Advances in Nanoscale Magnetism: Proceedings of the International Conference on Nanoscale Magnetism ICNM-2007, June 25 -29, Istanbul, Turkey

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springer proceedings in physics 105 Computer Simulation Studies in Condensed-Matter Physics XVIII Editors: D.P. Landau, S.P. Lewis, and H.-B. Sch¨uttler 106 Modern Trends in Geomechanics Editors: W. Wu and H.S. Yu 107 Microscopy of Semiconducting Materials Proceedings of the 14th Conference, April 11–14, 2005, Oxford, UK Editors: A.G. Cullis and J.L. Hutchison 108 Hadron Collider Physics 2005 Proceedings of the 1st Hadron Collider Physics Symposium, Les Diablerets, Switzerland, July 4–9, 2005 Editors: M. Campanelli, A. Clark, and X. Wu 109 Progress in Turbulence II Proceedings of the iTi Conference in Turbulence 2005 Editors: M. Oberlack, G. Khujadze, S. Guenther, T. Weller, M. Frewer, J. Peinke, S. Barth 110 Nonequilibrium Carrier Dynamics in Semiconductors Proceedings of the 14th International Conference, July 25–29, 2005, Chicago, USA Editors: M. Saraniti, U. Ravaioli 111 Vibration Problems ICOVP 2005 Editors: E. Inan, A. Kiris 112 Experimental Unsaturated Soil Mechanics Editor: T. Schanz 113 Theoretical and Numerical Unsaturated Soil Mechanics Editor: T. Schanz 114 Advances in Medical Engineering Editor: T.M. Buzug

115 X-Ray Lasers 2006 Proceedings of the 10th International Conference, August 20–25, 2006, Berlin, Germany Editors: P.V. Nickles, K.A. Janulewicz 116 Lasers in the Conservation of Artworks LACONA VI Proceedings, Vienna, Austria, Sept. 21–25, 2005 Editors: J. Nimmrichter, W. Kautek, M. Schreiner 117 Advances in Turbulence XI Proceedings of the 11th EUROMECH European Turbulence Conference, June 25–28, 2007, Porto, Portugal Editors: J.M.L.M. Palma and A. Silva Lopes 118 The Standard Model and Beyond Proceedings of the 2nd International Summer School in High Energy Physics, M¯gla, 25–30 September 2006 Editors: M. Serin, T. Aliev, N.K. Pak 119 Narrow Gap Semiconductors 2007 Proceedings of the 13th International Conference, 8–12 July, 2007, Guildford, UK Editors: B. Murdin, S. Clowes 120 Microscopy of Semiconducting Materials 2007 Proceedings of the 15th Conference, 2–5 April 2007, Cambridge, UK Editors: A.G. Cullis, P.A. Midgley 121 Time Domain Methods in Electrodynamics A Tribute to Wolfgang J. R. Hoefer Editors: P. Russer, U. Siart 122 Advances in Nanoscale Magnetism Proceedings of the International Conference on Nanoscale Magnetism ICNM-2007, June 25–29, Istanbul, Turkey Editors: B. Aktas, F. Mikailov

Volumes 80–104 are listed at the end of the book.

Bekir Aktas Faik Mikailov Editors

Advances in Nanoscale Magnetism Proceedings of the International Conference on Nanoscale Magnetism ICNM-2007 June 25–29, Istanbul, Turkey

With 203 Figures

123

Professor Dr. Bekir Aktas Dr. Faik Mikailov Gebze Institute of Technology, Department of Physics P.O.Box 141, 41400 Gebze-Kocaeli, Turkey E-mail: [email protected], [email protected]

Springer Proceddings in Physics ISBN 978-3-540-69881-4

ISSN 0930-8989 e-ISBN 978-3-540-69882-1

Library of Congress Control Number: 2008930848 © Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Prodcution: SPi Publisher Services Cover design: eStudio Calamar Steinen SPIN: 12250813 57/3180/SPi Printed on acid-free paper 987654321 springer.com

Preface

Intensive investigations on nanoscale magnetism have promoted remarkable progress in technological applications of magnetism in various areas. The technical progress of recent years in the preparations of multilayer thin films and nanowires led to the discovery of Giant Magnetoresistance (GMR), implying an extraordinary change in the resistivity of the material by varying the applied external magnetic field. The Nobel Prize for Physics in 2007 was awarded to Albert Fert and Peter Gr¨ unberg for their discovery of GMR. Applications of this phenomenon have revolutionized techniques for retrieving data from hard disks. The discovery also plays a major role in various magnetic sensors as well as the development of a new generation of electronics. The use of GMR can be regarded as one of the first major applications of nanotechnology. The GMR materials have already found applications as sensors of low magnetic field, a key component of computer hard disk heads, magnetoresistive RAM chips etc. The “read” heads for magnetic hard disk drives have allowed us to increase the storage density on a disk drive from 1 to 20 Gbit per square inch, merely by the incorporation of the new GMR materials. On the other hand, recently discovered giant magneto-impedance (GMI) materials look very promising in the development of a new generation of microwave band electronic devices (such as switches, attenuators, and antennas) which could be managed electrically. Magnetic data storage is one of the most promising applications of nanomagnetism. The physical size of the recording bits of hard disk drives is already in the nanometer regime, and continues decreasing due to the ever increasing demand for higher recording densities. Soon the dimension of the recording bit will reach the sub-10 nm regime. At this level, both the writing and reading processes will become extremely challenging, if not impossible. This is because the sensor must be made smaller or at least comparable to the bit size, and at the same time, its sensitivity must be improved continuously so as to compensate the loss in signal-to-noise ratio due to the decrease in the bit size. The former has to rely heavily on the advance of nanotechnology, and the latter on an emerging field called spintronics. The combination of these

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two fields has played an important role in advancing the areal density of magnetic recording from a few gigabits/in2 to the current level of more than 100 Gbits/in2 . In addition to hard disk drives, the developed technologies have also been applied to magnetic random access memories (MRAMs). Further advance in these fields is the key to realizing terabits/in2 hard disk drives and gigabit nonvolatile memories within this decade. Spintronics in its broader sense contains all types of electronics that make use of both charges and spins. It is a new and rapidly developing branch of commercial electronics, which utilizes spin degrees of freedom of the carriers to control an electric current. One may say that spintronics accumulates or synthesizes the frontier knowledge of the physics of spin and magnetism, electronics and optics, putting them together in the nanoscale range and realizing them in new multifunctional devices. The first generation of devices combines standard microelectronics with spin-dependent effects that arise from the interaction between the spin of the carrier and the magnetic properties of the material. Therefore, by controlling the spin alignment one can add an extra parameter to adjust the device characteristics. The magneto-optical effect is another important phenomenon for spin injection, detection and manipulation to bring electron-spin and photon in a single component of an electronic device. The ultimate goal is to use the collaborative effects of the magnetic, electric and optical nature to improve the performance of the devices. The magnetoresistive read heads for the computer hard disk drives give us the first example of successful long-term innovation, which resulted in total reorientation of the computer hard-disk industry on the magnetoresistive read heads. These metal-based spintronic devices are based primarily on the spatial modulation of electron spins through using layered structures of magnetic and non-magnetic materials. The lack of capability in charge modulation in these types of structures may eventually limit their ultimate performances in terms of both the magnetoresistance and other functionalities. To address this issue, recently, a great deal of effort has been devoted to the developments of magnetic semiconductors which allow the modulation of both the spins and charges. The advances in this field may eventually lead to spintronic devices with performances superior to their metal-based counterparts. In addition to pure metal-based or semiconductor-based spintronic devices, hybrid devices making use of both technologies also have been explored actively in recent years. The magnetic semiconductors are usually made by adding magnetic impurities to host semiconductors. It is not sufficient, however, that every semiconductor can be made magnetic using this approach because some of them still do not exhibit any magnetic properties even after they are doped with a substantial amount of magnetic impurities. Some of them, though magnetic, show a very low Curie temperature. However, the situation has changed drastically in recent years due to the intensive efforts made by researchers in this field in many research organizations. Several different types of magnetic semi-

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conductors having a Curie temperature higher than room temperature have already been found. It should be noted, however, that all these are based on preliminary experimental results; further experiments are required to verify the results. The progress was made not only in materials themselves, but also in the applications of these materials in creating new functional devices such as semiconductor-based magnetic tunnel junctions and spin-injection devices. Although the current technology for a read sensor is based on metallic spintronics, semiconductor-based spintronics has the potential to provide sensors or storage elements with superior performances for next-generation data storage devices. Although the read sensor for magnetic recording and storage cells for magnetic memory are based on spintronics, the storage of information in disk media is still based on classic physics, and it does not involve spintronics. However, as the bit size continues decreasing, it will approach atomic size in the near future. At this stage, a fundamental change in the information storage principle will be required. One of the possible scenarios is to store information in the reciprocal space or energy domain. In this case, the spin of electrons and nuclei instead of the magnetization of magnetic grains will play an important role. This is closely related to another emerging field called quantum information storage. At the present time the small-scale magnetic materials are used both in new recording media and, in particular, in the most critical elements of recording systems, in recording heads. Magnetic layers serving as recording medium in a storage media are becoming increasingly complex with increasing areal density of recording. The size of a single recorded bit in most recent hard disks is now in nanometric scale. Magnetic recording density is increasing close to 100 per cent every year. So, recording technology has already demonstrated recording density over 100 Gbit per square inch at the present time. However, even if all the technological problems could be solved there is still a theoretical limit for recording density. With the decreasing size of recorded bits, these magnetic particles have already been entering the superparamagnetic regime. The thermal agitation energy becomes comparable to the effective magnetic anisotropy (including the shape anisotropy) energy of a single particle. Then, magnetic moments start to flip within a finite time and therefore the recorded data are partly washed out. The stability of the bit exponentially depends both on the barrier energy (anisotropy energy) and on the particle volume. The superparamagnetic limit for some typical materials is of the order of 10 nm, which corresponds to a recording density of the order of Terabits per square inch. The important parameters are anisotropy, coercivity, size, preferential orientation and the density of magnetic particles. The density can be improved further by using perpendicular recording. In this case very thin and long enough magnetic wires are ordered perpendicular to the surface of the media. Even an individual wire or particle can be used as a bit unit in the near future. Production of magnetic fine particles with uniform physical properties can be achieved by some lithography techniques. Dimensions of

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magnetic components used as sensor and writer elements in recording heads are in nanometric scale and smaller than the limit of optical lithography. Thus, the sensitivity and the resolution of reading are going to increase drastically as the size of the particles is decreased down to nanometer scale. The sensitivity of magneto-restive device is defined as relative change of resistivity per Oersted. In a recording head, magneto-resistive elements based on GMR or spin tunneling effect are used. By using the spin tunneling effect, sensitivity can be an order of magnitude higher compared to conventional magneto-resistive Ni-Fe sensor. High sensitivity has been achieved by GMR materials. In order to enhance the sensitivity further, the current perpendicular to the surface mode is going to be implemented. Of late, magnetic and semiconducting elements are being combined into a chip (MRAM) in order to avoid the mechanical part from our lives (Speed of data transfer in hard disks is now close to 1 Gbit/s). Moreover, logic devices based on nano-magnetism are under investigation. Magnetic transistors have recently been demonstrated. The basic principles of this device depend on the injection of spin-polarized current between magnetic-nonmagnetic layers. Also, some researchers are trying to use magneto-optic, magneto-elastic, and other material properties to develop novel devices. This book is intended to provide a review of the latest developments and the fundamental concepts in the field of nanomagnetism, with an emphasis on the research and application of nanoscaled magnetic materials in high-density magnetic data storage and MRAMs and recent achievements in the emerging field of spintronics. The idea for this book was born at the Fourth International Conference on Nanoscale Magnetism (ICNM-2007) held in Istanbul (Turkey) from June 25-29, 2007. The contributions are focused on the magnetic properties of nanoscale magnetic materials, especially on fabrication and characterization as well as the physics behind the behavior of these structures. We would like to thank all the authors for their contributions. We also acknowledge the great efforts of Gebze Intitute of Technology (GIT), Istanbul Technical University (ITU) and The Scientific and Technological Research Council of Turkey (TUBITAK), and others who made major contributions to the organization of the ICNM-2007 Conference and made this publication possible. Gebze Institute of Technology, Turkey, June 2008

Bekir Akta¸s Faik Mikailov

Contents

1 Role of Defects and Disorder in the Half-Metallic Full-Heusler Compounds ¨ gan, and E. S I. Galanakis, K. Ozdo˜ ¸ a¸sıo˜glu . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Defects in Full-Heuslers Containing Co and Mn . . . . . . . . . . . . . . . . . 1.3 Defects Driven Half-Metallic Ferrimagnetism . . . . . . . . . . . . . . . . . . . 1.4 A Possible Route to Half-Metallic Antiferromagnetism . . . . . . . . . . . 1.5 Vacancies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 7 11 13 14 16

2 Clustering in Heusler Alloys N. Lakshmi, V. Sebastian, and K. Venugopalan . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 X-Ray Diffraction Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 M¨ ossbauer Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 DC Magnetization Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 24 24 24 28 31 34

3 Anisotropy of Ferromagnetic Heusler Alloys Thin Films R. Yilgin and B. Aktas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 TMR and GMR Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Critical Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Theory of Ferromagnetic Resonance . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Dynamics of Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Resonance Field for Polycrystalline Film . . . . . . . . . . . . . . . .

37 37 38 39 41 42 42 44

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3.2.3 Ferromagnetic Resonance in Single Crystalline Film . . . . . . 3.2.4 Line-Width of Resonance Absorption . . . . . . . . . . . . . . . . . . . 3.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Magnetic Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 FMR Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 47 50 50 53 55 62 64

4 Quantum Monte Carlo Study of Anderson Magnetic Impurities in Semiconductors N. Bulut, Y. Tomoda, K. Tanikawa, S. Takahashi, and S. Maekawa . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Two-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Magnetic Correlations Between the Impurities . . . . . . . . . . . 4.3.2 Impurity–Host Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Three-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Tight-Binding Model for a Mn d Orbital in GaAs . . . . . . . . . . . . . . . 4.6 Discussion and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 69 71 71 76 79 82 84 86

5 New Type of Nanomaterials: Doped Magnetic Semiconductors Contained Ferrons, Antiferrons, and Afmons L.I. Koroleva and D.M. Zashchirinskii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2 Ferrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.2.1 Giant Red Shift of Fundamental Absorption Edge Connected with Ferromagnetic Ordering . . . . . . . . . . . . . . . . . 90 5.2.2 Notion of Ferrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.2.3 Electrical Resistivity and Magnetoresistance of Nondegenerate Ferromagnetic Semiconducrors with n-Type of Electrical Conductivity . . . . . . . . . . . . . . . . . . 94 5.2.4 Magnetic Two-Phase Ferromagnetic–Antiferromagnetic State in Manganites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.2.5 Antiferrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.2.6 Afmons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6 Cerium-Doped Yttrium Iron Garnet Thin Films Prepared by Sol-Gel Process: Synthesis, Characterization, and Magnetic Properties ¨ urk, I. Avgın, M. Erol, and E. C Y. Ozt¨ ¸ elik . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.2 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

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6.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7 Tuning the Magnetic and Electronic Properties of Manganite Thin Films by Epitaxial Strain G.H. Aydogdu, Y. Kuru, and H.-U. Habermeier . . . . . . . . . . . . . . . . . . . . . . 131 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.2 Preparation and Analysis of Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.2.1 Deposition Technique and Film Growth . . . . . . . . . . . . . . . . . 135 7.2.2 Analysis of Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.3 Structural Characterization, Electrical and Magnetic Properties of Manganites Film . . . . . . . . . . . . . . . . . . . 138 7.3.1 Structural Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.3.2 Magnetic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.3.3 Electrical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.4 General Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 8 Radiation Nanostructuring of Magnetic Crystals V.A. Ageev, V.I. Kirischuk, Yu.V. Koblyanskiy, G.A. Melkov, L.V. Sadovnikov, A.N. Slavin, N.V. Strilchuk, V.I.Vasyuchka, and V.A. Zheltonozhsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.2 The Influence of Inhomogeneities upon the Properties of Ferrites and Ferrite Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 8.3 Wave-Front Reversal in a Medium with Inhomogeneities . . . . . . . . . 151 8.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 8.5 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 9 Electromagnetic Radiation of Micro and Nanomagnetic Structures with Magnetic Reversal B.A. Gurovich, K.E. Prikhodko, E.A. Kuleshova, A.G. Domantovsky, K.I. Maslakov, and E.Z. Meilikhov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 9.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 9.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 9.3.1 Coercivity Static Measurements of Magnetic Patterned Media . . . . . . . . . . . . . . . . . . . . . . . . . . 170 9.3.2 The Dependence of the Amplitude and Duration of Emitted Signal on the External Magnetic Field Amplitude . . . . . . . . . . . . . . 173

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9.3.3

Experimental Determination of Dynamic Emitted Parameters of Magnetic Patterned Media . . . . . . . . . . . . . . . . . . . . . . . . . . 174 9.3.4 Dependence of Dynamical Coercivity of Bit Arrays with Underlayer on the Bits Structural Geometry . . . . . . . . . 176 9.3.5 Peculiarities of the Magnetic Structure of Cobalt Bits with and without a Soft Magnetic Underlayer . . . . . . . . . . . . 177 9.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 10 Structural and Magnetic Properties and Preparation Techniques of Nanosized M-type Hexaferrite Powders T. Koutzarova, S. Kolev, C. Ghelev, K. Grigorov, and I. Nedkov . . . . . . . 183 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 10.2 Crystalline Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 10.3 Magnetic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 10.4 Methods for Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 10.5 Microemulsion Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 11 Nanocrystallization and Surface Magnetic Structure of Ferromagnetic Ribbons and Microwires A. Chizhik, A. Zhukov, V. Zhukova, C. Garcia, J.M. Blanco, J.J. del Val, L. Fernandez, N. Iturriza, and J. Gonzalez . . . . . . . . . . . . . . 205 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 11.2 Co-Rich Ribbons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 11.2.1 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 11.2.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 11.3 Ni-Rich Ribbons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 11.3.1 XRD and AFM Structural Results . . . . . . . . . . . . . . . . . . . . . . 210 11.3.2 Magnetic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 11.4 Microwires with Novel Composition Cu70 (Co70 Fe5 Si10 B15 )30 . . . . . 212 11.4.1 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 11.4.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 12 On Structural and Magnetic Properties of Fe73.5−x Si13.5 B9 Cu1 Nb3 Mnx Metal Alloys R. Brzozowski, M. Wasiak, P. Sov´ ak, and M. Moneta . . . . . . . . . . . . . . . . 219 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 12.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 12.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 12.3.1 As-Quenched Fe73.5−x Si13.5 B9 Cu1 Nb3 Mnx . . . . . . . . . . . . . . 220 12.3.2 Annealed Fe73.5−x Si13.5 B9 Cu1 Nb3 Mnx . . . . . . . . . . . . . . . . . 224 12.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

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XIII

13 FeCoZr–Al2 O3 Granular Nanocomposite Films with Tailored Structural, Electric, Magnetotransport and Magnetic Properties J.A. Fedotova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 13.1.1 Granular Nanocomposites for Electronics: Reasons of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 13.1.2 Preparation and Structure of Granular MMCs . . . . . . . . . . . 233 13.1.3 Percolation in Granular Nanocomposites . . . . . . . . . . . . . . . . 235 13.1.4 Carrier Transport in Granular MMCs around Metal–Insulator Transition . . . . . . . . . . . . . . . . . . . . . . 237 13.1.5 Magnetic Properties of Granular Nanocomposites . . . . . . . . . 242 13.2 Properties of FeCoZr–Al2 O3 Nanocomposite Films: Synthesis in Pure Ar and Mixed Ar + O Ambient . . . . . . . . . . . . . . . 243 13.2.1 Synthesis and Samples Preparation . . . . . . . . . . . . . . . . . . . . . 243 13.2.2 M¨ ossbauer Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 13.2.3 Alternation Grads- and SQUID-Magnetometry . . . . . . . . . . . 246 13.2.4 Atomic Force–Magnetic Force Microscopy . . . . . . . . . . . . . . . 249 13.2.5 Electric and Magnetotransport Properties . . . . . . . . . . . . . . . 253 13.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 14 Ferromagnetism of Nanostructures Consisting of Ferromagnetic Granules with Dipolar Magnetic Interaction E. Meilikhov and R. Farzetdinova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 14.2 Lattices of Point-like and Rod-like Ferromagnetic Granules with Dipole Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 14.2.1 3D Lattice of Point-Like Granules . . . . . . . . . . . . . . . . . . . . . . 275 14.2.2 2D Lattice of Point-Like Granules . . . . . . . . . . . . . . . . . . . . . . 277 14.2.3 3D Lattice of Rod-Like Granules . . . . . . . . . . . . . . . . . . . . . . . 279 14.2.4 2D Lattice of Rod-Like Granules . . . . . . . . . . . . . . . . . . . . . . . 279 14.3 Lattices of Ellipsoidal Granules with Dipole Interaction . . . . . . . . . . 280 14.3.1 Magnetic Field of the Ellipsoidal Granule . . . . . . . . . . . . . . . . 280 14.3.2 3D Lattice of Ellipsoidal Granules . . . . . . . . . . . . . . . . . . . . . . 282 14.3.3 2D Lattice of Prolate Ellipsoidal Granules . . . . . . . . . . . . . . . 282 14.3.4 2D Lattice with Oblate Ellipsoidal Granules . . . . . . . . . . . . . 284 14.3.5 2D Lattice of Oblate Ellipsoidal Granules in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 14.4 Partially Populated Lattices of Point-Like Ising Dipoles . . . . . . . . . . 287 14.4.1 Distribution of Local Magnetic Fields . . . . . . . . . . . . . . . . . . . 288 14.4.2 Magnetic Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 14.5 Random Systems of Point-Like and Rod-Like Ising Dipoles . . . . . . . 295 14.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

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14.5.2 Generalized Mean Field Theory for Point (Spherical) Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 14.5.3 Generalized Mean Field Theory for Rod-Like Dipoles . . . . . 303 14.5.4 Magnetic Properties of a Random System of Rod-Like Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 14.6 Experimental Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 14.6.1 Magnetism of Ultrathin Films . . . . . . . . . . . . . . . . . . . . . . . . . . 311 14.6.2 2D Lattices of Disk-Shaped Granules in a Magnetic Field . . 314 14.6.3 Magnetic Recording Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 14.6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 15 Magnetic Dipolar Interactions in Nanoparticle Systems: Theory, Simulations and Ferromagnetic Resonance D.S. Schmool and M. Schmalzl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 15.2 Theory of Dipole – Dipole Interactions in Magnetic Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 15.2.1 Dipolar Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 15.2.2 Simulations for Arrays of Nanoparticles . . . . . . . . . . . . . . . . . 323 15.3 Ferromagnetic Resonance in Magnetic Nanoparticles . . . . . . . . . . . . . 325 15.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

1 Role of Defects and Disorder in the Half-Metallic Full-Heusler Compounds ¨ I. Galanakis1, K. Ozdo˜ gan2 , and E. S¸a¸sıo˜ glu3,4 1

2

3

4

Department of Materials Science, School of Natural Sciences, University of Patras, GR-26504 Patra, Greece, [email protected] Department of Physics, Gebze Institute of Technology, Gebze, 41400, Kocaeli, Turkey, [email protected] Institut f¨ ur Festk¨ orperforschung, Forschungszentrum J¨ ulich, 52425 J¨ ulich, Germany, [email protected] ˙ Fatih University, Physics Department, 34500, B¨ uy¨ uk¸cekmece, Istanbul, Turkey

Summary. Half-metallic ferromagnets and especially the full-Heusler alloys containing Co are at the center of scientific research because of their potential applications in spintronics. For realistic devices, it is important to control accurately the creation of defects in these alloys. We review some of our late results on the role of defects and impurities in these compounds. More precisely we present results for the following cases (1) doping and disorder in Co2 Cr(Mn)Al(Si) alloys, (2) half-metallic ferrimagnetism appeared due to the creation of Cr(Mn) antisites in these alloys, (3) Co-doping in Mn2 VAl(Si) alloys leading to half-metallic antiferromagnetism, and finally (4) the occurrence of vacancies in the full-Heusler alloys containing Co and Mn. These results are susceptible of encouraging further theoretical and experimental research in the properties of these compounds.

1.1 Introduction During the last century, Heusler alloys [1] have attracted a great interest because of the possibility to study in the same family of alloys a series of interesting diverse magnetic phenomena such as itinerant and localized magnetism, antiferromagnetism, helimagnetism, etc. [2–7]. The first Heusler alloys studied were crystallizing in the L21 structure, which consists of 4 fcc sublattices. Afterwards, it was discovered that it is possible to leave one of the four sublattices unoccupied (C1b structure). The latter compounds are often called half or semi-Heusler alloys, while the L21 compounds are referred to as full-Heusler alloys. NiMnSb belongs to the half-Heusler alloys [8]. In 1983, de Groot and his collaborators [9] showed by using first-principles electronic structure calculations that this compound is in reality half-metallic, i.e., the minority band is semiconducting with a gap at the Fermi level EF , leading

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to 100% spin polarization at EF . Other known half-metallic materials except the half and full-Heusler alloys (see [10–13] and references therein) are some oxides (e.g., CrO2 and Fe3 O4 ) [14], the manganites (e.g., La0.7 Sr0.3 MnO3 ) [14], the double perovskites (e.g., Sr2 FeReO6 ) [15], the pyrites (e.g., CoS2 ) [16], the transition metal chalcogenides (e.g., CrSe) and pnictides (e.g., CrAs) in the zinc-blende or wurtzite structures [17–41], the europium chalcogenides (e.g., EuS) [42], and the diluted magnetic semiconductors (e.g., Mn impurities in Si or GaAs) [43,44]. Although thin films of CrO2 and La0.7 Sr0.3 MnO3 have been verified to present practically 100% spin-polarization at the Fermi level at low temperatures [14, 45], the Heusler alloys remain attractive for technical applications like spin-injection devices [46], spin-filters [47], tunnel junctions [48], or GMR devices [49, 50] because of their relatively high Curie temperature compared with these compounds [2]. Ishida and collaborators studied by means of ab-initio calculations the fullHeusler compounds of the type Co2 MnZ, where Z stands for Si and Ge, and have shown that they are half-metals [51]. Later the origin of half-metallicity in these compounds has been largely explained [52]. Many experimental groups during the last years have worked on these compounds and have tried to synthesize them mainly in the form of thin films and incorporate them in spintronic devices. The group of Westerholt has extensively studied the properties of Co2 MnGe films, and they have incorporated this alloy in the case of spin-valves and multilayer structures [53–55]. The group of Reiss managed to create magnetic tunnel junctions, based on Co2 MnSi [56, 57]. A similar study of Sakuraba and collaborators resulted in the fabrication of magnetic tunnel junctions using Co2 MnSi as one magnetic electrode and Al-O as the barrier (Co75 Fe25 is the other magnetic electrode), and their results are consistent with the presence of half-metallicity for Co2 MnSi [58]. Dong and collaborators recently managed to inject spin-polarized current from Co2 MnGe into a semiconducting structure [59]. Finally, Kallmayer et al. studied the effect of substituting Fe for Mn in Co2 MnSi films and have shown that the experimental extracted magnetic spin moments are compatible with the half-metallicity for small degrees of doping [60]. It is obvious from the experimental results that the full-Heusler compounds containing Co and Mn are of particular interest for spintronics. Not only they combine high Curie temperatures and coherent growth on top of semiconductors (they consist of four fcc sublattice with each one occupied by a single chemical element) but in real experimental situations they can preserve a high degree of spin-polarization at the Fermi level. To accurately control their properties, it is imperative to investigate the effect of defects, doping, and disorder on their properties. Recently Picozzi et al. published a study on the effect of defects in Co2 MnSi and Co2 MnGe [61] followed by an extensive review of the defects in these alloys [62]. Authors have studied in the recent years several aspects of these halfmetallic alloys like the origin of the gap [52, 63], properties of surfaces [64–66] and interfaces with semiconductors [67,68], the quaternary [69,70], the orbital

1 Role of Defects and Disorder

3

magnetism [71, 72], the exchange constants [73], the magneto-optical properties [74], the half-metallic ferrimagnetic Heusler alloys like Mn2 VAl, [75, 76], and the fully-compensated half-metallic ferrimagnets or simply half-metallic antiferromagnets [77]. In this chapter, we will overview some of our results on the defects in the half-metallic full Heusler alloys obtained using the fullpotential nonorthogonal local-orbital minimum-basis band structure scheme (FPLO) [78, 79] within the local density approximation (LDA) [80–82] and employing the coherent potential approximation (CPA) to simulate the disorder in a random way [79]. In Sect. 1.2, we present the physics of defects in the ferromagnetic Heusler alloys containing Co and Mn like Co2 MnSi [83,84]. In Sect. 1.3, we show the creation of half-metallic ferrimagnets on the basis of the creation of Cr and Mn antisites in Co2 (Cr or Mn)(Al or Si) alloys [85,86], and in Sect. 1.4, we expand this study to cover the case of Co defects in ferrimagnetic Mn2 VAl and Mn2 VSi alloys leading to half-metallic antiferromagnets [87]. In Sect. 1.5, we investigate a special case: the occurrence of vacancies in the full-Heusler compounds [88]. Finally in Sect. 1.6 we summarize and conclude.

1.2 Defects in Full-Heuslers Containing Co and Mn We will start our discussion presenting our results on the defects in the case of Co2 MnZ alloys, where Z is Al and Si [83,84]. The first part of our investigation in this section concerns the doping of Co2 MnSi. To simulate the doping by electrons, we substitute Fe for Mn whereas to simulate the doping of the alloys with holes, we substitute Cr for Mn. We have studied the cases of moderate doping substituting 5, 10, and 20% of the Mn atoms. We will start our discussion from Fig. 1.1, where we present the total density of states (DOS) for the Co2 [Mn1−x Fex ]Si and Co2 [Mn1−x Crx ]Si compounds. As discussed in reference [52], the gap is created between states located exclusively at the Co sites. The states low in energy (around −6 eV) originate from the low-lying p-states of the sp atoms (there is also an s-type state very low in energy which is not shown in the figure). The majority-spin occupied states form a common Mn-Co band, while the occupied minority states are mainly located at the Co sites and the minority unoccupied states at the Mn sites. Doping the perfect-ordered alloy with either Fe or Cr smoothens the valleys and peaks along the energy axis. This is a clear sign of the chemical disorder; Fe and Cr induce peaks at slightly different places than the Mn atoms resulting to this smoothening, and as the doping increases this phenomenon becomes more intense. The important detail is what happens around the Fermi level and in what extent is the gap in the minority band affected by the doping. So now we will concentrate only at the enlarged regions around the Fermi level. The blue dashed lines represent the Cr-doping, while the red dash-dotted lines are the Fe-doped alloys. Cr-doping has only marginal effects to the gap. Its width is narrower with respect to the perfect compounds but overall the

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I. Galanakis et al.

2

x = 0.05

DOS(states/eV)

0

Co2MnSi Co2[Mn1-xCrx]Si Co2[Mn1-xFe x]Si

−2 x = 0.1

2

x = 0.2

0

−2 −0.4 −0.2

0

0.2

0.4 −0.4 −0.2 Energy-EF (eV)

0

0.2

0.4

Fig. 1.1. (Color online) Spin-resolved total DOS for the case of Co2 [Mn1−x Crx ]Si and Co2 [Mn1−x Fex ]Si for three difference values of the doping concentration x. DOSs are compared with the one of the undoped Co2 MnSi alloy. In the insets, we present the DOS for a wider energy range. We have set the Fermi level as the zero of the Energy axis. Note that positive values of DOS refer to the majority-spin electrons and negative values to the minority-spin electrons

compounds retain their half-metallicity. In the case of Fe-doping, the situation is more complex. Adding electrons to the system means that, in order to retain the perfect half-metallicity, these electrons should occupy high-energy lying antibonding majority states. This is energetically not very favorable, and for these moderate degrees of doping, a new shoulder appears in the unoccupied states, which is close to the right-edge of the gap; a sign of a large change in the competition between the exchange splitting of the Mn majority and minority states and of the Coulomb repulsion. In the case of the 20% Fe doping, this new peak crosses the Fermi level, and the Fermi level is no more exactly in the gap but slightly above it. Further substitution should lead to the complete destruction of the half-metallicity as in the Quaternary Heusler alloys with a Mn-Fe disordered site [89]. Recent ab-initio calculations including the on-site Coulomb repulsion (the so-called Hubbard U ) have predicted that Co2 FeSi is in reality half-metallic reaching a total spin magnetic moment of 6 µB , which is the largest known spin moment for a half-metal [90, 91].

1 Role of Defects and Disorder

5

We expand our theoretical work to include also the case of Co2 MnAl compound, which has one valence electron less than Co2 MnSi. The extra electron in the the latter alloy occupies majority states leading to an increase of the exchange splitting between the occupied majority and the unoccupied minority states and thus to larger gap-width for the Si-based compound. In the case of Al-based alloy, the bonding and antibonding minority d-hybrids almost overlap and the gap is substituted by a region of very small minority density of states (DOS); we will call it a pseudogap. In both cases, the Fermi level falls within the gap (Co2 MnSi) or the pseudogap (Co2 MnAl) and an almost perfect spin-polarization at the Fermi level is preserved. We substitute either Fe or Cr for Mn to simulate the doping by electrons and holes, respectively, in Co2 MnAl. In Fig. 1.2, we present the total DOS for the Co2 [Mn1−x (Fe or Cr)x ]Al alloys to compare with the perfect Co2 MnAl alloys. As was the case also for the compounds in [83] and discussed earlier the majority-spin occupied states form a common Mn-Co band while the occupied minority states are mainly located at the Co sites and minority unoccupied at the Mn sites (note that the minority unoccupied states near the gap are Co-like but overall the Mn-weight is dominant). The situation is reversed with respect to the Co2 MnSi compound, and Cr-doping has significant effects

2

x=0.05

Co2MnAl Co2[Mn1-xCrx]Al Co2[Mn1-xFe x]Al

DOS(states/eV)

0

−2 2

x =0.1

x=0.2

0

−2 −0.4 −0.2

0

0.2

0.4 −0.4 −0.2 Energy-EF (eV)

0

0.2

0.4

Fig. 1.2. (Color online) Spin-resolved DOS for the case of Co2 [Mn1−x Crx ]Al and Co2 [Mn1−x Fex ]Al for three values of the doping concentration x. DOSs are compared with the one of the undoped Co2 MnAl alloy. Details as in Fig. 1.1

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I. Galanakis et al.

on the pseudogap. On the one hand, its width is larger with respect to the perfect compound and becomes slightly narrower as the degree of the doping increases. Fe-doping on the other hand, almost does not change the DOS around the Fermi level. The extra-electrons occupy high-energy lying antibonding majority states but since Co2 MnAl has one valence electron less than Co2 MnSi half-metallicity remains energetically favorable and no important changes occur upon Fe-doping, and further substitution of Fe for Mn should retain the half-metallicity even for the Co2 FeAl compound, although LDA-based ab-initio calculations predict that the limiting case of Co2 FeAl is almost half-metallic [89]. Finally, we shall briefly discuss the case of disorder simulated by the excess of the Mn or the sp atoms. In Table 1.1, we have gathered the total and atomic spin moments for all cases under study. Substituting 5, 10, 15, or 20% of the Mn atoms by Al or Si ones, corresponding to the negative values of x in the table, results in a decrease of 0.15, 0.30, 0.45, and 0.60 of the total number of valence electrons in the cell, while the inverse procedure results in a similar increase of the mean value of the number of valence electrons. Contrary to the Si compound, which retains the perfect half-metallicity, the Al-based compound is no more half-metallic. In the case of Co2 MnSi, disorder induces states at the edges of the gap keeping the half-metallic character but this is no more the case for Co2 MnAl compound where no real gap exists [83, 84]. Table 1.1. Total and atom-resolved spin magnetic moments for the case of excess of Mn (x positive) or sp atoms (x negative) atoms in µB a Co2 Mn1+x Al1−x x

Ideal

Total

Co

Mn

Al

−0.20 −0.10 −0.05 0.00 0.05 0.10 0.20

3.40 3.70 3.85 4.00 4.15 4.30 4.60

3.26 3.64 3.83 4.04 4.22 4.40 4.80

1.09 1.22 1.29 1.36 1.40 1.44 1.54

2.89 2.84 2.83 2.82 2.81 2.81 2.81

−0.12 −0.13 −0.13 −0.14 −0.14 −0.14 −0.15

3.19 3.15 3.14 3.13 3.10 3.09 3.05

−0.06 −0.08 −0.08 −0.09 −0.10 −0.10 −0.11

Co2 Mn1+x Si1−x −0.20 −0.10 −0.05 0.00 0.05 0.10 0.20 a

4.40 4.70 4.85 5.00 5.15 5.30 5.60

4.40 4.70 4.85 5.00 5.15 5.30 5.60

1.92 1.95 1.96 1.96 1.99 2.00 2.03

The spin moments have been scaled to one atom. In the second column the ideal total spin moment if the compound was half-metallic

1 Role of Defects and Disorder

7

1.3 Defects Driven Half-Metallic Ferrimagnetism In the previous section, we have examined the case of defects in half-metallic ferromagnets. But the ideal case for applications would be a half-metallic antiferromagnet (HMA), also known as fully-compensated ferrimagnet [92], since such a compound would not give rise to stray flux and thus would lead to smaller energy consumption in devices. Also half-metallic ferrimagnetism (HMFi) is highly desirable in the absence of HMA, since such compounds would yield lower total spin moments than the corresponding ferromagnets. Well-known HMFi are the perfect Heusler compounds FeMnSb and Mn2 VAl [93]. We will present in this section another route to HMFi on the basis of antisites created by the migration of Cr(Mn) atoms at Co sites in the case of Co2 CrAl, Co2 CrSi, Co2 MnAl, and Co2 MnSi alloys. We will start our discussion from the Cr-based alloys, and using Co2 CrAl and Co2 CrSi as parent compounds, we create a surplus of Cr atoms, which sit at the perfect Co sites. In Fig. 1.3, we present the DOS for the [Co1−x Crx ]2 CrAl

[Co1-xCrx]2 CrAl

4

DOS (states/eV)

0

−4 x=0 x=0.1 [Co1-xCrx]2 CrSi

4

0

−4 −4

−2 0 Energy-EF (eV)

2

Fig. 1.3. (Color online) Total DOS as a function of the concentration x for the [Co1−x Crx ]2 CrAl (upper panel ) and [Co1−x Crx ]2 CrSi (lower panel ) compounds

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Table 1.2. Atom-resolved spin magnetic moments for the [Co1−x Crx ]2 CrAl, [Co1−x Crx ]2 CrSi, [Co1−x Mnx ]2 MnAl, and [Co1−x Mnx ]2 MnSi compounds (moments have been scaled to one atom) [Co1−x Crx ]2 CrAl x

Co

Cr(imp)

Cr

Al

Total

Ideal

0 0.05 0.1 0.2

0.73 0.71 0.69 0.64

– −1.82 −1.85 −1.87

1.63 1.62 1.61 1.60

−0.09 −0.09 −0.08 0.06

3.00 2.70 2.40 1.80

3.00 2.70 2.40 1.80

[Co1−x Crx ]2 CrSi x

Co

Cr(imp)

Cr

Si

Total

Ideal

0 0.05 0.1 0.2

0.95 0.93 0.91 0.87

– −1.26 −1.26 −1.25

2.17 2.12 2.07 1.96

−0.06 −0.06 −0.05 −0.04

4.00 3.70 3.40 2.80

4.00 3.70 3.40 2.80

[Co1−x Mnx ]2 MnAl x

Co

Mn(imp)

Mn

Al

Total

Ideal

0 0.05 0.1 0.2

0.68 0.73 0.78 0.84

– −2.59 −2.49 −2.23

2.82 2.82 2.83 2.85

−0.14 −0.13 −0.12 −0.09

4.04 3.81 3.61 3.20

4.00 3.80 3.60 3.20

[Co1−x Mnx ]2 MnSi x

Co

Mn(imp)

Mn

Si

Total

Ideal

0.0 0.05 0.1 0.2

0.98 0.99 0.99 0.97

– −0.95 −0.84 −0.70

3.13 3.09 3.06 2.99

−0.09 −0.08 −0.07 −0.05

5.00 4.80 4.60 4.20

5.00 4.80 4.60 4.20

The two last columns are the total spin moment (Total) in the unit cell calculated as 2 × [(1 − x) ∗ mCo + x ∗ mCr or Mn(imp) ] + mCr(Mn) + mAl or Si and the ideal total spin moment predicted by the Slater– Pauling rule for half-metals [52]. With Cr(imp) or Mn(imp) we denote the Cr(Mn) atoms sitting at perfect Co sites

and [Co1−x Crx ]2 CrSi alloys for concentrations x = 0 and 0.1, and in Table 1.2, we have gathered the spin moments for the two compounds under study. We will start our discussion from the DOS. The perfect compounds show a gap in the minority-spin band and the Fermi level falls within this gap, and thus the compounds are half-metals. When the sp atom is Si instead of Al, the gap is larger because of the extra electron, which occupies majority states of the

1 Role of Defects and Disorder

9

transition metal atoms [52] and increases the exchange splitting between the majority occupied and the minority unoccupied states. This electron increases the Cr spin moment by ∼0.5 µB and the moment of each Co atom by ∼0.25 µB about. The Cr and Co majority states form a common band, and the weight at the Fermi level is mainly of Cr character. The minority occupied states are mainly of Co character. When we substitute Cr for Co, the effect on the atomic DOS of the Co and Cr atoms at the perfect sites is marginal. The DOS of the impurity Cr atoms has a completely different form from the Cr atoms at the perfect sites because of the different symmetry of the site where they sit. But although Cr impurity atoms at the antisites induce minority states within the gap, there is still a tiny gap and the Fermi level falls within this gap keeping the half-metallic character of the parent compounds even when we substitute 20% of the Co atoms by Cr ones. The discussion above on the conservation of the half-metallicity is confirmed when we compare the calculated total moments in Table 1.2 with the values predicted by the Slater Pauling rule for the ideal half-metals (the total spin moment in µB is the number of valence electrons in the unit cell minus 24) [52]. Since Cr is lighter than Co, substitution of Cr for Co decreases the total number of valence electrons and the total spin moment should also decrease. The interesting point is the way that the reduction of the total spin moment is achieved. Co and Cr atoms at the perfect sites show a small variation of their spin magnetic moments with the creation of defects and the total spin moment is reduced because of the antiferromagnetic coupling between the Cr impurity atoms and the Co and Cr ones at the ideal sites, which would have an important negative contribution to the total moment as confirmed by the results in Table 1.2. Thus the Cr-doped alloys are half-metallic ferrimagnets, and their total spin moment is considerable smaller than the perfect halfmetallic ferromagnetic parent compounds; in the case of [Co0.8 Cr0.2 ]2 CrAl it decreases down to 1.8 µB from the 3 µB of the perfect Co2 CrAl alloy. Here we have to mention that if also Co atoms migrate to Cr sites (case of atomic swaps) the half-metallity is lost, as it was shown by Miura et al. [94], because of the energy position of the Co states which have migrated at Cr sites. In the rest of this section, we will present results on the [Co1−x Mnx ]2 MnZ compounds varying the sp atom, Z, which is one of Al or Si. We have taken into account five different values for the concentration x; x = 0, 0.025, 0.05, 0.1, and 0.2. In Fig. 1.4, we have drawn the DOS for both families of compounds under study and for two different values of the concentration x: the perfect compounds (x = 0) and for one case with defects, x = 0.1. In the case of the perfect Co2 MnSi compound, there is a real gap in the minority spin band and the Fermi level falls within this gap, and this compound is a perfect halfmetal. Co2 MnAl presents in reality a region of tiny minority-spin DOS instead of a real gap, but the spin-polarization at the Fermi level only marginally deviates from the ideal 100%, and this alloy can be also considered as halfmetal. These results agree with previous electronic structure calculations on these compounds [52, 61, 70, 83, 95]. When we create a surplus of Mn atoms,

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6 3

DOS (states/eV)

0 −3 6

[Co1-xMnx]2 MnAl

x=0 x = 0.1

3 0 −3 −2

[Co1-xMnx]2 MnSi

−1

0

1

2

Energy-EF (eV) Fig. 1.4. (Color online) Total DOS around the gap region for the [Co1−x Mnx ]2 MnAl and [Co1−x Mnx ]2 MnSi alloys as a function of the concentration x: we denote x = 0 with the solid grey line with shaded region and x = 0.1 with a dashed thick blue line. In the insets, we present the DOS for a wider energy region and for a x = 0 concentration

which migrate at sites occupied by Co atoms in the perfect alloys, the gap persists and both compounds retain their half-metallic character as occurs also for the Cr-based alloys presented earlier. Especially for Co2 MnSi, the creation of Mn antisites does not alter the width of the gap and the halfmetallicity is extremely robust in these alloys with respect to the creation of Mn antisites. The atomic spin moments show behavior similar to the Cr alloys as can be seen in Table 1.2, and the spin moments of the Mn impurity atoms are antiferromagnetically coupled to the spin moments of the Co and Mn atoms at the perfect sites resulting to the desired half-metallic ferrimagnetism.

1 Role of Defects and Disorder

11

1.4 A Possible Route to Half-Metallic Antiferromagnetism Now, we will present a way to use the defects in perfect half-metallic ferrimagnets to create a half-metallic antiferromagnetic material: the doping with Co of the Mn2 VAl and Mn2 VSi alloys which are well known to be HMFi. The importance of this route stems from the existence of Mn2 VAl in the Heusler L21 phase as shown by several groups [96–98]. Each Mn atom has a spin moment of around −1.5 µB and V atom a moment of about 0.9 µB [96–98]. All theoretical studies on Mn2 VAl agree on the half-metallic character with a gap at the spin-up band instead of the spin-down band as for the other half-metallic Heusler alloys [52, 76, 99, 100]. Prior to the presentation of our results, we have to note that due to the Slater–Pauling rule [52], these compounds with less than 24 valence electrons have negative total spin moments, and the gap is located at the spin-up band. Moreover, the spin-up electrons correspond to the minority-spin electrons and the spin-down electrons to the majority electrons contrary to the other Heusler alloys [52]. We have substituted Co for Mn in Mn2 V(Al or Si) in a random way and in Fig. 1.5, we present the total and atom-resolved DOS in [Mn1−x Cox ]2 VAl (solid black line) and [Mn1−x Cox ]2 VSi (blue dashed line) alloys for x = 0.1. The perfect compounds

5 Total

Mn

DOS (states/eV)

0 −5

[Mn0.9Co0.1]2VAl [Mn0.9Co0.1]2VAi

5 Co

V

0 −5 −4

−2

0

2

−4

−2

0

2

Energy-EF (eV) Fig. 1.5. (Color online) Total and atom-resolved DOS for the [Mn0.9 Co0.1 ]2 VAl and [Mn0.9 Co0.1 ]2 VSi compounds. Note that the atomic DOS’s have been scaled to one atom. Positive values of DOS correspond to the spin-up (minority) electrons while negative values correspond to the spin-down (majority) electrons

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I. Galanakis et al.

show a region of low spin-up DOS (we will call it a “pseudogap”) instead of a real gap. Upon doping the pseudogap at the spin-up band persists and the quaternary alloys keep the half-metallic character of the perfect Mn2 VAl and Mn2 VSi compounds. Co atoms are strongly polarized by the Mn atoms, since they occupy the same sublattice and they form Co-Mn hybrids, which afterwards interact with the V and Al or Si states [52]. The spin-up Co states form a common band with the Mn ones and the spin-up DOS for both atoms has similar shape. Mn atoms have less weight in the spin-down band, since they accommodate less charge than the heavier Co atoms. In Table 1.3, we have gathered the total and atom-resolved spin moments for all the Co-doped compounds as a function of the concentration. We have gone up to a concentration that corresponds to 24 valence electrons in the unit cell, thus up to x = 0.5 for the [Mn1−x Cox ]2 VAl and x = 0.25 for the [Mn1−x Cox ]2 VSi alloys. In the last column, we have included the total spin moment predicted by the Slater–Pauling rule for the perfect half-metals [52]. A comparison between the calculated and ideal total spin moments reveals that all the compounds under study are half-metals with very small deviations because of the existence of a pseudogap instead of a real gap. Exactly for Table 1.3. Atom-resolved spin magnetic moments for the [Mn1−x Cox ]2 VAl and [Mn1−x Cox ]2 VSi compounds (moments have been scaled to one atom) [Mn1−x Cox ]2 VAl x 0 0.05 0.1 0.3 0.5

Mn

Co

V

Al

Total

Ideal

−1.573 −1.580 −1.564 −1.484 −1.388

– 0.403 0.398 0.456 0.586

1.082 1.090 1.067 0.953 0.782

0.064 0.073 0.069 0.047 0.019

−2.000 −1.799 −1.600 −0.804 ∼0

−2.0 −1.8 −1.6 −0.8 0

[Mn1−x Cox ]2 VSi x 0 0.05 0.1 0.2 0.25

Mn

Co

V

Si

Total

Ideal

−0.960 −0.944 −0.925 −0.905 −0.899

– 0.749 0.819 0.907 0.935

0.856 0.860 0.847 0.839 0.839

0.063 0.059 0.054 0.046 0.041

−1.000 −0.800 −0.600 −0.201 ∼0

−1.0 −0.8 −0.6 −0.2 0

The two last columns are the total spin moment (Total) in the unit cell calculated as 2 × [(1 − x) ∗ mM n + x ∗ mCo ] + mV + mAl or Si and the ideal total spin moment predicted by the Slater– Pauling rule for half-metals [52]. The lattice constants have been chosen 0.605 nm for Mn2 VAl and 0.6175 for Mn2 VSi for which both systems are half-metals [75] and have been kept constant upon Co doping

1 Role of Defects and Disorder

13

24 valence electrons, the total spin moment vanishes as we will discuss in the next paragraph. Co atoms have a spin moment parallel to the V one and antiparallel to the Mn moment, and thus the compounds retain their ferrimagnetic character. As we increase the concentration of the Co atoms in the alloys, each Co has more Co atoms as neighbors, it hybridizes stronger with them and its spin moment increases while the spin moment of the Mn atom decreases (these changes are not too drastic). The sp atoms have a spin moment antiparallel to the Mn atoms as already discussed in [75]. The most interesting point in this substitution procedure is revealed when we increase the Co concentration to a value corresponding to 24 valence electrons in the unit cell, thus the [Mn0.5 Co0.5 ]2 VAl and [Mn0.75 Co0.25 ]2 VSi alloys. The Slater-Pauling rule predicts for these compounds a zero total spin moment in the unit cell, and the electrons population is equally divided between the two spin-bands. Our first-principles calculations reveal that this is actually the case. The interest arises from the fact that although the total moment is zero, these two compounds are made up from strongly magnetic components. Mn atoms have a mean spin moment of ∼−1.4 µB in [Mn0.5 Co0.5 ]2 VAl and ∼−0.9 µB in [Mn0.75 Co0.25 ]2 VSi. Co and V have spin moments antiferromagnetically coupled to the Mn ones for which [Mn0.5 Co0.5 ]2 VAl are ∼0.6 and ∼0.8 µB , respectively, and for [Mn0.75 Co0.25 ]2 VSi ∼0.9 and ∼0.8 µB . Thus these two compounds are half-metallic fully-compensated ferrimagnets or as they are best known in literature as half-metallic antiferromagnets.

1.5 Vacancies Finally, we will also shortly discuss a special case: the occurrence of vacancies [88]. In experiments a vacancy can appear in all possible sites, and we will start our discussion by the case of vacancies (E) appearing at the sites occupied by Co atoms at the perfect compounds. In Fig. 1.6, we have gathered the total DOS for the four studied compounds, [Co1−x Ex ]2 YZ (Y is Cr or Mn and Z is Al or Si) when we replace 2.5% (x = 0.025) and 10% (x = 0.10) of the Co atoms. Although there is no visible effect on the properties of the gap with respect to the perfect compound [83, 86] when x is as small as 0.025, as we increase x to 0.1 impurity states located at the left edge of the gap appear. The vacancy carries very small charge with a vanishing DOS, and the induced minority states are mainly located at the Co sites (Mn and Si atoms have a very small DOS at the same energy range as a result of the polarization of these Co-located vacancy-induced minority-states). These states result in a considerable shrinking of the width of the gap of the order of 0.25–0.3 eV. In the case where the Fermi level is close to the left edge of the gap as in the Co2 MnAl alloy, the shrinking of the gap results in the loss of half-metallicity (the spin-polarization is reduced to ∼40% for x = 0.1). In the case when the vacancies occur at the Y site occupied in the perfect alloys by Cr or Mn atoms and the Z site occupied by Al or Si, they again

14

I. Galanakis et al.

2

[Co1-xEx]2CrSi

[Co1-xEx]2CrAl

DOS (states/eV)

0 −2 2

x=0.025 x=0.1

[Co1-xEx]2MnSi

[Co1-xEx]2MnAl

0 −2 −0.5

0

0.5

−0.5

0

0.5

Energy-EF (eV) Fig. 1.6. (Color online) Total DOS as a function of the concentration x for the four studied alloys [Co1−x Ex ]2 YZ where Y stands for Cr and Mn, Z for Al and Si, and E the vacant site

induce minority states within the left edge of the gap leading to a shrinking of its width as in the case discussed in the previous paragraph. But the effect is much more mild leading to a shrinking of the gap by 0.1 eV approximately when they substitute Y-type atoms and even less when they substitute the sp atoms. Moreover, not only the reduction of the width of the gap is smaller, but also the intensity of the minority-spin induced DOS within the gap is smaller with respect to the case discussed in the previous paragraph. This difference in the behavior of the vacancy-induced minority states has its routes to the fact that the gap is created between states exclusively localized at the Co sites, which due to symmetry reasons do not couple to Cr(Mn) or Al(Si) orbitals. When the vacancy substitutes Co atoms, the effect is much more intense around the Fermi level than when it substitutes Cr(Mn) or Al(Si) atoms, which have almost zero weight around the gap. This is also reflected on the change of the behavior of the atomic spin moments (see [88] for an extended discussion of spin moments).

1.6 Summary and Outlook In this chapter, we have reviewed our results on the defects in half-metallic Heusler alloys. First, we have studied the effect of doping and disorder on the magnetic properties of the Co2 MnAl(Si) full-Heusler alloys. Doping simulated

1 Role of Defects and Disorder

15

by the substitution of Cr and Fe for Mn overall keeps the half-metallicity. Both disorder and doping have little effect on the half-metallic properties of Co2 MnSi, and it keeps a high degree of spin-polarization. Co2 MnAl presents a region of low minority density of states instead of a real gap. Doping keeps the half-metallicity of Co2 MnAl while disorder simulated by excess of either the Mn or Al atoms completely destroys the almost perfect spin-polarization of the perfect compound contrary to Co2 MnSi. Afterwards, we have studied the effect of defects-driven appearance of halfmetallic ferrimagnetism in the case of the Co2 Cr(Mn)Al(Si) Heusler alloys. More precisely, on the basis of first-principles calculations, we have shown that when we create Cr(Mn) antisites at the Co sites, these impurity Cr(Mn) atoms couple antiferromagnetically with the Co and the Cr(Mn) atoms at the perfect sites while keeping the half-metallic character of the parent compounds. This is a promising alternative way to create robust half-metallic ferrimagnets, which are crucial for magnetoelectronic applications. Moreover we have also studied the effect of doping the half-metallic ferrimagnets, Mn2 VAl and Mn2 VSi. Co substitution for Mn keeps the half-metallic character of the parent compounds, and when the total number of valence electrons reaches the 24, the total spin moment vanishes as predicted by the Slater-Pauling rule and the ideal half-metallic antiferromagnetism is achieved. Thus, we can create half-metallic antiferromagnets simply by introducing Co atoms in the Mn2 VAl and Mn2 VSi half-metallic ferrimagnets. Since crystals and films of both Mn2 VAl and Co2 VAl alloys have been grown experimentally, such a compound maybe is feasible experimentally. Finally, we have studied the effect of vacancies in half-metallic Heusler alloys. On the one hand, we have shown using ab-initio electronic structure calculations that the occurrence of vacancies at the sites occupied by Co atoms in the perfect compounds seriously affects the stability of their half-metallic character, and the minority-spin gap is rapidly shrinking. On the other hand, vacancies at the other sites do not have an important impact on the electronic properties of the half-metallic full-Heusler compounds. Thus, it is crucial for spintronic applications to prevent creation of vacancies during the growth of the half-metallic full-Heusler alloys used as electrodes in the magnetoelectronic devices. Although we have presented several aspects of defects in Heusler alloys, many more calculations and experiments are needed; the aim is to find systems, which either do not lead to states in the gap (like the defects-driven half-metallic ferrimagnets presented in Sects. 1.3 and 1.4) or systems with particularly high defect formation energies or sufficiently low annealing temperatures (we have not discussed the energetics of defects in this contribution). Equally important for realistic applications is the control of surface and interface states in the gap, the latter are in particular important for interfaces to semiconductors, which we have not been examined in this chapter.

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18 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94.

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2 Clustering in Heusler Alloys N. Lakshmi1 , V. Sebastian2 , and K. Venugopalan1 1

2

Department of Physics, Mohanlal Sukhadia University, Udaipur, 313 001 Rajasthan, India, [email protected] Department of Physics, Nirmalagiri College, Nirmalagiri P.O., Kerala 670 701, India

Summary. The effect of disordering, particularly Cr clustering on the magnetic properties in bulk and nanosized Heusler alloy Fe2 CrAl has been studied using a combination of X-ray diffraction, 57 Fe M¨ ossbauer spectroscopy, and DC magnetization. The structural order/disorder has been confirmed using M¨ ossbauer spectroscopy, and the observed bulk magnetic properties have been correlated to these results. Mechanical milling of the bulk alloys result in a more even distribution of Cr, reducing the clustering and hence enhancing the bulk magnetic properties with nearly no change in the Curie temperature (TC ). Mechanical alloying of elemental Fe, Cr, and Al give rise to a highly disordered system with higher saturation moment, retentivity, and very enhanced TC .

2.1 Introduction Heusler alloys have been of interest since 1903 when Heusler [1] reported that ferromagnetic alloys could be made from nonferromagnetic constituents Cu, Mn, and main group elements such as Al and Sn. The ferromagnetic properties of these alloys were related to the chemical ordering and concentration of Mn atoms. In general, there are two types of Heusler systems. The first is the fullHeusler alloy, with chemical composition X2 YZ, which crystallizes in the L21 structure [Fm3m] and consists of four sets of interpenetrating fcc planes A, B, C, and D (Fig. 2.1). The A and C sublattices with Wyckoff coordinates (0,0,0) and (1/2,1/2,1/2) are occupied by X atoms, while the B and D sublattices with coordinates (1/4,1/4,1/4) and (3/4,3/4,3/4) are occupied by the Y and Z atoms, respectively. The X atoms at the two different sublattices have the same local environment rotated by 90◦ with respect to the (001) axis. The second type is the half-Heusler alloy with composition XYZ, which crystallizes in the Clb structure. This structure can be derived from the full Heusler structure by leaving one of the two equivalent (A,C) sites in Fig. 2.1 vacant. In both the Heusler systems, typical X-site elements are transition metal elements Cu, Pd, Au, Co, Ni, Fe, etc., Y-site elements are transition

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X(A)

X(C)

Y(B)

Z(D)

Fig. 2.1. Structure of Heusler alloys

metal elements Ti, Zr, Hf, V, Mn, Cr, etc., and Z is an s-p element like Al, Sn, In, Sb, Ge or Ga, etc. Most of these alloys, especially the X2 YZ type, are ferromagnetic at room temperature. In addition, many of these alloys also exhibit semiconducting [2], semi-metallic [3], Pauli paramagnetic [4], antiferromagnetic [5], or half-metallic [6] character. Heusler alloys are easily amenable to partial or complete substitution of one or more of the constituent elements. This flexibility in altering the composition can be exploited to develop new materials with superior physical properties such as electrical resistivity, thermal conductivity, Seebeck coefficients, heat capacity, and magnetic properties [7]. Initial studies on Heusler alloys were limited mainly to understanding their magnetic properties using techniques such as M¨ossbauer spectroscopy and perturbed angular correlation (PAC). However, in recent years, there has been a revival of interest in these alloys because of possible technological applications. A recent emerging area of technological applications of the Heusler systems is in shape memory alloys (SMA) [8, 9]. Shape memory alloys are materials that, after being strained, revert back to their original shape at a certain temperature. In Heusler systems like Ni2 MnGa, magnetic control of the structural transformation is exploited to prepare SMA systems, which are more efficient compared with conventional temperature driven SMA [10]. In recent years, Heusler systems have also been reported to be good candidates for possible applications in the emerging field of spintronics [11]. In spintronics systems, materials having nonequilibrium spin population are required for generation of spin polarization. Band structure calculations show that the gap in the minority spin band in Heusler systems can lead to 100% spin polarization of electronic states at the Fermi level [12, 13]. High Curie temperature in the range 200–1,100 K

2 Clustering in Heusler Alloys

23

and compatibility with compound and element semiconductors make these materials suitable for spin electronic applications. The physical properties of Heusler alloys are dependent, to a large extent, on the degree of disorder. Although many of the Heusler alloys are chemically well-ordered, chemical disorder of various degrees can coexist in these systems, depending on the constituent elements and heat treatment given to prepare these alloys. In general, ordering is easily possible in Heusler alloys X2 YZ with Pd as X. It is observed that there is more of a tendency for disorder in Co-based alloys when compared with Pd-based alloys, especially if Z site elements are Sn or Al [14]. When X is Fe, the tendency for disorder is greater. It is now well known that iron-based transition metal alloys show a great sensitivity to environmental effects [15]. For instance, Cr, Co, or V in Fe-Al or Fe-Si systems shows definite site preference even in the presence of disorder. Such preferential occupation of the different sites gives rise to clustering effects, which influence the magnetic properties of these systems. Such clustering has been observed in different Heusler systems with (Fe1−x Vx )3 Al [16] being a good example. Theoretical investigation [17] of atomic disorder-effects on half metallicity of the full-Heusler alloy Co2 (Cr1−x Fex )Al shows that disorder between Cr and Al does not significantly reduce the spin polarization while disorder between Co and Cr contributes to a considerable reduction of the spin polarization. Thus, the type of atomic disorders in Heusler systems play a crucial role in determining the spin polarization and hence their utility as spin injection sources. Atomic disorders and clustering effects also greatly influence the magnetic properties of Heusler alloys [18, 19]. Previous studies on bulk Fe2 CrAl using the M¨ ossbauer effect have shown that even in the presence of fairly good chemical ordering, clustering of Cr influences the microscopic magnetic properties [14, 18, 20]. Moreover, on reduction in the grain size to the nanoscale, vastly different magnetic properties are expected, as, for example, in the case of Al-rich, Fe-Al systems [21]. Coupled with the ease of tailoring of magnetic properties in Heusler systems by simple substitution – wholly or partially – of constituents, change in the magnetic properties due to induced disorder as well as reduction in the grain size can give rise to a whole new class of magnetic nanosystems with improved magnetic properties like enhanced saturation magnetization and high Curie temperatures. The present study has therefore been undertaken to better understand the effect of clustering and disorder on the magnetic properties of iso-structural full Heusler system Fe2 CrAl. The study also aims to investigate the effect of nanostructuring on the formation of clusters and the consequent changes in the bulk and microscopic magnetic properties of this Heusler alloy. The magnetic properties of nanostructured Fe2 CrAl alloys prepared through mechanical milling and mechanical alloying are compared with that obtained from the study of a bulk alloy prepared through arc melting. The nanosized samples prepared through ball milling are expected to be more disordered and could lead to an enhancement in the magnetic properties compared to the bulk alloy.

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Although mechanical milling and mechanical alloying lead to nanostructured alloys of the same nominal composition, the resultant alloys can have different microstructural and magnetic properties. This chapter reports the results of a comparative study of a bulk, mechanically milled (MM) and mecahnically alloyed (MA) Fe2 CrAl, characterized using X-ray diffraction (XRD) in conjunction with M¨ ossbauer and bulk magnetization techniques.

2.2 Experimental Methods To prepare the bulk Fe2 CrAl alloy, stoichiometric quantities of the constituent materials of at least 99.95% purity were arc melted in argon atmosphere and annealed in vacuum (≈10−5 torr) at first for 1,073 K for 72 h and further at 473 K for three days more, then cooled down to room temperature in the furnace itself. The MM and MA samples were prepared in a high-energy Spex 8000M mixer/mill using tungsten carbide balls and vials. The ball to powder ratio in both the cases was 20 : 1. The starting material for the MM sample was bulk Fe2 CrAl alloy prepared by arc melting and that for the MA sample was pure Fe, Al, and Cr taken in the required stoichiometric ratio. The MM sample was milled for 30 min and the MA sample for 15 h. Although the XRD of samples MA for 5, 10, and 15 h are nearly the same, M¨ ossbauer spectra showed the presence of a sextet corresponding to 33 T, characteristic of α-Fe, in the 5 and 10 h alloyed samples, indicating that all the iron had not been alloyed until 15 h of milling. The sample MA for 15 h (MA) has therefore been chosen for comparison with bulk and MM samples. X-ray diffractograms of the three samples were obtained using Cu Kα radiation. Room temperature M¨ ossbauer measurements were made using a 25 mCi 57 Co (Rh) source in the transmission mode. Fitting of M¨ ossbauer spectra were made using a hyperfine field distribution program. Magnetization measurements were made on a Lake Shore 7300 Vibrating Sample Magnetometer (VSM) up to a maximum field of 1 T.

2.3 Results and Discussion 2.3.1 X-Ray Diffraction Studies In Fm3m symmetry, Bragg reflections with nonzero structural amplitudes occur when Miller indices are all either odd or even. X-ray diffraction pattern will show three types of reflections with structure amplitudes F as follows [22] 1. h, k, l all odd, e.g., F(1,1,1) = 4|{(fA − fC )2 + (fB − fD )2}1/2| 2. h, k, l all even, which can be realized by two different conditions, which are

2 Clustering in Heusler Alloys

• •

25

(h + k + l)/2 = 2n + 1, e.g., F(200) = 4|fA − fC + fB − fD | (h + k + l)/2 = 2n, e.g., F(220) = 4|fA + fC + fB + fD |

In the above expressions, fA , fC , fB , and fD are the average scattering factors of atoms at the corresponding lattice sites. Reflections for which (h + k + l)/2 = 2n are the principal reflections. As seen from the above expression, all the average scattering factors simply add up and so any interchange in positions will not affect the structure amplitude F in this case. Thus, the intensity of the principal reflections are unaffected by the state of chemical ordering. However, the other two groups representing superlattice reflections are sensitive to chemical ordering. Measurement of the intensities of the superlattice reflections relative to that of the principal reflection can give the type of chemical disordering. Johnston and Hall [23] and Ziebeck and Webster [24] have discussed in detail the effects of different types of disorder on the structure amplitudes of ternary alloys of the Heusler type. Figure 2.2 shows the XRD patterns of bulk, MM, and MA samples. The Fe2 CrAl (220), (400), and (422) peaks are present in all the three samples.

q Fig. 2.2. X-ray diffraction patterns of bulk Fe2 CrAl prepared by arc melting, nanosized Fe2 CrAl prepared by high energy ball milling of the bulk sample for 30 min (MM) and 15 h MA Fe2 CrAl

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The value of lattice parameter a0 for the bulk sample is 0.5803 nm and is comparable to that reported by Zhang et al. [20] and earlier reported values [2,14]. Paduani et al. [25] have also reported similar values. Two low intensity peaks at 27.8◦ and 32.2◦ are present in the XRD spectrum of the bulk sample, which correspond to the (111) and (200) superlattice peaks. The degree of ordering in the sample can be measured by looking at the relative intensities of these two peaks with respect to the fundamental (220) peak. These two superlattice peaks disappear for the case of randomly occupied lattice sites (A2 structure). In the case of the Heusler alloy Co2 Mn1−x Fex Si, it has been pointed out that when Mn/Fe sites are randomly occupied by Mn/Fe and Si, the structure is B2 and only the (200) superlattice peak would be present, while the (111) peak vanishes [26]. When Co atoms are partly replaced by Mn/Fe atoms in Co2 Mn1−x Fex Si, the intensity of the (111) superlattice peak would have a higher intensity than the (200) peak. They have also pointed out that the intensity of the (111) and (200) peaks would be equal in the case of L21 structure. Since Fe2 CrAl is also a Heusler alloy with a similar structure, the degree of ordering and the nature of antisite disorders can be determined from the study of the relative intensities of the corresponding superlattice peaks. It is seen that the relative integrated intensity of the (111) peak is 10.1% with respect to the (220) peak, which is within expected limits of error of the reported value of 10%, while the (200) peak has a relative intensity of only 1% with respect to the (220) peak. In analogy with other Heusler alloys, this indicates that even in the bulk sample, there is an appreciable degree of antisite disorder, with Cr atoms replacing Fe atoms at the Fe sites. When the bulk sample is milled for 30 min, it is seen that the fundamental peaks are shifted to the left, indicating an expansion of the lattice. The lattice constant of the sample is 0.5951 nm, equivalent to an expansion of ≈2.5% compared with the bulk sample. High energy ball milling is known to produce lattice disorder and a high density of point defects like antisite atoms and vacancies, which leads to an expansion of the lattice [27, 28]. The relatively high value of lattice expansion in this sample points to the high density of antisite disorders. The presence of defects and vacancies in the present sample is also indicated by the decrease in intensity of the (111) superlattice peak and the disappearence of the (200) peak. The broadening of the peaks due to crystallite refinement and increase in lattice strain also contribute to the decreased intensity of these peaks. The crystallite size calculated using the Williamson–Hall [29] method shows that the crystallite size of the milled sample is ≈15 nm after 30 min of milling. XRD spectra of the elemental powders of Fe, Cr, and Al, MA for 5, 10, and 15 h, are given in Fig. 2.3. The samples are labeled MA 5, MA 10, and MA 15 in the figure. All the three samples have similar XRD spectra, with the (220), (400), and (422) peaks corresponding to Fe2 CrAl visible in all the spectra. Since the fundamental peaks of pure Fe also lie at nearly the same 2θ values, the presence of pure Fe phase cannot be detected by XRD. However, as the hyperfine field values of pure Fe and Fe2 CrAl are very different, the presence of

2 Clustering in Heusler Alloys

27

q Fig. 2.3. X-ray diffraction spectra of Fe2 CrAl prepared by mechanically alloying elemental powders of Fe, Cr, and Al powders for 5, 10, and 15 h. The sample denoted as MA 15 in this figure is denoted as MA in Fig. 2.2

unalloyed Fe can be detected through M¨ ossbauer studies. Presence of pure Fe phase is observed in the MA 5 and MA 10 samples, indicating that the alloying is not complete, whereas, M¨ossbauer studies on the MA 15 sample do not show evidence for the pure Fe phase. Thus, from XRD and M¨ ossbauer studies, it can be concluded that the elemental powders have completely alloyed to form Fe2 CrAl alloy after mechanical alloying for 15 h. Therefore, only the MA 15 sample, i.e., the one MA for 15 h, referred to as MA in text, has been used for further studies. From Fig. 2.3, it is evident that the fundamental peaks are gradually broadened with increase in MA time, indicating a reduction in the crystallite size and an increase in lattice strain. The crystallite size of the sample milled for 15 h (MA), calculated using the Williamson–Hall method is ≈8 nm. The lattice constant for this sample is 0.588 nm, corresponding to ≈1.3% expansion with respect to the bulk. However, it is smaller than that of the MM sample. The expansion of the lattice in this sample may be due to slight deviations from the stoichiometry, since the mechanical alloying process could produce other, nonstoichiometric, phases too. However, the fraction of other phases is too low since they have not been detected by XRD and M¨ ossbauer studies. The superlattice peaks are also absent in this sample (Fig. 2.2), indicating a high degree of disorder, as is expected for alloys prepared by mechanical alloying.

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Fig. 2.4. Room temperature M¨ ossbauer spectra and magnetic hyperfine field distributions of bulk Fe2 CrAl prepared by arc melting and nanosized Fe2 CrAl prepared by high energy ball milling of the bulk sample for 30 min (MM)

2.3.2 M¨ ossbauer Studies Room temperature M¨ossbauer spectra (Fig. 2.4) of the bulk and MM samples show the presence of a weak, unresolved magnetic sextet and a paramagnetic singlet. The isomer shift of the singlet is very near to zero in both the cases. On fitting a single sextet and singlet using discrete Lorentzians, the width of the peaks of the sextet is about two times that of natural iron. Hence, the spectra were fitted for a distribution of hyperfine fields using Windows programme [30] and the corresponding hyperfine field distribution (HFD) is also shown in Fig. 2.4. The line widths of the individual Gaussians were constrained to be equal to that of α-iron. The hyperfine field distribution of the bulk sample consists of a dominant paramagnetic component, a component centered on 9.5 T and two higher magnetic field components of almost equal intensity (Fig. 2.4). Temperaturedependent M¨ ossbauer studies [14] on bulk Fe2 CrAl has shown that the paramagnetic peak is present even at low temperatures. From these studies, it was concluded that the paramagnetic singlet in the bulk sample is due to Cr atoms replacing Fe at Fe sites, leading to the clustering of Cr atoms. This is consistent with the conclusions drawn from XRD studies that Fe-Cr type disorder is still present in the sample. Such antisite disorder has been observed in well-ordered samples of other Heusler alloys such as Co2 FeSi [31] also. The magnetic hyperfine field distribution (HFD) for the bulk sample shows two high field components of almost equal intensity centered at 28.6 and 19.8 T in addition to a low field component at 9.5 T. Fe at the equivalent A and C sites in a fully-ordered Heusler structure have the same environments and so magnetic HFD of the M¨ ossbauer spectrum of Fe2 CrAl should have only a single peak corresponding to one unique Fe site [14]. The low field peak at 9.5 T can be attributed to Fe atoms in the perfectly ordered environment.

