence of magnetic susceptibility. The ground state is settled to be Gð2Þ 8 (Oh) (0 K), however, there is no decisive evidence to select either model (III) or model (IV) at this stage.

4.5. Sm-based filled skutterudites 4.5.1. General trend of the Sm-based filled skutterudites Among the filled skutterudites, the Sm-based filled skutterudites have been most intensively investigated next to the Pr-based system. That is because some of them exhibit unprecedented features, such as the HF behavior robust against magnetic fields in SmOs4Sb12, the new type of multipolar ordered state in SmRu4P12, and the first Sm-based HF ferromagnetism in SmFe4P12. From those novel features, one might expect totally new type of mechanisms related with 4f-electrons. All of them exhibit unique features depending on the constituent elements A, T, and X, while they also exhibit a common feature: that is, most of them show ln(T)-dependence in the electrical resistivity at high temperatures as shown in Fig. 1.32, reminiscent of Kondo effect.

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(a)

Δρ (arb. unit)

0.8 Δρ=ρ(SmFe4P12)-ρ(LaFe4P12) 0.4

Δρ (arb. unit)

0

Δρ (arb. unit)

10

T (K) 30

300

100

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Δρ=ρ(SmRu4P12) -ρ(LaRu4P12) 0.10

Δρ (arb. unit)

300 (c)

Δρ=ρ(SmOs4Sb12) -ρ(LaOs4Sb12) 200

100

Δρ

0 1

10

100 T(K)

Figure 1.32 Temperature dependence of the magnetic part of electrical resistivity for the three Sm-based skutterudites (Sato et al., 2007a).

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In this section, the attractive part of behaviors found in SmFe4P12, SmRu4P12, and SmOs4Sb12 are first introduced, followed by the comparison with other Pr-based filled skutterudites. 4.5.2. Heavy-fermion ferromagnet SmFe4P12 SmFe4P12 was reported to be the first Sm-based Kondo-lattice compound exhibiting an FM transition at TC ¼ 1.6 K as shown in Fig. 1.33a (Takeda and Ishikawa, 2003). The magnetization M ¼ 0.16mB/Sm at 7 T and 2.0 K is far less than 0.71mB expected for the free Sm3 + ion, and the entropy of B0.15 J/K mol at TC is far less than R ln2, suggesting some novel mechanism to realize such FM state in SmFe4P12. To clarify the origin of such unusual FM state, Hachitani et al. (2006b) applied the zero-field mSR measurement to this system, and succeeded in observing an increase of the relaxation rate, indicating the development of a static internal field below TC. At the lowest temperature, they confirmed the oscillation in ZF-mSR spectra evidencing the bulk ferromagnetism in this material. The magnetic part of electrical resistivity rm, calculated by subtracting the phonon part of the electrical resistivity using r(T) for LaFe4P12, shows an evident ln(T)-dependence below RT down to a peak near 40 K, indicating the formation of a Kondo lattice with an effective Kondo temperature TKB40 K as shown in Fig. 1.32. Near TC ¼ 1.6 K, clear anomalies reflecting some phase transition are found in the bulk properties. Both the extrapolated specific heat coefficient gD0.37 J/K2 mol (Fig. 1.33b) and the coefficient AD0.23 mO cm/K2 of T 2-dependence of r indicate a remarkably HF state; for example, the second heaviest in Sm-based compounds next to SmOs4Sb12. In fact, a highly enhanced cyclotron effective mass of up to 1.5

500

(b)

(a)

C/T (J/mol•K2)

M (arb.units)

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200 SmFe4P12

1.0

0.5

100

0

0

2

1 T (K)

3

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T2 (K2)

Figure 1.33 Temperature dependences of (a) magnetization and (b) C/T under selected magnetic fields. (Reproduced with permission from Takeda and Ishikawa, 2003).

4


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B9m0 (m0: electron’s rest mass) has been directly detected in a dHvA experiment, although the expected heavier mass branches for the 48th open FS have not yet been detected (Kikuchi et al., 2007a). Matsuhira et al. (2005a) evaluated the CEF ground state to be G5 with the excited state G67 above B70 K, by analyzing the temperature dependence of specific heat, although they failed to reproduce the experimental result quantitatively. Hachitani et al. (2006b) found the typical behavior of the Kondo system under the CEF splitting of B70 K in the temperature dependence of the relaxation rate 1/T1 in the 31P-NMR measurement. At low temperatures, Takeda et al. (2008) found a clear metamagnetism in M at about HM ¼ 22 T, which smears out with increasing temperature (Fig. 1.34). They also investigated the La substitution effect for Sm (SmxLa1xFe4P12). They found the field (Hc) where the metamagnetism appeared to be increasing with increasing La concentration, suggesting the importance of the intersite interactions compared to the local interactions. They notified that the metamagnetic behavior appears even in the paramagnetic samples (x ¼ 0.7 and 0.8) and pointed out the similarity to the metamagnetic anomaly in CeRu2Si2. Note that the anomaly in CeRu2Si2 was inferred to reflect the crossover behavior of 4f-electrons from the itinerant state below Hc to the localized state above Hc.

Figure 1.34 Magnetization curves of SmFe4P12 at selected temperatures. The origin of vertical axis for each curve is shifted by 0.1lB/f.u. (Reproduced with permission from Takeda et al., 2008).

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However, the detected dHvA branch in SmFe4P12 below Hc has both magnitude and angular dependence close to those in LaFe4P12, suggesting the localized nature of 4f-electrons (Kikuchi et al., 2007a). Furthermore, Sugawara et al. (unpublished data) found no essential change in the dHvA frequency across Hc. A model to explain such a metamagnetic behavior has been proposed to explain the double peak structure in field dependence of specific heat in CeRu2Si2, assuming a sharp peak located close to EF by Aoki et al. (1998). In fact, the sharp peak in D(E) near EF is one of the typical features in metallic RFe4P12; however, the field-dependent specific heat measurement across Hc has not yet been done due to the large Hc. 4.5.3. Metal2insulator transition and multipole in SmRu4P12 SmRu4P12 was reported to show an M2I transition at TMI ¼ 16 K. The detailed measurements of r(T), C(T), and M(T) up to high magnetic fields revealed that the transition appears in two successive steps (Matsuhira et al., 2002b). Figure 1.35a shows the temperature dependence of C(T) under m0H ¼ 0 and 9 T. With increasing field, a round peak appears below TMI, which becomes evident at TX ¼ 13 K under 9 T. The H2T phase diagram is shown in Fig. 1.35b (Sekine et al., 2003). The magnetic entropy estimated in zero magnetic field reaches R log4 at TMI, suggesting that the CEF ground state is a quartet G67 (Matsuhira et al., 2005a). The l-type anomaly C(T) at TMI clearly observed in zero field moves to the higher temperatures with increasing magnetic field, while the anomaly at lower temperature TX (oTMI) indistinct below B5 T moves to the lower temperatures with increasing magnetic fields. The transition at TMI has been suggested to be an octupole ordering of 4f-electrons in Sm (Aoki et al., 2007a; Hachitani et al., 2006a; Ito et al., 2007; Yoshizawa et al., 2005). Even though any structural distortion has not been detected in lower temperatures, the Ru-NQR measurements have revealed that the crystal symmetry becomes lower and the quadrupole components develop under TMI (Masaki et al., 2007). However, the origin of the anomaly at TX, which was initially inferred to be an AFM transition, is still controversial. An evident gap opening (EgB0.01 eV) similar to that in PrRu4P12 was observed in the optical measurement (Matsunami et al., 2005); however, the M2I transitions have different features between the two compounds. Magnetic susceptibility shows a clear anomaly at TMI in SmRu4P12 while no anomaly was found at TMI in PrRu4P12 (Sekine et al., 1997). 4.5.4. Heavy-fermion state in SmOs4Sb12 robust against magnetic field SmOs4Sb12 is a rare Sm-ion-based HF material with the electronic specific heat coefficient g ¼ 0.82 J/K2 mol (Sanada et al., 2005; Yuhasz et al.,

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SmRu4P12

30

H (T)

25 20 15 I

II

III 10 5

TQ

TN 0 10

12

14

16

18

20

T(K)

Figure 1.35 (a) Temperature dependence of the specific heat in 0 and 9 T in SmRu4P12. (Reproduced with permission from Matsuhira et al., 2002b). (b) H2T phase diagram determined by various experiments in SmRu4P12. (Reproduced with permission from Sekine et al., 2003). Phase II corresponds to an octupole ordering. Phase III, which was initially inferred to be an antiferromagnetic ordered phase, is still controversial.

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Figure 1.36 Field dependences of the electronic specific heat coefficient c and the coefficient A of the quadratic T dependence in the electrical resistivity of SmOs4Sb12 (Sanada et al., 2005).

2005), which is the largest among the known Sm-based compounds (see Fig. 1.36). This value of g is 15 times larger than that for the no-4f-electron reference material LaOs4Sb12. In the electrical resistivity r(T), a quadratic temperature dependence dominates at low temperatures and its coefficient A is largely enhanced. However, as shown in Fig. 1.37, the ratio A/g2 ¼ 1.1 106 mO cm (mol K/mJ)2 is smaller than 1 105 mO cm (mol K/mJ)2, which is the Kadowaki2Woods ratio of HF materials. This discrepancy may be caused by the orbital degeneracy of the relevant 4f-electron states (Tsujii et al., 2005). The observed small but noticeable anisotropy in magnetization, suggesting a dominating quartet character of the 4f-electron state, is in agreement with this scenario (Aoki et al., 2006b). The coherence temperature of the HF state (T B20 K) is inferred from the temperature dependence of the 4f-component in r(T) (Sato et al., 2006), the nuclear spin-lattice relaxation rate 1/T1(T) in Sb-NQR (Kotegawa et al., 2005), and the temperature dependence of the electronic contribution in the specific heat (Matsuhira et al., 2007). The most remarkable feature of the HF state in SmOs4Sb12 is that g does not show any noticeable decrease in applied fields, in contrast with the well-known Ce-based HF compounds, suggesting an unconventional origin of the heavy quasiparticles. At low temperatures, magnetization M(T, H) behaves in a thermodynamically consistent manner with g(H) (Aoki et al., 2006b). In order to explain such a magnetically robust HF state, several theoretical models have been proposed, considering (i) rattling local


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Figure 1.37 Kadowaki--Woods plot with the data point for SmOs4Sb12 (Sanada et al., 2005).

vibrations of Sm ions and/or (ii) electronic charge fluctuations (Hattori et al., 2005; Hotta, 2007; Mitsumoto and Ono, 2005; Tanikawa et al., 2009). In Raman scattering spectra, second-order phonons are observed (Ogita et al., 2006), corresponding to the local vibrations of Sm ions. Large isotropic thermal atomic displacement parameter of the Sm ion, which shows strong temperature dependence, indicates the presence of an anharmonic vibration of Sm ions (Tsubota et al., 2008). The X-ray absorption spectra (Mizumaki et al., 2007) and X-ray photoemission spectra (Yamasaki et al., 2007) have revealed that the valence of Sm ions 2.83 at 300 K shows a significant decrease below 150 K to 2.76 at low temperatures, as shown in Fig. 1.38, indicating a strong mixed valence state. Synchrotron radiation powder diffraction (Tsubota et al., 2008) shows the Sm2Sb bond length decreases rapidly below 150 K, coinciding with the change in the valence of the Sm ion, suggesting that the c-f hybridization is enhanced at low temperatures. Calculated electronic

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2.84

averaged valence

2.82 TA 2.80 TB 2.78

2.76 0

50

100 150 200 Temperature (K)

250

300

Figure 1.38 Temperature dependence of the average valence (Vav) of SmOs4Sb12. The average valence of Sm ions in SmOs4Sb12 is 2.8370.01 at 300 K and is almost constant (temperature-independent) down to 150 K ( ¼ TA). The lowest temperature where Sm (2.83) is observed is defined as TA. Below TA, Vav decreases gradually and saturates to 2.7670.01 at around TB ¼ 20 K. The two hutched areas correspond to the temperature regions in which Vav is constant (Mizumaki et al., 2007).

band structure is consistent with the observed intermediate valence properties (Takegahara and Harima, 2008b). The calculated value of g is about 1/20 of the observed value, indicating strongly correlated electron effects. In 121,123Sb-NQR (Kotegawa et al., 2007, 2008a), a peak appears at around T in the nuclear spin2spin relaxation rate 1/T2, which is pressure dependent. From these observations, charge fluctuations are considered to play an important role in forming the HF state. A weak FM ordering sets in below TCD3 K in the HF state (Sanada et al., 2005; Yuhasz et al., 2005). A corresponding anomaly observed as anomalous Hall effect (Sanada et al., 2005), a broadening in the Sb-NQR spectra (Kotegawa et al., 2005), a softening in the elastic constants (Nakanishi et al., 2006), and a large-amplitude oscillating signal appearing in ZF-mSR spectra (Aoki et al., 2009) have confirmed that the weak FM anomaly is an intrinsic bulk property. From the ZF- and TF-mSR study, the size of the spontaneous magnetic moment Ms at 20 mK has been estimated to be 0.1(1)mB/Sm, which is much smaller than 0.714mB/Sm for the free Sm3 + -ion value. The largely suppressed Ms indicates that the weak FM moment is carried by itinerant heavy quasiparticles. This interpretation is in line with a small anomaly appearing in the specific heat at TBTC. The magnetic critical exponents obtained by modified Arrott plot analysis using the magnetization data differ significantly from those on the ordinary magnetic systems and cannot be adequately explained by classical or


63

quantum critical models (Yuhasz et al., 2005). The pressure dependence of TC suggests SmOs4Sb12 to be in the vicinity of its FM critical point (Kotegawa et al., 2005). In applied pressure, NQR signals are strongly suppressed and vanished below TC, suggesting that the ordered state is not simple ferromagnetism (Kotegawa et al., 2008b). In ambient pressure, magnetization at 1.3 K shows no tendency of saturation even in 45 T (MD0.4mB/Sm) (Yamada et al., 2007). 4.5.5. Other Sm-based skutterudites SmOs4P12 was reported to be an AFM metal with a Neel temperature TN ¼ 4.6 K (Giri et al., 2003). The electrical resistivity r decreases with decreasing temperature down to B40 K. After showing a broad minimum at around 40 K, r increases followed by a sharp drop at around 5 K, reflecting the AFM transition. The temperature dependence of magnetic susceptibility wm could be explained by the theory of Van Vleck2Frank based on the Sm3 + (Kasaya et al., 1980), and shows a distinct cusp at 4.6 K, which also indicates AFM transition. The specific heat shows a pronounced l-type peak at 4.6 K that agrees with temperatures of the cusp in wm and the sharp drop in r. The magnetic entropy estimated in zero field reaches R ln4 at TN, which suggests that the CEF ground state is quartet G67 (Giri et al., 2003; Matsuhira et al., 2005a). The AFM state was also investigated by mSR and P-NMR measurements in microscopic point of view (Hachitani et al., 2007). SmFe4As12 was reported to be an FM metal with a Curie temperature TC ¼ 39 K (Kikuchi et al., 2008a, 2008b). Magnetic part of electrical resistivity rm ( ¼ r(SmFe4As12)r(LaFe4As12)) decreases below room temperature and shows lnT-dependence around 100 K reminiscent of Kondo effect. The magnetization curve at 2 K shows typical FM field dependence with a saturation magnetization of B0.62mB/f.u. which is less than 0.71mB expected for the free Sm3 + ion. The enhanced TC and magnetization compared to those of LaFe4As12 indicate a cooperative contribution to the ferromagnetism in SmFe4As12 from the itinerant Fe 3dand the localized Sm 4f-electrons; the ratios of TC and magnetization to those in the weak itinerant ferromagnet LaFe4As12 are 39 K/5.2 K for TC and 0.62mB/0.23mB for M (2 K, 7 T). SmFe4Sb12 is an FM metal with a Curie temperature TC ¼ 45 K and saturation moment Ms ¼ 0.7mB/f.u. (at 5.5 T and 5 K) (Danebrock et al., 1996). TC and Ms estimated on a sample synthesized at high pressure are 43 K and 0.56mB/f.u. (at 7 T and 2 K) (Ueda et al., 2008), respectively. The origin of such sample dependence was inferred to be the difference in Sm filling fraction depending on the pressure in synthesizing process. The Sm valence was evaluated to be trivalent by an XAS measurement (Mizumaki et al., 2006). As an important experimental fact, the electronic specific heat

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coefficient gB72 mJ/K2 mol is strongly suppressed compared to g ¼ 195 mJ/K2 mol in LaFe4Sb12 (Ueda et al., 2008). That might be an indirect evidence for the large spin-fluctuation contribution to the estimated C/T at 0 K in LaFe4Sb12. In fact, remarkable suppression of C/T by applied magnetic fields was reported by Viennois et al. (2005). In the FM state of SmFe4Sb12, the spin-fluctuation contribution is expected to be suppressed but an internal magnetic state. SmRu4Sb12 has not been investigated except crystallographic measurements (Evers et al., 1995).

4.6. Eu-based filled skutterudites The lattice constants of Eu skutterudites tend to be larger than that expected for Eu3 + (Evers et al., 1995; Jeitschko and Braun 1977), which suggests that the valence of Eu ion deviates from Eu3 + to Eu2 + . EuFe4P12 is an FM metal having a Curie temperature TC ¼ 99 K (Grandjean et al., 1984). The effective magnetic moment 6.2mB/f.u. determined from the high-temperature magnetic susceptibility is smaller than 7.94mB/f.u. expected for Eu2 + , which suggests the valence of Eu to be about 2.4 (Legvold, 1980). The Mo¨ssbauer spectroscopy indicates the absence of Fe moment and suggests the Eu valence to be 2.5 (Ge`rard et al., 1983a, 1983b; Grandjean et al., 1984). EuRu4P12 is an FM metal with a Curie temperature TC ¼ 18 K (Grandjean et al., 1984; Sekine et al., 2000b). The temperature dependence of electrical resistivity r in a single crystalline sample shows a typical metallic behavior and a distinct anomaly reflecting the FM transition at 18 K (Sekine et al., 2000b). The effective magnetic moments determined from the 1/wm2T plots above 50 K are slightly anisotropic; 7.75mB/f.u. for H//[1 0 0], 7.81mB/f.u. for H//[1 1 0], and 7.69mB/f.u. for H//[1 1 1], which are slightly lower than the value of 7.94mB/f.u. expected for Eu2 + . The magnetization at 2 K shows typical FM field dependence and almost saturates near 5 T with the moment MsB6mB/Eu. These facts suggest that the valency of Eu is about 2.1. On the other hand, the Mo¨ssbauer spectroscopy performed on the polycrystalline sample suggests the valence of about 2.2 (Indoh et al., 2002). NMR measurements revealed that the nuclear spin-lattice relaxation rate 1/T1 is larger than that of LaFe4P12 and practically constant at higher temperatures above 50 K suggesting the large fluctuation of local moments at the Eu sites (Magishi et al., 2007a). EuOs4P12 is an FM metal with the Curie temperature TC ¼ 15 K (Kihou et al., 2004). The electrical resistivity r decreases with decreasing temperature, and shows a broad minimum at around 50 K. After showing minimum, r increases with decreasing temperature and shows a broad maximum at around 20 K, followed by a sharp drop at around 5 K, which is associated with the FM transition. The effective magnetic moment


65

estimated from the 1/wm2T plot is 7.81mB/f.u., which is slightly smaller than that expected for Eu2 + . EuFe4Sb12 was initially reported to be an FM metal with a Curie temperature TC ¼ 88 K (Bauer et al., 2004; Danebrock et al., 1996). Mo¨ssbauer measurements show that the Eu valence is divalent (Bauer et al., 2001a). However, the saturation moment Ms ¼ 4.9mB/f.u. at 5 K is smaller than Ms ¼ 7.0mB/Eu expected for Eu2 + . Moreover, the effective magnetic moment 6.8mB/Eu, estimated by subtracting the magnetic susceptibility of CaFe4Sb12 to account for the [Fe4Sb12] polyanion contribution (2.6mB/f.u.), is smaller than 7.94mB/Eu for Eu2 + . The X-ray magnetic circular dichroism (XMCD) and XAS investigation revealed that this compound is a ferrimagnet in which the Fe 3d moment and the Eu2 + 4f moment are magnetically ordered with dominant AFM coupling (Krishnamurthy et al., 2007). EuRu4Sb12 was reported to be an FM metal with a Curie temperature TC ¼ 4 K (Bauer et al., 2004; Takeda and Ishikawa, 2000a). The effective magnetic moment 8.0mB/Eu estimated from the magnetic susceptibility is close to 7.94mB/Eu expected for Eu2 + , which is also consistent with the XANES measurement (Bauer et al., 2004). The dHvA experiments revealed that both the magnitude and the angular dependence of the dHvA frequencies are close to those for the divalent compound BaRu4Sb12 (Sugawara et al., 2008b). EuOs4Sb12 was reported to be an FM metal with the Curie temperature TC ¼ 9 K (Bauer et al., 2004). The effective magnetic moment 7.3mB/Eu is slightly smaller than 7.94mB/Eu expected for Eu2 + . The XANES measurements show that Eu valence is nearly divalent with 10% of Eu3 + as an upper limit (Bauer et al., 2004).

4.7. Gd-based filled skutterudites Gd-filled skutterudites are only known for GdT4P12 (T ¼ Fe, Ru, Os). GdFe4P12 is an FM metal with TC ¼ 23 K (Jeitschko et al., 2000). The effective magnetic moment meff and the paramagnetic Curie temperature yp are 7.90mB/Gd and 17.5 K, respectively (Sekine et al., 2008b). GdRu4P12 undergoes AFM phase transition below 22 K (Sekine et al., 2000c). The resistivity shows metallic behavior in higher temperatures with a broad minimum at around 30 K, and increases rapidly below 22 K. After showing a maximum around 13 K, it decreases with decreasing temperature. The linear slope of w1 versus T from 150 to 300 K yields an effective magnetic moment meff of 8.04mB/Gd and the paramagnetic Curie temperature yp of 23 K. The optical conductivity measurement indicates that the pseudo-gap opened with AFM ordering (Matsunami et al., 2005). H2T phase diagram for the ordered phase has been determined by the electrical resistivity measurements under magnetic fields as shown in Fig. 1.39 (Matsuhira et al., 2006). The rapid increase of

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Figure 1.39 Electrical resistivity in GdRu4P12 under selected magnetic fields. Inset shows an H2T phase diagram determined by the measurements of electrical resistivity (closed circle) and magnetization (closed triangle), respectively. Solid line shows the fitting curve to TN(H) ¼ TN(0)[1(H/Hc)2]n, where TN ¼ 21.38 K, Hc ¼ 6.46 T, and n ¼ 0.58, respectively. (Reproduced with permission from Matsuhira et al., 2006).

electrical resistivity below 22 K is naturally ascribed to a gap opening on the FS triggered by the FS nesting, since FS nesting instability with q ¼ (1, 0, 0) is a common feature to RRu4P12 with a trivalent rare earth guest ion (Harima, 2008). GdOs4P12 is an FM metal with TC ¼ 5 K ( Jeitschko et al., 2000). The effective magnetic moment meff and the paramagnetic Curie temperature yp are 8.54mB/Gd and 2.9 K, respectively (Kihou et al., 2004).

4.8. Tb-, Dy-, Ho-, Er-, and Tm-based filled skutterudites All the heavy rare earth-filled skutterudites described below were synthesized under high pressures between 4 and 5 GPa. TbFe4P12 shows a metallic behavior and exhibits an FM transition at TC ¼ 10 K (Shirotani et al., 2006). Magnetic susceptibility wm(T) shows a Curie2Weiss behavior with ypD6 K and meff ¼ 9.48mB, which is close to 9.72mB expected for Tb3 + . P-NMR measurement (Magishi et al., 2008a,


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2008b) shows almost constant 1/T1 at high temperatures, suggesting that the relaxation mechanism is dominated by the interaction of the 31P nucleus with fluctuating Tb local moments. Below TC, 1/T1 decreases rapidly, indicating the suppression of the spin fluctuations by magnetic ordering. TbRu4P12 (Sekine et al., 2000c) indicates two successive magnetic transitions in the magnetization and specific heat at TN ¼ 20 K and T1 ¼ 10 K. A cusp in wm(T) appears at TN. Below T1, magnetization exhibits two-step metamagnetic transitions at 0.8 and 2.5 T and reaches 8.1mB/Tb at 18 T, which is close to Ms ¼ 9mB/Tb expected for the full moment of Tb3 + . A Curie2Weiss behavior observed at high temperatures indicates 9.76mB/Tb, which is close to 9.72mB/Tb for Tb3 + ion. The positive yp ( ¼ 8 K) indicates FM interaction, although the AFM ordering sets in. A sharp increase in r(T) around TN is probably due to a gap opening on the FS associated with the q ¼ (1, 0, 0) nesting instability. Below TN, complicated magnetic phase diagram appears as shown in Fig. 1.40, which includes at least four magnetic phases (Sekine et al., 2005). In an ultrasound measurement, corresponding anomalies have been observed (Fujino et al., 2008). In a neutron scattering experiment, q ¼ (1, 0, 0) AFM structure with antiparallel alignment of spins on the inequivalent Tb sites was observed, and the ordered magnetic moment on Tb ions is estimated to be

Figure 1.40 Magnetic phase diagram for TbRu4P12 determined by specific heat and magnetization measurements. (Reproduced with permission from Sekine et al., 2005).

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8.3mB/Tb (Kihou et al., 2005). Note that such a multiphase ordering also appears in SmRu4P12. For TbOs4P12, only the lattice constant has been reported (Kihou et al., 2004). The relationship between lattice constant and atomic number in the series of AOs4P12 indicates that the valence of Tb ion in TbOs4P12 is trivalent. In DyFe4P12 (Shirotani et al., 2003b), magnetic susceptibility follows a Curie2Weiss behavior, yielding meff ¼ 10.70mB/Dy, which is in good agreement with 10.63mB/Dy expected for Dy3 + . An FM ordering below 10 K is inferred. The electrical resistivity shows an anomaly at around 10 K. In DyOs4P12 (Kihou et al., 2004), small electrical anomaly is observed at around 10 K. The magnetic susceptibility follows a Curie2Weiss behavior down to 10 K, yielding meff ¼ 10.70mB/Dy, which is in good agreement with 10.63mB/Dy expected for Dy3 + . No magnetic ordering has been found down to 2 K. In HoFe4P12 (Shirotani et al., 2005a), a Curie2Weiss behavior in wm(T) at higher temperatures yields meff ¼ 10.43mB/Ho, which is in good agreement with 10.60mB/Ho expected for Ho3 + . At around 5 K ( ¼ TC), some indication of an FM ordering appears in wm(T). The electrical resistivity shows a metallic behavior below room temperature and a sudden decrease at TC. Elastic constant measurements show a softening below 10 K in the longitudinal mode in zero field and a shallow minimum at around 7 K in the transverse mode in high magnetic fields (Yoshizawa et al., 2007). These behaviors are attributable to CEF effects. Using the Jahn2Teller formula, a weak AFQ interaction is inferred. For HoOs4P12 (Kihou et al., 2004), only the lattice constant has been reported. In the series of AOs4P12, the relationship between lattice constant and atomic number indicates that the valence of Ho ion in HoOs4P12 is trivalent. For ErFe4P12 (Shirotani et al., 2005a), a Curie2Weiss behavior in wm(T) at higher temperatures yields meff ¼ 9.59mB/Er, which is in good agreement with 9.69mB/Er expected for Er3 + . The electrical resistivity decreases monotonically with decreasing temperature down to 12 K. Below this temperature, the resistivity slowly increases with decreasing temperature, although no magnetic anomaly has been observed at around 12 K. For TmFe4P12 (Shirotani et al., 2005a), a Curie2Weiss behavior in wm(T) from 100 to 300 K yields meff ¼ 6.58mB/Tm, which is between 4.54mB and 7.57mB for free Tm2 + and Tm3 + ions, respectively. The resistivity decreases linearly with decreasing temperature down to 10 K. No sign of phase transitions has been observed.

4.9. Yb-based filled skutterudites Only two filled skutterudites YbFe4P12 and YbFe4Sb12 have been investigated from the magnetic viewpoint. No report on YbOs4P12 has been made except the crystallographic data.


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YbFe4P12 was first synthesized in polycrystalline form by Shirotani et al. (2005a) using a wedge-type cubic-anvil high-pressure apparatus. From Curie2Weiss law above B100 K, they estimated an effective paramagnetic moment (meff) to be 3.58mB/Yb that is smaller than 4.54mB/Yb expected for Yb3 + and a paramagnetic Curie temperature yp to be 67 K. Furthermore, the lattice constant largely deviates upward from the R3 + line expected from the lanthanoid contraction. Based on these facts, they concluded that the Yb valence is intermediate between Yb2 + and Yb3 + . Wakeshima et al. (2005) reported two distinct anomalies in the specific heat; a l-type peak at B0.7 K and a Schottky-type peak near 30 K attributed to the CEF splitting. The origin of the former is not yet clarified. The electronic specific heat coefficient C/T was roughly estimated as W0.2 J/K2 mol at the lowest investigated temperature B0.3 K. In the measurement of 31P-NMR, Yamamoto et al. (2006) found NFL-like behaviors below B10 K in YbFe4P12 accompanying the magnetic (probably AFM) spin fluctuations. Furthermore, they asserted that a magnetic field of about 0.2 T could bring YbFe4P12 to a quantum critical point. YbFe4Sb12 first synthesized by Dilley et al. (1998) was initially thought to be an intermediate valence compound with a Yb valence of 2.68 (Leithe-Jasper et al., 1999), based on the lattice constant, magnetization, resistivity, and heat capacity and XANES measurements. However, the Yb valence is now settled to be basically Yb2 + in high-quality samples (Schnelle et al., 2005). This was further supported by the PES and XAS measurements on high-quality single crystals (Dedkov et al., 2007). In fact, Schnelle et al. (2008) demonstrated that the physical properties of YbFe4Sb12 and CaFe4Sb12 single crystals are very close to each other, which indicates the divalent state of Yb in YbFe4Sb12. Therefore, the large electronic specific heat coefficient B170 mJ/K2 mol should be ascribed not to Kondo effect, but to the high D(EF) mainly due to Fe 3d-electrons along with the associated spin fluctuation. As a common feature to AFe4Sb12 (Section 5.1), the magnetic state could be sensitive to the filling fraction of Yb sites (Bauer, et al., 2000; Alleno et al., 2006). Ikeno et al. (2007) reported systematic decrease of both the FM-like transition temperature (TC) and D(EF) in YbxFe4Sb12 with increasing x from 0.875 to 0.91.

4.10. Filled skutterudites with actinoid ions UFe4P12 was reported to be an FM semiconductor exhibiting an FM transition at 3.1 K (Meisner et al., 1985; Torikachvili et al., 1986). The semiconducting state has been further confirmed by the infrared reflectance spectroscopy measurements (Dordevic et al., 1999). Nakotte et al. (1999), by way of neutron diffraction experiments, have directly confirmed the FM ordering below 3.1 K, with magnetic moments on U ions only. Matsuda et al. (2004) have investigated the magnetic susceptibility up to 800 K as

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Figure 1.41 Temperature dependence of the inverse of magnetic susceptibility for H//[1 0 0]. (Reproduced with permission from Matsuda et al., 2004).

shown in Fig. 1.41. It is indispensable to evaluate the effective paramagnetic moment meff from the Curie2Weiss law. Furthermore, the magnetization measurements under magnetic fields up to 50 T have been made on a single crystal at 1.3 K. From the Curie2Weiss law at higher temperatures, effective paramagnetic moment and the paramagnetic Curie temperature are estimated as meff ¼ 3.0mB/U and yp ¼ 98 K, respectively. The meff value is close to 3.58mB/U expected for 5f2 (U4 + ) configuration. As shown in Fig. 1.42, the anisotropy in magnetization is evident in low fields below B2 T; the easy axis is H//[1 0 0] while the hard axis is [1 1 1]. The magnetization at 1.3 K nearly saturates to 1.3mB/U (far smaller than 3.2mB/U expected for a free U4 + ion) at around 10 T, above which the magnetization is almost constant. These facts indicate that only the CEF ground state and the first excited state are enough to explain the physical properties of this compound within the temperature range investigated, since the higher CEF excited states are largely separated from the first excited state. Taking into account all the experimental facts, the CEF level schemes, which well reproduce the experimental data, have been proposed: (a) G1 singlet and ð1Þ ð1Þ ð2Þ Gð2Þ 4 (or G4 ) triplet and (b) G23 doublet and G4 (or G4 ) triplet (for the ground and the first excited states), respectively. However, some additional experimental information is required to choose the final scheme. In ThFe4P12, the electrical resistivity was reported to show a monotonic metallic behavior, with a large residual resistivity compared to that for LaFe4P12 (Torikachvili et al., 1987). No SC transition has been found down to 1.2 K. The metallic behaviors have also been confirmed in the infrared reflectance spectroscopy measurement (Dordevic et al., 1999). The metallic state totally contradicts with the band-structure calculations,

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T = 2.0 K

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T = 1.3 K H//

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Figure 1.42 Anisotropy in the field dependence of magnetization. Inset shows the magnetization up to 50 T for H//[1 0 0]. (Reproduced with permission from Matsuda et al., 2004).

where semiconducting state with direct gap is predicted (Khenata et al., 2007; Takegahara and Harima, 2003b). NpFe4P12 is the first transuranium-filled skutterudite and exhibits an FM transition below TC ¼ 23 K with the magnetic easy axis along [1 0 0] and the saturation moment of 1.35mB/Np. The electronic specific heat coefficient g is B10 mJ/K2 mol. At low temperatures, the specific heat follows C ¼ gT + BT3 that contradicts with T3/2-dependence expected for magnon excitation in the three-dimensional ferromagnet. This is a common feature to the FM filled skutterudites, such as NdFe4P12, NdOs4Sb12, and UFe4P12, although the mechanism has not yet been clarified. The small g value along with the large electrical resistivity (r ¼ 65 mO cm at RT) suggests this compound to be a semimetal, where

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100

ρ (mΩ•cm)

0 kOe

50 NpFe4P12 J // [100] H // [010] H = 55 kOe

0

0

100

200

300

Temperature (K)

Figure 1.43 Temperature dependence of electrical resistivity in NpFe4P12 at 0 Oe and 55 kOe. (Reproduced with permission from Aoki et al., 2006a).

the 5f-electrons are localized with a 5f3 (Np4 + ) configuration. From room temperature, r first decreases with decreasing temperature, and starts to increase after showing a minimum at around 150 K (Fig. 1.43). Between 80 and 40 K, the resistivity shows lnT-dependence reminiscent of Kondo effect, and exhibits a broad peak at around 30 K. Below 30 K, the resistivity decreases steeply with decreasing temperature, exhibits a slight upturn below 4 K, and finally tends to saturate below 0.2 K. The resistivity shows no evident anomaly across TC.

5. Other Filled Skutterudites 5.1. AT4Sb12 (A ¼ alkaline and alkaline earth; T ¼ Fe, Ru, Os) In the Fe-based filled skutterudites with nonmagnetic guest ions (nonmagnetic rare earth, alkaline, alkaline earth, and monovalent thallium), magnetic properties are governed basically by the itinerant Fe 3d-electrons (Leithe-Jasper et al., 2003; Takabatake et al., 2006). From the magnetic point of view, the high electronic density of states at EF is realized only in Fe-based systems. The effect of the band structure on magnetic features is discussed in the references (Takegahara et al., 2008a; Schnelle et al., 2008). We first try to gain intuitive knowledge of the change in magnetic ground state of AFe4Sb12 with the guest ion’s valence. Most naively, assuming the rigid-band model, D(EF) changes with the valence of guest ions A through the position of EF with respect to D(E). In LaFe4X12, as X


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varied from P to Sb, the 3d-electrons’ contribution to D(EF) systematically increases (Harima and Takegahara, 2003a). For monovalent guest ion, EF is located at higher energy side of the main peak in D(E) dominated by Fe 3d-electrons. With increasing valence of guest ions, D(EF) continuously decreases and departs from the Stoner criterion, and the number of 3d holes decreases. It explains the systematic change of magnetic state from monovalent to trivalent guest ion in the experiments (Leithe-Jasper et al., 2003; Schnelle et al., 2008); that is, the monovalent guest systems are itinerant weak ferromagnets, the divalent ones are nearly FM metals, and the trivalent ones are exchange-enhanced paramagnets. 5.1.1. Fe4Sb12-based filled skutterudites with monovalent guests The finding of the weak ferromagnetism in NaFe4Sb12 and KFe4Sb12 by Leithe-Jasper et al. (2003, 2004) shown in Fig. 1.44 may be the most interesting result from the viewpoint of itinerant electron magnetism in Fe4Sb12-based filled skutterudites. Recently, TlFe4Sb12 was also reported to be a weak itinerant ferromagnet with Tl + ions (Leithe-Jasper et al., 2008). All the three compounds show a transition to an FM state below TCE80 K with almost the same magnetic moment of 0.420.5mB/Fe at 1.8 K under m0H ¼ 1.0 T. The cohesive force of B0.1 T is also almost the same. At higher fields, the magnetic moment of NaFe4Sb12 is 0.60mB/Fe at 14 T and

Figure 1.44 Magnetization M(T) per Fe atom of NaFe4Sb12. The inset shows isothermal hysteresis curves at 1.8 and 70 K. (Reprinted figure with permission from Leithe-Jasper et al. (2003), by the American Physical Society r 2009.)

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0.65mB/Fe when M is extrapolated to infinite magnetic field (Leithe-Jasper et al., 2004). From a fit to Curie2Weiss law, an effective magnetic moment in the paramagnetic range is 1.521.7mB/Fe and a paramagnetic Curie temperature yp is 87288 K, nearly identical with TC (Schnelle et al., 2008). All these features indicate that the three compounds are itinerant electron ferromagnets (Moriya, 1985). The specific heat coefficient g and Debye temperature yD for NaFe4Sb12, KFe4Sb12, TlFe4Sb12 are 116, 113, 104 mJ/K2 mol and 243, 252, 215 K, respectively (Leithe-Jasper et al., 2008; Schnelle et al., 2008).

5.1.2. Fe4Sb12-based filled skutterudites with divalent guests Magnetic properties of AFe4Sb12 (A ¼ Ca, Sr, Ba) have been investigated for the first time by Danebrock et al. (1996), and discussed from the viewpoint of localized electrons’ magnetism. Overall, no essential change in experimental data has been found in the new measurements (Matsuoka et al., 2005; Schnelle et al., 2008; Takabatake et al., 2006), where no phase transition has been found down to the lowest temperatures. Matsuoka et al. (2005) found a maximum or a shoulder-like behavior in magnetic susceptibility at 50 K and a large electronic specific heat coefficient of 100 mJ/K2 mol in all the three compounds. The thermoelectric power also exhibits a marked shoulder at 50 K which is totally suppressed by the applied field of 14 T as shown in Fig. 1.45. Based on these facts, they concluded the existence of strong spin fluctuation in these compounds. In fact, Yoshii et al. (2006) found a metamagnetic increase of magnetization near 13 T at 1.3 K in CaFe4Sb12 (Fig. 1.46). At the transition (B13 T), the magnetic moment is B0.2mB/Fe and reaches 0.42mB/Fe at 50 T of the highest field investigated. The metamagnetic anomaly was also found in the other two compounds, although the anomaly becomes less evident and the critical field decrease as A changes from Ca to Ba (11 T for BaFe4Sb12).

5.1.3. Alkaline earth-filled skutterudites with T ¼ Ru, Os The first report on magnetic properties of AT4Sb12 (T ¼ Ru, Os; A ¼ Sr, Ba) has been made by Takabatake et al. (2006). They made systematic measurements of physical properties in MT4Sb12, and compared with those in Fe-based homologs. The Ru compounds are metallic diamagnets with g ¼ 10 mJ/K2 mol, while the Os compounds are strongly enhanced Pauli paramagnets with g ¼ 45 mJ/K2 mol (Matsuoka et al., 2006). The change of magnetic state with transition element T is consistent with the bandstructure calculation (Takegahara et al., 2008a).

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30 0T 7T 10 T 15 T

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Figure 1.45 Thermoelectric power S(T) for CaFe4Sb12 in various constant magnetic fields up to 15 T with an inset showing the field dependence of S at selected temperatures. (Reproduced with permission from Takabatake et al., 2006).

5.2. Ge cage-forming filled skutterudite APt4Ge12 (A ¼ Sr, Ba, La, Ce, Pr, Nd, Eu, U, Th) From both scientific and technological points of view, new series of skutterudites with cage-forming elements other than P, As, and Sb has been explored. Bauer et al. (2007) have succeeded in synthesizing Ge cageforming filled skutterudite APt4Ge12 (A ¼ Sr, Ba). Independently, Gumeniuk et al. (2008a) reported the successful synthesis of APt4Ge12 (A ¼ La, Ce, Pr, Nd, Eu) in addition to Sr and Ba compounds. Th- and Ubased Pt-Ge skutterudites have also been synthesized (Bauer et al., 2008; Kaczorowski and Tran, 2008). SrPt4Ge12 and BaPt4Ge12 are superconductors with almost the same transition temperature TsB5 K (Bauer et al., 2007; Gumeniuk et al.,

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0.5

1.3 K

Magnetization (μB / Fe)

0.4

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SrFe4Sb12

0.3 Bc 0.2

BaFe4Sb12

0.1

0.0

0

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20 30 Magnetic field (T)

40

50

Figure 1.46 Magnetization of AFe4Sb12. (Reproduced with permission from Yoshii et al., 2006).

2008a). Overall features of the temperature dependences of electrical resistivity r and specific heat C are similar to each other except the difference of upper critical field Hc2. The SC states of these compounds are BCS-type with a strong coupling. The substitution effect of Au for Pt site on the SC properties of BaPt4Ge12 was investigated by Gumeniuk et al. (2008b). The SC transition temperature Ts increases up to 7 K in BaPt4xAuxGe12 with increasing x to 1, which is the maximum concentration to synthesize single phase samples. In contrast to the relatively large substitution effect on Ts, the effect on Hc2 is almost negligible for x up to 1. Temperature dependences of magnetic susceptibility in BaPt4xAuxGe12 change systematically with x, in which the susceptibility can be explained by the sum of temperature-independent core contributions and a weakly temperature-dependent Pauli susceptibility. The electronic specific heat coefficient g is also reported to increase systematically with increasing x. Band-structure calculation within the LDA of DFT indicates that the D(EF) increases with increasing x assuming a


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rigid-band picture (Gumeniuk et al., 2008b). Based on these results, they proposed the increase in D(EF) as a possible origin of the sizable increase in Ts, assuming a constant electron2phonon coupling. LaPt4Ge12 is a superconductor with an SC transition temperature Ts ¼ 8.2 K (Gumeniuk et al., 2008a; Toda et al., 2008). The temperature dependence of magnetic susceptibility wm can be explained as the sum of temperature-independent core contributions (wm ¼ 9.49 105 emu/ mol at 300 K) and a weakly temperature-dependent Pauli susceptibility. The specific heat measurement suggests that the superconductivity is BCStype with a strong coupling (Gumeniuk et al., 2008a), which is also confirmed microscopically by La- and Pt-NMR measurements (Toda et al., 2008). Interestingly, additional contribution ascribed to the rattling motion of La ions has been observed in the nuclear relaxation at high temperatures (Toda et al., 2008), as in LaOs4Sb12 (Nakai et al., 2008). CePt4Ge12 is a metal exhibiting no phase transition down to 0.4 K (Gumeniuk et al., 2008a; Toda et al., 2008). The temperature dependence of the magnetic susceptibility shows a broad peak around 80 K, which is a typical feature of valence fluctuation systems (Toda et al., 2008). The NMR measurements revealed that the nuclear spin-lattice relaxation rate 1/T1 shows an activation-type temperature dependence 1/T1pexp(D/kBT) with an energy gap D/kB ¼ 150 K above 100 K, whereas it is proportional to temperature below 20 K (Toda et al., 2008). These behaviors are similar to those of other Ce-based skutterudites, which is explained by a pseudo-gap model originating the c-f hybridization. PrPt4Ge12 is a superconductor with a transition temperature Ts ¼ 7.9 K, and appears to be a Van Vleck paramagnet, since wm saturates at low temperatures without any magnetic phase transition (Gumeniuk et al., 2008a; Toda et al., 2008). From the specific heat measurements, this compound is suggested to be a strong-coupling superconductor with a significantly large coupling constant l ¼ 2.9 (Gumeniuk et al., 2008a). Temperature dependence of magnetic susceptibility wm follows the Curie2Weiss law above B100 K with an effective magnetic moment meff ¼ 3.59mB/f.u. and paramagnetic Curie temperature yp ¼ 16.8 K. The CEF level scheme is suggested to consist of G1 singlet ground state, Gð2Þ 4 triplet at 134 K, G23 doublet at 230 K, and Gð1Þ 4 triplet at 520 K (Gumeniuk et al., 2008a; Toda et al., 2008). NdPt4Ge12 was reported to be a metal with an AFM transition at TN ¼ 0.67 K (Gumeniuk et al., 2008a; Toda et al., 2008). Temperature dependences of magnetic susceptibility wm follows the Curie2Weiss law above B100 K with an effective magnetic moment meff ¼ 3.72mB/f.u. and paramagnetic Curie temperature yp ¼ 31.1 K (Toda et al., 2008). 1/wm decreases with decreasing temperature exhibiting a broad hump around 30 K. Assuming the CW law at low temperatures between 2 and 3.8 K,

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yp ¼ 1.81 K is estimated at T ¼ 0 K, which indicates the existence of the AFM correlations at this temperature range, although the absolute value of yp is more than twice larger than that of AFM transition temperature (Toda et al., 2008). The CEF level scheme is suggested to consist of Gð2Þ 67 quartet ground state, Gð1Þ 67 quartet first excited state at 176 K, and G5 doublet state at 630 K (Toda et al., 2008). EuPt4Ge12 was reported to exhibit an AFM transition at TN ¼ 1.7 K (Gumeniuk et al., 2008a). ThPt4Ge12 is a superconductor with a transition temperature TsB4.7 K (Bauer et al., 2008; Kaczorowski and Tran, 2008). The specific heat measurements suggested this material to be a BCS-type superconductor with a strong coupling. UPt4Ge12 is a metal with a large electronic specific heat coefficient g ¼ 156 mJ/K2 mol (Bauer et al., 2008) compared to g ¼ 35 mJ/K2 mol for ThPt4Ge12. No phase transition down to 1.9 K has been confirmed. The pronounced effect of spin fluctuations is suggested as an origin of the large electronic specific heat coefficient and absence of superconductivity.

5.3. YT4P12 (T ¼ Fe, Ru, Os) and I0.9Rh4Sb12 In Y-based filled skutterudites, only three phosphides, YFe4P12, YRu4P12, and YOs4P12, have been synthesized by Shirotani and coworkers (Kihou et al., 2004; Shirotani et al., 2003a, 2005a) using the high-pressure technique. All of them exhibit SC transition at 7, 8.5, and 3 K, respectively. Their SC transition temperatures are higher than their La-homologs, as expected from the lighter mass of Y compared to La. Recently, a filled skutterudite I0.9Rh4Sb12 with an anion (I) as a guest ion has been synthesized for the first time (Fukuoka, unpublished data) under high pressure. That was reported to be a nonmagnetic metal.

6. Summary The filled skutterudites exhibit full variety of attractive behaviors, only by changing the constituent elements A, T, and X. In most of the cases, 4f-electrons play main roles. Why so many novel phenomena can be realized in only one cubic crystal structure with basically single-variable parameter of lattice constant except a slight change in X position? To answer the question, steady and intensive studies have been done by various experimental methods including several sophisticated state-of-the-art techniques. Combined with the innovative theoretical ideas, remarkable progress in finding important key concepts necessary to reach the final answer has been made.


(i)

(ii)

(iii) (iv)

(v)

(vi)

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One of the puzzles was the origin of the strong hybridization between conduction and 4f-electrons in compounds containing rare earth ions with plural number of 4f-electrons, which had been widely recognized to be so hard. The c-f hybridization is largely enhanced compared to the ordinary rare earth compounds, due to the large coordination number (12) of pnictogens surrounding a RE guest ion. Why novel features could be realized in the systems with a singlet G1 CEF ground state? Relatively small CEF splitting between the ground and the first excited states (compared to ordinary rare earth compounds) is realized by the nearly isotropic environment around a rare earth ion, since it is surrounded by a cage made of 12 pnictogens. As a result, the finite contribution of the excited states to the various properties can be expected even at low temperatures. More generally, an extra term in CEF Hamiltonian under Th symmetry compared to that in Oh symmetry plays an essential role in many of the novel features in the filled skutterudites (see Section 3.3). In correlation with (ii), ‘‘multipole’’ is indispensable to understand the filled skutterudites (see Section 3.2). As a clue specific to the phosphides, the FS nesting instability is important. The main FS is nearly a cube with a half volume of the Brillouin zone, leading to the instability with the nesting vector q ¼ (1, 0, 0). This might trigger various types of phase transitions in RT4P12 (R ¼ rare earth; T ¼ Fe, Ru). As additional ingredients, the positional degrees of freedom of guest ion within a cage, such as ‘‘rattling’’, are thought to play a key role in some of the attractive features that are hardly ascribable to the known mechanisms. At this stage, however, it has not yet been clarified whether such mechanisms really play essential role. In the Fe-based filled skutterudites, the high electronic density states (mainly Fe 3d-electrons) play a key role in their magnetic properties. When A is a magnetic rare earth ion, there appears a wide variety of magnetic states as a result of cooperative interaction between Fe 3dand 4f-electrons.

According to the intensive study by various experimental techniques, the understanding of this system has been considerably deepened; however, further studies from both experimental and theoretical viewpoints are necessary to answer the first question.

ACKNOWLEDGMENTS This work was supported by a Grant-in-Aid for Scientific Research Priority Area Skutterudite of the Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan. The authors thank all the members of the priority area ‘‘Skutterudite’’ for fruitful discussion and collaborations.

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Appendix A Magnetic and thermal properties of the filled skutterudite compounds are summarized in the following three tables. The detailed explanation is given in the main text. Table A1 (Rare earth or actinide-based pnictides) Table A2 (Alkaline or alkaline earth-based antimonides) Table A3 (Ge-based filled skutterudites)

Compound

a (Å)

GS

TTR (K) leff (lB/f.u.)

YFe4P12

7.7896

SC

7

YRu4P12

8.0298

SC

8.5

YOs4P12 LaFe4P12

8.0615 7.8316

SC SC

3 4.6

hp (K)

Ms (lB/f.u.) c (mJ/ hD (K) K2 mol) 27.2

553

Ref

PH

Shirotani et al. (2003a, 2003b) Shirotani et al. (2005a) Kihou et al. (2004) Dordevic et al. (1999), Jeitschko and Braun (1977), Meisner (1981), Meisner et al. (1984), Sato et al. (2000), Sugawara et al. (2000a, 2002), Torikachivili and Maple (1984), Torikachvili et al. (1987)

PH

PP: wm ¼ 628 104 emu/mol

57

580

PH SA

93

Remarks


Table A1. Magnetic properties of filled skutterudites AT4X12. Lattice constant a, ground state (GS) (SC: superconducting, M: metallic, semi-M: semimetallic, semi-C: semiconducting, I: insulating, NFL: non-Fermi-liquid, PP: Pauli paramagnetic, P: paramagnetic, FM: ferromagnetic, AFM: antiferromagnetic, IV: intermediate valence, SF: spin fluctuation, WFM: weak ferromagnetic, AFQ: antiferro-quadrupole order, OPO: octupole order, MPO: multipole order), the transition temperature to the ordered state TTR, effective magnetic moment meff, paramagnetic Curie temperature yp, saturation moment Ms, electronic specific heat coefficient g, and Debye temperature yD are listed. Abbreviations PA, PH, SA, and SH in remarks mean sample characteristics: polycrystalline samples synthesized at ambient pressure (PA) or high pressure (PH), single crystals synthesized at ambient pressure (SA) or high pressure (SH). Hc2 is the upper critical field of superconductibity. Twmax is the temperature where wm shows a broad peak. HA, HA1, and HA2 are the fields where magnetization shows some anomally.yE is Einstein temperature related with the guest ion’s motion within a cage.

94

Table A1. (Continued ) a (Å)

GS


LaRu4P12

8.0561

SC

7.2

26

LaOs4P12

8.0844

SC

1.82

LaFe4As12

8.3252

WFM

5.2

1.9

hp (K)

Ms (lB/f.u.) c (mJ/ hD (K) K2 mol)

0.12 (2 K)

Remarks

Ref

446

SA

21.6

532

SA

170

W322

SH

DeLong and Meisner (1985), Jeitschko and Braun (1977), Meisner (1981), Shirotani et al. (1996), Tsuda et al. (2006), Uchiumi et al. (1999) DeLong and Meisner (1985), Jeitschko and Braun (1977), Matsuhira et al. (2005b), Sugawara et al. (2009) Braun and Jeitschko (1980a), Tatsuoka et al. (2008)

H. Sato et al.

Compound

8.5081

SC

10.3

LaOs4As12

8.5437

SC

3.2

LaxFe4Sb12

9.1395

SF

W0.4

W73

2.023.0

55 to 34

0.67 (48 T)

195

233

PH

95

Braun and Jeitschko (1980a), Namiki et al. (2007), Shimizu et al. (2007), Shirotani et al. (1997) Single crystal Braun and grown by As- Jeitschko Cd flux (1980a), method at Henkie et al. BMPa (2008), Ho et al. (2007), Shirotani et al. (2000), Wawryk et al. (2008) 2602348 PA (x ¼ 0.83), Bauer et al. PH (xB1) (2002c), Braun and Jeitschko (1980b), Danebrock et al. (1996), Harima and Takegahara (2003), Mori et al. (2007), Tanaka et al. (2007), Uchiumi et al. (1999), Viennois et al. (2003, 2004, 2005), Yamada et al. (2007)


LaRu4As12

96

Table A1. (Continued ) hp (K)


Compound

a (Å)

GS


LaRu4Sb12

9.2700

SC

3.58

PP: wm ¼ 1.01.8 104 emu/mol

37

253

LaOs4Sb12

9.3029

SC

74, B1

PP: wm ¼ 1.12 1.4 103 emu/mol

36, 57

304, 291

CeFe4P12

7.7920

Semi-C

Remarks

Ref

H. Sato et al.

SA, Hc2B0.3 T Abe et al. (2002), Braun and Jeitschko (1980b), Takeda and Ishikawa (2000a, 2000b), Uchiumi et al. (1999), Yogi et al. (2003) SA, Hc2 ¼ Aoki et al. (2005a), 0.035 T, Bauer et al. 0.6 T (2001c), Braun and Jeitschko (1980b), Maple et al., (2003a, 2003b), Nakai et al. (2008), Sugawara et al. (2002b, 2005a) SA Dordevic et al. (1999), Grandjean et al. (1984), Jeitschko and Braun (1977), Meisner et al. (1985), Sato et al. (2000)

8.0376

Semi-C

PH

CeOs4P12

8.0626

Semi-C

PH

CeFe4As12

8.2959

Semi-C or semi-M

PH or single crystal grown by As-Cd flux method at BMPa

CeRu4As12

8.4908

Semi-C or semi-M (NFL)

CeOs4As12

8.5249

Semi-C

20

PH or single crystal grown by As-Cd flux method at BMPa PH or single crystal grown by As-Cd flux method at BMPa

Jeitschko and Braun (1977), Shirotani et al. (1999) Jeitschko and Braun (1977), Shirotani et al. (1999) Braun and Jeitschko (1980a), Grandjean et al. (1984), Maple et al. (2008) Braun and Jeitschko (1980a), Maple et al. (2008), Sekine et al. (2007) Braun and Jeitschko (1980a), Maple et al. (2008), Sekine et al. (2008a, 2008b)


CeRu4P12

97

98

Table A1. (Continued ) TTR (K) leff (lB/f.u.)

hp (K)


a (Å)

GS

CexFe4Sb12

9.1350

Semi-M

2.024.15

124 to 1

63.8

CeRu4Sb12

9.2657

NFL

2.35, 2.26

26, 37

73

CeOs4Sb12

9.3011

Semi-C, 0.8 unusual order

1.88

6

92, 180

Remarks

Ref

250

PA, SA, Twmax ¼ 1002150 K

262

SA, Twmax ¼ 100 K

Braun and Jeitschko (1980b), Danebrock et al. (1996), Gajewski et al. (1998), Grandjean et al. (1984), Morelli and Meisner (1995), Mori et al. (2007), Viennois et al. (2003, 2004) Abe et al. (2002), Bauer et al. (2001b), Braun and Jeitschko (1980b), Takeda and Ishikawa (2000a, 2000b) Bauer et al. (2001c), Braun and Jeitschko (1980b), Namiki et al. (2003b)

SA

H. Sato et al.

Compound

7.8149

MPO

6.5

PrRu4P12

8.0420

I, MPO

63

3.63

16.5

2.8 (H// 12002 [1 0 0], 2700 1.3 K, 50 T)

3.84

7

1.3 (0.06 K, 12.5 T)

3.63

17

(M2I transition)

PrOs4P12

8.0710

M

W0.05

o60

56, 26

285, 320

SA, fieldAoki et al. (2002a, induced order 2005b), at around Jeitschko and narrow field Braun (1977), directions Kuramoto et al. along H// (2008), Namiki [1 1 1] below et al. (2003a), 0.53 K; PrSato et al. based heavy (2000), fermion Sugawara et al. (2002), Sugiyama et al. (2005), Tayama et al. (2004), Torikachvili et al. (1987) SA, magnetic Jeitschko and states are Braun (1977), different at Iwasa et al. two different (2005a, 2005b), Pr sites below Matsuhira et al. TMI (2002a), Sekine et al. (1997) SA, absence of Jeitschko and phase Braun (1977), transition Matsuhira et al. down to (2005b), 0.05 K Sugawara et al. (2008a, 2009), Yuhasz et al. (2007)


PrFe4P12

99

Table A1. (Continued ) GS


PrFe4As12

8.3125

FM

18

3.98, 3.52 (o45 K)

PrRu4As12

8.4963

SC

2.4

3.30

PrOs4As12

8.5311

AFM

2.3 (TN2), 3.81, 2.77 (o20 K) 2.2 (TN1)

PrFe4Sb12 PrxFe4Sb12

9.1362 9.1351

M W0.5 FM, AFM, 4.124.6 or ferri-M

4.5 4.224.6

hp (K)

11


Remarks

2.3

Single crystal Braun and grown by As- Jeitschko Cd flux (1980a), Sayles method at et al. (2008) BMPa Braun and PH, Jeitschko Hc2 ¼ 0.62 T (1980a), Maple et al. (2008), Namiki et al. (2007), Shimizu et al. (2007), Shirotani et al. (1997) Single crystal Braun and grown by Jeitschko As-Cd flux (1980a), Chi method at et al. (2008), BMPa Maple et al. (2006), Wawryk et al. (2008), Yuhasz et al. (2006) PH Tanaka et al. (2007) PA (x ¼ 0.83), Bauer et al. SA (2002a), Braun (x ¼ 0.87) and Jeitschko (1980b), Butch et al. (2005), Danebrock et al. (1996)

340

356

B95

344

25.8

1.75 (0.425 K, 502200 260 5.5 T) (WTN), B1000 (o2 K)

B4 K 22

2 (2 K, 7 T) B300 6521.9 (2 K, B1000 5.5 T)

Ref

H. Sato et al.

a (Å)

100

Compound

9.2648

SC

1.3

3.58

11

PrOs4Sb12

9.2994

SC

1.85

2.97

16

NdFe4P12

7.8079

FM

1.94

3.53

NdRu4P12

8.0364

FM

1.7

3.68

59

232

SA

101

Bauer et al. (2001b), Braun and Jeitschko (1980b), Takeda and Ishikawa (2000a, 2000b) 2.3 (H// 3102750 1652320 SA, fieldAoki et al. (2002b, [1 0 0], induced AFQ 2003a), Bauer 0.1 K, 25 T) below 1.3 K, et al. (2002c), above 4.5 T; Braun and unconvenJeitschko tional super(1980b), Keller conductor et al. (2001), Tayama et al. (2003, 2004), Vollmer et al. (2003) 1.72 (1.4 K, SA, ordered Jeitschko and 5 T) moment Braun (1977), 1.61mB at Keller et al. 1.5 K was (2001), Meisner determined (1981), by neutron Nakanishi et al. diffraction (2004), Sato experiments et al. (2000), Torikachvili et al. (1987) SA Jeitschko and Braun (1977), Masaki et al. (2008), Meisner (1981), Sekine et al. (1998)


PrRu4Sb12

Table A1. (Continued ) GS


hp (K)


Remarks

Ref

NdOs4P12

8.0638

FM

1.15

3.03

23

1.7 (H// [1 0 0], 2 K, 7 T)

SA

NdFe4As12

8.309

FM

15

4.7

45

FM

2.3

1.9 (2 K, 7 T)

PH

1.8 (2 K, 7 T)

PH

Jeitschko and Braun (1977), Sugawara et al. (unpublished data) Jeitschko et al. (2000), Takeda et al. (unpublished data) Namiki et al. (unpublished data) Braun and Jeitschko (1980a), Namiki et al. (unpublished data) Bauer et al. (2002b), Danebrock et al. (1996), Evers et al. (1995), Ikeno et al. (2008) Braun and Jeitschko (1980b), Takeda and Ishikawa (2000a, 2000b)

NdRu4As12

SH

NdOs4As12

8.5291

FM

1.6

NdxFe4Sb12

9.130

FM

8.6216.5 3.8624.5

15 to 36

2.3 (2 K, 5.5 T)

PA (x ¼ 0.7720.90)

NdRu4Sb12

9.2642

FM

1.3

28

1.8 (2.2 K, 4.8 T)

SA

3.45

H. Sato et al.

a (Å)

102

Compound

9.2989

FM

0.9

SmFe4P12

7.8029

FM

SmRu4P12

8.0397

SmOs4P12

8.0752

SmFe4As12

SmxFe4Sb12

9.13

3.84

1.73 (0.4 K, 5.5 T)

520

1.6

0.41 (1.6 K, 40 T)

370

I, AFM, OPO

16.5 (M2I transi tion)

0.08 (15.5 K, 7 T)

13.5 (ordered state)

AF

4.6

FM

39

FM

45, 43

43

SA, yE ¼ 39 K

Braun and Jeitschko (1980b), Ho et al. (2005), Sugawara et al. (2005a) SA, metaJeitschko and Braun (1977), magnetic anomaly at Matsuhira et al. 22 T (2005a), Takeda and Ishikawa (2003), Takeda et al. (2008) SA, PH, Aoki et al. (2007a), ordered Kikuchi et al. moment (2007a), 0.29mB was Matsuhira et al. determined (2002b, 2005a), by nuclear Sekine et al. specific heat (1998), Yoshizawa et al. (2005) PH Kihou et al. (2004), Matsuhira et al. (2005a) PH Kikuchi et al. (2008a, 2008b) PA, PH (xB1) Danebrock et al. (1996), Evers et al. (1995), Ueda et al. (2008)

103

0.62 (2 K, 7 T) 170 (ordered state) 0.7 (5 K, 5.5 T), 0.56 (2 K, 7 T)

255


NdOs4Sb12


GS


hp (K)


SmRu4Sb12 SmOs4Sb12

9.259 9.3009

WFM

3

0.63

0.99

0.135 (2 K, 7 T)

EuFe4P12

7.8055

FM

99

6.2

EuRu4P12

8.0406

FM

17.8

7.75

18

6 (H//[1 0 0], 2 K, 5 T)

EuOs4P12 EuxFe4Sb12

8.0792 9.165

FM FM

15 82287

7.64 7.728.4

18 to 19

4.525.1

85, 100

348

EuRu4Sb12

9.2824

FM

3.3, 4

7.2, 8.0

6.1, 3

6.2, 7.3

73

232

820, 880

294

Ref

PA SA

Evers et al. (1995) Braun and Jeitschko (1980b), Sanada et al. (2005), Yuhasz et al. (2005) Jeitschko and Braun (1977), Grandjean et al. (1984) Jeitschko and Braun (1977), Sekine et al. (2000b) Kihou et al. (2004) Bauer et al. (2001a), Danebrock et al. (1996), Dilley et al. (1998), Evers et al. (1995) Bauer et al. (2004), Braun and Jeitschko (1980b), Takeda and Ishikawa (2000a, 2000b)

PA

SA

PH PA (x ¼ 0.83), SA (x ¼ 0.95), yE ¼ 84 K

SA, yE ¼ 78 K

H. Sato et al.

Remarks

104

Compound

135

304

FM

9

7.3

8

6

GdFe4P12

7.795

FM

22, 23

7.6, 8.19

25, 9.9

6.8 (4 K, 5.5 T)

PH

GdRu4P12

8.0375

AFM

22

8.04

23

7 (4.2 K, 18 T)

GdOs4P12 TbFe4P12

8.0657 7.7926

FM FM

5 10

8.54 9.48

2.9 10

6.6 (2 K, 1 T) 6.8 (2 K, 1 T)

PH, magnetic anomaly at HA ¼ 6.1 T (4.2 K) PH PH

TbRu4P12

8.0338

AFM

20

9.76

8

8.1 (4.2 K, 18 T)

TbOs4P12

8.0631

M

W2

10.8

3

6 (2 K, 5 T)

DyFe4P12

7.7891

FM

4

10.7

3

4.5 (2 K, 1 T)

DyRu4P12

8.0294

AFM

15

12.23

0.1

DyOs4P12

8.0601

FM

2

13.55

4.4

Bauer et al. (2004), Braun and Jeitschko (1980b) Jeitschko et al. (2000), Kihou et al. (2004) Sekine et al. (1998, 2000c)

Kihou et al. (2004) Sekine et al. (2008a, 2008b), Shirotani et al. (2006) PH, magnetic Sekine et al. anomaly at (2000c, 2008a, HA1 ¼ 0.8 T, 2008b) HA2 ¼ 2.5 T (4.2 K) PH Kihou et al. (2004), Sekine et al. (2008a, 2008b) PH Sekine et al. (2008a, 2008b), Shirotani et al. (2003b) PH Sekine et al. (2008a, 2008b) PH Kihou et al. (2004), Sekine et al. (2008a, 2008b)

105

9.3187


SA, yE ¼ 74 K

EuOs4Sb12


GS


hp (K)

HoFe4P12

7.7854

FM

5

10.43

HoOs4P12 ErFe4P12

8.0579 7.7832

M

W1.8

9.59

TmFe4P12

7.7802

M (IV)

W1.8

6.58

YbFe4P12

7.7832

M (NFL)

W1.8

3.58

67

YbxFe4Sb12

9.158

M

W0.4

2.8123.09

40270.3

YbOs4Sb12

9.316

LuFe4P12

7.7771

UFe4P12

7.7729

Semi-C, FM

3.15

2.25


Remarks

5.3 (2 K, 1 T)

PH

Ref

H. Sato et al.

Shirotani et al. (2005) PH Kihou et al. (2004) 3.1 (2 K, 1 T) PH Shirotani et al. (2005a) PH Shirotani et al. (2005a) B200 580 PH, resistivity Shirotani et al. minimum at (2005a), around 45 K Wakeshima et al. (2005), Yamamoto et al. (2006) 1402175 1902364 PA (x ¼ Dilley et al. (1998), 0.87520.97) Ikeno et al. (2007), Schnelle et al. (2008) PA Kaiser and Jeitschko (1999) PH Shirotani et al. (2003b) 1.3 (1.3 K, SA, upper limit Dordevic et al. 50 T) of ordered (1999), Guertin moment et al. (1987), 0.5mB at 2.8 K Matsuda et al. was (2004), Meisner determined et al. (1985), by neutron Nakotte et al. diffraction (1999), experiments Torikachvili et al. (1986)

106

Compound

7.7999

ThRu4P12

8.0461

ThOs4As12

8.5183

NpFe4P12

7.7702

M

W1.2

FM

23

SA

1.35 (5 K, 5.5 T)

10

Dordevic et al. (1999), Jeitschko and Braun (1980), Khenata et al. (2007), Takegahara et al. (2001), Torikachvili et al. (1987) PA Jeitschko and Braun (1980) PA Jeitschko and Braun (1980) SA, resistivity Aoki et al. (2006a), minimum at Tokunaga et al. around 160 K (2009)


ThFe4P12

107

108

Table A2. Magnetic properties of filled skutterudites AFe4Sb12. Lattice constant a, ground state (GS) (SC: superconducting, M: metallic, FM: ferromagnetic), the transition temperatures TTR to the ordered states, effective magnetic moment meff, paramagnetic Curie temperature yp, saturation moment Ms, electronic specific heat coefficient g, Debye temperature yD, and Einstein temperature yE are listed. a (Å)

GS

TTR (K)

leff hp (K) (lB/Fe)

Ms (lB/Fe) M (lB/Fe) c (mJ/ at 1.0 T, K2 mol) 1.8 K

NaFe4Sb12

9.1767

FM

B85

1.6

B88

KFe4Sb12

9.1994

FM

B85

1.67

B87

TlFe4Sb12

9.1973

FM

80

1.69

B88.3

CaFe4Sb12

9.156

M

W0.35

1.5, 1.521.7

54, 45

o0.43 (1.3 K, 50 T)

118, 109

280, 266

88

SrFe4Sb12

9.177

M

W0.35

1.5, 1.521.7

53, 41

o0.33 (1.3 K, 50 T)

87, 111

279, 273

94

6 (1.8 K, 14 T)

0.62 (1.8 K, 7 T)

hD (K)

hE (K)

0.4

116

243

81.8

0.4

113

252

86.1

0.5

103.5

215

52.4

References

Leithe-Jasper et al. (2003), Schnelle et al. (2008) Leithe-Jasper et al. (2003), Schnelle et al. (2008) Leithe-Jasper et al. (2008) Matsuoka et al. (2005), Schnelle et al. (2008), Takabatake et al. (2006), Yoshii et al. (2006) Matsuoka et al. (2005), Schnelle et al. (2008), Takabatake et al. (2006), Yoshii et al. (2006)

H. Sato et al.

Compound

9.199

M

W0.35

SrRu4Sb12

9.29

M

BaRu4Sb12

9.31

CaOs4Sb12

9.321

SrOs4Sb12

BaOs4Sb12

1.5, 1.521.7

31, 20

o0.27 (1.3 K, 50 T)

104, 98

298, 260

101

W0.35

11

270

93

M

W0.35

9

270

100

9.33

M

W0.35

44

258

90

9.36

M

W0.35

46

251

88

Matsuoka et al. (2005), Schnelle et al. (2008), Takabatake et al. (2006), Yoshii et al. (2006) Takabatake et al. (2006) Takabatake et al. (2006) Kaiser and Jeitschko (1999) Takabatake et al. (2006), Matsuoka et al. (2006) Takabatake et al. (2006), Matsuoka et al. (2006)


BaFe4Sb12

109


hp (K)

Ms (lB/f.u.) c (mJ/ K2 mol)

hD (K)

a (Å)

GS

SrPt4Ge12

8.6601

SC

5.1

41, 39.9

220

BaPt4Ge12

8.6928

SC

5.35

42, 34.1, 47.3

247, 215

LaPt4Ge12

8.6235

SC

8.27

75.8

CePt4Ge12

8.6156

M (IV)

PrPt4Ge12

8.6111

SC

NdPt4Ge12

8.6074

EuPt4Ge12 UPt4Ge12 ThPt4Ge12

W0.4 K

2.85

62

7.91

3.67, 3.59

16.4, 16.8

AFM

0.67

3.72

31.1

8.6363

AFM

1.7

8.5887 8.5924

SF SC

4.75, 4.62

87.1

1.6 (2 K, 7 T)

18.4 0.12 (at 2 K)

156 35, 40

260, 217

Remarks

References

Hc2 ¼ 1 T

Bauer et al. (2007), Gumeniuk et al. (2008a) Twmax ¼ 160 K, Bauer et al. (2007), Gumeniuk Hc2 ¼ 1.8 T et al. (2008a, 2008b) Hc2 ¼ 1.6 T Gumeniuk et al. (2008a), Toda et al. (2008) Twmax ¼ 80 K Gumeniuk et al. (2008a), Toda et al. (2008) Hc2 ¼ 2 T Gumeniuk et al. (2008a), Toda et al. (2008) Gumeniuk et al. (2008a), Toda et al. (2008) Gumeniuk et al. (2008a) Bauer et al. (2008) Hc2 ¼ 0.21 T, Bauer et al. (2008), 0.3 T Kaczorowski and Tran (2008)

H. Sato et al.

Compound

110

Table A3. Magnetic properties of filled skutterudites APt4Ge12. Lattice constant a, ground state (GS) (SC: superconducting, M: metallic, AFM: antiferromagnetic, IV: intermediate valence, SF: spin fluctuation), the transition temperatures to the ordered state TTR, effective magnetic moment meff, paramagnetic Curie temperature yp, saturation moment Ms, electronic specific heat coefficient g, and Debye temperature yD are listed. Abbreviations Hc2 in remarks is upper critical field of superconductibity. Twmax is the temperature where wm shows a broad peak. All these properties were evaluated on polycrystalline samples synthesized at ambient pressure.

CHAPTER TWO

Spin Dynamics in Nanometric Magnetic Systems David Schmool

Contents 1. Introduction 111 2. Ferromagnetic-Resonance Theory Applied to Nanometric Systems 116 2.1. Fundamentals of FMR theory 116 2.2. Free-energy density and magneto-crystalline anisotropies 119 2.3. Dipolar interactions between particles 124 2.4. The magnetic surface: surface anisotropy and boundary conditions 128 2.5. Standing spin-wave modes 131 2.6. Resonance linewidth and relaxation processes 139 3. Review of Simulations of Magnetodynamics in Nanometric Systems 147 3.1. Stoner2Wohlfarth model 148 3.2. Nanoparticle assemblies 151 3.3. Nanostructured arrays 164 4. Spin-Current-Induced Dynamics in Magnetic Nanostructures 186 4.1. Theoretical background 188 4.2. Experimental observation of spin-current-induced magnetisation dynamics 201 5. Superparamagnetic Effects in Magnetic Nanoparticles and FMR 218 6. Selected Review of Experimental Studies of Spin Dynamics in Nanometric Systems 227 6.1. Random nanoparticle assemblies 227 6.2. Regular Nanostructured Arrays 266 7. Summary 333 Acknowledgements 336 References 336

Corresponding author. Tel.: 351 22 04 02 337

E-mail address: [email protected] Departamento de Fı´sica and IFIMUP 2 IN, Faculdade de Cieˆncias, Universidade do Porto, Rua Campo Alegre 687, 4169-007 Porto, Portugal Handbook of Magnetic Materials, Volume 18 ISSN 1567-2719, DOI 10.1016/S1567-2719(09)01802-2


111

112

David Schmool

1. Introduction Recent research in magnetism has paid much attention to the manipulation of the properties of magnetic materials which can be, in part, controlled by the size and species of magnetic entities and their separation. Other factors include growth phase conditions and defects. In general, this means the size reduction in one or more dimensions of the magnetic body, for example, thin films (one-dimensional (1D) reduction) or nanoparticles (three-dimensional (3D) reduction), and proximity effects in the form of magnetic interactions between the magnetic entities. There are a number of factors that are of interest in such low-dimensional structures, where one or both of these confinement and proximity effects can operate. Firstly, edge effects can become dominant, where the number of boundary spins becomes a significant proportion of the entire magnetic body. Since boundary (which can be distinguished as corner, edge or face) spins have a reduced (magnetic) co-ordination, their magnetic environment (local exchange interactions between neighbouring spins) is altered with respect to bulk (interior) spins. As such their magnetic properties can be modified; in particular their magneto-crystalline anisotropy can be very different from that of bulk spins. This is usually defined as surface anisotropy. Such surface or boundary conditions can be very important in dynamic properties, such as measured by ferromagnetic resonance (FMR), since they can provide magnetic pinning which will allow standing spin waves to be excited (Kittel, 1958; Puszkarski, 1979). Such conditions are well known in FMR or more specifically spinwave resonance (SWR). Standing spin-wave excitation will be dependent on the size of the magnetic entity and its exchange stiffness constant. In addition to boundary effects, interaction between magnetic entities, which can contribute to the effective magnetic field of the spin system, can also be of great importance. Interactions in magnetic multilayers were seen to be of great importance in magnetic multilayer, where for example the Ruderman2Kittel2Kasuya2Yosida (RKKY) interaction was at the origin of the giant magnetoresistance (GMR) observed in these structures (Baibich et al., 1988; Gru¨nberg et al., 1987). This was the area which was awarded with the 2007 Nobel Prize in Physics (Nobel Prize Committee, 2007). GMR was subsequently observed in granular systems. The RKKY interaction is only of importance where the intervening non-magnetic material between magnetic entities in metallic and non-magnetic (Hathaway, 1994). Other forms of interaction such as dipolar can also be of importance depending on the nature and disposition of the materials under consideration, see for example Heinrich and Bland (1994). In nanoparticle, nanogranular and nanostructured systems the dipole2dipole interaction (DDI) is also observed to be important and must be taken into account when considering assemblies (Schmool and Schmalzl, 2007; Schmool et al., 2007).

113

Spin Dynamics in Nanometric Magnetic Systems

FMR is a powerful analytical tool in the measurement of magnetic properties of ferromagnetic (FM) materials and has been applied to the entire range of materials, from bulk ferromagnetic and ferrimagnetic crystals to magnetic thin films and multilayers, and in more recent times it has also been used to characterise nanogranular and nanoparticle systems (Heinrich and Cochran, 1993; Schmool et al., 1998, 2007; Trunova et al., 2008; Vonsovskii, 1966). The FMR technique has a relatively long history, dating from the 1940s with the introduction of microwave technologies and radar. Its rapid evolution in the 1950s saw the development of both the experimental technique as well as the fundamental theory, which later extended to the related phenomenon of SWR (Kittel, 1947a, 1947b, 1958; Landau and Lifshitz, 1935; Van Vleck, 1950). The latter also illustrated the importance of surface and interface properties in FM thin films (Kittel, 1958; Tannenwald and Seavey, 1957). FMR is an excellent method for the evaluation of magnetic properties in low-dimensional structures or nanometric systems. The FMR technique provides a simple method for measuring the effective magnetic field to which the spin system is subject. This effective field can be expressed as: H eff ¼

@E , @M

(1)

where E represents the free-energy density of the magnetic system. The energy density is a convenient way in which to consider the magnetic system and has been successfully applied to many types of sample. The effective field considers all contributions to the local magnetic field to which the spin system is subjected and will include sample-dependent terms such as demagnetising and anisotropy fields as well as externally applied fields (both static and dynamic). Using this concept, the resonant state will correspond to the local Larmor precession in which the applied magnetic field acts in conjunction (or compensates) the sample-dependent contributions. In Section 2, we will discuss these contributions in details and show how a full analysis of an FMR experiment requires us to account of all contributions to the magnetic free energy which must include all angularly dependent terms. As will be seen, these magnetic anisotropies are very important when considering the overall magnetic properties and behaviour of the material. It is one of the strong points of FMR that it is able to unambiguously measure and distinguish the various forms of magnetic anisotropy. FMR has been enormously successful in characterising the complex magnetic interactions which can arise in magnetic multilayers and nanostructures (Cochran et al., 1986; Lindner and Baberschke, 2003). Over the last decade, or so, a number of developments have been made in experimental techniques which have been used to measure magnetisation dynamics, these include: pump-probe or femtosecond spectroscopy (Beaurepaire et al., 1996; Freeman and Hiebert, 2002; Hiebert et al., 1997;

114

David Schmool

Hicken et al., 2002; Hohlfeld et al., 1997, Wu et al., 2002), pulse generation (Silva et al., 1999), scanning probe (cantilever) FMR (Midzor et al., 2000), network analyser FMR (NA-FMR, Mosendz et al., 2006), bolometric detection of FMR (Meckenstock, 2008) and high-frequency electrical measurements of magnetodynamics (Mecking et al., 2007; Tserkovnyak et al., 2005). Many of these techniques are easily adaptable or were developed to measure magnetisation dynamics in nanometric structures. These techniques are mainly based on the delivery of the highfrequency excitation signal to the sample, usually through a micro-stripline on which the sample is located. Much of these advances are a direct reflection of the need to study ever smaller-sized magnetic features and for evaluating the temporal response of a magnetic systems, with particular emphasis on ultrafast dynamics for modern application for example in telecommunications and data storage systems. Although a full overview of all the techniques available for the study of spin dynamics will not be given, some of these techniques will be discussed in our review. Nanometric systems can be classified as random arrays of magnetic nanoparticles and regular arrays of artificially structured materials, produced for example by electron beam lithographic (see e.g. Castanõ et al., 2003; Skomski, 2003; Stahlmecke and Dumpich, 2005). In the latter case, we have well-defined magnetic entities; well-defined shape anisotropy and possibly magneto-crystalline anisotropy axes. Such modern sample preparation techniques allow a good level of sample reproducibility, though structure sizes are generally on the larger end of the nanometre scale, typically being of the order of around 100 nm. However, the advantage of these preparation methods is the degree of control and shape definition, where recent work has allowed the study of such diverse structures as magnetic rings, dots, antidots and the so-called ‘spin-ice’ systems (Mo¨ller and Moessner, 2006; Remhof et al., 2007; Wang et al., 2006a, 2006b). The case of random arrays of nanoparticles is more complex since the preparation techniques typically lead to a log-normal size distribution of particle sizes and a random orientation of the anisotropy axes (including both shape and magneto-crystalline contributions). The details of this size dispersion depend greatly on the production method of which there are many available using both physical and chemical processing techniques, see for example Batlle and Labarta (2002) and references therein. The size distribution, or polydispersion, is usually expressed in terms of particle volume V in the form (Kliava and Berger, 1999): ( 2 ) lnðV =V 0 Þ 1 exp PðV Þ ¼ pffiffiffiffiffiffi , 2s2 2psV

(2)

where s is the standard deviation of ln V, being related to the width of the distribution, and V0 the average (mean) particle volume. The form of


Figure 2.1

115

Log-normal size distribution.

Eq. (2) is illustrated in Fig. 2.1. Systems with no distribution of particle size are said to be monodisperse, many authors consider that the only systems to be truly monodisperse are those which are lithographically defined. The treatment of the random anisotropies can be a complex problem and requires an effective summation over all directions of the magnetic axes or over all magnetic particles in an assembly. This can be assessed using the so-called random anisotropy model (Herzer, 1990, 2005) and will be discussed further in detail later in this review (see Section 2.2). In this article, we will discuss the elements of FMR theory which are relevant for a general understanding of this technique as applied to nanometric systems. In the following section, we shall outline the main elements which are necessary to deal with these systems, where we consider the various components which make up the free energy of the magnetic systems, and hence its effective field. The specific cases of surface anisotropy and interparticle interactions will also be discussed in detail since they are of particular importance in these systems. It will be noted that while the analysis of FMR theory is mainly concerned with the consideration of the resonance field, we will also discuss the specific points which are particular to nanometric systems with regards to the linewidth. There has been very extensive numerical calculations performed on nanometric systems and Section 3 will outline some of the main approaches that have been used. This has grown into quite a broad area of research in recent years and it is not possible to give an exhaustive review of the subject (though we have tried to give a flavour of some of the main elements). There has been an enormous research effort based on the dynamical effects of a spin current on the magnetisation state in nanostructured elements and multilayers. This arises from the effect of the transfer of spin angular momentum and spin and charge accumulation effects. We will discuss some of the main details of

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these from both an experimental and a theoretical point of view in Section 4. An important aspect of the magnetism in nanometric systems is directly related to their size and magnetic anisotropy; this concerns the stability of the particles’ magnetisation which can be subject to thermal fluctuations. This phenomenon is known as ‘superparamagnetism’ and will be the subject of Section 5, where we will outline its most important features and discuss how this can influence the resonance spectra in an FMR experiment. There have been many FMR studies in nanometric magnetic systems, more so than the theoretical simulations and we shall aim at reviewing some of the principal studies where typical features are presented; Section 6. This is not meant to be an exhaustive review, but our objective will be to outline the main features that one encounters in these systems. We have separated this Section 6 into two parts where we distinguish between randomly oriented magnetic nanoparticles and well-defined magnetic nanostructures since there are important differences. We will not specifically deal with domain effects since these are usually associated with larger elements, however, domain-like effects can occur in collective excitations between elements where strong coupling is present and this will be briefly discussed. As a general reference, the books edited by Hillebrands and Ounadjela (2002, 2003) provide an excellent outlook to recent progress in theory and techniques for spin dynamics in low-dimensional systems. While thin films and magnetic multilayers do class as nanometric (1D), we will not deal explicitly with this class of material since they have been considered in great detail by many authors (see e.g. Farle, 1998; Heinrich and Bland, 1994; Heinrich, 2005; Lindner and Baberschke, 2003 and references therein), we will limit ourselves to considering thin films in a general manner and where it is useful for the overall discussion, as for example in the case of spin waves.

2. Ferromagnetic-Resonance Theory Applied to Nanometric Systems The theory of FMR is well established; in the following we shall give a basic outline of the main components of this theory. In addition, we will discuss the various elements which should be considered in the application of this theory to the specific case of nanometric magnetic systems.

2.1. Fundamentals of FMR theory FMR is a dynamic technique whose theoretical treatment is generally based on the Landau2Lifshitz (LL) equation of motion; where the Gilbert form of damping is frequently used to describe the relaxation mechanism

117


(Heinrich and Cochran, 1993; Heinrich, 2005) (see Section 2.6). This equation takes the form: 1 @M a ¼ ðM Heff Þ þ g @t gM

@M M , @t

(3)

where g is the magneto-gyric ratio and a represents the Gilbert damping parameter. The magnetisation, M, is comprised of a dynamic and static component; M ¼ M0 þ m, which arise from the application of a microwave component to the applied magnetic field, hrf. The total (effective) magnetic field Heff, has several components, which we can express as: Heff ¼ H0 þ hrf þ Hin . For FM materials, we need to consider not only the static magnetic field, H0, and dynamic field, hrf, but also the internal magnetic field, Hin, which has a very important role to play in FMR. It will be this internal field component which will provide all the angular dependences due to the various magnetic anisotropies which may exist in the sample; magneto-crystalline anisotropy, shape anisotropy, surface and interface anisotropy. In a standard FMR experiment, we can make the assumption that the dynamic component of the magnetisation is much smaller than the static component; jM0 j jmj, such that we can neglect second-order effects. This is a valid assumption when the microwave power used is low as in the case of standard microwave spectrometers. Recent developments in high-frequency dynamical techniques may, however, require that the consideration of the higher order terms be necessary, for example, in the case of pump-probe, stripline and pulse generation techniques (see e.g. Freeman et al., 1998; Choi and Freeman, 2005; Hicken et al., 2002; Rasing et al., 2003; Russek and McMichael, 2003; Silva et al., 2002), where precessional angles exceed a few degrees from the equilibrium orientation. We will only consider the standard theory, which covers a majority of the experimentally studied cases. For high-power and non-linear effects, see for example An et al. (2004); Berteaud and Pascard (1966); Kuanr et al. (2007) and Suhl (1957). Essentially, Eq. (3) describes the damped precessional motion of the magnetisation vector, M, about the direction of the effective magnetic field, Heff (see Fig. 2.2). In general, this will be a circular precession, whereby the magnetisation vector describes a cone; however, anisotropies in the magnetic system can often produce a non-circular precession. Manipulation of Eq. (3), via the substitution of the various components of M and H and the expansion of the equilibrium conditions for small-angle deviations, leads to the so-called Smit2Beljers (SB, also known as Smit2Suhl) equation (Smit and Beljers, 1955; Vonsovskii, 1966): " 2 2 # 2 o ð1 þ a2 Þ @2 E @2 E @E ¼ 2 2 . 2 2 g @W@f M sin W @W @f

(4)

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z Heff

M x dM dt

- M x Heff

M

Figure 2.2

Coordinate system for the Landau--Lifshitz equation.

This equation uses the second derivatives of the free energy with respect to the polar and azimuthal angles, W and f, in conjunction with the equilibrium conditions which are defined by the first derivatives: @E ¼0 @W

and

@E ¼ 0. @f

(5)

These equilibrium conditions define the direction of the magnetisation vector under the specific conditions of the free-energy density, including all magnetic anisotropies and applied external magnetic fields. The equilibrium (orientation) angles are often designated as Weq and feq and the resonance condition will then be evaluated at this orientation. Contributions to the free energy will depend on the magnetic sample under consideration and will in effect be the same contributions that are considered in the effective magnetic field, where we have simply transferred from considering the effective field to considering the total energy of the system in the passage from Eqs. (3)2(4). In FMR, the Zeeman energy will always be a principle component due to static and dynamic (microwave) magnetic field that are required. Additional contributions will also be required and are typically due to magnetostatic (or shape) energy and magneto-crystalline anisotropies. By extracting the exchange energy from the other terms in the effective field (used in the torque term if Eq. (3)), we can rewrite the LL equation in the form (Rado and Weertman, 1959): 1 @M 2A a @M 2 M r ¼ ðM Heff Þ þ M , M þ g @t M2 gM @t

(6)

119


where A denotes the exchange stiffness constant, which depends on the exchange between spins (see also Section 2.5). From Eq. (6), it is possible to modify the resonance Eq. (4) to explicitly illustrate the spin-wave terms as given by (Maksymowicz, 1986; Schmool and Barandiarań, 1998a): 2 o 2A 2 2 1 @2 E 1 @2 E 2A 2 k k ¼ þ þ g M M @W2 M sin2 W @f2 M " # 2 2 1 @2 E @ 2 E @E þ 2 2 , @W@f M sin W @W2 @f2

(7)

where k represents the wave vector of the standing spin-wave mode whose allowed values are determined by the boundary (or pinning) conditions, where we have neglected the small correction due to the damping. Evidently for FMR we can set k ¼ 0, whereby Eq. (7) reduces to Eq. (4) for the uniform excitation mode. The extent to which spin-wave terms are important will be principally defined by the exchange stiffness constant, A, the boundary conditions (Rado and Weertman, 1959) and the size of the magnetic entity. In most nanosystems, the size constriction is such that volume modes will not be excited, though arguments for surface modes can be made depending on the boundary conditions permitting surface freedom (Puszkarski, 1979). We will further take up this argument in Section 2.5 where we will discuss the basic elements necessary in the consideration of SWR and higher modes of excitation. Further contributions to the freeenergy density, E, are given in the following sections.

2.2. Free-energy density and magneto-crystalline anisotropies The free-energy density approach is very versatile since we only include those terms which are relevant to magnetic system under consideration. For the case of assemblies of nanoparticles two approaches are possible. Netzelmann (1990) introduced the idea of separating the magnetostatic energy into the particle demagnetisation term and the sample demagnetisation term, where later corrections from Dubowik (1996) and Kakazei et al. (1999) give the magnetostatic energy: 2 2 1 1 2 ENP DM ¼ f ð1 f ÞM N p M þ f M N s M, 2 2

(8)

2

where N p;s represents the demagnetisation tensor of the particle (p) and sample (s), respectively, and f the volume fraction of magnetic nanoparticles in the assembly, which can be specified as: f ¼

Vm ¼ Vs

PN

i¼1 V i

Vs

,

(9)

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where we introduce the total volume of the sample, Vs. In the case of polydisperse systems we can also write V m ¼ NhV i, where the average particle volume is given by: R VPðV ÞdV . hV i ¼ R PðV ÞdV

(10)

In an alternative approach, Schmool et al. (2007) substitute the dipolar energy, see Section 2.3, for the second term in Eq. (8). This is seen to have a similar result to the Netzelmann approach, though the resulting physical interpretation is different. An important component of the free energy of the nanosystem is the individual magneto-crystalline energy of the nanoparticles or structures. Frequently a uniaxial anisotropy is used in nanoparticle systems, though cubic contributions can also be important, as discussed later for the case of weak surface anisotropy (see Section 2.3). These energies can be expressed, using the spherical coordinate system illustrated in Fig. 2.3, in the form (Gurevich and Melkov, 1996): EuK ¼ K u1 sin2 W þ K u2 sin4 W þ K u3 sin6 W þ

(11)

for uniaxial magneto-crystalline anisotropy and Ecub K ¼

K cub K cub 1 ðsin2 2W þ sin4 Wsin2 2fÞ þ 2 sin2 W sin2 2Wsin2 2f 4 16

(12)

for cubic magneto-crystalline anisotropy. Energy surfaces corresponding to these forms of magneto-crystalline anisotropy are illustrated in Fig. 2.4. Table 2.1 gives a summary of some of the more common forms of anisotropy. For nanosystems where the anisotropy axes are perfectly aligned, these energy expressions can be used directly in the total freeenergy density expression. In many cases where there are random

Figure 2.3 Spherical coordinate system used in the analysis of the resonance conditions.

