New Itinerant Electron Models of Magnetic Materials 9811612706, 9789811612701

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Gui-De Tang

New Itinerant Electron Models of Magnetic Materials

New Itinerant Electron Models of Magnetic Materials

Gui-De Tang

New Itinerant Electron Models of Magnetic Materials

Gui-De Tang Department of Physics Hebei Normal University Shijiazhuang City, China State Key Laboratory of Magnetism Institute of Physics Chinese Academy of Sciences Beijing, China

ISBN 978-981-16-1270-1 ISBN 978-981-16-1271-8 (eBook) https://doi.org/10.1007/978-981-16-1271-8 Jointly published with Science Press The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Science Press. ISBN of the Co-Publisher’s edition: 978-703-06-8719-7 © Science Press 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Foreword

The conventional magnetic ordering models, such as exchange interaction, superexchange (SE) interaction, and double-exchange (DE) interaction models related to the valence electron states of materials, were proposed before 1960 when there is little experimental evidence. Since the 1970s, many experimental studies on the valence electron states were performed. First, photoelectron spectroscopy studies indicated that O1− anions exist in addition to O2− anions in oxides, reaching up to 30% of oxygen in some cases. These results suggested that the SE and DE models, which were developed based on the assumption that all oxygen atoms form O2− anions in oxides, need to be improved. Secondly, many experimental results indicated that some 4s electrons entered 3d orbits and changed into 3d electrons during the formation of metals from free atoms. In addition, using the density functional theory (DFT) to investigate the physical properties of magnetic materials is challenging due to the magnetic ordering energy, which was included in the exchange-correlation energy and that has not been found a phenomenological expression in DFT calculations. These investigations suggest potential opportunities and challenges in improving the understanding of the structure of valence electrons in typical magnetic materials. Based on a lot of experimental and theoretical investigations, Tang et al. proposed three models for magnetic ordering in typical magnetic materials. Using these models to replace the conventional models, they studied the magnetic structures of spinel ferrites, perovskite manganites, and Fe, Co, and Ni metals, including not only the magnetic ordering experimental phenomena based on the conventional models but also several experimental phenomena, which have been the topic of ongoing disputes for many years. Furthermore, they explained the electrical transport property of perovskite manganites, Fe, Co, and Ni metals, and NiCu alloys using their magnetic ordering models. These studies assisted in improving the valence electron structure theory for typical magnetic materials.

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Foreword

In this book, the author introduces their work along with related studies from the literature. This provides new insight into the valence electron states of typical magnetic materials. December 2019

You-Wei Du Academician of Chinese Academy of Sciences Nanjing University Nanjing, China

Author’s Preface

Conventional magnetic ordering models were proposed before 1960, and since then, many experimental studies, which contributed to the improvement of these models, have been reported. In 2018, a review article titled “Three models of magnetic ordering in typical magnetic materials” was published in Physics Reports by our group and Profs. Guang-Heng Wu and Feng-Xia Hu of Institute of Physics, Chinese Academy of Sciences. The three models developed based on the experimental results include an O 2p itinerant electron model for magnetic oxides, a new itinerant electron model for magnetic metals, and a Weiss electron-pair model for the origin of magnetic ordering energy for magnetic metals and oxides. By using these models, we explained magnetic ordering phenomena of several series of typical magnetic materials, including those that could not be explained by conventional models. In this book, we introduce these recently developed models and related literature. This book will benefit readers who would like to gain new insight into the magnetic ordering phenomena of magnetic materials and the design of novel materials. I welcome the opportunity to receive feedback from the readers on the shortcomings of the book to improve our investigations. During our investigations on the three models, our group frequently exchanged ideas with Professors Guang-Heng WU and Feng-Xia HU, and several papers were published by our group in collaboration with them. I wish to express our sincere thanks to them. I wish to thank experts with whom we enjoyed the opportunity to discuss our work, including Profs. You-Wei Du, Ding-Yu Xing, Wei Zhong, and Jun-Ming Liu of Nanjing University; Profs. Zhao-Hua Cheng, Xiu-Feng Han, Chang-Qing Jin, and Wu-Ming Liu of Institute of Physics, Chinese Academy of Sciences; Prof. Ji-Fan Hu of Shandong University; Prof. Jin-Bo Yang of Peking University; Profs. Deng-Lu Hou, Ying Liu, Wei Chen, and Norm Davison of Hebei Normal University. I wish to thank the members of our group for their hard work, including Dr. Zhuang-Zhi LI, Profs. Li Ma and Cong-Mian Zhen, doctoral students Wei-Hua Qi, Deng-Hui Ji, Jing Xu, and Shi-Qiang Li, and all of my graduate students.

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Author’s Preface

I wish to thank Prof. Xiang-Fu Nie of Hebei Normal University, Profs. Bao-Shan Han and Bo-Zang Li of Institute of Physics, Chinese Academy of Sciences. Under their guidance, I started my research studies on magnetism and magnetic materials in 1983. The publication of this book and related work were supported by the National Natural Science Foundation of China (NSF-11174069). In addition, related research studies were also supported by the National Natural Science Foundation of China (NSF-10074013), the several items of the Natural Science Foundation of Hebei Province, China, the Key Item Science Foundation of Hebei Province, China, and the Young Scholar Science Foundation of the Education Department of Hebei Province, China. I wish to thank Drs. Jun Qian and Han Zhou for their hard work to publish this book, and my family members and friends for their endless support. The author would like to thank Enago (www.enago.cn) for the English language review. Shijiazhuang City, China December 2019

Gui-De Tang

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Electron Shell Structure of Free Atoms and Valence Electrons in Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Electron Shell Structure of Free Atoms . . . . . . . . . . . . . . . . . . . . . . 2.2 A Simple Introduction to Classical Crystal Binding Theory for Typical Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Effective Radii of Ions in Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Electron Binding Energy Originating from Ions in Crystals . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A Simple Introduction to Basic Knowledge of Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Classification of Matter Based on Magnetic Properties . . . . . . . . . 3.2 Magnetic Domain and Domain Wall . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Basic Parameters of Magnetic Materials . . . . . . . . . . . . . . . . . . . . . 3.4 Magnetic Ordering Models in Conventional Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Difficulties Faced by Conventional Magnetic Ordering Models . . . . 4.1 Disputes Over the Cation Distributions in Mn and Cr Spinel Ferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Normal, Inverse, and Mixed Spinel Structure . . . . . . . . . 4.1.2 Magnetic Moments of 3d Transition Metal Ions . . . . . . . 4.1.3 Magnetic Ordering of CrFe2 O4 and MnFe2 O4 . . . . . . . . . 4.2 Difficulties in Describing the Observed Magnetic Moments of Perovskite Manganites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 5 5 6 8 9 11 13 13 16 18 21 24 25 25 25 27 27 31

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4.3

Relationship Between Magnetic Moment and Resistivity in Typical Magnetic Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Puzzle for the Origin of Magnetic Ordering Energy . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

6

O 2p Itinerant Electron Model for Magnetic Oxides . . . . . . . . . . . . . . 5.1 A Simple Introduction to Early Investigations of Ionicity . . . . . . . 5.2 Study of the Ionicity of Spinel Ferrites . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Quantum-Mechanical Potential Barrier Model Used to Estimate Cation Distributions . . . . . . . . . . . . . . . 5.2.2 Study of the Ionicity of Group II–VI Compounds Using the Quantum-Mechanical Potential Barrier Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Study of Ionicity of Spinel Ferrite Fe3 O4 . . . . . . . . . . . . . 5.2.4 Estimation of the Ionicity of Spinel Ferrites M 3 O4 Using the Quantum- Mechanical Potential Barrier Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Experimental Studies of O 2p Holes in Oxides . . . . . . . . . . . . . . . . 5.3.1 O 2p Hole Studies Using Electron Energy Loss Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Several Other Experimental Investigations for O 2p Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Study of Negative Monovalent Oxygen Ions Using X-Ray Photoelectron Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Study of Ionicity of BaTiO3 and Several Monoxides Using O 1s XPS . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Effect of Argon Ion Etching on the O 1s Photoelectron Spectra of SrTiO3 . . . . . . . . . . . . . . . . . . . . 5.5 O 2p Itinerant Electron Model for Magnetic Oxides (IEO Model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Relationship Between the IEO Model and the Conventional Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38 38 39 43 43 45 46

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50 51 52 54 54 55 60 70 75 79

Magnetic Ordering of Typical Spinel Ferrites . . . . . . . . . . . . . . . . . . . . 81 6.1 Method Fitting Magnetic Moments of Typical Spinel Ferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.1.1 X-ray Diffraction Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.1.2 Magnetic Property Measurements . . . . . . . . . . . . . . . . . . . 84 6.1.3 Primary Factors that Affect Cation Distributions . . . . . . . 85 6.1.4 Fitting the Magnetic Moments of the Samples . . . . . . . . 88 6.1.5 Discussion on Cation Distributions . . . . . . . . . . . . . . . . . . 91 6.2 Cation Distribution Characteristics in Typical Spinel Ferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Contents

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Experimental Evidences of the IEO Model Obtained from Spinel Ferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Additional Antiferromagnetic Phase in Ti-Doped Spinel Ferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 X-ray Diffraction Spectra of the Samples . . . . . . . . . . . . . 7.1.2 X-ray Energy Dispersive Spectra of the Samples . . . . . . 7.1.3 Magnetic Measurements and Analysis of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Cation Distributions of the Three Series of Ti-Doped Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.5 Magnetic Ordering of Spinel Ferrites Tix M1−x Fe2 O4 (M = Co, Mn) . . . . . . . . . . . . . . . . . . . . . . 7.2 Amplification of Spinel Ferrite Magnetic Moment Due to Cu Substituting for Cr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 X-ray Energy Dispersive Spectrum Analysis . . . . . . . . . . 7.2.2 X-ray Diffraction Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Magnetic Measurement and Magnetic Moment Fitting Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Unusual Infrared Spectra of Cr Ferrite . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Infrared Spectra of Spinel Ferrites MFe2 O4 (M = Fe, Co, Ni, Cu, Cr) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Dependency of the Peak Position ν 2 on the Magnetic Moment (μM2 ) of Divalent M Cations in MFe2 O4 (M = Fe, Co, Ni, Cu, Cr) . . . . . . . . . 7.3.3 Infrared Spectra of Co1−x Crx Fe2 O4 and CoCrx Fe2−x O4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spinel Ferrites with Canted Magnetic Coupling . . . . . . . . . . . . . . . . . . 8.1 Spinel Ferrites with Fe Ratio Being Less Than 2.0 Per Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Spinel Ferrites Containing Nonmagnetic Cations . . . . . . . . . . . . . . 8.2.1 Disputation of Nonmagnetic Cation Distribution . . . . . . 8.2.2 Fitting Sample Magnetic Moments . . . . . . . . . . . . . . . . . . 8.2.3 Discussion on Cation Distributions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Ordering and Electrical Transport of Perovskite Manganites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Ferromagnetic and Antiferromagnetic Coupling in Typical Perovskite Manganites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Crystal Structure and Magnetic Measurement Results of La1–x Srx MnO3 Polycrystalline Powder Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Study of Valence and Ionicity of La1–x Srx MnO3 . . . . . . .