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Table 2.1. Bulk magnetic parameters of the bulk, MM, and MA samples Sample

Ms

Mr

Hc (mT)

Tc (K)

Hav (T)

Bulk MM MA

1.08 1.95 2.24

0.005 0.036 0.332

0.4 1.4 15.9

298 300 887

9.9 11.8 18.0

Saturation magnetization (Ms ) and retentivity (Mr ) are in units of µB /formula unit. The average field (Hav ) is obtained from M¨ ossbauer measurements

The presence of two additional peaks in the HFD can be attributed to the Cr clustering. The clustering of Cr atoms would, in the basic L21 lattice, lead to an excess of Fe atoms being displaced from their actual positions. Thus a fraction of Fe atoms would be in an Fe3 Al-like environment. This is supported by the fact that the hyperfine fields reported for disordered Fe3 Al alloy is very close to the two high field peaks, viz. at 19.8 T and 28.6 T obtained here [32]. The overall microscopic picture obtained from M¨ ossbauer studies is thus consistent with the conclusions drawn from XRD studies that Fe-Cr type of disorder persists in the bulk sample. However, since the integrated intensity of the peaks for the high field components are very small, the disorder in this sample can be assumed to be very small and is consistent with the (220) superlattice peak in XRD. There is a slight increase in the average magnetic hyperfine field for the MM sample, compared with the bulk. It increases from 9.9 T for the bulk sample to 11.8 T for the milled sample (Table 2.1). On milling, the alloy becomes more disordered as indicated by the disappearance of the superlattice peaks in XRD and the increase in width of the peaks in the M¨ ossbauer spectrum. Milling leads to a greater density of defects and vacancies in the sample, so that effects due to Cr screening and dilution effect due to Al nearest neighbors on Fe is decreased, resulting in a higher average magnetic field for the MM sample. According to the Bethe-Slater curve, an increase in the nearest neighbor distance, in the case of Fe, would lead to an increased magnetic hyperfine field [33, 34]. M¨ ossbauer spectrum of the MM sample (Fig. 2.4) shows that the paramagnetic peak in the MM sample has a much lower intensity, which is reflected in the HFD also, despite the fact that Curie temperatures of both, the bulk and MM samples, are nearly the same. The main difference between the bulk and MM samples as observed from HFD is that the most dominant peak for the MM sample is at ≈9.8 T while it is the zero field peak for the bulk sample. The two high field peaks present in the bulk sample are also present in the MM sample. However, since the most intense peak in the HFD is at ≈9.8 T, which corresponds to the expected value for well-ordered Fe2 CrAl, it appears that the disordering produced due to milling leads to a more even distribution of Cr atoms, resulting in the disappearance of Cr clusters present in the

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Fig. 2.5. The M¨ ossbauer spectrum and hyperfine field distribution (HFD) of the sample MA for 15 h (MA). M¨ ossbauer spectrum and hyperfine field distribution (HFD) of the bulk sample is also given for comparison

bulk sample. The decrease in intensity of the paramagnetic component also supports this conclusion that the Cr atoms are redistributed. Thus, hyperfine studies on bulk and milled Fe2 CrAl systems leads to the conclusion that the Fe-Cr type disorder, leading to Cr clustering is decreased on milling, although random disorder increases. Room temperature M¨ ossbauer spectra and the hyperfine field distributions of the MA sample is given in Fig. 2.5. The spectrum consists of a paramagnetic singlet and a more pronounced sextet. The isomer shift of the singlet is again close to zero, as in the case of the bulk sample. The M¨ ossbauer spectrum and HFD of the bulk sample is also shown in Fig. 2.5 for comparison. Consistent with the highly disordered nature of the sample, the M¨ ossbauer spectrum of the MA sample is broad indicative of the distributed nature of the hyperfine fields. The MA sample has a higher ferromagnetic component, as is evident from the more prominent shoulders corresponding to the sextet. The magnetic hyperfine field distributions of the MA sample does not show any evidence for unalloyed Fe, and is also considerably different from that of the bulk and MM samples. From the HFD, it is seen that the zero field peak corresponding to the paramagnetic part of the spectrum of the MA sample has a lower probability when compared with that of the bulk, pointing to a more random distribution of the Fe, Cr, and Al atoms. Also, the low field peak around 10.0 T, which was present in both the bulk and milled samples is

2 Clustering in Heusler Alloys

31

absent in the MA sample. The absence of this peak in the MA samples cannot be attributed to nanocrystallization, since in the nanosized MM sample, this peak is the most prominent. Thus, the disappearance of this peak is entirely due to the higher random disorder present in the MA systems. M¨ ossbauer studies therefore show that the Fe environment in the nanosized MA sample is different from that of the bulk. The MA sample has two HFD components at 23.3 T and 15.5 T. The integrated intensities of the high field peak of the MA samples around 24 T is much higher than the integrated intensity of the 28.6 T component of the bulk sample. Thus, the probability of Fe atoms in the MA samples having less than four nearest neighbor Al or Cr atoms is much higher than that for the bulk sample. 2.3.3 DC Magnetization Studies Hysteresis loops of bulk Fe2 CrAl, MM, and MA samples obtained at 288 K are shown in Fig. 2.6. The bulk sample is weakly ferromagnetic, with low values of coercivity and retentivity (Table 2.1). On milling the bulk sample for 30 min, it is seen that both the coercivity and the retentivity increases. On the whole, milling leads to an improvement of the ferromagnetic properties, although the Curie temperature does not change appreciably for this sample. The changes in the magnetic properties of the MM sample, compared with the bulk, can be attributed to the nanosized nature of the MM sample and the consequent changes in the microstructure. Because of nanocrystallization, the fraction of atoms on the grain boundaries and interfaces increase. As seen from M¨ ossbauer studies, there is an increase in the disorder along with a reduction of Cr clusters, leading to a reduction of the nonmagnetic contribution.

Fig. 2.6. Hysteresis loops of Bulk, MM, ASM, and ANM samples shown together to indicate the change in saturation magnetization

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The hysteresis loop of the MM sample is typical of nanostructured materials with small magnetic domains [36]. The hysteresis loss in both, bulk and MM, samples is very small, indicating soft magnetic properties. The Curie temperature of the MM sample is slightly greater than that of the bulk, although the increase is not appreciable (Table 2.1). However, there is a considerable increase in the saturation magnetization of the MM sample, compared with that of the bulk, consistent with results obtained from M¨ ossbauer studies. The saturation moment per formula unit of the bulk is 1.08 µB , while that of the MM sample is 1.95 µB . The increase in saturation magnetization indicates an increase in disorder because of the milling process. During milling, the crystallite size decreases and a high density of defects are produced. XRD and M¨ ossbauer studies also indicate a higher degree of disorder in the milled system. Such defects cause an expansion of the unit cell, leading to changes in the geometry of the atomic arrangement and interatomic distances. These changes in the lattice cause changes in the exchange interaction and hence the spin alignment. In Fe-Al systems, it has been reported that the saturation magnetization increases with lattice parameter [37]. This has been attributed to the modification of the density of states at the Fermi level. Theoretical calculations of the density of states based on the band theory of ferromagnetism show that Fe2 CrAl falls on the Slater–Pauling curve [13]. It is predicted that well-ordered Fe2 CrAl would be a half-metallic ferromagnet and would have a total spin magnetic moment Mt = Zt 24, where Zt is the total number of valence electrons. Here it is assumed that the occupancy of the spin down bands does not change and the extra or missing electrons are taken care of by the spin up states only. In all full Heusler alloys, there are 12 occupied spin down states. Then, the total moment, which is equal to the number of uncompensated spins, is given by the total valence electrons minus two times the number of minority electrons. The total number of valence electrons in Fe2 CrAl is 25. Hence the total moment would be equal to 1 µB /formula unit [13]. Thus, well-ordered Fe2 CrAl is expected to have a saturation moment of 1 µB /formula unit and should exhibit half metallic properties. However, half-metallicity has not been experimentally reported in Fe2 CrAl, since the clustering of Cr leads to the formation of highly disordered alloys [14]. In the present study, in spite of the presence of antisite disorders in the bulk sample, the saturation moment is very close to the theoretically predicted value, pointing to the possibility of realizing half metallic behavior in bulk Fe2 CrAl. A modification of the density of states at the Fermi level due nanocrystallization and the subsequent expansion of the lattice would account for the enhancement of the total magnetic moment of the MM sample to 1.95 µB / formula unit from the 1.08 µB /formula unit for the bulk sample. The saturation moment of the milled sample is also very close to an integer number (≈2 µB /formula unit) but does not conform to the Mt = Zt 24 rule for Fe2 CrAl. To follow the SP behavior, theoretical calculations show that the local moment of Fe in ordered Fe2 CrAl is almost zero, while Cr has a moment of ≈1 µB . The

2 Clustering in Heusler Alloys

33

higher magnetic moment in the MM sample also implies that Fe has a higher local moment in this sample when compared with the bulk. Two simultaneous effects in the MM sample are responsible for the overall enhancement of magnetic moment in this sample. One is that the local moment of Fe atoms increases as the size decreases. The second is that the Cr clusters are reduced – as seen from M¨ ossbauer studies, thus reducing the number of isolated Fe atoms – again contributing to an increase in the effective saturation magnetization. The M-H loop of the MA sample is significantly different from that of the bulk and MM samples in that it exhibits enhanced bulk magnetic properties. The coercivity and the retentivity of the MA sample is much higher (Table 2.1). Although the saturation magnetization of the MA sample is very close to that of Fe, M¨ossbauer studies have completely ruled out the presence of unalloyed Fe in this sample. Thus, it appears that the size of the crystallites play an important role in determining the saturation moment, since the size of this sample is half that of the MM sample, while the lattice parameter is smaller. Expectedly, the degree of disorder in this system is also higher than that in the MM sample. Thus, the increase in the saturation moment in the MA sample compared with the MM sample, which is also nano sized, can be attributed to a higher disorder in addition to size effects. Another major difference between the MA sample compared with bulk and MM samples is the dramatic enhancement in the Curie temperature (TC ) of the MA sample. Both the bulk and MM samples have Curie temperatures very close to room temperature, while the TC of the MA sample is 887 K (Fig. 2.7). Since the MM sample, which has a Curie temperature comparable to that of the bulk, is also nanostructured, this enhancement in Curie temperature cannot be attributed to the nanocrystalline nature of the samples alone.

Fig. 2.7. M-T curve of nanosized Fe2 CrAl powders prepared by mechanical alloying

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A comparison of the magnetic properties of MM, MA and bulk samples show that the magnetic parameters of the MA sample are the highest. M¨ ossbauer studies show that the MA sample has a dominant high field component (≈24 T), which indicates an 57 Fe environment with fewer Cr and Al ions. Presumably the ferromagnetic coupling between the Fe atoms of this phase is strong and is probably enhanced due to the small grain sizes. Since the exchange stiffness is directly proportional to the Curie temperature, the dramatic increase in TC in the MA samples indicate that the spins in these systems are more strongly aligned compared with the bulk and MM samples. In conclusion, it is seen that the magnetic properties of nanostructured Fe2 CrAl alloys are dependent on both the size of the crystallites and on the degree of ordering in the samples. The most dramatic change is observed between the bulk and MM samples, since the enhancement in Ms is nearly 80% in the MM sample compared with that of the bulk, while the Curie temperature of both samples are nearly the same and very close to room temperature. In comparison, the saturation magnetization of the MA samples is only about 15% larger than the MM samples. However, the much higher TC for these samples makes them better for possible use as soft ferromagnets. Also, the nearly integral values of the saturation magnetization (in units of Bohr magneton) of the samples indicate the possibility of making Fe2 CrAl alloys exhibiting half metallic properties. Although mechanical milling/alloying expectedly induces more random disorders, it is also accompanied by a reduction in Cr clustering, leading to superior magnetic properties in the nanosized alloys. Acknowledgment We acknowledge UGC-DRS and DST-FIST programs at the physics department, M.L. Sukhadia University, Udaipur, India.

References 1. 2. 3. 4. 5. 6. 7. 8.

F. Heusler, Verh. Dtsch. Pys. Ges. 5, 219 (1903) K.H.J. Bushow, P.G. Van Engen, J. Magn. Magn. Mater. 25, 10 (1981) C.S. Lue, Y.K. Kuo, Phys. Rev. B 66, 085121 (2002) K. Kobayshi, R.Y. Umetsu, R. Kainuma, K. Ishida, T. Oyamada, A. Fujita, Appl. Phys. Lett., 85, 4684 (2004) ´ A. Slebarski, J. Deniszczyk, W. Borgie, A. Jezicrski, M. Swatek, A. Winiarska, M.B. Maple, W.M. Yuhasz, Phys. Rev. B 69, 155118 (2004) R.A. De Groot, F.M. Mueller, P.G. van Engen, K.H.J. Bushow, Phys. Rev. Lett. 50, 2024 (1983) C.S. Lue, Y.K. Kuo, S.N. Horn, S.Y. Peng, C. Cheng, Phys. Rev. B 71, 064202 (2005) K. Ullako, J.K. Huang, C. Kantner, R.C. OHandley, V.V. Kokorin, Appl. Phys. Lett. 69, 1996 (1996)

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9. J. Marcos, A. Planes, L. Ma˜ nosa, A. Labarta, B.J. Hattink, IEEE Trans. Mag. 37, 2712 (2001) 10. S.J. Murray, M. Marioni, S.M. Allen, OHandley, T.A. Lograsso, Appl. Phys. Lett. 77, 886 (2000) 11. I. Zutic, J. Fabian, Das Sarma S, Rev. Mod. Phys. 76, 323 (2004) 12. J.M.D. Coey, M. Venkatesan, J. Appl. Phys. 91, 8345 (2002) 13. I. Galanakis, P.H. Dederichs, N. Papanikolaou, Phys. Rev. B 66, 174429 (2002) 14. N. Lakshmi, K. Venugopalan, J.P. Varma, J. Phys. 59, 531 (2002) 15. M.C. Cadeville, J.L. Moran-Lopez, Phys. Rep. 153, 1153 (1987) 16. T.K. Nielsen, P. Klaven, R.N. Shelton, Sol. Stat. Commun. 121, 29 (2002) 17. Y. Miura, K. Nagao, M. Shirai, Phys. Rev. B 69, 144413 (2004) 18. N. Lakshmi, R.K. Sharma, K. Venugopalan, Hyperfine Interact. 160, 227 (2005) ´ 19. A. Slebarski, J. Phys. D Appl. Phys. 39, 856 (2006) 20. M. Zhang, E. Br¨ uck, F.R. de Boer, G. Wu, J. Magn. Magn. Mater. 283, 409 (2004) 21. V. Sebastian, N. Lakshmi, K. Venugopalan, J. Magn. Magn. Mater. 309, 153 (2007) 22. R.A. Dunlap, G. Stroink, Cand. J. Phys. 60, 909 (1982) 23. G.B. Johnston, E.O. Hall, J. Phys. Chem. Solids 29, 193 (1968) 24. K.R.A. Ziebeck, P.J. Webster, J. Phys. Chem. Sol. 35, 1 (1974) 25. C. Paduani, W.E. P¨ ottker, J.D. Ardisson, J. Schaf, A.Y. Takeuchi, M.I. Yoshida, S. Soriano, M. Kalisz, J. Phys. Cond. Mat. 19, 156204 (2007) 26. M. Kallmayer, H.J. Elmers, B. Balke, S. Wurmehl, F. Emmerling, G.H. Fecher, C. Felser, J. Phys. D Appl. Phys. 39, 786 (2006) 27. H. Xiao, I. Baker, Acta Metall. Mater. 43, 391 (1995) 28. L.S.J. Peng, G.S. Collins, Mater. Sci. Forum 235–238, 537 (1997) 29. G.K. Williamson, W.H. Hall, Acta Metall. 1, 22 (1953) 30. B. Window, J. Phys. E 4, 401 (1984) 31. S. Wurmehl, G.H. Fecher, H.C. Kandpal, V. Ksenofontov, C. Felsr, Appl. Phys. Lett. 88, 032503 (2006) 32. N. Lakshmi, V.K. Varma, J. Phys. Rev. B 47, 14054 (1993) 33. U. Herr, J. Jing, R. Birringer, U. Gonser, H. Gleiter, Appl. Phys. Lett. 50, 472 (1987) 34. U. Gonser, Hyperfine Interact. 68, 71 (1991) 35. K. Fukamichi, Appl. Phys. Lett. 85, 4684 (2004) 36. S. Azzaza, S. Alleg, H. Moument, A.R. Nemamcha, J.L. Rehspringer, J.M. Greneche, J. Phys. Cond. Mat. 15, 7257 (2006) 37. X. Amils, J. Nogu´es, M. Suri˜ nach, M.D. Bar´ o, J.S. Mu˜ noz, IEEE Trans. Magn. 34, 1129 (1998) 38. S. Chikazumi, The Physics of Ferromagnetism (Clarendon, Oxford, 1997)

3 Anisotropy of Ferromagnetic Heusler Alloys Thin Films R. Yilgin and B. Aktas Department of Physics, Gebze Institute of Technology, Gebze, 41400, Kocaeli, Turkey, [email protected], [email protected]

Summary. The Gilbert damping constant was investigated in Co2 MnSi Heusler alloy thin films by ferromagnetic resonance (FMR) technique. Samples were prepared using the magnetron sputtering technique on SiO2 substrate for polycrystalline and on a MgO(100) substrate for single crystalline and than annealed at various temperatures to control the structure. The magnetic properties such as g-value, effective magnetization magneto-crystalline anisotropy constant and uniaxial anisotropy constant and intrinsic Gilbert damping constant were obtained from angular dependences of resonance field and peak-to-peak line widths fitting from FMR spectra using Landau-Lifshitz-Gilbert equation. The minimum-damping constant was obtained for epitaxially growth film annealed at 300◦ C.

3.1 Introduction The discovery of the giant magneto-resistance (GMR) effect [1] and the large tunnel magneto-resistance (TMR) effect [2] at room temperature have attracted great attention to the spintronics studies. The TMR and the GMR are both very sensitive to the spin polarization of the magnetic electrodes. After understanding high spin polarization in some of the Heusler alloys [3–8], many scientists predict a large potential for applications of these alloys in spinelectronics devices, especially for producing magnetic random access memory (MRAM), and present-day communication industries. Therefore, intensive experiments have been carried out to determine the magnetic properties of Heusler alloys in recent years [9, 10]. Spin-injection magnetization switching, which was theoretically proposed [11,12], is one of the promising candidates for new recording scheme and magnetic storage such as MRAM, because this can realize a magnetization reversal of a ferromagnetic thin film by passing a current perpendicular to plane (CPP) spin valve without any magnetic field. As a result of a lot of efforts, there is a consensus on the nanosize mechanism that the driving force for magnetization reversal is the transfer of spin angular momentum from conduction electrons

38

R. Yilgin and B. Aktas

to the magnetization. However, this topic is still under the discussion. Particularly, the operating frequency limitations and the correlated magnitude of the writing current, I have a big importance and the studies on this topic gradually increase day by day [13]. Besides, current-induced magnetization switching (CIMS) has opened the possibility for writing the bits in MRAM. Most of the CIMS experiments have been performed in small metallic GMR elements with small magneto-resistance (several percents). In practical use, it is required to realize the CIMS in magnetic tunnel junction (MTJ) with a large TMR [14]. Then it has been recognized that the damping constant of ferromagnetic materials could be one of the most important magnetic properties for achieving high-speed magnetization switching for MRAM or Spin-RAM technology and reduction of critical current density for spin-transfer-driven magnetic reversal [4, 11, 15]. This chapter will present a study on half metallic Heusler alloys that are most promising candidate for spintronic applications. 3.1.1 TMR and GMR Effects When the external magnetic field is applied to a ferromagnetic substance, the change of tunneling current from one layer to the other depends on a relative angle between magnetization vectors of each ferromagnetic layer. The TMR ratio is calculated by using Julliere’s formula and according to that formula TMR (or GMR) ratio is given for FM/insulator/FM tunneling junctions [16] as RAP − RP 2P1 P2 TMR = = (3.1) RP 1 + P1 P2 where RAP and RP represent the resistivities of the sample for the direction of magnetization antiparallel and parallel to each other, respectively. Here P1 and P2 are spin polarizations of the two ferromagnetic layers at the Fermi level and given as D↑ (EF ) − D↓ (EF ) (3.2) P = D↑ (EF ) + D↓ (EF ) where Di s are density of state at Fermi levels of magnetic layers. Figure 3.1 shows the constructed magnetic tunnel junctions and memory cells of a MRAM. Each cell consists of an MTJ and the states “0” and “1” of the cell, respectively, correspond to the parallel (low resistance) and antiparallel (high resistance) configurations of the magnetic moments of two magnetic layers separated by a nonmagnetic (insulator) layer. Furthermore, each cell is connected together in a point contact array. The conducting wire lines (bit and word lines) provide the current to the junctions and permit voltage measurements for storing information [17]. As for giant magnetoresistance effect (GMR), structure is arisen from two ferromagnetic layers and nonmagnetic substance is inserted in between them. The change of the resistivity in ferromagnetic films depends on the

3 Anisotropy of Ferromagnetic Heusler Alloys Thin Films

39

Tunnel barries

Ferromagnetic electrodes

“1”

“0” Low resistance state

High resistance state

“Bit” lines

“1” “World” lines

“0”

Fig. 3.1. Memory cells of an MRAM [18]

direction of the magnetization. The GMR structure is like ferromagnetic layer/nonmagnetic layer/ferromagnetic layer. In case, the relatively magnetization is antiparallel state of two ferromagnetic layer, the resistance becomes bigger and then if external magnetic field is applied so that relatively magnetization can be parallel, then resistance reduces. GMR elements generally consist of spin-valve structure. A new generation of technological application could be magnetic sensors, based on GMR and spin-valve technology. Although the density of memory media (tera bits) goes up, the success in this field has lead to rapidly increasing activity in the development of high-density nonvolatile MRAM in future. Not only in GMR effect but also in TMR effect, the variation of resistivity can be observed according to parallel or antiparallel of magnetization sate of ferromagnetic layer. To use these effects in technological applications, one should control the relative orientation of magnetization vectors of neighboring magnetic layers. This can be done by application of external magnetic fields of electrical currents from one layer to the other. Moreover in both TMR and GMR effects, applying of current onto substance can be done by two different ways. One of them is the situation of the parallel electric current in plane (CIP), and other one is the current perpendicular to surface plane (CPP) [19]. 3.1.2 Critical Current The analytical calculation of Landau-Lifshitz-Gilbert (LLG) equation, considering the torque term acting between magnetizations of both electrodes separated by an insulator, shows that the critical currents to switch magnetization are given by [12–14, 20–22],

40

R. Yilgin and B. Aktas ± Ic0 = αeγMS Ad [H ± Han ± 2πMS ] /µB g(0,π)

(3.3)

where ± sign denotes direction. When a voltage is applied between the front and the back of the junction, the flux of polarized charged (injected spins) is carried from layer 1 into layer 2. The current through the junction depends on the angle between the quantization axes of layers 1 and 2. So, + and − signs are the current for the parallel (P, angle = 0) and P→AP antiparallel (AP, angle = π) orientations of the magnetizations, Ic0 and AP→P . Here α is the phenomenological intrinsic Gilbert damping constant Ic0 usually measured by ferromagnetic resonance experiments. The e is the electron charge, γ is the gyro-magnetic ratio, MS is the saturation magnetization of the spin-transferred layer. The A and d represent the area and thickness of spin-transferred layer, respectively. H, Han and 2πMS are the external field, the anisotropy field of the spin-transferred layer, and the related demagnetizing field, respectively. µB is the Bohr magneton and g(0,π) is the geometrical function depending on the spin polarization of the relative orientation (0 = parallel, π = anti-parallel) of the spin-transferred layer. As can be easily understood that for sufficiently small critical current or the critical current density, one must select an electrode materials with small a and MS . Nevertheless, spin polarization should be large because of the geometrical function (g(0,π) ) depending on the spin polarization. As a result, suitable materials must be searched experimentally because of no theoretical predictive method to determine the Gilbert damping constant to obtain the low switching current. As mentioned earlier, fast-switching and low switching current for current-induced magnetic reversals are very important in these devices. As can be seen in Table 3.1, orbital magnetic moments of Heusler alloy compounds is sufficiently small, and magnetic moment is localized on the Y atom as spin magnetic moment. So intrinsic magnetic damping parameter (α) of the Heusler alloys can be small, since the magnetic orbital moment is very small. Therefore, the Heusler alloys are expected to have a small-value and saturation magnetization, and also high spin polarization. In another words, they have a high potential to spin-dependent electronic devices applications. Table 3.1. Spin (mspin ) and orbital (morbit ) magnetic moments in µB for the some ferromagnetic full-Heusler alloy compounds [8] Some ferromagnetic full-Heusler alloys X: Co; Y: Mn, Cr; Z: Al, Si, Ge, Sn

Co2 MnAl Co2 MnSi Co2 MnGe Co2 MnSn Co2 CrAl

mX spin

mX orbit

mY spin

mY orbit

mZspin mZorbit

0.745 0.994 0.950 0.905 0.702

0.012 0.029 0.030 0.038 0.012

2.599 3.022 3.095 3.257 1.644

0.013 0.017 0.020 0.025 0.008

−0.091 −0.078 −0.065 −0.079 −0.082

3 Anisotropy of Ferromagnetic Heusler Alloys Thin Films

41

3.1.3 Theoretical Background Ferromagnetic Resonance The ferromagnetic resonance (FMR) experimental technique is commonly used to characterize the magnetic properties of ferromagnetic materials. Therefore, FMR was one of the earliest techniques to determine the fundamental properties of magnetic thin films. FMR technique is a very powerful experimental technique for studying the magnetic properties such as the g-factor, the effective magnetization, the effective magneto-crystalline anisotropy field or coefficient, and the effective uniaxial anisotropy parameter by analyzing FMR spectra [23–27]. The microwave losses or the absorption power in the resonance cavity directly proportional to measured permeability of the ferromagnetic materials of which are the resonance field and resonance line-width [28]. The line-width was simply defined as the field width at half height of the resonance absorption curve. The half-power line-width is derived from the shape of the resonance absorption curve, which is just a plot of the imaginary part of the susceptibility as a function of static field. In the macroscopic theory of FMR, different phenomenological equations are proposed for precession describing the shape of absorption curve. The equations proposed by Landau and Lifshitz and Gilbert are usually employed in theoretical studies. The equation proposed by Bloch–Bloembergen is frequently used in the study of the microscopic processes; and the approximated Lorentzian formula is used in experimental studies [23]. The line-width of the FMR spectra is directly related to the damping of spin dynamics and in-homogeneity including the defects, impurity, variant, porosity, vacant, etc. in the films. The intrinsic damping constant is generally calculated by analyzing the line-width of the FMR spectra. Even though the a-damping constant is extremely important for achieving high-speed magnetization switching, there are very few studies on a-Gilbert damping constant of Heusler alloy films [29, 30]. Although the intrinsic damping constant has been studied by several groups for thin films of various magnetic structures [23,27,31,32], the mechanism of magnetic damping has not been clearly understood yet. The energy losses due to damping and higher rates of remagnetization for weak driving fields can cause some technical problems in access memory applications in computers. The rate of remagnetization in ferromagnetic materials is determined by damping mechanism. Therefore, any effect influencing the damping mechanism also affects the remagnetization mechanism. When the external magnetic field is not strong enough to eliminate all domain walls, the domain structure plays a dominant role in the damping. That is, the local rate of the energy loss may vary by large amounts from point to point within a ferromagnet. Therefore, although many researchers have done investigation to understand the damping mechanism, its dominant mechanism has not been clarified yet [33].

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The intrinsic damping mechanism originates in microscopic spin–orbit interaction in a ferromagnetic material [25–27, 34, 35]. The anisotropy in the spectroscopic splitting factor is due to spin-orbit interaction as well. Therefore, g-factor is determined by using the FMR technique by simulation of the angular dependence of the resonant field [36, 37]. The determination of the g value with mentioned technique in the ultra-thin ferromagnetic material films looks like to be straightforward. The spin-orbit interaction determines the deviation of the Lande g factor from the free electron value (∆g = g − 2) and plays a dominant role in the damping mechanism. Furthermore, the Lande g factor and a small value for Gilbert factor G relate to the Elliott’s suggestion [38] G ∼ (∆g)2 . Note that spectroscopic splitting factor generally becomes isotropic, but sometimes an anisotropic g-factor should be taken into account in FMR [39]. An anisotropic g-factor causes an anisotropic Gilbert damping constant as well [40–42].

3.2 The Theory of Ferromagnetic Resonance 3.2.1 Dynamics of Magnetization Magnetization dynamic can be simply given by Landau–Lifthitz equation of motion as λ dM = −γ (M × H) − 2 [M × (M × H)] . (3.4) dt MS Here the first term is torque on the magnetization vector, M , due to effective field, H. The second term account for the relaxation of M , where the parameter characterizes the relaxation parameter. Since the solution of this equation is difficult, modified equation is given by using Gilbert-type damping [23, 43]
 α dM dM = −γ (M × H) + M× (3.5) dt |M | dt with Gilbert damping parameter, α = λ/γM . The line-width arising from the intrinsic spin relaxation effect is very small for an ideal crystal of some certain materials. For example, the line-width of single crystal yttrium iron garnet (YIG) is about 1 G at X-band frequencies. However, the FMR linewidth in most of the magnetic material system is very big, around 200 G. In fact, as it will be discussed in the next section in detail; the broadening of the FMR line-width is affected mainly from magnetic inhomogeneity i.e., crystal imperfections and thickness fluctuations or surface roughness are dominant factors for average spin relaxation effect within magnetic films. The effective Dc magnetic field acted on the magnetization vector can be derived from the gradient of the magnetic free energy, F , with respect to magnetization vector. To derive the dispersion relation, the equation of motion (3.5) for uniform mode analysis will be solved.

3 Anisotropy of Ferromagnetic Heusler Alloys Thin Films

z

43

H M qH

q y h~

ϕ ϕ

H

X

Fig. 3.2. Relative orientation of the sample, applied static magnetic field H, Static magnetization M

Furthermore, the equation of motion for the magnetization without a damping term can be written as [44]: 1 ˙ M − M × H = τ = −nm × ∇F (θ,ϕ) γ

(3.6)

ˆ where nm = M /|M | is with ∇F (θ,ϕ) = −∂F/∂θ · θˆ + (1/ sin θ)∂F/∂ϕ · ϕ, unit directional vector for the magnetization, M is assumed to be constant in magnitude and also θˆ and ϕˆ are unit vectors in the θ and ϕ directions, respectively, in the polar coordinate system. ∂F/∂θ denotes the partial derivative with respect to θ. The total energy of ferromagnetic system F includes all magnetic energy. In the presence of a small ac magnetic field that is applied perpendicular to the DC field, the magnetization vector M precesses about effective DC field and precession amplitude shows a resonance at a frequency, ω, given by [28, 45–48]   1 ω 2 = Fθθ Fϕϕ − Fθϕ . (3.7) γ M sin θ Here θ and ϕ are the polar angles of M . The relative orientations of external field, magnetization M , and references coordinate system with respect to the magnetic film are shown in Fig. 3.2. The first derivatives of the total free energy is equal to zero in the equilibrium position and we can take the form: ∂F ∂F = = 0. ∂θ ∂ϕ

(3.8)

The damping parameter of the magnetization precession, α, which is usually 1, is omitted for simplicity. The resonance condition is reached either sweeping the external microwave frequency or external dc magnetic field, which is

44

R. Yilgin and B. Aktas

more frequently used to get field for resonance, HR . We give a detailed discussion on the uniform mode resonance theory in thin films; and in this mode, all spins are parallel to each other and make a precession uniformly. Therefore, to find the general equation for the resonance frequency using (3.7), it is necessary to define the total free energy expression for unit volume of the ferromagnetic material that depends on the orientation of the magnetization. The expression for free energy density of a ferromagnetic specimen can be expressed as (3.9) FT = FZ + FD + Fani + . . . Here, FZ is the Zeeman energy density, FD is the demagnetization energy density, and Fani = is the anisotropy energy density including all components, such as the magneto-crystalline perpendicular, and uniaxial, cubic anisotropy terms in addition to unidirectional exchange anisotropy and some anomalous anisotropies like oblique (geometric) anisotropy originating form deposition geometry. Commonly perpendicular (referred to the normal to the film plane) anisotropy constant is denoted as either KU or K⊥ [49–51]. We choose the last one in order to avoid confusing with the uniaxial anisotropy. Therefore, perpendicular anisotropy expression can be written as F⊥ = −K⊥ cos2 θ .

(3.10)

3.2.2 Resonance Field for Polycrystalline Film The coordinate system shown in Fig. 3.2 (out-of-plane geometry where the field rotated with respect to film normal) was used to obtain the resonance frequency for the out-of-plane geometry. The total energy density is assumed as F = −MS H[sin θH sin θ cos(ϕH − ϕ) + cos θH cos θ] + 2πMS2 cos2 θ − K⊥ cos2 θ .

(3.11)

In the out-of-plane geometry, the azimuthal angle set to zero, ϕ = ϕH = 0, and polar angle θH is varied. So in the equilibrium position, the first partial derivative of the energy density, ∂F/∂θ, must be equal to zero. Under that condition, we get the following equation: sin 2θ =

2HR sin (θ − θH ) . 4πMeff

(3.12)

In this equation, 4πMeff is defined by the saturation magnetization and perpendicular anisotropy K⊥ as 4πMeff = 4πMS −

2K⊥ . MS

(3.13)

3 Anisotropy of Ferromagnetic Heusler Alloys Thin Films

45

By calculation the Fθθ , Fϕϕ and Fθϕ using (3.11) and substituting them into (3.7), the resonance frequency can be obtained  2 ω = H1 × H2 , γ

(3.14)

where H1 and H2 are the abbreviations as H1 = HR cos (θH − θ) − 4πMeff cos2 θ

(a)

H2 = HR cos (θH − θ) − 4πMeff cos 2θ

(b) .

(3.15)

This expression should be modified to include all energy (anisotropy) terms. So, angular dependence of the resonance field can be found out using the written a computer program. 3.2.3 Ferromagnetic Resonance in Single Crystalline Film Magneto Crystalline Anisotropy Many magnetic phenomena in nanoscale structures are strongly affected from the interaction of orbital magnetic moments and spin moments. The spinorbit interaction bonds the electron spin and orbital momentum together and makes the energy of atomic magnetic moments depending on their orientation with respect to crystalline axes [52]. Actually, the exchange and the dipolar interaction could also contribute to the magneto crystalline anisotropy. However, the exchange interaction cannot affect the anisotropy, because exchange energy result from scalar product among the spin vectors. Therefore, exchange interaction is independent of the angle between spins and the crystal axes. In addition, dipolar interaction energy relates the orientation of the magnetization with the crystalline axes. However, interaction for cubic crystal can be shown by symmetry arguments where the sum of dipolar interaction energies is ignored. As a result, spin-orbit interaction primarily is responsible for the magneto-crystalline anisotropy for cubic crystal [53]. It is an experimental fact that ferromagnetic single crystals have easy and hard directions of magnetization; i.e., the energy required to magnetize the crystal depends upon the direction of the applied magnetic field relative to the crystalline axes. The difference between the energies required to magnetize the crystal in the hard and easy direction is called the anisotropy energy. Polycrystal ferromagnetic metals appears as isotropic but the single crystals such as nickel, iron, and their alloys prepared in the absence of a magnetic field has a cubic symmetry. Magneto crystalline anisotropy for cubic crystal (fourfold symmetry) can be defined by [49, 54] FK1 = K1 (α21 α22 + α22 α23 + α23 α21 ) .

(3.16)

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R. Yilgin and B. Aktas

In that equation α1 , α2 , α3 indicate the directional cosines with respect to the cubic axes and K1 is the magneto-crystalline anisotropy constant. In that equation the higher order parameters may be also added, but generally not so much necessary. The coefficient of K1 depends on the temperature and K1 > 0 means that the minimum anisotropy energy for cubic structure in the easy axis corresponds to the [100] easy axis direction of crystalline axis. K1 < 0 is is hard axis direction but perpendicular the easy plane [55]. Uniaxial Anisotropy Energy Actually so many effects may cause the uniaxial anisotropy, but one of them for magnetic thin films is the origin of this uniaxial magnetic anisotropy in which an easy axis is developed in the direction of the field give rise to an uniaxial anisotropy when a static magnetic field is applied parallel to the substrate during deposition of a thin film. The contributing sources to this uniaxial magnetic anisotropy are the following: (a) pair formation, (b) preferential imperfection orientation, (c) magnetostriction due to induced stress, (d) local variation (dislocation), (e) variants, and (f) the vapor beam angle of incidence effect [56]. Additionally for some of the films, when the substance makes a magnetic anneal at elevated temperatures under the enough length of time, their easy magnetization direction come from when the deposition changed. The origin of this rotatable anisotropy is of the former mentioned mechanism that is responsible for the uniaxial anisotropy in thin films. To determine the magnetic anisotropy, a lot of methods are used. The most common techniques that can be mentioned are the torque magnetometer, the hysteresis loop, and ferromagnetic resonance technique. The uniaxial anisotropy energy is given as [4, 51] Fu = Ku sin2 θ sin2 (ϕ − φu ) . (3.17) The general expression of the uniaxial anisotropy energy is a function of θ, ϕ, and phase angle φu . All angles were defined according to the Fig. 3.1. The subscript u refers to uniaxial and Ku is constant, which depends on the temperature. In most cases, higher-order coefficient is negligible and in many experiments, it can be analyzed using the first term only. The constant of Ku becomes sometimes negative, sometimes positive. Meaning of positive constant is the easy axis direction regarding [100] crystallographic axis. Angle of φu refers the deviation angle from [100] crystalline axis [55]. Resonance Field In single crystalline films, FMR measurement should be performed both inplane and out-of-plane geometry since they have a variation of resonant field anisotropy in the in-plane geometry position. Therefore, taking into account the cubic structure [57, 58], the density of free energy can be modeled as a following

3 Anisotropy of Ferromagnetic Heusler Alloys Thin Films

47

F = −MS H[sin θH sin θ cos(ϕH − ϕ) + cos θH cos θ]   + (2πMS2 − K⊥ ) cos2 θ + (K1 /4) sin4 θ sin2 2ϕ + sin2 2θ + Ku sin2 θ sin2 (ϕ − φu ) .

(3.18)

As mentioned above, last equation must be examined and determined as an inplane and out-of-plane. Therefore, the in-plane FMR equations can be derived easily using the simple configuration in single crystalline films having cubic structure. In case θ = θH = π/2, the ϕ and ϕ dependencies of the second derivative energy expression and regarding the effective magneto crystalline anisotropy field, HK1 = (2K1 )/MS , and the effective uniaxial anisotropy field, HKu = (2Ku )/MS , resonance condition can be further simplified as  2
  HK1  ω 2 2 2 − sin 2ϕ − HU sin ϕ = HR cos (ϕ − ϕH ) + 4πMeff + γ 2 × [HR cos (ϕ − ϕH ) + HK1 cos 4ϕ + HU cos 2ϕ] .

(3.19)

Resonance field, HR , depends on a few parameters as seen in that expression. In-plane angular dependence of resonant field experimental results will be fitted with the theoretical results calculated using (3.19). If we continue using the same procedure to get the theoretical solving for out-of-plane geometry, the dispersion equations follow that; in case of ϕ = ϕH = 0 or π/2  2

ω = HR cos(θ−θH )−(4πMeff −HK1 cos 2θ−HU ) cos 2θ−HU −HK1 sin2 2θ γ

× HR cos (θ − θH ) − (4πMeff − HK1 cos 2θ − HU ) cos2 θ + HK1 sin2 θ , (3.20)

and ϕ = ϕH = π/4    2  HK1 ω 2 (2 cos 2θ+sin θ) cos 2θ = HR cos(θ−θH )− 4πMeff − γ 2 HU 5 cos2 θ − HK1 sin2 2θ+ 4 2 



× HR cos(θ−θH )−(4πMeff −

 HK 1  2 cos 2θ+sin2 θ) cos2 θ 2

HU cos 2θ . −HK1 sin θ+ 2 2

(3.21)

3.2.4 Line-Width of Resonance Absorption The line width of the resonance absorption is one of the most important and difficult question in the ferromagnetic resonance. Knowledge of the origin

48

R. Yilgin and B. Aktas

of the relaxation mechanisms is too important for spintronics applications as well. The vast majority of such FMR studies have concentrated on the magnitude of the resonance field, HR , and its dependence on such variables as the field orientation, the sample thickness, the temperature, and so on. Relatively very few attentions have been paid to the peak-to-peak line-width of the FMR spectra, ∆HPP . Generally ∆HPP directly relate the sample’s structural and magnetic quality [59, 60]. The narrowest line width has been measured in single crystalline yttrium iron garnet with ≈0.1 Gauss. When analyzing the causes of the broadening of line width of the ferromagnetic substance, it is necessary to consider some parameters such as nonuniformity of internal magnetic field, structural inhomogeneity, porosity, and samples’ (surface) roughness. The effect of the porosity on FMR line width for same polycrystalline ferromagnetic materials was explained in detail in [61–63]. ∆HPP consists of homogeneous and in-homogenous parts. The line broadening usually depends on the details of the sample preparation, film thickness, and the growth temperature. For parallel FMR configuration case, where the static magnetization and the applied magnetic field are in the film plane, the angular independent FMR line-width (using the full-width at half maximum (FWHM) of the absorbed microwave power) is given by 2 G ω, ∆HPP = ∆Hin hom + √ 2 3 γ MS

(3.22)

where the frequency-independent line broadening, ∆Hin hom , is caused by the magnetic inhomogeneities and therefore its origin is not intrinsic but extrinsic. Here γ = gµB /
is the gyro-magnetic ratio, G denotes the Gilbert damping related to damping parameter as G = αγMS and ω = 2πf is the frequency of microwave. The Gilbert damping coefficient G in metals arises from the spin orbit interaction of itinerant electrons [31, 38, 41, 42, 64, 65]. The contribution to the peak-to-peak line width from intrinsic damping α , (for homogeneous contribution) is angle of magnetization precession, ∆HPP dependent and can be written in a general formula [47, 66, 67]  2  α 2 1 ∂2F ∂ F 1 α

, (3.23) ∆HPP (θ, ϕ) = √

+ sin2 θ ∂ϕ2 3 ∂(ω/γ)

MS ∂θ2 ∂HR

where θ and ϕ are the polar angles of magnetization vector, while F is the magnetic free energy. Homogeneous line broadening is closely related to the damping constant, α. To go further we can write (3.23) depending on H1 and H2 by using (3.11), (3.12), and (3.23), and homogeneous contribution to the experimentally observed line width is obtained as 



d(ω/γ) −1 1 α

∆HPP = √ α (H1 + H2 ) . (3.24) dHR 3

3 Anisotropy of Ferromagnetic Heusler Alloys Thin Films

49

The inhomogeneity part of the experimental peak-to-peak line width due to only isotropic magnetic media also angular dependent and given as [34,59,68]




dHR 1 dHR

in hom

∆ (4πM ∆θ (3.25) = √

) + ∆HPP eff

dθH H . 3 d (4πMeff ) The first and the second terms represent the dispersion of the magnitude and direction of 4πMeff , respectively. In √ this expression, the derivatives are evaluated at the resonance field HR . 1/ 3 is the correction factor of the difference between the FWHM and peak-to-peak line width for Lorentzian line shape. It should be noted that this correction factor has been omitted in some papers [62]. If magneto-crystalline cubic and/or uniaxial anisotropy is taken into account considering the either (3.20) or (3.21) for (ω/γ), it is necessary to add new in-homogeneity terms to the peak-to-peak line width, (3.25). That is, the general form of line-width expression should be given as [69–71] intrinsic in hom in hom ∆HPP = ∆H0 + ∆HPP + ∆HPP ∆ (4πMeff ) + ∆HPP ∆θH in hom in hom + ∆HPP ∆HK1 + ∆HPP ∆HU .

(3.26)

Here, ∆H0 is the so-called frequency-independent contribution to the linewidth and generally related to lattice defects, the thermal history of sample and preparation process [42, 69–71]. The additional terms, fifth and sixth, in expression 26 are inhomogeneous contribution to the line-width from possible variations in the magnitude of the magneto-crystalline cubic and axial anisotropies, respectively. The magnitude of the cubic crystalline anisotropy contribution can be formulated as

∂HR ∂ (ω/γ) in hom

∆HK = (3.27)

∂ (ω/γ) ∂HK ∆HK1 1 1 while contribution form axial anisotropy is also given by

∂HR ∂ (ω/γ) in hom

∆HU .

∆HU = ∂ (ω/γ) ∂HU

(3.28)

According to the Slonzcewski model [11] and Fig. 3.3, the spin current-induced magnetic switching equation for free layer magnetization vector is written as Ie g α ∂M 2 ∂M 2 = −γM 2 × H eff − M2 × + M 2 × (M 2 × M p ) . ∂t |M 2 | ∂t e |M 2 | (3.29) In that equation, the first term at the right-hand side is precession torque due to effective static field, the second and the third terms, respectively, represent the Gilbert type intrinsic damping in free layer and spin transfer torque from pinning layer through spin injection current. Here α is the intrinsic damping factor, Ie critical current, e charge of electron, and g is spin efficiency (the

50

R. Yilgin and B. Aktas

Ferromagnetic Layer

Ferromagnetic Layer

Current Driven Magnetization Switching

dM dt

= M

gM x H +

ga M

M x (M x H ) + G

H M

H

Spin Torque

Fig. 3.3. Spin-current induced magnetization switching between Ferro1/spacer/ Ferro2 layers

geometrical function of spin polarization). The M2 is the magnetization vector of free layer (with small anisotropy) in Fig. 3.3. In case of the negative sign, critical current, it contributes to the damping torque. The angular momentum carried by a spin-current passing through a thin magnetic film exerts a spintransfer torque on the film magnetization. If the torque is strong enough, the magnetization switches. Spin-current induced magnetization switching is observed in L1 /spacer/L2-structures in Fig. 3.3, where the ferromagnetic layer (L1 ) acts as polarizer, the nonmagnetic metal as a spacer to avoid direct exchange between the magnetic layers, and L2 is the layer being switched.

3.3 Introduction The thin films of Heusler alloy, Co2 MnSi, have been prepared by sputtering technique in both poly and single crystalline forms. Then the physical properties have been studied by using usual X-ray measurements to determine the crystalline structure, the SQUID and VSM measurements for getting saturation magnetization and coercivity, and ferromagnetic resonance measurements, to obtain the intrinsic Gilbert damping parameter in addition to dc magnetic properties such as magnetic anisotropies. 3.3.1 Sample Preparation Polycrystalline Films The 50-nm thick films of Co2 MnSi alloy were prepared on a surface treated (oxidized, SiO2 ) polycrystalline Si substrate. The base pressure was kept less than 5 × 10−7 Pa to improve the crystalline structure and atomic order. First, a magnetic Co2 MnSi layer of 50-nm thick layer and then a very thin (5 nm) Ta capping layer were deposited on Co2 MnSi alloy by using usual magnetron sputtering technique at 0.1 Pa Ar pressure. The “as-deposited” films were

3 Anisotropy of Ferromagnetic Heusler Alloys Thin Films

51

Θ Fig. 3.4. X-ray diffraction patterns of Co2 MnSi film annealed at different temperatures indicated on each curve [72]

annealed at various temperatures in a high vacuum (less than 10−7 Pa) furnace to control the structure and the atomic order between Co, Mn, and Si sites. Both “as grown” and annealed films at various temperatures were studied. Figure 3.4 shows X-ray diffraction patterns of Co2 MnSi films annealed at some selected temperatures, TA , indicated above relevant curves. Here X-ray is rotated from substrate plane toward normal to the substrate plane (out-ofplane geometry). The region of the curves at around 70◦ contains very strong peaks for SiO2 substrate and was removed for just clarity. The Co2 MnSi films, as deposited and annealed below 200◦ C, did not show any peak because of amorphous structure. X-ray diffraction method confirmed Heusler structures of the prepared films. The patterns for the film annealed above 300◦ C exhibit clear peaks indicate to B2 structure. The strong (200) peak of B2 structure, revealing partial disorder (interchange, that is a substitution) between Mn and Si sites, was observed in diffraction patterns of the films annealed at over 300◦ C as well. However, the (220) peak intensity is very small, which indicates a negligible mix from A2 phase. It should be noted that the peaks (111) for L21 structure were not observed. Figure 3.5 shows the size of the grains for polycrystalline Co2 MnSi films calculated using Scherrer law [73,74]. As can be seen from this figure the grain size mainly increases with increasing annealing temperature after 400◦ C. Single Crystalline Co2 MnSi Films Epitaxial 30 nm thick Co2 MnSi-(001) Heusler alloy films were prepared on MgO-(001) substrate with 40 nm thick Cr(001) seed layer by using magnetronsputtering technique. The epitaxial Co2 MnSi films were grown on selected single crystal MgO (001) substrate as an orientation relationship of MgO[100] Cr[110]Co2MnSi[110] direction at a–b plane as seen in Fig. 3.6. After deposition of Co2 MnSi films in ambient temperature, the films were annealed at

R. Yilgin and B. Aktas

⬚

52

⬚ Fig. 3.5. The size of the grain for polycrystalline Co2 MnSi films vs. annealing temperature. The size of the grain was calculated using Scherrer law

[010]

[100] Mg O

Mg-Mg, O-O = 2.98 Å

Co2MnSi a = 2.88 Å

Fig. 3.6. Geometry of Heusler alloy films on MgO substrate

300, 400, and 500◦ C to improve the crystal structure and the atomic order between Co, Mn, and Si sites. The XRD profiles for the as-deposited and temperature-annealed Co2 MnSi films are shown in Fig. 3.7. In this figure, only the (200) Cr peak and (200) and (400) Co2 MnSi peaks were detected, beside the peaks for MgO substrate. This indicates perfect (100)-preferred orientation of Heusler Co2 MnSi films. The (200) peak, which is clearly observed for all samples, corresponds to B2order structure in which Mn and Si are substituted by themselves randomly. The (200) peak intensities of annealed samples are slightly larger than that of the “as-deposited” sample. The small inset in Fig. 3.14 portrays the pole figure in-plane X-ray measurements (X-ray is rotated in sample plane) for 400◦ C annealed films, reveals completely atomic order among all Co, Mn, and Si sites. The “as-deposited” and annealed films at 300◦C do not exhibit (111)

3 Anisotropy of Ferromagnetic Heusler Alloys Thin Films

53

f

Q Fig. 3.7. XRD patterns of as-deposited and postannealed Co2 MnSi films at 500◦ C. Inset shows in-plane pole figure X-ray diffraction [27]

peaks. This means that crystal structure of films changes from B2 to the L21 structure beyond that temperature. The Co2 MnSi crystal is epitaxially grown along the normal to the substrate plane and as epitaxial relationship [110]Co2 MnSi[110]Cr[100]MgO as evidenced from in-plane pole-figure measurements [75, 76]. 3.3.2 Magnetic Characterizations DC Magnetization Measurements Hysteresis curve measured at room temperature (RT) using SQUID magnetometer for Co2 MnSi film. Hysteresis curve measured at RT using SQUID magnetometer for Co2 MnSi film annealed at different temperature were plotted in Fig. 3.8. From this curves, the ferromagnetic nature of the Co2 MnSi is evident. The curve is clearly temperature-dependent as well. At highest annealing temperature, the hysteresis curve is broaden and even exhibit a small step at lower field (about 10 Oe) dependence, indicating to an inclusion of an additional very soft magnetic phase probably originating from further growth of crystallites (grain) of Co2 MnSi in B2 structure. The step height relative to the saturation value of the magnetization might be taken as a measure of the percentage of this second magnetic phase. Hysteresis curves of Single Crystalline Co 2 MnSi Films were recorded by sweeping the external field in (110) direction, and it is plotted in Fig. 3.8b for different annealing temperatures. As seen in that figure, the rectangular hysteresis loop is much narrower compared with the polycrystalline films of the same compound. Figure 3.9 shows annealing temperature dependence of saturation magnetization (MS ) and coercivity (HC ) of Co2 MnSi films. In this figure, the rectangular and open circles represent saturation magnetization and coercive

R. Yilgin and B. Aktas

Ms (emu/cc)

54

Ms (emu/cc)

(a)

(b)

Magnetic Field (Oe)

Hc (Oe)

Ms (emu/cc)

Fig. 3.8. (a) Hysteresis curves of polycrystalline Co2 MnSi film annealed at some selected temperature. (b) Hysteresis curves of epitaxial Co2 MnSi film for easy and hard axis of film plane annealed at 300◦ C

Annealing Temp. (⬚C) Fig. 3.9. Annealing temperature dependence of saturation magnetization, the rectangular, and coercivity, the open circle, of polycrystalline Co2 MnSi film

3 Anisotropy of Ferromagnetic Heusler Alloys Thin Films

55

Fig. 3.10. Annealing temperature dependence of saturation magnetization (MS ) and coercive field (HC ) of epitaxial Co2 MnSi. The open up-triangle and open circle lines depict the coercivity and the saturation magnetization variations, respectively

field, respectively. Here the experimental data points are connected by continuous line for eye guide. As seen from this figure, the saturation magnetization increases with increasing annealing temperature except 400◦ C. However, the coercivity monotonically increases with increasing annealing temperature. The saturation magnetization Ms and coercivity HC of single crystalline Co2 MnSi films are plotted as a function of the annealing temperature in Fig. 3.10. As seen from this figure, the MS increases with increasing temperature, and passes a maximum around 300◦ C, even though it is smaller than the bulk value of 1,020 emu.cm−3 , and then decreases beyond this temperature. The small MS value of these films represents the disorder between Co and Mn sites. In addition, the chromium atoms from the seed layer are likely to diffuse into the Co2 MnSi film layer at 500◦C. In contrast, the curve for coercivity behaves opposite to the magnetization. This is consistent to the fact that any defect and inhomogeniety creating local magnetic anisotropy can reduce saturation magnetization as well. Obviously the temperature annealing improves or degrades the crystalline order depending on the annealing temperature. And thus the maximum in MS (or minimum in HC ) can be accounted for by this fact. 3.3.3 FMR Results FMR measurement was carried out using an X-band (9.7 GHz) microwave source, and the FMR spectra recorded for different orientation of applied magnetic filed with respect to sample normal in out-of-plane geometry (OPG), where the magnetic field is rotated from film normal toward film plane. Angular dependence of FMR spectra and numerical analyses were performed for the OPG to evaluate magnetic properties of Co2 MnSi films for different annealing temperatures. Used coordinate system and relative orientations of relevant

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Fig. 3.11. Typical out of plane angular dependence of resonance field of Co2 MnSi film annealed at 400◦ C. The inset shows peak-to-peak line width (∆HPP ). Open circles and solid lines, respectively, represent experimental and fitted values. The dashed line is obtained from intrinsic damping parameter as explained in the text

vectors are shown in Fig. 3.2. M , H, h indicate the vectors of magnetization, the external DC magnetic field, and the external microwave field, respectively. H lies in the Y–Z plane – its direction with respect to z-axis is denoted by θH . The direction of h is parallel to the X-direction. The FMR spectra were recorded at orientation (θH ) of the external field. The FMR spectra for each orientation mainly consist of a broad and strong resonance absorption. Figure 3.11 shows −θH dependences of the resonance field (HR ) and the peak-to-peak line-width (∆Hpp ) of FMR curves of Co2 MnSi film annealed at 400◦ C. The open circles denote experimental data, while the solid line represents fitted values in Fig. 3.11. As seen from this figure, there is a sufficiently good agreement between experimental and fitted values. The resonance field (HR ) was obtained from dispersion equation given by (3.15). Thus magnetic parameters such as ω/γ (where γ = gµB /
), effective magnetization Meff (which is abbreviated as MS − (K⊥ /2πMS ) were obtained. From the computer analysis of the experimental data by using the theoretical expression (3.7) for resonance field, the effective g values, dc magnetization, and perpendicular anisotropy were deduced, respectively. By comparison to the dc magnetization results, it was found that perpendicular anisotropy is almost negligible (zero), while the temperature-independent g-factor is about 1.91 (HR = 3530 G). The angular variation of the peak-to-peak line width (∆HPP ) of resonance curve of FMR spectra of Co2 MnSi films is given in the inset of the same figure. The circles represent experimental values, while the continuous curve denotes fitted values by using total peak-to-peak given by (3.24) and (3.25). There is almost perfect agreement between the experimental and the fitted values. As explained in theoretical section, the intrinsic damping parameter is related to

3 Anisotropy of Ferromagnetic Heusler Alloys Thin Films

57

×

×

Fig. 3.12. Annealing temperature dependence of damping parameter for polycrystalline Co2 MnSi films

the line-width of FMR spectra. Therefore, the α damping parameter was evaluated by analyzing experimental results of out-of-plane angular dependence of ∆HPP of FMR spectra. The dashed line in the inset of the Fig. 3.9 represents the intrinsic damping contribution to the line width. It was concluded that the other parameters representing the inhomogeneities might be practically taken as zero. We have deduced a value of 60 Oe for frequency independent extrinsic inhomogeneity parameter ∆H0 in (3.26). Figure 3.12 shows the annealing temperature dependency of the evaluated α-damping constant and α × MS for Co2 MnSi films. The damping parameters (α) slightly change with increasing annealing temperature but they were almost the same for polycrystalline films and quite large in comparison with those of polycrystal Co2 MnAl [77]. By increasing the annealing temperature, the damping constant decreased and showed a minimum value, 0.022, around 400◦ C. As can be seen the Gilbert damping constant expression G = αγMS in the reference [38, 39], the α-Gilbert damping constant is expected to be proportional to 1/Ms [35, 78]. However, Fig. 3.12 shows that α × MS also showed annealing temperature dependency as α does. Therefore, in this case, MS does not have considerable effect on the α-value. It should be noted that α-damping constant of Co2 MnSi film was increased by annealing temperature in parallel to grain size, which increases the inhomogeneity of magnetization. The large α-value can arise from the effect of inhomogeneity or porosity [79–81] because the pores at the grain boundary cause nonuniformity in the internal magnetizing field or nonuniform demagnetizing field. As can be inferred, the observed α-damping constant of Co2 MnSi resulted from not only intrinsic (small spin-orbit interaction) but also extrinsic contributions.

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Single Crystalline Co2 MnSi Heusler Alloy In this section, single crystalline Co2 MnSi Heusler alloy thin films will be touched. All FMR measurements were performed both in-plane and outof-plane geometry which is shown in Fig. 3.13a, b. The in-plane (ϕH ) and out-of-plane (θH ) angular dependences of the resonance field (HR ) and line width (∆HPP ) of FMR measurements were fulfilled and recorded for different orientation of applied magnetic field with respect to crystal planes. Analyses were carried out using the LLG equation regarding to the cubic crystalline and uniaxial magnetic anisotropy effect, which can be expressed in (3.18). The in-plane angular (ϕH ) dependence of HR was measured to determine the crystalline axis direction and to get exact magnetic properties especially correct g-value (splitting factor). Also it is possible to investigate the crystalline direction dependence of the damping constant. To reveal the magnetic axes with respect to crystalline (structural) axes, first, in-plane angular (ϕH ) variation of resonance field (HR ) was fulfilled. The axes definitions of MgO substrate and films are denoted in Fig. 3.6. Therefore, easy and hard axes are determined by providing the rotation of the sample around vertical axis in static dc field lying in horizontal plane between the two poles of the electromagnet. Therefore, FMR spectra were recorded as a function of azimuthal angle (ϕH ) of external magnetic field. The small inset of Fig. 3.13a depicts in-plane geometry for FMR measurements. Out-of angle θ and θH values of magnetization and external magnetic field vector, respectively, were assumed as π/2. Out-of-plane geometry for FMR

Fig. 3.13. Angular dependence of resonance field (HR ) of FMR spectra of single crystal Co2 MnSi film annealed at 300◦ C. (a) For in-plane measurements, (b) for the out of plane geometries. Two curves for out of plane geometries represent the angular variation of the Hr when the field is rotated from either from the easy [100] or hard axis [110] is sample plane as indicated on the relevant curves

3 Anisotropy of Ferromagnetic Heusler Alloys Thin Films

59

measurements is shown in the Fig. 3.13b. Moreover, in the case of out-of-plane geometry, the external magnetic field first was set to the easy axis direction in film plane and then the field was rotated toward normal to the film. Figure 3.13a shows the in-plane geometry angular dependence of the resonance field for the Co2 MnSi film annealed at 300◦ C. In this figure, the circles indicate experimental values and the solid line denotes fitted values. As seen from this figure, there is a very good agreement between the experimental and calculated values. From the fitting, we have deduced the g-value, effective K⊥ magnetization Meff = MS − 2πM , (were K⊥ represents any induced perS pendicular anisotropy), effective magneto-crystalline anisotropy components (fields) such as, cubic K1 , HK1 = 2K1 /MS , and in-plane uniaxial anisotropy, KU , HU = 2KU /MS . Figure 3.13b shows angular dependence of the resonance field of Co2 MnSi film for out-of-plane geometry where the external magnetic field is rotated from either crystallographic axis [100] or [110] toward film normal [001]. It should be mentioned that we have used the same deduced values to fit the experimental data for both geometries. The deduced values for different parameters are given in Table 3.2. Obtained g-value for “as-deposited” and annealed at 300◦C samples are 1.91, which is significantly smaller than 2.0023 of free electron. However, the value of the sample annealed at higher temperature is even greater than that of the free electron. This indicates an essential modification of spin-orbit interaction or a need to consider higherorder anisotropy terms that obviously can vary with change of crystalline microstructure due to temperature annealing. As seen from the Table 3.2, magnitude of the magneto crystalline anisotropy coefficient changes by annealing that relaxes the crystalline structure. First, the cubic anisotropy, K1 , initially increases with annealing until 300◦C, and then sharply decreases beyond that temperature at which the uniaxial Table 3.2. Magnetic parameters were obtained from fitting to the angular dependence of resonance fields and the angular dependence of line-width of both in-plane and out-of-plane FMR spectra

g-value MS (emu/cm3 ) Meff (Oe) HC (Oe) K1 (erg/cm3 ) KU (erg/cm3 ) φU (phase ang.) α of [100] α of [110]

as-depos.

TA = 300◦ C

TA = 400◦ C

TA = 500◦ C

1.91 458 950 61 0.8 × 105 0 0 0.0055 0.008

1.91 744 900 20 1.4 × 105 0 0 0.003 0.006

2.02 683 1055 18 0.1 × 105 0.08 × 105 17 0.008 0.009

2.00 421 920 31 −0.05 × 105 0.02 × 105 5 0.01 0.01

TA denotes the annealing temperature

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R. Yilgin and B. Aktas

Fig. 3.14. The angular dependence of line-width (∆HPP ) of FMR spectra in the out-of-plane geometry of annealed Co2 MnSi film at 300◦ C temperature. Here the external field is rotated from the film normal toward either easy [100] or hard [110] direction lying in film plane as indicated above the curves. The continuous curves were obtained from the theoretical fitting. The symbols are explained in the text

anisotropy, KU , effect appears. This behavior depends not only on annealing but also on the structural transformation to L21 in the temperature region above 300◦ C. It should be noted that the strain anisotropy in Co2 MnGa Heusler alloy, discussed in [9], increases by tension as well. So, one of the possible origins of the change of the anisotropy is the release of interface strain. It should also be noted in Table 3.2 that the crystalline axes changes with annealing at higher temperature, (500◦ C). In another word, easy and hard magnetic (not structural) axes seem to interchange at higher temperature. The uniaxial anisotropy might arise from various effects, such as, the latticemismatch between the sample and substrate, imperfection, dislocation, defect, vacant sites, etc. Because of annealing over at 300◦ C, the Cr at buffer layer possibly diffuse into the films, and this diffusion may cause some imperfections or dislocations as well. As can be seen in the Table 3.2, all obtained parameter are listed. Therefore, the in-plane phase angle, φU , was obtained by fitting and given in Table 3.2. This phase angle describes the deviation angle of the easy axis from the [100] direction of MgO substrate as modeled in (3.17). Figure 3.14a, b respectively depicts angular variations of line-width, (∆HPP ), of FMR spectra of Co2 MnSi for out-of-plane geometries. Here the field is rotated either from easy or hard axis in sample plane toward the film normal. The angle (θH ) is measured with respect to the film normal. Open circles denote experimental peak-to-peak (∆HPP ) values while the other symbols represent the fitted values for different contributions to the total peak-to-peak line width as will be explained below. To explain the angular dependence of the line-width, we assume that the lines are intrinsically narrower compared

3 Anisotropy of Ferromagnetic Heusler Alloys Thin Films

61

Fig. 3.15. The annealing temperature dependence of the α-value

with the experimental peak-to-peak values. That is the observed line-width is affected mainly by inhomogeneity. The exchange mechanism tries to suppress line-broadening originating from inhomogeneous local dipolar fields and from any other imperfections or from any other interactions. Two-magnon scattering was not considered here. The cross symbols in both Figs. 3.14a and b denote the calculated contribution of intrinsic damping to total peak-topeak line-width. The upward-pointing triangle symbols in both figures and the downward-pointing triangle symbols in Fig. 3.14a correspond to the inhomogeneity in 4πMeff and the contribution of in (3.26), respectively. The values for the other inhomogeneity parameters in (3.26) are assumed to be zero in calculation. We have achieved a good fit and determined the damping parameter using only ∆4πMeff and ∆HK inhomogeneity parameters for both axis directions at each annealing temperature. As a result, the crystalline axis dependence of the α-damping parameter was observed for all samples except for those annealed at 500◦C sample. Figure 3.15 shows the annealing temperature-dependency of the α-value. In that figure, the stars and the circles represent the variation of the damping parameter with annealing temperature for the [100] and the [110] directions, respectively. The experimental points are connected by continuous curves for eye-guide. The minimum values of 0.003 were obtained at 300◦C and significant increase of the damping parameter was observed for this sample. The increasing of damping for annealed films at 400 and 500◦ C may result from the Cr diffusion at higher temperature. We think that the extrinsic effects will be contributed to the damping parameter at those annealed temperatures. The possible microscopic origin of out-of-plane dependencies of the different α-damping parameters of the line width for the directions between [100] and [110] is spin-orbit interaction, which couples the spin to the lattice. As well known, the spin-orbit interaction energy is given as H = −λLS. Different orientations of the spin with respect to the lattice or orbit vector

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evidently yield different interaction energies [42]. In addition, an anisotropic g-factor causes anisotropy of the Gilbert damping factor as mentioned in the theoretical section.

3.4 Conclusions The “as-prepared” and/or low temperature-annealed Co2 MnSi thin films were found to be amorphous and did not demonstrate the magnetic properties. All films annealed above 300◦ C crystallize (polycrystalline) and showed the (200) X-ray peaks that correspond to B2 crystalline structure. It was found that in order to get a satisfactory peak-to-peak line-width fitting for polycrystalline films, the (∆H0 ) inhomogeneity parameter must be used in (3.26). This parameter is a frequency-independent inhomogeneity parameter arising from history of film, and depending on its structure. Furthermore, the value for the intrinsic damping parameter was obtained to be unexpectedly larger. This result is correlated with the size of the grains, which were calculated taking into Scherrer law for polycrystalline Co2 MnSi films as seen in Fig. 3.5. Obviously the larger grain size results in a porosity-based destruction in the uniformity of polycrystalline films and ultimately causes line broadening. Also an increase in coercive field of polycrystalline films can be attributed to this effect. The physical structures and magnetic properties of single crystalline Co2 MnSi films have been studied. The observed (200) peaks in XRD pattern of ferromagnetic material, which are characteristic for B2 type disordered structure between Mn and Si atoms, were detected for all films. However, the (111) peaks relation with L21 type ordered structure of Co, Mn, and Si atoms were observed only for the films annealed at 400 and 500◦C. The B2 type disordered structure transformed to L21 type ordered structure the films annealed over 400◦ C. The saturation magnetization decreases above 300◦C. This result could be attributed to the diffusion of chromium into ferromagnetic layer. The α-intrinsic damping constant has been evaluated with analysis of in and out-of-plane angular dependence of resonance field and out-of-plane angular dependence of peak-to-peak line width of FMR spectra. We obtained good fitting with experimental results using LLG equations. The temperature annealing influences structural transformations from B2 to L21 and as a result both g factor and magneto crystalline anisotropy change. As mentioned earlier, G and g-factor are two fundamental quantities, and meanwhile the (g −2) relates to 2(L/S), where L is orbital magnetic moment and S is spin magnetic moment. The minimum value for α was observed for the sample annealed at 300◦ C. The α-value is considerably anisotropic as seen in Fig. 3.15. According to the microscopic framework, the origin of magnetic damping mechanism relates to the spin-orbit interaction [35,82–84], and therefore the line-width is expected to be directly proportional to intrinsic damping factor for metallic ferromagnetic films.

3 Anisotropy of Ferromagnetic Heusler Alloys Thin Films

63

(X108) Fe-Co Fe-Ni Co-Ni

6

Ni

G (sec-1)

C oF eB C o2Mn(Al,S i)

4 Co

2 C o2MnS i (L 21) C o2MnAl (B 2) YIG

0

Fe

0

0.01 C o2MnS i (B 2)

0.02

0.03

0.04

2

(g-2)

Fig. 3.16. The Gilbert damping constants of in various ferromagnetic thin films as a function of [34]

In another words, the spin-orbit interaction reveals the deviation of the Lande g factor from the free electron value (∆g = g − 2) and plays a dominant role in the damping mechanism. This factor was determined correctly from angular dependence of the FMR spectra [36, 37]. In literature [85], it was reported that the line-width of the conductionelectron spin resonance (CESR) spectra in pure metals is proportional to the (g − 2)2 , where g is Lande g factor, which is commonly known as the spectroscopic splitting factor. Furthermore, up to now the mechanism of the magnetic damping in ferromagnetic materials has not been clarified due to lack of systematic results. Figure 3.16 shows the Gilbert damping constants of various ferromagnetic thin films as a function of (g − 2) for comparison to Heusler alloys studied in this work. It should be recalled that the spin-orbit interaction represents the degrees of deviation of g-factor from free-electron value of 2. Thus one can conclude that the damping constant of ferromagnetic materials increases with spin-orbit interaction [34]. In our case, deduced effective g-factor and a small value for Gilbert factor, G, were found to be comparable with other magnetic films and in accordance with the Elliott’s suggestion [38, 83] αγMS = G ∝ (∆g)2 . As can be seen in Table 3.1, there is very small orbital contribution from exited states to the total magnetic moment and the total magnetic moment localizes onto the Ysite atoms as spin magnetic moment in full-Heusler alloy. Since the orbital magnetic moments of Co and Mn in the Heusler alloys are very small, the small magnetic damping constants of the Heusler alloys Co2 MnSi films can be attributed to a week spin-orbit interaction.

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As proven by Sakuraba et al. the single crystalline films of Co2 MnSi/ insulator/Co2MnSi have a large-scale tunnel magnetic ratio at low temperature [75, 76]. At room temperature, TMR ratio polarization was calculated as 70%. The critical current can be minimized for this Heusler alloy if 100% of spin polarization can be experimentally achieved, which is theoretically projected for Co2 MnSi Heusler alloy at room temperature. For spintronic applications, the spin damping and anisotropy need to be relatively smaller, and spin polarization higher and thus, Co2 MnSi full Heusler alloy will become a big candidate for low critical current spintronic applications.

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4 Quantum Monte Carlo Study of Anderson Magnetic Impurities in Semiconductors N. Bulut1,2 , Y. Tomoda1 , K. Tanikawa1, S. Takahashi1 , and S. Maekawa1,2 1 2

Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan CREST, Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan, [email protected], [email protected], [email protected], [email protected], [email protected]

Summary. We use quantum Monte Carlo simulations to study the electronic properties of Anderson magnetic impurities in a semiconductor host. We find that in a semiconductor the magnetic impurities exhibit ferromagnetic correlations, which can have a much longer range than in a metallic host. In particular, the range is longest when the Fermi level is located between the top of the valence band and the impurity bound state. We study the dependence of the ferromagnetic correlations on the parameters of the Anderson model, and the dimensionality and band structure of the host material. Using the tight-binding approximation for calculating the host band structure and the impurity–host hybridization, we obtain an impurity bound state, which is located at ≈100 meV above the top of the valence band, which is in agreement with the transport measurements on GaAs with dilute Mn impurities.

4.1 Introduction The discovery of ferromagnetism in alloys of III–V semiconductors with Mn started an intense research activity in the field of dilute magnetic semiconductors (DMS) [1–3]. Within this context, it is important to understand the nature of the correlations that develop between magnetic impurities in semiconductors and how they differ from that in a metallic host. With this purpose, we present Quantum Monte Carlo (QMC) results on the two-impurity Anderson model for a semiconductor host. The Anderson model of magnetic impurities in a semiconductor host was previously studied using the Hartree-Fock (HF) approximation [4–6] and perturbative techniques [7]. It was shown that long-range ferromagnetic (FM) correlations develop between Anderson impurities in a semiconductor when the Fermi level is located between the top of the valence band and the impurity bound state (IBS) [5, 6], as illustrated in Fig. 4.1.

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Fig. 4.1. Semiconductor host bands εα k (solid curves) and the impurity bound states (thick arrows) obtained with HF in the semiconductor gap. The dashed line denotes the chemical potential µ

To study the multiple charge states of Au impurities in Ge, the singleimpurity Anderson model of a metallic host was extended to the case of a semiconductor host using the HF approximation [4]. After the discovery of DMS, the magnetic properties of this model were addressed within HF [5, 6], and it was shown that long-range FM correlations develop when the Fermi level is located between the top of the valence band and the IBS. The FM interaction between the impurities is mediated by the impurity-induced polarization of the valence electron spins, which are antiferromagnetically coupled to the impurity moments. The impurity-host hybridization also induces host split-off states at the same energy as the IBS. When the split-off state becomes occupied, the spin polarizations of the valence band and of the split-off state cancel. This causes the long-range FM correlations between the impurities to vanish. Within the context of DMS, the Anderson Hamiltonian for a semiconductor host was also considered by Krstaji´c et al. [7], and it was shown that an FM interaction is generated between the impurities because of kinematic exchange. In addition, this model was studied within HF for investigating the multiple charge and spin states of transition-metal atoms in hemoprotein [8]. Finally, the role of IBS in producing the FM interaction in DMS was also discussed within the “double resonance mechanism” using HF [9]. In this chapter, we present QMC results for the impurity–impurity and impurity–host magnetic correlations in the Anderson model for semiconductor and metallic hosts [10]. We found that, in a semiconductor host, the magnetic correlations do not exhibit Ruderman-Kittel-Kasuya-Yosida (RKKY) type oscillations. Instead, the impurity–impurity magnetic correlations are ferromagnetic with a range that can become strongly enhanced depending on the occupation of IBS. In agreement with the HF results [5, 6], the range is longest when the chemical potential is between the top of the valence band and the IBS. We also show that the FM correlations between the impurities

4 Quantum Monte Carlo Study of Anderson Magnetic Impurities

69

is generated by the antiferromagnetic (AFM) impurity–host coupling. Comparisons with the experiments on GaAs with dilute Mn impurities suggest that the physical picture presented here describes how magnetic correlations develop between dilute magnetic impurities in a semiconductor host. In addition, we determine the dependence of the FM correlations between the impurities on the parameters of the Anderson model. We also study the dependence of the FM correlations on the dimensionality and band structure of the host materials. In particular, we use the tight-binding approximation to calculate the host band structure and the impurity–host hybridization, and then use these as input for QMC simulations of the Anderson model. This way of combining the tight-binding approximation with the QMC technique allows us to study the material dependence of the magnetic and electronic properties. For instance, this approach yields an IBS energy of ≈100 meV , which is comparable to the experimental value. We think that these results are useful for understanding the properties of the DMS and dilute oxide ferromagnets. In Sect. 4.2, we introduce the Anderson model for a semiconductor host and describe the quantities, which we calculate with QMC, and the parameter regime, which we explore in the simulations. In Sect. 4.3, we show QMC results on the impurity–impurity and impurity–host correlations for the twodimensional host with a quadratic quasiparticle dispersion. Section 4.4 shows results for the 3D case, while in Sect. 4.5 the QMC results for a magnetic impurity in a GaAs host are discussed using the tight-binding approximation for the band-structure of the host material. Finally, Sect. 4.6 presents a discussion and summary of the numerical results.