121


1

(a)

0.9

0.9 0.4

0.7

0.2

0.8 0.7

0.5

0.6

0.6 0

N

0

N

1

0.8

0.5

0.5

−0.2 −0.4 1 0.5

1 0.5

0 y

−1

−1

−0.5

−0.5

0.3

−1 0.4

0.2

0.4 0.3 0.2

0.4 0.2

0

0.1

0

−0.5

0.4

y

x

−0.2 −0.4 −0.4

0.1

0 x

−0.2

0.2

1

(b) 0.9

0.95

1

0.4 0.25

0.9

0.5

0.2 N

N

0.2 0 0.15

−0.2

0.1

−0.4 0.4 0.4

0.2 0 y −0.2

−0.5

−0.4 −0.4

0.05

0.75 1

0.5 0.5

0

0 x

0.8

−1 1

0.2 −0.2

0.85

0

0

y

−0.5 −1 −1

−0.5

0

0.7

x

Figure 2.4 Energy surfaces for magneto-crystalline anisotropies. (a) Uniaxialanisotropy energy density. (Left) easy-axis anisotropy (Ko0) and (right) easy-plane anisotropy (KW0). (b) Cubic anisotropy energy density (left) coordinate axes are easy axes (Ko0) and (right) coordinate axes are easy axes (KW0).

distributions of the nanoparticles, it is necessary to sum over all directions of the anisotropy axes for all particles. For large assemblies of particles an average anisotropy will be homogeneous and will have a constant contribution to the overall effective field. An effective anisotropy can be introduced which incorporates both volume and surface effects, where it is assumed that the overall angular dependence is the same (Bødker et al., 1994): K eff ¼ K v þ

6 K s. D

(13)

with Kv,s being the volume and surface contributions to the total anisotropy, and the factor (6/D) arises from the surface-to-volume ratio for the case of a sphere. Herzer (2005) has considered the case of randomly oriented nanocrystals in and amorphous magnetic matrix. These systems exhibit very soft magnetic properties, see Herzer (1997) and references therein. In the random anisotropy model, the overall effect of the individual nanocrystal anisotropies is considered in the assembly. The model was initially

Crystal class

Typical example

MAE

Cubic a¼b¼c

Ni, Fe

F 100 ¼ K 4 sin2 y 18 K 4 ð7 þ cos 4jÞsin4 y þ F 110 ¼ 0:5K 4 sin2 y 0:75K 4 sin4 y þ 0:25K 4 ð10sin2 j 3sin4 jÞsin4 y 1:5K 4 sin2 ysin2 j pffiffiffi 2 F 111 ¼ 13 K 4 23 K 4 sin2 y þ 12 K 4 sin4 y þ 13 2K 4 sin 3j cos ysin3 y

a ¼ b ¼ g ¼ 901 Tetragonal a ¼ b6¼c

122

Table 2.1 The anisotropic part of the free-energy densities F for different crystal structures. F can be written differently for the same symmetry. In thin films, an additional uniaxial in-plane anisotropy ½(k2||cos 2ø sin 2y) is sometimes added. Note that with F ¼ K2siny + K4sin4y the MAE calculated from total energy differences FMAE ¼ F(901)F(01) ¼ K2 + K4 ¼ dF/dø|451. In this paper, the tetragonal expression () is used. Fijk in the first row are for different thin-film orientations (ø in-plane angle, y out-of-plane angle).

Ni, Co, Fe/Cu(0 0 1)

12 K 2 a2z 12 K 4 a2z 12 K 4 ða4z þ a4z Þ þ or K 2 cos2 y 12 K 4 cos4 y 12 K 4 14 ð3 þ cos 4jÞsin4 y þ ðÞ or ðK 2 þ K 4 Þsin2 y 12 ðk4 þ 34 K 4 Þsin4 y 12 K 4 cos4 jsin4 y þ ¼ k2 sin2 y þ k4 cos4 jsin4 y þ

a ¼ b ¼ g ¼ 901 Hexagonal a ¼ b6¼c a ¼ b ¼ 901 g ¼ 1201 Trigonal

Co

K 2 sin2 y þ K 4 sin4 y þ K 6 sin6 y þ K 6 cos6 jsin6 y þ

a-Fe2O3

K 2 sin2 y þ K 4 sin4 y þ K 6 sin6 y þ K 4 sin3 y cos ycos3 j þ K 6 cos6 jsin6 y þ K 6;3 sin3 ycos3 ycos3 j

Orthorhombic a6¼b6¼c a ¼ b ¼ g ¼ 901

sin2 yðk1 cos2 j þ k2 sin2 jÞ þ sin4 yðk3 cos2 y þ k4 sin2 jcos2 j þ k5 sin4 jÞ þ sin2 ycos2 yðk6 cos2 j þ k7 sin2 jÞ Fe3O4

aI are direction cosines with respect to the main cubic axes (Farle, 1998).

David Schmool

Rhomboedric a¼b¼c a ¼ b ¼ g6¼901 a ¼ b ¼ g 120


123

developed in order to explain the soft magnetic properties exhibited by amorphous FM materials (Alben et al., 1978) and further developed by Chudnovsky et al. (1986). By considering and assembly of exchangecoupled nanocrystals of diameter D and volume fraction vcr with uniaxial magneto-crystalline anisotropy K, the effect of random orientation on the anisotropy axes is obtained by averaging over the N nanocrystals within the FM correlation volume, where N ¼ v cr ðL ex =DÞ3 and the correlation volume is V ¼ L 3ex (see Fig. 2.5(a)). Statistical fluctuations within the exchange volume will give rise to an easy direction for the N crystals and the averaged anisotropy energy will be determined by the fluctuation amplitude. As such, the effective anisotropy constant can be expressed as: 3=2 v cr K pffiffiffiffiffi D hKi pffiffiffiffiffi ¼ v cr K . L ex N

(14)

Figure 2.5 (a) Schematic representation of the random anisotropy model, where arrows indicate the randomly fluctuating anisotropy axes and the hatched area represents the ferromagnetic (FM) correlation volume which is determined by the exchange length, Lex (Herzer, 1997). (b) Random average anisotropies, with kind permission of Springer Science and Business Media (Suzuki and Herzer, 2006).

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It is, therefore, seen that the averaging out of the local anisotropy depends on the number of particles within the exchange volume and will only be effective for coupled particles, that is where interactions between the particles exist. Figure 2.5(b) illustrates how the local anisotropies are averaged out over an assembly of nanocrystals, see Suzuki and Herzer (2006). The fact that there is a dependency between the averaging of the local anisotropy and the exchange interaction, the exchange length must be renormalized by substituting /KS for K, such pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi that Lex is self-consistent with the averaged anisotropy: L ex ¼ A=hKi, where A is the exchange stiffness constant. From this we can obtain the following relation: hK i

v 2cr K

D L0

6 ¼

v 2cr D6 K 4 , A3

(15)

where L0 is the FM exchange length and represents the minimum length scale over which the magnetic moments of the particles can vary. If the particle size, D, is less than this value, the magnetisation will not follow the randomly oriented easy axis and will be forced to align by the exchange interaction. The arguments used in obtaining this result are valid not only for uniaxial anisotropies and cubic but other symmetries will also apply. The final, homogeneous, component in the total energy density will be the Zeeman energy due to the applied magnetic field, which in spherical coordinates takes the form: EZ ¼ M H ¼ MH sin W sin yH cosðfH fÞ þ cos W cos yH ,

(16)

where yH and fH define the orientation of the applied magnetic field H. The final free-energy density can now be written as a summation of the relevant contributions: E ¼ E Z þ E K þ E DM þ E DDI þ ESA .

(17)

The relevant forms for these contributions will be based on the particular system under consideration. There are various examples in the literature which show explicit cases, see for example Farle (1998), which gives an extensive overview of the different forms of magneto-crystalline anisotropy and its effects in FMR. By way of example, we will consider the case of tetragonal symmetry which is often encountered in thin magnetic films (see Table 2.1 and Fig. 2.6). In the case where yH ¼ 451 and f ¼ 451 (i.e. H|| and M|| are parallel to [1 1 0] and K4||o0), the resonance equation can be written as (Farle et al., 1997a, 1997b, Schulz and

125


θH

[001]

H M [010]

θ [110] hmw

ϕ

ϕH [100]

Figure 2.6 A face-centred tetragonal (fct) crystal. The azimuthal / and polar h angle of the magnetisation M is shown. The microwave magnetic field hmw oscillates in the film plane. The sample can be mounted such that the magnetic field H can be rotated between the [1 1 0] and [0 0 1] or between the [1 0 0] and [0 0 1] directions (Farle, 1998).

Baberschke, 1994): 2 o 2K 2 K 4? K 4jj þ cos 2W ¼ H cosðyH WÞ þ 4pM þ M M M g

K 4? K 4jj (18) þ cos 4W H cosðyH WÞ þ M M

2K 2 K 4jj 2K 4? K 4jj 2K 4jj þ 4pM þ þ cos2 W þ þ cos4 W . M M M M M

The frequency-field characteristics can be studied from this relation for various cases of the direction of the applied field and anisotropy constants, as illustrated in Fig. 2.7. The horizontal lines show the intersections of the dispersion relation and indicate where in an FMR spectrum the resonance lines would be found for a fixed frequency. The temperature dependence of the anisotropy constants has been studied using FMR in thin magnetic films, see for example Schulz and Baberschke (1994), Farle et al. (1997a) and Farle (1998).

2.3. Dipolar interactions between particles Interactions between magnetic entities depend not only on their inherent properties but also on their average separation (particle density) and the intervening media between the magnetic particles. The theory of dipolar interactions is important for insulating matrices and has been extensively studied in recent years. A comprehensive method will account for all

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Figure 2.7 (a) x/c according to Eq. (18) for a thin film with l0M ¼ 0.528 T, K4i ¼ 0 and K2 ¼ 1.7 105 J/m3 (11.7 meV/atom) (dotted curve) and K2 ¼ 1.8 105 J/m3 (12.3 meV/atom) (full curve). The easy direction of M (H ¼ 0) is perpendicular to the film. Resonance occurs at the intersections of the 4 (9.3) GHz lines with the x/c(H) curves. The broken line is for M ¼ K2 ¼ 0, that is paramagnetic resonance. (b) x/c according to Eq. (18) for a thin film with l0M ¼ 0.528 T (cgs units: M/ 4p ¼ 420 G), K4i ¼ 0 and K2 ¼ 1.5 105 J/m3 (dotted line, full curve) and K2 ¼ 1.0 105 J/m3 (chain, broken lines). The easy direction of M is perpendicular (parallel) for the larger (smaller) K2. The magnetic field is applied either parallel (||) or perpendicular (>) to the film plane (Farle, 1998).

interactions between all particles in the nanosystem. The basic magnetic dipolar interaction between two magnetic nanoparticles or entities is governed by the equation: E DDI ij ðrij Þ

" # 3ðmi :rij Þðmj :rij Þ 1 ¼ mi m j ; 4pr 3ij r 2ij

(19)


127

where mi,j represents the magnetic moment of particle i, j and rij is the displacement vector between them. We are principally interested in the effects of interactions in assemblies under conditions of FMR. We consider the case in which the applied magnetic field is sufficient to saturate the sample and as such all magnetic moments will be aligned (as is the case for an FMR experiment). Under this state, we can simplify Eq. (19) using the spherical coordinate system as: EDDI ij ðW; j; y; fÞ ¼

mi mj f1 3½sin y sin W cosðf jÞ þ cos y cos W 2 g. 4pr 3ij

(20)

All angles are defined in Fig. 2.8. This equation can be further simplified for the case where the vector between magnetic moments is aligned along one of the principal axes. Importantly this equation demonstrates the introduction of a magnetic anisotropy into the system. The overall anisotropy will depend explicitly on the spatial distribution of the magnetic nanoparticle assembly. We can sum all interactions in an assembly using the following equation: EDDI TOT ¼

1 X X DDI E ðW; j; y; fÞ. 2 i jai ij

(21)

The factor of ½ is required so as not to count the interactions twice. In this form of the dipole interaction between particles, it is assumed that the direction of the magnetic moments, mi,j, are parallel due to the effect of an applied magnetic field. This is valid for the case of FMR measurements on samples with weak anisotropies (HaoHres). Equation (21) expresses the DDI in a spherical coordinate system where rij is the interparticle separation with polar and azimuthal angles, Yij and Fij , respectively (see Fig. 2.8). The angles W and f define the direction of the magnetisation vectors Pof the particles, Mi ¼ mi =V i . Vm is the total magnetic volume; V m ¼ i V i of the particle assembly.

Figure 2.8

Interaction geometry and coordinate system.

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Figure 2.9 Angular variation of DDI energy for an assembly of nanoparticles with slab-like spatial distribution.

The summation over particles can be complex and must be treated with care. Schmool and Schmalzl (2008) have considered the cases of regular and random particle arrays of rectangularly shaped assemblies. Cases for a variation of particle density and overall assembly shape have been considered. Figure 2.9 shows the angular variation of the DDI energy for such an assembly, which fits to a simple cosine-squared form (Schmool and Schmalzl, 2008): 2 ERect DDI ¼ E 0 cos W þ E 1 ,

(22)

where E0 represents the pre-factor in the summation in Eqs. (20) and (21), and will depend on the particles magnetic properties as well as on the spatial distribution while E1 is an offset which will also depend on the spatial distribution of the particles and the total number of particles in the assembly. This analysis shows the importance of the spatial distribution of the particles and the overall shape of the assembly, and can be seen as an overall sample shape function which introduces a magnetic anisotropy into the system. In addition to the variation of the DDI with particle shape distribution, the DDI anisotropy also exhibits a density squared variation which is independent of distribution details and is identical for regular or random particle arrays. In Fig. 2.10, the experimental data from FMR is shown with a fit to the FMR theory. This shows the same form of angular dependence as expected from DDI interactions. Other researchers have considered the DDI and its effects in nanoparticle assemblies. For example, Kakazei et al. (2005), Schmool et al. (2007), Majchra´k et al. (2007a, 2007b) have all applied the DDI contribution in the FMR analysis of discontinuous magnetic (DM) multilayer systems, where the island structure of the discontinuous layers was treated as a granular nanoparticle system. A fuller description of the analysis will be discussed in Section 6.1.1 along with some experimental and theoretical results of FMR experiments in discontinuous multilayer systems.

129


3200

Resonance Field (Oe)

3100 3000 2900 2800 2700 2600 2500 0

20

40 60 Angle (°)

80

100

Figure 2.10 Angular variation of resonance field for c-Fe2O3 nanoparticles (with a mean diameter of 4.6 nm), points correspond to experimental values and the line corresponds to the FMR-DDI theory (Schmool and Schmalzl, 2007).

2.4. The magnetic surface: surface anisotropy and boundary conditions Since a surface spin has a reduced magnetic coordination with respect to bulk spins, the magneto-crystalline anisotropy will be modified. In general, the treatment of surface anisotropy can be based on the extension of bulk anisotropies and can be expressed in terms of the surface energy (Kachkachi and Dimian, 2002): ESA ¼

X

K i ðSi :ei Þ2 .

(23)

i

Here Si refers to the unit vector spin on site i of the spin cluster (nanoparticle), ei defines the easy-axis unit vector and Ki the anisotropy constant. As represented in Eq. (23), the anisotropy of the particle can have two contributions stemming from the core and from the surface. In the case of a spherical particle for example, all core spins (with full coordination) will have the same core anisotropy constant, Kc, while all surface spins have a surface constant, Ks. The form of surface anisotropy represented in Eq. (23) is known as the transverse anisotropy model. Another approach to this problem uses the surface anisotropy introduced by Neél, which has the general form (Aharoni, 2000; Jamet et al., 2004; Neél, 1954): eel EN SA ¼ K s

XX ðSi :eij Þ2 . i

(24)

j

In this equation, the second summation is over nearest neighbours j of site i and eij ¼ rij =r ij is the unit vector connecting site i to its nearest neighbours. This model is more realistic in that there is no single direction of the anisotropy and will be explicitly dependent on the magnetic

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(100) feet 8 nearest neighbors (n.n)

(111) facet (111) facet 9 n.n

edge axis edge axis

(100)-(111) edge 7 n.n

(100) facet

(111)-(111) edge 7 n.n apex 6 n.n

Figure 2.11 Polyhedral (truncated octahedron) nanoparticle showing the various crystalline faces and atomic positions which will have different atomic coordination and consequently magneto-crystalline (surface) anisotropies. (Reprinted figure with permission from Jamet et al. (2004), r 2004 by the American Physical Society.)

environment of each surface spin. Such an approach while being more true to the nature of the particle surface comes at the cost of being more complex to analyse and treat. This is illustrated in Fig. 2.11. For weak surface anisotropies in magnetic nanoparticles, the surface component can be approximated as a cubic contribution (Garanin and Kachkachi, 2003). To account for such cases in FMR, a simple model has been proposed by Kachkachi and Schmool (2007) and Schmool and Kachkachi (2006). A more detailed approach is outlined in Section 3.2 below. A more standard approach in dealing with a magnetic surface, which has been applied extensively to cases of thin magnetic films and magnetic multilayers, is the consideration of the boundary conditions at the magnetic discontinuity of surface/interface. The so-called Rado2Weertman boundary condition for a free magnetic plane can be written as (Rado and Weertman, 1959):

2A @n M þ ssurf ¼ 0, M M2

(25)

where qn demotes the surface normal derivative. The first term represents the surface exchange torque, which will differ from that for the bulk spin


131

due to the change of exchange coordination. We can extend this to a nanoparticle by considering the individual conditions of each surface spin, i:

2Ai @n si þ ssurf si ¼ 0. i s2i

(26)

The corresponding condition for an interface spin which is subject to the exchange field of a neighbouring magnetic layer, in a bilayer or planar magnetic system for example, can be written in the form of the so-called Hoffman boundary conditions (Hoffmann, 1970):

and

2A1 @M1 ðx1 Þ Jx ½M1 ðx1 Þ M2 ðx2 Þ ¼ 0 M ðx Þ 1 1 2 M1 M 1M 2 @x

(27)

2A2 @M2 ðx2 Þ Jx ½M2 ðx2 Þ M1 ðx1 Þ ¼ 0, M2 ðx2 Þ M 22 M 1M 2 @x

(28)

where x1,2 defined the positions of the two interfaces and Jx is defined by the energy per unit area of the interface as:

E x ¼ J x

M1 ðx1 Þ M2 ðx2 Þ , M 1M 2

(29)

with J x ¼ 2J 12 S 1 S2 =a2 . Alternatively, we can represent the general interface condition with exchange coupling in the following form (Schmool and Barandiarań, 1998a):

2A1 2A12 @n M 1 M 1 @n M2 þ s12 ¼ 0. M1 M 21 M 1M 2

(30)

Here A12 represents the interphase coupling between the two magnetic layers or phases and s12 is the interface torque. For a nanoparticle system, we would have to convert to a spin-by-spin boundary condition analogous to Eq. (26). In a nanoparticle system, such boundary conditions should still apply and the interparticle interactions will certainly be expected to affect these conditions, as in the case of magnetic multilayers. Adapting the Rado2Weertman boundary condition, Eq. (25), we can write the nanoparticle boundary condition as:

M

2A @n M þ M HDDI þ ssurf ¼ 0. M2

(31)

The additional second term accounts for the dipole field contribution from neighbouring magnetic nanoparticles. It would be more appropriate to consider the nanoparticle as a collection of spins, as illustrated above, whereby each surface spin can be considered in its own effective field, including the local dipole field, HiDDI . In this way, we can write Eq. (26) in

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David Schmool

the following form: si

2Ai @n si þ si HiDDI þ ssurf ¼ 0, i s2i

(32)

where i labels the surface spin sites. The evaluation of the dipole field can be obtained from the DDI, which we considered in the previous section. The relationship between the surface anisotropy and boundary conditions is an important one and will determine, along with magnetic parameters and spatial confinement effects, the form of higher order (kW0) spin-wave excitations in magnetically confined systems, such as thin films and lithographically defined structures. In the next section, we will further discuss the implications of this in magnetic thin films.

2.5. Standing spin-wave modes The existence of higher order spin-wave modes (kW0) was indicated earlier in Section 2.1, where we introduced the dispersion relation for spin waves; Eq. (6). The spin-wave concept was originally envisaged by Bloch (1930) to consider the lowest lying magnetic states above the perfectly collinear spin ground state of a FM medium. Thermal excitation is envisaged as perturbing the spins from their equilibrium orientation which can then propagate through the magnetic material. The spin-wave theory of Bloch leads to the prediction that the magnetisation of a 3D ferromagnet falls off from its zero-temperature value with a T3/2 dependence for the lowtemperature regime and is generally considered valid in the limit of To0.5Tc. It should be noted that the rate of decrease of the magnetisation with temperature is determined by the ‘stiffness’ of the exchange coupling which tends to align the individual spins in the material. This same exchange stiffness constant, which in competition with magnetic anisotropy, determines the thickness and surface energy of domain walls (DWs) in FM materials. The exchange stiffness constant is a crucial parameter which, along with the characteristic length and boundary conditions of a confined magnetic system, will determine the allowed values for spin-wave vectors and thus eigenmodes of the system. In terms of physical quantities, the exchange stiffness constant is directly related to the exchange integral between neighbouring atomic spins and can be written as (Cochran et al., 1986): A¼Z

zS2 J , ann

(33)

where z is the coordination number, S the spin quantum number for the atom, J the exchange integral, ann the nearest-neighbour lattice distance and Z a factor determined by the crystalline structure (e.g. Z ¼ 1 for simple cubic,

133


pffiffiffi pffiffiffi Z ¼ 3 for bcc and Z ¼ 2 2 for fcc). The units of A are frequently given as erg/cm. As a starting point, the relevant part of the LL equation which deals with the precessional motion is considered (cf. Eq. (3)) as: 1 @M ¼ ðM Heff Þ, g @t

(34)

where Mðr; tÞ ¼ Ms þ mðr; tÞ is the total magnetisation, which can be vectorially represented by the saturation magnetisation vector and the variable component of the magnetisation, which is generally subject to the condition: Ms jmj. Since Heff depends on M in the right-hand side (RHS) of Eq. (34), this equation will be intrinsically non-linear, however, using the condition of small dynamic magnetisation, that is small-angle precession, the variable magnetisation can be expanded in a series of plane (spin) waves (with 3D wave vector k): mðr; tÞ ¼

X

mk ðtÞeik:r .

(35)

k

The linearised equation is then used to describe the spin waves. Using the contribution of the spatial variation of the magnetisation to the time derivative of the magnetisation (dM/dt), Herring and Kittel (1951) obtain the dipole-exchange spin-wave dispersion relation for an infinite FM medium, which can be written in the general form: 2 o ¼ ðH eff þ Dk2 ÞðH eff þ Dk2 þ 4pM s sin2 Wk Þ, g

(36)

where Wk is the angle between the directions of the wave vector and the static magnetisation and we have used D ¼ 2A=M is the spin-wave constant. For a magnetic thin film, this relationship will be modified due to the broken symmetry at the film surfaces. In this case, the boundary or pinning conditions at the film surfaces must be considered (Soohoo, 1965): dmx ðzÞ þ p1;2 cos 2Wmx ðzÞ ¼ 0; dz dmy ðzÞ þ p1;2 cos2 Wmy ðzÞ ¼ 0; dz

(37)

p1,2 being the pinning parameters of the two surfaces. The allowed wave vectors can then be determined from the pinning parameters and can be expressed in the general formula: ½ðkrn Þ2 pr1 pr2 tanðkrn LÞ ¼ krn ðpr1 þ pr2 Þ

(38)

in which L is the film thickness, px1;2 ¼ p1;2 cos 2W and py1;2 ¼ p1;2 cos2 W. Assuming the film normal to be in the z-direction and the external

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David Schmool

magnetic filed to be aligned along the x-direction, in the long wavelength limit (k||Lo1), an approximate expression for the spin-wave frequencies can be obtained, which is analogous to Eq. (36): 2 o ¼ ðH þ Dk2 Þ½H þ Dk2 þ 4pM s F qq ðkk LÞ . g

(39)

Here F qq ðkk LÞ is the matrix element of the magnetic dipole interaction. In the case of free pinning and perfect pinning, we can write the wave vectors as: k2 ¼ k2x þ k2y þ

np 2 L

¼ k2k þ

np 2 L

.

(40)

In which the quantisation of the perpendicular standing spin waves for free pinning is n ¼ 0, 1, 2, y, while for perfect pinning we have n ¼ 1, 2, 3, y , k|| is the in-plane continuously varying wave vector. For an arbitrary angle between k|| and Ms, the matrix elements of the DDI can be written as follows (Kalinikos and Slavin, 1986): 2! 2 ky 4pM s k F qq ðkx ; ky Þ ¼ 1 þ P qq ðkÞ½1 P qq ðkÞ

P qq ðkÞ x2 . 2 2 H þ Dk k k

(41)

Expressions for Pqq(k) for qW0 can be found in Kalinikos and Slavin (1986). In the case where the spin wave is propagating in the film plane, but perpendicular to the bias magnetic field (kx ¼ 0, k|| ¼ ky), Eq. (41) becomes:

4pM s . F qq ðkjj LÞ ¼ 1 þ P qq ðkÞ½1 P qq ðkÞ

H þ Dk2

(42)

In this case, the function Pqq(k||L) for the lowest thickness mode (q ¼ 0) is given by: P 00 ¼ 1 þ

ð1 ekjj L Þ . kjj L

(43)

If we consider the case of zero exchange (A ¼ 0), the dispersion relation for the lowest thickness mode can be obtained from Eqs. (39), (42) and (43) which results in the dispersion relation of the form of the Damon2Eshbach (DE-dipole surface) mode: oDE ¼ g½HðH þ 4pM s Þ þ ð2pM s Þ2 ð1 e2kjj L Þ 1=2 .

(44)

When the film is magnetised in the plane with kjj ? Ms , spin-wave modes can be divided into dipole dominated (q ¼ 0) with frequencies given by Eq. (44) and exchange dominated (qW0) with frequencies given by the


135

perpendicular standing spin-wave modes (PSSW): 2 np 2 o 4pM s =H 2 H þ DþH k2jj ¼ H þ D kjj þ g L np=L

np 2 þ 4pM s . þD L

(45)

In the general case, where k||LW1, a numerical solution is used to obtain the spin-wave frequencies. In Fig. 2.12, an illustration of various

Figure 2.12 Transformation of the dipole-exchange SW spectrum in tangentially magnetised FM film with the change in the direction of propagation of the spin wave with respect to the direction of equilibrium magnetisation M0, (kl>M0, k2||M0) (Kalinikos et al., 1990).

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David Schmool

types of spin-wave mode are illustrated along with their specific orientations of spin-wave-vector propagation direction and magnetisation. In Section 3.3, we will consider some simulations of spin-wave modes in confined magnetic structures to illustrate the effect of lateral confinement on the spin-wave excitation modes. In the above, we have only considered the case of small-amplitude oscillations. When this is no longer valid, the linearisation of the LL equation is not possible and the spin waves become non-linear, in which case the spin-wave spectrum will be amplitude dependent. There are many phenomena in which non-linear effects in the spin-wave spectrum become important. One particularly important case is when the system has a strong anisotropy for waves having different relative orientations of their in-plane wave vector and magnetisation. For a discussion of this and other related phenomena see for example, Demokritov et al. (2001), Slavin et al. (2002) and Demokritov (2008). Alternative approaches to this problem have been given by Puszkarski (1979) and Maksymowicz (1986). Puszkarski introduced the surface inhomogeneity (SI) model for magnetic thin films in which the film is considered as layers, which are divided into the sublattices containing the internal layers (l ¼ 1, 2, y, L2) and the two surface layers (l ¼ 0 and l ¼ L1), the film having L1 layers and thickness (L1)a, a being the lattice constant in the direction of the film normal. The system is described by the Heisenberg localised spin model: ^ ¼ 2 H

X

J g Slg Slþg;j0 gmB

lj;lþg;j0

X

Heff l Slj .

(46)

lj

The summations extend over different pairs of neighbouring spins, Jg is the exchange integral between nearest neighbours in layers l and l7g, the effective field, Heff l , contains both static and alternating components. The surface is considered as a defect to the structure of the finite body due to the absence of neighbours either above or below the surface layers and as such will experience a different effective field; Heff surf . The surface anisotropy is represented on either surface by this effective field and will take into account the difference in the ‘pinning’ in the direction of the magnetisation eff (c ¼ M=M ) of the surface and internal spins, where Heff surf ¼ Hsurf ðcÞ. For asymmetric pinning, the surface effective fields will be different for the two surfaces. The surface pinning parameter is defined as: A1

gmB ðHeff cÞ. 2SzJ surf

(47)

Care should be taken not to confuse this with the exchange stiffness constant given in Eq. (33). In Eq. (47) z is the surface coordination and J its exchange integral.

137


By considering the wave functions of the spin waves and the relevant boundary conditions, it is possible to determine the relationship between the wave vectors and the shape of the spin-wave modes and gives rise to the following characteristic equations: f ðtÞ ¼

cos½ðL þ 1Þ=2 t ¼ AðjÞ cos½ðL 1Þ=2 t

for symmetric waves

(48a)

for antisymmetric waves:

(48b)

and gðtÞ ¼

sin½ðL þ 1Þ=2 t ¼ AðjÞ sin½ðL 1Þ=2 t

In Eqs. (48a) and (48b), the quantity t is a quantum number which is in general complex, for which we can distinguish three types: (i) t ¼ k for bulk spin-wave modes; (ii) t ¼ it for acoustical surface spin waves and (iii) t ¼ p + it for optical surface spin waves, where k and t are real numbers. The effect of the surface parameter on the standing spin-wave mode profiles is shown in Fig. 2.13 for various values of A and mode numbers (n) for symmetric pinning conditions. It will be seen that surface localised modes naturally arise in this model, when the surface parameter is greater than unity and the degree of localisation increases with A. It will be noted that the wave vectors for the surface spin-wave modes are complex while those for bulk modes are real. For the lowest lying modes (n ¼ 1 and 2), the effect of changing A can be seen in Fig. 2.14, where a critical point is reached where the mode switches from a bulk mode to a surface mode. The natural limit occurs for t ¼ 0 (k ¼ t ¼ 0). In terms of spectral features, the existence of surface spin-wave modes will be manifest as absorption peaks at fields above that of the uniform mode (k ¼ 0) since they have complex and imaginary wave vectors, while their intensities will be smaller and reduce with the degree of localisation. Since the absorption intensity is proportional to the transverse magnetisation, only odd (symmetric) spinwave modes will have any absorption. Thus the intensity will decrease with mode number as well. Antisymmetric modes will not be observed. Asymmetric pinning conditions will provide a suitable imbalance in the antisymmetric modes to allow for some absorption, and the existence of alternating mode intensities should be seen as a sign of asymmetric pinning. For the case where the microwave driving field is in the xy plane, the modal intensities can be expressed as (Maksymowicz, 1986): hR In ¼ R 1 1

1 1

dzðmj =iÞ

i2

, dz m2W þ ðmj =iÞ2

(49)

where the microwave component of the magnetisation is written as m ¼ ^ W þ jm ^ j in the spherical coordinate system. From the variation from the Wm

138

A=-∞

A=0

A=1

A=+∞

A=2 SURFACE MODES

n=1

n=1

n=1

n=2 n=3

n=5

n=6

0 L-1

n=5

n=6

0 L-1

0 L-1

n=4

n=5

n=6

0

n=3 n=4

n=4

n=5

L-1

n=3

n=4

n=5

n=2

n=2

n=3

n=4

n=1

n=2

n=2

n=3

0

n=1

n=5

0 L-1

0 L-1

0 L-1

n=5

0 L-1

0 L-1

L-1

Figure 2.13 Shapes of spin-wave modes (with low n) for various values of the surface parameter A. Symmetric and antisymmetric modes correspond to odd and even n, respectively (Puszkarski, 1979). David Schmool

139


n=1

n=2

A=−∞

A=1083

A=−1

A=1125

A=0

A=2

A=1

A=+00

n=3

n=4

Figure 2.14 Shape of the symmetric spin-wave modes n ¼ 1;3 and antisymmetric modes n ¼ 2;4 versus the surface parameter A. At a well-defined limiting value of A the modes n ¼ 1 and n ¼ 2 change their shape from oscillatory to localised (Puszkarski, 1979).

limits of A ¼ 7N, it is seen that the first symmetric (antisymmetric) mode turns effectively into the next higher mode (see Fig. 2.14). Asymmetric pinning can also be readily incorporated into the SI model for a more general description. An illustration of the spin-wave mode profiles for asymmetric pinning, with pinning parameters a and b, are illustrated in Fig. 2.15 for the first two spin-wave modes. Spin-wave spectra for the case of symmetric and asymmetric pinning conditions are shown in Fig. 2.16 for various values of the pinning parameters. In the former, only odd modes are observed, while in the latter both are present, where it is noted that the even and odd modes have different envelopes. There has been much interest in recent years with regards to the interference of spin waves in nanometric structures. One of the earliest

140

David Schmool

b=1.4

b=1.4

SfM

n=2 a=1.5

a=1.5

n=1 b=1.5

b=1.5

SfM

n=2 a=0.5

a=0.5

n=1

b=1.2

b=1.2

a=1.5

b=1.12

SfM

SfM a=1.5

SfM

b=1.12

a=1.5 a=1.5

b=1.08 b=0.8

a=0.5

b=1.08 b=0.8

k=0

a=0.5

a=0.5

a=0.5

SfM

b=1.0

a=1.5

b=1.0 b=0.5

a=1.5 a=1.5

b=0.5

b=0.5 b=-5

SfM

a=1.5

b=0.5

a=0.5

b=-5

a=0.5

a=0.5

a=0.5

k=0

SfM

Figure 2.15 The shape of the quasi-symmetric mode n ¼ 1 and quasi-antisymmetric mode n ¼ 2 versus the pinning parameter b of one of the surfaces (pinning on the other is kept constant, i.e. a ¼ constant). Two cases are shown corresponding to a ¼ 0.5 and a ¼ 1.5. The parameter a is for the left-hand boundary, whereas b, the variable parameter, is for the right-hand one (we assume L ¼ 11). The case a ¼ 0.5 visualises the emergence of one surface mode (denoted SfM) and the case a ¼ 1.5, that of two surface modes (Puszkarski, 1979).

considerations of the propagation, decay and collision of spin waves was in the study of Tsankov et al. (1994), see also the studies by Kalinikos et al. (2000) and Slavin et al. (2003). Another example is Nitta et al. (1999) who proposed a spin-interference device based on the Aharonov2Bohm ring, and can be seen as analogous to the superconducting quantum interference device (SQUID). Hertel et al. (2004) also use a similar premise in their micromagnetic modelling simulations of spin-wave interference in a ringlike structure; some examples of the interference simulations are shown in Fig. 2.17. Choi et al. (2006) have also performed micromagnetic simulation of the interference of spin waves through pinholes in an analogy of the Young’s double slit experiment, where the interference pattern show much in common with the interference of other waveforms (see Fig. 2.18).

141


UM

SfM A=1 (Natural defect)

PINNING WEAKENS

9 7 5 3

NUM

SfM

PINNING GROWS

A=2.5

A = 0.5

A=2

7 5 3

5 3 NUM

SfM

A = 1.5

A=0

7 5 3

7 53 NUM

UM

A=1 (Natural defect)

A = -1.5 SfM 11 9 7 5 3

(a) Spectra for unpinned surface spins

(a)

intensity

1

2

2

1

(b) Spectra for pinned surface spins

1

1

1

1

1

1

1

1

10−1

2

4

4 3

6

5 8

10−3

8

10 7

3

10

9

8 9

11

10−5

(b)

9

9 11 11 a r=−1, p=1.5 r=−1, p=1 r=−1, p=3 a=0, b=1.5 a=0, b=1 a=0, b=3

11

2 6

11 10 11

11

11

9

4 9

8

10

10

7

6

5 7

9 9

8 10

10−4

7

7

7 7

4

7

7

5

5

5

5 6

3 11

5

5

5

2

4

3

6

10−2

3

3

3

3

3

9 10 9

8

8 6

6

6 4

10

4

4

10

r=−1, p=0.8 r=−1, p=0.5 a=0, b=0.8 a=0, b=0.5

2 r=−1, p=0 a=0, b=0

r=−1, p=−0.5 a=0, b=−0.5

r=−1, p=−1 a=0, b=−1

2 r=−1, p=−2 a=0, b=−2

1 Mode number

Figure 2.16 (a) SWR spectra calculated for various values of the surface parameter A, that is for various pinning of the surface spins. The spectra exhibit only peaks corresponding to symmetric modes (of odd number n ¼ 1, 3,y). The calculations are for the case of L ¼ 11 (11 layers in the film). UM, uniform mode, SfM, surface mode, NUM, non-uniform mode. For the case A ¼ 1.5, the peak with number n ¼ 11 corresponds to an ‘optical’ surface mode; in all other cases SfM denotes an ‘acoustical’ surface mode. Units on the horizontal axis are proportional to the wave number k (for space peaks) or to t (for surface peaks). (b) Calculated SWR spectra corresponding to various points on the straight line r ¼ 1 (a ¼ 0) and visualising the following effects: (i) ‘passing-over’ of the envelopes of even and odd peaks, (ii) absorption increasing with the peak number n. We draw attention to the limiting cases p ¼ 1, p ¼ 0 (even peaks are absent) and p ¼ 1. Empty circles denote space modes and full circles denote surface modes. Intensities (ordinates) and peak numbers (abscissae) are in a logarithmic scale (Puszkarski, 1979).

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David Schmool

time (b)

0 ps 0.005 0.004 0.003 0.002 0.001 0 −0.001 −0.002 −0.003 −0.004 −0.005 (e)

(c)

0.6 0.4 0.2 0 -0.2 -0.4 -0.6

660ps

no wall 360 deg wall

0

0.2

(f)

0 ps

(d)

200 ps

60 ps

mz[%]

(a)

0.4 0.6 time [ns] (g)

60 ps

0.8

150 ps

(h)

660 ps

time

Figure 2.17 Snap shots of the wave propagation and interference effects in a ring at times as indicated. The series (a)--(d) shows the propagating front that is magnetised in an ‘onion’ state, while the series (e)--(h) shows the evolution for a ring that contains a 3601 wall in one branch. The phase difference induced by the domain wall (DW) becomes visible in (h), where the merging wave fronts have opposite sign. This leads to destructive interference. Correspondingly, the oscillations of the magnetisation on the other end of the ring are strongly suppressed (inset). The colour-coding refers to the out-of-plane component as indicated by the colour scale (maximum 0.5%). (Reprinted figure with permission from Hertel et al. (2004), r 2004 by the American Physical Society.)

2.6. Resonance linewidth and relaxation processes Relaxation behaviour in magnetic nanostructures is an area of key interest and can have important implications in the specific response of magneticbased devices. The subject of spin relaxation is a highly topical one and should be an essential component in the understanding of the high-frequency behaviour of magnetic and spintronic devices. A recent review of the area by Heinrich (2005) covers much of the essential ingredients of this subject; see also Mills and Rezende (2003). As stated above, there are many components to the damping observed in spin dynamic measurements and importantly, the damping is a sample-dependent parameter. The intrinsic, material, processes (at finite temperature) depend on the scattering of spin waves or magnetic excitations with electrons and phonons. Other processes, which depend on defects and magnetic confinement of the sample, are extrinsic. In magnetically confined systems, the dynamic response is frequently affected


143

Figure 2.18 (a) Snapshot image taken at t ¼ 0.67 nanoseconds for the dynamic evolution of the local Mz/Ms distribution, illustrating the interference pattern of SWs diffracted through the two openings. (b) and (c) show their FFT power and phase for f ¼ 36.5 GHz, respectively. (Reprinted with permission from Choi et al. (2006), r 2006 American Institute of Physics.)

by mode coupling. While there has been much progress in recent years, there is much that was developed in the early years of FMR and SWR theory. Damping of magnetisation movement has important implications for the behaviour and functioning of magnetic devices and is a fundamental physical parameter in magnetic switching. The critical motion of magnetic nanoelements could be a key factor in magnetic storage technology in the future. In Eq. (3), we gave a form of the equation of motion for the magnetisation in which the dissipation is given in the Gilbert form

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David Schmool

(Landau2Lifshitz2Gilbert (LLG)) (Gilbert, 2004):

1 @M a @M M ¼ ðM Heff Þ þ . g @t gM s @t

(50)

However, Landau and Lifshitz (1935) introduced the phenomenological equation of motion in which the relaxation is described in the following form (LL): 1 @M l ¼ ðM Heff Þ ½M ðM Heff Þ , g @t gM 2s

(51)

where for small damping we can write the relation a ¼ l=M s 1. The damping parameter a is often written as G=gM s and G is called the Gilbert damping parameter. In general, we can say that the relaxation time of the magnetic system, t, is inversely proportional to the Gilbert and Landau2Lifshitz damping parameters. For a discussion on the differences between these two equations see Iida (1963). There are other forms of the dynamical equations of motion, such as Bloch2Bloembergen (Bloch, 1946, 1957; Bloembergen, 1950) and Callen (Callen, 1958). The Bloch2 Bloembergen dynamical equations can be written in the form: 1 @M x M x ðtÞ ¼ ðM Heff Þx , g @t T2

(52a)

M y ðtÞ 1 @M y ¼ ðM Heff Þy , T2 g @t

(52b)

1 @M z M z ðtÞ M 0 ¼ ðM Heff Þz . T1 g @t

(52c)

Here T1 is known as the longitudinal (or spin-lattice) relaxation time and T2 the transverse (or spin2spin) relaxation time. This is another phenomenological form of the dynamical equation of the motion of the magnetisation and is typically used in the treatment of nuclear magnetic resonance. In this formalism, the temporal dependence of the magnetisation (relaxation from initial orientation at time t ¼ 0) can be described by the following relations: M x;y ðtÞ ¼ M x;y ð0Þet=T 2

(53a)

M z ðtÞ ¼ M z;eq M z;eq M z ð0Þ et=T 1 .

(53b)

and

An overwhelming majority of authors use the LL and LLG forms and we shall mainly consider these here. As stated above, for small damping (a{1), the LL and LLG equations are identical, however, for larger


145

damping the LLG gives a physically more realistic description of the damping and is generally favoured over the LL form, see Heinrich (2005). General relaxation processes are also discussed in depth by Suhl (2007). Solving the Gilbert equation of motion it is possible to show that the relation between the peak-to-peak linewidth and the damping parameter is given by (Heinrich, 2005): DH G ¼ 1:16

G o o ¼ 1:16a . gM s g g

(54)

The factor 1.16 arises from the inflection points of a Lorentzian line. The characteristic of Gilbert damping is its linear dependence on microwave frequency and 1/Ms. Equation (54) shows that the linewidth will grow proportionately with the damping coefficient a. Gilbert damping exhibits no dependence with the applied field. With increasing damping coefficient the damping torque will dominate and @M=@t ! 0, as such the system will show a sluggish response and slowly approach the equilibrium state. Some of the explicit causes of the intrinsic linewidth in metallic systems can arise from eddy currents, phonon drag and spin2orbit relaxation. We shall not discuss these mechanisms here, for the interested reader a good starting point would be Heinrich (2005). The resonance linewidth in magnetic structures frequently manifest elevated levels of extrinsic broadening. The broadening mechanisms can be summed using the following expression (Schmool and Schmalzl, 2007; Vittoria et al., 1967): @H @H @H @H DH i þ Df þ DV þ DS. DH ¼ DH 0 þ @f @H i @V @S

(55)

The first term represents the intrinsic linewidth, which can be written as (Vonsovskii, 1966): a DH 0 ¼ M

2 @E 1 @2 E . þ @W2 sin2 W @f2

(56)

In this way, the angular dependence of the resonance linewidth can be explained as arising from the angular-dependent contributions to the freeenergy density. The second term in Eq. (55) arises from a spread in crystalline axes, f (which differs from the azimuthal orientation of the magnetisation, j). The third term is due to magnetic inhomogeneities in the sample, the fourth term results from the volume distribution of the magnetic particles (a monodisperse system will clearly not contribute since DV ¼ 0) and the last term will be due to the differences in resonance condition due to surface spins. Bulk systems will have negligible contributions because the number of surface spins will not be appreciable for large

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magnetic bodies. Equation (56) shows that there will be an explicit dependence of the linewidth on the direction of the applied field, where linewidth broadening will be expected in anisotropic systems. It will be noted that assemblies of nanoparticles with broad log-normal distributions can be expected to have very broad resonances; this is indeed the case, where measured linewidth can be as large as several kOe (see Sections 5 and 6.1). Defect scattering of magnons can occur in magnetically inhomogeneous materials, where the uniform mode (qB0) is scattered into non-uniform modes (qa0 magnons). This process is referred to as two-magnon scattering. A detailed discussion of the two-magnon scattering mechanism is given by Mills and Rezende (2003) and Heinrich (2005). An additional damping mechanism can be observed in magnetic multilayer samples due to the spin-pumping effect which occurs via the spin angular momentum transfer between magnetic layers (Urban et al., 2001). The additional linewidth due to this mechanism takes the form: DH add Dm þ _w,

(57)

where Dm ¼ Dm" Dm# is the difference in the shifts of the Fermi levels of the spin-up and spin-down electrons. A fuller discussion of the spin angular momentum transfer effect will be given in Section 4. Another approach to the relaxation phenomenon is that given by Skomski et al. (2005) who consider the relaxation process from the Liouville2von Neumann equation. For sufficiently slow magnetisation processes a separation of timescales leads to a LL precession term and a Langevin force which is responsible for the thermal activation over an energy barrier. Considering a magnetic nanostructure, the precession is a deterministic zero-temperature property and thermal forces are dependent on the temperature of the heat bath. Damping reflects the interaction between the magnetic and heat bath degrees of freedom and is described by Fermi’s golden rule: W ij ¼

2p jhCi jV jCj ij2 dðEi Ej Þ. _

(58)

Here Wij is the transition rate between two quantum states i and j, and hCi jV jCj i is the matrix element between the two states. These lead to the time-dependent decay of the original modes. The energy of the system is conserved and includes the energy redistribution between the different subsystems. The transition rates will determine the dynamics of the system, where there are three different type of equation; rate equations, Fokker2Planck or generalised diffusion equations and Langevin or random-force equations, which are essentially equivalent (Risken, 1989). The low-temperature solution of the Fokker2Planck equation, for example gives rise to a solution of the Arrhenius or Neél2Brown law

147


(Skomski et al., 1999): t ¼ t0 exp

EB , kB T

(59)

where t is the relaxation time of the system for thermal activation over the energy barrier, EB. EB depends on the anisotropy constant of the materials and the volume of the particle, which is typically given as KV, for the uniaxial case. t0 is the relaxation time in the absence of energy barriers. As for the damping parameters of Gilbert and Landau2Lifshitz, 1/t0 is essentially proportional to Wij. In the Fokker2Planck magnetic diffusion equation, the parameter t0 is explained in terms of spin diffusion. Skomski et al. (2005) then consider a magnetic nanoparticle as an ensemble of N spins which are exchange coupled and the particles’ magnetisation will be of the form of a macrospin with magnitude SN. The spin dynamics of the system will have two components: damped precession towards a well-defined equilibrium state and random thermal motion of the magnetisation vector. Figure 2.19 illustrates the cases for (a) damped precessional motion and (b) random thermal motion. It will be noted that the two are opposing processes, where (a) is essentially a low-temperature process and (b) concerns the case of weak effective fields (low anisotropies, spherical particles, no applied field). In the diffusive regime, we can choose the case where EB ! 0 and t ! t0 . For a general discussion of this method, see also Skomski (2008). From the experimental point of view, resonance line width broadening can be a sensitive probe of the magnetic state of the sample and this is particularly true of confined magnetic systems. Low-dimensional systems (a)

(b)

90° 120°

60°

150°

30°

180°

0°

210°

330° 240°

300° 270°

90° 120°

60°

150°

30°

180°

0°

210°

330° 240°

300° 270°

Figure 2.19 Magnetisation dynamics of a nanoparticle: (a) damped precession and (b) random thermal motion. The curves are simulations for typical but not critical parameters, covering a time of order 0.1 nanoseconds. In both polar plots, the direction of the motion is from the white circles to the black circles. (Reprinted with permission from Skomski et al. (2005), r 2005 American Institute of Physics.)

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can, however, suffer from difficulties in extensive extrinsic broadening and can be very difficult to interpret. Low-dimensional systems will very often display multipeaked spectra and overlapping resonance peaks can be difficult to disentangle even with sophisticated software packages.

3. Review of Simulations of Magnetodynamics in Nanometric Systems There are many theoretical works which report on the numerical simulations of FMR and spin dynamics in low-dimensional structures. Much of the theoretical modelling is based on the LL formalism, and essentially uses equations of the form of Eq. (3) with some form of damping. As we have discussed previously, one of the major concerns when dealing with assemblies of particles is to take into account the effects of interparticle interactions and the distribution of particle axes and sizes. A further problem concerns the effects of surface anisotropy. We have indicted some of the main components to the theoretical problem in previous sections and now will outline some of the important results from various theoretical studies and simulations. A majority of the simulations found in the literature deal with the situation of magnetic nanostructures in one of two ways; an isolated magnetic nanostructure treated as an ensemble of spins or an ensemble of nanoparticles treated as macrospins which are arranged in either regular or random arrays.

3.1. Stoner2Wohlfarth model One of the simplest and earliest models of the behaviour of single-domain particles (sdp) is that of Stoner and Wohlfarth (1948). This model has been the basis of many subsequent models of fine particle systems with sdp and is a good starting point for discussion. This model considers the case of an sdp with uniaxial anisotropy, K, in the presence of an external magnetic field, H. In the case where the magnetic field is applied at an angle j, to the easy axis, the magnetisation, M, will align at some equilibrium orientation between the directions of the applied field and easy axis. This equilibrium will depend on the relative strengths of the applied field and the magnetocrystalline anisotropy as well as the angle between them. This is essentially a two-dimensional (2D) problem and we can write the free-energy density of the particle as the sum of the anisotropy and Zeeman energies: E SW ¼ HM s cosðj yÞ ¼ Ksin2 y.

(60)

The equilibrium condition will be given by: dE SW ¼ 0. dy

(61)

149


From which we obtain: HM s sinðj yÞ ¼ K sinð2yÞ.

(62)

The solution of the equilibrium state will then be considered for various orientations of the applied field with respect to the easy axis. For example, when the applied field is applied perpendicular to the easy axis (j ¼ 901), we obtain sin y ¼

H HK

for HoH K ,

(63)

where HK ¼ 2K/Ms is the anisotropy field. In this case, the magnetisation in the direction of the applied field will be M s sin y and will be a linear function of H, where reversible (coherent) rotation of the magnetisation occurs. Saturation will occur at H ¼ Hs ¼ HK, and sin y ¼ 1 for HWHK where the magnetisation is saturated along the field direction. When the field is applied parallel to the direction of the easy-axis irreversible switching occurs when the field reaches the anisotropy field value and coercivity results, giving a square hysteresis loop. For the applied field applied at intermediate angles the critical or switching field will vary from zero (for (j ¼ 901) to HK (for (j ¼ 01)). This variation is illustrated in Fig. 2.20. 1.5

1

mII

0.5

0

90° 60°

−0.5

30° −1 0° −1.5 −1.5

−1

−0.5

0 h

0.5

1

1.5

Figure 2.20 Hysteresis loops for the Stoner--Wohlfarth (SW) model at various angles of the applied field with respect to the anisotropy easy axis, see text (Tannous and Gieraltowski, 2008).

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David Schmool

Modifications to the Stoner2Wohlfarth (SW) model will be required for the case of mixed shape and magneto-crystalline anisotropies with noncollinear easy axes. The SW model is essentially a macrospin model in which all spins within the particle rotate in phase. More sophisticated models are necessary to account for fanning and curling of the spin configuration and gives rise to non-coherent rotation, with the result that the switching fields are reduced with respect to those predicted by the SW model. One of the essential physical properties of this model is the role played by the applied magnetic field, which can effectively stabilise the magnetisation in a certain direction. In the absence of the applied field, the system has two energy minima separated by an energy barrier, which for a particle of volume V, has a height of KV. The system has two equivalent states where the magnetisation can relax, as defined by the uniaxial-anisotropy axis (see Fig. 2.21). Superparamagnetic (SPM) effects become important when the thermal energy becomes comparable to the height of the barrier and spontaneous fluctuation can occur between these minima, see Section 5 below. Applying a magnetic field along say the j ¼ 01 direction will help to stabilise the system in this direction, thus increasing the barrier height by an amount dependent on the applied magnetic field. The modification of the energy landscape is illustrated in Fig. 2.22. Shown are the modifications of the free-energy profile with applied magnetic field. The initial uniaxial symmetry is modified by the magnetic field, which for large fields will produce a stabilisation of the magnetisation in one direction (one energy minima). There we see that the energy minima are altered such that there is a difference of dE ¼ 2HMs

Energy

Hext = 0

Hext > 0

Hext > Hswitching

Magnetization angle

Figure 2.21 Uniaxial anisotropy in the SW model for zero applied field (Schrefl et al., 2006).

151


Energy vs angle θ for variable field h = H/Hk and fixed φ = 30° 2.5

h=0.1 0.2 0.3 0.4 0.5 1.0 2.0

E(θ) = sin2(θ) −h cos(θ − φ)

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 π/2

π

3π/2 θ

2π

5π/2

Figure 2.22 Variation of the energy landscape with angle h for different values of the normalised magnetic field h ¼ H/HK. The field orientation is held fixed at the value u ¼ 301 with the anisotropy axis (Tannous and Gieraltowski, 2008).

and the barrier height will now be given by (Aharoni, 2000): "

HM s B ¼ KV 1 þ 2K

2 # .

(64)

From this model, the number (probability) of particles passing from one minimum to the other can be evaluated as: 2

nab ¼ c ab eKV ð1H=H K Þ =kB T ,

(65)

where cab is a constant and ‘7’ refers to the direction of the jump, reflecting the difference in barrier height produced by the bias caused by the applied magnetic field. The boundaries of the hysteresis loop with the applied magnetic field can be determined from the nullification of the first and second derivatives of the free energy and give rise to the well-known Stoner2Wohlfarth asteroid, as illustrated in Fig. 2.23. The asteroid being described by: ðH ? =H K Þ2=3 þ ðH jj =H K Þ2=3 ¼ 1. Variations of this model will apply for cubic and other symmetries. For further details on the magnetic relaxation in magnetic particles, see Aharoni (2000), Suhl (2007) and references therein. The Stoner2Wohlfarth model has important implications in magnetisation switching in magnetic nanoparticles and nanostructures (Thirion et al., 2003).

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Stoner-Wohlfarth Asteroid 1 0.8 0.6

Field HII /Hk

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

−0.8 −0.6 −0.4 −0.2 0 0.2 Field H⊥/Hk

0.4

0.6

0.8

1

Figure 2.23 Inside the astroid domain is made of the field values for which a reversal of the magnetisation is possible. Outside the astroid domain, no reversal is possible (Tannous and Gieraltowski, 2008).

3.2. Nanoparticle assemblies Nanoparticle assemblies generally consist of smaller magnetic entities of single domains and are subject to the SPM effects. In this section, we consider some of the principal theoretical modelling of such particles. Thomas et al. (1995) use the SW model to consider the FMR response of interacting fine particle systems. The energy for each particle in the assembly is minimised in turn using an iterative process. This is necessary since the inclusion of interactions between particles means that the equilibrium state of one particle can alter that of a neighbour. Simulations were carried out for a system of 100 particles randomly placed on a cubic cell using periodic boundary conditions with a truncated interaction volume to reduce computational times. The energy for each particle is calculated based on Eq. (60) where the field is replaced by a local field which includes the dipolar field from those particles within the truncated volume. The strength of the local field will, therefore, be dependent on the size and direction of an applied field as well as the sum of the dipolar fields of the particles within the truncated volume and as such will be explicitly dependent of the packing fraction of the particles. The evaluation of the

153


resonance was performed by calculating the second derivatives of the free energy, as indicated in Eq. (4) of Section 2.1, in steps of 100 Oe from 10 kOe to zero, where in the region of resonance resolution was increased to 10 Oe. Anisotropy in the particles arise from particle shape anisotropy for Fe3O4 particles with a length to width ratio of 1:1.2 giving a uniaxial anisotropy of 5.45 104 erg, a damping parameter of 2 1010 seconds and a frequency of 10 GHz was also used. The authors consider in-plane and perpendicular spectra as well as spherical samples, in which the demagnetising factor will change. Shifts in the resonance field are predicted as a function of the sample packing fraction, as shown in Fig. 2.24, which will be due to the variations in the interaction field. Divergences between these calculations and the Netzelmann model occur for increasing packing fractions, which is suggested, arises from fluctuations in the local interaction fields. FMR in granular solids has been evaluated by micromagnetic simulations, where both dipolar interactions and exchange has been included for a log-normal size distributed assembly of N particles with randomly oriented easy axes, by Verdes et al. (2001, 2002). The total magnetic field acting on a magnetic particle is considered to be the sum of applied, dipolar and exchange fields; HT ¼ Ha þ

X

(66)

Hij

j¼1;N jai

where Hij ¼

3ðmj rij Þrij mj 3 þ C ij mj . r 5ij r ij

(67)

5000

Resonance Position (Oe)

Perpendicular results 4000 Spherical sample results 3000 In-plane results

2000

1000

0 0.00

0.05

0.10

0.15

0.20 0.25 0.30 Packing Frection

0.35

0.40

0.45

0.50

Figure 2.24 Packing fraction resonance position relationship for the various sample geometries (Thomas et al., 1995).

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David Schmool

C ij is an exchange coupling constant between particles i and j which has a fixed C for short distances and is zero otherwise. The free energy of the system is described by the SW model, which will define the equilibrium orientations, though a thermally activated transition between minima is permitted with a probability given by the expression: p ¼ 1 etm =t , in which tm is the measuring time and t the relaxation time. (These parameters are further discussed in Section 5.) The resonance field and linewidth are evaluated via Eqs. (4), (55) and (56), respectively. Simulations were made on ensembles of over 1000 particles of 60 nm, using a frequency of 16.3 GHz and saturation magnetisation of 1400 emu/cm3. The packing fractions were varied from 0 to 40% with the anisotropy also varied in the range 0.522.0 106 erg/cm3. Interaction strength was also varied between C ¼ 0.0 and 0.3. Results are illustrated in Fig. 2.25. These show that the dipolar interactions have only a weak influence on the resonance field,

K=2e6 erg/cc, C*=0.0 K=2e6 erg/cc, C*=0.3 K=1e6 erg/cc, C*=0.0 K=1e6 erg/cc, C*=0.3 K=5e5 erg/cc, C*=0.0 K=5e5 erg/cc, C*=0.3 6000 5500

Resonance field (Oe)

5000 4500 4000 3500 3000 2500 2000 1500 0.0

0.1

0.2

0.3

0.4

Packing density

Figure 2.25 Variation of resonance field with anisotropy and packing fraction. (Reprinted with permission from Verdes et al. (2001), r 2001 American Institute of Physics.)

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while the exchange coupling is more important as is the anisotropy. Dipolar effects are more important in the linewidth. Another approach is used by Dumitru et al. (2002) and Kodama and Berkowitz (1999), where the FMR of an interacting fine particle system is considered. Here the dynamic susceptibility is used to evaluate the resonance spectra as a function of frequency, which is expressed as: E Wj g2 ð1 þ a2 Þ 2 2 E jj , ioab þ 2lm w¼ 2 l ðE WW ioabÞ þ m sin W ores ioDo o2 sin2 W (68)

where Enm are the second derivatives of the free energy with respect to angles nm( ¼ j or W) and are evaluated at the equilibrium conditions; Eq. (5) and b ¼ M =½gð1 þ a2 Þ . Weighting factors are defined by (Netzelmann, 1990): l ¼ sin d sinðj lÞ and m ¼ cos W sin d cosðj lÞ cos d sin W;

(69)

where d and l define the orientation of the microwave field. ores is given by Eq. (4) and Do ¼ gDH given by Eq. (54). The numerical calculation uses the Preisach model for SW particles with aligned easy axes, where each particle is defined by switching fields Ha and Hb, with distribution functions given as: PðH a ; H b Þ ¼

1 ðH a þ H b Þ2 ðH a H b 2H 0 Þ2 exp . exp 4hsi 4hsc 2phsi hsc

(70)

hsi and hsc are standard deviations of the interaction and coercive fields, respectively, and H0 is the field of the maximum in the Preisach distribution. At equilibrium, the SW particles can be aligned in one of two positions ( + or , i.e. W ¼ 0 or p). These positions will give rise to two resonance conditions and susceptibilities; oresþ ; ores and wþ ; w . The susceptibility of the particle system is then evaluated as: ZZ

ZZ

PðH a ; H b Þwþ dH a dH b þ

w¼ Sþ

PðH a ; H b Þw dH a dH b .

(71)

S

By varying the frequency the numerical dependence of the susceptibility is calculated, thus giving the associated spectra of the system. Using a saturation magnetisation of 1400 emu/cm3 and a uniaxial anisotropy of 2.3 106 erg/cm3, the isothermal remnant magnetisation is evaluated from the Preisach model, giving H0 ¼ 3280 Oe; hsi ¼ 330 Oe and hsc ¼ 1150 Oe. The FMR spectrum is illustrated in Fig. 2.26. The two peaks correspond to the ‘ + ’ (higher resonance frequency) and ‘’ states of the particles. Increased static fields will eventually align the particle and a single

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David Schmool

5 H=1000 Oe 4

χ″

3

2

1

0 0

2

4

6 8 Frequency f(GHz)

10

12

14

Figure 2.26 Calculated in-plane FMR spectra for particle systems. (Reprinted figure with permission from Dumitru et al. (2002), r 2002 by the American Physical Society.)

resonance is observed. The effects of interactions are illustrated on the resonance field and linewidth are shown in Fig. 2.27. The maximum in the linewidth occurs when the static field is near the distribution field maximum and is explained by the dispersion of the interaction field. The problem of temperature effects in FMR of nanoparticle systems is twofold: there will be a thermal fluctuation of the magnetic moment of atoms in the particles, which results in a reduced magnetisation and there will be thermal fluctuations of the net magnetisation around its easy axis. The former will depend on the ratio T/J, J being the exchange interaction between atoms and T the temperature. The latter will be size dependent due to the relation between particle size and moment. Usadel (2006) considers the fluctuations of the net magnetisation of a nanoparticle in the ‘macrospin’ approximation which is coupled to a heat bath. The particle dynamics in this case are governed by the stochastic LLG equation which takes into account the temperature fluctuations. The Langevin dynamics in small magnetic particles was also studied by Garcıá-Palacios and La´zaro (1998) which is based on the Brown2Kubo2Hashitsume model for stochastic dynamics. Usadel and Garcıá-Palacios and La´zaro use rather similar approaches and we can express the LLG in the following manner: @S g S fBe ðtÞ þ a½S Be ðtÞ g, ¼ @t ð1 þ a2 Þ

(72)

157


Resonance frequency f(GHz)

20

hsi=330 Oe

18

hsi=0 Oe

16 14 12 10 8 6 4 2 0

1000

(a)

2000 3000 4000 Static field H(Oe)

5000

6000

Our Results hsi=0

FMR linewidth (GHz)

2.4

Verdes Results

2.2 2.0 1.8 1.6 1.4 1.2 1000

(b)

2000 3000 4000 Static field H(Oe)

5000

Figure 2.27 (a) Variation of resonance frequency as a function of the static field. (b) Variation of FMR linewidth as a function of the static field. (Reprinted figure with permission from Dumitru et al. (2002), r 2002 by the American Physical Society.)

where S ¼ ls/ms denotes the unit vector parallel to the magnetic moment ms and the effective field is given by: Be ðtÞ ¼

^ 1 @H þ zðtÞ. ms @S

(73)

Here we define the Hamiltonian of the nanoparticle as: ^ ¼ ms S B ms S hðtÞ DS 2z , H

(74)

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David Schmool

where B is the external magnetic field, h(t) the high-frequency field and D the anisotropy energy. In Eq. (73), z(t) denotes the thermal noise (stochastic field), which is Gaussian distributed with the following statistical properties: hzi ðtÞi ¼ 0;

hzi ð0Þzj ðtÞi ¼

dij dðtÞ2akB T , ms g

(75)

where i, j are Cartesian components and other symbols have their usual meanings. Sukhov et al. (2008) defines the power absorbed in the FMR experiment as: 1 P¼ t0

Z

SðtÞ

@hðtÞ dt. @t

(76)

Here t0 denotes the measuring time and the integral runs over this time interval in the stationary state. To obtain the FMR fixed frequency experimental conditions, the fixed frequency is written in the form of the resonance field at D ¼ 0, T ¼ 0 with BL ¼ o/g. Using reduced dimensions for the temperature as q ¼ 2akB T =ms BL simulations for FMR absorption were made as a function of reduced field for different temperatures with a Gilbert damping parameter of a ¼ 0.2 and a reduced anisotropy energy of d ¼ D=ms BL ¼ 0:2. Results are shown in Fig. 2.28(a). At zero temperature (q ¼ 0), an exact solution for the resonance is found for bmax;T ¼0 ¼ 1 2d. Increasing the temperature, a shift in resonance field is observed with a reduction of the maximum absorbed power. It will also be noted that with increasing temperature an increase in fluctuations is evident. In Fig. 2.28(b), the variation of the resonance field as a function of temperature is given for two values of the reduced anisotropy, where bmax tends to 1 with increasing temperature, which is interpreted as being due to the temperature dependence of the magnetic anisotropy (Usadel, 2006). The temperature shift 1bmax fits well the data and varies as the square of the equilibrium magnetisation, m0; 1 bmax ¼ 2dm20 , which is shown in solid lines in Fig. 2.28(b). This result implies that for FM nanoparticles, it is the direction of the magnetisation which thermally fluctuates and not spin fluctuations within the particles themselves. The variation of the resonance linewidth, which at T ¼ 0 is taken as 2a, increases with temperature as shown in Fig. 2.28(c) for three values of the anisotropy energy. Using a similar approach, Garcıá-Palacios and La´zaro (1998) have evaluated trajectories of the magnetic moment of the particles, where they observe a switching process in the absence of an applied field, as illustrated in Fig. 2.29. The magnetic anisotropy potential has been defined as DUðmz =mÞ2 , where DU ¼ KV. For the calculations, a time parameter is defined as 1=tK ¼ lgH K , where l is the damping parameter l ¼ agS=V ¼ 0:1 and HK is the anisotropy field. In addition, the authors also calculate the linear dynamic susceptibility as a function of temperature for various frequencies and orientations of an external probing field

159


q = 0.00 q = 0.06 q = 0.12 q = 0.18

PFMR

1

0.5

0 0

0.5

1 b

(a)

1.5

2

1

bmax

0.9 0.8 0.7 d = 0.1 d = 0.2

0.6 0

0.1

(b)

0.2

0.3

0.2

0.3

q 0.5

d = 0.0 d = 0.1 d = 0.2

w

0.4

0.3

0.2 (c)

0

0.1 q

Figure 2.28 (a) Absorbed FMR power (arbitrary units) versus reduced magnetic field b for different values of reduced temperature q. (b) Position of the maximum of the resonance curve versus reduced temperature for two values of the reduced anisotropy. (c) Width of the resonance curves versus temperature. (Reprinted figure with permission from Usadel (2006), r 2006 by the American Physical Society.)

DBðtÞ ¼ DB0 cosðotÞ. In Fig. 2.29 calculations for an assembly of magnetic moments with parallel anisotropy; DBðtÞ==n. ^ More recently, Sukhov et al. (2008) have extended simulations in FePt nanoparticle arrays by studying the effects due to the random distribution of anisotropy axes. The theoretical framework is the same as that based on

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David Schmool

anisotropy axis 1.2 mz/m

kBT/ΔU=0.2

0.8

0.4

1

4

0 3 5

−0.4

2

6 −0.8 displayed time ∼ 20τK

−1.2 −1.2

−0.8

−0.4

mx/m 0

0.4

0.8

1.2

1.2 my/m

4

0.8

5

0.4

0 6 −0.4

−0.8 displayed time ∼ 3τK

−1.2 −1.2

−0.8

−0.4

mx/m 0

0.4

0.8

1.2

Figure 2.29 Two-dimensional (2D) projections of the time evolution of the magnetic moment, as determined by numerical integration of the stochastic Landau--Lifshitz--Gilbert (LLG) equation. The magnetic-anisotropy potential is DU(mz/m)2, no magnetic field has been applied, and the damping coefficient in the dynamical equation is k ¼ 0.1. Upper panel: Projection of the trajectory onto a plane containing the anisotropy axis. Lower panel: Projection onto a plane perpendicular to the anisotropy axis of the first stages of the damped precession down to the m ¼ mz potential minimum, after the last potential--barrier crossing. The small dashes demarcate the unit circle. (Reprinted figure with permission from GarcıáPalacios and La´zaro (1998), r 1998 by the American Physical Society.)

161


the model of Usadel (2006) with some redefinition of the normalisation. In the first instance, the effect of the direction of the uniaxial axis of the NP array is assessed by aligning all anisotropy axes along different directions, where a shift of the resonance field will occur due to the compensation with the applied field direction, as shown in Fig. 2.30(a). In all cases, the quasi-static field is applied along the z-axis and the high-frequency field lies parallel to the x-axis. The simulations with random anisotropy axes 0.0015 no anisotropy anisotropy || z

anisotropy || x anisotropy || y

anisotropy || xz

anisotropy || xy

anisotropy || yz

PFMR

0.001

0.0005

0

0

0.5

(a)

1

1.5

d =0.06 d =0.12 d =0.18 d =0.24 d =0.30 d =0.36 d =0.42 d =0.48

0.0006

PFMR

2

b

0.0004

0.0002

0 (b)

0

0.5

1

1.5

2

b

Figure 2.30 (a) Absorbed FMR power (arbitrary units) versus reduced magnetic static field for different directions of anisotropy axes. (b) Absorbed FMR power (arbitrary units) versus reduced magnetic field at zero temperature for different values of reduced anisotropy d (Sukhov et al., 2008).

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are considered for an array of 2000 particles with no size distribution or interparticle interactions. There is a Gaussian distribution of the anisotropy energy with a 5% variance. In Fig. 2.30(b), a set of simulations at T ¼ 0 K is shown for the anisotropy energy in the range d ¼ 0.0620.48, where a ¼ 0.1. After an initial broadening of the resonance line with increasing d, two separate resonances develop. Comparison with experimental data of Antoniak et al. (2005) is made where the authors note that the double resonances are not observed. Although the specific reason is not clear, it seems probable that the effects of the distribution of the anisotropy energy and particle volume, as well as the dipolar interactions between the particles, might be important. The effect of damping is also studied, with only minor changes in the resonance field with increasing a, while clearly broadening occurs. The dependence of the FMR spectra were also studied as a function of temperature, where more important changes are evident, as shown in Fig. 2.31(a). The simulated data is quite noisy, see inset, and averaging and fitting were performed using three Lorentzian lines. While for lower temperatures three separate lines are observed, at 100 K only one peak is evident with a position close to the zero anisotropy line. A gradual reduction of absorption intensity is also observed with increasing temperature. For higher frequency simulations, the lines become weaker and the broadening is reduced. The effect of temperature on the mean resonance field is shown in Fig. 2.31(b) for 9.2 and 24 GHz. The results show a qualitative agreement with experiment. Recent simulations based on a new model by Sousa et al. (2009) aim to clarify the role of surface anisotropy in FM nanoparticle systems, where the theory has been developed to study the resonance conditions in a multispin system. In this model, a single nanoparticle is constructed of N2spins which are exchange coupled, the resonance condition for each spin (i) is then considered in its own specific effective field, H ieff . As such, the reduced free energy takes the form: i ¼

zi X î H ¼ hðeh si Þ þ ki Aðsi Þ þ si lij sj . J j¼1

(77)

In this equation, the first term represents the Zeeman energy, the second term is the reduced anisotropy energy, which can be modified to include any relevant symmetries including spins at the particle surface and the final term is the reduced exchange energy which accounts for the local coordination, zi. Using a similar approach to that when considering the classical FMR theory, we obtain the N-spin resonance equation, which is analogous to the SB equation, of the form: N X X b¼W;j k¼1

f½Hik Xik ðZÞ IZ ga;b dsk;b ð0Þ ¼ 0,

(78)


163

Figure 2.31 Fits of absorbed FMR power versus reduced static field of an ensemble of 2000 nanoparticles for different temperatures: (a) n1 ¼ 9 GHz; (b) n2 ¼ 24 GHz. The insets show the original data at T ¼ 5.0 K. (c) Position of the maximum of the resonance curve versus temperature for two frequencies (Sukhov et al., 2008).

164

David Schmool

where the following substitutions are used: 0

@2Wk Wi i

B Hik ði Þ B @ 1 2 @ i sin Wk jk Wi

1 1 2 @W k ji i C sin Wi C, A 1 @2jk ji i sin Wk sin Wi

Z 1 IZ ¼

1

(79)

!

Z

(80)

and Xik ðZÞ ¼ io

1 dik IZ . 1 þ Z2

(81)

Solutions for the resonance frequency are then obtained from the pure imaginary values of the matrix ½Hik IZ ab . It is note worthy that the form of the kernel has a particular symmetry; the leading diagonal block terms (of 2 2) are of the form of the SB equation for each spin, then offdiagonal (2 2) blocks will be interaction terms between the various spins, where the further off-diagonal the further spins are physically. Preliminary simulations illustrate the variation of the resonance (frequency) spectra with possible indications of the influence of surface spins, which are illustrated in Fig. 2.32 for cubic- and spherical-shaped particles. The spectral analysis is made as a function of the particle size (number of spins), this should show the influence of the surface spins which will be most important for the smaller particles. The direct computed spectrum is comprised of the resonance frequencies of the individual spins, which will be different for each spin environment (corner, edge, face, etc.). The corresponding correlation function is then evaluated to obtain the statistically most probable frequencies. The size dependence of the resonance frequency is illustrated in Figs. 2.32(c) and 2.32(d) for the cubic and spherical particles, respectively. As the particles increase in size the resonance frequency tends to saturate. It will be noted that the main resonance frequency saturates rapidly in the case of the cubic particles, which have a much larger surface area to volume ratio. Also, we note a small contribution (shoulder) which appears to correspond to the resonance of the smallest particle, which could be related to the surface spins resonance (Sousa et al., 2007).

3.3. Nanostructured arrays Micromagnetic calculations have been performed to try to illustrate the complex nature of spin-wave excitations in nanometric structures. Various authors have addressed this problem. These are typically based on the Landau2Lifshitz formulation with Gilbert damping. In the study by Rivkin


165

Figure 2.32 Simulated frequency spectra for (a) cubic and (b) spherical shaped Fe nanoparticles. The evolution of the dominant resonance peak for these are given in (c) and (d), respectively (Sousa et al., 2009).

166

Figure 2.32

David Schmool

(Continued)

167


and Ketterson (2006) dipole and exchange contributions are included in the evaluation of the spin-wave frequencies and modes. These authors considered disks, rings (vortex core removed) and square slab shaped structures. In Fig. 2.33(a), the calculated absorption spectra are illustrated 2.4

disk ring square slab

2.2 2.0 absorption, a.u.

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

5

10 ω/2π

(a)

15

20

+1.0

(i)

(ii)

(iv)

(v)

(vi)

(vii)

(iii)

0.0

-1.0

(viii)

(ix)

(x)

(b)

Figure 2.33 (a) Absorption spectrum for: a disk (D ¼ 175 nm, L ¼ 25 nm), ring D ¼ 175 nm, inner diameter 35 nm, L ¼ 25 nm) and a square (150 150 25 nm); the RF field is uniform and in-plane (along a principal axis for the square). (b) Z projection of the most strongly excited modes for: a disk, (I) f ¼ 1.05 GHz, (II) f ¼ 10.11 GHz, (III) f ¼ 12.34 GHz; a ring, (IV) and (V) f ¼ 10.5 GHz; a square slab, (VI) f ¼ 1.04 GHz, (VII) f ¼ 6.27 GHz, (VIII) f ¼ 10.34 GHz, (IX) f ¼ 12.22 GHz, (X) f ¼ 13.05 GHz (Rivkin and Ketterson, 2006).

168

David Schmool

for these shapes. The very low-frequency modes found in the squares and disks are related to the vortex-state resonances and are known as ‘gyroscopic resonances’. In Fig. 2.33(b), the perpendicular (z-axis) projections of the magnetisation are illustrated for the dominant excitation modes in the three structures. Extension of this work by Rivkin et al. (2007b) on square nanodot arrays of permalloy was performed to identify various experimentally observed resonance features. The experimental data indicate resonances in the regions of 1200 Oe (uniform mode) with smaller peaks around 110 Oe, 430 Oe and 1410 Oe. The main resonance displays a small increase from the continuous film peak of about 50 Oe (Jung et al., 2002a, 2002b). Simulations were performed using the materials parameters of magnetisation, Ms ¼ 795 erg/cm3 and the exchange stiffness constant, A ¼ 1.3 106 erg/cm. Calculation of equilibrium conditions (T ¼ 0 K) with and without random field perturbations illustrate that the vortex core, without perturbation, shifts in a direction governed by the applied field and reaches a saturated state with an applied field over 400 Oe. With perturbation, the saturated state requires higher static fields. FMR simulations are performed using an eigenvalue method (Rivkin et al., 2005) based on the LLG equation, where the microscopic field can include dipole2dipole and exchange interactions. The main resonance line for H0 ¼ 1181 Oe has a frequency of 9.81 GHz, which compares well with the experimental value of 9.75 GHz. In Fig. 2.34(a), the frequency spectrum is shown for H0 ¼ 1181 Oe, the x, y, and z projections of the various modes are shown in Fig. 2.34(b) (note that the radio-frequency (RF) field is applied along the z-axis). Additional work by Rivkin et al. (2007a) on arrays of circular nanodots uses the linearisation of the LL equation. The assumption that for sufficiently high external magnetic fields (around 300 Oe for permalloy) all nanoparticles have the same distribution of magnetisation (which may be non-uniform) in equilibrium is made. Also sufficiently large arrays are assumed to be infinite such that the eigenvectors of the dynamic magnetisation have a Bloch2Floquet form; Vqn ðiÞeiqR , where i denotes the site index in a given cell, q is a continuous Bloch wave vector, n is a band index which number the mode of an individual particle and R ¼ na a þ nb b is a set of all 2D real-space lattice vectors (see Fig. 2.35). Vqn(i) will be periodic and expresses the distribution of phases and amplitudes inside the nanoparticles. A mode frequency oqn is associated with each Bloch state. The static (s) and dynamic (d) fields are expressed as: ðs;dÞ hia ¼

XX2 A ab ðri rj na a nb bÞmðs;dÞ b ðrj ; na ; nb Þ.

(82)

b;rj na ;nb

2

Here A ab is a demagnetisation tensor that gives the field at point ri due to a dipole at rj in the same cell and, due to summation over na and nb,

169


1.0

absorption, a.u.

0.8 0.6 0.4 0.2 0.0 4

6

(a)

8 10 f, GHz

12

14

+1

0

-1 (a)

(b)

(c)

(d)

(e)

(f)

(b)

Figure 2.34 (a) Absorption spectrum for out-of-plane RF field, and applied directcurrent (DC) magnetic field H0 ¼ 1181 Oe. (b) Coordinate projections of the most strongly excited modes for DC applied field H0 ¼ 1181 Oe. Mode amplitudes are presented in terms of their three projections on coordinate axes (x on top, y in the middle, z on the bottom). Different colours correspond to the amplitude scale shown. Mode frequencies are: (a) f ¼ 4.05 GHz; (b) f ¼ 7.41 GHz; (c) f ¼ 8.17 GHz; (d) f ¼ 8.63 GHz; (e) f ¼ 8.95 GHz; (f) f ¼ 9.81 GHz (Rivkin et al., 2007b).

all other cells. Due to the large number of nanoparticles, periodic boundary conditions are imposed. The static magnetisation is strictly ðsÞ periodic; mðsÞ for arbitrary na and nb, while for b ðrj ; na ; nb Þ ¼ mb ðrj ; 0; 0Þ ðdÞ dynamic magnetisation we have: mb ðrj ; N a ; N b Þ ¼ Vqn ðrj Þeiq:ðN a aþN b bÞ , where Na and Nb are the number of unit cells in the a and b directions. A simplification is made by introducing an effective q-dependent demagnetisation tensor: q

Aab ðri rj Þ ¼

X2 A ab ðri rj na a nb bÞeiq:ðna aþnb bÞ , na ;nb

(83)

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David Schmool

na = 0, nb = 1

na = 1, nb = 1

b

a

na = 0, nb = 0

na = 1, nb = 0

Figure 2.35 Array of nanodots (Reprinted figure with permission from Rivkin et al. (2007a), r 2007a by the American Physical Society).

where for q ¼ 0, Eq. (83) corresponds to the static demagnetisation tensor. The eigenvalue problem for the dynamic magnetisation can then be written: (

ioVqn ðri Þ ¼ g

mðsÞ i

"

X

)

#

q

A ðri rj ÞVqn ðrj Þ þ Vqn ðri Þ

hðsÞ i

þ damping.

rj

(84)

This has the same form as for a single particle, where the static and dynamic magnetic fields are due to all nanoparticles in the array and depend on the value of the wave vector q. A comparison to experiment is made for measurements of permalloy nanoparticle square arrays with 500 nm diameter, 85 nm thick and 600 nm between nanodot centres, with the external field applied parallel and at 451 to the array axis. Measured and calculated FMR spectra are illustrated in Fig. 2.36. A reasonable agreement is observed for the 01 orientation, where the single nanodot calculation is shown for comparison. The difference is partially due to the interdot dipolar field (533 Oe). For the 451 spectrum, the uniform mode is fairly well predicted, while small discrepancies are evident for higher field modes. Static configurations for the dots with direct-current (DC) fields applied at 0 and 451 are shown in Fig. 2.37 and modal patterns are illustrated in Fig. 2.38. With the DC field at 01, the interdot interaction produces greater

171


numerics, array numerics, single dot experiment

absorption dervative ( arb. units)

1.0

0.5

0.0

-0.5

-1.0 1000

1500

(a)

2000

2500

3000

H (Oe)

1.0

absorption dervative ( arb. units)

numerics, array experiment 0.5

0.0

-0.5

-1.0 1000 (b)

1500

2000 H (Oe)

2500

3000

Figure 2.36 (a) The heavy line shows the experimental absorption derivative as a function of the external DC field. The simulations for isolated dot and the dot array are shown by the dashed and continuous lines, respectively; the DC field is applied at 01 with respect to the array axis. (b) Absorption derivative (heavy line) and the simulation (light line) as a function of the external DC field for the dot array (the saturation at the lower extreme is an instrumental effect); the DC field is applied at 451 with respect to the array axis. (Reprinted figure with permission from Rivkin et al. (2007a), r 2007a by the American Physical Society.)

172

David Schmool

Figure 2.37 Local-spin orientations arising from the external field and interdot fields (associated with other nanoparticles); external DC field is applied (a) at 01 and (b) at 451. (Reprinted figure with permission from Rivkin et al. (2007a), r 2007a by the American Physical Society.) +1

0

(a)

(b)

(c)

-1

Figure 2.38 Z projections of uniform modes for (a) an isolated dot (1530 Oe), (b) an array with the external field at 01 (1265 Oe), and (c) 451 (1170 Oe). The magnitude of excitations is colour coded. (Reprinted figure with permission from Rivkin et al. (2007a), r 2007a by the American Physical Society.)

non-uniformity at the dot perimeters, which pushes the uniform mode to higher frequencies (lower fields). With the DC field at 451, the interaction field produces a more non-uniform magnetisation distribution (see Fig. 2.38(c), DC spin configuration). This affects not only the uniform mode but also gives rise to a number of localised edge modes (see Fig. 2.39). Discrepancies between theory and experiment could arise from non-idealities in the sample, where edge mode can be expected to be very sensitive. Novosad et al. (2002) and Hertel and Kirschner (2004) also observe the motion of the vortex structure, which rotates around the centre of the nanodisk, however, reducing the frequency and field converts this mode into a quasi-static field-driven displacement of the vortex core. Higher frequency standing modes are predicted for various harmonics, where the frequency is too high for the vortex core to follow. Such works show broad

173


+1 0 (a)

(b)

(c)

(d)

(e)

-1

Figure 2.39 Z projections of low-frequency resonant modes (those responsible for high-field satellites). (a) An isolated dot (2500 Oe), (b) 01 (1375 Oe), (c) 451 (2097, 2345 and 2630 Oe) (d) 451 (2345 Oe), and (e) 451 (2630 Oe) arrays. The magnitude of excitations is colour coded. (Reprinted figure with permission from Rivkin et al. (2007a), r 2007a by the American Physical Society.)

agreement with above-mentioned simulations. Boust and Vukadinovic (2004) further predict a second vortex core mode whose resonance frequency decreases rapidly with increasing dot thickness. This mode originates in the non-uniform vortex structure along the dot normal. Figure 2.40 shows simulations of the zero-field dynamic susceptibility (frequency spectra) and modal patterns for permalloy dots of 20 and 80 nm thickness. Also shown are the resonance frequencies as a function of dot thickness. The effect of static applied field is illustrated in Fig. 2.41, where the two lowest vortex modes are given. The frequency gap between them remains fairly constant. For fields below 0.19 T, the vortex configuration is stabilised, where a decrease in field leads to a reduction of the core radius down to the vortex core switching at 0.37 T. For fields above 0.19 T, a vertical plane crossing the dot centre develops in the Mz component of the magnetisation. Increasing the static field shows an increased slope of the resonance frequency. Further increase of the DC field will eventually lead to saturation of the sample and a loss of the vortex modes. Removal of the central core leaves the disk samples in the form of a ring where no ventral vortex mode can exist. Such a system is considered by Nguyen and Cottam (2006), this is briefly discussed at the end of this section. The considerations of other forms of magnetic nanostructure have also been performed by various authors, typically of structures with rectangular shape. One of the specific complexities inherent in this type of structure is that the internal field (and hence magnetisation) can be strongly inhomogeneous and will change significantly with applied external fields. Depending on the direction of applied static fields and magnetic element aspect ratio, the conditions for the quantisation of spin wave (SW) modes will differ significantly. For long elements with longitudinal magnetisation, the SW wave vector has a single component of quantisation (1D), where the internal field in the stripe will be homogeneous and equal to the bias field and the weak inhomogeneous dynamic magnetisation leads to effective ‘pinning’ at the lateral edges of the stripe and can be described by effective dipolar boundary conditions (Guslienko et al., 2003). The lowest quantised modes correspond to the DE surface magnetostatic modes.

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David Schmool

150 Lz = 20 nm (1)

χ″xx

100

50 (2) (a)

(3)

0 Lz = 80 nm

40

χ″xx

(1)

(1′)

(1′)

20 (2) (3) 0

4

Resonance Frequency (GHz)

(b)

16

20

15 (3) 10

(2)

5 (1′) (1)

0 0

(c)

8 12 Frequency (GHz)

20

40 60 Dot Thickness (nm)

80

Figure 2.40 Three-dimensional dynamic micromagnetic simulations of nanodots. Zero-field susceptibility spectra (imaginary part w00xx ) for two dot thicknesses Lz ¼ 20 nm (a) and Lz ¼ 80 nm (b). The spatial distribution of resonant modes within the nanodots is displayed in the insets high level in dark grey and low level in light grey. (c) Thickness dependence of resonance frequencies for the four main modes. The dashed line corresponds to the model by Guslienko et al. (2002a). (Reprinted figure with permission from Boust and Vukadinovic (2004), r 2004 by the American Physical Society.)

Transversal magnetisation will lead to drastically different quantisation of the dynamic magnetisation, where the SW quantisation gives rise to dipoleexchange backward volume modes, having wave vectors parallel to the bias field. Also the internal bias field in the transverse case will be strongly non-uniform which gives quantisation of SW modes of fixed frequency

175


μ0H2=0 T

μ0H2=-0.2 T

μ0H2=0.19 T

(a)

Resonance Frequency (GHz)

12 10 Lz = 80 nm 8 6 4 (1′) 2 (1) 0 -0.4 (b)

-0.2

0.0

0.2

0.4

0.6

Magnetic Field (T)

Figure 2.41 Computed static and dynamic properties of the vortex core in the presence of a static magnetic field applied along the dot normal (z-axis). The dot radius is R ¼ 80 nm and the dot thickness is Lz ¼ 80 nm. (a) Cross-sectional view (vertical plane through the dot centre) of Mz. (b) Magnetic field evolution of the resonance frequencies for the two vortex core modes (symbols). The dashed line correspond to Kittel’s law for an uniformly perpendicular magnetized dot. (Reprinted figure with permission from Boust and Vukadinovic (2004), r 2004 by the American Physical Society.)

being composed of various magnitudes of wave vector. In this case, the lowest spin-wave modes are of exchange nature and localised in ‘potential wells’ formed by the strongly inhomogeneous internal bias magnetic field near the lateral edges of the stripe. The quantisation (of spin-wave modes) can be expressed in general terms as (Jorzick et al., 2002): nðqÞ ¼

g 2p

1=2 2A 2 2A 2 Hþ , Hþ q q þ 4pM F pp ðQdÞ M M

(85)

where Fpp(Qd) denotes the matrix element of the DDI (Kalinikos and Slavin, 1986), q ¼ qp ex þ Q with qp ¼ pp=d being the quantisation from thickness, d and in-plane component Q ¼ Qy ey þ Qz ez . All other symbols have their usual meaning. Assuming that the quantisation of the x and y

176

David Schmool

components takes place as in the longitudinally and transversally magnetised strip, respectively, a 2D approximation is made whereby the spin-wave eigenfunctions of the rectangular magnetic element are products of the 1D cases for longitudinal and transversal magnetisation. In this case, the x-quantisation of the wave vector takes the form: qmx

ðm þ 1Þp 2 1 ; ¼ w dðpÞ

m ¼ 0; 1; 2; . . . .

(86)

The effective dipolar pinning at the lateral edges of the rectangular element depends only on the length, L, to width, w, ratio; p ¼ L/w{1, and dðpÞ ¼ 2p=½pð1 2 ln pÞ . Quantisation along the y-axis is more complex due to the inhomogeneous internal field along this direction, for which an explicit determination of the internal field is necessary (Guslienko et al., 2003). In general, fulfilment of the quantisation condition: I

qðyÞdy ¼ 2ðn þ 1Þp

(87)

is sufficient. Guslienko et al. (2003) evaluate the approximate dipoleexchange dispersion relation for spin waves as: 2 mn 2 o2mn ¼ ðomn H þ ZoM kmn Þ½oH þ ZoM kmn þ oM F mn ðkmn Þ ,

(88)

where oH ¼ gH, oM ¼ 4pgM , Z is the exchange constant, k2mn ¼ q2mx þ q2ny . The demagnetisation is manifest in the frequency: omn H ¼ oH oM N mn , where Nmn is the demagnetisation factor for mode (m, n) given by: N mn

4 ¼ wlM 2

Z

d 2 rm2mn ðrÞN yy ðrÞ,

(89)

where mmn ðrÞ ¼ M cosðqmx xÞmn ðyÞ is the transverse variable magnetisation, r ¼ xex þ yey and mn(y) is the distribution of the quantised eigenmode along the y-direction. Fmn(kmn) is the quantised matrix element of the DDI and takes the form:

oM F mn ðkmn Þ ¼ 1 þ Pðkmn Þ½1 Pðkmn Þ mn oH þ ZoM k2mn

k2mx k2mn

Pðkmn Þ

k2my k2mn

!