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9.1.3

Fitting of the Curve of the Magnetic Moment Versus Sr Ratio for La1–x Srx MnO3 . . . . . . . . . . . . . . . . . . 9.2 Spin-Dependent and Spin-Independent Electrical Transport of Perovskite Manganites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 A Model with Two Channels of Electrical Transport for ABO3 Perovskite Manganites . . . . . . . . . . . 9.2.2 Fitting the Curves of Resistivity Versus Test Temperature of Single-Crystal La1−x Srx MnO3 . . . . . . . . 9.2.3 Fitting the Curves of Resistivity Versus Test Temperature of La0.60 Sr0.40 Fex Mn1−x O3 Polycrystalline Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Discussion on Factors Affecting Electrical Transport Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Experimental Evidence on the Canting Angle Magnetic Structure in Perovskite Manganites . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Analyses for the Crystal Structure of the Samples . . . . . 9.3.2 Magnetic Measurement Results . . . . . . . . . . . . . . . . . . . . . 9.3.3 Measurement Results of Electrical Transport Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Fitting of Sample Magnetic Moment Using the IEO Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.5 Effects of Thermal Excitation, Lattice Scattering, and Spin-Dependent Scattering on the Transition Probability of Itinerant Electrons . . . . . . . . . . . . . . . . . . . . 9.3.6 Effect of Canted Ferromagnetic Coupling on Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Magnetic Coupling Between the Two Sublattices in Perovskite Praseodymium Manganites . . . . . . . . . . . . . . . . . . . . 9.5 Substituting for Mn in Perovskite Praseodymium Manganites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Fitting of the Sample Magnetic Moment as the Function of Doped Level . . . . . . . . . . . . . . . . . . . . . 9.5.2 Influence of Canted Magnetic Structure on the Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Experimental Evidence for Antiferromagnetic Coupling Between Divalent and Trivalent Mn Ions in Perovskite Manganites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Preparation of the Samples . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2 Crystal Structure and Crystal Lattice Constants of the Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.3 Magnetic Measurement Results . . . . . . . . . . . . . . . . . . . . . 9.6.4 Discussion on the Magnetic Structures of the Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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158 160 166 167 169 174 179

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Contents

10 Antiferromagnetic Ordering in Oxides with Sodium Chloride Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Characteristics of Antiferromagnetic Oxides with Sodium Chloride Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Difference Between Magnetic Structures of Manganese Monoxide and Lanthanum Manganite . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Itinerant Electron Model for Magnetic Metals . . . . . . . . . . . . . . . . . . . 11.1 Experimental and Theoretical Studies for Atomic Magnetic Moments in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Itinerant Electron Model for Magnetic Metals (IEM Model) . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Study on the Origin of Magnetic Ordering Energy for Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Weiss Molecular Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Thermal Expansion of Perovskite Manganites Near the Curie Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Weiss Electron Pair (WEP) Model for Origin of Magnetic Ordering Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Explanation for the Curie Temperature Difference of Typical Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Explanation for Cu Ratio Dependence of Resistivity and Curie Temperature for NiCu Alloys . . . . . . . . . . . . . . . . . . . . . 12.5.1 Free and 3d Electron Ratios in NiCu Alloys . . . . . . . . . . 12.5.2 Electrical Transport Model with FE and IE Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.3 An Explanation of the Curie Temperature Using the WEP Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Prospects and Challenges for Future Work . . . . . . . . . . . . . . . . . . . . . . 13.1 Other Factors Affecting Magnetic Ordering Energy . . . . . . . . . . . 13.2 Magnetic Ordering Energy in DFT Calculations . . . . . . . . . . . . . . 13.3 Applications of the IEO and IEM Models . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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203 203 205 206 207 207 208 211 213 213 215 215 221 222 222 223 226 229 231 231 233 233 234

Appendix A: Electron Structure and Ionization Energies of Free Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Appendix B: Effective Ion Radii Reported by Shannon . . . . . . . . . . . . . . . 239 Appendix C: Symbol Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Chapter 1

Introduction

One of the oldest applications of magnetic materials is the use of compass. In modern times, the applications of magnetic materials have benefited many fields, such as aviation, spaceflight, military affairs, radio, television, communication, and medicine, in the form of magnetic memory devices, magnets, transformers, and microwave devices. However, some of the challenging problems on magnetic ordering phenomena have not been reasonably explained because of the lack of phenomenological expression of the magnetic ordering energy, or the energy of the Weiss molecular field. In 1907, Weiss proposed the presence of small regions in magnetic materials called magnetic domains. In each magnetic domain, atomic magnetic moments arrange in a certain order subjected to a “molecular field.” Magnetic domains have been observed in many experimental studies. However, the origin of the molecular field is yet to be explained satisfactorily. Several different models for the magnetic ordering mechanism were introduced in the textbooks [1–4], including phenomenological spontaneous magnetization theory, exchange interaction theory for spontaneous magnetization, spin-wave theory, and metal energy band theory. These theories are based on different assumptions and rely on different theoretical systems. Since they fail to explain several experimental phenomena, developing ferromagnetism theory is challenging. In classical ferromagnetism, the origin of magnetic ordering energy was explained by using exchange interactions of electrons between ions, called direct exchange interaction in magnetic metals and alloys, superexchange (SE) interaction for the antiferromagnetic coupling between magnetic cations in an oxide, and double-exchange (DE) interaction for the ferromagnetic coupling between magnetic cations in an oxide. Because nearly a thousand times difference for magnetic ordering energy between estimated (using the Curie temperature) and calculated (using classical electromagnetism model) values exist, the origin of magnetic ordering energy is considered to be a pure quantum-mechanical effect, independent of the classical electromagnetism model. However, magnetic material calculation using the density

© Science Press 2021 G.-D. Tang, New Itinerant Electron Models of Magnetic Materials, https://doi.org/10.1007/978-981-16-1271-8_1

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1 Introduction

functional theory (DFT) based on quantum mechanics is challenging because the expression to calculate the magnetic ordering energy has not been developed. No report has addressed the valence electron spectrum when the classical ferromagnetism models were proposed before 1960. Since the 1970s, many studies have reported electron spectra of magnetic materials, and an improved understanding of the electrical transport mechanism for magnetic perovskite manganites was provided. The magnetic DE interaction was firstly used to explain the ferromagnetic coupling between Mn cations in ABO3 perovskite manganites, in particular, La1–x Srx MnO3 . In the classical view [5, 6], all oxygen anions are assumed to be O2− in these materials. With increasing Sr2+ ratio (x), an equal number of Mn4+ ions exist in the system. The DE interaction of 3d electrons between Mn3+ and Mn4+ ions mediated by O2− ions, was used to explain the magnetic ordering and the electrical transport phenomena in La1–x Srx MnO3 . However, based on the electron energy loss spectra and other electron spectrum experimental results, Alexandrov et al. [7] pointed out that the DE model contradicts these experimental results, which clearly showed that the current carriers are oxygen p holes rather than d electrons of ferromagnetic manganites. Studies have shown that O1− ions may constitute 30% or more of oxygen ions in oxides. The outer electron shell of an O1− ion exists a p hole, which affects the magnetic and electrical transport properties of oxides. In fact, the effect of oxygen p holes was accurately considered in the investigation of superconductor oxides [8] but has not been widely accepted in studies concerning magnetic oxides. Our group cooperated with Professors Wu and Hu of State Key Laboratory of Magnetism, Institute of Physics, Chinese Academy of Sciences, and published a series of articles about the new magnetic ordering models, including a review article in Physics Reports [9] titled “Three models of magnetic ordering in typical magnetic materials”. These models include an O 2p itinerant electron model for magnetic oxides (IEO model) [10, 11], a new itinerant electron model for magnetic metals (IEM model) [12], and a Weiss electron-pair (WEP) model for the origin of magnetic ordering energy [13]. By using the IEO model that replaces the SE and DE models, the magnetic structures of not only Co-, Ni-, or Cu-doped spinel ferrites but also Cr-, Mn-, or Ti-doped spinel ferrites could be explained, moreover, the dependence of the magnetic moments on the Sr ratio in perovskite manganites (such as La1-x Srx MnO3 ) can be explained, for which there have been many ongoing disputes regarding the cation distributions of these materials in conventional views [9–11]. Using the IEM model, we qualitatively explained the relationship between the magnetic moments and resistivities of Fe, Co, Ni, and Cu metals [12] and fitted the curves of the resistivity of NiCu alloys with the different Cu doping level versus the test temperature [14]. Using the WEP model, we explained why Fe, Co, Ni metals, NiCu alloys, Fe3 O4 and La0.7 Sr0.3 MnO3 oxides have different Curie temperatures [14, 15]. All the IEO, IEM, and WEP models have unambiguous physical mechanisms and are based on the experimental results of the electron spectra and the basic principles of atomic physics. This is the unique character of this book, making it different from a classical ferromagnetism textbook.

1 Introduction

3

According to the classical ferromagnetism textbooks [1–4], the classical itinerant electron model is different from the local electron model. In fact, these two models belong to different schools of thought. The local electron model was used to explain the magnetic properties of oxides but is unsuitable to explain the magnetic properties of metals. The classical itinerant electron model was used to explain the magnetic properties of metals and alloys in which all s and d electrons in 3d transition metal atoms were assumed to be neither local electrons nor free electrons but itinerant electrons. The new itinerant electron model proposed by our group has five characters: First, s electrons in free 3d transition metal atoms divide two parts in a metal or an alloy: one part enter the d orbits and change into the d electrons, another part act as free electrons. Second, only one or two d electrons occupying the outer d orbit of a cation may form itinerant electron in a certain probability, and other d electrons are local electrons. Third, the itinerant electrons undergo a spin-dependent transition below the Curie temperature, and the transition probability decreases with increasing test temperature; but it is spin-independent transition above Curie temperature, and the transition probability increases with increasing test temperature. Fourth, the transition probability of the itinerant electrons decreases with increasing the distance between adjacent ions. Fifth, the itinerant electrons in magnetic metals and alloys are similar in character to those in magnetic oxides. In this book, although we propose the IEO and IEM models for magnetic metals and oxides, respectively, the WEP model is related to both of them. So, the three models may be referred to improve the theory of ferromagnetism. We expect that these models could be widely used for studies of magnetic materials and theoretically improved further.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Dai DS, Qian KM (1987) Ferromagnetism (in Chinese). Science Press, Beijing Coey JMD (2010) Magnetism and magnetic materials. Cambridge University Press, Cambridge Chikazumi S (1997) Physics of Ferromagnetism (2e). Oxford University Press, London Stöhr J, Siegmann HC (2006) Magnetism: from fundamentals to nanoscale dynamics. Springer, Berlin Heidelberg, New York Salamon MB, Jaime M (2001) Rev Moder Phys 73:583 Dagotto E, Hotta T, Moreo A (2001) Phys Rep 344:1 Alexandrov AS, Bratkovsky AM, Kabanov VV (2006) Phys Rev Lett 96:117003 Han RS (1998) Physics of high temperature superconductor (in Chinese). Peking University Press, Beijing Tang GD, Li ZZ, Ma L, Qi WH, Wu LQ, Ge XS, Wu GH, Hu FX (2018) Phys Rep 758:1 Xu J, Ma L, Li ZZ, Lang LL, Qi WH, Tang GD, Wu LQ, Xue LC, Wu GH (2015) Phys Status Solidi B 252:2820 Wu LQ, Qi WH, Ge XS, Ji DH, Li ZZ, Tang GD, Zhong W (2017) Europhys Lett 120:27001 Qi WH, Ma L, Li ZZ, Tang GD, Wu GH (2017) Acta Phys Sin 66:027101 Qi WH, Li ZZ, Ma L, Tang GD, Wu GH, Hu FX (2017) Acta Phys Sin 66:067501 Li ZZ, Qi WH, Ma L, Tang GD, Wu GH, Hu FX (2019) J Magn Magn Mater 482:173–177 Qi WH, Li ZZ, Ma L, Tang GD, Wu GH (2018) AIP Adv 8:065105

Chapter 2

Electron Shell Structure of Free Atoms and Valence Electrons in Crystals

A crystalline (single-crystal or polycrystalline) material is made up of atoms. The magnetic property of a material originates from its atomic properties and crystal structure. In this chapter, the knowledge of atomic physics related to the magnetic property is introduced, including the electron shell structure of an atom, the atomic magnetic property, electronic affinity energy, ionization energy of free atoms, the binding of magnetic crystals, effective radii of ions in a crystal, and the electronic binding energy in crystals.

2.1 Electron Shell Structure of Free Atoms According to the scattering experiment of α particles by Rutherford et al., an atom is made up of a nucleus and electrons [1]. The size of an atom is on the order of 10−10 m, while a nucleus has a size on the order of 10−15 –10−14 m. Based on experimental and theoretical findings, electrons distributed outer of the nucleus in different shells are in a stable state, obeying the quantum mechanics principles. The electronic shells consist of main shells and subshells. The quantum number of the main shell is represented by n. In the periodic table, the element in the nth period has n main shells. The main shell may have several subshells represented by s, p, d, and f , which can be occupied by 2, 6, 10, and 14 electrons, respectively. Electron distributions in different shells can be found in the tables of atomic physics textbooks and ferromagnetism textbooks. In general, the electronic number in a shell is represented by an agreed method, such as the valence electronic state of Fe atom is represented by 3d 6 4s2 , indicating that there are six electrons in the third subshell of the third main shell, and two electrons in the first subshell of the fourth main shell. The electronic shell structures of free atoms are shown in Appendix A [2] in which the electronic shell of Fe atom is [Ar]3d 6 4s2 , representing it has the same electronic structure as Ar atom, in addition to the electrons in the outer shells.