4.2 Model The two-impurity Anderson model for a semiconductor host is defined by   † (εα (Vki c†kασ diσ + H.c.) H= k − µ)ckασ ckασ + k,α,σ

+ (Ed − µ)

 i,σ

k,i,α,σ

d†iσ diσ

+U



d†i↑ di↑ d†i↓ di↓ ,

(4.1)

i

where c†kασ (ckασ ) creates (annihilates) a host electron with wavevector k and spin σ in the valence (α = v) or conduction (α = c) band, and d†iσ (diσ ) is the creation (annihilation) operator for a localized electron at impurity site i. The hybridization matrix element is Vkj = V eik·Rj ,

(4.2)

where Rj is the coordinate of the impurity site j. As usual, Ed is the dlevel energy, U is the onsite Coulomb repulsion, and µ the chemical potential. In Sects. 4.3 and 4.4, we will use quadratic dispersion for the valence and conduction bands

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εvk = −D (k/k0 ) εck

2

= D (k/k0 ) + ∆G ,

(4.3) (4.4)

with D the bandwidth, k0 the maximum wavevector, and ∆G the semiconductor gap. In Sect. 4.3, we consider a two-dimensional (2D) semiconductor host with a constant density of states ρ0 , and in Sect. 4.4, the three-dimensional (3D) case. Here, the energy scale is determined by setting D = 12.0. In Sect. 4.5, we use the tight-binding approximation to calculate the host band structure. In addition, in these calculations we use U = 4.0 and Ed = µ− U/2, so that the impurity sites develop large moments both in the metallic and semiconductor cases. We report results for ∆G = 2.0, and inverse temperature β ≡ 1/T from 4 to 32 for the 2D and 3D cases. The numerical results presented here were obtained with the HirschFye QMC technique [11]. In the following, we show results on the equaltime impurity–impurity magnetic correlation function M1z M2z , where the impurity magnetization operator is Miz = d†i↑ di↑ − d†i↓ di↓ ,

(4.5)

for the two-impurity Anderson Hamiltonian, (4.1). In addition, we will discuss the results on the impurity–host correlation function M z mz (r) for the singleimpurity Anderson model. Here, the host magnetization at a distance r away from the impurity site is given by  † (cα↑ (r)cα↑ (r) − c†α↓ (r)cα↓ (r)). (4.6) mz (r) = α=v,c

For the metallic case, M1z M2z and M z mz (r) were previously studied by using QMC [11–14]. We also present results on the impurity single-particle spectral weight A(ω) = −(1/π)Im Gσii (ω), which is obtained with the maximum-entropy analytic continuation technique [15] from the QMC data on the impurity Green’s function (4.7) Gσii (τ ) = − Tτ diσ (τ )d†iσ (0) . Here, Tτ is the Matsubara time-ordering operator and diσ (τ ) = eHτ diσ e−Hτ . Since the maximum-entropy procedure requires QMC data with very good statistics, our results on A(ω) will be limited to the high-temperature β = 8 case. In addition, we present data on the zero-frequency inter-impurity magnetic susceptibility defined by 

β

dτ M1z (τ )M2z (0)

χ12 =

(4.8)

0

for the two-impurity case. The following results were obtained using Matsubara time step ∆τ = 0.25 except for A(ω) which was obtained with ∆τ = 0.125.

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At β = 16, M1z M2z varies by a few percent as ∆τ decreases from 0.25 to 0.125. For the single-impurity Anderson model, we present QMC data on the square of the impurity moment, (M z )2 , and the impurity susceptibility χ defined by  β dτ M z (τ )M z (0) . (4.9) χ= 0

The effects of IBS are clearly visible in these quantities. We also present QMC data on the charge distribution of the host material around the impurity for the single-impurity case. As expected, we will find that the quantitative results depend on the dimensionality and the band structure of the host material. In the following, we first show results for the 2D and 3D host materials with simple quadratic band dispersions. Then we discuss the case for a GaAs host using the tight binding approximation.

4.3 Two-Dimensional Case In this section, we show results for the 2D case. Here, the density of states of the pure host, ρ0 , is a constant with a sharp cutoff at the semiconductor gap edge, which leads to stronger impurity–host coupling compared with the 3D case. Here, we present results for hybridization ∆ ≡ πρ0 V 2 varying from 1 to 4. 4.3.1 Magnetic Correlations Between the Impurities Figures 4.2a–c show the impurity magnetic correlation function M1z M2z vs. k0 R, where R = |R1 − R2 | is the impurity separation, at β = 16 for µ varying from −6.0 to 1.0. Figure 4.2a is for hybridization ∆ = 1.0. At µ = −6.0, we observe oscillations in the R dependence due to an RKKY-type effective interaction between the impurities. These results are similar to what has been obtained previously with QMC for a half-filled metallic band [11–13]. The wavelength of the oscillations increases when µ moves to −1.0, because of the shortening of the Fermi wavevector. When µ = 0.0, the impurity spins exhibit long-range FM correlations at this temperature. We observe that upon further increasing µ to 0.5 or 1.0, the FM correlations become weaker. This is because the IBS becomes occupied as µ changes from 0.0 to 0.5, as will be seen in Fig. 4.3a. In Figs. 4.2b and c, results on M1z M2z are shown for ∆ = 2.0 and 4.0, respectively. In Fig. 4.2b, we observe that M1z M2z has the slowest decay for µ = 0.5, while in Fig. 4.2c this occurs for µ = 1.0. We find that the impurity occupation nd increases between µ = 0.5 and 1.0 for ∆ = 2.0 and β = 16. Hence, 0.5 < ωIBS < 1.0 for ∆ = 2. In addition, for ∆ = 4.0 and β = 8, the maximum-entropy image of A(ω) shows that the IBS is located

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Fig. 4.2. M1z M2z  vs. k0 R plotted at β = 16 and various µ for hybridization (a) ∆ = 1.0, (b) 2.0, and (c) 4.0

at ω ≈ 1.0. Hence, we observe that the range of the FM correlations for the semiconductor is determined by the occupation of the IBS in agreement with the HF predictions [5,6]. In Figs. 2a–c, it is also seen that the maximum range increases with decreasing ∆. In Fig. 4.3, we discuss the ∆ = 1.0 case in more detail. In Fig. 4.3a, the impurity spectral weight A(ω) vs. ω is plotted for β = 8, µ = 0.1, and k0 R = 5 and 10. Here, the ω-axis has been shifted so that the top of the valence band is located at ω = 0. For k0 R = 10, we observe a peak at ωBS ≈ 0.1 in the semiconductor gap, which we identify as the IBS. For k0 R = 5, the bound state is broader due to stronger correlations between the impurities. However, we also find that A(ω) exhibits significant T dependence at β = 8, and Fig. 4.3a does not yet represent the low-T limit. Next, in Figs. 4.3b and c, M1z M2z

evaluated at k0 R = 5 and 10 is plotted as a function of µ. Figure 4.3b shows that, at low T for k0 R = 5, M1z M2z decreases when µ > ∼ 0.25. For this value of k0 R and β = 32, we find that the impurity occupation nd develops a step discontinuity at µ ≈ 0.25, which is consistent with the decrease of M1z M2z

z z when µ > ∼ 0.25. For k0 R = 10 and β = 32, both M1 M2 and nd exhibit significant T dependence in the vicinity of the semiconductor gap edge. These results show that M1z M2z depends strongly on the value of µ.

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Fig. 4.3. (a) Impurity single-particle spectral weight A(ω) vs ω for k0 R = 5 and 10 at β = 8. Here, the vertical dashed line denotes µ, and the top of the valence band is located at ω = 0. In (b) and (c), M1z M2z  vs µ is plotted for k0 R = 5 and 10, respectively, at various β. These results are for ∆ = 1.0

Figures 4.4a and b show M1z M2z vs. k0 R, where R is the impurity separation, for the two-impurity Anderson model for half-filled metallic (µ = −6.0) and semiconductor (µ = 0.1) cases. On the one hand, in the metallic case, M1z M2z exhibits both FM and AFM correlations depending on the value of k0 R. On the other hand, for µ = 0.1, we observe FM correlations of which range increases with β. Figure 4.5 shows the inter-impurity magnetic susceptibility χ12 vs. the inverse temperature for various values of µ near the semiconductor gap edge. These results are for impurity separation R = 10k0−1 . We observe that the FM correlations are strongest when µ is between the gap edge and the IBS.

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Fig. 4.4. Impurity-impurity magnetic correlation function M1z M2z  vs k0 R at various β for (a) µ = −6.0 and (b) µ = 0.1 for the two-impurity Anderson model

Fig. 4.5. Inter-impurity magnetic susceptibility χ12 (ω = 0) vs β for k0 R = 10 at various µ. These results are for ∆ = 1.0

The remainder of the data shown for the 2D case in this section are for the single-impurity Anderson model. In Fig. 4.6, we show results on (M z )2

vs. µ for various values of ∆. As T is lowered, a step discontinuity develops in (M z )2 near the gap edge. The location of the discontinuity coincides with the location of IBS deduced from data on M1z M2z , A(ω), and χ12 discussed

4 Quantum Monte Carlo Study of Anderson Magnetic Impurities β β=8 β = 16 β = 32

〈( Mz )2〉

0.9

0.8

75

∆=1

∆=2 0.7

∆=4

0.6 -1

-0.5

0

0.5

1

µ

Fig. 4.6. Impurity magnetic moment square (Mz )2  vs. µ at various β for ∆ = 1, 2 and 4 for the single-impurity Anderson model 0.8

Tc

0.6 D=1

0.4

b=4 b=8 b = 16 b = 32

0.2

0 -1

-0.5

0

0.5

1

m

Fig. 4.7. T χ vs. µ at various β for ∆ = 1 for the single-impurity Anderson model

above. Hence, Fig. 4.6 shows that the local moment increases rapidly as the IBS becomes occupied. Here, we also observe that the moment size decreases with increasing hybridization, as expected. Figure 4.7 shows T χ vs. µ for ∆ = 1, where we observe that a step discontinuity develops at the same location as in (M z )2 shown in Fig. 4.6a. For µ < ∼ ωIBS , T χ decreases with decreasing T because of the screening of the impurity moment by the valence electrons. However, for µ > ∼ ωIBS , the impurity susceptibility exhibits freemoment behavior in agreement with the role of the IBS discussed earlier. We note that determining the location of the IBS from A(ω) is costly in terms of computation time. For this reason, in the single-impurity case, it is more convenient to determine ωIBS from data on (M z )2 vs. µ. In the remaining sections, we will use this approach for determining ωIBS . Finally, we note that within HF, ωIBS = 0.017 for ∆ = 1. Hence, the HF approximation underestimates the value of ωIBS by about an order of

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0 D=1 (a) m = -6.0 b=4 b=8 b = 16 b = 32

-1 -1.5 0

5

10

15

D=1 (b) m = 0.1 b=4 b=8 b = 16 b = 32

-0.2 s(r)

s(r)

-0.5

-0.4 -0.6

20

0

5

10 k0r

k0r

15

20

0 ∆=1 (c) µ = 0.5 β=4 β=8 β = 16 β = 32

s(r)

-0.2 -0.4 -0.6 0

5

10 k0r

15

20

Fig. 4.8. Impurity–host magnetic correlation function s(r) vs. k0 r at various β for (a) µ = −6.0, (b) µ = 0.1, and (c) µ = 0.5 for the single-impurity Anderson model

magnitude. Hence, for ∆ = 1, the long-range FM correlations would be restricted to a narrow range of µ within the HF approximation. 4.3.2 Impurity–Host Correlations In this section, we discuss the magnetic correlations between the impurity and the host. In addition, we show results on the induced charge oscillations around the impurity site. Figures 4.8a–c show the impurity–host magnetic correlation function s(r) defined by 2πk0 r M z mz (r)

(4.10) s(r) = n0 vs. k0 r for the single-impurity Anderson model for µ = −6.0 (half-filled metallic), µ = 0.1 (semiconductor with IBS unoccupied), and µ = 0.5 (semiconductor with IBS occupied). Here, n0 is the electron density and r is the distance from the impurity site. For µ = 0.1, the coupling between the impurity and host spins is AFM for all values of k0 r, while for µ = −6.0, s(r) exhibits RKKY-type 2kF oscillations. We also note that, for the metallic case, the magnetic correlations saturate before reaching β = 32. Comparison of Fig. 4.8b and Fig. 4.4b for µ = 0.1 show that the AFM impurity-host coupling produces the FM correlations between the impurities.

4 Quantum Monte Carlo Study of Anderson Magnetic Impurities

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0

S(k0r = 25)

-0.2 D=1 b=4 b=8 b =16 b = 32

-0.4 -0.6 -0.8 -1

-0.5

0 m

0.5

1

Fig. 4.9. S(k0 r = 25) vs. µ at various β for the single-impurity Anderson model

As seen in Fig. 4.8c, when µ is increased to 0.5, the AFM correlations between the impurity and host spins become much weaker. This is because the IBS becomes occupied for µ > 0.1. Within the HF approximation [5,6], the FM interaction between the impurities is mediated by the impurity-induced polarization of the valence electron spins, which exhibit an AFM coupling to the impurity moments. The impurity–host hybridization also induces host split-off states at the same energy as the IBS. When the split-off state becomes occupied, the spin polarizations of the valence band and the split-off state cancel. This causes the long-range FM correlations between the impurities to vanish. These QMC and HF results emphasize the role of the IBS in determining the range of the magnetic correlations for a semiconductor host. The total magnetic coupling of the impurity magnetic moment to the host is obtained from  k0 r

S(k0 r) =

d(k0 r )s(r )

(4.11)

0

Figure 4.9 shows S(k0 r = 25) vs. µ for ∆ = 1 at various β. Here, we see that the impurity becomes magnetically decoupled from the host when µ > ∼ ωIBS . Next, in Fig. 4.10, we show the modulation of the charge density around the impurity. Here, we plot p(r) vs. k0 r, where p(r) is defined by p(r) =

 2πk0 r (nα (r) − nα (r = ∞)) n0 α=v,c

(4.12)

 with nα (r) = σ c†ασ (r)cασ (r) . For the metallic case of µ = −6.0, we observe long-range RKKY-type of oscillations in p(r). When µ = 0.1, the charge density around the impurity is depleted up to k0 R ≈ 20 at these temperatures. This depletion represents the extended valence hole, which forms around the impurity. However, for µ = 0.5, the induced charge density decreases significantly as T is lowered, because now the IBS is occupied. We next integrate p(r),  k0 r P (k0 r) = d(k0 r )p(r ), (4.13) 0

N. Bulut et al.

p(r)

0

D=1 (a)m = -6.0

1

0 b=4 b=8 b = 16 b = 32 15 20

-1 0

5

10 k0r

-0.2 p(r)

78

D=1 (b)m = 0.1

-0.4

b=4 b=8 b = 16 b = 32

-0.6 -0.8 0

5

10 k0r

15

20

0 -0.2

D=1 (c) m = 0.5 b=4 b=8 b = 16 b = 32

S(r)

-0.4 -0.6 -0.8 0

5

10 k 0r

15

20

Fig. 4.10. p(r) vs. k0 r for ∆ = 1 and (a) µ = −6.0, (b) 0.1, and (c) 0.5 0

P(k0r = 25)

-0.2 -0.4

∆=1 β=4 β=8 β = 16 β = 32

-0.6 -0.8 -1 -1

-0.5

0

0.5

1

µ

Fig. 4.11. P (k0 r = 25) vs. µ at various β for the single-impurity Anderson model

and plot P (k0 r = 25) vs µ in Fig. 4.11. We observe that the total charge < density around the impurity is most depleted when 0 < ∼ µ ∼ ωIBS , which is due to the valence hole induced around the impurity. In the metallic case, the induced charge density is oscillatory and has a long range as we have seen in Fig. 4.10a. As µ approaches the gap edge, the electron density around the impurity is depleted. However, when µ > ∼ ωIBS , we see that this depletion is canceled by the extended charge density of the split-off state.

4 Quantum Monte Carlo Study of Anderson Magnetic Impurities

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4.4 Three-Dimensional Case In this section, we discuss the 3D case, where the hybridization parameter πV 2 N (0) vanishes at the gap edge because of the vanishing of N (0) of the pure host. Hence, the impurity–host coupling near the gap edge is much weaker compared with the 2D case. Consequently, the FM correlations between the impurities have a shorter range. We find that the dimensionality of the host material strongly influences the magnetic correlations. In particular, we see that the IBS does not exist in 3D unless the hybridization is sufficiently large. Here, we define hybridization as ∆ = πρ∗0 V 2 where√ρ∗0 is the density of states when the valence band is half-filled, ρ∗0 = k03 /(4 2π 2 D). This choice allows us to use comparable values for the hybridization matrix element V , when we compare the 2D and 3D results. In the following, we present results for ∆ = 1, 2, and 4. Figures 4.12a–c show M1z M2z vs. k0 R at various values of µ for ∆ = 1, 2, and 4. These results are for µ = −6.0, µ = −1.0, and µ = 0.0. On the one hand, we observe that, for ∆ = 1, the FM correlations between the impurities weaken as µ approaches the top of the valence band. On the other hand, for ∆ = 2 and 4, the FM correlations are stronger for µ = 0.0. This is because, for ∆ = 1, the IBS does not exist in a 3D host, as we will see later in Fig. 4.15, which shows results on (M z )2 vs. µ.

β = 16 (a) ∆ = 1 µ = 0.0 µ = -1.0 µ = -6.0

0 0

β = 16 (b)∆ = 2

z z

0.1

0.2 〈M1 M2 〉

z

z

〈M1 M2 〉

0.2

µ = 0.0 µ = -1.0 µ = -6.0

0.1

0 5

10 k0R

15

20

z z

5

10 k0R

15

20

β = 16 (c) ∆ = 4 µ = 0.0 µ = -1.0 µ = -6.0

0.2 〈M1 M2 〉

0

0.1

0 0

5

10 k0R

15

20

Fig. 4.12. M1z M2z  vs. k0 R at β = 16 and various µ for hybridization (a) ∆ = 1.0, (b) 2.0, and (c) 4.0 in the 3D case

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β=4 β=8 β = 16

z

β=4 β=8 β = 16

0.1

0 0

∆=2 (b)µ = 0.0

z

0.1

0.2

z

∆=2 (a) µ = − 0.1 〈M1 M2 〉

0.2 z

〈M1 M2 〉

80

0

5

10 k0R

15

20

0

z z

10 k0R

15

20

∆=2 (c)µ = 0.1

0.2 〈M1 M2 〉

5

β=4 β=8 β = 16

0.1

0 0

5

10 k0R

15

20

Fig. 4.13. M1z M2z  vs. k0 R at various β for ∆ = 1, and (a) µ = −0.1, (b) µ = 0.0, and (c) µ = 0.1 in the 3D case

Next, in Fig. 4.13, we show M1z M2z vs. k0 R for ∆ = 2. We will see that for this value of ∆, and IBS exists with ωIBS ≈ 0.0. Here results are shown at various values of β for µ = −0.1, 0.0 and 0.1. We observe that the FM correlations weaken as µ mover through µ = 0.0. Figure 4.14 shows the impurity-host magnetic correlation function s(r) vs. k0 r for the single-impurity case for the same parameters as in Fig. 4.13. In 3D, s(r) is defined by 4π(k0 r)2 M z mz (r) . (4.14) s(r) = n0 We see that the impurity-host coupling weakens rapidly for µ > ∼ 0.0. These figures show that, in 3D and for ∆ = 2, the IBS is located at ωIBS ≈ 0.0, which is consistent with the results on (M z )2 shown in Fig. 4.15. In Fig. 4.15, for ∆ = 1, we do not observe the development of a discontinuity for temperatures down to β = 32. For ∆ = 2, we observe the development of a discontinuity centered at µ ≈ 0.0, as β increases. For ∆ = 4, a step discontinuity at µ ≈ 0.3 is clearly observed. So, in 3D the IBS exists only for sufficiently large values of the hybridization matrix element V . This is consistent with the dependence of the IBS on the dimensionality in the U = 0 case. These results show that the dimensionality of the host material strongly influences the magnetic properties.

4 Quantum Monte Carlo Study of Anderson Magnetic Impurities 0 D=2 (a) m = -0.1 b=4 b=8 b = 16 b = 32

-1

-2 0

5

10 k0r

15

s(r)

s(r)

0

81

D=2 (b) m = 0.0 b=4 b=8 b =16 b = 32

-1

-2 20

0

5

10 k0r

15

20

0 D=2 (c) m = 0.1 b=4 b=8 b = 16 b = 32

s(r)

-1

-2 0

5

10 k0r

15

20

Fig. 4.14. s(k0 r) vs. k0 r at various β for ∆ = 2 and for (a) µ = −0.1, (b) 0.0, and (c) 0.1 for the single-impurity Anderson model in a 3D host 1 ∆=1

〈(M z )2 〉

0.9

∆=2

0.8

∆=4

0.7

β=4 β =8 β =16 β =32

0.6 0.5

-1

-0.5

0 µ

0.5

1

Fig. 4.15. Impurity magnetic moment square (Mz )2  vs. µ at various β for (a) ∆ = 1, 2, and 4 for the single-impurity Anderson model in a 3D host

The results presented in Sects. 4.3 and 4.4 show that the density of states of the pure host at the gap edge and the value of the hybridization matrix element are crucial in determining the magnetic properties when transition metal impurities are substituted into a semiconductor host. This means that the band structure of the pure host material will also be crucial in determining the magnetic properties. In the next section, we explore the consequences

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of a more realistic band structure for a GaAs host using the tight-binding approximation.

4.5 Tight-Binding Model for a Mn d Orbital in GaAs In this section, we study the magnetic correlations that develop when one dxy orbital is added to a GaAs host using the tight-binding approximation to calculate the host electronic structure and the host–impurity hybridization. Here, the tight-binding band structure of GaAs host is obtained by keeping the s and the three p orbitals at each site of the zincblende crystal structure of GaAs and by taking into account only the nearest-neighbor hoppings. We perform the QMC calculations for the case of a single Mn dxy orbital added to the GaAs host. Hence, we are assuming that the sp3 orbitals of the Mn impurity are the same as those of Ga of the host material. We are also assuming that GaAs electronic structure is not modified by the additional d orbital at the impurity site. In addition, here we consider only one of the three t2g orbitals of the Mn impurity, hence we neglect the spin-orbit interaction. Clearly, this is a simple model for a transition metal impurity substituted into GaAs. However, this approach allows us to take into account the effects of the host band structure beyond the quadratic dispersion used in the previous sections. Furthermore, in Sects. 4.3 and 4.4, we had treated the hybridization ∆ as a free parameter. However, within this approach, we perform the calculations using realistic parameters for the hybridization of the d orbital with the host.

10 8 6 4

e [eV ] a

2 0 -2 -4 -6 -8 -10 -12 L

G

X

W

L

Fig. 4.16. Band structure of GaAs within the tight-binding approximation

4 Quantum Monte Carlo Study of Anderson Magnetic Impurities

2

|V a=c.xy| [ eV ]

|Va=v.xy| [ eV]

2

1

0 L

83

G

X

W

1

0 L

L

G

X

W

L

Fig. 4.17. Hybridization matrix element Vα,xy (k) of the dxy orbital of a Mn impurity with (a) the valence and (b) the conduction bands of GaAs vs. k obtained using the tight-binding approximation

0.8

0.9

0.8

0.7 -0.3 -0.2 -0.1 0 m [eV ]

T=2900K T=1500K T=730 K T=360 K T=180 K 0.1 0.2 0.3

Tc

z 2

<(M ) >

0.6 0.4 0.2 0 -0.3 -0.2 -0.1 0 0.1 m [eV ]

0.2

0.3

Fig. 4.18. (a) (M z )2  and (b) T χ vs. µ near the gap edge for a single dxy orbital added to the GaAs host

Figure 4.16 shows the tight-binding band structure of GaAs, which consists of eight bands originating from the sp3 orbitals at each Ga and As site. Here, the bands εα (k) vs. k are plotted in the first Brillouin zone of the zincblende crystal structure. These bands were obtained by using the SlaterKoster parameters from Chadi and Cohen [16]. We note that the top of the valence band is located at the Γ point, where the semiconductor gap is about 1.6 eV , consistent with the experimental value. Next, Figs. 4.17a and b show the hybridization matrix element Vα,xy (k) of the dxy orbital with the valence and the conduction bands, respectively. In obtaining these results, we have taken the Slater-Koster parameter for d − s hybridization, c(ds), to be 0. We have also taken the ratio of the SlaterKoster parameters for the d − pσ hybridization to the d − pπ hybridization, c(dpσ )/c(dpπ ), to be −2.16 [17]. The results shown in Fig. 4.17 are for c(dpσ ) = −1.1 eV. The estimate for c(dpσ ) from the photoemission experiments is −1 eV [18]. Hence, the tight-binding model parameters used in producing εα (k) and

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Vα,xy (k) are reasonable. The values of Vα,xy (k) at the top of the valence and at the bottom of the conduction band will be particularly important for producing the IBS. Here, we observe that at the Γ point, the impurity dxy orbital exhibits significant hybridization with the top of the valence point. However, the hybridization with the lowest-lying conduction band vanishes at the Γ point. Using the tight-binding results for εα (k) and Vα,xy (k) as input, we have performed the QMC simulations to study the magnetic properties. In Figs. 4.18a and b we show the resulting data on (M z )2 and T χ vs µ. Here, we observe that a step discontinuity develops at ωIBS ≈ 100 meV , which is close to the experimental value of 110 meV for dilute Mn impurities in GaAs [19]. We see that by using realistic parameters for the host band structure and the host–impurity hybridization, it is possible to obtain an accurate value for ωIBS . Such quantitative agreement supports the physical picture described in this chapter for the origin of the FM correlations in DMS.

4.6 Discussion and Summary In the QMC and HF calculations, the location of the Fermi level with respect to the IBS is important; the FM correlations weaken as the IBS becomes occupied. Photoemission [20] and optical measurements [21] on Ga1−x Mnx As suggested the existence of an IBS above the top of the valence band in this prototypical DMS ferromagnet. Photoemission experiments [20] observed an Mn-induced state above the valence band and right below the Fermi level in Ga1−x Mnx As. Clearly, inverse photoemission experiments are required to detect the unoccupied portion of the Mn-induced impurity band. STM experiments also observed the impurity band in this compound [22]. The transport measurements [23] on insulating samples of (Ga,Mn)As with less than 1% Mn impurities also suggest the existence of an IBS in the semiconductor gap. However, in metallic samples with more than 2% Mn, no activated contribution from an IBS to dc transport has been observed. We think that the physical picture presented in this chapter is consistent with the experiments in (Ga,Mn)As in the dilute impurity limit, and provides a mechanism for the origin of ferromagnetism. In particular, by using the tight-binding results for the host electronic structure and the impurity–host hybridization as input for the QMC calculations, we have obtained ωIBS ≈ 100 meV , which is close to the experimental value. These comparisons suggest that the Anderson Hamiltonian for a semiconductor host provides a basic electronic model for the DMS ferromagnets. Alternative ways of enhancing the FM correlations in this model is provided by varying the hybridization parameter ∆ or the semiconductor gap ∆G . The QMC simulations show that ξ increases as ∆ goes from 4.0 to 1.0. Within HF, ξ can take very large values as ∆/∆G decreases. Hence, new DMS compounds with weaker hybridization or a larger semiconductor gap might lead

4 Quantum Monte Carlo Study of Anderson Magnetic Impurities

85

to higher Curie temperatures. However, it is necessary to keep in mind that the two-impurity Anderson model for a semiconductor host might be oversimplified for describing the ferromagnetism of the DMS. Our calculations are for spin-1/2 Anderson impurities, and we have neglected the multiorbital structure of the spin-5/2 Mn impurities. We are currently performing calculations for the multiorbital case, which take into account the Hund’s rule coupling between the three Mn t2g orbitals. Various other theoretical approaches have been used to describe the ferromagnetism of DMS and to predict the Curie temperature as a function of the Mn concentration. These include the coherent potential approximation (CPA) and LDA+U techniques, which describe the host band structure accurately but treat the Coulomb correlations at mean-field level [24]. In particular, we note that the CPA or LDA+U type calculations cannot be relied upon to accurately predict the dependence of the FM correlations on µ or ∆/∆G . In this respect, the results presented in this chapter are valuable because here the many-body effects due to the Coulomb correlations at the impurity site in a semiconductor host have been treated exactly. For example, we have observed that, within the HF approximation, the long-range FM correlations are restricted to a narrow range of µ. We also point out that the dependence of the Curie temperature on the carrier concentration has been studied for DMS using a disordered RKKY model [25]. However, our results show that the RKKY form is not appropriate for describing the interimpurity magnetic correlations when the host is a semiconductor. Recently, Tc ’s exceeding the room temperature have been reported in dilute oxides such as ZnO and TiO2 with transition metal impurities [26, 27]. The importance of the oxygen vacancies in producing the ferromagnetism has been pointed out experimentally [28]. Perturbative and LSDA+U mean-field calculations have been performed to describe the ferromagnetism induced by oxygen vacancies in (Ti,Co)O2 [29, 30]. We think that it would also be useful to perform exact numerical calculations to study the role of the vacancy band in producing the FM correlations in this compound. In summary, we have presented QMC results to show that long-range FM correlations develop between magnetic impurities in semiconductors. In particular, the FM correlations have the longest range when the Fermi level is located above the top of the valence band, and they weaken as the IBS becomes occupied. Hence, the position of the Fermi level with respect to the IBS plays a crucial role in determining the range of the FM correlations in qualitative agreement with HF. These results also show that the RKKY form is not appropriate for describing the inter-impurity magnetic correlations when the host is a semiconductor. Finally, comparisons with the experimental data suggest that the Anderson model for a semiconductor host captures the basic electronic structure of the DMS ferromagnets.

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Acknowledgment This work was supported by the NAREGI Nanoscience Project and a Grant-in Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and NEDO.

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26. Y. Matsumoto, M. Murakami, T. Shono, T. Hasegawa, T. Fukumura, M. Kawasaki, P. Ahmet, T. Chikyow, S. Koshihara, H. Koinuma, Science 291, 854 (2001) 27. P. Sharma, A. Gupta, K.V. Rao, F.J. Owens, R. Sharma, R. Ahuja, J.M.O. Guillen, B. Johansson, G.A. Gehring, Nature Mater. 2, 673 (2003) 28. K.A. Griffin, A.B. Pakhomov, C.M. Wang, S.M. Heald, K.M. Krishnan, Phys. Rev. Lett. 94, 157204 (2005) 29. K. Kikoin, V. Fleurov, Phys. Rev. B 74, 174407 (2006) 30. V.I. Anisimov et al., J. Phys. Cond. Mat. 18, 1695 (2006)

5 New Type of Nanomaterials: Doped Magnetic Semiconductors Contained Ferrons, Antiferrons and Afmons L.I. Koroleva and D.M. Zashchirinskii M.V. Lomonosov Moscow State University, Vorobevy gory, Moscow, 119992 Russia Summary. New type of magnetic nanomaterials is described. These are the ferromagnetic nanoregions (ferrons) in antiferromagnetic and ferromagnetic semiconductors near Curie point with a giant red shift of the band-gap, the nanoregions with destructive magnetic ordering in ferromagnetic semiconductors with the giant blue shift of the band-gap (antiferrons) and the nanoregions with layered antiferromagnetic order in semiconductors in which the most part of spins has antiferromagnetic order (afmons). Namely these nanoscale regions hold responsible for the giant effects of a magnitoresistance and a volume magnetostriction and for peculiarities of magnetic and another properties.

5.1 Introduction A specific feature of the magnetic semiconductors is a strong dependence of a conduction electron energy on a magnetization of the crystal due to the exchange interaction between the mobile electrons or holes and localized d(f )-spins. To describe this interaction, the s–d model is usually used. This model assumes that all the electrons of the crystal can be subdivided into those located inside the partially occupied d - or f -shells, and into the mobile electrons. The former is called the d -electrons and the latter the s-electrons. On the one hand, in this the term “s-electrons” is used for mobile electrons that are not necessary in the states of the s-type but can be in the states of other types. Moreover, this term is applicable for the holes. On the other hand, the localized spins are denoted as “d -spins,” though, in reality, they can correspond not to partially-filled d -shells but to partially-filled f -shells. The very important properties of s–d model consists in the following: the minimal charge carrier energy is attained at a complete ferromagnetic ordering. Nagaev, Yanase, Kasuya, Dagotto, Hotta and Moreo [1–6] have shown that the doped magnetic semiconductors contain peculiar magneto-impurity states due to strong s–d exchange in which the charge carriers are localized. These are the ferromagnetic nanoscale regions (ferrons) in antiferromagnetic

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and ferromagnetic semiconductors near Curie point TC with a giant red shift of the band-gap, the nanoscale regions with destructive magnetic ordering in ferromagnetic semiconductors with the giant blue shift of the band-gap (antiferrons) and the nanoscale regions with layered antiferromagnetic ordering in semiconductors in which the most part of spins is ordered by antiferromagnetic manner (afmons). Namely these nanoscale regions hold responsible for such interest effects in magnetic semiconductors as giant magnetoresistance, a giant volume magnetostriction and others that are described in this review.

5.2 Ferrons 5.2.1 Giant Red Shift of Fundamental Absorption Edge Connected with Ferromagnetic Ordering Optical spectra of magnetic semiconductors have the peculiarities. So, in several magnetic semiconductors, the fundamental absorption edge shifts toward more low energy with a temperature drop as opposed to nonmagnetic semiconductors in which the one shifts toward more high energy. This shift is produced by shift of bottom of the conductive band in EuO, EuS, HgCr2 Se4 , CdCr2 Se4 and top of valence band in La0.9 Sr0.1 MnO3 . Figure 5.1 shows this effect for latter compound from article [1]. It is evident that this shift has the giant amount 0.4 eV in the direction of lower energies as the temperature decreases from TC = 155 K to 141 K. This shift has a giant value 0.2–0.5 eV in EuO, EuS, EuSe, CdCr2 Se4 , HgCr2 Se4 , HgCr2 S4 and ZnCr2 Se4 as the temperature decreases from a paramagnetic region to 4.2 K. This phenomenon gave a name as giant red shift of the absorption edge. On the contrary to magnetic semiconductors, the absorption edge

1 0.9 0.8 0.7 0.6 0.5 130

140

150

T, K Fig. 5.1. Temperature dependence of the fundamental absorption edge of the La0.9 Sr0.1 MnO3 composition [7]

5 New Type of Nanomaterials (a)

(b) Shift of absorption edge, eV

Energe of absorption edge, eV

0,8

0,6

0,4

Tc

0,2

0

91

100

200

T,K

300

_ 0,04

130 122

_0,02

108

115 0

2000

4000

H, Oe

Fig. 5.2. Temperature dependence of the fundamental absorption edge (a) and the shift of the fundamental absorption edge from the magnetic field (b) of the HgCr2 Se4 single crystal sample [7]

of nonmagnetic semiconductors is shifted toward the more high energy with temperature drop (blue shift), and a rate of this shift is lower on 2–3 order than the rate of the above-mentioned giant red shift. We illustrate this phenomenon on HgCr2 Se4 , in which a crack width is decreasing by a factor of three at the temperature drop from 200 to 4.2 K [7]. Figure 5.2 shows the temperature dependence of the edge of optical absorption of this compound. As Fig. 5.2 suggests, the edge of optical absorption shifts most sharply in the Curie temperature region. Its energy is proportional to magnetization at T < TC . The red shift takes place in the paramagnetic region too. There it is going on the same value as in ferromagnetic region, i.e., the red shift of the absorption edge depends not only from the long ferromagnetic ordering but from a near ferromagnetic ordering as well. As will be seen from Fig. 5.3, an external magnetic field shifts the absorption edge to more low energy, and the rate of this edge is maximum at the Curie point region. By this means the giant red shift of the absorption edge take place in that case when ferromagnetic ordering increases in crystal under the action of magnetic field or the temperature drop. Giant red shift of the absorption edge in EuO, EuS, EuSe, CdCr2 Se4 , HgCr2 Se4 , and HgCr2 S4 is provoked by the shift of a bottom of a conduction band since the electrons are spread on magnetic atoms. That is the conduction band bottom is formed between swave functions of the magnetic ions which are heavily overlaped with d -wave functions of magnetic ions from strong s–d exchange. At the same time, a top of the valence band is formed by p-wave functions of nonmagnetic anions that are weak overlapped with d -wave functions of magnetic ions. On the contrary, in La0.9 Sr0.1 MnO3 , the top of the valence band is formed by s-wave

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F

AF

Fig. 5.3. Magnetic two-phase insulating state of degenerate antiferromagnetic semiconductor. Ferromagnetic droplets (ferrons), in which the charge carriers are localized, are shadded ; antiferromagnetic part is not

functions of magnetic cations that are heavily overlapped with d -wave functions from strong s–d exchange. Therefore, in this compound, giant red shift of the absorption edge is provoked by shift of the top of the valence band. 5.2.2 Notion of Ferrons Giant red shift of the fundamental absorption edge, observed in several magnetic semiconductors, means that the energy of the conduction electrons (holes) decreases with increasing degree of ferromagnetic order. This is an experimental confirmation by basis conclusion of s–d model that the minimum charge carrier energy is attained at a complete ferromagnetic ordering. For this reason, it was energetically favorable for conduction electrons with not too high density to become localized near impurities maintaining ferromagnetic order around them. Here their localization, besides a gain in s–d exchange, gave rise to Coulomb attraction of electrons to donors (holes to acceptors). The radius of ferron depends from temperature and magnetic field and have a size of several nanometers. The number of the charge carriers per ferron can reach several tens, which emphasizes the cooperative nature of the self-trapping. These bound ferrons exist in the ferromagnetic semiconductors near Curie point and in the antiferromagnetic semiconductors starting with T = 0 K. In nondegenerate semiconductor, the one electron is localized in ferron whereas in degenerate semiconductor a lot of electrons are localized. In degenerate antiferromagnetic semiconductor with not too high density of the charge carriers, all the charge carriers are locked inside the ferromagnetic droplets (ferrons), which disposed in antiferromagnetic host lost of charge carriers. Hence, as a whole, the crystal is insulating. So the phase separation means also the charge separation. The Coulomb interaction between each droplet and the rest of the crystal together with interface energy determine a topology of a magnetic two-phase state, which is shown in Fig. 5.4. Presence of ferrons in magnetic semiconductors ensures similarities of their properties which will be described further.

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Ig(p, Ohm.cm)

9 1 2

9

(b) 8 3 4

Ig(p, Ohm.cm)

8 5 6

7 20

7

40

H, kOe

60 (a)

6

5 0

50

100 T, K

150

200

Fig. 5.4. Cd0.983 Ga0.017 Cr2 Se4 single crystal. (a) Temperature dependence of the logarithm of the electrical resistivity in magnetic field H equal to 0 kOe (curve 1), 5 kOe (2), 10 kOe (3), 20 kOe (4), 30 kOe (5), 50 kOe (6). Temperature of the maximal negative magnetoresistance is noted by dotted line. (b) Dependence of logarithm from magnetic field at temperature equal to 124.5 K (lower curve), 128 K (circles), 132 K (crosses) [8]

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5.2.3 Electrical Resistivity and Magnetoresistance of Nondegenerate Ferromagnetic Semiconducrors with n-Type of Electrical Conductivity Figure 5.5 shows the temperature dependence of the logarithm of the electrical resistivity ρ and the influence on it of the magnetic field of Cd0.983 Ga0.017 Cr2 Se4 single crystal [8]. As illustrated in this figure, nonmonotonous dependence of lg ρ(T ) takes place, which is characteristic for all the nondegenerate ferromagnetic semiconductors. Electrical resistivity is dropping with temperature increasing on an exponential law, typical for nondegenerate nonmagnetic semiconductors, at T > TC only. In low-temperature region at T < 50 K, electrical resistivity is abrupt dropping with temperature increase, after that the lg ρ(T ) curve have a minimum and then the sharp increasing on its is beginning. At T ∼ 150 K a giant maximum of resistivity is observed (TC = 130 K). In maximum the ρ–value is on 4–5 order higher than in minimum. In Eu–monochalcogenides, this difference achieves of 17 order. Very strong action of magnetic field on ρ–value is observed in the region of ρ–maximum as illustrated in Fig. 5.5. This is a giant negative magnetoresistance, which is isotropic. The module of it has maximum at T = 130 K that is in Curie point. Magnetic field smoothes out a giant maximum of resistivity and shifts of it toward more high temperatures. In more low magnetic fields,

0.6

r/r 100

(b) 200

8

T,K

p, 10 Ohm.cm

0.2

(a)

0.5

5 kOe

160k 165

0.3

178

13

1.0

0

10

5 H, kOe

Fig. 5.5. Cd0.987 In0.013 Cr2 Se4 single crystal. (a) Temperature dependence of magnetoresistance ∆ρ/ρ = (ρH − ρ)/ρ. (Here ρH is electrical resistivity of the sample in magnetic field H). (b) Electrical resistivity dependence from magnetic field in that temperature range in which the positive magnetoresistance is observed [9]

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σ, emu/g 40 125H 130,S 133

30

137

143

20

148 151

10

158

0

5

10

H, kOe

Fig. 5.6. Dependence of magnetization σ from magnetic field at T ≥ TC of the CdCr2 Se4 single crystal with the Se-deficit ∼1.5% [10]

a positive magnetoresistance is observed at T > TC in addition to the negative magnetoresistance at T = TC [9]. Figure 5.6 shows the magnetoresistance ∆ρ/ρ = (ρH − ρ)/ρ of the Cd0.987 In0.013 Cr2 Se4 single crystal as function of temperature and magnetic field. (Here ρH is electrical resistivity of the sample in magnetic field H.) One can see the large negative magnetoresistance in the Curie point, but above the Curie point in the region 140–200 K, the magnetoresistance is positive in a certain range of magnetic fields. The dependence of the positive magnetoresistance on the temperature has a maximum. In the maximum, the magnetoresistance achieves of high value equal to ∼0.1, and in minimum on the {∆ρ/ρ}(T ) curve the negative magnetoresistance |∆ρ/ρ| ∼ 0.9. These are the giant negative and positive magnetoresistances. The both are isotropic. Mentioned above, the peculiarities of resistivity and magnetoresistance are explained by the presence in nondegenerate ferromagnetic semiconductors of bound ferrons. So the non-monotone temperature dependence of resistivity is provoked basically by non-monotone temperature dependence of the conduction electron density that is determined by the dependence of an ionizational energy of donors from temperature. At low temperature, where ρ drops with the rise of temperature, the high ferromagnetic order is in crystal and electron energy is minimal. Behavior of the conductance electrons is identical with those of nonmagnetic semiconductors. Ferron formation is finished with the

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temperature of minimum on the ρ(T ) curve, where the long ferromagnetic order is partly destroyed by heating and the difference is present between a ferromagnetic order near donors heightened from s–d exchange and the host. At T < TC , the temperature shift of the donor level is considerably less than the one of the conduction band bottom caused by the giant red shift of the latter. This is evident from the fact that ferromagnetic order near donor is higher from s–d exchange than an average of crystal; therefore, the donor level depends few from the average magnetization of crystal. Hence ionizational energy of donors rises with the temperature increasing in this temperature region. At T > TC , the ferromagnetic order far from donor gets completely destroyed, and destruction of the order close to the donor begins. For this reason, the depth of the donor level decreases with increasing temperature now. Hence, the ionizational energy of donors passes a maximum in the Curie point region provoking the maximum of the electron density. Maximum on the ρ(T ) curve is largely ensured by the electron density, although a mobility of electrons make a several contribution too. Giant negative and positive magnetoresistance are basically determined by the temperature dependence of a ionizational energy of donors too. At first consider the magnetoresistance in that temperature region of TC , where the magnetization near donor achieves of maximal value. The magnetic field does not function practically on the donor level of only decreasing of its energy at the cost of the orientation and their magnetic moments on magnetic field direction. In this time, the magnetic field splits the conduction band on two spin subbands. As a result, the minimal energy of the conduction electron is decreased. Therefore, magnetic field decreases the activation energy of donors and effect of giant negative magnetoresistance offers by increasing the density of conduction electrons. However, in that temperature region of TC , where the magnetization near donors is considerably less than the maximum one, magnetic field rises the ferromagnetic order near donors stronger than an average of crystal as its action increases by s–d exchange. In consequence of this, magnetic field decreases the donor level energy as at the cost of the increasing the magnetic moments of ferrons as the orientation of the ones. On the contrary, magnetic field decreases the minimal energy of the conduction electrons from the split of the conduction band. In this case, the decreasing of donor level by magnetic field is comparable with the decreasing of the minimal energy of the conduction electrons. However, the dependence of the energy of the conduction electrons from magnetic field is different than the one of the electrons localized on donors why the negative and positive magnetoresistance can observe at various magnetic field and an inversion of its sign. Mobility of the conduction electrons make a contribution in the maximum of the magnetoresistance near TC too but it is considerably smaller than from the density of the conduction electrons. Renewal of ferrons of magnetic field at T > TC is conform by experimental study of magnetization and photoconductivity spectra of nondegenerate ferromagnetic semiconductors CdCr2 Se4 doped by Ga or with Se–deficit.

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∆σ

1

2 A 12

3 B

4 C

8 4 0

1.0

1.2

1.4

1.6

1.8

2.0

2.2

E, eV Fig. 5.7. The ∆σ relative photoconductivity ∆σ spectra of Cd0.99 Ga0.01 Cr2 Se4 single crystal sample at temperatures: 166 K (curve 1 ), 149 K (2), 140 K (3), 112 K (4); the lower part of figure shows the influence on ∆σ-spectrum at 152 K of magnetic field H: 0 kOe (flooded circles), 3 kOe (crosses), 7 kOe (circles) [11]

Figure 5.7 shows the dependence of magnetization M from magnetic field H at T > TC of CdCr2 Se4−y single crystal sample [10]. From this figure, it follows that a jump on M (T )-curves is observed at T ≥ 143 K i.e., in that temperature region where the positive magnetoresistance was found. These jumps provoke probably by the removal of ferrons

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1

0 _1 ω ⋅10

4

H = 3.6 k0e 6.2

_2

(a)

8.2

_3 _4 0 _4

∆ r / r, %

_8 _12

(b)

_16 _20 _24 100

150

200

250

300

350

400

T.K Fig. 5.8. The isotherms of the volume magnetostriction ω (a) and the magnetoresistance ∆ρ/ρ (b) in the selected temperatures near the Curie point of the La0.7 Ba0.3 MnO3 single crystal [12]

by magnetic field. Figure 5.8 shows the relative photoconductivity spectra of Cd0.99 Ga0.01 Cr2 Se4 single crystal sample [11]. It is seen that a B–maximum at energy of a light quantum E = 1.23 eV disappears at T ∼ 150 K and it is restored by magnetic field. This maximum can attribute to an electron transition from top of valence band on ferron level corresponded to donor level of Ga–ion since it has not of a temperature shift.

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5.2.4 Magnetic Two-Phase Ferromagnetic–Antiferromagnetic State in Manganites Now consider the giant magnetoresistence and the giant volume magnetostriction ω. Giant volume magnetoresistance was found by us in manganites La1−x Mx MnO3 (M = Sr, Ca, Ba, Ag) and Re1−x Srx MnO3 (Re = Sm, Tb0.25 Nd0.3 , Eu0.4 Nd0.15 and Eu) [12, 13]. Here ω = λ + 2λ⊥ , where λ and λ⊥ are parallel and perpendicular to relation of magnetic field direction correspondingly magnitostriction. In two compounds La1−x My MnO3 at M = Ag, x = y = 0.15, and M = Ba, x = y = 0.3, we observed |ω| ∼ 2 × 10−4 , 2.5 × 10−4 and magnetoresistance |∆ρ/ρ| ∼ 10%, 12% in H = 8.2 kOe correspondingly at room temperature. This is highest values of ω that can use in various magnetomechanics devices. In Figs. 5.9 and 5.10 temperature and magnetic field dependence of ω and ∆ρ/ρ of La0.7 Ba0.3 MnO3 single crystal is shown. It will be noted that the behavior of ω and ∆ρ/ρ is similar near TC , as shown in Figs. 5.9 and 5.10. So, ω and ∆ρ/ρ are negative, their isotherms are not saturated up to the maximum field of measurement 8.2 kOe, the curves of |ω|(T ) and |∆ρ/ρ|(T ) have the maxima and in the maximum |ω| and |∆ρ/ρ| achieve of giant values 4 × 10−4 and 22.7 %, correspondingly. Because of this it may be supposed that the peculiarities of both near TC are caused by the presence in crystals of ferrons. Giant magnetoresistance is explained by the rise of nanoregion radii and the alignment of their magnetic moments in the magnetic field that facilitates the charge carriers tunneling between nanoregions. At last, the field tends to destroy the nanoregions [3]. Yanase and Kasuya showed [14] that inside ferrons the lattice constants are reduced. They are named as “Giant quasimolecule” to ferron. The reason is that in ferron the spacing between an impurity ion and its nearest magnetic ions is shortened to screen the new charge distribution and to lower the energy of ferron by increasing the overlap between the valence electron shells of the impurity and d-shells of the nearest magnetic ions. Hence it follows that near TC , an extra thermal expansion may be observed what is connected with the ferron destruction. A switching on of magnetic field at T ≥ TC would lead to a stronger increase of ferromagnetic ordering in the vicinity of impurities than on average over the crystal because its action is enhanced by s–d exchange. In other words, a magnetic field would reconstruct nanoregions, destroyed by heating and the lattice compression, inherent in them. This is the giant volume magnetostriction. Obvious, the giant volume magnetostriction due to strong s–d may manifest in the soft lattice only where s–d exchange interaction is compared with the electrostatic interaction provided with the existence of crystal. So, the giant volume magnetostriction is not found in monochalcogenides of Eu and chalcospinels in which the crystalline lattice is harder than in manganites. The presence of ferromagnetic–antiferromagnetic state in manganites is confirmed by magnetic properties. For example, a spontaneous magnetization

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0 330 K

−1

θ ⋅ 10 4

325 320

−2

(a)

−3

315

305

−4 0

350

∆ r / r, %

−4 −8

330

−12

325

−16

300

(b)

−20 315

−24 0

2

4

6

8

10

H, kOe Fig. 5.9. The temperature dependence of volume magnetostriction ω (a) and magnetoresistance ∆ρ/ρ (b) in the selected magnetic field near Curie point of the La0.7 Ba0.3 MnO3 single crystal [12]

portion on the magnetization isotherms M (H) is appeared at doping of the antiferromagnetic semiconductor EuMnO3 by Sr or Ca [15]. It is caused by ferrons (Fig. 5.11). For La0.9 Sr0.1 MnO3 single crystal, Eu0.7 A0.3 MnO3 (A = Ca, Sr) ceramics, La0.1 Pr0.6 Ca0.3 MnO3 and La0.84 Sr0.16 MnO3 epitaxial films we observed distinction between the values of magnetization of sample cooled in magnetic field (FC sample) and without field (ZFC sample) shown in Fig. 5.12 for

5 New Type of Nanomaterials 3

18.5K 4.4

Magnetization (mB/f.u.)

Eu0.7Sr0.3MnO3 2

101

5 4

1

3

11.3 27.8 35.8 43.6 49.6 53.7

0 -1

2 1

-2 10

0

40 20 30 Applied field (kOe)

50

0 60

Magnetization, 4TTM (Gs)

Fig. 5.10. Magnetization isotherms of Eu0.7 Sr0.3 MnO3 [15] FC

20

15 ZFC 10

5

0 0

50

100

150

Temperature, T(K) Fig. 5.11. Temperature dependence of the magnetization in a field of 0.6 kOe obtained under different cooling conditions for La0.1 Pr0.6 Ca0.3 MnO3 thin film on (001)SrTiO3 substrate. ZFC curve: the film is cooled in zero field. FC curve: the film is cooled in magnetic field of 0.6 kOe [15]

La0.1 Pr0.6 Ca0.3 MnO3 sample, and displacement of hysteresis loop of FC sample with respect to H–axis (Fig. 5.13). Figure 5.12 shows that M (T )–curve for ZFC sample has a maximum at Tf whereas such maximum is absent for FC sample. It was found for all the listed compounds that the remanence of FC sample was higher than the one of ZFC sample. It seems that the magnetic properties mentioned earlier are similar to those of the cluster spin glasses. Shifted hysteresis loops, similar to Fig. 5.13, were first observed in [16] in partly oxidized Co and were connected to the exchange interaction between ferromagnetic Co

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40

Arbitrary Units

Magnetization, 4TTM (GS)

60

FC

ZFC

20 −200 −100

0 100 200 H, oe

0 20 −40 −4000

−2000

0

2000

4000

H, Oe Fig. 5.12. Hysteresis loop for the La0.1 Pr0.6 Ca0.3 MnO3 thin film on (001)SrTiO3 substrate at 5 K obtained under different cooling conductions. ZFC curve: the film is cooled in zero field. FC curve: the film is cooled in magnetic field of 4 kOe. Inset: hysteresis loop for bulk sample La0.9 Sr0.1 O3 at 5 K after cooling in magnetic field of 20 kOe [15]

Ea, eV 1.15

1.10

1.05

1.00 100

200

300

Tc 400

T, K

Fig. 5.13. Sample of CuCr2 S1.5 Se0.5 . The temperature dependence of energy of the fundamental absorption edge Eg [22]

5 New Type of Nanomaterials

103

particles and their shells from antiferromagnetic CoO. This phenomenon is given the title, an exchange anisotropy. Later on the shift of hysteresis loops is considered as a prove of spin-glass state [17]. Meanwhile this phenomenon is only explained for a cluster spin glasses. The presence of the exchange anisotropy in manganites is unambiguously connected to magnetic two-phase ferro–antiferromagnetic state in them. From the displacement of hysteresis loop (5.1) ∆H = Ku / Ms , where Ku is the constant of exchange anisotropy and Ms is the saturation magnetization, and Ku -values are calculated. They turn out that Ku ∼ 103– 104 erg cm−3 . From Ku -value, the negative exchange integral J, written the one Mn–O–Mn connection on the ferron/host interface, was estimated [15]. |J|-value is equal to 10−7 − 10−6 that is on 2–3 order smaller than the |J2 | = 5.8 × 10−4 eV. Here J2 is the exchange integral between ferromagnetic layers in LaMnO3 , received from neutron skattering experiments [18]. By this it is meant that on the droplet/host interface the presence of layer with a canted spins is unlikely. 5.2.5 Antiferrons Nagaev [19] gave an explanation for the large blue shift of the width of the forbidden gap in ferromagnetic semiconductors on the basis of the assumption that interband s–d exchange plays a dominant role in them. Interband s–d exchange is generally weaker than s–d exchange in the band formed by d-wave functions, although it is stronger than s–d exchange in band formed by p-wave functions. In Eu-monoshalcogenides and chalcospinels, the conduction band is formed by d-wave functions while in manganites the valence band is formed by d-wave functions. This is attributable to the fact that in magnetic semiconductors, the electron move mainly from one magnetic cation to another. However, for some reason the s–d exchange in the band, formed by dwave functions, may be weaker (for example, if it appears as a result of the hybridization of states with opposite signs of the exchange integral with localized d electrons). Interband s–d exchange causes virtual transitions of electrons from the valence band to the conduction band. The interaction between two energy terms is known to result in repulsion between them. Like intraband exchange, interband s–d exchange is enhanced when ferromagnetic ordering is established. Therefore, the repulsion between the valence band and the conduction band caused by it becomes stronger, i.e., the gap between them increases. The blue shift of forbidden gap width associated with ferromagnetic ordering should be observed in just such cases. Ferromagnetic semiconductors with a blue absorption edge shift have specific properties that differ sharply from the properties of magnetic semiconductors with a red absorption edge shift.

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Further we will consider the chalcospinels. The fact that the bottom of the conduction band is descended when the ferromagnetic ordering is disrupted “antiferron” states of the conduction electrons in crystals with a blue shift is possible, in principle. They are characterized by the fact that conduction electrons create regions with disrupted ferromagnetic ordering in a ferromagnet and stabilize these regions by being trapped in them. The superexchange energy expended to create the nonferromagnetic regions is compensated by lowering of the electronic energy, since such regions are potential wells for conduction electrons. Such nonferromagnetic regions can also form around defects. In this case, the trapping of an electron around a donor (hole around an acceptor) is promoted by the electrostatic attraction of the electron to the donor (the hole to the acceptor) in addition to the s–d exchange. Several experimental facts attest to the existence of antiferron nanoregions in feromagnetic In or Ga-doped CdCr2 S4 in which the giant blue shift of the band-gap connected with ferromagnetic ordering is observed. Nanoregions with disrupted F ordering were discovered in In or Ga-doped CdCr2 S4 using Cd2+ , Ga3+ , and In3+ NMR [20]. In the opinion of authors [20], this disruption of the ferromagnetic ordering is caused by the exchange interaction of a valence electron of an impurity with the 3d electrons of Cr3+ ions surrounding it, since the spin density of this electron, which derives from the hyperfine interaction, is unusually high. Therefore, these nanoregions are essentially antiferrons. In this compound doped by Ga or with deficiency of S the giant magnetoresistance, decreased values of the magnetic moment and paramagnetic Curie point θ (without a change in TC ) with increasing of doping were observed [21]. In fact, since θ is determined by the sum of the exchange interaction occurring in the crystal, the presence of antiferrons should lower θ, since exchange is suppressed in these nanoregions. The lack of a magnetic moment in the antiferrons diminishes the magnetic moment of the crystal as a whole. It should be noted that lattice constant does not vary in response to doping and is equal to the lattice constant of the undoped compound, i.e., there are no variations in the exchange interaction due to variation of the lattice constant. Another compound with the giant blue shift of the band-gap connected with ferromagnetic ordering is CuCr2 S1−x Sex with 0.5 ≤ x ≤ 1.5. Figure 5.14 shows the temperature dependence of band-gap of compound CuCr2 S3.5 Se0.5 from article [22]. It is seen that the band-gap undergoes a large blue shift of 0.15 eV in the 130 ≤ T ≤ 410 K temperature range and that the rate of this shift is greatest in the vicinity of the Curie point. This allowed the existence in this compound of antiferron nanoregions (nanoregions with disrupted ferromagnetic ordering). In this compound, a semiconductor type of conduction has been discovered while in the two extreme compounds of CuCr2 S1−x Sex system, namely in ferromagnetics CuCr2 S4 and CuCr2 Se4 , the metallic type of conduction is observed. Figure 5.15 shows that the semiconducting samples of this system with 0.5 ≤ x ≤ 1.5 exhibit decreased values of the magnetic

5 New Type of Nanomaterials

105

5

n 4.2 K

4

np 3

450

400

Tc,q, K

Tc

2

q 1

350

0

300 1

2 x

3

Fig. 5.14. CuCr2 S4−x Sex system. Dependence of the magnetic moment per chemical formula unit determined from the Curie–Weiss constant np , the Curie point TC , and the paramagnetic temperature θ on the composition [22]

moments, the Curie temperature and paramagnetic Curie point in comparison with the extreme compounds that is explained by the presence in them of antiferrons. It was shown in [23] that the presence of two kinds of anions in CdCr2 S1−x Sex lattice results in a sharp decrease in the intraband s–d

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µ,cm2 V.S

R0, cm3/c

20

p

15

µ

0.7

10

0.5

0.010

p•10−20,cm−3

0.015

0.3 0.005

0.1 0

0 1

2

3

4

x Fig. 5.15. Dependence of the normal Hall constant R0 , the mobility µ, and the density of charges carriers p on the composition in the CuCr2 S4−x Sex system at 100 K [22]

exchange. It is completely possible that a similar phenomenon also occurs in the CuCr2 S1−x Sex system. This permits the existence of antiferron states in this system, whose presence can account for the semiconductor type of conduction and the lowering of the Curie temperatures and the magnetic moments in the samples with 0.5 ≤ x ≤ 1.5 in comparison with the extreme compounds and the samples with x ≥ 2. The samples with x = 0 and x ≥ 2 exhibit higher Curie points and a metallic type of conduction. It is assumed that in the samples with metallic conduction, there is exchange by means of the charge carriers with resultant significant rising of the Curie point in comparison with the semiconducting chalcogenide spinels, such as CdCr2 S4 , where TC = 130 K. Trapping of the charge carriers in antiferron nanoregions produces semiconductor conduction in the sample with 0.5 ≤ x ≤ 1.5. Because of the sharp

5 New Type of Nanomaterials

107

1 2 3

σ, µB

0.8

0.4

0

10

20

30

H, kOe Fig. 5.16. Field dependence of the magnetization σ at 4.2 K for xCoCr2 S4 – (1 − x)CuCr1.5 Sb0.5 S4 with x = 0 (1), 0.25 (2), 0.5 (3) [24]

decrease in the concentration of charge carriers (Fig. 5.16), exchange by means of the charge carriers is suppressed with the resultant lowering of TC and θ. Since there is not magnetic moment in antiferrons, the total magnetic moment in the crystal is also lower than in the metallic samples of this system. The samples with 0.5 ≤ x ≤ 1.5 are also characterized by anomalous relative positions for the ferro- and paramagnetic Curie points (TC > θ). It is also interesting that the magnetic moment np calculated for these samples from the Curie constant in the Curie-Weiss law is significantly higher than the magnetic moment at 4.2 K (n4.2 K ). For x ≥ 2 and x = 0, np < n4.2 K (Fig. 5.15). Ferromagnets usually satisfy the relation np < n4.2 K , which is generally attributed to the temperature dependence of the exchange interactions. The unusual relationship between np and n4.2 K for the semiconducting

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samples can be explained in the following manner. The value of np is determined from data on the paramagnetic susceptibility at temperatures above TC . At such temperatures, the antiferron states are thermally destroyed in part, i.e., the lowering of the magnetic moment of the crystal as whole, which can be observed, for example, at 4.2 K, decreases on their account. Since the paramagnetic Curie point is determined by the sum of the exchange interactions in the crystal, the contribution to the total exchange from the antiferron nanoregions, where ferromagnetic exchange is disrupted, lowers θ. At the same time, TC is determined mainly by the connected ferromagnetic phase of the crystal and therefore TC > θ. 5.2.6 Afmons An “afmon” is a new type of self-trapped of charge carrier in antiferromagnetic semiconductor with a sufficiently high Neel temperature TN , in which the energy conditions are unfavorable for ferrons. Nagaev showed [25] that an antiferromagnetic phase different from that which exists in a crystal may serve as a potential well for a charge carrier in an antiferromagnetic host. For example, the energy of a charge carriers in a layered antiferromagnetic semiconductor is lower than in checkerboard antiferromagnetic phase; their difference can reach several tenths of electronvolts. In semiconductor with a checkerboard, antiferromagnetic ordering the charge carrier may thus become self-trapped in nanoregion with the layered antiferromagnetic ordering. The energy of such nanoregion (afmon) can be lowered even further if the moments of the sublattices are skewed in it, so that afmon acquires a magnetic moment. However, the greatly skewed sublattice moments rules out by the requirement of the energy disadvantage of ferrons. Magnetic and galvanomagnetic properties of the system xCoCr2 S4 – (1 − x)CuCr1.5 Sb0.5 S4 are evidenced about the afmon presence in it [24]. This system is a solid solution of two semiconducting compounds with spinel structure: CuCr1.5 Sb0.5 S4 (mineral, florensovit) and ferrimagnetic CoCr2 S4 . The magnetic properties of compounds with x = 0 and 0.25 are typical for antiferromagnet: the magnetization is a linear function of the magnetic field (Fig. 5.17) and the susceptibility goes through a maximum at the Neel temperature TN (23.7 K for the former compound and 32 K for the latter). Figure 5.17 shows the dependence of magnetization σ from H at 4.2 K for all the considered compounds. For compound with x = 0.5, curve σ(H) illustrate the superposition of a linear part, which is inherent to the antiferromagnet, and a small spontaneous part σs , which is equal to 0.27µB to chemical formula. Value TN = 48.3 K is obtained for this compound as the temperature of the maximum of the susceptibility of a linear part of σ(H) dependence vs. temperature. Paramagnetic susceptibility of the florensovit and compounds with Co obeys the Curie-Weiss law. The paramagnetic Curie point θ in compounds with Co is rather increased in comparison with the value θ = −156 K for florensovit, so θ = 45 K for the compound with x = 0.25 and θ = 109 K

5 New Type of Nanomaterials

0

109

1

∆r/r

2 −0.1 3

−0.2

0

10

20

30

40

T, K Fig. 5.17. Temperature dependence of the magnetoresistance ∆ρ/ρ of the composition xCoCr2 S4 – (1 − x)CuCr1.5 Sb0.5 S4 for external magnetic fields H = 3 kOe (1), 15 kOe (2), and 30 kOe (3) [24]

for x = 0.5. It has been known that the value of θ is defined by the sum of the exchange interactions that take place in the crystal. Since the paramagnetic Curie point in CuCr1.5 Sb0.5 S4 is negative and large in magnitude, antiferromagnetic ordering in this compound apparently has a checkerboard antiferromagnetic structure. It is possible that a more complicated magnetic structure, within which antiferromagnetic interactions dominated, take place. The addition of CoCr2 S4 in this compound apparently gives rise to nanoregions with layered antiferromagnetic structure whose existence causes a substantial rise in the value of θ. A giant negative magnetoresistance has been observed in compound with x = 0.25 and 0.5. Figure 5.17 shows the temperature dependence of magnetoresistance of sample with x = 0.5. Maximum of ∆ρ/ρ ∼ 16 % and 24 % in H = 30 kOe for compound with x = 0.25 and 0.5 correspondingly and the isotherms of ∆ρ/ρ are still far from saturation. In florensovit and CoCr2 S4 , there is essentially no magnetoresistance at the error of measurement, ∼0.01 %. The giant negative magnetoresistance and sharp increasing of θ in compounds with Co are provoked by afmon presence. In the CoCr2 S4 ferrimagnet, the moments of the Cr3+ ions occupying octahedral positions have a ferromagnetic order. They form a sublattice whose moment is ordered antiferromagnetically with respect to the moment of the sublattice of Co2+ ions, which occupy tetrahedral positions in the spinel structure. The addition of CoCr2 S4 to CuCr1.5 Sb0.5 S4 with the checkerboard antiferromagnetic structure creates favorable conductions for the formation, near Co2+ ions, nanorigions with a layered antiferromagnetic structure.

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Localization of charge carriers in these nanoscale regions promotes their stabilization if their energy is lower in this layered structure than in the main checkboard structure. We have determined the band gap in CoCr2 S4 and CuCr1.5 Sb0.5 S4 from measurements of the spectra of the diffuse-reflection coefficient. The width of the band gap turned out to be 0.9 eV in the former and 1.45 eV in the latter. The carrier energy in CoCr2 S4 is thus lower than in CuCr1.5 Sb0.5 S4 . Consequently, a localization of charge carriers near Co2+ ions will stabilize afmon states in compounds under consideration with Co dopes. Existence of the afmons within them involves a rise of θ, which is described above. The imposition of an external magnetic field destroys the afmons and delocalizes the charge carriers in them; i.e., it leads to a giant negative magnetoresistance, as observed in the compounds with Co. As noted above, the compound with x = 0.5 has a small spontaneous magnetic moment σs − 0.27 µB /f.u. It can be explained by means of uncomplete compensation of magnetic moments of layers in afmons. Compensation is not complete since the number of layers is odd or since squares are dissimilar or both two factors take place. Simultaneously, the magnitude of σs is too small to be explained by the CoCr2 S4 cluster magnetisation. Indeed, in this situation the value of spontaneous magnetization must exceed the experimentally observed value: namely, for the compound with x = 0.25, 0.62 µB/f.u., and for x = 0.5, 1.25 µB/f.u., Here we have accounted that the spontaneous magnetization of CoCr2 S4 is about 2.5 µB /f.u.

References 1. E.L. Nagaev, Physics of Magnetic Semiconductors, (Nauka, Moscow,1979; Mir, Moscow, 1973) 2. E.L. Nagaev, Giant magnetoresistance and Phase Separation in Magnetic Semiconductors (Imperial College Press, London, 2002) 3. E.L. Nagaev, Phys. Rep. 346, 387 (2001) 4. E. Dagotto, T. Hotta, A. Moreo, Phys. Rep. 344, 1 (2001) 5. E. Dagotto, Nanoscale Phase Separation and giant magnetoresistance: The Physics of Manganites and Related Compounds (Springer, Berlin, 2003) 6. T. Kasuya, A. Yanase, Rev. Mod. Phys. 40, 684 (1968) 7. T. Arai, M. Wakaki, S. Onari, K. Kudo, T. Saton, T. Tsushima, J. Phys. Soc. Jpn. 34, 68 (1973) 8. K.P. Belov, L.I. Koroleva, L.N. Tovmasyan, J. Eksp, Theor. Fiziki 73, 2309 (1977) 9. K.P. Belov, L.I. Koroleva, C.D. Batorova, V.T. Kalinnikov, T.G. Aminov, G.G. Shabunina, ZhETF (Pisma) 22, 304 (1975) 10. K.P. Belov, L.I. Koroleva, M.A. Shalimova, V.T. Kalinnikov, ZhETF 72, 1994 (1977) 11. K.P. Belov, L.I. Koroleva, V.Yu. Pavlov, FTT 27, 626 (1986) 12. L.I. Koroleva, R.V. Demin, A.V. Kozlov, D.M. Zashcherinskii, Ya.M. Mukovskii, JETP 104, 76 (2007) 13. A.I. Abramovich, L.I. Koroleva, A.V. Michurin, JETP 122, 1063 (2002)

5 New Type of Nanomaterials 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

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A. Yanase T. Kasuya, J. Phys. Soc. Jpn. 25, 1025 (1968) L.I. Koroleva, R. Szymczak, J. Phys. Chem. Solids 64, 1565 (2003) W.H. Meiklejohn, C.P. Bean, Phys. Rev. 105, 904 (1957) (a) J.S. Kouvel, J. Phys. Chem. Sol. 21, 57; (b) J. Phys. Chem. Sol., 24, 795 (1965) F. Moussa, M. Hennion, J. Rodriguez-Carvajal, H. Moudden, L. Pinsard, A. Revcolevschi, Phys. Rev. B54, 15149 (1996) E.L. Nagaev, JETP Lett. 25, 76 (1977) M.C. Mery, P. Veilet, K. Le Dang, Phys. Rev. B31, 2656 (1985) K.P. Belov, S.D. Batorova, L.I. Koroleva, M.A. Shalimova, JETP Lett. 26, 62 (1977) L.I. Koroleva, JETP 79, 153 (1994) Z.E. Kunkova, T.G. Aminov, L.L. Golik, Sov. Phys. Sol. Stat. 18, 1212 (1976) L.I. Koroleva, M.Kh. Mashaev, D.A. Saifullaeva, J. Magn Magn. Mat. 140–144, 2045 (1995) E.L. Nagaev, JETP Lett. 55, 675 (1992)

6 Cerium-Doped Yttrium Iron Garnet Thin Films Prepared by Sol-Gel Process: Synthesis, Characterization, and Magnetic Properties ¨ urk1 , I. Avgın1 , M. Erol2 , and E. C Y. Ozt¨ ¸ elik2 1

2

Electrical and Electronics Engineering Department, Ege University, 35100 Izmir, Turkey, [email protected] Metallurgical and Materials Engineering Department, Dokuz Eylul University, 35160 Izmir, Turkey

Summary. We studied here synthesis, characterization, and magnetic properties of YIG (yttrium-iron-garnet, Y3 Fe5 O12 ) and Ce-doped YIG (Cex Y3−x Fe5 O12 ) thin films prepared by using a sol–gel technique for magneto-optical applications. Pure YIG and Ce-doped YIG films were deposited on a glass and Si (100) substrates out of a solution prepared from Ce, Y, and Fe-based precursors, solvent, and chelating agent at low temperature using the sol–gel technique. Prior to coating process, solution characteristics that influence the intended thin film structure were determined using turbidimeter, pH meter, and rheometer machines. Film thickness was monitored with varying sol–gel solution’s properties and spin coating’s process parameters. Since we mainly want to improve the magnetic properties of our films, an optimum sol–gel solution containing cerium, yttrium, and iron precursors were found, and a garnet phase was formed after annealing at temperatures between 700 and 1,000◦ C for 2 h in air. The thermal, structural, and microstructural properties of the films were characterized using DTA/TG, XRD, and SEM-EDS. The magnetic properties of the films produced by doping with Ce with an optimal process conditions were investigated through VSM device. The films include micro and nanosize CeO2 regions because of using partially dissolved Ce precursor in the solution. Our preliminary study revealed that a significant improvement in magnetic properties of polycrystalline YIG thin films have been achieved through the substitution of Ce.