,

(90)

where Pðkmn Þ ¼ 1

1 ekmn L . kmn L

(91)

The second term in Eq. (90) describes the increase in SW frequency with increase in wave vector perpendicular to the bias field (DE


177

magnetostatic surface wave) and the last term in Eq. (90) determines the decrease of SW frequency with increase of the quantised component parallel to the bias field (backward volume magnetostatic wave (BWVMSW)). The demagnetisation factor takes into account the nonellipsoidal shape of the magnetic element. Using this formalism, Guslienko et al. (2003) predict the modal patterns for the (0, 0) and (0, 6) SW modes, with m0 ðyÞ ¼ cosðpy=lÞ, as shown in Fig. 2.42. Modal frequencies are given in Table 2.2 and compare well to experimental data of Tamaru et al. (2002). Grimsditch et al. (2004b) use a micromagnetic calculation of the LL equation to evaluate the normal modes of spin excitations in nanosized particles. In this calculation, the particle is divided into cells whose dimensions are smaller than the exchange length, such that the magnetisation inside the cell is essentially uniform. In this way, the LL maintains a constant magnetisation at all times. In addition to this the dipolar (far-field) contribution is derived from a scalar magnetic potential, which means no artificial boundary conditions are introduced. The authors consider the energy of the system as comprising of the exchange, dipolar and Zeeman energies. The equation of motion for each spin vector is monitored at regular time intervals and the time series of each component in each cell is then Fourier analysed. The normal modes are then identified by correlating the Fourier transforms throughout the sample. Alternatively, a dynamical matrix approach is used by Grimsditch et al. (2004a) to simulate the normal modes of oscillation for Fe parallelepiped structures. Again the sample is divided into a large number of cells. The advantages of this method are characterised as: (i) a single calculation leads to the frequencies and eigenvectors of all modes of any symmetry; (ii) particles of any shape can be considered; (iii) no a priori limitation is imposed on the form and pinning of the modes and (iv) the computational time is affordable. The theoretical model for the simulations employs the coupling of the N cells, into which the rectangular magnetic element is divided, via the SB equations for each cell. In essence, this will produce a 2N 2N matrix in a similar fashion to that indicated above, Eqs. (77)2(81). If the cells are decoupled, the matrix will consist of block diagonal 2 2 submatrices of a form analogous to Eq. (79). Off-diagonal 2 2 blocks will be non-zero when coupling is present. Mode profiles are then defined by the dynamical magnetisation: dmk ¼ ð sin Wk sin jk djk þ cos Wk cos jk dWk ; sin Wk cos jk djk þ cos Wk sin jk dWk ; sin Wk dWk Þ,

(92)

where k denotes the cell site position. The energy density of the system is defined considering the sum of the Zeeman, exchange and dipolar

178

Coordinate perp. to bias field, x/w

David Schmool

Coordinate parallel to bias field, y/l

Coordinate perp. to bias field, x/w

(a)

(b)

Coordinate parallel to bias field, y/l

Figure 2.42 Calculated dynamic magnetisation distributions (contour plots) using mnm(q), see text (to compare with experiment see Tamaru et al. (2002)), for the (a) SW mode with m ¼ 0, n ¼ 0 and (b) SW mode with m ¼ 6, n ¼ 0. (Reprinted figure with permission from Guslienko et al. (2003), r 2003 by the American Physical Society.)

179


Table 2.2 Comparison of experimental (Tamaru et al., 2002) and calculated resonance fields of square FeNi element. L ¼ w ¼ 50 mm, L ¼ 104 nm, m ¼ 0, 2, 4, 6, 8; n ¼ 0. Ms ¼ 860 G, y/2p ¼ 3.04 GHz/kOe (Guslienko et al., 2003).

The SW mode number Experimental resonance fields, Oe Calculated (m, 0)-fields by Eq. (87), Oe

(0,0) 469 466

(2,0) 392 396

(4,0) 324 330

(6,0) 261 266

(8,0) 196 204

energies, where EZ ¼ M s H

N X

mj .

(93)

j¼1

H being the external field. E exch ¼

N X AX ð1 mj mn Þ. d 2 j¼1 n

(94)

The second summation is over nearest neighbours of cell j: E dip ¼ 2

N X N 2 M 2s X mk N ðk; jÞmj . 2 k¼1 j¼1

(95)

Here N ðk; jÞ represents the demagnetising tensor. The demagnetising field in each cell is constant. Two methods are employed to assess the demagnetising tensor (A and B). In method A, the interactions between magnetic surface charges from all cells are evaluated (a similar approach used in the OOMMF code, Donahue and Porter). Method B assumes that the x are given by point dipole interactions, with the dipoles at the centre of each cell. The magnetic parallelepiped is obtained as a stack of cubic cells in the z-direction. The magnetisation of each cell is constant (where mk is independent of z). Simulations were made comparing the two methods and the results of other calculations (from Grimsditch et al., 2004a and OOMMF). The comparison is generally good, with results of mode frequencies given in Table 2.3. The method of calculation only detects modes with hdmi ia0 and the modes correspond to the fundamental symmetric end mode and lowest order symmetric SSW modes. In general, it was found that method A agrees best with the full simulation (Grimsditch et al., 2004a) and this provides agreement with OOMMF calculations to within 0.5 GHz. The dynamical approach also predicts additional modes with variations along the direction perpendicular to the applied field; two such modes are illustrated in Fig. 2.43. The dispersion (oq) behaviour of these modes is consistent with DE-like modes (see Fig. 2.44). A family of end modes is observed, with oq also illustrated in Fig. 2.44, modal patterns are shown in Fig. 2.45. The results are also in agreement with the

180

David Schmool

Table 2.3 Frequencies, in GHz, of normal modes of a 116 60 20 nm Fe particle with a field of 10 kOe applied along the long axis. The symmetric and antisymmetric end modes are essentially degenerate at this field. The modes labelled as ‘n-nodes’ correspond to the standing wave-like modes, with wave vectors along the field direction, reported in Grimsditch et al. (2004a, 2004b). Method (number of cells)

Cell size (nm)

Funda- End Mode 10-nodes 9 2-nodes mental mode 6-nodes

Ref. 14 (29 15 5) A (25 15 1) A (58 30 1) B (29 15 5) B (58 30 10) OOMMF (29 15 1)

444

52.9

30.2

55.6

89.3

94.4

4 4 20 2 2 20 444 222 4 4 20

53.1 52.4 52.9 52.7 52.6

30.0 29.3 28.3 28.5 29.5

55.5 55.5 54.8 55.6 55.1

86.5 90.4 85.2 90.3

91.7 94.2 90.7 94.3

findings of Guslienko et al. (2003). Hybridisation of modes was also observed; the patterns for modal frequencies 64.6 and 66.1 GHz are shown in Fig. 2.46(a), with the sum and difference illustrated in Fig. 2.46(b). Using the OOMMF software, Gubbiotti et al. (2007) have analysed Brillouin light scattering (BLS) data on square permalloy dots with good overall agreement between experiment and simulations. The SW mode patterns are similar to those obtained by Grimsditch et al. (2004a), with both volume and edge modes being predicted. Comparison is also made with analytics of Guslienko et al. (2002b) and agreement is also good. Another micromagnetic simulator is that of Scheinfein and Price (2001), which Mohler and Harter (2005) have applied to study the resonance frequencies in FM particles. The aspect ratio of the particles was seen to be an important factor in the existence of one or more resonance peaks, as illustrated in Fig. 2.47 for rectangular prisms of Co. The variation of resonance frequency with particle length is shown in Fig. 2.48. For shorter particles, the lower resonance line agrees with the Kittel equation, while for lengths over 50 nm the second resonance appears and the Kittel equations agrees with neither resonance. For very long particles (B500 nm), the Kittel theory asymptotically approaches the second resonance frequency. The micromagnetic simulations show that the second resonance arises from an edge SW mode with an extension of about 20 nm into the particle. The decoupling in the longer particles occurs when the central region of the particle no longer acts under the influence of the long axis demagnetisation field, where the resonance frequency can be obtained from the relation: o ¼ gðH K þ M s =2Þ. The decoupling of the central region is believed to be due to the presence of a strongly inhomogeneous demagnetising field at the


181

Figure 2.43 Mode profiles of two Damon--Eshbach (DE)-like modes; (a) 1-node mode with O ¼ 58.6 GHz, (b) 6-node mode with O ¼ 138.2 GHz. Method A with 58,330 cells. We plot the real part of dmz (arbitrary units) as a function of the cell position. As explained in the text, the imaginary part is zero. (Reprinted figure with permission from Grimsditch et al. (2004a), r 2004a by the American Physical Society.)

sample edges. The authors use a harmonic oscillator model to demonstrate the decoupling effect. Other forms of nanostructures have also been considered from a theoretical point of view. For example, Nguyen and Cottam (2006) have made numerical calculations of spin-wave excitations in FM nanorings. Such systems are readily prepared experimentally and their stable magnetic states have been studied quite extensively (see e.g., Castanõ et al., 2003; Vaz et al., 2007 and references therein). One of the principal parameters which determine their magnetic spin configuration is the dimension of their inner and outer diameter, also an applied magnetic field will play an important

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Figure 2.44 Variation of the normal mode frequencies with the wave vector. The wave vector is perpendicular to H and is plotted in units of p/w. Dots: DE-like modes; squares: end modes. (Reprinted figure with permission from Grimsditch et al. (2004a), r 2004a by the American Physical Society.)

role in the stable magnetic configuration of the system. Variation of the dimensions of the nanoring, including its height, can give rise to vortex, bidomain, bottleneck or twisted configurations. The non-uniform magnetisation in these systems makes them a challenging problem from a theoretical point of view. A robust switching between stable magnetic states is possible with the application of an applied external field and makes the nanoring system an interesting candidate for high-density magnetic storage devices. A brief discussion of FMR measurements in such systems will be outlined in Section 6.2.4. In the study of Nguyen and Cottam (2006), the nanoring structure is approximated as a hollow hexagon of L layers of spins stacked vertically with a separation of a. Inner and outer radii are designated as ri and ro in units of a, respectively. Each layer will have N ¼ 3½r o ðr o þ 1Þ r i ðr i 1Þ

spins which are arranged in a triangle. The Hamiltonian for the system is then described by: 1 H^ ¼ 2

XX

a b V ab nm S n S m gmB

n;m a;b

X

H n Sn ,

(96)

n

where n and m refer to the indices of the spins and a and b are Cartesian components; x, y and z. The coupling tensor between the spins Sn and Sm include both dipole and exchange interactions: 2 V ab nm ¼ J nm dab þ ðgmB Þ

jrnm j2 dab 3r anm r bnm , jrnm j5

(97)


183

Figure 2.45 Mode profiles of three end modes: (a) 1-node mode with O ¼ 37.5 GHz, (b) 2-node mode with O ¼ 46.3 GHz, (c) 6-node mode with O ¼ 122.0 GHz. (Reprinted figure with permission from Grimsditch et al. (2004a), r 2004a by the American Physical Society.)

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Cell index

David Schmool

58

58

50

50

40

40

30

30

20

20

10

10

1

1 1

(a)

10

20

30

1

10

20

30

(b) Cell index

Figure 2.46 Contour plot of two mode profiles of mixed character; (a) O ¼ 66.1 GHz, (b) O ¼ 64.6 GHz. Method A with 58,330 cells. We plot the real part of dmz (arbitrary units) as a function of the cell position. (Reprinted figure with permission from Grimsditch et al. (2004a), r 2004a by the American Physical Society.)

where only nearest-neighbour interactions were considered, J. The second term of Eq. (96) represents the combined effect of Zeeman energy, due to an applied field H0 and the single ion anisotropy, represented by a field Han, which may be site dependent. Equilibrium states were calculated by minimising energy at each spin site, different initial configurations lead to different local minima. The spin configuration is used as the initial state of the system from which SW excitations are calculated via a bosonisation technique combined with the Green’s function method. The system was chosen with ro ¼ 5 and ri ¼ 3 and varying the number of layer L between 3 and 13, where an applied field can be either transversal or longitudinal. The relative strength of dipolar and exchange interactions can be expressed as Rd ¼ g2 m2B =Ja3 . Equilibrium configurations for the vortex and bidomain states are favoured for low aspect ratios (low L) while twisted and bottleneck configurations are found for larger L. The states are illustrated in Fig. 2.49. Setting the H0 ¼ 0 and Rd ¼ 0.09, the variation of energy minima are shown as a function of L for these states in Fig. 2.50. The vortex state is the most stable, though switching to other states is possible


185

Figure 2.47 Imaginary susceptibility as a function of frequency for a variety of simulated Cobalt rectangular prisms, specified by aspect ratios. (Reprinted with permission from Mohler and Harter (2005), r 2005 American Institute of Physics.)

Figure 2.48 Observed resonance frequencies as a function of prism length for Cobalt data presented in Fig. 2.47. The black solid line denotes the Kittel prediction. Nearly identical behaviour was seen for NiFe. (Reprinted with permission from Mohler and Harter (2005), r 2005 American Institute of Physics.)

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Figure 2.49 The geometry of a nanoring with ro ¼ 5, ri ¼ 3, and L ¼ 5, showing the local equilibrium spin configurations in the middle layer for (a) vortex state and (b) bidomain (or ‘onion’) state (in z-direction). (c) Twisted state and (d) bottleneck (or ‘flower’) state. Here the top layer of the nanoring is shown. (Reprinted with permission from Nguyen and Cottam (2006), r 2006 American Institute of Physics.)

via the application of an applied magnetic field. Spin-wave excitations for various spin states were calculated, results for transverse and longitudinal fields are illustrated in Fig. 2.51, transitions from the vortex state are evident in increasing fields, where a complex behaviour with hysteresis is encountered (shaded regions).

4. Spin-Current-Induced Dynamics in Magnetic Nanostructures It is well over a decade that Slonczewski (1996) and Berger (1996) gave their theoretical predictions of the effects of a spin current on the state of the magnetisation of a FM material. These effects were recognised to be of fundamental importance to the physical basis of magnetic device operation and underlie dynamical phenomena such as magnetisation switching and the excitation of coherent spin wave (SW) modes (magnons).

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Vortex state Bottleneck state Bidomain (z-direction) Bidomain (x-direction) Twisted state

Energy per spin (units of S2J)

-2.9

-3.0

-3.1

-3.2

-3.3

4

6

8

10

12

Number of Layers

Figure 2.50 The free energy per spin versus the number of layers L for the different configurations, taking H0 ¼ 0, ro ¼ 5, and ri ¼ 3. (Reprinted with permission from Nguyen and Cottam (2006), r 2006 American Institute of Physics.)

The future potential of such devices is enormous, where manipulation of the state of magnetisation on a magnetic nanostructure could have important implications in information technology and is a realistic prospect. While it is not the intention of the present work to give an extensive review of this here, it is worthwhile noting the relevance and importance of this area of modern research and we will give a brief summary of some of the more salient points and some general results of theoretical and experimental work. The spin-torque transfer effect is observed in magnetic structures where an electrical current passes through a layered structure (e.g. in a magnetic multilayer system, where at least two FM layers are separated by a nonmagnetic layer) in a direction perpendicular to the layers (CPP geometry). In this way, the current passing through the first FM layer becomes spin polarised and then transfers spin angular momentum to a subsequent FM layer. This transferred momentum acts as a torque on the magnetisation of this layer. For this torque to be sufficient to act on the second magnetic layer, the current density needs to be relatively large and typically W1062107 A/cm2 (Tsoi et al., 1998). When two stable equilibria exist for the magnetisation (due to uniaxial anisotropy for example), it is possible for the spin-torque transfer to effectuate a magnetisation reversal switching from one equilibrium to the other; this effect is generally referred to as

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25

Frequency (GHz)

20

15

10

5 Vortex

Bottleneck

0 (a)

0.0

0.1

0.2

0.3

0.4

0.5

25

Frequency (GHz)

20

15

10

Vortex

5

Bidomain

0 0.0 (b)

0.1

0.2

0.3

0.4

0.5

Magnetic Field (T)

Figure 2.51 SW frequencies (lowest ten branches only) versus H0 for a nickel nanoring: (a) longitudinal field and (b) transverse field. (Reprinted with permission from Nguyen and Cottam (2006), r 2006 American Institute of Physics.)

‘current-induced magnetisation switching’. For the spin-torque effect to be present the thickness of the non-magnetic interlayer must be smaller than the spin-diffusion length to allow some degree of spin polarisation to emerge from the non-magnetic layer into the second FM film; usually thicknesses of less than 100 nm are required (Johnson, 1995).

189


4.1. Theoretical background There are a number of issues which are related to the material properties of the FM switching layer which must be considered; the critical current for switching and the speed of switching. The principal properties which will govern the switching dynamics are the magnetic-anisotropy strength, magnetisation, damping constant and dimensions of the switching layer. The model proposed by Slonczewski (1996, 1999) considers the effects of spin transfer by which a steady precession can be driven by a constant current and switching can be induced by a pulsed current. The magnetic layers are considered as macrospins, within which the magnetisation is uniform. The transverse component of the spin torque is then written as (Sun, 2006):

_ ZI ½ðns mÞ m . G ¼ gðnm ; ns Þ 2e m2

(98)

Here m is the magnetisation vector of magnitude m and direction nm, ns is the direction of the spin polarisation entering the FM layer and Z ¼ ðI " I # Þ=ðI " þ I # Þ is the spin polarisation; Im and Ik being the majority and minority spin currents, respectively. The factor gðnm ; ns Þ is a numerical pre-factor describing the angular dependence of the spin angular momentum transfer efficiency which originates from the quantum mechanical nature of the interaction between the spin-polarised current and the macrospin. An alternative approach was considered by Berger (2002), where the introduction of the difference between the spin-up and spin-down Fermi levels; Dm ¼ m" m# , is taken into account. The Fermi distributions (near T ¼ 0) for spin-up and spin-down electrons are shown in Fig. 2.52. ε μ

|Δμ|

ε

|Δμ|-hω

ε

hω

hω

μ hω

f

f

Figure 2.52 Fermi distributions f " ðÞ, f # ðÞ for spin-up and spin-down electrons of energy, with different Fermi levels lm, lk at TE0. The oblique arrow represents a spin--flip process where a magnon of energy _o is emitted. (Reprinted with permission from Berger (2002), r 2002 American Institute of Physics.)

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The spin-wave energy for a magnon (in the second FM layer of a bilayer system) can be written as _o ¼ " # , where o is the frequency of precession of S2, the spin value of the second FM layer, and " # represents the energy of a spin-flip process. The Pauli exclusion principle must be taken into account, where the initial state must be filled and the end state initially empty, such that the only suitable initial states are those located in the region of energy thickness given by jDmj _o ¼ ðDm þ _oÞ from the top of the spin down Fermi electron sea (see Fig. 2.52). This will mean that the resulting spin-torque drive will be proportional to Dm þ _o, and can be written as (Berger, 2002): G ¼ as ðWÞðDm þ _oÞ

S2 ðS1 S2 Þ . _S 1 S 2

(99)

Since Dm will be proportional to the intensity of the spin current and will thus be consistent with Eq. (98). The additional term related to _o represents a surface damping torque, as ðWÞ, which is independent of the current and is a result of the Pauli exclusion principle. The main idea behind the current-driven FMR is the effect of an electrical current on the magnetisation dynamics. In this case, the generalised LLG equation for the motion of the magnetisation will be augmented by the current-dependent spin-transfer torque, which can be written as (Kovalev et al., 2007): @M a ¼ gðM Heff Þ þ @t M

@M _ IðtÞ M þg ½Z m ðmfixed mÞ @t 2e V m 1 (100)

þ Z2 ðmfixed mÞ ,

where I(t) is the time-dependent current through the system, Z1 denotes the efficiency of conventional spin transfer and Z2 parameterises an effective spintransfer exchange field. In general, this approach is used to describe the spintransfer torque between magnetic layers in a multilayer system. In Eq. (100), it has been assumed that one of the FM layers is fixed; mfixed ¼ M1 =M s and the dynamics considered are due to the second (free or soft) layer: m ¼ M2 =M s ¼ M=M s . Urban et al. (2001) found that this spin-pumping effect has an appreciable effect on the linewidth of resonance curves and can only occur in all metallic systems. When the intervening non-magnetic layer is a normal metal (NM), the LLG equation of the free magnetic layer can be written in the form: @M a ¼ gðM Heff Þ þ @t M

@M Zg_ m ½ðIs1 þ Is2 Þ m , M @t 2eV m

(101)

where Is1,2 are the spin currents at the ferromagnetic2metal interfaces and depend on the interface mixing conductances, chemical potential and spin accumulation in the NM (Brataas et al., 2001; Brataas et al., 2006). The form


191

of Eq. (101) shows that the effect of the spin current can be opposed to that of the damping (Gilbert) term and can hence amplify the magnetisation precession amplitude; this has obvious implications in switching applications. For simple geometries under the macrospin approximation, the LLG can be linearised and solved. For a thin free layer nanomagnet in collinear geometry, where the uniaxial easy axis is aligned in-plane with the applied field a stability threshold can be established, for which a critical current is evaluated as (Sun, 2000, 2006): 2e a Ic ¼ mðH þ H k þ 2pM s Þ. _ Z

(102)

Ms being the saturation magnetisation of the free layer, Hk the anisotropy field and m ¼ abtM s ¼ V m M s , a and b being the lateral dimensions of the nanomagnet and t its thickness. In experimental situations, the switching current can often be affected by other, sometimes significant, factors: temperature effects (which can lead to lower values of Ic), spin-transport mechanisms (diffusive or ballistic), interface effects (spin-flip scattering). Additional care must be taken with regards to the initial cone angle increase, since an increase in cone angle dictated by the instability threshold is not necessarily accompanied by a magnetic reversal. Berger (2002) also considered the instability conditions from damping processes, where when the current exceeds a critical value, Ic, the drive torque will overcome the natural (Gilbert) damping and produce a magnetic reversal. Such considerations lead to the relation: 1 a _o. 1þ Ic ¼ K as ðWÞ

(103)

This will mean that the critical current will be proportional to o. When considering the effect of a current with a magnetic moment there are two possible causes that should be considered: (i) current-induced magnetic (or Oersted) field and (ii) spin-polarised current-induced spin torque. Distinction between these mechanisms can be made via the size dependence of the current; in the former case Maxwell’s equations lead to the relation: I ¼ ðc=2ÞrH (in Gaussian units, where c is the speed of light). The corresponding relation for the spin-torque effect can be deduced from Eq. (102) as I c ð2e=_Þða=ZÞð4r 2 tÞM s ðH þ H k þ 2pM s Þ. Thus the distinction is made via an r and r2 dependence. It is, therefore, evident that the threshold current for the spin-torque effect is more significant for smaller structures, where a crossover can be established as: rc

c_ Z 1 H . 4e a M s t H þ H k þ 2pM s

(104)

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For typical values of a Co pillar-structured spin valve this gives a crossover size of around 0.1 mm. Therefore, we see that the spin-torque effect will be important for nanostructured devices. The speed of magnetisation reversal is an important factor in the study of current-induced magnetisation dynamics. In the simplest case of a uniaxial material, the time dependence of motion will be directly related to the damping coefficient, and for subcritical currents will produce a classical spiral motion of the magnetisation vector from its initial position, W0 , to the equilibrium position. Once the spin current reaches a supercritical value, the dynamics will profoundly alter the motion of the magnetisation vector and we need to consider the balance between the damping parameter and the spin current. Such a situation will be described by a firstorder differential equation in terms of the polar and azimuthal angles of the magnetisation vector (Sun, 2000), the solution of which can be expressed as: WðtÞ ¼ W0 et=t1 ,

(105)

where t¼

gH k t 1 þ a2

and

1 H t1 ¼ bI þ a 1 þ Hk

(106)

with b ¼ ð_=2eÞZ=abtM s H k and the critical current is given by Eq. (102). In Fig. 2.53, the effect of the current is illustrated for different values of the spin current, hs ¼ ð_=2eÞZI=abtM s H k , where the shape anisotropy was neglected. The influence of shape anisotropy will be to produce an elliptical precession, as shown in Fig. 2.54. To obtain the switching time, Eq. (105) can be manipulated, for which the uniaxial anisotropy with shape anisotropy can be expressed as: tðWÞ ¼ H k

_ ZI aðH k þ H þ 2pM s Þ 2e tM s

1 W . ln W0

(107)

For the case of a large overdrive current, the damping becomes negligible and the switching speed will be determined by the spin current. The effect of spin currents on the magnetisation state in magnetic layers was reported by Heide (2001) and Heide et al. (2001), where the influence of the conduction electron spin on the local moment in the magnetic layer due to the s2d type exchange is considered. The case of a magnetic bilayer system is studied and is described in terms of a non-equilibrium exchange interaction (NEXI), being different to the RKKY interaction since it deals with the effect of the non-equilibrium magnetisation on the local moments and not spin density oscillations. Account is also made for the relaxation of the injected spin-polarised electrons due to spin-flip processes and can lead to Stoner excitations and the generation of spin waves. The treatment of


193

Figure 2.53 The precession of magnetisation under the influence of a spin current. Uniaxial anisotropy alone. (a) Time dependence of Mz. (b) Time dependence of Mx. (c) A 3D portrait of the spiral motion of the tip of M. North pole is ez direction. (Reprinted figure with permission from Sun (2000), r 2000 by the American Physical Society.)

the magnetisation dynamics with the current perpendicular to the magnet layers requires a simultaneous solution of the equations of motion for the local magnetisation and the spin-polarised charge carriers. The FM layers are comprised of the core (d) electrons characterised by the local magnetisation, Md, and a paramagnetic subsystem of almost free electrons (s) with magnetisation Ms. Since the s and d electron are exchange coupled (Eex ¼ aMs Md, where a is the s2d exchange parameter) the equations of motion for the two magnetic subsystems will be coupled in the form of LL equations (Langreth and Wilkins, 1972): @Ms dMd dMs gðMs Heff ;s Þ þ r JM ¼ @t tds ts

(108a)

@Md dMs dMd gðMd Heff ;d Þ ¼ . @t tsd td

(108b)

and

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Figure 2.54 The precession of magnetisation under the influence of a spin current. Uniaxial anisotropy plus an easy-plane anisotropy of hp ¼ 5. The uniaxial-anisotropy-alone trace of a ¼ 0.01 is included for comparison. The elliptical precession is apparent here, with the cone angle being compressed in the direction normal to the easy plane. Panels have the same definition as in Fig. 2.53. (Reprinted figure with permission from Sun (2000), r 2000 by the American Physical Society.)

Spin diffusion of the s electrons is described by the divergence of the spin current, JM. The effective fields are given by: Heff ;sðdÞ ¼ H0eff ;sðdÞ þ aMdðsÞ , the first term holds those contributions which are relevant. In the case of the s electrons this will be the applied external field, while the d electrons will have additional contributions due to anisotropy, dipolar interactions, etc. The relaxation of the system is determined by 1 1 RHS of Eqs. (108a) and (108b), where t1 sðdÞ ¼ tsdðdsÞ þ tslðdlÞ , with the first term characterising the spin angular momentum transfer between the s and d electrons and the second term describes the spin2lattice relaxation. The non-equilibrium magnetisation being given by: dMsðdÞ ¼ MsðdÞ wsðdÞ Heff ;sðdÞ ,

(109)

where the free electron (s) susceptibility ws ¼ ð_gÞ2 NðF Þ=4, is just the Pauli susceptibility at the Fermi energy and NðF Þ is the density of states. The d electron susceptibility will depend on the applied field, and can be

195


approximated as (Heide, 2001): wd ¼

2 td M d dMs Md . Heff ;d Md td tsd

(110)

The scattering between s and d electrons will be balanced to conserve angular momentum, M ¼ Ms + Md. This will mean that the total torque will be conserved such that @Md =@t ¼ @Ms =@t. The detailed balance of the electron scattering gives rise to the condition: wd tsd ¼ ws tds . The nonlocal character of the conduction electrons gives rise to the r JM term in Eq. (108a) and consideration of the transport equations allows the spin current to be determined (Johnson and Silsbee, 1987; Valet and Fert, 1993): JM ¼ ms M0s rV DrdMs .

(111)

ms is the mobility and D the diffusion constant of the conduction electrons. Using a steady-state pffiffiffiffiffiffiffiffi approximation, it is found that rdMs ¼ dMs =lsf , where lsf ¼ Dtsl is the spin-diffusion length, describes the spin accumulation. Providing then that the magnetisation of the layers is uniform the non-equilibrium magnetisation due to the s electrons will read: n

n

dMns ðxÞ ¼ mn1 ex=lsf þ mn2 ex=lsf .

(112)

mn1ð2Þ are constant vectors for layer n, determined by the appropriate boundary conditions. Given that MdcMs, the temporal dependence of the conduction electrons on the rotation of the localised moments will be negligible. However, for large currents the non-local magnetisation will be of the order of the time-dependent fluctuations, dMd . To obtain the correct form of the equations of motion for the magnetisation vectors of the two layers, the contributions of dMs which enter the effective fields of the layers must be derived from the same energy function; Enexi ¼ aeffMA MB, where aeff is the effective NEXI between the two magnetic layers A and B. This allows the non-local effects of the spin current to be accounted for in the effective field: HAðBÞ ¼ H0eff ;AðBÞ þ aeff MBðAÞ þ aðM0s;AðBÞ þ mAðBÞ Þ and the equations of motion will read: MAðBÞ wAðBÞ ðHAðBÞ þ dHÞ @MAðBÞ gðMAðBÞ HAðBÞ Þ ¼ , td @t

(113)

where dH ¼ ðmBðAÞ þ mAðBÞ Þtd =ðtsd wAðBÞ Þ is the effective field due to the spin transfer between the layers. Significant effects will be observed when there is an asymmetry in the spin transport; otherwise there will be a cancellation of the torques generated in each layer. Eq. (113) exhibits two instabilities driven by the spin current: one from the precession of the first layer and the second from the spin transfer on the second magnetic layer. Asymmetries in the magnetic layers can be arrived at by making the layers of different thicknesses; one fixed layer (thickness Wlsf ) and one free layer

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(thickness olsf ). Switching of the free layer will occur from the NEXI in the case where aeff MFixed MFree a0. The critical current for switching will then be derived from the switching field, which for Co films will be of the order of 1062107 A/cm2 and is similar to values predicted by other models. The second instability occurs when the magnetisation is removed from the instantaneous field, which for a thin film has the approximate form: H H appl 4pM oH =g. The instability condition will then read: _oH þ DmðxB Þ

tdl o0, tdl þ tds

(114)

where DmðxÞ ¼ _g½mA ðxÞ þ mB ðxÞ =ws is the difference in chemical potential for spin-up and spin-down electrons. The onset of the instability is characterised by a uniform precessional motion; stronger currents will lead to higher order excitations. The model presented by Heide (2001) and (Heide et al., 2001) does not require the presence of a second magnetic layer to predict current-induced magnetisation dynamics. A further model of current-driven magnetisation switching which includes the effect of spin diffusion has been presented by Zhang et al. (2002). This model builds on that of Heide et al. (2001) and the influence of transversal spin accumulation is taken into account. In this approach, a consideration of the spin transport leads to the equations of motion for spin accumulation, m, and the local magnetisation, Md. With the interaction between these being given by: Hint ¼ Jm Md, the equation of motion for the spin accumulation takes the form: dm J m ðm Md Þ ¼ . dt _ tsf

(115)

tsf is the spin2flip relaxation time. The second term represents the precessional motion of the accumulation due to the s2d interaction, which occurs when the s and d moments are not collinear. Since the conduction electrons transport spin dm=dt ! @m=@t þ @jm =@t, and the spin accumulation equation of motion takes the form: 1 @m @2 m @2 m m m Md . ¼ 2 bb0 m 2 2D0 @t @x @x lsf l2J

(116)

pffiffiffiffiffiffiffiffiffiffiffi lsf and lJ p areffiffiffiffiffiffiffiffiffiffiffiffi related ffi to the diffusion constant via relations: lsf ¼ 2Dtsf and lJ ¼ 2hD=J , respectively. Where the latter length scale holds only for toh=J. The constants b and bu are the spin polarisation parameters for the spin current and diffusion, respectively. The equation of motion for the local magnetisation is given by: dMd dMd ¼ gMd ðHeff þ jmÞ þ aMd , dt dt

(117)


197

where the effective field, Heff, will include external and internal field contributions. The two sets of equations of motion can be considered independently since their timescales differ by about three orders of magnitude; spin accumulation: tsf e1012 second; local moment (for fields on the order of 0.1 T) the timescale is of the order of nanoseconds. Separating the spin accumulation into longitudinal and transverse components reveals that former decays on a length scale of around 60 nm (in Co) compared to the transverse spin accumulation which is only a few nanometres. In considering the transverse component of the spin ð1Þ accumulation in a two FM layer system, the term jm? ¼ aMð2Þ d Md þ ð1Þ ð2Þ ð1Þ bðMd Md Þ Md is used, where a and b are constants related to geometric factors of the multilayer. This gives the equation of motion: dMð1Þ dMð1Þ ð2Þ ð1Þ ð2Þ ð1Þ ð1Þ d d ¼ gMð1Þ . d ðHeff þ bMd Þ gaMd ðMd Md Þ þ aMd dt dt (118)

The term in bðMð1Þ Mð2Þ Þ relates to the torque due to the effective d ð2Þ ð1Þ d ð1Þ field from bMd and aMd ðMð2Þ d Md Þ is the spin torque. The findings of this model show that the longitudinal spin accumulations plays no role in magnetisation dynamics and only the transversal component has importance in magnetisation switching as does the effective field bMð2Þ d . Evaluation of the constants a and b reveal that the former is about twice the value of the latter (Zhang et al., 2002). Xi and Shi (2004) studied the transient behaviour of the magnetisation under the action of a spin-polarised current for which they derive a phase diagram of magnetisation states. The LL equation of the form of Eq. (101) is rewritten as: 1 þ a2 @m ¼ m F am ðm FÞ, g @t

(119)

where F ¼ Heff m I. In the case where aHeff ¼ I the second term vanishes and the magnetisation precession will persist without damping. Thus the spin current will effectively counter balance the damping. Inversion will then occur for the condition: I4aHeff . These authors consider the simple case where there is an easy-plane anisotropy with z being the hard axis. This formulation uses a spin-current vector of the form: I¼

_ ZI _ ZJ gðWÞ^z gðWÞ^z ¼ 2e M s V m 2e M s l

(120)

and has a similar form to that given above. The factor gðWÞ is the angular dependence of the torque contribution from the spin current and has a similar function to gðnm ; ns Þ in Eq. (98). The functional dependence of this factor is taken from Slonczewski (2002) who provides a generalised model

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for the spin-torque effect in magnetic multilayers (from which the critical currents can also be derived) and can be written in the form: gðW; LÞ ¼

L2 . L cos2 ðW=2Þ þ sin2 ðW=2Þ 2

(121)

This is consistent with the model of Stiles and Zangwill (2002). The pconstant L is related to the ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Sharvin pffiffiffi conductance: L ¼ AGðRþ þ R =2Þ, where G ¼ e2 k2F = 3ph is 2= 3 of the Sharvin ballistic conductance per unit area of the point contact, A is the area and R are the resistances of the two spin channels. Since the spin-torque effect is proportional to this parameter, the effect will increase for increased polar angle in the case where L41. Transforming the LLG equation into spherical coordinates: 1 þ a2 dW _ ZI ¼ aH eff sin W cos W þ gðW; LÞ sin W; g dt 2e M s V m 1 þ a2 dj _ ZI ¼ H eff cos W þ a gðW; LÞ: g dt 2e M s V m

(122)

These can then be solved to study the magnetisation dynamics due to the spin current. Examples are illustrated in Fig. 2.55. In the absence of a spin current, the magnetisation will lie in the easy direction, somewhere in the x2y plane and the effect of a spin current will be to shift it to higher polar angles, W, away from the equatorial initial position. A phase diagram of the magnetisation state is shown in Fig. 2.56. This depicts the parameters B ¼ ZI_=2eM s V m against L. The inset of this figure shows the variation of the critical field with L. We note pffiffiffi that the magnetisation will lie along the z-direction (W ¼ p) for L 3 and will be a stationary state above the critical line. Below this line, the magnetisation is in a precessional steady state. This model allows the evaluation of the transition time from the stationary state, before the spin current is applied, to the stable precessional state. Defining this transition time as the time that the magnetisation takes to make a 95% of the movement from equilibrium, the transition time will be dependent on the parameter B, the effective field and the damping parameter as well as the constant L. For the case of L ¼ 1, the transition times are approximated as: 1 þ a2 aH eff ts ¼ 3 g ðaH eff Þ2 B2

for

BoaH eff

(123a)

and ts ¼

5 1 þ a2 1 g 2 B aH eff

for

B4aH eff

(123b)

199


(a) 0 -0.2

z

-0.4 -0.6 -0.8 -1 1 1

0.5 0 y

0.5 0

-0.5 -1

-0.5

x

-1

(b) 0 -0.2

z

-0.4 -0.6 -0.8 -1 1 0.5 y

0 -0.5 -1

-1

-0.5

0.5

0

1

x

Figure 2.55 Trajectories of the magnetisation (a) from a stationary state in the x--y plane to a steady precessional state when a current is on and (b) from the precessional state back to a stationary state when the current is off. The difference between these two transitions is clearly indicated. (Reprinted with permission from Xi and Shi (2004), r 2004 American Institute of Physics.)

More complex relations will exist for the case where La1. When B is not too near the critical current value, see arrows in Fig. 2.57, where a divergence occurs and is intrinsic to the LLG equation. These authors report further on this model in Xi and Lin (2004) and Xi et al. (2004), where the application to microwave generation is considered as being a direct result of the spin-polarised current-induced magnetisation dynamics. Spin-wave theory related to the current-driven magnetisation dynamics has been studied by Rezende et al. (2006), who have shown that the inclusion of non-linear effects due to magnon interactions can account for the stabilisation of the magnetisation precession as well as the frequency shifts

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David Schmool

Figure 2.56 Phase diagram of the magnetisation states. The inset shows the critical angle Bc as a function of L. (Reprinted with permission from Xi and Shi (2004), r 2004 American Institute of Physics.)

Figure 2.57 Transition time for the magnetisation to take from a stationary state in the x--y plane to a final steady state since a current is turned on. The numbers shown in the figure are the values of L. The ticks above the figure show the critical spin currents where the transition time diverges. For a ¼ 0.01 and Heff ¼ 5000 Oe, a unit of time in the figure is 1.08 nanoseconds. (Reprinted with permission from Xi and Shi (2004), r 2004 American Institute of Physics.)

201


that occur with the increase of the DC. This predicts a shift to lower frequencies (red shift) with increasing currents when the external field is applied in the plane of the film, while a shift to higher frequencies (blue shift) when the field is applied perpendicular to the film plane. Such shifts can be interpreted as a simple reduction of the magnetisation due to the excitation of magnons. See also calculations by Slavin and Kabos (2005) who discuss the theory of microwave generation by current-induced magnetisation dynamics.

4.2. Experimental observation of spin-current-induced magnetisation dynamics Many groups have reported on the experimental observation of currentinduced magnetisation dynamics in magnetic nanostructured layered systems. We can divide experimental measurements into two types with regards to whether the current applied is alternating or continuous. We will consider some of the representative data in the literature in the following text. Probably the first experimental confirmation of current-induced magnetisation dynamics was performed by Tsoi et al. (1998) using Co/Cu multilayers. Peaks in the dV/dI(V) curve shift with the variation of an applied magnetic field in the direction perpendicular to the layers, see Fig. 2.58, where HWHs (the saturation field). The magnetodynamic origin 45

-24

V (8T) 44

(7T)

-18

dV/dl(Ω)

43

V (mV)

(6T) (5T)

42

(3T) (2T)

41

-12 -6 0

0

2

4 μ0H(T)

6

8

40 39 38 -0.03

-0.02

-0.01

0.00 V(V)

0.01

0.02

0.03

Figure 2.58 The point contact dV/dI(V) spectra for a series of magnetic fields (2, 3, 5, 6, 7 and 8 T) revealing an upward step and a corresponding peak in dV/dI at a certain negative bias voltage V(H). The inset shows that V(H) increases linearly with the applied magnetic field H. (Reprinted figure with permission from Tsoi et al. (1998), r 1998 by the American Physical Society.)

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of this measurement was confirmed using microwave resonance studies. The observations are considered in light of the conservation of energy and angular momentum. To simplify the situation the authors justify using a single magnetic layer since HWHs and the magnetic moments will be parallel. The electron transport being diffusive, the chemical potential, m, may be defined at any point across the contact. For the two (spin) current model, where spin-up and spin-down electrons carry current independently in parallel, two chemical potentials are defined for the two spin channels. When a current passes from the NM to the FM layer, its distribution must change since there is a difference (spin gap) in the FM metal: Dm ¼ m" m# , and will have a dependence on the spin-diffusion length, which can be expressed as (van Son et al., 1987): Dm ¼ eJ

2ð2aF 1ÞðLN =sN ÞðLF =sF Þ , ðLF =sF Þ þ 4aF ð1 aF ÞðLN =sN Þ

(124)

where J is the current density, sN;F are the conductivities of the NM and FM layers with spin-diffusion lengths LN;F and aF is the bulk spin asymmetry in the FM layer. Using the constants for Co and Cu, DmðeVÞ ¼ 1:4 1012 J ðA=cm2 Þ. Only above a critical current density, where DmðJÞ _oðHÞ, will the emission of a spin wave be possible. In this case very high values of H will lead to a suppression of excitations since DmðJÞo_oðHÞ. The critical current density was obtained as JcE1.4 108 A/cm2. The asymmetry of the I(V) relation is understood in terms of the conservation of energy, which requires m" 4m# , the resulting sign of Dm ¼ m" m# o0 requires that Jo0. A year later Wegrowe et al. (1999) reported on the current-induced magnetisation reversal in single Ni magnetic nanowires of diameter B80 nm and length 6000 nm. Magnetoresistance (MR) measurements in the absence of a current show a switching field (Hsw) of 0.43 kOe. The same MR measurements were then performed using 200 nanoseconds current pulses of 0.15 mA (107 A/cm2) and a lower switching field of, H isw , 0.33 kOe is obtained. When a current is injected at a field between Hsw and H isw a jump in the magnetisation is observed. The maximum switching field variation DH max ¼ jH isw H sw j was measured as a function of the direction of the applied field. The results are interpreted in terms of the effect of a spin-polarised current on the magnetisation, where it is noted that when the conduction electrons experience a change in the magnetisation over a distance smaller than the spin-diffusion length, LF , then the polarised conduction electron spins relax (Gregg et al., 1996; Levy and Zhang, 1997) and the current will produce a rotation, Dj, of the magnetisation from its position, j, in the absence of the current (Bazaliy et al., 1998). In terms of the magnetisation reversal, the magnetic moment of the nanowire reversibly follows the external field up to the switching

203


field where a jump in the magnetisation occurs, corresponding to a critical angle, jc. If a current is injected at fields H isw corresponding to the magnetic state defined by angle j ¼ jc Dj, then a jump occurs. The maximum distance DH max ¼ jH isw H sw j will then be a measure of Dj produced by the current. The nature of the switching can have important consequences on the switching fields measured experimentally and a consideration of whether this occurs uniformly or in a more complex manner is required. To study the switching field, Wegrowe et al. (1999) use the following expression (Aharoni, 2000): aða þ 1Þ hsw ðWÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 a ð2a þ 1Þcos2 W

(125)

in which the switching field has been normalised: hsw ðWÞ ¼ HðWÞ=H K and a is a fitting parameter given by: a ¼ AðR0 =rÞ2 , where A is a geometrical parameter and R0 is the exchange length. Using this model, the relationship between the direction of the applied field, W, and that of the magnetisation, j, can be written as: tan W ¼ ½ða þ 1Þ=a tan j. The effect of a current is then envisaged as causing a variation Dj in the magnetisation and if this is sufficient to cause switching (j þ Dj jc ), hsw is shifted to hisw , such that hisw ¼ hsw ðWÞ þ Dhsw ðWÞ. Such considerations then lead to the variation of H sw ðWÞ for the Ni nanowire system studied as shown in Fig. 2.59. where aE0.28, corresponding to an activation radius of r ¼ 30 nm. In another early study, Katine et al. (2000) considered a nanopillar system consisting of a thin (free) Co layer (Co1) of thickness 2.5 nm and a thicker 1.4

1.2 Hsw (kOe)

Hsw curling fit 1 Hlsw measured

0.8

Δϕ=1.1′

0.6

0.4 180

200

220 θ′

240

260

Figure 2.59 Angular dependence H sw ðWÞ of the switching field without pulse and angular dependence H isw ðWÞ of the minimum switching field with pulsed current. Grey line: H sw ðWÞ fitted with Eq. (125). Dashed line: H isw ðWÞ fitted with hisw ¼ hsw ðWÞ þ Dhsw ðWÞ assuming Df ¼ 1:1 (Wegrowe et al., 1999).

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David Schmool

V−

I− Cu Au

e− (positive bias)

Co 1 Co 2 Cu I+

V+

Figure 2.60 Schematic of pillar device with Co (dark) layers separated by a 60 Å Cu (light) layer. At positive bias, electrons flow from the thin (1) to the thick (2) Co layer. (Reprinted figure with permission from Katine et al. (2000), r 2000 by the American Physical Society.)

(fixed) Co layer (Co2) of 10 nm thickness, which are separated by a 6 nm Cu spacer layer. The lateral dimension of the nanopillar was 130 nm. A fourpoint probe electrical measurement set-up was used, as illustrated in Fig. 2.60. The lateral confinement allows the current density to attain relatively high values (1072108 A/cm2) and thus establish a steady constant current non-equilibrium condition in the structure. The results are typically given as a differential resistance, dV/dI, which is then measured as a function of the applied current and magnetic field (see Fig. 2.61). For a negative bias, electrons will flow from the fixed Co layer to the free Co layer and stabilise a collinear magnetisation alignment of the two FM layers and will lead to a low differential resistance. For positive biases, the parallel alignment can be destabilised and Co1 can switch to an antiparallel alignment with Co2. This occurs for large currents and is manifest by an increase of dV/dI. Hysteretic behaviour is observed on a reduction of the current, as seen Fig. 2.61, where the Co1 layer switches back to the parallel alignment at low current densities. An external magnetic field is applied to align the fixed magnetisation direction of the Co2 layer. It is noted that the upper curve shows two stable states at zero current, which means that positive and negative current pulses can be used to switch between the parallel and antiparallel states. Further studies by Albert et al. (2000) display this hysteretic behaviour more clearly, where positive and negative critical fields can be observed ðI þ c ; Ic Þ corresponding to the high-resistance and low-resistance state critical currents (see Fig. 2.62(a)). These critical currents can be expressed as: ae M s V m ðH þ H k þ 2pM s Þ, hgð0Þ ae M s V m ðH H k 2pM s Þ. I c ¼ hgðpÞ

Iþ c ¼

ð126Þ

205


H (kOe) 0.6 1.0 1.4 1.8

dV/dl (Ω)

1.9

1.8

I– (mA)

I+ (mA)

-7.9 -1.4 -1.0 3.1

-3.8 -0.2 8.2 9.3

1200 Oe 1.7 1600 Oe 1.6 -20

-10

(a)

0 I (mA)

10

20

R (Ω)

1.64

1.60

-1.0 (b)

0.0

1.0

H (kOe)

Figure 2.61 (a) dV/dI of a pillar device exhibits hysteretic jumps as the current is swept. The current sweeps begin at zero; light and dark lines indicate increasing and decreasing currents, respectively. The traces lie on top of one another at high bias, so the 1200 Oe trace has been offset vertically. The inset table lists the critical currents at which the device begins to depart from the fully parallel configuration (I + ) and begins to return to the fully aligned state (I ). (b) Zero-bias magnetoresistive hysteresis loop for the same sample. (Reprinted figure with permission from Katine et al. (2000), r 2000 by the American Physical Society.)

Here g(0) and g(p) depend on the spin-dependent transmission probabilities of the FM2NM interfaces, its relative orientation (0 ¼ parallel, p ¼ antiparallel) of the nanopillar and any spin-flip scattering that may occur in the system. The authors have found a good fit to experimental data using g(0) ¼ g(p) ¼ 0.25. The variation of the critical currents with applied field is shown in Fig. 2.62(b). The critical current I c , for switching from antiparallel to parallel alignment shows a weak dependence on the field until it reaches a value that forces the system into alignment even in the absence of a current. The other critical field, I þ c , shows a more linear variation with H. The high- and low-resistance states can also be observed by scanning the magnetic field, see Fig. 2.63, which shows the differential resistance versus H curve. Starting from zero field, as the field increases, the nanopillar begins in a low-resistance state. When the field reaches a critical field Hc1 (B750 Oe), there is a jump to a high-resistance state, which corresponds

206

David Schmool

Figure 2.62 (a) dV/dI of a nanopillar spin-transfer device as a function of the applied current through the device. The current is defined as positive when the spinpolarized electrons are flowing from the nanomagnet to the thick Co film. (b) The critical currents I þ c and I c that switch the nanomagnet into, respectively, antiparallel and parallel alignment with the Co film as a function of H. This data was taken with H increasing from zero to its maximum value of 1100 Oe. (Reprinted with permission from Albert et al. (2000), r 2000 American Institute of Physics.)

to a reversal of the Co1 layer, leaving the nanopillar system in the antiparallel configuration. With increasing magnetic field, the resistance gradually decreases, indicating the nucleation of reversal domains. At a second critical field, Hc2 (B7900 Oe), a second jump in the differential resistance is observed. This corresponds to the reversal of the Co2 layer, bringing the nanopillar into the parallel, low-resistance state. From these results, it is seen that the resistance response is in general hysteretic with respect to both the applied magnetic field and the bias current. Sun et al. (2003) have shown this is an unequivocal way in nanostructured spin valves. Figure 2.64 shows grey-scale images of the current-field characteristics, where dark regions correspond to low resistance and light regions to high resistance, clear boundaries are seen between the high- and low-resistance states. Directions of the current sweep and field stepping as indicated with arrows in the upper

207

dV/dl (Ω)


1.58 1.56 1.54 -1000

0

1000

H (Oe)

Figure 2.63 The differential resistance dV/dI as a function of magnetic field H applied parallel to the long axis of the nanomagnet. (Reprinted with permission from Albert et al. (2000), r 2000 American Institute of Physics.)

10

10

5

5

Ic (mA)

Ic (mA)

0.05×0.10mm2, easy axis field. Ambient temperature

0 −5

−5

−10

−10

−1500 −1000 −500

(a)

0

500

1000

−1500 −1000 −500

1500

(b)

H (Oe)

10

5

5

0

−5

−10

−10

−1500 −1000

−500

0

H (Oe)

500

1000

500

1000

1500

0

−5

(c)

0

H (Oe)

10

Ic (mA)

Ic (mA)

0

−1500 −1000

1500

(d)

−500

0

500

1000

1500

H (Oe)

Figure 2.64 Current-induced switching as a function of magnetic fields. The grey scales represent the value of junction resistance. The light colour corresponds to high resistance, and the dark colour to low resistance. It is a representation of the junction R(I,H), where R is the junction resistance, I the bias current and H the applied magnetic field. These measurements were done by sweeping the bias current I for one complete cycle in a fixed applied field H, and then stepping H to complete a full cycle from Hmin to Hmax back to Hmin. The arrows in each plot describe the direction of current and field sweeps. Data taken from sample with junction size: 0.05 0.10 mm2. (Reprinted with permission from Sun et al. (2003), r 2003 American Institute of Physics.)

208

David Schmool

40

3

0

20

P/I2, pW/(mA2GHz)

f, GHz

f, GHz

12

2.4 H,kOe

0.15

0 0

10 H,kOe

(a)

0 40

H=7.5 kOe

20

0

0

10 2.2

log10[(P/PJN)(1mA/I)2] 40

H=4.2 kOe

20

I,mA

-0.8 (d)

H=4.2 kOe

20

0

0 1

I,mA

-0.4 40

0

10 0.6

log10[(P/PJN)(1mA/I)2]

(e)

40

H=2.8 kOe

20

I,mA

10

-0.8 2.2 log10[(P/PJN)(1mA/I)2]

(f)

f, GHz

f, GHz

H=7.5 kOe

20

10

f, GHz

f, GHz

I,mA

-1.4 0.6 log10[(P/PJN)(1mA/I)2] 40

H=2.8 kOe

20

0

0 1

(g)

18

0 1

(c)

9 f, GHz

(b)

f, GHz

f, GHz

40

0

20

I,mA

10

-0.8 0.8 log10[(P/PJN)(1mA/I)2]

0

(h)

I,mA

10

-0.8 2.2 log10[(P/PJN)(1mA/I)2]


209

right-hand corners of the figures. It is seen that the current switching threshold has a dependence on the value of the bias field but is independent of the sample bias-field history. There are three main regions in these plots: (i) hysteretic switching between parallel and antiparallel states, (ii) telegraph noise, typically involving two-level fluctuations, which arise from thermally activated transitions between two metastable states and (iii) a reversible step in the IV curve is seen (high-field and high-current density). The dynamics in nanopillars of Co/Cu/NiFe due to spin torque were studied as a function of current and magnetic field by Kiselev et al. (2004). The dynamics are measured from the resistance changes measured as a voltage variation from a constant DC injected through the sample. Since the precessional motion of the magnetisation causes changes in the sample resistance, the output signal will be a time-varying voltage with microwave frequencies. This can be used for the basis of a nanoscale microwave generator. The frequency of the output can be varied by the application of an external magnetic field, as expected from FMR theory and is shown in Fig. 2.65(a). Small-amplitude oscillations agree well with the Kittel model. A series of illustrations are shown in Figs. 2.65(c)22.65(h) comparing experimental results and simulations for selected values of the applied field. The signals are normalised by subtracting the Johnson noise background (see Foros et al., 2005) and dividing by I2. By plotting the positions of steps and kinks in the frequency and peaks in dV/dI the authors produce a phase diagram, which is compared with the simulation model of their system (Fig. 2.66). Similar results using Co/Cu/Co layered nanostructures are reported by Kiselev et al. (2003). Again small-amplitude oscillations agree with the Kittel formula and estimates of the precessional angles are made based on the integrated microwave power measured about f and 2f (Pf and P2f). Using the assumption WðtÞ ¼ Wmis þ Wmax sinðotÞ for the time-varying polar angle, where Wmis is the precession axis and the magnetic moment axis and Wmax is the pressional angle and that the angular dependence of the resistance change is give by: DRðWÞ ¼ DRmax ð1 cos WÞ=2, the anisotropic Figure 2.65 (a) Measured small-amplitude signal frequency for perpendicular H (circles) compared to numerical simulations of the LLG equation (black curve). The red line is a linear fit to the Kittel formula. Inset: The small-amplitude signal frequency for in-plane H, with a fit to f ¼ gmB ðH 4pM Py Þ=2p. (b) Spectra for H ¼ 18 kOe perpendicular to the sample plane, for I ¼ 2.29.2 mA, in increments of 1 mA (offset vertically). (c), (e), (g) Colour scale: Measured spectra as a function of I at selected values of H perpendicular to the sample plane. The spectra are normalised as discussed in the text. PJN is the magnitude of room-temperature Johnson noise. White lines show dV ¼ dI, plotted using different vertical scales. (d), (f), (h) Spectra predicted by the simulation described in the text. The diagrams illustrate the dynamical modes of the free layer within the simulations, for the values of I marked by arrows. The vertical direction is normal to the sample plane. (Reprinted figure with permission from Kiselev et al. (2004), r 2004 by the American Physical Society.)

210

David Schmool

20

20

H, kOe

SLR

SLR

SHR

10

10

0

0 0

(a)

SHR

2

4

6

8

10

0 (b)

I, mA

2

4

6 I, mA

8

10

Figure 2.66 Experimental room-temperature stability diagram for perpendicular H. Filled symbols indicate features that are reproducible between samples. Open symbols show transitions that can differ in number and position between samples. SLR (SHR) denotes a region of static low (high) resistance. Squares indicate the onset of small-angle precession. Triangles denote the position of frequency steps. Diamonds show kinks in the dependence of f on I. Circles denote points where the microwave signal drops below measurable values. Lines indicate peaks in dV ¼ dI. (b) Theoretical room-temperature stability diagram obtained by numerical solution of the LLG equation with the Slonczewski spin-torque term (Slonczewski, 1996). Solid lines separate SLR, SHR and dynamical states (precession angles W21). Dotted and dashed lines show the transition from small-angle precession about I ¼ 0 equilibrium angle to large-angle precession about H, associated with a step in f (dashed line) or a kink in f versus I (dotted line). At the 1 microseconds timescale in the simulations, transitions between the SLR and SHR states are hysteretic ¨ zyilmaz et al., 2003), with the boundary given by the black line for increasing (O I and the grey line for decreasing I. (Reprinted figure with permission from Kiselev et al. (2004), r 2004 by the American Physical Society.)

magnetoresistance (AMR), where jWmis Wmax j 1, the authors obtain: W4max

512P 2f R DR2max I 2

and

W2mis

32P f R , DR2max I 2 W2max

(127)

where DRmax corresponds to the resistance change from the parallel to the antiparallel alignments. Rippard et al. (2004) have studied spin-valve structures [Ta(2.5 nm)/ Cu(50 nm)/Co90Fe10(20 nm)/Cu(5 nm)/Ni80Fe20(5 nm)/Cu(1.5 nm)/Au(2 nm)] attached to lithographically defined point contacts of nominal diameter 40 nm. The CoFe layer is fixed with respect to the NiFe layer, due to its larger volume, exchange stiffness and magnetisation. The sample is electrically connected via a 50 O planar waveguide to microwave probes and a DC can be injected via a bias tee along with a 20 mA alternating current (AC) (500 Hz). This allows the simultaneous measurement of DC resistance, differential resistance and microwave output. Detection of the


211

Figure 2.67 (a) dV ¼ dI versus I with l0H ¼ 0.1 T. (b) High-frequency spectra taken at several different values of current through the device, corresponding to the symbols in (a). Variation of f with I (inset). (Reprinted figure with permission from Rippard et al. (2004), r 2004 by the American Physical Society.)

alignment changes of the two FM layers is made using a current bias to give a voltage change across the point contact. The differential resistance versus current, with an applied magnetic field of 0.1 T, is shown in Fig. 2.67(a). The peak is interpreted as evidence of current-induced magnetisation dynamics (Katine et al., 2000; Myers et al., 1999; Wegrowe et al., 1999). On varying the current through the spin valve, the frequency of the response undergoes a linear (red) shift, as shown in Fig. 2.67(b). The authors evaluate that if the signal results from an MR response, the excursion angle between the magnetic layers is around 201. Frequency2field measurements have been made, where the data fit well with the Kittel equation: nðHÞ ¼

gmB m0 ½ðH þ H sw þ H h þ M eff ÞðH þ H sw þ H h Þ 1=2 h

(128)

and H sw ¼ Dk2 =ðgmB m0 Þ, where D is the spin-wave constant which depends on the exchange stiffness, k ¼ 2p=l is the magnon wave vector and Meff is the effective magnetisation. The fit is shown in Fig. 2.68(a) where the

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David Schmool

fitting parameters are given. Spectra taken over a wide frequency range display a further peak at 2n. This results from a misalignment of a few degrees between the layers. Amplitude ratios of the n and 2n lines are shown in the inset of Fig. 2.68(b). Fuchs et al. (2007) have studied the effects of the current on the damping parameter, a, which built on previous studies of spin transferdriven FMR (Sankey et al., 2006). In these measurements, the experiments are performed by passing an RF and a DC through the magnetic multilayers, with a resistance change being recorded as a function of the frequency of the RF current. The resonance lines are then fit using a twoterm Lorentzian-containing symmetric and antisymmetric terms. The latter arises when the spin-torque vector is not in a principal plane of the anisotropy tensor or from a component of the effective field component of

Figure 2.68 (a) In-plane f versus H dispersion curve along with a fit to Eq. (128). Error bars (FWHM) are smaller than the data points. (b) Frequency spectra for I ¼ 5--9 mA in 0.5 mA steps with l0H ¼ 0.06 T showing responses at both f and 2f. Inset: f to 2f amplitude ratios as a function of I for two different fields. (Reprinted figure with permission from Rippard et al. (2004), r 2004 by the American Physical Society.)

213


the spin torque perpendicular to the magnetisation of one or both magnetic layers. The relative strength of the two components will be a measure of the line asymmetry. The damping coefficient is given as a ¼ D0 =Dk , where D0 is the linewidth of the Lorentzian lineshape, obtained from fitting the data, and Dk ¼ gM s ðN 0y N 0z Þ, which are related to the demagnetising factors Nx, Ny, Nz by: N 0y ¼ ðN y N z Þðcos2 j sin2 jÞ þ

cosðc jÞH ext 4pM s

and

(129)

cosðc jÞH ext : N z ¼ ðN z N x cos2 j N y sin2 jÞ þ 4pM s

For the thin film geometry used in this study N x N y 0 and N z 1. Equations (129) describe the in-plane and out-of-plane anisotropy factors, respectively. Experimental results show a linear behaviour of the damping parameter with the DC strength (see Fig. 2.69). Damping constants of a ¼ 0.014 for CoFeB and 0.01 for permalloy agree broadly with FMR measurements of thin films of these materials showing

Figure 2.69 (a) ST-FMR data for the CoFeB sample taken at Happl ¼ 200 Oe, Irf ¼ 0.18 mA and IDC ¼ --2.0 mA. The solid line is a Lorentzian fit. (b) Plot of the effective damping a versus IDC for the CoFeB sample. The dashed line is a linear fit. (c) ST-FMR data for the Py sample taken at Happl ¼ 200 Oe, Irf ¼ --0.035 mA and IDC ¼ --0.5 mA. (d) Effective damping for the Py sample. (Reprinted with permission from Fuchs et al. (2007), r 2007 American Institute of Physics.)

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3.5

320

3.0

300

2.5

280

2.0

260

1.5

240

1.0

220

0.5

200

0.0

−6

−4

−2

0 2 i (mA)

4

6

−6

−4

−2

0 2 i (mA)

4

Linewidth (MHz)

Amplitude (10−15 Hz−1)

that the ST-FMR measurements only have additional damping due to the spin-torque effect with the size of the DC. Such results indicate the potential for using the switching process from the spin-torque effect to control the data storage for magnetic random access memory (MRAM) devices (Cui et al., 2008). More recent studies have considered the effects of spin torque in magnetic tunnel junctions (MTJ). In one study, Petit et al. (2007) have observed spin-transfer torque effects at current densities well below the critical value; where excitations are observed in the range: jc =20 j jc =4. The magnetic fluctuations are either suppressed or enhanced by a DC bias, depending on the sign of the bias. These effects are related to the spin angular momentum transfer torque of the FMR and have a Kittel-like behaviour; linear relationship between the square of the frequency and the applied field. Similar results were also obtained by Synogatch et al. (2003) and Stutzke et al. (2003). There is a strong asymmetry of line amplitude with the sign of the direct bias current, which is related to the spin torque favouring the parallel magnetic configuration. The linewidth shows a linear dependence with the DC; Fig. 2.70 shows both the amplitude and linewidth dependence on the direct bias current for the antiparallel state, where the extrapolation of the linewidth to zero will yield the critical current, which corresponds to a current density of 4 107 A/cm2. An asymmetry in the frequency shift about the zero current is related to Joule heating effects in the MTJ. Sankey et al. (2008) have also studied the spin-torque effect in MTJ. Spin transfer has also been found to produce the vortex dynamics discussed in the previous section in metallic nanocontacts (Mistral et al., 2008). Such orbital motion of the vortex core of the nanocontact shows

6

Figure 2.70 Peak amplitude (left) and linewidth (right) as a function of the current bias in the AP state (200 Oe). The critical current is estimated both from the amplitude divergence and from the linewidth extrapolation to zero. Both methods lead to 2075 mA. (Reprinted figure with permission from Petit et al. (2007), r 2007 by the American Physical Society.)


215

Figure 2.71 (a) Colour map of experimental PSD for l0H ¼ 350 mT. Solid squares are results of micromagnetics simulations. (b) Top view of the vortex magnetisation profile in the free layer obtained from simulation. (c) In-plane component of magnetisation (black for my ¼ + 1 and white for my ¼ 1) for the entire simulation area for I ¼ 30 mA. Also shown are the contact and the vortex orbit (158 nm in radius). (Reprinted figure with permission from Mistral et al. (2008), r 2008 by the American Physical Society.)

a frequency dependence on the applied current and magnetic field, as seen in Figs. 2.71 and 2.72. The voltage (electrical) detection of magnetisation dynamics offers another method for the study of magnetisation dynamics in nanoscale magnetic systems and is derived from the spin-pumping effect (Costache

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David Schmool

0.5

0.45

Frequency (GHz)

30 mA 33 mA 36 mA

105 mT 210 mT 315 mT

0.4

0.35

0.3

0.25

0.2 (a)

28

30

32

34

36

Current (mA)

38

100

200

(b)

Field (mT)

300

Figure 2.72 Vortex oscillation frequency as a function of (a) applied current and (b) applied field. Dots are experimental data, and solid lines are fits to Thiele’s model (Thiele, 1973). (Reprinted figure with permission from Mistral et al. (2008), r 2008 by the American Physical Society.)

et al., 2006). The voltage signal is related to the precessional angle, W, and arises from the AMR, giving an approximate (small-angle precession) value of (Wang et al., 2006b): V dc ¼

Zg"# o W2 _o, 2eð1 Z2 Þgo

(130)

where Z is the spin polarisation and g"# o defines the spin-up and spin-down conductances with go ¼ g"o þ g#o . Good agreement between the model and experiment are found for the permalloy2aluminium interface studied by Costache et al. (2006). Further to the ferromagnetic2metal contact, Moriyama et al. (2008) have studied a tunnel barrier structure (AlOx/NiFe), where an unexpectedly large voltage output was observed (of the order of microvolts). This is a surprising result at face value, since the insulating barrier might be expected to suppress the spin-pumping mechanism which will depend on the interface transparency. The electrical measurements display FMR in the single magnetic element which fit nicely into the Kittel scheme and is manifest as a DC voltage which arises only from the AlOx/NiFe interface. To interpret their results, Moriyama et al. (2008) consider the spin current pumped into a NM layer at the interface with a FM metal, due to the steady-state precession of the magnetisation, which


217

takes the following form (Brataas et al., 2002; Tserkovnyak et al., 2005): Ipump s

_ dm "# "# dm Reg m þ Img . ¼ 4p dt dt

(131)

For transparent FM/NM contacts, the imaginary part of the spinmixing conductance can be neglected, while this is not the case for opaque contacts, where Reg"# =g 0:5, see also Turek and Carva (2007). For the case where the spin2flip relaxation rate is smaller than the spin injection rate, the injected spin current is accumulated in the NM layer (at the FM/NM interface) driving a back flowing spin current into the (precessing) FM layer. The spin accumulation will then give rise to a voltage drop across the FM/NM interface. In the studied structure Al/AlOx/NiFe/Cu tunnel junction, the spin-mixing conductance will be governed by the I/FM interface and spin accumulation will occur in the Al layer. While the experimental results show the correct behaviour as expected by the theory, some discrepancies in values are obtained in the fitting process. The authors speculate as to the role of dynamic processes in the FM layer and effects of electron2electron interactions in producing the unexpectedly large voltage drop in the tunnel junction as compared to an FM/NM interface. The role of charge pumping to explain the voltage generation through tunnel structures (FM/I/FM and FM/I/NM) are considered by Tserkovnyak et al. (2008). Since electrons carry both charge and spin, the pumping of these quantities may occur across an interface when the FM layer precesses, where the pumping strength (in the adiabatic limit) is proportional to the precessional frequency (Xiao et al., 2008). It turns out that for a FM/I/NM structure, the pumping shows a purely spin character, while for the FM/I/FM multilayer there is an admixture of spin and charge pumping, although since the spin relaxation is typically greater than the tunnelling injection rate, spin pumping can be neglected. Using a spin-rotation transformation of the Hamiltonian for itinerant electrons in a FM/I/FM structure, the spin and charge pumping are seen to arise from the difference in the spin splitting in the two FM layers; one layer has a splitting of _o, while the other has a reduced splitting of _o cos W, due to the precessional motion of its magnetisation (with precession angle W). The charge current due to non-equilibrium spin accumulation (from one layer to the next) is given by: I 1!2 ¼ ejT j2

_o 2 ðD" D2# Þcos2 ðW=2Þ, 2

(132)

where T is related to the orbital component of the tunnelling matrix element and D"ð#Þ is the density of states for the spin-up (-down) electrons. The voltage relation is then obtained as: V ¼

_o Zsin2 W , 2e 1 þ Z2 cos W

(133)

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David Schmool

where Z is the spin polarisation. The voltage can be seen when Z ¼ 1 and W ! p, that is for antiparallel alignment between the magnetic layers. For the FM/I/NM device, the corresponding tunnel current is slightly modified, having the form: I 1!2 ¼ ejT j2

_o DNM ðD" D# Þ cos W, 2

(134)

where DNM is the density of states in the NM. The Bloch equation of motion is used to assess the spin density and hence spin accumulation: m

ð_o sin WÞ2 , D

(135)

where D is the exchange splitting along the spin direction, m. The measured voltage is then given as V Zm=2e. Tunnel barriers of the FM/I/FM type support charge pumping and will change sign on magnetisation reversal of one of the FM layers. However, the FM/I/NM structures produce spin pumping and accumulation and are symmetric under magnetisation reversal of one of the FM layers.

5. Superparamagnetic Effects in Magnetic Nanoparticles and FMR It is well known that fine particle systems suffer from SPM effects, which depend on the interdependence of the particle size and anisotropy strengths. Such behaviour is characterised by the thermal fluctuation of the spontaneous magnetisation. As such the SPM properties of nanoparticles and structures will be intimately related to their relaxation properties. An outline of relaxation and its relation to the resonance linewidth is discussed in Section 2.6. As the particle temperature increases, the thermal energy, kBT, becomes sufficient to overcome the energy barrier, EB, defined by the magnetic anisotropies and particle size. (The physical basis of this effect is outlined above, Section 3.1, in the SW model for sdp.) Thermal excitations induce rapid fluctuations of the particle magnetic moment, in the simplest case, are described by the Arrhenius law (Neél, 1949):

EB . t ¼ t0 exp kB T

(136)

Typically EB is taken to be KeffV, where Keff is an effective anisotropy and V the particle volume. The characteristic time t0 depends on various parameters (this can be related to Fermi’s golden rule, as discussed in Section 2.6); temperature magneto-gyric ratio, magnetisation, energy

219


barrier, direction of applied field and damping constant. Most authors take t0 to be constant for simplicity (with a value of B1011 seconds), where this assumption along with Eq. (136) is known as the Neél2Brown model. If the time window of an experimental measurement, tm, is shorter than t at a fixed temperature, the particles magnetic moment remains blocked during the observation, this is the so-called blocked regime, where an assembly of identical non-interacting particles would appear ferromagnetic. Since this regime is temperature dependent, a blocking temperature, TB, is defined as that temperature below which the sample appears ferromagnetic, and will be specific to the measurement technique. At this temperature, t ¼ tm and the blocking temperature can be defined from Eq. (136) as: TB ¼

EB 1 EB 0:43 , kB lnðtm =t0 Þ kB

(137)

where we have used tm ¼ tFMR 1010 seconds. In Fig. 2.73, the effect of size distribution is illustrated where a distribution of blocking temperatures is predicted for FMR (SQUID magnetometry by Antoniak et al., 2005). It will be noted that this distribution arises from the volume distribution given in Eq. (2) and that TBWTmax, which arises from polydispersion effects. Above the blocking temperature, the rapid fluctuations produced by thermal excitations will mean that the particles’ magnetic moment is reversing between local minima so rapidly that its behaviour mimics atomic paramagnetism, and the particle is said to be in the SPM state.

fraction

SQUID

(TBFMR ) FMR 10 nm

0

(TBSQUID )

100

200 TB (K)

300

400

Figure 2.73 Distribution of blocking temperatures for two different time windows. The fraction of particles was calculated for the time windows sSQUIDE102 seconds, sFMRE1010 seconds using the size distribution determined from TEM image analyses. The inset shows a typical TEM image of the nanoparticles (Antoniak et al., 2005).

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David Schmool

Introducing interactions between the particles can be shown to modify Eq. (136) to the form of a Vogel2Fulcher law (Dormann et al., 1988):

EB t ¼ t0 exp kB ðT B T 0 Þ

(138)

in which T0 is an effective temperature proportional to H 2i , and increases with interaction strength in the weak interaction limit. Hi is the effective interaction field. For further discussion on the effects of interactions see the review by Batlle and Labarta (2002). The application of a magnetic field should also modify the Arrhenius relationship, which according to Chantrell and Wohlfarth (1985) yields: EB ð1 H 0 =H K Þ , t ¼ t0 exp kB T B

(139)

where HK is the anisotropy field and H0 a DC applied magnetic field. Models of FMR in nanoparticle assemblies by de Biasi and Devezas (1978), Berger et al. (1997) and Berger et al. (1998) which are valid for strong fields or weak anisotropies (magnetic moment parallel to the applied field, H) in spherical nanoparticle systems represent the resonance condition in the form: o ¼ H þ H A ðcÞ. g

(140)

With the anisotropy field being given as: H A ðcÞ ¼

K ð3cos2 c 1Þ, M

(141)

c being the angle between the anisotropy axis and the magnetisation. In the SPM regime, thermal fluctuations are active and produce a dynamic narrowing of the resonance spectra in which H A ! H SPM ¼ H A ðcÞhP 2 ðcos WÞi ¼ H A ðcÞ A

½1 3LðxÞ=x

LðxÞ

(142)

hP 2 ðcos WÞi is the second-order Legendre polynomial (P 2 ðyÞ ¼ ð3y2 1Þ=2) and L(x) is the Langevin function: LðxÞ ¼ cothðxÞ 1=x, where x ¼ MVH=kB T ¼ mH=kB T . A realistic model of the SPM regime requires that the magnetisation of the NP assembly be defined so as to take into account the effects of size distribution, such that: M ! M SPM ¼

M bulk s

Z

1

LðxÞPðmÞ dm;

(143)

0

where m ¼ MV for the particle, P(m) follows the log-normal variation of Eq. (2) and L(x) is the Langevin function. The model of Kliava and Berger (1999) predicts the spectral function for an assembly of SPM NPs

221


of the form: dP ¼ dH

Z Z Z F½H H r ðV ; W; fÞ; DH PðV Þ sin W dVdWdf. f

W

(144)

V

Dm = 2.0

P(D,nz )

The lineshape function, F ½H H r ðV ; W; fÞ; DH , can be Gaussian or Lorentzian, with Hr being the resonance field and DH the individual linewidth for a particle of a given size. In Eq. (144), P represents the resonant absorption of a particular line, whose intensity can be expressed as R I n ¼ P n dH. The intensity is often expressed in the empirical form: ðnÞ 2 I n ¼ I ðnÞ pp ðDH pp Þ , where pp denotes the peak-to-peak extrema in the derivative absorption spectra (dP/dH vs. H). Figure 2.74 shows some predicted

Dm = 4.0

200 (a)

250

300 350 400 Magnetic Field (mT)

-0.3

nz

-0.3

0.0 nz

-0.3

0.0 nz

0.3

P(D,nz )

5 D (n10 15 m) 20

Dm = 6.0

0.3 0.0

P(D,nz )

Derivative of Absorption

2 4 D (n 6 8 m) 10

10 D (n 20 m)

450

0.3 30

(b)

Figure 2.74 Series of computer-generated X-band (9.288 GHz) SPR spectra for different D, indicated alongside the curves (a) and the corresponding joint distribution densities of diameters and demagnetising factors (b). The simulation parameters are: g ¼ 2.00, M ¼ 310, DM ¼ 45 (kA/m), K1 ¼ 8.0, DK1 ¼ 2.0 (kJ/m3), sD ¼ 0.37, n0 ¼ 0.0, sn ¼ 0.06, q ¼ 0.5 (Kliava and Berger, 1999).

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David Schmool

and experimental curves. The spectra in general display a sharp and a broad resonance which arise from SPM and blocked particles in the polydisperse samples. Dubowik and Baszynski (1986) present an earlier model based on the de Biasi2Devezas SPM function: H spm ¼ Ha a

1 10x1 LðxÞ þ 35x3 LðxÞ . LðxÞ

(145)

Here the resonance field is expressed as: o 1 spm 3 5 ¼ H0 þ Ha þ cos 4j , g 4 2 2

(146)

where j is the angle between the applied magnetic field and the easy axis of the nanoparticles. Berger et al. (2001) have extended their model, again based on the LL equations, where the normalised lineshape takes the form: LLðH; H 0 ; DH Þ ¼

H 20 DH ½ðH 20 þ D2H ÞH 2 þ H 40

. p½H 20 ðH H 0 Þ2 þ D2H H 2 ½H 20 ðH þ H 0 Þ2 þ D2H H 2

(147)

Here the linewidth parameter is given by DH ¼ lH 0 =jgM j. This function differs from the Lorentzian (or Gaussian) function in having a linewidth-dependent resonance field as given by: Hr ¼

H 20 ½2H 0 ðH 20 þ D2H Þ1=2 H 20 D2H 1=2 . H 20 þ D2H

(148)

As such, the resonance field, Hr, will increase with DH. Thermal fluctuations of the magnetic moments will reduce the effect of anisotropic contributions and at elevated temperatures narrow resonance lines are observed. At low temperatures, the thermal fluctuations are gradually frozen out and spectral lines will broaden. In the case of cubic symmetry, the anisotropy field is approximated as: Ha ¼

2K 1 GðxÞ½5ðsin2 Wcos2 W þ sin4 Wsin2 jcos2 jÞ 1 . M

(149)

where the SPM averaging factor is given by (de Biasi and Devezas, 1978): GðxÞ ¼

1 10 35 105 3 . þ LðxÞ x x2 LðxÞ x

(150)

L(x) being the Langevin function defined above. The thermally averaged demagnetising field is given by: Hd ¼

m0 M LðxÞðN == N ? Þð3cos2 W 1Þ. 2

(151)


223

The ellipsoidal particles are considered to have parallel and perpendicular demagnetising factors N// and N>, respectively. The linewidth for an individual nanoparticle has a volume and temperature dependence described by: MV H eff . DH ¼ DT L kB T

(152)

Here DT is a saturation linewidth at temperature T, corresponding to the individual linewidth of the largest particles and the effective field will be the sum of applied, anisotropy and demagnetising fields; H eff ¼ H appl þ H a þ H d . In addition to this the temperature dependence of DT must be taken into account, which arises from the thermal fluctuation-induced modulation of the magneto-crystalline anisotropy, and results in the following dependence: DH ¼ D0 L

MV H eff K 1V G , kB T kB T

(153)

Here D0 is the saturation linewidth at 0 K and G is the function given in Eq. (150). A comparison of theory and experiment (using g-Fe2O3 nanoparticles) for the resonance field and linewidth as a function of temperature is illustrated in Fig. 2.75. A direct comparison of experimental and computer-simulated spectra are shown in Fig. 2.76. The problem of thermal fluctuations in FMR modelling of nanoparticles with uniaxial anisotropy has also been also studied by de Biasi et al. (2003). In this case, the authors consider only shape anisotropy for particles which are ellipsoids of rotation. As such only the Zeeman and shape energy are taken into account, where the free energy is given as: 1 1 E ¼ M s HL 1 ðxÞ sin W cos j þ ðN == N ? ÞM 2s L 2 ðxÞ sin2 Wcos2 ðj jn Þ . 2 3 (154)

Here the n subscript refers to the anisotropy axis direction, L1(x) is the Langevin function and L 2 ðxÞ ¼ 1 ð3=xÞL 1 ðxÞ. It is assumed that the applied magnetic field lies along the x-axis, where the anisotropy axis forms an angle jn with this direction and lies in the xy plane. The effect of thermal fluctuations is thus contained in this free energy via the Langevin function. In this case, the equilibrium magnetisation has Weq ¼ p=2 and the resonance equation is expressed as: 2 o 2 eff ¼ ½H r cos j þ H eff A cos ðj jn Þ ½H r cos j þ H A cosð2ðj jn ÞÞ , (155) g

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David Schmool

0.36

+

+ Apparent resonance field (T)

0.32

+ 0.28

+ experimental simulations

+

0.24

+ 0.20

+ 0.16 0

100

(a)

200

300

Temperature (K) 0.4

Peak-to-peak linewidth (T)

0.3

++ +

+ experimental simulations

+ 0.2

0.1

+ + 0 0 (b)

100

200

300

Temperature (K)

Figure 2.75 Temperature dependence of the apparent resonance field (a) and of the peak-to-peak linewidth (b) in the experimental and computer-simulated spectra. The full curve of (a) is the dependence of Br given by Eq. (148), and that of (b) shows the theoretical DT(T) dependence, see Eqs. (152) and (153) (Berger et al., 2001).

225


300 K

200 K

100 K 50 K 25 K 15 K 5K

0.0

0.2

0.4

Magnetic field (T)

0.6

0.0

0.2

0.4

0.6

Magnetic field (T)

Figure 2.76 Experimental (left) and computer-generated (right) SPR spectra of the sol--gel glass at different temperatures. All spectra are displayed with the same peak-to-peak amplitude (Berger et al., 2001).

where H eff A ¼ ðN ? N == ÞM s L 2 ðxÞ=L 1 ðxÞ. For very weak anisotropy M is parallel to the applied field and in this limit the resonance field reduces to: Hr ¼

o H eff A 3cos2 jn 1 . 2 g

(156)

This has the same form as Eq. (140) and also agrees with the model of Raikher and Stepanov (1994) and Shilov et al. (1999). The latter is a more rigorous derivation of the FMR condition in SPM nanoparticle systems. Based on their model, de Biasi et al. (2003) make numerical simulations of FMR spectra and compare their linearised model (LM) with the normal SB model. The calculations of both models are based on randomly oriented particles (2.9 nm, Ms ¼ 640 G and particle magnetic moment, m ¼ 900 mB).

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David Schmool

SB LM

(a)

SB LM

(b)

300K

dI/dH (a.u.)

300K

x1.3

x1.3 77K

77K ΔHPP x2

x2

4K

4K

2

3

4

5

2

3

4

5

H (kOe)

Figure 2.77 FMR simulation using the LM and the SB models (see text) at different temperatures using a constant intrinsic linewidth of DH0 ¼ 200 Oe: (a) prolate ellipsoid and (b) oblate ellipsoid (de Biasi et al., 2003).

Results for prolate and oblate spheroids are shown in Fig. 2.77 for three different temperatures. The models give qualitatively similar results; the most relevant result is the calculation of the peak-to-peak linewidth, DHpp, as a function of temperature. This is shown in Fig. 2.78 for three different values of DH pffiffi0ffi , where at high T it approaches the asymptotic intrinsic value of 2DH 0 = 3, corresponding to a Lorentzian line shape. From an exact solution of the LM in the high T limit it should be possible to extract m: DH pp

pffiffiffi 2 8 3 ðH EA Þ2 ¼ pffiffiffi DH 0 þ . 5 DH 0 3

(157)

The second term comes from the random easy-axis contribution which has a dependence on the magnetic moment of the particle. At low temperatures, the thermal fluctuations drop off and the anisotropy contribution to the linewidth will approach its maximum value, which for the LM is (3/2)HA. For this case, the intrinsic linewidth becomes negligible and according to the LM, a measure of the linewidth should give a measure of the anisotropy field. Raikher and Stepanov (1994) considered the case where the anisotropy energy of the particles is smaller than the Zeeman energy associated with the applied external magnetic field, obtaining the resonance condition for sdp as: o L 1 ðxÞ P 2 ðcos WÞ. ¼ H þ HK g L 2 ðxÞ

(158)

227


1/T (K-1) 0.05

0.10

0.15

0.20

0.25

1.6

ΔHpp (kOe)

1.2

0.8

0.4

0.0 0

50

100

150 T (K)

200

250

300

Figure 2.78 DHpp as a function of temperature for different intrinsic linewidths, DH0: The thick line in the 200--300 K region is given by Eq. (157). The 1/T scale shows the low-temperature saturation behaviour of DHpp; calculated using a constant DH0 for each curve. DH0 ¼ 0.2; 0.4 and 0.6 kOe for the solid-, dot-, and dash lines, respectively (de Biasi et al., 2003).

HK is the anisotropy field (2K/Ms), L1(x) and L2(x) are as given in Eq. (154) and P 2 ðcos WÞ is the Legendre polynomial of second order, see Eq. (142), W is the angle between the magnetisation and the anisotropy axis.

6. Selected Review of Experimental Studies of Spin Dynamics in Nanometric Systems There are many experimental studies of nanosystems by FMR; we will review some of the main typical features of these studies, where a distinction will be made between regular and random assemblies of particles. It should be pointed out that given the extent of research in this area, it is not possible to give an exhaustive overview. We do not explicitly discuss the case of amorphous and nanocrystalline materials in this review. A recent review of these materials has recently been prepared by Schmool (2009).

6.1. Random nanoparticle assemblies As we have indicated in the opening sections, the treatment of magnetic nanoparticle assemblies is fraught with complications. These arise from the specific variations of materials’ parameters in the samples. For example, the

228

David Schmool

distribution of particle sizes (polydispersion) will provide a range of conditions for which resonance can occur. The range of sizes will give a corresponding spread in magnetic moments of the particles. In addition to this we will also, in general, expect a random orientation of the magnetocrystalline anisotropy axes, which again will provide more inhomogeneity in the system. To further complicate matters, SPM effects will be important and the size and orientation variation of the crystalline axes of the particles will also produce a spread in the blocking temperatures of the particles. Yet another parameter which can be very important for NP systems, and is also a poorly defined quantity in these systems. This is surface anisotropy, which can provide further anisotropies in the system and give rise to pinning conditions which can severely affect resulting FMR spectra. FMR in granular systems was considered by Netzelmann (1990). The Netzelmann theory uses an effective magnetostatic energy density function, which combines the cases of isolated magnetic particles, with a certain demagnetisation tensor and the ‘homogeneous’ magnetised entity with demagnetisation tensor. Subsequent corrections to this approach have been made by Dubowik (1996) and Kakazei et al. (1999). It is important to note that the authors in this paper proposed using the magnetisation for the resonance field and not the saturation magnetisation since the magnetic particles in consideration are well below saturation in fields of a few kOe. The packing fraction, f, defines the relative quantity of the magnetic particles in the sample. Such considerations lead to a free-energy expression, where we exclude any intrinsic form of magneto-crystalline anisotropy of the individual particles, which takes the following form: 2 2 1 1 E ¼ f ð1 f ÞM N p M þ f 2 M N s M f M H. 2 2

(159)

Similar such approaches have been applied by various authors, see for example Rubinstein et al. (1994), Butera et al. (1999), Pogorelov et al. (1999) and Pujada et al. (2001). The first term on the RHS of Eq. (159) is the magnetostatic energy due to the shape of the particles, the second term is the film magnetostatic energy, arising from the overall film geometry and the final term is the Zeeman energy. Using the coordinate system indicated in Fig. 2.79, where the z-axes is perpendicular to the film plane and considering flattened particles (oblate spheroids) with major axis parallel to 2

the film plane, we obtain a diagonal N p tensor, where Nx ¼ Ny ¼ N|| and Nz ¼ N>. Substituting the free energy into the resonance Eq. (4), a quadratic equation in the applied field (where only the positive solution has physical meaning and is thus considered) is obtained (Schmool et al., 2007): 2 sin y cosðy W0 Þ 2 sin y cos 2W0 o H a 2C Ha ¼0 sin W0 sin W0 g

(160)

229


→ H

z

→ M θ

ϑ

φ

y

Φ

X

Figure 2.79 Coordinate system used, where the z-axis is considered to be normal to the film plane. (Reprinted with permission from Schmool et al. (2007), r 2007 American Institute of Physics.)

and C sin 2W0 ¼ H a sinðy W0 Þ

(161)

is the equilibrium condition. The constant C depends on sample material constants: magnetisation, volume fraction and the particle shape factor, being given by: C ¼ 2pfM ½ð1 f ÞðN jj N ? Þ f ¼ 2pf ð1 f ÞM ðN jj N ? Þ 2pf 2 M . (162)

Thus we see that the angular dependence of the resonance field in the Netzelmann theory is directly related to the shape factors and also on the volume fraction of the ferromagnetic content of the sample. 6.1.1. Discontinuous multilayers We indicated in Section 2.3 the importance of considering the DDI in NP assemblies. In the approach of Schmool et al. (2007), the formulation for the theory of a DM system, we consider contributions which arise from dipolar interactions within each (discontinuous) layer 2 in-plane (or intraplanar interactions, IP) and those between the various planes (out of plane 2 OP), or interplanar interactions. Within the layer, the total (in-plane 2 IP) dipolar energy, EIP, is the sum over all the magnetic moments, where Yij ¼ 90, which gives: E IP dip ¼

N X N 1X IP 2 ij ¼ GIP 1 þ G2 sin W. 2 i¼1 j¼1 jai

(163)

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David Schmool

Here GIP 1;2 are constants depending on the distribution of the magnetic moments in the layer: GIP 1 ¼

N X N N X N mi mj 3mi mj 2 1 X 1 X and GIP sin Fij . 2 ¼ 3 2V mag i¼1 j¼1 4pr ij 2V mag i¼1 j¼1 4pr 3ij jai

(164)

jai

The summation over j considers one particles’ interaction with all the other (N1) particles within the layer, while the summation over i considers all of these interactions for all N particles. The value of the constants (mi, mj and hr ij i) will determine the energy scale, or in the FMR spectra, the field scale, of the total angular variation. Vmag represents the total magnetic volume of the particles, which can be expressed as: V mag ¼

N X

V i.

(165)

i¼1

We note that the magnetic volume and volume fraction are related via the expression, f ¼ Vmag/V, where V is the total volume of the sample. With regards to the dipolar interaction between layers; only interactions between adjacent layers are considered, while those between non-adjacent layers are considered as negligible. In this case, we consider mi and mk as two magnetic moments from two different but adjacent layers. Summing over all the magnetic moments of the two layers, we obtain, OP OP 2 OP 2 EOP dip ¼ G1 þ G2 sin W þ G3 cos W,

(166)

where GOP 1;2;3 are constants depending on the distribution of the magnetic moments inside the two layers. These are given by: 0

GOP 1 ¼

N X N 1 X mi mk ; 2V mag i¼1 k¼1 4pr 3ik 0

GOP 2 ¼

N X N 1 X 3mi mk 2 sin Yik sin2 Fik ; 2V mag i¼1 k¼1 4pr 3ik

(167)

0

GOP 3

N X N 1 X 3mi mk 2 ¼ cos Yik : 2V mag i¼1 k¼1 4pr 3ik

The sum over k corresponds to the combined interactions of a magnetic particle in one layer with all Nu particles in an adjacent layer. The sum over i is required to consider all the (N) particles of the layer (see Fig. 2.80). The total dipolar energy for n layers will be, E dip ¼

n n1 X X eff eff 2 OP 2 ðEIP ðEIP dip ÞL þ dip ÞL 0 ¼ G1 þ G2 sin W þ ðn 1ÞG3 cos W L¼1

L 0 ¼1

(168)

231


Figure 2.80 Illustration of (a) intraplanar interaction geometry and (b) interplanar particle interaction geometry. (Reprinted with permission from Schmool et al. (2007), r 2007 American Institute of Physics.) IP OP IP OP with Geff and Geff 1 ¼ nG1 þ ðn 1ÞG1 2 ¼ nG2 þ ðn 1ÞG2 . We note that the summation for the first term is over the n layers (intraplanar term) while the second summation (interplanar term) is only between adjacent layers, of which there are (n1) pairs, care has been taken not to count interactions twice. In this consideration, it is assumed that the interplanar and intraplanar interactions are effectively identical for each (discontinuous) layer. Now, redefining the energy density as: * 1 E ¼ f ð1 f ÞM Np M þ E dip f M H, 2

(169)

we obtain the same results for the resonance condition as given in expression (159), however, in this case the constant C will be different. In this second model, we obtain the constant: C ¼ 2pf ð1 f ÞM ðN ? N == Þ þ

1 OP ½nGIP 2 ðn 1ÞG3 . M

(170)

We notice that the IP and OP terms have opposite signs as should be expected since these will be effectively different anisotropies; the OP interactions will give a perpendicular anisotropy. The IP interactions will give in-plane anisotropy. As we noted above, there are some difficulties associated with the filling factor interpretation, however, we should realise that we strictly need a shape factor for the particles but this complexity will be removed if we consider spherical particles. In this case, the first terms of Eqs. (162) and (170) will vanish and the angular variation of the resonance field will depend only on the dipolar interaction, or the volume fraction, in the Netzelmann approach. The angular dependence of the resonance field can be fit using the resonance condition as expressed in Eq. (160). The size of this field variation will be directly dependent on the constant C, given in Eqs. (162)

232

David Schmool

and (170), for the two theoretical models. As can be seen from the analysis, C has the dimensions of field (Oe). In both approaches, we have a term which depends on the shape anisotropy of the individual particles, which essentially depends on the term ðN jj N ? Þ, for ellipsoidal particles. In general, we would expect any particle shape to be arbitrarily random and can easily be assessed as a random anisotropy, as given in the random anisotropy model (see Herzer, 1990, 2005), which will give an isotropic response and can thus be excluded from the angular variation that we are concerned with. In the case where the particles are spherical, this first term will vanish, effectively eliminating any shape anisotropy contribution. In such a situation the C constant will only have one term. In the case of the Netzelmann theory, it will thus depend on the particle density or particle volume fraction f, where C ¼ 2pf2M. For the dipolar interaction approach, this will depend directly on the average strengths of both interplanar and intraplanar interactions. By varying the interlayer separation, it is effectively possible to distinguish between these two contributions. The angular variation constant in this case can be written explicitly as: 2 C¼

3MV mag 6 n 4 8p NðN 1Þ

3 N X N0 X ðn 1Þ 1 7 sin2 Fij cos2 Yik 5. r 3ij NN 0 i¼1 k¼1 r 3ik

N X N X 1 i¼1

j¼1 jai

(171)

Here it has been assumed that all particles have the same volume (monodispersion) and therefore P P magnetic moment; Vi ¼ Vj ¼ V, such that i V i ¼ NV ¼ V mag ; j V j ¼ ðN 1ÞV V mag , and M ¼ mi/Vi. We see more clearly from this expression the geometric factors which must be evaluated in order to estimate the expected angular variation of the resonance field. If we take a simple example we can explicitly evaluate this field. As stated above, the contributions IP and OP have opposite signs. This means that the largest variations of resonance field would be expected for non-interacting planes (E OP dip ¼ 0), while in-plane will depend on the average in-plane particle separation (particle density). In Eq. (171) above, the first term in square brackets will correspond to an inverse effective average interaction volume; this represents the region of influence for the average DDI. The second term in brackets will be related to the average interaction between planes. In the case where the separation between planes is greater than that between particles in the plane, the IP (first) term will dominate. From Eq. (171), we see that the terms in the summations are purely geometric factors which depend solely on the positional distribution of the nanoparticles. Assuming that the number of particles in each layer is equal, we can write N(N1)ENNuEN2. From a further consideration of the geometric factors, we can write cos Yik ¼ d2 =r 2ik , where d is the interplanar separation between adjacent DM layers and r 2ik ¼ a2ik þ d2 ,

233


where aik is the in-plane (xy) projection of rik where a2ik ¼ X 2ik þ Y 2ik . We can then express Eq. (171) in the form: C¼

3MV mag 8pN 2

N X i¼1

2

6 4n

N X j¼1 jai

Y 2ij ðX 2ij

þ

Y 2ij Þ5=2

N0 X

3

ðn 1Þ 1 7 5 3 5=2 2 2 2 d f½ðX þ Y Þ=d

þ 1g k¼1 ik ik

2 3 N N Y2 N0 X X X 3MV mag ðn 1Þ 1 ij 6 7 ¼ 4n 5. 5 3 2 5=2 8pN 2 i¼1 a d fða =dÞ þ 1g j¼1 ij ik k¼1

ð172Þ

jai

We can further simplify this expression if we recognise that the anm factors are essentially very similar once we allow that each discontinuous layer is practically identical. As such we can then rewrite Eq. (172) in the following form: " # N X N Y 2ij ðn 1Þ 3MV mag X 1 C¼ n 5 . 8pN 2 i¼1 j¼1 aij d3 fðaij =dÞ2 þ 1g5=2

(173)

It is important to remember that this approximation is valid for the case where N is large, which in nanoparticle systems in generally the case. Since the terms in the argument of the summations refer to random positions, we cannot use convergence relations which may apply to regular series and as such the above expression must be computed for a known array of particles. In Fig. 2.81, a comparison between theory and experiment is shown for various DM structures with changing effective thicknesses for the magnetic CoFe layers separated by 4 nm Al2O3. The agreement is quite good in most cases, though some discrepancies are evident for the thinner magnetic layers. The FMR in DM systems has been considered by Kakazei et al. (2001, 2005). In the former, the authors study the transition from the continuous to the discontinuous regime by studying the variation of the effective magnetic layer (CoFe) thickness, which changes the average particle size and separation. The effective field was obtained using a Kittel analysis of the FMR, which is given as: H eff ¼ 4pM

2K V 4K S 1 , M M t

(174)

where KV,S denote the surface (Neél) and volume anisotropies. The variation of the effective field with the thickness displays three regions, which the authors designate as SPM, for thicknesses below 1.3 nm, FM for thickness above 1.7 nm and a mixed SPM-FM phase in between. Kakazei et al. (2005) use the dipolar interactions between magnetic nanoparticles in their DM systems to interpret the existence of double peaked FMR spectra, where the number of bilayers, n, is varied (see Fig. 2.82).