© Science Press 2021 G.-D. Tang, New Itinerant Electron Models of Magnetic Materials, https://doi.org/10.1007/978-981-16-1271-8_2

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2 Electron Shell Structure of Free Atoms and Valence …

When an atom with electric neutrality obtains an electron and forms a negative monovalent ion, part of the energy released is known as the electronic affinity energy. For example, the second electronic affinity energy of oxygen equals 8.08 eV. An O2− ion can be formed when an oxygen atom obtains two electrons, which has a stable outer shell structure with 2s2 2p6 . An atom with electric neutrality may lose an electron and change into a positive monovalent ion when this electron obtains enough energy called the first ionization energy. Similarly, the requisite energy when a free atom loses its nth electron is called the nth ionization energy. In Appendix A, the data from the first to fifth ionization energy (V 1 to V 5 ) [2] can be found. Figure 2.1a–e shows the dependencies of V 1 –V 5 on the atomic number, N. It can be seen from Fig. 2.1 that, in general, ionization energy increases with increasing N in the same period but decreases with increasing the period number. For a given atom, the ionization energy of nth electron is far higher than that of (n−1)th electron. For instance, the ionization energies from the first to the fifth electron of Fe atom are 7.90, 16.19, 30.65, 54.8, and 75.0 eV, respectively. The values of ionization energy show the binding ability of the nucleus for its electrons. These ionization energy values are important for the ability of an ion to obtain or lose an electron in a crystal, although the valence electronic binding energy in a crystal is different from the ionization energy for a free atom. In Sect. 6.1, an application of ionization energy data for cation distribution in oxides by the quantum- mechanical potential barrier model is introduced.

2.2 A Simple Introduction to Classical Crystal Binding Theory for Typical Magnetic Materials According to the classical crystal binding theory [3, 4], the binding types include ionic, covalent, metallic, molecular, and mixed bonds. The final states of crystal binding may be single crystal, polycrystal, nanocrystal bulk, thin film, nanoline, and so on. The interaction energy (U 0 ) between all ions in a solid at the equilibrium state is negative. The cohesive energy (E B ) of a crystal at 0 K and 1 atm is the energy required to disassemble it into neutral atoms with the ground electronic state; then, E B = |U 0 |. The first ionization energy values of alkali metal-free atoms are very low, between 5.39 and 5.89 eV. The electronic affinity energies of halogens are between 3.06 and 3.61 eV. Therefore, a typical ionic compound can be made from the alkali metal and the halogen. For example, Na and Cl are positive and negative ions, respectively, in NaCl crystal. The binding of Na+ and Cl− ions is due to the Coulomb energy, while the Pauli repulsive energy also plays a role in forming the equilibrium spacing between Na+ and Cl− ions in the stable ionic crystal. The magnetic oxides made up of oxygen and 3d transition metal ions are ionic compounds. However, O1− ions exist in addition to O2− ions in an oxide, resulting in the average valence states of the anions and cations to be lower than the values in the conventional view. The reason may be discussed using the electronic affinity

2.2 A Simple Introduction to Classical Crystal …

7

(b)

80

(a)

25 20

V2 (eV)

60

V1 (eV)

15

40

10 20

5 0

0

20

40

60

N

80

0

100

160

40

60

80

300

100

(d)

250

120 100

V4 (eV)

V3 (eV)

20

N

(c)

140

0

80 60

200 150 100

40

50

20 0 0

20

40

60

80

0

100

0

20

40

N

400

N

60

80

100

(e)

350

V5 (eV)

300 250 200 150 100 50 0

0

20

40 N

60

80

Fig. 2.1 Dependences on the atom number (N) of the ionization energy (V N ) of nth electron in a free atom. The figures (a) to (e) correspond to V 1 to V 5

energy and ionization energy of the free atoms: The second electronic affinity energy of an oxygen ion is 8.08 eV, and the second ionization energies of 3d transition metal ions are between 12.80 and 20.30 eV. Thus, an oxygen anion cannot easily obtain the second electron. This phenomenon is explained using ionicity (f i ): covalent bonds exist in addition to the ionic bonds in an oxide. When all chemical bonds are ionic bonds in an oxide, the ionicity is defined as f i = 1.0 [5]. Therefore, if there are covalent bonds in an oxide, f i < 1.0. In 1970, Phillips published a review article [5] on ionicity, including the ionicity study reported by Pauling in 1932 and 1939. Phillips presented many ionicity data. For

8

2 Electron Shell Structure of Free Atoms and Valence …

example, the ionicity (f i ) values of BeO, MgO, CaO, and SrO are 0.60, 0.814, 0.913, and 0.926, respectively. The second ionization energies (V 2 ) of Be, Mg, Ca, and Sr are 18.21, 15.04, 11.87, and 11.0 eV, respectively, suggesting that f i increases with decreasing V 2 and that the ionization energy of the free atom influences obtaining or losing electrons in a compound. The metals and alloys possess metallic bonds including free electrons. For example, according to the theoretical and experimental investigations, each Cu atom ([Ar]3d10 4s1 ) contributes a free electron to Cu metal, which has good electrical conductivity. In a metal crystal, the ion cores form the periodic potential field. On the one hand, the Coulomb attraction energy between the ion cores and free electrons results in the decrease of distance between ion cores; on the other hand, there is the Pauli repulsive energy between the outer electronic shells of adjacent ion cores. These two energies form the stable ion core distance and the stable crystal structure for metal crystals. For convenience, we simply use “ion” instead of “ion core” in metals in the following sections.

2.3 Effective Radii of Ions in Crystals According to quantum mechanics, the state of the electrons in an atom is described as a wave function. The probability of an electron occupying a position is proportional to the intensity of the wave function. This suggests that an atom has no stable radius. However, the distance between ions in a crystal is stable, according to X-ray diffraction data, by which an ion may have an effective radius. This is especially important for material science and chemistry researchers to discuss material properties. Shannon [6] reported a set of data of effective radii for various ions with different valence states and coordination numbers as listed in Appendix B, indicating that the effective radii of cations decrease with increasing the valence and decreasing the coordination number. Table 2.1 shows the effective radii of several divalent and trivalent cations with coordination number 6, r 2+ and r 3+ , and their difference r 2+ −r 3+ . It can be seen that Table 2.1 Effective radii of several divalent and trivalent cations with the coordination number 6, r 2+ and r 3+ , and their difference r 2+ −r 3+ [6]

Element

r 2+ (Å)

r 3+ (Å)

r 2+ −r 3+ (Å)

Cr

0.80

0.615

0.185

Mn

0.83

0.645

0.185

Fe

0.78

0.645

0.135

Co

0.745

0.61

0.135

Ni

0.69

0.60

0.09

Ag

0.94

0.75

0.19

Note The effective radius of O2− ion with the coordination number 6 is 1.40 Å [6]

2.3 Effective Radii of Ions in Crystals

9

the difference ranges from 0.09 Å to 0.19 Å, suggesting that more or less one electron has an obvious effect on the ionic radius and that most of the valence electrons of an ion are local electrons.

2.4 Electron Binding Energy Originating from Ions in Crystals The nth ionization energy (V N ) of the free atoms is the requisite energy when a free atom loses its nth electron. The electronic binding energy (E b ) is the energy difference between the electronic energy level of the ion and the Fermi energy level in a crystal. Therefore, there is a difference between E b and V N for the same element. However, E b and V N are related. The value of E b may be measured using the X-ray photoelectron spectroscopy (XPS) [7]. Using X-rays to shine a crystal sample, two phenomena occur: (1) When the photons are scattered by the orbital electrons of the ions, the scattered photons with energy being lower than the incident photons may be accepted using a detector, forming Compton scattering spectra. (2) When the orbital electrons obtain the whole energy of the incident photons, they form the photoelectrons. When the photoelectrons move to the surface of the sample and are transmitted by overcoming the work function, they can be detected by XPS. The XPS measurement is based on the Einstein photoemission equation. For an isolated atom, the kinetic energy of a photoelectron is E k = hν − E b ,

(2.1)

where hν is the energy of the incident photon. E k may be measured using the energy analyzer. Then, the binding energy E b of the photoelectron can be obtained. For an element, E b of the electrons at different energy levels have different values, resulting in distinct XPS peaks in which the peak with the highest intensity is called the main peak. For different elements, E b and E k have different values. Therefore, XPS may be used to analyze the surface composition of materials. For a crystal sample, since photoelectrons need energy from production to transmittance, only the photoelectrons near the surface may be transmitted. The probe depth of the XPS may be represented by the mean free path, λ. The value of λ is about 0.5–3 nm for a metal and about 2–4 nm for an oxide. The probe depth is about 3λcosθ, where θ is the angle between the detector and the normal of the sample surface. If the sample surface was etched using an ionic beam, different XPS results are obtained for different etching durations, and the inner photoelectron information of a material may be probed. Notably, in XPS analyses, more ions with small atomic numbers are lost than ions with larger atomic numbers during etching.

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2 Electron Shell Structure of Free Atoms and Valence …

Fig. 2.2 Dependences on the atom number (N) of the binding energy (E b ) of 1s electron in a crystal and the ionization energy (V N ) of 1s electron in a free atom

The effect of the work function and other factors may be reduced by the software of advanced XPS. The XPS data below the Fermi energy level are directly presented. The data of the electron binding energy of the ions in a crystal may be found in the XPS handbook [8]. These data are binding energy values from the electronic energy level of the ion to the Fermi energy level in a crystal. Figure 2.2 shows the dependencies on the atomic number (N) of the binding energy (E b ) of 1s electron in a crystal and the ionization energy (V N ) of 1s electron in a free atom. Both E b and V N increase rapidly with increasing N, although E b is lower than V N , suggesting that E b increases with increasing V N , which is why we analyze the valence of the ions in a material using the ionization energy of the electron in a free atom (see Sect. 6.1). The valence band (VB) photoelectron spectra of a crystal can be obtained using XPS. The VB spectra of 3d transition metals from Sc to Fe were reported by Ley et al. [9]. They showed that the valence electrons of these metals are distributed in a range of about 12 eV below Fermi energy level, E F . We reported the VB spectra of CaO, ZnO, MnFe2 O4 , and ZnFe2 O4 [10]. The valence electrons of these oxides are distributed in a range about 11.5 eV below E F , as shown in Fig. 2.3. Interestingly, the Zn 3d electrons distribute in a narrow region from 8.2 to 11.5 eV below E F because every Zn2+ ion has ten 3d electrons with full subshell structure, and the third ionization energy of Zn is 39.72 eV; thus, the third electron of Zn ion is very hard to be ionized. These XPS results are useful to understand the new itinerant electron model in this book.

2.4 Electron Binding Energy Originating from Ions in Crystals

11

Fig. 2.3 Valence band photoelectron spectra of CaO, ZnO, MnFe2 O4 , ZnFe2 O4 [10]

References 1. Chu SL (1979) Atomic physics (in Chinese). People’s Education Press, Beijing 2. Author’s group of Practical handbook of Chemistry (2001) Practical handbook of Chemistry (in Chinese). Science Press, Beijing 3. Achcroft NW, Mermin ND (1976) Solid state physics. Reinhart and Winston inc., New York 4. Kittel C (2005) Introduction to Solid State Physics. John Wiley & Sons Inc., New York 5. Phillips JC (1970) Rev Mod Phys 42:317 6. Shannon RD (1976) Acta Cryst. A. 32:751 7. Lu JH, Chen CY (1995) Modern analysis technology (in Chinese). Tsinghua Nuniversity Press, Beijing 8. Wagner CD, Davis WM, Moulder JF, Muilenberg GE (1979) Handbook of X-ray photoelectron spectroscopy. Perkin-Elmer coporation, Eden Prairie, Minnesota, USA 9. Ley L, Dabbousi OB, Kowalczyk SP, McFeely FR, Shiley DA (1977) Phys Rew B 16:5372 10. Ding LL, Wu LQ, Ge XS, Du YN, Qian JJ, Tang GD, Zhong W (2018) Results Phys 9:866

Chapter 3

A Simple Introduction to Basic Knowledge of Magnetic Materials

It may be considered that all matters possess magnetic property because electrons and nuclei in atoms possess magnetic property. However, the magnetic property intensity differences between different materials are very large. In this chapter, we introduce basic knowledge of magnetic materials and conventional magnetic ordering models.

3.1 Classification of Matter Based on Magnetic Properties The magnetization, M, of a material is the total magnetic moment per unit volume in Am−1 , which relates to the applied magnetic field, H, in Am−1 , and magnetic induction, B, in Tesla (T): M = χ H,

(3.1)

B = μ0 (H + M) = μ0 (1 + χ )H = μ0 μH.