Abbreviations YIG: Yttrium iron garnet(Y3 Fe5 O12 ) Ce:YIG: Cerium doped (Cex Y3−x Fe5 O12 ) DTA/TG: Differential thermal analysis-thermogravimetry XRD: X-Ray Diffraction SEM: Scanning electron microscope

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EDS: Energy dispersive spectrometer VSM: Vibrating sample magnetometer DC: Direct Current AC: Alternating Current Ms : Saturation magnetization

6.1 Introduction New technological applications such as magnetic sensor, optical wave-guides, magneto-optical modulator, and integrated magneto-optic devices require improved sensitivity, smaller size, and compatibility with electronic systems. For such applications, materials with a good magneto-optic property are of a significant issue. Number of magneto-optic materials is available e.g., flint glass (SF6 ), BSO (Bi12 SiO2 ), BGO (Bi12 GeO20 ), and garnets [1–5]. Garnets have more appeal because of their large magneto-optical response. Yttrium iron garnet (Y3 Fe5 O12 , YIG) having promising magnetic and magneto-optic properties is the most suitable material for these applications [1, 2]. YIG has a complex cubic structure, wherein nonmagnetic Y3+ ions occupy dodecahedral (c) sites and magnetic Fe3+ ions occupy octahedral (a) and tetrahedral (d) sites as shown in Fig. 6.1. To improve magneto-optic properties of pure YIG, yttrium may be substituted by one of the lanthanides e.g., lanthanum, cerium, neodymium, gadolinium, and so on. Its unit cell includes different magnetic ions, iron, and one of the rare earth groups. Its magnetic property arises from the antiparallel ordering between Fe3+ ions in the a-site and d-site as a result of exchange couplings between the ions but the c-site

[


[

 0]


 0]






 ]

1.88A

[

A

37

2. 2.44A

[ 000]

2.00 A

[


0
 ]

y

16 a ° Fe3+ at [00 
] 24 d • Fe3+ at [0
  ] 24 c ° Y3+ at [ 


 ] and at [0
  ] O o2− Commom to polyhedra [ y,z,
 + x ]

x z

Fig. 6.1. Cubic crystal structure of YIG [7]

6 Cerium-Doped Yttrium Iron Garnet Thin Films

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ions couple weekly leading to cant-parallel to the d-site ion [6, 7]. The net magnetic moment of YIG per unit cell is 40 Bohr magnetrons [8, 9]. Its saturation magnetization is 136 kA m−1 at room temperature. With Ce addition, paramagnetic trivalent Ce3+ ions are replaced with nonmagnetic Y3+ ions in c-sites [3]. Magneto-optic effect arises after the change of the state of the polarization of light due to interaction with a magnetic material discovered by M. Faraday, who found that a polarized light was rotated after passing through a glass under an external magnetic field along the direction of propagation of incoming light. For a magnetic field perpendicular to incoming light similar rotation was also observed now known as Voigt effect. Magneto-optical effects for the YIG type and its variants can be explained macroscopically via difference in the refraction indices for right and left circularly polarized light hence it is quite often called circular birefringence. There is also similar effect of the circular dichroism arises from the absorbance differences for left and right circularly polarized light. Yet both effect can be attributed usually to the Zeeman effect, i.e., two degenerate electronic state split into two circular components, which worked well for Bi-YIG [10]. Transitions involving with these states produces usual absorption and dispersion line shapes [10]. YIG has very large figure of merit (rotation angle over the absorption) near infrared; however, in the visible band, the absorption becomes very large compared with the increase in the Faraday rotation angle making the material hard to use in a desired application. Hence there are many efforts to increase the magneto-optic properties of YIG, e.g, for Bi-YIG material this increase was observed as a function of the Bi concentration [10]. The most suitable ions expected to increase the magneto-optical properties are those of having a right radii, e.g., Bi, Pb, Ce, Pr, Nd, Ru, Rh, Ir, and Co [10, 11]. The phenomenological theory of magneto-optic effects can be obtained by assuming a dielectric tensor with a cubic symmetry and with an off-diagonal element after solving the Maxwell’s equations where the interested reader can refer to Refs. [10,12]. Furthermore, some of the studies have focused on to enhance the magneto-optical properties [13], and the magnetic properties [14] revealed that the most promising candidate to enhance magneto-optical activity strongly in iron garnets in the visible and near infrared regions are cerium and bismuth substituted materials. Faraday rotation of the polarization in YIG-based materials has very important technological applications. In this case, the polarization rotation angle can be formulated for magneto-optic materials. It can be written by θ(ω) = V (ω)Hd, where H is applied field, d is the thickness of the sample, and V is the Verdet’s constant as a function radial frequency ω of the incoming light. The large Verdet’s constant finds technological applications as found in certain magnetic materials, i.e., the Faraday rotation of 104 ∼ 106◦ /cm for magnetic fields nearly 104 Oe. This feature of the rotation has a nonreciprocal behavior that was exploited in many devices like isolators. In as much as the rotation depends upon the magnetic field strength of the incoming wave mak-

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ing a certain angle after passing the sample, if it is reflected back and passing through the material again, it will be rotated once more with the same angle and so reflected twice the initial rotation angle with respect to incoming wave polarization. This feature may be called optical “diode.” Optical or RF isolators are two-port unidirectional devices that transmit light in one direction and block it in the opposite direction. They are used reliably after eliminating the light back-reflected into lasers as required in optical telecommunication systems. Hence hard work is still required for further improvements in terms of greater power demand and efficiency for some photonic devices or integration with a nano/microelectronic circuitry [15]. Magnetic photonic crystals have been advanced into an area known as magnetophotonics. Inclusion of magnetic materials in photonic structures introduces some peculiar feature in the electromagnetic wave propagation i.e., the above-mentioned nonreciprocity and unidirectionality, controlled by an external magnetic field. Numerous works exist to augment the Faraday rotation in photonic band gap structures composed of magnetic thin film stacks [16]. Briefly in magnetic thin film stacks the augmented rotation is due to trapping the light in a cavity. Although this is a right approach, the resulting transmission is weakened unfortunately owing to the absorption. To avoid this, one exploits interactions of defect splitting and magnetic splitting to form random resonances and produce a rotation with unaffected/better transmission. And this leads to a narrowband solution [17]. To obtain broadband, M. Inoue et al. [18] have studied one-dimensional photonic crystals in Ce-YIG with multiple defects at 1.55 µm wavelength, a favorite wavelength by telecommunication workers, and it was revealed that the defect spacing can be tuned resulting in a flat broadband response, producing nearly total transmission and very large Faraday angle through a stacked film with defects. A linear relation between number of defects and bandwidth was found. A hybrid incorporation of Faraday rotators and polarizing elements is now available where extinction ratios of about 30 dB for a band of around 1.55 µm was achieved by attaching half-wave plates. For other applications, there are also several works on the behavior of yttrium iron garnet mainly on the valence-uncompensated doping or the substitution of iron in tetrahedral or octahedral sites [5–7], or the substitution of yttrium in dodecahedral sites by different other metallic cations [3, 8–11, 19]; whereas, some others focused on possible application of YIG and substituted YIG as Higuchi et al. [20, 21] who obtained Ce-YIG thin films for magnetic sensor applications. Their materials displayed greater Faraday rotation in Ce substitution than that of in Bi substitution. The magnetic-field sensitivity of Ce0.24 Y2.76 Fe5 O12 was about 0.0048% m/A larger than that of (BiGdLaY)3 (FeGa)5 O12 . Gomi et al. [21] has made a single crystal thin film Ce-YIG for optical memory devices, and Mino et al. have grown Ce-YIG films for optical waveguides [22]. They both employed the usual R.F. diode sputtering or the pulse laser deposition to prepare the cerium-substituted YIG films. Ce-YIG single crystals have already been widely studied because the addition

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of cerium oxide can significantly enhance the Faraday rotation and reduce the optical propagation loss [9]. Sekijima et al. [6] produced fibrous single crystals Ce-YIG by floating zone (FZ) method changing atmosphere to nitrogen atmosphere as they increased the solubility limit of Ce. Another application of Ce-YIG based on Faraday rotation is successful fabrication of nonreciprocal planar light-wave circuits of the thin films on an amorphous substrate having good magneto-optical properties made by Uno and Noge [23]. As mentioned, several techniques can be employed to make YIG-based materials such as RF magnetron sputtering, sol–gel and pulse laser deposition technique [1–6]. Of these, the sol-gel processing has a number of advantages. To illustrate this, it is possible to synthesize quite good polycrystalline ferrites with the sol–gel method. The sol–gel process offers considerable advantages such as better mixing of the starting materials and excellent chemical homogeneity in the final product. Moreover, the molecular level mixing aids the structure evolution lowering the crystallization temperature [7], and the sol–gel layer can be deposited in desired thickness in one step because this thickness depends only on precursor’s concentration. The available Ce-YIG material research is mainly on single thin film crystals [5]. Here a polycrystalline Ce-YIG, studied infrequently, have been produced using the sol–gel method, and eventually its magneto-optical properties will be studied. This chapter is devoted to the research on YIG and Ce-YIG films prepared on pyrex glass and Si (100) substrates from solutions of Ce, Y, and Fe alkoxide preqursors, 2,4-pentanedionate, propionic acid, glacial acetic acid, and hydrochloric acid by using the sol–gel technique for magneto-optical technologies. Along this aim, turbidity, pH measurement, and rheological properties of the prepared solutions were resolved. To define chemical structure and reaction type of intermediate temperature products and to use suitable process regime, differential thermal analysis-thermogravimetry (DTA-TG) device was used in the film production. The structural and microstructural properties of the coatings were extensively characterized using X-ray diffractometry (XRD), profilometer and scanning electron microscopy (SEM) plus energy dispersive spectroscopy (EDS). The magnetic properties of thin films measured trough vibrating sample magnetometer (VSM).

6.2 Experimental Details The solutions were separately prepared from Ce, Y, and Fe-based precursor materials as shown in Fig. 6.2. The procedure include the following: Ce, Y, and Fe-based precursor materials were dissolved in the 2,4-pentanedionate used as solvent. An adequate amount of propionic acid, glacial acetic acid, and hydrochloric acid was also mixed to the solutions to serve as a stabilizing agent by forming a chelate complex. The solutions were formed through the hydrolysis of Ce, Y, and Fe alkoxides after vigorously stirring at room temperature for a 30-min period at 100 rpm. Transparent precursor solutions

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Ce, Y and Fe alcoxides Dilution with 2,4 pentanedionate Dissolution with propionic acid Hydrochloric acid o Mixing 25 C, 30 min, air Transparent solutions

Spin coating / Dipping

o

Drying (300 C, 10 min)

o

Heat treatment (500 C, 5 min)

o

Annealing (700 - 1000 C, 2 h, air)

YIG based coatings Fig. 6.2. The diagram of flow chart for sol–gel process

can remain at least 7 days without precipitation. Two different solutions were prepared for the Ce-YIG thin film production. We observe that Ce precursor was not dissolved completely in the base YIG solution. To avoid this problem, hydrochloric acid was added into Ce-YIG based solutions. To determine solution characteristics that affect thin film structure, turbidity, pH values, and rheological properties of the prepared solutions were measured by turbidimeter, pH meter, and rheometer machines, respectively, before the coating processes. Turbidity properties of the solutions were also measured using standard solutions for coating process by TB 1 Velp Scientifica Model turbidimeter according to ISO 7027 nepheometric method. The sample ´ mm and height of 50 mm. Forwas placed in a vessel with a dimension of Ø25

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mazin has been recognized through out the world as a primary standard. Thus formazin solution was used to calibrate the turbidity in the range of 0 and 1,000 ntu (nephelometric turbidity unit). Having prepared transparent solutions, we measured pH values of the solutions to check their acidic and basis characteristics, using a standard pH meter with Mettler Tolede electrode. The observed pH values of the transparent solutions varied from 3 to 4. Besides pH and turbidity measurements, rheological behavior of the solutions including viscosity was also measured using CVO 100 Digital Rheometer (Bohlin Instrument). Commercial Pyrex glass and Si (100) substrates were used as a substrate. The substrate surface was cleaned with acetone just before the coating process. The solutions were used for coatings on the Pyrex glass and Si (100) substrates by dip and spin coating systems at ambient conditions. The solutions were deposited on the substrates by using a dip coating process with a withdrawal speed of 0.3 cm s−1 and with staying in the solution for 2 min. The temperature of solutions was kept at room temperature in our laboratory. As reported [24], the dip coating system involves formation of a film through a liquid entrainment process that may be either batch or continuous in nature. Usual steps include immersion of the substrate into the dip-coating solution, start-up, where withdrawal of the substrate from the solution begins, film deposition, solvent evaporation, and continued drainage as the substrate is completely removed from the liquid bath. The film thickness formed in dip coating is mainly governed by viscous drag, gravitational forces, and the surface tension. The spin coating system has been used for several decades in thin film preparation process. A typical process entails depositing a small puddle of a gel onto the center of a substrate and then spinning the substrate at high speed. Centripetal acceleration causes the gel to spread to, and eventually off, the edge of the substrate leaving a thin film of gel on the surface. Hence the film thickness and their properties depend on the nature of the gel (viscosity, drying rate, percent solids, surface tension, etc.) and the parameters chosen for the spin process. Factors such as final rotational speed, acceleration, and fume exhaust contribute to how the properties of coated films are defined. For most resin materials, the final film thickness will be inversely proportional to the spin speed and spin time. Final thickness will be also be proportional to the exhaust volume although uniformity will suffer if the exhaust flow is too high, since turbulence will cause nonuniform drying of the film in the spin process [25]. YIG and Ce-YIG solutions were spin coated on the glass and Si (100) substrates with 3,000 rpm for 20 s at room temperature. After dip and spin coating processes, the obtained gel coatings were dried at 300◦C for 10 min, heat treated at 500◦ C for 5 min, and subsequently annealed at temperatures between 700 and 1,000◦C for 2 h in air in an electric furnace. Thicker coatings were obtained by repeating this process over and over again before annealing process and also by increasing viscosity of solution. Six kinds of the prepared samples, called as A, B, C, D, E, and F, were represented in this study.

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¨ urk et al. Y. Ozt¨ Table 6.1. Sample descriptions of thin films

Parameters

A

B

C

D

E

F

Solution Coating process Substrate

Ce-YIG Spin coating Si (100)

Ce-YIG Spin coating Si (100)

Ce-YIG Spin coating Si (100)

YIG Spin coating Si (100)

YIG Spin coating Si (100)

800

Ce-YIG Dip coating Pyrex glass 900

900

800

1,000

250

1,000

900

700

1,000

Annealing 800 temp. (◦ C) Thickness 900 (nm) Ce precursor Completely solved

Not Not Completely – completely completely solved solved solved

–

Solution differences, coating processes, substrates, annealing temperatures, film thicknesses, and Ce condition in the solution are listed in Table 6.1. Thermal behavior of Ce, Y, and Fe-based xerogels, which were dried at 200◦ C for 30 min in air, was evaluated at a heating rate of 10◦ C min−1 under oxygen atmosphere by using DTA/TG machine (DTG-60H Shimadzu) to gain decomposition and phase formation, and to obtain an optimum heat regimes for drying, heat treatment, and annealing processes. The experiments were conducted in oxygen flow (60 ml min−1 ) in the temperature range from ambient to 700◦C at a heating rate of 10 C min−1 . XRD patterns of the coatings were determined to identify phase structure by means of a Rigaku (D/MAX-2200/PC) diffractometer with a CuKα irradiation (wavelength, λ = 0.15418 nm) by both θ–2θ mode and 2θ scan mode. Thin-film XRD geometry where incident angle was fixed at 1◦ was used to collect data from only thin films. Thicknesses of the films were measured by profilometer. The surface topographies of YIG and Ce-YIG coatings were examined by using SEM (JEOL JSM 6060) attached with energy dispersive spectroscopy (EDS). The magnetic properties of YIG films were measured at room temperature in a vibrating sample magnetometer (VSM, Lakeshore 736, 7400 Series) in a maximum applied field of 1,000 Gauss. From the obtained hysteresis loops, the saturation magnetization (Ms ) and coercivity (Hc ) were determined.

6.3 Results and Discussion The dissolving quality of the solutions can be determined via turbidimetric measurements. As indicated previously, the range is set between 0 and 1,000 ntu, while measuring turbidity. It is interpreted that powder-based precursors are completely dissolved as turbidity value approaches to 0 ntu and they

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are not dissolved, and some powder particles are suspended in a solution near 1,000 ntu. The fabrication of homogeneous and continuous thin film is directly related to turbidity value, which is 0 ntu. In our experiment, turbidity values of the YIG and Ce-YIG solutions were measured as 188 and 396 ntu, respectively. According to the turbidity values, it can be pointed that powder-based precursors are completely or partially dissolved in the solutions. Having high turbidity values are owing to red color of iron. Moreover, these values indicated an important clue for further processing carried out. However, we note that films prepared from completely undissolved solutions are not homogeneous and continuous. This is a worthwhile observation that optimum thermal, structural, microstructural, magnetic, and magneto-optical properties are not obtained by using undissolved solutions. The pH values of the solutions YIG and Ce-YIG were found to be 3.31 and 3.81, respectively. As pH value of the solution is an important factor influencing the formation of the polymeric three-dimensional structure of the gel during the gelation process, it should be taken into consideration during solution preparation. Ramified structure is randomly formed in acidic conditions, while separated clusters are formed from the solutions showing basic characters. The other factor is dilution of the solution, using solvent. The excess solvent affects physically the structure of the gel, because the liquid phase during the aging procedures mainly consists of the excess solvent. A characteristic feature of many sol–gel solutions is the shear-rate or test time dependence of the viscosity. It was determined that the viscosity values of the YIG and Ce-YIG solutions were approximately equal to 1.93 and 2 mPa s, respectively. We have to point here that the viscosity values of the solutions are almost similar that shows a key factor in controlling film thickness. The obtained results are reasonable for sol–gel processing since thin films are formed by diluted solutions. Thus, YIG and Ce-YIG films were obtained by using diluted solutions having low viscosity. Furthermore, since there is practically no change in viscosity upon addition of Ce into the solution, the network is not significantly reinforced. Likewise we can see parallel results with a turbidity study, where we observed indeed small difference among the solutions. Depending on test time, a small decrease of the viscosity in the solutions probably signals fragmentation of the network as strong associated complexes are formed [20]. However, to observe a strong decrease in the solution’s viscosity or a gel point as a result of this, test time should be prolonged. Thermal behavior of Ce, Fe, Y-based xerogels dried at 200◦ C for 30 min in air was shown in Fig. 6.3. DTA curve revealed that endothermic and exothermic reactions taken place at temperatures between 30◦ C and 1,200◦ C as seen in Fig. 6.3. Four thermal phenomena were occurred during the process, since physical water and solvent evaporated and also carbon-based materials coming from alkoxides, solvent, and chelating agent burnt out in the xerogel samples. The first thermal phenomenon was the solvent removal in the temperature range of approximately 90 and 120◦ C. At these temperatures, the

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first peaks of YIG and Ce-YIG xerogels were determined at 100 and 110◦C, respectively, and the endothermic reaction is mostly owing to evaporation of volatile organic components. The second phenomenon was combustion of OR groups at nearly 300◦ C. An exothermic peak was determined at this temperature range owing to combustion of carbon-based materials. The third stage was the formation of ceramic oxides between 398 and 412◦C for YIG and CeYIG, respectively. However, when Ce was added in to the solutions, oxidation temperature slightly decreased. In the last step, the formation of garnet phase was occurred between 700 and 720◦ C for YIG and Ce-YIG, respectively. As mentioned in oxidation peaks, similar behavior was observed in the formation of garnet phase. According to the TG curve in Fig. 6.3, weight losses of the YIG and Ce-YIG xerogel powders were determined to be ∼64% and ∼77%, respectively, where the progress continues until approximately 500◦ C. On the basis of these results, a heat regime for coating process was determined at temperatures such as 300, 500, and 700–1,000◦C, which indicate combustion or drying, oxidation or heat treatment, and annealing processes prior to not using a sol–gel coating processes, respectively. Figure 6.4 shows XRD patterns of selected samples. All produced samples contains YIG phase. However, there are FeYO3 (YIP) and orthorhombic YIG or Ce-YIG phases in our films (which is shown with square symbols). But the fraction of these phases is so small in our B and C films, which were prepared with partially-solved Ce precursor. Lattice constant of YIG increases with Ce substitution causing the peaks of YIG slight shift, since the radius of Ce3+ is 0.118 nm whereas the radius of Y3+ is 0.106 nm. Nevertheless, the Cedoped YIG kept its garnet structure, rather than inducing a new structure [9]. Although there are same amount of cerium in all the solutions, there are CeO2 domains in B and C films different from A and D because of undissolved cerium in the solutions of B and C. Clearly the peaks corresponding to the CeO2 phase were not observed, since the ratio of this phase must be too weak to be observed. Rastogi and Moothy reported that Ce doping causes change in the growth kinetics and improve crystallization [11]. We obtained similar results

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A:800°C Ce: YIG/Si(100) B:800°C Ce: YIG/Si(100) C:900°C Ce: YIG/Glass

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with their crystallization findings. Comparing phase formation temperatures from Figs. 6.4a and b, with those of temperatures for the Ce-YIG films, we have lower temperature ranges than those of the YIG films. The XRD results in Fig. 6.4 show that CeO2 domains in YIG films change the crystallization and phase formation positively. We also observed that, the iron composition in produced films is different from the solutions used, since some of the iron content in the structure migrates with organic materials during the thermal treatment. Because of a broken stochiometry, YIP formation in our films was formed. Having determined the optimal conditions for producing the desired phases, we believe that we will eliminate these undesired components with more refined experimental conditions. The SEM micrograph of YIG and Ce-YIG films (Fig. 6.5a and c) on Si (100) substrate shows that there are cracks on the surface. This surface morphology occurred because of thermal treatment effects of our production steps. Figure 6.5b discloses the SEM micrograph of Ce-YIG film on Si (100) substrate (sample B). The surface of sample B has Ce domains (white area). Sizes of regions containing Ce (Fig. 6.5b) vary from 100 nm (or less) to 1,000 nm. EDS counts of white areas shows that these regions contain mainly Ce and O and some faction of Y and Fe. Gray areas have very small fraction of Ce. In addition, some Ce ions tend to exist in a diamagnetic tetravalent state Ce4+ without free electrons while some precipitates as a CeO2 . The nonmagnetic CeO2 particles in the Ce-YIG or YIG will affect the intended magnetic characteristics [9]. As a result, there were nano and micro size CeO2 weighted regions on the Ce-YIG surface. CeO2 regions incredibly increased the surface quality. This may be the result of absorbed stress of films occurred while thermal treatments by this regions. All the prepared samples show magnetization both in and out of plane directions. The figures of M-H curves show qualitatively that the films have inplane magnetic anisotropy. In-plane anisotropy is suitable for the fabrication

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(c) Fig. 6.5. (a) SEM micrograph of YIG film on Si (100) substrate (sample E) with 10 µm scale bar. (b) SEM micrograph of Ce-YIG film on Si (100) substrate (sample B) with 5 µm scale bar. (c) SEM micrograph of Ce-YIG film on Si (100) substrate (sample A) with 10 µm scale bar

of planar waveguides and for magnetic biasing with magnetized films [26]. The saturation magnetization of YIG is given by the magnetic Fe3+ ions in the d-sites. YIG is a cubic ferrimagnet, with magneto crystalline anisotropy constants K1 = −610 J m−3 and K2 = 5.1 J m−3 [27]. The < 111 > directions are the easy axes for YIG, one easy direction lies normal to the (111) film and will be unflavored due to the high demagnetization factor but there are three easy directions from the film plane [28]. The small value of Ms suggests that the crystallization of the YIG phase is low. Figures 6.6 and 6.7 shows the M–H curves measured at room temperature of the YIG films on Si (100) substrates prepared at 800 and 1,000◦C (samples E and F). Saturation magnetization (Ms ) of the sample E was about 8.6 kA m−1 for in-plane and 5.7 kA m−1 for perpendicular case. By increasing annealing temperature, Ms value was increased up to 35.6 for our sample F. But the coercivity value changed 4.7 − 4 kA m−1 . The reduced Ms from the bulk value for polycrystalline YIG is also obtained by other studies e.g., the films produced under the low pressure chemical vapor deposition (LPCVD) [11].

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Fig. 6.6. M–H curves of the YIG film on Si (100) substrate at room temperature (sample E)

Fig. 6.7. M–H curve of the YIG film on Si (100) substrate at room temperature (sample F)

The magnetic moment of Ce3+ can be parallel to the magnetic moment of Fe in the d-sites, causing that the saturation magnetization of Ce-YIG is higher than that of pure YIG (greater than the 5 µB per formula since Ce ions substituting Y ions now have nonzero magnetic moment). But, this contribution of Ce in the crystal structure introduces a small change. Also nonmagnetic CeO2 regions are reducing the saturation magnetization. Figures 6.8 and 6.9 shows M-H curves of the sample A and B prepared with Si (100) substrate and annealed at 800◦ C. Sample B prepared to include nano and micro size CeO2 regions different from sample A. Saturation magnetization (Ms ) of samples B and A were about 56 kA m−1 and 25 kA m−1 for in plane. Hence increase in the saturation magnetization can be attributed to the increased crystalinity of Ce-YIG. Having increased the annealing temperature 800–900◦C, we observed that Ms values also increased for both cases (CeO2 include film and not include film). The Ms value of the sample D is measured as 30 kA m−1 . This is higher than Ms value of sample A. Similar observation is made between sample B 3+

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and C. Sample C has highest Ms value (59 kA m−1 ) and so better XRD results among all films but its coercivity is lower (2.5 kA m−1 ) nearly same coercivity of the sample B. The coercivity of the sample D was not changed (4.6 kA m−1 ) (Figs. 6.10 and 6.11). We can conclude that CeO2 regions containing films has lower coercivity. There are some ways to improve the coercivity of the films, which is important for memory applications. Several studies show that substituting cobalt with iron at the octahedral sites improves coercivity [29]. To replace Fe3+ ion with Co2+ ion, one need to add charge compensating ions such as Ge4+ [30]. These diamagnetic ions reduce the ferromagnetic coupling within the octahedral Fe-sublattice. Therefore, this method reduces the saturation magnetization needed for sufficient recording signal levels [31]. Another method is to use CoO overlayer to YIG. With the overlayer prepared films by a low pressure chemical vapor deposition, Rastagi et al. [11] managed to improve coercivity values of YIG to 282.6 kA m−1 .

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Fig. 6.11. M–H curves at room temperature of the Ce-YIG film on Si (100) substrate (sample D)

The measured saturation magnetization values of samples are smaller in comparison to bulk YIG (136 kA m−1 ) [8]. Although the other researchers generally obtained nearly same values for Ms but some produced films have values over 159 kA m−1 greater than bulk value [32]. Dorsey et al. [33] reported values greater than 198 kA m−1 and suggested that the level was affected by strain-induced anisotropy. Finally, remnant magnetization of films was obtained between 3.4 (sample A) and 26 kA m−1 (sample C) for in plane case.

6.4 Summary and Conclusions YIG and Ce-YIG thin films were synthesized on a glass and Si (100) substrates from the solutions prepared from Ce, Y, and Fe-based precursors, methanol pentanedionate, glacial acetic acid, and propionic acid, using the sol-gel process for magneto-optical applications. We carried out our initial experiments in nonrefined conditions such as in air just to make sure we got the right

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phases which we succeeded. We will make subsequent study to fabricate our films in more refined conditions. The DTA curve revealed that endothermic and exothermic reactions happen at temperatures between 30 and 1,200◦C. The formation of garnet phase occurred between 700 and 720◦ C for YIG and Ce-YIG, respectively. XRD patterns and SEM micrographs disclosed that Ce-YIG film were successfully prepared with and without CeO2 regions. We found also that Ce doping causes the change in the growth kinetics and improve crystallization. Magnetization curves shows that prepared samples show both in and out plane anisotropy. The saturation magnetization values could be increased from low values of 8.6 kA m−1 for polycrystalline YIG, to 56 kA m−1 for Ce-YIG prepared with same conditions. By increasing the annealing temperature, 59 kA m−1 Ms value is obtained. But the coercivity of films decreased with CeO2 substitution. Saturation magnetization of YIG generally depends on substrate, thermal processes, and film thickness [14,34]. It is possible to obtain greater Ms values by optimizing these parameters. We conclude that it is possible to prepare thin garnet films with suitable magnetic properties by the sol–gel technique with alkoxide route. Having fabricated optimal films produced in a textured and refined conditions, we will undertake magneto optical measurements. We would like to produce nanoparticles using the same techniques for other applications. Acknowledgments The authors acknowledge to Dr. S. Tari at Dept. of Physics of Izmir Institute of Technology, for supporting of the magnetic characterization studies and Ms. I. Kayatekin at Dept. of Metallurgical and Materials Engineering of Dokuz Eylul University for helping XRD and SEM studies. We also thank to Dr. G. Kahraman for valuable discussions. This work has been supported by The Scientific and Technological Research Council of Turkey (TUBITAK).

References 1. T. Shintaku, T. Uno, Jpn. J. Appl. Phys. 35, 4689–4691 (1996) 2. A. Tate, T. Uno, S. Mino, A. Shibukawa, T. Shintaku, Jpn. J. Appl. Phys., 35, 3419–3425 (1996) 3. K.A. Wickersheim, R.A. Buchanan, J. Appl. Phys. 38, 1048 (1967) 4. A. DiBiccari, M.S. Thesis, Virginia Polytechnic Institute and State University, Materials Science and Engineering Department, Virginia (2002) 5. N. Inoue, K. Yamasawa, Elect. Eng. Jpn. 117, 1 (1996) 6. T. Sekijima, H. Kishimoto, T. Fujii, K. Wakino, M. Okada, Jpn. J. Appl. Phys. 38, 5874 (1999) 7. L.L. Hench, J.K. West, Principles of Electronic Ceramics, (Wiley, New York, 1990)

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8. A.J. Moulson, J.M. Herbert, Electroceramics, Chap. 9, (Wiley, West Sussex, 2003) 9. T.-C. Mao, J.-C. Chen, J. Magn. Magn. Mater. 302, 74–81 (2006) 10. K. Shinagawa, Magneto-optics, Chap. 5, (Springer, Berlin, 1999) 11. A.C. Rastogi, V.N. Moothy, Mater. Sci. Eng. B95, 131–136 (2002) 12. G. Traeger, L. Wenzel, A. Hubert, IEEE Trans. Magn. 29, 3408 (1993) 13. M. Huang, S.-Y. Zhang, Appl. Phys. A 74, 177–180 (2002) 14. X. Zhou, W. Cheng, F. Lin, X. Ma, W. Shi, Appl. Surf. Sci. 253, 2108–2112 (2006) 15. G.F. Dionne, G.A. Allen, P.R. Haddad, C.A. Ross, B. Lax, Lincoln Lab. J., 15, 2 (2005) 16. M.J. Steel, M. Levy, R.M. Osgood Jr., Photon. Technol. Lett., 12, 1171–1173 (2000) 17. M. Inoue, T. Yamamoto, L.P. Boey, K. Nishimura, T. Fujii, IEEE. Trans. Magn. 33, 1564 (1997) 18. M. Inoue, R. Fujikawa, A. Raryshev, A. Khanikaev, P.B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, A. Granovsky, J. Phys. D 39, R151 (2006) 19. E. Garskaite, K. Gibson, A. Leleckaite, J. Glaser, D. Niznansky, A. Kareiva, H.-J. Meyer, Chem. Phys. 323, 204–210 (2006) 20. S. Higuchi, K. Ueda, F. Yahiro, Y. Nakata, H. Uetsuhara, T. Okada, M. Maeda, IEEE Trans. Magn. 37, 2451–2453 (2001) 21. M. Gomi, K. Satoh, M. Abe, Jpn. J. Appl. Phys. 27, 1536 (1988) 22. S. Mino, A. Tate, T. Uno, T. Shintaku, A. Shibukawa, Jpn. J. Appl. Phys. 32, 3154–3159 (1993) 23. T. Uno, S. Noge, J. Eur. Ceram. Soc. 21, 1957–1960 (2001) 24. M.S. Bhuiyan, M. Paranthaman, K. Salama, Supercond. Sci. Technol. 19, R1–R21 (2006) 25. M.A. Uddin, H.P. Chan, C.K. Chow, Y.C. Chan, J. Electron. Mater. 33, 224–228 (2004) 26. K.L. Saegner, in Pulsed Laser Deposition of Thin Films, ed. by D.B. Chrisey, G.K. Hubler, (Wiley, New York, 1994) 27. K.H. Hellwege, A.M. Hellwege, Science & Technology, New Series, Group III 12 Part A, Garnets and Perovskites, (Springer, Berlin, 1978) 28. N.B. Ibrahim, C. Edwards, S.B. Palmer, J. Magn. Magn. Mater. 220, 183–194 (2000) 29. S. Dhara, A.C. Rastogi, B.K. Das, J. Appl. Phys. 79, 2, 953–956 (1996) 30. B.M. Simon, R. Ramesh, V.G. Keramidas, G. Thomas, E. Marinero, J. Appl. Phys. 76, 6287 (1994) 31. P.F. Carcia, J. Appl. Phys. 63, 5066–5073 (1988) 32. M. Kucera, J. Bok, K. Nitsch, Solid State Comm. 69, 1117–1121 (1989) 33. P.C. Dorsey, S.E. Bushnell, R.G. Seed, C. Vittoria, J. Appl. Phys. 74, 1242 (1993) 34. N. Kumar, D.S. Misra, N. Venkataramani, S. Prasad, R. Krishman, J. Magn. Magn. Mater. 272–276, e899–e900 (2004)

7 Tuning the Magnetic and Electronic Properties of Manganite Thin Films by Epitaxial Strain G.H. Aydogdu1 , Y. Kuru2 , and H.-U. Habermeier1 1 2

Max Planck Institute for Solid State Research, Stuttgart D-70569, Germany Max Planck Institute for Metals Research, Stuttgart D-70569, Germany

Summary. Electrical and magnetic properties of manganites are governed by a delicate balance between several mechanisms such as charge, orbital, and spin ordering superimposed to lattice effect that can cause mesoscopic phase separation. Manganites have generally a rich phase diagram, and their properties are very sensitive to external perturbations (e.g., electrical and magnetic fields, X-ray illumination, hydrostatic pressure, and epitaxial strain), which can cause phase separation at a given temperature. The growing interest in the manganites, in both, bulk and thin film form, is due to their possible device applications and, particularly, the new physics, based on strong electron–electron interaction. The observed peculiarities like colossal magnetoresistance (CMR) and metal to insulator (MI) transitions may serve as examples. In this work, first an overview about the general properties of manganites, important mechanisms controlling phase separation, and some of the key observations about the modification of electrical and magnetic properties by external effects is given. Subsequently, the consequence of epitaxial strain is elaborated in more detail and results regarding the epitaxial La0.5 Ca0.5 MnO3 (LCMO) thin films, grown on planar (100), (111) SrTiO3 (STO), (001) SrLaGaO4 (SLGO) substrates by pulsed laser deposition technique (PLD), are presented.

7.1 Introduction In recent years, many studies have focused on manganites with the general formula RE(1−x) Ax MnO3 (where RE is a rare earth element e.g. Pr, La, Y; A is a divalent alkaline earth element e.g. Ba, Ca, Sr; see Fig. 7.1 for the crystal structure [1]), in the form of bulk or thin film, mainly due to two important reasons: (1) They exhibit a very interesting phenomena called colossal magnetoresistance (CMR [2, 3]; a large variation in the electrical conductivity of the specimen caused by the application of a magnetic field), which opens the application of these materials as switches and sensors in microelectronics [4]. (2) They are considered as a subgroup of strongly correlated electron systems, for which interaction between electrons is pronounced and responsible for many peculiar properties that they demonstrate. Understanding the new

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Fig. 7.1. Structure of LCMO

Fig. 7.2. Phase diagram of La(1−x) Cax MnO3

physics introduced by manganites is of cardinal importance as a fundamental research problem. Besides, it is clear that information obtained from manganites can be transferred to other subgroups of strongly correlated electron systems by analogy and in turn will shed more light on other hot topics like high-Tc superconductors. Electrical and magnetic properties of manganites are resulting from a competition between several interesting mechanisms like charge, orbital and spin ordering [5–7], and relation to lattice effects. Their rich phase diagram [8] (Fig. 7.2, as an example phase diagram of La(1−x) Cax MnO3 system is shown) can be considered as a consequence of this competition. Despite the ideas proposed to explain the observations such as CMR, inhomogeneous character, ferromagnetic metallic phase, and metal-insulator transition, most of these

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Fig. 7.3. Double-exchange mechanism

Fig. 7.4. Charge, spin, and orbital ordering in manganites

points could not be clarified entirely. There are still many open questions in addition to many clues obtained from both experiments and theoretical investigations. Some of the basic ideas mentioned above are double exchange, electron–phonon interaction due to Jahn-Teller distortion, mesoscopic phase separation, and percolation of these phases. Without going into detail main points of these ideas will be underlined in the following part of this section. The double-exchange model, proposed first by Zener [9], is crucial to explain the electron transfer in the ferromagnetic-metallic phase in Fig. 7.2. This mechanism is demonstrated by a simple sketch in Fig. 7.3. Although eg electron of Mn3+ is transferred to the O 2p orbital, an electron goes from O 2p orbital to Mn4+ simultaneously. Because of strong Hund coupling the probability of electron transfer is larger when spins of neighboring Mn3+ and Mn4+ ions are parallel. Consequently, strong ferromagnetic interaction is ensured. Double exchange generally competes with the super exchange mechanism representing the electron transfer between two Mn4+ ions, which favors antiferromagnetism. Charge ordering is the periodic distribution of the Mn3+ and Mn4+ ions in the lattice. This rearrangement is thought to stem from strong Coulomb interaction. In addition to charge ordering, there are two more degrees of freedom namely orbital andspin ordering [10]. These three mechanisms are shown in Fig. 7.4 in a simple sketch. For instance, it is seen that a sublattice

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Fig. 7.5. Jahn-Teller distortion in manganites

can be constructed from the ordered arrangement of dz 2 orbital of Mn+3 ions. Charge ordering leads to localization of the mobile electrons in certain positions in the lattice and reduction of the electrical conductivity whereas orbital and spin ordering give rise to anisotropic electron transfer. The Jahn-Teller effect [11] is the distortion of a nonlinear molecule with a degenerate electronic state to remove the degeneracy and to reduce the overall energy. In an octahedral environment, as in the case of Mn3+ and Mn4+ ions in manganites, two eg orbitals have larger energy than the remaining t2g orbitals (Fig. 7.5). A large distortion is caused by the Mn3+ ion, since the number of electrons in the eg orbital is odd. However, no Jahn-Teller effect is expected for Mn4+ ion, which has only three electrons in the t2g orbitals. Pronounced Jahn-Teller distortions are generally associated with the insulating behavior because charge carriers are trapped and electron-lattice polarons are formed in the vicinity of these local distortions. It has also been demonstrated employing several manganite systems that Jahn-Teller distortion gradually decreases as the temperature is decreased. It has the lowest value at the insulator to metal transition and stays practically constant if temperature is reduced further [12]. Another interesting finding related to manganites is the coexistence of different electronic phases in a nominally chemically homogeneous system. The transport properties of the specimen are determined by the evolution of percolation path. An insulator to metal transition can be monitored if metallic regions are connected to each other and constitute a continuous network. Various kinds of external perturbations can be employed to trigger the electronic phase separation, change the dominant phase, and tune the electronic and magnetic properties. Hydrostatic pressure, electrical, magnetic fields [13–17], laser [18], and X-ray irradiations [19] and epitaxial strain [20,21] imposed by the substrates are some of the tools highlighted in the recent studies. In this chapter, the latter possibility is discussed in detail employing La0.5 Ca0.5 MnO3 thin films on (100) SrTiO3 , (111) SrTiO3 , and (001) SrLaGaO4 substrates as a case study.

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7.2 Preparation and Analysis of Films 7.2.1 Deposition Technique and Film Growth Pulsed laser deposition (PLD) is one of the most convenient and unique methods to produce epitaxial multicomponent oxide thin films with very low defect density [22]. In most of the cases, one-to-one composition transfer from target to the film is easily achieved. The main components and working principle of the PLD system are described in Fig. 7.6. The laser beam enters to the vacuum chamber and is focused on the rotating target, a pellet with a diameter of 15 mm. The angle between the laser beam and the target surface is adjusted to 45◦ . The surface of the target material vaporizes, and the atoms separated from the target are condensed (having the required structure and composition) on the surface of the substrate, placed approximately at a distance of 4 cm from the target. The distance between target and substrate, oxygen pressure, wavelength of the excimer laser, laser fluence, pulse frequency, substrate temperature, annealing time and temperature are some of the important parameters optimized to produce high-quality thin films. In this study, an excimer laser with KrF gas mixture emitting ultraviolet radiation with a wavelength of 248 nm was used in the PLD system. Laser fluence and pulse frequency were fixed to 1.6 J cm−2 and 5 Hz, respectively. The LCMO

Fig. 7.6. The basic components of the PLD system

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Fig. 7.7. X-ray diffraction data for bulk LCMO

films were deposited on single crystal substrates at 1,073 K and with an oxygen pressure of 0.4 mbar. After deposition, films were annealed at 1,173 K for 30 min in ambient oxygen pressure. The thicknesses of the films were adjusted by counting the pulses after several calibration runs to determine the growth rate. Before deposition, single crystal substrates were cleaned in ultrasonic baths of acetone and ethanol; then, mounted to a sapphire sample holder, heated radiatively. Lattice parameters of the LCMO target were obtained from the measured X-ray diffraction (XRD) data (Fig. 7.7) using the FULLPROF software [23]. It belongs to the Pnma space group (FeGdO3 type structure). The refined lattice parameters for the orthorhombic cell are a = 0.54093 nm, b = 0.76493 nm, c = 0.54101 nm, and α = β = γ = 90◦ . Magnetization of the bulk target was measured under an applied field of 0.01T while decreasing the temperature from 300 to 5 K. Figure 7.8 shows that, there are two magnetic transitions: the first one is a paramagnetic to ferromagnetic transition at around 270 K, and the second is a ferromagnetic to antiferromagnetic transition when 150 K is reached [24]. The saturation magnetization is approximately 0.18 µb /Mn ion. The substrate material is also crucial for the epitaxial growth of oxide films. There should be a low lattice mismatch (ξ) [25] between the material that will be deposited and the substrate. Moreover, their coefficients of thermal expansions (CTEs) should be similar to preserve this low mismatch even at elevated temperatures. Absence of a phase transition between the deposition temperature and the application temperature, mechanical and chemical stability are

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Fig. 7.8. Magnetization vs. temperature graph of bulk material Table 7.1. Mismatch strain Substrate

Crystal structure

Lattice parameter (nm)

ξ 1 (%)

ξ 2 (%)

(100) STO (111) STO (001) SLGO

Cubic Cubic Tetragonal

a = 0.3905 [27] a = 0.3905 a = 0.3844 [28] c = 1.2688

1.92 1.92 0.490

1.86 2.12 0.475

the other requirements for a suitable substrate [26]. (100) SrTiO3 (STO), (111) SrTiO3 , and (001) SrLaGaO4 (SLGO) are the three single crystalline substrates employed in this study. The lattice parameters of these substrates and corresponding lattice mismatch values calculated for two in-plane directions (ξ1 and ξ2 ) are summarized in Table 7.1. 7.2.2 Analysis of Films After the deposition, the surface morphology of the films was investigated by atomic force microscopy (AFM) in tapping mode. The determination of the epitaxial relationship between the film and the substrate and phase analysis were carried out by XRD using Cu Kα radiation. Pole figures were measured in a Philips X’Pert MRD diffractometer equipped with an Eulerian cradle and operating in parallel beam geometry. The rocking curves and 2θ−ω (where 2θ is the angle between the incident and the diffracted X-ray beams, and ω is the angle between the incident X-ray beam and the

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sample surface) scans between 10◦ and 120◦ at various inclination angles (ψ) were performed with a four-circle Bruker D8 Discover diffractometer. The temperature dependence of magnetization was investigated by a Quantum Design MPMS superconducting quantum interference device (SQUID) magnetometer. The magnetic field (0.01 T and 1 T) was oriented parallel to the film surface during field cooling (FC) and zero field cooling (ZFC). During ZFC, samples first were cooled to 5 K then field was applied to sample. Afterwards, magnetization was measured in the heating process up to 300 K. During FC, samples were cooled from 300 K down to 5 K in the same field. Resistivities of the films were measured within a temperature range between 5 K and 300 K by four-point probe method. Evaporated chromiumgold pads and silver epoxy were used to attach the gold wires to the specimen. DC current was varied between 10−7 and 10−4 A during the transport measurements.

7.3 Structural Characterization, Electrical and Magnetic Properties of Manganites Film 7.3.1 Structural Characterization Three exemplary AFM images for films on (100) STO, (111) STO, and (001) SLGO are shown in Fig. 7.9. An island-growth type surface morphology was observed for all films. La0.5 Ca0.5 MnO3 films (145-nm thick) show root mean square roughness values of 8, 0.6, and 9.8 nm (over 25 µm2 area), respectively. The 2θ-ω scans in Fig. 7.10 show that all samples are single phase and have orthorhombic crystal structure. Two different kinds of {121} pole figures are presented in Fig. 7.11. On the one hand, the films deposited on (100) STO and (001) SLGO display fourfold symmetry. All four peaks are at ψ = 43.9◦, which is the angle between (121) and (020) planes of La0.5 Ca0.5 MnO3 . On the other hand, threefold symmetry is observed for the films deposited on (111) STO substrates. Inclination angles of the peaks in this pole figure (ψ = 34.7◦ ) are consistent with the angle between (121) and (022) planes of La0.5 Ca0.5 MnO3 . Epitaxial notation of the films on (100) STO and (001) SLGO can be described by (100) STO or (001) SLGO/(020) LCMO: [110] STO or [110] SLGO//[100] LCMO and [−110] STO or [−110] SLGO//[001] LCMO relations. For the films on (111) STO substrates, (111) STO//(022) LCMO: [1–10] STO//[100] LCMO and [11–2] STO//[011] LCMO relations hold. Rocking curve measurements reveal that full width at half of the maximum intensity (FWHM), inversely proportional to the degree of crystallinity, decreases with increasing film thickness for the films on (001) SLGO whereas a monotonous trend cannot be observed for the films deposited on (100) STO (Fig. 7.12a and b). However, the situation is rather different for the films on (111) STO; two shoulders are identified to develop on both sides of the main peak with increasing film

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Fig. 7.9. AFM images for 145 nm thickness film on (a) (100) STO, (b) (111) STO, (c) (001) SLGO

thickness (Fig. 7.12c). Mosaic spread increases and orientation of the (044) planes deviates more from their initial orientation, parallel to the substrate. In addition, (044) diffraction peak becomes more asymmetric (Fig. 7.12d). A strain gradient through the film thickness and asymmetry in the diffraction peaks can be caused by the above-mentioned mosaic structure, which may act as a relaxation mechanism in the surface region [29]. 7.3.2 Magnetic Properties The results of the magnetization measurements for the LCMO films on (100) STO, (111) STO, and (001) SLGO are presented in Fig. 7.13. Saturation magnetization (Ms ) values of the films on (100) STO substrate are quite low. The variation of magnetization with temperature for LCMO films on (100) STO is almost linear in the entire temperature range and determination of the Curie temperature (Tc ) is difficult because of the absence of a sharp ferromagnetic transition. The films on (111) STO substrate have considerably larger Ms values than the films on (100) STO. Tc of the films on (111) STO is

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Fig. 7.10. 2θ − ω scans for films on (a) (100) STO, (b) (111) STO, (c) (001) SLGO

Fig. 7.11. Pole figures of films on (a) (100) STO and (001) SLGO, (b) (111) STO

increasing with thickness and reaches to approximately 230 K for the 290 nm thick film. This value is still lower than the Tc of the bulk ceramic (270 K). Both the Ms and Tc values are found to be the highest for the films deposited on (001) SLGO substrates (Fig. 7.13c). One common feature about the magnetization measurements of all films is the difference between the FC and ZFC magnetization curves. The thermomagnetic irreversibility, ∆M [30], related to the degree of magnetic inhomogeneity in the films, can be expressed by ∆M = 100(MFC − MZFC )/MFC ,

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Fig. 7.12. Rocking curve results of films on (a) (001) SLGO, (b) (100) STO, (c) (111) STO, and (d) (044) diffraction peaks of films on (111) STO

Fig. 7.13. Magnetization vs. temperature graphs of films on (a) (100) STO, (b) (111) STO, (c) (001) SLGO. Open symbols show zero field cooling (ZFC) and closed symbols represent field cooling (FC)

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Fig. 7.14. Magnetization vs. temperature graphs of 290 nm thick film on (111) STO. Open symbols show zero field cooling (ZFC) and closed symbols represent field cooling (FC)

where MFC and MZFC are the magnetization values at a certain temperature during FC and ZFC, respectively. High ∆M values indicate the existence of spin glass like behavior (SG), which stems from magnetic randomness and short range spin ordering. Application of low magnetic fields is not sufficient to rotate the spins in the direction of the field. ∆M values are around 90% if applied field is 0.01 T. ZFC and FC magnetization curves of films on (111) STO for 0.01 T and 1 T magnetic fields can be seen in Fig. 7.14. When the magnetic field is increased to 1T, rotation of spins to the direction of field (long range ordering of spins) is easier; spin glass behavior is suppressed. Hence, ∆M decreases with increasing magnetic field [31–33]. Figure 7.14 also demonstrates that there is a pronounced anisotropy at low magnetic fields (ex. 0.01 T); magnetization values of the films decrease significantly if the field is applied perpendicular to the film surface [34]. However, magnetization values and shape of the curves converge for the parallel and perpendicular alignments of the magnetic field with respect to film surface if magnetic field is increased to 1 T. 7.3.3 Electrical Properties Transport measurements show that electrical resistivities of the LCMO films on (100) STO substrates increase with decreasing temperature; only insulator behavior can be observed in the temperature range between 5 and 300 K.

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Fig. 7.15. Resistivity vs. temperature graphs of films on (a) (100) STO, (b) (111) STO, (c) (001) SLGO

However, the films on (111) STO substrates have an insulator to metal transition if the film thickness exceeds 145 nm (Fig. 7.15). All films on (001) SLGO exhibit an insulator to metal transition when the temperature is decreased below a certain value, TMI . Moreover, TMI is very close to room temperature for 80 and 145 nm thicknesses. For understanding the electrical transport, resistivity data above Tc are analyzed and fitted according to variable range hopping (VRH) model of polarons [35], which is given in equation ρ(T ) = ρ0 exp (T0 /T )1/4 , where ρ0 is the residual resistivity and T0 is the characteristics VRH temperature. The parameter T0 can be regarded as a measure of strength of Jahn-Teller distortion [36]. If T0 increases, the localization length (1/α) and average hopping distance are reduced, and resistivity increases. Figure 7.16 demonstrates that data above Tc follow a straight line and can well be described by the VRH model [37–40]. T0 values are evaluated from the linear fit to the lnρ vs. T −0.25 plots. It is found that T0 values of the films on (100) STO are relatively high for all thicknesses (around 108 ) whereas T0 values of the films on (111) STO are smaller and inversely proportional to the film thickness. For instance, T0 values of the 80 and 220 nm thick films are 1.42 × 108 K and 3.08 × 107 K, respectively. Among all LCMO films, the ones deposited on (001) SLGO have the smallest T0 values.

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Fig. 7.16. lnρ vs. T −0.25 plots for (a) 80 nm thick film and (b) 220 nm thick film on (111) STO

7.4 General Discussion Figures 7.8 and 7.13–7.15 indicate that electrical and magnetic properties of LCMO can be modified drastically in thin films. Magnetization per Mn ion values can be decreased or increased more than an order of magnitude. It is possible to have both an insulator in the entire temperature range between 5 and 300 K (films on (100) STO substrates) and a room temperature metal (for example the 80 nm thick LCMO film on (001) SLGO). Furthermore, large magnetic anisotropy has been monitored at low magnetic fields although intrinsic magnetoanisotropy of LCMO with a distorted cubic structure is negligible [41]. The magnitude of epitaxial strain imposed by different substrates varies according to their lattice parameters. However, the total strain that the film experiences is determined by the combined effect of epitaxial strain, variation of epitaxial strain with temperature according to CTEs of the film and the substrate, growth strain and possibly some relaxation processes. For the present case, the first, third and fourth points are dominant, and the effect of CTEs on the observed variations in the properties of LCMO films when deposited on different substrates can be neglected, since CTEs of STO and SLGO are quite close to each other, 10.3 × 10−6 and 10.1 × 10−6 1/K, respectively [42]. The bond lengths and angles are directly influenced from the total strain, and the observed modifications in the electrical and magnetic properties can be correlated with these structural modifications. The strain tensor can be separated into two parts. Hydrostatic strain represents the volume change, while the second part (deviatoric component) is related to the structural distortions and does not affect the volume. Jahn-Teller strain (εJ−T ) is proposed to evaluate this distortion and it is proportional to electron lattice coupling. √ εJ−T = 1/6(2εzz − εxx − εyy ),

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Fig. 7.17. Jahn-Teller strain, εJ−T , vs. thickness graphs for films on (100), (111) STO and (001) SLGO

where εzz is out-of-plane and εxx and εyy are in-plane strain components [43, 44]. εJ−T values calculated from the measured lattice parameters of the films are plotted against film thickness in Fig. 7.17 for the three substrates. LCMO films on (100) STO have large compressive εJ−T values. On the one hand, this large distortion can explain why films on (100) STO are insulators for all thicknesses. On the other hand, low εJ−T values are found for the films on (111) STO and (001) SLGO, in agreement with the relatively higher magnetizations and metallic behavior observed for the films coated on these substrates. The T0 values, obtained from application of the VRH model to the electrical transport data, are also consistent with the above discussion. The LCMO films on (100) STO have the largest T0 values, T0 is low and decreases with increasing film thickness for the films on (111) STO and the lowest T0 values are obtained for films on (001) SLGO substrates. As already mentioned in Sect. 7.1, there are various factors that are, probably, incorporated in the phase separation phenomena, simultaneously. A clear correlation between the lattice distortion and the electrical and magnetic properties of the LCMO films is observed for the experimental results presented earlier. However, the other mechanisms such as charge, orbital, and spin ordering cannot be excluded from the story. It is possible that these processes are not independent from the distortion but somehow coupled together. Consequently, both the processes hindering the conductivity may be triggered by a distortion in the structure and distortion may be increased or decreased dependent on the result of the competition between the different mechanisms.

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7.5 Conclusions Epitaxial LCMO thin films were deposited successfully on (100) STO, (111) STO, and (001) SLGO substrates by PLD method. Electrical and magnetic properties of these films are completely different from the bulk material. By using the substrate material as a knob to tune the film properties, a wide range of phases from an insulator in the entire temperature range between 5 and 300 K to a metallic, even at temperatures close to room temperature, can be stabilized, although chemical compositions, deposition conditions, and thicknesses are kept constant. Finally, it is observed that metallic behavior generally dominates in samples with low structural distortions while large distortions are accompanied by the insulating behavior. Acknowledgments The authors thank Mr. G. Maier and Dr. U. Welzel of Central Scientific Facility X-ray Diffraction and department of Prof. Dr. Ir. E. J. Mittemeijer in Max Planck Institute for Metals Research for assistance with the pole figure measurements. G. H. A. also gratefully thanks Mrs. E. Bruecher for the magnetization measurements. This work was supported by NMP4-CT-2005517039 controlling mesoscopic phase separation (COMEPHS) project.

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8 Radiation Nanostructuring of Magnetic Crystals V.A. Ageev1 , V.I. Kirischuk1, Yu.V. Koblyanskiy2 , G.A. Melkov2 , L.V. Sadovnikov1, A.N. Slavin3 , N.V. Strilchuk1 , V.I. Vasyuchka2 , and V.A. Zheltonozhsky1 1

2

3

Institute for Nuclear Research, National Academy of Sciences, Prospect Nauky 47, 03680, Kiev, Ukraine, [email protected] Taras Shevchenko National University, Volodymyrska Str., 64, 01033, Kiev, Ukraine Department of Physics, Oakland University, Rochester, Michigan, USA

Summary. The influence of irradiations (reactor neutrons, 3 MeV protons and Ar+ ions with the energy of 125 keV) upon ferrite YIG films and devices properties has been investigated. Qualititative similarity of such influence for neutrons and protons has been established. Because of the homogeneous distribution of radiation defects in both cases, there is a simultaneous broadening of homogeneous FMR linewidth, magnetostatic oscillations and modes, and short dipole-exchange spin waves. In the case of Ar+ irradiation due to small path length, there is an inhomogeneous nanostructuring over the thickness of the film with characteristic layer size ∼0.1 µm. The presence of nanolayers influences in the different way upon the properties of different film oscillations and modes. The decrease of FMR linewidth by 40% at the fluence of 3 × 1016 cm−2 has been observed for the first time. At the same time, spin wave linewidth has practically not changed allowing to increase the efficiency of wave front reversal in YIG films irradiated by Ar+ ions. All discovered experimental facts are explained in the frame of multilayer model of the film consisting of alternate magnetic and nonmagnetic layers.

8.1 Introduction One of the most important goals of modern material science is to find simple and reliable methods to control the materials’ structure and properties at nanoscale. This includes the creation of artificial nanostructures, modification of parameters of natural nano-granular materials, and control of the number of nano-sized defects in the conventional materials. All such objectives can be reached making use the irradiations of initial materials by different projectiles (particles). It is well-known that world key manufacturers use radiation technologies to produce the most qualitative

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highly homogeneous semiconductor monocrystals with a diameter up to 200 mm. First of all it concerns Si GaAs [1, 2]. In our present work, we provide the experimental evidence demonstrating that irradiation of single crystal ferrite films and bulk ferrite samples with ions, protons, and neutrons leads to dramatically different results in terms of the increase in the number of nano-sized defects in these materials, and their FMR and spin wave linewidth broadening, that is directly related to the number of the magnetic nano-sized defects.

8.2 The Influence of Inhomogeneities upon the Properties of Ferrites and Ferrite Devices The presence of different inhomogeneities produced during the irradiation of monocrystals results in the occurrence of additional noninherent relaxation processes [3]. The main of them – two-magnon relaxation that results in the scattering of initial spin wave with wave vector k at the inhomogeneity and its transformation into another spin wave with wave vector ks = k, ks ≈ k ± 2π/a, where a is the inhomogeneity size. Because of two-magnon processes, there is observed the increasing of ferromagnetic resonance linewidth of magnetostatic waves (MSW) ∆H0 and spin waves ∆Hk : ∆H0 = ∆H0i + ∆H0i + ∆H0r , ∆Hk = ∆Hki + ∆Hki + ∆Hkr , where ∆H0i , ∆Hki – intrinsic linewidths of the homogeneous resonance and spin waves with wave vector k, correspondingly. These parameters are determined by nonlinear many-magnon and magnon–phonon processes. ∆H0r , ∆Hkr – radiation linewidths defined by radiation effects and ∆H0i , ∆Hki take into account the influence of twomagnon scattering at initial inhomogeneities that are available in the crystals before the irradiation and have been produced during their growth, mechanical processing, and so on. However, the influence of inhomogeneities and two-magnon scattering upon the broadening results not only in the intensification of relaxation processes. In the general, there are changes in the lattice parameters, saturation magnetization, exchange constants, Curie temperature, crystal anisotropy field, degree of spin pinning at the surface [4]. The layer nanostructing of ferrite films is possible as a result of controlled irradiations. So, several (up to 4) nanolayers with the thickness of 10–100 nm have been produced, each of which is independent spin region with its own ferromagnetic resonance conditions and magnetostatic dispersion [5]. And at last, as it is understood from the above, using the inhomogeneities produced with controlled parameters (in particular, with given size a) the objective of excitation of dipole-exchange spin waves (DESW) with wave vector k ≈ 104 −105 cm−1 can be reached, while their wideband excitation by traditional methods is impossible. These waves have much lower relaxation frequencies and group velocities, than ordinary MSW with k ≤ 102 cm−1 that are widely used in the ferrite devices. So the devices using DESW should have a number of advantages: less sizes

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and bias magnetic field volume, long delays, less losses, etc. It has been shown experimentally and theoretically that the use of initial inhomogeneities for the parametric reversal of DESW wave front allows to increase by an order of magnitude the delay of ferrite time-delay lines [6]. The main attention in this work has been paid to study the influence of radiation inhomogeneities produced in ferrite films during the irradiation by neutrons, protons, and heavy ions upon DESW wave front reversal (WFR).

8.3 Wave-Front Reversal in a Medium with Inhomogeneities The phenomenon of WFR under the influence of parametric pumping is well known in both optics and acoustics [7]. WFR was also observed for dipolar spin waves in the ferrite films [8, 9]. In all examples mentioned above, the WFR process involved the waves having almost equal frequencies, wave numbers, and velocities. For example, in the experiments [8, 9], all the waves participating in the WFR process were backward volume magnetostatic waves (BVMSW) having wave numbers k ∼ 102 cm−1 . Hereafter we shall consider a different situation when the waves participating in the WFR process have equal frequencies, ω, but wave numbers kand velocities that could differ by many orders of magnitude. This situation is impossible for optic and acoustic waves as their spectra are practically isotropic, but can be easily realized for spin waves in tangentially magnetized magnetic films where the wave spectrum is substantially anisotropic. The wave number of a spin wave propagating in magnetic film depends strongly on the direction of the wave propagation. The spin wave spectrum in the ω − k space forms a zone [3] where every frequency ω corresponds to the wide range of wave numbers k starting from k ∼ 102 cm−1 (dipolar spin waves or MSW) and ending with k ∼ 106 cm−1 (exchange spin waves (ESW)). The group velocities vk of spin waves corresponding to the same frequency in this case could vary from vk = 107 cm s−1 (BVMSW with k ∼ 102 cm−1 ) to zero at the inflection point of the dispersion curve in the region k ∼ 104 cm−1 , where the negative “dipolar” dispersion is compensated by the positive “exchange” dispersion. So in this region of intermediate wave number values k ∼ 104 cm−1 , the spin wave dispersion is influenced by both dipolar and exchange interactions and the waves are called DESW. Thus, when a spin wave propagating in the magnetic film experiences elastic (when frequency is conserved) scattering on defects and inhomogeneities that are always present in even the best samples of ferrite (e.g., yttrium–iron garnet (YIG)) films, not only the direction of the wave propagation is changed (as it happens in optics), but also the wave itself undergoes the substantial change. As a result, the wave number, group velocity, and dissipation parameter of a new wave created in the scattering process depend on the size and nature of the inhomogeneity. Of course, all these new waves will interact with

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Fig. 8.1. Propagation and scattering of spin waves in a ferrite film with inhomogeneities: 1 – ferrite (YIG) film, 2 – inhomogeneity, 3 – strip-line antenna that excites signal MSW, 4 – shadowed region of parametric pumping localization, 5 – the arrow shows the position of the MSW that did not scatter on inhomogeneities, 6 – the arrow shows the position of the DESW created as a result of scattering of the signal MSW on the inhomogeneity 2

parametric pumping and will, therefore, substantially change the character of WFR process in ferrite films in comparison to the well known case of WFR involving only similar waves known in optics and acoustic [7–10]. Figure 8.1 demonstrates one of the possible situations of WFR for spin waves in a tangentially magnetized ferrite film 1 with inhomogeneities 2. Antenna 3 at the time t = 0 excites an input signal MSW having wave number k = ks ∼ 102 cm−1 and group velocity vs , which propagates toward the region of parametric pumping localization 4 of the width l situated at the distance L from the antenna 3. A certain part of the signal, MSW is scattered on the inhomogeneities 2 and, as a result of this scattering, their wave numbers are changed by the amount ∼2π/a, where a is the linear size of the inhomogeneity [3, 11]. A typical size of the inhomogeneities in high-quality samples of YIG films is of the order of 1 µm, which means that, as a result of signal MSW scattering on these inhomogeneities, spin waves having k ∼ 104 cm−1 are created. As it was mentioned earlier, these waves are slow (vk << vs ) DESW 6 (Fig. 8.1). Since the number of inhomogeneities in the YIG film is not large, the main part of the signal MSW 5 does not interact with them and continues to propagate with their original velocity vs . Figure 8.1 demonstrates the position of waves at the time t > t0 = (L + l)/vs , when nonscattered MSW 5 have already passed the region of parametric pumping localization 4, while the slow DESW formed as a result of scattering of a part of the MSW on inhomogeneity are still inside this region. If at the time t = tp > t0 , a pumping pulse is supplied to the region 4, the slow DESW will be reversed, interacts again with the inhomogeneity at which they were created, and gets converted in this interaction process into a MSW having wave vector −ks , frequency ωs , and propagates toward the input antenna 3. This reversed MSW will excite a delayed electromagnetic signal in the antenna 3 at the time 2tp . Note, that the input signal MSW will not be reversed

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because at the time tp > t0 when the pumping is applied this MSW have already passed the region of pumping localization 4. The elastic scattering of spin waves on inhomogeneities, which is necessary for the excitation and revesal of DESW always take place in real ferrite samples. In this scattering process (named two-magnon scattering [11]), a primary magnon having wave vector k is scattered on the inhomogeneity of size a, and forms a secondary magnon of the same frequency and the wave vector k  . The probability of this scattering process Rkk has a maximum value when k = k  ± 2π/a [3, 11], and we took this fact into account while evaluating the wave vector of scattered DESW. The theoretical analysis of WFR in ferromagnetic medium with inhomogeneities has been done elsewhere [6]. The final expression for the reversal coefficient K, equals to the ratio of output magnetostatic wave amplitude A− , produced in the parametric pumping region 4 and propagating to the input antenna 3, and input wave amplitude A+ , propagating from the antenna 3 to the region 4, looks as follows: K=

A− ; A+

K=

1 ∆H0i + ∆H0r hp Vk τp −γ∆Hk tp e e , 2 ∆H0

(8.1)

where Vk is the coupling coefficient for the pumping and DESW [3], hp is the magnetic field amplitude of the parallel pumping [3], and τp is the pumping pulse duration. One can see from (8.1) that to achieve a maximum value of the reversal coefficient K, it is necessary to have optimum amplitude of the two-magnon scattering (that is roughly characterized by the magnitude of the additional relaxation caused by two-magnon scattering). When the scattering is extremely small (ideal film with no defects: ∆H0r → 0, Rkk → 0) the front reversal effect is vanishing, as the efficiency of the DESW creation on defects is very small. In the opposite limiting case (imperfect film with a large number of defects), the reversal coefficient K is exponentially decreasing with the increase of the probability of two-magnon scattering due to the increase of the DESW relaxation parameter ∆Hk . It is also clear from (8.1) that the reversal coefficient K exponentially increases with the increase of the amplitude and duration of the pumping pulse, and exponentially decreases with the increase of the time interval tp between the application of the signal pulse and the application of the pumping pulse that causes the reversal process. Thus, having changed purposefully both ∆H0r and ∆Hkr using the irradiation one can not only control the efficiency of DESW WFR, but improve it as well.

8.4 Experimental Results The experimental investigation of the phenomenon of parametric WFR was performed on the system of spin waves propagating in a ferrite (YIG) films (1 in Fig. 8.2). Our YIG film sample was epitaxially grown on a substrate

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Fig. 8.2. Experimental setup: 1 – ferrite (YIG) film; 2 – substrate; 3 – input antenna exciting and receiving spin wave signals; 4 – pumping dielectric resonator

of gallium-gadolinium garnet (8.2). The YIG film sample was tangentially magnetized by a constant bias magnetic field H0 directed along the long side of the sample and parallel to the direction of propagation of the primary signal MSW (H0 ||ks ). The signal MSW (or BVMSW [3]) were excited and received by a microstrip antenna 3 of the width W = 50 µm. The inputpulsed microwave signal supplied to the input antenna to excite the signal MSW had the following parameters: carrier frequency ωs /2π ∼ 4.7 GHz, pulse duration τs = 50 ns, power Ps < 0.3 mW. This input electromagnetic pulse excited in the film a packet of signal BVMSW having the carrier wave number ks ∼ = 102 cm−1 and group velocity vs ∼ = 3 × 106 cm s−1 . To supply pulsed microwave pumping of carrier frequency ωp ∼ = 2ωs to the ferrite film, the film sample was placed inside a rectangular opening in a pumping dielectric resonator 4 (Fig. 8.2), which is made of thermostable ceramics having a dielectric constant ε ∼ = 80. The length of the resonator determining the size of the pumping localization region along the direction of the MSW propagation (Fig. 8.1) was l = 3.5 mm; oscillation type of the resonator was H11δ . Pumping pulses of duration τp = 50 ns and power Pp = 4.5 W were supplied to the dielectric resonator (3) from a magnetron generator via a standard rectangular waveguide. The microwave pumping magnetic field of the dielectric resonator hp was parallel to the bias magnetic field H0 , i.e. the case of parallel pumping (Chap. 11 in [3]) described by the (8.1) was realized in our experiment. Under the influence of the input signal power Ps and the pumping power Pp , a reversed signal MSW of the amplitude A− was formed in the YIG film. This reversed MSW induced in the microstrip antenna 3 an output signal power Pout (Fig. 8.2) that was transmitted via a circulator to a measurement circuit: Pout = K 2 Ps . During the experiments it was measured output, reversal, and delayed power Pout , which is proportional to the square of reversal coefficient K 2 (8.1). According to (8.1), the dependence Pout (tp ) in logarithmic scale should look as a straight line with the slope defined by doubled DESW linewidth 2∆Hk that take part in WFR processes. The intersection of this line with Y axis at tp = 0, assigned here as P2m characterizes the efficiency of two-magnon MSW scattering, so in accordance to (8.1) P2m ∼ [(∆H0i + ∆H0r )/∆H0 ]2 . If at different values of tp , a number of DESW with different ∆Hk participate in

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Fig. 8.3. The dependence of reversal power Pout for YIG film with a thickness 5.1 µm before (filled square) and after (open circle) the irradiation by reactor neutrons to the fluence of 6 × 1018 cm−2 . Here H0 = 937 Oe, Ps = 0.1 mW; 0 dB at the scale of Pout corresponds to the power of 56 dBm

the reversal process, then the dependence Pout (tp ) would have several linear regions with different slopes that correspond to the different linewidths of spin waves. For each of these straight lines, the efficiency of two-magnon MSW scattering P2m can differ as well. The typical experimental dependence Pout (tp ) presented in logarithmic scale for YIG film before and after the irradiation by reactor neutrons to the fluence of 6 × 1018 cm−2 is shown in Fig. 8.3. One can clearly see that after the irradiation DESW linewidth has increased from 0.45 Oe to 0.78 Oe (∆Hki = 0.33 Oe) or, in other words, by 70%. At the same time, the efficiency of two-magnon MSW scattering has increased by 6 dB. It means that due to the scattering at inhomogeneities produced during the irradiation, MSW FMR linewidth has increased by ∼2 : ∆H0r ≈ ∆H0i . Nevertheless, such increased efficiency of MSW scattering at inhomogeneities is not enough for the increase of DESW WFR efficiency: at any tp output power Pout for not irradiated film was higher than corresponding value of Pout after the irradiation. The change of external irradiation times (and corresponding change of the fluences) results of essential changes in ∆H0r and ∆Hkr . At the fluence less than 1018 cm−2 , any visible changes of MSW and DESW properties have not been detected. At the fluence higher than 1019 cm−2 , sharp degradation of ferrite film microwave properties has been observed. The field dependencies of WFR efficiency for YIG film before and after the irradiation are presented at Fig. 8.4. As one can see there is a shift of region where the WFR was observed toward higher magnetic field (∼12 Oe) and some expanding of this region as well.

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Fig. 8.4. The output power dependence of DESW WFR for YIG film with a thickness 5.1 µm upon bias magnetic field before (filled square) and after (open circle) the irradiation by reactor neutrons to the fluence of 6 × 1018 cm−2

Fig. 8.5. The dependence of the power of reversed signal Pout for the YIG film with a thickness 20 µm before (open square) and after the irradiation by 3 MeV protons up to the fluence of 7 × 1016 cm−2 (open circle) and 1.2 × 1017 cm−2 (open triangle)

Experimental data for WFR in YIG films irradiated by 3 MeV protons are shown in Fig. 8.5. After the first irradiation (current – 0.8 mA, fluence – 7 × 1016 cm−2 ), the effects were qualitatively similar to the effects observed under the neutron irradiations: DESW linewidth increased slightly from 0.37 to 0.40 Oe, and the efficiency of MSW scattering at inhomogeneities has increased by ∼2 dB. In general, it has been found that for fluences more than 1017 cm−2 both spin wave linewidth and efficiency of MSW scattering at inhomogeneities are constantly increased.

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Further increasing of 3 MeV proton fluences acquired by YIG films results in the nonlinear changes of DESW WFR parameters. The dependence of Pout (tp ) at the fluence of 1.2 × 1017 cm−2 (experimental points marked by open triangle at Fig. 8.5) illustrates it clearly. This dependence in logarithmic scale consists of three linear regions characterized by different values of the spin wave linewidth ∆Hk and the efficiency of MSW two-magnon scattering P2m . According to Fig. 8.5 for tp < 220 ns: ∆Hk = 0.53 Oe, P2m = 43 dB; for tp > 450 ns: ∆Hk = 0.52 Oe, P2m = 53 dB, and for the intermediate region of tp : ∆Hk = 0.24 Oe, P2m = 33 dB that is less, than in the case of nonirradiated film. It should be mentioned that at the fluences higher than 1017 cm−2 the dependence of Pout (tp ) with several linear regions of different slope has been observed for all investigated YIG films, moreover for some of them, just as it is shown at Fig. 8.5, some decreasing of DESW linewidth compared with the nonirradiated film has been observed. Just as for neutron irradiation, for fast protons irradiation it is also observed the increase of bias magnetic field but essentially lower and not exceed several Oe. Both for neutrons and 3 MeV protons, the path lengths (or ranges) exceeded significantly the thickness of YIG films, so in both cases the radiation defects were distributed practically homogeneously over the film thickness. On the contrary, when YIG films have been irradiated by Ar+ ions with the energy of 125 keV, the path length consists of several tenths of micrometer and greatly small that YIG film thickness. The obtained experimental results differ completely from the ones mentioned above. The main difference is that FMR linewidth for YIG films measured by standard EPR spectrometer has shown a tendency to the decrease when the acquired fluence increased. The changes have been observed from the minimum fluence of 1014 cm−2 and increased constantly up to the maximal value of about 3 × 1016 cm−2 in our experiments. It is clear from Fig. 8.6 where FMR lines for two YIG films with dimension 7.5 µm × 1.4 mm × 5 mm before and after irradiations by Ar+ ions to different fluences are presented. Really, for lowest fluence of 5 × 1014 cm−2 FMR linewidth has decreased from 0.72 to 0.66 Oe (i.e., ∼8%), and for highest fluence of 3 × 1016 cm−2 FMR linewidth has decreased from 0.98 to 0.59 Oe (i.e. ∼40%). Just as in the previous cases with neutrons and protons after the irradiations by Ar+ ions along with FMR linewidth decrease, the small (less than 4 Oe) increase of FMR field has been detected as well. The influence of Ar+ ion irradiation on MSW and DESW linewidths, on the contrary to FMR linewidth, has never resulted in the linewidth decrease. The values ∆Hkr for different films and fluences were 0−0.05 Oe, the increase of ∆H0r has not exceeded 0.5 Oe. In the case of ∆Hkr > 0, experimental dependences Pout (tp ) were quite similar to the corresponding dependences for the films irradiated by reactor neutrons. It is well illustrated by Fig. 8.7 showing FWR in YIG film irradiated by Ar+ ions with the energy of 125 keV to the fluence of 3.3 × 1016 cm−1 . According to the figure, the DESW linewidth due to the radiation addition ∆Hkr = 0, 04 Oe has increased from 0.4 to

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Fig. 8.6. EPR spectrograms for two YIG film samples before (dashed) and after (solid) irradiation by Ar+ of 125 keV for different fluences. The film thickness is 7.5 µm

Fig. 8.7. The dependence of the power of reversed signal Pout for YIG film with thickness 7 µm before (open triangle) and after (open square) irradiation by Ar+ ions with the energy of 125 keV to the fluence of 5 × 1014 cm−2

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Fig. 8.8. The dependence of the power of reversed signal Pout for YIG film with a thickness 7.54 µm before (open square) and after (inverted triangle) the irradiation by Ar+ ions with the energy of 125 keV to the fluence of 3.3 × 1016 cm−2

0.44 Oe, and the efficiency of MSW scattering at inhomogeneities has increased by 2 dB. It means that at small delays (τp < 180 ns) WFR efficiency for the irradiated film has exceeded by ∼1 dB the efficiency for the nonirradiated film. However, due to the increase of ∆Hk at tp ≥ 250 ns WFR process happened to be much more efficient in nonirradiated films. So, from practical point of view, the most interesting are YIG films for which after the irradiation no increasing in ∆Hk is observed: ∆Hkr = 0 (as mentioned above, in all the experiments, on the contrary to FMR, ∆Hkr ≥ 0). The dependence of WFR power Pout on tp for YIG film with a thickness 7.54 µm is shown in Fig. 8.8. In this case due to the equal slopes of two dependencies measured before and after the irradiation, one can state that DESW linewidth happened to be constant, i.e. ∆Hkr = 0. However, since the efficiency of MSW scattering at inhomogeneities increased by more than 2 dB, the magnitude of the power of reversed signal Pout for the irradiated film is higher by the same value than for the nonirradiated film in all range of pumping pulse delays Tp . This fact allows hopping for the increase of efficiency of ferrite active delay lines when the irradiated YIG films are used.