234

David Schmool

8 7 6 5 4 3 2 1

5

Hres (kOe)

Hres (kOe)

6

4 3 2 0

(a)

30 60 Angle (degrees)

90

Hres (kOe)

Hres (kOe)

9 7 5 3 1 0 (c)


0


90

0


90

(b) 11 10 9 8 7 6 5 4 3 2 1

90 (d)

Figure 2.81 Angular variation of the resonance field from out-of-plane (01) to in-plane (901) for (a) t ¼ 7 Å, (b) t ¼ 9 Å, (c) t ¼ 11 Å and (d) t ¼ 13 Å. (Reprinted with permission from Schmool et al. (2007), r 2007 American Institute of Physics.)

In the perpendicular geometry considered, where all moments (of magnitude m ¼ pD3 M =6, D being the particle diameter) in a square mesh (separated by a) are assumed to be aligned with the applied perpendicular field, the dipolar field in the kth layer is given as: h? k ¼

k1 X l¼1

h? ðldÞ þ

nk X

h? ðldÞ þ h? 0.

(175)

l¼1

d being the interlayer separation and h? ðzÞ is the cumulative dipolar field from another layer a perpendicular distance z away, which can be written as: 1 2 z 2 5=2 2m X z 2 2 2 n n þ m þ a3 n;m¼1 a a 3 D Meabz=a . a

h? ðzÞ ¼

ð176Þ

With the latter being true for zWa and a ¼ 5.31 and b ¼ 6.35. The summation in Eq. (175) can be restricted to nearest-neighbour coupling,

235


dl/dH (a.u.)

(a) 0

1 layer

dl/dH (a.u.)

(b) 2hc 0 5 layers: Iin /Iout ∼ 3/2

hc

dl/dH (a.u.)

(c)

0 8 layers: Iin /Iout ∼ 3

10000

12000

14000

Magnetic field II (Oe)

Figure 2.82 FMR signal in granular multilayered Al2O3 (3 nm) [Co80Fe20 (1.3 nm)/Al2O3 (3 nm)]n films with different number of layers n. Single absorption line at n ¼ 1 splits at nZ3 into two shifted lines whose intensities are roughly proportional to the numbers of outer and inner layers in the stack. (Reprinted with permission from Kakazei et al. (2005), r 2005 American Institute of Physics.)

hc ¼ h? ðdÞ; which is single for the outer layer and double for all other layers. The last term in Eq. (175) represents the intralayer dipolar field produced by the other particles of the same layer and can be written as: h? 0

3 m X D 2 2 3=2 ¼ 3 ðn þ m Þ 1:506p M. a n2 þm2 a0 a

(177)

This will give a demagnetisation field which is weaker than in the Kittel model (4pM) in a continuous film. As such the outer layers have h? k ¼ ? ? h? þ h (i.e. for k ¼ 1 and n) while the inner layers have h ¼ h þ 2hc . c 0 k 0

236

David Schmool

The difference between inner and outer layers defines the splitting found in the spectra (Fig. 2.82). This model is employed by Majchra´k et al. (2007a, 2007b) in the study of the [Fe97Si3/SiO2]5 DM system. Solving the coupled LL equations for the local field for the nanoparticles, which is comprised of external and static and dynamic fields due to other particles, the authors obtain the normal configuration resonance equation for a single layer as: H? res ¼

o 3 ? þ h . g 2 0

(178)

? Which differ for the inner and outer layers; H ? out ¼ H res hc and ? ¼ H res 2hc . The amplitudes of each resonance are proportional to the number of inner and outer layers in the DM as found by Kakazei et al. (2005). Majchra´k et al. (2007a) also consider the exchange coupling between layers via electron tunnelling through the oxide layer. Acoustic (in-phase) and optic (antiphase) resonance modes are observed, as shown in Fig. 2.83, which are typical for exchange-coupled systems. These have resonance conditions:

H? in

3 oac ¼ g H res h0 , 2

oop

2J 3 ¼ g H res h0 , dM 2

(179)

(180)

where J is the exchange coupling strength. Relatively high values of FM coupling were observed, with J ranging from 2.9 to 3.5 erg/cm2. Only the optical modes display any splitting of the resonance lines due to the dipolar effects. Pires et al. (2006a) have studied the Co/SiO2 multilayer system for different thicknesses of the two components which determine the metallic of insulating behaviour of the samples. Evidence for both fcc and hcp Co is found in the FMR analysis of a percolated sample. Both exchange and dipolar interactions are considered as competing, where the former will be more important for metallic films while DDI are dominant in insulating samples. Strong shape effects are present in the FMR measurements where the Dubowik model is used (Dubowik, 1996, 2000). 6.1.2. Three-dimensional assemblies In effect the DM systems can be treated in a quasi-2D manner, separating the interplanar and intraplanar effects, which can be varied relatively by altering the intervening non-magnetic (typically insulating) layer. In a majority of studies, the nanoparticle assembly is fully random in three dimensions and typically comprises the effects of polydispersion, random orientations of

237


optical modes

49 GHz

Absorption Derivative (a.u.)

(a) acoustic modes

(b) HNC-hc (c) HNC-2hc 18

20

22

24

25 GHz

Absorption Derivative (a.u.)

(a)

optical modes

acoustic modes (b)

(c)

12

14

16

Magnetic field H [kOe]

Figure 2.83 Room-temperature measurements of FMR curves at 25 and 49 GHz for three spots on the DM sample of varied FeSi and oxide spacer thickness: spot (a) with the FeSi layer thickness equal to 3.2 nm and the SiO2 layers were thick about 2.3 nm, (b) the 4.8 nm FeSi layer and 1.8 nm SiO2, and (c) 8.5 nm Fe and 1.4 nm SiO2. Due to stronger dipolar interactions (higher D/a ratio), the splitting (corresponding resonance fields for one of the samples are indicated by arrows) of optical modes in (b) and (c) is more evident. (Reprinted with permission from Majchra´k et al. (2007a), r 2007a American Institute of Physics.)

particles and SPM effects. The number of studies of such systems is rather extensive and we shall try to limit our review to some of the more important ones and indicate the main features of FMR spectra in these systems. A majority of experimental studies by FMR appear to have been performed on g-Fe2O3 and Co based nanoparticle assemblies, we will discuss the main results of these studies and also some selected studies on other materials.

238

David Schmool

One of the earliest studies of fine particle systems by FMR was that of Valstyn et al. (1962). The study was performed on particles of g-Fe2O3, which although not of nanometre dimensions were single domain and have much in common with our present interest in particulate media. The theory and experimental considerations are historically important since they have not changed much to the more recent approaches. The theory was based on the LL equation and analysed via the SB resonance condition. The scalar susceptibility is evaluated, much as indicated in Eq. (68). The free-energy considerations account for the Zeeman, demagnetising and magneto-crystalline contributions (see Section 2). Calculations and measurements of the susceptibility are made with and without an external applied field and particles are considered as ellipsoidal in shape. Experimental spectra display a low-frequency hump, which are more prominent the less acicular the particles and occur only when the crystalline anisotropy is comparable to the shape anisotropy. When the crystalline anisotropy is negligible the resonance condition depends only on the shape anisotropy in the absence of an applied field; o ¼ gM s ðN a N b Þ, alternatively for only crystalline anisotropy the resonance condition reads: o ¼ ð4g=3ÞðjK 1 j=M s Þ. When both contributions are present there will be dependences on the shape factor, K1 and on the orientation of the particle with respect to the crystalline axes. Multiple resonances are also evident. In the absence of an applied field, the authors consider the interactions between particles to be negligible since the particles will have randomly oriented magnetic moments. The agreement between theory and experiment in the absence of an applied field is qualitatively reasonable (i.e. the SW model is a reasonable approximation in the case for noninteracting particles) and the authors state that improvements would be expected if interactions were taken into account. The disagreement between theory and experiment also arises because the shape and crystalline anisotropies are comparable. In an applied field, the approximate resonance condition yields: H res ¼

o K1 þ . g 2M s

(181)

Results at two frequencies (9.48 and 24.12 GHz) allowed the authors to obtain a lower limit of relaxation time (from the linewidth) as t ¼ 1.2 1010 seconds as well as other material constants: Ms ¼ 370 emu/ cm3, g ¼ 1.97 and K1 ¼ 2.9 105 erg/cm3. The ferrimagnetic g-Fe2O3 nanoparticle system was studied more recently by several authors, for example Hseih et al. (2002), Dutta et al. (2004), Noginova et al. (2007), Schmool and Schmalzl (2007, 2008) and Ortega et al. (2008). In Fig. 2.84, the representative spectra are shown for g-Fe2O3 nanoparticles with an average diameter of 6.4 nm (Dutta et al., 2004). The temperature variation of the FMR resonance lines and

239


γ-Fe2O3 NP

T = 300 K

f = 9.28GHz

f = 9.28GHz

198 K bulk γ-Fe2O3

132 K γ-Fe2O3 NP 98 K 46 K

Suspended γ-Fe2O3 NP 14 K A

50 (a)

B

2550

5050

7550

Magnetic field (Oe)

10050

50 (b)

2550

5050

7550

10050

Magnetic field (Oe)

Figure 2.84 Absorption derivative FMR spectra of (a) c-Fe2O3 NP at several temperatures and (b) bulk c-Fe2O3, c-Fe2O3 NP and c-Fe2O3 NP suspension in ethanol at room temperature. The magnetic field scan starts from 50 Oe (rather than zero), because for H, 50 Oe the field of the electromagnet is not stable. (Reprinted figure with permission from Dutta et al. (2004), r 2004 by the American Physical Society.)

linewidth, are given in Fig. 2.85. Multipeaked spectra are common in FMR studies of NP systems and have been observed by many authors. The highand low-field resonance (HFR and LFR) lines have typically very different temperature dependences, where the high field lines are generally much less sensitive to changes in temperature. Despite this the linewidth variations are quite similar. It is noted that the transition through the blocking temperature (TBB100 K) does not show any marked behaviour in the resonance field, though the linewidth does noticeably increase below this transition. For higher temperatures, the two resonance lines tend to merge producing strongly overlapping lines. Such results are typical of NP assemblies and it is frequently necessary to use a fitting procedure to separate the various contributions. The low field line arises from particles with their major axis aligned parallel to the applied field, while the high field line is to the particles with their major axis perpendicular to the applied field. In the study of Hseih et al. (2002), sol2gel prepared Fe2O3 particles are studied as a function of annealing temperature. In Fig. 2.86, the spectra for

240

David Schmool

Figure 2.85 (a) Temperature variations of the resonance field Hr for the two FMR lines of c-Fe2O3 NP and for the single line of bulk c-Fe2O3. (b) Temperature variations of the linewidth DH of the two FMR lines (line A and line B) of c-Fe2O3 NP and that of the single line of bulk c-Fe2O3. (Reprinted figure with permission from Dutta et al. (2004), r 2004 by the American Physical Society.)

samples at various states of annealing are shown. Lines for g 2 and g 4:2 are observed at around 3400 and 1700 Oe, respectively. In the un-annealed sample only these two lines are present, however, after annealing to 6001C for 1 hour the spectrum changes significantly, with six hyperfine lines superposed on to the broad gE2 line. With further


241

Figure 2.86 The ESR spectra for samples of (a) un-annealed with molar concentration of Fe2O3 being 1.94% showing a very broad line without the presence of hyperfine field lines. (b) Annealed at 6001C in air showing a broad line superposed on hyperfine lines. (c) Annealed at 8001C in air showing a weakening of the hyperfine lines. (d) 9001C in air with molar concentration of Fe2O3 being 2.39% and (e) 10001C in air of Fe12H (Hseih et al., 2002).

annealing, the spectra changes again, with the sextet disappearing and the gE2 line exchange narrowing. Increased intensity is assigned to growth of the grains. The hyperfine structure is believed to be due to the presence of Mn + + ions, though the nuclear spin of 5/2 for O17 is not excluded. The data are fit using the Raikher2Stepanov model (Raikher and Stepanov, 1994) from which the linewidth can also be deduced. This shows the typical increase with reduced temperatures. The presence of haematite (a-Fe2O3) was also noted. De Biasi and Gondim (2006) have used FMR to determine the size distribution in g-Fe2O3 nanoparticles. They performed 9.5 GHz measurements where the influence of surface and shape effects are neglected since TEM results showed that the particles were practically spherical. The anisotropy field of the samples was evaluated using FMR and H SP A ¼ HA

1 10x1 cothðxÞ þ 45x2 105x3 cothðxÞ þ 105x4 , cothðxÞ x1

(182)

242

David Schmool

where x ¼ pM s hDi3 H FM 0 =6kB T and is an extension of that given in Eq. (145). Here H FM is the (FM) resonance field. Fitting the experimental 0 data to Eq. (182), the authors obtain a value of hDi. Additional analysis of the temperature dependence of the intensity of the FMR line and fitting to the relationship: " 1=3 #1 @I SP pM s H SP pM s H SP 0 0 ¼ CPðDC ðT ÞÞ , @T 2xc kB 6xc kB T

(183)

with C being a constant (weighting), P(D) is the particle size distribution as given in Eq. (2), DC(T) is the critical diameter below which the magnetic moment of the particles is unblocked and H SP 0 is the (SP) resonance field. The parameter xc is the critical value of x; xc ¼ pM s hDi3 H FM 0 =6kB hT C i, where hT C i is the average blocking temperature and R1 TQðT ÞdT hT C i ¼ R0 1 , 0 QðT ÞdT

(184)

1=3 1 @I SP . QðT Þ ¼ C @T T

(185)

with

The fitting yields a value of the most probable diameter; Dm ¼ 8.1 nm as compared to 11.2 nm from TEM measurements of the same system. The discrepancy is attributed to the existence of a disordered layer in the surface which reduces the effective magnetic size of the nanoparticles. The effects of particle volume fraction on the resonance line position and linewidth were studied by Pereira et al. (2006) in g-Fe2O3 nanoparticle assemblies. The linewidth is based on the Van Vleck (1948) method of moments where the Hamiltonian takes into account the Zeeman, exchange and dipolar interaction energies. The linewidth has two contributions which arise from dipolar interaction between the atomic spins and another between the magnetic nanoparticles. The linewidth (peak-to-peak) is then given as: DH pp ¼ gb

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi zint pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi zext SðS þ 1Þ þ f S ðS þ 1Þ . eff eff a3 w 3 D3

(186)

zint is a constant related to the atomic spin distribution in the nanoparticles, and depends on the nanocrystals’ symmetry, while zext and w are constants which depend on the nanoparticle arrangement. Seff ¼ nS is the effective spin of the nanoparticle. The product gbS describes the magnetic moment of each centre. The parameter f denotes the particle volume fraction (NpD3XR =6, N is the number of particles per unit volume and DXR is the particle diameter). The evaluation of the mean value of the magnetic moment under the action of an effective field at finite


243

temperature can be calculated and the same averaging can be used for the intraparticle term. After using S ¼ 1/2 and making some mathematical manipulations the linewidth is given by the relation: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 þ tanhðB þ CfÞ=2

DH ¼ Af tanhðB þ CfÞ þ D tanhðB þ CfÞ . 2

(187)

Here the following substitutions are used: A ¼ zext gbnS=2ðwRÞ3 , B ¼ mðH ext H dem þ H K Þ=2kB T , C ¼ p2 M 2s D3XR =36kB Tw 3 and D ¼ zint gb=a3 . The first term in Eq. (187) represents the interparticle dipolar term while the second term is related to the intraparticle atomic spin dipolar interaction. The parameter r ¼ 2wR=f1=3 represents the mean particle distance, with radius R. The fit of the linewidth to the experimental data is shown in Fig. 2.87, for a particle diameter of 5 nm and shows a good agreement to the theory. Further fits at different temperatures are shown in Fig. 2.88. The resonance-field position is described in terms of the dipolar interactions, where the resonance field is found to be: dip H res ¼ o=g 2K eff =2M s H int , where Keff is the effective anisotropy dip and H int ¼ 2m=r 3 is the dipolar interaction field, with r being the centreto-centre distance between particles. The value of Hres at f ¼ 0 can be evaluated by extrapolation of Fig. 2.88 and as such we can write H res ðf ¼ 0Þ H res ðfÞ ¼ H int . From this the interparticle separation can be calculated using r ¼ ½pM s =3H int 1=3 DXR . However, the point DDI is not a good approximation for closely separated particles and a corrected interaction field must be used which accounts particle size dimensions; this dip is given as: H int ¼ 2mr=ðr 2 D2XR =4Þ2 . Noginova et al. (2007) also performed temperature-dependent resonance measurements on g-Fe2O3 nanoparticles which were coated with organic molecules and suspended in liquid (toluene) and solid (polystyrene) matrices. The temperature-dependent spectra for the two sets of samples are shown in Fig. 2.89 and show sharp and broad superposed resonance lines. The broad line shifts to lower values with temperature decrease while the linewidth increase. The narrow line does not shift in field with decreasing temperature, but its intensity reduces following an Arrhenius variation, from which Ea/kB ¼ 850 K is obtained. In addition to this field freezing (FF), measurements were made in which the diluted toluene (freezing temperature Tfr ¼ 180 K) samples were cooled in the presence of an applied field, Bfr. Angular measurements were performed with respect to the direction of Bfr and show a sine-squared dependence. The difference between the 0 and 901 directions was then plotted as a function of Bfr, as shown in Fig. 2.90, and appears to saturate for BE 0:5 T. No FF effect was observed for the nanoparticle samples dispersed in the solid polymer matrix. Noginova et al. (2007) discuss the inaccuracies of the Raikher2Stepanov model, particularly at lower

244

David Schmool

γ-Fe2O3-silica nanocomposite 800

model

760

φ = 1.15% Intencity (l.u.)

Resonance linewidth (Oe)

DXR =5.0 nm 300 K

720

φ = 0.92% φ = 0.38% x3

φ = 0.22%

x10

φ = 0.09%

x15

φ = 0.03%

680 0

0.0

0.4

1 2 3 4 5 6 Magnetic filed (kOe)

0.8 1.2 Particle volume fraction (%)

1.6

7

2.0

Figure 2.87 Room-temperature magnetic resonance linewidth versus particle volume fraction for the maghemite--silica nanocomposites. The solid line is the best fit of the experimental data using the theoretical model. The inset shows the magnetic resonance spectrum for different nanoparticle concentrations. (Reprinted with permission from Pereira et al. (2006), r 2006 American Institute of Physics.)

temperatures and propose a temperature dependence of the line intensity based on the magnetic susceptibility: wðT Þ ¼ CLðxÞ.

(188)

C being a temperature-independent constant and L(x) the Langevin function; x ¼ mB=kB T . Fitting the intensity of the main resonance line as a function of temperature yields; B0 ¼ 3:44 kG; m ¼ 3:4 103 mB or a total

245


3540 100 K

3520

Resonance field (Oe)

3500

Resonance linewidth (Oe)

1400

140 K 1200

180 K 220 K

1000 260 K 300 K

800

3480

600 0.0 3460

0.5

1.0

1.5

2.0

Particle volume fraction (%)

3440 DKR =5.0 nm 300 K 3420 0.0

0.4

0.8

1.2

1.6

2.0

Particle volume fraction (%)

Figure 2.88 Room-temperature magnetic resonance field versus particle volume fraction for the maghemite--silica nanocomposites. The inset shows magnetic resonance linewidth versus particle volume fraction for different temperatures. The solid line is the best fit of the experimental data using the theoretical model. (Reprinted with permission from Pereira et al. (2006), r 2006 American Institute of Physics.)

spin S of 1700, and using a magnetisation (bulk) of 400 emu/cm3 gives a nanoparticle diameter of about 5.4 nm, which compares well with TEM measurements. The narrow resonance linewidth analysis (non-Lorentzian) indicates the importance of dipolar interactions in the formation of aggregates and only about 10220% of the sample consists in ‘free particles’. Noginova et al. (2007) adapt the Raikher2Stepanov model to include deviations of the magnetic moment, m, of the nanoparticles from the ground state, c ¼ 0, due to thermal excitations, which they denote as the ‘quantisation approach,’ and obtain an expression which is similar to that of Eq. (158) o ¼ H þ H K cos cP 2 ðcos WÞ. g

(189)

246

David Schmool

(a)

(b) 295 K

340 K

252 K 290 K 217 K 168 K

226 K

136 K

185 K

90 K

0

2000

4000 B [G]

6000

140 K 0

2000

4000 B [G]

6000

Figure 2.89 FMR in c-Fe2O3 suspensions in the polymer matrix (a) and toluene (b) at different temperatures (Noginova et al., 2007).

70

B(90) - B(0) [mT]

60 50 40 30 20 10 0 0.2

0.4

0.6

0.8

6

7

8

B+ [T]

Figure 2.90 Difference between the resonance fields B0 (b ¼ 901) and B0 (b ¼ 0) after the FF procedure versus the freezing field Bfr. Black squares, experiment at 77 K; curves, calculation based on Eq. (195) with glB0/kB ¼ 4300 K, Ba ¼ 800 G (solid trace); g ¼ 800 K, Ba ¼ 2040 G (dotted); and g ¼ 800 K, Ba ¼ 2380 G (dashed) (Noginova et al., 2007).

247


Using the definition of cos c ¼ m=S, where m is the magnetic quantum number and the parameter D, defined through gH K ¼ 2DS, the Hamiltonian is obtained for the systems as: H^ ¼ gðH SÞ þ DS 2n . _

(190)

From this the resonance field in the high-field approximation (HcHK) take the form: 1 H m;W ¼ H 0 þ ð2m þ 1ÞDP 2 ðcos WÞ. g

(191)

To calculate the spectra, the transition probabilities, associated with the intensity of the resonance line (m, m + 1) were taken as: W m;W ¼ AgðH H m;W Þ½SðS þ 1Þ mðm þ 1Þ ,

(192)

where A is a coefficient of proportionality, gðH H m;W Þ the form factor of the resonance line and the equilibrium distribution of the populations of the magnetic sublevels were evaluated as: E m;W =kB T rm;W ¼ Z 1 , r e

(193)

where Zr is the partition function. The resonance absorption is proportional to the population difference in adjacent levels (@rm;W =@m) and for an assembly of nanoparticles with random distribution of anisotropy axes leads to the expression: Z

Z

p

S

sin WdW

GðH H 0 Þ ¼ 0

W m;W S

@rm;W dm. @m

(194)

The difference between the quantisation approach and the Raikher2 Stepanov model is the averaging performed over c. The latter of which gives a single relaxation time for the LL model, while quantisation approach sums over all m, m + 1 transitions giving relaxation times for the longitudinal and transverse relaxation (T1 and T2) according to the Bloch equations. The values for the two relaxation times were obtained as 10220 and B2 nanoseconds, respectively. The appearance of the narrow line in the resonance spectra in the quantisation approach comes from states in the vicinity of c ¼ p=2 which are not affected by the anisotropy term and are not broadened by the random anisotropy distribution. Since these states are well above the ground state the intensity of this line exponentially decreases with temperature decrease (see Fig. 2.91). The angular dependence of the resonance lines are shown in Fig. 2.92. The effect of surface spins, in the model of Noginova et al. (2007), Koksharov et al. (2000) and Koksharov et al. (2001) is also discussed. Here the shift of the resonance field of the broad line is expected to be proportional to dH ð1Þ ¼ Ds mP 2 ðcos W0 Þ in a first-order approximation,

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David Schmool

(b) 3

A [arb. units]

A [arb. units]

(a)

100

2

1

10 0 0.00 0.01 0.02 Volume content of nanoparticles

0.003 0.004 0.005 0.006 0.007 1/T[1/K]

Figure 2.91 Properties of the narrow spectral component. (a) Magnitude of the narrow component in polymer (squares) and toluene (triangles) matrices. The solid line is exp(Ea/kBT), with Ea/kB ¼ 850 K. (b) Magnitude of the narrow line versus nanoparticle concentration in toluene (T ¼ 295 K). The solid line connects the experimental points. The dotted line shows a linear dependence at low concentration (Noginova et al., 2007).

3300

1

HD [G]

2

3100 2900 0 3

0

2000

4000

50

100 angle [deg]

150

6000

H [G]

Figure 2.92 The EMR at 77 K in the samples diluted in toluene after ZFF (trace 2) and FF with Bfr ¼ 7 kG and b ¼ 0 (trace 1) and b ¼ p/2 (trace 3). Inset: The line position in dependence on the orientation of the measuring field Bm relative to the direction of the freezing field Bfr. Dots are experimental points and the solid trace corresponds to cos2 b (Noginova et al., 2007).

249


where Wu is the angle between the field and the local anisotropy axis and Ds is the surface anisotropy parameter. A second-order approximation is proportional to D2s ðm=SÞ2 and the relative number of surface spins can be quite significant for particles of a few nanometres. In order to explain the FF phenomenon, the authors consider a distribution, f(W), of the anisotropy axes around the freezing field Bfr, which arises from the competition between magnetic and anisotropy energies of the nanoparticle with thermal energy kBT. As such the difference in the resonance field for the 0 and 901 orientations will be given as: H 0 ð90Þ H 0 ð0Þ ¼

3 2

Z

p=2

f ðWÞP 2 ðcos WÞ sin WdW.

(195)

0

A fit to the experimental data, shown in Fig. 2.90, yields Z mB0 =kB ¼ 4300 400 K and H K ¼ 800 100 Oe. The value of Z is very large and a more realistic value of 800 K is suggested and the discrepancy is possibly due to interparticle interactions. The thermal behaviour of FMR in nanoparticles of g-Fe2O3 embedded in a matrix of SiO2 was investigated by Ortega et al. (2008). The samples were produced using the sol2gel technique and results indicated the presence of a-Fe2O3, which are antiferromagnetic. The size distribution for their samples appears to be bimodal, where most of the smaller particles are of the a-Fe2O3 type and the larger are of the g-Fe2O3 phase. The spectra exhibit up to four resonance lines at low temperature, the spectra at various temperatures are illustrated in Fig. 2.93. The designation of the lines is as follows: the 3350 Oe peak, which remains roughly unchanged in value throughout the temperature range studied, is due to a uniform resonance mode; with gE2. This is the dominant resonance and is that of the majority of the SPM g-Fe2O3 particles. In fact this peak separates into a double resonance, though this seems to be at odds with the general increase expected with reduced temperature and the expected randomising effects on freezing which should increase the linewidth, see Eq. (55). Using the effects of dipolar interactions between nanoparticles, Schmool and Schmalzl (2007) have explained the angular dependence of FMR using Eq. (160) where for this case of a 3D nanoparticle assembly the C constant takes the form: C¼

phri3 M p V mag ¼ hri3 MV . 6f 6

(196)

f is the volume fraction of particles which we define as; f ¼ Vmag/V, V being the total volume of the sample and hri is the average particle radius. For non-spherical particles this constant will have an additional term related to the shape anisotropy, but is not considered. The angular dependence is shown in Fig. 2.94 which shows a good agreement between experiment and theory. Further measurements at low temperature indicate variations in

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David Schmool

Figure 2.93 FMR spectra at selected temperatures. The left-side graphs are magnifications of the evolution followed by the line 3 (Ortega et al., 2008).

the anisotropy constant, which varies from 0.238 105 J/m3 at room temperature to 3.034 105 J/m3 at 5 K. The temperature variation of the resonance is a reflection of the variation of magnetisation M in the SPM regime, which can be expressed using a weighted Langevin function: Z M ¼ Ms

HMVmag PðV ÞdV . L kB T

(197)

P(V) represents the log-normal distribution. The comparison of the Langevin function with the experimental data is shown in Fig. 2.95 for these g-Fe2O3 samples with hDi ¼ 4:6 nm.

251


3200

Resonance Field (Oe)

3100 3000 2900 2800 2700 2600 2500 0

20

40

60

80

100

Angle (°)

Figure 2.94 Angular variation of the resonance field for rectangular lamina samples. Points refer to experimental data while line is a fit to resonance Eq. (160) (Schmool and Schmalzl, 2007). 1 T=5K T = 100 K

M(x)/Ms

0.8 0.6 0.4

T = 295 K 0.2 0 0.01

0.1

1 log (x)

10

100

Figure 2.95 Experimental data points at the measured temperatures with the Langevin function (line) (Schmool and Schmalzl, 2007).

Zohar et al. (2008) have measured maghemite particles deposited on GaAs substrates using a broadband technique in which a co-planar waveguide (CPW) is used to provide the excitation field for both parallel (pc) and normal (nc) configurations. The authors show that the Kittel equations for a thin film will significantly underestimate (pc) or overestimate (nc) the resonance frequencies. For higher frequencies, the experimental and Kittel model tend to converge. To account for the

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David Schmool

significantly unsaturated magnetisation of the NP assembly, the authors use the Langevin function and approximate the assembly as a uniformly magnetised thin film (in xy plane), where the demagnetising field is approximated as Hd ¼ 4pM z z^ and the local field is H0 ¼ H þ Hd . For the parallel configuration, the local field if Hu ¼ H and the Kittel equation will take the form: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mH . ojj ðHÞ ¼ g H H þ 4pM s L kB T

(198)

The perpendicular case has a local field of H 0 ¼ H 4pM s and can only be obtained numerically and the resonance frequency, o? ¼ gH 0 , is a transcendental function of H: o? ðHÞ m o? ðHÞ ¼ H 4pM s L . g kB T g

(199)

The results of fitting experimental data to Eqs. (198) and (199) are illustrated in Fig. 2.96 along with the Kittel equations. L(x) in these equations represents the Langevin function. The Co nanoparticle system has been studied by many researchers. Some of the earlier studies by FMR were of Co nanoparticles embedded in an Ag matrix Kakazei et al. (1999), Pogorelov et al. (1999), Sańchez et al. (1999), though Respaud et al. (1999) performed their measurements on Co nanoparticles dispersed in a polymer matrix. In this latter, the FMR measurements were performed at very high frequencies (3002700 GHz). Such high frequency measurements require high saturating fields in which thermofluctuation effects become negligible; that is HcHK, and the model of Raikher and Stepanov is applicable. Solving the LL equation via the SB method, Respaud et al. (1999) obtain a resonance equation of the form: 2 o ¼ ð1 þ a2 Þ½H cos W þ H K cos2 ðWK WÞ ½H cos W þ H K cos 2ðWK WÞ , g (200)

where WK represents the direction of the anisotropy with respect to the z-axis. Simulations show the importance of the damping parameter, which can affect the form of the spectra as single or double peaks, with the transversal relaxation time, t>, varying from 1 to 10 1011 seconds. These results concur with those of Raikher and Stepanov (1994) and others. Fitting of experimental spectra to the model yield the following parameters; g, Keff and t>. Of particular interest are the reduced relaxation times (7215 1012 seconds), which are about two orders of magnitude lower than bulk Co, and reduce with particle size. The enhanced damping parameter is given by a ¼ 1/(t>o0), where o0 ¼ gHK, giving a in the


253

Figure 2.96 (a) Resonance frequencies ojj ðHÞ, for different field orientations /, in-plane (pc) configuration. Model calculations for thin-film superparamagnetic (SPM) resonance Eq. (198), varying volume fraction x from TEM-measured x ¼ 0.27. (b) Resonance frequencies o? ðHÞ for normal configuration (nc), calculated according to Eq. (199). (Reprinted with permission from Zohar et al. (2008), r 2008 American Institute of Physics.)

range 0.320.55 for particles with about 310 and 150 atoms. The g values of about 2.20 agree fairly well with bulk values, while enhanced anisotropy (7.529 106 erg/cm3) are believed to arise from surface anisotropy effects, which is also the believed source of enhanced damping. Sańchez et al. (1999) also find g factors of the order of 2.1 and use the models of de Biasi and Devezas (1978) and Berger et al. (1997), as outlined above in Section 5. These authors also note that from their analysis the energy barrier hKV iFMR 6:4 7 1015 erg which can be as high as about three times that obtained by magnetisation data, M(H, T), at low temperature.

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David Schmool

Enhanced anisotropy was also concluded, where KB7 107 erg/cm3 from the resonance measurements. Rubinstein et al. (1994), Schmool et al. (1999), Lachowicz et al. (2000) and Pujada et al. (2003) studied cobalt2copper alloys in ribbon form, in which the Co phase forms nanoparticles. In the former study of Lachowicz et al. (2000), the authors use an alloy of Cu90Co10. From the FMR spectra, they note that the asymmetry of the resonance lines can be explained by the deformation of the (fcc) Co nanoparticles under thermal contraction in the Cu matrix, and shifts in the resonance field may result from the stresses there induced. However, these changes are minor compared to the effects of sample temperature on the FMR spectra. The effective anisotropy of the Co particles is mainly due to uniaxial magneto-crystalline anisotropy where contributions due to particle shape, magnetostriction and surface anisotropy are negligible. The variation of the resonance field (parallel configuration) and linewidth as a function of sample temperature is shown in Fig. 2.97. The Kittel equation is used to evaluate a g-factor of 2.09. The resonance

Figure 2.97 Temperature dependences of the FMR resonant field (applied DC field required for the resonance) and FMR linewidth. (Reprinted with permission from Lachowicz et al. (2000), r 2000 American Institute of Physics.)

255


field is constant above 160 K, indicating that the sample is in the SPM state at higher temperatures. The reduction in Hres is due to the blocking of the particles and will be proportional to the magnetic anisotropy of the sample. The broadening of the resonance line is also a reflection of the anisotropy in the sample. Using the modified blocking temperature under an applied field, see Eq. (139), the authors estimate the characteristic time of about 1013 seconds for their particles of 6.7 nm and a blocking temperature for FMR of 166 K, which appears to be in agreement with the variation of Hres in Fig. 2.97. This characteristic time is suggested for the SPM state instead of the usually cited 109 seconds value for FMR. The system studied by Pujada et al. (2003) is weaker in Co; Co5Cu95. A reduction in Hres and an increase of DH with decrease in temperature were also noted. The authors use the model of de Biasi and Devezas (1978) to evaluate the temperature dependence of the linewidth, which varies as: DH ¼ DH 0 þ sH eff

1 ð3=xÞLðxÞ . LðxÞ

(201)

The fitting of this function with 1/T for the as cast and annealed (5001C) samples is shown in Fig. 2.98 and show excellent agreement. From this, it is possible to obtain the mean grain diameter, Dm, and the term proportional to the effective uniaxial anisotropy (sHeff). The effective anisotropy Keff versus 1/Dm is shown in Fig. 2.99 for these samples which shows a good agreement with the expected variation given in Eq. (13) from which the volume and surface anisotropies are evaluated as 1.61 10 and 0.18 103 J/m2, respectively. More recently Pires et al. (2006b) have studied CoxCu100x granular co-deposited films for x ¼ 10 and 30, where the latter is above the percolation limit. Measurements were performed as a function of sample temperature and annealing temperature. For the Co10Cu90 sample, the perpendicular measurement shows the existence of at least three SW modes as for a thin film. Heat treatment of the sample causes the modes to further separate and generally broaden. Parallel and perpendicular measurements allowed the assessment of the effective anisotropy, which increased with heat treatment up to 3001C. An SPM contribution is distinguished using the model of Viegas et al., 1998), and the magnetisation is given by: M SPM ¼ M Co s

Z

1

L 0

mH f ðvÞdv; kB T

(202)

where f ðvÞ ¼ cvebv is an approximation of the log-normal volume, v, distribution with c and b being constants of the distribution. M Co s is the Co saturation magnetisation and m ¼ M Co v is the magnetic moment of the s

256

David Schmool

Figure 2.98 Experimental FMR linewidth (filled triangles) versus inverse temperature. The dashed lines are computer fits using Eq. (201). (Reprinted with permission from Pujada et al. (2003), r 2003 American Institute of Physics.)

particles. From this the magnetisation can be approximated as: M SPM ¼

2 ðM Co s Þ Hc 3kB T

Z

1

v 2 ebv dv ¼ 0

2 2ðM Co s Þ Hc . 3kB Tb3

(203)

The sample is estimated to be almost entirely SPM and, therefore, Eq. (202) is essentially the total magnetisation. The Co30Cu70 sample displays a much more thin film character since there is a much larger FM component and has more rich SWR spectra. In the latter Co-rich sample which is beyond the percolation limit, interactions between the magnetic clusters severely affect the SWR spectra.


257

Figure 2.99 Effective anisotropy constant as a function of the reciprocal mean diameter for the Co5Cu95 alloys. The dashed line is a computer fit to Eq. (13). (Reprinted with permission from Pujada et al. (2003), r 2003 American Institute of Physics.).

Monodisperse fcc Co arrays were studied by FMR with in-plane (azimuthal) and out-of-plane (polar) angular measurements to the effective magnetisation and in-plane anisotropy field by Spasova et al. (2002). Regular arrays of Co particles of about 12 nm were obtained by drying a solution of the NP in an applied field of 0.35 T on a grid. The resulting assembly consisted of stripes of regular triangular Co Nanocrystals with a width of around 2002250 nm. The lowest resonance field was obtained when the external field was applied along the direction of the stripes; (Hres)min ¼ 0.233 T which is lower than the EPR field of o=g ¼ 0:3085 T, showing that an additional intrinsic magnetic field due to an effective magnetisation and an easy-axis magnetisation in the film plane is evident. Three considerations are made to the interpretation of the resonance field: (i) shape anisotropy due to stripes, (ii) magnetic anisotropy due to interparticle magnetostatic coupling in an fcc-like lattice inside the stripes and (iii) the effective magnetic anisotropy of the individual particles (including shape, volume and surface anisotropy contributions). Assuming as a first approximation that all anisotropies due to spin2orbit coupling vanish, with only shape being present, a resonance equation is obtained as given by, see Vonsovskii (1966): 2 o ¼ ½H res þ ðN x N z Þ4pM s ½H res þ ðN y N z Þ4pM s . g

(204)

This function is not sufficient to explain the angular dependence of the FMR and further contributions are required. By introducing a cubic and

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David Schmool

uniaxial symmetry, the in-plane resonance field could be obtained as: 2 o 2== ¼ ½H res þ 2H 4== an cos 4j H an cos 2ðj ju Þ

g ½H res þ H eff þ

H 4== an ð2

2

sin 2jÞ

2 H 2== an cos ðj

(205) ju Þ ,

where ju is the angle of the applied field with the axis of the uniaxial 4== anisotropy, H 2== an and H an are the effective uniaxial and fourfold anisotropy fields and the effective field is given by H eff ¼ ð2K 2? =M s Þ þ 4pfM s , where f is the volume fraction. The fit, shown in Fig. 2.100(a) shows good agreement with experiment where the following values were 2== obtained: H 4== an ¼ 0, H an ¼ 0:037 T and H eff ¼ 0:127 T, that is only uniaxial anisotropy. For the polar dependence the following resonance equation was used: 2 o ¼ ½H res cosðW WH Þ H eff cos 2W ½H res cosðW WH Þ H eff cos2 W þ H 2== an . g (206)

The fit, Fig. 2.100(b), yields Heff ¼ 0.13 T and H 2== an ¼ 0:037 T, which is in good agreement with the previous fit and experiment. The difference between the expected 4pfM s ¼ 0:222 T for f ¼ 0.31 and Heff is accounted for using the perpendicular anisotropy field 2K 2? =M s and/or the possible existence of an antiferromagnetic CoO outer layer which would reduce Ms. Such core2shell Co2CoO nanoparticles have been further studied by Wiedwald et al. (2003) using FMR and X-ray magnetic circular dichroism (XMCD) techniques to study the ratio of orbital-to-spin magnetic moment. Essentially, XMCD yields a value of mL =meff S ¼ 0:24 0:06 which is over three times the value obtained from g-factor analysis of g ¼ 2:15 0:015 corresponding to mL =meff S ¼ 0:075 0:008. The difference is explained as being due to the presence of uncompensated Co magnetic moments at Co2CoO core2shell interface. High-resolution TEM corroborates the existence of the CoO shell. The existence of oxide shells in the FMR in nanoparticle systems is demonstrated in Fig. 2.101 on Fe nanocubes, where a plasma treatment is used to remove the oxide shell in vacuum, with FMR being performed in situ, Trunova et al. (2008). The explanation of the lineshapes are illustrated in Fig. 2.102, and essentially arises from a consideration of the distribution of particle axes and summing the various contributions in the final spectrum. The fit shows excellent agreement with the model. Tomita et al. (2004) have studied the FMR in Fe nanogranular films, where Fe NPs are dispersed in an SiO2 matrix with variable concentrations. For concentrations of 5% no angular dependence was observed for the FMR, while at 15% a shift of around 600 Oe at 9.1 GHz is observed. This could be

259


0.30

μ0HR (T)

0.28

0.26

0.24 -135°

-90°

-45°

0° ϕH

45°

-30°

0° θH

30°

(a)

90°

135°

0.45

0HR

(T)

0.40 0.35 0.30 0.25 0.20 -90° (b)

-60°

60°

90°

Figure 2.100 Dependence of HR on the direction of the external magnetic field: (a) in plane /H (measured from the stripe long axis) and (b) out-of-plane hH (measured from the normal to the sample). The error bar is on the order of the symbol size (Spasova et al., 2002).

attributed to shape effects or more likely to interparticle interactions, which would be expected to be significant for the more concentrated sample. The temperature dependence of the FMR in these two samples were also studied, where for the 5% sample no significant variation was observed as opposed to the 15% sample, which for the perpendicular resonance showed an increase in value with reduced temperatures, while the parallel resonance shows a small downward shift with decreasing temperature. The magnetic percolation is expected with increased concentration and the authors use the Kittel equations to study the more concentrated sample. The shift in the FMR for this sample with decreasing temperature fits to an increase in 4pMs as measured by SQUID. Sarmiento et al. (2007) measured Fe50Ag50 granular films by FMR. These samples are at the percolation limit and the

260

David Schmool

Figure 2.101 Conventional FMR spectra of iron/iron oxide core/shell cubes and oxide free Fe cubes after H-plasma treatment and covered by Ag/Pt matrix. (Reprinted with permission from Trunova et al. (2008), r 2008 American Institute of Physics.)

complex FMR data show three well-defined resonances and are explained in terms of Fe nanoclusters. It would be expected that strong coupling between the particles exist and may give rise to spin-wave modes in a cooperative manner since the particles are so strongly coupled. Measurements on Ni nanoparticles for various concentrations were also performed by Tomita et al. (2005) where enhancement of the angular dependence of the FMR is related to the increased magnetic dipolar interaction in the more concentrated samples. Ni nanoparticles in a single-walled carbon nanotube (SWCNT) were studied as a function of concentration and temperature by Konchits et al. (2006). The resonance condition was taken as: H res ¼ o=g H K H D H DDI , where HK and HDDI are the fields due to anisotropy and DDI while the demagnetising field is suggested as being written as: ^ NP þ f N M ^ s , cf. Eq. (8), where N ^ NP and N ^ are the H D ¼ ½ð1 f 0 ÞN demagnetising factors for the nanoparticles and the sample, f ¼ f 0 =ð1 þ kÞ, with f0 being the volumetric filling factor and k the degree of dilution. The existence of multipeaked spectra is described as being due to the regions of the sample with different densities of SWCNT bundle agglomerates and magnetic NPs. With increased dilution (k 30) the resonance lines become narrower and more Lorentzian, whereby the randomising is


261

Figure 2.102 On top: Theoretical simulation of crystalline anisotropy distribution from easy to hard in-plane direction (shown as grey cubes). The black curve is the sum of all coloured resonance modes. On bottom: Theoretical fits (lines) to the in-plane and out-of-plane experimental FMR spectra (symbols). (Reprinted with permission from Trunova et al. (2008), r 2008 American Institute of Physics.)

reduced (see Fig. 2.103). The inset shows the decrease of linewidth with increase of k, which is empirically fitted to the relation; DH ¼ 930=k þ 350 in units of G. For sample of different sizes and shapes, etc., the resonance spectra will consist of unblocked (SPM) and blocked (FM) nanoparticle resonances. For higher temperatures, the number of SPM particles will increase, while at low temperature the FM will dominate. However, the experimental data show that this transition is not abrupt and for the dilute case is not monotonic; the resonance-field peaks at 130 K at the point where there is a minimum in the linewidth, where g ¼ 2.1 and DHppB200 Oe. This corresponds to the intrinsic value of the

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David Schmool

Hpp, G

1200

2000

800 400 0

A

20

40

60

dχ″/dH, a.u.

A1 0 A6 A30

-2000 B

B30

-4000 0

2000

4000

6000

Magnetic Field, G

Figure 2.103 FMR spectra for the initial SWCNT bundles (sample A) and for those diluted in paraffin wax (samples A1, A6 and A30, ratios of 1:1, 1:6 and 1:30, respectively). The dotted line (near the curve A30) is the derivative of Lorentzian curve. The inset shows the linewidth DHpp as function of the dilution parameter j. The solid line is the computation one taking into account such contributions: (DHpp)1 + (DHpp)2, where (DHpp)1 ¼ (930/j) G and (DHpp)2 ¼ 350 G. The spectra B and B30 correspond to the initial SWCNT bundles (sample B) and for that diluted in paraffin wax (sample B30, ratio 1:30). All the line amplitudes are normalised. T ¼ 300 K, n ¼ 9600 MHz. (Reprinted with permission from Konchits et al. (2006), r 2006 American Institute of Physics.)

pffiffiffi linewidth: DH 0 ¼ 2Go=ð 3g2 M s Þ, where G ¼ 5 108 per second is the obtained Gilbert damping parameter. The temperature dependence of the resonance field is depicted in Fig. 2.104 for two samples; one as prepared and the other diluted (A30). Explanation of the variation for the dilute samples is based on the temperature-dependent effects of the dipolar interaction and is based on a variant of the de Biasi-Devezas (1978) and Berger et al. (1997) approaches, a fitting of which provides an estimate for the mean size of the particles (5 nm). An analysis of the blocking

263


3200

Hres G

2800

2400

2000

1600

0

100

200

300

T, K

Figure 2.104 Temperature dependence of the resonance field Hres for the initial sample A (full circles) and the diluted sample A30 (open circles). The maximum value of Hres at T ¼ 130 K corresponds to g ¼ 2.1. The error bar corresponds approximately to the symbol size. (Reprinted with permission from Konchits et al. (2006), r 2006 American Institute of Physics.)

temperature for these Ni nanoparticles allowed the evaluation of the effective anisotropy; K eff V 23kB T B , which gave KeffB10.2 105 erg/ cm3. Temperature dependence of the linewidth in the dilute sample is determined by the effective anisotropy field dependence, with the functional dependence taken from de Biasi et al. (2003): 2DH 0 ðH eff Þ2 DH pp pffiffiffi þ C A . DH 0 3

(207)

A fitting of which provides the parameters; DH 0 ¼ 126 Oe and C ¼ 2.9, cf. Eq. (157). Differences between the dilute and initial samples are related to the influence of the demagnetising fields which are probably due to the dipolar fields between the magnetic nanoparticles. In fact these differences are also evident in the linewidth dependences of the resonance fields and the linewidths. The initial samples have the linewidth dependence as governed by: DH pp

K 1V LðxÞ. DH pp ð0ÞG kB T

(208)

DHpp(0) is the saturation linewidth at T ¼ 0 K and G(x) is a function of the form of (150) and describes the thermal fluctuation-induced modulation of the magneto-crystalline anisotropy energy. Fitting again provides

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David Schmool

the governing parameters; DHpp(0) ¼ 3.4 kOe and K1 ¼ KeffE10.2 105 erg/cm3. Fits for the linewidth for the two samples are shown in Fig. 2.105. The temperature dependence of the magnetic anisotropy is seen to be an important parameter in determining the magnetic behaviour of nanoparticle assemblies. In the study of Antoniak et al. (2005), FMR and SQUID data are jointly used in the evaluation of t0 and Keff in FePt nanoparticles from the two difference blocking temperatures for these two methods. From the analysis of the FMR using the Kittel equation: 2 o ¼ ½H res cosðW WH Þ þ m0 H A cos2 W ½H res cosðW WH Þ þ m0 H A cos 2W . g (209)

HA ¼ 2Keff/M is the effective anisotropy field, which is the uniaxial contribution due to small deviations from the spherical shape, as well as surface and step anisotropies at the particle surface which are not averaged out. Averaging over the angles WH of the external magnetic field gives the numerical relation: " H res ¼

H 0res

HA 1 H 0res

1:25 #0:44 .

(210)

3000

Hpp, G

Hpp, G

3000 2500 2000 1500

2000

1000 0

100 T, K

200

1000

0 0

100

200

300

T, K

Figure 2.105 The EMR absorption linewidth as function of temperature for the sample A. The squares (inset) are the experimental data. The open circles are the same data without the contribution from the demagnetisation fields. The solid line corresponds to the function (208) fitted to the experimental data. The triangles are the experimental data for the diluted sample A30. (Reprinted with permission from Konchits et al. (2006), r 2006 American Institute of Physics.)


265

Here H 0res ¼ _o=gm0 mB and g is the g-factor, which is obtained as 2.05470.010 from frequency-dependent measurements. Since the intensity of the FMR line is proportional to the magnetisation, the blocking temperature was evaluated analysing the intensity versus temperature. This will give a higher value than SQUID measurements since the time windows for the two methods are very different; tFMR 1010 and tSQUID 102 seconds. By comparing the two blocking temperatures, see Figs. 2.73 and 2.106, and using the Arrhenius relationship, Eq. (136), the effective anisotropy constant can be written: K eff ðhT SQUID iÞ B

!1 27kB 1 a , V m hT SQUID i i hT FMR B B

(211)

where a ¼ K eff ðhT FMR iÞ=K eff ðhT SQUID iÞ ¼ H A ðhT FMR iÞ=H A ðhT SQUID iÞ, B B B B for which a temperature-dependent Keff, a ¼ 1 and Vm is the mean volume. A small deviation of TB from Tmax arises from a distribution of sizes. From the FMR data hT FMR i 110 K, this is about five times that B from the magnetisation measurements. The anisotropy constant is shown in Fig. 2.107 as a function of temperature, with a ¼ 0.8 and using Eq. (211) K eff ¼ ð8:4 0:9Þ 105 J=m3 and from Eq. (136) t0 1:7 1012 seconds. The experimental values of Keff are found to follow a Bloch law-like dependence, with a power of 2.1; that is K eff ðT Þ / ½M s ð1 ðT =HÞ3=2 Þ 2:1 (Antoniak et al., 2006). The anisotropy was found to be about an order of magnitude higher than in the bulk.

Figure 2.106 ZFC measurements of the total magnetic moment using the superconducting quantum interference device (SQUID, open circles) and FMR (full circles) technique (Antoniak et al., 2005).

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10

Keff (105 J/m3)

8

6

4

2

0 0

100

200 T (K)

300

400

Figure 2.107 Temperature-dependent anisotropy constant of Fe70Pt30 nanoparticles (Antoniak et al., 2005).

In general, experimental measurements of nanoparticle systems the resonance linewidths are typically very large (B122 kOe). The extensive linewidth broadening is principally due to extrinsic effects, being mainly due to the spread of crystalline and consequently anisotropy axes, see Eqs. (55) and (56). The characteristic line narrowing observed at temperatures above the blocking temperature is due to the alignment of the magnetic moment (or magnetisation vector) of the individual particles under the influence of the external magnetic field, where the thermal energy is sufficient to overcome the anisotropy energy barrier. The resonance field also shows strong temperature dependence and is also due to SPM effects and the variation of magnetic anisotropies with temperature. The effects of surface anisotropy have been indicated by some authors, Winkler et al. (2005) and Schmool et al. (2006), though confirmation should be made in a more rigorous manner.

6.2. Regular Nanostructured Arrays Regular structured arrays present a simpler magnetic system, in that the shape and dimensions are well defined in form and orientation of the anisotropy axes (including surface, bulk and shape contributions). Only in such systems can we truly consider an assembly of nanoparticles to be monodisperse. One of the other major differences with the magnetic entities discussed in the previous section is the size involved. Due to preparation techniques nanopatterned materials tend to be of a larger


267

overall size. Frequently, we are dealing with magnetic structures of the order of hundreds or at best tens of nanometres. Magnetic nanostructures come in a large variety of shapes and patterns, for a recent review of the subject see Adeyeye and Singh (2008). In this section, we will consider experimental studies on various types of patterned arrays as studied using FMR and related techniques. While we recognise that we are at the limits of what we can define as nanomagnetism, much of the physics involved will still be relevant to our study and as such it is worthwhile taking this into account. It is worth bearing in mind that the size limitations of the magnetic elements we are dealing with in this section will effectively mean that SPM effects will not be in general important and in some cases even domain effects will be of relevance since the objects in question will be above the single-domain limit and DW energies can become favourable for ground state configurations. Of course size, shape and other magnetic-anisotropy energy contributions will also be of importance in these considerations. We will separate the study into nanodots, antidots, nanorings, nanowires and other magnetic structures. In the case of nanodots, we will deal with circular, square and rectangular shaped bodies that may or may not be interacting, depending on the arrays involved. Interactions will typically only involve considerations of the DDI. Dipolar interactions are also important in the case of antidot structures and are responsible for the appearance of multipeaked spectra; we will consider both circular and square antidot structures. Nanorings are a special case of dot structure and have a complex ground state, as discussed in Section 3.3. 6.2.1. Nanodots: square, rectangular and circular In ‘dot’ samples, we have seen that vortex states can be important in determining the magnetic behaviour of the sample with applied field, see Section 3.3. Such vortex states are strongly size dependent, where for larger disk structures multiple vortex structures can exist. In the FMR study of permalloy disks by Pechan et al. (2006), the resonance spectra are complex and show differences in sweep-up and sweep-down measurements, as illustrated in Fig. 2.108. For the larger disks (1000 nm), the spectra show single resonances for the higher frequency measurements since resonance conditions are satisfied at fields above the vortex annihilation field. For lower frequency, resonances can be seen near the vortex nucleation field. Further resonances are explained in terms of non-uniform modes (DE modes) associated with spin waves in the unsaturated (vortex) state, while edge effects are also evident in the appearance of higher field modes. For disks with o200 nm diameter, the resonance spectra are typically characterized by a single dominant resonance with no resonance mode observed in the vortex state. By sweeping the field up or down, it is possible to highlight the annihilation or nucleation of the vortex state, respectively.

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H sweeps up

H sweeps down

Signal (a.u.)

16 GHz

0

1000

2000 H (Oe)

3000

4000 0

1000

2000 H (Oe)

3000

4000

Signal (a.u.)

7.5 GHz

6.75 GHz

3.5 GHz

Han 0

250

500 750 H (Oe)

Hn 1000

0

250

500 750 H (Oe)

1000

Figure 2.108 FMR spectra for the 1000 nm dot array. Left-hand panels are for fields swept from low to high. Right-hand panels are for fields swept from high to low (Reprinted with permission from Pechan et al. (2006), r 2006 American Institute of Physics.)

Keupper et al. (2007) also report on the vortex dynamics in permalloy disks (in micrometres) and the suppression of the gyrotropic mode. Using spatially resolved (SR) FMR and scanning transmission X-ray microscopy (STXM) these authors demonstrate the temporal evolution of the vortex state. By introducing a ‘hole’ (defect) in the disks, the authors show that the gyrotropic mode can be suppressed when the defect is close enough to the core and effectively captures it. Nucleation of the core can be induced by increasing the microwave field amplitude or by the application of a suitable static field in the plane of the disk, whose trajectory will have an increased radius with the increase of the static field. Elliptical motion of the vortex core was also observed for higher static fields. In elliptical shaped FM dots, pairs of vortices can interact and annihilate, as illustrated by Buchanan et al. (2005) and (2006) (see Fig. 2.109(a)). The frequency of vortex (gyrotropic) motion will be dependent on applied magnetic field and the aspect ratio of

269


Absorption Derivative (arb. units)

single pair

p1p2 =+1

20

p1p2 =-1 0

100

200 Frequency (MHz)

(a)

300

300 experiment

Frequency (MHz)

simulations 200

300 100

p1p2 =-1

200 p1p2 =+1 100

0.02 (b)

0.04

0.04

0.06 0.08 0.10 Vertical Aspect Ratio, β

0.06 0.12

Figure 2.109 (a) Representative impedance derivative spectra (real part) obtained for an in-plane field of 25 Oe applied along the long axis of sample B. The lowest frequency peak is associated with a single-vortex state (solid), whereas two higher frequency peaks are observed for the vortex pair (dashed). The inset shows an optical microscope image of the sample. (b) The experimental (solid) and simulated (open) resonance frequencies as a function of ellipse aspect ratio and corresponding linear fits (dotted and dashed lines, respectively) and analytical calculation according to o ¼ ð5=9pÞoM bðab=rzc ðc=2rc ÞÞ (solid line) with xM ¼ 4pcM ¼ 26 GHz. The dot parameters are 2a ¼ 3.1, 2.1 and 1.1 mm, 2b ¼ 1.7, 1.2, and 0.7 mm, and L ¼ 40 nm. The inset shows the measured frequencies versus b ( ¼ 2L/(a + b)) for the vortex pairs with linear fits through zero. (Reprinted with permission from Buchanan et al. (2006), r 2006 American Institute of Physics.)

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the ellipse, as shown in Fig. 2.109(b). The interaction of vortices can induce polarisations of the same and opposing directions. As well as vortex states and dynamics in circular arrays of magnetic dots, a fourfold anisotropy is also evident in FMR and Brillouin light scattering (BLS) measurements Mathieu et al. (1997). The stray field from unsaturated parts of the sample magnetisation inside the dots was deemed responsible. Using high frequency (X-band) FMR to study square arrays, so as to remove the possibility of vortex domains (i.e. high fields are required, so that the vortex state will be eliminated), Kakazei et al. (2006) have shown that the dipolar forces create non-uniform magnetisation in the dots when the interdot separation is comparable to the diameter of the dot. Demokritov and Hillebrands (1999) and Jorzick et al. (2000) have also performed BLS measurements of square lattice arrays of micron sized dots. They observed that the lowest energy spin-wave modes are dispersionless (insensitive to wave vector) and are also generally insensitive to separation of the dots; only the lowest two modes showed any shift in frequency and is probably due to the dipolar coupling in the lowest modes since their mode profiles are closest to that of the uniform mode. Similar results were observed by Gubbiotti et al. (2003), where quantisation of the lowest lying spin-wave modes are observed using BLS on elliptical nanodots arrays (450 150 nm) of permalloy. The BLS study by Bayer et al. (2003) has also shown multiple localised spin-wave states in micron sized permalloy stripes. Rastei et al. (2004) and (2005) performed FMR on square arrays of 200 nm diameter circular dots of height 200 and 400 nm. The dots were produced by making holes in a Si wafer by focused ion beam (FIB) patterning. The resonance fields were above the saturation and switching fields for the dots. The aspect ratio of the dots, which are well separated (B2 mm), is seen to define the direction of the easy axis, where dots with an aspect ratio of 2 have perpendicular anisotropy, while dots of aspect ratio 1 have in-plane easy axes. This indicated that the shape anisotropy is responsible for the perpendicular anisotropy in the larger aspect ratio dots. No in-plane anisotropy was noted since the interparticle separation is too large for any interactions to be appreciable. A larger in-plane linewidth supports the hypothesis that there will be a random in-plane distribution of dot magnetisation. For the case where the holes were over filled, producing mushroom shaped magnetic bodies, extra resonances were observed. Rectangular arrays of circular (Ni) submicron dots have also been studied (Kakazei et al., 2003). The dot diameter was set to 1 mm and one axis of the dot separation (dy) was also set to this value while the other axis separation (dx) was varied between 50 and 800 nm. A continuous film was used as a reference. For widely separated dots (dx ¼ 0.85 mm), a small decrease in the out-of-plane anisotropy was observed which can be mostly accounted for by the changes in demagnetisation factors between the thin film and dot arrays (cylindrical shape). The in-plane anisotropy was


271

negligible for both the film and these dots. For reduced dx, an induced in-plane anisotropy becomes evident, as shown in Fig. 2.110(a), which increases with decreasing dx. To explain this induced magnetic anisotropy due to the interdot magnetostatic coupling, the authors use the model of Guslienko et al. (2002b) in which the magnetostatic energy density (in units

Figure 2.110 (a) The dependence of the uniaxial in-plane anisotropy induced by dipolar interactions on interdot distance dx. Points are experimental data and the solid line represents the theoretical calculation using Eqs. (212) and (213) with parameters Ms ¼ 484 G, R ¼ 0.5 mm, t ¼ 70 nm, dy ¼ 1 mm. (b) The in-plane FMR angular dependencies Hr(/) for Ni dot arrays with different interdot distances dx: (a) dx ¼ 0.05 mm; (b) dx ¼ 0.25 mm; (c) dx ¼ 0.85 mm. Points are experimental data and the solid lines represent the theoretical fit using the algorithm described in Golub et al. (1997). Fitting parameters are: (a) Hperp ¼ 3350 Oe, Hin plane ¼ 131.4 Oe, g ¼ 2.07; (b) Hperp ¼ 3250 Oe, Hin plane ¼ 61.8 Oe, g ¼ 2.07. (Reprinted with permission from Kakazei et al. (2003), r 2003 American Institute of Physics.)

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of M2 and normalised to the dot volume) can be expressed as: em ðjH Þ ¼ em

p 2

þ K u cos2 jH

(212)

and Ku ¼

8p2 X J 2 ðkRÞ f ðbkRÞ 1 2 cos2 2jk . T xT y k k

(213)

where b ¼ t/R, J1(x) is the Bessel function, jH and jk are the polar angles of the applied field and reciprocal lattice vectors k; k ¼ ðkx ; ky Þ ¼ 2pðm=T x ; n=T y Þ, respectively. m and n are integers with T x;y ¼ 2R þ dx;y . This gives a uniaxial anisotropy via the interdot coupling with an easy axis of magnetisation parallel to the shortest period, Tx of the rectangular array. The variation of the in-plane anisotropy field; H inplane ¼ 2K u =M , shows good agreement between theory and experiment as shown in Fig. 2.110(b). Perpendicular measurements were also performed in similar structures (Kakazei et al., 2004) in which additional SWR modes are observed with respect to the continuous film of the same thickness in Ni and NiFe samples, see Fig. 2.111. The additional modes arise from the confined in-plane geometry. It was found that dot separation affects only absolute and not relative positions of the resonance fields implying that the DDI between dots creates an additional effective field perpendicular to the sample. The spectral separation between resonance modes is governed by the size, shape and magnetic properties of the dots. Using dipole-exchange theory (Kalinikos et al., 1990), the spectra are accounted for, where the in-plane wave vectors are quantised due to the lateral spatial confinement. For a perpendicularly magnetised film, the dispersion relation for spin waves takes the following form (Kalinikos et al., 1990): 2 ok ¼ ðH i þ Dk2 Þ½H i þ Dk2 þ 4pM s f ðkLÞ . g

(214)

D ¼ 2A/Ms is the spin-wave constant (A being the exchange stiffness constant), k the modulus of the in-plane spin-wave wave vector and f ðkLÞ ¼ 1 ð1 ekL Þ=kL, where L is the film thickness. This is analogous to the classical Herring2Kittel dispersion relation, where f(kL) replaces sin2 Wk (Herring and Kittel, 1951). Due to lateral confinement, the in-plane k vectors will now be quantised; k ! km ; m ¼ 1; 2; 3::: and an inhomogeneous demagnetising field can be expected for a non-ellipsoidal shape, for which the internal field will be: H i ðrÞ ¼ H 0 4pM s NðrÞ þ H K? ,

(215)

where N(r) is the effective demagnetising factor and HK> is the perpendicular anisotropy field. For strong dipolar pinning at the edges,

273


Continuous Ni film

(a)

Microwave power absorbtion (a. u.)

5000

6000

7000

8000

6000

7000

8000

Patterned Ni film

(b) 5000

Patterned permalloy film

(c) 10000

11000

12000

Magnetic Field (Oe)

Figure 2.111 FMR resonance spectra of FeNi and Ni circular dots: (a) continuous Ni film; (b) patterned Ni film; (c) patterned FeNi film. (Reprinted with permission from Kakazei et al. (2004), r 2004 American Institute of Physics.)

the eigenmodes of the disk shaped dot will be expected to have the form of zeroth-order Bessel functions; that is the mode profiles will be of the form: mm ðrÞ ¼ J 0 ðkm rÞ, where km ¼ bm =R, R being the dot radius and bm are the roots of the zeroth-order Bessel function J0(bm) ¼ 0. Since the internal field is inhomogeneous, being dependent on the coordinate-dependent effective demagnetising field we can expect H i ! H i;m will be quantised

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where N(r)-Nm and 1 Nm ¼ Am

Z 0

R

NðrÞJ 20 ðkm rÞrdr

(216)

with Am ¼ R2 J 21 ðkm Þ=2. Resonance-field values evaluated compare well with experiment, as shown in Fig. 2.112. Comparison of nanodots and 20 mm disk of permalloy were made using a broadband FMR and standard FMR by Schneider et al. (2007). The nanodots were slightly elliptical (65 71 nm), with a pitch of 100 nm, and have a small in-plane anisotropy due to the ellipticity. In Figs. 2.113(a) and 2.113(b), the scattering transmission parameter, S21, which is proportional to the absorption, is shown as a function of frequency for various applied fields for both the nanomagnets and 20 mm disk. The frequency2field plots are shown in Fig. 2.113(c), with the corresponding FMR linewidths given 7

experimental data calculations

6

Resonance field (kOe)

5

(a) Ni patterned film 4 1

2

3

4

5

6

7

8

experimental data calculations

12

11

10

(b) Permalloy patterned filma 1

2

3 4 5 6 Resonance mode number

7

8

Figure 2.112 Comparison of measured and calculated FMR resonance peak positions: (a) patterned Ni film, parameters used in calculation are Ms ¼ 484 G, L ¼ 70 nm, R ¼ 500 nm, Hperp ¼ 1.84 kOe, A ¼ 8 107 erg/cm; (b) patterned NiFe film, parameters used in calculation are Ms ¼ 830 G, L ¼ 50 nm, R ¼ 500 nm, Hperp ¼ 0, A ¼ 1.4 10--6 erg/cm. (Reprinted with permission from Kakazei et al. (2004), r 2004 American Institute of Physics.)

275


S21 (Arb. Units)

0.14 0.11 0.09 0.06

S21 (Arb. Units)

24kA/m 40 56 72 88 104 120

0.17

0.03 0.00 0

2

(a)

4

6

8

10

1.00 0.86 0.71 0.57 0.43 0.29 0.14 0.00

12

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0

4

6

8

10

12

f (GHz) 18 nanodots fit 20 μm disk fit

15

6 nanodots fit 20 μm disk fit

4 2

ΔH (kA/m)

8 f0 (GHz)

2

(b)

10

12 9 6 3 0

0 0 (c)

8 kA/m 24 40 56 72 88 104

20 40 60 80 100 120 H (kA/m)

0 (d)

2

4

6

8

10

12

frequency (GHz)

Figure 2.113 (a) Raw FR-MOKE data (open circles) and fits to Eq. (68) (solid lines) versus frequency for the nanomagnet array at several field values. (b) Raw data (open circles) and fits to Eq. (68). Solid lines for a 20 mm diameter and 10 nm thick permalloy disk. (c) Resonance frequency versus applied field for 65 71 nm2 dots (open squares) and 20 mm diameter 10 nm thick continuous disk (open circles); fits are given by solid lines. (d) Effective field-swept linewidth versus frequency for an array of 65 71 nm2 dots (open squares) and 10 nm continuous disk (open circles); fits are shown as solid lines. (Reprinted with permission from Schneider et al. (2007), r 2007 American Institute of Physics.)

in Fig. 2.113(d). The fits of the resonance frequency2field curves were made using the following relation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oM o2 þ ðboÞ2 , jS 21 j / 2 y 2 o o ioDo

(217)

0

where the following definitions are used: oM ¼ jgjm0 M s ; o0 ¼ _ pffiffiffiffiffiffiffiffiffiffiffi ox oy ; ox;y ¼ j g j m0 H x;y ; H x;y ¼ H þ H K þ ðN x;y N z ÞM s ; Do ¼ _ aðox þ oy Þ and b is the ratio of polar to longitudinal MOKE sensitivities used in the measurements and is equal to 1.5. Fitting for the nanodots yielded the following: Ms ¼ 800 kA/m, HK ¼ 0.4 kA/m, N 0x ¼ 0:172, N 0y ¼ 0:666 and N 0z ¼ 0:162, where the in-plane values are almost equal and the out-of-plane factor is over three times larger. The linewidths for the nanodots are larger than in the disk sample, but with a similar

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slope, giving damping parameters of 0.011 and 0.008 for the nanodots and disk, respectively. The larger values in the nanodots are ascribed to the fluctuations in dot dimensions, which gives the non-zero intercept in the DH plot. The linear increase with frequency is indicative of Gilbert damping, Eq. (54). Square and rectangular dot arrays show strongly anisotropic spectra, where demagnetising field, proximity and confinement effects are all present. Multiple resonance spectra are understood in terms of spin-wave excitation, some examples are illustrated in the study of square dot arrays by Maeda and Kume (1996) shown in Fig. 2.114. Here we see that the configuration of the RF and static fields have an important influence on the modes that are excited, though these authors did not go much further than to state the importance of dipolar fields. Pardavi-Horvath et al. (2005) studied arrays of permalloy rectangular nanodots of varying length to width ratios using FMR. In-plane and out-of-plane angular measurements are fit using the following energy expression: E ¼ m0 M H þ

m0 ðH x M 2x þ H z M 2z Þ. 2M s

(218)

Here Hx,z are the in-plane and out-of-plane anisotropy fields. The fits to the data show good agreement, as shown in Fig. 2.115 for the 50 700 nm rectangular sample. For the out-of-plane measurements a second, weaker resonance line is also observed, which arises from the inhomogeneous internal field distribution present in non-ellipsoidal H

RF

0

2500 Magnetic field (G)

(a) H

RF H

RF

0 (b)

5000

2500 Magnetic field (G)

5000

7000 (c)

9500

12000

Magnetic field (G)

Figure 2.114 Microwave absorption of 60 nm-thick NiFe, box array with a width and a spacing of 1.0 mm (Maeda and Kume, 1996).

277


900 Out-of-plane fit Out-of-plane Out-of-plane

800

Resonance Field (mT)

700 600 500 400 300 200 100

In-plane fit In-plane

0 0

50

100

150 200 Field angle (°)

250

300

350

Figure 2.115 Angular dependence of the measured and fitted to Eq. (218) in-plane (filled circles) and out-of-plane resonance fields (strong resonance lines -- filled squares, weak resonance lines -- open squares) for a 2D array of 5-nm-thick, 50 700 nm rectangular permalloy particles (sample c) (Pardavi-Horvath et al., 2005, r 2005 IEEE).

magnetic elements. The intensity of this feature is dependent on the aspect ratio of the element and is strongest for longer wire-like shapes. Zhai et al. (2002) also performed in-plane angular FMR on rectangular patterned arrays with varying aspect ratio. For the square elements, a fourfold symmetry is observed, in which the maximum of Hres is along the diagonal direction of the squares. Rectangular elements are uniaxial in character and have a larger anisotropy field, which increases with the aspect ratio, as would be expected. The film thickness also has a bearing on the size of the anisotropy field. Multipeaked spectra are observed in the perpendicular configuration and are related to the confined lateral dimensions and nonuniform internal field and bear much similarity to the results in disk samples reported by Kakazei et al. (2004). The in-plane anisotropy of well spaced (B1 mm) rectangular permalloy nanodots with varying aspect ratios (300 300 nm to 3 and 100 mm) have also been reported by Malkinski et al. (2007). The large interdot spacing means that interdot interactions can be neglected. The in-plane angular plot of the resonance field is shown in Fig. 2.116, where the largest anisotropy is evident for the largest aspect ratio. The authors note that the 100 mm sample is almost identical to that of the 3 mm sample. Multipeaked spectra were also reported, though data was taken from the dominant mode. Kuanr et al. (2008) have used a similar set of samples in a study using NA-FMR, where frequency sweep measurements are possible in fixed applied static fields which can be varied,

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1:1

2400

1:2

Resonant Filed [Oe]

1:3 2000

1:5 1:10

1600

1200

800

0

30

60

90

120

150

180

ANGLE (Deg)

Figure 2.116 Dependencies of the resonant field on the angle between the long axis of the stripes and the in-plane applied field for the five patterned arrays of stripes with different aspect ratios. The smallest resonant fields occur for the direction of the field along the stripes, whereas maximum resonant fields were found for the transverse direction of the field. The difference between the smallest and the largest resonant fields, for a given array, is related to the magnitude of the shape anisotropy of the stripes. (Reprinted with permission from Malkinski et al. (2007), r 2007 American Institute of Physics.)

which in this case is applied along the long axis of the magnetic elements. Differences in resonance frequency are due to the self-demagnetising fields produced by surface magnetic charges during precession. Uniaxial anisotropy is also found due to the shape of the elements. In Fig. 2.117, the frequency-field data for the samples of different aspect ratio are shown for the field applied along and across the width of the strips. For the field parallel (||) to the strips, all data shows an increase of resonance frequency with applied field, while with the field perpendicular (>) to the strips, the largest aspect ratio samples show a minimum, which is related to the anisotropy field, Hu, due to sample shape, the resonance for HoHu is called a soft mode. In fact the increasing values of the resonant frequency in the parallel orientation is also related to the shape anisotropy, where f ðH ¼ 0Þ ¼ f 0 ¼ ðg=2gÞH u . Frequency linewidths are seen to increase with the aspect ratio and are significantly greater than the value of the corresponding continuous layer, see Fig. 2.118, where the resonance frequencies for parallel and perpendicular geometries are also shown as a function of the shape anisotropy field.


279

Figure 2.117 Resonance frequency versus applied magnetic field along the length and the width of the strip for structures with different aspect ratios and for a continuous Py film. The theoretical results from micromagnetics calculations are shown in the lower panel. For the simulations Ms ¼ 0.68 kG. The parallel and perpendicular signs indicate the direction of the applied field with respect to the length of the strip. (Reprinted with permission from Kuanr et al. (2008), r 2008 American Institute of Physics.)

Square nanodot arrays (50 50 nm) of Fe were studied by pump-probe measurements by Lepadatu et al. (2007). The samples were measured with an applied variable static field in-plane along the square easy axis. The frequency versus field plots and damping with field are shown in Fig. 2.119, where p the fit for the frequency-field data is made with the Kittel formula: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o ¼ g HðH þ 4pM eff Þ, where an effective field of 800 emu/cm3 was obtained and is much reduced from the bulk value of 1700 emu/cm3. This reduction cannot be explained simply by shape effects and a large perpendicular anisotropy of about 107 erg/cm3 is believed to be responsible. The damping constant, given as: a ¼ 1/ot reduces with increased applied

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Figure 2.118 The resonance frequencies (easy axis and hard axis), measured by NA-FMR and linewidth as a function of shape anisotropy (HU) at a magnetic field of 3 kOe. The top of the x-axis shows the aspect ratios of the strips (1:1, 1:2, 1:3, 1:5 and 1:10). (Reprinted with permission from Kuanr et al. (2008), r 2008 American Institute of Physics.)

static field. Comparison to a continuous thin film is made, where no precessional motion of the magnetisation is observed, showing that the modified magnetic anisotropies in the patterned sample play an important role in the magnetisation precession. Edge effects in rectangular shaped samples are known to have an effect in magnetic samples (Schmool and Barandiarań, 1998b). Maranville et al. (2007) have specifically studied these effects in patterned permalloy stripes of 240 nm width, where the sidewall angle is varied. The microwave resonance properties are measured using a CPW and detecting the changes in the transmission coefficient (S21), where the static applied field, H0, is applied perpendicularly to the stripes. The spectrum and field-frequency dispersion are shown in Fig. 2.120, where the bulk and edge modes are identified with the LFR and HFR. The minimum in the resonance frequency at 0.09 T corresponds to the saturation magnetisation in the centre of the stripes (parallel to H0). The deeper minimum at 0.15 T occurs at Hsat, where the edge magnetisation is saturated perpendicular to the edges. The variation of the edgep mode can be modelled ffiwith a Kittel-like ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi behaviour for HWHsat: f ¼ gm0 ðH 0 H sat ÞðH 0 þ H 2 Þ, where H2 is an effective stiffness field for the out-of-plane motion of the magnetisation near the edges of the magnetic element. The variation of the edge saturation field versus side wall angle is shown in Fig. 2.121. Micromagnetic calculations were performed using the OOMMF software and the dashed

281


735Oe Longitudinal MOKE Signal

830Oe 920Oe 1.1kOe 1.39kOe 1.84kOe Hard Axis

Dot Array H Easy Axis

(a)

Polar MOKE Signal

735Oe

920Oe

1.84kOe

(b) 0

100

200

300

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500

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Time (ps) 14

0.10 Damping constant

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2 0.02 0 0

(c)

300

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900 1200 1500 1800

Magnetic Field (Oe)

600

(d)

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1200

1500

1800

Magnetic Field (Oe)

Figure 2.119 (a) Longitudinal TRMOKE responses of the Fe dot array with magnetic field applied along the easy magnetic axis. The different responses are spaced out for clarity with the horizontal line indicating the initial magnetisation state. (b) Polar TRMOKE responses obtained simultaneously with the longitudinal responses above. (c) Measured precession frequency shown as solid squares together with fitting using the Kittel formula shown as a solid line. (d) Damping constant as a function of bias field. (Reprinted with permission from Lepadatu et al. (2007), r 2007 American Institute of Physics.)

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dχ″/dH (arb)

H0 + hmod(t) 10 GHz rf Bulk mode

Edge mode

in

500 μm

hrf

to det.

Frequency (GHz)

25 20 15 10 5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Applied field μ0H0 (T)

Figure 2.120 Top: FMR spectrum of the stripe array with 39 nm etch depth measured at 10 GHz. The high-field resonance (HFR) corresponds to the edge mode. The inset shows the experimental arrangement with slowly swept field H0 perpendicular to the stripes. Bottom: FMR map showing field dependence of resonance frequencies. For fields above Hsat ¼ 0.15 T, the lower frequency resonance is the edge mode. (Reprinted with permission from Maranville et al. (2007), r 2007 American Institute of Physics.)

Figure 2.121 Edge saturation field versus sidewall angle. The model values reproduce the trend observed in the measured values. The difference may be due to edge defects other than edge surface tilting. Inset: Measured values of Hsat and H2 have a linear relationship. The line represents the prediction of the macrospin model: H 2 ¼ M s 3H sat . (Reprinted with permission from Maranville et al. (2007), r 2007 American Institute of Physics.)


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line shows the comparison. Double edge modes are reported for lower angles. The inset shows the linear relationship between Hsat and H2, which is empirically modelled as: H 2 ¼ M s 3H E . Relatively large micron size elements will support the existence of DWs and the dynamics related to their motion can also be of importance. In the studies of Stoll et al. (2004) and Puzic et al. (2005), micron-sized elements were imaged in real time to observe SR-XMCD for element specific FMR. The technique is a combination of magnetic transmission X-ray microscopy (MTXM) and a stroboscopic pump-probe method, which allows good spatiotemporal resolution (around 70 picoseconds and 20 nm). In the ground state, the square magnetic elements (permalloy 4 4 mm2) exhibit a Landau pattern, where the magnetisation lies in the plane forming closure domains with 901 Neél walls. A short and weak magnetic pulse is applied perpendicular to the element and the time evolution of the response is recorded as the element relaxes back to the ground state (see Fig. 2.122). The magnetic contrast shows only the z-component of the magnetisation, hence at Dt ¼ 400 picoseconds there is no out-of-plane magnetisation. Micromagnetic simulations are shown for comparison, where qualitative agreement is seen. Two main frequencies are observed; one between 1 and 2 GHz, which originates in the DW and another at 5.6 GHz which lies inside the magnetic domains. In smaller elements, Puzic et al. (2005) show that the size of the element has an important effect in the domain dynamics, where the vortex movement is slower in larger elements, see Fig. 2.123, where

Figure 2.122 Scanning electron microscope image of the patterned 4 4 mm2 element and the microcoil. The inset shows a line scan across the boundary of the patterned element from a MTXM image indicating a resolution of 32 nm for these experiments due to the zone plate used here. (Reprinted with permission from Stoll et al. (2004), r 2004 American Institute of Physics.)

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(b)

(a) I

30°

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1μm Hpulse I

-100ps

400ps

1000ps

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Figure 2.123 (a) Permalloy patterns, 1 1 mm2 and 2 2 mm2, 50 nm thick on a Cu--Al stripline (150 nm thick). The orientations of the Landau structures of the magnetic samples are indicated by arrows. For pump-and-probe experiments, an in-plane magnetic field pulse Hpulse (pump) is generated by a pulsed electrical current I. (b) Schematic of the stripline tilted by 301 with respect to a plane perpendicular to the photon beam. In this configuration, the time dependence of the in-plane magnetisation of the samples can be detected by the X-ray flashes sprobed using the XMCD effect. (c) Differential images of the magnetisation dynamics at different time delays between the magnetic pump-pulses and the probing X-ray flashes. The ‘dots’ and ‘crosses’ are the result of a differential noise reduction technique suppressing all ‘non-dynamic’ effects. At a delay time of 100 picoseconds (first picture) the X-ray flashes arrive before the magnetic pump-pulses, thus no dynamic effects are observed. For the smaller 1 1 mm2 sample, the contrast of the cross is inverted at 2500 picoseconds delay (indicating a half turn of the gyrating vortex movement) and restored at 4000 picoseconds delay (full turn). An inversion of the contrast of the cross (a half turn) is observed in case of the larger 232 mm2 pattern at a later delay time showing a vortex movement of about half the speed of the smaller sample. (Reprinted with permission from Puzic et al. (2005), r 2005 American Institute of Physics.)


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Figure 2.124 Explanation of the origin of the ‘dots’ and ‘crosses’ observed in Fig. 6.45. The pictures in columns I and II differ by a 1801 phase change of the vortex gyration due to an inverted electronic pump-pulse polarity (for pump-and-probe measurements) or a 1801 phase change in the sine excitation (spatially resolved (SR) FMR measurements). The differences of both pictures (most right column) result in the same black and white dots and crosses as observed in the experimental result (pictures in the most left column, cf. also Fig. 2.123). Examples are given for following vortex movements (a) to the right, (b) up, (c) to the left, (d) down. (Reprinted with permission from Puzic et al. (2005), r 2005 American Institute of Physics.)

the contrast inversion is observed much later in the larger permalloy square. The initial configuration is again the same Landau pattern as before. The magnetic contrast is obtained using a differential method as illustrated in Fig. 2.124, and the variable patterns can be seen to be produced by the vortex motion of the magnetic domain structure. Dynamics of the magnetisation of magnetic elements have been studied by several authors using optical pump-probe measurements. For example, Hicken et al. (2003) have shown the fourfold symmetry induced by nonuniform demagnetisation effects in 10 mm permalloy squares. Images were also taken at various times after the initial excitation pulse, Fig. 2.125, for different values of the static field, applied along the easy axis (vertical), for the case of H0 ¼ 288 Oe a stripe pattern is seen to evolve from the sample edges. A non-uniform precession is observed for 76 Oe, though some indication of domain structure is evident. Electronic detection of magnetisation dynamics in single element strips has been performed by Grollier et al. (2006). The electrical resistance between two voltage probes is measured to detect changes that occur

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Figure 2.125 An intensity image and series of time resolved polar Kerr images obtained from the square element are shown for two values of the static field. The static field was applied in the vertical direction and parallel to the uniaxial easy axis of the sample (Hicken et al., 2003).

upon magnetisation reversal, which are due to the DW nucleation and propagation in a Co strip. Additionally microwave assisted switching was also detected as a function of microwave frequency and amplitude. Resonances at the switching field were observed at 4.2 and 6.6 GHz, for which there is a linear decrease in switching field, HSW, with microwave field strength, HMW. Conditions of pinning of the DW can also be important for the dynamic response of magnetisation reversal. The magnetisation reversal was always seen to be initiated by DW dynamics and not by the dynamics of the uniform mode. The DW resonance


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frequency and switching fields were found to be: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W H DH C o¼g xc

(219)

and W H MW , H SW ¼ H C 1 xc aH D

(220)

where W is the DW width, a is Gilbert damping parameter, xc the width of the quadratic pinning centre of strength, HC and HD are the demagnetising fields. The latter equation shows the switching field dependence on the microwave field strength. While it is not our intention to review local probe techniques for the detection of FMR, it is worth pointing out that such techniques can provide excellent opportunities to observe the dynamic magnetic behaviour in nanometric magnetic elements. For more information on the development of such techniques see Mo and Patton (2008) and references therein. One such technique discussed, which shows great promise for single-element FMR detection, is scanning thermal microscopy ferromagnetic resonance (SThM-FMR). Essentially the technique employs a microwave cavity which is used to excite the magnetic response in the sample and a thermal sensor is then used to detect small changes in local temperature which occur when the sample absorbs microwave power as the sample is driven through resonance. The spatial resolution is quoted as 30 nmoDxo100 nm with a thermal sensitivity of around 1 mK and an overall sensitivity of about 106 spins (Meckenstock et al., 2006, 2007). A review of this and related techniques has been recently published by Meckenstock (2008). By way of illustration, we will consider the study of locally resolved FMR in 1.5 mm Co stripes using this technique (Meckenstock et al., 2007). A schematic illustration of the experimental set-up with sample orientation is shown in Fig. 2.126. The sample demagnetisation tensor will produce a uniaxial anisotropy in the plane of the sample with non-uniform modes of excitation expected due to non-ellipsoidal stripe shape. These may take the form of rim (edge) and backward volume spinwave modes. Conventional FMR and SThM-FMR measurements exhibit at least three SW modes (I, II and III), as shown in Fig. 2.127. These have been identified as (I) rim mode, (II) backward volume modes and (III) uniform modes, with the aid of OOMMF micromagnetic calculations. This is shown by the thermal images in Fig. 2.128, where for applied fields of 85 and 120 mT, different areas of the Co stripe are heated. This means that the heat generated due to the absorption of microwave power can be spatially resolved and can be seen to locally drive the spin dynamics of the system.

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y Z

Si

Co Thermal nanoprobe

Co

B ϕ

X

Synthesizer TWTA

Cavity Detector Diode

Sample and cavity in external field

Figure 2.126 Schematic diagram of SThM-FMR set-up and stripe orientation. (Reprinted with permission from Meckenstock et al. (2007), r 2007 American Institute of Physics.)

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Figure 2.127 FMR spectrum of the Co stripe taken in the SThM-FMR set-up with microwave amplitude modulation. Inset: In-plane angle-dependent conventional FMR showing the amplitude of the FMR field derivative as grey scale versus applied field angle (x-axis) and external field (y-axis). I--III mark the resonances further discussed by OOMMF and locally resolved SThM-FMR (Meckenstock et al., 2007). (Reprinted with permission from Meckenstock et al. (2007), r 2007 American Institute of Physics.)


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Figure 2.128 Topography and thermal amplitude image of SThM-FMR with external field perpendicular to the Co stripes at 85 mT (upper images corresponding to mode III in Fig. 2.127) and topography thermal amplitude and phase images for rim resonance with external field perpendicular at 120 mT (lower images corresponding to mode I in Fig. 2.127). (Reprinted with permission from Meckenstock et al. (2007), r 2007 American Institute of Physics.)

This is particularly evident for the rim mode, which is restricted to a region of about 150 nm at the edge of the Co stripe. In general, for disks, only odd symmetry modes should be excited, however, the vortex core couples strongly to the RF field and other modes become possible. For the square slabs, the main difference occurs in the SW spectra (with respect to the disks) due to low-frequency modes being excited in the inhomogeneous demagnetisation field of the structure as corner modes. 6.2.2. Antidots In the previous section, we saw that there are a number of issues that will determine the spin-wave mode excitations. This will be governed principally by the shape, size and strength of interdot coupling. In the case of antidot arrays, the continuous media are now perforated by a regular array of holes. This will introduce a periodic demagnetising distribution which can drastically affect the static and dynamic magnetic properties of the film. There are a number of structural parameters which will be important in determining the explicit modifications to the continuous thin film properties; size, shape and separation of the antidots themselves. As with the case of the nano and microdots, the size of the antidots will be limited by the method of fabrication, which in many cases will be in the micron scale. In recent years, the use of nanoporous alumina (NPA) as a template has provided a new method of producing antidot arrays. Antidot systems offer a unique type of magnetic systems in which long-range

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(dipolar interactions) and short-range (confinement) effects can be studied in the sample. Experimental results typically give multipeaked resonance spectra with strong angular dependences. In the study of Yu et al. (2003), samples with 1.5 mm diameter holes in square and rectangular arrays separated by 327 mm were measured. The resulting spectra were always double peaked, with intensity and amplitude of the angular variation dependent on the separation (see Fig. 2.129). The symmetries show an in-plane twofold symmetry, which is related to the symmetry of the arrays, however, the two resonance lines display an opposing phase in terms of the easy and hard magnetic axes. Simulations (OOMMF) of the antidot structure show that there are two regions for the distribution of the demagnetising dipolar fields, as shown in Figs. 2.130(a) and 2.130(b) for a 3 4 mm antidot mesh. 8500 Absorption (a.u.)

0° 8300 45°

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Figure 2.129 FMR spectra of a (a) rectangular (3 4 mm) and (b) square (3 3 mm) antidot array with applied in-plane field at various angles, f, from the long axis; (c)--(e) the in-plane angular dependence of resonance fields extracted from FMR spectra for different hole mesh. The solid and open circles are the data from the weak and main resonance peaks respectively, and the solid lines represent theoretical fits. (Reprinted with permission from Yu et al. (2003), r 2003 American Institute of Physics.)