(3.2)

Or

where χ is the susceptibility per unit volume, μ0 = 4 π × 10−7 H/m is the permeability of vacuum and μ = 1 + χ is the relative permeability. In literature including this book, μ0 H is also used to represent the magnetic field, in Tesla (T). The specific magnetization, σ, is the magnetic moment per mass in Am2 kg−1 . Mass susceptibility, χm , molar susceptibility, χ A , and volume susceptibility χ are related: χm = χ /d

© Science Press 2021 G.-D. Tang, New Itinerant Electron Models of Magnetic Materials, https://doi.org/10.1007/978-981-16-1271-8_3

(3.3)

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and χ A = χm A = χ A/d,

(3.4)

where d is the density of the material and A is the mass per mole of a material. Materials are conventionally classified as diamagnet, paramagnet, and ferromagnet. They have the following characteristics: When a ferromagnet is exposed to an external magnetic field, its magnetic field strength is greatly enhanced, whereas that of the paramagnet is only slightly enhanced. On the contrary, when a diamagnet is exposed to an external magnetic field, its magnetic field strength is slightly decreased. The above classification of magnetic materials is explained by using atomic magnetic moments: Metal Cu is a typical diamagnetic material, and the reason for that is the following: The free Cu atom has valence electronic state 3d 10 4 s1 . In metal Cu, 4s electrons become the free electrons, and the whole five energy levels of 3d subshell are filled by five electrons with up-spin and five electrons with down-spin, resulting in the total spin magnetic moment and the atomic magnetic moment being zero. When Cu is exposed to an external magnetic field, the magnetic field strength is slightly decreased due to the movement of electrons subjected to the electromagnetic induction in the external magnetic field. In fact, the diamagnetic property exists in all materials, but it is concealed in paramagnets and ferromagnets. In paramagnets and ferromagnets, a portion of or whole atoms have subshells with holes. These atoms have nonzero intrinsic magnetic moments. A distinct characteristic of ferromagnets is that they have Curie temperatures above which they become paramagnets. To explain the ferromagnetism of the materials, in 1907, Weiss proposed an assumption: Many small regions existing in a ferromagnetic material have spontaneous magnetization, even in the absence of an external magnetic field. These small regions that exhibit spontaneous magnetization are called magnetic domains. This spontaneous magnetization is caused by a high inner field in the crystal, known as the molecular field [1, 2]. It arranges the atomic magnetic moments that are parallel to each other. The magnetic domains had been confirmed by experiments; however, the molecular field has not been explained using a phenomenological model so far and is assumed to be caused by the quantum-mechanical effect [1, 2]. Furthermore, the ferromagnetic materials are classified as ferromagnetic, ferrimagnetic, and antiferromagnetic materials. Their differences can be explained using several examples: Fe, Co, and Ni are typical ferromagnetic materials. In a magnetic domain of these materials, the magnetic moments of all atoms are parallel to each other. MnO, FeO, and NiO are typical antiferromagnetic materials. In their magnetic domains, cation magnetic moments are antiparallel to each other, and the sum of the cation magnetic moments in one direction is equal to that in another direction; therefore, the total magnetic moments of the material are zero or very small. Fe3 O4 is a typical ferrimagnetic material. In a domain of Fe3 O4 , the magnetic moments of Fe ions are aligned antiparallel to each other. However, the number of Fe ions aligned in one direction is more than that in another direction, resulting in the total magnetic moment of the material being nonzero.

3.1 Classification of Matter Based on Magnetic Properties

15

For the five types of materials, Table 3.1 shows several examples and the order of magnitude and the temperature dependencies of magnetic susceptibilities χ . Table 3.2 shows whether the ions (for metal, ionic core excluding free electrons) have intrinsic magnetic moments and how they arrange in a magnetic domain. Figure 3.1 shows the dependencies of χ −1 on the test temperature for paramagnetic, ferromagnetic, and antiferromagnetic materials [2]. Here, T C represents the Curie temperature of ferromagnetic materials, T N and θ p represent the Néel temperature and paramagnetic Curie temperature of antiferromagnetic materials, respectively. Typical ferrimagnetic materials have two magnetic sublattices with magnetic moments represented by M A and M B , respectively. The sample saturation magnetic moment M S is equal to the difference between M A and M B . Therefore, the dependence of M S on the test temperature T may show three phenomena [1]: (a) it is similar to that of the ferromagnetic materials; (b) there is a maximum M S at a certain temperature below T C ; (c) M S value may be equal to zero at a certain temperature below T C . Table 3.1 Classification, magnetic susceptibility characteristics, and examples of materials. Where T C and T N represent Curie temperature and Néel temperature, respectively Examples

χ (Order of magnitude, SI)

Temperature dependency

Diamagnet

Cu, Ag

−10−5

Independent temperature

Paramagnet

Al, Ti

10−3 —10−5

χ = C/T (Curie law)

Ferromagnet

Fe, Co, Ni,

106

Becoming paramagnetism above T C

Ferrimagnet

Fe3 O4 , CoFe2 O4

0, it represents ferromagnetic coupling. When the two spins are antiparallel, J < 0, it represents antiferromagnetic coupling. Therefore, the exchange interaction energy has a negative value, suggesting that the exchange interaction energy lowers the system energy. According to the value given by Chikazumi [10], for Fe, J = 2.16×10−21 J = 13.5 meV. The SE interaction model was explained by Chikazumi [10] and Coey [2] using the antiferromagnetic coupling between Mn2+ (3d 5 )−O2− −Mn2+ (3d 5 ) in MnO, as shown in Fig. 3.7. O2− ion has the valence electrons 2s2 2p6 , and two 2p electrons in the outer orbit may enter the outer 3d orbits of the adjacent two Mn2+ cations. There

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3 A Simple Introduction to Basic Knowledge …

Fig. 3.7 Diagrammatic sketch of SE interaction model in MnO [2]

are five 3d electrons per Mn2+ cation with a half-full shell. Subjected to the limit of Hund’s rule, after the p electron enters the 3d orbit of Mn2+ cation, its spin must be opposite to that of the other 3d electrons. The two p electrons in the outer orbit of O2− anion have opposite spins, resulting in the antiparallel magnetic moments of the two Mn2+ cations. The DE interaction model was explained using the ferromagnetic coupling of Mn3+ –O2− –Mn4+ cations, as shown in Fig. 3.8, in perovskite manganites [12–14], 3 1 3 0 eg ) and Mn4+ (3d 3 : t2g eg ) at such as La0.7 Sr0.3 MnO3 . There are the Mn3+ (3d 4 : t2g 0 the same time, where eg represents the empty eg orbit. The orbit of t 2g electron with lower energy has little overlap with the p orbit of O2− anions, resulting in the three t 2g electrons to form local electrons. The eg electron with higher energy has more overlap with the p orbits of O2− anion, resulting in a p electron of O2− anion may enter the empty eg0 orbit of Mn4+ cation. At the same time, the eg electron of Mn3+ cation transits to the p orbit of O2− anion. In this process, the eg electron jumps to the Mn4+ cation from the Mn3+ cation, and the system energy is constant. The spin of the itinerant eg electron must be parallel with those of the local t 2g electrons when subjected to the limit of Hund’s rule, leading to the ferromagnetic coupling of Mn cations. The direct exchange model along with the metal energy band theory, including the density functional theory (DFT), was used to explain the valence electron structure of magnetic and nonmagnetic metals. However, a phenomenological expression for the exchange-correlation energy (including the exchange interaction energy) in DFT has not been found so far, which has to be fit using various models.

3.4 Magnetic Ordering Models in Conventional Ferromagnetism

23

Fig. 3.8 Diagrammatic sketch of DE interaction model

Several works for calculation using the metal energy band theory were reported by Grechnev et al. [15], Sánchez-Barriga et al. [16] and Kvashnin et al. [17]. In 2007, Grechnev et al. [15] investigated the quasiparticle energy band structure of Fe, Co, and Ni, using local density approximation and dynamical mean-field theory (LDA + DMFT). In 2012, SSánchez-Barriga et al. [16] investigated the spin-dependent quasiparticle lifetimes and the strength of electron correlation effects in the ferromagnetic 3d transition metals Fe, Co, and Ni by spin- and angle-resolved photoemission spectroscopy. The experimental data were accompanied by state-of-the-art many-body calculations within the dynamical mean-field theory and the three-body scattering approximation, including fully relativistic calculations of the photoemission process within the one-step model. The values of the Coulomb interaction parameter (U) used were 1.5, 2.5, and 2.8 for Fe, Co, and Ni, respectively. Comparing these U values with 1.2, 2.4, and 3.7 for Fe, Co, and Ni, respectively, reported by Steiner et al. [18] in 1992, distinct differences in U values may be noticed for Ni. In 2016, Kvashnin et al. [17] calculated the nearest neighbor exchange integral J 1 of Fe by

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3 A Simple Introduction to Basic Knowledge …

density functional theory plus dynamical mean-field theory (DFT + DMFT). They found that the J 1 value of Fe is 1.0 mRy (= 13.61 meV). This value is close to 13.5 meV given by Chikazumi [10] but is far lower than 730 meV given by Steiner et al. [18].

References 1. Dai DS, Qian KM (1987) Ferromagnetism (in Chinese). Science Press, Beijing 2. Coey JMD (2010) Magnetism and magnetic materials. Cambridge University Press, Cambridge, UK 3. Han BS (1991) J Magn Magn Mater 100:455 4. Magnetic bubble writing group (1986) Mangetic buble, (in Chinese). Science Press, Beijing 5. Han BS, Ling JW, Li BZ, Nie XF, Tang GD (1986) Acta Phys Sin 35:130 6. Tang GD, Ma CS, Yang LX, Ma LM (2003) Modern physics experiments (in Chinese). Hebei Science & Technology Publishing House, Shijiazhuagn City 7. Qi WH, Li ZZ, Ma L, Tang GD, Wu GH, Hu FX (2017) Acta Phys Sin 66:067501 8. Tang GD, Liu Y, Hu HN, Liu YP, Sun HY, Nie XF (2003) Phys Stat Sol (b) 240(1):201 9. Nie XF, Tang GD, Niu XD, Han BS (1991) J Magn Magn Mater 95:231 10. Chikazumi S (1997) Physics of Ferromagnetism, 2nd edn. Oxford University Press, London 11. Stöhr J, Siegmann HC (2006) Magnetism: From Fundamentals to Nanoscale Dynamics. Springer, Berlin, Heidelberg 12. Salamon MB, Jaime M (2001) Rev Moder Phys 73:583 13. Dagotto E, Hotta T, Moreo A (2001) Phys Rep 344:1 14. Dai DS, Xiong GC, Wu SC (1997) Progress Phys 17(2):201 15. Grechnev A, Di Marco I, Katsnelson MI, Lichtenstein AI, Wills J, Eriksson O (2007) Phys Rev B 76:035107 16. Sánchez-Barriga J, Braun J, Minár J, Di Marco I, Varykhalov A, Rader O, Boni V, Bellini V, Manghi F, Ebert H, Katsnelson MI, Lichtenstein AI, Eriksson O, Eberhardt W, Dürr HA, Fink J (2012) Phys Rev B 85:205109 17. Kvashnin YO, Cardias R, Szilva A, Di Marco I, Katsnelson MI, Lichtenstein AI, Nordström L, Klautau AB, Eriksson O (2016) Phys Rev Lett 116:217202 18. Steiner MM, Albers RC, Sham LJ (1992) Phys Rev B 45(23):13272

Chapter 4

Difficulties Faced by Conventional Magnetic Ordering Models

Although the conventional magnetic material theory has been developed for more than 110 years since the molecular field assumption was proposed by Weiss, the explanation of material magnetic property is still challenging due to (i) conflicting views regarding the Mn and Cr cation distributions in (A)[B]2 O4 spinel ferrites; (ii) the failure to explain dependence of the magnetic moments observed in R1-x T x MnO3 perovskite manganites on doped divalent cation ratio x; (iii) the failure to describe the relationship between the average ionic magnetic moments and electrical resistivities of Fe, Co, and Ni; (iv) no consistent model exists that explains the magnetic ordering in both oxides and metals. In this chapter, we briefly introduce these difficulties faced by the conventional magnetic ordering models.