8.5 Discussions Physical processes in solid state under the reactor neutron irradiation are well known [4, 13]. At low fluences (<5 × 1018 cm−2 ), noninteracting simple and complex (cluster) defects, the number of which is proportional to the fluence,

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are produced almost homogeneously over ferrite film thickness. There are also the effects of Fe3+ ion recharging resulting in Fe2+ and Fe4+ formation; however, in the case of YIG films such effects are important only at the low temperatures. For YIG films at fluences less than 5 × 1018 cm−2 FMR linear broadening ∆H with the fluence increasing is observed. Uniaxial crystallographic anisotropy, due to lattice constant a0 increasing, has been detected as well. With further fluence increasing the defects are grouped interacting with each other, the amorphisation is starting and, as a result, at the fluences over 1019 cm−2 ∆H increases sharply. At last, at the fluences over 1020 cm−2 , it results in amorphous spin glass. The films can be returned to their initial state making use the long term and high temperature annealing. Regardless to well and commonly excepted knowledge of all the processes in YIG films under the reactor neutron irradiation, there are no theories that may quantitatively explain the experimental data. For instant, now it is experimentally confirmed that the scattering of MSW and oscillations at neutron produced defects results in short ESW appearance [13]. Taking into account the size of these defects (1 nm–10 nm), the same result is given by the existing theories of ferromagnetic relaxation [3], though the broadening values ∆H happen to be underestimated by the order of magnitude. One of the explanations, from our point of view, for such fact is that all the theories do not take into account the interaction of magnetization with optical magnons and phonons, as it has been done only once when the linewidth of dipole spin waves with wave vector k → 0 has been calculated [14]. So all our further discussions have generally qualitative character. As one can see from Fig. 8.3 in full accordance with mentioned above under the reactor neutron irradiation, MSW and DESW linewidth broadening is observed. At the same time, MSW and DESW linewidth broadenings (∆Hkr and ∆H0r , correspondingly) are of the same order of magnitude and at the fluence of 6 × 1018 cm−2 are 0.33 and 0.6 Oe, respectively. Such result is quite understandable qualitatively while for both MSW with k ∼ 102 cm−1 and DESW (k ∼ 104 cm−1 ) wavelengths λ = 2π/k essentially exceed (by three orders of magnitude and more) the defect size produced by the reactor neutron irradiation at low fluences. Thus in both events there is rather weak diffraction of spin waves at the magnetic inhomogeneities. The shift of resonance fields (Fig. 8.4) can be resulted from the increase of lattice constant a0 under reactor neutron irradiation. The increase of lattice constant by the value ∆a0 results in elastic tensions [15]: δj =

E ∆a 1−µ a0 ,

(8.2)

where E is the Young’s module and µ is the Poisson coefficient. In turn the elastic tension results in the uniaxial anisotropy [3] changing all the characteristic fields when compared with nonirradiated film. According to Fig. 8.4, the value of induced anisotropy at the fluence of 6 × 1018 cm−2 is ∼10 Oe.

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By completing the analysis of YIG films irradiated by reactor neutrons, it is worth to mention that their use for the improvement of characteristics of spin wave devices, in particular active delay lines, is hardly possible. Because of DESW linewidth broadening ∆Hk according to (8.1) the wave losses increase exponentially when pumping pulse delay tp (that defines the delay Td ∼ = 2tp ) is increased. Small increase of the transformation coefficient of MSW into DESW due to the increasing of preexponential factor in (8.1) gives principally an opportunity to increase WFR efficiency coefficient K only at small tp . Really, for some YIG films, on the contrary to Fig. 8.3, we obtained the increase of efficiency coefficient K after the irradiation. However, this increase was just 2–3 dB and has been registered only at small delays Tp < 200 ns. At last it should be mentioned that reactor neutron irradiation influences much stronger upon homogeneous FMR parameters, than upon MSW and DESW. At the fluence of 6 × 1018 cm−2 FMR linewidth broaden from 0.73 to 4.6 Oe (by factor of 6, while only ∼2 times in the case of MSW), resonance field has increased by 50 Oe. It may be caused the FMR excitation character – the resonance is homogeneous all over the film while MSW, propagating from antenna with dissipation more than 10 dBcm−1 , are really influenced only by the defects localized not far from the antenna. Experimental results obtained for the films irradiated by 3 MeV protons at low fluences (<7 × 1017 cm−2 ) happened to be qualitatively similar to the results obtained for reactor neutrons (Fig. 8.5). It is quite expected since in both cases the irradiation produces the similar nano-defects with the size of 1–10 nm homogeneously distributed over the film thickness. However, the picture changes significantly with fluence increasing: at the fluence of 1.2 × 1017 cm−2 , taking into account the slopes of experimental curves Pout (tp ), spin waves with linewidth ∆Hk as broader (what should be expected for the increase of defects during the irradiations) as narrower, than the initial linewidth ∆Hk0 , take part in WFR processes in the film. In some cases the curve slope Pout (tp ) corresponded to the waves with extremely small value ∆Hk ∼ 0.2 Oe. One of the explanations of this effect is the following. Parametric parallel pumping influences on the spin wave system in such a way that parametric spin waves (PSW) with minimal dispersion and polar angle Θk = 90◦ have the biggest increment [3]. Namely PSW are known to be excited from the thermal level because of the instability during parallel pumping and their linewidth is ∆Hk0 ≤ 0.2 Oe [3]. However, due to the large value of wave vector (k ≥ 5 × 104 cm−1 ), PSW are excited quite weakly during MSW scattering at the inhomogeneities. One can see from Fig. 8.5 that for PSW the interaction coefficient with MSW P2m is substantially lower than for DESW (up to 20 dB). So their contribution to Pout is essential only at big Tp when the amplitude of high excited DESW due to the high damping with time will be less than the amplitude of low excited, but at the same time weakly damping PSW. According to Fig. 8.5, it takes place at Tp > 200 ns. However, at Tp > 500 ns, DESW play the main role in the output signal formation again since PSW

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have greater noise component due to their parametrical excitation by pumping from the thermal level, the role if which is higher at bigger Tp and which does not recover due to PSW backward scattering at the inhomogeneities. As a result, PSW presence has only negative effect upon WFR process. One can see from Fig. 8.5 that at the fluence of 1.3 × 1017 cm−2 , the exponential increase of WFR efficiency at the decrease of tp is observed for tp ≥ 500 ns. At tp < 500 ns PSW, which absorb energy from the pumping and decrease WFR efficiency, play more important role. So in the event of YIG films irradiated by 3 MeV protons the improvement of spin wave device parameters is impossible. In the case of YIG films irradiated by heavy ions with the energy of ∼105 eV the situation is completely different. The main feature here is the small path length of the heavy ions, the values of which are essentially less than the film thickness and not exceed several tenths of micrometer. In such event, there is inhomogeneous distribution of admixtures in the film that corresponds approximately to Gaussian distribution [15]. In the simplest case when the path length is h there is amorphous ionized layer at the distance from the surface d, so the thickness of this layer is h − d. Thus the irradiation produces a three-layer system in the film consisting of lower nondamaged layer (at the depth h and more), amorphous paramagnetic layer with the thickness of h − d and upper nondamaged layer of the film with the thickness of d. The depth h is defined by the type and energy of heavy ions, though the distance from the surface d, by the fluence. In our case (Ar+ ions with the energy of 125 keV), the depth h ∼ 0.15 µm at the fluence of 1014 cm−2 the thickness of amorphous layer d ∼ 0.05 µm, though at fluence 1015 cm−2 −d = h. Therefore, the amorphous layer has reached the surface and the film becomes a two-layer system – the amorphous paramagnetic layer with the thickness of h covers the nondamaged film with the thickness by h less compared with the initial thickness. The described above picture is ideal and can be real only for low and high fluences. In the case of intermediate fluences (1015 −1016 cm−2 ), the film can have not two, but up to four nondamaged layers, each of them giving the exact FMR line with its own linewidth ∆H0 and anisotropy field that defines the shift of additional line from the main resonance, which is determined by part of the film that is situated under the layer with the thickness of h [5]. It is quite understandable that inhomogeneous distribution of defects over the thickness of the film influences in a different way on the magnetostatic oscillations and waves. Let us first discuss the possibility to decrease the FMR linewidth under heavy ion irradiations (Fig. 8.6). One can suppose that FMR curves at Fig. 8.6 are mainly defined by nondamaged part of the film at the depth of h ∼ 0, 15 and having the thickness ∼7.5 µm − 0.15 µm = 7.35 µm. First, the signal from this part of the film will be much higher, than from ionized layers covering about 2% of the total volume. Second, due to the induced crystallography anisotropy the ionized layer FMR will be shifted by the value significantly bigger than FMR linewidth ∆H0 < 1 Oe. So in fact FMR curves at Fig. 8.6 present the results of resonance investigation in YIG films exposed

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to the ion etching. In this event, just as in the case of chemical etching, the decrease of FMR linewidth is quite understandable, since the deformed surface magnetic layer of YIG film, having elastic tensions produced during its growth that result in the homogeneous procession scattering for short spin waves, is eliminated from the consideration [16]. As for the introduced above the total FMR linewidth ∆H0 = ∆H0i + ∆H0i + ∆H0r , as a result of Ar+ ion implantation the intrinsic linewidth ∆H0i remains stable, radiation linewidth ∆H0r = 0, since FMR is observed for nondamaged part of the film, and linewidth ∆H0i defined by scattering at initial inhomogeneities decreases due to the fact that the deformed surface magnetic layer is eliminated from the consideration. For DESW due to low path lengths the influence of near surface layer is negligible, so one should expect rather small changes of ∆Hk , and it has been confirmed by the experiment (Fig. 8.8).

Fig. 8.9. Relief change of YIG film surface before (top picture) and after (bottom picture) the irradiation by Ar+ ions with the energy of 125 keV to the fluence around 3 × 1016 cm−2

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Finally, let us consider the influence of Ar+ ion implantation upon MSW properties with wave vector value of k ≤ 102 cm−1 . In BVMSW case used by us in the experiment, the influence of Ar+ ion implantation must be similar to the influence on homogeneous FMR, though weaker due to magnetization inhomogeneous distribution over the thickness of the film, limited both wave vector and MSW losses. However, because of the above-mentioned features of MSW excitation by the microstrip antenna, the alternate magnetic field amplitude in the film will be inhomogeneous with a maximum exactly in the near surface layer. Thus the role of inhomogeneities in this layer cannot be ignored as in the event of homogeneous FMR. Moreover, after irradiations the film surface is more inhomogeneous than before irradiations. One can see it clearly at Fig. 8.9, where the photo image of film surfaces before and after the irradiation by Ar+ ions with the energy of 125 keV to the fluence around 3 × 1016 cm−2 have been made by the atomic force microscope. After the irradiation at the practically ideal surface of the film, there are geometrical inhomogeneities with the size up to 50 nm at horizontal level and over 3 nm in vertical direction. The experimental investigation of MSW dispersion influence upon the propagation from the input to output antenna shows that in Ar+ implanted films the total linewidth, on the contrary to homogeneous FMR, is always increasing and it means that two-magnon linewidth ∆H0i + ∆H0r increases as well. So in Ar+ implanted YIG films, it is quite possible to increase the efficiency of MSW scattering at the inhomogeneities and it has been confirmed experimentally. Namely this fact at the stable parameter of DESW dispersion ∆Hk contributes to the increase of WFR coefficient in YIG films implanted by Ar+ ions.

8.6 Conclusions The influence of reactor neutrons on the YIG film magnetic properties has been detected for the fluences ≥1018 cm−2 . At the fluences less than 6×1018 cm−2 , the linear broadening of MSW and DESW linewidths by several tenths of Oe has been observed. At the same time, the efficiency of MSW scattering at the inhomogeneities increases by 6 dB. The linewidth for homogeneous FMR increases much more than for MSW (by 6 times compared with ∼2 times in the case of MSW). There is a field of single axis crystallographic anisotropy at ∼10 Oe that changes the values of all characteristic magnetic fields. The DESW WFR efficiency decreases due to the exponential increase of spin wave losses at their linewidth broadening. At low fluences (<1017 cm−2 ) the irradiation by 3 MeV protons produces the effects analogous to ones observed in the event of reactor neutrons. At the higher fluences (up to 1.3 × 1017 cm−2 ) the WFR efficiency decreases sharply. It could be explained by the pumping excitation of short PSW with the wave vector value of k ≥ 104 cm−1 weakly bounded with signal MSW due to large value of the wave vector.

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The decrease of homogeneous FMR linewidth at the fluence increase has been observed when YIG films were irradiated by Ar+ ions with the energy of 125 keV. For the fluence of 3 × 1016 cm−2 the linewidth decreased by 40% from 0.98 to 0.59 Oe. At the same time, short DESW linewidth has not practically changed, though long MSW linewidth has increased by several tenths of percent. All observed phenomena result from the inhomogeneous distribution of admixtures over the film thickness. The presence of implanted layer of submicrometer thickness in the YIG film influences in different ways upon properties of different magnetostatic oscillations and waves. Because of the efficiency increase of MSW two-magnon scattering in this layer at DESW dispersion parameter being stable we have managed to increase by 2 dB the WFR coefficient in the YIG films irradiated by Ar+ ions. Acknowledgments This work has been supported by the Scientific and Technical Center of Ukraine (STCU) under STCU Grant No. 3066.

References 1. N.G. Kolin, High School Izv., Phys. 6, 12 (2003) 2. J.M. Meese, (ed.), Neutron Transmission Doping in Semiconductors. (Plenum, New York, 1979) 3. A.G. Gurevich, G.A. Melkov, Magnetization Oscillations and Waves (CRC, New York, 1996) 4. S.B. Ubizskii et al., Phys. Stat. Sol. (a) 177, 349 (2000) 5. A.P. Bedyukh et al., Solid State Phys. (Rus) 31, 63 (1989) 6. G.A. Melkov, V.I. Vasyuchka, Y.V. Kobljanskyj, A.N. Slavin, Phys. Rev. B, 70, 224407, (2004) 7. B.Y. Zel’dovich, R.F. Pilipetskii, V.V. Shkunov, Principles of Phase Conjugation, (Springer, Berlin, 1985); A. Brysev, L. Krutyansky, V. Preobrazhensky, Phys. Sci. Usp. (Rus) 41, 793 (1998) 8. A.L. Gordon, G.A. Melkov, A.A. Serga, V.S. Tiberkevich, A.V. Bagada, A.N. Slavin, JETP Lett., 67, 913 (1998) 9. G.A. Melkov, A.A. Serga, V.S. Tiberkevich, A.N. Oliynyk, A.V. Bagada, A.N. Slavin, J. Exp. Theor. Phys. (Rus) 89, 1189 (1999) 10. G.A. Melkov, Y.V. Kobljanskyj, A.A. Serga, V.S. Tiberkevich, A.N. Slavin, Phys. Rev. Lett. 86, 4918 (2001) 11. M. Sparks, Ferromagnetic Relaxation Theory (McGraw-Hill, New York, 1964) 12. A. Okaya, L.F. Barash, IRE Proc. 50, 2081 (1962) 13. B.M. Lebed, L.Y. Mukha, V.I. Mosel, U.A. Ulmanis, Solid State Phys. (Rus) 31, 1708 (1989) 14. I.V. Kolokolov, V.S. Lvov, V.B. Cherepanov, J. Exp. Theor. Phys. (Rus) 86, 1946 (1984) 15. X. Rassel, I. Ruge, Ion Implantation (Nauka, Moscow, 1983) 16. Y.M. Yakovlev, S.Sh. Gendelev, Ferrite Monocrystals in radioelectronics (Sov. Radio, Moscow, 1975)

9 Electromagnetic Radiation of Micro and Nanomagnetic Structures with Magnetic Reversal B.A. Gurovich, K.E. Prikhodko, E.A. Kuleshova, A.G. Domantovsky, K.I. Maslakov, and E.Z. Meilikhov Russian Research Center Kurchatov Institute, 1, Kurchatov sq., Moscow 123182, Russia, [email protected] Summary. By means of the selective removal of atoms from the cobalt oxide under ion beam irradiation, micro and nanopatterned magnetic Co structures have been produced consisting of pseudo-single domain “bits” with different coercivity. With the magnetization reversal of those structures, electromagnetic radiation pulses arise whose characteristics have been investigated in detail. It was experimentally revealed about the huge increasing of the radiation intensity (up to ≈100 times) because of the synergetic effects depending on bit coercivities (varying with their shape anisotropy) and the distance between bits. In addition, there have been discovered the strong influence of the soft magnetic underlayer (of 50 nm thickness) on the process of magnetization reversal. Peculiar properties of the magnetic structure with and without the soft magnetic underlayer have been studied by magnetic force microscopy.

9.1 Introduction The present study investigates magnetic properties (a hysteresis, domain structure) of the 2D periodic magnetic nanopatterned structures consisting of the ordered magnetic cobalt bit arrays as well as the properties of a signal emitted from bits at magnetization reversal in an external magnetic field (Barkhausen effect). The objective has both scientific and applied significance, as Barkhausen effect in ferromagnetic materials is used commercially for the development of anti-theft labels and for the generation of chipless remote multibit multipurpose labels, for example, for electromagnetic identification (EMID-labels), for remote reading of the information from magnetic carriers, etc. It is obvious that the potential of possible applications predetermines a heightened interest in magnetic patterned structures study since traditional continuous magnetic environments are unsuitable for such applications.

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The most important research directions are the following: • Research the opportunities for controlled coercivity change of magnetic patterned media from the same material. • Search the opportunities for increasing the amplitude of a signal emitted from magnetic patterned media at magnetization reversal. • Research the factors influencing the form and quality of a signal emitted from magnetic patterned media at magnetization reversal. Let us consider possible ways of bits coercivity manipulation. Traditional techniques for changing magnetic materials coercivity are mainly based on the variation of their structure and/or a texture. In case of magnetic patterned media, the theory predicts a possible coercivity change due to the variation of bit shape anisotropy. However, the existing theories predict fairly well coercivity and influence of the shape anisotropy for small (single domain with the maximal sizes of no more than ≈200–300 nm) bits [1] and qualitatively correctly describe properties of one multi domain bits, but have little to say in situations where bits are interacting with each other and with a substrate. The value of bit coercivity can be adjusted basically by selecting bit sizes, by orientation with regard to axes of easy magnetization, and by a degree of their interaction controlled both by relative bits positioning and magnetic properties of a substrate (underlayer) [2]. Specific limits of this regulation are to be determined experimentally, which was the subject of the current study. One of the ways to manipulate bit coercivity and generate signals is to use the so-called crystal magnetic anisotropy. Energy of the magnetized pseudosingle domain bit depends on the direction of its vector of magnetization in relation to the crystallography directions and geometrical axes of a bit. Therefore, magnetic energy of a crystal is anisotropic. Though the energy of such magnetic anisotropy (which is defined as magnetic energies at magnetization along hard and easy axes of magnetization) is, as a rule, significantly lower than the isotropic exchange energy (i.e., the energy of interaction of ordered spins with each other), it is this energy that determines the direction of magnetization. Magnetic anisotropy constant for cobalt is larger, by an order of magnitude, than the appropriate constants for such ferromagnetic materials as iron and nickel. The range of coercivity regulation is proportional to a constant of anisotropy that has caused a choice of this material as the research one.

9.2 Experimental For the controlled change of the coercivity of the same material, the present study used patterned magnetic media: thin-film structures with a matrix of the nonmagnetic cobalt oxide, which accommodated periodic arrays from the cobalt rectangular bits. Coercivity was changed by the variation of anisotropy of the bits form.

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The above-mentioned structures were formed using selective removal of atoms (SRA) method [3] in combination with optical or electron-beam lithography. The latter helped form 2D-periodic structures of penetrating apertures in a resistive mask. The sizes, geometry, and characteristic values of the aperture periods corresponded to the parameters of magnetic patterned structures. After resistive masks formation on the cobalt oxide films, the films were irradiated by the accelerated protons for selective removal of oxygen atoms. As a result, the arrays of cobalt bits were formed in the matrix from the cobalt oxide film. In its essence, the method of SRA makes it possible to directly alter the solid atomic composition under exposure to an accelerated ion beam of a specific energy. This modification of atomic composition is not a result of any chemical or nuclear reaction, but is completely caused by the selective removal of atoms of a specific type from two or multiatomic compounds as a result of atomic displacements induced by accelerated ions. Such modification can result in a radical change of the material’ physical properties and, in particular, in the transformation of insulators into metals or semiconductors, nonmagnetic materials into magnetic ones, in changed optical properties, etc., [3–5]. It is important to note that SRA allows simultaneously to change physical properties of each of the layers in a multilayered structure. The ability of SRA method to considerably modify physical properties of the irradiated sites of materials, in particular, to transform nonmagnetic materials into magnetic one predetermined its use as the basic method for manufacturing various patterned magnetic media. Measurements of sample magnetic properties were carried out with BHV50 vibrating sample magnetometer in the lengthways direction of a bit axis and also in a cross direction due to magnetic anisotropy of patterned magnetic media composed of bits with anisotropy of the form. Magnetometer measurements, where speed of change of a magnetic field is small, allow to receive important information about magnetic properties and also to predict the possibility of generating samples with differing properties. However, this method precludes the study of emitting characteristics of patterned magnetic media at dynamic magnetization reversal in variable magnetic fields. To study the dynamic characteristics of bits, a specific experimental measuring installation and appropriate techniques of dynamic measurements were developed. Main dynamic characteristics of magnetic patterned media include the following: dynamic coercivity determined as instant value of an external field, where a turn of a bit magnetization vector occurs; duration of the pulse arising at magnetization reversal of a bit, determined as half of width of the signal peak registered by the antenna; and also the intensity of a pulse determined as the amplitude of a magnetization reversal pulse registered by the antenna. The possibility of discriminating signals from two and more arrays is defined by a difference of dynamic coercivities of various arrays in view of duration of the signals of the bits forming them at their magnetization reversal.

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The technique of experimental definition of magnetic patterned media dynamic characteristics consists in studying the form and parameters of the signal formed by bit arrays at the moment when the magnetization vector changes direction in a variable magnetic field. The research required building experimental installations for generation of a variable magnetic field with a frequency of 50 Hz and amplitude up to 500 Oe, as well as generation of fields with frequency from 50 Hz up to 200 Hz and the maximal amplitude up to 60 Oe. It also required antennas of various configurations, allowing registering a pulse of magnetization reversal of magnetic patterned structures with the cross size of up to 15 mm and the length of up to 50 mm. A hardware complex for registration of a signal of magnetization reversal from magnetic patterned structures consisting of the antenna, the broadband amplifier, and a digital 12 bits ADC oscillograph was developed. It allowed to measure the value of an external field at the moment of magnetization reversal of array with bits and thus to determine its coercivity. The features of bit magnetic structures were investigated with the Solver P47 atomic force microscope (AFM) manufactured by NT–MDT (Russia). Images were acquired in magnetic force microscopy (MFM) mode, using cobalt-coated tips. Before the experiment, the tips were magnetized along the axis in 5 kOe magnetic field.

9.3 Results and Discussion 9.3.1 Coercivity Static Measurements of Magnetic Patterned Media Static magnetic measurements of coercivity were performed using BHV– 50 vibrating sample magnetometer at room temperature. The effects of anisotropy ratio (the ratio of bit length to bit width), mutual arrangements of bits on the coercivity of single-layered samples are listed in Table 9.1. A clear trend is obvious: the higher bits anisotropy ratio, the bigger bits coercivity is. The analysis of static hysteresis loops makes it possible to estimate the emitting capacity of the sample in the course of dynamic magnetization reversal. In terms of physics, it is evident that the emitting capacity is mainly determined by the square-law characteristics of the hysteresises, K. The squareness is, by definition, the ratio of residual magnetization to saturation magnetization. The bigger K, the faster magnetization vector rotation, the higher the signal amplitude emitted in a variable magnetic field and shorter its width. As an example, in Fig. 9.1 are shown in-plane static hysteresis loops for deposited 50 nm pure Co film measured along easy magnetization axis (a) and perpendicularly to easy magnetization axis (b). Low value of K in the direction perpendicularly to easy magnetization axis (Fig. 9.1b) prevents the recording of the emitting signal in the course of dynamic magnetization reversal, whereas a short intense signal is easy to record in the direction along

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Table 9.1. Experimental static coercivity of single-layred samples Bit size (µm2 )

Anisotropy ratio

Bit spacing (µm) Along

Across

Coercivity (Oe) Along

Initial cobalt oxide thickness (nm)

Across

0.3 × 0.6 0.3 × 0.9 0.3 × 1.8

2 3 6

Set 1 (Substrate – Si+SiO2 ) 2 2 142 104 2 2 210 191 2 2 319 276

0.5 × 1.0 0.5 × 2.0 0.5 × 3.0 0.5 × 3.0

2 4 6 6

2 2 2 2

Set 2 (Substrate – Si) 2 138 99 2 159 78 2 318 241 2 356 300

70 70 70 70

5.0 × 5.0 5.0 × 10 5.0 × 15 5.0 × 20 5.0 × 30

1 2 3 4 6

7 7 7 7 7

Set 3 (Substrate – Si) 7 91 7 96 7 108 7 108 7 112

86 70 77 75 76

80 80 80 80 80

10 × 10 10 × 20 10 × 30 10 × 40

1 2 3 4

7 7 7 7

Set 4 (Substrate – Si) 7 81 7 107 7 112 7 115

74 75 80 68

80 80 80 80

10 × 10 10 × 20 10 × 20 10 × 30 10 × 40 10 × 60 10 × 60

1 2 2 3 4 6 6

Set 5 (Substrate – Si+SiO2 ) 7 7 76 86 7 7 84 65 7 7 88 72 7 7 152 120 7 7 160 113 7 7 262 241 7 7 275 238

80 80 80 80 80 80 80

50 × 50 50 × 30 50 × 20

1 1.7 2.5

Set 6 (Substrate – Si+SiO2 ) 7 7 90 78 7 7 217 166 7 7 242 187

80 80 80

70 70 70

Proton irradiation dose – 1020 H+ /cm2

easy magnetization axis (Fig. 9.1a). Figure 9.2 shows in-plane static hysteresis loops for 30 nm Co bits array with 30 × 10 µm2 bit sizes and 7 µm bit spacing measured with an applied field parallel to the bits long sides (a) and perpendicular to the bits long side (b). It has been found experimentally that in spite of sufficiently large values of K in the direction along bits’ long sides, the recording equipment applied

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Fig. 9.1. In-plane hysteresis loops for deposited 50 nm pure Co film measured along easy magnetization axis (solid curve) (K = 0.97) and perpendicularly to easy magnetization axis (dashed curve) (K = 0.35)

Fig. 9.2. In-plane hysteresis loops for 30-nm Co bits array with 30 × 10 µm2 bit sizes and 7 µm bit spacing measured with an applied field parallel to the bits long sides (solid curve) (K = 0.95) and perpendicular to the bits long side (dashed curve) (K = 0.77). Initial cobalt oxide thickness was 80 nm

is not sensitive to record the emitted signal from single-layered samples. At the same time, the emitted signal from deposited pure Co film (“underlayer”) along easy magnetization axis can be easy recorded. This experimental fact forms the basis of the principle of the signal magnetic amplification (discussed below) from bit arrays. Signal amplitude from single-layered patterned media is weak because of the small amount of magnetic matter (bits’ spacing are big whereas their thickness is small). Therefore, a signal recording is a rather difficult experimental task. Nevertheless, the signals from single-layered patterned media had been recorded in a few cases. Therefore, the specific signal intensity (per unit magnetic volume or per unit area) emitted by a patterned magnetic array had to be significantly increased. To solve this problem, the present study was the first to propose and experimentally implement the concept of magnetic amplification, i.e., significant enhancement of the signal from patterned magnetic media by applying

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soft magnetic underlayer. The main idea implied that hard magnetic array placed on soft magnetic media could manage the switching process of the underlayer. The concept has been proved experimentally. 9.3.2 The Dependence of the Amplitude and Duration of Emitted Signal on the External Magnetic Field Amplitude We investigated the dependence of parameters of the signal emitted by arrays of bits and cobalt underlayer on the amplitude of external sinusoidal magnetic field for fixed orientation between bits and magnetic field. As the rate of external field alteration influences the emitted signal parameters, increased field amplitude at fixed frequency is equivalent to the increased frequency at fixed amplitude due to the field alteration rate increase at the instant of magnetization reversal. Figure 9.3 shows the emitted signal amplitude in relation to the external field amplitude for two-layer structures of 5 × 5 µm2 bits array and 50 nm soft cobalt underlayer (a) and 60 × 10 µm2 bits array and 100 nm soft cobalt underlayer (b). Figure 9.4 gives the duration of emitted signal in relation to the external field amplitude for a two-layer structure of 5 × 5 µm2 bits array and 50 nm soft cobalt underlayer. It is clear from Fig. 9.3–9.4 that: • The emitted signal amplitude for two layer structures of patterned magnetic media and cobalt underlayer increases with the growth of external field amplitude without saturation. • The emitted signal amplitude from two-layer structures exceeds the corresponding signal from the soft underlayer by about 1.5–2 times. At the same time, there is no significant difference in dynamic coercivity for these structures. The higher the coercivity difference is the lower the emitted amplitude difference will be for two-layer structures and underlayer for fixed frequency and amplitude of the external field. It takes place due to

Fig. 9.3. Dependence of the emitted signal amplitude on external magnetic field amplitude for: 5 × 5 µm2 bits array and 50 nm soft cobalt underlayer; 60 × 10 µm2 bits array and 100 nm soft cobalt underlayer; 50 nm soft cobalt underlayer

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Fig. 9.4. Duration of emitted signal in relation to external field amplitude for two-layer structure of 5 × 5 µm2 bits array and 50 nm soft cobalt underlayer

the lower rate of external field alteration at the instance of magnetization reversal of sandwich structure, while this magnetization moment shifts closer to the maximum of the external field. • Our experimental data showed that amplitude of a signal from the 50 nm soft cobalt underlayer tends to become saturated with external field growth and lags behind the amplitude of a signal from the sandwich array with bits; the lagging magnitude correlated with the amplitude of the external field. • Data in Fig. 9.4 show that the duration of the emitted signal drops with the growth of external field amplitude. This is equivalent to the magnetization reversal rate increase. We think that the saturation of signal duration in Fig. 9.4 was a method-induced artifact due to the low frequency of digitization (250 kHz). One of the most important results obtained in this work was a linear dependence of the dynamical coercivity and rate of field growth at magnetization reversal moment on the external sinusoidal field amplitude. These linear dependencies made it possible to recalculate emitted signals parameter as if they were formed in accordance with the linear law of external field alteration. The linear law of external field alteration was not used in this work though it was believed to be very promising since it could yield uniform emitted signal intensity in a wide range of dynamic coercivity. We investigated the emitted signal for two-layer structures of magnetic bits and underlayer at 200 Hz external field frequency. We found signal intensity growth and drop of its duration that was similar to the effect of external field amplitude increase at 50 Hz. 9.3.3 Experimental Determination of Dynamic Emitted Parameters of Magnetic Patterned Media Table 9.2 shows the experimental results for the emitted signal parameters for single layer 30 × 10 µm2 bits array, single layer 55 × 10 µm2 bits array, soft

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Table 9.2. Emitted signal parameters of dynamical magnetization reversal Bit sizea 2

(µm ) 30 × 10 30 × 10+Coc 55 × 10 55 × 10+Co Co

Experimental parameters

Calculated parametersb

Amplitude Duration Dynamic coercivity Amplitude (mV) (s) (Oe) (mV) 21 255 5 340 212

160 90 100 60 90

33.9 29.2 39.4 32.9 24.2

23 267 6 372 212

Duration (s) 146 86 83 55 90

a

Bit size in patterned magnetic bits array Calculated parameters for the linear law of external field alteration c Soft 50 nm Co underlayer b

50 nm cobalt underlayer and also for two-layer structures of above-mentioned bits arrays and cobalt underlayer. We investigated the emitted signal from bits array with a long side of bits aligned with the external magnetic field (this case was called “parallel orientation”). The sensitivity of the experimental equipment used did not allowed to measure the emitted signal with a short side aligned with the external field (this case was called “perpendicular orientation”). It means that the amplitude of the emitted signal at the latter case was at least one order of magnitude less than the amplitude of the emitted signal in parallel orientation. Table 9.2 also contains the data of calculated signal parameters at a fixed rate of external field alteration of 17 Oe s−1 (the rate at the instant of magnetization reversal of soft underlayer). These data could be used as a reference for the case of linear law of external field alteration. Table 9.2 clearly shows that the dynamical coercivity of sandwich structures increases relative to the soft underlayer. It is also obvious the growth of emitted signal amplitude of sandwich relative to one-layer magnetic patterned structures. On the basis of the calculated values, the effect of magnetic amplification would be greater when the linear law of external field alteration is used, while the peaks positions (instance of magnetization reversal) could be determined by the corresponding coercivity. In general, it was very difficult to measure the emitted signal from onelayer patterned media due to the low signal amplitude. Taking into account our equipment characteristics, it means that a combination of soft underlayer and patterned media with different bits geometry allowed us to amplify emitted signal amplitude about 10–100 times.

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Fig. 9.5. Time sequence of emitted signals from four different 2D arrays of bits at the same underlayer

The measurements of emitted signals from magnetic patterned media showed: • Most of the sandwich structures with bit patterned layer and soft underlayer revealed ten times more intensive emitted signal in parallel orientation than in perpendicular one. • At optimal underlayer thickness, the amplitude of the signal emitted by sandwich structures exceeded the sum of amplitudes from independent patterned layer and underlayer. • For the same type of sandwich structures in a certain range of underlayer thickness, samples with greater underlayer thickness demonstrated higher signal amplitude. Further increase of underlayer thickness led to a decrease of amplification effect due to the limited volume of underlayer influenced by magnetic bits magnetization reversal. Thus, it was shown that the optimal underlayer thickness existed and could helpfully utilize the emitted signal amplification for dynamical magnetization reversal of patterned arrays media. We also made a prototype of multibit EMID labels (“smart labels”) by SRA technique. Figure 9.5 shows the time sequence of emitted signals from four different 2D arrays of bits with the same underlayer. Every bit array had an unique coercivity. It is clear from Fig. 9.5 that this structure can be considered as a prototype of multibit EMID labels (‘smart labels’). 9.3.4 Dependence of Dynamical Coercivity of Bit Arrays with Underlayer on the Bits Structural Geometry For the demonstration of the influence of bits arrangement on a coercivity, samples of patterned arrays of 40 × 10 µm2 bits were produced with different distance between bits along the big bit size (a parameter) and along a small bit size (b parameter). The array dimensional sizes were 35 mm in length and 100 µm in width. One sample was made with the shift of adjacent bit rows

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Fig. 9.6. Dependence of dynamical coercivity of bit arrays with 50 nm cobalt underlayer on the distance between bits along big size (dB ), along small size (dS ) and also for the sample with shifted structure. Bits size: 40 × 10 µm2 ; arrays area width: 100 µm (for shifted sample: 300 µm); arrays area length: 35 mm

along big bit size by ≈10 µm. The width of bits array in this sample was 300 µm. The whole size of the samples was 40 mm long and 4 mm wide. Soft cobalt underlayer was sputtered on the whole sample, thus the lager part of the sample was covered with the substrate without bits. The qualitative result of the influence of bits arrangement geometry on the coercivity was obtained (Fig. 9.6). Analysis of Fig. 9.6 shows that the coercivity of bit arrays increases with the growth of distances between bits for selected range of their alteration. The distance b along the small bit size had more influence on the coercivity. But the coercivity increase was not great and made up ≈1 Oe. The more significant coercivity increase was found in the sample with shifted bits structure (≈4.5 sOe). 9.3.5 Peculiarities of the Magnetic Structure of Cobalt Bits with and without a Soft Magnetic Underlayer The influence of bits shape and shape anisotropy on the coercivity of patterned magnetic media is more pronounced for the size of single magnetic bits provided they are of a single domain nature. The maximum bits size of single domain bits is currently unknown, but it apparently lies within 0.2–0.3 µm region with lager bits to be multi domain. We are not aware of any experimental and theoretical works, which study the influence of magnetic bits shape anisotropy on the coercivity of magnetic media with bits size increase and after the decay of their single domain structure. The magnetic structure of patterned cobalt magnetic media with bits size from 80 × 400 nm2 to 10 × 40 µm2 produced by proton irradiation of cobalt oxide was studied in this work. Figure 9.7 demonstrates nanopatterned magnetic media with 80× 400 nm2 magnetic bits. Judging by MFM images, these bits have a single domain both in the initial state and after the magnetization reversal.

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Fig. 9.7. The examples of nanopatterned magnetic media with bit size of 80 × 400 nm2 . (a) AFM topography image and (b, c) MFM images

Fig. 9.8. AFM topography (a) MFM (b) images of 2 × 12 µm2 magnetic bits after proton irradiation (without additional magnetization). Initial cobalt oxide thickness was 70 nm

The study of micrometer-sized magnetic bits after the irradiation showed that they have a multidomain magnetic structure, which was expected according to the data of maximum single domain particles size. Topographic and MFM images of the magnetic media with 2 × 12 µm2 bit size just after irradiation are shown in Fig. 9.8. The multidomain nature of demagnetized magnetic bits is clearly seen. These bits were also studied after the magnetization in the external magnetic field along the short and long bit axes. MFM images in this case demonstrate pseudo single domain contrast (pseudo means here single domain in the magnetized state only), i.e., the MFM image of each bit has two pronounced poles with no addition structure inside the bit (Fig. 9.9). Thus magnetized bits were ageing for some days to test the stability of their magnetic state. No change of magnetic bits MFM images were found, which testified to the stability of the magnetic state of bits.

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Fig. 9.9. MFM images of 2 × 12 µm2 magnetic bits after magnetization: (a) along the long bit axis, (b) along the short bit axis

Fig. 9.10. AFM topography (a) MFM (b–d) images of 8 × 48 µm2 magnetic bits after irradiation (b) and after magnetization along the long (c) and short (d) bit axes. Initial cobalt oxide thickness was 70 nm

Similar results were obtained for 8×48 µm2 bits (Fig. 9.10). After magnetization these bits also demonstrate pseudo single domain structure. But in this case nonuniformity of the magnetic contrast of poles can be seen after magnetization along the short bit axis, especially for the lowest bit in Fig. 9.10d, which is probably explained by the shot axis magnetized bit being less stable with respect to decay into multidomain state. Larger bits (with the size of 10 × 40 µm2 and initial cobalt oxide thickness 120 nm) were found to be multidomain after magnetization along both the long and short bit axes (Fig. 9.11). It is worth noting that in this case the

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Fig. 9.11. MFM images of 10 × 40 µm2 magnetic bits after magnetization: (a) along the long bit axis, (b) along the short bit axis. Initial cobalt oxide thickness was 120 nm

Fig. 9.12. MFM images of 10×40 µm2 magnetic bits over 50 nm cobalt soft magnetic underlayer after magnetization: (a) along the long bit axis, (b) along the short bit axis. Initial cobalt oxide thickness was 120 nm

poles are more pronounced after magnetization of bits along the short bit axis. MFM images for the same bits produced over cobalt soft magnetic underlayer significantly differ from those without the underlayer. Magnetization of these bits along the long axis reveals a magnetic contrast close to pseudo single domain (It is single-domain when magnetized) (Fig. 9.12a). Every bit has two clearly delineated poles. Besides, no domain-related structure has been observed. But after short axis magnetization pronounced multidomain magnetic contrast appeared (Fig. 9.12b). So, the presence of soft magnetic underlayer was shown to stabilize pseudo single domain magnetic structure along the long bit axis.

9.4 Conclusions It was experimentally shown that SRA technique makes it possible to produce pseudo-single domain magnetic bits and arrays with a wide range of coercivity values. There is an obvious tendency of bit coercivity growth with increase of

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shape anisotropy factor and of distance between the bits. Soft magnetic underlayer (of ≈50 nm thickness) allows to receive a number of potential advantages when compared with single-layered magnetic patterned media: 100-fold intensity amplification of the signal emitted from bits at magnetization reversal; keeping stable pseudo-single-domain condition for bit magnetized along its large size; increase of the registration distance; reduction of the level of magnetic device requirements, including power supply. Amplitude of the signal emitted by sandwich structures at magnetization reversal along the longitudinal direction exceeds the amplitude of the signal from magnetization reversal across a longitudinal bit direction by one order of magnitude. The amplitude of the emitted signal from two-layer magnetic patterned media grows with the increase of amplitude of an external magnetic field without saturation. The amplitude of the emitted signal of two-layer magnetic patterned media at magnetization reversal grows with the increase of external magnetic field frequency. The emitted signal duration drops both with the increase of external field amplitude or frequency. In any case, this is equivalent to the increase in the magnetization reversal rate (or external field alteration). It was shown that it is possible to produce a multibit “smart label” on the basis of the time sequence of emitted signals at magnetization reversal of different 2D periodic bit arrays at the same soft magnetic underlayer due to the different dynamic coercivity for every array. Acknowledgment This work was supported in part by the Russian Foundation for Basic Research (RFBR) within the framework of the Grant 06–080–08046–ofi and by Russian Federal Agency on Science and Innovations (FASI) through the Contract 02.513.11.3192.

References 1. C. Kittel, Phys. Rev. 70, 965–971 (1946) 2. J. Nogues, I.K. Schuller, J. Magn. Magn. Mat. 192, 203–232 (1999) 3. B.A. Gurovich, D.I. Dolgy, E.A. Kuleshova, E.P. Velikhov, E.D. Ol’shansky, E.D. Domantovsky, J. Magn. Magn. Mat. 44, 95–105 (2001) [Usp. Fiz. Nauk. 171, 105–117 (2001)] 4. B. Gurovich, E. Kuleshova, E. Meilikhov, K. Maslakov, J. Magn. Magn. Mat. 272–276, 1629–1630 (2004) 5. B. Gurovich, E. Kuleshova, D. Dolgy, K. Prikhodko, A. Domantovsky, K. Maslakov, E. Meilikhov, in Nanostructured Magnetic Materials and their Applications, ed. by B. Aktas, L. Tagirov, F. Mikailov, (Kluwer Academic Publishers, Dordecht, 2002), pp. 13–22

10 Structural and Magnetic Properties and Preparation Techniques of Nanosized M-type Hexaferrite Powders T. Koutzarova, S. Kolev, C. Ghelev, K. Grigorov, and I. Nedkov Institute of Electronics, Bulgarian Academy of Sciences, 72 Tzarigradsko Chaussee buld., 1784 Sofia, Bulgaria, [email protected], [email protected], [email protected], [email protected], [email protected] Summary. In recent years, the scientific efforts of a large number of research teams have been concentrating on developing, exploring and applying nanosized magnetic ferroxides. In this review, we consider the fundamental structural and magnetic characteristics of nanosized particles of barium hexaferrite. We discuss in some detail the most common techniques for preparation of nanosized ferroxide powders. Finally, we present original results on applying a promising chemical technique, namely, the single microemulsion technique, for the synthesis of barium hexaferrite powders consisting of homogeneous in shape and size particles.

10.1 Introduction In the past decades, magnetic nanoparticles have been the focus of intense research activities not only because of their unusual behaviour compared to the bulk materials but also for their wide applications in the practical world. The scientific and technological importance of magnetic nanostructures has three main reasons [1]: • There is an overwhelming variety of structures with interesting physical properties, ranging from naturally occurring nanomagnets and comparatively easy-to-produce bulk nanocomposites to demanding artificial nanostructures, • The involvement of nanoscale effects in the explanation and improvement of the properties of advanced magnetic materials, and • Nanomagnetism has opened the door for completely new technologies. Hard magnetic hexagonal ferrites have been extensively used as permanent magnets [2–4], magnetic recording media [5], magnetic tapes and floppy disks [6], magneto-optic materials, microelectromechanical systems [7] and

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microwave filters and devices [8–11] and in recent years, as materials with potential bio-medical applications [12]. Their great attraction is mainly due to the abundance of the raw materials and low production costs. The development of radar electronics and wireless technologies requires planar and low loss magnetic microwave devices (isolators, filters, phase shifters, and circulators, etc.) [13–17], which can be realized by the integration of a ferrite material with semiconductor platforms. Barium hexaferrite with M-type hexagonal crystalline structure (BaFe12 O19 or BaM) has attracted a great deal of attention for microwave device applications because of its bulk properties, namely, high permeability, low conductive losses, and, particularly, large uniaxial anisotropy (HA = 17 kOe) with the easy magnetization direction along the c-axis [18]. These materials exhibit high magneto-crystalline anisotropy, high Curie temperature, high coercivity and relatively high saturation magnetization, as well as excellent chemical stability and corrosion resistivity required for many applications [19,20]. The interest in these nanosized particles lies in our ability to affect their physical properties through manipulation of size, composition and aspect ratio to produce changes in the overall physical properties [20]. The properties of the nanoparticles are of interest for the investigation of nanowires, dot arrays, thin films and bulk composites [1]. It is also well known that the magnetic properties strongly depend on the particles’ microstructure [21, 22].

10.2 Crystalline Structure The hexaferrites form a group of complex oxides in the system AO–Fe2 O3 – MeO, where A is a large divalent cation, i.e. Ba, Sr, Ca, and Me are a small divalent cations, i.e. Mn, Fe, Co, Ni, Cu, Zn. They can be classified on the basis of chemical composition by varying the A–Me combination and, respectively, the crystal structure. Thus, they are subdivided into five fundamental, simplest structural types: M, W, Y, X, U and Z [23–25] . Figure 10.1 shows the known hexaferrite types, while the most common types are summarized in Table 10.1 [26]. We will now consider in detail the structure of the M-type hexaferrites. Barium hexaferrite (BaFe12 O19 ) is the M-type hexaferrite family’s best known compound. It has the crystal structure of the mineral magneto-plumbite. The crystallographic unit cell corresponds to the space group P63 /mmc and contains two molecules of the chemical composition BaFe12 O19 [27]. The dimensions of the unit cell are a = b = 5.88 ˚ A and c = 23.20 ˚ A [26]. The basic structure of the unit cell is built up by ten layers of oxygen ions that are formed by a close packing of cubic or hexagonal stacked layers alternately along the [001] direction. One O2− ion is replaced by barium, which has a similar ionic radius in every fifth layer. The crystal structure can be divided into several blocks. The S-block (Fe6 O2+ 8 ) contains two oxygen layers forming a spinel structure, where the R-block (MFe6 O2+ 11 ) is a three layer-block containing the

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Fe 2O 3

M Fe 2O 3 Ba 2Fe 8O 14

MeO U

X W

Z Y

BaO

BaFe 2O 4

S = Me2Fe4O8

Fig. 10.1. Phase diagram of AO–Fe2 O3 –MeO system

Table 10.1. The most well-known hexaferrite types with their compositions and a description of their crystal structures. Me stands for Mn, Fe, Co, Ni, Cu, Zn and ∗ denotes a rotation of 180◦ around the c-hexagonal axis Type

Nominal composition

Nominal composition

M W X Y Z U

BaFe12 O19 BaMe2 Fe16 O27 Ba2 Me2 Fe28 O46 Ba2 Me2 Fe12 O22 Ba3 Me2 Fe24 O41 Ba4 Me2 Fe36 O60

RSR*S* RS2 R*S*2 (RSR*S*2 )3 (TS)3 RSTSR*S*T*S* RSR*S*T*S*

layer with the barium ion. The whole structure can be symbolically described as RSR*S*, where the R*- and S*-blocks are built up by rotation of 180◦ around the hexagonal c-axis. Within the basic structure the Fe3+ ions occupy five different interstitial sites. Three sites, named 12k, 2a and 4f2 , have octahedral coordination, one site (4f1 ) has tetrahedral coordination and the 2b site has a fivefold coordination [28, 29]. The iron ions in the trigonal bipyramid are not in a symmetry plane but are displaced along the threefold/L3 axis and occupy randomly one of two equivalent position separated by 0.156 ˚ A from the symmetry plane of the bipyramid (Fig. 10.2) [30]. The 4f1 positions and the 2a octahedral positions are occupied by Fe3+ in the S block. Fe3+ in the R block occupies octahedral sites in the octahedra shared by common faces (4f2 ), in octahedra at the interface of adjacent blocks (12k), and trigonal bipyramidal sites (2b). The presence of magnetic Fe3+ cations in these positions is responsible for the barium hexaferrite’s magnetic properties and for its magneto-crystalline anisotropy (Table 10.2) (K1 = 3.3 × 105 J m−3 ) [31].

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2+

Ba O

2-

Fe (12k) Fe (4f 1 ) Fe (4f 2 ) Fe (2a) Fe (2b)

Fig. 10.2. M-type barium hexaferrite structure [32] Table 10.2. Crystallographic and magnetic properties for the various cation sublattices of M-type hexaferrite [28] Sublattice

Coordination

Block

12k 4f1 4f2 2a 2b

Octahedral Tetrahedral Octahedral Octahedral Fivefold coordination (trigonal bipyramidal)

R–S S R S R

Ions per formula unit

Spin direction

6 2 2 1 1

↑ ↓ ↓ ↑ ↑

10.3 Magnetic Properties The fundamental properties of magnetic materials are the saturation magnetization, the coercivity, the magneto-crystalline anisotropy constant and the Curie temperature. Intrinsic properties, such as the spontaneous magnetization Ms , the first uniaxial anisotropy constant K1 and the exchange stiffness A, refer to the atomic origin of magnetism. As a rule, the intrinsic properties are realized on length scales of at most a few inter atomic distances and tend to approach their bulk values on a length scale of less than 1 nm [1]. Extrinsic properties, such as the remanence Mr and the coercivity Hc , are non-equilibrium properties-related to magnetic hysteresis- and exhibit a pronounced real-structure dependence [1, 33]. The position of the magnetic ions and orientation of the spins in the crystal lattice were determined by Gorter by considering exchange interactions in barium hexaferrite [34]. The magnetic moments of the iron ions are arranged

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parallel to the hexagonal c-axis, but with opposite spin directions of the sublattices. The iron ions in the 12k, 2a and 2b sites have their spins aligned parallel to each other and the crystallographic c-axis, whereas those of 4f2 and 4f1 point in the opposite direction [35]. The resulting magnetization M at a temperature T of BaFe12 O19 per formula unit can be approximated by simple summation according to the formula M (T ) = 6σ12k (T ) − 2σ4f1 (T ) − 2σ4f12 (T ) + σ2a (T ) + σ2b (T )

(10.1)

where σi stands for the magnetic moment of the i-Fe3+ ion. Assuming a magnetic moment of 5 µB per Fe3+ ion at 0 K (µB is the Bohr magneton) the net magnetization is of 20 µB per formula unit of barium hexaferrite [28]. M¨ ossbauer spectroscopy is a basic technique for exploring the fine magnetic structure of magnetic materials. The M¨ossbauer spectrum of barium hexaferrite below the Curie point contains a superposition of five magnetically split subspectra associated with the five different iron sites [36]. Figure 10.3 presents a typical spectrum of nanosize barium hexaferrite; the data thus obtained is summarized in Table 10.3 [37]. The spectrum was fitted with five six-line sub-patterns. The five six-line sub-patterns were assigned to the 12k, 4f2 , 4f1 , 2a and 2b sites of the hexagonal crystal structure.

Fig. 10.3. M¨ ossbauer spectrum of BaFe12 O19 powder at room temperature [37] Table 10.3. Hyperfine parameters of BaFe12 O19 with average particle size 80 nm Hhf , hyperfine magnetic field; δF e , isomer shift; 2ε, quadrupole splitting; RA, relative area) [37]

12k 4f2 4f1 2a 2b

Hhf 107 (A m−1 )

δFe (mm s−1 )

2ε (mm s−1 )

RA (%)

3.28 4.10 3.89 4.03 3.19

0.35 0.38 0.26 0.34 0.27

0.42 0.20 0.24 0.06 2.23

50 16 19 10 5

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The energy of a magnetic material depends on the orientation of the magnetization with respect to the crystal axes, which is known as magnetic anisotropy. The magnetic anisotropy affects strongly the hysteresis loop shape and the values of the coercivity and the remanence. It is, therefore, of considerable importance for the practical applications of magnetic materials in, e.g., magnetic recording media. For example, permanent magnets need high magnetic anisotropy to keep the magnetization in a desired direction. The magneto-crystalline anisotropy is an intrinsic property of the ferrimagnetic materials which does not depend on the particles’ shape and size. For a single crystal, it is the energy necessary to re-orient the magnetic moment of the crystal from the easy magnetization axis of to the hard magnetization axis. The existence of these two axes of magnetization arises from the interaction between the spin magnetic moment and the crystal lattice (spin–orbital coupling). Generally, ferrites with hexagonal structure have two types of anisotropy, namely c-axis anisotropy and c-plane anisotropy, which are associated with the easy magnetization along the c-axis and in the c plane, respectively. In the barium ferrite family, only the Y-type barium ferrite has c-plane anisotropy, while the others have c-axis anisotropy [38, 39]. The BaFe12 O19 exhibits one of the highest values of the magneto-crystalline anisotropy – K1 = 3.3 × 105 J m−3 [31]. The energy EK per unit volume of the magneto-crystalline anisotropy for uniaxial anisotropy can be written as follows [40]: EK = K1 sin2 θ + K2 sin4 θ + · · · ,

(10.2)

where θ is the angle between the magnetization and the c-axis. K1 and K2 are the first and the second anisotropy constant. The direction along which EK has an absolute minimum is called the easy magnetization axis. The easy axis is determined by the sign and relative value of K1 , and when K1 > 0 it coincides with the hexagonal axis of symmetry (001), while for K1 < 0 it lies in the basic plane [41]. It is often convenient to express anisotropies in terms of anisotropy fields Ha . The law of approach to saturation is often used to estimate the anisotropy field Ha and the magneto-crystalline anisotropy K1 [42]. M = Ms (1 −

B A − 2 · · · ) + χp H, H H

(10.3)

where A is the inhomogeneity parameter, B is the anisotropy parameter and χp , the high-field differential susceptibility. The factor B is proportional to K 2 , where K denotes the effective anisotropy constant. In the spatial case of BaFe12 O19 , which possesses uniaxial crystalline anisotropy along the c-axis and K2  K1 , the factor B may be expressed as [43]: B=

4K12 Ha2 = . 15 15Ms2

(10.4)

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Coercivity is one of the most important characteristics of the hexaferrites in what concerns their potential applications. It describes the stability of the remanent state and gives rise to the classification of magnets into hard magnetic materials. A widely used phenomenological coercivity expression is [44] 2K1 − Deff Ms − H(T, η), (10.5) Hc = αK µ0 Ms where αK is the real-structure-dependent Kronmuller parameter [45, 46], Deff is a magneto-static interaction parameter and ∆H is a fluctuationfield correction due to thermal activation and η = dH/dt is a sweep rate [1, 33, 44, 47]. A fundamental characteristic of the coercivity is its dependence on the particles’ size, which explains the unceasing development of techniques for preparation of hexaferrite powders with high homogeneity and ever smaller particles’ size. Below a certain critical size (Dcrit ) the particle become monodomain; due to the hexaferrites’ magneto-crystalline anisotropy, this size is significantly higher than that of ferrites with a spinel structures. Figure 10.4 presents schematically Hc as a function of the size D of superparamagnetic (SPM), monodomain (MD) and polydomain (PD) particles [48]. The critical size for monodomain BaFe12 O19 particles can be calculated by the following expression [26]: Dcrit =

9σw 2πMs2

(10.6)

where σw = (2kB Tc |K1 |/a)1/2 is the energy density of the domain wall, |K 1 | is the magneto-crystalline anisotropy constant, Tc is the Curie temperature, Ms is the saturation magnetization, kB is Boltzmann constant and a is the crystal lattice constant. In particles with size D > Dcrit one observes a polydomain state. Below this critical size, the particles exhibit only one zone of spontaneous magnetization and absence of domain wall, i.e., they become

Fig. 10.4. Schematic presentation of the coercivity Hc dependence on the particles’ domain structure at room temperature

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monodomain. For barium hexaferrite, using the values of the single crystal parameters [49], one calculates the value Dcrit 460 nm. When a monodomain particle is very small, the anisotropy energy becomes comparable to or less than the thermal energy kB T ≥ Keff V ; the magnetic state of the particles is then defined as superparamagnetic [50]. Keff is the constant of effective anisotropy, which includes the magneto-crystalline anisotropy and the anisotropy of shape [51]. Thus, there exists a specific limiting size, Ds , for a particle to be monodomain under which the coercivity of a particle is zero. The initial rise in Hc as the particle’s size rises (above Ds ) (Fig. 10.4) can, therefore, be explained by the rise in the number of monodomain particles. As the particles’ size increases further, the coercivity reaches a maximum and then drops down again. This coercivity reduction for sizes exceeding Dcrit is related to the appearance of domain walls. The transition from a monodomain to a polydomain state results in a decrease of Hc , since the magnetization mechanism changes, namely, shifting the domain walls becomes energetically more advantageous than rotating the individual atomic spins. Another important parameter used to describe the properties of hexaferrites is the saturation magnetization Ms . The relation between the domain state and the saturation magnetization can be divided into four regions [52]: • For very small superparamagnetic particles (D < Ds ), the variation in Ms is due to thermal processes • For particles with sizes (Ds < D < Dtrans ) the variation in Ms is independent of the particles’ size and is related to rotational processes; • In larger particles (Ds < D < Dcrit ; processes of inhomogeneous magnetization arise and the coercivity decreases • As the particles’ size is increased further (D > Dcrit ) the monodomain particles become polydomain, where the variation of the saturation magnetization has to do with domain wall motion. Table 10.4 presents data on the magnetic characteristics of single crystal BaFe12 O19 [2, 26, 53]. The most important micromagnetic phenomenon is magnetic hysteresis, which refers to the dependence of the magnetization as a function of the external magnetic field. Hysteresis is a complex non-linear, non-equilibrium and non-local phenomenon, reflecting the existence of anisotropy-related metastable energy minima separated by field-dependent energy barriers. On an atomic scale, the barriers are easily overcome by thermal fluctuations, but on nanoscale or macroscopic length scales the excitations are usually Table 10.4. Magnetic characteristics of single crystal BaFe12 O19

BaFe12 O19

Tc (◦ C)

Hc (A m−1 )

Ms (emu g−1 )

K1 (J m−3 )

Ha (A m−1 )

450

5.3 × 105

72

3.3 × 105

1.35 × 108

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Fig. 10.5. Hysteresis loop of nanosized barium hexaferrite

Fig. 10.6. Determination of the saturation magnetization value

too weak to overcome the barriers. The determination of the local magnetization M (r), from which the hysteresis loop is obtained by averaging, is complicated by the influence of the magnet’s real structure (defect structure, morphology, metallurgical ‘microstructure’) [1]. Figure 10.5 presents a typical hysteresis loop of nanosized barium hexaferrite in high magnetic fields up to 2.5 × 106 A m−1 . In this case the magnetization curve does not reach saturation, so that data on the remanent magnetization (Mr ) and coercivity field (H c ) can only be obtained. The saturation magnetization value can be estimated by extrapolating the curve for H → ∝. Barium hexaferrite being a hard magnetic material, it reaches saturation at very high magnetic fields, where one can determine the saturation magnetization value (Fig. 10.6). Figure 10.7 illustrates the magnetization variation of barium hexaferrite with particles’ size of 80 nm with ellipsoidal shape as the magnetic field is raised

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Fig. 10.7. Magnetization variation of barium hexaferrite with particles’ size of 80 nm [37]

to 2.4 × 107 A m−1 [37]. As is seen, no saturation is reached; this behavior is related to the relative increase of the surface as the particle size is decreased and, respectively, to the increased role of the disordered magnetic structure of the surface layer. This effect should be the object of further studies, since one might thus be able to clarify the contribution of the various types of anisotropies on the magnetic properties of this type of particles.

10.4 Methods for Preparation It is well known that the electrical, optical and magnetic properties of materials vary widely with the particle sizes and shape and with the degree of crystallinity. At present, tremendous efforts have been made in improving their magnetic capabilities by using different synthesis methods [35]. At the same time, the research on their structural and physical properties has continued [4, 35, 54, 55]. Recent studies have shown that physical properties of nanoparticles are influenced significantly by the processing techniques [56]. Since crystallite size, particle size distribution and inter particle spacing have the greatest impact on magnetic properties, the ideal synthesis technique must provide superior control over these parameters [57]. A variety of techniques have been employed for the synthesis of nanoparticles with definite shapes and sizes [20,58–60] . A typical method of obtaining ferrimagnetic hexagonal oxide particles in general is the solid-state reaction. The conventional solid-state method for preparing BaFe12 O19 is to fire an appropriate mixture of α-Fe2 O3 and BaCO3 at very high temperatures (1,150–1,250◦C). The resulting powder is then ground to reduce the particles’ size. Although high-temperature firing assures the formation of the required ferrite phase, larger particles (>1 µm) are often obtained in this firing process. It has been shown that the theoretical intrinsic coercivities of ferrites can be approached only when the particle sizes are

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below 1 µm [61]. On the other hand, grinding may introduce impurities into the powder and cause strains in the crystal lattices, which has unfavorable effect on the magnetic properties [62]. To overcome these problems, variuos soft chemical methods have been developed in order to reduce the particle size and obtain highly homogeneous ultra fine single-domain particles of barium hexaferrite. Among the most popular techniques we should mention: the glass-ceramic method [63,64], chemical co-precipitation [65–68], hydrothermal processes [69–71], the ammonium nitrate melt method [72], sol–gel [73–77], pyrolisis of aerosol [78,79], the mechanochemical method [80,81], auto combustion [20,82,83]. In all these processes, precursors are used that have ultra-fine size and high surface area; thus conventional restrictions of phase equilibria and kinetics can be easily overcome, which leads to lowering of sintering and solid-state reaction temperatures and increased sintering rate [57]. These methods are widely known and commonly used in the synthesis of magnetic oxides. We will now consider some of them in more detail. In the sol–gel synthesis the term sol refers to a suspension or dispersion of discrete colloidal particles, while gel represents a colloidal or polymeric solid containing a fluid component, which has an internal network structure wherein both the solid and fluid components are highly dispersed. The cations first form a sol of either hydroxides or citrates or acetates. The discrete colloidal particles slowly coalesce together to form a rigid gel. Since the particle sizes are very fine, these gels can be calcined at much lower temperatures than the conventionally derived powders to obtain a homogeneous product. Atomic level mixing of constituents in the sol–gel process leads to the formation of single-phase products much more easily than by other process. The purity, microstructure and properties of the product can be controlled by the proper selection of starting precursors, solvent, pH, of sol, calcinations temperature and processing environment. The main problems in the hexaferrite preparation by the sol–gel technique are the gel formation and the deviation of measured and expected values of the specific saturation magnetization [84]. The citric acid precursor method originated from the Pechini method. Pechini developed this method in 1967 and applied for patent in the United States (Patent No. 3 330 697). In the precursor method, the metallic salts are dissolved in water to have the required metallic ions well mixed. The metallic ions are then chelated by a poly-acid (e.g., citric acid), and esterification of chelated cations is carried out by adding poly-alcohol (e.g., ethylene glycol) at appropriate temperatures. After dehydration, a solid ester precursor with well-mixed metallic ions can be obtained. The solid precursor is subjected to proper heat treatment to form the final ceramic particles. Lucchini et al. [85] showed that using pectic acid to chelate barium and iron ions in an aqueous solution of nitrates and heating in air at 700◦ C can produce crystalline barium ferrite with particle sizes less than 1 µm in diameter [62]. The hydrothermal process is used to synthesize pure, ultra-fine, stressfree barium hexaferrite powder with a narrow size distribution at relatively low temperature (200–300◦C). This synthesis uses different precupsors as

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Ba(NO3 )2 and Fe(NO3 )3 .9H2 O mixtures in the presence of NaOH/KOH/ NH4 OH, (C2 H5 )4 NOH [69]; FeOOH and Ba(OH)2 mixture; αFe2 O3 and Ba(OH)2 mixtures; FeCl3 and Ba(OH)2 mixtures. The low temperature combustion route is based on the gelling and subsequent combustion of an aqueous solution containing salts of the desired metals and some organic fuel, giving a voluminous and fluffy product with large surface are. This method has been proved to be a novel, extremely facile, time-saving and energy-efficient route for synthesis of ultra-fine powders [86]. Using this method, Huang et al. [86] synthesized barium hexaferrite powders based on the combustion of nitrate-citrate gels due to an exothermic redox reaction between nitrate and citrate ions. The particles have sizes between 80 and 120 nm and Ms = 59.36 emu g−1 and Hc = 4.4 × 105 A m−1 . In the aerosol process, a solution of the cations is passed trough an aerosol generator in the form of fine droplets, which are subsequently dried to form fine powders on passage through vacuum. The particles are than carried through a heated reactor tube in which the precursor compounds react to yield fine particulates, which are then collected on a filter. Monosized spherical particles can also be obtained by controlling the droplet size and contamination can be avoided to a large extent by this method; powders having various size distributions can also be synthesized. The chemical co-precipitation method is a cheap and easy choice for mass production [43]. In this process, the cations are generally precipitated from solutions, such as hydroxides or carbonates. Co-precipitation of multivalent cations in a multicomponent system is difficult because the precipitating agent (OH− , CO3 2− ) form insoluble species with cations, which can have approximately the same solubility product only under very narrow boundary condition of pH, temperature, dielectric constant of solvent. In the hydroxide process, the cations are precipitated from the solutions by using NaOH/KOH or NH4 OH as precipitating agent. The carbonates are precipitated from the metal salts solution by adding Na/K-carbonate or (NH4 )2 CO3 . Jacobao et al. [87] and Roos [88] used the coprecipitation method to prepare barium ferrite and showed that by heating the coprecipitates at relatively low temperatures (≤800◦C), submicron BaFe12 O19 particles can be obtained. W. Ng et al. [67] studied in detail the influence of the heat treatment temperature on barium hexaferrite’s magnetic properties. In general, this method does not allow one to control the size and size distribution of the particles [89]. In order to overcome these difficulties, the microemulsion method was proposed [90–93], which will be discussed in more detail later. Table 10.5 summarizes the magnetic parameters of barium hexaferrite produced by different soft-chemical techniques. In all cases listed in the table, the values of the magnetic parameters are lower than the theoretical ones calculated for single-crystal barium hexaferrite. This is most probably the result of the presence of magnetic and structural defects on the particles surface and, in some cases, due to the worse size homogeneity in the former samples. For particles with size of about 100 nm, the lower values of Ms and Hc are

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Table 10.5. Summarizes the magnetic parameters of barium hexaferrite produced by different soft-chemical techniques Synthesis method Low temperature combustion [86] Ion-exchange resin [94] Co-precipitation [95] Sol–gel [74] Ultrasonic spray pirolysis [96] Sol–gel [97] Co-precipitation [67] Sol–gel [98] Sol–gel [99] Ammonium nitrate melt [72] Ammonium nitrate melt [72] Co-precipitation [100] High-energy milling [101] Aerosol route [102] Self-propagation high temperature [103] Co-precipitation [104] Microemulsion [57] Microemulsion [105] Co-precipitation [106] Co-precipitation [105] Aerosol pyrolysis [79] Mechanical alloying [107] Microemulsion [108] Spark plasma sintering [109] Co-precipitation [110]

Temperature Average particle Ms (emu g−1 ) Hc (kA m−1 ) size (nm) (◦ C) 850

120

59.36

440.8

850

220

71

302.4

900

130 300

63.6 70 51

381.9 473.4 401.6

800 950 1,000 850

130 85 200

60.6 57 61.62 58.4 36.7

399 450 442.7 405.8 203.6

900

300

45

243.1

800

220

43 60.9

358 381.1

1,000

108

50.8 49

290 190.9

800 925 925 1,000 900

50–100 100 100 50–100 100 50–70 100

67.8 61.2 60.48 67 <50 42.6 68

436.7 429.4 342.9 413.8 <238.7 469.5 477.4 413.8 111.4

800 800

100

58 65.52

830

500 (a small fraction of 10 nm)

52

2.38

also due to the fact that the particles have not achieved the perfect hexagonal shape typical for barium hexaferrite. The low saturation magnetization values can be explained by the fact that the particles are smaller than the critical diameter for barium hexaferrite and should possess non-compensated magnetic moments on the surface.

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10.5 Microemulsion Technique We will now consider the use of the aqueous cores of water-in-oil microemulsions as reactors for the synthesis of barium hexaferrite nanoparticles. One of the reasons to explore this technique more closely is that the precipitation reactions in microemulsions offer a novel and versatile technique for synthesis of a wide variety of magnetic nanoparticles with the ability to control precisely the size and shape of the particles formed, as well as a unique method to control the kinetics of particle formation and growth by varying the physicochemical characteristics of the microemulsion [57]. The microemulsion system consists of an oil phase, a surfactant phase and an aqueous phase. The reverse micelles are water-in-oil droplets stabilized by a surfactant. The high homogeneity of the nanosized precipitate particles produced is due to fact that each of the aqueous droplets acts as a nanosized reactor for nanoparticles formation [111, 112]. One of the advantages of this technique is the preparation of very uniform particles (<10% variability) [113]. A microemulsion system exhibits a dynamic structure of nanosized aqueous droplets, which are in constant formation, breakdown, and coalescence. This result in a continuous exchange of solvent. If a nanoparticle is nucleated within the water droplet, its growth is limited by the size constraint of the water droplet [114]. The size of these aqueous droplets is in the range 5–100 nm depending on the water/surfactant ratio: Rw =

3Vaq [H2 O] , σ[S]

(10.7)

where Rw is the water droplet radius, Vaq is the volume of the water molecule, σ is the area per polar head of surfactant, [S] is the concentration of surfactant [115]. An increase in the ratio increases the size of the water pool inside the inverse micelle, and therefore allows bigger particles to form [116]. Thus, the surfactants not only reduce the surface energy, but also control the growth and shape of the particles and act against aggregation. The surfactants are of three types – non-ionic, anionic and cationic. Various surfactants have been employed in the synthesis of hexaferrites, with cetyltrimethylammonium bromide CH3 –(CH2 )15 –N(CH3 )3 Br (CTAB), a cationic surfactant, being most commonly used. Usually, the synthesis of precursors for oxide particles formation is carried out by way of mixing two microemulsion systems with identical compositions but different aqueous-phase types – the one containing metal ions, the other, a precipitating agent (NH4 OH, NaOH, KOH, etc.). The co-precipitation reactions are expected to take place when aqueous droplets containing the desirable reactants collide with one another, coalesce and break apart. The collision process depends upon the diffusion of the aqueous droplets in the continuous media, i.e. oil, while the exchange process depends on the attractive interactions between the surfactant tail and the rigidity of the interface, as the aqueous droplets approach closely each other [57, 117]. One of the

10 Structural and Magnetic Properties Microemulsion system one

NaOH

197

Microemulsion system two

Aqueous phase Ba(NO3)2 FeCl3

-

OH

-

OH

Precipitate

Single microemulsion

Double microemulsion

Fig. 10.8. Schematic diagram of the microemulsion techniques

many microemulsion systems employed to produce magnetic oxides consists of cetyltrimethylammonium bromide (CTAB) as a cationic surfactant; nbutanol as a co-surfactant; n-hexanol as a continuous oil phase and an aqueous phase [37,118]. An advantage of using CTAB as a surfactant is the possibility of free passage of OH− ions through the water droplet walls in both directions. This fact allows one to use a single microemulsion system to produce nanosized particles when the precipitating agent is NH4 OH, NaOH, or KOH. The single microemulsion method is characterized by the presence of only one microemulsion system whose aqueous phase contains metal ions only. One of the advantages of the single microemulsion technique is that it is much less expensive than the classical double microemulsion method. Figure 10.8 presents schematically the two microemulsion techniques. The XRD spectrum of the synthesized BaFe12 O19 powder is presented in Fig. 10.9. It shows the characteristic peaks corresponding to the barium hexaferrite structure. Scanning electron microscopy is widely used to determine the grain size and morphology of powders. Figure 10.10 shows the morphology of the BaFe12 O19 powder obtained by single microemulsion. It exhibits a narrow grain-size distribution, with the average particle size being 130 nm. The particles have an irregular shape between spherical and hexagonal. The process of forming the platelet shape typical for BaFe12 O19 hexahedral has not been completed due to the small particle size. The critical diameter for

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Fig. 10.9. X-ray diffraction pattern of barium hexaferrite powder obtained by single microemulsion technique

Fig. 10.10. SEM image of barium hexaferrite powder sample with average particle size of 130 nm prepared via single microemulsion

Fig. 10.11. Hysteresis loop of barium hexaferrite powder sample with average particle size of 130 nm prepared via single microemulsion

single-domain barium hexaferrite particles is about 460 nm [64], so that the particles are single domain. The hysteresis loop of the powder sample at room temperature and a maximum applied field of 2.3 × 106 A m−1 is shown in Fig. 10.11. The satu-

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Fig. 10.12. Magnetization curve barium hexaferrite powder sample with average particle size of 130 nm prepared via single microemulsion

ration magnetization value (Ms ) was obtained from the magnetization curve in high magnetic fields up to 1 × 107 A m−1 , which is presented in Fig. 10.12. The results of the magnetic measurements, namely, saturation magnetization Ms of 62 emu g−1 and coercivity field (Hc ) of 3.9 × 105 A m−1 at room temperature are comparable to the best results for powders prepared via the double microemulsion method [105, 119]. Such high Ms and Hc values may be attributed to the high phase purity, the well-defined crystallinity and the homogeneity with respect to the BaFe12 O19 particles size. It was thus demonstrated that the single microemulsion method, which is less expensive than the classical double microemulsion method, may be used to prepare powders of monodomain barium hexaferrite nanoparticles with high size-homogeneity and good magnetic properties in view of possible applications. Acknowledgement T. Koutzarova was supported by NATO Reintegration Grant (EAP.RIG.981472). The work was supported in part by research agreements between the Bulgarian Academy of Sciences and Bulgarian Scientific Fund under grant HT-1/01.

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49. 50. 51. 52.

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11 Nanocrystallization and Surface Magnetic Structure of Ferromagnetic Ribbons and Microwires A. Chizhik1 , A. Zhukov1, V. Zhukova1, C. Garcia1 , J.M. Blanco2 , J.J. del Val1 , L. Fernandez1 , N. Iturriza1 , and J. Gonzalez1 1

2

Departamento de F´ısica de Materiales, Facultad de Qu´ımica, University of the Basque Country, San Sebastian, Spain, [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected] Departamento F´ısica Aplicada I, EUPDS, UPV/EHU, 20018, San Sebastian, Spain, [email protected]

Summary. The nano-crystals nucleation and growth mechanism in the bulk and at the surface of alloys under the application of a magnetic field or a constant tensile stress during thermal treatment are studied. Clear differences have been found in the nano-crystals growth mechanisms and nano-crystallization of the bulk and surface of the ribbons depending on the treatment in ribbons. The use of a tensile stress is found to activate the nucleation rate, while the use of applied field during thermal annealing produces a completely different effect which does not activate the grain growth in the bulk, but produces ordered arrangements of nano-grains at the surface of the ribbon. In micro-wires the studies show an unusual temperature dependence of the coercive field, consequence of a coexistence of blocked and unblocked particles, and the typical decreasing behaviour of the remanence increasing temperature.