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Figure 2.130 Micromagnetic simulation of the 3 4 mm sample with magnetisation along (a) long axis (/ ¼ 01) and (b) short axis (/ ¼ 901). Shading and arrow size indicate the induced dipole field strength. The antidot sample is approximately divided into two regions in terms of the demagnetisation field orientation and amplitude. (c) Angular variation of the dipolar field for this structure. (Reprinted with permission from Yu et al. (2003), r 2003 American Institute of Physics.)

When the magnetisation is saturated along the long axis (a) a larger portion of the film (region A) has an additional internal field which opposes the applied field causing the resonance field to shift to higher values, while the region between the short axis (region B) tend to align along the applied field giving a lower resonance field. When the field saturates the sample along the short axis direction, (b) the smaller region along the short axis opposes the applied field, while the larger region also opposes the applied field but with a smaller intensity. The angular variation of the dipolar field, Hd, is shown in Fig. 2.130(c) and displays a very similar variation to the resonance field. Given this variation, the authors associate the larger resonance field with the larger region (curve A) and the weaker resonance with the smaller region (curve B). The latter has a larger amplitude variation due to the closer proximity of the antidots. As such this explains also the origin of the orthogonal uniaxial anisotropies of the two resonances.

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The results for the square mesh will thus appear fourfold. The fact that the resonances cross and are independent implies that they are uncoupled modes and as such will resonate in their local dipole fields. This is borne out by the fact that the square sample has two resonances of roughly equal intensity while the rectangular arrays have different amplitudes which arises from the disparity in the two different zones. The data were modelled using the LL equation and a free-energy density of the form: E ¼ MH sin W cosðj jH Þ K ? sin2 W K u sin2 Wcos2 j,

(221)

where all symbols have their usual meaning. The fits are shown in Figs. 2.129(c)22.129(f). The short direction is unchanged in all samples measured and has a uniaxial-anisotropy field of H K ¼ 2K u =M ¼ 190 Oe. The second uniaxial component due to the larger spacing in the rectangular mesh has an anisotropy field which varies as 1/r3, indicating the dipolar origin of this effect. Yu et al. (2004) extended this work to study the perpendicular FMR for 40 nm thick films with rectangular antidot arrays with fixed separation (3 mm) in one direction and 4, 5 and 7 mm in the other orthogonal direction. Resonance spectra for the various antidot arrays along with the continuous film reference sample are shown in Fig. 2.131. The latter shows just a single uniform mode (given by the Kittel formula: o0 =g ¼ H 4 pM s ), all other samples show many spin waves at the low field side of the uniform mode. The density of the spin-wave manifold increases with separation along the long axis, with a wave vector, k ¼ np/L, that scales with the antidot mesh size. (Note L is the characteristic length along the wave direction.) This implies that the spin waves are laterally bound by the antidot holes, much in the way we saw with the magnetic dots of Kakazei et al. (2004) (see Section 6.2.1). The shift in spectra towards lower fields is related to the hole density which directly affects the effective perpendicular demagnetisation factor (4pMs). Using the model of Arias and Mills (1999), the dispersion relation for the transverse in-plane spin wave is expressed as:

oðkjj Þ g

2

2A 2 ¼ ðH 4pM s Þ þ 2ðH 4pM s Þ pM s kjj d þ k , M s jj 2

(222)

where the in-plane spin-wave vector is k2jj ¼ k2x þ k2y . Simplifying to k|| ¼ np/L, the authors fit the data to experiment, as shown in Fig. 2.132. Again time-dependent OOMMF simulations support these findings, where a snapshot of My is shown in Fig. 2.133(a), with a simulated spectrum shown in Fig. 2.133(b). This shows the propagation and interaction of spin waves in the cross-like structure used to simulate a portion of the antidot array. Using time-resolved Kerr microscopy (TRKM, which is essentially a pump-probe technique) and FMR, Pechan et al. (2005) have studied 1.5 mm diameter antidots in a rectangular mesh of 3 5 mm2. The FMR


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Figure 2.131 FMR spectra in a perpendicular configuration (H normal to the film plane) for a permalloy sheet film (a), and antidots arrays (b)--(d). Arrows in (b) indicate positions of spin-wave resonances and are indexed sequentially from high to low field. (Reprinted with permission from Yu et al. (2004), r 2004 American Institute of Physics.)

data are very much of the same form as given in Yu et al. (2003). The TRKM measurements were made by focusing the 300 nm diametre probe beam on different areas of the antidot sample, as shown in Fig. 2.134(a), subsequent spectra are illustrated for the four positions A2D in Figs. 2.134(b)22.134(e). A Fourier transform of the time-domain data gives the power spectrum. The data shown are all for an external field of 200 Oe applied along the short axis of the array. Various different mode frequencies are observed in the different positions: A, 5.0 GHz; B, 3.2 GHz; C, 4.2 GHz, and D 1.4 and 2.6 GHz, all of which are seen in the power spectrum, Fig. 2.134(f), which was scanned over the entire area. The two dominant modes (5.0 and 3.2 GHz) correspond to the main FMR modes

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Figure 2.132 Resonance fields of spin-wave modes as a function of index number for the three antidot arrays. The solid lines are theoretical fittings from the Arias and Mills (1999) dipole-exchange theory. (Reprinted with permission from Yu et al. (2004), r 2004 American Institute of Physics.)

discussed previously. Mapping the modes as a function of position, it is possible to see the localisation of the various modes, as shown in Figs. 2.135 and 2.136, which were variously taken at the mode frequencies indicated for positions A2D. The frequency-field characteristics for the two dominant modes are illustrated in Fig. 2.137. The antidots lead to the additional dipole field in the plane of the film (Hd) and the introduction of a non-zero spin-wave vector, k, parallel to the shortest dimension of the two regions A and B. In regions A, M is practically perpendicular to k, while being parallel to k in region B. These situations correspond to DE modes (k ? M) and BWVMS modes (kjjM). These can be jointly described with the following general dispersion relation: o ¼ g

1=2 2A 2 2A 2 H0 þ Hd þ k k þ 4pM s F 00 ðkdÞ . H0 þ Hd þ Ms Ms

(223)

Here F00(kd) is the dipole2dipole matrix element which depends on the angle between M and k, where for uniform modes k ¼ 0 and F00(kd) ¼ 1 and Eq. (223) reduces to the usual Kittel formula. Corrections to the Kittel formula can be important, especially for small k and DE modes. The full function form of F00(kd) is given in Eq. (42) and are further discussed in Kalinikos and Slavin (1986) and Demokritov et al. (2001). Edge localised modes are believed to arise from the effective field profile which drops off strongly at the antidot edges. Two edge modes are predicted from the Bohr2Sommerfeld quantisation condition: R k½o; HðxÞ dx ¼ np in 1D.


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Figure 2.133 Time-dependent micromagnetic simulations of the antidot arrays (3 4 mm2). (a) Snapshot of the spatial magnetisation My distribution. (b) Simulated FMR spectra converted from the frequency spectra (not shown). The arrows indicate the spin-wave modes. The inset shows My as a function of simulation time (note MzEMsat is normal to the page). (Reprinted with permission from Yu et al. (2004), r 2004 American Institute of Physics.)

Square arrays of Co circular antidots were studied by FMR using a Co film as a reference (Martyanov et al., 2007). The ratio of radius to antidot separation (r/a) was maintained constant with two samples studied with a ¼ 1.2 and 0.8 mm. In-plane angular FMR measurements were performed,

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Figure 2.135 (a) Grey-scale map of the observed power in an applied field of 200 Oe at a frequency of 5.0 GHz. Note that the power is concentrated in region A. (b) The dominant mode in region B is at 3.2 GHz, which corresponds to the lower frequency mode observed in the cavity FMR. This mode extends almost continuously in the x-direction to connect to the adjacent B regions. (Reprinted with permission from Pechan et al. (2005), r 2005 American Institute of Physics.)

where the FMR spectra for the field applied along one of the square axes is shown for the two samples and the film reference sample in Fig. 2.138. While spectra in the antidot samples show a small resonance close to the uniform film mode, the main resonance lines are shifted to higher values. Large changes in the spectra are observed on in-plane rotation of the external field for the antidot samples, while the film sample shows virtually no change. Micromagnetic simulations are performed to elucidate the experimental observations. Static configurations in a 1000 Oe field (of the order of magnitude of the resonance field and sufficient to bring about technical saturation) along the antidot array axis yields an ‘antionion’ or ‘butterfly’ state, see Fig. 2.139, which tends to give a magnetisation which Figure 2.134 Polar Kerr rotation as a function of time at several fixed positions. (a) SEM image of the sample with labels indicating the positions for which time scans are shown. (b) Time scan at position A (region A) with a frequency of 5.0 GHz. (c) Time scan at position B (region B) with a frequency of 3.2 GHz. (d) Time scan at position C with a frequency of 4.2 GHz. (e) Time scan of the edge mode at position D with a frequency of 1.4 GHz. (f) Power spectrum averaged over the entire scan area. (Reprinted with permission from Pechan et al. (2005), r 2005 American Institute of Physics.)

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Figure 2.136 (a) Spectral power image at 4.2 GHz. (b) Image of the 2.6-GHz mode that is closer to the edges of the antidots. (c) The lowest frequency edge modes appear between 1.4 and 1.8 GHz. The 0.4-GHz spread in frequency is likely due to imperfections in the lithographically defined edges. (Reprinted with permission from Pechan et al. (2005), r 2005 American Institute of Physics.)

encloses the antidot hole. The antidot butterfly state for the field along the axis do not interconnect, however, a rotation of 301 gives a new static configuration in which there is cross-talk among the magnetisation inhomogeneities creating an effective stripe domain-like structure. This shows that the response even in the static case is anisotropic with the applied field. Dynamic simulations were made by perturbing the magnetic system with a short magnetic pulse in a direction perpendicular to the 100 Oe static applied field, after which the magnetic system will oscillate incoherently and a Fourier analysis is used to distinguish well-defined modes of the magnetisation dynamics. Two principal modes are obtained with frequencies of 10.5 and 21 GHz, which shift slightly (11.5 and

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Figure 2.137 FMR frequency as a function of applied field for the different modes: region A (solid dots) and region B (open dots). Data above 600 Oe were obtained from ferromagnetic-resonance measurements. The curves are fits to Kittel dispersions (see text) with varied demagnetisation fields at each applied field . (Reprinted with permission from Pechan et al. (2005), r 2005 American Institute of Physics.)

x 0.5 intensity, arb.units

(a)

(b)

(c)

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500

1000 magnetic field, G

1500

2000

Figure 2.138 (a) The FMR spectra in parallel orientation (h ¼ 01) of the original continuous film and 2D periodic Co antidote arrays with round-shaped holes with radius r ¼ 0.3a and period length (b) a ¼ 0.8 and (c) 1.2 mm. (Reprinted figure with permission from Martyanov et al. (2007), r 2007 by the American Physical Society.)

23 GHz) on a 301 rotation of the static field from the antidot array axis. Mode mapping of the antidot array are shown for these resonances in Fig. 2.140, where the z and y components are shown to be dominant in different regions of the array. Further simulations in a static field of 350 Oe

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1.0

0.5

0

-0.5

-1.0 (a)

y

0

z (b)

Figure 2.139 (a) Onion state in a disk-shaped Co disk with r ¼ 360 nm. (b) ‘Antionion’ or ‘butterfly’ state of the magnetisation in the vicinity of a circular antidot of the same size in a Co film. In both cases, the film thickness is 20 nm. An in-plane field of 1000 G is applied along the z-axis. The grey scale and the contour lines represent the y component of the magnetisation my. The local orientation of the magnetisation is schematically represented by the arrows. (Reprinted figure with permission from Martyanov et al. (2007), r 2007 by the American Physical Society.)

Figure 2.140 Resonant modes of the antidot array in the case of H ¼ 1000 G applied parallel to the array axis. The main oscillation mode mz and my at 10.5 GHz is displayed in the panels (a) and (b), respectively. Panels (c) and (d) describe the second resonant mode mz and my, occurring at a resonance frequency of 21 GHz. These images were obtained by means of a Fourier transform of the simulated magnetisation dynamics, followed by a windowed Fourier back transformation from frequency space into the time domain. The frequencies are selected by applying a small window around the frequency of interest prior to the back transformation. The grey scale and the contour lines represent the region of largest oscillation


301

were performed and the main resonance modes are found at around 5.5 and 16 GHz for the field applied along the array axis. These results are in broad agreement with those of Yu et al. (2003, 2004) and Pechan et al. (2005), where the main resonance modes arise from different regions of the sample. As well as circular antidot arrays, square hole antidot arrays have also been studied, where there is a more strict control on the confinement of the magnetic layer in the lateral dimensions. McPhail et al. (2005) studied permalloy square antidot arrays with a square mesh, defined with dimensions [(2a, 2b), (dz, dy)] as illustrated in Fig. 2.141, which also shows an AFM image for the studied [(1, 1), (1, 1)] system, where thicknesses varied between 15 and 120 nm. The angle Z is defined as that between the in-plane z-axis and the applied external field. Measurements were performed using BLS and compared with a continuous film of the same thickness. For the thicker films (120 nm), a splitting of the peaks is observed, while for the thinner film little or no splitting is evident. The variation of the surface mode frequencies for the 120 nm sample (film and antidot) as a function of the angle of incidence, f(q||), and applied field orientation, Z, are shown in Fig. 2.142. The increase of these mode frequencies is contrary to the dispersionless behaviour observed in other confined magnetic structures (see Section 6.2.1). The mode splitting is seen to reduce as a function of the applied field angle, collapsing at an angle of 451. Simulations of the demagnetising magnetic field were performed using the OOMF package, as shown in Fig. 2.143, for Z ¼ 0 and 451 (120 nm) and 01 (15 nm). This compares reasonably well with the spatial variations of the demagnetising field in the circular antidots, where two regions can be distinguished, cf. Fig. 2.130, Yu et al. (2003). Modelling of the spectra were performed assuming that the modes in the main regions (P, Q and R, see Fig. 2.141), where the demagnetising fields are almost constant, do not interact. This implies that the spin-wave behaviour is mainly dominated by the demagnetising field rather than the boundary conditions.

Figure 2.141 (a) AFM picture of the [(1,1), (1,1)]-mm 120-nm array, (b) diagram of unit cell showing dimensions and significant points. (Reprinted figure with permission from McPhail et al. (2005), r 2005 by the American Physical Society.)

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Figure 2.142 Splitting of surface mode as function of /(q||) and g (applied field direction) in the 120-nm-thick [(1,1), (1,1)]-mm structure with H ¼ 80 kA/m. The square open symbols are the continuous film surface mode, crosses indicate the two prominent peaks of the surface mode in the patterned film. The solid lines are plots of the solutions to the Landau--Lifshitz equation for the continuous film surface mode, the dotted lines are guides to the eye. (Reprinted figure with permission from McPhail et al. (2005), r 2005 by the American Physical Society.)

Figure 2.143 Demagnetising field (normalised by Ms) in the direction opposing the external field as produced by the OOMMF simulation. (a) The 120-nm film with g ¼ 01, (b) the 120-nm film with g ¼ 451, and (c) the 15-nm film with g ¼ 01. (Reprinted figure with permission from McPhail et al. (2005), r 2005 by the American Physical Society.)

The difference in the demagnetisation field is predicted, by simulation, to be thickness dependent and the lack of variation in the 15 nm sample (regions P and R) and hence no splitting occurs in the modes for this sample. So despite the extended nature of the antidot structure and the unbroken exchange coupling throughout the layer distinct mode frequencies exist and appear to be independent, since a strong coupling would not expect to give split modes. Neusser et al. (2008) obtain similar modal spin-wave patterns in square antidots in square arrays of permalloy, where they use the model of Guslienko et al. (2003) with the dispersion


303

relation as indicated in Eq. (88). Localised and non-localised mode patterns are illustrated in Fig. 2.144 (Table 2.4). Yu et al. (2007) have also performed dynamic magnetic studies, by FMR, on square-holed antidot arrays in which the size of the hole is varied while maintaining the separation between holes constant (400 nm). The hole sizes studied were: 1200, 800 and 400 nm. The latter show a slight rounding at the corners compared with the larger holes. In-plane to out-ofplane measurements were made, see Fig. 2.145, which shows that the inplane spectrum consists of two strong (almost equal intensity) resonance lines, which have their origin in the demagnetising field modulation in the in-plane configuration with the field applied along the side of the holes, as discussed previously (Fig. 2.145 inset shows the demagnetising field distribution). The perpendicular measurement shows the existence of several spin-wave modes, which arise from the perpendicular and lateral confinements. For the in-plane orientation, the spectrum can be understood in terms of the splitting of the uniform mode (from the continuous film) into the two modes by a shift of the resonance line in the antidot structure created by the differences in the internal effective field of the different regions (I and II) of the antidot; the mode from region I shifts to higher fields, while that from region II to lower fields. FMR spectra of the continuous and antidot structures, Fig. 2.146, illustrate this point. Differences in the perpendicular configuration spectra are also shown. For the in-plane spectra, the smaller antidot structures exhibit further localised spin-wave structure due to localised edge modes, as discussed above, see Pechan et al. (2005). In the perpendicular spectra, the continuous film shows spin-wave structure due to surface pinning, while these are shifted in the antidot structures, and gradually move with hole size due to changes in the out-of-plane demagnetisation field. In-plane variations of the main (two) resonance modes for the antidot structures show very similar variations to those in the circular antidots, having the same origin in the variation of the demagnetising field distributions. The use of NPA as a template for thin films provides a more economic route to producing antidot arrays. Size and separation of pores can be controlled by growth conditions (current density of anodisation and time), where regions of local hexagonal order are frequently found, for an overview of the production of NPA, see for example Li et al. (2006) and Marsal et al. (2009). Figure 2.147 shows a typical image of the surface of the NPA where regions of hexagonal order are apparent. One of the main differences between these structures and the previously considered arrays is that the latter permits pore (antidot) sizes at the scale of tens of nanometres, which is much smaller and more in the range of the nanomagnetic properties we are concerned with. Vovk et al. (2005) have studied permalloy films sputtered onto NPA templates with a pore diameter of 100 nm and a pore separation of 100 nm, where films were deposited with

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Figure 2.144 Colour-coded mode profiles for lattice 3 (a)--(f) and lattice 1 (g)--(j). Red (bright) means high-precession amplitude, dark blue reflects zero amplitude. The orientation of H is indicated by the white arrow. Broken lines illustrate highsymmetry axes along which mode analysis is performed. Lattice 3: Eigenfrequencies at 60 mT are (a) 5.68 GHz (mode 0Deg-2), (b) 13.1 GHz (mode 0Deg-4), (c) 8.5 GHz (mode 45Deg-2), (d) 10.65 GHz (mode 45Deg-3), (e) 2.58 GHz (mode 0Deg-1, edge mode, (f) 8.95 GHz (mode 0Deg-3). Lattice 1: (g) 8.05 GHz (mode 0Deg-2 at 40 mT), (h) 12.2 GHz (mode 0Deg-4 at 40 mT), (i) 8.0 GHz (mode 0Deg-2 at 110 mT) and (j) 13.8 GHz (mode 0Deg-4 at 110 mT). See Table 2.4 for explanation of lattices. (Reprinted figure with permission from Neusser et al. (2008), r 2008 by the American Physical Society.)

305


Table 2.4

Parameters of antidot lattices (Neusser et al., 2008).

Lattice

Unit cell size (nm)

Hole diameter (nm)

Hole shape

Hole area/ Thickness unit cell area (nm) (%)

1 2 3 4

206 206 272 272 272 272 1100 1100

85 85 136 550

Square Square Circular Square

17.4 9.8 19.6 25

24 24 24 16

90°

Absorption (a.u.)

80° 70° 60° 40° H

RF

20° 0°

H

2

4 H (kOe)

6

8

Figure 2.145 FMR spectra of permalloy antidot arrays with square hole width of 1200 nm; out-of-plane field is applied at different angles (h) from the array plane. The insets show the configuration of FMR measurement and the schematic demagnetisation field distribution. (Reprinted with permission from Yu et al. (2007), r 2007 American Institute of Physics.)

thicknesses in the range 102500 nm. With films up to 100 nm, the pore size and shape are unaltered, however, for thicknesses above this the pore size reduces and a continuous layer is obtained at 200 nm. FMR measurements in various NPA and continuous film samples are shown in Fig. 2.148, the spectra for the nanoporous films are multipeaked and the main resonance is shifted with respect to the continuous layer, however, for the thicker samples, the nanoporous spectra appear to tend towards that of the continuous film. The resonance peaks are also broadened with respect to the continuous film counterparts. The appearance of the multipeaked

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David Schmool

1200 nm

Absorption (a.u.)

3

800 nm 400 nm RF 1

2

3

HRES (KOe )

H (kOe)

H

2

1

0

45

90

135

180

φ (Deg.)

Figure 2.146 In-plane angular dependence of resonance fields extracted from FMR spectra for the two distinct peaks of square antidot arrays with different hole widths. The insets show the FMR spectra of antidot array (the hole width of 1200 nm) with applied in-plane field at three typical angles / with respective to one side of the hole, and the configuration of FMR measurement. (Reprinted with permission from Yu et al. (2007), r 2007 American Institute of Physics.)

Figure 2.147 SEM images of nanoporous alumina (NPA) films prepared in 0.3 M oxalic acid: (a) top surface, (b) large scan of top surface (Marsal et al., 2009).

spectra can be expected to be related to the lateral confinement of the film, giving rise to various spin-wave modes, and also the demagnetising field distribution should also play an important role. Vassallo Brigneti et al. (2008) have also measured permalloy films deposited onto NPA. Pore enlargement was achieved using a phosphoric acid for times from 0 to 30 minutes. With film thicknesses of around 100 nm, sample morphology was analysed by SEM and AFM; arrays

307


1.0 0.5 0.0 -0.5 -1.0 -1.5

(a)

Intensity, Arb. units

1.0 0.5 0.0 -0.5 -1.0

(b)

1.0 0.5 0.0 -1.5 -1.0 -1.5

(c) 0

2000 H, Oe

4000

Figure 2.148 FMR spectra of permalloy films of thicknesses 10 nm (a), 100 nm (b), and 500 nm (c) deposited on Si wafers (solid square) and on NPA membranes (open circle). Magnetic field was applied in the film planes. (Reprinted with permission from Vovk et al. (2005), r 2005 American Institute of Physics.)

showed hexagonal local symmetry, with lattice parameter of SB108 nm and hole diameter of DB42 nm, giving an estimated filling factor of f ¼ 1 0:91ðD=SÞ2 ¼ 0:86. Again a continuous film is used as a reference. The FMR of this layer shows a single line with the classical Kittel behaviour. The antidots in this study again show multiple peaks consistent with lateral magnetic confinement. Other recent studies (Madureira et al., 2008) have confirmed these findings, where in-plane to out-of plane FMR measurements show the complexity of the spectra particularly in the perpendicular configuration. Here many overlapping resonance modes make fitting and interpretation difficult. It would indeed seem likely that the variations in the

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inhomogeneous demagnetisation field and the lateral confinement combine to produce rich spin-wave spectra. In addition, the variation of the grains of local hexagonal organisation will compound the problem in which orientational differences will produce a superposition of resonance conditions thus providing a distribution of resonance modes. Extrinsic linewidth broadening can also be expected, much as is observed in polycrystalline samples with respect to single crystals. 6.2.3. Nanowires Magnetic nanowires are another example of easy to produce nanostructures which can offer large area 2D arrays for magnetic applications. As with the case of antidots, nanowire structures can be produced in parallel arrays by electrodepositing into nanometre wide cylindrical pores, such as track etched polymer membranes and anodised alumina filters. These techniques offer the means to prepare nanowires of uniform width and large aspect ratios. For an overview see Fert and Piraux (1999). The lateral confinement in wire arrays can be expected to have some resemblance to that observed in nano and microdot structures. Indeed, Mathieu et al. (1998) using the BLS technique observed quantisation of spin-wave modes in 1.8 mm diameter width wires, with standing spin-wave patterns governed by the wave vectors: kjj;n 2p=ln ¼ ðp=nÞw; n ¼ 0, 1, 2,y and where w is the wire width. Using the DE dispersion relation: on ¼ fHðH þ 4pM s Þ þ ð2pM s Þ2 ½1 e2kjj;n d g1=2 , g

(224)

good agreement is observed with experiment, as shown in Fig. 2.149. Permalloy wires of 2 and 5 mm width separated by 15 mm were deposited onto GaAs wafers by electron beam lithography and studied using TRKM by Park et al. (2002). Measurements were made in the BWVMS and DE geometries as illustrated in Fig. 2.150, with some data shown for the 5 mm wire at three positions on the wire in the BWVMS geometry. Spatiotemporal images illustrate more clearly the time evolution of the signal (Fig. 2.151) for different applied fields, with frequency domain data also illustrated. The DE geometry measurement is also given. It seen that localised edge modes play an important part in the overall spin-wave spectrum. Edge modes are more pronounced in thinner wires where for low fields (o75 Oe) a crossover to a single mode is observed. For spin waves propagating along the direction of the magnetic field, with in-plane wave vector k, the dispersion relation is given as: oðkÞ ¼ g

1=2 1 ekd ðH þ Dk2 Þ H þ Dk2 þ 4pM s . kd

(225)


309

Figure 2.149 Comparison of the measured mode frequencies for the wire arrays with separations of 0.7 mm (full symbols) and 2.2 mm (open symbols) with the calculated frequencies (horizontal lines), shown for the five lowest order quantised surface spin-wave modes. The grey bars indicate the calculated wave-vector regions of observability of the discrete modes. (Reprinted figure with permission from Mathieu et al. (1998), r 1998 by the American Physical Society.)

Out of plane dispersion has been neglected. The correction to the demagnetising energy has the important consequence of giving a minimum in the dispersion relation at non-zero k in the BWVMS geometry. As such, the uniform mode is not favoured when shorter wavelength spin-wave modes are available by some means of momentum conservation, such as multimagnon scattering. The edge mode spin waves are then evaluated using the Bohr2Sommerfeld quantisation condition discussed previously in Section 6.2.2. Uniform and edge mode calculations give reasonably good agreement for the high-field regime (W300 Oe). The edge modes fall off more quickly than the model predicts, where below 75 Oe no edge modes are evident. Better agreement is given by LLG (OOMMF) simulations in the low-field regime. Results for 2 and 5 mm wires are presented in Fig. 2.152, illustrating the effect of the wire radius on the mode frequencies. NA-FMR measurements were used by Encinas-Oropesa et al. (2001a, 2001b) to measure Ni nanowires arrays, with diameters ranging from 56 to

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David Schmool

Figure 2.150 Top: The experimental configuration. The orientation of the wires with respect to the stripline is shown for the BWVMS and DE geometries. Bottom: Time response (left) and frequency response (right) of the polar Kerr rotation after a 120 picoseconds field pulse for a 5 mm wide wire with an external field of 100 Oe applied perpendicular to its long axis. The three panels show the response measured at three different locations on the wire: (a) at the centre, (b) 1.5 mm from the centre, and (c) 2.3 mm from the centre, which corresponds to the edge of the wire. (Reprinted figure with permission from Park et al. (2002), r 2002 by the American Physical Society.)

250 nm for variable membrane porosities (4238%). Data were obtained by operating in frequency sweep mode and varying the external static field, which was applied in both parallel and perpendicular directions with respect to the wires (0.528 kOe). Results for a wire of diameter 105 nm are shown in Fig. 2.153. The resonance frequencies are plotted on the dispersion curve, see Fig. 2.154 for data plotted at various orientations, where at high fields there appears to be a linear decrease with field which then becomes constant below a certain field (which is orientation dependent). In the high-field regime, the sample is in the saturated state (M||H). Varying the sample membrane porosity will alter the dipolar coupling between the wires, data for which is shown in Fig. 2.155. Modelling the system with an effective anisotropy (shape, magnetocrystalline, etc.), the dispersion relation can be obtained as: 2 o 2 eff ¼ ½H res cosðW WH Þ þ H eff K cos W ½H res cosðW WH Þ þ H K cos 2W (226) g

and from which the theoretical lines are calculated in the figures. Strong deviations from the theoretical behaviour are observed for increased porosity (larger dipolar interactions). The approximation of the perpendicular and


311

Figure 2.151 The out-of-plane component of the magnetisation is shown in time-domain images of a cross section of a 5 mm wire. The three lower rows show images in fields of 50, 100 and 200 Oe applied perpendicular to the long axis of the wire. Black and white indicate positive and negative magnetisation. The right-hand panels show the same cross section in the frequency domain. The top panels, labelled DE, show images at 200 Oe in the DE geometry, with the applied field along the axis of the wire. Note that the edge modes do not appear in the DE geometry. (Reprinted figure with permission from Park et al. (2002), r 2002 by the American Physical Society.)

parallel branches illustrate the reduction of the effective anisotropy field, H eff K , which results from the dipole interactions. The DDI can be taken into account by writing H eff K ¼ 2pM s H u , where Hu is a positive uniaxialanisotropy field due to the DDI between the wires. Taking this into account the theory agrees better with experiment, as shown by the dashed lines in Figs. 2.155(b) and 2.155(c). The value of Hu appears to vary linearly with the porosity, and gives an empirical relationship: H u ¼ 6pM s P, where P represents the porosity. In the absence of an applied magnetic field, the zerofield resonance should follow the simple relationship: o ¼ 2pM s ð1 3PÞ. g

(227)

Although this is valid only in the saturated state (Encinas et al., 2002). In general when the field is reduced from the saturated state, the wire will split up into domains in order to minimise the stray field and thus dipolar interactions. At zero field, the dipolar interactions should therefore be minimised, only magnetic shape anisotropy will give some alignment inside the wire, and the presence of a zero-field peak in the resonance frequency

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David Schmool

Figure 2.152 Frequency domain images of cross sections of the 2 and 5 mm wires at various magnetic fields. Note the crossover to a single mode in the 2 mm wire in applied fields below75 Oe. (Reprinted figure with permission from Park et al. (2002), r 2002 by the American Physical Society.)

will arise from this and the resonance frequency will be given by: ðo=gÞ ¼ 2pM s . This is borne out by experiment, where there is no variation in zero-field resonance frequency and is independent of wire diameter and density. This is further supported by the variation of the zerofield resonance frequency as a function of magnetisation, which is shown in Fig. 2.156 for a variety of alloys. The sample deposition current and the pH of the electrolytic solution can have a very strong influence on the resulting magnetic anisotropies of the magnetic wires, which can be both positive and negative (Darques et al., 2004, 2005). FMR of Ni nanowires of diameters ranging from 35 to 500 nm were performed by Ebels et al. (2001), where fits to Eq. (226) show excellent

313


0

(a)

-10

db

-20 -30

2 3

-40

4

5

6

-50

7

8

0 (b) -10

db

-20 -30

5

6

7

-40

8

-50 0

10

20

30

Frequency (GHz)

Figure 2.153 Microwave absorption spectra as a function of the applied field intensity measured with the field applied (a) parallel and (b) perpendicular to the wires for an array of Ni nanowires of diameter 105 nm and membrane porosity of 12-15%. Measurements were done by decreasing field steps of 500 Oe from 8 kOe down to zero. Continuous lines correspond to integer multiples of the applied field. (Reprinted figure with permission from Encinas-Oropesa et al. (2001a), r 2001a by the American Physical Society.)

agreement with experimental data (Fig. 2.157). Linewidth angular dependence is also analysed and shows a complex response (see Fig. 2.158). The main contributions are given by the first three terms in Eq. (55), where the spread of wire axes replaces the spread of crystalline axes in Eq. (55). Assuming that the saturation magnetisation (and hence demagnetising field) to be homogeneous, a spread in effective field will arise from the distribution of the magnetic anisotropy. The variation of the uniaxial anisotropy, Hu, will bring about changes in H res ðyH Þ, see Fig. 2.158(b) where a set of curves for various values of Hu are shown. These cross at an angle between the parallel and perpendicular orientations, which will give a minimum in DH u ðyH Þ. In addition to this any misalignment of the wires will also provide variations to DHðyH Þ, this is taken into account by using a top-hat distribution (yH Dyw ). Various curves are shown in Fig. 2.158(a), and the overall

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David Schmool

40 0° 30° 60° 90°

F (GHz)

30

20

10 (a) 0 0°

1

M/Ms

90°

0.5

(b) 0 0

2

4 H (kOe)

6

8

Figure 2.154 Measurements performed on an array of Ni nanowires with diameter of 105 nm and membrane porosity of 12--15%. (a) The measured resonance frequencies as a function of the intensity of the field applied at several directions from the parallel to the wires (01), the theoretical resonance-field dispersion of isolated wires with the field applied parallel (//) and perpendicular 90 1(>) to the wires (continuous lines) as well as 89 and 851 (dashed and dashed-dotted lines, respectively) and, (b) corresponding (normalised) hysteresis loops measured at room temperature with the field applied parallel (open circles) and perpendicular (filled circles) to the wires, lines are just a guide to the eye. (Reprinted figure with permission from EncinasOropesa et al. (2001a), r 2001a by the American Physical Society.)

angular variation of the linewidth is found by fitting DHu and Dyw, for which the best fit gives; DHu ¼ 1.5 kOe and Dyw ¼ 3.51. The large value of DHu is an artefact resulting from a substructure in the FMR spectra. However, some samples show a clear double resonance line. The origin of the second line is not entirely clear, though according to the theory of Arias and Mills (2001) it could be due to exchange-dipole spin-wave mode excitation. FMR was used to study dipolar interactions in hexagonal arrays of Ni nanowires of length 1210 mm, radius 30240 nm and separation B100 nm grown by electrodeposition onto NPA were studied by Ramos et al.

315


40

(a)

F (GHz)

30 20 10 0 40

(b)

F (GHz)

30 20 10 0 40

(c)

F (GHz)

30 20 10 0 0

2

4 H(kOe)

6

8

Figure 2.155 Resonance frequencies measured in arrays of Ni nanowires with the field applied parallel (filled symbols) and perpendicular (open symbols) to the wires. (a) Two samples having a porosity of P ¼ 4--5% and diameters of 56 nm (circles) and 180 nm (diamonds), (b) diameter 95 nm and membrane porosity of 25--27%, (c) diameter is 250 nm and the porosity of the membrane is 35--38%. Continuous lines correspond to the theoretical resonance-field dispersion of isolated wires with the field applied parallel (//) and perpendicular (>) to the wires. (Reprinted figure with permission from Encinas-Oropesa et al. (2001a), r 2001a by the American Physical Society.)

(2004). Angular studies show that the FMR conforms to Eq. (226), where anisotropy fields of 2.122.5 kOe were obtained. The linewidth angular dependence displays fourfold symmetry, contrary to the uniaxial angular dependence of the resonance field (with easy axis along the wire axis). Maxima occur in DH along the parallel and perpendicular directions of the wire. The uniaxial anisotropy arises from shape factors (2pM s ¼ 3:05 kOe)

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David Schmool

Frequency (GHz)

40

30

CoFe Co

20

CoFeNi NiFe alloys

10

Ni NiCu

0 0

500

1000 1500 Magnetization (emu/cm3)

2000

Figure 2.156 Measured zero-field resonance frequency versus the extrapolated value of the effective magnetisation. The straight line was calculated from ðo=gÞ ¼ 2pM s with g ¼ 3.0 GHz/kOe. (Reprinted with permission from Encinas et al. (2002), r 2002 American Institute of Physics.)

35nm 80nm 270nm

Hres (kOe)

2 kOe

7.5 -30

0

30

60

90

120

150

180

210

Field Angle θΗ (°)

Figure 2.157 The experimentally determined angular variation of the resonance field Hres at f ¼ 34.4 GHz, for three different wire arrays having diameters of 35 nm (full squares), 80 nm (open squares) and 270 nm (full circles). The different data set are offset vertically by 1 kOe with respect to each other, with the horizontal bar denoting 7.5 kOe for each spectrum. The full lines are fits of the data to Eq. (218). (Reprinted figure with permission from Ebels et al. (2001), r 2001 by the American Physical Society.)

317


H

2.0

H Δθw = 10°

(a) 1.6 ΔHres (kOe)

8° 1.2

Experiment 270nm

6°

0.8 3°

intrinsicLW

0°

0.4

0.0 13

(b)

Hres (kOe)

12 11 10 0.0kOe 9 0.3kOe

8

0.6kOe 7

ΔHu

1.6

Δθw = 0°

1.2 ΔHres (kOe)

(c) 1.5kOe

1.0 0.8

0.8 0.4 0.2

0.4

intrinsic 0.0 -30

0

30 60 90 Field Angle θH (°)

120

Figure 2.158 (a) The experimentally determined angular variation of the resonance-field linewidth DHres(hH) for the 270-nm wire array (full squares) as well as the calculated linewidths DHres(hH) (full lines) for different values of the wire orientation distribution Dhw ¼ 0 to 101 (in steps of 11). Here the frequency is f ¼ 34.4 GHz, the g value is g ¼ 2.19, the uniaxial-anisotropy field is Hu ¼ 0.3 kOe, and the uniaxial-anisotropy field distribution DHu ¼ 1.5 kOe. The (almost) horizontal line is the calculated intrinsic linewidth. (b) The calculated angular variation of the resonance field Hres for three different uniaxial-anisotropy values: 0 kOe (dotted), 0.3 kOe (dashed), and 0.6 kOe (full line). (c) The calculated angular dependence of DHres for different distributions of the anisotropy DHu ¼ 0.2--1.5 kOe and for Dhw ¼ 0. (Reprinted figure with permission from Ebels et al. (2001), r 2001 by the American Physical Society.)

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and is larger than the measured anisotropy. The dipolar interactions will tend to oppose shape anisotropy, where the dipolar field is calculated pffiffiffi as: H ddi ¼ 4pM s f , where f is the filling factor given by f ¼ 2pr 2 = 3S2 , r being the wire radius and S the interwire separation. From this the effective anisotropy field takes the form: H eff K ¼ 2pM s ð1 2f Þ, which indicates that changes in H eff are only due to the saturation magnetisation and the filling K factor. For the samples studied the filling factor was calculated as about 0.1, giving an effective anisotropy field of 2.4 kOe, which agrees fairly well with the experimental value. In addition to the bulk resonance mode, surface spin-wave modes have been reported in nanowires arrays, which arise due to the surface anisotropy of the 12 nm diameter Co nanowires (Yalc- in et al., 2004), which has a hard axis perpendicular to the wire axis. The effective anisotropy now reads: H eff K ¼ 2K u =M s þ 2pM s ð1 3PÞ, where the authors model the angular behaviour with Eq. (226) plus an additional term; ð1=gT 2 Þ2 , where T2 ( ¼ 0.2 108 seconds) is the spin2spin relaxation time and Ku ¼ 3.8 106 erg/cm3. The FMR study of Li et al. (2005) concludes that there are discrepancies in the model of Encinas-Oropesa et al. (2001a, 2001b) for their study of 20 and 35 nm wire arrays. They cite the importance of stray fields inside the nanowires caused by complex domain structures when they are not completely saturated. The deconvolution of the FMR spectra as a function of angle show a high-field mode, which is considered as the uniform mode (for simplicity), which moves up in field with increasing angle, revealing a lowfield mode which is almost field independent (Fig. 2.159). It is noted that this LFR is weaker for the larger radius wire. The LFR importantly occurs in the unsaturated state. In the zero-field configuration domains can exist in the wire, where a 1801 DW is expected to separate them. Hertel (2001) showed that two types of structures are possible, with either head on domains or tail on domains, where the spins point towards or away from the DW (see Fig. 2.160). Regarding the DW structure as a transverse spin domain (TSD), it is easy to imagine how this can propagate on the application of a transverse field. Li et al. (2005) propose that the LFR arises from a resonant mode associated with the TSD. They state that the fact that the LFR is relatively stronger in the thinner wire also supports this thesis since the larger packing density gives rise to a smaller effective anisotropy field (2pM s ð1 3PÞ) thus making it less liable to confine the spin arrangement. We would, therefore, expect that as the field moves away from the parallel direction the TSD should broaden and the LFR should grow in strength. Furthermore, defects should favour the nucleation of TSDs and hence enhance the LFR. Further measurements using parallel pumping, where the LFR seems to be suppressed, adding weight to the argument. By making FMR measurements at low (L-band; 1.2 GHz), medium (X-band; 9.4 GHz) and high (Q-band; 34 GHz) frequencies, Ramos et al.

319


LFR

LFR

HFR

HFR

90°

Resonance absorption (arb. units)

Resonance absorption (arb. units)

90° 75°

60°

30°

75°

60°

30°

0° 0°

0 (a)

2000 4000 6000 8000 10000 H0 (Oe) φ-20nm sample

0

2000

(b)

4000 6000 H0 (Oe)

8000

φ-35nm sample

Figure 2.159 The deconvolved resonance peaks from the derivation spectra of both 20 and 35 nm diameter samples in different field orientations (Li et al., 2005).

Longitudinal spin domain (LSD)

Domain wall (DW)

(DW) Transverse spin domain (TSD)

(LSD)

Figure 2.160 Scheme of two different zero-field states after the wire was technically saturated perpendicular to its hard axis: head on DW with the domain orientation (a) towards and (b) away from the wall. Another definition of the domains is drawn schematically with two 901 DWs (Li et al., 2005).

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David Schmool

15 QB

ω/γ (kOe)

10 energy

H // - Mi H // Mi

H // Mi

5 applied field angle XB H // - Mi

LB

0 0

5

10 H (kOe)

Figure 2.161 Dispersion relation for H along the easy axis. The arrow schematically represents the reversal of the magnetisation occurring at H ¼ Hirr. The inset represents the energy as a function of the applied field direction for an easy-axis anisotropy system characterized by an anisotropy field HA under the condition HoHA. The lowest minimum has associated a higher resonant frequency, while the other metastable minimum has associated a decreasing branch, both linear in H (Ramos et al., 2007).

(2007) have explicitly measured different regimes in the microwave response of magnetic nanowires. Since the field is applied along the anisotropy axis of the wires, two possible states can be found, depending on the initial state of the wire; H||Mi or H||2Mi. The dispersion relation for both cases is schematically illustrated in Fig. 2.161, where the excitation fields for the three microwave bands used are also indicated. In the FMR field-swept experiment, as the field increases the population of magnetically aligned wires increases and will revert to the H||M branch at the irreversibility field, Hirr, as indicated by the arrow; this was obtained from SQUID magnetometry with a value of 2.1 kOe for the applied field in the parallel configuration. FMR spectra for the L, X and Q bands are shown in Fig. 2.161 for both H||Mi and H||2Mi configurations, where the samples were measured after saturation, with the sweeping field applied parallel or antiparallel to the initial saturating field. The resonance condition for the parallel ( + ) and antiparallel (2) can be expressed as (Dumitru et al., 2002): 2 o ¼ ð1 þ a2 Þ½HðH þ H K H i H d Þ , g

(228)

321


where Hi is the interaction field and Hd the demagnetisation field, other symbols have their usual meaning. In Fig. 2.162, the fits were made using this formalism. Because of the high fields required for measurements at the Q-band, the resulting spectra were unchanged by the initial direction of

34Ghz (QB)

Absorption derivative (a.u)

Hirr

0

5

10

15

H(kOe)

9.4GHx(XB)

H//Mi model H//Mi H//-Mi model H//-Mi

1.2GHx (LB)

Hirr DPPH 0

1

2

3

4

5

6

H (kOe)

Figure 2.162 From top to bottom: QB, XB and LB spectra obtained saturating along the direction of measurement (H||Mi, full triangles) and in the opposite direction to the measurement (H|| Mi, open circles). The asterisk corresponds to an additional line needed to best reproduce the XB spectrum (see text). The solid and dash lines correspond to the model (see text) (Ramos et al., 2007).

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magnetisation. In the X-band measurement, the parallel spectrum (H||Mi) has an initial positive signal, reaching an absorption maximum (dw00 =dH ¼ 0) when the upward branch of the dispersion relationship crosses the excitation field. A second weaker resonance is observed at higher fields and could be related to SW excitation due to lateral confinement, or could be related to the field crossing Hirr, though no explicit explanation is given (an addition peak was added in the fitting procedure to match this resonance). For the reverse initial magnetisation spectrum (H||2Mi), the initial sign of dw00 =dH is negative, as the model predicts. However, the fit is rather poor for this case. In the L-band measurement, the mode fits fairly well the data, showing opposite signs for dw00 =dH for the two initial magnetisation states. The authors make the analogy to nanoparticle systems, whereby the wires are considered as blocked particles and the response observed should give an absorption maximum as the field crosses the excitation field of the microwaves. No blocking should be expected in a singly populated minimum (M6¼0). If the anisoptropy field is greater than the excitation field, the absorption of microwave energy will decrease leading to intensity reduction, which should not be mistaken for blocking phenomena, but rather as a measurement of the anisotropy gap at that field and temperature. 6.2.4. Nanorings The FM ring structure has been seen to exhibit some fairly unique magnetic properties due to the new ground state spin configuration. Onion and vortex states have been observed and simulated depending on the applied magnetic field; see Vaz et al. (2007) and references therein. The former is characterised by the presence of two opposite head-to-head and tail-to-tail DWs, while the latter comprises of the in-plane magnetisation following the ring around in a loop. This will have profound effects on the spin dynamics associated with them over continuous media and other nanostructures considered above. For example, fewer spin-wave states will be available due to lack of central core and vortex core dynamics can be suppressed. One of the earliest FMR studies of FM ring samples was made by Xu et al. (2004). In this study, samples of square arrays of permalloy rings with thickness (t) 25 nm, outer diameter (Do) 700 nm, lateral spacing (d ) 1 mm and inner diameters (Di) of 0, 40, 80 and 300 nm were measured as a function of the in-plane orientation of the external magnetic field. SEM images and X-band spectra are shown in Fig. 2.163, where a continuous film of the same thickness was also measured for comparison. The reference sample displays a single resonance at around 1150 Oe. In the first sample (A), the arrays have no central hole (Di ¼ 0). In this case, the disk array sample displays a small upward shift in the resonance field of the uniform resonance line and additional smaller resonances at B110 and 1410 Oe;


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Figure 2.163 Room-temperature derivative FMR spectra of the nanoring lattices of 25 nm thickness, 1 mm spacing, 700 nm outer diameter and 0 (a), 40 (b), 80 (c), and 300 nm (d) inner diameters, respectively. h is the angle between the in-plane applied DC magnetic field and the square ring lattice symmetry axis. (e) Equivalent reference spectrum of a 25 nm thick plane permalloy film (Reprinted with permission from Xu et al. (2004), r 2004 American Institute of Physics.)

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a further weak resonance is also seen at around 430 Oe. The peaks at 430 and 1410 Oe were also reported for disks (Jung et al., 2002b), arising from exchange-coupled spin-wave modes and a hybrid coupling of exchange and dipolar couplings, respectively. With an inner diameter of 40 nm, the rings display a uniform resonance which is further shifted up in field to 1330 Oe, with small resonances at 180 and 1610 Oe. Further resonances are also observed at the low field side of the uniform mode. In the sample with Di ¼ 80 nm, the uniform mode appears at 1340 Oe, with more resonances at the low field side of the main resonance. Finally, in sample D with Di ¼ 300 nm, a strong LFR appears at 275 Oe, the uniform modes is said to have reduced in intensity (B1800 Oe), though it has probably been transformed into a localised mode with a weaker intensity, with further resonance peaks at 710 and 2110 Oe. As a series of samples, we see a shift of the uniform mode to higher fields and a weakening intensity as the inner diameter of the rings increase. It seems likely, from these measurements, that as the inner diameter increases and thus the ring width decreases, localisation effects and confinement in the plane become more important, where boundary conditions lead to pinning. The number of edge spins also increases and what was the uniform mode transform into a spin-wave mode of fundamental wave vector and requires higher fields for the resonance condition to be satisfied. At the same time, the lower field resonances become stronger with increased inner diameter. Broadband FMR measurements in square arrays of permalloy rings with dimensions: inner/outer ring diameters of 0.85/2.1 mm, thickness of 25 nm and centre-to-centre spacing of 4 mm, were performed by Zhu et al. (2005). In the static magnetisation state, the rings show no internal DW structure, where Lorentz microscopy indicates a circulation state with both clockwise and anticlockwise orientations. The broadband FMR was performed using a pump-probe technique, see Fig. 2.164(a), where the probe beam has been focused at various points on the ring structure to show the positional dependence of spin-wave excitation modes (see Fig. 2.164(b)22.164(d)). Fourier transforms allow the evaluation of the spin-wave frequencies, which give the spectrum amplitudes as a function of frequency, Fig. 2.165. Three principal modes (3.8, 4.3 and 7 GHz) are evident in the spectra and depend on the spatial uniformity and in-plane component of the transient excitation field. Mode identification was performed using micromagnetic simulations using different initial pulse fields: perpendicular, in-plane and a combination of the two. Temporal and frequency responses are illustrated in Fig. 2.166, which also indicate the modal patterns. For perpendicular pulsed fields, only radial modes are excited, while parallel pulse excitation leads to a low-frequency mode with a pair of nodes and mirror symmetry. Higher frequency modes, with much reduced amplitudes, are also evident in both cases and correspond to radial modes. The combination of in-plane and out-of-plane pumping leads to all modes being excited. It is noted that

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Probe Beam

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(b)

1x10-4 0

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-1x10-4 1x10-4

(c)

5x10-5 0 -5x10-5 1x10-4

(d)

5x10-5 0 -5x10-5 0

1000

2000

3000

4000

Time (ps)

Figure 2.164 Experiment set-up and time-domain ferromagnetic resonance (FMR) curves of permalloy rings in the circulation state. (a) Schematic drawing of the experimental set-up (not to scale). The ring array (not to scale) is placed on top of the co-planar transmission line. The probe beam focuses on the rings through the backside of a transparent substrate patterned with transmission lines. Close to the edge of the transmission lines, only partial rings are exposed to the view of the probe beam. (b) The time-domain FMR curve averaged over an entire ring near the centre of the transmission line structure. (c) Time-domain curve corresponding to (b), but locally probed within the ring (as indicated by the inset cartoon). (d) A locally probed response for a ring positioned close to the edge of the transmission line gap (as schematically shown in the inset). (Reprinted with permission from Zhu et al. (2005), r 2005 American Institute of Physics.)

only those modes with asymmetry in the magnetisation will be observed experimentally. Permalloy nanoring structures were studied using both static (DC) and high-frequency measurement techniques by Podbielski et al. (2005) and

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Figure 2.165 Spectrum amplitudes as a function of the frequency: (a) Fourier transform curve of an entire ring; (b) for focusing part of a ring and (c) for a ring close to the transmission line. The spectrum amplitude is magnified by a factor of 10 for the frequency between 6 and 10 GHz. (Reprinted with permission from Zhu et al. (2005), r 2005 American Institute of Physics.)

Giesen et al. (2005a). The hysteresis loop of Fig. 2.167, shows the static configurations in low and high fields where the vortex and onion states dominate, respectively. The transitions between these spin configurations are sharp with the onion state being characterised by a large stray field, while the vortex state has a vanishing stray field. Again broadband dynamic measurements are made using a CPW, where the static field is applied in the plane of the rings. At high applied external fields, two resonance modes are observed, while at remanence only a single mode is evident. The spatial separation between nanorings and the number of rings in the array are not important for the HFR mode (which persists at remanence). When the applied field is reduced and inverted through zero, this resonance mode apparently branches into two modes. For high fields, that is in the onion state, spins are predominantly aligned along the external field and the RF field can efficiently exert a torque on the spins. To explain the existence of the HFR, the authors consider the ring structure as being made up of two segments, as shown in Fig. 2.168, which differ in their effective internal fields, Hint. The resonance frequency for the uniform mode in segment I is


327

Figure 2.166 Simulation results of time-domain magnetisation (a)--(c) and corresponding Fourier transformation (d)--(f) of permalloy ring in respond to different pulsed field. (a) and (d) Transient field applied perpendicular to the ring; (b) and (e) transient field applied parallel to the ring; (c) and (f) transient field applied with an angle of 101 to the normal direction of the ring. The spectrum amplitude is magnified by a factor of 10 for the frequency from 5 to 10 GHz. Insets show the spectrum amplitude map of the ring with dominant spectrum amplitude. (Reprinted with permission from Zhu et al. (2005), r 2005 American Institute of Physics.)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi given by the Kittel relation: f res / ðH int þ M ÞH int , where the internal field can be expressed as H int ¼ H ext þ H anis . In the high field situation (HextZ90 mT), the internal field is large in segment I and small in segment II. This is understood as arising from the large anisotropy field, Hanis, due to a small ring width for segment I. The low-frequency mode arises from segment II where the demagnetising field must be subtracted from the external field, Hext. This argument is valid only for the sample in the onion state. As the external field is reduced the spin configuration will alter, in accordance with the hysteresis loop, and the spins turn gradually towards the perimeter and DWs form. The lower resonance mode tends to broaden and disappears at an external field of r40 mT. For intermediate fields two branches of the main resonance are observed, which reflects the vortex state. Spins in segment II are aligned with the perimeter perpendicular to the CPW such that Hhf exerts no torque. However, excitation in segment I is still possible. Now the two sides of the ring will have oppositely aligned magnetisations with respect to the applied external field and the resonance mode splits. The resonance frequency increases with Hext in segment I which is aligned along the field such that H int ¼ H ext þ H anis . However, the resonance frequency will decrease with the increase of Hext in the

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Figure 2.167 Simultaneous measurements of (a) the stray field and (b) the magnetoresistance (MR) of an individual nanostructured ring (D ¼ 2 mm, w ¼ 200 nm, t ¼ 37 nm). An offset has been subtracted from the stray-field curves. In (a), the characteristic magnetisation configurations are schematically indicated. Long horizontal arrows show the sweep direction of Hext. Irreversible jumps occur at the same fields in (a) and (b) as highlighted by dotted vertical lines. The abrupt increase in Rring can be correlated directly with the vortex state as evidenced by the intermediate plateau regions in the stray-field data. The tiny jump at H ¼ 0 in the grey curve in (b) is not reproducible and is most likely due to noise in the set-up (Podbielski et al., 2005).

portion of segment I with the field aligned antiparallel, where we have H int ¼ H anis H E . This forms the lower branch with negative dispersion and will disappear when the ring irreversibly jumps to the reversed onion state, leading to H int ¼ H E þ H anis in both segments I of the ring. Interaction between rings due to the stray field of neighbouring rings is expected for the onion states and segments II will be more sensitive to these effects. In Fig. 2.169, the dispersion relations are shown for two different arrays of nanorings with the same ring dimensions. These are very similar and differences are minimal. The effect of ring width was also investigated by these authors, which shows that as the width of the ring increases, while maintaining the outer diameter constant, the resonance modes tend to collapse, that is the separation between modes reduces and tend towards the thin film resonance frequency.

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Hext

(a)

Hext

Hext

(b)

(c)

Hext

(d) D/2+d

segment I

segment II

Figure 2.168 (a) Schematic representation of the onion state as calculated by OOMMF for a ring of width w ¼ 500 nm, thickness t ¼ 40 nm and outer diameter D ¼ 2330 nm at 100 mT. Arrows indicate the in-plane orientation of local magnetic moments. (b) Spatial profile of the internal field Hint derived from (a): white -- high field, dark -- low field. (c) Simulated spin configuration at + 10 mT which represents the vortex state. (d) MFM picture of rings (D ¼ 2 mm) at 100 mT: black and white colour indicates high stray field of positive and of negative polarisation, respectively. Stray-field lines are sketched (Podbielski et al., 2005).

Figure 2.169 Magnetic field dispersion of the main FMR mode A at high frequencies and the nearest lower lying weak mode measured on two further arrays: (a) stepwise increase and (b) stepwise decrease of Hext. In both samples, the width w was about 300 nm. Open and closed symbols are for the different arrays as depicted in (c) (scanning electron micrographs) (Podbielski et al., 2005).

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Further studies on permalloy ring structures have been made by taking spectra after applying a saturation field ( + 90 mT), Podbielski et al. (2006). The rings exhibit a reversal behaviour as discussed above, with three regimes which are characterized as: the onion state, the vortex state and the reversed onion state. These regions are demarcated by the switching processes. In addition to the resonances discussed above, a stepwise behaviour in the vortex state has been observed, which are labelled as In, where n is an integer starting at unity. Here the spin-wave eigenfrequencies are discrete and form plateaus which are only weakly dependent on the applied magnetic field. This complex behaviour is illustrated in Fig. 2.170, where frequency gaps of around 500 MHz are observed for the 250 nm ring width structure. The discrete spin-wave behaviour is more clearly observed in Fig. 2.171, where the rings are initially saturated at 90 mT and then a field of 14 mT is applied to produce the vortex state. The vortex state is characterised by spin-wave modes which have mirror symmetry with respect to H ¼ 0, where only the I1 mode is observed. Each spin-wave excitation is observed within a certain field range, upto a critical field, Hc,n, which increases with n. At the same time, the eigenfrequency decreases in value. To calculate the spin dynamics of the ring system, use is made of the circular magnetisation configuration illustrated in Fig. 2.172(a) and spinwave modes are indexed by integer modal lines m and n along the radial ~ er and azimuthal~ ef directions, respectively. The total wave vector K can then be expressed as K ¼ k2r þ k2f , which are defined in Fig. 2.172(b).

Figure 2.170 Grey-scale plot of absorption spectra taken at successively decreased magnetic field (cf., dashed arrow) after saturation at l0H ¼ + 90 mT (onion state). Dark represents strong absorption. In the onion state modes, A and B are detected consistent with Giesen et al. (2005b). In the vortex state discrete spin-wave eigenfrequencies I1; I2; I3; I4 (see labels) and Au are resolved (cf. also Fig. 2.169). Black solid vertical lines indicate switching fields. (Reprinted figure with permission from Podbielski et al. (2006), r 2006 by the American Physical Society.)


331

Figure 2.171 (a) Grey-scale plot of spectra taken in the minor loop starting at 14 mT. Dark represents strong absorption. The contrast for 10 GHzrfr13:2 GHz is increased. (b) Full symbols are eigenfrequencies extracted from the individual absorption spectra. Open circles refer to the quantized modes calculated via Eq. (229). Numbers in brackets label the modes (m; n) from the semianalytical calculations. (Reprinted figure with permission from Podbielski et al. (2006), r 2006 by the American Physical Society.)

Figure 2.172 (a) Symmetry of the experiment: Hrf and the external field H are orthogonal. Small arrows indicate the local magnetisation m in the ring’s vortex state. (b) Definition of parameters. (c) Spatial profile of the internal field Hint as a function of /. H6¼0 leads to a spatially oscillating field Hint, calculated by micromagnetic simulations using OOMMF (open symbols) and modelled by Hint ¼ Hcos / (solid line). (Reprinted figure with permission from Podbielski et al. (2006), r 2006 by the American Physical Society.)

Quantisation will be expected in the radial direction much in the same way as a magnetised wire, where discrete values are given as km ¼ kmr. The azimuthal component of the wave vector, which is parallel to the ring magnetisation in the vortex state, represents a BVMSW and is subject to the following quantisation rule: I

2pn ¼

kf ðf ; H int ðfÞÞrdf,

(229)

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where r is the radius of the integration path and Hint is the internal magnetic field, which includes external, demagnetising and effective exchange fields. For non-zero fields, the internal field varies along f and so consequently will kf. Neglecting the exchange energy in the remanent vortex state is mathematically equivalent to a longitudinally magnetised wire with no demagnetisation field which exhibits periodic boundary conditions along the axis. The calculation then proceeds as given by Guslienko et al. (2003). An important difference is that if a magnetic field is applied to the ring, the internal field becomes a spatially oscillating function. Using the simplifying condition of r ¼ RC, the simulation shows good agreement with experiment for H ¼ H cos f, as seen in Fig. 2.172(c). The quantisation condition will require that the spin-wave modes satisfy constructive interference. The experimental findings are in broad agreement with the BLS measurements of Gubbiotti et al. (2006) who consider a square array of nanorings with an outer radius of 355 nm, a width of w ¼ 200 nm and a separation of 330 nm. Gubbiotti et al. (2006) and Montoncello et al. (2008) have performed OOMMF simulations of this system to interpret their results. Results of BLS measurements and calculated spin mode frequencies are illustrated in Fig. 2.173. The transition

Figure 2.173 Calculated spin mode frequencies (continuous and dotted lines) of the ring magnetized in the vortex and saturated (onion) states. The continuous lines represent modes of appreciable BLS intensity and compare with the experimental points taken from Gubbiotti et al. (2006). Dotted lines are modes of vanishing or small BLS intensity. The inset shows the vortex state geometry. (Reprinted with permission from Montoncello et al. (2008), r 2008 American Institute of Physics.)


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of the vortex to onion states is also shown. Calculations are performed using a dynamical matrix approach of the normal component of the dynamical magnetisation mz (real) for the various modes are shown in Fig. 2.174 under various in-plane applied magnetic fields. The splitting of the (71, 0) backward modes (BA) and transformation of modal patterns in the transition between the vortex and onion states is shown in more detail in Fig. 2.175, which is accompanied by a discontinuity in the frequency spectrum. It is noted that the splitting of (7m, 0) modes arises from their symmetry and that the corresponding splitting magnitudes are smaller than in disk samples since there is no interaction with the vortex core, which is absent in the nanorings. For higher applied fields, the modes become more localised and turn into end modes excitations. These authors interpret the modes as localised excitations for all values of the applied magnetic field.

7. Summary Ever since the pioneering work of Kittel, Seavey and Tannenwald, Rado, etc. of the 1950s, the technique of ferromagnetic (and spin wave) resonance has been an important method of the characterisation of magnetic materials and in particular thin films. In recent years, there has been an important multiplication of experimental methods for the observation of FM excitations, such as the vector network analyser, pump 2 probe spectroscopy and the electrical detection methods. This development is a direct reflection of the increased interest in the dynamic properties of magnetic materials and with a view to their potential applications in spintronic and magnetic data storage devices. The magnetisation dynamics in small magnetic entities suffer modifications to the excitations that would be present in bulk materials due to the spatial restrictions imposed as boundary conditions. As we have seen these can cause very significant changes in the dynamic response of the material. Some of the main ingredients in FMR in nanometric magnetic structures can be listed as follows: 1. Ratio of surface to bulk spins: structure size dependence and relative

surface and bulk anisotropy contributions. 2. Interactions between nanostructures: DDI can affect overall effective

fields experienced by both surface and bulk spins. 3. Size (volume) dispersion: monodispersion typically occurs in artificial

nanostructures while NP assemblies typically always display some form of polydispersion. 4. SPM: interplay between size and effective particle anisotropy.

(1,0)

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Figure 2.174 Calculated normal component of the dynamical magnetisation mz (real part) for different modes observed in the BLS spectra at different values of the field H, applied along the vertical direction. The last row corresponds to the saturated state at H ¼ 600 Oe. The horizontal brackets indicate the splitting of the (0,1) and (0,2) modes. The inset (at the bottom of the first column) refers to what has been labelled as F-loc in Fig. 2.172. (Reprinted figure with permission from Gubbiotti et al. (2006), r 2006 by the American Physical Society.)

David Schmool

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9.5 9.0 8.5 8.0 7.5 7.0 0

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300 400 H (Oe)

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Figure 2.175 Evolution of the (71, 0) doublet into backward-like modes (1-BA and 2-BA) versus the applied magnetic field H. The bold curves indicate the frequency of the BLS active modes, symmetric with respect to the scattering plane, while the dotted curves correspond to antisymmetric modes. The profiles of the dynamic magnetisation around the transition field are shown. (Reprinted figure with permission from Gubbiotti et al. (2006), r 2006 by the American Physical Society.)

5. Reduced magnetisation from the saturated value: significant variations of

the nanoparticle magnetisation have frequently been reported and the use of the Langevin function and derivatives thereof are common in accounting for this. FMR provides a powerful experimental tool in the investigation of the magnetic properties of magnetic nanostructures. In the case of randomly oriented and dispersed particles, the characteristic spectra display broad resonances which are sensitive to small changes in sample temperature. Much of the temperature variation is due to SPM effects which dominate their properties at high temperatures (TWTB). Distinct resonances appear from blocked and unblocked particles having differing resonance conditions. In addition, further resonances can occur from surface effects. Interactions, principally due to dipolar effects, provide a homogenisation of the effective field and can also significantly modify the magnetic properties of the assembly. The spectra observed in regular arrays of nanostructures do not suffer from the extrinsic broadening of random arrays since shape and anisotropy axes are aligned and provide a well-defined magnetic system. The latter display rich excitation spectra, whose nature is intimately related to the symmetry of the magnetic entities. Filled forms can exhibit a strong dependence and coupling to the vortex states common in these systems. Interparticle interactions will also allow the coupling of excitations to form cooperative effects between the particles.

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Interest in the dynamical process in magnetic systems is of enormous technological interest due mainly to the possible applications in spintronic devices and magnetisation reversal processes for data storage technologies. Indeed much current research is directed at the understanding of the reversal dynamics and the possibility of optimising the temporal response via an understanding of the relaxation process. With the recent development of electrical detection and current-induced magnetisation dynamics via the spin angular momentum transfer and spin accumulation, the manipulation of the magnetisation can be performed in a convenient manner through the application of electrical signals. This offers a usable and easily applicable method for magnetisation reversal in magnetic elements which can be employed in data storage devices, for example. If recent trends are to continue, we can expect further development of dynamical techniques for the study of magnetisation processes in magnetic nanoparticles and nanostructures.

ACKNOWLEDGEMENTS I am indebted to my close collaborators Professor Hamid Kachkachi and Dr Jose Garitaonandia for useful discussions and exchanges on the general topic of magnetic nanoparticles. I am also grateful to Professor Michael Farle and the members of his research group, in particular Dr Ju¨rgen Lindner and Dr Ralf Meckenstock. I would also like to express my gratitude to Professor Bret Heinrich. Several of my students have worked in the area of FMR and nanoparticles over recent years and I would like to express my gratitude to Nuno Sousa, Arlete Apolina´rio, Paulo Madureira, Ana Silva, Pedro Monteiro, Markus Schmalzl and Rui Rocha for their hard work.

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CHAPTER THREE

Magnetic Sensors: Principles and Applications Pavel Ripka1, and Karel Za´veˇta2 Contents 1. Introduction 2. Materials for Magnetic Sensors 2.1. Semiconductors 2.2. Soft magnetic materials 2.3. Hard magnetic materials 3. Important Effects and Parameters of Magnetic Sensors 3.1. Perming 3.2. Crossfield effect 3.3. Anisotropy 3.4. Noise 3.5. Temperature stability 4. Magnetic Field Sensors 4.1. Hall sensors 4.2. Flux concentrators 4.3. Semiconductor magnetoresistors 4.4. Ferromagnetic magnetoresistors: AMR 4.5. GMR and SDT sensors 4.6. Fluxgate sensors 4.7. SQUID 4.8. Resonant sensors and magnetometers 4.9. Induction magnetometers 4.10.GMI 4.11. Other devices 5. Magnetic Sensors of Position and Distance 5.1. Inductance and transformer sensors

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Corresponding author. Tel.: + 420 736 760 601

E-mail address: [email protected] 1 2

Czech Technical University, Department of Measurement, Technicka 2, 166 27 Prague, 6, Czech Republic. Institute of Physics, Academy of Sciences of the Czech Republic, v. v. i., Na Slovance 2, 182 21 Prague, Czech Republic.

Handbook of Magnetic Materials, Volume 18 ISSN 1567-2719, DOI 10.1016/S1567-2719(09)01803-4


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6.

7.

8.

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5.2. Magnetostrictive position sensors 5.3. Position sensor with permanent magnets 5.4. Magnetic proximity switches 5.5. Speed and flow sensors Force and Torque Sensors 6.1. Force sensors 6.2. Torque sensors Electric-Current Sensors 7.1. Instrument current transformers 7.2. Rogowski coil 7.3. DC current transformers 7.4. Hall current sensors 7.5. AMR current sensors 7.6. Other principles for current sensing 7.7. Current clamps Applications of Magnetic Sensors 8.1. Position measurement 8.2. Position tracking 8.3. Navigation 8.4. Antitheft systems 8.5. Detection of ferromagnetic objects 8.6. Space research and geophysics 8.7. Medical distance and position sensors 8.8. Nondestructive testing (NDT) Conclusions References

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1. Introduction The term magnetic sensors is actually used in at least two different senses. The most common one concerns the sensors of magnetic field of various origins, from the Earth’s magnetic field to the stray fields produced by bits of magnetically stored information. The magnetic sensors of the second type comprise various sensors that use magnetic materials or principles and may be exploited for measuring either magnetic or nonmagnetic quantities. Usually the first meaning of the term magnetic sensor is used without explicitly specifying that we speak about sensing or measuring the magnetic field. By far most of the produced magnetic sensors are devices based on the Hall effect. These semiconductor sensors are cheap and can be made small, but their resolution and stability is very limited. If higher accuracy is required, soft magnetic material should be employed in the sensor 2 as either yoke, field concentrator, or functional

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element. The fast development of ferromagnetic magnetoresistors (AMR, GMR, and SDT) for magnetic reading heads brought these devices also on the sensor market. Although anisotropic magnetoresistors (AMR) and other sensors based on magnetic thin films are much more sensitive than Hall sensors, their performance is still limited. Sensors with cores, yokes, or field concentrators made of bulk magnetic material are more sensitive and stable than thin-film sensors. What are the most critical parameters for magnetic sensor? It is not sensitivity for sure, as amplification is so cheap. Nonlinearity and temperature dependence of sensitivity can often be suppressed by a feedback. Noise matters, but usually the most serious problem of sensors containing magnetic material is remanence, crossfield sensitivity, and temperature stability of offset. The general trend for miniaturization and integration of electronic elements was reflected in long-time effort to miniaturize the fluxgates, which, however, led only to a few practical designs. For flat sensors (either PCB or CMOS) the core etched from amorphous tape gives better properties than electrodeposited or sputtered core. We shall specifically compare traditional miniature fluxgates using wire cores based on longitudinal fluxgate effect with those using transverse fluxgate effect and also with GMI sensors. Well-designed field concentrators or yokes can improve the parameters of any magnetic sensor. The achievable stable amplification factor is from 10 to 100. Having a means to demagnetize the field concentrator is desirable. An overview of magnetic sensors for mechanical quantities will also be given with special focus given to torque sensors. In each section we shall shortly review the main principles and then discuss the recent developments since the following reference books appeared: the overview of Boll and Overshott (1989) and Ripka (2001a) deal with all magnetic sensors, while the book of Tumanski (2001) is concentrated on magnetoresistors, Popovic (2004) on Hall sensors, and Clarke and Braginski (2004, 2006) on SQUID. Recent journal reviews on magnetic sensors include Edelstein (2007), Lenz and Edelstein (2006), and Ripka (2008). For general books on sensors we refer to Fraden (2004), Ripka and Tipek (2007), and Webster (1999).

2. Materials for Magnetic Sensors Predominant functional materials for magnetic sensors are semiconductors; smaller part of magnetic sensors use soft magnetic materials, and hard magnetic materials are mainly used as targets or to create a bias field.

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2.1. Semiconductors Semiconductors are used in Hall sensors (Section 4.1) and semiconductor magnetoresistors (Section 4.2). Traditional silicon is used for integrated Hall sensors, which are typically made by CMOS technology. Another popular material is GaAs. High mobility semiconductors, such as InSb and InAs, display higher sensitivity. The working temperatures of traditional Hall sensors range between 100 and + 1001C, but some sensors work from milli-Kelvin temperatures and other up to 1801C (SOI structures and InSb). Silicon carbide 4H-SiC Hall sensors may work up to 5001C, but the smallest achievable thickness is too large. AlGaN/GaN heterostructures have wide bandgap and they are stable even in small thicknesses and at elevated temperatures; Hall sensors in this technology work up to 6001C (Yamamura et al., 2006).

2.2. Soft magnetic materials Crystalline, amorphous, and nanocrystalline magnetically soft alloys are used for ferromagnetic magnetoresistors (AMR, GMR, and SDT: Sections 4.4 and 4.5), fluxgates (Section 4.6), and more exotic sensors (e.g., GMI in Section 4.10; Jiles and Lo, 2003). Soft magnetic alloys are also used in flux concentrators to enhance the sensitivity of some Hall sensors and magnetoresistors (Section 4.2) and also as cores of induction coils (Section 4.9). Magnetically soft magnetic shields are used in giant magnetoresistor (GMR) sensors. Yokes for position sensors based on reluctance change are another application (Section 5.4). The requirements on magnetic materials for these sensors are rather diverse: while minimum Bs and minimum ls are usually required for fluxgate sensors, maximum Bs is needed for most of other sensors and large ls is required for magnetoelastic devices (Jiles and Lo, 2003). Minimum coercivity, high or constant permeability with minimum temperature dependence, and high electrical resistivity are general requirements, but in special cases such as bistable proximity sensors square-loop materials are utilized. Most of the magnetically soft magnetic materials used in sensors are crystalline; less often amorphous alloys are used. Nanocrystalline soft magnetic alloys are rarely used because of their brittleness. Nanocrystalline cores for instrument current transformers are one of the favorable applications (Draxler and Styblikova, 1996). Sixteen percent of Si Finemet was used by Butvin et al. (2003) for fluxgate sensor of relaxation type. The material has a Curie temperature of 6001C compared to 2101C for commonly used Vitrovac 6025. The saturation magnetization is 1.12 T, which is normally regarded as too high for fluxgates, as such material requires higher excitation power. However in this case of sensor working in open-loop (uncompensated) mode, higher saturation increases the sensor


351

range. All the mentioned materials are used in bulk form and as thin films made by sputtering, electroplating, or laser deposition. Bulk materials are produced in the form of thin tape or also a wire. The magnetic properties of the bulk material are usually better than those of thin films: minimum achieved coercivity for a Permalloy film electroplated onto a sputtered buffer layer of Cu was 280 A/m (Yi et al., 2007). Most common sensors based on soft magnetic alloys are AMR sensors, which reached 100 times higher field resolution than Hall sensors with the same size and power consumption. They are usually based on sputtered Permalloy and work up to 2251C.

2.3. Hard magnetic materials Hard magnets are rarely used in magnetic field sensors: if they serve as biasing magnets, problems arise with the time and temperature stability of such field. Magnetic field sensors based on force between the measured field and the permanent magnet belong to the history, but they reappeared in MEMS technology (Section 4.11). More often permanent magnets are used as a field source or target for position sensors. Hard ferrites and NdFeB are most often used in these applications, but SmCo magnets are also used. Permanent magnets from isotropic material can be magnetized in a multipole pattern. Most often ferrite rings are magnetized in an n pole pattern on the circumference with a multipole pulse coil. Recently, many of these magnets have been substituted by polymer-bonded anisotropic magnets. By field-assisted injection molding of permanent magnet powder in a polymer matrix it is possible to orientate the powder particles during the injection process and thus to form a magnet with multipole anisotropy (Gro¨nefeld, 2007).

3. Important Effects and Parameters of Magnetic Sensors Similarly as other sensors and devices, important parameters for magnetic sensors are temperature coefficient of offset and sensitivity, noise, linearity, and hysteresis. Errors specific for magnetic sensors are perming and crossfield error. We also discuss anisotropy, magnetostriction, and other effects utilized in magnetic sensors.

3.1. Perming Perming (sometimes referred as remanence) is a change of the sensor offset after the shock of strong pulse of magnetic field. It occurs in all sensors that

352


contain functional magnetic material (either as core of fluxgate or as GMI sensors, field concentrator, or shields used for Hall sensors and magnetoresistors) or other ferromagnetic parts (such as nickel-plated wires or ceramics containing magnetic particles). Efficient way how to suppress the perming is periodic remagnetization of functional magnetic parts and magnetic cleanness of the rest of the sensor.

3.2. Crossfield effect Crossfield effect is a nonlinear influence of the magnetic fields that are perpendicular to the sensing direction. Crossfield appears in fluxgates and AMR sensors. Careful design of fluxgate sensor can reduce crossfield error caused by the Earth’s field below 1 nT (Ripka and Billingsley, 2000). In case of AMR sensor crossfield error may be severe. It can be reduced by shape anisotropy, corrected by iterative algorithm; but the best way to get rid of crossfield error is to use the AMR sensor in feedback compensated mode (Kubı´k et al., 2006a).

3.3. Anisotropy Magnetocrystalline anisotropy is usually averaged off in polycrystalline materials. However, induced and shape anisotropies may be present, which are important for the sensor performance. They regularly are of uniaxial type and the density of anisotropy energy Eu may be written in the first approximation as Eu ¼ K u sin2 y,

where Ku is the constant of uniaxial anisotropy and y measures the angle from the easy direction. This anisotropy may also be characterized by the anisotropy field Ha, which is given by Ha ¼

2K u , Ms

where Ms is saturation magnetization. If we apply magnetic field perpendicular to the easy direction, the component of magnetization in the field direction M is proportional to the applied field H being equal to M ¼ Ms

H Ha

for H H a ,

and the magnetization is rotated into the direction of hard axis at H ¼ Ha. Two uniaxial anisotropies with anisotropy fields Ha1 and Ha2 result in uniaxial anisotropy in the same plane. If they have the same direction, the resulting characteristic field H0 ¼ H1 + H2. In general case, when e is the


353

angle between easy axes, the direction of the resulting anisotropy is inclined by a from H2: 1 H 1 sin 2 , a ¼ arctan 2 H 1 þ H 2 cos 2

and its characteristic field is (Sczaniecki et al., 1974) H0 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H 21 þ H 22 þ 2H 1 H 2 cos 2.

In particular, for e ¼ 901, the angle a is either 0 or 901 depending on the relative magnitudes of Ha1 and Ha2, with easy axis along the larger of the two anisotropy fields and resulting Ha given by their difference. For Ha1 ¼ Ha2 where the angle a is not defined by the given relation, we obtain Ha ¼ 0, that is, isotropic case. 3.3.1. Induced anisotropy and magnetic aftereffect Anisotropy may be induced by a magnetic field applied during the deposition process or by subsequent field annealing. In order to induce homogeneous anisotropy in the whole volume of the material, the field must be sufficiently high to practically saturate the sample, as the anisotropy is actually induced not by the field, but by the corresponding distribution of local magnetization. This induced anisotropy is often exploited in AMR sensors. The sensor volume becomes essentially single domain and the magnetization process along the hard direction is then rotational instead by domain wall movement. The characteristic field of this anisotropy is typically 2502300 A/m (Tumanski, 2001). Nielsen et al. have used stress annealing of the low-magnetostriction Co-based amorphous tape to produce a creep-induced anisotropy with easy axis perpendicular to the excitation field direction. This treatment was shown to decrease the sensor noise (Nielsen et al., 1990). The anisotropy is in principle induced at any temperature, but the rate with which it approaches its equilibrium value increases at elevated temperatures; during the process the direction of local magnetization becomes the easy direction. This also applies to the case of non-saturated or demagnetized sample with domain structure where the direction of the magnetization in the domain becomes the easy direction and similarly the continuously changing direction of the local magnetization within the volume of domain walls. The domain walls are thus stabilized in their positions by the induced anisotropy, this process being particularly effective for 1801 walls, as the easy direction is identical on both sides of the wall. This stabilization leads on one hand to special types of hysteresis curves (perminvar, constricted loops) and on the other hand to a time decrease of differential permeability during stabilization of the domain walls in their

354


new positions after changes of magnetization. It is worth noting that, in particular, the amorphous alloys are rather susceptible to the manifestation of this phenomenon even at room temperatures, and demagnetization, which helped against perming, may lead to rather large changes of permeability with time. 3.3.2. Shape anisotropy Due to demagnetization, the magnetic field Hi inside the magnetic sample with finite dimensions is smaller than the external field Hext H i ¼ H ext DM c ¼

H ext , 1 þ Dðm 1Þ

where D is the dimensionless demagnetization factor and Mc ¼ kHc is the magnetization. From that B¼

m0 mr H 0 ¼ m0 m a H 0 , 1 þ Dðmr 1Þ

where ma is the apparent relative permeability, ma ¼ mr/(1 + D(mr1)). For very high mr: ma-1/D. In case of non-ellipsoidal samples, a local or global (averaged, magnetometric) demagnetization factor can be defined. Formulae for rectangular prisms are given in a paper of Aharoni (1998). For long strips one can use an estimate D¼

t t , tþw w

where w is the width and t the thickness of the strip (Tumanski, 2001). The effective demagnetization factor for long rods (m ¼ l/dW10, where l is the rod length and d its diameter) was approximated for mr ¼ N by the Neumann2Warmuth empirical formula: D

2:01 logðmÞ 0:46 . m2

For finite permeability D is decreased. Demagnetization factors for prolate ellipsoids and cylinders are plotted versus m in an extremely important graph in Fig. 3.1, which is in various forms reproduced in many handbooks. We can see that for high permeability and moderate values of the apparent permeability, the demagnetization is given by the shape. This is valid for any shape and important for the design of cores for induction sensors (Section 4.9) and flux concentrators (Section 4.2). The demagnetization factor for ring cores and racetracks is discussed in Section 4.6.

355


105

10-1 μr =3.104 D 10-2

μa 104

Del ∞

104 3.103

Dcyl

10-3

103

10-4

102 μr =104 μr = ∞

10-5 1

10

100

1000

m

Figure 3.1 Demagnetization factors and apparent permeability of the prolate ellipsoids and cylindrical rods, as a function of m ¼ l/d. While in case of ellipsoids D is independent of mr, in case of rods mr is a parameter -- after Boll and Overshott (1989). Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission.

Let us mention that the demagnetization field is homogeneous just for ellipsoidal bodies or shapes that are limiting cases of ellipsoids (thin strips, films, wires, and long rods). The demagnetization factor in any direction is then constant over the whole body and is a tensor of second order. Such a tensor may be always transformed to the principal axes with Dx, Dy, and Dz being the only nonzero components of the tensor. The useful relation Dx þ Dy þ Dz ¼ 1

may be conveniently exploited for some (limiting) cases; for example, the demagnetization factor for a sphere is obviously equal to 1/3. For a planar sample the factor is 0 for any direction in the plane and equal to 1 in the direction perpendicular to it, and for a thin and long rod (wire) D is zero along the rod and equal to 0.5 in the direction perpendicular to it. The mentioned restriction to ellipsoidal bodies means that only for them the demagnetization factor is constant for a given direction over the whole volume of the sample. In such case in homogeneous external field

356


the inner field is also homogeneous. On the other hand the bodies, whose shapes are rather different from ellipsoids or their limiting cases, possess demagnetization factors varying with position, and thus the inner field is not homogeneous. As an example let us compare the homogeneous case of a sphere with a cube, where in its corners the demagnetization factor is rather different from the expected average value of 1/3 and thus the inner field is also different. This fact, together with the dependence of permeability on field, leads to the annoying phenomenon that various parts of such bodies are saturated at rather different (external) fields. The important design tool for magnetic devices is finite-element simulation (FEM). Anisotropy was added into the model of the hysteresis loop that can be used in FEM studies (Szewczyk, 2007). 3.3.3. Susceptibility tensor in magnetic wires In general the axial and circular magnetization components are given by: Mz Mf

! ¼

kzz kfz

kzf kff

!

Hz Hf

! .

The diagonal components correspond to axial and circular magnetization loops. The Mz2Hf loop is related to the inverse Wiedemann effect (IWE) and the Mf2Hz loop to the Matteucci effect. These two effects appear only in the presence of helical magnetic anisotropy caused by stress and strain caused by applied torque or torsion annealing (Knobel et al., 2003; Kraus et al., 1994).

3.4. Noise The low-frequency noise of magnetic sensors is usually of 1/f type. In order to be able to quantitatively compare various cases, the noise should be given in a reliable way. The best form is the power spectrum density P(f) (PSD) at 1 Hz given in field units. P has a unit of nT2/Hz and for 1/f noise we may write P(f) ¼ P(1)/f. The noise is usually expressed as OP in the units of nT/ OHz or pT/OHz. For 1/f noise such noise density changes with 1/Of. The other possibility is to give rms noise in a given frequency interval fL to fH. With only small inaccuracy, fL is usually given by the observation time and fH as a frequency band of the data acquisition circuits. Rms noise can be calculated either from the collected time-domain data or by integration of the power noise spectrum, which for 1/f noise is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi f Pðf Þ dt ¼ Pð1Þ ln H . fL

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z fH

N rms ¼ fL

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Y = 2.919 (pTrms/√Hz)

1.03760 Hz 1000 noise PSD (pTrms/√Hz) 100 10 1 0.1 0.1

1

10

frequency (Hz) 40

sensor noise (pT)

20 0 -20 -40 0

5

10

15

20

25

time (s)

Figure 3.2 Noise of Billingsley Magnetics fluxgate sensor. The sensor core is a 17 mm diameter amorphous ring -- from Ripka and Billingsley (1998).

1/f noise at low frequencies was observed in fluxgate sensors (Ripka, 2001a), in GMR multilayers (Gijs et al., 1996), and also at Hall sensors (Kunets et al., 2005). An example of noise PSD of a fluxgate sensor is given in Fig. 3.2. The value 3pT/OHz@1Hz corresponds to 6.7 pT rms in the frequency range of 60 mHz210 Hz. The peak-to-peak value is not a reliable variable for expressing the sensor noise 2 it is, however, practical as can be seen directly from the plot. In this case the p2p noise is about 35 pT. Some authors give the noise power density at some higher frequency, which obviously gives more favorable results. However, it is often possible to recalculate the noise to 1 Hz using the 1/f rule, which seems to be practically universal for magnetic sensors at low frequencies. Sometimes the declared white noise of several pT/OHz corresponds to tens of nT/OHz@1 Hz.

3.5. Temperature stability The temperature stability of the sensor output is often the most important parameter for sensor application. This parameter is repeatedly underestimated both in the scientific literature and in the datasheets of some manufacturers. Temperature parameters are difficult to measure and may

358


show that theoretical resolution can only be reached at stable temperature (which is the case, e.g., for underwater magnetometers). First of all, the sensitivity S should be given in units of V/T (or more practical units such as mV/mT). Sensitivity should not be mixed with resolution, which is defined as the smallest field step observable in noise (this definition is very subjective, but 10 times noise density at 1 Hz is a good estimate). The temperature coefficient of sensitivity (sensitivity tempco) is defined as dS/du, where u is the temperature. If this dependence is not linear, it is a good practice to give both dS/du at room temperature and the highest value in a given temperature range. If the temperature dependence has no dominant source (such as in high-quality sensors, having all major error sources compensated by design), it is better to give maximum sensitivity deviation in the given temperature interval (such as 7100 ppm in 40 to + 801C temperature range). The sensitivity tempco can be effectively reduced by using field-compensation techniques. The most usual one is the analog negative feedback. With sufficient gain in the feedback loop, the resulting tempco is given by the properties of the feedback coil. The temperature coefficient of offset (offset tempco) is defined as dB/du (nT/K). Some authors give this parameter as a percentage from the full scale, which is misleading. Offset tempco should be measured in zero field to eliminate the possible influence of sensitivity tempco. The best sensors again do not have a dominant source for the temperature dependence of this quantity, so their offset may change with temperature in a complicated manner. In this case it is better to define the range of offset changes for the given temperature interval. This range should include possible irreversible changes of the offset with temperature and its temperature hysteresis.

4. Magnetic Field Sensors 4.1. Hall sensors More than 90% of all sensors of magnetic field nowadays in use are Hall sensors. An excellent reference on physics and technology of Hall sensors is the book of Popovic (2004). The sensors are supplied either by stabilized DC voltage or current. The Hall sensitivity for silicon sensors is typically 1 mV/mT for 1 mA current. Traditional current supply is often replaced by stabilizing the supply voltage, which gives lower temperature dependence in a wide temperature range. A five times higher sensitivity is achievable with InSb. Thin-film InSb sensors are made by molecular beam epitaxy (MBE). Proper design and use of voltage supply may reduce the temperature coefficient of sensitivity from 2 to 71%/K, but only in the 10 to + 601C range.


359

InAs sensors work in the temperature range of 40 to + 1501C required by automotive applications. Promising for future applications are 2D quantum-well multilayer structures based on InSb and GaAs. Noise of 100 nT/OHz@1 Hz was achieved using these structures and external spinning-current electronics (Karlain and Mosser, 2007). Also promising are Hall sensors made by silicon-on-insulator (SOI) technology: 1 mT/OHz@1 Hz noise was achieved for 80 mm wide, 50 nm thick cross Hall sensor (Haddab et al., 2003). Hall sensor with magnetic antenna may have 10 nT resolution (Quasimi et al., 2004). The integrated CMOS micro-Hall plate sensor with an active area of 2.4 mm2.4 mm uses spinning current to suppress the sensor 1/f noise by one order of magnitude down to 300 nT/OHz@1 Hz (Kejik et al., 2006). Most of the Hall sensors are used for position sensors (either linear or angular) in automotive applications (ignition control, AntiBlock System, ABS) and for the control of stator current in contactless DC motors with permanent magnet on rotor (from computer drives and fans up to 1 kW motors). Integrated Hall sensors are made of silicon, most often using CMOS technology. Such chip contains electronic circuits for the excitation, amplification, and other signal processing. They may have analog or twostate output, some of them even have digital output. The basic disadvantage of CMOS amplifiers are their large offsets. Other offsets originate in the imperfection of the Hall element itself. Effective offset reduction is achieved by rotation current technique: the symmetrical Hall element has four (or more) electrical contacts, whose role is periodically switched over from current supply to voltage sensing. The switching scheme includes commutation of the polarity and thus the nonsymmetry of the element (which causes the basic offset) is averaged off.

4.2. Flux concentrators Soft magnetic alloys are used in flux concentrators that are designed to enhance the sensitivity of Hall sensors and magnetoresistors. They can also be used to change the direction of the measured field: by this technique Hall IC sensitive to in-plane field can be made (Popovic et al., 2006). Shields of magnetically soft materials are used in some GMR sensors. The drawbacks of flux concentrators are mainly: remanence; nonlinearity and danger of saturation; temperature sensitivity of the gain factor.

Remanence of field concentrators is a very serious issue, although it is ignored in the majority of publications. Only few concentrators are equipped with demagnetization coils. The same coils can be used for the

360


feedback compensation to increase the linearity and avoid saturation. In order not to compromise the sensor bandwidth, the feedback may only reduce the DC component of the field. For the conservatively designed concentrators, the gain factor is reasonably low and the core permeability is high, thus the gain is mainly given by the geometry and is very stable. For high gains the temperature coefficient of permeability plays an important role. This is a well-known problem especially for ferrites at low temperatures (LTs): available materials typically experience 50% permeability drop between + 40 and 201C. The mechanical design should guarantee that the airgap variations due to the temperature dilatations are small. Concentrators 20 cm long with 100 mm airgap have a gain of 600 and the achieved noise level is 100 pT/ OHz@1 Hz with special low-noise Hall sensor (Leroy et al., 2007). Much smaller magnetometer described in Quasimi paper uses modulation of the permeability of the 5 mm long wire concentrator. This shifts the signal frequency out of the 1/f noise of the Hall sensor. The achieved noise level is 8 nT/OHz@1 Hz (Quasimi et al., 2004). Another possibility to modulate the field in the airgap is to periodically move the concentrator by a MEMS electrostatic machine (Edelstein, 2007; Ozbay et al., 2006), but compared to the field modulation of permeability, this approach is complicated and unlikely to bring a real advantage. An InSb Hall element with ferrite field concentrators (HW series) produced by Asahi Kasei Electronic is shown in Fig. 3.3. The thin-film Hall element is sandwiched between ferrite substrate and chip. The concentration of magnetic flux by ferrite pieces is shown by FEM simulation of magnetic force lines in Fig. 3.4. A magnetometer that uses four Hall sensors to measure the field near a rotationally magnetized disk made of amorphous ribbon is described in Hristoforou (2006). The device is based on the fluxgate principle with Hall sensing instead of a pick-up coil. The achieved sensitivity is 300 V/T and the field resolution is 1 nT. The device has by far better performance than a similar thin-film sensor with internal galvanic reading (based on anisotropic magnetoresistance effect), described in the same paper. Flux concentrators based on superconducting loop (flux to field transformer) may have gain of 1000 (Pannetier-Lecoeur et al., 2007b).

4.3. Semiconductor magnetoresistors They are less common than Hall sensors, but they are still used by automotive industry. Modern semiconductor magnetoresistors are fabricated as a serial connection of many miniature elements on one chip (Murata). In 1002200 mT field of permanent magnet, such structure may have very good temperature stability (Heremans, 1993). The main disadvantage of these sensors is their quadratic characteristics, which does

361


Figure 3.3 InSb Hall element with ferrite field concentrator (Asahi Kasei Electronic: HW series).