4.1 Disputes Over the Cation Distributions in Mn and Cr Spinel Ferrites 4.1.1 Normal, Inverse, and Mixed Spinel Structure Figure 4.1 shows the crystal structure of the (A)[B]2 O4 spinel ferrites [1] with space ¯ The O ions arrange in close-packed, face-centered cubic (fcc) lattice group Fd3m. structure. Two types of interstitial sites are occupied by metal cations: tetrahedral (8a) or (A) sites and octahedral (16d) or [B] sites. Four and six nearest adjacent O ions are present around an (A) and a [B] site metal cation, as shown in Fig. 4.1b, c, respectively. Therefore, the O ion number ratio near the (A):[B] sites is 2:3. If all (A) sites are occupied by divalent cations and all [B] sites are occupied by trivalent cations, the positive and negative charge densities in the crystal are balanced and the crystal structure is called the normal spinel structure, which is represented by (M 2+ )[N 3+ ]2 O2− 4 .

© Science Press 2021 G.-D. Tang, New Itinerant Electron Models of Magnetic Materials, https://doi.org/10.1007/978-981-16-1271-8_4

25

26

4 Difficulties Faced by Conventional Magnetic Ordering Models

Fig. 4.1 Crystal structure diagrammatic sketch of (A)[B]2 O4 spinel ferrite

The crystal lattice constant, a, the bond length between the O anions and the cations at the (A) and [B] sites, d AO and d BO , respectively, and the bond length between the cations at the (A) sites and those at [B] sites, d AB , can all be obtained by analyzing X-ray diffraction (XRD) patterns. √The ideal values√of d AO , d BO , and d AB for the cubic spinel structure are known to be 3a/8, a/4, and 11a/8, respectively. This implies that the volume of space surrounding an (A) site is smaller than that √ surrounding a [B] site. In practice, the observed average value of d AO is larger than 3a/8, while that of d BO √ is smaller than a/4. However, the observed value of d AB is almost always equal to 11a/8. From these results, we can infer that, to lower the Pauli repulsion energy, all the (A) sites should be occupied by trivalent cations with small ionic radii and all the [B] sites should be occupied by the rest of the divalent and trivalent cations. This system is called an inverse spinel structure, which is represented by (N 3+ )[M 2+ N 3+ ]O2− 4 .

4.1 Disputes Over the Cation Distributions in Mn and Cr Spinel Ferrites

27

In fact, the balance tendency of charge density and lowering of the Pauli repulsion energy both affect the cation distribution, resulting in the mixed spinel structure, 2+ 3+ N x3+ )[Mx2+ N2−x ]O2− represented by (M1−x 4 , where 0.0 < x < 1.0. When x = 0.0, a normal spinel structure is afforded; when x = 1.0, an inverse spinel structure is afforded. In the spinel ferrites MFe2 O4 (M = Fe, Co, Ni, or Cu), all the cation magnetic moments at the (A) or [B] sites couple ferromagnetically at low temperatures, but the cation magnetic moments at the (A) sites couple antiferromagnetically with those of [B] sites. This magnetic ordering was theoretically explained using the magnetic superexchange interaction model [2–5].

4.1.2 Magnetic Moments of 3d Transition Metal Ions As is well known, a 3d subshell has five energy levels. For divalent Cr and Mn cations, 3d electron number nd is equal to 4 and 5, respectively. Since nd is less than or equal to the energy level number 5, one electron is present per energy level, all electron spins arrange in one direction, and every electron contributes 1 μB magnetic moment to the ionic magnetic moment. Therefore, the magnetic moments of divalent Cr and Mn cations are μM2 = 4 and 5μB , respectively. For divalent Fe, Co, Ni, and Cu, nd = 6, 7, 8, and 9, respectively. Since nd is greater than the energy level number 5, the spins of extra electrons subjected to Hund’s rule must arrange in opposite directions. Thus, the energy levels having two electrons with opposite spins have no contribution to the ionic magnetic moment. Therefore, for the magnetic moments of divalent Fe, Co, Ni, and Cu, μM2 = (10 − nd ) μB = 4, 3, 2, and 1 μB , respectively.

4.1.3 Magnetic Ordering of CrFe2 O4 and MnFe2 O4 If the MFe2 O4 ferrites with (A)[B]2 O4 spinel structure have an inverse spinel structure, where all the (A) sites are occupied by Fe3+ cations and the [B] sites are occupied by equal ratio Fe3+ and M 2+ cations, the sum of the magnetic moments of Fe3+ cations is zero. This is due to the fact that the Fe3+ cations at the [B] sites couple antiferromagnetically with equal ratio Fe3+ cations at the (A) sites. Therefore, the calculated average molecular magnetic moment is equal to the magnetic moment, μM2 , of the M 2+ cations occupying [B] sites. The observed values for the average molecular magnetic moments of MFe2 O4 for M = Fe, Co, Ni, and Cu are μobs = 4.2, 3.3, 2.3, and 1.3 μB , respectively [2–5], which are all slightly greater than those of μM2 (4, 3, 2, and 1 μB ). For MnFe2 O4 , however, μobs is 4.6 μB , which is slightly less than that of Mn2+ (5 μB ). Furthermore, for CrFe2 O4 , μobs is 2.0 μB , which is only half that of Cr2+ (4 μB ). Figure 4.2 shows the values of μobs and μM2 as the functions of the number, nd , of 3d electrons in M 2+ cations. The obvious different magnetic moment change trend of Mn and Cr ferrites from other materials in Fig. 4.2 affords obvious different

28

4 Difficulties Faced by Conventional Magnetic Ordering Models

Fig. 4.2 Dependences of divalent M-ion magnetic moment, μM2 (❚), and observed average molecular magnetic moment, μobs (▲), of (A)[B]2 O4 spinel ferrites MFe2 O4 (M = Cr, Mn, Fe, Co, Ni, and Cu), on the number, nd , of 3d electrons in the divalent M-ion

cation distributions in Mn and Cr ferrites that were reported by different authors [6–20]. These issues have not been addressed in classical ferromagnetism textbooks [2–5] since they are difficult to explain on the basis of conventional magnetic ordering models. For Cr-doped spinel ferrites, some authors claim that all Cr cations enter [B] sites, while others believe that Cr cations may enter both (A) and [B] sites. Kim et al. [6] fabricated spinel ferrite films Crx Fe3−x O4 (x ≤ 0.95) with a thickness of 700–800 nm. They measured the magnetic hysteresis loops of the samples at room temperature and observed that the saturation magnetization of the samples decreased as the Cr ratio (x) increased. They, therefore, assumed that all Cr were trivalent cations, substituted for trivalent Fe cations at [B] sites. They fitted the curve of the saturation magnetization of the samples versus x and found that the fitted result agrees with the observed data. Lee et al. [7] synthesized Cr-substituted Mg-ferrites MgFe2−x Crx O4 (x = 0.0, 0.1, 0.3, 0.5, 0.7) and believed that all Cr cations entered [B] sites. Singhal et al. [8] prepared Cr-substituted Co-ferrites CoCrx Fe2−x O4 (x = 0.2, 0.4, 0.6, 0.8, 1.0) using sol–gel auto combustion method and studied the crystal structure as well as the magnetic, electrical, and optical properties. They assumed that all Cr cations were trivalent with magnetic moment 3 μB and substituted Fe3+ cations at [B] sites; the sample magnetic moment decreased with increasing Cr ratio (x). Birajdar et al. [9] fabricated Cr-substituted Ni-ferrites Ni0.7 Zn0.3 Crx Fe2−x O4 (x = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5). With increasing Cr ratio (x), they observed that the density and crystallites decreased, and the specific magnetization linearly decreased from 58.31 to 42.90 Am2 /kg. They assumed that all Ni2+ cations occupied [B] sites, that all Zn2+ cations occupied (A) sites, and that Cr3+ cations preferentially occupied [B] sites. Based on this ion distribution assumption, the calculated molecular magnetic

4.1 Disputes Over the Cation Distributions in Mn and Cr Spinel Ferrites

29

moments of their samples are obviously higher than the observed values. Gismelseed et al. [10] synthesized NiCrx Fe2−x O4 (0.0 ≤ x ≤ 1.4) using the conventional doublesintering ceramic technique and investigated the structural and magnetic properties of the samples by XRD and Mössbauer spectroscopy techniques. They believed that all Cr cations occupied [B] sites when the Cr ratio x is less than 0.9. Liang et al. [11] fabricated Fe3−x Crx O4 (0.00 ≤ x ≤ 0.67) via a precipitation–oxidation method. By analyzing X-ray absorption near-edge structure (XANES) spectra of the samples, they assumed that all Cr cations occupied [B] sites. Mane et al. [12] fabricated CoAlx Crx Fe2−2x O4 (0.0 ≤ x ≤ 0.5) ferrite samples via the usual ceramic method. They thought that all the Cr ions, in the form of Cr3+ , occupied octahedral [B] sites, while the Co ions were distributed across (A) and [B] sites. Hashim et al. [13] prepared ferrite samples Ni0.5 Mg0.5 Fe2−x Crx O4 (0.0 ≤ x ≤ 1.0). They measured the Mössbauer spectroscopy and XRD patterns of the samples and concluded that all the Cr cations entered [B] sites when x ≤ 0.6, while 0.05 of Cr cations entered (A) sites only when x = 0.8 and 1.0. Fayek et al. [14] synthesized ferrite samples NiCrx Fe2−x O4 and analyzed the Mössbauer spectra of the samples. They concluded that, when x ≤ 0.6, all the Cr cations entered [B] sites; when x = 0.8 and 1.0, 0.2 and 0.4 of the Cr cations entered (A) sites, respectively. Kadam et al. [15] synthesized Co0.5 Ni0.5 Crx Fe2−x O4 samples and obtained the cation distributions by analyzing XRD data. They considered that the number of Cr cations that entered the [B] sites is four times the number of cations that entered the (A) sites in all Cr-doped samples. Magalhães et al. [16] prepared ferrite samples Fe3−x Crx O4 (x = 0.00, 0.07, 0.26, 0.42, and 0.51) and analyzed the Mössbauer spectroscopy and XRD patterns of the samples. They concluded that, when x is lower, the Fe3+ ions at [B] sites were replaced by Cr3+ ions, and when x is higher, both Fe2+ ions at [B] sites and Fe3+ ions at (A) sites were also replaced by Cr ions. Ghatage et al. [17] synthesized NiCrx Fe2−x O4 (0.2 ≤ x ≤ 1.0). They analyzed the neutron diffraction data and determined that the Cr ion ratio entered (A) sites increased from 0.10 to 0.30 when x increased from 0.2 to 1.0. Table 4.1 shows the content ratios of Cr cations occupying (A)/[B] sites in these studies. For Mn-doped spinel ferrites, some authors claimed that all Mn cations entered (A) sites, while others claimed that all Mn cations entered [B] sites or both (A) and [B] sites. Zhao et al. [18] prepared a nanocrystalline Ni0.7 Mn0.3 Nd0.1 Fe1.9 O4 ferrite sample, analyzed the Mössbauer spectra, and considered that all the Mn2+ cations entered the 3+ 2+ 2+ 3+ 3+ (A) sites, forming the cation distributions (Mn2+ 0.3 Fe0.41 Ni0.29 )[Ni0.41 Nd0.1 Fe1.49 ]O4 . Li et al. [19] fabricated ferrite samples of Fe3−x Mnx O4 , analyzed the Mössbauer spectra and concluded that all the Mn ions occupied [B] sites. Fayek et al. [20] prepared ferrite samples of CoMnx Fe2−x O4 (0.0 ≤ x ≤ 1.0), analyzed the Mössbauer spectra and neutron diffraction data, and considered that all the Mn ions (with Mn3+ state) entered the [B] sites. Lee et al. [21] prepared spinel ferrites of Co1−x Mnx Fe2 O4 and measured the magnetic moments, conductivities, and Mössbauer spectra. They concluded that, when x was low, the Co2+ ions at [B] sites were replaced by Mn2+ ions; when x ≥ 0.6, some Fe3+ ions at (A) sites were replaced by Mn2+ ions. Roumaih [22] synthesized ferrite samples with compositions Ni1−x Cux Fe2−y Mny O4 (x = 0.2, 0.5,

30

4 Difficulties Faced by Conventional Magnetic Ordering Models

Table 4.1 Content ratios of Cr cations occupying (A)/[B] sites of spinel ferrites in literature Material

Content ratio of Cr ions occupying the (A)/[B] sites

References

Crx Fe3-x O4 (x ≤ 0.95)

0.0/x

Kim et al. [6]