11.1 Introduction The soft magnetic properties exhibited by nano-crystalline alloys have attracted a great interest in scientific and industrial community [1]. The nanograins are produced by devitrification of the amorphous alloy after thermal treatment. The small ultrafine crystallites, surrounded by a remaining ferromagnetic amorphous matrix, being the reduced grain size compared with the ferromagnetic correlation length the key for the extremely low effective magnetocrystalline anisotropy. In the present work, the grain nucleation and growth mechanism in the bulk and at the in the presence of a magnetic field and a tensile stress during thermal treatment are studied. Atomic Force Microscopy (AFM) has been used to study the crystallization state in the surface of the

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ribbons. A structural analysis has been performed by X-ray diffraction (XRD), providing information about the crystallization state in the bulk of the ribbons and the size of the grown nano-crystals. The magneto-optical Kerr effect [2] measurements have been performed to investigate the magnetization reversal in the surface of the ribbons.

11.2 Co-Rich Ribbons Different kinds of magnetic anisotropies have been induced during the nanocrystallization process of Co-rich amorphous ferromagnetic ribbons (Co77 Si13.5 B9.5 )90 Fe7 Nb3 using diverse procedures like the application of a constant stress or an axial magnetic field during the annealing process. Magnetization measurements have evidenced the anisotropy of the treated samples. The main goal of this study has been the structural and microstructural analysis of the treated ribbons. 11.2.1 Experimental Details The structural analysis of the ribbons was carried out by XRD and AFM measurements in order to respectively analyse the bulk and the surface of the ribbons. XRD patterns have been normalized in the high q-range where the intensity cannot vary between the different samples owing to the very short structural distances responsible for the scattering in this q region. In the case of crystallized samples, the evaluation of the crystalline phase has been performed by the subtraction of the amorphous halo of the as-cast sample (Ia) to the measurement performed on the crystallized ribbon (I). The resulted intensity pattern can be used to estimate the crystallinity ratio, c, defined as following:  2 q (I − Ia)dq  χ= , q = 4π sin θ/λ, (11.1) q 2 I dq where (I − Ia) denotes the resulting intensity pattern that corresponds to the crystalline fraction of the sample, q is the scattering vector and 2q is the scattering angle. The crystallite size, D, has been evaluated from the Scherrer’s formula. Surface hysteresis loops were measured by magneto-optical Kerr effect. An axial magnetic field was applied to the sample, while a polarized light was focused on the ribbon surface. With the applied magnetic field and the laser light aligned, the rotation of the polarization plane of the reflected light was proportional to the magnetization, which is in the plane of the surface and parallel to the plane of the incident light. Measurements were carried out on the ribbons keeping magnetic field and light positions, but rotating the sample in the plane. Amorphous ribbons were prepared by melt-spinning technique with dimensions: 3.1 mm wide and 20 mm thick. The amorphous

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character of the as-cast ribbons was confirmed by XRD and Transmission Electron Microscopy analysis. It was found that thermally treated ribbons show the formation of bcc-(Fe, Co[Si]) and fcc-Co phases [3]. The thermal treatments on the as-cast ribbons were carried out in air by Joule-heating. The samples were submitted to the same annealing current (32.5 A mm−2 , 120 s), applying simultaneously an external tensile stress (50 MPa), or an axial magnetic field (750 A m−1 ) during annealing and cooling down process to room temperature. The sample treated with stress will be identified as sample SA, and the one treated with magnetic field will be called FA. 11.2.2 Results and Discussion Figure 11.1 shows an example of XRD patterns corresponding to different treatments and compared to the as-cast ones. SA sample presents an incipient but remarkable crystalline peak corresponding to the crystallization of the bcc-(Fe, Co[Si]) phase that is overlapped with the amorphous halo of the still present amorphous matrix. A relative crystalline volume of 6% and grain size of 9 nm are obtained. This grain size results to be relatively small compared to the typical grain size (≈15 nm) found in Finemet alloys after conventional thermal annealing, and it is thought to be linked with the application of a tensile stress during the annealing process due to modifications in the grain nucleation, which could avoid the growth of the grains to larger sizes. The XRD measurements corresponding to FA sample show the absence of crystallization. The scattering excess of this sample is a sign of structural relaxation or other possible mechanism like an incipient nucleation. The AFM analysis of the surface of SA ribbons shows a strong crystallization without signs of amorphous matrix (see Fig. 11.2). The nano-grains are visible with a typical diameter of 9 nm. Such strong surface crystallization contrasts with the low 6% of relative crystalline volume obtained by XRD in this sample, showing that the kinetic of grains nucleation and growth at the surface of the ribbons is completely different to the process that takes

Intensity (a.u.)

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Fig. 11.2. AFM micro-graphs for samples treated with stress (a) and field (b)

place in the bulk. These different crystallization states could be explained considering the higher energy of the surface that leads to a stronger instability of the surface morphology, resulting in an earlier crystallization than in the bulk. The “quasi-amorphous” state detected in the bulk of the FA ribbon contrasts with the AFM micro-graphs measured on the surface of the FA sample, which display a crystallized surface with typical grain sizes around 20 nm (see Fig. 11.2). Furthermore, they present the formation of large agglomerates similar to inlands with typical sizes of approximately 125 nm that are locally arranged with an apparent ordering suggesting an organization of these inlands at the surface of the ribbon during the thermal treatment under the effect of the applied magnetic field. The longitudinal Kerr-effect surface hysteresis loops show differences depending on the orientation of the ribbon respect to the applied magnetic field, which indicates anisotropy. For the SA sample, strong differences are found in the surface hysteresis loops measured in geometries where the magnetic field was applied parallel (orientation of 0◦ ), or perpendicular to the axis of the ribbon (orientation of 90◦ ). For an orientation of 0◦ , the surface hysteresis loops behave similarly to the bulk hysteresis loop of SA showing a clear inclined shape. It was not possible to saturate in this geometry the surface hysteresis loop with the application of the maximum magnetic field available in the experimental set up. For an orientation of 90◦ the strong inclination commented before is not detected, and it is possible to saturate the surface hysteresis loop (see Fig. 11.3a). This fact proves the presence of an induced easy axis transverse to the axis of the ribbon. For the directions different of 90◦ or 0◦ the surface hysteresis loops are generally not saturated, and show an inclination that varies depending of the proximity to the orientation of 0◦ . The jumps found in the hysteresis loop at 90◦ orientation, reveal the possible presence of two different, soft and hard, ferromagnetic phases in the ribbon surface. The Kerr-effect surface hysteresis loops measured on sample FA show a behaviour that is not expected, because an induced easy axis at an orientation of 30◦ respect to the ribbon axis is detected (Fig. 11.3b). Hysteresis loops, measured at angles around this direction (70◦ ), show an abrupt change of behaviour compared to the measurements at 30◦ . This result contrasts with the result obtained in previous bulk measurements, where the easy axis

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Fig. 11.3. Kerr-effect hysteresis loops: (a) sample SA and (b) sample FA

is oriented along the axis of the ribbon at 0◦ . The existence of an easy axis that is 30◦ disoriented relatively to the ribbon axis could be assumed considering the effects of the magnetic field induced by the annealing current that could interact with the applied magnetic field during annealing treatment, turning aside its initial orientation. Consequently, an easy axis at 30◦ could be induced. The thermal annealing in the presence of tensile stress or magnetic field is a well-known technique to induce anisotropy in Finemet ribbons. In the present work, clear differences have been found in the nano-grain growth mechanisms and nano-crystallization of the bulk and surface of the ribbons depending on the performed treatment. The use of a tensile stress is found to activate the nucleation rate, while grains with a reduced grain size compared to standard treated Finemet ribbons are found. On the other hand, the use of applied field during thermal annealing seems to produce a completely different effect that does not activate the nano-grain growth in the bulk, but it produces ordered arrangements of nano-grain agglomerates at the surface of the ribbon. The Kerr-effect surface hysteresis loops evidence the presence of anisotropies at the surface of the ribbons. In the case of SA sample, it is transverse to the ribbon axis, as it is expected. The measurement carried out on FA sample shows an easy axis 30◦ oriented respect to the expected induced easy axis.

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Fig. 11.4. XRD patterns without treatment (as-cast) and treated with field (FA) and stress (SA)

11.3 Ni-Rich Ribbons 11.3.1 XRD and AFM Structural Results Usually, the X-ray reflexions of the amorphous ribbon become higher and narrower with longer time and temperature treatments, indicating the beginning of the crystallization process. Differences in the diffraction patterns of the treated samples are detected depending on the kind of annealing process that has been performed. Figure 11.4 shows an example of XRD patterns corresponding to Ni5 Fe68.5 Nb3 Si13.5 B9 Cu1 (NIF) samples with different treatments and compared to the as-cast ones. SA samples present an incipient but remarkable crystalline peak corresponding to the crystallization of the bcc-(Fe, Ni[Si]) that is overlapped with the amorphous halo of the still present amorphous matrix. A crystallinity percentage between 6% and 20% with 11 nm grain size are obtained for NIF–SA ribbons. Similar results of small grain sizes (≈8 nm) have been also obtained by other authors using flash-stress annealing on Finemet ribbons [4]. Such modification in the grain nucleation of sample SA is detected in the comparison between the XRD patterns of samples SA and CA. While the XRD pattern of sample SA displays an incipient crystallization peak, the pattern of samples CA do not show any sign of crystallization being similar to the amorphous halo of the as-cast ribbons. Therefore, it seems that the use of stress annealing enhance the nano-crystallization of Finemet ribbons. 11.3.2 Magnetic Results Bulk magnetization measurements in the NIF ribbons, by means of classic fluxmetric method in quasi-static conditions have evidenced the anisotropy of the treated samples. As an example, Fig. 11.5 shows the evolution of the coercive field with the treatment time, giving a clear proof of the induction of the anisotropy by the stress. Surface hysteresis loops were measured by

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Fig. 11.5. Variation of the coercive field (bulk) with the treatment time for Ni-rich Finemet (NIF) samples treated with current (CA) and stress (SA)

Fig. 11.6. Surface hysteresis loops for Ni-rich Finemet (NIF) samples treated with stress (SA)

magneto-optical Kerr effect. The measured surface hysteresis loops display clear differences depending on the orientation of the ribbon respect to the applied magnetic field, which indicates anisotropy. The difference is found in the surface hysteresis loops measured in different geometries (see Fig. 11.6). The interesting remarks are the jumps found in the hysteresis loop at 90◦ orientation, related to the presence of two ferromagnetic phases in the ribbon surface. The surface hysteresis loop measurements reveal a magnetic hardening of the ribbon surface respect to the bulk, which is evidenced in the coercive values detected in these measurements that are approximately two orders of magnitude higher than the obtained ones in the bulk magnetization measurements. The application of a tensile stress or a magnetic field during thermal annealing is a well-known technique to induce anisotropy in Ni-rich Finemet ribbons. Surface hysteresis loops evidence the presence of anisotropies at the surface of the ribbons. A strong hardening of the ribbon surface is detected compared to the bulk, which could be explained considering the high density of inhomogeneities present at the surface that difficult the movement of the magnetic domains.

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11.4 Microwires with Novel Composition Cu70 (Co70 Fe5 Si10 B15 )30 Studies of thin glass-coated micro-wires produced by Taylor–Ulitovski technique gained special attention within the last few years owing to unusual magnetic properties and low dimensionality of these materials. The Taylor– Ulitovski method allows the fabrication of few kilometer long glass-coated metallic micro-wires with typical radius of the metallic nucleus ranging from 1 and 15 µm and the thickness of the insulating glass coating between 1 and 10 µm. High enough quenching rate can be achieved while producing such micro-wires, allowing fabrication of amorphous, nano-crystalline, microcrystalline or granular samples [5]. On the other hand, granular alloys formed by immiscible elements (Co, Fe, Ni)–(Cu, Pt, Au, Ag) attracted recently special attention mostly because of the giant magneto-resistance (GMR) observed in these alloys [6, 7]. These materials consist of ferromagnetic nano-particles, initially of Fe or Co embedded in a non-magnetic matrix (typically Cu, Ag, Au, or Pt). At zero applied field, when magnetic moments of the particles are not aligned (e.g., oriented randomly), the resistivity of the material is high, and when an external magnetic field aligns these moments, the resistivity decreases, as in the previously discovered GMR multilayered materials [8]. The field-dependent resistivity in granular and multilayer materials is related to a spin-dependent scattering of conduction electrons within the magnetic particles as well as the interfaces between magnetic and non-magnetic regions (however, it is assumed that the interfacial effect contributes to a dominant extent to GMR [9–11]). Because of the complex structure of the granular material, the relationship between microstructure and galvanomagnetic properties is still not fully understood. Initially, main attention has been paid to studies of magnetically soft micro-wires with amorphous and/or nano-crystalline structure. Recently, few attempts have been made to obtain granular or nano-crystalline micro-wires with unusual properties [5, 12, 13]. Thus, semi-hard magnetic properties and GMR effect have been reported in Fe–Ni–Cu and Co–Cu micro-wires with coercivities of about 700 Oe and GMR up to 18% [12, 13]. Other new compositions of granular sample where part of the alloy composition is commonly used on amorphous soft magnetic materials and the other part is based on Cu has been recently reported [14]. Unusual multiphase structure consisting of Cu, small amount of a-Fe and amorphous phase has been observed in micro-wires with CuFeSiBC composition, exhibiting non-regular hysteresis loop. In this work, we report structural, magnetic and magnetotransport properties of a new family of glass coated micro-wire of nominal composition Cu70 (Co70 F e5 Si10 B15 )30 where initially Co70 F e5 Si10 B15 alloy was mixture with Cu with the proportions indicated.

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11.4.1 Experimental Details Thin glass coated micro-wires with total diameter of 28.2 µm and metallic nucleus diameter of 15.2 µm of nominal composition Cu70 (Co70 Fe5 Si10 B15 )30 were produced by the Taylor–Ulitovski method containing 30% of alloy composition commonly used amorphous soft magnetic material and 70% Cu [12]. The master alloy of the composition was prepared by arc melting of the pure elements in Ar atmosphere. Subsequently, when the metallic alloy and the Pyrex glass coating was molten, it was drawn and rolled onto a rotating cylinder and quenched to room temperature. The sample obtained was in the form of a tiny metallic wire with the dimensions above mentioned. Structural analysis by X-ray diffraction (XRD) has been carried out in a powder diffractometer from Philips, provided with an automatic divergence slit and a graphite monochromator, with CuKα radiation. The measurements were carried out using the step scanning technique between 5◦ and 90◦ (2θ) in steps of 0.05◦ (2θ) with accumulation times of 5 s at each point. Magnetic properties (coercive field and magnetization) were deduced from the hysteresis loop of the micro-wire using a vibrating sample magnetometer (VSM) in the range of temperature 5–300 K. The hysteresis loops M(H) of these samples were measured in the temperature range from 10 to 300 K and maximum field of 90 kOe. Magnetoresistance response (MR) was evaluated using a Physical Properties of Measurements System (PPMS) device operating from 5 to 300 K. The magnetoresistance response is defined as the change of the electrical resistance ρ under an applied magnetic field, Happ , as: MR(%) = ((ρ(H) − ρ(0))/ρ(0)) × 100,

(11.2)

where ρ(H) and ρ(0) is the resistivity at 70 and 0 kOe applied magnetic field. 11.4.2 Results and Discussion Figure 11.7 shows the XRD pattern for the as-prepared Cu70 (Co70 Fe5 Si10 B15 )30 micro-wire. We can observe the following: (a) The three major peaks with atomic spacings of 2.05, 1.78 and 1.29 ˚ A corresponding to the (111), (200) and (220) reflections of ffc Cu lattice. (b) The first peak at around 44◦ (2θ) can also have a contribution of the hcp Co phase (in (002) reflection). Nevertheless, this contribution must be lower that the one from Cu phase owing to the lower Co content in the core. (c) A contribution of the amorphous halo arising from both, the pyrex coating and the non-crystallized part of the metallic core. Structural information for the micro-wires is extracted once the crystalline peak of each patterns is identified. The grain size of the crystals formed in is derived from Scherrer’s equation: D = Kλ/ cos θm

(11.3)

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Fig. 11.7. WAXS patterns for Cu70 (Co70 Fe5 Si10 B15 )30 as cast glass coated microwire with total diameter of 28.2 µm and metallic nucleus diameter of 15.2 µm

Fig. 11.8. Hysteresis loop for the Cu70 (Co70 Fe5 Si10 B15 )30 as cast micro-wires with total diameter of 28.2 µm and metallic nucleus diameter of 15.2 µm at room temperature

where 2θm is the scattering angle corresponding to maximum of the peak and  is its half height width. The parameter K is assumed to be around 0.9 or closer to 1 in certain cases. Calculations show the existence of nano-crystalline grains of around 40 nm in grain size. The M (H) curve (hysteresis loop) of the glass coating micro-wires at room temperature shown in Fig. 11.8 exhibits a like-super-paramagnetic shape. Similar behaviour was observed at different measuring temperatures. From the hysteresis loop at different temperatures, the thermal behaviour of the coercivity and remanence of the glass coating micro-wire were deduced (Fig. 11.9). The slight increase of both coercivity and remanence are inside the error bars.

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Fig. 11.9. Coercive field Hc vs. temperature and remanent magnetization Mr vs. temperature for the sample investigated

0.0014

ZFC-FC

M (emu)

0.0012

50 Oe

0.0010 0.0008 0.0006 0.0004 0.0002 0

50

100 150 200 250 300 350 400

T (k)

Fig. 11.10. Zero field cooled and field cooled curves for the Cu70 (Co70 Fe5 Si10 B15 )30 as-cast measured at HDC = 100 Oe up to 300 K

The monotonic decrease observed in the remanence curve follows the expected behaviour [15]. On the other hand, temperature dependence of magnetization can be explained in terms of the coexistence of both blocked and superparamagnetic particles [16,17]. The initial increase of the coercive field, which is in the margin of error, could be attributed to the thermal fluctuations, it continue until the large particles start to unblock and HC(T ) decrease again. The distribution of particle sizes is in the origin of the observed decrease of the coercive field. Figure 11.10 presents the temperature dependence of magnetization of field cooling (FC) curve (50 Oe applied magnetic field) and zero-field cooling (ZFC) curve. Magnetization measured under 50 and 200 Oe applied magnetic field (FC) field and without magnetic field (ZFC) shows a

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R (%)

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300 k 100 k 5k -60000 -30000

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60000

Fig. 11.11. Magnetoresistance of the as-cast micro-wires taken at 300, 100 and 5 K from −7 to 7 T. Magnetoresistance reaches significant values around 1.6% in magnetic field H = 7 T at 5 K

similar behaviour. The blocking temperature was found around 25 K which correspond with a critic diameter around 36 ˚ A, which correspond with the value obtained by XRD. Figure 11.11 shows the magnetoresistance response of the as-cast micro-wire measured at 5, 100 and 300 K. Measurements have been carried out with the current parallel to the applied field and how it was showed this orientation shows negative magnetoresistance. MR observed in the micro-wires originates from the spin-dependent scattering of electrons by the Co, Fe, Si and B nano-particles imbedded in the crystalline paramagnetic Cu matrix. The magnetoresistance is about 1.6% at 5 K in a field of 7 T, being the resistivity as insensible to the applied magnetic field at high temperatures (100 and 300 K), which could be attributed to the small particles content of super-paramagnetic behaviour. Structure and magnetic properties of Cu70 (Co70 Fe5 Si10 B15 )30 micro-wires have been studied. The structure consists of crystalline Cu, and mixture of ferromagnetic phases. The as-prepared Cu70 (Co70 Fe5 Si10 B15 )30 micro-wires present coercivity of about 85 Oe. Temperature dependence of coercivity is measured. Significant increase of coercivity till 135 Oe is observed at about 50 K. Magnetization measured under applied magnetic field (FC) and without magnetic field (ZFC) exhibit significant difference. The inhomogeneity of as-prepared samples consisting of crystalline paramagnetic Cu matrix with ferromagnetic phases gives rise also to the significant difference in ZFC–FC curves and allows us to consider the possibility to observe the magnetoresistance, MR in this sample. Finally, indeed a magnetoresistance is also observed at 5 K. Acknowledgement This work was supported by MEyC under project PCI2005-A7-0230.

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References 1. J. Gonzalez, O. Chubykalo, J.M. Gonzalez, Enciclopedia of Nanoscience and Nanotechnology 10, 1–26 (2004) 2. A. Chizhik, A. Zhukov, J. Gonzalez and J.M. Blanco, J. Appl. Phys. 97, 073912 (2005) 3. N. Iturriza, L. Fernandez, J.J. del Val, C. Garcia, J.M. Blanco, J. Gonzalez, G. Vara, A.R. Pierna, J. Appl. Phys. 99, 08F104 (2006) 4. F. Alves, J. Magn. Magn. Mat. 226–230, 1490 (2001) 5. A. Zhukov, J. Gonz´ alez, J.M. Blanco, M. V´ azquez and V. Larin, J. Mater. Res., 15, 2107 (2000) 6. J.Q. Xiao, J.S. Jiang and C.L. Chien, Phys. Rev. Let., 68, 3749 (1992) 7. A.E. Berkowitz, J.R. Mitchell, M.J. Carey, A.P. Young, S. Zhang, F.E. Spada, F.T. Parker, A. Hutten and G. Thomas, Phys. Rev. Lett., 68, 3745 (1992). 8. M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petron, P. Etienne, G. Creuzer, A. Friederich and J. Chazelas, Phys. Rev. Lett., 61, 2472 (1988) 9. C.L. Chien, J.Q. Xiao and J. Samuel Jiang, J. Appl. Phys., 73, 5309 (1993) 10. S. Zhang and P.M. Levy, J. Appl. Phys., 73, 5315 (1993) 11. T.M. Rabelau, M.F. Toney, R.F. Marks, S.S.P. Parkin, R.F.C. Farrow and G.R. Marp, Phys. Rev. B, 48, 16810 (1993) 12. A. Zhukov, J. Gonzalez and V. Zhukova, J. Magn. Magn. Mater., 294, 165 (2005) 13. A. Zhukov, D. Martn y Marero, F. Batallan, J.J. del Val, V. Zhukova, J.L. Martinez, C. Luna, J. Gonzlez S. Kaloshkin and M. V´ azquez, Phys. Stat. Sol.(c) 1, 12, 3717 (2004) 14. A. Zhukov, C. Garcia, V. Zhukova, V. Larin, J. Gonzalez, J.J. del Val, M. Knobel and J.M. Blanco, J. Non-Cryst. Solids, 353, 922 (2007) 15. E.F. Ferrari, F.C.S. Silva and M. Knobel, Phys. Rev. B, 56, 6086 (1997) 16. W.C. Nunes, W.S.D. Folly, J.P. Sinnecker and M.A. Novak, Phys. Rev. B, 70, 014419 (2004) 17. C. Garcia, A. Zhukov, V. Zhukova, V. Larin, J. Gonzalez, J.J. del Val and M. Knobel, J. Magn. Magn. Mat., 316, e71 (2007)

12 On Structural and Magnetic Properties of Fe73.5−xSi13.5B9Cu1 Nb3 Mnx Metal Alloys R. Brzozowski1, M. Wasiak2 , P. Sov´ ak3 , and M. Moneta1 1

2

3

Uniwersytet L ´ odzki, Katedra Fizyki Ciala Stalego, Pomorska 149, PL 90-236 L ´ od´z, Poland, [email protected], marek [email protected] Uniwersytet L ´ odzki, Katedra Chemii Fizycznej, Pomorska 155, PL 90-236 L ´ od´z, Poland, [email protected] ´ arika v Ko´siciach, Prirod. Fakulta, Park Angelinum 9, 04-101 Univerzita P.J. Saf´ Ko´sice, Slovakia, [email protected]

Summary. Structural and magnetic properties of Fe73.5−x Si13.5 B9 Cu1 Nb3 Mnx=1,3−15 amorphous alloys were studied by means of M¨ossbauer spectroscopy, differential scanning calorimetry, X-ray diffraction pattern and energy dispersive X-rays, as-quenched and after annealing. The magnetic ordering of alloys substantially decreases with increase of Mn content x. The initially amorphous phase becomes nanocrystalline structure as the annealing temperature Ta and time ta increase. The alloys were shown to pass several different phase transformations, when heated.

12.1 Introduction FeSiB finemets are known as soft magnetic materials of high permeability (∼105 at 1 kHz), large magnetization (∼1.2 T), low magnetostriction (∼2 × 10−6 ), low coercivity, and power loss. The as quenched (a–q) amorphous metal alloys, like Fe73.5 Si13.5 B9 Cu1 Nb3 , consist primarily of amorphousmagnetic phase which, after appropriate thermal or mechanical treatment, are transformed first into amorphous-paramagnetic phase and next into the phase where iron-silicates, α-Fe(Si), and boride (Fe3 B and Fe2 B) magnetic nanocrystals are embedded in a paramagnetic amorphous residual matrix [1–3]. The temperature dependence of structure and magnetic properties of the finemet, x = 0, were analysed in many previous papers [4–6]. XRD patterns, MS spectroscopy, and DSC scans allow us to determine phase transitions and structure composition. After annealing MS spectra show multiple narrow lines, which represent structurally different crystallographic sites assigned to the ironsilicate grains, superimposed on a broadened feature ascribed to the residual amorphous phase [7–10].

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It was shown that Cu enhances nucleation of nanocrystals during annealing. Nb reduces atomic mobility, retards grain grows, and decreases the Curie temperature, from ∼350 to ∼200◦ C as its content increases from 1 to 8%. Also, on the other hand recent results suggest that Mn addition to finemet leads to an improved exchange coupling between grains and to an enhancement of the magnetic softness [11]. The resultant stronger giant magnetoimpedance (GMI) predestinates the alloy material for GMI sensors. On the other hand, selected experimental data related to the Mn-doped finemet (x = 1–5) published recently [12, 13], show decrease of Curie temperature, the activation energy and suggest magnetic hardening of the nanocrystalline state with increase of Mn concentration.

12.2 Experiment The amorphous ribbons of Fe73.5−x Si13.5 B9 Cu1 Nb3 Mnx=1,3−15 , of approximately 2 mm × 30 µm, were quenched from basic alloys by a melt-spinning technique, commonly used for producing metal glasses. The Cu-wheel surface speed over 25 m s−1 and the cooling rate of 10−7 K s−1 , assure amorphous alloy. For lower speeds we get nano and microcrystalline structure determined by cooling conditions [14]. The MS experiments were performed in transmission geometry and the constant acceleration mode with 57 Co:Rh (14.4 keV) source of 100 mCi activity. The measurements were complemented by X-ray diffraction (XRD) patterns measured with Philips PW 1830 with Cu Kα generator (Eγ = 8.05 keV, λ = 0.15405 nm). The differential scanning calorimetry (DSC) with Setaram TG DSC-111 with the heating rate up to 10 K per min and maximum temperature 900◦C in N2 atmosphere was used after calibration with CrO4 K2 standard. Also we used the electron diffraction spectroscopy (EDX) with Link300 ISIS Oxford Instruments and Zeiss optical microscopy with up to 1,000× magnification. Thermal stability of the sub-surface region and the elemental composition was also observed with selected ion-beam techniques; particle induced X-ray emission (PIXE) analysis, Rutherford backscattering spectrometry (RBS), and elastic recoil detection (ERD) analysis. As a matter of fact, the present set of Fe73.5−x Si13.5 B9 Cu1 Nb3 Mnx alloys is treated a possible variation of the basic finemet, x = 0.

12.3 Results and Discussion 12.3.1 As-Quenched Fe73.5−xSi13.5B9 Cu1 Nb3 Mnx Optical microscopy inspection of both the (a–q) and the annealed (a) samples shows irregular and porous interface suitable rather for bulk analysis of the sample, than for surface analysis. Topological objects of the 10 µm size can be found on the inner surface and chemical inhomogeneity on the outer surface,

12 On Structural and Magnetic Properties

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Fig. 12.1. Example of OM pictures of finemet alloys with Mn. (a-left) outer and b-right) inner side of the a–q foil. Scale of 15 µm is given by distance between dashes

Table 12.1. Fe73.5−x Si13.5 B9 Cu1 Nb3 Mnx EDX average composition [at.%], at T0 , a–q x=

0

1

3

5

9

11

13

15

Fe Mn Si B Cu Nb

73.50 0.00 13.50 9.00 1.00 3.00

72.50 0.90 13.12 9.01 0.98 3.48

70.50 3.09 13.55 8.15 1.21 3.51

68.50 5.00 13.45 8.84 0.86 3.36

64.50 9.05 14.35 7.54 1.22 3.34

62.50 11.12 14.35 7.54 1.05 3.63

60.50 10.97 15.13 9.15 0.97 3.29

58.50 12.89 15.33 9.18 0.99 3.21

as can be seen in Fig. 12.1. However, the averaged surface composition of the alloys, determined at room temperature T0 by EDX and scaled to Fe nominal atomic content, are reasonably well correlated with nominal composition of the glasses, except for samples with high Mn content, as shown in Table 12.1. The result was also confirmed by analysis of bulk material by PIXE and RBS with 0.1–1 MeV protons. The XRD distributions of initially a–q alloys are typical for the amorphous structure up to TxFeSi ≈ 450◦ C – the Fe(Si) crystallisation temperature. XRD spectra for all x = 1–15 samples display broad maxima in the range of 2θ ∼ 45◦ , which is characteristic for Fe amorphous alloys. The TxFeSi was shown to depend on Mn content x and decreases from ca. 480◦ C at x = 0 to ca. 420◦C at x = 7 and then slightly increases to ca. 450◦ C at x = 15, as was measured by DSC and depicted in Figs. 12.2 and 12.3. To define phase transformations regions, the DSC scans, at the 10 K per min heating rate, were performed with sample masses of ∼1 mg. The complete DSC scans for the specified parameters are shown in Fig. 12.2. They allow us resolve in the heat flow, 5 peak positions Ti−v corresponding to enthalpy change and exothermic phase transformations, presented in Fig. 12.3 and in Fig 12.4 for all x. Black symbols at x = 0 mark the reference crystallisation

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vi

v ii

i

ii

iv

1

DSC heat flow [mW]

2.0 3 5

0.0 7 15

−2.0

9 −4.0

11 13

a-q at T0,10 K/min

−6.0

Fe73,5-xSi13.5B9Nb3Cu1Mnx=

100

200

300

400

500

600

700

o

Temperature[ C]

Fig. 12.2. DSC scans at 10 K per min of ∼1 µg a–q samples of Fe73.5−x Si13.5 B9 Cu1 Nb3 Mnx with identification of the peak positions i−v 700

f Tcryst a

Fe73,5-xSi13.5B9Nb3Cu1Mnx

d e

10 K/min, a−q

650

Ti-FeB-cryst.

DSC peaks Ti−v[o C]

600 550 500

e

d

f b Tcryst a

Tii-FeSi-cryst. Tiii-FeSi

c 450

Tiv-FeSi-cryst.

400 Tv 350

TCurie b a,c

300

d 0 1

3

5

7

9

11

13

15

Mn substitution of Fe − x

Fig. 12.3. DSC i−v peak positions as functions of Fe replacement by Mn − x. The reference data: a - [3], b - [7], c - [5], d - [13], e - [12], f - [6]

12 On Structural and Magnetic Properties

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DSC

Ti−v

peaks enthalpy [J/g]

Fe73,5x−Si13.5 B9 Nb3 Cu 1Mnx 10 K/min, a−q T0at

100

80

[c] [b]

60 Tii-Fe(Si) Ti-Fe(B) 40

Tv Tiv.

20 [b]

Tiii

0 0 1

3

5

7

9

11

13

15

Mn substitution of Fe − x

Fig. 12.4. Enthalpy change corresponding to i−v DSC peaks as function of Fe replacement by Mn – x. The reference data symbols: [b] - [19], [c] - [1]

temperature Tcr and the Curie temperature TCu from [3, 5–7] and the set for x = 1, 3, 5 is taken from [13]. We can precisely separate Tv - not clearly identified characteristic temperature for the amorphous phase, which appears also in a subsequent heating of the same sample. It slightly increases as a function of Mn content x, and for small x it is close to the Curie temperature of the finemet. TxCu is shown in the figure by black diamonds. We can also separate Ti , which disappears in subsequent DSC heating, so it is interpreted as a final crystallisation temperature for iron-silicates and also for boride. It decreases with replacement of Fe by Mn, contrary to the results from [15, 16], shown in the Fig. 12.2 by black , which slightly increase up to x = 5. Decrease of the crystallization temperatures Ti with increasing Mn doping can be interpreted in terms of Mn diffusion into the grains, which should reduce of their magnetisation and the Curie temperature TCu for the phase. The additional temperatures – Tii , Tiii , and Tiv are interpreted as region of prolonged crystallisation of α-Fe3 Si phase. This is consistent with the reference data shown in the figure for x = 0 and with XRD monitoring of crystallisation process as a function of T ; it can really start at T ∼ 420◦ C, [17]. The MS distributions of initially a-q alloys, shown in Fig. 12.5, are typical for the amorphous structure up to TxFeSi ≈ 450◦ C – the crystallisation temperature. The numerical analysis of the spectra, performed with RECOIL code [18] by fitting sextets with distribution of hyperfine field Hhf , presented in Fig. 12.6 and associated with distribution of not equivalent Fe-sites

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relative transmission [au]

15 13 11 9 7 5 3 1

a-q ata T0

-8

-6

-4

-2

0

2

4

6

8

velocity [mm/s]

Fig. 12.5. MS transmission spectra for a–q Fe73.5−x Si13.5 B9 Cu1 Nb3 Mnx=1,3−15 alloys at T0

shows in Fig. 12.7 that the a-q samples analysed at T0 became less and less ferromagnetic as x increases. The average hyperfine field Hhf and the average centre shift CS are shown in Fig. 12.7 as functions of Mn content x, by diamonds  and circles ◦, respectively. The Hhf tends to 0 at x ≈ 17, whereas CS only slightly decreases. From MS spectra, we can determine and draw in Figs. 12.6 and 12.7 a strong-field component - HH (), which becomes negligible at larger x and a weak-field contribution - HL (∇), which becomes dominant at large x, but both fields rapidly decrease with addition of Mn. The weaker fields may be attributed to more diluted, amorphous Fe solution, whereas the stronger field component may probably be related to sites surrounded by higher Fe concentration, a precursor of the nanocrystalline fractions precipitated in annealing. Shape of the Hhf distributions suggests magnetic inhomogeneity of the structurally amorphous initial phase. 12.3.2 Annealed Fe73.5−xSi13.5B9 Cu1 Nb3 Mnx In the heated basic finemet, two fractions namely amorphous and crystalline were identified, and their characteristics were determined [3, 5, 6]. The sample remains ferromagnetic and amorphous up to the Curie temperatures

12 On Structural and Magnetic Properties 0.20

225

Fe73,5-xSi13.5B9Nb3Cu1Mnx= a-q atT0

15

DistributionhfP(H)

0.15

0.10 13

11

0.05 9

7 5 3 1

0.00 0

5

10

15 Hhf [T]

20

25

30

Fig. 12.6. Distributions of hyperfine field Hhf , a–q, T0 , in dependence on Mn content x = 1, 3−15 30 Fe73,5-xSi13.5 B9 Nb3Cu1Mnx= a−q atT0,foil transmission

Hhf[T]/CS*130 [mm/s]

25

20

b c a

HH



15 a

10 HL 5

0 0 1

3

5

7

9

11

13

15

17

Mn substitution of Fe − x

Fig. 12.7. Hyperfine fields for a–q foils measured at T0 : high HH ( ), low HL (∇), average Hhf (line and diamonds) and centre shift CS (line and circles), in dependence on Mn content x. The reference data for x = 0 are shown by a - [2], b - [5], c - [6]

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TCam ≈ 328◦ C. It maintains paramagnetism up to crystallisation temperature of Tcryst ≈ 480◦ C (XRD) [6] or 513◦ C (VSM) [3] or 483◦C (MS), 527◦C (DSC), 532◦ C (VSM), when magnetisation appears again as a consequence of crystallisation to (bcc) α-Fe3 Si crystal-ferromagnetic phase. The alloy with x = 0 annealed at Ta = 550◦C displays dominant XRD peak at 2θ = 45.0◦ corresponding to reflection from (110) plane of (bcc) α-Fe(Si) solid solution accompanied by reflections from (200), (211), and (220) planes at 65.5◦ , 82.8◦ , and 102.3◦ , respectively. Reflections evidencing the DO3 superlattice structure [4] from (111) and (200) planes appear at 2θ ∼ 28.6◦ and ∼32◦ , respectively. Annealing at Ta = 700◦ C gives additional 8 reflections [1] from tetragonal Fe3B and bcc Fe23B6 structures, which start to appear at ∼600◦ C. With XRD we can determine change of the crystalline grain size d with annealing conditions: the temperature Ta and the time ta of annealing. The average size d can be determined with the Scherrer relation [16]: d = λ/σ cos θ, where λ is the X-ray wavelength, 2θ is the angle of the dominant Bragg maximum, and σ is the half-width of the peak. The grain size starts from d ≈ 10 nm at ca. 480◦ C and remains constant up to ca. 630◦C when the size increasers rapidly [6]. Also the volume ratio of crystalline to amorphous fraction increases from 0% at 480◦ C up to 70% at 630◦ C when Fe(B) phases are formed. Kinetics of the primary crystallisation of the selected alloys, represented by the first heat flow maximum (Tii ) in the DSC scan, was studied. This crystallization corresponds to formation of ferromagnetic phase and soft magnetic properties of the alloys. To determine the activation energy Exii the a–q amorphous alloys, with x = 1, 3–15, were DSC-scanned with heating rates: r = 1 and 10 K per min. It enables calculation of the Exii for the first Fe(Si) crystallization at peak temperature Txii by using the Kissinger model [13] 2 ) = const - Exii /RTxii , where R = 8.31 J molK−1 . with the equation: ln(r/Txii For x = 0, 1, 3, 5 from fitting we can get Exii = 3.75, 3.6, 3.3, 3.1 eV, respectively. This should be related to the reference crystallization activation energy per molecule for finemet: E0ii = 4.0 eV [1] and E0i = 4.5 eV from [19], with relaxation E0ii = 1.5 eV and crystallization activation energy E0ii = 4.5 eV from [20]. DSC can also be used to estimate crystalline volume fraction γ in the annealed alloys [15], by using the crystallization enthalpy change Haq for a–q alloy and the crystallization enthalpy change Ha for alloy annealed at Ta within ta , with the equation: γ = 1 − Ha /Haq . For primary crystallisation of a–q finemet, x = 0, it was accepted that T0ii ≈ 550◦ C and Haq ≈ 73 J g−1 . In this work the a-q alloys with x = 3, 9, 15 were initially DSC-scanned with 10 K per min heating rate, whereas another samples were annealed at Ta = 550 and 700◦C for ta = 1 h and then DSC-scanned. The crystalline fraction γ for annealing at 550◦ C was ∼55% for x = 3 and raised to ∼75% and to ∼80% for x = 9 and x = 15, respectively. This suggests easier crystallisation of the alloys with Mn doping and larger crystalline grains and thus enhanced exchange

12 On Structural and Magnetic Properties

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Fe73,5-xSi13.5 B9 Nb3 Cu1 Mnx=

relative transmission [au]

5

3

1

a-q at T0 Ta=700oC,ta=1h

-8

-6

-4

-2

0

2

4

6

8

velocity [mm/s]

Fig. 12.8. MS velocity distributions measured at T0 for a–q alloys (thin lines) and after annealing at Ta = 700◦ C for ta = 1h (thick lines) for x = 1, 3, 5 Fe73,5-x Si13.5 B Nb3 Cu1Mnx= 9

relative transmission [au]

15

13

11

9

a-q at T0 Ta=700oC,ta =1h

-8

-6

-4

-2

0

2

4

6

8

velocity [mm/s]

Fig. 12.9. MS velocity distributions measured at T0 for a–q alloys (thin lines) and after annealing at Ta = 700◦ C for ta = 1 h (thick lines) for x = 9, 11, 13, 15

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coupling between grains leading in turn to the softer magnetic properties of the sample. For Ta = 700◦C, the amorphous fraction was ∼18% almost independent of x, what is less than the MS spectroscopy result. For the M¨ ossbauer analysis, annealing of the samples was performed at Ta = 550◦C and Ta = 700◦C for the time ta = 1 h. The first one is the precipitation temperature for silicates while, the latter one is placed above the highest phase transformation temperature for the alloys, shown in Fig. 12.3, and is sufficient for creation of boride. Figures 12.8 and 12.9 show transmission MS spectra for the samples annealed at 700◦ C and measured at T0 : Figure 12.8 covers the x = 1–5 and Fig. 12.9 x = 9–15 range. The thick lines correspond to the annealed samples, whereas thin lines correspond to a–q samples. The MS spectra from annealed samples show multiple narrow lines, which represent structurally different crystallographic sites assigned to nanocrystalline structure of Fe(Si) and Fe(B) grains superimposed on a broadened background ascribed to the residual amorphous phase. The hyperfine fields, much higher than in the amorphous case, were identified from the correspondence to x = 0 case [8]. The MS spectrum parameters for a given x were taken as a starting parameters for x + 2 in the RECOIL fitting procedure. In the x = 0 case five sextets were used to reproduce discrete lines in the spectra, corresponding to 35 8 30

7

hyperfine fieldhf [TH]

6 2

25

3

5 20

4 1

15



10



o

Ta=700 C,ta=1h

5

Fe73,5-x Si13.5 B9 Nb3Cu1Mnx 0 1

3

5

7

9

11

13

15

Mn substitution of Fe - x

Fig. 12.10. Averaged hyperfine fields < Ham > of the amorphous matrix (filled circle – annealed alloys and open diamond – a–q glasses) and site hyperfine fields Hhf for 8 crystalline components (Fe(B): thin lines 1–3 and Fe(Si): dots 4–8), in case of alloys annealed at Ta = 700◦ C for ta = 1 h, as functions of Mn concentration x. The 3-points regression curves are drawn to guide the eye

12 On Structural and Magnetic Properties

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o

30

Ta=700 C,ta = 1h

site components [%]

Fe73,5-x Si13.5B9Nb3Cu1Mnx

am . 20

4 7

10

3

8 6 1

5

2

Tc

0 0 1

3

5

7

9

Mn substitution of

11

13

15

Fe − x

Fig. 12.11. Site contribution for the amorphous matrix (open diamond) and for 8 crystalline components (Fe(B): thin lines 1–3 and Fe(Si): dots 4–8) in case of alloys annealed at Ta = 700◦ C for ta = 1 h, as functions of Mn concentration x. The 3-points regression curves are drawn to guide the eye

bcc α-Fe(Si) structure sites where each Fe atom can be surrounded by 8–4 Fe atoms as the nearest neighbors, and additionally three sextets corresponding to Fe(B) sites. The average Ham-a determined for the amorphous matrix in the annealed alloys, closely corresponds to the average fields Ham-a–q encountered in a–q alloys, as shown in Fig. 12.10. It means that the average structural and magnetic properties of the amorphous phase changes in the same way with Mn content x. Also the hyperfine fields assigned to different Fe-sites are roughly independent of x, Fig. 12.10. It means that the crystallites formed at low x appear also at higher x. At this Ta they consume ca. 80% of the basic material leaving ca. 20% for amorphous phase and this proportion is roughly independent on x. What seems to substantially change with x is the relative contribution of the crystalline fractions, shown in Fig. 12.11.

12.4 Conclusions The alloys of Fe73.5−x Si13.5 B9 Cu1 Nb3 Mnx=1,3−15 were analysed by means of MS, DSC, XRD, EDX as-quenched and after annealing. Also, selected ion-beam methods were used for complementary analysis, but results were

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not presented in detail in this report. Partial replacement of Fe by Mn changes much hyperfine properties of the amorphous matrix and substantially alters the phase transformation temperatures and relative contributions of nanocrystalline fractions. Acknowledgement Much thanks to Mrs. Zosia Fijarczyk for help in preparing the manuscript.

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13 FeCoZr–Al2 O3 Granular Nanocomposite Films with Tailored Structural, Electric, Magnetotransport and Magnetic Properties J.A. Fedotova NC PHEP Bearusian State University 153 M. Bogdanovich, Minsk 220040, Belarus, [email protected] Summary. Unique physical properties of granular soft ferromagnetic metal–dielectric matrix nanocomposite materials (SF MMCs) make them very attractive in designing of magnetoelectronic devices. The most essential characteristics of GNMs are discussed in the chapter within the frame of percolation models, that establish correlation between microstructure and carrier transport mechanisms in MMCs. The chapter summarizes experimental results on tailoring of structure, magnetic, transport, and magnetotransport properties of (FeCoZr)x (Al2 O3 )1−x (17 at.% < x < 65 at.%) nanocomposite films by variation of metallic fraction and oxygen incorporation. Applied research techniques are alternation grads- and SQUIDmagnetometry, atomic force microscopy/magnetic force microscopy (AFM/MFM), M¨ ossbauer spectroscopy (MS), DC conductivity and magnetoresistance. Variations of the magnetic characteristics (coercivity fields, magnetization, MFM magnetic contrast) and carrier transport mechanisms due to incorporation of oxygen into FeCoZr–Al2 O3 are discussed with regard to the formation of complex semiconductive FeCo-oxide interlayers and/or shells possessing different types of magnetic ordering.

13.1 Introduction Composite materials containing at least one phase with constituents of less than 100 nm (usually from 1 to some tenth of nanometers) in size can be termed as nanocomposites. Nanosize of phases in composites gives rise to very large area of active surface which promotes many of physical and chemical processes resulting in enhanced or specific mechanical [1–3], electric [4, 5], magnetic [2], high-temperature [6], optical [7–9] properties, as well as excellent catalytic properties [11]. However, unusual properties of nanocomposites originate not only from the growth of nanoparticles surface but also from the “switching in” of new physical phenomena which are characteristic for nanoworld – for example,

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quantum dimensional effects. Just these effects determine specific discretization of energy spectrum of electrons and, as a consequence, unique electronic properties of nanogranular systems allowing to develop superminiaturized magnetic memory cells [12], sensors and detectors of magnetic as well as electric fields [13], etc. 13.1.1 Granular Nanocomposites for Electronics: Reasons of Interest As is known, one nanoparticle, containing a few (denumerable) number of atoms of one sort, is not a stable system at usual temperatures. From this point of view, creation of devices or any other electronic facilities on the basis of single nanoparticles or their noninteracting assemblies represents a great challenge. So, now scientists actively retrieve such configurations and structural states of nanoparticles which would make them more stable but with conservation of that advantages which inhere to nanoworld. One such perspective direction is related to the development of MMCs that consist of metallic nanoparticles (filler) randomly or orderly distributed in matrix of other, noninteracting with nanoparticles, material: metal, semiconductor or insulator. Such compositions allow to stabilize nanoparticles at moderate temperatures. MMCs demonstrate a number of physical properties which strongly differ from usual (single-phase) nanograin materials. Depending on the concentration xMe of metallic filler, the electrical properties of such nanocomposites can be varied between those of the matrix and those of the filler. In the typical binary composite materials a critical concentration, so-called percolation threshold xC [14], is reached when a continuous current-conducting cluster of filler particles is formed through the sample. At present, great interest for magnetoelectronics, including spintronics, is shown in MMCs, that contain FeCo-based soft ferromagnetic nanoparticles, embedded into dielectric matrix [15–18]. Such granular MMCs possess unique physical properties, including giant magnetoresistance [19, 21, 24], a considerable magnetorefractive effect [25], good magnetooptical characteristics [26], high absorbance of electromagnetic radiation in the RF and microwave ranges [27], a broad range of electrical resistivity variation [22, 27–29], etc. Their low coercivity along with high saturation magnetization, high inplane anisotropy field, high resistivity and giant positive (GMR) or tunnel negative (TMR) magnetoresistive effects are beneficial in designing of different magnetoelectronic applications such as memory media with super-high density of magnetic recording, RF and SHF shielding, etc. Note also that most of the electrical, magnetic and other properties of such MMCs have the highest sensitivity to the influence of external subjections just at concentration of alloy phase xMe close to percolation threshold xC [20, 23, 30, 31]. Thus, granular MMCs form a new class of artificial materials with tuneable electronic properties controlled at the nanoscale and composed of close-packed

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granules varying in size and embedded into dielectric matrix. But we should note that the intense interest to such materials is motivated not only with the important technical promise but by the appeal of dealing with the experimentally accessible model system that is governed by tuneable cooperative effects of inhomogeneity, disorder, electron correlations and quantum phenomena [10, 33, 34]. 13.1.2 Preparation and Structure of Granular MMCs Manufacturing of MMCs is not the trivial technological task assuming that metallic particles should have dimensions of the order of a few (3–10) nanometers and matrix should separate them by “strata” with 0, 5–2 nm thickness. The commercial applications of MMCs as an engineering material rely on the successful consolidation of these materials into bulk-sized components while preserving their nanostructure. Traditional consolidation techniques include such as cold pressing and sintering at high temperatures, hot pressing and hot isostatic pressing of metal and dielectric nanoparticles. But these techniques have strong limitations of not being able to retain the nanoscale grain (granular) size due to their excessive growth during processing. Among the techniques successfully applied for preparation of the abovementioned granular MMCs are vapour deposition [35], RF sputtering [36], reducing from metal oxides [37, 38], hydrothermal precipitation [39], sol–gel processing of Fe–SiO2 [40, 41], Ni–SiO2 [42], Fe–Al2 O3 [43], ball-milling of FeCo–SiO2 [38], Fe–SiO2 [44, 45], wet chemistry method [46], etc. Particular attention is paid for thermal evaporation and sputtering techniques of granular MMCs synthesis, especially for electronic and magnetoelectronic applications [47]. During those processes metallic and insulating components are simultaneously evaporated or sputtered onto a substrate. For example, to manufacture a metal–insulator composite material by ion-beam sputtering technique [48] (see Fig. 13.1), it is necessary to atomize targets 3 and 4 of correspondent compositions and to direct atoms towards any substrate 8 where composite film is formed. Evaporation of the target material is carried out using ionized argon gas beam directed to the sputtered targets. Two Ar sources 5 serve for sputtering of metallic and dielectric constituents of the material. Still one source 6 is used for preliminary cleaning (etching) of substrate. The holder 2 of substrates 8, which can be revolved with the rate up to 2 rev min−1 , is situated by the perimeter of vacuum chamber 1. When depositing insulating materials, positive potential, which appears on dielectric surface, is neutralized by a special compensator 6 – the source of intensive electronic emission. Granular MMCs are manufactured using, as a rule, composite targets which present a fused metallic plates of necessary composition with several strips of dielectric substances (silica, alumina, etc.) fixed at its surface with the variable distances between them from one to another edge of the target (see Fig. 13.2). By changing the number of dielectric strips and the distances

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Fig. 13.1. Installation for ion-beam sputtering: 1 vacuum chamber; 2 rotating support holder; 3,4 water-cooled targets; 5 ion-beam sputtering Ar source; 6 ionized Ar etching source; 7 compensator; 8 substrates [48]

Fig. 13.2. Schematic diagram of composite target from metallic alloy base (1) and dielectric strips (2) [48]

between them, it is possible to vary the volume ratio of metallic and dielectric components, thus modifying both composition and properties (for example, resistivity) along the substrate. In this case of the sputtering of the compound target by Ar ions, atoms of both components with strongly different surface energies will be deposited simultaneously on the substrate and, diffusing by substrate and being noninteracting to each other, they will be bunched around atoms of the same sort in the “self-organized” manner. In doing so, constituents with the greatest value of the surface energy (metal or alloy) will tend to form nearly spherical granules in matrix of another (dielectric) constituent. Figures 13.3 and 13.4 show examples of microstructures of some MMCs sintered by ion-beam sputtering to a water-cooled substrate [48]. Dark regions in micrographs carried

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Fig. 13.3. Scanning electron microscopy images and electron diffraction images of (Co41 Fe39 B20 )x (SiOn)100−x granular nanocomposites: (a) (Co41 Fe39 B20 )30 (SiOn)70 ; (b) (Co41 Fe39 B20 )44 (SiOn)56 ; (c) (Co41 Fe39 B20 )52 (SiOn)48 [49]

Fig. 13.4. Scanning electron microscopy images and electron diffraction images of (Co86 Ta12 Nb2 )x (SiOn)100−x granular nanocomposites: (a) (Co86 Ta12 Nb2 )34 (SiOn)66 ; (b) (Co86 Ta12 Nb2 )42 (SiOn)58 ; (c) (Co86 Ta12 Nb2 )50 (SiOn)50 [49]

out in scanning electron microscopy correspond to metallic granules, since they contain heavier elements, which are less permeable for electrons. As is seen, size of the granules varies within the range 2–10 nm and grows with the increase of metallic phase content. Micrographs show that the metallic granules are surrounded by brighter regions, which correspond to insulating layers (SiO2 ) [49]. 13.1.3 Percolation in Granular Nanocomposites Inasmuch as granular MMCs are mixtures of conducting and insulating or poorly conducting nanoparticles, they are widely discussed in the scientific literature within the frame of so-called percolation models [50]. These models study properties of granular systems depending on the ratio between xMe of conducting nanoparticles and the total volume of the system. In accordance with these models, at some threshold volume fraction xMe = xC of the conductive (metallic) filler particles a transition from nonconducting to conducting state is observed so that at xMe > xC (beyond the percolation

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threshold) MMC is characterized by metallic conductance and at xMe < xC (below the percolation threshold) MMC is in dielectric state with bad conductivity [51–53]. This metal–insulator transition (MIT) is the most essential characteristics of the electric behavior of granular MMCs around xC . It must be remembered once again that the most of carrier transport and magnetic properties of MMCs containing soft magnetic alloy nanoparticles have the highest sensitivity to the influence of magnetic and electric fields just near the xC values [10, 20, 23, 30]. Thus, the major characteristics of granular composites metal–insulator is the value of percolation threshold xC . For a squared lattice of granules, the theory gives xC ≈ 0.5 [50]. In the framework of the effective field theory, for 3D composites of spherical particles (balls) of metal and dielectric substances the value of xC is close to 0.33 [53–55]. The results of numerical computations of xC for a site task at simple cubic lattice gives xC ≈ 0.32 and for other lattices and a bonds task smaller values [53–55]. Unfortunately, existing theoretical percolating models allow us to make only approximate calculations of xC and other parameters of real percolating systems. These models do not take into account the structural organization and interaction of particles as well as the finite-size scaling effects in percolation. A critical analysis of the different models shows that they fail to predict the xC values for a given particle size ratio λ and the percolation threshold falls within the relatively large range [56]. As experiments have shown, in real metal–insulator composite materials the value of xC is a complex function of the space dimensionality, shape and size of the particles, local variations in their concentration, their spatial arrangement and many other factors [57,58]. For example, the percolation values of xC for carbon black fillers in the polymer matrixes fall within the range of 0.05–0.12. In experiments with spherical powder, particles of one size the value xC ≈ 0.27–0.05 was found [59–61] that is close to theories. Another situation is observed in evaporated granular films where xC lies in the range 0.5–0.6 [59–61]. It is believed that differences in the observed values of xC are caused by structure differences [62,63]. The percolation values of xC may be significantly lower for filler particles of an elongated geometry (rods, ellipsoids or discs) [64–68] and multiple percolation thresholds can occur in this case [69]. The low xC were observed also for the composite media where the insulating particles considerably exceed the conducting particles in size [63, 70–74]. It was shown that the percolation threshold volume fraction decreases with the increase in the ratio of the matrix (R) to filler particle size (r), λ = R/r [75–83]. Generally, we can conclude that the percolation behavior of a nanocomposite material can be controlled by the microstructure of the system [84, 85] and the percolation threshold xC and the critical indexes can be controlled not only by the particle size ratio λ = R/r, but also by the width of channels filled with the conducting particles.

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13.1.4 Carrier Transport in Granular MMCs around Metal–Insulator Transition Some Common Remarks. It has been realized for quite some time that granularity and/or heterogeneity brings new physics extending already wealthy list of remarkable effects exhibited by different systems near metal–insulator transition (MIT). In this paper, we shall deal with the composite system which can be well modeled by an array of metallic nanoparticles with different sizes, shape and properties where the intergranular electron coupling can be realized via the tunneling over dielectric matrix (below the xC ) or via electron diffusion and/or tunneling through the metallic networks formed by electrically contacting metallic granules. The effect of irregularities in the particle positions and in the strengths of the tunneling coupling on the physical properties of such systems is different for metallic and insulating sides of MIT in nanocomposites. If the coupling between the metallic nanoparticles is sufficiently strong and the system is well conducting (nanocomposite is beyond the xC ), the irregularities are not very important. On the contrary, irregularities become crucial in the limit of low coupling where the system is an insulator side of MIT (strongly below the xC ) [86]. At the same time, considering transport of electrons in granular composites, we should also take into account the disorder inside of the individual particles and/or at the interface between metallic nanoparticles and insulating matrix and the presence of charged defects and impurities in the matrix strata which are unavoidable. Even in metallic granular networks with the mean free path exceeding size of granules, the electrons that move ballistically within the granules yet scatter at the interface irregularities. In this case, the electron motion becomes chaotic with the resulting effect equivalent to the action of the intergranular disorder [87]. This interface chaotization results in additional degeneracy of the energy levels which would have led to singularities in physical quantities. Moreover, the slightest deviations (even of the order of 0.1 nm or of the order of the electron wavelength) of the shape of the granules from the ideal spheres or cubes also would have lifted the accidental degeneracy of the levels that is also equivalent to adding an intergranular disorder. This means that in real composite materials, the particles are always microscopically irregular. Besides, the coupling between the nanoparticles is the source of the additional irregularities. Therefore, such materials around and beyond the percolation threshold can be described by the models with the diffusive electron motion within the each particle and tunneling between nanoparticles without any loss of generality. Conductivity in Granular Systems. The key parameter that determines carrier transport in the granular systems is the average tunneling conductance G between the neighboring granules. It is convenient to introduce the dimensionless conductance g (corresponding to one spin component) measured in the units of the quantum conductance e2 /
:

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g = G/(2e2 /
),

(13.1)

where e is electron charge and
is Plank’s constant. The samples with g ≥ 1 exhibit metallic transport properties, while those with g ≤ 1 should show an insulating behavior. For the arrays of interconnected granules coupled to a bulk metallic network (beyond the xC ), when g  1, electrons propagate easily through the network and the Coulomb interaction is screened. In the opposite limit of the low coupling, g  1 (below the xC ), the charge on each granule is quantized as in the standard Coulomb blockade behavior. In this case, at temperatures T , when kT < Ec , electrons perform hops between the neighboring granules overcoming the electrostatic barriers Ec . Another important thing is that in the most granular systems the intergranular (tunneling) conductance g is much smaller than the intragranular conductance g0 : g  g0 . (13.2) The intergranular conductance g0 in nanocomposites is brought by scattering on impurities or at the interface granule-matrix and the inequality (1.4) means that the metallic granules are not very disordered by defects and impurities. Thus, the case g ∼ g0 can be viewed as a homogeneously disordered system and we do dwell on this limit here. The single granule conductance g0 can be most easily defined as the physical conductance of a granule of a cubic geometry measured in the units of the quantum conductance e2 /
(13.1). Insulating Regime of Conductance. Many earliest observations of temperature dependences of conductance σ(T ) in granular systems, including MMCs being at the insulating side of MIT (strongly below the xC ), showed the stretched exponential temperature behavior of the conductivity [37, 88]:    −n T0 , (13.3) σ(T ) = σ0 · exp − T where T0 is a material-dependent constant and n is exponent which is equal either 0.25 or 0.5 for 3D samples. This behavior resembled the Mott–Efros– Shklovskii variable range hopping (VRH) conductivity [51] and works for granular arrays both irregular [37] and strictly periodic [89–91]. VRH is a very general conduction mechanism in Anderson localized systems at sufficiently low temperatures. As was originally pointed out by Mott [92], the VRH conductivity σ(T ) can be obtained by optimizing (minimizing) the exponent (13.4) η = 2r/ξ + e/kb T in the hopping probability, p ∝ exp(−η), where ξ is the localization length, r the hopping distance and E is the hopping energy which are related to each other through the one-particle density of states N (EF ) by the 3D normalization expression

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4 3 πr 3



239

E

g(ε)dε = 1, 0

where E is the one-particle energy measured from EF . Assuming that the N (EF ) is constant near the Fermi level EF , so that N (EF ) = N0 , such an optimization derivation leads to the famous Mott [92] law for the temperature dependence of conductivity   −0.25 T0M σ(T ) = σ0 · exp − (13.5) T with T0M = βM /(kB ξ 3 N (EF )),

(13.6)

where kB is Boltzmann’s constant. Pollak [93] was the first to suggest that the long-range nature of the Coulomb interactions should lead to a dip (soft gap) in the N (EF ) at the Fermi energy. Efros and Shklovskii [94] showed that in this case the N (EF ) vanishes at EF for T = 0 and has the parabolic form in the immediate vicinity of EF , N (EF ) = α3D E 2 (13.7) with α3D = (3/π)(κ3 /e6 ),

(13.8)

where E is the one-particle energy measured from EF , κ = κr κ0 is the dielectric constant (κr is the relative dielectric constant and κ0 = 6.85 × 10−12 C2 J m−1 is the permittivity constant). Efros and Shklovskii have argued that the most observable manifestation of the Coulomb gap, expressed by (13.10), should be the temperature dependence of the VRH conductivity σ(T ) which has the form [94]   −0.5  T0ES . (13.9) σ(T ) = σ0 exp − T with T0ES = e2 βES /(kB κξ 3 N (EF )),

(13.10)

Metallic Regime of Conductance. Now we consider carrier transport by the conducting metallic networks formed by electrically contacting metallic granules in MMCs being beyond the xC when diffusion and/or tunneling of electrons can be realized depending on the state of granules and intergranular contacts (ordered or disordered). As for any granular system, the key parameter that determines the transport properties of the nanocomposite materials beyond the xC is the tunneling conductance g. As before, for g  1 regime a granular array has metallic properties, while in the opposite case, g  1, the highly disordered array shows

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an insulating properties. The insulating state of percolation network appears at low temperatures as a result of the strong Coulomb correlations that block ballistic or diffusive low-temperature electron transport. Generally speaking, apart from the Coulomb interaction effects one has also to consider the effects of quantum interference that may play an important role in the low conducting samples. In homogeneously disordered metallic systems the interference effects play an important role leading to the localization of electron states in the absence of interaction [95]. In such a state, as experiments show, temperature dependences of conductivity σ(T ) are described by Mott-like equation (13.3) [54, 92, 96] but with very low values of T0 (of the order of some units or tenths of K while for nanocomposites below the percolation threshold T0 approaches 103 –106 K). In the metallic regime and at high enough temperatures both Coulomb correlation and interference effects are weak. In this case, for a cubic granular array with an intergranular disorder, when the electron motion inside the granules is chaotic, the global sample conductivity per unit volume σ0 is given by the classical Drude formula, in particular, σ0 = 2e2 ga2−D ,

(13.11)

where a is the size of the granules, D the dimensionality of the sample and g is the conductance of the intergranular contact. When hopping from granule to granule, the electron momentum is not conserved and this leads to the finite conductivity σ0 (13.11). The conductance of the contact g depends on its microscopic properties, and we will consider it in most cases as a phenomenological dimensionless parameter controlling behavior of the system. A most recent incitement to intense and deep study of physics of granular materials was given by the influential works [97, 98] where a logarithmic

or root-like [99]

σ(T ) = a + b ln T,

(13.12)

√ σ(T ) = c + d T ,

(13.13)

dependence of the conductivity σ(T ), where a, b, c, and d are material dependent constants, was observed in the metallic side of MIT. This logarithmic or root-like behavior, has been observed in both 2D and 3D disordered metallic samples thus ruling out the tempting explanation in terms of the weak localization [100, 101] or interaction corrections [102–104]. Experimental Demonstration of Percolating Network Presence. As follows from earlier text, electron transport in MMCs occurs mainly via conducting networks created by electrically contacted metallic nanoparticles if the concentration of metal phase exceeds the percolation threshold xC . Efforts have been made to map such networks in MMCs composites by electric force microscopy (EFM) [105, 106] and conducting atomic force microscopy (C-AFM) which is a modification of normal AFM [107–110]. It has advantages because it allows

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not only to map conducting networks, but it can also be used to study electron transport down to the scale of individual conducting paths [104, 106]. When the AFM tip rasters a surface of MMC, the tip contacts either the metallic or the insulating phase. If the tip contacts the insulating phase, there should be no current between the tip and the sample. If, however, the tip contacts a metal particle, that is part of a conducting network at x > xC and extends to the surface, then electrons can flow from the sample to the tip when a bias is applied. Therefore, the electric current image maps the conducting network extending to the surface. By combining the current and topographic images, which can be obtained simultaneously, one can identify the metal particles from the insulating matrix. Authors of paper [104] have demonstrated some examples of such mapping of conducting networks and identified the metallic particles in Nix (SiO2 )1−x composites by combining the current and topographic images obtained from the same C-AFM scanning (see Fig. 13.5). The metallic phase (dark spots) is clearly distinguished from the insulating matrix (bright background) on this image. The advantages of this technique are its high resolution (nanometer scale), that it is nondestructive and that sample preparation is simple. It should be pointed out, however, that the typical feature size of the metal particles shown in Fig. 13.5 is in the range of 30–50 nm, which is larger than the true size of the metal particles, whose size is about 3–6 nm as determined by high-resolution transmission electron microscopy [111]. This was attributed in [105] to two effects in C-AFM: one is the finite tip size and the other is the tunneling effect [105, 107]. Nevertheless, the described measurements have shown that information obtained from C-AFM studies is helpful in understanding the peculiar properties in percolating systems.

Fig. 13.5. Current image by C-AFM at a bias of 0.45 V on an Nix (SiO2 )1−x sample with x = 0.576. The scan size 800 × 800 nm. The vertical scale is 40 nm

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13.1.5 Magnetic Properties of Granular Nanocomposites From the viewpoint of magnetism, basically noninteracting metallic granules in MMCs at xMe < xC are single-domain structures. Generally, the critical granular size for single-domain state depends on the chemical nature of metal (or metallic alloy) and shape of nanogranules but does not exceeds few tens of nanometers. Magnetic axis of single-domain nanoparticle is determined with the energy of magnetic anisotropy (Ea V ), where Ea is magnetic anisotropy energy per volume unit and V is the particle volume. At low temperatures, magnetic axis of such nanoparticle is sporadically oriented and “frozen” that leads to the zero magnetization of granular material at the absence of external applied magnetic field. At much higher temperatures (for example, room temperature) when Ea V  kB T the barriers originating from the magnetic anisotropy energy could be overcame by the thermal energy. In this case, due to thermal fluctuations magnetic moment of nanoparticles changes its direction between different energy minima’s. This causes the so-called superparamagnetic state of MMCs. In its simplest case, the superparamagnetic relaxation at zero magnetic field could be approximated by the Arrhenius relationship: 1 = f0 e−KT /kB T , (13.14) τ where τ is relaxation rate, f0 the natural gyromagnetic frequency of the particle, K the anisotropy constant, and T the absolute temperature [112]. Nanoparticles generally have very low relaxation times (10−7 –10−10 s [113]). Due to that, at close to zero external fields observed average magnetization value is also close to zero. In the case when the thermal energy kB T is much greater than the anisotropy energy Ea V magnetization directions are almost energetically equal, so that magnetization could be described by the Langevine function:  
 kB T µH − , (13.15) M = Nµ coth kB T µH where µ is the magnetic moment of each metallic granular and H is the magnetic field induction [113]. The temperature at which magnetization of nanoparticles fluctuates randomly at zero field is defined as blocking temperature TB . Two experimentally observed features are characteristic for the superparamagnetic state of nanocomposites. The first one is absence of hysteresis in magnetization curves. The second is the dependency of magnetization on temperature as µH/kT [113]. Observation of magnetic hysteresis at temperatures higher than TB is attributed to the effects of magnetic coupling between nanoparticles. This effect occurs at xMe higher and within xC when continuous current conducting net of metallic nanoparticles is forming. It is noteworthy that generally xMe correspondent to the ferromagmetically interacting net of nanoparticles does

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not coincide with the percolation threshold xC . Interpretation of magnetic properties in such systems with cooperative magnetic ordering is a very complicated task. The reason is in their multifunctional dependency on particle size distributions, coexistence of nanoparticles with different phase composition, specific local anisotropy of nanoparticles and the interaction between them [114]. However, experimental research allowed to establish correlation between some magnetic characteristics (coercivity Hc , saturation magnetization) of nanocomposites and the size and concentration of metallic granules. It is experimentally observed that at room temperature Hc values of nanocomposites initially increases with the volume fraction of metallic nanoparticles due to the increase of the particle size [113]. Coercivity value reaches its maximum close to the percolation threshold xC due to the formation of magnetically interacting net of metallic nanoparticles. While further increase of metallic fraction draw to the decrease of Hc because of the change from single-domain to multi-domain structure of nanoparticles. Below the blocking temperature TB , coercivity of single-domain nanoparticles could be much higher than in the bulk materials. Assuming that temperature dependency of Hc is governed by superparamagnetic relaxation the following relation Hc (T ) could be drawn:    T Hc = Hc (0) 1 − . (13.16) T0 The temperature dependence HC ∼ T 0.5 was experimentally confirmed for many MMCs.

13.2 Properties of FeCoZr–Al2 O3 Nanocomposite Films: Synthesis in Pure Ar and Mixed Ar + O Ambient 13.2.1 Synthesis and Samples Preparation Synthesis of nanocomposite films (FeCoZr)xMe (Al2 O3 )1−xMe (17 at.% < xMe < 65 at.%) was performed by ion-beam sputtering of the compound target onto the motionless water cooled substrate. The details of deposition technique were described in Part 13.1.2 of the present review. For preparation of FeCoZr– Al2 O3 films the compound target contained Al2 O3 strips (see Fig. 13.2) layered with the variable distances (from 3 mm on one side of the target to 24 mm on another one) between them on Fe45 Co45 Zr10 alloy plate-foundation. Such original configuration of the target enabled to prepare composite film with different metallic-to-dielectric phases ratio in one technological process. The thickness of films varied between 1 and 6 µm. Sputtering was carried out in two different regimes allowing to obtain two series of samples: in pure argon ambient (Ar series) and Ar + O mixed ambient (Ar + O series) in the vacuum chamber. Ar pressure in the chamber was 8.0 × 10−4 Pa. The pressure of the

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Ar + O gas mixture in the chamber was 9.6 × 10−2 Pa at the oxygen partial pressure of 4.41 × 10−2 Pa. Nanocomposite films were deposited on different substrates in accordance with experimental techniques applied for investigation. The nanocomposite films sputtered onto glass–ceramic substrates were cut into rectangular strips of 10 mm long and 2 mm wide (for SQUID-magnetometry, AC/DC magnetoresistivity measurements) and squared-like 10 mm × 10 mm samples (for measurements by room-temperature alternating-grad magnetometry). Aluminium foil substrate (5 cm × 25 cm) was used for preparation of films intended for M¨ ossbauer spectroscopy. Due to specific preparation method of nanocomposites the concentration of metallic alloy was not constant along the samples’ surface. However, although different applied experimental techniques demanded different sizes and shapes of samples, the mean metal–dielectric ratio was chosen to be similar or at least close for the samples. This made it possible to establish correlations between the results obtained by different experimental methods. Concentration of chemical elements in the composites was controlled by special microprobe X-Ray analyser in SEM LEO 1455VP with accuracy better than 1 at.%. The thicknesses of the films were estimated by SEM on the plane chips of the samples with the accuracy not worse than 3–4% that allowed to have ratio error of about 5%. 13.2.2 M¨ ossbauer Spectroscopy M¨ ossbauer spectra have been recorded at room temperature in transmission geometry using 57 Co/Rh source (25 mCi). The fitting procedure has been performed with the use of MOSMOD program assuming the distribution of hyperfine magnetic fields (Hhf ) and quadrupole splitting (QS). All isomer shifts (IS) were given with respect to α-Fe. Lorentzian width (FWHM) of fitted spectral lines was equal to 0.15 mm s−1 . The M¨ossbauer spectra recorded for the granular films of Ar series (see Fig. 13.6) strongly depend on the atomic fraction xMe of CoFeZr. As evidences from Fig. 13.6, the spectra look like totally nonferromagnetic at xMe ≤ 42 at.%. They could be fitted with two Fe3+ doublets (D1: IS = 0.13–0.14 mm s−1 , QS = 0.88–1.12 mm s−1 ; D2: IS = 0.04–0.05 mm s−1 , QS = 0.49–0.52 mm s−1 ). At xMe approaching the percolation threshold xC ≈ 43–47 at.% (estimated from the electric properties of the composites [115]), the appearance of a wide partially collapsed magnetic subspectrum with IS close to 0.08 mm s−1 has been observed in addition to the above-mentioned two doublets (see Fig. 13.6). Spectra of the films with xMe > xC are magnetically split and could be evaluated in the assumption of two magnetic sextets (Hhf1 = 16.0–30.0 T; Hhf2 = 28.2–33.8 T). It is noteworthy that D2 doublet could be eventually fitted in the spectra of films with xMe > xC but its relative contribution was found to be about 3% (i.e. within experimental error).

13 FeCoZr–Al2 O3 Granular Nanocomposite Films

245

Fig. 13.6. RT M¨ ossbauer spectra recorded for the FeCoZr–Al2 O3 films sputtered in pure Ar ambient

Contrary to the previous case, M¨ossbauer spectra of the samples deposited in Ar + O mixture (see Fig. 13.7) show nonmagnetic behavior independently on the CoFeZr content (both before and beyond percolation threshold xC ≈ 55 at.% as estimated in [115]). However, the local Fe states observed in this case also demonstrated a threshold character. In particular, at xMe ≤ 55 at.% M¨ ossbauer spectra were evaluated assuming three local Fe states in the films: two doublets (D3: IS = 0.08–0.11 mm s−1 , QS = 0.41–0.49 mm s−1 ; D4: IS = 0.40–0.46 mm s−1 , QS = 0.77–0.94 mm s−1 ) associated with Fe3+ ions and one doublet assigned to Fe2+ state (D5: IS = 0.90–0.96 mm s−1 , QS = 1.81– 1.91 mm s−1 ). Starting from xMe values above 59 at.% (i.e. far beyond xC ), the M¨ ossbauer spectra could be fitted with only D4 and D5 doublets, while

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Fig. 13.7. RT M¨ ossbauer spectra recorded for the FeCoZr–Al2 O3 films sputtered in Ar + O2 ambient

D3 disappeared. It should be noticed that a relative contribution of D4 subspectrum increased with the value of x at the expense of the D5 contribution. A fraction of D3 before xC was constant within the experimental error. 13.2.3 Alternation Grads- and SQUID-Magnetometry The magnetoresponse measurements were carried out by alternation grads magnetometer at 300 K with the applied magnetic induction B ≤ 600 mT. The pick-up coil signal recorded at different B was proportional to the value of magnetization M (B) and reduced to the volume of the film.