InSb Hall film Ferrite chip

Ferrite substrate

Figure 3.4 Magnetic force lines of field concentrators for a thin-film Hall sensor (FEM simulation) -- courtesy of Asahi Kasei Electronic.

362


not allow their use in small fields. Both magnetoresistors and Hall sensors are sensitive to the magnetic field perpendicular to their surface.

4.4. Ferromagnetic magnetoresistors: AMR AMR sensors are much more sensitive than Hall sensors and they are much more resistant to mechanical stress (do not exhibit any piezo-effect, which is a problem for semiconductors devices). AMR sensors have many industrial applications in vehicle detection, contactless measurement of electrical currents, measurements of position and rotational speed in machinery, and low-end electronic compasses. At present, AMRs have increasing importance in the automotive industry with applications such as measurement of the pedal position, wheel-speed sensors for ABS, and engine management systems, where they are used to measure position to tenths of a millimeter and crankshaft angle for electronic ignition timing. AMR sensors were originally developed for the reading heads in hard disks, but they were later replaced in this application by GMR and spindependent tunneling (SDT) sensors that allow processing of higher storage densities because of their small size (Li, 2007). At present linear AMR sensors are produced mainly by Philips, Honeywell, and Sensitec. They are more sensitive than Hall sensors and allow to measure with 10 nT resolution, but often require more complicated electronics that cannot be integrated on the same chip. Best AMR sensors have 200 pT/OHz@1 Hz noise (Zimmermann et al., 2005). An excellent book on magnetoresistors is Tumanski (2001). A valuable source of material for this section is also Vopa´lensky´ (2006). 4.4.1. Technology and sensitivity AMRs are made of thin-film (typically 50 nm) strips of Permalloy (Fig. 3.5). The general material requirements are low magnetostriction and low coercivity; the deposition is mostly made by DC magnetron sputtering. The resistance changes by about 2% with the magnetic field due to spindependent scattering. The sensing direction is in the film plane, perpendicular to the strip axis (y in Fig. 3.5). The basic AMR response is an even (y-axis symmetrical) ‘‘hat’’ curve: the resistivity reaches maximum for zero field (Fig. 3.6). As already mentioned in Section 3.3, AMR sensors display anisotropies induced by annealing with characteristic anisotropy fields of 2502300 A/m, which makes them single domain for the given geometry. Another reason for annealing is to decrease the number of grains in the structure and thus excessive resistance caused by electron scattering at the grain boundaries. The direction of the annealing field and the resulting easy axis is along the strip length (x) so that the induced

363


Permalloy Hy

+ y

Θ Current

x

Magnetization

Figure 3.5 Basic principle of an AMR sensor. H is the measured field -- from Ripka and Tipek (2007).

R(Hy) a

ΔR b

R0(Hy=H0)

0 -1 -0.5

0 0.5

1

Hy H0

Figure 3.6 Basic characteristics of an AMR strip (a) and using barber poles (b) -- from Ripka and Tipek (2007).

anisotropy is added to the shape anisotropy. Supposing that the external field is in y direction, the magnetization rotates out of the x direction by an angle Y for which sin Y ¼

Hy . H0

(1)

The simplified formula for the resistance of AMR element is

Hy RðH y Þ ¼ R0 þ DR 1 H0

2 !

¼ R0 þ DR cos2 Y,

(2)

where H0 is the anisotropy characteristic field, DR the maximum resistance change, and Y the angle between the magnetization and the easy axis (x).

364


4.4.2. AMR bridge and barber poles Almost all AMR sensors in the market adopted ‘‘barber pole’’ structure: aluminum stripes sputtered on the Permalloy strips deflect the direction of the current by 451 and make the characteristics linear. This results in a change of the rotation angle of the magnetization M relative to the current from y to y451. Then the resistance equation becomes: R ¼ R0 þ DR cos2 ðY þ 45 Þ.

(3)

pffiffiffi Using cosð45 þ YÞ ¼ ð 2=2Þðcos Y sin YÞ and sin2Y + cos2Y ¼ 1 we obtain Hy R ¼ R0 DR H0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi Hy 1 , H0

(4)

where ‘‘7’’ represents two different possible orientations of the aluminum stripes on top of the Py strip. In the case that Hy{H0, we obtain a linear dependence: R ¼ R0 DR

Hy . H0

(5)

Four such meander-shaped elements are connected into a Wheatstone bridge. In order to get the output from the bridge, the sensitivity of two branches should have reversed characteristics. This is made by changing the direction of the strips from + 45 to 451 as shown in Fig. 3.7. The sensitive direction is again y. This arrangement reduces the temperature drift and gives bipolar characteristics. When the bridge is supplied by a constant current I, the sensor output V is V ¼ 2I DR

Hy . H0

(6)

Vout1 current

permalloy

shorting bars

V+

V-

y easy axis

x Vout2

Figure 3.7

Basic structure of an AMR bridge with barber poles.

365


4.4.3. Crossfield error When the applied field also has a perpendicular (crossfield) component Hx, the situation becomes complex. However, for Hx, Hy{H0 we can write: sin Y ¼

V ¼ 2I DR

Hy , H xH 0 Hy . Hx þ H0

From the previous equation we see that the crossfield error (response to the field Hx perpendicular to the sensing direction) is zero for Hy ¼ 0. This can be achieved by feedback that automatically compensates Hy (Section 4.4.5). However, in some cases this is not possible due to the limitations for the power, circuit complexity, or speed. Crossfield error may cause 2.41 azimuth error for two-axial AMR compass on the Equator, if we use Honeywell HMC 1002 sensors with H0 ¼ 640 A/m (Kubı´k et al., 2006a). This error can be numerically compensated using an iterative process described in Pant and Caruso (1996) with negligible residual error. The algorithm requires the knowledge of the value of H0, but fortunately it is not much sensitive to the uncertainty of this value: 10% error in H0 estimation causes only 0.051 error in azimuth. It is not easy to precisely measure H0: a detailed analysis is given in Kubı´k et al. (2006a). Another possibility to suppress the crossfield error is to use periodical flipping that changes the polarity of H0. Honeywell released new sensors with ‘‘reduced crossfield error’’ by increasing H0. This unfortunately decreased the field sensitivity and increased the sensor noise. 4.4.4. Flipping The proper function of an AMR sensor is based on complete magnetization of the magnetic layer so that it forms a single domain. The sensors should be magnetized before use and this magnetization should be maintained during its lifetime. The sensor should be protected from large external fields that may change this magnetization and thus change the sensor characteristics (possibly completely reverse the response). Two forms of protection are used: bias and flipping. Biasing is used mainly in AMR position and current sensors: a biasing permanent magnet attached to AMR sensor produces field in the strip direction (x). Flipping is periodical remagnetization of the structure by short pulses into the coil (which is usually integrated on the chip). The flipping field has x direction (Figs. 3.3 and 3.5). Flipping is used for low-field sensors, because it also reduces the sensor offset and crossfield error. Typical values of the flipping magnetic field are of the order of 300 A/m with 10-microsecond duration. Such field can be easily achieved by

366


discharging an electrolytic capacitor through the coil wound around the sensor or integrated inside the sensor chip. As flipping changes the sign of the sensor response, it is a good idea to use periodical flipping by alternating polarities and processing of the sensor output by synchronous detector. The flipping frequency is typically about 200 Hz21 kHz to give a bandwidth of one-tenth of the flipping frequency. For some applications much lower flipping rate is used to lower the power consumption. Sensor parameters depend on the amplitude and shape of the flipping pulse.

4.4.5. Magnetic feedback Another technique to improve the accuracy of AMR sensors is magnetic feedback, also often using another integrated coil. Feedback reduces the temperature dependence of the sensor sensitivity and largely improves its linearity. AMR magnetometers using these techniques can reach the accuracy level required for a compass with 11 accuracy (Vcelak et al., 2006). Figure 3.8 shows the structure of a precise AMR magnetometer working within the Earth’s field range (750 mT). Figure 3.9 shows the total error of

Figure 3.8 A typical circuit design for a complete precise AMR magnetometer -from Vopa´lensky´ et al. (2003).

367

Total Error [nT]


5

50 40 4

30 20

3

10 8 0 -50 2

-40

-30

-20

-10

-10

0

10

-20 -30 -40 -50

20

30

40

6

50

7 1

9 Measured field [µT]

Figure 3.9 Overall accuracy of flipped and compensated Philips AMR sensor KMZ 51 -- from Vopa´lensky´ et al. (2003).

that magnetometer. The testing field was swept several times across the fullscale range (FSR) and deviations from linear response were plotted. The maximum error was 750 nT when proper flipping and feedback techniques were used. By using the negative feedback with sufficient gain, the temperature coefficient of sensitivity is reduced from 0.25 to 0.01%/K (100 ppm), which is the temperature coefficient of the field factor of the built-in flat compensation coil. The temperature coefficient of the offset remained the same (typically 10 nT/K, but varying from piece to piece even between sensors from the same batch), as feedback has no effect on this parameter. Stutzke et al. (2005) have measured the noise of AMR, GMR, and SDT magnetoresistors: the lowest noise at 1 Hz was found for a Honeywell HMC 1001 AMR sensor: 400 pT/OHz (Fig. 3.10). Recently, three orthogonal AMR sensors were integrated inside one 3 mm3 mm1.4 mm chip. Although the performance is compromised by dimension, the sensors achieve 0.1% linearity in 7100 mT range (1.8% in the full 7100 mT range; Honeywell).

4.5. GMR and SDT sensors GMRs and SDTs are made of multilayer structures. The most common GMR structure is the ‘‘spin valve’’: two thin, soft ferromagnetic layers are separated by a nonmagnetic metallic layer. In a commonly used ‘‘spin valve’’, the magnetization of one layer is fixed, while the magnetization of the other layer rotates with external field. The resistance of such structure

368


Figure 3.10 Noise of various commercial magnetic sensors at various DC magnetic fields -- from Stutzke et al. (2005).

depends on the angle between the two magnetizations. More about GMR physics can be found in Barthele´my et al. (1999) and Coehoorn (2003). Also GMR sensors are made as Wheatstone bridges, but there is no trick similar to barber poles. The sign of response of the GMR bridge branches can be modified by DC bias or in case of spin valves by orientation of the magnetization of the magnetically hard pinning layer (Pannetier-Lecoeur et al., 2007a). Commercially available Nonvolatile Electronics (NVE) sensors use another technique: magnetically soft layer shields two branches, which become unaffected by the measured field and serve only as temperature compensation. The sensors are unpinned GMR multilayers, which are more temperature stable than spin valves. However, the latter technique leads to nonlinear sensor with even characteristics (Fig. 3.11). One possibility to achieve linear response is DC bias. Noise of 43 nT/ OHz@1 Hz and 5% hysteresis in the 300 mT range was attained with an unpinned multilayer of Si/Si3N4 2000 Å substrate/Ta 30 Å/NiFeCo 40 Å/ CoFe 15 Å/Cu 40 Å/CoFe 15 Å/NiFeCo 40 Å/Ta 200 Å (Ripka et al., 1999). This GMR sandwich film operates at temperatures up to 2001C and has a temperature coefficient of sensitivity of about 1500 ppm/K. The

369


250 50 200

150 30 100 20 50

Output (mV/V)

Voltage (mV)

40

10

0

0

-2.5

-1.5

-0.5

0.5

1.5

2.5

Applied Field (mTesla)

Figure 3.11

Field response of a bridge GMR sensor -- courtesy NVE.

thickness of the Cu interlayer of 40 Å is large enough to suppress the exchange coupling and significant is only the magnetostatic coupling. Four such 1.8 mm wide resistors are connected into a bridge. Over the top of the bridge is the integrated biasing flat coil with individual resistors positioned in the peripheral part so that they are biased in opposite directions. The field constant of this coil is 0.1 mT/mA in the plane of the sandwich. Without a biasing field, the four magnetoresistors are symmetrical, so the bridge has, in the ideal case, zero output, independent of the external field. An on-chip biasing field causes a shift of resistor characteristics so that the bridge output is proportional to the measured field. Figure 3.12 shows the sensor characteristics with two polarities of the bias current. Unlike the flipping in an AMR sensor, the bias field must be present during the measurement period of the GMR sensor. The DC biased GMR sensor has 5% hysteresis in the 300 mT range. The best performance was achieved when AC biasing scheme was applied (Fig. 3.13). For 10 kHz/5 mA rms bias the hysteresis was reduced to 1% and the offset to 1 mT; the noise dropped from 45 to 16 nT/OHz@1 Hz. Recent developments in GMR technology report an increase of the temperature stability: 20% sensitivity change between 40 and + 1201C and 30 minutes survival at 2501C was reported by Hitachi. GMR spin valves can be destroyed by large magnetic fields, especially at elevated temperatures due to changes in magnetization of the pinning layer.

370


output (mV)

150

100 -12 mA

12 mA

50

0 -4

-2

0

2

4

B (mT)

-50

-100

-150

output (mV)

Figure 3.12 Large-field characteristics of the bridge of unpinned GMR sensors biased by 12 mA DC current into the flat coil integrated in the chip -- from Ripka et al. (1999).

100 80 60 40 20 0 -4

-3

-2

-1

0 -20

1

2

3

4

B (mT)

-40 -60 -80 -100

Figure 3.13 Large-field characteristics of the bridge of unpinned GMR sensors biased by 10 kHz/5 mA sinewave -- from Ripka et al. (1999).


371

The measuring current of GMR sensors flows in the plane of the multilayer structure. In SDT sensors the current flows perpendicularly to the layer plane and tunnels through the nonconducting separation layer. The advantage of SDT sensors is their large resistance and thus low-power consumption, but the insulation layer can easily be destroyed by electrostatic discharge. SDT sensors can be made smaller than GMR sensors. Both GMR and SDT were developed for two-state reading heads (Li, 2007). A lot of effort was spent to make them suitable for linear sensing, but with only partial success. Therefore up to now they found only isolated application such as counting of magnetic particles. A matrix of 1616 GMR elements with 300 mm pitch is described in Cardoso et al. (2006). The individual sensors are of 2 mm10 mm size and they are able to detect a single magnetic bead of 250 nm diameter. GMR and SDT sensors have 1/f noise with the cutoff frequency in megahertz. The reported noise levels are quite high (Ferreira et al., 2006; Gijs et al., 1996). Picotesla detection predictions were based only on thermal noise and did not take the 1/f noise into account (Tondra et al., 1998). Various efforts to employ chopping techniques did not significantly suppress the SDT sensor noise, which was 0.5 nT/OHz@1 Hz for a 2 mm2 mm device. The sensor consists of 12 parallel junctions with a total junction area of 19,800 mm2 and two 1 mm long NiFe flux concentrators, which give an amplification factor of 10. Although thanks to the high junction area and field concentrators the noise was decreased to a level only two to three times higher than that of AMR sensors, high coercivity and low linearity still remain a serious problem of SDT sensors (Jander et al., 2003). GMR and SDT sensors with proper feedback field compensation can be used, if larger field range is required and the demand for field resolution is not high. With an SDT sensor in the digital delta-sigma loop, a linear range of 71 mT and 1 mT resolutions was achieved (Deak et al., 2006).

4.6. Fluxgate sensors Fluxgates are classical precise sensors developed in 1930s. They can measure DC and low-frequency AC fields up to approximately 1 mT with a resolution of 100 pT and linearity better than 10 ppm (Ripka, 2003). The measured field creates magnetic flux in the sensor’s core, which is inside the multiturn pick-up coil. This flux is modulated by using the fluxgate effect: the core is periodically switched off by saturation caused by periodical bipolar current pulses into the excitation coil. Gated flux induces voltage in the pick-up coil. The output voltage has double excitation frequency as gating occurs twice in each period. In fact the sensor exhibits sensitivity at

372


all even harmonics. Fluxgate performance can be analytically calculated using a simple hysteresis model, which is fitted to the curves measured at working frequency (Perez et al., 2006), or more elaborated arctan model (Geiler et al., 2006). The basic formula for the fluxgate output voltage is V i ¼ NAm0 H

dmr ðtÞ , dt

where N is number of turns of the pick-up coil, A the cross-section of the core, H the measured field, and mr the relative permeability of the core. If demagnetization is considered (Section 3.3.2), the equation for the output voltage from the fluxgate sensor becomes more complex: V i ¼ NAm0 H 0

1D dmr ðtÞ . 2 ð1 þ Dðmr ðtÞ 1ÞÞ dt

4.6.1. Core shapes Vacquier-type fluxgate. The sensitivity and noise for a Vacquier type of fluxgate, which has two straight cores and solenoidal windings, were studied in Moldovanu et al. (2000). The core material was 1 mm wide, 25 mm thick stress-annealed Vitrovac 6025. It was shown that in order to receive the same 11 pT rms noise (64 mHz210 Hz) as a 17 mm diameter ring-core sensor from the same material, the Vacquier sensor should be 65 mm long. We can expect a good temperature stability of the offset from these sensors: for similar sensors the 0.04 nT/1C offset drift was achieved in 275 to + 251C temperature range (Ioan et al., 2002). Despite the higher noise, the Vacquier type of fluxgate has important advantages: (1) due to very low demagnetization, the sensor is insensitive to external fields, and (2) unlike the ring-core sensors, the sensing direction is well defined by the direction of the core. This is utilized in gradiometers, which require very high directional stability. The Foerster gradiometer is based on a nonmagnetic tension wire with two attached magnetic cores of fluxgates. The excitation and sensing coils are wound around these cores, but should not touch them. Thus the core directions are given by the absolutely straight tension wire, not dependent on temperature changes. The sensor core is made of low-noise, low-magnetostriction magnetically soft magnetic material 2 either crystalline Permalloy or amorphous Co-based alloy. The fluxgate sensitivity highly depends on the shape of the sensor core.


373

Ring core. Based on the measured data, the empirical formula for the global demagnetization factor of a ring core was derived in Primdahl et al. (2002): D ¼ 3:66

A 1 , þ d 2 mr þ 1

where A is the cross-sectional area of the ring (for excitation field the area is A, for external field 2A), d its mean diameter and mr the material relative permeability. Notice that in the original paper Primdahl gives a formula for local demagnetization field. The formula was later confirmed by analytical calculation in De Graef and Beleggia (2006). Racetrack. The empirical formula for a racetrack core was derived using FEM and confirmed by experimental data in Kubı´k and Ripka (2008): D ¼ 6:58

2Tt 0:2 þ 15 106 , 2þ mr ðl þ 1:8cÞ

where t is the thickness, T the track width, l the length, and c the core width. Study of sensitivity and noise of tape-wound racetrack fluxgates made of Vitrovac 6525X confirmed that the sensitivity depends on mefAcore. The lowest noise achieved with these sensors was 2 pT/OHz@1 Hz for 1.7 cm long FG4 (see Fig. 3.14), which surprisingly was one of those having the lowest sensitivity (Hinnrichs et al., 2001). The sensor core is usually made of tape. Ring core and racetrack cores are usually wound, but they can also be etched from a wider sheet (Ripka, 2001a). Another form of core material is magnetic wire. Wire shape is suitable for small fluxgates, as the available diameters are from 20 mm, which allows achieving low demagnetization factor even for short core. The disadvantage of amorphous low-magnetostriction wires is that they usually have an inner core with longitudinal easy axis anisotropy and an outer shell with circular or helical anisotropy. Thus there is a central region with single domain or low number of long domains, while the outer shell consists of domains oriented circumferentially (Chizhik et al., 2006; Va´zquez and Hernando, 1996). Fluxgates made of these wires thus have large noise. Koch observed that this noise can be suppressed by DC current through the wire. With 1 in. diameter core consisting of 16 turns of 100 mm diameter amorphous wire they achieved noise 1.5 pT/OHz@1 Hz, which was 30 times less than without the DC current (Fig. 3.15). The explanation is that the DC hardaxis bias makes the remagnetization process more rotational, predictable,

374


Figure 3.14 Sensitivity of racetrack sensors as a function of a product of effective permeability and cross-sectional area -- after Hinnrichs et al. (2001). r 2001 IEEE, reproduced with permission.

Figure 3.15 Ring-core fluxgate with core made of amorphous magnetic wire. The noise power spectrum is given without and with DC current through the core. Reprinted with permission from Koch and Rozen (2001), r 2001 American Institute of Physics.

375


and less hysteretic, which is expected to reduce the nonequilibrium noise (Koch and Rozen, 2001). Ioan et al. used an excitation current both into the solenoid coil and through the magnetic wire core. The two cores of their 40 mm long Vacquier sensor were made of Co68.18Fe4.32Si12.5B15 amorphous wire with 30 mm in length and 130 mm in diameter, with thin copper leads soldered at the ends. Using this double excitation they reduced the noise from 500 pT p2p (for excitation only by solenoids) to 30 pT p2p (for combined excitation). They used 1.2 A p2p/15 kHz solenoid current and 50 mA p2p core current with the same frequency (Ioan et al., 2004). The noise was measured for 30 seconds in 10 Hz bandwidth. The decrease of the noise was proportional to the amplitude of the wire current up to 50 mA. Important is the observation that if the frequency of the current flowing through the wire was changed so that it was different from the main excitation frequency, the effect of noise reduction disappeared. The noise reduction was not observed in Permalloy wires. Although the sensor size is usually about 2 cm, it can be used to measure inhomogeneous fields (Pavo et al., 2004). Fluxgates have nonlinear field response; thus they are usually feedback compensated. In such a case the nonlinearity error can be below 10 ppm. Care should be taken that the field of the feedback coil is homogeneous. Too short coil degrades linearity, as shown in Fig. 3.16.

60 Linearity error [nT]

3Amod IIA IIB

40

20

0 -50

-40

-30

-20

-10

0 -20

10

20

30

40

50

Measured field [μT]

-40

-60

Figure 3.16 Nonlinearity error of race-track PCB fluxgate for three feedback coils: 3Amod: wire-wound, length 80% of the core, IIA: pcb, 80% length, IIB: pcb, 50% from Kubı´k et al. (2007).

376


Fluxgate magnetometers. The most frequently used method of signal processing for fluxgates is the detection of the second harmonic of the output voltage. The other detection principles have not brought any significant advantages. Fluxgate may also work in the short-circuited mode (with current output; Primdahl et al., 1991). This mode is used in magnetometers made at the Danish Technical University (Primdahl et al., 2007), which are probably the most precise fluxgate magnetometers ever made. However, it seems that the current output is not always the best solution (Ripka and Billingsley, 2000). Short-circuiting the pick-up coil suppresses the self-capacitance, but tuning is more difficult (Ripka and Primdahl, 2000). The main advantage may be that the optimum number of turns for the current output is much lower than for the voltage output, which may be of benefit for small-size fluxgates. The most common feedback-type fluxgate magnetometer usually demodulates the second harmonic to DC using phase-sensitive detector (PSD) and lowpass filter. Analog integrator gives a large feedback gain, which is necessary to have a low feedback error. The feedback current is sensed by the differential amplifier, and serves as the magnetometer output. The generator usually first produces 2f frequency for the detector and divides it by 2 for excitation. Excitation signal is usually square wave at a frequency typically between 5 and 15 kHz. The excitation circuit is often tuned to increase its efficiency 2 the current waveform then has short, large peaks, which give deep core saturation required for low perming, while the power consumption is low (Ripka and Hurley, 2006). A pick-up coil can also serve for feedback and this is the case in the majority of fluxgate sensors. However, this brings some drawbacks and a compromise must be found: the ideal pick-up coil is shorter than the core (to maximize the sensitivity) and the ideal feedback coil is longer (to maximize the homogeneity of the compensation field). The most precise fluxgate magnetometers use the compact spherical coil (Fig. 3.17) with three orthogonal windings for the feedback (Nielsen et al., 1995). The magnetic field inside the coil is completely compensated so that the three orthogonal fluxgate sensors inside are in magnetic vacuum. The offset stability in the 18 to + 631C range is 70.6 nT (Primdahl et al., 2006). Sensitivity, offset, and noise in a wide temperature range was investigated by Nishio et al. (2007). With ceramic core bobbin the offset variation was below 73 nT in the 180 to + 2201C range. Nonlinear tuning at the output is in fact parametric amplification. In most cases this not only amplifies the signal, but also reduces noise (Ripka and Billingsley, 1998). The resonance frequency depends on the excitation amplitude and inductance of the pick-up coil, so that increased temperature dependence may result. Therefore this type of fluxgate should be used only in feedback mode. Note that parametric amplification is a fully


377

Figure 3.17 The compact spherical coil with three orthogonal windings for the feedback (Danish Space Research Institute).

deterministic process. ‘‘Stochastic resonance’’ is a hypothetical process to reduce the noise, which was never proven to work for magnetic sensors. All high-quality fluxgates process the output signal by PSDs. However, some alternative techniques appeared based mainly on time-domain measurement of asymmetry of the output signal or on sampling the instantaneous value or the excitation current at the moment of saturation (Dimitropoulos and Avaritsiotis, 2001). Similar devices such as ‘‘residence time difference’’ measure saturation time (Ando et al., 2005). Up to now, all these efforts resulted, however, in noisy devices. Digital fluxgate magnetometers. The best analog feedback fluxgate magnetometers reach 120 dB dynamic range and o10 ppm FSR linearity. For the most common 7100 mT range this represents 200 pT resolution and o2 nT linearity error. The output of an analog magnetometer is always digitized. Would it be better to make the magnetometer directly fully digital? Efforts to make this are reviewed in Cerman et al. (2005). The first step is to make the PSD digital, and the second step is to digitize the feedback loop: either to employ a D/A converter inside the feedback loop, or by using delta-sigma modulation. The first digital fluxgate magnetometer was reported in Auster et al. (1995). The best digital magnetometer was made for the Astrid-2 satellite (Pedersen et al., 1999). The instrument samples the sensor output by 12-bit/128 kHz ADC, makes all signal processing digitally, and controls the feedback current by DAC, which is then the weak point of the whole

378


system. The magnetometer hardly reached the parameters of its analog sibling onboard Oersted satellite, but with higher power consumption. The only advantage of the digital solution is that the magnetometer can be very easily reconfigured, which is extremely useful during some missions, when the spacecraft travels through vastly varying field regions. Utilizing delta-sigma modulator inside the feedback loop is a very attractive idea and theoretically it should bring significant advantage over other solutions. However, despite the effort of several research groups, the results are not promising (Kawahito et al., 2003; Magnes et al., 2003). A simple digital fluxgate magnetometer based on a 16-bit PC plug-in card is described in Platil et al. (2007). In the 7100 mT range the theoretical resolution corresponding to 1 bit is 200,000/216 ¼ 3 nT. The achieved noise was 2 nT rms (0.12125 Hz), and the noise density was 0.3 nT/OHz@1 Hz. Here the noise was completely given by the electronics, not by the sensor itself. Miniature fluxgates. Fluxgates are expensive, bulky, and powerconsuming devices. Recent effort is to develop miniaturized fluxgate sensors to fill the gap between fluxgate and AMR. Three basic paths for this development are: 1. CMOS-based devices with flat coils; 2. PCB-based devices with solenoids made by tracks and vias; 3. sensors with thin film or microfabricated solenoids.

Cores etched from amorphous alloy (such as Vitrovac 6025 from Vacuumschmelze) are preferred as they have much better magnetic properties than thin-film cores made by sputtering, laser ablation, or electroplating (Choi et al., 2004; Perez et al., 2004). CMOS microfluxgate may have low-power sensor electronics integrated on the same chip. Two-axis sensor for watch compass reached 15 nT/ OHz@1 Hz noise and 92 V/T sensitivity with 10 mW power consumption and 4 mm4 mm chip size (Drljaca et al., 2005). The excitation squarewave current of 350 kHz, 14 mAp had a duty ratio of 1/8 in order to reduce the power consumption (excitation current is only 1.75 mA rms). Figure 3.18 shows the principle of the flat coils: even if the coil axis is perpendicular to the core, a properly positioned coil pair can magnetize the core strip perpendicularly to that axis. Similar design using a sputtered 1 mm thick Vitrovac (Hc ¼ 100 A/m) achieved 7.4 nT/OHz@1 Hz with 100 kHz, 18 mAp triangle excitation (which gives 9 mA rms). The sensitivity was 450 V/T and the linearity was 1.15% in the 750 mT range (Baschirotto et al., 2006, 2007). Both Drljaca and Baschirotto use crossshape core; the principle difference in design is in the shape of the excitation coils. While Drljaca uses four coils around the core ends,


379

Core

B0

Flat coils

Figure 3.18

Principle of a flat core.

Baschirotto uses only one big excitation coil; in order to achieve proper saturation, there is a window in the central part of the core. The disadvantage of the CMOS process is that the available metal thickness is only several units of micrometers, which leads to high resistance of the coils. Therefore the coils cannot be tuned, as the quality factor of the resonant circuit is very low. A flat coil has in principle poor magnetic coupling with the core. One possible solution is to use a double-sided core with an excitation coil in the middle. Thus the magnetic path becomes more closed. This technique was used for both CMOS (Ripka et al., 2001) and electroplated coils (Almazan et al., 2003). Generating the magnetic field by solenoids is much better in this aspect, but the latter are more difficult to manufacture. The UV-LIGA process makes it possible to produce MEMS single-layer solenoids with 25 turns/ mm. This technology was used to build field-sensitive LC resonant circuits (Kim et al., 2006). By using copper micromolding with 18 mm thick photoresist, an 8 mm wide and 10 mm high conductor was fabricated, with 10 mm spacing. The corresponding turn density is 55 turns/mm (Woytasik et al., 2006). Printed-circuit board (PCB) technology for fluxgate was first used in Dezuari et al. (2000). PCB-based fluxgates achieved low noise and good temperature stability (20 nT in the 20 to + 701C range shown in Fig. 3.19), but the minimum size achievable with this low-cost technology is about 10 mm (Kubı´k et al., 2006b). The disadvantage of these sensors is their high power consumption; it may be lowered by using short current pulses for excitation, but this brings challenges to the signal processing (Kubı´k et al., 2007). Smaller solenoids can be made by thin-film technology (Joisten et al., 2005). The 1 mm long sensor has 20 nT/K offset drift, 100 ppm/K sensitivity drift, 1% linearity error in the 72 nT range, and 100 nT perming

380


Offset [nT]

15

Double-core sensor fexc = 5 kHz, sine

10

Iexc,p-p = 300 mA D A-B-C-D cycle

5 C

B 0

-30

-20 A

-10

0 -5

10

20

30

40

50

60

70

T [°C]

-10

Figure 3.19 (2006b).

Temperature offset stability of PCB fluxgate -- from Kubı´k et al.

for 100 mT field shock. In order to suppress the noise to 1 nT/OHz@1 Hz, the authors used auxiliary AC perpendicular field. One can speculate if it was not possible to achieve a similar effect by inducing anisotropy in the core material with an easy axis perpendicular to the sensing direction. A 2.6 mm sensor of similar type developed for compass application was described in Park et al. (2004). The disadvantage of that sensor, which has micromachined coils and electrodeposited NiFe core, is 1.6 mT perming for 6 mT field shocks; this value is rather high, but we should consider that many authors do not measure this parameter. Significant improvement was made by using 1.1 mm core of sputtered amorphous Co85Nb12Zr3 with DC permeability of 10,000 and low coercivity of 3 A/m. Unfortunately the only data given are a sensitivity of 60 V/T for 5 MHz excitation frequency and 20 turns of the pick-up coil (Na et al., 2006). Another Korean group made double-layer electroplated solenoids around electroplated NiFe. The parameters of the coil winding are 10 mm width, 4 mm space, and 6 mm thickness (Choi and Kim, 2006). The two-axis fluxgate on a single 3.1 mm rectangular core was developed for compasses; the sensitivity is 280 V/T for 2 MHz excitation. Liu (2006) suggested achieving small power consumption by using a closed core with a wide part for the excitation coil and short, narrow part for the pick-up coil. The core is saturated only in this narrow part, which has a cross-section area of 1/10 of the wide part. The disadvantage of this device is the remanence caused by the fact that the wider parts are not saturated by the excitation field. Another factor is that the wide core has a


381

high demagnetization factor, which lowers the sensitivity. A similar device was later tested by Wu. The achieved sensitivity is 650 V/T for 70 mA sinewave excitation current (Wu and Ahn, 2007). In similar devices it is always possible to demagnetize the core. Another possibility is to use flipping pulses, which create defined remanence of opposite polarity, which can be canceled by averaging. Such flipping pulses may have low frequency, thus the necessary power is low. Orthogonal fluxgates. Besides the mainstream, a lot of modified fluxgate sensors have appeared. One of them uses the orthogonal (sometimes also called transverse) fluxgate effect, which may find application in miniature sensors. The main advantage of this type of sensor is that it needs no excitation coil 2 the sensor is excited by the current flowing through the core. This however needs high current levels for saturation. As the magnetic field in the inner part of the conductor is low, which causes perming, the favorable design is a nonmagnetic conductor covered by a magnetically soft, thick layer (Fan et al., 2006). The sensitivity achieved for an 18 mm long, 20 mm diameter copper core with 2.5 mm ferromagnetic layer was 20 mV/mT. The excitation current was 10 mA rms/600 kHz; the 9 mm/1000-turn pick-up coil was tuned at second harmonics to selfresonance by its parasitic capacitance. However, electrodeposited Permalloy layers on wires exhibit a large coercivity of 75 A/m (Petridis et al., 2007). The diameter should be kept low in order to reduce the required excitation current: a sensor having 0.3 mm diameter copper core requires already 250 mA excitation (Garcıá et al., 2007). The orthogonal fluxgate works in the second harmonic mode, which is more sensitive than the ‘‘off-diagonal GMI’’ (or better IWE) mode 2 see Section 4.10 (Kollu et al., 2007). Also, here the output coil may be tuned to enhance the sensitivity by parametric amplification. Notice that an essential condition for parametric amplification is nonlinearity. Reported tuned sensitivities for amorphous microwires are 25 mV/mT for 170 kHz excitation frequency and 1000-turn pick-up coil, and 15 mV/mT for 2.3 MHz and 100 turns (Zhao et al., 2007). The orthogonal fluxgate with rod core formed by a Permalloy layer electrodeposited on a rectangular copper conductor is reported in Zorlu et al. (2007). The sensor core is only 1 mm long and the sensor has two flat 60-turn pick-up coils. The overall dimension of the sensor chip is 1.8 mm0.8 mm. With 100 mAp, 100 kHz excitation current, the sensitivity was 510 mV/mT and the noise 95 nT/OHz@1 Hz with 8 mW net excitation power consumption. The ‘‘fundamental mode’’ transverse fluxgate uses unipolar excitation; the output is on the excitation frequency. As this sensor is saturated only in one polarity, the offset stability is poor. This was improved by periodical switching of the excitation bias (Sasada, 2002). By adjusting the proper

382


excitation and bias amplitudes, the achieved noise was as low as 20 pT/ OHz@1 Hz, which was lower than the noise of the same sensor in the second harmonic mode (Paperno, 2004). The achieved sensitivity for a 5 cm double Metglas ribbon core and 350 turns of pick-up coil is 5 mV/mT for 108 kHz, 20 mA rms excitation plus 100 mA DC bias current. The sensitivity for an amorphous wire was 1.8 mV/mT. The offset stability was 1.2 nT/hour and the noise level was about 100 pT/OHz@1 Hz with periodical bias switching (20 pT without it; Goleman and Sasada, 2006). Triaxial sensor using a single wire is described in Goleman and Sasada (2007). Compensation can reduce the temperature coefficient of sensitivity from 6500 to 100 ppm/K (Plotkin et al., 2006). A similar triaxial device with a core made of amorphous wire (125 mm diameter, type UNITIKA AC 20) achieved 360 pT/OHz@1 Hz with 4 mA/100 kHz rms excitation and 20 mA DC bias. The length of each core branch was 38 mm and the pick-up coils were 17 mm/250 turns (Goleman and Sasada, 2007). A double-axis fluxgate sensor with a disk core magnetized by a rotating field is another experimental device. The core material is Metglas 2705 M with 45 mm diameter and 20 mm thickness. The authors believe that the core is a single domain, which is for this thickness and diameter highly questionable. The maximum achieved sensitivity was 1.5 mV/mT. A combination of fluxgate and AMR sensor is proposed by Petrou et al. (2006). The device uses a single-domain AMR core that is also magnetized by a rotating field. In this case the core material is much thinner and a single-domain state may possibly be achieved (Garcia and Moron, 2002). Similar devices using garnets already appeared, but have never shown application potential.

4.7. SQUID SQUID magnetometers are based on a superconducting Josephson junction and a flux antenna (Clarke and Braginski, 2004, 2006; Fagaly, 2006). These extremely sensitive devices measure magnetic field changes rather than absolute field values. The LT SQUIDs work at liquid helium temperatures and may have 1 fT field resolution. The high-temperature (HT) SQUIDs work at liquid nitrogen temperatures and have about 50 fT resolution. Such a high sensitivity does not allow direct operation, as the sensor would be overloaded by the Earth’s field. Thus gradient coils and magnetic shielding should be used. SQUIDS are at present used for measuring very weak magnetic fields produced by various organs of living objects (e.g., human brain or heart) in biomagnetic experiments. Another widespread use is for measurements of magnetic properties of samples that are either small or magnetically very weak. Direct measurements of magnetic properties of thin films are one of the applications of SQUID magnetometers relevant to our subject. SQUIDs were recently reviewed in Robbes (2006).


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4.8. Resonant sensors and magnetometers Unlike all the other magnetic sensors, resonant magnetometers usually measure the (scalar) magnitude of the total field rather than the vectorial quantity. This means that the reading is not dependent on the field direction (with an exception of possible dead zones). This may be an advantage in cases when the directional information is not required or can be derived using other sensors: in such case the field measurement is easy, as the orientation of the magnetometer sensing element may be arbitrary. The main advantage of resonant magnetometers is that they are absolute instruments with very small uncertainty. Resonant magnetometer coupled to a three-axial fluxgate is a popular combination to achieve long-term stability of the vector magnetometer. The disadvantage of resonant magnetometers is that they usually have limited field range and that some of them fail at low fields and some have dead zones.

4.8.1. Proton magnetometers These devices are based on the gyromagnetic effect. The volume of a liquid is polarized by a strong magnetic field. After the polarization field is switched off, magnetic moments move toward the direction of the measured field B. During this transit, the molecules exhibit precession with an angular frequency of o ¼ cB, where c is the gyromagnetic ratio of the proton. The corresponding frequency constant is 42 MHz/T, which means that 1 nT change corresponds to only 42 mHz frequency change. Classical proton magnetometers are sensitive to field gradients and interferences, and thus they generally cannot be used in ordinary buildings. They require a relatively large sensor volume (102500 ml) in order to pick up a measurable signal. However, they are practically absolute instruments 2 only Doppler shift, chemical shift, and paramagnetic shift slightly affect their precision. The last shift takes place only for nonspherical shape of the sample. Toroidal samples possess no dead zones (Primdahl et al., 2005).

4.8.2. Overhauser magnetometers Overhauser effect (dynamic nuclear polarization) is a transfer of energy from large electron magnetic moments to protons in the same sample. Electrons from free radicals in the sample are continuously excited by RF field from the megahertz range. In this way protons can be polarized easier than using DC magnetic field. Free precession frequency of protons can be observed after the RF signal is switched off, but the instrument can also work in continuous mode. Overhauser is more resistant to field gradient than classical proton magnetometer and it also measures faster. The instrument can have 0.1 nT resolution and 0.5 nT absolute accuracy (Duret et al., 1995).

384


4.8.3. Optically pumped resonance magnetometers This type of scalar magnetometer measures fields from true zero. It is based on the Zeeman splitting of the electron energy levels, which is proportional to the magnetic field. Vapor of alkali metal (cesium, rubidium, or potassium) in the resonance cell is excited by monochromatic light (from a discharge lamp that is also filled with the same vapors). Thus the photons have just the right energy to excite the electrons into a higher energy level and the cell has maximum absorption of light. But this happens only in zero magnetic field. If a field is present, the associated Zeeman splitting creates new energy levels, from which the electrons cannot be excited by that monochromatic light. After some time all electrons stay at these new (m ¼ 71) levels and the cell becomes transparent. By applying an AC field with appropriate frequency, electrons are excited from the new m ¼ 71 levels to the basic m ¼ 0 level, from which they can be again light excited. The frequency of that AC field depends on the measured DC field. A laser can also be used for excitation instead of the discharge lamp. Noise of 10 fT/OHz@10 Hz for 7 cm3 potassium cell at 1901C was reported in Allred et al. (2002). Probably the lowest noise achieved for this type of magnetometer was 1 fT/OHz@10 Hz for the SERF magnetometer (Kominis et al., 2003). SERF is a vector magnetometer that can only work in fields below 10 nT. The cell is illuminated by a circularly polarized excitation laser beam and a perpendicular linearly polarized detector beam. The magnetic field is detected by rotation of the polarization plane of the detector beam. The sensitivity of reading this rotation is 108 rad/fT. These are experimental devices requiring a complicated instrumentation around the miniature cell and they are far from the market and applications. A miniature atomic magnetometer of this type is based on a rubidium vapor cell, heated to 1201C. The sensor is 4 mm high and has a volume of 12 mm3. The power consumption is 160 mW for heating and the sensor requires 3.4 GHz oscillator and control circuits. The achieved noise is 360 pT/OHz@1 Hz (Schwindt et al., 2004). The advantage may be that the sensor does not produce any DC or low-frequency fields. A recent review of these ‘‘optical atomic magnetometers’’ is presented by Budker and Romalis (2007). Potassium magnetometers are manufactured by GEM, and cesium magnetometers by Geometrics and Scintrex. A laser-pumped He4 system with a noise of 0.5 pT is produced by POLATOMIC.

4.9. Induction magnetometers Induction coils are based on the Faraday induction law, which means that the voltage sensitivity is proportional to the frequency of the measured field. In order to obtain flat frequency characteristics one can use integrators


385

at their output. Another possibility is to use current output (either openloop or feedback). Current-output induction coils have flat frequency characteristics for frequencies higher than R/2pL. Induction coils are used for AC magnetic fields in the frequency range of 110 mHz21 MHz. At low frequencies, ferromagnetic cores are used to increase the sensitivity. Optimum shape of such core is a long rod and the coils are slim solenoids (Dehmel, 1989; Lukoschus, 1979; Ripka, 2001b; Seran and Fergeau, 2005). While rod made of amorphous laminations is preferable core for large coils working to about 100 kHz, ferrites are cheaper and work at higher frequencies. Cores made of ferrite may have more complicated shape that gives higher field homogeneity for the coil. Thus the optimum length of the coil is up to 90% of the core length, while for simple cylindrical cores the optimum length may be only 75%. A small sensor of this type described in Coillot et al. (2007) is 10 cm long and weighs only 11 g; its noise is 2 pT/OHz@1 Hz. For higher frequencies air cores are used, and the optimum shape of the coil is a large loop. A single loop connected to a transformer with feedback compensation gives a flat frequency response: an example of such device with 100 kHz230 MHz frequency range and noise of 0.2 fT/OHz is described in Cavoit (2006). An induction coil can measure DC fields only if the coil flux is modulated 2 either by coil rotation or by vibrational movement in a gradient field. It is more convenient to modulate the flux by changing the core permeability 2 this is a principle of fluxgate (Section 3.6). Induction magnetometers were recently reviewed in Tumanski (2007).

4.10. GMI Giant magnetoimpedance (GMI) is based on the field-dependent change of the penetration depth (Knobel et al., 2003). The effect has only few practical applications as it gives weak, temperature dependent signals and the characteristics are nonlinear and unipolar. The noise values are rarely available 2 in Ding et al. (2007), the lowest noise level is 20 pT/ OHz@100 Hz. For 1/f noise it would correspond to 200 pT/OHz@1 Hz. Similarly we may extrapolate the noise level of 100 pT/OHz@1 Hz from the data given for a 10 mm long device in Robbes et al. (2002). The temperature change of GMI sensitivity is large, but it can be compensated by feedback. The large temperature offset drift is usually due to the temperature dependence of DC resistivity. GMI curves of anisotropic cores have double-peak character, with maximum GMI around the anisotropy field (Hauser et al., 2007). Figure 3.20a shows a single-peak GMI characteristics at 1 MHz on highquality GMI amorphous tape 2 the detail for small fields region is shown in Fig. 3.20b (Mala´tek et al., 2008). In order to achieve bipolar response, we

386


250 (a) 200

GMI(%)

150

100

50

0 -2400

-1600

-800

0 H (A/m)

800

1600

2400

250 (b) 200 GMI (%)

Decreasing H Increasing H 150

100

50 -50

-30

-10

10

30

50

H(A/m)

Figure 3.20 (a) GMI curve of amorphous tape -- from Mala´tek et al. (2008). (b) Detail of the GMI curve -- from Mala´tek et al. (2008).

should use about 5 A/m biasing. The sensitivity is then about 8%/A/m, that is, 6.5%/mT. The zero-field impedance of the 10 cm long sensor is 23 O, which means that the sensitivity in impedance units is 0.065 23 ¼ 1.5 O/mT. With 10 mA measuring current the voltage sensitivity is 0.01 1.5 ¼ 15 mV/mT. Using proper alloy composition, the temperature coefficient of DC resistance was very small and the achieved offset drift of 30 nT/1C was caused mainly by the temperature dependence of circular permeability (Mala´tek et al., 2008). The temperature dependence of the DC resistance and total impedance is shown in Fig. 3.21.

387


23

5.79

22

5.78

21

5.77

|Z| (Ω)

RDC (Ω)

Impedance and resistance dependence on temperature 5.8

20 Rdc

5.76

|Z|

5.75 -25

0

19

18 25 temp (°C)

50

75

Figure 3.21 Temperature dependence of a GMI sensor DC resistance and total AC impedance at 1 MHz -- from Mala´tek et al. (2008).

Detailed design and testing results for a GMI magnetometer are described in Boukhenoufa et al. (1995). The sensing element was a 30 mm diameter, 1 cm long amorphous wire (MXT Inc.) manufactured by melt extraction (Strom-Olsen, 1994). For 16 MHz square-wave measuring current the sensitivity was 4200 V/T. Square-wave biasing of 780 mT, 50 kHz suppressed the noise from 900 to 300 pT/OHz@1 Hz. The sensitivity of GMI sensors is roughly linearly proportional to the sensor length. This gives a potential advantage of GMI over fluxgate, as the fluxgate sensitivity drops more rapidly with downsizing. A 300 mm long and 3 mm wide strip made of CoFeB trilayer has an inductance of only 10 nH, so the measurement frequency should be high. However, such strips can be structured as meander, possibly leading to a practical sensor (Giouroudi et al., 2006). The only available GMI sensors in the market are manufactured by Aichi Micro Intelligent. These sensors are used in low-end compasses for mobile phones. They have amorphous UNITIKA wire cores and sputtered feedback coils, the sensor size being 0.5 mm1 mm (Mohri and Honkura, 2007). The nonlinearity of the highly linear type in the 300 mT range is 1%, but no data about temperature stability and noise are available from the manufacturer (Aichi, 2007). The Aichi MGM-1DS Milli-Gaussmeter has 72% accuracy and 10 nT resolution. Aichi also manufactures a hybrid 5.5 mm5.5 mm chip, which contains a three-axial GMI magnetometer and a two-axial inclination sensor based on two cantilevers with permanent magnets and two other GMI sensors that monitor position of these cantilevers. The asymmetric GMI gives a linear response, but the effect is based on potentially unstable self-biasing (Knobel et al., 2003).

388


The IWE is of similar origin as the GMI, but instead of measurement of a voltage drop across the core, the output is formed by the voltage induced into the pick-up solenoid (Kraus and Mala´tek, 2007). The effect is antisymmetrical and requires helical or circular anisotropy and a DC component in the AC excitation. Although the output is voltage, some authors call this effect ‘‘off-diagonal magnetoimpedance’’ (Fry et al., 2004). The sensor is exactly the same as for transverse fluxgate, but the substantial difference is that in the IWE mode the current does not saturate the core and the output signal has the same frequency as the excitation. Using helical anisotropy and 23 mA DC bias in a double-core 8 cm long IWE sensor with a 400-turn pick-up coil, the sensitivity with 100 kHz/5 mA excitation was 28 mV/mT. The achieved temperature coefficient of sensitivity was 0.5%/K, which is significantly worse than with a similar GMI sensor. However, this can be suppressed by feedback. The most important advantage is reasonable offset stability of 5 nT/K, more than 10 times better than for GMI sensor made of the same core. The offset long-term stability was 3 nT/12 hours (Kraus and Mala´tek, 2007). The sensitivity can be increased by tuning the coil with a parallel capacitor, or just by using the self-resonance caused by a parasitic capacitance. However, the use of resonance can degrade the temperature stability. The common disadvantage of IWE and GMI sensors is the perming effect, because the ferromagnetic core is not demagnetized. This problem is often ignored in literature. Contrary to IWE sensors, in a fluxgate the current should saturate the core and, as the output, the signal on the second harmonics of the excitation frequency is usually used. The perming is removed by the excitation field. A system of multiple wires connected in parallel decreased the sensitivity at 10 MHz; the sensitivity at 500 MHz was increased, but the MI effect at such a high frequency can hardly find an application (Garcıá, 2007).

4.11. Other devices Here we present various sensing principles that are interesting, but did not reach parameters of the above-described classical sensors. The colossal magnetoresistance is observed mainly in manganese-based perovskite oxides. As this effect is connected to a structural change, it is also sensitive to temperature. A huge magnetoresistance is a geometrical effect in a very large magnetic field. These effects are extremely interesting, but their future application in sensors is questionable. Magnetostrictive magnetometers: The dimensional change of a magnetostrictive material due to the external field is measured by optical fibers, or piezoelectric layer. A sandwich made from a magnetostrictive and


389

piezoelectric material can be used for a synthetic magnetoelectric device with 130 V/T sensitivity (Duc and Huong, 2008). Other sensors are based on changing the resonance frequency of free-standing elements manufactured by MEMS technology. The preliminary results on large-scale models of a ‘‘xylophone magnetometer’’ were promising, as the noise was of the order of 0.1 nT, but such low noise was not achieved in MEMS polysilicon technology (Wickenden et al., 2003). Another device type uses a piezoelectric layer for periodical excitation of the magnetostrictive layer (similarly to fluxgate, where the excitation is by magnetic field). Tensile or compressive stress developed by a piezoelectric component is coupled to the magnetostrictive core, which changes its permeability with double frequency. When a measured DC field is present, the core flux is modulated by this second harmonic frequency. This is sensed as a voltage in the pick-up coil. The hysteresis can be suppressed by a superimposed alternating magnetic field or by passing a current through the ribbon. A resolution of 10 nT was demonstrated using viscous interface in Prieto et al. (2000). Using a 60 mm long piezoelectric strip actuator a linear and hysteresis free response with 250 V/T sensitivity was obtained (Michelena et al., 2002). Magnetostatic devices (Ciudad et al., 2004) are based on the repulsion force between the magnetized plates. The detectable field is 300 nT using optical readout. A torque exerted on a permanent magnet in a magnetic field was used in an optically interrogated magnetometer. With a 1.3 mm3 sensor the field range was 820 A/m26 kA/m with an uncertainty of 90 A/m (Va´zquez and Hernando, 1996). Similar device in MEMS technology is described in Vasquez and Judy (2007). The device described in DiLella et al. (2000) is by far the most sensitive. A micromachined device detects magnetic fields by the changes in the torque on a suspended magnet. The position of the torque element is sensed by a tunneling tip current and kept by electrostatic feedback 2 330 V voltage is necessary for that. The range is 200 mT and the noise is 300 pT/OHz@80 Hz. A micromachined Lorentz force magnetic sensor achieved field resolution of 10 nT/OHz for 100 mA current (Kyyna¨raïnen et al., 2007). The advantage of Lorentz force magnetometers is their high linearity and wide range selection by changing the measuring current. The sensor can work up to 50 T (Keplinger et al., 2004). Also magnetooptical sensors based on Faraday effect, that is, on the rotation of the polarization plane of light passing through a transparent material exposed to magnetic field, may be used to measure high and very high fields. This method is used in particular for extremely high fields of short duration (Fowler, 1973). The sensitivity of the method depends both on the material and on the possibility to precisely measure small angles. In Machado Gama (1975), the resolution of B1.5 mT was reached at a field of 2 T with optical glass (SF57) as the magnetooptically active material

390


of 1 cm length, a HeNe laser as the source of 632.8 nm wavelength and with B1-minute resolution of the angle measurement. A comparison of the field intensity determined by this method and by NMR confirmed the claimed sensitivity and no systematic error was found. The Faraday effect was also used for measuring magnetic fields of the order 100 T in laserproduced plasma (Stamper and Ripin, 1975). Some modifications of these sensors have higher sensitivity, but their range is limited: 1.5 mT/OHz@80 Hz was reported for a thin-film garnet sensor with in-plane magnetization, but the range is only 715 mT (Holthaus et al., 2006). The magnetooptic Kerr effect (MOKE) consists in the rotation of the polarization plane by a certain angle, depending on the orientation and magnitude of magnetization of the material, from whose surface the linearly polarized light beam is reflected. This effect is also used to visualize magnetic domains. The Faraday effect in magnetooptical films is used in eddy current imaging for nondestructive evaluation (NDE). The Faraday shift can be doubled when the light beam is reflected by a mirror and reflected back through the same material (Joubert and Pinassaud, 2006; Drexler and Fiala, 2008). Magnetotransistors and magnetodiodes are rather exotic devices, which periodically appear in literature, but do not show any advantage over simple Hall sensors. The most advanced one is probably the integrated circuit based on a MAGFET transistor, which makes corrections to nonlinearity and offset: the resolution is 16 mT in the 10 mT range, the offset is less than this resolution, and the nonlinearity error is below 0.4% (Kuo et al., 2006). The temperature stability of the offset and sensitivity, which is weak point of these devices, was not specified. The following table gives an overview of low-noise magnetic field sensors: Principle

Size (mm)

Fluxgate Ring core Racetrack PCB

25 70 3081.8

Noise Power (pT/OHz@1 Hz) (mW)

3.8 2.5 17 13

Thin film

33

1000

CMOS micro

1.5 (44) 15,000

50 70 20 (pulse excitation) 50 (sine excitation) 10 (100) (10)

Reference

Ripka (2003) Ripka (2003) Kubı´k et al. (2006b)

Joisten et al. (2005) Drljaca et al. (2005)

391


(Continued ) Principle

AMR HMC 1001

Size (mm) Noise Power (pT/OHz@1 Hz) (mW)

4111.7

200 300

HMC 1023 844

1000

HMC 1021 451.7

3000

KMZ 51

2000

541.7

1600 + Flipping

Extra power

Reference

30 (5 V, Zimmermann 0.86 kO) et al. (2005) 1.7 (1.2 V, Stutzke et al. 0.86 kO) (2005) 30 (each axis) Zimmermann et al. (2005) 1.5 (1.2 V, Stutzke et al. 1 kO) (2005) 15 Zimmermann et al. (2005) 15(5 V, Vopa´lensky´ 1.7 kO) et al. (2003) 25 mW/1 kHz

Including electronics Three-axis Two-axis

5. Magnetic Sensors of Position and Distance Traditional sensors like LVDT, inductosyns, synchros and resolvers, eddy current sensors, variable reluctance sensors, magnetic encoders, and PLCD sensors (Ripka, 2001a) are well known and widely used in industry. In these cases, recent improvements were mainly made by redesigning the sensor geometry based on improved CAD models rather than by further development of the materials used.

5.1. Inductance and transformer sensors A linear variable differential transformer (LVDT) is a differential transformer with movable core (Fig. 3.22). The primary winding is supplied from the voltage source and the voltages on two symmetrical secondary windings are monitored. If the core is displaced from the central position, one secondary voltage is larger. LVDT is one of the most popular position sensors. It is available in measurement ranges from 200 mm to 50 cm, the resolution is from 1 mm, and linearity up to 0.05%.

392


Inductosyn is a position-sensing transformer with movable flat meander coils. It combines an analog output (within one coil pitch) and an incremental output (counting of pitch numbers). Inductosyns are often used in large machine tools and due to their ruggedness also in military and aerospace applications. Standard linear accuracy is 1 mm, or 1 arc second for the rotary type. Synchros and resolvers are rugged rotational transformers that are similar to electric machines. Their typical application is heavy industry and military. Eddy current sensors allow to measure the distance to a conducting target, which need not be a part of the sensor. If the excitation frequency is large enough so that the penetration depth is very small, the reading does not depend on target width and its conductivity. Most of the proximity switches are based on this principle. The basic circuit consists of an LC oscillator, which is ‘‘killed’’ by losses due to the eddy currents (Fericean and Droxler, 2007). Variable reluctance sensors measure the changes of the airgap in a magnetic circuit. They need AC excitation and can be made as transformers or variable inductors. Differential sensors of this type have improved linearity. PLCD sensors use a magnetically soft core, which is saturated in one point by permanent magnet attached to the target. They were developed for automotive applications, but they found only limited use. Although the output reading is theoretically independent of the size and distance to the permanent magnet, large changes of the magnet distance cause distortion of the sensor characteristics (Fig. 3.23; Reininger et al., 2006). When the distance is too large, the core is not properly saturated. When the distance is too small, two side saturation zones appear that result from the longitudinal field distribution of the permanent magnet, which has one maximum and two side lobes with opposite polarity. ±Δl M1 S1

L1

+Δl -Δl L2′ V2

P V1 V3 S2 I1(jω) a)

Figure 3.22

LVDT position sensor.

M2

L2′′ b)

Vout = V2-V3

393


Influence due to distance 10 min

9 8 Output voltage U [V]

typical 7 6 max 5 4 3 2 1 0 10

15

20

25

30

35

40

45

50

55

60

Position x [mm]

Figure 3.23 Output signal of a PLCD position sensor for different distances to the permanent magnet -- from Reininger et al. (2006) reproduced with permission from Elsevier.

5.2. Magnetostrictive position sensors These sensors use sonic waveguide made of magnetostrictive wire or tube. When a movable permanent magnet saturates a small region of such waveguide, a traveling strain pulse is partly reflected from this region back to the source. The time-of-flight is then proportional to the distance between the source and the magnet. Another reflection from the waveguide end is also sometimes measured and used to compensate for the sound velocity, which is temperature dependent. The length of these sensors is limited by attenuation to about 426 m. Resolution can be as low as 0.4 mm and uncorrected nonlinearity 0.02% FS. Magnetostrictive delay lines also allow to measure other physical quantities at multiple points (Hristoforou, 2003).

5.3. Position sensor with permanent magnets Permanent magnets and stationary sensors are used in rotary speed sensors and to signalize the end position of the piston. These two-state sensors have achievable repeatability of 70.1 mm and as a type of proximity switches they are discussed in Section 5.4. However, permanent magnets are also used to precisely measure, for example, the position of pneumatic pistons with required resolution of 0.01 mm and high speeds. Position sensors with permanent magnets have the following problems: 1. temperature dependence of the magnet properties; 2. sensitivity to magnet shape and its fine adjustment;

394


Figure 3.24 Method for sensing the absolute position of a permanent magnet with an array of sensors. The exact position of the magnet is calculated using the output signals in the vicinity of the crossover point -- from Reininger et al. (2006) reproduced with permission from Elsevier.

3. sensitivity to mechanical tolerances and magnetic properties of the

holder; 4. ambiguity caused by the symmetry of the field distribution.

These problems can be elegantly solved by using an array of Hall or AMR sensors. The configuration in Fig. 3.24 shows an array of sensors that is used to interpolate the zero-crossing position, which is independent of the mentioned influences (Reininger et al., 2006). An SOI Hall sensor array with 0.25 mm pitch suitable for similar application is described in Gruger (2003). Contactless potentiometers: The basic configuration is a permanent magnet attached at the end of the shaft and a sensor of magnetic field mounted under this shaft. The required sensing direction is perpendicular to the shaft: this means that the ordinary Hall sensors, which are sensitive to the field perpendicular to its plane, cannot be used. One possibility is to use vertical Hall sensors, but up to now they are not very practical devices. 2SA-10 developed by the former Sentron has integrated ferromagnetic pieces, which internally change the field direction (Popovic et al., 2006). The device has integrated amplifiers and provides sin and cos signal output for 3601 of mechanical rotation of a magnet over the sensor (Melexis). AMR sensors of this type have only 901 angular range. The range can be increased to 1801 by using another AMR that is rotated by 451. A 3601 range can be achieved by using a switched auxiliary field. The GMR spin


395

valve also offers a 3601 range; once the device is saturated, it is dependent only on the angular position of the magnet, no longer on its distance (Treutler, 2001). The first commercial product of this type, Infineon GMR-C6, was not successful and has been discontinued. After many years of further development, new devices to be released have improved temperature characteristics (401C to + 1501C) and on-chip digital electronics (Infineon). GMR sensors can be destroyed by large magnetic fields. The maximum allowed field for TLE5010 is 125 mT for 5 minutes (Infineon). This is a serious limitation compared to Hall sensors and AMRs. The device consists of two GMR full bridges (Fig. 3.25a) giving sin and cos response. From these two signals the digital processor calculates the angular position (Fig. 3.25b). An elegant solution that gives a 1801 range and a triangular response to the angular position was described in Gonzalez et al. (2001). The device uses colossal magnetoresistance (CMR). The disadvantage is its limited temperature range: the temperature dependence is normally linear so that it allows for electronic compensation. However, at 701C the sensitivity abruptly drops down. Magnetic encoders are available as incremental or absolute position sensors either in linear or in rotational form. They use magnetic marks created in rulers or wheels made of hard magnetic material. Absolute magnetic encoders need multiple tracks and associated sensors. Rotational magnetic encoders may have up to 10 bit resolution.

5.4. Magnetic proximity switches A proximity switch can be made from any distance sensor and Schmidt trigger. Hall sensors with permanent magnets and eddy current sensors are often used for this application. The permanent magnet is attached either to the measured object (or it is a natural part of it as, e.g., the rotor of permanent magnet machines or generators) or more often to the stationary sensor and serves as a field source. The sensor then measures the field change caused by the presence of the object, which should be ferromagnetic. Gradient sensors that have two internal Hall plates are very popular in this application, as they compensate for the temperature effects and external fields. The following two magnetic sensors give natural bipolar output: magnetic reeds and Wiegand wires. Reed contacts are cheap and robust and need no power. They are based on attractive or repulsive force between two magnetically soft magnetic strips, which are magnetized by a permanent magnet. Both normally open and normally closed contacts are available. The position of the moving permanent magnet should be carefully selected to avoid multiple switching regions.

396


90° GMR Resistors

VX

VY

0° S

N

ADCX+ ADCX-

GND

ADCY+ ADCY-

VDEG

Y Component (SIN)

VY V

X Component (COS) VX VX (COS)

0°

90°

180°

270°

360° Angle α

VY (SIN)

Figure 3.25 (a) Sensitive bridges of the GMR sensor -- the arrows in the resistor symbols show the direction of the reference layer -- from Infineon. (b) Ideal output of the GMR sensor -- from Infineon.

397


4

U(V)

3

2

1

0

0

10

20

30

40

50

t (μs)

Figure 3.26

N

S

S

N

Wiegand wire (after VAC brochure).

Wiegand wires are also called ‘‘pulse wires’’. They are based on an asymmetric hysteresis loop of the composite material with large Barkhausen jump. The characteristics are biased either by piece of permanent magnet in form of a parallel wire or by an external magnetically hard layer on the same wire. As a result the large Barkhausen jump appears only on one side of the loop. These sensors produce a voltage pulse, which is to some extent independent of the speed of the moving permanent magnet (Fig. 3.26). The advantage is that the sensor needs no power, disadvantage is that its function can be destroyed by a strong magnetic field.

5.5. Speed and flow sensors Speed sensors usually use the induction law either in the Blv or in the dF/dt form. The field source is usually a permanent magnet. Sensor for measuring the rotational rate of the bullet uses the Earth’s field as a source and a small coil as a sensor (Yoon et al., 2006). Magnetic flowmeters measure the speed of liquid flow and calculate the volume flow. They use an electric field created in a conductive liquid flowing in a magnetic field. The field is usually square wave created by

398


saddle coils around the pipe. The electric field is commonly detected as the voltage between two conducting electrodes immersed in the liquid, but it can also be measured contactlessly by two electrodes on the outer surface of the nonconducting pipe.

6. Force and Torque Sensors 6.1. Force sensors Many force and pressure sensors use deformable elastic elements together with a magnetic position sensor. Magnetic load cells (pressductors) were developed by ASEA (later ABB) in 1950s. These sensors are based on a transformer with perpendicular windings. In the unloaded state the coupling between the primary and the secondary windings is zero, and there is no output voltage. In the loaded state the stress-induced anisotropy causes an asymmetry of the fluxlines and an output voltage appears. Strain sensors using magnetoelastic amorphous ribbons are shown in Bydzovsky et al. (2004). The sensor performance is based on the changes of permeability. With double-sensor arrangement it is theoretically possible to suppress the sensitivity of these sensors to external magnetic field and temperature changes, but in practical applications this is very difficult and prevents these sensors from wider use. The advantage of these sensors is contactless reading. Bilayers or multilayers of magnetic and nonmagnetic materials are very sensitive to bending, but again the affected parameter is permeability (Pfutzner et al., 2006). Another possibility is to use magnetostrictive material as a free layer in a spin valve. Such a sensor was recently fabricated on a flexible polyimide substrate. Despite their offset and nonlinearity, these sensors have one advantage over traditional piezoresistors: they may work for large elongations up to 2.5% (Uhrmann et al., 2006). Magnetostrictive nanowires of 102200 nm diameter and up to 60 mm length were grown for artificial cilia to sense acoustic signals. However the sensing of the field changes is a challenge, since the required sensor size is very small (McGary et al., 2006).

6.2. Torque sensors Magnetoelastic torque sensors become very compact and inexpensive, so that in many applications they are more popular than traditional torque meters based on strain gauges measuring the shaft deformation. They are used in mass applications: to measure cyclist effort in power-assisted


399

bicycles and exercise bikes, in electric power steering systems, to measure engine and transmission torque in automatic car transmissions, and individual wheel torques in four-wheel vehicles with automatically controlled differential, and also in handheld tools to measure the tightening torque. Magnetoelastic torque sensors are based on stress-induced anisotropy of the magnetic shaft or magnetic layer on nonmagnetic shaft. The torque on rotating shaft can be measured without contact by external coils. There are two basic types of these sensors: (1) ‘‘Permeability-based’’ sensors, which use changes of permeability of a

surface layer of magnetic shaft caused by stress. The permeability is sensed by coupling between the source and the pick-up coil. (2) ‘‘Polarized band’’ sensors.

6.2.1. Cross-type permeability-based torque sensors The traditional sensor of this type uses two orthogonal U-shaped cores (with excitation and sensing windings) directed toward the shaft. If the permeability of the shaft becomes anisotropic, the symmetry is broken and voltage appears on the sensing coil. The phase of this voltage depends on the torque direction. A new thin pick-up head was proposed in Sasada et al. (2006). The new head consists of a stacked pair of figure-eight coils, in which one of the coils is rotated in the coil plane by 901 from the other. These coils face the shaft through an airgap and they are directed in 7451 directions with respect to the shaft axis. The applied torque can be detected from the difference in self-inductances of these coils, as each figure-eight coil sees the permeability of the shaft along one of the 7451 directions, in which the applied torque creates surface stress. The disadvantage of the cross-design is that the local variations in the magnetic properties of the shaft, which are unavoidable, as well as the geometrical imperfections (causing variations of the airgap) cause distortions in the output signal. This may be partially averaged by multiple coil systems. Frequency dependence of the permeability causes dependence of the torque sensor on the rotation speed, even if the excitation frequency is stable. The reason for this is the mentioned AC signals caused by rotational imperfections, which modulate the amplitude of excitation frequency. That is why these sensors were almost replaced by the solenoidal type ones, in which the field has axial direction. However, precise cross-type torque sensors are used to monitor the radial vibrations of the shaft.

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6.2.2. Solenoidal permeability-based torque sensors These sensors use a shaft with 7451 grooves. The axial permeability increases when the easy axis of the stress-induced magnetic anisotropy is parallel to these grooves and decreases when it is perpendicular to them. The changes of the axial permeability are therefore proportional to the measured torque. Axial permeability is measured by a pair of axial excitation/sensing solenoids. The grooves can be machined or formed, or the same effect can be reached by electrodeposited strips of copper. The advantage of the coaxial (solenoidal) design is that the output is free of rotational artifacts. A sensor of this type can be made differential, so the output for zero torque may be zero, but to maintain this balance is difficult. An industrial example of this type of torque meter is the Torductor S by ABB. Sensors of this type achieve a measuring error o2% and a repeatability error o0.1%. The common disadvantage of both types of the permeability-based torque sensors is the fact that permeability is not an intrinsic property of the material. It changes with temperature, frequency, and is also sensitive to the amplitude of the AC field and presence of external DC magnetic fields. Moreover, permeability is irreversibly changed each time after the sensor is overloaded. In automotive environment, which is characterized by high temperature changes and mechanical vibrations, the permeability gradually decreases in time due to microstructural changes of the material. Experiments with the use of magnetic tapes attached to the surface of a nonmagnetic shaft gave promising results, but the performance of these devices was always degraded by the glue between the shaft and the tape (Pasquale, 2003; Umbach et al., 2002). 6.2.3. Polarized band sensors These sensors use a thin ring of magnetoelastic material rigidly attached to the shaft (or a part of the shaft material itself). The ring is circumferentially magnetized. The operation is based on rotation of the ring magnetization toward the direction of the stress induced by the torque. The magnetization rotation is measured by a DC magnetic field sensor (Garshelis et al., 1997). This is a DC type sensor and works without excitation. The torque sensor usually consists of two rings polarized in opposite directions and one or more magnetic sensors (Hall, AMR, but most often fluxgate). Sensors of this type are manufactured by Magna-Lastic Devices, Inc. (www.mdisensor.com), Magnova, Inc. (http://magnova.com), and by Siemens VDO (Kilmartin, 2003). The Siemens sensor has the shaft covered by a 0.5 mm layer of magnetostrictive material made by thermal spray. The radial magnetic field is sensed by a large fluxgate sensor that is encompassing the shaft. This brings also a shielding of external fields. These sensors achieved high temperature stability of 70.04% FS/1C, while the


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hysteresis/nonlinearity error is below 0.5% FS and repeatability below 0.25% FS. A sensor of this type applied on a 0.8 mm shaft has mNm resolution (Garshelis and Jones, 1999). 6.2.4. Other magnetic torque sensors For even thinner shafts, the torsion impedance effect can be employed: the high-frequency impedance of a magnetostrictive Fe-based wire changes with applied torsion by up to 200% with negligible hysteresis (Sańchez et al., 2007). Diagnostics of motors require monitoring very small variations in the torque. This is possible simply by replacing a DC magnetic sensor by an induction coil (Garshelis et al., 2007). Amorphous tapes and wires can also be used to sense torque. However to the author’s knowledge no such device reached the market; temperature dependence, nonlinearity, and sensitivity to magnetic field are obvious reasons. Finally we should mention another type of magnetic torque sensor, which is based on a deformation element and a rotational position sensor. A modification of such sensor for a rotation shaft that needs no rotational contacts can be found in Froehlich and Jerems (2008).

7. Electric-Current Sensors Almost all contactless electric-current sensors use magnetic principles (Ripka, 2004).

7.1. Instrument current transformers Precise current transformers use high-permeability ring cores to scale down the measured current and convert it to voltage drop on a load resistor. They are also made with openable core most often as AC current clamps. Current transformers were used in electronic energy meters, but it turned out that they could easily be disabled by saturation 2 either from a strong permanent magnet or by a DC component in the measured current. One way to avoid this is to use flat-loop magnetic material, which has a very high saturation field H; another approach is to use a Rogowski coil, which has no magnetic core. Some manufacturers use composite cores, consisting of a highpermeability ring, which gives low phase and amplitude error, and a low-permeability ring, which is DC resistant. However, some cheap lowpermeability rings can do the job as well: they have some amplitude error and phase shift, but these are constant over a wide range of measured currents and therefore can be compensated (Mlejnek et al., 2007). Nanocrystalline alloys are prospective materials for very precise small-size

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current transformers (Draxler and Styblikova, 1996). They may have the same permeability as NiFe crystalline or Co-based amorphous alloys, but have much larger saturation induction, so that the instrument current transformers are smaller. Fe-based nanocrystalline ring cores may have almost constant permeability over a wide range of induction. This gives a low phase error and constant amplitude error in a wide range of the measured current. As such amplitude error can also be compensated; this is extremely useful for current transformers.

7.2. Rogowski coil The Rogowski coil for the measurement of current is a toroid wound around the measured current conductor. The basic operating principle is given by the mutual inductance between the primary (single turn) and the secondary (many turns that should be homogeneous to suppress external fields). It measures dI/dt, so normally it is used together with an integrator. Single-chip digital integrators are used to process the signal of Rogowski coils (also called dI/dt sensors) for energy meters (Analog Devices).

7.3. DC current transformers Basic current transformers use only AC currents. DC current transformers and DC current comparators use the fluxgate effect in a core that is periodically saturated by the excitation field. The first sensors of this type were transductors (Bera and Chattopadhyay, 2003). To lower the noise and increase the linearity, new designs use PSDs instead of the traditional peak detection. The most precise sensors of this type are DC current comparators (Moore and Miljanic, 1988). Fluxgate DC current sensors or ‘‘DC transformers’’ are similar to DC comparators but of a much simpler design. The accuracy of a typical commercial 40 A module is 0.5%, linearity 0.1%, and current temperature drift o30 mA (25 to + 7001C; Ripka, 2004). Self-oscillating sensor of similar type is described by Ponjavic and Duric (2007). The first fluxgate current sensor in PCB technology was described by Gijs et al. (Belloy et al., 2000). Their sensor had a single winding of 36 turns over toroidal core made of amorphous magnetic tape. They reached 10 mV/A sensitivity and range up to 5 A. A prototype of the fluxgate current sensor with an electroplated core in PCB technology was described in Ripka et al. (2005). Short excitation pulses should be used in order to lower the excitation power. This can be achieved using a tank circuit with an external saturable inductor (Tang et al., 2004). Another geometry for a PCB fluxgate current sensor with an electroplated core was used by Tipek et al. (2006).


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Because of their low offset drift, fluxgate-based ‘‘DC current transformers’’ are superior to the current sensors having Hall sensor in the airgap. The problem is a large power consumption of these devices.

7.4. Hall current sensors Many DC current sensors use a Hall element mounted in the airgap of a magnetic core. The yoke has two effects: (1) increase of the sensitivity and (2) shielding of the external fields. The sensor linearity can be increased by using the feedback principle. A multiturn feedback coil allows that the feedback current is much lower than the measured current. Feedback-compensated devices can achieve 0.02% error and sensitivity tempco of only 50 ppm/K. The main problem of these sensors is their limited zero stability given by the Hall sensor offset: the typical offset drift of a 50 A sensor is 600 mA in the 02701C range. This parameter is 20 times worse than that of fluxgate-type current sensor modules. Even when using a magnetic yoke, Hall current sensors are sensitive to external magnetic fields and close currents due to the nonhomogeneity associated with the airgap. Another DC error is caused by the remanence of the magnetic core 2 only a few Hall current meters have an AC demagnetization circuit to erase perming after the sensor was exposed to large DC current or external field. A low-cost current sensor based on a highly sensitive Hall sensor with integrated flux concentrators is described in Popovic et al. (2006). The sensor now manufactured by Sentron has only a simple ferromagnetic circuit, and it has 1% accuracy in the 712 A range. Yokeless current transducer with six Hall probes around the rectangular bus bar achieved 0.5% linearity and 0.2% temperature stability in the 100 kA range (Scoville and Petersen, 1991). The same 100 kA return current at 50 cm distance was suppressed by a factor of 100. The crosstalk error can be further suppressed by the algorithm described in DiRienzo et al. (2001). Gapped current transformer with Hall sensor in the airgap is described in Dalessandro (2007). Signals from the DC and AC sensors are added without an electronic stage. The frequency range of 30 MHz was achieved for 40 A device.

7.5. AMR current sensors AMR sensors are more sensitive and stable than Hall sensors, but they cannot be used in the narrow airgap of the magnetic core, as they measure field in the sensor plane and usually require W1 mm sensor length in the field direction. Such large airgap would cause field leakage and result in sensor nonlinearity and sensitivity to an external magnetic field. Current

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Vout

Imeas

Figure 3.27 AMR current sensor -- from Ripka (2004).

sensors based on an AMR magnetoresistor are thus yokeless. The most developed configuration is a bridge measuring the magnetic gradient from the current (Fig. 3.27). This type of sensor usually has the current conductor integrated with sensors in one device. The AMR bridge has all the barber poles in the same direction, which means it is made insensitive to an external field, but sensitive to measured current through the primary bus bar (Lemme and Friedrich, 2000). The measured current is compensated by a feedback current through a compensation conductor. A typical application is galvanically isolated current sensing in a PWM regulated brushless motor. These sensors are manufactured by Sensitec (also under F.W. Bell label) with ranges from 5 to 220 A. The achieved linearity is 0.1%, temperature coefficient of sensitivity is 100 ppm/K, and offset drift in the (45 to + 851C) range is 1.4% FS (Sensitec). Another possible configuration is a multisensor arrangement in a circular pattern around the measured conductor. This can be used to measure large currents.

7.6. Other principles for current sensing A GMI current sensor was reported in Mala´tek et al. (2005). An amorphous Co67Fe4Cr7Si8B14 strip was annealed to have 230% GMI at 20 MHz. The schematic diagram of the current sensor is shown in Fig 3.28. The strip was wound around the measured conductor and DC biased by an external coil to achieve linear response. A double-core structure was used in order to improve the temperature stability. The temperature coefficient of sensitivity


Figure 3.28

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GMI current sensor -- from Mala´tek et al. (2005).

and offset drift were reduced to one-half compared to the single-core sensor. Such a low improvement was due to the fact that it is difficult to find pieces of amorphous tape that would have exactly the same characteristics. The linearity error was decreased by a factor of three. Further utilization of AC biasing (up to 200 Hz) of the double-core sensor suppressed the temperature offset drift by a factor of 30 (down to 0.6 mA/K) and increased the openloop linearity to 0.5% for the 2 A range. GMI current sensor reported by Zhao et al. (2007) has only a single core, but the reported zero stability is 0.25 mA/K in 72.5 A range. The accuracy is 0.45%. An asymmetric GMI was also used for current sensing. The current sensitivity for a single-strip sensor is 0.13 V/A at 100 kHz operating frequency and increases up to 0.94 V/A at 1 MHz. However the DC stability is questionable (Rheema et al., 2003). The mentioned papers show that the GMI effect can be utilized for current sensing, but with parameters that hardly compete to the wellestablished technologies. Magnetooptical current sensors are suitable for high-voltage, high-current applications, but the reported errors are larger than 1% even after temperature compensation (Cruden et al., 1998; Didosyan et al., 2000). Magnetooptical current clamps were described in Yi et al. (2001). They do not use optical fibers, but bulk-optic glass. Achieved accuracy was 1% for a 50 Hz AC current in a 1000 A range; the sensitivity was 4.45105 rad/A, which is double the Verdet constant of the SF-6 glass. By using bulk flint glass optical detector in the 20 mm wide airgap of ferromagnetic yoke a noise level of 1.6 mA/OHz@280 Hz was achieved. However such a large airgap should significantly reduce the geometrical selectivity.

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7.6.1. Magnetometric location and measurement of hidden currents The magnetic field of a long straight conductor decreases with 1/r. Underground electric currents in single conductors can be located and measured by measuring the magnetic field in several points. This technique was used for location of underwater optical cables that also contain metallic conductors delivering a DC current of about 1 A to supply the repeaters. The field distribution was measured by two three-axial fluxgate magnetometers. The cables were detected from 40 m distance, and their position was determined with 0.1 m accuracy from 4 m distance (Takagi et al. 1996). The magnetometer method is also used to measure the currents in constructions such as bridges and in pipelines. Changes of natural magnetic fields may induce large currents in long conductors: a 70 A current was observed to flow in the Alaska Oil Pipeline (Campbell and Zimmerman, 1980). Current balances measure the Lorentz force. This classical device is still used in metrology.

7.7. Current clamps Current clamps consist of an openable magnetic circuit, which ensures that the reading is not dependent on the actual position of the clamped conductor and the device is insensitive to unclamped conductors. AC current clamps are usually based on openable instrument current transformer. The measured conductor forms the primary winding, the secondary winding being terminated by a small frequency-independent resistor, or connected to current-to-voltage converter. Very accurate clamp current transformers use electronic compensation of the magnetization current and achieve errors of 0.05% from the measured value in 12100% FS. High-current AC and AC/DC openable current transformer clamps (So et al., 1993) and low-current multistage clamp-on current transformer with ratio errors below 50 ppm (So and Bennet, 1997) were developed by So and coworkers. Some of the available DC current clamps based on a Hall sensor may have 10 mA resolution, but the maximum achieved accuracy is 30 mA, even if they are of the compensated type. Their main disadvantage is unwanted sensitivity to the external fields due to the airgap in the magnetic circuit, which is necessary for the Hall sensor. Even the change of the position with respect to the Earth’s field causes significant error and the offset should be manually readjusted. Precise DC/AC current clamps based on a shielded fluxgate sensor were described in Kejı´k et al. (1996). The device has rectangular ferrite core consisting of two symmetrical L-shaped halves. Permalloy shielding decreases the effect of the residual airgap at the clamp joint. Single winding serves for the excitation (by 1 kHz square-wave voltage), sensing


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(second harmonics in the excitation current), and feedback. The sensor linearity and hysteresis error is less than 0.3% of the 40 A full scale. The noise is 10 mA p2p, and long-term zero stability is 1 mA. The main advantage is high suppression of the external currents and fields due to virtually zero airgap.

8. Applications of Magnetic Sensors Magnetic position sensors are cheap and robust. They can work in dirty environment (e.g., under an oil film), where optical sensors fail. The magnetic proximity switch is probably the most used sensor type globally.

8.1. Position measurement Position measurement is the main application of magnetic sensors. The advantage of magnetic sensors is low cost and ruggedness. Rotational speed and position sensors are used in automotive industry (e.g., for ignition timing and ABS). They are most often based on a Hall sensor integrated with a small permanent magnet. A ferromagnetic toothed wheel changes the magnetic path for the magnet. Thus if the wheel rotates, the Hall sensor observes the changing field. AMR sensors start to penetrate this market: due to their higher sensitivity they allow to use larger airgap between the sensor and the wheel.

8.2. Position tracking Magnetic position trackers use a multiaxial artificial magnetic field created by the coil system. Three-axial magnetic sensors (usually induction coils, AMR sensors, or fluxgates) are attached to the tracked object. Trackers are used in virtual reality and entertainment, but they also have many industrial applications 2 one example is the drilling and mining industry, where trackers measure distance and position between drilling paths and underground tunnels. Sophisticated techniques are used to detect and correct field distortion caused by metal objects. Another possibility of position sensing consists in the use of passive targets such as LC resonant markers. LC resonators with a 10 mm long NiZn ferrite core are used in Hashi et al. (2007). The system comprises one excitation and multiple detection coils. It is able to localize multiple targets that have different resonant frequencies with accuracy below 1 mm for a distance of 100 mm between the marker and the sensor array. Some navigation systems are based on artificial source or magnetic field. This can be considered as a large-scale position tracking. A system

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consisting of one transmitter coil and an array of 79 flat receiving coils is used to navigate the tunneling robot. The drilling device is 325 m below the ground. The source coil is gimbaled in vertical position, and the sensing coils are built inside a 5 mm thick rubber mat. The achieved resolution was in the centimeter range and the final accuracy after 75 m long drill was 1.5 cm (Tsujimura and Tetsuya, 2002). Localization of the magnetic dipole (or more complicated object) is called the inversion problem of magnetometry. The solution for a single dipole is analytical if we know the three components of magnetic field and the six components of the field gradient in one point (the gradient tensor is symmetrical, as long as curlB is zero; Nara et al., 2006).

8.3. Navigation The traditional magnetic compass has a moving needle leveled in the horizontal plane. It is possible to detect the position of this needle by Hall sensors, however this is very inaccurate. The precise electronic magnetic compasses traditionally use fluxgate sensors and they reach 0.11 accuracy. Keeping sensors in the horizontal plane is not practical especially for fast moving platforms. In this case it is possible to use triaxial magnetic sensors and two inclinometers and recalculate the correct azimuth from the known pitch and roll. This is called strap-down compass. AMR sensors give less precision for this case. The main source of error here is the crossfield effect: Fig. 3.29 shows the corresponding azimuth error. After all corrections, a 11 azimuth error is typical for an AMR compass, but it is sufficient for many portable applications, as the compass is usually used together with GPS and the Earth’s field is often distorted by the presence of ferromagnetic bodies anyway. If better accuracy is required, miniature fluxgate sensors are the option (Vcela´k et al., 2007).

8.4. Antitheft systems These systems use permanent magnets or other targets attached to the monitored objects. The most popular are magnetostrictive targets: a strip of a magnetostrictive material has a high absorption of an AC magnetic field at its mechanical resonance frequency. Some systems use a pulsed mode: after decay of the excitation pulse the signal from vibrating magnetostrictive labels is detected. The label can be deactivated by demagnetization of the attached strip of magnetically hard material (Herzer, 2007). Another type uses a nonlinear magnetic element that produces rich harmonics (Hasegawa, 2004).

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Azimuth Error due to Cross Axis Azimuth Error (deg)

3 2 1 0 0

90

180

270

360

-1 -2 -3 Azimuth (deg)

Figure 3.29 Azimuth error of an AMR compass caused by the crossfield effect -from Kubı´k et al. (2006a).

8.5. Detection of ferromagnetic objects Induction and DC magnetic sensors (fluxgates and AMRs) are being used to detect and recognize vehicles. This is utilized for traffic control and for security and military purposes (Kang et al., 2005). Multiaxis gradiometers are used to detect submarines. Detectors for antipersonnel mines use eddy currents in conductive targets. The detector head is a flat coil, often a gradient one. Continuous type detectors work typically at 10 kHz frequency, pulse types sample in 35280 microseconds time after the abrupt change of a large current step. These detectors are optimized to find low-metal content mines in small depths. There are two main problems of mine detection: high false alarm rate and signals from magnetic soils (Ripka et al., 2007a). The latest models combine the eddy current method with ground-penetrating radar (GPR) that allows to observe the shape of the nonmetallic part of the mine using its contrast in permittivity. Modern detectors can compensate magnetic soils, but problems occur when the soil has superparamagnetic nanoparticles, which exhibit magnetic viscosity, that is, frequency dependence of the susceptibility or non-instant response to the field step. Detectors for large and deeper objects (anti-tank mines, bombs, and unexploded ordnance, UXO) use similar principles, but with much larger loops, or they are based on DC magnetic gradiometers. Magnetometers can detect large bombs as deep as 6 m. They measure either the vertical gradient using two fluxgates or the scalar vertical gradient using two Overhauser or cesium magnetometers. An extremely precise triaxial fluxgate gradiometer is described in Merayo et al. (2005). Vectorial sensors give more information about the target, but they are sensitive to angular mismatch and positioning errors (Ripka et al., 2007b). The Forster gradiometer measures the vertical vectorial gradient by two uniaxial

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fluxgate sensors. Using sensor cores mounted on a wire, they reached an impressive degree of angular stability. Method for estimation of the position, orientation, and size of the magnetic moment is described in Sheinker et al. (2007). Static magnetic target is scanned along a track by three-axial fluxgate magnetometer mounted on a moving platform. The simulated annealing algorithm is used to solve the set of nonlinear equations with resulting accuracy of 10%.

8.6. Space research and geophysics The instruments and methods to observe the Earth’s field from surface stations and satellites are described in Mandea and Purucker (2005). At present we experience a fast drift of the magnetic poles (40 km/year). Space DC magnetometers use three orthogonally mounted fluxgate sensors together with a resonant magnetometer (Acuna, 2002). A precise fluxgate sensor on board of the Oersted satellite had o0.28 nT rms offset deviation over 6.5 years in the orbit (Primdahl et al., 2006). O’Brien et al. (2007) describe radiation tolerant fluxgate magnetometer for deep space missions. Similar instruments mounted on stable pillars are used in the Earth’s field variation network. The daily data from this network are available from www.intermagnet.org. Satellites with a scientific magnetometer on board should be made from carefully selected, if possible nonmagnetic, materials and the magnetometer should be mounted far from the satellite body so that the field from the satellite itself is sufficiently small. AC fields are usually measured by iron-cored induction coils. Geophysical and archeological prospecting methods include DC magnetometry and measurement of magnetic properties of samples. Instrumentation for DC magnetometry is similar to that used for bomb location (see previous section). Measurement on samples includes evaluation of the tensor of susceptibility and remanence. The susceptibility is usually measured by AC methods. The direction of the remanent magnetization is frozen when the sample is cooled to its Curie temperature. As the historical drift of magnetic poles is known, when the sample orientation is documented, the remanent magnetization can be used for dating of archeological (bricks) or rock samples. The remanent magnetization of rock, soil samples, and archeological artifacts is measured either by a fluxgate gradiometer or by an induction coil when the sample is rotating. As both the archeological and the rock samples may display rather low magnetic moments, special care has to be taken to perform the measurements in well magnetically shielded space (‘‘magnetic vacuum’’).


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8.7. Medical distance and position sensors Magnetic trackers are used to navigate catheters inside the body: 1 mm precision is achievable for sensors of 2 mm diameter. The main operational principle is a simple wire-wound induction coil. As the body liquids are highly conductive, the field frequency should be small, typically 1 kHz. A simple system measuring the distance between two coils was used to monitor the movements and size of the stomach (Tomek et al., 2007). The pick-up coil has a ferrite core to increase its sensitivity; using a core for the small source coil is contraproductive 2 the only effect is that the coil impedance is increased, which can increase the necessary excitation voltage. A position system based on a rare-earth magnet and a Hall sensor was fused with accelerometers to monitor relative jaw movements with 0.3 mm resolution (Flavel et al., 2002). The relative joint angle was measured in O’Donovan et al. (2007) by using a pair of triaxial AMR magnetometers fused with accelerometers and angular rate sensors. The Earth’s field was the reference for the magnetometers. The advantage is that after adding the magnetic sensors the system is resistant to all translatory and angular body movements and drifts of inertial sensors. A bilayer thin-film curvature (bending) sensor attached to the neck was used to measure cardiorespiratory activity (Katranas et al., 2007). Glass-covered magnetic wires were used as targets to monitor the movement of the heart valve (Rivero et al., 2007). Magnetic biscuits are employed to monitor the digestion tract. These biscuits are swallowed and their movement is monitored by external magnetic sensors. They are based on the same technologies as magnetic trackers with passive marker, which may be a hard magnet, magnetically soft magnetic material, Wiegand wire, an LC resonator (Hashi et al., 2007), or an RF transponder. Measurements of the human colon motility using 3 mm diameter NdBFe permanent magnet and a pair of three-axial fluxgate magnetometers are described in Cordova-Fraga et al. (2006). The observed field changes are up to 5 mT. Even though the system was not calibrated, the measured contractile activity frequencies give information for the diagnostics of disorders. Magnetic beads are used to carry drugs and diagnostic molecules. SDT sensors are suitable for detection of individual beads as the sensor can be made very small and sensitive and linearity is not required. Magnetic labeling and detection using microparticles: Superparamagnetic magnetic particles are ideal labels for biosensing. They can be manipulated by field gradient and detected by field sensors. GMR and especially SDT sensors are suitable for this application because of their small size. The nonlinearity of these sensors does not matter in such counting application.

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8.8. Nondestructive testing (NDT) Magnetic NDT and evaluation methods include DC and AC tests (Vertesy and Gasparics, 2003). The DC flux leakage method is used to find cracks in ferromagnetic parts and also to detect corrosion in pipelines. These methods use fluxgate sensors or magnetoresistors as magnetic field detectors or the field is visualized using a ferrofluid: a liquid that contains colloid ferromagnetic particles. GMR arrays (Butin et al., 2005), or arrays of CMOS fluxgates with 0.5 mm pitch (Gruger, 2003), can be used for this purpose. AC methods based on eddy currents can be used for every conducting part. Crack or another discontinuity causes a change in the eddy current path that results in an impedance change of the testing coil (or in the amplitude of the induced voltage). Magnetooptic methods also allow noncontact evaluation of the materials (Joubert and Pinassaud, 2006).