MgFe2−x Crx O4 (x = 0.0, 0.1, 0.3, 0.5, 0.7)

0.0/x

Lee et al. [7]

CoCrx Fe2-x O4 (x = 0.2, 0.4, 0.6, 0.8, 1.0)

0.0/x

Singhal et al. [8]

Fe3−x Crx O4 (0 ≤ x ≤ 0.67)

0.0/x

Liang et al. [11]

CoAlx Crx Fe2-2x O4 (0.0≤ x ≤ 0.5)

0.0/x

Mane et al. [12]

Ni0.5 Mg0.5 Fe2−x Crx O4 (x = 0.2, 0.4, 0.6)

0.0/x

Hashim et al. [13]

Ni0.5 Mg0.5 Fe2−x Crx O4 (x = 0.8)

0.05/0.75

Hashim et al. [13]

Ni0.5 Mg0.5 Fe2−x Crx O4 (x = 1.0)

0.05/0.95

Hashim et al. [13]

Co0.5 Ni0.5 Crx Fe2-x O4 (x = 0.25, 0.5, 0.75, 1.0)

1/4

Kadam et al. [15]

NiCrx Fe2−x O4 (x = 0.2)

0.1/0.1

Ghatage et al. [17]

NiCrx Fe2−x O4 (x = 0.4)

0.15/0.25

Ghatage et al. [17]

NiCrx Fe2−x O4 (x = 0.8)

0.25/0.55

Ghatage et al. [17]

NiCrx Fe2−x O4 (x = 1.0)

0.3/0.7

Ghatage et al. [17]

and 0.8; 0.0 ≤ y ≤ 1.0). Based on the magnetic measurements and the electroneutrality condition, they considered that, when y ≤ 0.75, all the Mn cations entered the [B] sites; when y = 1.0, 12% (for x = 0.2) and 25% (for x = 0.5 and 0.8) of the Mn cations entered the (A) sites. Sakurai et al. [23] grew a single-crystal Mn0.80 Zn0.18 Fe2.02 O4 sample, and based on their analysis of XANES spectra and X-ray magnetic circular dichroism spectra, they concluded that the Mn2+ cations entered both (A) and [B] 2+ 3+ 2+ 2+ 3+ sites and formed the chemical formula (Mn2+ 0.71 Zn0.10 Fe0.19 )[Mn0.09 Zn0.08 Fe1.83 ]O4 . Harrison et al. [24] grew a single-crystal ferrite sample Mn0.972 Fe1.992 O4 . They assumed that 0.787 of the Mn cations entered the (A) sites by fitting the sample magnetic moment. Gabal et al. [25] synthesized ferrites of Mn1−x Znx Fe2 O4 (0.2 ≤ x ≤ 0.8) and assumed that all Mn ions were divalent cations and occupied both (A) and [B] sites. Hemeda [26] prepared ferrite samples of Co0.6 Zn0.4 Mnx Fe2−x O4 (x = 0.0, 0.1, 0.2, 0.3, 0.4, and 0.5) and analyzed the electron spin resonance spectra. They concluded that 80% of Mn ions entered (A) sites with Mn3+ state and 20% of Mn ions entered [B] sites with Mn2+ state. The ratios of the Mn cations that entered the (A)/[B] sites in these studies are shown in Table 4.2. The obvious differences of Cr or Mn cation distributions in Tables 4.1 and 4.2 suggest that the conventional SE and DE interaction models need to be improved.

4.2 Difficulties in Describing the Observed Magnetic Moments …

31

Table 4.2 Content ratio of Mn cations entered (A)/[B] sites of spinel ferrites in literature Material

Mn content ratio occupying the (A)/[B] sites

References

Ni0.7 Mn0.3 Nd0.1 Fe1.9 O4

0.3/0.0

Zhao et al. [18]

Fe3−x Mnx O4 (x = 0, 0.25, 0.5, 0.75 and 1.0)

0.0/x

Li et al. [19]

CoMnx Fe2-x O4 (0.0≤x≤1.0)

0.0/x

Fayek et al. [20]

Co1−x Mnx Fe2 O4 (x = 0.2, 0.4)

0.0/x

Lee et al. [21]

Ni1−x Cux Fe2-y Mny O4 (x = 0.2, 0.5, 0.8; y = 0.25, 0.50, 0.75)

0.00/y

Roumaih [22]

MnFe2 O4

0.8/0.2

Dai et al. [2]

Co1−x Mnx Fe2 O4 (x = 0.6)

0.40/0.19

Lee et al. [21]

Ni1−x Cux Fe2-y Mny O4 (x = 0.5, 0.8; y 0.25/0.75 = 1.0)

Roumaih [22]

Co1−x Mnx Fe2 O4 (x = 0.8)

0.22/0.54

Lee et al. [21]

Mn0.80 Zn0.18 Fe2.02 O4

0.71/0.09

Sakurai et al. [23]

Mn0.972 Fe1.992 O4

0.787/0.185

Harrison et al. [24]

Co0.6 Zn0.4 Mnx Fe2-x O4 (x = 0.0≤x≤0.5)

0.8/0.2

Hemeda [26]

4.2 Difficulties in Describing the Observed Magnetic Moments of Perovskite Manganites Figure 4.3 shows an ideal cubic crystal cell with an ABO3 perovskite crystal structure [1, 27], which belongs to the space group Pm3m, such as CaTiO3 wherein Ca cations with large effective radius occupy A sites and Ti cations with small effective radius occupy B sites. The crystal lattice constant are a = b = c = a0 , and the angles between

Fig. 4.3 Diagrammatic sketch of a cubic crystal cell of ABO3 perovskite oxide

32

4 Difficulties Faced by Conventional Magnetic Ordering Models

the three crystal axes are α = β = γ = 90°. One A cation, one B cation, and three O anions  √are present per crystal cell. The bond lengths between A–O and B–O ions 2 and a0 /2, respectively. Therefore, the space of an A site is larger than are a0 that of a B site. In a crystal cell of perovskite manganite Re1−x Aex MnO3 (Re = La, Pr, and Nd; Ae = Ca, Sr, Ba, and Pb), rare earth (Re) cations and alkaline-earth (Ae) cations with large effective radii occupy A sites and Mn cations with small effective radii occupy B sites [27–39]. The large difference between cation effective radii at A and B sites may cause crystal lattice distortion. An actual crystal lattice may have a rhombohedral or orthogonal structure. An orthogonal crystal cell includes four molecules that belong to the Pbnm space group, as shown in Fig. 4.4. A rhombohedral crystal cell includes six molecules and belongs to the R3c space group, as shown in Fig. 4.5. Figures 4.4 and 4.5 show the relations of orthogonal and rhombohedral structures to the cubic structure. In general, the deviation of orthogonal and rhombohedral structures from the cubic structure is small [27]. For convenience, the equivalent cubic crystal cell of the orthogonal and rhombohedral structures was used sometimes. ABO3 perovskite manganites Re1−x Aex MnO3 have been extensively researched due to their potential applications in magnetic devices [27–40]. When 0.2 < x < 0.4, the materials possess the best ferromagnetic and conductive properties. According to the review by Tokura et al. [39], Fig. 4.6 shows the magnetic phase diagrams of ABO3 perovskite manganites La1−x Srx MnO3 , where T C and T N represent the Curie and Néel temperatures, respectively. The PM, FM, and AFM denote the paramagnetic, ferromagnetic, and antiferromagnetic states, respectively. For the magnetic and electrical transport properties of perovskite manganites, Zener [40] proposed an explanation using a double-exchange (DE) interaction model

Fig. 4.4 √ Diagrammatic sketch of an orthogonal crystal cell of ABO3 perovskite manganite. If a = b = 2a0 , c = 2a0 , then it become into cubic structure [27]

4.2 Difficulties in Describing the Observed Magnetic Moments …

33

Fig. √ 4.5 Diagrammatic sketch of a rhombohedral crystal cell of ABO3 perovskite manganite. If √ c = 6a, a = 2a0 , then it become into cubic structure [27]

34

4 Difficulties Faced by Conventional Magnetic Ordering Models

400 350

PM

300

TC La1-xSrxMnO3

T (K)

250 200 150

TN

100

FM

50 AFM 0 0.0

0.1

0.2

0.3 x

AFM 0.4

0.5

0.6

Fig. 4.6 The magnetic phase diagrams of ABO3 perovskite manganites La1−x Srx MnO3 [39], where T C and T N represent the Curie temperature and Néel temperature. The PM denotes the paramagnetic state. FM and AFM denote the ferromagnetic and antiferromagnetic states, respectively

in 1951. For La1−x Srx MnO3 , all O ions are assumed to be O2− , and the magnetic properties and electronic transport properties are correlated via the superexchange (SE) interaction along the Mn3+ –O2− –Mn3+ ionic chain and the DE interaction along the Mn3+ –O2− –Mn4+ ionic chain. The itinerant electrons, originated from 3d electrons of Mn cations, hop between the Mn cations mediated by O2− anions. To quantitatively explain why the crystal structures for these materials differ with different compositions, Goodenough et al. [41] proposed a covalent bond theory in 1955. In this theory, orbital hybridization takes place between 3d electrons of Mn cations and 2p electrons of O anions. Wollan and Koehler [42] studied the crystal structure of La1−x Cax MnO3 using neutron diffraction in 1955. Later, numerous investigations for perovskite manganites were performed [43–50]. Millis et al. [43] assumed that the Jahn–Teller effect should be considered in addition to the DE interaction to explain the magnetic and electrical transport properties of perovskite manganites. In 1995, Xiong et al. [51] investigated the magnetic and electrical transport properties of Nd0.7 Sr0.3 MnO3 and found that the magnetoresistance of the sample reached 106 %. The physical mechanism of perovskite manganites was explored via several methods, such as Mössbauer spectroscopy, electron paramagnetic resonance, and Raman and infrared spectroscopies [52–58]. Although many investigations have been conducted for the magnetic and electrical transport properties of perovskite manganites, some issues have not been sufficiently explained. For example: (1)

Divalent cation doping level dependence of the sample magnetization. The polycrystalline bulk and the single-crystal film La1−x Srx MnO3 were prepared

4.2 Difficulties in Describing the Observed Magnetic Moments … Table 4.3 Magnetic moments of perovskite manganites La1−x Srx MnO3

(2)

x

Urushibara et al. [30] μobs (μB ) (4.2 K)

35 Jonker et al. [36] μobs (μB ) (90 K)

0.00 –

0.00

0.10 3.6

3.08

0.15 4.2

–

0.20 3.9

3.73

0.25 3.9

–

0.30 3.5

3.71

0.35 –

3.66

0.40 3.4

3.51

and explored by Jonker and Van Santen [36], and Urushibara et al. [30]. They observed that the average magnetic moments per molecule, μobs , rapidly increases with increasing Sr ratio x when 0.0 ≤ x ≤ 0.15 and slowly decreases when 0.15 ≤ x ≤ 0.40, and μobs = 4.2 μB when x = 0.15, as shown in Table 4.3. Many similar works were reported for Re1−x Aex MnO3 [27–40]. However, a reasonable explanation has not yet been found for why μobs of La1−x Srx MnO3 rapidly increases from 0.0 (x = 0.0) to 4.2 μB (x = 0.15). The magnetic coupling mechanism between Mn and other 3d cations. Using 3d transition elements, M = Cr, Fe, Co, and Ni, to partially substitute for Mn, perovskite oxides Re1−y Aey Mn1−x M x O3 have been investigated by many groups [59–62]. They found that the μobs of the samples decreases with increasing x and explained it using DE and SE interactions models. However, no systematic explanation was provided for the curves of μobs versus x of these materials; e.g., why does Cr ferromagnetic couple with Mn cations and why do Fe, Co, and Ni cations antiferromagnetic couple with Mn cations.