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247

(a)

(b) Fig. 13.8. Alternation grad-magnetometry at RT recorded for the FeCoZr-Al2 O3 nanocomposites sputtered in pure Ar (a) and Ar + O2 (b) ambient

As a whole, the behavior of the magnetization curves shown in Fig. 13.8a for the Ar series correlated with the data obtained by M¨ ossbauer spectroscopy. As has been found, dielectric samples far from the percolation threshold (xMe ≤ 35 at.%) revealed a nearly linear increase of M with B growth, which looks like a paramagnetic behavior. At higher concentrations of the metallic phase up to the percolation threshold xC ≈ 43–47 at.% the room temperature magnetization curves show no saturation and hysteresis features to confirm the M¨ ossbauer data concerning the fact that the metallic alloy nanoparticles

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are in the superparamagnetic state. Both the remanence and coercivity in this case are zero at 300 K. Above the percolation threshold xC the shape of magnetization curves is changed (Fig. 13.8a) approaching saturation without any hysteresis that is characteristic for soft amorphous ferromagnetic materials and confirmed by the magnetically split M¨ ossbauer spectra. For the Ar + O series the behavior of magnetization curves M (B) also correlated with M¨ ossbauer data. As is seen from Fig. 13.8b, formally M (B) curves for this series are similar to those for the samples of Ar series. Also at xMe  55 at.% (far from the percolation threshold xC ) FeCoZr–Al2O3 films exhibit the paramagnetic behavior of M (B). However, as seen from Fig. 13.8b, the M (B) curves show neither saturation nor hysteresis both before and above the xC , indicating the superparamagnetic behavior of the nanoparticles for the whole concentration range. SQUID-magnetometry at 10 K in magnetic fields of 50 kOe was applied to characterize more precisely the field dependence of magnetization for the studied nanocomposites. The obtained curves were normalized by maximum value of magnetization M and film thickness. SQUID magnetization M/Ms curves for the selected nanocomposites with varying xMe are presented in Fig. 13.9. In contrary to the RT case where

(a)

(b)

Fig. 13.9. SQUID-magnetometry at 10 K recorded for the FeCoZr–Al2 O3 nanocomposites sputtered in pure Ar (a) and Ar + O2 (b) ambient

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zero coercivity, no saturation and no hysteresis features were observed for nanocomposite films sintered in pure Ar the field dependence of normalized magnetization at 10 K demonstrated hysteresis loops. Generally, coercivity values from 30 to 40 Oe were relatively small and typical for soft ferromagnetic FeCo-containing amorphous alloy. The coercivity showed the tendency to decrease with the growth of FeCoZr fraction in the nanocomposites. Magnetization curves for (FeCoZr)54 (Al2 O3 )46 and (FeCoZr)59 (Al2 O3 )41 (compositions close to and above the xC ) were almost saturated. However, for the sample of (FeCoZr)17 (Al2 O3 )83 composition (below the xC ) the magnetization showed no saturation indicating the availability of superparamagnetic nanoparticles. The presence of metallic nanoparticles with sizes of 2–10 nm in the studied samples was previously confirmed by TEM microscopy [117]. Normalized M/Ms curves obtained at 10 K for the nanocomposites of Ar + O series (see Fig. 13.9b) demonstrated very different shape as compared to the previous case. The shape of magnetization hysteresis curves confirmed the ferromagnetic nature of at least some part of the formed complex iron oxides indicated by M¨ ossbauer spectroscopy study. However, it should be emphasized that none of these curves were completely saturated even at the applied magnetic field of 5 kOe. This could be explained by either of the two main reasons: the formation of nonmagnetic phases (oxides) due to the incorporation of additional oxygen or the superparamagnetic state of ferromagnetic nanoparticles (oxides or FeCoZr alloy). Actually, the increasing coercivity with growth of FeCoZr fraction was observed. It is interesting to note that at comparable compositions the coercivity values of the nanocomposites sintered in Ar + O mixture were one order of magnitude larger than that of the Ar series (see Figs. 13.9). At the same time, the magnetization for Ar + O series at magnetic field of 5 kOe (close to saturation state of Ar series samples) appeared to be about 2.5 times lower as compared to pure Ar series. The observed variations of the coercivity and magnetization for the composites when additional oxygen was incorporated were in favor of complex FeCo-based oxides formation generally possessing higher coercivity fields [118, 119] but smaller magnetization as compared to the soft amorphous alloys. Nevertheless, the additional contribution of paramagnetic FeAl2 O4 formation (indicated by M¨ ossbauer spectroscopy [116]) as a reason of the decrease of magnetization value also cannot be excluded. It should be emphasized that the formed FeCobased ferromagnetic oxides were of nanometer scale as it was confirmed by the nonsaturated magnetization curves. 13.2.4 Atomic Force–Magnetic Force Microscopy Complementary study of magnetic structure of the (FeCoZr)x (Al2 O3 )100−x covering the range of low, medium (percolation threshold) and high concentrations of metallic phase was performed using AFM and MFM methods. For AFM measurements in the tapping mode (i.e., measuring of frequency and phase shifts at the resonance frequency 341 kHz) an atomic force

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(a)

(b)

Fig. 13.10. AFM PC images of (FeCoZr)33 (Al2 O3 )67 film for scanning areas measuring 100 × 100 nm2 (a) and 2200 × 2200 nm2 (b)

microscope Nanotop-204M was used with the standard commercial V-shaped NSC11cantilevers (stiffness of 48 N m−1 ). As follows from the detailed analysis of phase contrast images in Fig. 13.10, the metallic nanoparticles can create more or less dense conglomerates. Below the xC , these conglomerates probably look like “clouds” (shown by arrow) of magnetically noninteracting nanoparticles in the superparamagnetic state detected by M¨ ossbauer spectroscopy [116]. These “clouds” are linked by “bridges” of nanoparticles embedded into the alumina matrix (see Fig. 13.10b). Beyond the xC , these clouds are probably transformed into greater ferromagnetic particles forming (together with the above-mentioned “bridges”) the net-like metallic percolative structure in agreement with M¨ ossbauer spectroscopy and magnetometry results. Some part of these “clouds” can form anisotropic regions looked like elongated rings (see Fig. 13.10b) of 200–600 nm in size with alumina particles at their centers and surrounded by metallic nanoparticles with different density. AFM/MFM images for nanocomposites of Ar and Ar + O series covering FeCoZr concentrations below, within and above the xC are summarized in Figs. 13.11 and 13.12, respectively. AFM topography images of the films sintered in pure Ar atmosphere with three studied compositions (FeCoZr)33 (Al2 O3 )67 , (FeCoZr)45 (Al2 O3 )55 and (FeCoZr)64 (Al2 O3 )36 (see Figs. 13.11a, c, e), showed rather weak dependence on FeCoZr fraction. However, MFM study demonstrated very pronounced change of magnetic contrast with the increase of metallic fraction. It is essential that for (FeCoZr)33 (Al2 O3 )67 composition (lower than xC ) MFM image (see Fig. 13.11b) is just the inversed image of the topography due to the absence of magnetic interactions between the isolated nanoparticles. For composite films of (FeCoZr)45 (Al2 O3 )55 (just at the xC ) labyrinthlike structure of ferromagnetically interacting nanoparticles was detected (see Fig. 13.11d). This structure was very similar to that observed with TEM

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251

Fig. 13.11. Images of AFM (a, c, e) and MFM (b, d, f ) for the FeCoZr–Al2 O3 composites sintered in pure Ar ambient with the increase of metallic fraction

images for such composition. Magnetic contrast image for the (FeCoZr)64 (Al2 O3 )36 sample (above the percolation threshold) showed the stripe-like structure (see Fig. 13.11f) confirming the formation of magnetically ordered net of the metallic granules described in [115].

252

J.A. Fedotova (FeCoZr)33 (Al2 O3 )67

(a)

(b) (FeCoZr)45 (Al2 O3 )55

(c)

(d) (FeCoZr)64 (Al2 O3 )36

(e)

(f)

Fig. 13.12. Images of AFM (a, c, e) and MFM (b, d, f ) for the FeCoZr–Al2 O3 composites sintered in Ar + O mixed ambient with the increase of metallic fraction

Figure 13.12d, f shows the MFM images of the composites sputtered in Ar + O mixture indicating a dramatic difference from those obtained for the samples deposited in pure Ar. Absence of the magnetic contrast was an evidence for the Ar + O series at x ≥ xC to reveal the oxidation of

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253

the nanoparticles and the precipitation of nonmagnetic hercinite (observed by M¨ ossbauer spectroscopy and confirmed with magnetization experiments) suppressing both agglomeration of the nanoparticles and the ferromagnetic interaction between them. 13.2.5 Electric and Magnetotransport Properties Mechanisms of carrier transport in the nanocomposite samples with different concentration of xMe were studied using temperature dependences of DC conductivity and low-frequency (0.1–1000 Hz) impedance at the temperature range 4–300 K. To make these measurements, the nanocomposite films sputtered onto glass–ceramic substrates were cut into rectangular strips with dimensions of 10 mm × 2 mm × d µm (d is thickness of the films), and four indium contacts were deposited using ultrasound soldering. These samples were subjected to conductance measurements in flow cryostat system with electric field intensities E up to 105 V m−1 and magnetic fields with induction B up to 600 mT using the four probe potentiometric method. Ratio error of electric conductance measurements was better than 5%. The details of experiments were published in [10, 20, 120]. As was mentioned in Sect. 13.1, the character of DC carrier transport in binary metal–dielectric composites is strongly dependent on the position of the sample relative to metal–dielectric transition (MIT) point xC . The latter is determined by atomic fraction and composition of metallic phase xMe in a composite and some other parameters of metallic and dielectric constituents (i.e., type of matrix, dimensions, shape, distribution topology of nanoparticles, etc.). The dependences of room temperature conductivity σ and low frequency (0.1 ≤ f ≤ 5 kHz) reciprocal real part of impedance 1/Z  on concentration xMe for the films of Ar and Ar + O series are shown in Fig. 13.13 (see also [120]). At the comparable metallic phase concentrations, the σ and 1/Z  values of the films of Ar + O series are approximately two orders of magnitude lower than that of the Ar series. As follows from the M¨ ossbauer spectroscopy results (see Part 13.2.2 of the paper and also [116, 121]) this is related to the partial oxidation of metallic granules due to the presence of oxygen in the gas mixture in the chamber during target sputtering. Figure 13.14 shows conductivity as a function of temperature for the samples of the Ar series in the x range between 30 and 63 at.% and also for the film containing pure FeCiZr alloy deposited at the same conditions as composites. As evidenced from our measurements, for the most samples of the Ar series temperature dependences σ(T ) are characterized with activational behavior but with the changing energy of activation (see Fig. 13.14a in Arrhenius scale). Comparison of the curves 10 and 11 in Fig. 13.14b in normal scale shows that for the samples with the values of xMe > 60 at.% σ(T ) exhibit crossover from activational ln(σ) ∼ (1/T )n to power-like σ(T ) ∼ T k dependence observed just as it is in amorphous FeCiZr alloy films deposited in pure

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Fig. 13.13. Dependences of room temperature conductivity σ (1, 2) and reciprocal real part of impedance 1/Z  (1 , 2 ) for the samples of Ar series (1) and Ar + O series (2) in the ratio of metallic fraction xMe . Percolation threshold xC for the films of Ar series is shown by arrow

(a)

(b)

Fig. 13.14. Temperature dependences of conductivity for the samples of Ar series at x ≈ 31 (1), 33 (2), 35 (3), 38 (4), 43 (5), 46 (6), 48 (7), 50 (8), 52 (9), 56 (10), 63 (11) and 100 at.% (12) in Arrhenius (a) and normal (b) scales. In (b) σ(T ) curves are normalized to the conductivity at 285 K

Ar at the same conditions (see curve 12). Underline that for the films of Ar series even beyond the percolation threshold (xMe > xC ) σ(T ) dependences keep activational shape up to 60 at.% but with much lower values of activation energies as compared to ones at xMe < xC [10, 20, 120]. The described behavior of σ(T ) for the samples belonging to dielectric side of MIT (xMe < xC ), according to the model approach described above in Part 13.1.4, is characteristic for thermo-stimulated tunneling or VRH conductivity. Conservation of exponential form of σ(T ) for the samples belonging to metallic side of MIT at xMe > xC up to 60 at.% can be attributed

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255

Fig. 13.15. Temperature dependences of conductivity for the samples of Ar + O series at x ≈ 31.2 (1), 36.6 (2), 37.4 (3), 38.6 (4), 40.7 (5), 49 (6), 53.6 (7), 62.5 (8), 63.7 (9), and 100 at.% (10)

(a)

(b)

Fig. 13.16. Temperature dependences of conductivity σ for nanocomposites of the Ar series with x = 31.8 at.% (a) and x = 37.7 at.% (b)

to amorphous state of FeCiZr nanoparticles due to the presence of Zr as amorphizer. Temperature dependences of DC conductivity for the films of the Ar + O series also display, as seen in Fig. 13.15, activational laws ln(σ) ∼ (1/T )n with the “sliding” activation energies at the whole range of metallic phase concentration (from 31 to 64 at. [10, 20, 120]. The σ(T ) corresponding to the curve 10 is the same as one in curve 12 in Fig. 13.14 and shown for comparison. To study mechanisms of carrier transport of both series of films depending on their composition and conditions of sputtering, we subjected temperature dependencies of DC conductivity to more detailed analysis. As was shown in [122] and seen from Fig. 13.16, σ(T ) dependences of the most samples studied were close to VRH behavior     −n T0 σ(T ) = σ0 · exp − , (13.17) T where σ0 is an equilibrium conductivity at T → ∞. In doing so, experiments yielded either the Mott law [54], ln(σ) ∼ (T0M /T )−0,25 , or the Shklovski–Efros

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Table 13.1. T0 parameter, equilibrium conductivity σ0 , energy density N (EF ) of localized states at the Fermi level in dielectric matrix and the localized wave function damping length a for the samples of the Ar series x (at.%) 31.2 31.4 31.8 32.5 34.7

Temperature range (K) T T T T T T T T T T

> T∗ < T∗ < T∗ > T∗ < T∗ > T∗ < T∗ > T∗ < T∗ > T∗

T0 × 10−3 , (K)

σ0 × 10−3 , (S m−1 )

a (nm)

N (EF ) × 10−18 , (eV−1 cm−3 )

279 638 289 631 256 601 246 601 159 380

547 105 644 135 612 115 681 119 527 116

12 8 11 7 12 7 11 7 11 7

3 6 4 7 4 8 5 8 9 12

law, ln(σ) ∼ (T0SE /T )−0,5 , where parameters T0M and T0SE determine average activation energies for carrier hopping by localized states around Fermi level in dielectric matrix for Mott and Shklovskii–Efros VRH regimes, respectively. The conducted detailed analysis of approximation errors by fitting these two models made it possible to assert that the Mott law was valid in the whole temperature region for the samples of the Ar series at xMe < xC . Besides, for these samples we observed σ(T ) dependencies with two different slopes in the Mott scale at T ∗ ≈ 110–120 K (Fig. 13.16a) and therefore sharp change of T0M when crossing T ∗ (see Fig. 13.16 and Table 13.1). Such behavior was interpreted in [122] as an evidence for the presence of two different systems of routes for carrier transport in the studied temperature range. As was seen from AFM PC images in Part 13.2.4 (see Fig. 13.10), these two kinds of routes could be connected with inhomogeneous distribution of FeCoZr nanoparticles in dielectric matrix, i.e. the presence of the dense “clouds” of magnetically noninteracting nanoparticles and loose “bridges” of nanoparticles embedded into the alumina matrix between the “clouds” and separated by greater distances between each other. So we can ascribe to these two systems of routes movement of electrons either only along “bridges” (at the temperatures higher than T ∗ ) or by the routes including both dense “clouds” and loose “bridges” (at T < T ∗ ). This is also confirmed indirectly by M¨ ossbauer experiments [118], where comparison of M¨ ossbauer spectra of the studied composites allowed to fix two types of nanoparticles with different sets of hyperfine parameters. Note the next important peculiarities of the σ(T ) behavior in the films of Ar series at crossing of the percolation threshold. The first one is the disappearance of high-temperature part of Mott-like contribution in carrier transport at xMe > xC and the second is the tendency to saturation of σ(T )

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(a)

257

(b)

(c) Fig. 13.17. Temperature dependences of conductivity σ for nanocomposites of Ar series with x = 43 (a), 58.9 (b) and 63.2 at.% (c)

with the lowering of the temperature below 10 K (Fig. 13.17). Moreover, the estimations of the slopes of linear parts of σ(T ) in ln σ(T ) − (1/T )0,25 scale have proved that T0 values became less than the temperatures of measurements for the samples with xMe > 50 at.%. This means that in the samples of Ar series at xMe > 50 at.% VRH mechanism of carrier transport practically disappears and linear parts of curves in Fig. 13.17b, c should be considered as transition between above mentioned metallic power-like contribution (see high temperature contribution in these curves) and quantum corrections contribution and/or contribution of the rest resistivity due to defect scattering of electrons at the lowest temperatures. The fitting procedure for the composites of Ar + O series showed that the Mott law holds only at low temperatures while at higher temperatures Tco > 160–170 K we observed the crossover from Mott low to the Shklovski– Efros law (Fig. 13.18) describing the hopping transport of the charge carriers at the Coulomb gap in the localized states density near Fermi level [122]. The activation energies estimated from the σ(T ) showed that the T0M values are essentially lower by the order of magnitude for the samples of Ar series than those for the samples of Ar + O series (Fig. 13.19). This confirms that in the latter case, the oxide “shells”, that separate nanoparticles in the currentconducting routes, created additional barriers for electron transport between nanoparticles.

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Fig. 13.18. Temperature dependences of conductivity σ for nanocomposites of Ar series (a) and Ar + O series (b) with x = 31.2 at.% (1) and x = 45 at.% (2)

(a) T0M

(b)

T0SE

Fig. 13.19. and dependencies on the films composition for the samples of Ar series (a) and Ar + O series (b) for different temperature regions

Therefore the Shklovski–Efros behavior of σ(T ) in the samples of Ar series can be naturally attributed to the influence of these oxide “shells”, which electric properties are close to semiconducting [122]. The temperature dependences of nanocomposite conductivity allowed us to estimate principally the changes in the localized state density N (Ef ) and localized electron wave function damping length a in alumina matrix depending on the metallic phase content using experimentally extracted T0 and σ0 values (see [10, 20, 120]). When the Mott hopping mechanism of VRH carrier transport dominating the semi-empirical Kirkpatrick formula [123]

13 FeCoZr–Al2 O3 Granular Nanocomposite Films

   0,25  c   T 0,35 T0 0 σ = 0, 0217 exp − d T T    0,25 T0 0 , = σAC exp − T

259

(13.18)

was used to fit T0 and σ 0 values, where  0 σAC ≈ σ0

T0 T

0,35 (13.19)

and  C=

 

E12 π 4 × d × s5 ×
4

·



2e3 3 × 4 × π × ε × ε 0 × a2

(13.20)

Here a is the localized wave function damping length (in cm). The parameters values used in this relations for Al2 O3 were: deformational potential E1 = 10 eV, density d = 3.97 g cm−3 , sound velocity s = 3 × 103 cm s−1 , dielectric constant κ0 = 10 [124]. The σ0 is constant characterizing the sample (with dimensionality Ω −1 cm−1 ) and N (EF ) – energy density of localized states (eV−1 cm−3 ) at the Fermi level in dielectric matrix. Using these parameters in (13.18)–(13.20), we can connect the T0 and σ0 values with N (EF ) and a: σ0 =

6.95 × 10−15 a3

(13.21)

T0 =

8.65 × 10−4 N (EF ) × a3

(13.22)

The estimations of N (EF ) and a values for dielectric side of MIT of the sets 1 and 2 samples on the base of (13.21,13.22) are shown in Tables 13.1 and 13.2 correspondingly. As follows from Table 13.1 in the range of 31 < xMe < 40 at.% (below the percolation threshold xC ≈ 43–45 at.% for binary composites deposited Table 13.2. T0 parameter, equilibrium conductivity σ0 , energy density N (EF ) of localized states at the Fermi level in dielectric matrix and the localized wave function damping length a for the samples of the set Ar + O series at T < Toc where the Mott law is applied x (at.%) 31.2 36.6 37.4

T0 × 10−3 , (K)

σ0 × 10−3 , (S m−1 )

a (nm)

N (EF ) × 10−18 , (eV−1 cm−3 )

1990 795 323

12390 5318 976

4 5 8

144 213 131

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in pure Ar) radius of localization of electrons is about 9 ± 2 nm and N (EF ) is about (3 − −12) × 1018 eV−1 cm−3 . The addition of oxygen to Ar ambient of sputtering chamber results in some lowering of a down to 6 ± 2 nm and the increase of N (EF ) up to (1 − −2) × 1020 eV−1 cm−3 (see Table 13.1). This increase of N (EF ) nearly two orders in the Ar + O series films can be attributed to the influence of oxide “shells” around the alloy nanoparticles. Additional confirmation of VRH mechanism of carrier transport was obtained from the investigation of magnetoresistive effect in the studied nanocomposite films. The DC measurements at room temperature for both sets of samples have showed (see Fig. 13.20) that the magnetoresistance (MR) r(B) = (ρ(B)/ρ(0)) of the most studied nanocomposites films was negative (except the samples for x > xC for the Ar series) and obeying the relationship of r(B) ∝ −B k as the magnetic induction B increased. Note also that dependences of negative MR on composition x at constant B displayed principally different behavior for the samples deposited in Ar and in Ar–O mixture (see Fig. 13.21). For the Ar series samples negative MR sharply decreases at the approaching the percolation threshold (confirming the tunneling character of carrier transport between metallic nanoparticles at x < xC ) turning to

(a)

(b)

Fig. 13.20. Reduced resistance vs. magnetic induction B at RT for nanocomposite films of the Ar series (a) and set Ar + O series (b)

Fig. 13.21. DC magnetoresistance of the composite films of Ar series (a) and Ar + O series (b) measured at RT temperature and B = 300 mT

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positive MR regime for metallic percolating net beyond the xC . As to samples of Ar + O series, they display negative sign of MR even at the electrical contacting of metallic nanoparticles covered with oxide shells with its maximal value at x ≈ 55–56 at.%. The negative sign of MR and the exponential factor k ∼ 2 confirmed the VRH regime of the carrier transport in the studied nanocomposites [125]. Note also that maximal sensitivity of ρ to the change of magnetic field was very closely to percolation thresholds xC in the Ar series estimated from the conductivity measurements.

13.3 Concluding Remarks The paper summarized the experimental results obtained on the (FeCoZr)xMe (Al2 O3 )1−xMe nanocomposites (17 at.% < xMe < 65 at.%), reflecting the correlations between structure, magnetic state, carrier transport and magnetotransport mechanisms due to the change of FeCoZr nanoparticles’ fraction and incorporation of oxygen ions. Structural transformations with the increase of FeCoZr concentration included the transition from the system of noninteracting superparamagnetic FeCoZr nanoparticles at x below 45–47 at.% to the net of FeCoZr nanoparticles ferromagnetically interacting and electrically contacting at higher x values. Theoretically expected replacement of hopping conductance over localized states in dielectric Al2 O3 matrix with metallic conductance through percolative net of nanoparticles has been observed at xC ≈ 45–47 at.% FeCoZr. Correlation between structure, magnetic state and carrier transport mechanisms are summarized in Fig. 13.22. X < XC

Matrix

X = XC Nanoparticles

X

>X

C

Fig. 13.22. Correlation between structure and carrier transport for (FeCoZr)x (Al2 O3 )1−x sintered in Ar ambient. Dashed lines correspond to the currentconductive routes along the percolation cluster

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J.A. Fedotova X < XC

Matrix

X = XC Oxidized nanoparticles

X > XC

Fig. 13.23. Correlation between structure and carrier transport for (FeCoZr)x (Al2 O3 )1−x sintered in mixed Ar + O ambient. Dashed lines correspond to the current-conductive routes along the percolation cluster

It should be emphasized that for the studied nanocomposites FeCoZr nanoparticles concentrations correspondent to the percolation threshold xC (where the change of carrier transport mechanism is observed) and to the formation of magnetically interacting net were very close although that is not the general case. Sufficient change of structure and properties of (FeCoZr)–(Al2 O3 ) nanocomposites due to incorporation of oxygen could be related to the sequential oxidation of FeCoZr nanoparticles. The change of structure, magnetic state and carrier transport mechanisms originating from the oxygen incorporation is presented in Fig. 13.23. Complex semi-insulating interlayers containing ferromagnetically ordered nanosized FeCo-based oxides and paramagnetic FeAl2 O4 compound prevented the formation of ferromagnetically interacting net of pure FeCoZr nanoparticles. In this case deviation from simple metal–dielectric state of nanocomposites (due to more complex phase composition) makes impossible to consider the carrier transport mechanisms within the above-mentioned simple percolative models applicable only for binary composites. Acknowledgement The author acknowledges the financial support from VISBY Program of the Swedish Institute, grants of Belarusian Fundamental Research Foundation (Contracts F06P-128 and F05K-015) and grant of the Belarusian State Program “Nanotech” (project 3.07).

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The help of S. Kovaleva (Joint Institute of Mechanical Engineering of NASB, Minsk) and L. Baran (Belarusian State University, Minsk) for AFMMFM measurements, M. Marszalek (Niewodniczanski Institute of Nuclear Physics PAN, Cracow) for X-ray reflectometry as well as discussion and comments on paper of A. Fedotov (Belarusian State University, Minsk), Yu.E. Kalinin (Voronez State University, Voronez) and other colleagues are gratefully acknowledged.

References 1. 2. 3. 4.

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14 Ferromagnetism of Nanostructures Consisting of Ferromagnetic Granules with Dipolar Magnetic Interaction E. Meilikhov and R. Farzetdinova Kurchatov Institute, 123182 Moscow, Russia, [email protected], [email protected] Summary. We study the magnetic state of Ising-type structures consisting of ferromagnetic nano-granules (dipoles) with dipolar magnetic interaction. Two types of the structures are considered: (1) 3D and 2D lattices populated fully or partially by ferromagnetic granules, and (2) 3D and 2D spatially random systems of granules. A variety of granules’ forms are also have been taken into account: we consider point and spherical dipoles, rod-like dipoles, and ellipsoidal (oblate and prolate) dipoles. For all those systems, the ground magnetic state of the systems, their magnetic properties, and Curie temperatures have been calculated precisely or approximately. In the latter case, the results have been obtained in the framework of the generalized mean-field theory, taking into account spatial variations of the local effective field by numerically and/or analytically calculating its distribution functions. Various practical realizations of the systems and their properties are discussed.

14.1 Introduction Whether ferromagnetism can exist at zero temperature in a system of magnetic moments featuring dipole–dipole interaction? An answer to this question, requiring exact calculation of the energy of a long-range magnetic interaction involving all dipoles in the system, is not easy to obtain even for a system composed of point magnetic moments. For a pair of parallelpoint magnetic moments µ at a distance of r from each other, the magnetic interaction energy is given by the formula w=

µ2 (1 − 3 cos2 θ), r3

where θ is the angle between the direction of moments and the line connecting the two points. The sign of this energy varies depending on the angle θ: √ for |θ < θ0 or |π − θ| < θ0 , where θ0 = arccos(1/ 3) ≈ 55◦ , the interaction retains parallelism of the magnetic moments (i.e., favors ferromagnetic

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ordering), otherwise the antiparallel (antiferromagnetic) dipole configuration becomes energetically favorable. The state of a system containing a large number of dipoles is determined by the competition of these trends. Calculations show that the ground magnetic state of one-dimensional chains [1], two-dimensional (2D) square [2] and rectangular [3] lattices, as well as 3D cubic lattices [4, 5] composed of point magnetic dipoles is not ferromagnetic: the dipolar (non-exchange) ferromagnetism in such systems is impossible. This conclusion is also valid for the systems composed of homogeneously magnetized (single-domain) spherical granules, because the field of each granule coincides with the field of an equivalent point dipole placed at the center and the dipolar magnetic interaction energy of a pair of such granules coincides with that of two equivalent point dipoles [6]. It was found, however, that lowering of the symmetry of any element of the system (lattice type or the shape of granules) may facilitate its transition to a magnetic state more advantageous from the energy point of view. A different situation is observed for cubic lattices partially populated or consisting of homogeneously magnetized granules of nonspherical shape and for tetragonal lattices (obtained by the uniaxial deformation of cubic ones). We assume that the exchange interaction establishes only a ferromagnetic ordering of elementary magnetic moments in granules, while the interaction between the granules is of the pure dipole type (this means that the separation between the granules is larger than the characteristic range of the exchange interaction). In addition, we confine our analysis to granules in the form of (prolate and oblate) ellipsoids of revolution or their extreme forms – point-like (spherical), rod-like, and disk-shaped granules. In all cases (except for granules in the form of oblate ellipsoids whose magnetic moment lies in the equatorial plane), we assume that the magnetic moment of a granule is directed along its easy-magnetization axis determined either by the geometry (rods and prolate ellipsoids) or crystal anisotropy (point-like or spherical granules). When the shape of a granule differs considerably from a sphere, its field at small distances from the surface differs from the field of an equivalent dipole placed at its center. For a prolate ellipsoid, its field at points close to the equatorial plane of the granule is much weaker than the field of an equivalent dipole; the opposite situation is observed for a granule in the form of an oblate ellipsoid.1 Consequently, a 2D lattice of nonspherical granules whose major axes lie in the lattice plane has a larger tendency to ferromagnetism (than for a spherical shape of granules) in the first case and to antiferromagnetism in the second case [7]. This situation is illustrated in Fig. 14.1. As can be seen, in the points close to the equatorial plane (in which the other granules preferring antiferromag1

The magnitude of the field in the equatorial (relative to the direction of the magnetic moment) plane constitutes 50% of the field at its axis for a spherical granule, 30% for a granule in the form of a strongly prolate ellipsoid of revolution, and 85% for a granule in the form of a strongly oblate ellipsoid of revolution.

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1.0

e=0 0.5 0.7 0.8 0.9 0.99

0.8

H

/H

0.6 0.4 0.2 0 -0.2

R/a = 2.2

-0.4 0

10

20

30

40

50

60

70

80

90

q, deg Fig. 14.1. The angular distribution of the magnetic field at a distance R = 2.2a from the center of an ellipsoidal granule (θ is the angle between the long axis of the ellipsoid and the direction to the point of observation, a is the semimajor axis, and e is the eccentricity of the ellipsoid)

netic ordering would occur), the field is significantly smaller than that of the equivalent dipole. This implies that the system exhibits a greater tendency to ferromagnetism as compared to the case of spherical granules. Therefore, we may expect that an increase in the nonsphericity of granules will inspire a transition from antiferromagnetic to ferromagnetic ground state of the system. In view of various fundamental, technological, and technical circumstances, real systems are virtually never symmetric (in the above sense), and their asymmetry is characterized by a variety of parameters. There are numerous examples of such systems, including planar periodic structures of nonspherical magnetic granules, which are being extensively investigated at present and are treated as the media with considerable potentialities for elevating the magnetic recording density [8]. Typical structures of this kind have the form of 2D rectangular lattices of one-domain extended magnetic granules with uniaxial geometric anisotropy. The shape of such granules resembles an ellipsoid of revolution with an axes ratio 3–5; the period of such lattices is comparable with the granule size. Another example related to the problem under investigation is magnetic dielectric nanocomposites. The electrical conduction of such a (3D) system is due to tunnel electron transitions between the granules [9], whose probability is determined by the mutual orientation of magnetic moments of adjacent granules. Consequently, it is clear that the resistance of such a medium depends directly on its magnetic state. The same refers to the (giant) magnetoresistance of such a system [10]. Finally, it is appropriate to mention a slightly unexpected object in the context of our discussion, namely, ultrathin films of ferromagnetic metals on monocrystalline substrates. It was found that for a certain effective thickness of such films, a long-range

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ferromagnetic order is established in them [11]. It was shown in a recent publication [12] that, under certain conditions, such films grow through natural lithography, that is, via the formation of nuclei which distributed more or less uniformly over the substrate area and which are transformed, during their subsequent growth, into ellipsoidal granules whose major axes are parallel to one another. Thus, systems of the type under investigation are quite diverse. This work aims at an analysis of their magnetic properties such as the ground state, the magnetic phase diagram, and magnetic phase transitions. In contrast to known publications (see, e.g., [4, 5, 13]) in which similar systems were investigated, we consider more general oblique lattices formed by nonspherical (or, which is the same, non-point-like) dipoles. It is shown that the inclusion of these new circumstances modifies the properties of such systems considerably. Another problem is originated from the fact that the traditional mean-field theory of magnets, while taking into account thermal fluctuations of interacting magnetic moments, does assume that their local fields are identical. This assumption is not true for random systems with configurational disorder. The latter leads to spatial fluctuations of a local field, which, in contrast to thermal fluctuations, precludes the establishing of magnetic order even at zero temperature. In the systems with long-range interactions, to which our random three-dimensional Ising point-dipole lattice belongs, this is supplemented by the necessity of including the anisotropy of this interaction. Thus, to adequately describe disordered magnetic systems consisting of ferromagnetic granules, the traditional mean-field theory must be generalized. The character of generalization depends on the type of random process. In liquid random systems, the point dipoles can be situated at any site of the space, and, to a first approximation, the correlation in their spatial arrangement need not be taken into account. The corresponding random process is Markovian, and, to determine the distribution function for random magnetic fields in such a system, one can use the Markov theory [14]. The magnetic properties of such systems have been considered in [15]. In random lattice structures, dipoles can (randomly) occupy only the sites of a certain regular (crystal) array. The method [14] does not apply to such systems, because they are not Markovian. It is the purpose of this work to consider the magnetic properties of a non-Markovian three-dimensional random lattice. The properties of Ising ferromagnets with short-range (exchange) interaction between magnetic moments have been studied by the statistical [16] and percolation [17] methods. Unfortunately, these methods are unsuitable for the systems with long-range (in particular, dipolar) interactions. In this case, it is necessary to appropriately modify the mean-field theory for random systems. The model of a random dipole lattice described in this work can be used to study the properties of various physical objects. Among these are materials with giant magnetoresistance (magnetic nanogranules in a nonmagnetic metallic matrix), transparent ferromagnets (including those in a polymeric

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matrix), frozen ferromagnetic fluids, and crystalline systems with partially substituted magnetic ions. In the final section of this paper, the obtained results are used to describe the properties of a number of real systems.

14.2 Lattices of Point-like and Rod-like Ferromagnetic Granules with Dipole Interaction Let us consider a system of identical one-domain ferromagnetic granules whose centers are at the sites of a 3D tetragonal lattice with periods lx , ly , and lz , and whose magnetic moments µ are identical and parallel to one another. Such a magnetic anisotropy may be due to crystal anisotropy of granules or anisotropy in their shape, or is manifested (in the absence of the former anisotropies) in a weak external magnetic field (and become decisive at a low temperature). Further, we assume that these moments form angle θ (latitude) with the x axis and are turned through an angle φ (longitude) about the x axis relative to the xy plane. On account of the long-range dipole–dipole interaction, the  magnetic state of such a system is determined by the magnetic field Hd = m, l, n Hmln created in the volume of an individual granule (located, for definiteness, at the origin of coordinates) by all the remaining granules. Here, Hmln is the field component created at the origin by a granule with the center at point (mlx , lly , nlz ), where m, l, and n are integers, and parallel to magnetic moments of the granules; the sum does not contain the term with m = l = n = 0. The energy of interaction between the chosen granule and the magnetic field Hd is given by  Wd = −(µ/V ) Hd (r) dV, (14.1) V

where integration is carried out over the volume of the central granule. The ground state of the system corresponds to the configuration of the magnetic moments µmln of the granules, for which the energy Wd attains its maximum value. In our case, there exists only one ferromagnetic configuration (the magnetic moments of granules are directed along their major axes and are parallel to one another); for the antiferromagnetic state of the system, we confine the analysis to the situation when the magnetic moments of granules form two identical magnetic sublattices of antiparallel magnetic moments. In this case, µmln = µ exp[iπ(φx m + φy l + φz n)], where phases φx , φy , φz may assume the values 0 or 1; for the ferromagnetic state, φx = φy = φz = 0 or φx = φy = φz = 1, while six different antiferromagnetic states correspond to different combinations of these phases, in

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which at least one phase is equal to zero and at least one phase is equal to unity.2 Let i1 be the absolute value of magnetization of each of the sublattices (the total magnetization of the system is i = 2i1 in the ferromagnetic state and i = 0 in the antiferromagnetic state). In accordance with the mean-field theory, the energy of the dipole–dipole interaction is proportional to magnetization i1 : Wd = λi1 . From all the states under investigation, the one that is most advantageous from the energy point of view corresponds to the highest value of the dipole interaction parameter λ. Thus, to determine the type of the ground state of the system under investigation, it is sufficient to find and compare the values of these parameter corresponding to the ferromagnetic and various antiferromagnetic configurations of the magnetic moments of the granules. As regards the magnetization i1 of the sublattices, it is defined by the self-consistent equation of the mean-field theory, i1 = i0 tanh [λi1 /kT ] ,

(14.2)

where i0 = (1/2)µNg , Ng = 1/(lx ly lz ) being the granule concentration. This equation determines, as usual, the Curie temperature TC = λi0 /k of the system. It should be noted, however, that it is applicable only in the case when the difference in the energies Wd of the ferromagnetic and antiferromagnetic states of the system is larger than kT . Otherwise, we must take into account thermal fluctuations of the magnetic order. While calculating the dipole interaction energy, the following circumstance should be borne in mind. It was noted in [4] that the energy WF of the ferromagnetic state strongly depends on the sample shape: −WF ∝ λF ∝ (N − 4/3), where N is the factor of demagnetization in the direction of the magnetic moment (the energy of any antiferromagnetic state is independent of N ). All the results described below correspond to spherical (actually, cubic) samples for which N = 4π/3 irrespective of parameters βy and βz of lattice extension along the y and z axes. We consider below the following three situations: (1) point-like (zerodimensional), but anisotropic granules, (2) rod-like (one-dimensional) granules magnetized along their axes, and (3) three-dimensional granules in the form of prolate and oblate ellipsoids of revolution (including the extreme case of an oblate ellipsoid, viz., two-dimensional disks). The first two situations are interesting since, on the one hand, they permit an exact solution, and on the other hand, are limiting cases for the third, much more real situation, which can be described only numerically. 2

Any other set of integral phases φx , φy , φz is identical to one of the eight sets enumerated above.

14 Ferromagnetism of Nanostructures

275

14.2.1 3D Lattice of Point-Like Granules In this case (which is realized when the separation between the granules is much larger than their size), the magnetic field component for each granule at the origin under investigation is given by 
1 3(e0 Rmln )2 iπ(φx m+φy l+φz n) Hmln = µ − 3 + e , (14.3) 5 Rmln Rmln where e0 = (cos θ, sin θ sin φ, sin θ cos φ) is the unit vector parallel to the magnetic moments of the granules, Rmln = (mlx , lly , nlz ) is the radius vector of the granule (m, l, n). In this case, the magnetic interaction energy has the form   
1 3(e0 Rmln )2 − 3 + Hmln = µ Wd = −µHd , Hd = 5 Rmln Rmln m,l,n

m,l,n

× eiπ(φx m+φy l+φz n) .

(14.4)

Taking into account the fact that all sums whose terms are odd relative to the summation variables m, l, n are equal to zero for a tetragonal lattice, we can present the dipole interaction parameter λ in the form [18]   λ = 2µβy βz λ0 cos2 θ + λπ/2 sin2 θ + 3λϕ sin2 θ sin2 ϕ , (14.5) where βy = ly /lx and βz = lz /lx are the lattice extension coefficients along the y and z axis, respectively, and  λ0 =  λπ/2 =  λϕ =

4π 3βy βz 4π 3βy βz 4π 3βy βz



∞ 

(2Xmn − Ymn − Zmn ) ,

m,n=−∞   ∞ m,n=−∞   ∞

(2Zmn − Ymn − Xmn ) , (Ymn − Zmn ) ,

m,n=−∞

where Xmn =

∞ 

(x)

−αmn l (1/l + α(x) , mn ) cos(lπφx ) e

l=1



α(x) mn

= 2π

Ymn =

∞  l=1

m + φy /2 βy

2

 +

n + φz /2 βz (y)

2 1/2

−αmn l (1/l + α(y) , mn ) cos(lπφy ) e

,

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E. Meilikhov and R. Farzetdinova

 α(y) mn

 2

= 2π (m + φx /2) + ∞ 

Zmn =

n + φz /2 βz

2 1/2 ,

(z)

−αmn l (1/l + α(z) , mn ) cos(lπφz ) e

l=1



 2

α(z) mn = 2π (m + φx /2) +

n + φy /2 βy

2 1/2 .

The evaluation and comparison of the values of the dipole interaction parameter λ for different magnetic configurations makes it possible to construct the magnetic phase diagram of the system. In the case under investigation, when there exist six simple two-sublattice antiferromagnetic configurations along with the ferromagnetic configuration, such a diagram may be quite complicated. Its form is determined by four parameters corresponding to lattice extension (βy , βz ) and the direction of the magnetic moment of the dipoles (θ, φ). Figure 14.2a, b shows, by way of an example, the angular dependences of the dipole interaction parameters for two different magnetic configurations, one of which corresponds to a cubic lattice of dipoles with magnetic moments perpendicular to the y axis (φ = 0), while the other corresponds to a tetragonal lattice with dipoles oriented so that φ = 0. It can be seen that the ground state of the simple cubic lattice of point-like dipoles is antiferromagnetic, although the specific form of the corresponding magnetic configuration is determined by the values of the above parameters. A more detailed analysis confirms the general nature of this conclusion for any tetragonal lattice of point-like dipoles. 6 λ

λ 010

011

λ 110

4

30

λ/2µβyβz

0

λF

λ 111

-2 -4 -6

λ 101

-8 -10 -12

λ 100 AF(011) 0

15

φ=0 βy=βz=1

λ 001

AF(010)

AF(110)

30

45

θ, deg

60

75

90

λ/2µβyβz

2

AF(001) AF(010) AF(100)

20 λ

001

10 λ

101

0 λ

λF

010

-10

λ 110

-20

-30 λ 0

λ 011

βz=0.5 o φ=25

100

15

λ 111

βy=1.5

30

45

60

75

90

θ, deg

Fig. 14.2. Angular dependences λ(θ) of the dipole interaction parameters for a cubic (a) and tetragonal (b) lattice of point-like dipoles. The domains of ferromagnetic (F) and antiferromagnetic (AF) states are indicated. Subscript F correspond to the ferromagnetic configuration, and subscripts 100, 110, etc. correspond to the values of phases φx , φy , and φz

14 Ferromagnetism of Nanostructures

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14.2.2 2D Lattice of Point-Like Granules For obvious reasons, it is more convenient to analyze the properties of the system under investigation for a 2D lattice of anisotropic point-like magnetic dipoles. To obtain the required relations, it is sufficient to set l = 0 in all the above formulas. In this case, we obtain the following expression for the dipole interaction parameter:3    µβ  (3 cos2 θ − 1)Σm + β 2 (3 sin2 θ − 1)Σn , (14.6) λ=2 x where β ≡ βz , and Σm =

 m2 eiπ(φx m+φz n) , (m2 + β 2 n2 )5/2 m,n

Σn =

 n2 eiπ(φx m+φz n) . (m2 + β 2 n2 )5/2 m,n

(14.7)

Simple approximate expressions for these sums, which correspond to different magnetic configurations, have the following form [18]: (Ferromagnetic configuration: φx = 0, φz = 0) Σm = (4π 2 /9β){1 + (24/β 2 )[K2 (2π/β) + 5K2 (4π/β) + 10K2(6π/β)]}, Σn = (4π 2 /9β 4 ){1 + 24β 2 [K2 (2πβ) + 5K2 (4πβ) + 10K2(6πβ)]}, (Configuration S01 : φx = 0, φz = 1) Σm = (8π 2 /3β 3 )[K2 (π/β) + K2 (2π/β) + 10K2(3π/β)], Σn = −(2π 2 /9β 4 ){1 + 48β 2 [K2 (2πβ) + 3K2 (4πβ) + 10K2 (6πβ)]}, (Configuration S10 : φx = 1, φz = 0) Σm = −(2π 2 /9β){1 + (48/β 2 )[K2 (2π/β) + 3K2 (4π/β) + 10K2 (6π/β)]}, Σn = (8π 2 /3β 2 )[K2 (πβ) + K2 (2πβ) + 10K2(3πβ)], (Configuration S11 : φx = 1, φz = 1) Σm = −(8π 2 /3β 3 )[K2 (π/β) − K2 (2π/β) + 10K2 (3π/β)], Σn = −(8π 2 /3β 2 )[K2 (πβ) − K2 (2πβ) + 10K2 (3πβ)]. where K2 is the Macdonald function. The angular dependences λ(θ) of the dipole interaction parameter calculated using formula (14.6) using the exact formulas (14.7) and the approximate relations (ferromagnetic and antiferromagnetic S01 , S10 , S11 magnetic configurations, respectively) are presented in Fig. 14.3a, b. For a moderately deformed 3

Relation (14.11) can be written in the symmetric form λ = λ0 cos2 θ +λπ/2 sin2 θ, where λ0 = 2(µβ/x )(2Σm − β 2 Σn ), λπ/2 = 2(µβ/x )(2β 2 Σn − Σm ) are λ-values corresponding to the angles θ = 0 and θ = π/2, respectively.

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E. Meilikhov and R. Farzetdinova 6

40 AF(01)

λF

0

λ 11

-2 -4

AF(01)

AF(10)

F

0

15

30

45

20

λ 01

10

λ 11

λF

λ 10

0 -10 -20

β=1

-6

AF(10)

F

30

λ 10

λ 01

2

λ /(2µβ/lx)

λ /(2µβ /lx )

4

β=0.5

-30

60

75

90

0

15

30

θ, deg

45

60

75

90

θ, deg

Fig. 14.3. Angular dependences λF (θ), λ10 (θ), λ01 (θ), and λ11 (θ) of the dipole interaction parameters for a (a) quadratic and (b) rectangular lattice of pointlike dipoles. Solid curves are calculated using exact formulas (14), and symbols correspond to approximate relations (17)–(24). The domains of ferromagnetic (F) and antiferromagnetic (AF) states are indicated 90 80

AF (10)

q, deg

70 q

60 50

F

40 30

lx

20 10 0,1

AF (01)

lz 1

10

lz / lx Fig. 14.4. Magnetic phase diagram of a rectangular lattice of point-like dipoles parallel to one another. The ground state of the system corresponds to the antiferromagnetic configuration S01 for small values of the angle of inclination θ, and to the antiferromagnetic configuration S10 for large angles. The hatched region corresponds to the ferromagnetic state

rectangular lattices (0.5 < β < 2), relations (17)–(24) provide a result that practically does not differ from the exact result; however, the error increases with the strain (the error becomes equal to 4% for β = 2.5 or β = 0.4). The evaluation and comparison of the values of the dipole interaction parameter λ for different magnetic configurations makes it possible to construct the magnetic phase diagram of the system (see above). Figure 14.4 shows such a diagram for a rectangular lattice of point-like coordinate system, and the center of another dipole like (bur anisotropic) dipoles. It can be seen that for a slight deviation of the direction of such dipoles from the sides

14 Ferromagnetism of Nanostructures

279

of the lattice, its ground state is always antiferromagnetic (irrespective of the ratio of the lattice periods). On the contrary, for large angles of inclination of the dipoles, the ground state of the lattice becomes ferromagnetic. 14.2.3 3D Lattice of Rod-Like Granules An analysis of a lattice formed by rod-like granules is interesting since the parameters of the magnetic phase diagram of such a lattice can be determined to a high degree of accuracy on the basis of the well-known simple analytic expression for the energy of interaction of such dipoles and can be used for estimating the errors in a rough model of lattices formed by 3D ellipsoidal granules (see below). If the center of one of such dipoles is at the origin of the coordination system, and the center of another dipole is determined by the radius vector Rmln , and if such dipoles are parallel, the energy of their interaction can be written in the form [7]   1 1 µ2 2 − − , (14.8) wd = 2 a |Rmln | |Rmln + 2ae0 | |Rmln − 2ae0 | where µ and 2a are the magnetic moment and the length (rod-like dipoles of a finite length) of each dipole, respectively. It follows hence that λ = 2µβy βz γ 2



 eiπ(φx m+φy l+φz n)

m,l,n

− −

2 (m2 + βy2 l2 + βz2 n2 )1/2

(14.9)

1 [(m + 2γ −1 cos θ)2 + (βy l + 2γ −1 sin θ sin ϕ)2 + (βz n + 2γ −1 sin θ cos ϕ)2 ]1/2 [(m − 2γ −1 cos θ)2 + (βy l − 2γ −1

 1 , sin θ sin ϕ)2 + (βz n − 2γ −1 sin θ cos ϕ)2 ]1/2

where γ = lx /a. 14.2.4 2D Lattice of Rod-Like Granules In this case, the general formula (14.9) is simplified as follows:    µβ λ=2 eiπ(φx m+φz n) (14.10) x m,n  
1 1 × (m2 + β 2 n2 )1/2 [(m + 2γ −1 cos θ)2 + (βn + 2γ −1 sin θ)2 ]1/2 
1 1 . + (m2 + β 2 n2 )1/2 [(m − 2γ −1 cos θ)2 + (βn − 2γ −1 sin θ)2 ]1/2

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E. Meilikhov and R. Farzetdinova

14.3 Lattices of Ellipsoidal Granules with Dipole Interaction 14.3.1 Magnetic Field of the Ellipsoidal Granule The method of approximate calculation of the energy Wd of intergranule dipole interaction for a lattice of 3D ellipsoidal granules was proposed in [19, 20]. This method is applicable for strongly prolate or strongly oblate uniformly magnetized ellipsoids of revolution and makes it possible to present the expression for the energy wmln of their pair interaction in the form of a series in the derivatives of potential ψ of the magnetic field created by such a granule (H = −∇ψ). Let the center of one of the grains be at the origin, while the center of another grain be at the point with coordinates x0 , y0 , z0 . If the magnetic moments µ of the grain are directed along the z axis, wmln = −µ H mln , where  1 [∂ψ(x, y, z)/∂x] dxdydz H mln = − V  1 [∂ψ(x , y  , z  )/∂x ] dx dy  dz  , =− (14.11) V V is the grain volume, and integration is carried out over the volume of the second grain. The coordinates x = x − x0 , y  = y − y0 , z  = z − z0 correspond to the coordinate system obtained by parallel translation of the initial system with the origin at the center of the second grain (x0 = mlx , y0 = lly ≡ βy lx , z0 = nlz ≡ βnlx ). In view of the geometrical anisotropy, the magnetic moments of the grain are always directed along their major axis, which will be assumed to be parallel to the x axis (it is the only major axis for a prolate ellipsoid of revolution and any of the major axes for an oblate ellipsoid). In this case, expanding potential ψ(x , y  , z  ) into the Taylor series we finally obtain for a prolate ellipsoid [18]  
 1  3 ψ0 (a, 0, 0) − ψ0 (−a, 0, 0) − ψ0x Hx mln = − 2 2a 2   4 2 3 2b b  + a2 1 + 2 ψ0xxx + O(ψ V ), (14.12) 20 a 3a where O(ψ V ) is the sum of terms proportional to the fifth-order derivatives of the magnetic potential. Similarly, we find that the central oblate ellipsoid with the equatorial plane xy creates in this plane a magnetic field H mln whose components are defined by the relations

14 Ferromagnetism of Nanostructures

Hx mln

281

 
 1  3 ψ0 (a, 0, 0) − ψ0 (−a, 0, 0) − ψ0x =− 2 2a 2
    2c2   2  3 2   ψ0xxx − ψ0xyy + 2 ψ0xxx + ψ0xyy + O(ψ V ), + a 20 3 3a (14.13)  
 1  3 ψ0 (0, a, 0) − ψ0 (0, −a, 0) − ψ0y 2 2a 2
    2c2   2  3   + 2 ψ0yyy + O(ψ V ). − ψ0xxy + ψ0yxx + a2 ψ0yyy 20 3 3a (14.14)

Hy mln = −

Numerical calculations show that the contribution of the term O(ψ V ) is negligibly small (less than 1%) in all cases of practical importance. Thus, the application of expressions (14.12)–(14.14) taking into account explicitly written terms ensures an accuracy not worse than 1% in the calculation of magnetic energy. As to the magnetic field potential which is generated by the central granule in the second granule volume, the relevant relations are known [21]: for a prolate ellipsoid    3µx ψ= (Arth t − t), t = e/ 1 + ξ, (14.15) 3 3 e a where e = (1 − b2 /a2 )1/2 is the ellipsoid eccentricity, ξ is the larger root of the equation [(y 2 + z 2 )/a2 ]/(1 − e2 + ξ) + (x/a)2 /(1 + ξ) = 1, while for an oblate ellipsoid, we have  
  t 3µx ψ= Arctg t − , t = e/ 1 + ξ, (14.16) e 3 c3 1 + t2 where e = [(a2 /c2 −1)1/2 ], ξ is the larger root of the equation [(x2 +y 2 )/c2 ]/(1+ e2 + ξ) + (z/c)2 /(1 + ξ) = 1. For a strongly elongated ellipsoidal granule (e → 1), (14.15) and (14.16) give a simple relationship between the magnetic field strengths H⊥ and H at the points situated in the equatorial plane and on the axis at a distance R from the center: √ √ H⊥ Arth(1/ 1 + r3 ) − (1/ 1 + r3 ) , (14.17) = H Arth(1/r) − [r/(r2 − 1)] where r = R/a. For R  a, this relationship yields a well-known result for the point dipole (H⊥ /H = 1/2), while for R = 2a (the case of contacting granules) we obtain H⊥ /H = −0.29. Thus, the relative magnitude of the field in the equatorial plane is significantly smaller for an ellipsoidal magnetic granule than for the point dipole or a spherical particle.

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E. Meilikhov and R. Farzetdinova

14.3.2 3D Lattice of Ellipsoidal Granules Relations (14.12)–(14.14), which determine the mean magnetic field H mln of the central granule, are written in the coordinate system in which the x axis coincides with the direction of magnetic moments (θ = 0). Using a linear coordinate transformation, we can easily generalize these equations to the case when the direction of these moments is characterized by angles θ = 0 and φ = 0. However, for a 3D lattice, we confine our analysis to a simpler situation with θ = 0. For a tetragonal lattice of ellipsoidal granules with magnetic moments parallel to the lattice side directed along the x axis, parameter λ determining the type of magnetic order is given by λ = 2µβy βz



[ H mln /H0 ] ei(φx m+φy l+φz n) ,

H0 = µ/3x .

(14.18)

m,l,n

14.3.3 2D Lattice of Prolate Ellipsoidal Granules For prolate ellipsoidal granules whose major axes lie in the plane of a 2D rectangular lattice (xz plane), in the general case (θ = 0), we obtain    µβ ˜ mn /H0 ) ei(φx m+φz n) , λ=2 [ H

(14.19) x m,n ˜ mn should be calculated using formulas (14.12) in where the mean field H

the coordinate system x ˜ = x cos θ + z sin θ, z˜ = z cos θ − x sin θ. This expression for the dipole interaction parameter λ was used for determining the limits of applicability of our approximate model for prolate ellipsoidal granules. A comparison of the results corresponding to a rectangular lattice and obtained on the basis of the exact formula (14.10) for rod-like dipoles and the approximate relation (14.19) for similar strongly prolate ellipsoids with the eccentricity e = 0.9999 (the ratios of the axes a/b ≈ 70) is illustrated in Fig. 14.5. It can be seen that the approximate model correctly reflects all features of the complex phase diagram of the system and leads to qualitatively correct results in the entire range of parameters, except in the situation close to contact of granules occurring due to their finite transverse dimensions. As before, the evaluation and comparison of the values of the dipole interaction parameter λ for different magnetic configurations of magnetic moments of granules makes it possible to construct the magnetic phase diagram of the system. The dependences of these parameters for a square lattice of ellipsoidal granules on the angle of inclination of their magnetic moments are presented in Fig. 14.6. Examples of magnetic phase diagrams for two lattices of ellipsoidal granules with different scaling ratios lx /a are shown in Fig. 14.7a, b. For a small value of this ratio, there exist lattices whose ground state is ferromagnetic even for θ = 0 (e.g., all lattices with β < 1 are of this type for

14 Ferromagnetism of Nanostructures 90

lx/a=3, e=0.9999 - ellipsoid - rod

80 70

q, deg

60

AF(10)

50 40

283

AF(11)

F

30 20 10

AF(01)

0.3

1

lz /lx

3

10

Fig. 14.5. Comparison of magnetic phase diagrams of a rectangular lattice of linear dipoles (solid curves) and strongly prolate (e = 0.9999) ellipsoids (points) of the same length 2a = (2/3)lx . The hatched region corresponds to contacting ellipsoidal grains

3

e=0.95, lx/a=3

2

lF

0 -1

l 01

-2

l 11

l 10

AF(10)

1

AF(01)

l /(2mb/lx )

4

F

-3

b=1

-4 0

15

30

45

60

75

90

q, deg

Fig. 14.6. Dependences of the dipole interaction parameters for a square lattice of ellipsoidal grains on the angle of inclination of their magnetic moments (cf. Fig. 14.3a). Ellipsoidal grains: e = 0.95, lx /a = 3

lx /a = 2.5). In addition, a comparison of Fig. 14.7a, b shows that upon a decrease in this ratio, the antiferromagnetic phase vanishes in the range of small angles θ, indicating the possibility for a transition of the system from the antiferromagnetic to the ferromagnetic state. It will be shown below that magnetic phase transitions in thin films of magnetic metals may be associated precisely with this feature of the phase diagrams of lattices formed by ellipsoidal granules. Unfortunately, the approximate nature of the model does not permit the exact determination of individual boundaries on the phase diagrams in all cases. For example, in the cases illustrated in Fig. 14.7a, b, the difference in the energies WF and W01 of the corresponding states in the range of parameters θ < 10◦ and β > 1 amounts to less than 0.1%. Consequently, it is impossible to

284

E. Meilikhov and R. Farzetdinova 90

90

e=0.95; lx/a =2.5

80 70

70 60

AF (10)

50

θ, deg

θ, deg

60

40

e=0.95; lx/a =3

80

AF(11)

30

F

40

20

10

10

1

3

lz/lx (a)

AF(11)

F

30

20

0.3

AF (10)

50

0

AF (01)

0,3

1

3

10

lz /lx (b)

Fig. 14.7. Magnetic phase diagrams of rectangular lattices of ellipsoidal grains with eccentricity e = 0.95 and lx /a = 2.5 (a) and 3 (b). The notation is the same as in Fig. 14.5. The boundaries of the upper hatched regions “forbidden” by the geometry of the problem correspond to contacting grains. Phase boundaries for θ  10◦ , β > 1 lie within the lower hatched regions

determine the actual type of magnetic ordering in this region in the framework of our model. We can only state that the boundary between these phases probably lies in the hatched rectangular regions. 14.3.4 2D Lattice with Oblate Ellipsoidal Granules Let us suppose that the equatorial planes of oblate ellipsoidal granules coincide with a plane of a 2D rectangular lattice (xy plane). In the absence of crystal anisotropy, their magnetic moments also lie in this plane, although are not attached to a certain direction. It would be unjustified from the physical point of view to assume that the ground magnetic state in this case corresponds to one of collinear antiferromagnetic configurations, say, of the type S01 studied in the previous analysis, in which the magnetic moments of the granules are parallel to one another. Among candidates for the ground state, we can consider, for example, the “fan” configuration of magnetic moments, where the latter are turned through angles ±χ relative to one of the lattice sides (see the inset to Fig. 14.8). Such states for a 2D square lattice of point-like magnetic dipoles were studied in [2], where it was proved that the energies of all fan configurations are identical irrespective of angle χ. It was found, however, that such a degeneracy is typical only of point-like dipoles and is removed as we pass to oblate ellipsoidal granules. In the latter case, the energy of a fan configuration can be calculated by the formula   µβ λ(χ) = 2 Λ(χ), (14.20) x

14 Ferromagnetism of Nanostructures 1,0

lx /a=5 2.5

0,8

lFan(c)/lFan(0)

285

c

c

lx

0,6 0,4 0,2 0

lx=ly

2a 0

15

30

45

60

75

90

c, deg

Fig. 14.8. Dependences of the energy of fan antiferromagnetic configurations for a square lattice of disk-shaped grains on the angle χ of rotation of their magnetic moments (see inset) for various distances between the grains. The inset shows the representation of a fan configuration in the form of wavy magnetic field lines

Λ(χ) =



eiπl [( Hx ml /H0 ) cos(χ − χml ) − ( Hy ml /H0 ) sin(χ − χml )],

m,l

where χml = χ eiπ(m+l) , the fields Hx ml and Hy ml are defined by relations (14.13) and (14.14) in the coordinate system x ˜ = x cos χ + y sin χ, y˜ = y cos χ−x sin χ, and the potential of the field created by the central granule is defined by relation (14.16), in which we must put ξ = (e2 +1)[(x2 +y 2 )/a2 −1]. Obviously, a noticeable difference from the lattice of pointlike dipoles can be expected only in the case when the shape of granules differs considerably from spherical and their size is comparable with the lattice period (l ∼ a). Figure 14.8 shows an example of the dependence of the energy of a fan-like antiferromagnetic configuration on angle χ, which is obtained in this way for very flat (c/a = 104 ) and closely spaced granules (disks) in a square lattice. It follows that any vortex configuration in this case is inferior in energy relative to the collinear configuration S01 , which represents the ground state. Calculations show that this conclusion remains in force for any rectangular lattice of disk-shaped granules. Similarly, the “wavy” configurations for which χml = χeiπl is also less advantageous from the energy point of view (see Fig. 14.9). 14.3.5 2D Lattice of Oblate Ellipsoidal Granules in a Magnetic Field The energy disadvantage of the ferromagnetic configuration of the ground state can be suppressed by applying an external magnetic field, which inevitably leads to a transition to the ferromagnetic state (if the field is strong enough). Depending on whether this magnetic field is directed along or across the magnetic moments of the sublattices of the initial antiferromagnetic state,

286

E. Meilikhov and R. Farzetdinova 1,0

1,0 1

i /2i0

0,5

-1,0

4

ly=6a

2a

lx

0

-0,5

2 3

0,5

Hy , Oe

Hy

0

ly

-0,5

-15 -10

-5

0

5

10

15

-1,0

Hy /(m/ly ) 3

Fig. 14.9. Field dependences of magnetization for various rectangular lattices of disk-shaped grains of diameter 2a. The magnetic field is directed along the longer lattice period ly = 6a. The smaller lattice period lx is equal to 6a (curve 1), 4a (curve 2), 3.33a (curve 3), and 3a (curve 4). The left inset shows a wavy configuration of the magnetic moments of grains in the field parallel to the longer side of the lattice. The right inset presents the experimental field dependences of magnetization of lattices of flat grains [22]

such a transition follows different scenarios. In one case, the phase transition occurs through the formation, growth, and coalescence of nuclei of the other (ferromagnetic) phase; this is the process that should be considered in the general theory of kinetics of phase transitions. In another case, the magnetic moment of granules rotate freely and coherently in their plane (under the assumption that there is no magnetic anisotropy in the crystal, as, for example, in the case of granules of a soft magnetic material). The moments of different sublattices have a tendency to align themselves in the direction of the magnetic field by rotating in the opposite directions. For the ground antiferromagnetic state S01 , such a transition occurs though a wavy configuration (which is disadvantageous in zero field). When the magnetic moments of the granules rotate through angle χ, the energy of their dipole interaction increases by ∆Wd (χ) = −∆λ(χ)i0 ≡ [λ(0) − λ(χ)]i0 , where the dipole interaction parameter λ is calculated by formula (14.20) for χml = eiπl . This energy increase for each granule is compensated by the decrease in its Zeeman energy, ∆Wd (χ) = µH sin χ, which gives H=

µ ∆Λ(χ) , lx3 sin χ

where ∆Λ(χ) = Λ(0) − Λ(χ). At the same time, the lattice magnetization is given by i = 2i0 sin χ =

µβ sin χ. lx0

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The last two relations determine the parametric relation between the magnetic field H and the equilibrium magnetization I in this field. Such a magnetization could be observed upon an infinitely slow variation of the field in the absence of magnetic anisotropy of any kind. A series of such field dependences for 2D lattices of disk-shaped granules is presented in Fig. 14.9. As the small lattice period (along which the moments of the granules in the initial antiferromagnetic state are directed) decreases, the magnetization saturation field of the system increases.

14.4 Partially Populated Lattices of Point-Like Ising Dipoles In this section we consider a three-dimensional cubic lattice, whose sites are randomly occupied by Ising point magnetic dipoles [23]. The fraction of filled sites is p and the dipole magnetic moments m are parallel to one of the lattice faces and take only two values: m = ±me0 , where e0 is a unit vector in the dipole direction. In the traditional theory, each dipole is subjected to the same local field H0 , which determines the mean magnetic dipole moment mikl T (angular brackets stand for ensemble averaging, and the subscript T denotes thermodynamic averaging). For Ising dipoles, H0  e0 , and one has  m exp(mH0 /kT ) m=±me0  = me0 th(mH0 /kT ), I = n mikl T , mikl T = exp(mH0 /kT ) m=±me0

(14.21) where I is the magnetization of the system and n is the dipole concentration. For a sample shaped like a prolate cylinder with its axis parallel to the direction of magnetic dipoles, the local field H0 is also parallel to its axis and equals 4π I + H3 . (14.22) H0 = 3 Here, (4π/3)I is the Lorentz field produced by the polarization magnetic charges at the surface of a sufficiently large sphere, H3 = m

 3 cos2 αikl − 1 , ρ3ikl

(14.23)

ikl

where the summation goes over the sites occupied by the dipoles inside the sphere, ρikl is the distance between a certain dipole (placed at the origin of coordinates) and the dipole at the site (ikl), and αikl is the angle between the line connecting these dipoles and the direction e0 .

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14.4.1 Distribution of Local Magnetic Fields Let the system with randomly distributed dipoles be in the state with mean magnetization I  e0 . The local magnetic fields H3 are different for different lattice sites and characterized by the distribution function Fp (η; H3 ), which depends on the fraction p of occupied lattice sites and the fraction η of dipoles whose average moments are directed along the magnetization I of the system. Let us consider the possibility of magnetic ordering in the random system of interest. The magnetization is I = pn mi T , where the average magnetic moment mi T should be calculated with allowance made for the scatter of random fields H3 through the generalization of regular (14.21). The corresponding mean random-field equation for the magnetization j = I/pnm has the form 
 ∞ H3 1 4π pj + Fp (j; H3 ) dH3 , tanh (14.24) j= Θ 3 mn −∞

where Θ = kT /m n is the reduced temperature (the magnetization j = 2η − 1 is one of the arguments of the function Fp ). The ground state (T = 0) of the system is ferromagnetic (j0 ≡ j(T = 0) = 1) if the integral on the right-hand side of (14.24) equals unity at j = 1. With allowance for the normalization of the distribution function, this condition means that the random fields H0 must be positive at all lattice sites. It is fulfilled if Fp (1; 4πI/3 + H3 < 0) = 0. The limiting case of a lattice whose sites are completely filled with dipoles (p = 1) was considered in [4], where it was shown that the character of the ground state of a long sample with a cubic dipole lattice depends on the lattice type; a simple lattice is always antiferromagnetic, whereas body-centered and face-centered lattices are ferromagnetic. In the two latter cases, the magnetic field H3 is zero at all lattice sites if p = 1. According to the abovesaid, this corresponds to the distribution function Fp (1; H3 ) = δ(0), which, clearly, satisfies the above-mentioned condition for ferromagnetism. To answer the question of whether the ferromagnetic state can be the ground state of a partially filled random lattice (p < 1), one must know how the distribution function Fp (j; H3 ) changes its form with a decrease in p. Inasmuch as no exact methods for determining the form of this function are presently available, we calculated it numerically for a body-centered cubic lattice of Ising point dipoles, whose magnetic moments were directed along one of the lattice faces. The dipoles were distributed uniformly and randomly (with probability p) over the sites of a 21 × 21 × 21 lattice, and the magnetic field H3 was calculated for the central site. The functions Fp (1; H3 ) were found by exhausting a large number (about 104 ) of realizations for this system. The resulting distribution functions Fp (1; H3 ) are shown in Fig. 14.10. At p ≈ 1, the distribution function has a quasi-deltalike form and is centered near zero. As the fraction p of filled sites decreases, the width of the distribution first increases and then decreases. This follows from the fact that 2

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6 4 2 0 1.2 0.9 0.6

p=0.01 -2

-1

0

1

2

-1

0

1

2

-1

0

1

2

-1

0

1

2

0

1

2

p=0.1

mnFp(1; H3)

0.3 0

-2

0.6 0.4

p=0.5

0.2 0

-2

0.9 0.6

p=0.9

0.3 -2 6 4

p=0.99

2 0 -3

-2

-1

3

H3/mn

Fig. 14.10. Local-magnetic-field distribution functions Fp (1; H0 ) over the sites of a body-centered cubic Ising-dipole lattice for dipoles of the same sign (j = 1) and various fractions p of randomly occupied sites

the distribution functions Fp (1; H3 ) and F1−p (1; H3 ) for the complementary systems with, respectively, the fractions p and (1 − p) of filled sites are related to each other as Fp (1; H3 ) = F1−p (1; −H3 ). One can see from Fig. 14.10 that, when p is close to unity, the distribution function corresponds to the ferromagnetic ground state. However, at relatively small p values, this function becomes nonzero in a broader range of H3 fields, which, according to the abovesaid, precludes the formation of a ferromagnetic state with j0 = 1. Therefore, with a decrease in p, the system undergoes a percolation magnetic phase transition from the ferromagnetic state to the magnetic (spin) glass state with j0 < 1. To determine the percolation threshold pc1 for this transition, we introduce the parameter Ω(p) characterizing the degree of penetration of the distribution function Fp (1; H0 ) into the region with H3 < 4πI/3. It is equal to the probability of a dipole configuration with such an H3 value occurring among various configurations with a given p value. In other words, Ω(p) is the fraction of configurations with H3 < 4πI/3 among all studied (for a given p) dipole configurations. The corresponding (numerically found) dependence is presented in Fig. 14.11. As in the standard percolation theory [17], the Ω(p) dependence follows the power law (it is linear in our case): Ω(p) ∝ (pc1 − p) with the percolation threshold pc1 = 0.55 ± 0.02. This value considerably (more than two fold) exceeds the value pc = 0.25 corresponding to the percolation magnetic phase transition from the ferromagnetic to paramagnetic state in a body-centered cubic magnetic-moment lattice with short-range (pair) interactions [17]. This is so because the net result of the long-range dipole–dipole interactions is more sensitive to the disorder in the system.

290

E. Meilikhov and R. Farzetdinova 0.30

Ω( p)

0.25 0.20 0.15 0.10 Ω ( p) µ ( p c1- p),

0.05

pc1 =0.55

0

0.1

0.2

0.3

0.4

0.5

0.6

p Fig. 14.11. Extrapolation method for determining the percolation threshold. Points are for the results of numerical calculations, and the dashed line represents the linear dependence Ω(p) ∝ (pc1 − p) (pc1 = 0.55)

Analogous calculations suggest that the face-centered cubic Ising-dipole lattice is even more sensitive to disorder; in the ground state, it remains a genuine ferromagnet (i.e., characterized by the magnetization j0 = 1) only in a very narrow interval pc1 < p < 1, where pc1 ≈ 0.95 (recall, in this connection, that the ground state of a simple cubic Ising-dipole lattice is not ferromagnetic at p = 1). 14.4.2 Magnetic Phase Diagram It was established above that, at p > pc1 , the ground state (T = 0) of our system corresponds to a magnetic glass with magnetization j0 < 1. The temperature dependence of magnetization and the Curie temperature of the system for a chosen p value is determined by (14.24). To find its solution, one should trace how the corresponding distribution function Fp (j; H0 ) changes upon varying j. An example of such evolution is shown in Fig. 14.12 for the distribution function F0.9 (j, H3 ). In the numerical calculations, the sign of the dipole magnetic moment was chosen at random on the condition that the fraction of dipoles with upward directed moments is η = (1 + j)/2. It is seen that the distribution width increases monotonically with decreasing magnetization (cf. Fig. 14.10 relating to the systems with j = 1 but various p), leading to a decrease in the Curie temperature. Using the calculated distribution functions Fp , one can evaluate the integral on the right-hand side of (14.24) and find its solution for different values of magnetization j and temperature. The temperature dependences of magnetization determined for two p values, one of which (p = 0.9 > pc1 ) corresponds to the ferromagnetic phase and the other (p = 0.5 < pc1 ) to the glass phase, are shown in Fig. 14.13. The width of the temperature range (0 < Θ < ΘC ) where the random system of interest possesses spontaneous magnetization j > 0 decreases with

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291

0.5 0.4

mnF0.9(0.1; H3)

0.3 0.2 0.1 0 0.5

-5

0.4

-4

-3

-2

-1

0

1

2

3

4

5

-1

0

1

2

3

4

5

-1

0

1

2

3

4

5

-1

0

1

2

3

4

5

0

1

2

3

4

5

mnF0.9(0.3; H3)

0.3 0.2 0.1 0 0.6 0.5 0.4 0.3 0.2 0.1 0 1.0

-5

-4

-3

-2

mnF0.9(0.5; H3)

-5

0.8

-4

-3

-2

mnF0.9(0.7; H3)

0.6 0.4 0.2 0 1.6

-5

1.2

-4

-3

-2

mnF0.9(1; H3)

0.8 0.4 0 -5

-4

-3

-2

-1

H3 /mn

Fig. 14.12. Local-magnetic-field distribution functions F0.9 (j, H0 ) over the sites of a random (p = 0.9) body-centered cubic Ising-dipole lattice for dipoles of different sign 1.0 0.8 0.6

0.9

p =0.5

j 0.4 0.2

0

0.5

1.0

1.5

2.0

2.5

3.0

Θ

Fig. 14.13. Temperature dependences of the magnetization of random (p = 0.9, 0.5) body-centered cubic Ising-dipole lattices. The dashed lines indicate the corresponding Curie temperatures calculated by (14.28)

decreasing p, and the system becomes paramagnetic in its ground state at dipole concentrations smaller than a certain critical value (p < pc2 ; see below). To describe the properties of the paramagnetic system in an external magnetic field He , one can again use the mean-field equation (14.24) with the

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E. Meilikhov and R. Farzetdinova

replacement H3 → He + H3 in the argument of tanh, following which this equation takes the form


∞ tanh

j= −∞

1 Θ



He + H3 4π pj + 3 mn

 Fp (j; H3 ) dH3 .

(14.25)

As before, of greatest interest is the low-field magnetic susceptibility   ∂I

, χ = lim ∂He He →0 for which one obtains from (14.25) χ=

pJ1 (p, Θ) , Θ − [(4πp/3)J1 (p, Θ) + ΘJ2 (p, Θ)]

(14.26)

where ∞ J1 (p, Θ) = −∞

Fp (0; H3 ) dH3 , ch2 (H3 /Θmn) 

∞ J2 (p, Θ) =

tanh(H3 /Θmn) −∞

∂Fp (j; H3 ) ∂j

 dH3 .