9. Conclusions Our treatment of magnetic sensors was focussed on describing principles of the detection of magnetic fields and giving examples of the characteristics of concrete experimental realizations of the sensors. We have to bear in mind that the range of real magnetic fields is enormous and covers unbelievable 25226 orders of magnitude. The range of interest starts at fT fields, which are produced by neural currents. These are the lowest presently detectable fields. On the other hand, the best experimentally achievable ‘‘magnetic vacuum’’ (i.e., volume shielded from an external magnetic field) in the laboratory is only in the range of picoteslas so that the most sensitive magnetic sensors can be used only as gradiometers. After passing galactic magnetic fields of the order of nanoteslas, we come to the magnetic field at the Earth surface of about 50 mT and only several times higher fields at the solar surface. The magnetic field on the surface of massive stars is estimated by astronomers to be approximately 10 mT, which is incidentally comparable with the field on the surface of a cheap toy magnet. The magnetic field on the surface of Jupiter is about 0.1 T, roughly equal to fields produced by our solenoids and permanent magnets. At the surface of magnetic stars the field reaches about 1 T and in a laboratory, working with iron-core electromagnets, we attain fields in the vicinity of 2 T. If the material of the solenoid is replaced by a superconducting alloy or compound, we can reach fields in excess of 15 T or even 30 T with special materials cooled to liquid helium temperatures. Still higher sustainable laboratory fields are the domain of specially constructed metallic coils (Bitter magnets) requiring excitation


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power and consequently cooling capacity of the order of tens of megawatts. Laboratory pulse fields, and especially fields produced by flux compression, are of short duration and fields up to several hundreds of teslas were achieved. This is comparable with the field on the surface of white dwarfs. The magnetic field on the surface of a neutron star is still approximately six orders of magnitude higher and the objects with the highest known magnetic fields at their surface are magnetars, where the field intensity probably acquires values of 100 GT. These cosmic objects were theoretically predicted and later observed as soft gamma repeaters and anomalous X-ray pulsars (Ibrahim et al., 2002). Our discussion did not cover the whole range of magnetic fields and we restricted ourselves to terrestrial exploitations of magnetic sensors, with some exceptions concerning measurements of magnetic fields in the outer space by space missions. The large extent of field intensities forces us to utilize not only various physical principles for their measurements but also, more importantly for our treatise, various materials in the sensors. If we look on the development of the employment of magnetic materials, fields, principles, and phenomena in practice, we see an accelerating transition from the traditional use in ‘‘heavy’’ industry connected with production, processing, and transmission of electrical energy to more sophisticated utilization: in information technologies, in particular for storing and reading information, biology and medicine, and last but not least in automobile industries. The push of information technologies for ever denser data storage leads to miniaturization of the field-reading elements and massive incorporation of accomplished semiconductor fabrication processes. The medical and biological uses of magnetic methods, from the point of view of sensors, usually require very high sensitivity and sometimes extremely well-screened space, as the magnetic fields produced by living species (heart, brain) are very feeble and their detection and measurement may easily be deteriorated or even rendered impossible by external fields. The fast proceeding automation and computerization of motor cars needs sensing and measuring of many variables and magnetic sensors of various types are of increasing importance for this use. Typical for this field is the measurement of nonmagnetic quantities, such as force, torque, acceleration, speed, distance, etc., by magnetic methods and many sophisticated solutions of this problem were realized. With the enormous and still increasing mass production of this industry, the market for magnetic sensors in this field is probably the most promising and the fastest growing one. On the other hand the price of the new or improved elements or methods is decisive for their introduction into manufacturing much more than in other sectors. The mentioned tendency to miniaturization led to a vast progress in the rather fashionable field of nanotechnology. This has not missed magnetism

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and the use of quantum effects in magnetic phenomena is actually of older date as documented by the use of SQUIDs in measuring the magnetic flux. Let us mention here that the progress to smaller measured quantities and higher sensitivity may hit the limit set by the magnetic flux quantum. In our overview of the progress made in the field of magnetic sensors during the first years of the 21st century we tried to pay attention to as many of the mentioned aspects as possible. We believe that the readers will get the impression that this field is rather active and the use of magnetic sensors in industries and science is perpetually growing.

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AUTHOR INDEX Aarts, J., see Steglich, F. 37 Abad, E., see Schmool, D.S. 254 Abd-Elmeguid, M.M., see Bauer, E. 65, 104 Abe, K. 24, 50, 53, 96, 98 Abe, K., see Aoki, Y. 99 Abe, K., see Keller, L. 29, 51, 101 Abe, K., see Matsuda, T.D. 32, 50 Abe, K., see Sato, H. 18, 21–22, 27, 30, 50–51, 53, 93, 96, 99, 101 Abe, K., see Sugawara, H. 5, 16, 24, 32–33, 42–43, 93, 99 Abe, R., see Sekine, C. 21, 23, 97, 105 Abe, Y., see Sato, H. 18, 21–22, 27, 30, 50–51, 53, 93, 96, 99, 101 Abe, Y., see Sugawara, H. 5, 9–10, 16, 35, 51, 93 Abraham, D.W., see Maranville, B.B. 280, 282 Abraham, D.W., see Sun, J.Z. 206–207 Acker, H., see Umbach, F. 400 Acuna, M.H. 410 Adachi, T., see Shirotani, I. 4, 18, 94 Adeyeye, A.O. 267 Adeyeye, A.O., see Gubbiotti, G. 180 Adriensen, S., see Haddab, Y. 359 Adroja, D.T. 21, 24, 46 Adroja, D.T., see Park, J.-G. 34 Adroja, D.T., see Viennois, R. 21, 24 Adroja, T.D. 21, 25 Aguayo, A. 17 Aguiar, F.M., see Rezende, S.M. 199 Aharoni, A. 129, 151, 203, 354 Ahmed, R., see Khenata, R. 22, 71, 107 Ahn, C.H., see Wu, P.M. 381 Akamatsu, K., see Tomita, S. 260 Akatsu, M., see Goto, T. 18, 46 Akhmadaliev, see Keupper, K. 268 Akhter, M., see Fry, N. 388 Akita, H., see Ochiai, A. 44 Aktas- , B., see Yalc- ın, O. 318 Alben, R. 123 Albert, F.J. 204, 206–207 Albert, F.J., see Katine, J.A. 203–205, 211 Albertini, J.B., see Joisten, H. 379, 390 Al-Douri, Y., see Khenata, R. 22, 71, 107 Aleksonis, J.A., see Garshelis, I.J. 400 Allen, W., see Gregg, J.F. 202 Alleno, E. 69 Allred, J.C. 384 Almazan, R.P. 379 Alvarez, L., see Rivero, G. 411 Alvarez, P., see Sańchez, M.L. 401

Amanuma, H., see Hachitani, K. 63 Amanuma, H., see Shimizu, M. 49, 95, 100 Amiens, C., see Respaud, M. 252 Amisuka, H., see Sekine, C. 34, 64–65, 67 Amitsuka, H., see Giri, R. 63 Amitsuka, H., see Matsuhira, K. 103 Amitsuka, H., see Sakakibara, T. 45 An, S.Y. 117 Anderson, J.M., see Jander, A. 371 Ando, B. 377 Ando, H., see Sekine, C. 65, 97, 105 Ando, S., see Nara, T. 408 Andonovic, I., see Cruden, A. 405 Andraka, B., see Rotundu, C.R. 38–39, 41, 44 Andreev, A.V., see Aoki, Y. 58 Anisimov, A.-N., see Farle, M. 124–125 Ansermet, J.Ph., see Wegrowe, J.E. 202–203, 211 Antoniak, C. 162, 219, 264–266 Antonio, M.R., see Xue, J.S. 22 Aoki, D. 72, 107 Aoki, D., see Tokunaga, Y. 107 Aoki, H., see Aoki, Y. 60 Aoki, H., see Kikuchi, D. 57–58, 63, 103 Aoki, H., see Sanada, S. 58, 60–62, 104 Aoki, H., see Sato, H. 29, 34, 60 Aoki, H., see Tanaka, K. 7, 49–50, 95, 100 Aoki, H., see Tatsuoka, S. 7, 15–17, 94 Aoki, H., see Ueda, M. 63–64, 103 Aoki, Y. 39, 41, 43–44, 46, 58, 60, 62, 96, 99, 101, 103 Aoki, Y., see Abe, K. 24, 50, 53, 96, 98 Aoki, Y., see Hao, L. 29, 42 Aoki, Y., see Higemoto, W. 46 Aoki, Y., see Ikeno, T. 6, 52–53, 102 Aoki, Y., see Ishida, K. 33 Aoki, Y., see Ito, T.U. 58 Aoki, Y., see Iwasa, K. 10, 29, 33–34, 36–37, 42, 99 Aoki, Y., see Keller, L. 29, 51, 101 Aoki, Y., see Kikuchi, D. 57–58, 63, 103 Aoki, Y., see Kobayashi, M. 45 Aoki, Y., see Kohgi, M. 38, 40–41 Aoki, Y., see Kotegawa, H. 44–45 Aoki, Y., see Kuwahara, K. 28, 38, 42–43, 54 Aoki, Y., see Matsuda, T.D. 32, 34, 50 Aoki, Y., see Namiki, T. 25, 27, 43, 49, 95, 98–100 Aoki, Y., see Pourret, A. 30 Aoki, Y., see Rotundu, C.R. 38–39, 41 Aoki, Y., see Saha, S.R. 18, 36

421

422 Aoki, Y., see Sakakibara, T. 45 Aoki, Y., see Sanada, S. 58, 60–62, 104 Aoki, Y., see Sato, H. 18, 21–22, 27, 29–30, 34, 50–51, 53, 60, 93, 96, 99, 101 Aoki, Y., see Shiina, R. 41 Aoki, Y., see Shu, L. 46 Aoki, Y., see Sugawara, H. 5, 9–10, 16, 18, 22, 24, 32–33, 35, 38–39, 42–43, 51, 93, 96, 99 Aoki, Y., see Sugiyama, K. 99 Aoki, Y., see Tanaka, K. 7, 49–50, 95, 100 Aoki, Y., see Tatsuoka, S. 7, 15–17, 94 Aoki, Y., see Tayama, T. 38–39, 41, 101 Aoki, Y., see Toda, M. 6, 77–78, 110 Aoki, Y., see Ueda, M. 63–64, 103 Aoki, Y., see Yamada, T. 16, 49, 63, 95 Aoki, Y., see Yogi, M. 44, 46 Arai, K.I., see Hashi, S. 407, 411 Araki, S., see Kohgi, M. 38, 40–41 Araki, S., see Kuwahara, K. 38, 43 Aramaki, K., see Kawahito, S. 378 Araseki, N., see Shirotani, I. 6, 68–69, 78 Arias, R. 292, 294, 314 Aroca, C., see Almazan, R.P. 379 Aroca, C., see Ciudad, D. 389 Aroca, C., see Michelena, M.D. 389 Aroca, C., see Perez, L. 372, 378 Aroca, C., see Prieto, J.L. 389 Arrott, A.S., see Cochran, J.F. 113, 132 Arvanitis, D., see Wiedwald, U. 258 Aso, N., see Iwasa, K. 25 Atsarkin, V.A., see Noginova, N. 238, 243, 245–248 Audoin, M., see Joisten, H. 379, 390 Auster, H. 377 Avaritsiotis, J.N., see Dimitropoulos, P.D. 377 Averous, M., see Viennois, R. 95, 98 Azevedo, A., see Rezende, S.M. 199 Baberschke, K., see Farle, M. 124–125 Baberschke, K., see Lindner, J. 113, 116 Baberschke, K., see Schulz, B. 125 Back, C.H., see Keupper, K. 268 Back, C.H., see Puzic, A. 283–285 Back, C.H., see Stoll, H. 283 Backes, D., see Vaz, C.A.F. 181, 322 Bacri, J.-C., see Shilov, V.P. 225 Bader, S.D., see Buchanan, K.S. 268–269 Bader, S.D., see Novosad, V. 172 Baenitz, M., see Leithe-Jasper, A. 72–74, 108 Baıëtto, V., see Berger, R. 220 Baffa, O., see Cordova-Fraga, T. 411 Baglio, S., see Ando, B. 377 Baibich, M.N. 112 Bailey, W.E., see Zohar, S. 251, 253 Bailleul, M., see Jorzick, J. 175 Bain, J.A., see Tamaru, S. 177–179 Bakuzis, A.F., see Pereira, A.R. 242 Bal, M., see Yalc- ın, O. 318

Author Index

Baraduc, C., see Petit, S. 214 Barandiaran, J.M., see Schmool, D.S. 254 Barandiarań, M., see Schmool, D.S. 113, 119, 131, 280 Barker, R.C., see Vittoria, C. 145 Barman, A., see Hicken, R.J. 285–286 Barnard, J.A., see Butera, A. 228 Barnes, T., see Chi, S. 47, 100 Barrera-Solano, C., see Ortega, D. 238, 249–250 Barrois, G., see Joisten, H. 379, 390 Barsukov, I., see Meckenstock, R. 287–289 Barsukov, I., see Trunova, A.V. 113, 258, 260–261 Bartenlian, B., see Jorzick, J. 270 Bartenlian, B., see Mathieu, C. 270, 308–309 Barthele´my, A. 368 Baschirotto, A. 378 Basov, D.N., see Dordevic, S.V. 21, 24, 69–70, 93, 96, 106–107 Bass, J., see Tsoi, M. 187, 201 Baszynski, J., see Dubowik, J. 222 Batlle, X. 114, 220 Bauer, E. 6, 49, 65, 69, 75, 78, 104, 110 Bauer, E., see Reissner, M. 49 Bauer, E.D. 5, 18–19, 21, 24, 28, 38, 43–44, 52, 65, 95–96, 98, 100–102, 104–105 Bauer, E.D., see Cao, D. 36 Bauer, E.D., see Dilley, N.R. 69, 104, 106 Bauer, E.D., see Dordevic, S.V. 21, 24, 69–70, 93, 96, 106–107 Bauer, E.D., see Gajewski, D.A. 23, 98 Bauer, E.D., see Goremychkin, E.A. 28, 38, 43 Bauer, E.D., see Grube, K. 44 Bauer, E.D., see MacLaughlin, D.E. 44–45 Bauer, E.D., see Maple, M.B. 39, 42, 44, 96 Bauer, E.D., see Oeschler, N. 39 Bauer, E.D., see Tenya, K. 39 Bauer, E.D., see Vollmer, R. 28, 38, 44, 101 Bauer, G.E.W., see Brataas, A. 190, 217 Bauer, G.E.W., see Foros, J. 209 Bauer, G.E.W., see Kovalev, A.A. 190 Bauer, G.E.W., see Tserkovnyak, Y. 217 Bauer, G.E.W., see Wang, X. 114 Bauer, G.E.W., see Xiao, J. 217 Bauer, M., see Mathieu, C. 270 Bauer, M., see Slavin, A.N. 140 Baumbach, R.E. 7 Baumbach, R.E., see Henkie, Z. 6, 23, 48–49, 95 Baumbach, R.E., see Maple, M.B. 21, 23, 48, 97, 100 Baumjohann, W., see Magnes, W. 378 Bayer, C. 270 Bayle-Guillemaud, P., see Darques, M. 312 Bazaliy, Ya.B. 202 Bazzocchi, R., see DiRienzo, L. 403 Bødker, F. 121

Author Index

Beaurepaire, E. 113 Becker, J.J., see Alben, R. 123 Beech, R., see Ripka, P. 368, 370 Beech, R.S., see Tondra, M. 371 Beek, T., see O’Brien, H. 410 Behnia, K., see Pourret, A. 30 Beleggia, M., see De Graef, M. 373 Belie¨n, P., see Gijs, M.A.M. 357, 371 Beljers, H.G., see Smit, J. 117 Belloy, E. 402 Belloy, E., see Dezuari, O. 379 Beltran, M., see Koksharov, Yu.A. 247 Bennemann, K.H., see Hohlfeld, J. 114 Bennet, D.A., see So, E. 406 Bentien, A., see Viennois, R. 15–16, 64, 95 Bera, S.C. 402 Berardan, D., see Alleno, E. 69 Berezovsky, J., see Park, J.P. 308, 310–312 Berger, L. 186, 189–191 Berger, R. 220, 222, 224–225, 253, 262 Berger, R., see Kliava, J. 114, 220–221 Berger, St., see Bauer, E. 49, 65, 104 Berginski, M., see Podbielski, J. 325, 328–329 Berkov, D.V., see Jorzick, J. 175 Berkowitz, E.A., see Kodama, R.H. 155 Bermu´dez Urenã, E., see Vassallo Brigneti, E. 306 Bernal, O.O., see MacLaughlin, D.E. 44–45 Bernfoeft, N., see Kohgi, M. 38, 40–41 Berteaud, A.J. 117 Bessais, L., see Dormann, J.L. 220 Betts, J.B., see Butch, N.P. 6, 24, 28, 49, 100 Betts, J.B., see Ho, P.-C. 44, 50, 53–54, 103 Betts, J.B., see Yuhasz, W.M. 60, 62–63, 104 Beuss, M., see Puzic, A. 283–285 Beuss, M., see Stoll, H. 283 Bewley, R., see Chi, S. 47, 100 Bewley, R.I., see Viennois, R. 21, 24 Beyermann, W.P., see Yatskar, A. 26 Bigot, J.-Y., see Beaurepaire, E. 113 Bilc, D.I., see Krishnamurthy, V.V. 65 Billingsley, W., see Ripka, P. 352, 357, 376 Bircan, C., see Meckenstock, R. 287 Bischoff, L., see Keupper, K. 268 Bissey, J.-C., see Berger, R. 220, 222, 224–225, 253, 262 Biziere, N., see Pannetier-Lecoeur, M. 360 Black, W.T., see Kunets, V.P. 357 Blanco, J.M., see Chizhik, A. 373 Blanco, J.M., see Garcıá, C. 388 Bland, J.A.C., see McPhail, S. 301–302 Bland, J.A.C., see Vaz, C.A.F. 181, 322 Blazek, J., see Butvin, P. 350 Bloch, D., see Joisten, H. 379, 390 Bloch, F. 132, 144 Bloembergen, N. 144 Bloom, S.H., see Guertin, R.P. 106 Bloyet, D., see Quasimi, A. 359

423 Boero, G., see Kejik, P. 359 Bogner, J., see Leithe-Jasper, A. 69 Bonville, P., see Alleno, E. 69 Boonk, L., see Evers, C.B.H. 64, 102–104 Bordallo, H.N., see Nakotte, H. 69, 106 Borghetti, F., see Baschirotto, A. 378 Borrmann, H., see Leithe-Jasper, A. 73–74 Borrmann, H., see Schnelle, W. 69 Borza, F., see Katranas, G.S. 411 Botters, B., see Neusser, S. 302, 304–305 Bouhafs, B., see Khenata, R. 22, 71, 107 Bouhemadou, A., see Khenata, R. 22, 71, 107 Boukhenoufa, A. 387 Boulet, P., see Bauer, E. 69 Bourlinos, A.B., see Noginova, N. 238, 243, 245–248 Boust, F. 173–175 Ba¨r, L., see Uhrmann, T. 398 Braganca, P.M., see Fuchs, G.D. 212–213 Braganca, P.M., see Sankey, J.C. 212 Braginski, A.I., see Clarke, J. 382 Braithwaite, D., see Measson, M.-A. 44, 46 Braithwaite, D., see Seyfarth, G. 44 Brataas, A. 190, 217 Brataas, A., see Foros, J. 209 Brataas, A., see Kovalev, A.A. 190 Brataas, A., see Tserkovnyak, Y. 217 Brataas, A., see Wang, X. 114 Brataas, A., see Xiao, J. 217 Brauer, P., see Merayo, J.M.G. 409 Brauer, P., see Moldovanu, C. 372 Brauer, P., see Nielsen, O.V. 376 Brauer, P., see Pedersen, E.B. 377 Brauer, P., see Primdahl, F. 373, 376, 383, 410 Braum, D.J., see Grandjean, F. 19, 21–23, 64, 96–98, 104 Braun, D.J. 3, 5, 94–98, 100–105 Braun, D.J., see Evers, C.B.H. 64, 102–104 Braun, D.J., see Ge`rard, A. 64 Braun, D.J., see Jeitschko, W. 2–3, 22–23, 29, 52, 64, 107 Bredl, C.D., see Steglich, F. 37 Briaire, J., see Gijs, M.A.M. 357, 371 Bridges, F., see Bauer, E.D. 65, 104–105 Bridges, F., see Cao, D. 36 Brion, S.D.E., see Sekine, C. 58–59 Brison, J.-P. 44 Brison, J.P., see Measson, M.-A. 44, 46 Brison, J.P., see Seyfarth, G. 44–45 Bru¨kl, H., see Puzic, A. 283–285 Brodsky, M.B., see Gru¨nberg, P. 112 Broto, J.M., see Baibich, M.N. 112 Broto, J.M., see Respaud, M. 252 Brown, P., see O’Brien, H. 410 Brownell, D., see Deak, J. 371 Bu¨ttner, O., see Mathieu, C. 270 Bu¨ttner, O., see Slavin, A.N. 140 Buchanan, K.S. 268–269

424 Bucher, J.P., see Rastei, M.V. 270 Buda, L.D., see Ebels, U. 312, 316–317 Budker, D. 384 Bud’ko, S., see Suderow, H. 44–45 Buhrman, R.A., see Albert, F.J. 204, 206–207 Buhrman, R.A., see Cui, Y.-T. 214 Buhrman, R.A., see Fuchs, G.D. 212–213 Buhrman, R.A., see Katine, J.A. 203–205, 211 Buhrman, R.A., see Kiselev, S.I. 209–210 Buhrman, R.A., see Myers, E.B. 211 Buhrman, R.A., see Sankey, J.C. 212, 214 Bulsara, A.R., see Ando, B. 377 Bunce, C., see Lepadatu, S. 279, 281 Burgei, W.A., see Yu, C. 292–295, 301 Burkett, S.L., see Stutzke, N. 214 Burkhardt, U., see Leithe-Jasper, A. 73–74 Burkhardt, U., see Schnelle, W. 69 Butch, N.P. 6, 24, 28, 49, 100 Butch, N.P., see Frederick, N.A. 44 Butch, N.P., see Ho, P.-C. 44, 50, 53–54, 103 Butch, N.P., see Maple, M.B. 21, 23, 29, 47–48, 50, 97, 100 Butch, N.P., see Wawryk, R. 47, 95, 100 Butch, N.P., see Yuhasz, M.W. 27, 47, 100 Butch, N.P., see Yuhasz, W.M. 60, 62–63, 104 Butera, A. 228 Butin, L. 412 Butko, A., see Meckenstock, R. 287–289 Butvin, P. 350 Butvinova, B., see Butvin, P. 350 Bydzovsky, J. 398 Callen, H.B. 144 Cambril, E., see Jorzick, J. 270 Cambril, E., see Mathieu, C. 270, 308–309 Camley, R.E., see Grimsditch, M. 177, 180 Camley, R.E., see Kuanr, B.K. 117, 277, 279–280 Campbell, W.C. 406 Canfield, P.C. 5 Canfield, P.C., see Suderow, H. 44–45 Canfield, P.C., see Yatskar, A. 26 Cao, D. 36 Cao, D., see Bauer, E.D. 65, 104–105 Cao, R., see Moriyama, T. 216 Carabias, I., see Rivero, G. 411 Cardoso, F.A. 371 Cardoso, S., see Kakazei, G.N. 128, 233, 235–236 Cardoso-Gil, R., see Leithe-Jasper, A. 73–74, 108 Cardoso-Gil, R., see Schnelle, W. 16, 69, 72–74, 106, 108–109 Carey, M., see Pechan, M. 267–268 Carlotti, G., see Gubbiotti, G. 180, 270, 332, 334–335 Carneiro, A.A.O., see Cordova-Fraga, T. 411 Carr, C., see O’Brien, H. 410

Author Index

Caruso, M., see Pant, B. 365 Carva, K., see Turek, I. 217 Castano, E., see Gonzalez, O.J. 395 Castanõ, F.J. 114, 181 Castanõ, F.J., see Vaz, C.A.F. 181, 322 Castellano, J.C., see Gonzalez, O.J. 395 Cavoit, C. 385 Celinski, Z., see Kuanr, B.K. 277, 279–280 Celinski, Z.J., see Kuanr, B.K. 117 Cerman, A. 377 Cerman, A., see Kawahito, S. 378 Chainani, A., see Matsunami, M. 11, 21–22, 25 Champion, J.L., see Wickenden, D.K. 389 Chandrasekhar, V., see Jung, S. 168, 324 Chang, S., see Nakotte, H. 69, 106 Chanteur, G., see Coillot, C. 385 Chanteur, G., see Leroy, P. 360 Chantrell, R., see Lepadatu, S. 279, 281 Chantrell, R.W. 220 Chantrell, R.W., see Guslienko, K.Yu. 173, 176–180, 302, 332 Chantrell, R.W., see Kakazei, G.N. 270–271 Chantrell, R.W., see Thomas, A.H. 152–153 Chantrell, R.W., see Verdes, C.G. 153–154 Chapon, L.C., see Viennois, R. 21, 24 Chappert, C., see Jorzick, J. 270 Chappert, C., see Mathieu, C. 270, 308–309 Chappert, C., see Mistral, Q. 214–216 Charar, S., see Viennois, R. 15–16, 23, 64, 95, 98 Chattopadhyay, S., see Bera, S.C. 402 Chau, R., see Gajewski, D.A. 23, 98 Chaudret, B., see Respaud, M. 252 Chazelas, J., see Baibich, M.N. 112 Chen, C.-T., see Matsunami, M. 11, 21, 25 Chen, C.T., see Tsuda, S. 94 Chen, F., see Noginova, N. 238, 243, 245–248 Chen, J., see Tomek, J. 411 Chen, J.W., see Torikachvili, M.S. 27, 29, 50–51, 53, 70, 93, 99, 101, 107 Chen, L., see Matsunami, M. 35, 58, 65 Chen, M., see Tsankov, M.A. 140 Chen, S.L., see Kuo, C.H. 390 Chen, W., see Midzor, M.M. 114 Chen, X.-Q., see Bauer, E. 6, 75, 78, 110 Cheng, J., see Zhao, Z. 405 Cheng, S.F., see Ozbay, A. 360 Cherifi, S., see Vaz, C.A.F. 181, 322 Chi, M.C., see Alben, R. 123 Chi, S. 47, 100 Chi, S., see Maple, M.B. 47, 100 Chia, E.E.M. 44–45, 50 Chiang, K., see Yi, B. 405 Chiang, W.-C., see Tsoi, M. 187, 201 Childress, J., see Synogatch, V. 214 Chiriac, H., see Ioan, C. 372, 375 Chizhik, A. 373 Choi, B.C. 117

425

Author Index

Choi, B.W., see Kang, M.H. 409 Choi, S. 140, 143 Choi, S.O., see Choi, W.Y. 378 Choi, S.O., see Na, K.W. 380 Choi, S.O., see Park, H.S. 380 Choi, S.O., see Ripka, P. 379 Choi, W.Y. 378, 380 Choi, W.Y., see Park, H.S. 380 Chou, K.W., see Keupper, K. 268 Chou, K.W., see Puzic, A. 283–285 Chouteau, G., see Sekine, C. 58–59 Chrisey, D.B., see Rubinstein, M. 228 Christianson, A., see Nakotte, H. 69, 106 Christianson, A.D., see Maple, M.B. 39, 42, 96 Chu, B., see Yi, B. 405 Chu, V., see Cardoso, F.A. 371 Chudnovsky, E.M. 123 Chudo, H., see Tokunaga, Y. 107 Cichorek, T. 41 Cichorek, T., see Baumbach, R.E. 7 Cichorek, T., see Henkie, Z. 6, 23, 48–49, 95 Cichorek, T., see Maple, M.B. 21, 23, 48, 97, 100 Cichorek, T., see Sayles, T.A. 27, 48, 100 Cichorek, T., see Wawryk, R. 47, 95, 100 Ciudad, D. 389 Clarke, J. 382 Cochran, J.F. 113, 132 Cochran, J.F., see Heinrich, B. 113 Coehoorn, R. 368 Coey, J.M.D., see Gregg, J.F. 202 Coillot, C. 385 Coillot, C., see Leroy, P. 360 Comprehensive Survey. vol. 5, ch. 6, VCH, see Dehmel, G. 385 Compton, R.L., see Pechan, M. 292, 297–299, 301, 303 Conde, J.P., see Cardoso, F.A. 371 Connell, A., see Tipek, A. 402 Cooley, B.J., see Wang, R.F. 114 Cordova-Fraga, T. 411 Costa, B.A., see Cardoso, F.A. 371 Costa, M.D., see Kakazei, G.N. 128, 233, 235–236, 270 Costache, M.V. 216 Costache, M.V., see Grollier, J. 285 Cottam, M.G., see Nguyen, T.M. 173, 181–182, 186–188 Covington, M., see Tamaru, S. 177–179 Cox, D.L. 26, 46 Crawford, T.M., see Silva, T.J. 114 Crawford, T.M., see Tamaru, S. 177–179 Crespi, V.H., see Wang, R.F. 114 Creuzet, G., see Baibich, M.N. 112 Crowell, P.A., see Park, J.P. 308, 310–312 Crowell, P.A., see Pechan, M. 292, 297–299, 301, 303 Crozat, P., see Mistral, Q. 214–216

Crozat, P., see Woytasik, M. 379 Cruden, A. 405 Cubitt, R., see Huxley, A.D. 45 Cuchet, R., see Joisten, H. 379, 390 Cui, Y.-T. 214 Cui, Y.-T., see Sankey, J.C. 214 Cupido, E., see O’Brien, H. 410 Curnoe, S.H. 35–36 Curnoe, S.H., see Harima, H. 36 Curson, N.J., see McPhail, S. 301–302 Custers, J., see Sakakibara, T. 45 Custers, J., see Tayama, T. 32, 99, 101 da Silva, E.C., see Pires, M.J.M. 236, 255 Dai, P., see Chi, S. 47, 100 Dai, P., see Maple, M.B. 47, 100 Dalessandro, L. 403 Dalichaouch, Y., see Torikachvili, M.S. 27, 29, 50–51, 53, 70, 93, 99, 101, 107 Dallago, E., see Baschirotto, A. 378 Danebrock, M.E. 6, 23, 52, 63, 65, 74, 95, 98, 100, 102–104 Darques, M. 312 Das, B.N., see Rubinstein, M. 228 Daubric, H., see Berger, R. 222, 224–225 Daughton, J.M., see Tondra, M. 371 Daunois, A., see Beaurepaire, E. 113 Davis, Z., see Malkinski, L.M. 277–278 de Araujo, D.B., see Cordova-Fraga, T. 411 De Biasi, E. 223, 225–227, 263 de Biasi, E., see Ramos, C.A. 320–321 De Biasi, R.S. 220, 222, 241, 253, 255, 262 De Graef, M. 373 De Long, L., see Jung, S. 168, 324 de Vismes, A., see Pannetier-Lecoeur, M. 368 Deak, J. 371 Decanini, D., see Jorzick, J. 270 Decanini, D., see Mathieu, C. 270, 308–309 Dedkov, Yu.S. 69 Deen, D.A., see Kunets, V.P. 357 Degiorgi, L., see Dordevic, S.V. 69–70, 93, 96, 106–107 Dehmel, G. 385 DeLong, L.E. 94 DeLong, L.E., see Rivkin, K. 168–173 DeLong, L.E., see Xu, W. 322–323 Demand, M., see Encinas, A. 311, 316 Demand, M., see Encinas-Oropesa, A. 309, 313–315, 318 Demeus, L., see Haddab, Y. 359 Demierre, M., see Kejik, P. 359 Demokritov, S.E., see Slavin, A.N. 136 Demokritov, S.O. 136, 270, 294 Demokritov, S.O., see Bayer, C. 270 Demokritov, S.O., see Jorzick, J. 175, 270 Demokritov, S.O., see Mathieu, C. 270, 308–309 Demokritov, S.O., see Slavin, A.N. 140

426 Denardin, J.C., see Pires, M.J.M. 236 Denbeaux, G., see Stoll, H. 283 Devaux, E., see Rastei, M.V. 270 Devezas, T.C., see De Biasi, R.S. 220, 222, 253, 255, 262 Devolder, T., see Mistral, Q. 214–216 Dewalsky, M.V., see Jeitschko, W. 65–66, 102, 105 Dewhurst, C.D., see Huxley, A.D. 45 Dezuari, O. 379 Dezuari, O., see Belloy, E. 402 Diaconu, E.D., see Ioan, C. 372 Dickey, R.P., see Bauer, E.D. 19, 24, 98, 101 Didosyan, Y.S. 405 Dieny, B., see Petit, S. 214 Dietzel, D., see Meckenstock, R. 287 DiLella, D. 389 Dilley, N.R. 69, 104, 106 Dilley, N.R., see Bauer, E.D. 19, 24, 98, 101 Dilley, N.R., see Dordevic, S.V. 21, 24, 69–70, 93, 96, 106–107 Dilley, N.R., see Gajewski, D.A. 23, 98 Dilley, N.R., see Nakotte, H. 69, 106 Dimian, M., see Kachkachi, H. 129 Dimitropoulos, P.D. 377 Dimitropoulos, P.D., see Petrou, J. 382 Dimopoulos, T., see Uhrmann, T. 398 Ding, J., see Yi, J.B. 351 Ding, L. 385 DiRienzo, L. 403 Do¨nni, A., see Keller, L. 29, 51, 101 Do, T.D., see Frederick, N.A. 44 Do, T.D., see MacLaughlin, D.E. 44–45 Doi, M., see Tou, H. 38 Doi, Y., see Giri, R. 63 Doi, Y., see Matsuhira, K. 94, 99, 103 Doi, Y., see Sekine, C. 67 Dolabdjian, C., see Butin, L. 412 Dolabdjian, C., see Ding, L. 385 Dolabdjian, C., see Quasimi, A. 359 Dolabdjian, C., see Robbes, D. 385 Dolabdjian, C.P., see Boukhenoufa, A. 387 Dolgos, D., see Keupper, K. 268 Domıńguez, M., see Ortega, D. 238, 249–250 Dordevic, S.V. 21, 24, 69–70, 93, 96, 106–107 Dormann, J.L. 220 Downey, P.R., see McGary, P.D. 398 Draxler, K. 350, 402 Draxler, K., see Mlejnek, P. 401 De´rer, J., see Majchra´k, P. 128, 236–237 Drexler, P. 390 Drljaca, P.M. 378, 390 Drljaca, P.M., see Popovic, R.S. 359, 394, 403 Drobnik, S., see Grube, K. 44 Droxler, R., see Fericean, S. 392 Du, Y., see Li, T. 318–319 Dubowik, J. 119, 222, 228, 236 Duffy, M., see Ripka, P. 402

Author Index

Duffy, M.C., see Tang, S.C. 402 Dufour-Gergam, E., see Woytasik, M. 379 Dumitru, I. 155–157, 320 Dumpich, G., see Stahlmecke, B. 114 Dupuis, V., see Jamet, M. 129–130 Duret, D.N. 383 Duric, R.M., see Ponjavic, M.M. 402 Dutta, P. 238–240 Duvail, J.-L., see Ebels, U. 312, 316–317 Eames, P., see Park, J.P. 308, 310–312 Ebbesen, Th., see Rastei, M.V. 270 Ebel, T., see Evers, C.B.H. 64, 102–104 Ebels, U. 312, 316–317 Ebels, U., see Darques, M. 312 Ebels, U., see Encinas-Oropesa, A. 309, 313–315, 318 Ebels, U., see Petit, S. 214 Edelstein, A. 349, 360 Edelstein, A.S., see Ozbay, A. 360 Edelstein, S., see Lenz, J. 349 Eguchi, R., see Matsunami, M. 11, 21, 25 Eilez, A., see Castanõ, F.J. 114, 181 El-Hilo, M., see Thomas, A.H. 152–153 Elliott, R.J., see Heide, C. 192, 196 Elmers, H.J., see Martyanov, O.N. 295, 299–300 Emley, N.C., see Kiselev, S.I. 209–210 Encinas, A. 311, 316 Encinas, A., see Darques, M. 312 Encinas-Oropesa, A. 309, 313–315, 318 Endo, T., see Sugawara, H. 18, 27, 47–48, 94, 99 Engebretson, D.M., see Park, J.P. 308, 310–312 Enrı´quez, J.L., see Garcıá, A. 381 Errebbahi, A., see Viennois, R. 95, 98 Estourne`s, C., see Berger, R. 222, 224–225 Etienne, P., see Baibich, M.N. 112 Etoh, Y., see Sasada, I. 399 Etournean, J., see Kasaya, M. 63 Evers, C.B.H. 64, 102–104 Evers, C.B.H., see Danebrock, M.E. 6, 23, 52, 63, 65, 74, 95, 98, 100, 102–104 Evers, C.B.H., see Jeitschko, W. 65–66, 102, 105 Fagaly, R. 382 Faini, G., see Vaz, C.A.F. 181, 322 Faist, A., see Vollmer, R. 28, 38, 44, 101 Fan, J. 381 Fan, J., see Zhao, Z.J. 381 Fan, X., see Moriyama, T. 216 Farle, M. 116, 122, 124–126 Farle, M., see Antoniak, C. 162, 219, 264–266 Farle, M., see Spasova, M. 257, 259 Farle, M., see Trunova, A.V. 113, 258, 260–261 Farle, M., see Wiedwald, U. 258 Fassbender, J., see Keupper, K. 268

Author Index

Fauth, K., see Keupper, K. 268 Fdez-Gubieda, M.L., see Sarmiento, G. 259 Fdez-Gubieda, M.L., see Schmool, D.S. 254 Fergeau, P., see Seran, H.C. 385 Fericean, S. 392 Fermon, C., see Jorzick, J. 175 Fermon, C., see Pannetier-Lecoeur, M. 360, 368 Fernandez, A., see Nielsen, O.V. 376 Ferrari, E.F., see Viegas, A.D.C. 255 Ferre´-Borrull, J., see Marsal, L.F. 303, 306 Ferreira, H.A., see Cardoso, F.A. 371 Ferreira, R. 371 Fert, A. 308 Fert, A., see Baibich, M.N. 112 Fert, A., see Barthele´my, A. 368 Fert, A., see Valet, T. 195 Fert, A., see Zhang, S. 196–197 Fiala, P., see Drexler, P. 390 Fink, A., see Tondra, M. 371 Fiorani, D., see Dormann, J.L. 220 Fiorani, D., see Winkler, E. 266 Fischer, G., see Bauer, E. 69 Fischer, G.A., see Ozbay, A. 360 Fischer, P., see Bauer, E. 49 Fischer, P., see Keller, L. 29, 51, 101 Fischer, P., see Puzic, A. 283–285 Fischer, P., see Stoll, H. 283 Fisk, Z. 5 Fisk, Z., see Canfield, P.C. 5 Flandre, D., see Haddab, Y. 359 Flatau, A.B., see McGary, P.D. 398 Flavel, S.C. 411 Flouquet, J., see Brison, J.-P. 44 Flouquet, J., see Huxley, A.D. 45 Flouquet, J., see Measson, M.-A. 44, 46 Flouquet, J., see Seyfarth, G. 44–45 Fluss, M.J., see Maple, M.B. 47, 100 Fluss, M.J., see Yuhasz, M.W. 27, 47, 100 Foecker, A.J., see Jeitschko, W. 65–66, 102, 105 Folks, L., see Pechan, M. 267–268 Formentin, P., see Marsal, L.F. 303, 306 Foros, J. 209 Foulquet, J., see Kuwahara, K. 28, 42 Fowler, C.M. 389 Fraden, J. 349 Fradin, F.Y., see Buchanan, K.S. 268–269 Fraga, E., see Kraus, L. 356 Fraile Rodriguez, A., see Wiedwald, U. 258 Frait, Z., see Majchra´k, P. 128, 236–237 Frait, Z., see Wiedwald, U. 258 Frandsen, C., see Castanõ, F.J. 114, 181 Frank, A., see Mathieu, C. 308–309 Franz, W., see Steglich, F. 37 Frederick, N.A. 38, 44 Frederick, N.A., see Bauer, E.D. 18, 24, 28, 38, 43–44, 52, 65, 95, 100–102, 104–105

427 Frederick, N.A., see Butch, N.P. 6, 24, 28, 49, 100 Frederick, N.A., see Cichorek, T. 41 Frederick, N.A., see Goremychkin, E.A. 28, 38, 43 Frederick, N.A., see Ho, P.-C. 44, 50, 53–54, 103 Frederick, N.A., see Maple, M.B. 39, 42, 44, 47, 50, 96, 100 Frederick, N.A., see Oeschler, N. 39 Frederick, N.A., see Shu, L. 46 Frederick, N.A., see Tenya, K. 39 Frederick, N.A., see Yuhasz, M.W. 27, 47–48, 99–100 Frederick, N.A., see Yuhasz, W.M. 60, 62–63, 104 Freeman, E.J., see Bauer, E.D. 5, 19, 21, 24, 96, 98, 101 Freeman, E.J., see Dilley, N.R. 69, 104, 106 Freeman, M.R. 113, 117 Freeman, M.R., see Choi, B.C. 117 Freeman, M.R., see Hiebert, W.K. 113 Freeman, M.R., see Zhu, X. 324–327 Freemany, E.J., see Gajewski, D.A. 23, 98 Freitas, P.P., see Cardoso, F.A. 371 Freitas, P.P., see Ferreira, R. 371 Freitas, P.P., see Kakazei, G.N. 128, 233, 235–236 Freitas, R.S., see Wang, R.F. 114 Freye, D.M., see Mathur, N.D. 46 Friederich, A., see Baibich, M.N. 112 Friedrich, A.P., see Lemme, H. 404 Fry, N. 388 Fuchs, G.D. 212–213 Fudamoto, Y., see Luke, G.M. 46 Fujii, M., see Tomita, S. 258 Fujimoto, T., see Ito, T.U. 58 Fujino, T. 67 Fujino, T., see Nakanishi, Y. 62 Fujino, T., see Yoshizawa, M. 68 Fujita, T. 25 Fujiwara, H., see Kobayashi, M. 45 Fujiwara, H., see Yamasaki, A. 61 Fujiwara, K. 22 Fukamichi, K., see Guslienko, K.Yu. 174, 271 Fukamichi, K., see Kakazei, G.N. 270–271 Fukazawa, H., see Hachitani, K. 56–58, 63 Fukazawa, H., see Shimizu, M. 49, 95, 100 Gai, K.-Z., see Xi, H. 199 Gajewski, D.A. 23, 98 Galatanu, A., see Bauer, E. 49, 65, 69, 104 Galatanu, A., see Matsuda, T.D. 69–71, 106 Galli, M., see Bauer, E. 65, 104 Garanin, D. 130 Garcıá, A. 381 Garcıá, C. 388 Garcıá-Bastida, A.J., see Sańchez, R.D. 252–253

428 Garcıá-Palacios, J.L. 156, 158, 160 Garcıá-Paéz, J.M., see Rivero, G. 411 Garcia, A. 382 Garcia, A.G.F., see Fuchs, G.D. 212–213 Garcia, A.G.F., see Sankey, J.C. 212 Garcia-Arribas, A., see Schmool, D.S. 254 Garitaonandia, J.S., see Ortega, D. 238, 249–250 Garitaonandia, J.S., see Schmool, D.S. 112–113, 120, 128, 228–229, 231, 234, 254 Garshelis, I.J. 400–401 Gasparics, A., see Pavo, J. 375 Gasparics, A., see Vertesy, G. 412 Gaud, P., see Joisten, H. 379, 390 Gazeau, F., see Shilov, V.P. 225 Gegenwart, P., see Oeschler, N. 39 Gegenwart, P., see Tenya, K. 39 Geiler, A.L. 372 Ge`rard, A. 64 Ge`rard, A., see Grandjean, F. 19, 21–23, 64, 96–98, 104 Germano, J., see Cardoso, F.A. 371 Gerrits, T., see Rasing, T. 117 Gerrits, Th., see Schneider, M.L. 274–275 Geshev, J., see Viegas, A.D.C. 255 Giannelis, E.P., see Noginova, N. 238, 243, 245–248 Gieraltowski, J., see Ding, L. 385 Gieraltowski, J., see Tannous, C. 149, 151–152 Gier"owski, P., see Lachowicz, H. 254 Giersig, M., see Spasova, M. 257, 259 Giersig, M., see Wiedwald, U. 258 Giesbers, J.B., see Gijs, M.A.M. 357, 371 Giesen, F. 326, 330 Giesen, F., see Podbielski, J. 325, 328–331 Giester, G., see Bauer, E. 6, 75, 78, 110 Giester, G., see Bauer, E.D. 65, 104–105 Giester, G., see Butch, N.P. 6, 24, 28, 49, 100 Giester, G., see Ho, P.-C. 50, 53–54, 103 Giester, G., see Yuhasz, M.W. 27, 47–48, 99 Giester, G., see Yuhasz, W.M. 60, 62–63, 104 Gijs, M.A.M. 357, 371 Gijs, M.A.M., see Belloy, E. 402 Gijs, M.A.M., see Dezuari, O. 379 Gil, D., see Castanõ, F.J. 114, 181 Gilbert, S.E., see Belloy, E. 402 Gilbert, S.E., see Dezuari, O. 379 Gilbert, T.L. 144 Gilles, J.-P., see Woytasik, M. 379 Giouroudi, I. 387 Giouroudi, I., see Hauser, H. 385 Giovannini, L., see Grimsditch, M. 177, 180–184 Giovannini, L., see Gubbiotti, G. 332, 334–335 Giovannini, L., see Montoncello, F. 332 Gippius, A., see Leithe-Jasper, A. 72–74, 108 Gippius, A.A., see Leithe-Jasper, A. 73–74 Girard, L., see Viennois, R. 21, 24

Author Index

Giri, R. 63 Giri, R., see Hachitani, K. 56–57 Giri, R., see Matsuhira, K. 103 Givens, R.B., see Wickenden, D.K. 389 Glaas, W., see Zimmermann, E. 362, 391 Glemot, L., see Brison, J.-P. 44 Godart, C., see Alleno, E. 69 Godart, C., see Leithe-Jasper, A. 69 Goddard, P.A., see Ho, P.-C. 47, 95 Goel, N., see Kunets, V.P. 357 Goiran, M., see Respaud, M. 252 Goleman, K. 382 Golub, V. 271 Golub, V.O., see Kakazei, G.N. 128, 233, 235–236, 270, 272–274, 277, 292 Golub, V.O., see Vovk, A. 303, 307 Gonc- alves, A.P., see Bauer, E. 75, 78, 110 Gondim, E.C., see De Biasi, R.S. 241 Gontarz, R., see Golub, V. 271 Gonzalez, J., see Garcıá, C. 388 Gonzalez, O.J. 395 Gonza´lez-Penedo, A., see Sańchez, R.D. 252–253 Goolaup, S., see Gubbiotti, G. 180 Goremychkin, E.A. 28, 38, 43 Goremychkin, E.A., see Adroja, D.T. 46 Goremychkin, E.A., see Adroja, T.D. 21, 25 Gorn, N.L., see Jorzick, J. 175 Gorrıá, P., see Sańchez, M.L. 401 Goryo, J., see Izawa, K. 41, 45 Goto, T. 18, 46 Goto, T., see Sayles, T.A. 27, 48, 100 Goto, T., see Sekine, C. 105 Goto, T., see Yanagisawa, T. 47 Graber, G., see Magnes, W. 378 Gracia, F.J., see Gonzalez, O.J. 395 Grandchamp, J.-P., see Woytasik, M. 379 Grandjean, F. 19, 21–23, 64, 96–98, 104 Grandjean, F., see Ge`rard, A. 64 Grandjean, F., see Long, G.J. 23 Gregg, J.F. 202 Grenier, B., see Huxley, A.D. 45 Grewe, N. 37 Grimalsky, V.V., see Slavin, A.N. 140 Grimsditch, M. 177, 180–184 Grimsditch, M., see Buchanan, K.S. 268–269 Grimsditch, M., see Novosad, V. 172 Grin, Y., see Dedkov, Yu.S. 69 Grin, Yu., see Gumeniuk, R. 6, 75–78, 110 Grin, Yu., see Leithe-Jasper, A. 72–74, 108 Grin, Yu., see Schnelle, W. 16, 69, 72–74, 106, 108–109 Gro¨nefeld, M. 351 Grollier, J. 285 Grosche, F.M., see Mathur, N.D. 46 Grube, K. 44 Gruger, H. 412 Gru¨nberg, P. 112

Author Index

Grundler, D., see Giesen, F. 326, 330 Grundler, D., see Neusser, S. 302, 304–305 Grundler, D., see Podbielski, J. 325, 328–331 Grytsiv, A., see Bauer, E. 6, 49, 65, 75, 104, 110 Gu, B., see Li, T. 318–319 Gubbiotti, G. 180, 270, 332, 334–335 Gubin, S.P., see Koksharov, Yu.A. 247 Guertin, R.P. 106 Guertin, R.P., see Meisner, G.P. 19, 21–23, 69, 96, 106 Guertin, R.P., see Torikachvili, M.S. 27, 29, 50–51, 53, 69–70, 93, 99, 101, 106–107 Gui, Y.S., see Mecking, N. 114 Guilhamat, B., see Joisten, H. 379, 390 Guimara˜es, A.P., see Pujada, B.R. 228, 254–257 Guittienne, Ph., see Wegrowe, J.E. 202–203, 211 Gukasov, A., see Kohgi, M. 38, 40–41 Gumeniuk, R. 6, 75–78, 110 Gumeniuk, R., see Schnelle, W. 16, 69, 72–74, 106, 108–109 Gurevich, A.G. 120 Gu¨rtler, C.M., see McPhail, S. 301–302 Guslienko, K.Y., see Jorzick, J. 175 Guslienko, K.Y., see Kakazei, G.N. 270–271 Guslienko, K.Yu. 173–174, 176–180, 271, 302, 332 Guslienko, K.Yu., see Buchanan, K.S. 268–269 Guslienko, K.Yu., see Kakazei, G.N. 272–274, 277, 292 Guslienko, K.Yu., see Novosad, V. 172 Guzun, D., see Kunets, V.P. 357 Hachitani, K. 56–58, 63 Hachitani, K., see Shimizu, M. 49, 95, 100 Haddab, Y. 359 Haen, P., see Sekine, C. 58–59 Haen, P., see Viennois, R. 15–16, 23, 64, 95, 98 Haga, Y., see Aoki, D. 72, 107 Haga, Y., see Aoki, Y. 58 Haga, Y., see Matsuda, T.D. 69–71, 106 Haga, Y., see Namiki, T. 27, 43, 49, 95, 100 Haga, Y., see Tokunaga, Y. 107 Hagiwara, M., see Tomita, S. 258, 260 Hagiwara, M., see Yamada, T. 16, 49, 63, 95 Hagiwara, M., see Yoshii, S. 74, 76, 108–109 Halperin, B.I., see Brataas, A. 217 Halperin, B.I., see Tserkovnyak, Y. 217 Hammel, P.C., see Kakazei, G.N. 270 Hammel, P.C., see Midzor, M.M. 114 Hanson, M., see Gubbiotti, G. 270 Hanton, J.P., see Valstyn, E.P. 238 Hao, L. 29, 42 Hao, L., see Iwasa, K. 29, 33–34, 36–37, 50, 99 Harima, H. 10–12, 15–16, 18, 32, 35–36, 52, 66, 73 Harima, H., see Aoki, Y. 99 Harima, H., see Curnoe, S.H. 35–36

429 Harima, H., see Hao, L. 42 Harima, H., see Ho, P.-C. 47, 95 Harima, H., see Ishida, K. 33 Harima, H., see Kikuchi, D. 63, 103 Harima, H., see Masaki, S. 58 Harima, H., see Matsunami, M. 11, 21–22, 25 Harima, H., see Saha, S.R. 18, 36 Harima, H., see Sato, H. 34 Harima, H., see Sugawara, H. 5, 9–10, 18, 27, 35, 47–48, 51, 93–94, 96, 99 Harima, H., see Takegahara, K. 9, 12, 62, 72 Harima, H., see Tatsuoka, S. 7, 15–17, 94 Harima, H., see Tsuda, S. 94 Harima, H., see Yamada, T. 16, 49, 63, 95 Harima, H., see Yogi, M. 44, 46 Harris, V.G., see Geiler, A.L. 372 Harter, A.W., see Mohler, G. 180, 185 Hartmann, C., see Mathieu, C. 270 Hartmann, U., see Mathieu, C. 270 Hasegawa, R. 408 Hasegawa, T., see Iwasa, K. 36, 99 Hasegawa, T., see Ogita, N. 18, 46, 50, 61 Haselwimmer, R.K.W., see Mathur, N.D. 46 Hashi, S. 407, 411 Haskel, D., see Krishnamurthy, V.V. 65 Hassel, C., see Trunova, A.V. 113, 258, 260–261 Hathaway, K.B. 112 Hattori, K. 61 Haug, T., see Puzic, A. 283–285 Haug, T., see Stoll, H. 283 Hauser, H. 385 Hauser, H., see Didosyan, Y.S. 405 Hauser, H., see Giouroudi, I. 387 Hauser, H., see Keplinger, F. 389 Hautot, D., see Long, G.J. 23 Havela, L., see Aoki, Y. 58 Hayashi, J., see Shirotani, I. 66, 105 Hayashi, K., see Matsuoka, E. 74, 108–109 Hayashi, K., see Takabatake, T. 72, 74–75, 108–109 Hayashi, K., see Yoshii, S. 74, 76, 108–109 Hayashi, S., see Tomita, S. 258 Haycock, P.W., see Thomas, A.H. 152–153 Hayward, T.J., see Vaz, C.A.F. 181, 322 Hazama, H., see Goto, T. 18, 46 Hedo, M., see Matsuda, T.D. 69–71, 106 Hedo, M., see Sugawara, H. 5, 9–10, 16, 35, 51, 93 Heffner, R.H., see Aoki, Y. 62 Heffner, R.H., see Cao, D. 36 Heffner, R.H., see Ito, T.U. 58 Heffner, R.H., see MacLaughlin, D.E. 44–45 Heffner, R.H., see Shu, L. 46 Hehn, M., see Gregg, J.F. 202 Heide, C. 192, 195–196 Heinrich, B. 113, 116–117, 142, 145–146 Heinrich, B., see Cochran, J.F. 113, 132 Heinrich, B., see Mosendz, O. 114

430 Heinrich, B., see Urban, R. 146, 190 Henkie, Z. 6, 23, 48–49, 95 Henkie, Z., see Baumbach, R.E. 7 Henkie, Z., see Chi, S. 47, 100 Henkie, Z., see Ho, P.-C. 47, 95 Henkie, Z., see Maple, M.B. 21, 23, 29, 47–48, 97, 100 Henkie, Z., see Sayles, T.A. 27, 48, 100 Henkie, Z., see Wawryk, R. 47, 95, 100 Henkie, Z., see Yanagisawa, T. 47 Henkie, Z., see Yuhasz, M.W. 27, 47, 100 Heremans, J. 360 Hermannsdorferd, T., see Bauer, E. 49 Hernandez, C., see Viennois, R. 23, 95, 98 Hernando, A., see Kraus, L. 356 Hernando, A., see Rivero, G. 411 Hernando, A., see Va´zquez, M. 373, 389 Hernando, B., see Nielsen, O.V. 376 Hernando, B., see Sańchez, M.L. 401 Herring, C. 133, 272 Herrmannsdo¨rfer, T., see Keller, L. 29, 51, 101 Hertel, R. 140, 142, 172, 318 Hertel, R., see Martyanov, O.N. 295, 299–300 Herzer, G. 115, 121, 123, 232, 408 Heyderman, L.J., see Vaz, C.A.F. 181, 322 Hicken, R.J. 114, 117, 285–286 Hicken, R.J., see Wu, J. 114 Hidaka, H., see Kotegawa, H. 50, 60, 62–63 Hiebert, W.K. 113 Hiebert, W.K., see Freeman, M.R. 113, 117 Higashimichi, H., see Imada, S. 53 Higashimichi, H., see Yamasaki, A. 61 Higashiwaki, M., see Yamamura, T. 350 Higashiya, A., see Yamasaki, A. 61 Higemoto, W. 46 Higemoto, W., see Aoki, Y. 39, 41, 43–44, 46, 62, 96 Higemoto, W., see Ito, T.U. 58 Higemoto, W., see Shu, L. 46 Hijiri, K., see Kihou, K. 68 Hilgendorff, M., see Spasova, M. 257, 259 Hilgendorff, M., see Wiedwald, U. 258 Hill, R.W. 50 Hill, R.W., see Rahimi, S. 50 Hillebrands, B., see Bayer, C. 270 Hillebrands, B., see Demokritov, S.O. 136, 270, 294 Hillebrands, B., see Jorzick, J. 175, 270 Hillebrands, B., see Mathieu, C. 270, 308–309 Hillebrands, B., see Slavin, A.N. 136, 140 Hillier, A.D., see Adroja, D.T. 46 Hilscher, G., see Bauer, E. 6, 49, 65, 75, 78, 104, 110 Hinatsu, Y., see Giri, R. 63 Hinatsu, Y., see Matsuhira, K. 60, 65, 94, 99, 103 Hinatsu, Y., see Matsuhira, M. 103 Hinatsu, Y., see Sekine, C. 7, 21, 23, 67, 97

Author Index

Hinnrichs, C. 373–374 Hirayama, T., see Lee, C.H. 10, 35, 42 Hirayama, Y., see Hattori, K. 61 Hiro, M., see Shirotani, I. 19, 21–23, 97 Ho, P.-C. 44, 47, 50, 53–54, 95, 103 Ho, P.-C., see Bauer, E.D. 18, 24, 28, 38, 43–44, 52, 95, 100–102 Ho, P.-C., see Baumbach, R.E. 7 Ho, P.-C., see Butch, N.P. 6, 24, 28, 49, 100 Ho, P.-C., see Frederick, N.A. 44 Ho, P.-C., see Henkie, Z. 6, 23, 48–49, 95 Ho, P.-C., see Maple, M.B. 21, 23, 29, 39, 42, 44, 47–48, 50, 96–97, 100 Ho, P.-C., see Vollmer, R. 28, 38, 44, 101 Ho, P.-C., see Wawryk, R. 47, 95, 100 Ho, P.-C., see Yanagisawa, T. 47 Ho, P.-C., see Yuhasz, M.W. 27, 47–48, 99–100 Ho, P.-C., see Yuhasz, W.M. 60, 62–63, 104 Hodges, J.A., see Ge`rard, A. 64 Hoffmann, B., see Mathieu, C. 270 Hoffmann, F. 131 Hohlfeld, J. 114 Hohlfeld, J., see Rasing, T. 117 Hollberg, L., see Schwindt, P.D.D. 384 Hollinger, R., see Stoll, H. 283 Holthaus, C. 390 Homma, Y., see Aoki, D. 72, 107 Homma, Y., see Tokunaga, Y. 107 Honda, D., see Sugiyama, K. 99 Honkura, Y., see Mohri, K. 387 Horiba, K., see Matsunami, M. 11, 21–22, 25 Horimoto, T., see Magishi, K. 64 Horiuchi, K., see Iwasa, K. 36, 99 Horva´th, D., see Majchra´k, P. 128, 236–237 Hoshi, N., see Sekine, C. 7, 21, 23, 97 Hotta, T. 61 Howitz, J., see Rubinstein, M. 228 Hoyer, N., see Podbielski, J. 325, 328–329 Hristoforou, E. 360 Hristoforou, E., see Petrou, J. 382 Hrkac, G., see Mistral, Q. 214–216 Hseih, C.T. 238–239, 241 Hu, C.-M., see Mecking, N. 114 Huang, H.B., see Zhai, Y. 277 Huang, W.L., see Hseih, C.T. 238–239, 241 Huang, Z., see Li, T. 318–319 Huffman, G.P., see Dutta, P. 238–240 Hughes, N.D., see Hicken, R.J. 114, 117 Hughes, N.D., see Wu, J. 114 Hultman, K., see Zohar, S. 251, 253 Humphrey, F.B., see Castanõ, F.J. 114, 181 Hurley, W.G., see Ripka, P. 376, 402 Hurley, W.G., see Tang, S.C. 402 Hutching, M.T. 13 Hutchison, A., see Kuanr, B.K. 117 Huxley, A., see Brison, J.-P. 44 Huxley, A.D. 45 Huynen, I., see Encinas, A. 311, 316

Author Index

Huynen, I., see Encinas-Oropesa, A. 309, 313–315, 318 Hwang, J.S., see Choi, W.Y. 378 Hwang, J.S., see Park, H.S. 380 Ibberson, R.M., see Adroja, D.T. 46 Ibrahim, A.I. 413 Ichihara, M., see Tanaka, K. 7, 49, 95, 100 Ihara, Y., see Ishida, K. 33 Iida, S. 144 Ikeda, A., see Matsuoka, E. 74, 108–109 Ikeda, H., see Aoki, Y. 58 Ikeda, S., see Aoki, D. 72, 107 Ikeda, S., see Matsuda, T.D. 69–71, 106 Ikeda, S., see Tomita, S. 260 Ikenaga, E., see Matsunami, M. 22 Ikeno, T. 6–7, 52–53, 69, 102, 106 Ikeno, T., see Ogita, N. 18, 50 Ikeno, T., see Sato, H. 34 Ikeno, T., see Tatsuoka, S. 7, 15–17, 94 Ilic, B., see Rivkin, K. 168–169 Imada, S. 53 Imada, S., see Yamasaki, A. 61 Imamura, Y., see Kotegawa, H. 44–45 Imamura, Y., see Yogi, M. 44, 46, 50, 96 Inaba, T., see Sekine, C. 34, 64–65, 67, 104 Inada, Y., see Matsuda, T.D. 32, 50 Inada, Y., see Saha, S.R. 18, 36 Inada, Y., see Sugawara, H. 32, 96 Inagawa, I., see Shirotani, I. 19 Indoh, K. 64 Inoue, M., see Sekine, C. 104 Ioan, C. 372, 375 Ipatov, M., see Garcıá, C. 388 Irie, Y., see Kotegawa, H. 50, 62–63 Ishida, K. 33 Ishida, K., see Nakai, Y. 18, 77, 96 Ishida, K., see Shu, L. 46 Ishida, M., see Ripka, P. 379 Ishihara, K., see Fujiwara, K. 22 Ishii, M., see Lee, C.H. 22, 34 Ishikawa, I., see Ogita, N. 50 Ishikawa, K., see Ogita, N. 18, 46, 61 Ishikawa, M., see Adroja, D.T. 21, 24 Ishikawa, M., see Kanai, K. 24 Ishikawa, M., see Shirotani, I. 93 Ishikawa, M., see Takeda, N. 24, 56, 65, 96, 98, 101–102, 104 Ishikawa, T., see Matsunami, M. 22 Ishikawa, T., see Yamasaki, A. 61 Ishikawa, Y., see Ikeno, T. 6–7, 52–53, 69, 102, 106 Ishiyama, K., see Hashi, S. 407, 411 Ito, K., see Fujino, T. 67 Ito, T.U. 58 Ito, T.U., see Aoki, Y. 62 Itobe, S., see Iwasa, K. 25 Itobe, S., see Kuwahara, K. 54

431 Ivanov, B.A., see Guslienko, K.Yu. 174 Iwahashi, Y., see Magishi, K. 64, 67 Iwahashi, Y., see Sugawara, H. 18, 27, 47–48, 94, 99 Iwasa, K. 10, 25, 29, 33–34, 36–37, 42, 46, 50, 99 Iwasa, K., see Aoki, Y. 39, 41, 43 Iwasa, K., see Hao, L. 29, 42 Iwasa, K., see Kohgi, M. 38, 40–41 Iwasa, K., see Kuwahara, K. 28, 38, 42–43, 54 Iwasa, K., see Park, J.-G. 34 Iwasa, K., see Sato, H. 29, 34 Iwasa, K., see Yang, C. 25 Izawa, K. 41, 45 Izawa, K., see Huxley, A.D. 45 Izawa, K., see Seyfarth, G. 45 Jaccard, Y., see Wegrowe, J.E. 202–203, 211 Jachimowicz, A., see Keplinger, F. 389 Jamet, M. 129–130 Jander, A. 371 Jander, A., see Deak, J. 371 Janosˇek, M., see Kubı´k, J. 375, 379 Jana´sek, V., see Tomek, J. 411 Jansen, A.G.M., see Tsoi, M. 187, 201 Jeffries, J.R., see Butch, N.P. 6, 24, 28, 49, 100 Jeffries, J.R., see Ho, P.-C. 50, 53–54, 103 Jeffries, J.R., see Maple, M.B. 29, 47, 50, 100 Jeffries, J.R., see Sayles, T.A. 27, 48, 100 Jeffries, J.R., see Yuhasz, M.W. 27, 47, 100 Jeitschko, W. 2–3, 22–23, 29, 52, 64–66, 102, 105, 107 Jeitschko, W., see Braun, D.J. 3, 5, 94–98, 100–105 Jeitschko, W., see Danebrock, M.E. 6, 23, 52, 63, 65, 74, 95, 98, 100, 102–104 Jeitschko, W., see Evers, C.B.H. 64, 102–104 Jeitschko, W., see Grandjean, F. 19, 21–23, 64, 96–98, 104 Jeitschko, W., see Ge`rard, A. 64 Jeitschko, W., see Kaiser, J.W. 3 Jeong, I.-K., see Cao, D. 36 Ji, J.H., see Na, K.W. 380 Jiles, D.C. 350 Johnson, M. 188, 195 Joisten, H. 379, 390 Jones, B.A., see Bazaliy, Ya.B. 202 Jones, C.A., see Garshelis, I.J. 400 Jorzick, J. 175, 270 Jorzick, J., see Mathieu, C. 308–309 Joynt, R. 44 Ju, G.P., see Lepadatu, S. 279, 281 Judy, J.W., see Vasquez, D.J. 389 Julian, S.R., see Mathur, N.D. 46 Jung, J.-S., see Malkinski, L.M. 277–278 Jung, J.-S., see Vovk, A. 303, 307 Jung, S. 168, 324

432 Kabos, P., see Silva, T.J. 117 Kabos, P., see Slavin, A.N. 201 Kachkachi, H. 129–130 Kachkachi, H., see Garanin, D. 130 Kachkachi, H., see Schmool, D.S. 130 Kachkachi, H., see Sousa, N. 162 Kaczorowski, D. 75, 78, 110 Kaczorowski, D., see Bauer, E. 49 Kaczorowski, D., see Leithe-Jasper, A. 69 Kadono, R., see Aoki, Y. 44, 46, 96 Kadono, R., see Higemoto, W. 46 Kadono, R., see Shu, L. 46 Kaiser, J.W. 3 Kaka, S., see Rippard, W.H. 210–212 Kakazei, G.N. 119, 128, 228, 233, 235–236, 252, 270–274, 277, 292 Kakazei, G.N., see Golub, V. 271 Kakazei, G.N., see Pogorelov, Yu.G. 228, 252 Kakazei, G.N., see Schmool, D.S. 112–113, 120, 128, 228–229, 231, 234, 266 Kaldarar, H., see Bauer, E. 6, 75, 110 Kalinikos, B.A. 134–135, 140, 175, 272, 294 Kalinikos, B.A., see Slavin, A.N. 140 Kambe, S., see Brison, J.-P. 44 Kambe, S., see Tokunaga, Y. 107 Kanai, K. 24 Kanayama, T., see Aoki, Y. 44, 46 Kane, S.N., see Kraus, L. 356 Kaneko, K. 46 Kaneko, K., see Kuwahara, K. 28, 38, 42–43 Kang, H.J., see Chi, S. 47, 100 Kang, H.J., see Maple, M.B. 47, 100 Kang, M.H. 409 Kaniusas, E., see Pfutzner, H. 398 Kanoda, K., see Shirotani, I. 6, 18, 27, 43, 49, 95, 100 Kanoda, K., see Uchiumi, T. 18, 94–96 Kasˇpar, P., see Mlejnek, P. 401 Kasˇpar, P., see Tomek, J. 411 Kasˇpar, P., see Vcelak, J. 366 Kasˇpar, P., see Vcela´k, J. 408 Kaper, H.G., see Grimsditch, M. 177, 180–184 Kardasz, B., see Mosendz, O. 114 Kari, R.J., see Garshelis, I.J. 401 Karlain, A. 359 Karpeev, A., see Grimsditch, M. 177, 180 Karrer, N., see Dalessandro, L. 403 Kasaya, M. 63 Kashiwagi, T., see Tomita, S. 258, 260 Kaspar, P., see Platil, A. 378 Kaspar, P., see Ripka, P. 409 Kasuya, T., see Fujita, T. 25 Katine, J., see Pechan, M. 267–268 Katine, J.A. 203–205, 211 Katine, J.A., see Albert, F.J. 204, 206–207 Katine, J.A., see Myers, E.B. 211 Kato, K., see Sasada, I. 399 Katranas, G.S. 411

Author Index

Kattelus, H., see Kyyna¨raïnen, J. 389 Kawahito, S. 378 Kawahito, S., see Ripka, P. 379 Kawahito, Y., see Kikuchi, D. 63, 103 Kawahito, Y., see Tanaka, K. 7, 49–50, 95, 100 Kawahito, Y., see Ueda, M. 63–64, 103 Kawakami, T., see Shirotani, I. 95 Kawana, D., see Hao, L. 29 Kawasaki, Y., see Kotegawa, H. 44–45 Kazakova, O., see Gubbiotti, G. 270 Keavney, D.J., see Krishnamurthy, V.V. 65 Kejik, P. 359 Kejik, P., see Drljaca, P.M. 378, 390 Kejik, P., see Popovic, R.S. 359, 394, 403 Kejik, P., see Zorlu, O. 381 Keller, L. 29, 51, 101 Keller, L., see Bauer, E. 49 Kelly, D., see Wegrowe, J.E. 202–203, 211 Kelly, P.J., see Brataas, A. 190 Kemna, A., see Zimmermann, E. 362, 391 Kent, A.D., see Sun, J.Z. 206–207 ¨ zyilmaz, B. 210 Kent, A.D., see O Keplinger, F. 389 Kerr, E., see Pannetier-Lecoeur, M. 368 Ketterson, J.B., see Jung, S. 168, 324 Ketterson, J.B., see Rivkin, K. 167–173 Ketterson, J.B., see Xu, W. 322–323 Keupper, K. 268 Khenata, R. 22, 71, 107 Khivintsev, Y.V., see Kuanr, B.K. 117 Khodorkovsky, Y., see Koksharov, Yu.A. 247 Kido, G., see Matsuhira, M. 103 Kihou, K. 64, 66, 68, 78, 93, 103–106 Kihou, K., see Lee, C.H. 36 Kihou, K., see Matsuhira, K. 65, 94, 99 Kihou, K., see Sekine, C. 67 Kihou, K., see Shirotani, I. 6, 66, 68–69, 78, 93, 105–106 Kikuchi, D. 57–58, 63, 103 Kikuchi, D., see Aoki, Y. 58, 60, 62, 103 Kikuchi, D., see Ikeno, T. 6, 52–53, 102 Kikuchi, D., see Imada, S. 53 Kikuchi, D., see Ito, T.U. 58 Kikuchi, D., see Kotegawa, H. 50, 60, 62–63 Kikuchi, D., see Kuwahara, K. 54 Kikuchi, D., see Magishi, K. 67 Kikuchi, D., see Masaki, S. 52, 58, 101 Kikuchi, D., see Matsuhira, K. 60 Kikuchi, D., see Mizumaki, M. 61–63 Kikuchi, D., see Mori, I. 6, 21, 23, 95, 98 Kikuchi, D., see Nakai, Y. 18, 77, 96 Kikuchi, D., see Nakanishi, Y. 62 Kikuchi, D., see Ogita, N. 18, 46, 50, 61 Kikuchi, D., see Pourret, A. 30 Kikuchi, D., see Sanada, S. 58, 60–62, 104 Kikuchi, D., see Sato, H. 29, 34, 60 Kikuchi, D., see Shu, L. 46

Author Index

Kikuchi, D., see Sugawara, H. 18, 27, 47–48, 94, 99 Kikuchi, D., see Tanaka, K. 7, 49–50, 95, 100 Kikuchi, D., see Tatsuoka, S. 7, 15–17, 94 Kikuchi, D., see Tsubota, M. 61 Kikuchi, D., see Ueda, M. 63–64, 103 Kikuchi, D., see Yamada, T. 16, 49, 63, 95 Kikuchi, D., see Yamasaki, A. 61 Kikuchi, D., see Yogi, M. 44, 46 Kikuchi, J. 34 Kikuchi, J., see Sakai, O. 34 Kim, C., see Kollu, P. 381 Kim, J.O., see Choi, W.Y. 380 Kim, S.-K., see Choi, S. 140, 143 Kim, S.K., see Ho, P.-C. 44 Kim, Y.-S. 379 Kima, C.G., see Rheema, Y.W. 405 Kima, C.O., see Rheema, Y.W. 405 Kimura, S., see Shirotani, I. 19, 21–23, 97 Kindo, K., see Matsuda, T.D. 69–71, 106 Kindo, K., see Sugiyama, K. 99 Kindo, K., see Takeda, N. 57 Kindo, K., see Yamada, T. 16, 49, 63, 95 Kindo, K., see Yoshii, S. 74, 76, 108–109 Kinoshita, M., see Shirotani, I. 4, 18, 94 Kirby, R.D., see Skomski, R. 147 Kirschner, J., see Hertel, R. 140, 142, 172 Kiselev, S.I. 209–210 Kiss, A. 34 Kiss, A., see Kuramoto, Y. 27, 34 Kiss, T., see Matsunami, M. 11, 21, 25 Kiss, T., see Tsuda, S. 94 Kistenmacher, T.J., see Wickenden, D.K. 389 Kitagawa, K., see Ishida, K. 33 Kitaoka, Y., see Ishida, K. 33 Kitaoka, Y., see Kotegawa, H. 44–45 Kitaoka, Y., see Yogi, M. 21, 24–25, 44, 46, 50, 96 Kitazawa, H., see Matsuhira, M. 103 Kitazawa, H., see Sekine, C. 58–59 Kitching, J., see Schwindt, P.D.D. 384 Kittel, C. 112–113 Kittel, C., see Herring, C. 133, 272 Kliava, J. 114, 220–221 Kliava, J., see Berger, R. 220, 222, 224–225, 253, 262 Klaüi, M., see Vaz, C.A.F. 181, 322 Knappe, S., see Schwindt, P.D.D. 384 Knight, K.S., see Adroja, T.D. 21, 25 Ku¨nnen, B., see Jeitschko, W. 65–66, 102, 105 Knobel, K. 356, 385, 387 Knobel, M., see Pires, M.J.M. 236 Knobel, M., see Schmool, D.S. 254 Knorren, R., see Hohlfeld, J. 114 Kobayashi, K., see Matsunami, M. 22 Kobayashi, M. 45 Kobayashi, M., see Kikuchi, D. 63, 103 Kobayashi, M., see Sugawara, H. 18, 38–39

433 Kobayashi, T.C., see Kotegawa, H. 50, 60, 62–63 Kobayashi, T.C., see Sugawara, H. 18, 99 Koch, R.H. 374–375 Koch, R.H., see Sun, J.Z. 206–207 ¨ zyilmaz, B. 210 Koch, R.H., see O Koda, A., see Aoki, Y. 44, 46, 62, 96 Koda, A., see Higemoto, W. 46 Koda, A., see Shu, L. 46 Kodama, R.H. 155 Koga, M., see Shiina, R. 43 Koh, K.C., see Kang, M.H. 409 Kohgi, M. 38, 40–41 Kohgi, M., see Aoki, Y. 39, 41, 43 Kohgi, M., see Hao, L. 29, 42 Kohgi, M., see Iwasa, K. 10, 25, 29, 33–34, 36–37, 42, 46, 50, 99 Kohgi, M., see Kaneko, K. 46 Kohgi, M., see Kuwahara, K. 28, 38, 42–43, 54 Kohgi, M., see Park, J.-G. 34 Kohgi, M., see Sato, H. 29, 34, 60 Kohgi, M., see Yang, C. 25 Kohl, F., see Keplinger, F. 389 Kohori, Y., see Hachitani, K. 56–58, 63 Kohori, Y., see Shimizu, M. 49, 95, 100 Kojima, K.M., see Luke, G.M. 46 Kojima, R., see Ogita, N. 18, 50 Koksharov, Yu.A. 247 Kolar, J.W., see Dalessandro, L. 403 Kolesnik, S.P., see Konchits, A.A. 260, 262–264 Kollar, M., see Bydzovsky, J. 398 Kollu, P. 381 Kolodzey, J., see Moriyama, T. 216 Komatsubara, T., see Fujita, T. 25 Konchits, A.A. 260, 262–264 Kondo, T., see Ogita, N. 18, 46, 50, 61 Kontani, H., see Tsujii, N. 60 Koon, N.C., see Rubinstein, M. 228 Korn, T., see Giesen, F. 326, 330 Kornack, T.W., see Allred, J.C. 384 Kos, A.B., see Schneider, M.L. 274–275 Kosel, J., see Pfutzner, H. 398 Ko¨seog˘lu, Y., see Yalc- ın, O. 318 Kosobudsky, I.D., see Koksharov, Yu.A. 247 Koster, G.F., see Slater, J.C. 9 Kostylev, M.P., see Kalinikos, B.A. 135, 272 Kostylev, M.P., see Slavin, A.N. 140 Kotegawa, H. 44–45, 50, 60, 62–63 Kotegawa, H., see Ishida, K. 33 Kotegawa, H., see Sugawara, H. 18, 99 Kotegawa, H., see Yogi, M. 21, 25, 44, 50, 96 Kotzyba, G., see Jeitschko, W. 65–66, 102, 105 Kovalev, A.A. 190 Koyama, K., see Hachitani, K. 63 Koyama, K., see Magishi, K. 21–22, 24, 64, 67 Koyama, K., see Mori, I. 6, 21, 23, 95, 98 Koyama, K., see Sugawara, H. 18, 27, 47–48, 94, 99

434 Koyama, K., see Toda, M. 6, 77–78, 110 Kozhus, N.V., see Kalinikos, B.A. 135, 272 Kraemer, M.A., see An, S.Y. 117 Krafft, C., see Holthaus, C. 390 Kraus, L. 356, 388 Kraus, L., see Bydzovsky, J. 398 Kraus, L., see Knobel, K. 356, 385, 387 Kraus, L., see Mala´tek, M. 404–405 Kravets, A.F., see Kakazei, G.N. 119, 228, 252 Kravets, A.F., see Pogorelov, Yu.G. 228, 252 Krishnamurthy, V.V. 65 Krivorotov, I.N., see Kiselev, S.I. 209–210 Krivorotov, I.N., see Sankey, J.C. 212 Krivosik, P., see An, S.Y. 117 Kruglyak, V.V., see Hicken, R.J. 285–286 Kryder, M.H., see Tamaru, S. 177–179 Ksenofontov, V., see Leithe-Jasper, A. 73–74 Ktena, A., see Petridis, C. 381 Kuan, T.S., see Sun, J.Z. 206–207 Kuanr, B.K. 117, 277, 279–280 Kubı´k, J. 352, 375, 379–380, 409 Kubı´k, J., see Ripka, P. 402, 409 Kubı´k, J., see Vcelak, J. 366 Kuchenbrandt, K., see Hinnrichs, C. 373–374 Kudoh, M., see Takegahara, K. 72 Kuisma, H., see Kyyna¨raïnen, J. 389 Kumagai, K., see Hachitani, K. 56–57, 63 Kumagai, T. 50 Kumagai, T., see Nakanishi, Y. 21, 26, 101 Kumar, P., see Rotundu, C.R. 44 Kume, M., see Maeda, A. 276 Kuna, A., see Cerman, A. 377 Kunets, V.P. 357 Kunii, S., see Fujita, T. 25 Kunze, U., see Remhof, A. 114 Kuo, C.H. 390 Kuramochi, E., see Sugawara, H. 38–39 Kuramoto, Y. 27, 34 Kuramoto, Y., see Kiss, A. 34 Kuramoto, Y., see Otsuki, J. 34 Kuric, M.V., see Guertin, R.P. 106 Kusunose, H., see Kuramoto, Y. 34 Kusunose, H., see Otsuki, J. 34 Kuwahara, K. 28, 38, 42–43, 54 Kuwahara, K., see Aoki, Y. 39, 41, 43 Kuwahara, K., see Hao, L. 29, 42 Kuwahara, K., see Ikeno, T. 6, 52–53, 102 Kuwahara, K., see Iwasa, K. 10, 25, 36–37, 42, 99 Kuwahara, K., see Kikuchi, D. 57–58, 63, 103 Kuwahara, K., see Sato, H. 29, 34, 60 Kuwahara, K., see Tanaka, K. 7, 49–50, 95, 100 Kuwahara, K., see Tatsuoka, S. 7, 15–17, 94 Kuwahara, K., see Ueda, M. 63–64, 103 Kuwai, T., see Ikeno, T. 7, 69, 106 Kvasnica, S., see Keplinger, F. 389 Kyyna¨raïnen, J. 389

Author Index

Labarta, A., see Batlle, X. 114, 220 Lacerda, A.H., see Butch, N.P. 6, 24, 28, 49, 100 Lacerda, A.H., see Gajewski, D.A. 23, 98 Lacerda, A.H., see Ho, P.-C. 44, 50, 53–54, 103 Lacerda, A.H., see Maple, M.B. 39, 42, 96 Lacerda, A.H., see Yuhasz, W.M. 60, 62–63, 104 Lachowicz, H. 254 Ladak, S., see Hicken, R.J. 285–286 Lagae, L., see Mistral, Q. 214–216 Lamb, J.L., see Wickenden, D.K. 389 Landau, L. 113, 144 Lang, A., see Jeitschko, W. 65–66, 102, 105 Lang, J.C., see Krishnamurthy, V.V. 65 Lange, E., see Deak, J. 371 Langer, J., see Ferreira, R. 371 Langheinrich, W., see Umbach, F. 400 Langreth, D.C. 193 Lapertot, G., see Seyfarth, G. 44 Larkin, M.I., see Luke, G.M. 46 Laskaris, E., see Petridis, C. 381 Last, T., see Remhof, A. 114 Laufenberg, M., see Vaz, C.A.F. 181, 322 Laursen, I., see Primdahl, F. 383 Lea, K.R. 13 Leaf, G.K., see Grimsditch, M. 177, 180–184 Leask, M.G.M., see Lea, K.R. 13 Lechner, A., see Uhrmann, T. 398 Lee, C.H. 10, 22, 34–36, 42 Lee, C.H., see Hao, L. 42 Lee, C.-H., see Kihou, K. 68 Lee, C.S., see Silva, T.J. 114 Lee, H., see Kim, Y.-S. 379 Lee, J.-B., see Kim, Y.-S. 379 Lee, J.H., see Kang, M.H. 409 Lee, K.-S., see Choi, S. 140, 143 Lee, R.N., see Martyanov, O.N. 295, 299–300 Lee, S.W., see Yoon, S.H. 397 Lee, Y.H., see Yoon, S.H. 397 Leger, J.M., see Joisten, H. 379, 390 Legvold, S. 64 Leighton, C., see Wang, R.F. 114 Leithe-Jasper, A. 69, 72–74, 108 Leithe-Jasper, A., see Dedkov, Yu.S. 69 Leithe-Jasper, A., see Gumeniuk, R. 6, 75–78, 110 Leithe-Jasper, A., see Schnelle, W. 16, 69, 72–74, 106, 108–109 Lemme, H. 404 Lemos, J.M., see Cardoso, F.A. 371 Lenz, J. 349 Lepadatu, S. 279, 281 Lerner, B., see Sheinker, A. 410 Leroy, P. 360 Leroy, P., see Coillot, C. 385 Lesnik, N.A., see Golub, V. 271

Author Index

Lesnik, N.A., see Kakazei, G.N. 119, 228, 233, 252, 270–274, 277, 292 Lesnik, N.A., see Pogorelov, Yu.G. 228, 252 Levy, P.M. 202 Levy, P.M., see Zhang, S. 196–197 Lewis, A.M., see Ripka, P. 409 Lezama, L., see Sarmiento, G. 259 Lezama, L., see Schmool, D.S. 112–113, 120, 128, 228–229, 231, 234 Lhotel, E., see Measson, M.-A. 44, 46 Li, J., see Wang, R.F. 114 Li, M., see Petit, S. 214 Li, S., see Hill, R.W. 50 Li, T. 318–319 Li, X.P., see Fan, J. 381 Li, X.P., see Yi, J.B. 351 Li, X.P., see Zhao, Z.J. 381 Li, Y. 303, 362, 371 Li, Y., see Zhao, Z. 405 Li, Z.-P., see Cui, Y.-T. 214 Li, Z.-P., see Fuchs, G.D. 212–213 Lieke, W., see Steglich, F. 37 Liew, L.-A., see Schwindt, P.D.D. 384 Lifshitz, E.M., see Landau, L. 113, 144 Lin, Z., see Xi, H. 199 Linderoth, S., see Bødker, F. 121 Lindner, J. 113, 116 Lindner, J., see Antoniak, C. 162, 219, 264–266 Lindner, J., see Meckenstock, R. 287–289 Lindner, J., see Trunova, A.V. 113, 258, 260–261 Liu, S. 380 Liu, S.I., see Kuo, C.H. 390 Liu, Y., see Petit, S. 214 Liu, Z., see Zhu, X. 324–327 Llandro, J., see Vaz, C.A.F. 181, 322 Lo, C.C.H., see Jiles, D.C. 350 Lobotka, P., see Majchra´k, P. 128, 236–237 Locatelli, A., see Vaz, C.A.F. 181, 322 Lohneysen, H.v., see Grube, K. 44 Lohneysen, H.v., see Vollmer, R. 28, 38, 44, 101 Long, G.J. 23 Lonzarich, G.G., see Mathur, N.D. 46 Lopez, E., see Almazan, R.P. 379 Lopez, E., see Ciudad, D. 389 Lopez, E., see Michelena, M.D. 389 Lopez, E., see Perez, L. 378 Lopez, E., see Prieto, J.L. 389 Lopusnik, R., see Kuanr, B.K. 277, 279–280 Louie, R.N., see Myers, E.B. 211 Lo´pez-Quintela, M.A., see Sańchez, R.D. 252–253 Lu, Z.H., see Zhai, Y. 277 Lucas, I., see Perez, L. 372 Lue, J.T., see Hseih, C.T. 238–239, 241 Luke, G.M. 46 Lukoschus, D.G. 385

435 Lund, M.S., see Wang, R.F. 114 Lyman, R.N., see Allred, J.C. 384 Lynn, J.W., see Butch, N.P. 6, 24, 28, 49, 100 Lynn, J.W., see Chi, S. 47, 100 Lynn, J.W., see Goremychkin, E.A. 28, 38, 43 Lynn, J.W., see Maple, M.B. 44, 47, 100 Lysowec, M., see Haddab, Y. 359 La´zaro, F.J., see Garcıá-Palacios, J.L. 156, 158, 160 Ma, L., see Li, Y. 303 Maass, W., see Ferreira, R. 371 Machado Gama, M.A. 389 MacLaughlin, D.E. 44–45 MacLaughlin, D.E., see Aoki, Y. 39, 41, 43 MacLaughlin, D.E., see Ishida, K. 33 MacLaughlin, D.E., see Shu, L. 46 Macovei, C., see Ioan, C. 372 Madami, M., see Gubbiotti, G. 180, 332, 334–335 Madden, I., see Cruden, A. 405 Maeda, A. 276 Maeno, Y., see Luke, G.M. 46 Maganto, F.J., see Garcıá, A. 381 Magishi, K. 21–22, 24, 64, 67 Magishi, K., see Mori, I. 6, 21, 23, 95, 98 Magishi, K., see Sugawara, H. 18, 27, 47–48, 94, 99 Magishi, K., see Toda, M. 6, 77–78, 110 Magnes, W. 378 Mailly, D., see Jamet, M. 129–130 Mailly, D., see Thirion, C. 151 Majchra´k, P. 128, 236–237 Majkrzak, C.F., see Nakotte, H. 69, 106 Makhnovskiy, D.P., see Fry, N. 388 Maki, H., see Matsuhira, M. 103 Maki, K., see Izawa, K. 41, 45 Maksymowicz, A. 119, 137 Malac, M., see Zhu, X. 324–327 Malcovati, P., see Baschirotto, A. 378 Malinowska, M., see Pogorelov, Yu.G. 228, 252 Malkinksi, L.M., see Kuanr, B.K. 277, 279–280 Malkinski, L., see Vovk, A. 303, 307 Malkinski, L., see Yu, M. 303, 305–306 Malkinski, L.M. 277–278 Mala´tek, M. 404–405 Mala´tek, M., see Kraus, L. 388 Manara, A., see DiRienzo, L. 403 Mandea, M. 410 Mandrus, D., see Gajewski, D.A. 23, 98 Mandrus, D.G., see Krishnamurthy, V.V. 65 Manivannan, A., see Dutta, P. 238–240 Mankey, G.J., see Yu, C. 290–295, 301 Mansanares, A.M., see Pires, M.J.M. 255 Mao, Z.Q., see Luke, G.M. 46 Maple, M.B. 21, 23, 29, 39, 42, 44, 47–48, 50, 96–97, 100

436 Maple, M.B., see Bauer, E.D. 5, 18–19, 21, 24, 28, 38, 43–44, 52, 65, 95–96, 98, 100–102, 104–105 Maple, M.B., see Baumbach, R.E. 7 Maple, M.B., see Butch, N.P. 6, 24, 28, 49, 100 Maple, M.B., see Cao, D. 36 Maple, M.B., see Chi, S. 47, 100 Maple, M.B., see Cichorek, T. 41 Maple, M.B., see Cox, D.L. 46 Maple, M.B., see Dilley, N.R. 69, 104, 106 Maple, M.B., see Dordevic, S.V. 21, 24, 69–70, 93, 96, 106–107 Maple, M.B., see Frederick, N.A. 38, 44 Maple, M.B., see Gajewski, D.A. 23, 98 Maple, M.B., see Goremychkin, E.A. 28, 38, 43 Maple, M.B., see Grube, K. 44 Maple, M.B., see Guertin, R.P. 106 Maple, M.B., see Henkie, Z. 6, 23, 48–49, 95 Maple, M.B., see Hill, R.W. 50 Maple, M.B., see Ho, P.-C. 44, 47, 50, 53–54, 95, 103 Maple, M.B., see MacLaughlin, D.E. 44–45 Maple, M.B., see Meisner, G.P. 19, 21–23, 69, 96, 106 Maple, M.B., see Nakotte, H. 69, 106 Maple, M.B., see Oeschler, N. 39 Maple, M.B., see Rahimi, S. 50 Maple, M.B., see Sayles, T.A. 27, 48, 100 Maple, M.B., see Shu, L. 46 Maple, M.B., see Tenya, K. 39 Maple, M.B., see Torikachvili, M.S. 27, 29, 50–51, 53, 69–70, 93, 99, 101, 106–107 Maple, M.B., see Vollmer, R. 28, 38, 44, 101 Maple, M.B., see Wawryk, R. 47, 95, 100 Maple, M.B., see Yanagisawa, T. 47 Maple, M.B., see Yuhasz, M.W. 27, 47–48, 99–100 Maple, M.B., see Yuhasz, W.M. 60, 62–63, 104 Mapps, D.J., see Fry, N. 388 Maranville, B.B. 280, 282 Margeat, O., see Trunova, A.V. 113, 258, 260–261 Marquardt, B., see Pfutzner, H. 398 Marsal, L.F. 303, 306 Martincic, E., see Woytasik, M. 379 Martyanov, O.N. 295, 299–300 Masaharu, T., see Hashi, S. 407, 411 Masaki, S. 52, 58, 101 Mathias, H., see Woytasik, M. 379 Mathieu, C. 270, 308–309 Mathur, N.D. 46 Matsuda, T.D. 32, 34, 50, 69–71, 106 Matsuda, T.D., see Abe, K. 24, 50, 53, 96, 98 Matsuda, T.D., see Aoki, Y. 58, 99 Matsuda, T.D., see Hao, L. 29 Matsuda, T.D., see Ishida, K. 33 Matsuda, T.D., see Iwasa, K. 10, 29, 33–34, 42 Matsuda, T.D., see Kaneko, K. 46