Wang et al. [59] synthesized La0.67 Sr0.33 (Mn1−x Nix )O3 (0 ≤ x ≤ 0.2) via the conventional solid-state reaction method. They observed that the saturation magnetization and the crystal cell volumes of the samples decrease with increasing x. They assumed that Ni cations were in the Ni2+ state by XPS analysis and that “ferromagnetic superexchange interaction” occurred between Ni2+ and Mn3+(4+) . Gupta et al. [60] studied the magnetic, transport, and magnetoresistance behaviors of La0.67 Sr0.33 Mn1−x Nix O3 (0 ≤ x ≤ 0.09). They also observed that the saturation magnetization of the samples decreases with increasing x and inferred that Ni2+ aligns antiferromagnetically with Mn cations. Creel et al. [61] studied the structural and magnetic properties of manganites La0.7 Sr0.3 Mn1−x Nix O3 (0 ≤ x ≤ 0.4) using X-ray diffraction, neutron diffraction, and magnetic measurements. They assumed that Ni cations are in Ni3+ state rather than the Ni2+ state and that the magnetic moments of Ni3+ cations aligned antiferromagnetically to those of Mn cations. Selmi et al. [62] studied the structural and magnetic properties of Pr0.7 Sr0.3 Mn1−x Nix O3 (0 ≤ x ≤ 0.1) powder samples. They considered that antiferromagnetic interaction exists between Ni2+ and Mn4+ ions.

36

4 Difficulties Faced by Conventional Magnetic Ordering Models

Sun et al. [63] synthesized polycrystalline La0.67 Ca0.33 Mn1−x Crx O3 (0 ≤ x ≤ 0.30) using the conventional solid-state reaction method and measured its magnetization, resistivity, colossal magnetoresistance (CMR), and thermopower. They found that the CMR temperature range is greatly broadened around x = 0.1 and that the Curie temperatures of the samples decrease with increasing x. Furthermore, they considered the presence of DE interactions between Cr3+ and Mn4+ ions. Ammar et al. [64] investigated the physical properties of Cr-doped perovskite manganites Pr0.5 Sr0.5 Mn1−x Crx O3 (0 ≤ x ≤ 0.30). They found that the Curie temperature and the magnetization decrease with increasing Cr ratio x. They assumed that the DE interaction between Mn3+ and Mn4+ ions was interrupted by Cr3+ ions. Xavier et al. [65] prepared the polycrystalline samples of La0.7 Sr0.3 Mn1−x Fex O3 and studied the XRD, resistivity, ac susceptibility, magnetization, and magnetoresistance of the samples. They considered that the Fe ion couples antiferromagnetically with its Mn neighbors based on the experimental results of the Mössbauer spectra. Baazaoui et al. [66] studied the polycrystalline samples La0.67 Ba0.33 Mn1−x Fex O3 (0 ≤ x ≤ 0.2). They reported that the magnetic coupling between Fe3+ and Mn3+ ions is canted antiferromagnetic. Figueroa et al. [67] studied the magnetic properties of La2/3 Ca1/3 Mn0.97 Fe0.03 O3 thin films using the element-selective technique of X-ray magnetic circular dichroism (XMCD). They concluded that the magnetic moments of Mn and Fe have antiparallel alignment and that Mn–O–Fe bonds belong to superexchange coupling. Nasri et al. [68] prepared Pr0.6 Sr0.4 Fe1−x Mnx O3 (0 ≤ x ≤ 0.3) compounds and measured the dependencies of the magnetization on temperature and applied magnetic field. They concluded that the substitution of Mn3+ ions by Fe3+ ions triggers antiferromagnetic interactions between Fe3+ and Mn4+ . Dhahri et al. [69] synthesized La0.67 Pb0.33 Mn1−x Cox O3 (0 ≤ x ≤ 0.3) powders and observed that the Curie temperature decreases with increasing Co ratio. Yoshimatsu et al. [70] prepared Pr0.8 Ca0.2 Mn1−y Coy O3 (0 ≤ x ≤ 0.3) thin films. They showed the divalent state of Co ions by X-ray absorption and hard X-ray photoemission spectra and showed that Co2+ ion undergoes ferromagnetic superexchange coupling with Mn4+ ion. Chen et al. [71] fabricated La0.7 Sr0.3 Mn1−x Cox O3 (0 ≤ x ≤ 1) polycrystalline samples. They observed an interesting phenomenon: the increase of the Co ratio first decreases the specific saturation magnetization, σ S , and then increases it; the value of σ S when x = 1.0 reached about half of that when x = 0.0. Likewise, the resistivity, ρ, first increased and then decreased; the value of ρ when x = 1.0 was very close to that when x = 0.0. They considered that there is DE ferromagnetic interaction between Mn3+ and Mn4+ (Co3+ and Co4+ ) ions, while a superexchange antiferromagnetic interaction occurs between Mn3+ and Mn3+ (Co3+ and Co3+ ) ions. Table 4.4 lists the valence states of substituted ions and their magnetic coupling states with Mn ions for the above-mentioned references. These studies afford differing views. (3)

Different magnetic coupling mechanisms between LaMnO3 and MnO. In the conventional view, antiferromagnetic coupling occurs between Mn3+ ions in LaMnO3 . Töpfer et al. [72] and Prado et al. [73] studied the effects of thermal treatment on the crystal lattice constants and magnetic moments of LaMnO3 .

4.2 Difficulties in Describing the Observed Magnetic Moments …

37

Table 4.4 Substituted cation valence and magnetic couple with Mn in B sites of perovskite manganites reported by different authors Substituted cation

Composition

Magnetic couple

References

Cr

La0.67 Ca0.33 Mn1-x Crx O3 (0 ≤ x ≤ 0.30)

Double-exchange interaction between Cr3+ and Mn3+

[63]

Cr

Pr0.5 Sr0.5 Mn1-x Crx O3 (0 ≤ x ≤ 0.30)

Double-exchange interaction between Mn3+ and Mn4+ is interrupted by Cr3+

[64]

Fe

La0.7 Sr0.3 Mn1-x Fex O3 (0 ≤ x ≤ 0.10)

Fe ion couples antiferromagnetically with its Mn neighbors

[65]

Fe

La0.67 Ba0.33 Mn1−x Fex O3 (0 ≤ x ≤ Canting antiferromagnetic 0.2) state between Fe3+ and Mn3+ ions

[66]

Fe

La2/3 Ca1/3 Mn0.97 Fe0.03 O3

Magnetic moments of Mn and Fe align antiparallel

[67]

Fe

Pr0.6 Sr0.4 Fe1-x Mnx O3 (0 ≤ x ≤ 0.3)

Antiferromagnetic interactions between the Fe3+ and Mn4+

[68]

Co

Pr0.8 Ca0.2 Mn1−y Coy O3 (0 ≤ y ≤ 0.3)

Co2+ ion is ferromagnetic superexchange coupling with Mn4+ ion.

[70]

Co

La0.7 Sr0.3 Mn1−x Cox O3 (0.0 ≤ x ≤ Double-exchange 1.0) ferromagnetic interaction between Mn3+ and Mn4+ (Co3+ and Co4+ ) ions, superexchange antiferromagnetic interaction between Mn3+ and Mn3+ (Co3+ and Co3+ ) ions

[71]

Ni

La0.67 Sr0.33 (Mn1−x Nix ) O3 (0 ≤ x Ferromagnetic ≤ 0.2) superexchange interaction between the Ni2+ and Mn3+(4+)

[59]

Ni

La0.67 Sr0.33 Mn1−x Nix O3 (0 ≤ x ≤ 0.09)

Ni2+ cations are [60] antiferromagnetic alignments with Mn cations

Ni

La0.7 Sr0.3 Mn1−x Nix O3 (x ≤ 0.4)

Magnetic moments of Ni3+ cations aligned antiferromagnetically to those of Mn cations

[61]

Ni

Pr0.7 Ca0.3 Mn1-y Niy O3 (0 ≤ y ≤ 0.1)

Antiferromagnetic interaction between Ni2+ and Mn4+ ions

[62]

38

4 Difficulties Faced by Conventional Magnetic Ordering Models

They found that the average molecular magnetic moments of LaMnO3 may vary from 0 to 3 μB for the samples with different preparation conditions. However, the magnetic moment of a typical antiferromagnetic material (MnO) is always very small, irrespective of the thermal treatment conditions. Thus, LaMnO3 and MnO should belong to two kinds of antiferromagnetic materials with different properties. However, different magnetic coupling mechanisms for these materials have not been reported in the conventional view.

4.3 Relationship Between Magnetic Moment and Resistivity in Typical Magnetic Metals The observed average ionic magnetic moments for Fe, Co, and Ni metals are μobs = 2.22, 1.72, and 0.62μB , respectively [2, 74, 75]. The valence electronic configuration of free Fe, Co, and Ni free atoms are 3d 6 4s2 , 3d 7 4s2 , and 3d 8 4s2 , respectively. If the effect of 4s electrons is absent, the ionic magnetic moments of Fe, Co, and Ni metals should be 4, 3, and 2μB , which are far higher than μobs . The observed values of electrical resistivities are 8.6, 6.14, 5.57, and 1.55 µ cm for Fe, Ni, Co, and Cu, respectively [75]. The valence electronic configuration of free Cu atom is 3d 10 4s1 . As is well known, Cu is a typical diamagnetic material with a magnetic moment of zero, implying that there are ten 3d electrons in the 3d subshell per Cu ion in a Cu crystal and that the free-electron concentration in Cu is 1.0 per Cu atom. Therefore, we can reasonably assume that the free-electron concentrations of Fe, Ni, and Co are less than 1.0 per atom and that the remaining 4s electrons enter the 3d orbits and cause the magnetic moment per ion to decrease. Thus, these data about magnetic and electrical properties should be related to valence electron states; however, no such relationship has yet been reported in conventional theories.

4.4 Puzzle for the Origin of Magnetic Ordering Energy The average atomic magnetic moments of Fe, Co, Ni metals and average molecular magnetic moments of several typical magnetic oxides, μobs , their Curie temperature, T C , and their crystal structures are shown in Table 4.5 [30, 74–77]. Why are the T C values of these materials so different? In particular, why are the T C value of Ni far lower than that of Co metal? Satisfactory explanations for these questions still need to be provided. These explanations should be related to the origin of magnetic ordering energy, that is, the energy of the Weiss molecular field. In conventional ferromagnetism, the magnetic ordering energy is called the exchange interaction energy.

4.4 Puzzle for the Origin of Magnetic Ordering Energy

39

Table 4.5 Crystal structure, observed average magnetic moments per molecule for oxide (per atom for metal), μobs , and Curie temperature, T C , of several metals and oxides μobs (μB )

T C (K)

References

BCC

2.22

1043

[74, 75]

HCP

1.72

1404

[74, 75]

Ni

FCC

0.62

631

[74, 75]

MnFe2 O4

Spinel

4.6

570

[75]

FeFe2 O4

Spinel

4.2

860

[75]

CoFe2 O4

Spinel

3.3

793

[75]

NiFe2 O4

Spinel

2.3

863

[75]

CuFe2 O4

Spinel

1.3

766

[75]

La0.8 Ca0.2 MnO3

Perovskite

3.76

198

[76]

La0.75 Ca0.25 MnO3

Perovskite

3.13

240

[77]

La0.85 Sr0.15 MnO3

Perovskite

4.2

238

[30]

La0.7 Sr0.3 MnO3

Perovskite

3.5

369

[30]

Material

Crystal structure

Fe Co

In current magnetic material investigations, the energy of a magnetic material system was calculated using DFT [4], which is usually implemented using the Kohn– Sham method, where the problem of strongly interacting electrons moving in the potential of the nuclei is reduced to the more tractable problem of noninteracting electrons in an effective single-particle potential, which is written as 1 V = VN + 2



  e2 n r  d 3r  + Vxc [n(r )]. 4π ε0 |r − r  |

(4.1)

The first term is the Coulomb interaction of the electrons with the nuclei, the second term describes the electron–electron Coulomb repulsion, and the third one represents the exchange-correlation potential, which includes the exchange energyrelated magnetic ordering and all the many-electron correlations. Unfortunately, the expression of this exchange-correlation potential has unknown and needs to be explored using various approximation models, such as the local spin density approximation plus Hubbard U (LSDA+ U). Therefore, a phenomenological model is required to calculate the magnetic ordering energy.