(14.27)

j=0

Analysis of the calculated distribution functions Fp (j; H3 ) showed that J2  J1 ; this greatly simplifies (14.26) for the susceptibility and the Curie temperature of the system. The Curie temperature is found from the condition that the denominator in (14.26) turn to zero; that is, it is the root of the equation ΘC = (4πp/3)J1 (p, Θ).

(14.28)

The concentration dependence of the reduced Curie temperature ΘC (p) calculated by (14.28) is shown in Fig. 14.14. It fits well the power law ΘC ∝ (p − pc2)1/2 , where pc2 = 0.35. The value ΘC ≈ 3 found for the maximal Curie temperature for p = 1 corresponds to the characteristic dipole–dipole interaction energy kTC ∼ m2 n (for LiHoF4 , m ≈ 13 µB and n ≈ 1.4 × 1022 cm−3 [24], which gives TC ∼ 1 K, in agreement with the experiment). These results can be used to construct the magnetic phase diagram for the random system considered (Fig. 14.15). In this diagram, the regions of ferromagnetic, glass, and paramagnetic states are plotted in the coordinates “dipole concentration–temperature.” To what degree this diagram is correct in describing the magnetic behavior of the systems of interest, one need only compare it with the experimental data on the properties of the LiHox Y1−x F4 compound. The experimental magnetic phase diagram for LiHox Y1−x F4 (inset in Fig. 14.15) closely resembles

14 Ferromagnetism of Nanostructures

35 0. = 2

ΘC 2 2 1/ c2

µ ΘC

1

0.6 0.03

-p (p

)

0.1

;

293

pc

r−rc2

0.3

1

Fig. 14.14. Concentration dependence of the Curie temperature of a random bodycentered cubic Ising-dipole lattice 3.0 2.5 2.0

Θ 1.5 1.0

PM SG ( j0 = 0) ( j0 < 1)

0.5 0

0.25

0.5

pc2

pc1

FM ( j0=1) 0.75

1.0

p

Fig. 14.15. Magnetic phase diagram of a random body-centered Ising-dipole lattice (FM ferromagnet, SG spin glass, PM paramagnet). Inset: experimental magnetic phase diagram of the LiHox Y1−x F4 compound

the diagram obtained in this work. One can thus believe that the magnetic properties of LiHox Y1−x F4 can be adequately described by the model of a random Ising-dipole lattice considered in this work. The temperature dependence of the susceptibility in the paramagnetic phase is given by (14.26). The corresponding χ(T ) curves shown in Fig. 14.16 indicate that the temperature behavior of the susceptibility at Θ  ΘC obeys the Curie law. However, the susceptibility of a system with nonzero groundstate magnetization does not fit the standard linear law χ−1 ∝ (T − TC ); as T → TC , it increases more slowly because of the destructive effect of configurational disorder. In conclusion, for random systems with magnetic interactions, the meanfield theory requires generalization and the taking into account the nonequivalence of individual magnetic moments, because their surroundings are different (random). As the fraction p of the sites occupied by dipoles decreases in a

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E. Meilikhov and R. Farzetdinova

10

p =1

8

1/c

6

p =0.2

0.5

4

1

0.5

Q 0

c

1

2 2

4

0.1

0.2 0.01 1

10

100

Q

Fig. 14.16. Temperature dependences of low-field magnetic susceptibility in the paramagnetic state of random (p = 0.99, 0.5, 0.2) body-centered Ising-dipole lattices. The vertical dashed lines indicate the corresponding Curie temperatures calculated by (14.28). Inset: the same for the reciprocal of magnetic susceptibility; the inclined dashed straight lines are the extrapolations of the linear high-temperature curves χ−1 ∝ (Θ − ΘC )

configurationally disordered three-dimensional Ising point-dipole lattice, the system undergoes a percolation transition from the ferromagnetic state to the glass state and then to the paramagnetic state. In contrast to the percolation phase transition in a system with short-range exchange interactions, our system provides an example of long-range percolation. The percolation thresholds for these two cases are substantially different. All the results of this section were obtained within the framework of a modified mean-field theory, which, like the traditional approximation, disregards spin correlations. In a system with long-range interactions, to which the threedimensional dipole–dipole interaction belongs, only long-range correlations are possible. The latter can be taken into account using the Ginzburg–Landau (GL) theory. As known in [25], the GL theory brings about results that differ from the mean-field results only in the temperature range near the critical temperature and which change only slightly the mean-field TC value and, hence, the magnetic phase diagram obtained in this work. One of the possible manifestations of spin correlations is that, in certain local configurations, the moments of some dipoles are in opposition to the magnetization of the system (i.e., j0 < 1 even at pc1 < p < 1). The magnetization of the system differs from unity if the distribution function Fp (1; H0 ) is nonzero in the region of large negative magnetic fields. Numerical calculations could have provided the answer to the question of how much this function is different from zero and how fast it decreases in this region. However, any numerical calculation based on exhausting a finite number of random realizations cannot give an answer to the question about the degree of penetration of the distribution function Fp (1; H0 ) into the region of large negative fields. It is seen only that this degree is exceedingly small, because the functions Fp (1; H0 )

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295

resemble Gaussian functions (Figs. 14.11 and 14.13), for which the system magnetization is close to unity with exponential accuracy.

14.5 Random Systems of Point-Like and Rod-Like Ising Dipoles 14.5.1 Introduction The magnetic state of a random system of interacting magnetic dipoles cannot be described within the framework of the traditional mean field theory since the latter is applicable only for homogeneous systems in which local fields are identical for all interacting magnetic fields. In contrast to thermal fluctuations, spatial fluctuations of a local field, which exist in a random system, prevent the establishment of a magnetic order even at zero temperature. For this reason, a correct description of such systems requires a more general approach. Zhang and Widom [26] used one of the possible ways for generalizing the mean field theory to analyze a random system of Ising spherical dipoles of a finite diameter.4 They proved that the ground state of such a system becomes ferromagnetic only when the mean distance between spherical dipoles is comparable to their size; otherwise (in the case of large distances between dipoles), the system is paramagnetic even at T = 0. In other words, the ferromagnetic state is possible only in a system in which the dipole concentration exceeds a certain critical value; this is generally in accordance with tendencies in the behavior of dipole system with configuration disorder [27]. However, the above result was obtained under two assumptions, both of which are dubious. First, the spatial dipole distribution function was factorized as is usually done in an analysis of random systems. This function was represented by the product of two identical one-particle distribution functions g(r) each of which depends on only one spatial coordinate, viz., the distance r between a given particle (dipole) and the other particle placed at the origin: g(r) ∝ r2 for r > a (a is the particle diameter) and g(r) = 0 for r < a. This is justified for a system of point particles (a = 0) but is not observed in the case considered here since the arrangement of a finite-size particle is determined by the position of not one, but many neighboring particles even in the absence of magnetic correlations. One could hope that this approximation would not lead to considerable errors in the case of strongly rarefied systems, for which the probability of a close neighborhood of more than two particles is low (na3  1, n being the particle concentration), but it was used in [26] for na3 ∼ 1. Second, the approximation used is equivalent to the assumption that the correlation length of spatial fluctuations of the magnetic moment 4

The energy of interaction of uniformly magnetized spherical dipoles (one-domain spherical granules) coincides with the energy of interaction of two equivalent point dipoles [6].

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directions of dipoles is larger than the dipole diameter, which can also only be justified for strongly rarefied systems. These assumptions set a limit on the applicability of the obtained conclusions. In this study, another approach is proposed, which makes it possible to obtain an exact result for point Ising dipoles and to trace the modification of this result for finite-size dipoles. Using this approach, a magnetic phase diagram could be obtained for a random system of Ising dipoles and the magnetic parameters of individual phases of this system are determined. 14.5.2 Generalized Mean Field Theory for Point (Spherical) Dipoles Let the magnetic moment mi of Ising dipoles assume only two values: mi = ±me0 , where e0 is the unit vector parallel to the direction of the dipoles, and let us suppose that a system with a random arrangement of dipoles is in the ferromagnetic state characterized by the average magnetization I  e0 . Local magnetic fields H3 are different for different dipoles and can be characterized by the distribution function F (j; H3 ), which generally depends on the relative magnetization j ≡ I/mn of the system determined by the fraction η = (1/2)(1 + j) of dipoles whose average magnetic moments are directed along the magnetization I of the system [F (1; H3 ) is the distribution function in the case when all the dipoles in the Lorentz sphere are parallel to one another]. This function can be determined exactly using a simple but quite reasonable model. For this purpose, we apply the Markov method for determining the probability of the sum of a large number of random quantities [14], according to which ∞ 1 F (1; H3 ) = A(q) exp(−iqH3 ) ds, (14.29) 2π −∞

⎡ ⎢ A(s) = lim ⎣



ρmax

N →∞

⎤N

⎥ exp[iqh(ρ, α)]τ (ρ, α) dαdρ⎦ ,

(α) ρ=0

where h(ρ, α) = m(3 cos2 α − 1)/ρ3 is the effective magnetic field produced by a point dipole with coordinates ρ, α at the origin (in the case considered here, this field is equal to the component of the magnetic field of the dipole along its direction), τ (ρ, α) is the distribution function for these coordinates, and N is the number of dipoles in a sphere of radius ρmax over which integration is carried out. If we assume further that (1) the random nature of arrangement of dipoles does not change their average concentration n (i.e., N → 4ρ3max n/3 for ρmax → ∞); (2) the distribution of angles α is uniform; and (3) coordinates ρ and α are not correlated, then

14 Ferromagnetism of Nanostructures

τ (ρ, α) dρdα = (4πρ3max /3)−1 ρ2 dρ 2π sin α dα.

297

(14.30)

Substituting (14.30) in (14.29), we obtain  A(q) = exp[−nC(q)],

C(q) = 2π (α)

∞ sin α dα

{1 − exp[iqH(ρ, α)]}ρ2 dρ.

ρ=0

(14.31) While integrating with respect to ρ, we must use the above expression for H(ρ, α) and take into account the fact that  ∞ [1 − exp(iu)] du/u2 = π/2. u=0

This gives  nC(q) = 2

π 2 mnq 3

 π/2 sin α|3 cos2 α − 1| dα = sh0 ,

(14.32)

0

√ where h0 = 8π 2 mn/9 3 is the characteristic field approximately equal to the field of a dipole separated from the origin by a distance on the order of the mean distance n1/3 between dipoles; factor 2 takes into account the existence of two regions equivalent to the angular interval 0 < α < π/2. For the distribution function F (1; H3 ), we finally obtain 1 F (1; H3 ) = 2π

∞ −∞

1 cos(|H3 |q) exp(−|q|h0 ) ds = π



h0 h20 + H32

 .

(14.33)

This is a Lorentzian distribution5 of width h0 , which is centered at the mean field H3 = 0. The result obtained, which differs in principle from the Gaussian distribution predicted in [26] for a similar system (in the approximate model), can be easily interpreted if we assume that strong fields H are mainly produced by nearest neighbors. The law wρ (ρ) for distribution of distances ρ to the nearest neighbor is defined by the formula [14] wρ (ρ) = 4πρ2 n exp(−4πρ3 n/3) ∝ ρ2 , while the field H produced by this distribution is proportional to 1/ρ3 . It follows hence that 5

The fact that distribution (8) predicts that H32  = ∞ is obviously connected with the assumption that the distance between dipoles can be infinitely small. Deviations from this distribution should be expected for fields H3 > m/ρ3min , where ρmin is the minimal possible spacing of dipoles (e.g., their characteristic size).

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E. Meilikhov and R. Farzetdinova

F (H) = wρ [ρ(H)]|dρ/dH| ∝ 1/H 2 , which correctly describes the wings of the Lorentzian distribution in the range of strong fields. A remarkable property of the obtained distribution, which is not obvious at first glance, is that it predicts equal probabilities for fields H3 of opposite polarities.6 It will be shown below that such a symmetric spread of Lorentzian fields hampers the emergence of magnetic ordering in the system under investigation. As a matter of fact, the noted property is preserved for any random distribution of these directions in spite of the fact that distribution (8) was obtained for a random system in which the directions of the magnetic moments of all dipoles coincide. In other words, the emergence of a magnetic order in a random system of dipoles does not in any way affect the form of the distribution function for the Lorentzian field; that is, F (j; H3 ) ≡ F (1; H3 ). Let us now demonstrate that magnetic ordering cannot exist in the random system of point Ising magnetic moments considered here. The magnetization of the system is defined as I = n mi T , where the average mi magnetic moment mi T must be calculated taking into account the spread in Lorentzian fields H3 using the obvious generalization of relation (14.21) pertaining to a regular system: ∞ I = n mi T = mn −∞

 m(4πI/3 + H3 ) F (j; H3 ) dH3 . tanh kT


(14.34)

Using expression (14.33) for the distribution function F (1; H3 ) ≡ F (j; H3 ), we derive the generalizing equation for mean field: 1 j= π

∞ −∞




1 tanh Θ

  ∞ 4π A arctan u du du 1 , Au + j ≡− 3 1 + u2 πΘ ch2 Θ (Au + 4πj/3) −∞

(14.35) where j = I/Is (Is = nm) is the reduced magnetization of the system, Θ = kT /m2 n is the reduced temperature (equal to the ratio of the thermal energy to the characteristic energy of magnetic interaction between dipoles), and √ A = 8π 2 /9 3 ≈ 5.065. The paramagnetic state (j = 0) is obviously a solution to this equation. Analysis shows that this equation has no other solutions,7 and this obviously corresponds to the absence of ferromagnetic ordering in a random system of point Ising dipoles. This conclusion coincides with the result 6

7

This is formally connected with the specific form of the angular dependence of the field of a point dipole, h(ρ, α) ∝ (3 cos2 α − 1), which, together with the angular dependence of distribution τ (ρ, α) ∝ sin α, ensures the equality of the angular factors for the regions 3 cos2 α − 1 < 0 and 3 cos2 α − 1 > 0 corresponding to negative and positive values of field H3 . As Θ → 0, the right-hand side of (10) asymptotically approaches (2/π) arctan(4πj/3A) < 8j/3A = 0.526j < j.

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299

obtained in [26], according to which the system could become ferromagnetic only for a high dipole concentration nρ3min > 0.300, where ρmin is the minimal spacing of spherical dipoles.8 In our model, we assume that min = 0 (not spherical but point-like dipoles); for this reason, the dipole concentration is always low. The impossibility of unlimited decrease in the distance between dipoles leads to a truncation of the values of magnetic field from above by a value of Hmax ∼ m/ρ3min . For this reason, we can easily pass to spherical dipoles in the framework of our model by truncating the wings of distribution function (14.33), which correspond to fields |H3 | > Hmax = m/ρ3min : ⎧   ⎨ h0 [2arctan(Hmax /h0 )]−1 , |H3 | ≤ Hmax . (14.36) f (1; H3 ) = h20 + H32 ⎩ 0, |H3 | > Hmax In this case, instead of (14.35), we obtain the following equation for the mean field: 1 j= 2arctan(Hmax /h0 )

Hmax  /h0




1 tanh Θ

  4π du Au + j . 3 1 + u2

(14.37)

−Hmax /h0

An analysis shows that this equation acquires the second (nonzero) root corresponding to spontaneous magnetization of the system for Hmax /h0 < 1.08 (at zero temperature). This condition can be written in the form of a criterion for the concentration of spherical dipoles (which is required for the emergence of spontaneous magnetization): nρ3min > 0.183; this criterion differs significantly from that obtained in [26] (n3min > 0.300). To ensure a ferromagnetic ordering at T > 0, the truncation of the distribution function F (1; H3 ) upon an increase in temperature must be more radical; at a constant concentration of dipoles, this can be ensured only by increasing the minimal dipole spacing. (Parameter ρmin is bounded from above by the natural geometrical condition nρmin ≤ 1.) However, at a high temperature (Θ ≥ 4.2), the system remains paramagnetic even for ρmin → ∞. The corresponding temperature dependences of spontaneous magnetization and threshold values of nρ3min and Hmax are shown in Figs. 14.17 and 14.18, respectively. The magnetization of the system in question in the paramagnetic state is determined by the external magnetic field He . To find the magnetization, it is sufficient to carry out the substitution 4πI/3 → 4πI/3 + He in (14.34), after which the argument of the hyperbolic tangent in (14.35) is supplemented with the term (He /h0 )Θ−1 . The solution of the equation obtained in this way 8

In [26], the criterion for the emergence of spontaneous magnetization in a random system of rigid spherical Ising dipoles of diameter a is written in the form (π/6)na3 > 0.157. The minimal distance between the centers of dipoles in this system is ρmin = a, which leads precisely to the above condition n3min > 0.300.

300

E. Meilikhov and R. Farzetdinova 1.0

He=0

0.8

j

0.6 0.4 0.2

1

Hmax/h0=1.05 0

1

0.5

0.8

2

3

4

Q

Fig. 14.17. Temperature dependences of spontaneous magnetization j of a system with a random field distribution truncated from above for different values of the truncation parameter (external field He = 0)

Hmax /h0

1.0

nr3 min

He=0

1.2 1.0

0.8

0.8

0.6

0.6

0.4

Hmax /h0

1.2

0.4

nr3 min

0.2 0

1

0.2 2

3

4

0

Q

Fig. 14.18. Temperature dependences of the threshold value of parameter nρ3min for which a spontaneous magnetic moment arises in a system of random Ising dipoles and of the maximum value Hmax = m/ρ3min of random field H3 corresponding to this value (He = 0)

describes the temperature dependence of the system magnetization in the applied magnetic field. Figure 14.19 shows a series of such dependences for various magnetic fields. As expected, these dependences differ considerably from the Langevin dependence, according to which magnetization I = Is under saturation (i.e., at low temperatures), which means that j = 1. It can be seen from Fig. 14.19 that the saturation of the magnetization of a random system always corresponds to j = j0 < 1.9 The field dependence of the low-temperature magnetization j0 (He ) is shown in the inset to Fig. 14.19.

9

For Θ → 0, (14.37) assumes the form j = (2/π)arctan(4j/3A + He /Ah0 ); for j  1, we have j0 = (2/π)(A − 8/3)−1 He = 0.265He .

14 Ferromagnetism of Nanostructures

301

-1 3

10

0.03

[m n f (H3)]

mn F (1; H3 )

104

0.04

2

10

0.02

10

2

3

10 104 2 (H3 /m n)

0.01

0 -60 -40 -20

0

20

40

60

H/mn Fig. 14.19. Temperature dependences of magnetization j of a system in the paramagnetic state for different values of the external magnetic field He . The inset shows the field dependence of magnetization at T = 0 (the dashed straight line depicts the linear dependence j0 = 0.265He )

Numerical Experiment: Comparison with Theory The magnetic behavior of random systems of point Ising dipoles is obviously determined to a considerable extent by the distribution function F (1; H3 ) of local magnetic fields. In this connection, it is expedient to obtain an additional proof of the correctness of the theoretical scheme used for determining this function. For this reason, in addition to the derived analytic expression (14.36) for function F (1; H3 ), we calculated this function by simulating a random system of Ising dipoles. The system was “created” using a random uniform arrangement of dipoles in a sphere (the total number of dipoles is 5 × 103 or 4 × 104 ), after which the magnetic field H3 was calculated at the center of this sphere. Function F (1; H3 ) was found by sorting out a large number (approximately 104 ) of realizations of such a system. The distribution function determined in this way is shown in Fig. 14.20. It can be seen that the shape of this function is close to the Lorentzian (solid curve), especially on the wings, where the function remains Lorentzian up to very high values of field H3 = 100 mn (see the inset to Fig. 14.20). At the same time, an attempt to approximate function F (1; H3 ) by a Gaussian function proved futile (dashed curve in Fig. 14.20). Since strong fields are mainly created by nearest neighbors, the shape of the distribution function in this region is not sensitive to the choice of the total number of dipoles taken into account in calculations. This, however, does not apply to the range of weak magnetic fields (H ≤ h0 ) produced by a more or less symmetric aggregate of a large number of neighboring dipoles. Verification (involving an increase in the number of dipoles taken into consideration) confirms the correctness of the results depicted in Fig. 14.20.

302

E. Meilikhov and R. Farzetdinova 1.0 He/h0=100 0.8 1

j0

0.6 0,1

j

10 0.4

He /h0

0,01 0,1

1 10 100

0.2

0 0.1

1

0.1 1

Q

10

100

Fig. 14.20. Distribution function F (1; H3 ) for a random system of point Ising dipoles, approximated by the Lorentzian (solid ) and the Gaussian (dashed ) curves. The inset gives a representation of function F (1; H3 ) demonstrating the correctness of the Lorentzian approximation in the range of strong fields

Conclusions We have proved that the system studied here can be in the ferromagnetic state in the case of appropriate truncation of the region of strong random fields (which is equivalent to a transition from point dipoles to finite-size dipoles). To find out whether this is the ground state, note that the energy density of the system at T = 0 is W = n w , where the mean energy of a dipole is given by w = − mH0 = [−ηm + (1 − η)m] (4π/3)I + H3

(averaging is carried out over the ensemble of particles), and I = mn(12η), η being the fraction of dipoles with magnetic moments directed along the magnetization. Since the distribution function F (1; H3 ) is even, we obtain

4π I + H3 = 3

∞  −∞

 4π 4π 4π I + H3 F (1; H3 ) dH3 = I= mn(1 − 2η) 3 3 3

and finally W = −m2 n2 (2η − 1)2 ≤ 0. The paramagnetic state (η = 1/2) corresponds to W = 0; in the ferromagnetic state, we have W < 0; consequently, the ferromagnetic phase predominates at low temperatures. It should be emphasized that all the results obtained in this study pertain exclusively to a liquid random system, in which magnetic dipoles can be arranged in any spatial region. For this reason, it would be incorrect to apply

14 Ferromagnetism of Nanostructures

303

these results for describing the properties of crystalline random systems, in which dipoles can occupy (with a certain probability) only quite definite positions in a certain crystalline structure. The method applied here is unsuitable for studying such systems since these are not random systems in the Markovian sense. In this case, the initial crystal should be divided into nonequivalent sublattices and the distribution functions for random fields in each sublattice should be calculated numerically. 14.5.3 Generalized Mean Field Theory for Rod-Like Dipoles Many real systems of this type (e.g., ferroliquids or patterned media, which are promising materials for magnetic recording [8]) consist of essentially nonspherical dipoles. For this reason, for a high concentration of such dipoles (when the mean distance between the dipoles is comparable with their size), the type of their interaction differs considerably from the interaction of point dipoles. We illustrate this statement for a system of rod-like uniformly magnetized dipoles with magnetic moments m. The interaction energy for two such dipoles with parallel magnetic moments (the center of one dipole is at the origin and the center of the other dipole is determined by radius vector R) is given by the relation [7]   1 1 m2 2 wd = 2 − − , (14.38) 4a |R| |R + 2ae0 | |R − 2ae0 | where e0 is the unit vector in the direction of the magnetic moment of the dipoles and 2a is the length of each dipole. If the angle between vectors e0 and R is equal to α, (14.38) can be written in the form  1 m2 2 1 , (14.39) − − wd = 3 4a ρ ρ2 + 4 + 4ρ cos α ρ2 + 4 − 4ρ cos α where ρ = |R|/a. For ρ  1, relation (14.39) is transformed into the wellknown expression for the energy of interaction between point dipoles: wd = m2 (1 − 3 cos2 α)/|R|3 . The sign of energy wd determines the type of the ground state of a system of two dipoles: if wd < 0, the magnetic moments of the dipoles are parallel (ferromagnetic state); otherwise (wd > 0), these moments are antiparallel (antiferromagnetic state). The plane containing both dipoles under investigation (for definiteness, the xz plane; the magnetic moments are directed along the z axis) splits into two regions with different signs of wd (and, hence, with different types of the magnetic ground state). Figure 14.21 shows these regions for point- and rod-like dipoles. For point dipoles, the relation between the sizes of these regions is independent of the distance between the dipoles, while the fraction of space corresponding to the antiferromagnetic ground state of a pair  of closely spaced rod-like dipoles (ρ = (x/a)2 + (z/a)2 ≤ 1) is much larger.

304

E. Meilikhov and R. Farzetdinova 4

x/a

2

-4

-2

FM

0

0

2

AFM

+1

0

-1

-4

-2

0

FM

0

-2 -4

4

z/a

+1

2

4

Fig. 14.21. Boundaries of the regions of ferromagnetic (FM) and antiferromagnetic √ (AFM) ground states of a system of two Ising dipoles. Dashed lines x = ±z 2 correspond to point dipoles and solid curves are linear dipoles of length 2a. Solid curves correspond to a constant energy of interaction of linear dipoles (the digits on the curves indicate energy wd in units of m2 /4a3 )

However, this does not lead to a conclusion on a strong tendency of a system of a large number of rod-like dipoles toward antiferromagnetism, since the degree of anisotropy of the interaction of rod-like dipoles differs significantly from that for point dipoles. Thus, nonsphericity of dipoles must considerably affect the magnetic state of a dipole system. This study is devoted to generalization of the mean field theory for a random system of rod-like dipoles. To determine the distribution function of local magnetic fields, we again apply the Markov method [14] with (see (14.29)) hζ (ρ, α, ζ) = ζh(ρ, α),

(14.40)

 1 2 m 1 h(ρ, α) = 3 − +  + 4a ρ ρ2 + 4 + 4ρ cos α ρ2 + 4 − 4ρ cos α is the effective magnetic field produced by a point dipole with random coordinates ρ, α at the origin (in the case considered here, this field is equal to the component of the magnetic field of the dipole along its direction); the random parameter ζ assumes values +1 and −1 (with a probability η and 1 − η, respectively) and determines the direction of the magnetic moment of the dipole, τ (ρ, α, ζ) is the distribution function for random values of coordinates and parameter ζ, and N is the number of dipoles in a sphere of radius ρmax , over whose volume the integration is carried out. If we assume further that (1) the random nature of the arrangement of dipoles does not change their average concentration n (i.e., N → 4π(aρmax )3 n/3 for ρmax → ∞); (2) the distributions of coordinates and parameter ζ are uniform; and (3) correlations are absent, then

14 Ferromagnetism of Nanostructures



305



3 [(1 − η)δ(ζ + 1) + ηδ(ζ − 1)] sin α dρdζdα. 4πρ3max (14.41) In that case, relations (14.31) do not lead to a simple analytic expression for the distribution function F (j; H3 ). For this reason, we determined this function using two mutually complementing methods: (1) the small q approximation based on the fact that the region of large values of q is insignificant in the inverse Fourier transformation of (3) and (2) numerical calculations for a model random system of rod-like Ising dipoles. In the first approach, function exp[qh(ρ, α)], which is to be integrated in (14.31), is replaced by its approximate power expansion in the small argument qh (up to the first nonvanishing term in qh). After this, it becomes possible to obtain the following simple expression for the distribution function [15] 
1 (H3 + jHj )2 F (j; H3 ) = √ , (14.42) exp − 2σ 2 2πσ τ (ρ, α, ζ) dρdαdζ = 2πρ2

where Hj /h0 = (4π/3) ≈ 4.2,

√ σ/h0 = ( π/2)(na3 )−1/2 ≈ 0.89(na3 )−1/2 ,

h0 = mn.

Thus, in the small q approximation, distribution functions F (j; H3 ) have the form of Gaussian functions whose maximum (H3 = −jHj ) is shifted linearly upon an increase of magnetization to the region of negative values of magnetic field H3 .10 The approximation used here is valid if the probability of the emergence of strong local fields for which qh ≥ 1 is low. The Gaussian distribution function obtained above (with exponentially decreasing wings) does not possess such a property. However, as the concentration of rod-like dipoles decreases, their finite size becomes less and less significant, and such dipoles can be treated as point dipoles for na3 ≤ 1. In this case, the system is known to be characterized by a Lorentzian distribution function [28] with long wings decreasing according to a power law. Consequently, distribution (8) holds only in the concentration region na3 ≤ 1. On the contrary, strong fields are mainly created by the nearest neighbors [14], and the probability of their production increases with the dipole concentration n, when the mean distance between dipoles becomes smaller than the dipole size a. Consequently, the approximation used becomes inapplicable for na3 ≥ 1. To determine the limits of its applicability more precisely, we numerically calculated the local field distribution functions in a model random system of rod-like Ising dipoles. The system was “created” via a uniform random arrangement of dipoles over a sphere (the total number of dipoles is 104 ), after which the magnetic field H3 at the center of this sphere was calculated. Functions F (j; H3 ) were 10

Formally, this is connected with the specific form of the angular dependence (4) of the rod-like dipole field. As was mentioned earlier, there is an increase (as compared to point dipoles) in the fraction of space corresponding to the antiferromagnetic interaction of a pair of closely spaced rod-like dipoles (ρ ≤ α).

306

E. Meilikhov and R. Farzetdinova

h0 F( j; H3)

4

na3 = 0.01

6

h0 F(1; H3)

0.3

0.2 2 0 −0.2

0.1

0 −15

−10

−5

0

5

0

0.2

H3/h0

10

15

H3/h0 Fig. 14.22. Distribution functions F (0; H3 ) and F (1; H3 ) of local fields for a model system of randomly distributed linear Ising dipoles with na3 = 1. Solid curves correspond to approximated Gaussian functions. The discrepancies are most noticeable on the left wings of the distributions. The inset shows the distribution function F (1; H3 ) for a system with na3 = 0.01 close to a system of point dipoles (the solid curve corresponds to the Lorentzian approximation, and the dashed curve, to the Gaussian function)

determined by running through a large number (about 104 ) of realizations of such a system. Figure 14.22 shows the distribution functions F (j; H3 ) determined as a result of such a numerical calculation for a system with na3 = 1. It can be seen that these functions are close to Gaussian functions, their widths and the positions of peaks being close to those predicted by relations (8). Figure 14.22 also shows that the shape of the distribution functions in the wings differs considerably from the Gaussian shape (especially for fields lying to the left of the peak). These region correspond to strong (in absolute magnitude) fields, which cannot in principle be described correctly by the approximation of small q. The shape of the distribution function F (1; H3 ) in this region has been determined in [28]:   4 3   π na3 π 3 H1 H1 (H < 0), (14.43) F (1; H) ∝ exp − na 2 H1 H 6 |H|    32πna3 sh[4πna3 (H1 /H)] (H > 0). exp − 3 4πna3 (H1 /H) (14.44) It follows from (14.43) and (14.44) that the left wing of the distribution function F (1; H) must be higher than its right-hand wing. This can clearly be seen in the inset to Fig. 14.22. For a low dipole concentration (na3  1), both wings of the distribution function have, in accordance with (14.43) and (14.44), identical power form π F (1; H) ∝ 2



na3 H1



H1 H

4

14 Ferromagnetism of Nanostructures

 F (1; H) ∝

na3 H1



H1 H

307

4 ,

(14.45)

which considerably differs, however, from the Lorentzian profile.11 As regards the region of weak magnetic fields (H < H1 ), such fields are produced by a more or less symmetric set of all neighboring dipoles and it is impossible to determine the form of the corresponding distribution function f (H) using the nearest neighbors approximation. In this case, an appropriate statistical method (e.g., the Markov method) has to be used. As the dipole concentration decreases to values na3  1 and the system becomes equivalent to a random system of point dipoles, the shape of the distribution indeed becomes close to Lorentzian shape. This can be seen from the inset to Fig. 14.22, corresponding to a system with na3 = 0.01: the right wing of the distribution function is close to the Gaussian distribution, while its left wing is already Lorentzian. Figure 14.23 shows the dependences of parameters Hj and σ of the distribution function F (1; H3 ) on the dipole concentration determined by the value of na3 . It can be seen that the functional dependences Hj /h0 = const and σ/h0 ∝ (na3 )−1/2 , which are predicted by the small q approximation, are observed in the concentration range 1 ≤ na3 ≤ 103 , that is, for systems in which the mean distance between dipoles is 1 − 10 times smaller than their size. 10

Hj /h0 Hj /h0 , s/h0

1 10 10 10

10

s /h0

-1

µ

-2

(n a3 )

1/2

-3

FM

PM

-4

-4

-3

-2

-1

2

3

4

5

10 10 10 10 1 10 10 10 10 10 10

6

3

na

Fig. 14.23. Concentration dependences of calculation parameters (position of the Hj peak and width σ) of distribution F (1; H3 ) for a model system. The inclined √ dashed line corresponds to the na3 power dependence, while the vertical dashed line is the left boundary between the ferromagnetic (FM) and paramagnetic (PM) phases 11

In contrast to (14.45), the wings of the Lorentzian distribution decrease at a much slower rate (according to the law 1/H 2 ), that is, so slowly that the root-mean square value H 2  of the random field is infinitely large. It is for this reason that ferromagnetism is impossible in the system of point dipoles; consequently, the wings of the Lorentzian distribution corresponding to |H| ≥ 5h0 must be trimmed for the emergence of spontaneous magnetization [28].

308

E. Meilikhov and R. Farzetdinova

14.5.4 Magnetic Properties of a Random System of Rod-Like Dipoles The considerable shift of the distribution function F (1; H3 ) to the range of negative magnetic fields for na3 ≥ 1 prevents the formation of magnetic ordering in the system considered here. The system magnetization I = n mi T is determined by (14.34). Obviously, that equation may have the solution j = 1 at T = 0 only under the necessary condition F (1; H3 < 4πI/3) = 0, which can never be realized in a system of dipoles arranged absolutely at random. As in the case of point dipoles [28], the emergence of ferromagnetism can be facilitated by limiting the configuration randomness (e.g., by appropriately setting the lower limit on dipole spacing, which is equivalent to trimming the wings of the random field distribution function). Using expression (14.41) for the distribution function F (j; H3 ), we arrive at an equation generalizing the standard equation for mean field: 1 j= √ 2π(σ/h0 )

∞ −∞




1 tanh Θ



4π j+u 3






(u + jHj /h0 )2 exp − 2(σ/h0 )2

 du,

(14.46) where Θ = kT /m2 n is the reduced temperature (equal to the ratio of the thermal energy to the characteristic energy of magnetic interaction between dipoles). The paramagnetic state (j = 0) is obviously a solution to this equation since its right-hand side is equal to zero for j = 0. To find out whether this equation has a solution corresponding to the ferromagnetic state (j = 0), we note that this equation for Θ → 0 assumes the form j = −sign(z0 )Φ(|z0 |j), (14.47) √ √ x where z0 = (Hj /h0 4π/3)(h0 / 2σ) and Φ(x) = (2/ π) 0 exp(−x2 ) dx is the probability integral.√Equation (14.47) has a solution j = 0 only in the case when z0 < 0, |z0 | > π/2, that is, for  (14.48) (4π/3)mn > Hj + σ π/2. The meaning of the last condition is that the collective field (4π/3)mn created by dipoles on the surface of the Lorentz sphere must “overcome” local fields of the reverse sign with the mean value Hj and a spread ±σ. It can be seen from Fig. 14.23 that this condition holds only for na3 ≥ 5 × 102 , that is, for a fairly high dipole concentration. In this case, the system is ferromagnetic and (14.46) can be used for deriving the temperature dependence of its magnetization, which is shown in Fig. 14.24. The temperature range in which the magnetization differs from zero expands upon an increase in the dipole concentration na3 . The corresponding concentration dependence of the Curie temperature is given in Fig. 14.25. It can be seen from Fig. 14.23 that

14 Ferromagnetism of Nanostructures

309

1.0 0.8 0.6

j 0.4 0.2

103 0

0

na3 = 10

1

2

4

3

105 4

Q

Fig. 14.24. Temperature dependences of the system magnetization in the ferromagnetic state for different dipole concentrations 4π/3

4

χ

4

3

3

0 2

2 1

1

PM 0 −3 −2 −1 10 10 10 1

ΘC

FM

0 3 5 2 4 6 10 10 10 10 10 10

na3 Fig. 14.25. Concentration dependences of the low-temperature susceptibility of the paramagnetic phase and the Curie temperature of the ferromagnetic phase of a random system of linear Ising dipoles. As the dipole concentration increases (na3 → ∞), the Curie temperature attains saturation (ΘC → 4π/3). The vertical dashed line is the left boundary of the ferromagnetism region

Hj , σ → 0 for na3 → ∞. This means that the distribution function F (j; H3 ) approaches a delta function centered in the vicinity of H3 = 0. In this case, (10) can be reduced to the equation j = tanh(4πj/3Θ); for Θ ≈ ΘC (for j  1), this equation shows that ΘC = 4π/3 for high dipole concentrations. In the limit of low dipole concentrations (na3  1), the system under study resembles a random system of point dipoles without ferromagnetic ordering [28].12 Nevertheless, complete equivalence between the systems of rod-like and point dipoles is attained only in the limit of their vanishingly low concentration. 12

We are speaking of a system in which dipoles can be arranged at infinitely small distances relative to one another and the distribution function is Lorentzian. In such a system, strong random fields corresponding to long wings characteristic of such functions suppress ferromagnetism.

E. Meilikhov and R. Farzetdinova R ig ht w in g

z

L eft w ing

L or

10-4 10-5

Lor ent z

h 0 F (1; H 3 )

10-3

ent

310

10-6 3

10

-7

10

na =10 4

10

3

10

2

-3

10

10 2

10 3

10 4

|H3|/h0 Fig. 14.26. Left (a) and right (b) wings of the distribution function F (1; H3 ) of local fields for linear dipoles with a concentration corresponding to na3 = 10−3 : Lorentzian function (1) and power dependences 1/H 4 (2)

As a matter of fact, the distribution function in the region of strong fields for any finite dipole concentration decreases much more rapidly than the Lorentz function, namely, according to the law 1/H 4 (see (14.45)) (naturally, the lowfield boundary of this region depends on the dipole concentration and is shifted towards stronger fields upon a decrease in the concentration). One of the model distribution functions corresponding to the dipole concentration na3 = 103 is shown in Fig. 14.26, demonstrating the wings of this function rapidly decreasing according to the 1/H 4 law corresponding to formula (14.45). In the range of low concentrations (na3 ≤ 5 × 102), the system is paramagnetic. To describe the properties of this system in an external magnetic field He , it is sufficient to carry out the substitution H3 → H3 + He in the argument of the hyperbolic tangent in (14.46). This will lead to the substitution u → u + He /h0 in (14.46). In a weak external field (He /h0  1), the magnetization of the paramagnetic system is low (j  1). Expanding the functions appearing in (14.46) in He and j, we obtain   He Hj 4π j+ − I2 j, (14.49) j = I1 3 h0 h0 where 1 I1 = √ 2π

∞ −∞

2

1 e−x /2 dx, I2 = √ 2 Θch [(σ/h0 Θ)x] 2π

∞ 2 x tanh[(σ/h0 Θ)x] e−x /2 dx. −∞

(14.50) Relation (14.47) makes it possible to find the low-field magnetic susceptibility of the system χ = I/He = j/(He /h0 ) and its temperature dependence. At high temperatures (Θ  1), we have I1 = 1/Θ → 0, I2 → 0 and it follows from (14.47) that χ(Θ  1) = 1/Θ, or I = mn(mHe /kT ) is the conventional Curie law for noninteracting Ising dipoles.

14 Ferromagnetism of Nanostructures

311



At low temperatures (Θ  1), we have I1 = I2 = 2/π(h0 /σ), which gives  χ0 ≡ χ(Θ  1) = [ π/2(σ/h0 ) + (Hj /h0 ) − 4π/3]−1 , that is, a temperature-independent susceptibility. In the vicinity of the ferromagnetic transition (whose boundary is determined by condition (14.46)), it increases indefinitely. The concentration dependence of χ0 is shown in Fig. 14.25.

14.6 Experimental Examples 14.6.1 Magnetism of Ultrathin Films Experiments with ultrathin films of iron, cobalt, and nickel revealed that, for a certain effective thickness, a long-range ferromagnetic order is established in such films [11], but the origin and mechanisms of this phenomenon remain not quite clear. The control of the topology of such films and, in particular, an analysis of the geometry and magnetic properties of islands consisting of Co (Fe, Ni) atoms, the fraction of the substrate covered by these atoms, the structure of the film itself, and also the evolution of relevant parameters in the course of the film growth are very important for the development of physical ideas concerning the mechanism of establishment of the magnetic order in such films. In this connection, we can mention a recent publication [12] in which it is shown that, for a high (room) temperature of monocrystalline (110)-oriented Cu substrate, the Co film deposited on it grows through natural lithography, that is, through the formation of nuclei (distributed more or less uniformly over the substrate area), which are transformed into ellipsoidal granules with major axes oriented (with a small spread of 5◦ –10◦) along the [001] axes of the substrate. It was found using scanning tunnel microscopy (STM) that the size of this granules increases in the course of film growth, but the distance between their centers as well as the shape of the granules (i.e., their eccentricity) remain unchanged. Experiments show that such films become ferromagnetic only if their thickness is large enough. (The effective thickness d of an inhomogeneous film consisting of granules is equal to the thickness of a homogeneous film containing the same number of atoms and is measured by the number of effective monatomic layers.) Gu et al. [12] believe that the critical thickness dc of the film is the thickness for which its hysteresis loop exhibits a nonzero coercivity for the first time. According to their measurements, dc ≈ 4.6 monolayers, although noticeable nonlinearity of the magnetic-field dependence of the film magnetization, which is a consequence of intergranule interaction, appears even for d ≈ 4 monolayers.

312

E. Meilikhov and R. Farzetdinova

We assume that the described process of transition to the ferromagnetic state can be attributed to the change in the magnetic state of a 2D system of ellipsoidal Co granules with the long-range dipole interaction in accordance with the magnetic phase diagram whose examples were given above. In the framework of this model [30], the film growth is reduced only to a change in the scaling factor, whose role in the problem is played by the size a of the semi-major axis of an ellipsoidal granule. To pass to a quantitative description of the process, we must connect the parameters characterizing the size and shape of the granules (the semiaxis length a and eccentricity e) with the effective thickness d of the film. For this purpose, we can use the experimentally determined (for the same system) relation  = (d) between the fraction of the open surface of the substrate (which is not occupied by Co granules) and the effective thickness of the film [29]. Obviously, the fraction of the surface covered by granules is 1 −  = πab/(lx lz ); this leads to the required expression for the parameter lx /a determining the nature of the magnetic phase diagram: lx /a = [lx(0) /a(0) ][(1 − (d(0) )/(1 − (d))],

1,0

1,0

0,8

0,8 0,6

0,6 0,4

d

0,2 0

(0)

0,4

[M. T. Kief, e.a., 1993] Exponential fit: ε=exp(-d/2.56) 1

2

3

4

5

0,2

ε (free Cu-surface)

(0)

(0)

[lx /a] / [ lx /a ]

where the parameters labeled by the superscript (0) correspond to the thinnest film (in experiments [12], d(0) = 0.9 monolayers). Figure 14.27 shows such a dependence plotted on the basis of the experimental data [29] for a Co film on a (110) oriented Cu substrate. According to Gu et al. [12], the granule shape is close to ellipsoidal with the eccentricity e ≈ 0.95, and the ratio of the average distances between the

0

d, ML Fig. 14.27. Experimental dependence  = (d) of the fraction of open (not occupied by Co grains) (100)-oriented surface of the Cu substrate on the effective thickness of the Co film [29] (lower curve) and corresponding theoretical values of parameter lx /a (upper curve). The parameters marked by the superscript (0) correspond to the thinnest film of thickness d0 ≈ 0.9 monolayers. The experimental results are approximated by the exponential dependence  = exp(d/2.56) (lower dashed curve), while the results of calculations are approximated by the hyperbolic dependence (0) [lx /a]/[lx /a(0) ] = [1 − (d(0) )]/[1 − (d)] = 0.971/d0.289 (upper dashed curve)

14 Ferromagnetism of Nanostructures (0)

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(0)

lx /a =4, e=0.95, β =0.5 [E.Gu, e.a., 1999]

12

8

2

3

kTC /(2µ /lx )

10

6 4

λ 01 λF

2 0

1

2

3

4

5

6

7

8

9 10

d, ML Fig. 14.28. Dependence of the Curie temperature of the film on its effective thickness. Major axes of grains are parallel to one another (θ = 0), eccentricity e = 0.95; (0) grain lattice parameters: β = 0.5, lx /a(0) = 4

granules (lattice periods) is β = lz /lx = 0.51. As regards the angles θ of granule orientation, it follows from the STM images presented in [12] that these angles are distributed in a narrow interval near θ = 0. With increasing effective thickness of the film, the granule size increases, while lattice periods remain practically unchanged (right up to the contact between the granules). In this case, the ratio lx /a, which determines the magnetic state of the system in accordance with Fig. 14.27, decreases monotonically, which leads to the magnetic phase transition from the initially antiferromagnetic state to the ferromagnetic state. The critical thickness dc of the film, at which this transition occurs, and subsequent dependence of the Curie temperature TC of the emerging ferromagnetic state are determined (see above) by the value of the dipole interaction parameter λ for various magnetic states of the system. Proceeding from the STM images of Co films presented in [12], we assumed in our calculations that (1) the axes of all granules are parallel to one another (0) (θ = 0), (2) β ≡ lz /lx = 0.5, and (3) lx /a(0) = 4. Figure 14.28 shows the dependence TC (d) = λ(d)i0 /k of the Curie temperature of the films on their thickness, determined for the above values of the parameters. For the chosen set of parameters, a transition of the system of granules from the antiferromagnetic to the ferromagnetic state occurs at a critical thickness dc ≈ 4.3 of monolayers, which is close to the value indicated in [12]. After the attainment of such a thickness, the Curie temperature of the formed ferromagnetic state immediately assumes a finite value and then increases with the film thickness. The scale of the initial Curie temperature is determined by the value of µi0 for µ = 300 µB , i0 = 10−4 G cm (which corresponds to the volume of granules of 40 nm3 and their concentration N = 10−14 cm−2 ) and amounts to TC ∼ 300 K.

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14.6.2 2D Lattices of Disk-Shaped Granules in a Magnetic Field Modern electron lithography makes it possible to create artificial periodic magnetic structures with a size of individual elements (granules) up to tens of nanometers. On the one hand, the sizes of such granules are large enough for their ferromagnetic properties were virtually identical to the properties of large objects made of the same material, but on the other hand, these are so small that the granules themselves are one-domain ferromagnets. These structures with the dipole interaction between granules are just the objects studied by us here. To analyze the experimental manifestation of the rearrangement of the magnetic configuration for this type of structures, we consider the results obtained in [22], where 2D rectangular lattices of circular plane granules made of a magnetically soft material (Superalloy Ni80 Fe14 Mo5 ) were studied. The granule size was as follows: diameter 2a = 60 nm and thickness h = 7 nm, which corresponds to the granule magnetic moment µ = (πa2 h)Is = 1.6 × 10−14 G cm3 , where Is = 800 ± 60 G is the saturation magnetization of the granule material. One of the lattice periods (ly = 180 nm) remained unchanged, while the other (lx ) varied in the range 180–190 nm. Since lx < ly , the ground state of the system corresponded to the magnetic configuration S01 . The magnetic field applied along the y axis must transform the system to the ferromagnetic state via an intermediate wavy magnetic configuration (see above). The characteristic scale of the field required for this purpose is defined by the quantity µ/ly3 , which is equal to 2.7 Oe in our case. In accordance with the dependences presented in Fig. 14.9, the saturation magnetization of the system must take place in the fields Hy = 5.5 and 35 Oe, respectively, for lattices lx = 180 and 90 nm. The experimentally measured values of these fields [22] Hy ≈ 6 and 40 Oe are in satisfactory agreement with the results of computations (see the inset to Fig. 14.9). 14.6.3 Magnetic Recording Density The principle of magnetic recording is that the magnetic state (magnitude and direction of magnetization) of a small region of a magnetic medium deposited on the surface of a disk (tape) is memorized. Usually, the medium consists of small granules (of size 100–1,000 ˚ A) of a magnetic material (e.g., Fe–Co alloy). A special device known as the head can be positioned over any region of the disk and change the magnetic state of this region (information recording) or determine the state of this region (information readout). The latest advancement in this field was the demonstration of a disk with the recording density of about 5 Gbit cm−2 . A bit of information on the disk is recorded on a region containing approximately 100 granules. The extremely high density of information storage can obviously be attained by recording one bit on a single magnetic granule. For this purpose, it is necessary to create a special magnetic carrier containing individual regularly arranged magnetic nanoparticles of the same size, shape, and orientation. The

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periodicity of the arrangement is required for information recording/readout, while the uniformity in the above geometrical parameters is required for storing the recorded information for a long time (of the order of 10 years). The main mechanism leading to a reversal of the magnetic moment (and, hence, to a loss of information) in small granules is a thermal excitation of such a process. The time of information storage is determined by the characteristic time τ of rotation of the magnetic moment, which is defined as τ = τ0 exp(∆0 /kT ),

(14.51)

where τ0 = 10−10 –10−9 s and ∆0 is the height of the energy barrier separating two stable states of the granule magnetization. The time τ of information storage exhibits a very strong (exponential) dependence on ∆0 , and to obtain τ ≥ 10 years, it is necessary to satisfy the condition ∆0 ≥ 40. kT

(14.52)

It is well known [31] that ∆0 =

1 νV0 Is2 , 2

ν = Nb − Na ,

(14.53)

where Na and Nb are the demagnetization coefficients for a granule along the a and b axes, respectively, V0 is the granule volume, and Is is the magnetization of its material. In our subsequent estimates, we consider Fe granules for which Is = 1, 700 G and assume that their temperature is T = 300 K (room temperature). Substituting relations (14.53) into (14.52), we obtain V0 ≥ Vmin ,

Vmin , nm3 =

1000 , ν

(14.54)

Vmin being the minimal volume of Fe granules for which the magnetic moment preserves its direction for 10 years. It should be emphasized that the obtained estimate corresponds to isolated (noninteracting) magnetic granules. Relations (14.54) imply that in order to reduce Vmin , we must take granules with large values of parameter ν, that is, granules in the form of a strongly prolate ellipsoid (rod-like granules). It is sufficient to confine our analysis to the ratio of the granule axes a/b = 4 − 10 ensuring the value of ν = 5 − 6. A further increase of this ratio (i.e., transition to rod-like granules) does not increase parameter ν appreciably. Assuming that ν = 5.5, we obtain the final estimate V0 ≥ Vmin ,

Vmin = 180 nm3 .

(14.55)

The maximum attainable value of information storage density corresponds to close packing of such granules. For the granules of the shape under investigation (ellipsoid of revolution with the axes ratio a/b = 6), the recording density is about 1,000 Gbit cm−2 .

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However, the magnetic interaction of closely spaced granules, which reduces the information storage time like thermal excitation, may obviously reduce the obtained estimate significantly. To demonstrate the strength of the latter effect, we use the relation [15] ∆H /∆0 = (1 − η)2 ,

η=

H HV0 , = νIs νµ

(14.56)

which describes the lowering of the height ∆H of the energy barrier separating two stable states of a granule due to the magnetic field H created by all the remaining granules in the region of its location. This relation is based on the assumption that the magnetic moment of the chosen granule and the external magnetic field are antiparallel. Let the major axes of all granules be parallel to the x axis (θ = 0). Then, in the approximation of rod-like granules, the x component of the resultant field of all granules at the origin (the site of the chosen granule) is determined by a relation similar to (14.10):
1  iπφmn 2 H/H0 = 3 e 2 γ m,n (m + β 2 n2 )1/2 −

 1 1 , − [(m + 2γ −1 )2 + β 2 n2 ]1/2 [(m − 2γ −1 )2 + β 2 n2 ]1/2

(14.57)

where H0 = µ/a3 , φmn = 1 if the direction of the magnetic moment of the granule located in the point (mx , nz ) coincides with a positive direction of the axis x and φmn = −1 otherwise (it should be recalled that γ = x /a, β = z /x ). It is clear from geometrical considerations that x > 2a and z > 2b. Obviously, the magnitude of the field H depends on the sign distribution of magnetic moments of surrounding granules, that is, on the information recorded in the vicinity of the granule under investigation. While calculating the information storage time, we must proceed from the most unfavorable distribution of these moments (ensuring the maximum possible magnetic field). Such (the worst) configuration can easily be established: if the magnetic moment of the chosen granule (located at the origin) is negative (opposite to the positive direction of the x axis), the signs of the magnetic moments of surrounding granules in such a configuration √ depend on whether or not they fall into a cone with the angle 2arccos(1/ 3) and the axis parallel to the x axis). In the former case, the magnetic moment is positive and in the latter case it is negative. In particular, the magnetic moments of the granules located on the x and z axes are positive and negative, respectively. The dependences of field H on the geometrical parameters β and γ of the lattice formed by the granules calculated for this case are shown in Fig. 14.29. To compensate the effect of magnetic interaction between granules, their volume must be increased as compared to the minimum volume Vmin . It follows

14 Ferromagnetism of Nanostructures 500

V min= 1 8 0 n m a /b = 6

400

Storag e D ens ity, Gb it/ cm

10

3

H / H0

300 200 100

1 10

0,1 2

5

V / V min

100

2

1

0.2 0.3 0.5 lz/lx = 1.0

317

10

lx/a

Fig. 14.29. Dependences of “magnetization reversal” field H on the geometrical parameters β = lz /lx and γ = lx /a of a lattice of rod-like grains of length 2a and magnetic moment µ (H0 = µ/a3 ). The inset shows the dependence of attainable density of magnetic recording on the volume V0 of Fe grains in the form of ellipsoids of revolution with the axes ratio a/b = 6 (Vmin = 180 nm3 ). The dashed line corresponds to condition (14.58) for X = 4

from relations (54)–(56) that the required volume X = V0 /Vmin must satisfy the condition X ≥ 1/(1 − η)2 , where η = 0.125(H/H0) for a/b = 6 and ν = 5.5. Thus, the admissible values of periods lx and lz of the lattice formed by granules must satisfy the relation13   1 H(x , z )/H0 ≤ 8 1 − √ . (14.58) X Among these values, we must select those corresponding to the maximum surface density of granules, equal to 1/(lx lz ). The horizontal straight line H/H0 = const drawn in Fig. 14.29 corresponds to a certain value of X. It can be seen that the optimum values of lattice periods are always those corresponding to the point of intersection of this line with the curve H(lx , lz ) for the minimum ratio of the periods β = lz /lx dictated by geometrical considerations (in our case, β = b/a ≈ 0.2). The dependence of the maximum attainable density of magnetic recording determined in this way on the volume of granules is shown in the inset to Fig. 14.29. It can be seen that, although the dipole interaction between granules considerably reduces this density (in our case, to half the rated value), it can still be high enough and attain values 500 Gbit cm2 , which is two orders of magnitude higher than the density of recording in the best modern magnetic disks. The approximation of rod-like dipoles used in the previous analysis is completely applicable only for granules separated by large distances (as compared to the granule size) from the origin (the site of the selected molecule). Obviously, the largest errors appear in this approximation when we calculate the 13

Condition (14.58) indicates, in particular, that H must be equal to zero for V = Vmin (X = 1), which corresponds to granules separated by infinitely long distances.

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field created by the nearest granules. However, special calculations made by us proved that the resultant action of remote but numerous granules exceeds the action of a few nearest granules (this is a consequence of the long-range nature of the field of a magnetic dipole). Thus, the approximation of rod-like granules correctly describes the effect of magnetic interaction between the granules. 14.6.4 Conclusions We have described the methods for analyzing the magnetic properties of 2D and 3D lattices of nonspherical ferromagnetic granules with the intergranule dipole interaction: the ground state, the magnetic phase diagram, and the change in the magnetic state under the action of an external magnetic field. The obtained results can be used for describing the properties of a number of real systems, including 2D periodic structures of magnetic granules suitable for creating magnetic memory systems with a high recording density, ultrathin films of ferromagnetic metals on monocrystalline substrates, and rectangular lattices of disk-shaped magnetically soft granules. The methods developed for describing the properties of such systems are in good agreement with the results of relevant experiments. Acknowledgement If you want to include acknowledgments of assistance and the like at the end of an individual chapter please use the acknowledgement environment – it will automatically render Springer’s preferred layout.

References 1. R. Brout, in Magnetism, vol. 2, Part A, ed. by G.T. Rado, H. Suhl (Academic, New York, 1965) 2. P.I. Belobrov, R.S. Gekht, V.A. Ignatchenko, Sov. Phys. JETP 57, 636 (1983); P.I. Belobrov, V.A. Voevodin, V.A. Ignatchenko, Sov. Phys. JETP 61, 522 (1985) 3. M.D. Costa, Y.G. Pogorelov, in Proceedings of the Seeheim Conference on Magnetism, Seeheim, 2001 4. J.A. Sauer, Phys. Rev. 57, 142 (1940) 5. J.M. Luttinger, L. Tisza, Phys. Rev. 70, 954 (1946) 6. J.D. Jackson, Classical Electrodynamics (Wiley, New York, 1999) 7. M.D. Costa, Y.G. Pogorelov, Phys. Stat. Sol. A 189, 923 (2001) 8. R.L. White, J. Magn. Magn. Mater. 209, 1 (2000) 9. J.S. Moodera, G. Mathon, J. Magn. Magn. Mater. 209, 248 (2000) 10. B.A. Aronzon, S.V. Kapelnitsky, A.S. Lagutin, in Physico-Chemical Phenomena in Thin Films and at Solid Surfaces, vol. 34, ed. by L. Trakhtenberg, S. Lin, O. Ilegbusi (Elsevier, New York, 2007); E.Z. Meilikhov, JETP 89, 1184 (1999)

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11. W. Durr, M. Taborelli, O. Paul et al., Phys. Rev. Lett. 62, 206 (1989); C.M. Schneider, P. Dressler, P. Schuster et al., Phys. Rev. Lett. 64, 1059 (1990); J. de la Figuera, J.E. Pietro, C. Okal, R. Miranda, Phys. Rev. B 47, 13043 (1993); A.K. Schmid, D. Atlan, H. Itoh et al., Phys. Rev. B 48, 2855 (1993); F. Huang, M.T. Kief, G.J. Mankey, R.F. Willis, Phys. Rev. B 49, 3962 (1994); F.O. Shumann, M.E. Buckley, J.A.C. Bland, Phys. Rev. B 50, 16424 (1994); H.J. Elmers, J. Haushild, H. Hohe et al., Phys. Rev. Lett. 73, 898 (1994) 12. E. Gu, S. Hope, M. Tselepi, J.A.C. Bland, Phys. Rev. B 60, 4092 (1999) 13. P.I. Belobrov, R.S. Gekht, V.I. Ignatchenko, Sov. Phys. JETP 57, 636 (1983) 14. S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943) 15. E.Z. Meilikhov, R.M. Farzetdinova, JETP 97, 593 (2003) 16. T.L. Hill, Statistical Mechanics: Principles and Selected Applications. (McGrawHill, New York, 1956) 17. B.I. Shklovskii, A.L. Efros, Electronic Properties of Doped Semiconductors. (Springer, New York, 1984) 18. E.Z. Meilikhov, R.M. Farzetdinova, JETP 95, 886 (2002) 19. E.Z. Meilikhov, R.M. Farzetdinova, JETP 94, 751 (2002) 20. E.Z. Meilikhov, R.M. Farzetdinova, Microelectr. Eng. 69, 288 (2003) 21. L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics, vol. 8. Electrodynamics of Continuous Media (Pergamon, New York, 1984) 22. R.P. Cowburn, A.O. Adeyeye, M.E. Welland, New J. Phys. 1, 161 (1999) 23. E.Z. Meilikhov, JETP Lett. 77, 571 (2003) 24. D.H. Reich, B. Ellman, J. Yang et al., Phys. Rev. B 42, 4631 (1990) 25. V.S. Dotsenko, Phys. Usp. 38, 457 (1995) 26. H. Zhang, M. Widom, Phys. Rev. B 51, 8951 (1995) 27. B.E. Vugmeister, M.D. Glinchuk, Rev. Mod. Phys. 62, 993 (1990) 28. E.Z. Meilikhov, JETP 97, 587 (2003) 29. M.T. Kief, W.F. Egelhoff, Phys. Rev. B 47, 10785 (1993) 30. E.Z. Meilikhov, R.M. Farzetdinova, JETP Lett. 75, 142 (2002) 31. R.M. White (ed.), Introduction to Magnetic Recording (IEEE Press, New York, 1985)

15 Magnetic Dipolar Interactions in Nanoparticle Systems: Theory, Simulations and Ferromagnetic Resonance D.S. Schmool1 and M. Schmalzl2 1

2

Departamento de F´ısica and IFIMUP, Faculdade de Ciˆencias, Universidade do Porto, Porto, Portugal Bayerische Julius-Maximilians-Universit¨ at W¨ urzburg, Germany, [email protected]

Summary. Magnetic nanoparticle assemblies present novel magnetic properties with respect to their bulk constituent components. In addition to the surface effects produced by the modified atomic symmetry in such low dimensional systems, the magnetic coupling between the particles also plays a significant role in determining the overall magnetic behavior of a magnetic nanoparticle assembly. In this Chapter, we describe a theoretical model that accounts for the dipolar magnetic interaction between particles. There are two fundamental aspects of interest in our studies: the spatial distribution of the particles and density of the particle. These aspects have been addressed in our simulations, where we have performed simulations for regular and random arrays of particles. We will discuss the general theory of ferromagnetic resonance (FMR) applied to such systems and how the specific dipolar interactions can be incorporated for nanoparticle systems. The spatial distribution of particles can give valuable information of the strength of the dipolar interaction between them and we will demonstrate how this can be used in real systems. We have performed FMR experiments on nanoparticle assemblies of γ − Fe2 O3 nanoparticles with different average particle sizes (2.7–7.3 nm) and particle densities, where samples are in rectangular slab shape. Measurements were performed as a function of the angle between the sample plane and applied magnetic field. Recent studies have shown that the incorporation of the dipolar interactions into FMR theory can explain the experimental results for angular studies in magnetic nanoparticle assemblies [1].

15.1 Introduction The physics of magnetic nanoparticle (NP) systems is of broad scientific interest and has attracted much attention in recent years due to their potential for device and sensor applications. The magnetic properties exhibited by NP assemblies, can differ significantly from their bulk component systems due to

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the effects of surface anisotropies, superparamagnetic effects and the magnetic interactions between the magnetic particles themselves. In this Chapter, we present the basic study of the effects of the magnetic dipolar interaction in an assembly of ferromagnetic nanoparticles. We show the basic physics of the interaction and illustrate the overall effects using computer simulations of regular and random arrays of particles. In Sect. 15.2, the theory has been presented including an explanation of the nature of the simulations for arrays of particles. In Sect. 15.3, the relation between interparticle interactions in NP assemblies and angular studies by ferromagnetic resonance (FMR) have been discussed.

15.2 Theory of Dipole – Dipole Interactions in Magnetic Nanoparticles 15.2.1 Dipolar Interactions The basic magnetic dipolar interaction is governed by the equation:  1 3(mi · rij )(mj · rij ) DDI Eij (rij ) = mi · mj − , 3 2 4πrij rij

(15.1)

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 ferromagnetic resonance. 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 in the case of an FMR experiment). Under this state, we can simplify (15.1) using the spherical coordinate system as: & mi mj % 2 DDI 1 − 3 [sin θ sin ϑ cos(φ − ϕ) + cos θ cos ϑ] Eij (ϑ, ϕ, θ, φ) = 3 4πrij (15.2) All angles are defined in Fig. 15.1 (a) and (b). 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: 1   DDI ETDDI Eij (ϑ, ϕ, θ, φ) (15.3) OT = 2 i j =i

The factor of

1/2

is required so as to not count the interactions twice.

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Fig. 15.1. (a) Basic coordinate geometry for the dipolar interaction between two magnetic particles. (b) Spherical coordinate system for the magnetic sample. (c) Schematic illustration of a section of the nanoparticle assembly (see text)

15.2.2 Simulations for Arrays of Nanoparticles To perform the simulations, we have constructed a regular simple cubic lattice, with lattice constant d, as illustrated in Fig. 15.1(c). For all simulations, we use a slab shaped spatial distribution (Fig. 15.1(b)), this being the shape of the magnetic NP samples measured by FMR (see Sect. 15.3). We have performed two sets of simulations, which will be reported here; (1) Variation of the aspect ratio, R, (thickness/length; R = z/x; where we maintain x = y) of the particle distribution, i.e. varying the number of particles in the z−and x-directions. (2) Variation of the particle density, which can be controlled by simply varying the lattice parameter d, where the particle density can be expressed as: ρN = N/V = 1/d3 . For the former, we have used random arrays of particles, while in the latter we have performed simulations for both regular and random arrays of NPs. To assess these energy variations, we consider the angular dependence of (15.3), maintaining the azimuthal orientation constant. This illustrates the magnetic anisotropy introduced by the DDI, which is also proportional to the average interaction. In Fig. 15.2(a), we show this variation for a regular array of (50 × 50 × 5) particles. In this case the DDI variation can be written in the form: DDI ESA (ϑ) = E0 cos2 ϑ + E1

(15.4)

The size of the variation will be given by E0 and an offset E1 is also included. Both constants will depend on the magnetic moment (magnetisation) of the

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Fig. 15.2. (a) Angular variation of the DDI energy for a regular cubic array of NPs (50 × 50 × 5) as evaluated from (3) (points) and a fit to (4) (solid line). (b) Aspect ratio (sample shape) dependence of E0 on R = z/x, the dashed line corresponds to the demagnetising energy of an uniformly magnetised sample, see text

Fig. 15.3. Variation of DDI energy E0 as a function of particle density, ρN , for (a) random particle arrays and (b) regular NP assemblies

particles as well as their average effective separation. The energy E0 is a measure of the maximum to minimum energy of the angular dependence. In a series of simulations we have varied the aspect ratio R, from a thin layer (R ∼ 0; R < 1) through a cube (R = 1) to a columnar shape (R > 1). For such a shape variation, we see the dependence as illustrated in Fig. 15.2 (b). We note that for slab type geometries, E(0) > (π/2), i.e. positive E0 , while for the cube shaped distribution, we have E0 = 0 and E0 becomes negative for columnar samples distributions, i.e. DDI energy curve becomes inverted. The dashed line is based on the demagnetising energy with the dimensions of the NP distribution; E(R) = K(1 − R)/(1 + R), where K is a constant [2]. In Fig. 15.3, we show the results of simulations for both regular and random arrays of NPs, where we have varied the density (or equivalently the filling factor, f = Vmag /VT OT ) of NPs in the assembly. For the case of the random array calculations, a computer generated random number program partially “occupies” a set number of lattice sites (depending on the density to be evaluated). In these calculations, we repeat the simulation 20 times and obtain an average to eliminate any distorted assembly distributions that

15 Magnetic Dipolar Interactions in Nanoparticle Systems

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may occur; the error in the random distributions reflects the variance in these results. In both cases, we observe a density squared dependence.

15.3 Ferromagnetic Resonance in Magnetic Nanoparticles Ferromagnetic resonance is a powerful experimental technique, which can be used to directly measure the effective magnetic field to which a spin system is subject during resonance. As such, it is an appropriate method for directly measuring the DDI in magnetic nanoparticle assemblies. Considering the resonance condition, we can use the Smit–Beljers equation:   2  2 2 ω 1 ∂2E ∂2E ∂ E (15.5) = 2 2 − 2 2 γ ∂ϑ∂ϕ M0 sin ϑ ∂ϑ ∂ϕ In this formalism, we use the free energy density E, which will include all relevant contributions to the effective magnetic field, such as the Zeeman energy, magnetic anisotropy energies and any shape effects that may be of relevance. Equation (15.5) must be used in conjunction with the equilibrium conditions (first derivatives with respect to polar and azimuthal angles set to zero) to assess the resonance condition. In the case of spherical nanoparticles with randomly oriented weak magnetocrystalline anisotropy and DDI, the resonance equation takes the form [3]: sin ϑ cos(ϑ − ϑ0 ) 2 sin ϑ cos 2ϑ0 Hr + 2C Hr − sin ϑ0 sin ϑ0

 2 ω 2 − HK,eff =0 γ

(15.6)

where C is a constant, which depends only on parameters related to the DDI, is the equilibrium orientation and the final term is the effective anisotropy field, 2Keff /M , which is an average of over all random orientations of the local anisotropy axes of the individual particles. Samples of γ − Fe2 O3 NP assemblies, with mean particle diameters of 2.7, 4.6 and 7.3 nm, were measured by FMR as a function of the direction of the applied magnetic field, from the perpendicular (0o ) to the parallel (90o ) orientations. In Fig. 15.4, we show the angular dependence of the FMR resonance field, Hr , of NPs with mean diameter 4.6 nm. Also shown is the theoretical variation of the resonance field as predicted by (15.6) and has only an angular dependence due to the DDI, where we assume the particles are spherical. The points refer to experimental data and the line is the theoretical fit. This curve is the representative of all samples measured, with other sizes. We see an excellent agreement between the theory and experimental measurements. This implies that the main influence on the angular dependent magnetic properties is due to DDI.

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Fig. 15.4. Angular dependence of the resonance field for samples with an average diameter of 4.6 nm. The line is a fit to (6)

15.4 Conclusions We have shown that the dipole–dipole interaction in ferromagnetic nanoparticle assemblies is of great importance when considering the magnetic properties of the assembly as a whole. By summing all interactions in the assembly, it is possible to illustrate how this introduces a magnetic anisotropy into the system. Our results show that the spatial distribution will have a great influence on this overall anisotropy and that random arrays of particles can be easily approximated to regular arrays of NPs as they exhibit virtually identical results for the same particle density. This is much simpler to treat from a theoretical point of view. The important factor is the particle density and through this, the magnetic filling factor. Measurements by FMR on NP assemblies of γ − Fe2 O3 illustrate that the theory presented here shows an excellent agreement with experiment. We are currently investigating the effects of the volume dispersion on the DDI and its influence in FMR. Acknowledgement We would like to thank E. Tronc for supplying the samples used in this study.

References 1. D.S. Schmool, R. Rocha, J.B. Sousa, J.A.M. Santos, G.N. Kakazei, J.S. Garitaonandia, L. Lezama, J. Appl. Phys. 101, 103907 (2007). 2. D.S. Schmool, J.M. Barandiar´ an, J. Magn. Magn. Mater. 191, 211 (1999). 3. D.S. Schmool, M. Schmalzl, J. Non-Cryst. Solids 353, 738 (2007).

Contributors

Ageev, V.A., 149 Aktas, B., 37 Avgin, I., 113 Aydogdu, G.H., 131 Blanco, J.M., 205 Brzozowski, R., 219 Bulut, N., 67 C ¸ elik, E., 113 Chizhik, A., 205 del Val, J.J., 205 Domantovsky, A.G., 167 Erol, M., 113 Farzetdinova, R., 269 Fedotova, J.A., 231 Fernandez, L., 205 Galanakis, I., 1 Garcia, C., 205 Ghelev, C., 183 Gonzalez, J., 205 Grigorov, K., 183 Gurovich, B.A., 167 Habermeier, H.-U., 131 Iturriza, N., 205 Kirischuk, V.I., 149 Koblyanskiy, Y.V., 149

Kolev, S., 183 Koroleva, L.I., 89 Koutzarova, T., 183 Kuleshova, E.A., 167 Kuru, Y., 131 Lakshmi, N., 21 Maekawa, S., 67 Maslakov, K.I., 167 Meilikhov, E.Z., 167, 269 Melkov, G.A., 149 Moneta, M., 219 Nedkov, I., 183 ¨ Ozdo˜ gan, K., 1 Prikhodko, K.E., 167 Sadovnikov, L.V., 149 S ¸ a¸sıo˜ glu, E., 1 Schmalzl, M., 321 Schmool, D.S., 321 Sebastian, V., 21 Slavin, A.N., 149 Sov´ ak, P., 219 Strilchuk, N.V., 149 Takahashi, S., 67 Tanikawa, K., 67 Tomoda, Y., 67 Vasyuchka, V.I., 149 Venugopalan, K., 21 Wasiak, M., 219

328

Contributors

Yilgin, R., 37 Zashchirinskii, D.M., 89

Zheltonozhsky, V.A., 149 Zhukov, A., 205 Zhukova, V., 205

springer proceedings in physics 80 Computer Simulation Studies in Condensed-Matter Physics VIII Editors: D.P. Landau, K.K. Mon, and H.-B. Sch¨uttler 81 Materials and Measurements in Molecular Electronics Editors: K. Kajimura and S. Kuroda 82 Computer Simulation Studies in Condensed-Matter Physics IX Editors: D.P. Landau, K.K. Mon, and H.-B. Sch¨uttler 83 Computer Simulation Studies in Condensed-Matter Physics X Editors: D.P. Landau, K.K. Mon, and H.-B. Sch¨uttler

93 ISSMGE Experimental Studies Editor: T. Schanz 94 ISSMGE Numerical and Theoretical Approaches Editor: T. Schanz 95 Computer Simulation Studies in Condensed-Matter Physics XVI Editors: D.P. Landau, S.P. Lewis, and H.-B. Sch¨uttler 96 Electromagnetics in a Complex World Editors: I.M. Pinto, V. Galdi, and L.B. Felsen

84 Computer Simulation Studies in Condensed-Matter Physics XI Editors: D.P. Landau and H.-B. Sch¨uttler

97 Fields, Networks, Computational Methods and Systems in Modern Electrodynamics A Tribute to Leopold B. Felsen Editors: P. Russer and M. Mongiardo

85 Computer Simulation Studies in Condensed-Matter Physics XII Editors: D.P. Landau, S.P. Lewis, and H.-B. Sch¨uttler

98 Particle Physics and the Universe Proceedings of the 9th Adriatic Meeting, Sept. 2003, Dubrovnik Editors: J. Trampeti´c and J. Wess

86 Computer Simulation Studies in Condensed-Matter Physics XIII Editors: D.P. Landau, S.P. Lewis, and H.-B. Sch¨uttler

99 Cosmic Explosions On the 10th Anniversary of SN1993J (IAU Colloquium 192) Editors: J. M. Marcaide and K. W. Weiler

87 Proceedings of the 25th International Conference on the Physics of Semiconductors Editors: N. Miura and T. Ando 88 Starburst Galaxies Near and Far Editors: L. Tacconi and D. Lutz 89 Computer Simulation Studies in Condensed-Matter Physics XIV Editors: D.P. Landau, S.P. Lewis, and H.-B. Sch¨uttler 90 Computer Simulation Studies in Condensed-Matter Physics XV Editors: D.P. Landau, S.P. Lewis, and H.-B. Sch¨uttler 91 The Dense Interstellar Medium in Galaxies Editors: S. Pfalzner, C. Kramer, C. Straubmeier, and A. Heithausen 92 Beyond the Standard Model 2003 Editor: H.V. Klapdor-Kleingrothaus

100 Lasers in the Conservation of Artworks LACONA V Proceedings, Osnabr¨uck, Germany, Sept. 15–18, 2003 Editors: K. Dickmann, C. Fotakis, and J.F. Asmus 101 Progress in Turbulence Editors: J. Peinke, A. Kittel, S. Barth, and M. Oberlack 102 Adaptive Optics for Industry and Medicine Proceedings of the 4th International Workshop Editor: U. Wittrock 103 Computer Simulation Studies in Condensed-Matter Physics XVII Editors: D.P. Landau, S.P. Lewis, and H.-B. Sch¨uttler 104 Complex Computing-Networks Brain-like and Wave-oriented Electrodynamic Algorithms Editors: I.C. G¨oknar and L. Sevgi