Author Index

Matsuda, T.D., see Keller, L. 29, 51, 101 Matsuda, T.D., see Namiki, T. 27, 43, 49, 95, 99–100 Matsuda, T.D., see Sato, H. 18, 21–22, 27, 30, 50–51, 53, 93, 96, 99, 101 Matsuda, T.D., see Sugawara, H. 5, 16, 24, 32–33, 42–43, 93, 99 Matsuda, T.D., see Tokunaga, Y. 107 Matsuda, Y., see Huxley, A.D. 45 Matsuda, Y., see Izawa, K. 41, 45 Matsuda, Y., see Seyfarth, G. 45 Matsuhata, H., see Lee, C.H. 10, 35, 42 Matsuhira, H., see Lee, C.H. 36 Matsuhira, K. 35, 60, 65, 94, 99, 103 Matsuhira, K., see Fujino, T. 67 Matsuhira, K., see Giri, R. 63 Matsuhira, K., see Sekine, C. 7, 21, 23, 58–59, 65, 67, 97, 105 Matsuhira, M. 103 Matsui, T., see Yamamura, T. 350 Matsui, Y., see Tomita, S. 258 Matsumoto, M., see Shiina, R. 43 Matsumoto, T., see Shirotani, I. 95 Matsumura, M., see Matsuoka, E. 74, 108–109 Matsunami, M. 11, 21–22, 24–25, 35, 58, 65 Matsuoka, E. 74, 108–109 Matsuoka, E., see Takabatake, T. 72, 74–75, 108–109 Matsuoka, E., see Yoshii, S. 74, 76, 108–109 Matsushita, A., see Shirotani, I. 95 Matsuura, H., see Tanikawa, S. 61 Matthias, E., see Hohlfeld, J. 114 Mauger, A., see Viennois, R. 15–16, 64, 95 Mayergoyz, I.D., see Holthaus, C. 390 Mazin, I.I., see Aguayo, A. 17 Mazur, Y.I., see Kunets, V.P. 357 McCall, S.K., see Maple, M.B. 47, 100 McCall, S.K., see Yuhasz, M.W. 27, 47, 100 McCloskey, P., see Tipek, A. 402 McConville, W., see Wang, R.F. 114 McDonald, J.R., see Cruden, A. 405 McElfresh, M.W., see Guertin, R.P. 106 McElfresh, M.W., see Maple, M.B. 47, 100 McElfresh, M.W., see Torikachvili, M.S. 27, 29, 50–51, 53, 69–70, 93, 99, 101, 106–107 McElfresh, M.W., see Yuhasz, M.W. 27, 47, 100 McEwen, K., see Park, J.-G. 34 McEwen, K.A., see Adroja, D.T. 21, 24, 46 McEwen, K.A., see Adroja, T.D. 21, 25 McGary, P.D. 398 McMichael, R.D., see Maranville, B.B. 280, 282 McMichael, R.D., see Pardavi-Horvath, M. 276–277 McMichael, R.D., see Russek, S.E. 117 McMichael, R.D., see Schneider, M.L. 274–275 McPhail, S. 301–302 Mea, M.D., see Bauer, E. 49 Means, J., see Magnes, W. 378

Author Index

Measson, M.-A. 44, 46 Measson, M.-A., see Huxley, A.D. 45 Measson, M.-A., see Kuwahara, K. 28, 42 Measson, M.A., see Seyfarth, G. 44–45 Meckenstock, R. 114, 287–289 Meckenstock, R., see Pires, M.J.M. 255 Meckenstock, R., see Rastei, M.V. 270 Meckenstock, R., see Trunova, A.V. 113, 258, 260–261 Mecking, N. 114 Megherbi, S., see Woytasik, M. 379 Mehnen, L., see Pfutzner, H. 398 Meijer, F.E., see Nitta, J. 140 Meinander, T., see Kyyna¨raïnen, J. 389 Meisner, G.P. 19, 21–23, 69, 93–94, 96, 101, 106 Meisner, G.P., see DeLong, L.E. 94 Meisner, G.P., see Guertin, R.P. 106 Meisner, G.P., see Long, G.J. 23 Meisner, G.P., see Morelli, D.T. 5–6, 19, 23, 98 Meisner, G.P., see Shenoy, G.K. 16 Meisner, G.P., see Torikachvili, M.S. 27, 29, 50–51, 53, 69–70, 93, 99, 101, 106–107 Me´linon, P., see Jamet, M. 129–130 Melkov, G.A., see Gurevich, A.G. 120 Melnychenko-Koblyuk, N., see Bauer, E. 6, 75, 110 Merayo, J.M.G. 409 Merayo, J.M.G., see Cerman, A. 377 Merayo, J.M.G., see Nielsen, O.V. 376 Merayo, J.M.G., see Pedersen, E.B. 377 Merayo, J.M.G., see Primdahl, F. 373, 376, 383, 410 Merle, J.-C., see Beaurepaire, E. 113 Merlo, A.M., see Pfutzner, H. 398 Merrin, J., see Luke, G.M. 46 Meschede, D., see Steglich, F. 37 Metlushko, V., see Kakazei, G.N. 270–271 Metlushko, V., see Zhu, X. 324–327 Metlushko, V.V., see Rivkin, K. 168–169 Metlushko, V.V., see Xu, W. 322–323 Metoki, N., see Kaneko, K. 46 Metoki, N., see Kohgi, M. 38, 40–41 Metoki, N., see Kuwahara, K. 28, 38, 42–43 Mewes, T., see Kakazei, G.N. 270 Meydan, T., see Katranas, G.S. 411 Meydan, T., see Pfutzner, H. 398 Michelena, M.D. 389 Michie, C., see Cruden, A. 405 Michor, H., see Bauer, E. 6, 49, 65, 69, 75, 78, 104, 110 Midzor, M.M. 114 Mignot, J.M., see Kohgi, M. 38, 40–41 Mikheev, M.G., see Koksharov, Yu.A. 247 Miki, T., see Kotegawa, H. 60, 62–63 Miles, T.S., see Flavel, S.C. 411 Miljanic, P.N., see Moore, W.J.M. 402 Mills, D.L. 142, 146

437 Mills, D.L., see Arias, R. 292, 294, 314 Min, S.-H., see Vovk, A. 303, 307 Miragliotta, J.A., see Wickenden, D.K. 389 Miranda, K.L.C., see Pereira, A.R. 242 Mirwald-Schulz, B., see Farle, M. 124–125 Mishima, T.D., see Kunets, V.P. 357 Mistral, Q. 214–216 Mitamura, H., see Takeda, N. 57 Mito, T., see Masaki, S. 52, 58, 101 Mitsuda, A., see Ikeno, T. 7, 69, 106 Mitsumata, C., see Tomita, S. 260 Mitsumoto, K. 61 Miwa, D., see Matsunami, M. 22 Miyake, K., see Aoki, Y. 58 Miyake, K., see Hattori, K. 61 Miyake, K., see Tanikawa, S. 61 Miyamachi, T., see Yamasaki, A. 61 Miyoshi, K., see Fujiwara, K. 22 Mizumaki, M. 61–63 Mizushima, T., see Ikeno, T. 7, 69, 106 Mlejnek, P. 401 Mlejnek, P., see Tomek, J. 411 Mo, N. 287 Moessner, R., see Mo¨ller, G. 114 Mohler, G. 180, 185 Mohri, K. 387 Moldovanu, A., see Ioan, C. 372 Moldovanu, C. 372 Moldovanu, E., see Ioan, C. 372 Mo¨ller, G. 114 Mo¨ller, M.H., see Jeitschko, W. 65–66, 102, 105 Molodtsov, S.L., see Dedkov, Yu.S. 69 Monfort, Y., see Robbes, D. 385 ¨ zyilmaz, B. 210 Monsma, D., see O Monsma, D.J., see Sun, J.Z. 206–207 Montoncello, F. 332 Montoncello, F., see Grimsditch, M. 177, 180–184 Montoncello, F., see Gubbiotti, G. 332, 334–335 Moore, J., see Wu, J. 114 Moore, J.R., see Hicken, R.J. 114, 117 Moore, W.J.M. 402 Morais, P.C., see Pereira, A.R. 242 Morecroft, D., see Vaz, C.A.F. 181, 322 Moreland, J., see Schwindt, P.D.D. 384 Morelli, D.T. 5–6, 19, 23, 98 Morelli, D.T., see Long, G.J. 23 Mori, I. 6, 21, 23, 95, 98 Mori, I., see Magishi, K. 21, 24 Mori, Y., see Iwasa, K. 36, 50, 99 Mori, Y., see Luke, G.M. 46 Morimoto, S., see Takabatake, T. 72, 74–75, 108–109 Moriya, T. 16, 74 Moriyama, T. 216 Moriyama, T., see Tserkovnyak, Y. 217 Moroń, C., see Garcıá, A. 381

438 Moron, C., see Garcia, A. 382 Morozova, E., see Leithe-Jasper, A. 72–74, 108 Morozova, E.N., see Leithe-Jasper, A. 73–74 Morris, G.D., see MacLaughlin, D.E. 44–45 Morrish, A.H., see Valstyn, E.P. 238 Mørup, S., see Bødker, F. 121 Mosendz, O. 114 Mosser, V., see Haddab, Y. 359 Mosser, V., see Karlain, A. 359 Mosser, V., see Leroy, P. 360 Mosser, V., see Quasimi, A. 359 Mota, A.C., see Cichorek, T. 41 Motsnyi, F.V., see Konchits, A.A. 260, 262–264 Moutoussamy, J., see Coillot, C. 385 Movshovich, R., see Yatskar, A. 26 Mukuda, H., see Yogi, M. 21, 24, 44, 46 Mu¨ller, A., see Mathieu, C. 270 Multigner, M., see Rivero, G. 411 Mundon, S., see Deak, J. 371 Murakami, Y., see Iwasa, K. 25, 36, 50, 99 Murakami, Y., see Tsubota, M. 61 Murakawa, H., see Ishida, K. 33 Muro, T., see Imada, S. 53 Murphy, S.Q., see Kunets, V.P. 357 Musiejovsky, L., see Giouroudi, I. 387 Musiejovsky, L., see Hauser, H. 385 Mydosh, J.A., see Leithe-Jasper, A. 72–74, 108 Mydosh, J.A., see Schnelle, W. 16, 69, 72–74, 106, 108–109 Myers, E.B. 211 Myers, E.B., see Katine, J.A. 203–205, 211 Na, K.W. 380 Na, K.W., see Park, H.S. 380 Nabily, S., see Ding, L. 385 Nachumi, B., see Luke, G.M. 46 Nagai, T., see Yogi, M. 44, 46 Nagano, S., see Matsunami, M. 21, 24 Nakada, R., see Shirotani, I. 66, 105 Nakai, Y. 18, 77, 96 Nakajima, M., see Hao, L. 29 Nakajima, M., see Iwasa, K. 29, 33–34 Nakajima, M., see Kohgi, M. 38, 40–41 Nakajima, Y., see Izawa, K. 41, 45 Nakamura, A., see Aoki, D. 72, 107 Nakamura, A., see Matsuhira, K. 65 Nakamura, D., see Yamamura, T. 350 Nakamura, H., see Luke, G.M. 46 Nakamura, M., see Fujino, T. 67 Nakamura, M., see Nakanishi, Y. 62 Nakamura, N., see Sugiyama, K. 99 Nakamura, S., see Ochiai, A. 44 Nakanishi, T., see Shirotani, I. 95 Nakanishi, Y. 21, 26, 62, 101 Nakanishi, Y., see Fujino, T. 67 Nakanishi, Y., see Kumagai, T. 50 Nakanishi, Y., see Yoshizawa, M. 58, 68, 103 Nakashima, H., see Yamada, T. 16, 49, 63, 95

Author Index

Nakata, R., see Shirotani, I. 6, 68–69, 78 Nakazawa, Y., see Shirotani, I. 6, 18, 27, 43, 49, 95, 100 Nakazawa, Y., see Uchiumi, T. 18, 94–96 Nakotte, H. 69, 106 Namiki, T. 25, 27, 43, 49, 95, 98–100 Namiki, T., see Abe, K. 24, 50, 53, 96, 98 Namiki, T., see Aoki, Y. 44, 99, 101 Namiki, T., see Matsuda, T.D. 34 Namiki, T., see Sato, H. 51 Namiki, T., see Sugawara, H. 18, 38–39 Nanba, T., see Matsunami, M. 21, 24–25, 35, 58, 65 Nara, T. 408 Narazu, S., see Matsuoka, E. 74 Narazu, S., see Ogita, N. 50 Narazu, S., see Takabatake, T. 72, 74–75, 108–109 Nawafune, H., see Tomita, S. 260 Nazarov, A.V., see An, S.Y. 117 Nazarov, Y.V., see Brataas, A. 190 Neagu, M., see Petrou, J. 382 Neél, L. 129, 218 Nemoto, Y., see Goto, T. 18, 46 Nemoto, Y., see Sayles, T.A. 27, 48, 100 Nemoto, Y., see Yanagisawa, T. 47 Nepijko, S.A., see Martyanov, O.N. 295, 299–300 Netzelmann, U. 119, 155, 228 Neudecker, I., see Keupper, K. 268 Neudecker, I., see Puzic, A. 283–285 Neusser, S. 302, 304–305 Nguyen, T.M. 173, 181–182, 186–188 Nguyen Van Dau, F., see Baibich, M.N. 112 Ni, B., see Bauer, E. 65, 104 Nicklas, M., see Gumeniuk, R. 6, 75–78, 110 Nicolics, J., see Didosyan, Y.S. 405 Nicolics, J., see Hauser, H. 385 Nielsen, O.V. 353, 376 Nielsen, O.V., see Moldovanu, C. 372 Nielsen, O.V., see Pedersen, E.B. 377 Nielsen, O.V., see Primdahl, F. 373, 376 Niewczas, P., see Cruden, A. 405 Niki, H., see Yogi, M. 21, 24 Nikolic, B.K., see Moriyama, T. 216 Nishida, N., see Ito, T.U. 58 Nishino, Y., see Matsunami, M. 22 Nishio, Y. 376 Nishiyama, K., see Aoki, Y. 44, 46, 96 Nisoli, C., see Wang, R.F. 114 Nistor, I., see Holthaus, C. 390 Nitta, J. 140 Niu, D., see Lepadatu, S. 279, 281 Nizzoli, F., see Grimsditch, M. 177, 180–184 Nizzoli, F., see Gubbiotti, G. 332, 334–335 Nizzoli, F., see Montoncello, F. 332 Noakes, D.R., see Shenoy, G.K. 16 Noel, H., see Bauer, E. 69

Author Index

Noginova, N. 238, 243, 245–248 Nojima, T., see Ochiai, A. 44 Nojiri, S., see Sugawara, H. 32 Nolting, F., see Vaz, C.A.F. 181, 322 Nordman, C.A., see Jander, A. 371 Nordman, C.A., see Ozbay, A. 360 Nordstrom, L. 22 Nordstrom, M.A., see Flavel, S.C. 411 Noro, T., see Lee, C.H. 36 Novosad, V. 172 Novosad, V., see Buchanan, K.S. 268–269 Novosad, V., see Guslienko, K.Yu. 174, 271 Novosad, V., see Kakazei, G.N. 270–274, 277, 292 Nowak, E.R., see Ozbay, A. 360 Nowak, U., see Sukhov, A. 159, 161, 163 Nozawa, K., see Shirotani, I. 4, 18, 94 Nozawa, S., see Kanai, K. 24 O’Brien, H. 410 O’Brien, S., see Zohar, S. 251, 253 Ochiai, A. 44 Ocker, B., see Ferreira, R. 371 O’Connor, C., see Vovk, A. 303, 307 Oddy, T., see O’Brien, H. 410 O’Donnell, T., see Tipek, A. 402 Oeschler, N. 39 Oeschler, N., see Tenya, K. 39 Oezyilmaz, B., see Sun, J.Z. 206–207 Oftedal, I. 2 Ogita, N. 18, 46, 50, 61 O’Grady, K., see Thomas, A.H. 152–153 Ohashi, H., see Matsunami, M. 11, 21–22, 25 Ohishi, K., see Aoki, Y. 44, 46, 62, 96 Ohishi, K., see Higemoto, W. 46 Ohishi, K., see Ito, T.U. 58 Ohishi, K., see Shu, L. 46 Ohishi, Y., see Shirotani, I. 66, 105 Ohkuni, H., see Aoki, Y. 58 Ohsaki, S., see Aoki, Y. 44, 101 Ohsaki, S., see Kotegawa, H. 44–45 Ohsaki, S., see Yogi, M. 21, 25 Ohta, T., see Lee, C.H. 10, 35, 42 Ohtsuka, T., see Fujita, T. 25 Ohya, M., see Hashi, S. 407, 411 Oikawa, K., see Sugawara, H. 96 Oikawa, M., see Nakanishi, Y. 21, 26 Oikawa, M., see Yoshizawa, M. 58, 103 Oja, A., see Kyyna¨raïnen, J. 389 Okada, H., see Sato, H. 18, 21–22, 27, 30, 50–51, 53, 93, 96, 99, 101 Okamura, H., see Matsunami, M. 21, 24–25, 35, 58, 65 Okazaki, Y., see Hashi, S. 407, 411 Okuno, T., see Kakazei, G.N. 270 Oliveira, R.B., see Cordova-Fraga, T. 411 Olivera, J., see Sańchez, M.L. 401 Olson, H.M., see An, S.Y. 117

439 O’Mathuna, S.C., see Tipek, A. 402 Onimaru, T., see Sato, H. 55 Onishi, N., see Nishio, Y. 376 Ono, K., see Shirotani, I. 6, 18, 27, 43, 49, 95, 100 Ono, T., see Gubbiotti, G. 332, 334–335 Ono, Y., see Mitsumoto, K. 61 Onodera, H., see Indoh, K. 64 Onuki, K., see Goto, T. 18, 46 Onuki, Y., see Aoki, D. 72, 107 Onuki, Y., see Aoki, Y. 58 Onuki, Y., see Kikuchi, D. 63, 103 Onuki, Y., see Matsuda, T.D. 32, 50, 69–71, 106 Onuki, Y., see Saha, S.R. 18, 36 Onuki, Y., see Sugawara, H. 5, 9–10, 16, 18, 24, 27, 32–33, 35, 42–43, 47–48, 51, 65, 93–94, 96, 99 Onuki, Y., see Sugiyama, K. 99 Onuki, Y., see Tokunaga, Y. 107 Onuki, Y., see Yamada, T. 16, 49, 63, 95 O’Reilly, S., see Ripka, P. 402 Orlanducci, S., see Konchits, A.A. 260, 262–264 Ortega, D. 238, 249–250 Orue, I., see Sarmiento, G. 259 Osaki, S., see Izawa, K. 41, 45 Osaki, S., see Sugawara, H. 18, 38–39, 96 Osborn, R., see Adroja, D.T. 46 Osborn, R., see Adroja, T.D. 21, 25 Osborn, R., see Goremychkin, E.A. 28, 38, 43 Osiander, R., see Wickenden, D.K. 389 Otani, Y., see Guslienko, K.Yu. 174, 271 Otani, Y., see Kakazei, G.N. 270–274, 277, 292 Otani, Y., see Novosad, V. 172 Otsuki, J. 34 Otsuki, J., see Kuramoto, Y. 34 Ould Ely, T., see Respaud, M. 252 Ounadjela, K., see Ebels, U. 312, 316–317 Ounadjela, K., see Gregg, J.F. 202 Oursler, D.A., see Wickenden, D.K. 389 Ovari, T.A., see Katranas, G.S. 411 Owen, D., see Pechan, M. 267–268 Oyanagi, H., see Lee, C.H. 22, 34 Ozatay, O., see Fuchs, G.D. 212–213 Ozbay, A. 360 ¨ zdemir, M., see Yalc- ın, O. 318 O ¨ zyilmaz, B. 210 O Paglione, J., see Sayles, T.A. 27, 48, 100 Pallare´s, J., see Marsal, L.F. 303, 306 Paluch, S., see Wawryk, R. 47, 95, 100 Pang, Y., see Gru¨nberg, P. 112 Panina, L.V., see Fry, N. 388 Panissod, P., see Pogorelov, Yu.G. 228, 252 Pankratov, D.A., see Koksharov, Yu.A. 247 Pannetier-Lecoeur, M. 360, 368 Pant, B. 365 Paperno, E. 382

440 Paperno, E., see Plotkin, A. 382 Pappas, D.P., see Stutzke, N.A. 367–368, 391 Pardavi-Horvath, M. 276–277 Park, G.T., see Kang, M.H. 409 Park, H.S. 380 Park, J., see Adroja, T.D. 21, 25 Park, J.-G. 34 Park, J.-G., see Adroja, D.T. 21, 24, 46 Park, J.-G., see Adroja, T.D. 21, 25 Park, J.P. 308, 310–312 Park, J.P., see Pechan, M. 292, 297–299, 301, 303 Parke, W., see Ibrahim, A.I. 413 Pascard, H., see Berteaud, A.J. 117 Paschen, S., see Viennois, R. 15–16, 21, 23–24, 64, 95, 98 Paschke, D., see Jeitschko, W. 65–66, 102, 105 Pasquale, M. 400 Pasquale, M., see Bydzovsky, J. 398 Patton, C.E., see An, S.Y. 117 Patton, C.E., see Kalinikos, B.A. 140 Patton, C.E., see Mo, N. 287 Patton, C.E., see Tsankov, M.A. 140 Paul, Ch., see Bauer, E. 49, 65, 104 Paulsen, C., see Measson, M.-A. 44, 46 Pavel, L., see Kubı´k, J. 379–380 Pavo, J. 375 Pechan, M. 267–268, 292, 297–299, 301, 303 Pechan, M.J., see Yu, C. 290–295, 301 Pedersen, E.B. 377 Pekko, P., see Kyyna¨raïnen, J. 389 Pelekhov, D., see Midzor, M.M. 114 Pelzl, J., see Pires, M.J.M. 255 Pereira, A.R. 242 Pereira de Azevedo, M.M., see Kakazei, G.N. 119, 228, 252 Pe´rez, A., see Jamet, M. 129–130 Perez, C., see Kiselev, S.I. 209–210 Perez, L. 372, 378 Perez, L., see Almazan, R.P. 379 Perez, L., see Butin, L. 412 Pe´rez, M.J., see Sańchez, M.L. 401 Periera de Azevedo, M.M., see Pogorelov, Yu.G. 228, 252 Perzynski, R., see Shilov, V.P. 225 Petersen, J.R., see Moldovanu, C. 372 Petersen, J.R., see Nielsen, O.V. 376 Petersen, J.R., see Pedersen, E.B. 377 Petersen, J.R., see Primdahl, F. 376 Petersen, P.I., see Scoville, J.T. 403 Petit, S. 214 Petridis, C. 381 Petroff, F., see Baibich, M.N. 112 Petroff, F., see Barthele´my, A. 368 Petrou, J. 382 Petrov Yu, N., see Konchits, A.A. 260, 262–264 Petrucha, V., see Vcela´k, J. 408 Pfleiderer, C., see Grube, K. 44

Author Index

Pfleiderer, C., see Vollmer, R. 28, 38, 44, 101 Pfutzner, H. 398 Piedade, M.S., see Cardoso, F.A. 371 Pierce, D., see Magnes, W. 378 Pietraszko, A., see Baumbach, R.E. 7 Pietraszko, A., see Chi, S. 47, 100 Pietraszko, A., see Henkie, Z. 6, 23, 48–49, 95 Pietraszko, A., see Ho, P.-C. 47, 95 Pietraszko, A., see Maple, M.B. 21, 23, 29, 47–48, 97, 100 Pietraszko, A., see Sayles, T.A. 27, 48, 100 Pietraszko, A., see Wawryk, R. 47, 95, 100 Pietraszko, A., see Yanagisawa, T. 47 Pietraszko, A., see Yuhasz, M.W. 27, 47, 100 Piguet, D., see Drljaca, P.M. 378, 390 Piraux, L., see Darques, M. 312 Piraux, L., see Ebels, U. 312, 316–317 Piraux, L., see Encinas, A. 311, 316 Piraux, L., see Encinas-Oropesa, A. 309, 313–315, 318 Piraux, L., see Fert, A. 308 Piraux, see Encinas-Oropesa, A. 309, 318 Pires, M.J.M. 236, 255 Pirota, K., see Vassallo Brigneti, E. 306 Platil, A. 378 Platil, A., see Vcelak, J. 366 Platil, A., see Vopa´lensky´, M. 366–367, 391 Platow, W., see Farle, M. 124–125 Plotkin, A. 382 Podbielski, J. 325, 328–331 Podbielski, J., see Giesen, F. 326, 330 Podloucky, R., see Bauer, E. 6, 75, 78, 110 Pogorelov, Yu.G. 228, 252 Pogorelov, Yu.G., see Kakazei, G.N. 119, 228, 233, 235–236, 252, 270 Pohm, A.V., see Jander, A. 371 Ponjavic, M.M. 402 Ponomarenko, L.A., see Koksharov, Yu.A. 247 Popa, A., see Darques, M. 312 Popovic, R.S. 349, 359, 394, 403 Popovic, R.S., see Drljaca, P.M. 378, 390 Popovic, R.S., see Kejik, P. 359 Popovic, R.S., see Zorlu, O. 381 Posth, O., see Meckenstock, R. 287–289 Pourret, A. 30 Praslicka, D., see Butvin, P. 350 Pribiag, V.S., see Fuchs, G.D. 212–213 Prieto, J.L. 389 Prieto, P., see Vassallo Brigneti, E. 306 Primdahl, F. 373, 376, 383, 410 Primdahl, F., see Merayo, J.M.G. 409 Primdahl, F., see Nielsen, O.V. 376 Primdahl, F., see Pedersen, E.B. 377 Primdahl, F., see Ripka, P. 376 Pufall, M.R., see Rippard, W.H. 210–212 Puffall, M.R., see Silva, T.J. 117 Pujada, B.R. 228, 254–257 Purucker, M., see Mandea, M. 410

Author Index

Puszkarski, H. 112, 119, 136, 138–141 Puzic, A. 283–285 Puzic, A., see Keupper, K. 268 Puzic, A., see Stoll, H. 283 Qian, H., see Zhu, X. 324–327 Qian, L., see Fuchs, G.D. 212–213 Quasimi, A. 359 Raabe, J., see Stoll, H. 283 Rabis, A., see Leithe-Jasper, A. 72–74, 108 Rached, D., see Khenata, R. 22, 71, 107 Rado, G.T. 118–119, 130 Rahimi, S. 50 Raikher, Yu.L. 225–226, 241, 252 Raikher, Yu.L., see Shilov, V.P. 225 Rainford, B.D., see Adroja, D.T. 46 Rainford, B.D., see Adroja, T.D. 21, 25 Ralph, D.C., see Albert, F.J. 204, 206–207 Ralph, D.C., see Cui, Y.-T. 214 Ralph, D.C., see Fuchs, G.D. 212–213 Ralph, D.C., see Katine, J.A. 203–205, 211 Ralph, D.C., see Kiselev, S.I. 209–210 Ralph, D.C., see Myers, E.B. 211 Ralph, D.C., see Sankey, J.C. 212, 214 Ramchal, R., see Spasova, M. 257, 259 Ramlau, R., see Leithe-Jasper, A. 73–74 Ramos, C.A. 315, 320–321 Ramos, C.A., see De Biasi, E. 223, 225–227, 263 Ramos, C.A., see Sańchez, R.D. 252–253 Ramos, C.A., see Vassallo Brigneti, E. 306 Rapoport, Yu., see Slavin, A.N. 140 Rasing, T. 117 Rastei, M.V. 270 Ravot, D., see Viennois, R. 15–16, 21, 23–24, 64, 95, 98 Raymond, S., see Kuwahara, K. 28, 42 Ru¨diger, U., see Vaz, C.A.F. 181, 322 Redjdal, M., see Castanõ, F.J. 114, 181 Reiman, S., see Leithe-Jasper, A. 73–74 Reininger, T. 392–394 Reiss, G., see Puzic, A. 283–285 Reissner, M. 49 Reissner, M., see Bauer, E. 49 Reissner, M., see Leithe-Jasper, A. 69 Remeika, J.P., see Fisk, Z. 5 Remhof, A. 114 Remhoff, A., see Meckenstock, R. 287 Ren, S., see So, E. 406 Renaux, C., see Haddab, Y. 359 Renaux, P., see Joisten, H. 379, 390 Reshak, A.H., see Khenata, R. 22, 71, 107 Respaud, M. 252 Rezende, S.M. 199 Rezende, S.M., see Mills, D.L. 142, 146 Rheema, Y.W. 405 Rhodes, P. 16

441 Ru¨hrig, M., see Uhrmann, T. 398 Ribeiro Guevara, S., see Sańchez, R.D. 252–253 Riedling, S., see Mathieu, C. 270 Rinkoski, M., see Kiselev, S.I. 209–210 Ripin, B.H., see Stamper, J.A. 390 Ripka, P. 352, 357, 368, 370–371, 376, 379, 385, 390, 401–402, 409 Ripka, P., see Cerman, A. 377 Ripka, P., see Fan, J. 381 Ripka, P., see Kubı´k, J. 352, 375, 379–380, 409 Ripka, P., see Mala´tek, M. 404–405 Ripka, P., see Nielsen, O.V. 376 Ripka, P., see Primdahl, F. 376 Ripka, P., see Tang, S.C. 402 Ripka, P., see Tomek, J. 411 Ripka, P., see Vcelak, J. 366 Ripka, P., see Vopa´lensky´, M. 366–367, 391 Rippard, W.H. 210–212 Risbo, T., see Primdahl, F. 376, 383, 410 Riseborough, P.S., see Adroja, T.D. 21, 25 Riseborough, P.S., see Viennois, R. 21, 24 Risken, H. 146 Rivas, J., see Sańchez, R.D. 252–253 Rivero, G. 411 Rivero, G., see Kraus, L. 356 Rivkin, K. 167–173 Rivkin, K., see Xu, W. 322–323 Robbes, D. 382, 385 Robbes, D., see Boukhenoufa, A. 387 Robertson, J.L., see Krishnamurthy, V.V. 65 Rocha, R., see Schmool, D.S. 112–113, 120, 128, 228–229, 231, 234, 266 Rodewald, U.C., see Jeitschko, W. 65–66, 102, 105 Rogers, C.T., see Silva, T.J. 114 Rogl, P., see Bauer, E. 6, 49, 65, 75, 78, 104, 110 Rogl, P., see Bauer, E.D. 65, 104–105 Rogl, P., see Butch, N.P. 6, 24, 28, 49, 100 Rogl, P., see Ho, P.-C. 50, 53–54, 103 Rogl, P., see Leithe-Jasper, A. 69 Rogl, P., see Reissner, M. 49 Rogl, P., see Yuhasz, M.W. 27, 47–48, 99 Rogl, P., see Yuhasz, W.M. 60, 62–63, 104 Rohn, M., see Pfutzner, H. 398 Romalis, M.V., see Allred, J.C. 384 Romalis, M.V., see Budker, D. 384 Rooks, M.J., see Sun, J.Z. 206–207 ¨ zyilmaz, B. 210 Rooks, M.J., see O Roos, B., see Mathieu, C. 270 Rose, M.S., see Ishida, K. 33 Rose, M.S., see MacLaughlin, D.E. 44–45 Rosner, H., see Dedkov, Yu.S. 69 Rosner, H., see Gumeniuk, R. 6, 75–78, 110 Rosner, H., see Leithe-Jasper, A. 72–74, 108 Rosner, H., see Schnelle, W. 16, 69, 72–74, 106, 108–109

442 Ross, C.A., see Castanõ, F.J. 114, 181 Ross, C.A., see Pardavi-Horvath, M. 276–277 Ross, C.A., see Vaz, C.A.F. 181, 322 Rossel, C., see Guertin, R.P. 106 Rossel, C., see Torikachvili, M.S. 27, 29, 50–51, 53, 69–70, 93, 99, 101, 106–107 Rossi, A.M., see Pujada, B.R. 228, 254–257 Rossi, M., see Konchits, A.A. 260, 262–264 Rotay, R.M., see Garshelis, I.J. 400 Rott, K., see Puzic, A. 283–285 Rotter, M., see Bauer, E. 6, 75, 110 Rotundu, C.R. 38–39, 41, 44 Roukes, M.L., see Midzor, M.M. 114 Rousseaux, F., see Jorzick, J. 270 Rousseaux, F., see Mathieu, C. 270, 308–309 Roux, A., see Coillot, C. 385 Roux, A., see Leroy, P. 360 Roy, P.E., see Buchanan, K.S. 268–269 Royanian, E., see Bauer, E. 6, 75, 78, 110 Rozen, J.R., see Koch, R.H. 374–375 Re´rat, M., see Khenata, R. 22, 71, 107 Rubinstein, M. 228 Ruiz-Diaz, B., see Verdes, C.G. 153–154 Ruotsalainen, S., see Kyyna¨raïnen, J. 389 Russek, S., see Stutzke, N. 214 Russek, S.E. 117 Russek, S.E., see Rippard, W.H. 210–212 Russek, S.E., see Stutzke, N.A. 367–368, 391 Russell, C.T., see Magnes, W. 378 Ryan, E.M., see Fuchs, G.D. 212–213 Saarilahti, J., see Kyyna¨raïnen, J. 389 Sacco, V., see Ando, B. 377 Saez, S., see Ding, L. 385 Safi-Harb, S., see Ibrahim, A.I. 413 Saha, S.R. 18, 36 Saha, S.R., see Aoki, Y. 44, 46, 96, 101 Saha, S.R., see Hao, L. 42 Saha, S.R., see Higemoto, W. 46 Saha, S.R., see Iwasa, K. 36–37, 99 Saha, S.R., see Matsuda, T.D. 34 Saha, S.R., see Sugawara, H. 18, 38–39, 96 Saha, S.R., see Yoshizawa, M. 58, 103 Saita, T., see Yamasaki, A. 61 Saito, H., see Sekine, C. 35, 101, 103, 105 Saito, K., see Sugawara, H. 18, 99 Saito, T., see Magishi, K. 21–22, 24, 64, 67 Saito, T., see Mori, I. 6, 21, 23, 95, 98 Saito, T., see Sugawara, H. 27, 47–48, 94, 99 Saito, T., see Toda, M. 6, 77–78, 110 Sakai, A., see Sekine, C. 35, 101, 103, 105 Sakai, H., see Aoki, D. 72, 107 Sakai, H., see Tokunaga, Y. 107 Sakai, K., see Goto, T. 18, 46 Sakai, O. 34 Sakakibara, T. 45 Sakakibara, T., see Aoki, Y. 39, 41, 43 Sakakibara, T., see Iwasa, K. 25, 37, 99

Author Index

Sakakibara, T., see Matsuhira, K. 35 Sakakibara, T., see Sato, H. 55 Sakakibara, T., see Sekine, C. 34, 64–65, 67, 105 Sakakibara, T., see Tayama, T. 32, 38–39, 41, 99, 101 Salabas, E.L., see Wiedwald, U. 258 Salamo, G.J., see Kunets, V.P. 357 Salamon, M.B., see Chia, E.E.M. 44–45, 50 Sales, B.C. 3, 24 Sales, B.C., see Gajewski, D.A. 23, 98 Sales, B.C., see Krishnamurthy, V.V. 65 Salgueirinõ-Maceira, V., see Antoniak, C. 265 Salomonski, N., see Sheinker, A. 410 Samarth, N., see Wang, R.F. 114 Samohin, A., see Plotkin, A. 382 Sanada, S. 58, 60–62, 104 Sanada, S., see Aoki, Y. 58, 60, 96, 103 Sanada, S., see Kikuchi, D. 63, 103 Sanada, S., see Sato, H. 60 Sanada, S., see Shu, L. 46 Sanchez, M.C., see Almazan, R.P. 379 Sanchez, M.C., see Ciudad, D. 389 Sanchez, M.C., see Michelena, M.D. 389 Sanchez, M.C., see Perez, L. 378 Sanchez, M.C., see Prieto, J.L. 389 Sanchez, P., see Almazan, R.P. 379 Sanchez, P., see Ciudad, D. 389 Sanchez, P., see Michelena, M.D. 389 Sanchez, P., see Perez, L. 378 Sanchez, P., see Prieto, J.L. 389 Sancho, M., see Belloy, E. 402 Sandacci, S.I., see Fry, N. 388 Sandhu, A., see Yamamura, T. 350 Sandu, D.D., see Dumitru, I. 155–157, 320 Sankey, J.C. 212, 214 Sankey, J.C., see Cui, Y.-T. 214 Sankey, J.C., see Fuchs, G.D. 212–213 Sankey, J.C., see Kiselev, S.I. 209–210 Santos, J.A.M., see Schmool, D.S. 112–113, 120, 128, 228–229, 231, 234, 266 Santos, J.D., see Sańchez, M.L. 401 Santos, M.B., see Kunets, V.P. 357 Sanz, J.M., see Vassallo Brigneti, E. 306 Sarmiento, G. 259 Sartoratto, P.P.C., see Pereira, A.R. 242 Sasada, I. 381, 399 Sasada, I., see Goleman, K. 382 Sasada, I., see Plotkin, A. 382 Sasakawa, T., see Sugawara, H. 22 Sasakawa, T., see Takabatake, T. 72, 74–75, 108–109 Saslow, W., see Rivkin, K. 168, 170–173 Saslow, W.M., see Chudnovsky, E.M. 123 Sato, H. 18, 21–22, 27, 29–30, 34, 50–51, 53, 55, 60, 93, 96, 99, 101 Sato, H., see Abe, K. 24, 50, 53, 96, 98

Author Index

Sato, H., see Aoki, Y. 39, 41, 43–44, 46, 58, 60, 62, 96, 99, 101, 103 Sato, H., see Chia, E.E.M. 44–45, 50 Sato, H., see Goto, T. 18, 46 Sato, H., see Hao, L. 29, 42 Sato, H., see Higemoto, W. 46 Sato, H., see Huxley, A.D. 45 Sato, H., see Ikeno, T. 6, 52–53, 102 Sato, H., see Imada, S. 53 Sato, H., see Ishida, K. 33 Sato, H., see Ito, T.U. 58 Sato, H., see Iwasa, K. 10, 25, 29, 33–34, 36–37, 42, 46, 50, 99 Sato, H., see Izawa, K. 41, 45 Sato, H., see Keller, L. 29, 51, 101 Sato, H., see Kikuchi, D. 57–58, 63, 103 Sato, H., see Kikuchi, J. 34 Sato, H., see Kobayashi, M. 45 Sato, H., see Kohgi, M. 38, 40–41 Sato, H., see Kotegawa, H. 44–45, 50, 60, 62–63 Sato, H., see Kumagai, T. 50 Sato, H., see Kuwahara, K. 28, 38, 42–43, 54 Sato, H., see Magishi, K. 21–22, 67 Sato, H., see Masaki, S. 52, 58, 101 Sato, H., see Matsuda, T.D. 32, 34, 50 Sato, H., see Matsuhira, K. 60 Sato, H., see Matsunami, M. 11, 21–22, 24–25 Sato, H., see Measson, M.-A. 44, 46 Sato, H., see Mizumaki, M. 61–63 Sato, H., see Mori, I. 6, 21, 23, 95, 98 Sato, H., see Nakai, Y. 18, 77, 96 Sato, H., see Nakanishi, Y. 21, 26, 62, 101 Sato, H., see Namiki, T. 25, 27, 43, 49, 95, 98–100 Sato, H., see Ogita, N. 18, 46, 50, 61 Sato, H., see Pourret, A. 30 Sato, H., see Rotundu, C.R. 38–39, 41 Sato, H., see Saha, S.R. 18, 36 Sato, H., see Sakai, O. 34 Sato, H., see Sakakibara, T. 45 Sato, H., see Sanada, S. 58, 60–62, 104 Sato, H., see Seyfarth, G. 45 Sato, H., see Shu, L. 46 Sato, H., see Sugawara, H. 5, 9–10, 16, 18, 22, 24, 27, 32–33, 35, 38–39, 42–43, 47–48, 51, 93–94, 96, 99 Sato, H., see Sugiyama, K. 99 Sato, H., see Tanaka, K. 7, 49–50, 95, 100 Sato, H., see Tatsuoka, S. 7, 15–17, 94 Sato, H., see Tayama, T. 32, 38–39, 41, 99, 101 Sato, H., see Toda, M. 6, 77–78, 110 Sato, H., see Tou, H. 38 Sato, H., see Tsubota, M. 61 Sato, H., see Tsuda, S. 94 Sato, H., see Ueda, M. 63–64, 103 Sato, H., see Yamada, T. 16, 49, 63, 95 Sato, H., see Yamasaki, A. 61

443 Sato, H., see Yang, C. 25 Sato, H., see Yogi, M. 21, 24–25, 44, 46, 50, 96 Sato, H., see Yoshizawa, M. 58, 103 Sato, S., see Shirotani, I. 19 Satoh, K., see Ito, T.U. 58 Satoh, K.H., see Aoki, Y. 62 Sayles, T.A. 27, 48, 100 Sayles, T.A., see Baumbach, R.E. 7 Sayles, T.A., see Butch, N.P. 6, 24, 28, 49, 100 Sayles, T.A., see Ho, P.-C. 50, 53–54, 103 Sayles, T.A., see Maple, M.B. 21, 23, 47–48, 50, 97, 100 Sayles, T.A., see Wawryk, R. 47, 95, 100 Sayles, T.A., see Yanagisawa, T. 47 Sayles, T.A., see Yuhasz, M.W. 27, 47–48, 99–100 Sayles, T.A., see Yuhasz, W.M. 60, 62–63, 104 Saylesa, T.A., see Maple, M.B. 29 Schackert, F., see Vaz, C.A.F. 181, 322 Schafer, H., see Steglich, F. 37 Scheider, C.M., see Martyanov, O.N. 295, 299–300 Scheidt, E.W., see Bauer, E. 49 Scherer, D.J., see Malkinski, L.M. 277–278 Scherrer, D., see Kuanr, B.K. 277, 279–280 Schiffer, P., see Wang, R.F. 114 Schilling, M., see Hinnrichs, C. 373–374 Schmalzl, M., see Schmool, D.S. 112, 128–129, 145, 238, 249, 251 Schmidt, J.E., see Pires, M.J.M. 255 Schmidt, J.E., see Viegas, A.D.C. 255 Schmidt, M., see Dedkov, Yu.S. 69 Schmidt, M., see Schnelle, W. 69 Schmool, D.S. 112–113, 119–120, 128–131, 145, 228–229, 231, 234, 238, 249, 251, 254, 266, 280 Schmool, D.S., see Hicken, R.J. 114, 117 Schmool, D.S., see Kachkachi, H. 130 Schmool, D.S., see Mosendz, O. 114 Schmool, D.S., see Sousa, N. 162 Schmool, D.S., see Wu, J. 114 Schneider, M.L. 274–275 Schnelle, W. 16, 69, 72–74, 106, 108–109 Schnelle, W., see Dedkov, Yu.S. 69 Schnelle, W., see Gumeniuk, R. 6, 75–78, 110 Schnelle, W., see Leithe-Jasper, A. 72–74, 108 Scho¨nhense, G., see Martyanov, O.N. 295, 299–300 Schoelkopf, R.J., see Kiselev, S.I. 209 Scholz, U.D., see Evers, C.B.H. 64, 102–104 Schrefl, T., see Mistral, Q. 214–216 Schreiber, R., see Gru¨nberg, P. 112 Schultz, A.J., see Nakotte, H. 69, 106 Schulz, B. 125 Schulz, B., see Farle, M. 124 Schumann, A., see Remhof, A. 114 Schwarz, U., see Leithe-Jasper, A. 73–74 Schwindt, P.D.D. 384

444 Schwingenschuh, K., see Magnes, W. 378 Scott, M.M., see Kalinikos, B.A. 140 Scoville, J.T. 403 Seavey, M.H., see Tannenwald, P.E. 113 Sebestyen, I., see Pavo, J. 375 Sechovsky, V., see Aoki, Y. 58 Seck, M., see Tsoi, M. 187, 201 Seehra, M.S., see Dutta, P. 238–240 Sei, Y., see Matsuhira, K. 65 Sekine, C. 7, 18, 21, 23, 27, 34–35, 47, 58–59, 64–65, 67, 97, 99, 101, 103–105 Sekine, C., see Fujino, T. 67 Sekine, C., see Fujiwara, K. 22 Sekine, C., see Giri, R. 63 Sekine, C., see Hachitani, K. 56–58, 63 Sekine, C., see Hao, L. 42 Sekine, C., see Indoh, K. 64 Sekine, C., see Kihou, K. 64, 66, 68, 78, 93, 103–106 Sekine, C., see Lee, C.H. 10, 22, 34–36, 42 Sekine, C., see Matsuhira, K. 35, 60, 65, 94, 99, 103 Sekine, C., see Matsuhira, M. 103 Sekine, C., see Matsunami, M. 21, 24, 35, 58, 65 Sekine, C., see Namiki, T. 27, 43, 49, 95, 100 Sekine, C., see Ogita, N. 18, 46, 50, 61 Sekine, C., see Shimizu, M. 49, 95, 100 Sekine, C., see Shirotani, I. 6, 18–19, 21–23, 27, 43, 49, 66, 68–69, 78, 93, 95, 97, 100, 105–106 Sekine, C., see Viennois, R. 95, 98 Sekine, C., see Yamamoto, A. 69, 106 Sekine, C., see Yoshizawa, M. 58, 68, 103 Sekinea, C., see Uchiumi, T. 18, 94–96 Sekinek, C., see Magishi, K. 67 Sekiyama, A., see Imada, S. 53 Sekiyama, A., see Yamasaki, A. 61 Sellmyer, D.J., see Skomski, R. 146–147 Senba, Y., see Matsunami, M. 11, 21–22, 25 Senoo, K., see Matsunami, M. 21, 24 Senthilkumaran, N., see Leithe-Jasper, A. 72–73, 108 Seppa¨, H., see Kyyna¨raïnen, J. 389 Sera, M., see Takabatake, T. 72, 74–75, 108–109 Sera, M., see Tou, H. 38 Seran, H.C. 385 Serota, R.A., see Chudnovsky, E.M. 123 Sessa, V., see Konchits, A.A. 260, 262–264 Settai, R., see Aoki, Y. 58 Settai, R., see Kikuchi, D. 63, 103 Settai, R., see Matsuda, T.D. 32, 50, 69–71, 106 Settai, R., see Saha, S.R. 18, 36 Settai, R., see Sugawara, H. 5, 9–10, 16, 18, 24, 27, 32–33, 35, 42–43, 47–48, 51, 65, 93–94, 96, 99 Settai, R., see Sugiyama, K. 99

Author Index

Settai, R., see Yamada, T. 16, 49, 63, 95 Seyfarth, G. 44–45 Seyfarth, G., see Measson, M.-A. 44, 46 Shah, N., see Dutta, P. 238–240 Shah, V., see Schwindt, P.D.D. 384 Shaw, J.M., see Schneider, M.L. 274–275 Sheinker, A. 410 Shen, W., see Li, Y. 303 Shenoy, G.K. 16 Shi, J., see Zhai, Y. 277 Shi, L., see Zhai, Y. 277 Shi, Y., see Xi, H. 197, 199–200 Shiba, H., see Sakai, O. 34 Shiba, H., see Shiina, R. 41 Shiina, R. 41, 43 Shiina, R., see Sakai, O. 34 Shiina, R., see Tou, H. 38 Shilov, V.P. 225 Shilton, J.M., see McPhail, S. 301–302 Shim, D.S., see Park, H.S. 380 Shima, H., see Guslienko, K.Yu. 174, 271 Shima, H., see Kakazei, G.N. 270–271 Shimaoka, Y., see Kotegawa, H. 60, 62–63 Shimaya, Y., see Giri, R. 63 Shimaya, Y., see Kihou, K. 64, 66, 68, 78, 93, 103–106 Shimaya, Y., see Matsuhira, K. 103 Shimaya, Y., see Shirotani, I. 6, 68–69, 78, 93, 105–106 Shimizu, M. 49, 95, 100 Shimojima, T., see Tsuda, S. 94 Shin, S., see Kanai, K. 24 Shin, S., see Matsunami, M. 11, 21–22, 25 Shin, S., see Tsuda, S. 94 Shinkai, H., see Tomita, S. 260 Shinozawa, A., see Kikuchi, D. 63, 103 Shiokawa, Y., see Aoki, D. 72, 107 Shiokawa, Y., see Tokunaga, Y. 107 Shirane, G., see Nakotte, H. 69, 106 Shirotani, I. 4, 6, 18–19, 21–23, 27, 43, 49, 66, 68–69, 78, 93–95, 97, 100, 105–106 Shirotani, I., see Fujino, T. 67 Shirotani, I., see Giri, R. 63 Shirotani, I., see Hachitani, K. 56–58, 63 Shirotani, I., see Indoh, K. 64 Shirotani, I., see Kihou, K. 64, 66, 68, 78, 93, 103–106 Shirotani, I., see Lee, C.H. 10, 22, 34–36, 42 Shirotani, I., see Magishi, K. 67 Shirotani, I., see Matsuhira, K. 35, 60, 65, 94, 99, 103 Shirotani, I., see Matsuhira, M. 103 Shirotani, I., see Matsunami, M. 21, 24, 35, 58, 65 Shirotani, I., see Namiki, T. 27, 43, 49, 95, 100 Shirotani, I., see Ogita, N. 18, 46, 50, 61 Shirotani, I., see Sekine, C. 18, 27, 34–35, 47, 58–59, 64–65, 67, 97, 99, 101, 103–105

Author Index

Shirotani, I., see Shimizu, M. 49, 95, 100 Shirotani, I., see Uchiumi, T. 18, 94–96 Shirotani, I., see Yamamoto, A. 69, 106 Shirotani, I., see Yoshizawa, M. 58, 68, 103 Shirotania, I., see Sekine, C. 7, 21, 23, 97 Shishido, H., see Kikuchi, D. 63, 103 Shishido, H., see Saha, S.R. 18, 36 Shishido, H., see Sugawara, H. 96 Shu, L. 46 Shu¨tz, G., see Keupper, K. 268 Shu¨tz, G., see Puzic, A. 283–285 Sienkiewicz, A., see Lachowicz, H. 254 Sigrist, M. 44 Sigrist, M., see Luke, G.M. 46 Silsbee, R.H., see Johnson, M. 195 Silva, T.J. 114, 117 Silva, T.J., see Rippard, W.H. 210–212 Silva, T.J., see Schneider, M.L. 274–275 Singh, D.J., see Aguayo, A. 17 Singh, D.J., see Krishnamurthy, V.V. 65 Singh, D.J., see Nordstrom, L. 22 Singh, N., see Adeyeye, A.O. 267 Singh, N., see Gubbiotti, G. 180 Singleton, J., see Ho, P.-C. 47, 95 Sinnecker, E.H.C.P., see Pujada, B.R. 228, 254–257 Sirotani, I., see Fujiwara, K. 22 Siruguri, V., see Sarmiento, G. 259 Sirvent, C., see Bauer, E.D. 5, 19, 21, 24, 96, 98, 101 Skomski, R. 114, 146–147 Sladkov, M., see Costache, M.V. 216 Slater, J.C. 9 Slawska-Waniewska, A., see Lachowicz, H. 254 Slavin, A.N. 136, 140, 201 Slavin, A.N., see Bayer, C. 270 Slavin, A.N., see Demokritov, S.O. 136, 294 Slavin, A.N., see Gubbiotti, G. 180 Slavin, A.N., see Guslienko, K.Yu. 173, 176–180, 302, 332 Slavin, A.N., see Jorzick, J. 175 Slavin, A.N., see Kakazei, G.N. 272–274, 277, 292 Slavin, A.N., see Kalinikos, B.A. 134–135, 175, 272, 294 Slavin, A.N., see Mathieu, C. 308–309 Slebarski, A., see Bauer, E.D. 5, 19, 21, 24, 65, 96, 98, 101, 104–105 Slonczewski, J.C. 186, 189, 197, 210 Slonczewski, J.C., see Sankey, J.C. 214 Smit, J. 117 Smith, H.I., see Castanõ, F.J. 114, 181 Smith, N., see Synogatch, V. 214 Sańchez, M.L. 401 Sańchez, R.D. 252–253 So, E. 406 So, J.-Y., see Adroja, D.T. 21, 24 Sobal, N.S., see Wiedwald, U. 258

445 Soderhom, L., see Xue, J.S. 22 Sologub, O., see Bauer, E. 75, 78, 110 Sonier, J.E., see MacLaughlin, D.E. 44–45 Soohoo, R.F. 133 Sosa, M., see Cordova-Fraga, T. 411 Soulard, B., see Berger, R. 220, 253, 262 Sousa, J.B., see Kakazei, G.N. 119, 128, 228, 233, 235–236, 252 Sousa, J.B., see Pogorelov, Yu.G. 228, 252 Sousa, J.B., see Schmool, D.S. 112–113, 120, 128, 228–229, 231, 234, 266 Sousa, L., see Cardoso, F.A. 371 Sousa, N. 162 Sowers, H., see Gru¨nberg, P. 112 Spasova, M. 257, 259 Spasova, M., see Trunova, A.V. 113, 258, 260–261 Spasova, M., see Wiedwald, U. 258 Spinu, L., see Malkinski, L.M. 277–278 Spinu, L., see Yu, M. 303, 305–306 Spoddig, D., see Meckenstock, R. 287–289 Spottorno, J., see Rivero, G. 411 Srajer, G., see Krishnamurthy, V.V. 65 Stadler, B.J.H., see McGary, P.D. 398 Stahl, J., see Hinnrichs, C. 373–374 Stahlmecke, B. 114 Stamper, J.A. 390 Stancu, Al., see Verdes, C.G. 153–154 Stankiewicz, A., see Hiebert, W.K. 113 Stankiewicz, see Freeman, M.R. 117 Steglich, F. 37 Steglich, F., see Cichorek, T. 41 Steglich, F., see Grewe, N. 37 Steglich, F., see Oeschler, N. 39 Steglich, F., see Tenya, K. 39 Steglich, F., see Viennois, R. 15–16, 23, 64, 95, 98 Steiner, M., see Giesen, F. 330 Steiner, W., see Bauer, E. 49 Steiner, W., see Leithe-Jasper, A. 69 Steiner, W., see Reissner, M. 49 Stepanov, V.I., see Raikher, Yu.L. 225–226, 241, 252 Steurer, J., see Giouroudi, I. 387 Steurer, J., see Hauser, H. 385 Steurer, J., see Keplinger, F. 389 Stewart, G.R. 37 Stiles, M.D. 198 Stokes, J., see Ripka, P. 368, 370 Stoll, H. 283 Stoll, H., see Keupper, K. 268 Stoll, H., see Puzic, A. 283–285 Stoner, E.C. 148 Strand, J.D., see Suderow, H. 44–45 Strom-Olsen, J. 387 Stutzke, N. 214 Stutzke, N.A. 367–368, 391 Styblikova, R., see Draxler, K. 350, 402

446 Suderow, H. 44–45 Suderow, H., see Brison, J.-P. 44 Suga, S., see Imada, S. 53 Suga, S., see Yamasaki, A. 61 Sugawara, H. 5, 9–10, 16, 18, 22, 24, 27, 32–33, 35, 38–39, 42–43, 47–48, 51, 65, 93–94, 96, 99 Sugawara, H., see Abe, K. 24, 50, 53, 96, 98 Sugawara, H., see Aoki, Y. 39, 41, 43–44, 46, 58, 60, 96, 99, 101, 103 Sugawara, H., see Chia, E.E.M. 44–45, 50 Sugawara, H., see Goto, T. 18, 46 Sugawara, H., see Hao, L. 29, 42 Sugawara, H., see Higemoto, W. 46 Sugawara, H., see Huxley, A.D. 45 Sugawara, H., see Imada, S. 53 Sugawara, H., see Ishida, K. 33 Sugawara, H., see Ito, T.U. 58 Sugawara, H., see Iwasa, K. 10, 25, 29, 33–34, 36–37, 42, 46, 50, 99 Sugawara, H., see Izawa, K. 41, 45 Sugawara, H., see Keller, L. 29, 51, 101 Sugawara, H., see Kikuchi, D. 57–58, 63, 103 Sugawara, H., see Kikuchi, J. 34 Sugawara, H., see Kobayashi, M. 45 Sugawara, H., see Kohgi, M. 38, 40–41 Sugawara, H., see Kotegawa, H. 44–45, 50, 60, 62–63 Sugawara, H., see Kumagai, T. 50 Sugawara, H., see Kuwahara, K. 28, 38, 42–43, 54 Sugawara, H., see Magishi, K. 21–22, 24, 64, 67 Sugawara, H., see Masaki, S. 52, 58, 101 Sugawara, H., see Matsuda, T.D. 32, 34, 50 Sugawara, H., see Matsuhira, K. 60 Sugawara, H., see Matsunami, M. 11, 21–22, 24–25 Sugawara, H., see Measson, M.-A. 44, 46 Sugawara, H., see Mizumaki, M. 61–63 Sugawara, H., see Mori, I. 6, 21, 23, 95, 98 Sugawara, H., see Nakai, Y. 18, 77, 96 Sugawara, H., see Nakanishi, Y. 21, 26, 62, 101 Sugawara, H., see Namiki, T. 25, 98–99 Sugawara, H., see Ogita, N. 18, 46, 50, 61 Sugawara, H., see Pourret, A. 30 Sugawara, H., see Rotundu, C.R. 38–39, 41 Sugawara, H., see Saha, S.R. 18, 36 Sugawara, H., see Sakai, O. 34 Sugawara, H., see Sakakibara, T. 45 Sugawara, H., see Sanada, S. 58, 60–62, 104 Sugawara, H., see Sato, H. 18, 21–22, 27, 29–30, 34, 50–51, 53, 55, 60, 93, 96, 99, 101 Sugawara, H., see Seyfarth, G. 45 Sugawara, H., see Shu, L. 46 Sugawara, H., see Sugiyama, K. 99 Sugawara, H., see Tanaka, K. 7, 49–50, 95, 100 Sugawara, H., see Tatsuoka, S. 7, 15–17, 94

Author Index

Sugawara, H., see Tayama, T. 32, 38–39, 41, 99, 101 Sugawara, H., see Toda, M. 6, 77–78, 110 Sugawara, H., see Tou, H. 38 Sugawara, H., see Tsubota, M. 61 Sugawara, H., see Tsuda, S. 94 Sugawara, H., see Ueda, M. 63–64, 103 Sugawara, H., see Yamada, T. 16, 49, 63, 95 Sugawara, H., see Yamasaki, A. 61 Sugawara, H., see Yang, C. 25 Sugawara, H., see Yogi, M. 21, 24–25, 44, 46, 50, 96 Sugawara, H., see Yoshizawa, M. 58, 103 Sugiuchi, Y., see Sekine, C. 65, 97, 105 Sugiyama, K. 99 Sugiyama, K., see Matsuda, T.D. 69–71, 106 Sugiyama, K., see Yamada, T. 16, 49, 63, 95 Suhl, H. 117, 145, 151 Suhonen, M., see Kyyna¨raïnen, J. 389 Sui, X., see Li, T. 318–319 Sukhov, A. 159, 161, 163 Sun, J.Z. 189, 191–194, 206–207 Sun, J.Z., see Sankey, J.C. 214 ¨ zyilmaz, B. 210 Sun, J.Z., see O Sun, N.X., see Geiler, A.L. 372 Sun, P., see Nakanishi, Y. 62 Sun, P., see Yoshizawa, M. 68 Suski, J., see Haddab, Y. 359 Suzuki, H., see Sekine, C. 58–59 Suzuki, M., see Fujita, T. 25 Suzuki, S., see Nara, T. 408 Suzuki, T., see Lee, C.H. 36 Svec, P., see Bydzovsky, J. 398 Swank, J.H., see Ibrahim, A.I. 413 Synogatch, V. 214 Szewczyk, R. 356 Tabira, K., see Kotegawa, H. 50, 62–63 Tacchi, S., see Gubbiotti, G. 180, 332, 334–335 Tachi, K., see Shirotani, I. 4, 18, 94 Tadokoro, Y., see Kawahito, S. 378 Taguchi, M., see Matsunami, M. 11, 21–22, 25 Taillefer, L., see Hill, R.W. 50 Taillefer, L., see Joynt, R. 44 Takabatake, T. 72, 74–75, 108–109 Takabatake, T., see Matsuoka, E. 74, 108–109 Takabatake, T., see Ogita, N. 50 Takabatake, T., see Sugawara, H. 22 Takabatake, T., see Yoshii, S. 74, 76, 108–109 Takagi, M., see Kuwahara, K. 54 Takagi, S., see Fujino, T. 67 Takagi, S., see Matsuhira, K. 65 Takagi, T., see Iwasa, K. 36, 99 Takahashi, H., see Shirotani, I. 95 Takamatsu, T., see Matsuhira, M. 103 Takano, Y., see Rotundu, C.R. 38–39, 41 Takasu, Y., see Ogita, N. 18, 46, 50, 61 Takata, Y., see Matsunami, M. 22

Author Index

Takayanagi, H., see Nitta, J. 140 Takazawa, H., see Lee, C.H. 10, 35, 42 Takeda, K., see Kihou, K. 68 Takeda, K., see Magishi, K. 67 Takeda, K., see Sekine, C. 7, 21, 23, 97, 105 Takeda, K., see Shirotani, I. 19, 66, 105 Takeda, N. 24, 56–57, 65, 96, 98, 101–102, 104 Takeda, N., see Adroja, D.T. 21, 24, 46 Takeda, N., see Adroja, T.D. 21, 25 Takeda, N., see Kanai, K. 24 Takeda, N., see Masaki, S. 58 Takeda, N., see Ogita, N. 18, 46, 50, 61 Takeda, N., see Shirotani, I. 93 Takeda, N., see Yogi, M. 21, 24 Takegahara, K. 9, 12, 62, 72 Takegahara, K., see Curnoe, S.H. 35–36 Takegahara, K., see Harima, H. 10–11, 15, 32, 35–36, 73 Takemura, M., see Masaki, S. 58 Takeuchi, J., see Fujiwara, K. 22 Takeuchi, T., see Matsuda, T.D. 69–71, 106 Takeuchi, T., see Sugiyama, K. 99 Takigawa, M., see Kikuchi, J. 34 Takigawa, M., see Sakai, O. 34 Takikawa, T., see Matsuhira, K. 35 Takimoto, M., see Matsunami, M. 35, 58, 65 Takimoto, T. 36 Tamaru, S. 177–179 Tamasaku, K., see Matsunami, M. 22 Tamburri, E., see Konchits, A.A. 260, 262–264 Tamura, I., see Ikeno, T. 7, 69, 106 Tan, L., see McGary, P.D. 398 Tanaka, K. 7, 49–50, 95, 100 Tanaka, K., see Ikeno, T. 6, 52–53, 102 Tanaka, K., see Kikuchi, D. 57–58, 63, 103 Tanaka, K., see Kotegawa, H. 50, 62–63 Tanaka, K., see Magishi, K. 67 Tanaka, K., see Matsuoka, E. 74, 108–109 Tanaka, K., see Mori, I. 6, 21, 23, 95, 98 Tanaka, K., see Sato, H. 29, 34, 60 Tanaka, K., see Tatsuoka, S. 7, 15–17, 94 Tanaka, K., see Ueda, M. 63–64, 103 Tanaka, K., see Yamada, T. 16, 49, 63, 95 Tang, J., see Shirotani, I. 95 Tang, S.C. 402 Tanida, H., see Mizumaki, M. 61–63 Tanigawa, H., see Gubbiotti, G. 332, 334–335 Tanikawa, S. 61 Tanizawa, T., see Nakanishi, Y. 21, 26, 62 Tannenwald, P.E. 113 Tannous, C. 149, 151–152 Tarascon, J.M., see Kasaya, M. 63 Tatsuoka, S. 7, 15–17, 94 Tatsuoka, S., see Kikuchi, D. 63, 103 Tatsuoka, S., see Kotegawa, H. 50, 62–63 Tatsuoka, S., see Sato, H. 34

447 Tayama, T. 32, 38–39, 41, 99, 101 Tayama, T., see Aoki, Y. 39, 41, 43 Tayama, T., see Iwasa, K. 25, 37, 99 Tayama, T., see Sakakibara, T. 45 Tayama, T., see Sato, H. 55 Taylo, B.J., see Maple, M.B. 50 Taylor, B.J., see Yuhasz, W.M. 60, 62–63, 104 Taylor, J.A., see Tondra, M. 371 Taylor, J.W., see Adroja, T.D. 21, 25 Tedenac, J.C., see Viennois, R. 95, 98 Tenya, K. 39 Terki, F., see Viennois, R. 23, 95, 98 Terranova, M.L., see Konchits, A.A. 260, 262–264 Tetsuya, M., see Tsujimura, T. 408 Tøffner-Clausen, L., see Primdahl, F. 376, 410 Thadani, K.V., see Cui, Y.-T. 214 Thalmeier, P., see Izawa, K. 41, 45 Thalmeier, P., see Oeschler, N. 39 Thalmeier, P., see Shiina, R. 41 Thiele, A.A. 216 Thirion, C. 151 Thirion, C., see Jamet, M. 129–130 Thirion, C., see Petit, S. 214 Thomas, A.H. 152–153 Thompson, S.M., see Gregg, J.F. 202 Thompson, S.M., see Verdes, C.G. 153–154 Tibu, M., see Ioan, C. 375 Tilli, M., see Kyyna¨raïnen, J. 389 Tillmann, A., see Zimmermann, E. 362, 391 Tipek, A. 402 Tipek, A., see Ripka, P. 379 Tishin, A.M., see Koksharov, Yu.A. 247 Toda, M. 6, 77–78, 110 Todo, S., see Shirotani, I. 4, 6, 18, 27, 43, 49, 94–95, 100 Todo, S., see Uchiumi, T. 18, 94–96 Togashi, T., see Matsuhira, M. 103 Togashi, T., see Matsunami, M. 11, 21, 25 Togasi, T., see Tsuda, S. 94 Tohyama, F., see Nishio, Y. 376 Tokunaga, Y. 107 Tollens, S.P.L., see Garshelis, I.J. 401 Tomek, J. 411 Tomek, J., see Platil, A. 378 Tomita, S. 258, 260 Tondra, M. 371 Tondra, M., see Ripka, P. 368, 370 Torikachvili, M.S. 27, 29, 50–51, 53, 69–70, 93, 99, 101, 106–107 Torikachvili, M.S., see Guertin, R.P. 106 Torikachvili, M.S., see Meisner, G.P. 19, 21–23, 69, 96, 106 Torikachvili, M.S., see Nakotte, H. 69, 106 Tou, H. 38 Tran, L., see Deak, J. 371 Tran, V.H., see Bauer, E. 65, 104

448 Tran, V.H., see Kaczorowski, D. 75, 78, 110 Tremps, E., see Garcıá, A. 381 Treutler, C.P.O. 395 Trots, D., see Schnelle, W. 16, 69, 72–74, 106, 108–109 Trunova, A.V. 113, 258, 260–261 Tsankov, M.A. 140 Tserkovnyak, Y. 217 Tserkovnyak, Y., see Brataas, A. 217 Tserkovnyak, Y., see Foros, J. 209 Tserkovnyak, Y., see Moriyama, T. 216 Tserkovnyak, Y., see Wang, X. 114 Tsoi, M. 187, 201 Tsoi, V., see Tsoi, M. 187, 201 Tsubota, M. 61 Tsuchiya, A., see Aoki, Y. 44, 46 Tsuchiya, A., see Sanada, S. 58, 60–62, 104 Tsuchiya, A., see Sugawara, H. 22 Tsuda, S. 94 Tsujii, H., see Rotundu, C.R. 38–39, 41 Tsujii, N. 60 Tsujimura, T. 408 Tsunashima, Y., see Aoki, Y. 62 Tsuruta, C., see Tomita, S. 258 Tsutsui, S., see Mizumaki, M. 61–63 Tsutsui, S., see Tsubota, M. 61 Tumanski, S. 349, 353–354, 362, 385 Tunashima, Y., see Shu, L. 46 Tuominen, M.T., see Yalc- ın, O. 318 Turek, I. 217 Turolla, R., see Ibrahim, A.I. 413 Tyliszczak, T., see Puzic, A. 283–285 Uchiumi, T. 18, 94–96 Uchiumi, T., see Sekine, C. 18, 27, 34–35, 47, 99, 101, 103, 105 Uchiumi, T., see Shirotani, I. 6, 18–19, 21–23, 27, 43, 49, 95, 97, 100 Udagawa, M., see Ogita, N. 18, 46, 50, 61 Ueda, K., see Curnoe, S.H. 35–36 Ueda, K., see Harima, H. 36 Ueda, K., see Sigrist, M. 44 Ueda, M. 63–64, 103 Ueda, M., see Kikuchi, D. 63, 103 Ueda, M., see Sato, H. 34 Ueda, M., see Tatsuoka, S. 7, 15–17, 94 Ueda, U., see Tanaka, K. 50 Uemura, Y.J., see Luke, G.M. 46 Ueno, K., see Lee, C.H. 10, 35, 42 Uher, C. 3 Uhrmann, T. 398 Ulmeanu, M., see Wiedwald, U. 258 Umbach, F. 400 Umeo, K., see Matsuoka, E. 74 Umeo, K., see Takabatake, T. 72, 74–75, 108–109 Urban, R. 146, 190 Uruga, T., see Mizumaki, M. 61–63

Author Index

Usadel, K.D. 156, 158–159, 161 Usadel, K.D., see Sukhov, A. 159, 161, 163 Usov, N., see Garcıá, C. 388 Uwatoko, Y., see Matsuda, T.D. 69–71, 106 Valavanoglou, A., see Magnes, W. 378 Valde´s, J., see Rivero, G. 411 Valet, T. 195 Valstyn, E.P. 238 van den Berg, H., see Rasing, T. 117 van der Veerdonk, R., see Tamaru, S. 177–179 van der Wal, C.H., see Costache, M.V. 216 van der Wal, C.H., see Grollier, J. 285 van Est, J.W., see Gijs, M.A.M. 357, 371 van Kampen, M., see Mistral, Q. 214–216 van Kempen, H., see Van Son, P.C. 202 Van Son, P.C. 202 van Staa, A., see Giesen, F. 330 Van Vleck, J.H. 113, 242 Van Waeyenberge, B., see Keupper, K. 268 Van Waeyenberge, B., see Puzic, A. 283–285 Van Waeyenberge, B., see Stoll, H. 283 van Wees, B.J., see Costache, M.V. 216 van Wees, B.J., see Grollier, J. 285 van Wees, B.J., see Wang, X. 114 Vandamme, L.K.J., see Gijs, M.A.M. 357, 371 Vasquez, D.J. 389 Vasquez Mansilla, M., see Winkler, E. 266 Vassallo Brigneti, E. 306 Vassallo Brigneti, E., see Ramos, C.A. 315, 320–321 Vavassori, P., see Gubbiotti, G. 270 Vavassori, P., see Novosad, V. 172 Vaz, C.A.F. 181, 322 Vazquez, M., see Kraus, L. 356 Vazquez, M., see Pfutzner, H. 398 Vcelak, J. 366 Vcelak, J., see Kubı´k, J. 352, 409 Vcelak, J., see Ripka, P. 409 Vcela´k, J. 408 Verdes, C.G. 153–154 Verdes, C.G., see Dumitru, I. 155–157, 320 Vertesy, G. 412 Vertesy, G., see Pavo, J. 375 Verweerd, A., see Zimmermann, E. 362, 391 Viala, B., see Joisten, H. 379, 390 Vidal, D., see Cardoso, F.A. 371 Viegas, A.D.C. 255 Vieira, S., see Suderow, H. 44–45 Viennois, R. 15–16, 21, 23–24, 64, 95, 98 Vieux-Rochaz, L., see Pannetier-Lecoeur, M. 368 Vila, L., see Darques, M. 312 Vila, L., see Encinas, A. 311, 316 Vincent, F., see Drljaca, P.M. 378, 390 Viret, M., see Gregg, J.F. 202 Vittoria, C. 145 Vittoria, C., see Geiler, A.L. 372

449

Author Index

Vojkuvka, L., see Marsal, L.F. 303, 306 Vollmer, R. 28, 38, 44, 101 von Kim, J., see Mistral, Q. 214–216 von Kluge, J., see Umbach, F. 400 von Zeppelin, M., see Reininger, T. 392–394 Vonsovskii, S.V. 113, 117, 145, 257 Vopa´lensky´, M. 366–367, 391 Vovk, A. 303, 307 Vovk, A.Y., see Malkinski, L.M. 277–278 Vukadinovic, N., see Boust, F. 173–175 Va´vra, I., see Majchra´k, P. 128, 236–237 Va´zquez, M. 373, 389 Va´zquez, M., see Knobel, K. 356, 385, 387 Va´zquez, M., see Ramos, C.A. 315, 320–321 Va´zquez, M., see Vassallo Brigneti, E. 306 Wache, G., see Butin, L. 412 Wada, N., see Sugawara, H. 18, 99 Wada, S., see Masaki, S. 52, 58, 101 Wada, S., see Yamamoto, A. 69, 106 Wakeshima, M., see Matsuhira, K. 60, 65, 94, 99, 103 Wakeshima, M., see Sekine, C. 7, 21, 23, 65, 67, 97, 105 Walker, H.C., see Adroja, T.D. 21, 25 Walker, I.R., see Mathur, N.D. 46 Walz, U., see Gru¨nberg, P. 112 Wang, C., see Cui, Y.-T. 214 Wang, D., see Tondra, M. 371 Wang, P., see Petit, S. 214 Wang, R.F. 114 Wang, X. 114 Wang, X.-Y., see Matsunami, M. 11, 21, 25 Watanabe, H., see Yanagisawa, T. 47 Watanabe, I., see Hachitani, K. 56–58, 63 Watanabe, S., see Matsunami, M. 11, 21, 25 Watanabe, S., see Tsuda, S. 94 Watanabe, Y., see Iwasa, K. 10, 42 Watanuki, F., see Matsuda, T.D. 32, 50 Watkins, B., see Jung, S. 168, 324 Watkins, D.B., see Xu, W. 322–323 Watts, S.M., see Costache, M.V. 216 Wawryk, R. 47, 95, 100 Wawryk, R., see Baumbach, R.E. 7 Wawryk, R., see Henkie, Z. 6, 23, 48–49, 95 Wawryk, R., see Maple, M.B. 21, 23, 48, 97, 100 Wawryk, R., see Sayles, T.A. 27, 48, 100 Weaver, T., see Noginova, N. 238, 243, 245–248 Weertman, J.R., see Rado, G.T. 118–119, 130 Wegrowe, J.E. 202–203, 211 Weickert, F., see Oeschler, N. 39 Welker, F., see Reininger, T. 392–394 Wenger, M., see Kuanr, B.K. 277, 279–280 Wentao, Xu., see Rivkin, K. 168–169 Wernsdorfer, W., see Jamet, M. 129–130 Wernsdorfer, W., see Thirion, C. 151

Wernsdorfer, W., see Vaz, C.A.F. 181, 322 Westphalen, A., see Remhof, A. 114 White, W.T., see Xue, J.S. 22 Whittenburg, S., see Malkinski, L.M. 277–278 Whittenburg, S., see Vovk, A. 303, 307 Whittenburg, S., see Yu, M. 303, 305–306 Wickenden, D.K. 389 Wiedwald, U. 258 Wiedwald, U., see Spasova, M. 257, 259 Wiese, N., see Uhrmann, T. 398 Wiesinger, G., see Leithe-Jasper, A. 69 Wigen, P.E., see Ebels, U. 312, 316–317 Wigen, P.E., see Kakazei, G.N. 128, 233, 235–236, 270–274, 277, 292 Wigen, P.E., see Midzor, M.M. 114 Wilkins, J.W., see Langreth, D.C. 193 Wilks, R., see Hicken, R.J. 114, 117 Winkler, E. 266 Wisniowski, P., see Ferreira, R. 371 Wohlfarth, E.P., see Chantrell, R.W. 220 Wohlfarth, E.P., see Rhodes, P. 16 Wohlfarth, E.P., see Stoner, E.C. 148 Wolf, W.P., see Lea, K.R. 13 Woltersdorf, G., see Keupper, K. 268 Woltersdorf, G., see Urban, R. 146, 190 Woodward, F.M., see Butch, N.P. 6, 24, 28, 49, 100 Woodward, F.M., see Goremychkin, E.A. 28, 38, 43 Woodward, F.M., see Maple, M.B. 44 Woytasik, M. 379 Wu, J. 114 Wu, J., see Hicken, R.J. 114, 117 Wu, J., see Lepadatu, S. 279, 281 Wu, P.M. 381 Wulfhekel, W., see Hertel, R. 140, 142 Wyder, P., see Tsoi, M. 187, 201 Wyder, P., see Van Son, P.C. 202 Xi, H. 197, 199–200 Xiao, J. 217 Xiao, J.Q., see Moriyama, T. 216 Xiao, J.Q., see Tserkovnyak, Y. 217 Xu, W. 322–323 Xu, Y., see Zhao, Z. 405 Xu, Y.B., see Lepadatu, S. 279, 281 Xu, Y.X., see Zhai, Y. 277 Xuan, G., see Moriyama, T. 216 Xue, J.S. 22 Yabashi, M., see Matsunami, M. 22 Yabashi, M., see Yamasaki, A. 61 Yabukami, S., see Hashi, S. 407, 411 Yaegashi, T., see Fujino, T. 67 Yagi, T., see Kihou, K. 64, 66, 68, 78, 93, 103–106 Yagi, T., see Namiki, T. 27, 43, 49, 95, 100

450 Yagi, T., see Sekine, C. 18, 27, 34–35, 47, 99, 105 Yagi, T., see Shirotani, I. 4, 6, 18–19, 27, 43, 49, 66, 68–69, 78, 93–95, 100, 105–106 Yagi, T., see Uchiumi, T. 18, 94–96 Yalc- ın, O. 318 Yamada, A., see Sakakibara, T. 45 Yamada, T. 16, 49, 63, 95 Yamaguchi, H., see Lee, C.H. 36 Yamaguchi, T., see Goto, T. 18, 46 Yamaguchi, Y., see Indoh, K. 64 Yamamoto, A. 69, 106 Yamamoto, A., see Lee, C.H. 10, 35, 42 Yamamoto, E., see Aoki, D. 72, 107 Yamamoto, E., see Aoki, Y. 58 Yamamoto, E., see Matsuda, T.D. 69–71, 106 Yamamoto, K., see Matsunami, M. 11, 21–22, 25 Yamamoto, T., see Sugiyama, K. 99 Yamamura, T. 350 Yamasaki, A. 61 Yamasaki, A., see Imada, S. 53 Yanagisawa, T. 47 Yanagisawa, T., see Goto, T. 18, 46 Yanagisawa, T., see Ho, P.-C. 44 Yanagisawa, T., see Maple, M.B. 21, 23, 29, 47–48, 97, 100 Yanagisawa, T., see Sayles, T.A. 27, 48, 100 Yanagisawa, T., see Wawryk, R. 47, 95, 100 Yanagisawa, T., see Yuhasz, M.W. 27, 47–48, 99–100 Yanase, A., see Takegahara, K. 12 Yang, C. 25 Yang, C., see Iwasa, K. 25 Yang, F.H., see Respaud, M. 252 Yang, K.N., see Meisner, G.P. 19, 21–23, 69, 96, 106 Yang, S., see Li, T. 318–319 Yano, K., see Sakakibara, T. 45 Yano, M., see Imada, S. 53 Yao, Y.S., see Guertin, R.P. 106 Yasumoto, Y., see Yanagisawa, T. 47 Yasuoka, H., see Tokunaga, Y. 107 Yatskar, A. 26 Ye, F., see Chi, S. 47, 100 Yefanov, V.S., see Konchits, A.A. 260, 262–264 Yelon, A., see Vittoria, C. 145 Yi, B. 405 Yi, J.B. 351 Yıldız, F., see Yalc- ın, O. 318 Yogi, M. 21, 24–25, 44, 46, 50, 96 Yogi, M., see Ishida, K. 33 Yogi, M., see Kotegawa, H. 44–45 Yogi, M., see Tou, H. 38 Yokoya, M., see Giri, R. 63 Yokoya, T., see Kanai, K. 24 Yokoya, T., see Tsuda, S. 94 Yokoyama, M., see Sakakibara, T. 45

Author Index

Yokoyama, M., see Sekine, C. 34, 64–65, 67 Yonezawa, Y., see Aoki, Y. 62 Yonezawa, Y., see Shu, L. 46 Yonezawa, Y., see Tanaka, K. 7, 49, 95, 100 Yoon, S.H. 397 Yoon, S.S., see Kollu, P. 381 Yoon, S.S., see Rheema, Y.W. 405 Yoshida, T., see Sekine, C. 7, 21, 23, 97 Yoshii, S. 74, 76, 108–109 Yoshimitsu, Y., see Hachitani, K. 56–57 Yoshimura, K., see Tsujii, N. 60 Yoshino, G., see Ochiai, A. 44 Yoshizawa, M. 58, 68, 103 Yoshizawa, M., see Fujino, T. 67 Yoshizawa, M., see Kumagai, T. 50 Yoshizawa, M., see Nakanishi, Y. 21, 26, 62, 101 Young, B.-L., see Ishida, K. 33 Young, B.-L., see MacLaughlin, D.E. 44–45 Yu, C. 290–295, 301 Yu, C., see Pechan, M. 267–268, 292, 297–299, 301, 303 Yu, M. 303, 305–306 Yu, M., see Kuanr, B.K. 277, 279–280 Yu, M., see Malkinski, L.M. 277–278 Yu, S.-C., see Kim, Y.-S. 379 Yuan, J.L., see Na, K.W. 380 Yuasa, S., see Sugawara, H. 22 Yudanov, V.F., see Martyanov, O.N. 295, 299–300 Yuhasz, M.W. 27, 47–48, 99–100 Yuhasz, W.M. 60, 62–63, 104 Yuhasz, W.M., see Bauer, E.D. 65, 104–105 Yuhasz, W.M., see Butch, N.P. 6, 24, 28, 49, 100 Yuhasz, W.M., see Cao, D. 36 Yuhasz, W.M., see Cichorek, T. 41 Yuhasz, W.M., see Goremychkin, E.A. 28, 38, 43 Yuhasz, W.M., see Henkie, Z. 6, 23, 48–49, 95 Yuhasz, W.M., see Ho, P.-C. 44, 50, 53–54, 103 Yuhasz, W.M., see Maple, M.B. 21, 23, 39, 42, 44, 47–48, 50, 96–97, 100 Yuhasz, W.M., see Sayles, T.A. 27, 48, 100 Yuhasz, W.M., see Shu, L. 46 Yuhasz, W.M., see Wawryk, R. 47, 95, 100 Yuhasz, W.M., see Yanagisawa, T. 47 Yuhasza, W.M., see Maple, M.B. 29 Yurkov, G.Yu., see Koksharov, Yu.A. 247 Zabel, H., see Remhof, A. 114 Zane, S., see Ibrahim, A.I. 413 Zangwill, A., see Stiles, M.D. 198 Zapf, V., see Vollmer, R. 28, 38, 44, 101 Zapf, V.S., see Bauer, E.D. 18–19, 24, 28, 38, 43–44, 52, 95, 98, 100–102 Zapf, V.S., see Frederick, N.A. 44 Zapf, V.S., see Ho, P.-C. 44

Author Index

Zapf, V.S., see Maple, M.B. 39, 42, 44, 96 Zawadowski, A., see Cox, D.L. 26 Zerec, I., see Oeschler, N. 39 Zhai, H.R., see Zhai, Y. 277 Zhai, Y. 277 Zhang, C.Q., see Tsuda, S. 94 Zhang, S. 196–197 Zhang, S., see Levy, P.M. 202 Zhang, S.-C., see Bazaliy, Ya.B. 202 Zhang, X.Y., see Zhai, Y. 277 Zhao, Z. 405 Zhao, Z.J. 381 Zhen, G.-q., see Masaki, S. 58 Zheng, G.-q., see Kotegawa, H. 44–45 Zheng, G.-q., see Yogi, M. 21, 25, 44, 50, 96 Zheng, M., see Li, Y. 303 Zheng, X.G., see Butch, N.P. 6, 24, 28, 49, 100 Zhou, J., see Skomski, R. 146–147 Zhou, J.N., see Butera, A. 228

451 Zhou, W., see Malkinski, L.M. 277–278 Zhou, W., see Yu, M. 303, 305–306 Zhu, X. 324–327 Zhukov, A., see Chizhik, A. 373 Zhukov, A., see Garcıá, C. 388 Zhukova, V., see Garcıá, C. 388 Zilberman, P.E., see Heide, C. 192, 196 Zimmerman, J.E., see Campbell, W.C. 406 Zimmermann, E. 362, 391 Zogal, O., see Wawryk, R. 47, 95, 100 Zohar, S. 251, 253 Zorlu, O. 381 Zou, J., see McGary, P.D. 398 Zou, X., see Lepadatu, S. 279, 281 Zysler, R.D., see De Biasi, E. 223, 225–227, 263 Zysler, R.D., see Ramos, C.A. 320–321 Zysler, R.D., see Sańchez, R.D. 252–253 Zysler, R.D., see Winkler, E. 266

SUBJECT INDEX AFQ ordering, 35 alkaline or alkaline earth-based antimonides, 92 AMR current sensors, 348, 404 antidot arrays, 289 anti-quadrupole ordering, 12 antitheft systems, 348, 409 atomic magnetometer, 384 band-structure calculations for La-filled skutterudites, 9 barber pole, 364 contactless electric-current sensors, 401 contactless potentiometers, 394 crossfield effect, 352 crossfield error, 365 crystalline electric field, 13 current clamps, 406 current-driven magnetisation switching, 196 current-induced magnetisation dynamics, 201 DC current comparators, 403 dipole-exchange spin-wave dispersion, 133 eddy current sensors, 392 electronic magnetic compasses, 408 flipping, 365 flux concentrators, 359 fluxgate current sensor, 403 effect, 381 magnetometer, 376 sensors, 347, 371, 376 fluxgates, 350 FMR, 113 in granular solids, 153 in nanoparticle assemblies, 220 of an interacting fine particle system, 155 of Ni nanowires, 312 studies of FM ring samples, 322 technique, 113 force sensors, 348, 398 Ge-based filled skutterudites, 92 giant magnetoresistor, 350 Gilbert and Landau–Lifshitz damping parameters, 144 Gilbert damping parameter, 117

GMI current sensor, 405 magnetometer, 387 sensors, 387 GMR sensors, 371 Hall current sensors, 403 Hall sensors, 358 heavy-fermion ferromagnet, 56 heavy-fermion superconductivity, 37 hexacontatetrapole, 12 hexadecapole, 12 Hoffman boundary conditions, 131 induced anisotropy, 353 induction magnetometers, 385 inductosyn, 392 instrument current transformers, 348, 402 inverse Wiedemann effect, 356 Landau–Lifshitz (LL) equation, 116 LLG equation, 144, 156 Lorentz force magnetic sensor, 389 magnetic beads, 412 magnetic dipolar interaction, 126 magnetic encoders, 395 magnetic feedback, 366 magnetic flowmeters, 397 magnetic nanowires, 308 magnetic position sensors, 407 magnetic position trackers, 408 magnetically robust HF state, 60 magnetooptical current sensors, 406 magnetooptical sensors, 389 magnetostrictive magnetometers, 388 magnetostrictive position sensors, 348, 393 materials for magnetic sensors, 349 medical distance and position sensors, 348, 411 metal–insulator transition, 34, 58 miniaturized fluxgate sensors, 378 nanodots, 267 nanometric systems, 111, 116 nanorings, 322 nanostructured arrays, 111, 164 nesting property of the FS, 35 noise of magnetic sensors, 356

453

454 nondestructive testing, 348, 412 non-equilibrium exchange interaction (NEXI), 192 optically pumped resonance magnetometers, 384 orthogonal, 381 overhauser magnetometers, 383 permeability, 354 perming, 351 perpendicular standing spin-wave modes, 135 position measurement, 407 position sensor with permanent magnets, 348, 393 proton magnetometers, 383 quadrupolar excitons, 43 quadrupolar ordering, 37 racetrack core, 373 Rado–Weertman boundary condition, 131 random anisotropy model, 121 random nanoparticle assemblies, 111, 227 rare earth or actinide-based pnictides, 92 reed contacts, 395 resonance line width broadening, 147 ring core, 373 Rogowski coil, 402 semiconductor magnetoresistors, 360 sensors, 362 shape anisotropy, 354 simulations of magnetodynamics in nanometric systems, 111, 148

Subject Index

size dispersion, 114 speed of magnetisation reversal, 192 spin dynamics in nanometric systems, 111, 227 spin-current-induced dynamics in magnetic nanostructures, 111, 186 spin-dependent tunneling (SDT) sensors, 362 spin-torque transfer effect, 187 square and rectangular dot arrays, 276 SQUID magnetometers, 382 standing spin-wave modes, 111, 132 STD sensors, 371 superparamagnetic effects, 111, 218 surface anisotropy, 111, 129 susceptibility tensor, 356 switching time, 192 synchros, 392 synthesis of filled skutterudites, 1, 3 synthesis under high pressures, 1, 6 temperature coefficient of offset, 358 temperature coefficient of sensitivity, 358 temperature effects in FMR, 156 temperature stability of the sensor, 357 torque sensors, 398 transverse anisotropy model, 129 unconventional superconducting properties, 43 vacquier sensor, 372 variable reluctance sensors, 392 wiegand wires, 397

MATERIALS INDEX g-Fe2O3 nanoparticle system, 238 Al/AlOx/NiFe/Cu tunnel junction, 217 AlOx/NiFe, 216 APt4Ge12 (A=Sr, Ba, La, Ce, Pr, Nd, Eu, U, Th), 75 AT4Sb12 (T=Ru, Os; A=Sr, Ba), 74 BaFe4Sb12, 74 BaPt4Ge12, 75 BaPt4 xAuxGe12, 76 CaFe4Sb12, 74 Ce-filled skutterudites, 1, 19 CeFe4As12, 23 CeFe4P12, 22 CeFe4Sb12, 23 CeOs4As12, 23 CeOs4P12, 22 CeOs4Sb12, 24 CePt4Ge12, 77 CeRu4As12, 7, 23 CeRu4P12, 22 CeRu4Sb12, 5, 24 CoxCu100 x granular co-deposited films, 255 CoFeB, 213 Co/SiO2, 236 DyFe4P12, 68 Eu-based filled skutterudite, 1, 64 ErFe4P12, 68 EuFe4P12, 64 EuFe4Sb12, 65 EuOs4P12, 64 EuOs4Sb12, 65 EuPt4Ge12, 78 EuRu4P12, 64 EuRu4Sb12, 65 filled skutterudites, 2 Filled skutterudites with actinoid ions, 1, 69 Gd-based filled skutterudites, 1, 65 GdFe4P12, 65 GdOs4P12, 66 GdRu4P12, 65

HoFe4P12, 68 HoOs4P12, 68 I0.9Rh4Sb12, 78 KFe4Sb12, 73 LaFe4As12, 52 LaFe4P12, 4, 32 La-filled skutterudites, 1, 15 LaOs4Sb12, 38, 44 LaxRh4P12, 19 LaFe4As12, 16 LaFe4P12, 16 LaFe4Sb12, 16 LaOs4Sb12, 18 LaPt4Ge12, 77 LaRu4As12, 18 LaRu4P12, 18 LaRu4Sb12, 18 NaFe4Sb12, 73 nanopillars of Co/Cu/NiFe, 209 Nd-filled skutterudites, 1, 50 NdFe4As12, 52 NdFe4P12, 51 NdFe4Sb12, 52 NdOs4As12, 52 NdOs4P12, 52 NdOs4Sb12, 50, 53 NdPt4Ge12, 77 NdRu4As12, 52 NdRu4P12, 52 NdRu4Sb12, 50, 52 Ni nanoparticles in a single-walled carbon nanotube, 260 NpFe4P12, 71 Permalloy nanoring structures, 326 Pr-based filled skutterudites, 1, 26 (Pr1 xLax)Os4Sb12, 44 (Pr1 yLay)Os4Sb12, 46 PrFe4As12, 48 PrFe4P12, 29 PrFe4P12, PrRu4P12, PrOs4As12 and PrOs4Sb12, 29 PrFe4Sb12, 49 PrInAg2, 26 Pr(Os1 xRux)4Sb12, 46

455

456 PrOs4As12, 47 PrOs4Sb12, 37–38, 44 PrOs4P12, 47 PrPt4Ge12, 77 PrRu4As12, 49 PrRu4P12, 34 PrRu4Sb12, 50 RFe4As12 (R=La, Ce, Pr), 8 RFe4Sb12, 5 ROs4Sb12, 5 RRu4Sb12, 5 skutterudites, 2 Sm-based filled skutterudites, 1, 54 SmFe4As12, 63 SmFe4P12, 56 SmFe4Sb12, 63 SmOs4P12, 63 SmOs4Sb12, 56, 58 SmRu4P12, 58 SmRu4Sb12, 64

Materials Index

SmxLa1 xFe4P12, 57 SrPt4Ge12, 75 [Ta(2.5 nm)/Cu(50 nm)/Co90Fe10(20 nm)/ Cu(5 nm)/Ni80Fe20(5 nm)/Cu(1.5 nm)/ Au(2 nm)], 210 Tb-, Dy-, Ho-, Er-, and Tm-based filled skutterudites, 1, 66 TbFe4P12, 66 TbOs4P12, 68 ThFe4P12, 70 ThPt4Ge12, 78 TmFe4P12, 68 UFe4P12, 69 UPt4Ge12, 78 Yb-based filled skutterudites, 1, 68 YbFe4P12, 68 YbFe4Sb12, 68 YFe4P12, YRu4P12, 78 YOs4P12, 78

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