References 1. Fang JX, Lu D (1980) Solid state physics (in Chinese). Shanghai Scientific & Technical Publishers, Shanghai 2. Dai DS, Qian KM (1987) Ferromagnetism (in Chinese). Science Press, Beijing 3. Chikazumi S (1997) Physics of Ferromagnetism, 2nd edn. Oxford University Press, London

40

4 Difficulties Faced by Conventional Magnetic Ordering Models

4. Coey JMD (2010) Magnetism and magnetic materials. Cambridge University Press, Cambridge 5. Stöhr J, Siegmann HC (2006) Magnetism: From fundamentals to nanoscale dynamics. Springer, Berlin Heidelberg 6. Kim KJ, Lee HJ, Lee JH, Lee S, Kim CS (2008) J Appl Phys 104:083912 7. Lee SW, An SY, Ahn GY, Kim CS (2000) J Appl Phys 87:6238 8. Singhal S, Jauhar S, Singh J, Chandra K, Bansal S (2012) J Mol Struct 1012:182 9. Birajdar AA, Shirsath SE, Kadam RH, Patange SM, Lohar KS, Mane DR, Shitre AR (2012) J Alloy Compd 512:316 10. Gismelseed AM, Yousif AA (2005) Phys B 370:215 11. Liang XL, Zhong YH, Zhu SY, He HP, Yuan P, Zhu JX, Jiang Z (2013) Solid State Sci 15:115 12. Mane DR, Devatwal UN, Jadhav KM (2000) Meter. Lett. 44:91 13. Hashim M, Alimuddin, Kumar S, Shirsath S E, Kotnala RK, Chung H, Kumar R (2012) Powder Technol 229: 37 14. Fayek MK, Ata-Allah SS (2003) Phys Stat Sol A 198:457 15. Kadam RH, Birajdar AP, Alone ST, Shirsath SE (2013) J Magn Magn Mater 327:167 16. Magalhães F, Pereira MC, Botrel SEC, Fabris JD, Macedo WA, Mendonca R, Lago RM, Oliverira LCA (2007) Appl Catal A 332(1):115 17. Ghatage AK, Patil SA, Paranjpe SK (1996) Solid State Commun 98:885 18. Zhao LJ, Xu W, Yang H, Yu LX (2008) Curr Appl Phys 8:36 19. Li YH, Kouh T, Shim IB, Kim CS (2012) J Appl Phys 111:07B544 20. Fayek MK, Sayed Ahmed FM, Ata-Allah SS (1992) J Mater Sci 27:4813 21. Lee DH, Kim HS, Yo CH, Ahn K, Kim KH (1998) Mater Chem Phys 57:169 22. Roumaih K (2011) J Mol Struct 1004:1 23. Sakurai S, Sasaki S, Okube M, Ohara H, Toyoda T (2008) Phys B 403:3589 24. Harrison FW, Osmqnd WP, Teale W (1957) Phys Rev 106:865 25. Gabal MA, Al-Luhaibi RS, Al Angari YM (2013) J Magn Magn Mater 348:107 26. Hemeda OM (2002) J Magn Magn Mater 251:50 27. Tang GD (2007) Study of cohesive energy and ionic valence in several ABO3 perovskite manganites (Doctoral dissertation). Hebei Normal University, Shijiazhuang 28. Helmolt RV, Wocker J, Holzapfel B, Schultz L, Samwer K (1993) Phys Rev Lett 71:2331 29. Ju HL, Kwon C, Li Q, Greene RL, Venkatesan T (1994) Appl Phys Lett 65:2108 30. Urushibara A, Moritomo Y, Arima T, Asamitsu A, Kido G, Tokura Y (1995) Phys. Rev. B. 51:14103 31. Tang GD, Hou DL, Li ZZ, Zhao X, Qi WH, Liu SP, Zhao FW (2006) Appl Phys Lett 89:261919 32. Tang GD, Hou DL, Chen W, Zhao X, Qi WH (2007) Appl Phys Lett 90:144101 33. Tang GD, Hou DL, Chen W, Hao P, Liu GH, Liu SP, Zhang XL, Xu LQ (2007) Appl Phys Lett 91:152503 34. Tang GD, Liu SP, Zhao X, Zhang YG, Ji DH, Li YF, Qi WH, Chen W, Hou DL (2009) Appl Phys Lett 95:121906 35. Hong F, Cheng ZX, Wang JL, Wang XL, Dou SX (2012) Appl Phys Lett 101:102411 36. Jonker GH, Van Santen JH (1950) Physica 16(3):337 37. Jonker GH, Van Santen JH (1953) Physica 19(1):120 38. Jonker GH (1954) Physica 20(7):1118 39. Tokura Y, Tomioka Y (1999) J Magn Magn Mater 200(1):1 40. Zener C (1951) Phys Rev 82(3):403 41. Goodenough JB (1955) Phys Rev 100(2):564 42. Wollan EO, Koehler WC (1955) Phys Rev 100(2):545 43. Millis AJ, Shraiman BI, Mueller R (1996) Phys Rev Lett 77:175 44. Sun JR, Rao GH, Gao XR, Liang JK, Wong HK, Shen BG (1999) J Appl Phys 85(7):3619 45. De Gennes PG (1960) Phys Rev 118(1):141 46. Searle CW, Wang ST (1970) Canadian J Phy 48(17):2023 47. Dabrowski B, Xiong X, Bukowski Z, Dybzinski R, Klamut PW, Siewenie JE, Chmaissem O, Shaffer J, Kimball CW (1999) Phys Rev B 60(10):7006 48. Kusters RM, Singleton J, Keen DA, Mcgreevy R, Hayes W (1989) Phys B 155(1):362

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49. Hibble SJ, Cooper SP, Hannon AC, Fawcett ID, Greenblatt M (1999) J Phys: Condens Matter 11(47):9221 50. Hwang HY, Cheng SW, Radaelli PG, Marezio M, Batlogg B (1995) Phys Rev Lett 75(5):914 51. Xiong GC, Li Q, Ju HL, Mao SN, Senapati L, Xi XX, Greene RL, Venkatesan T (1995) Appl Phys Lett 66(11):1427 52. Roy C, Budhani RC (1999) J Appl Phys 85(6):3124 53. Iliev MN, Abrashev MV (2001) J Raman Spectrosc 32(10):805 54. Wang X, Cui Q, Pan Y, Zou GT (2003) J Alloy Compd 354(1):91 55. De Marzi G, Popovi´c ZV, Cantarero A, Dohcevic-Mitrovic Z, Paunovic N, Bok J, Sapinã F (2003) Phys Rev B 68:064302 56. Souza Filho AG, Faria JLB, Guedes I, Sasaki JM, Freire PTC, Freire VN, Mendes Filho J (2003) Phys Rev B 67:052405 57. Autret C, Gervais M, Gervais F, Raimboux N, Simon P (2004) Solid State Sci 6(8):815 58. Mostafa AG, Abdel-Khalek EK, Daoush WM, Moustfa SF (2008) J Magn Magn Mater 320:3356 59. Wang ZH, Cai JW, Shen BG, Chen X, Zhan WS (2000) J Phys: Condens Matter 12:601 60. Gupta M, Kotnala RK, Khan W, Azam A, Naqvi AH (2013) J Solid State Chem 204:205 61. Creel TF, Yang J, Kahveci M, Malik SK, Quezado S, Pringle OA, Yelon WB, James WJ (2013) J Appl Phys 114:013911 62. Selmi A, Cheikhrouhou-Koubaa W, Koubaa M, Cheikhrouhou A (2013) J Supercond Novel Magn 26(5):1421 63. Sun Y, Xu XJ, Zhang YH (2000) Phys Rev B 63:054404 64. Ammar A, Zouari S, Cheikhrouhou A (2003) J Alloys Compd 354:85 65. Xavier MM Jr, Cabral FAO, Araújo J H de, Chesman C (2000) Dumelow. T Phys Rev B 63:012408 66. Baazaoui M, Zemni S, Boudard M, Rahmouni H, Gasmi A, Selmi A, Oumezzine M (2009) Mater Lett 63:2167 67. Figueroa AI, Campillo GE, Baker AA, Osorio JA, Arnache OL, Laan GVD (2015) Superlattices Microstruct 87:42 68. Nasri A, Zouari S, Ellouze M, Rehspringer JL, Lehlooh AF, Elhalouani F (2014) J Supercond Novel Magn 27(2):443 69. Dhahri N, Dhahri A, Cherif K, Dhahria J, Taibib K, Dhahric E (2010) J Alloy Compd 496:69 70. Yoshimatsu K, Wadati H, Sakai E, Harada T, Takahashi Y, Harano T, Shibata G, Ishigami K, Kadono T, Koide T, Sugiyama T, Ikenaga E, Kumigashira H, Lippmaa M, Oshima M, Fujimori A (2013) Phys Rev B 88:174423 71. Chen XG, Fu JB, Yun C, Zhao H, Yang YB, Du HL, Han JZ, Wang CS, Liu SQ, Zhang Y, Yang YC, Yang JB (2014) J Appl Phys 116:103907 72. Töpfer J, Goodenough JB (1997) J Solid State Chem 130:117 73. Prado F, Sanchez RD, Caneiro A, Causa MT, Tovar MJ, Solid State Chem 146:418 74. Chen CW (1977) Magnetism and metallurgy of soft magnetic materials. North-Holland Publishing Company, Amsterdam 75. Ida SK, Ono K, Kozaki H (1979) (Translated by Z. X. Zhang), Data on Physics in Common Use (in Chinese), Beijing: Science Press 76. Hibble SJ, Cooper SP, Hannon AC, Fawcett ID, Greenblatt M (1999) J Phys: Condens Matter 11:9921 77. Radaelli PG, Cox DE, Marezio M, Cheong SW, Shiffer PE, Ramirez AP (1995) Phys Rev Lett 75:4488

Chapter 5

O 2p Itinerant Electron Model for Magnetic Oxides

According to the conventional solid-state physics theory [1, 2], mixing bonds, including ionic and covalent bonds, can exist in some compounds. Ionicity (f i ), first reported by Pauling in 1932, is used to represent the ratio between covalent and ionic bonds. If all bonds are covalent bonds, f i = 0.0; if all bonds are ionic bonds, f i = 1.0. The f i value (0.0 < f i < 1.0) in an oxide can be interpreted as the transit of some valence electrons between cations and anions, resulting in the average valence absolute value of the oxygen ions being less than 2.0. That is, O1− anions are present in addition to O2− anions. An O2− anion has a full outer electron shell with eight electrons 2s2 2p6 . An O1− anion has a hole in its outer electron shell. In this chapter, first, we introduce the conventional ionicity investigations and the ionicity study for the spinel ferrites reported by our group; then we highlight the experimental and theoretical studies for O 2p holes and O1− anions; and finally, we explain the O 2p itinerant electron model for magnetic oxides (IEO model).

5.1 A Simple Introduction to Early Investigations of Ionicity In 1932, Pauling proposed the definition of ionicity as [2]    fi = 1 − exp −(xA − xB )2 4 ,

(5.1)

where x A and x B are the electronegativity values of A and B atoms [1, 2]. The electronegativity represents the ability of a neutral atom to attract an electron, which may be defined by the ionization energy and electronic affinity, Electronegativity = 0.18(ionization energy + electronic affinity),

© Science Press 2021 G.-D. Tang, New Itinerant Electron Models of Magnetic Materials, https://doi.org/10.1007/978-981-16-1271-8_5

(5.2)

43

44

5 O 2p Itinerant Electron Model for Magnetic Oxides

Table 5.1 Electronegativity of elements (After Introduction to inorganic chemical basis of Energetic material [3]) H 2.1

He

Li Be 1.0 1.5

B C N O F Ne 2.0 2.5 3.0 3.5 4.0

Na Mg 0.9 1.2

Al Si P S Cl Ar 1.5 1.8 2.1 2.5 3.0

K Ca 0.8 1.0

Sc Ti V Cr 1.3 1.6 1.6 1.6

Mn Fe Co Ni Cu Zn 1.5 1.8 1.9 1.8 1.9 1.6

Ga Ge As Se Br Kr 1.6 1.8 2.0 2.4 2.8 3.0

Rb Sr 0.8 1.0

Y Zr Nb Mo Tc 1.2 1.4 1.6 1.8 1.9

Ru Rh Pd Ag Cd 2.2 2.2 2.2 1.9 1.7

In Sn Sb Te I Xe 1.7 1.8 1.9 2.1 2.5 2.6

Cs Ba 0.7 0.9

La Hf Ta W 1.1 1.3 1.5 1.7

Re 1.9

Os Ir Pt Au Hg 2.2 2.2 2.2 2.4 1.9

Tl Pb Bi Po At Rn 1.8 1.8 1.9 2.0 2.2 2.4

Fr Ra 0.7 0.7

Ac 1.1

Ce Pr 1.1 1.1

Nd Pm Sm Eu 1.1 1.1 1.1 1.1

Gd 1.1

Tb Dy Ho Er Tm Yb Lu 1.1 1.1 1.1 1.1 1.1 1.1 1.2

Th Pa 1.3 1.5

U Np Pu Am Cm Bk Cf Es Fm Md No Lr 1.7 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3

and is expressed in eV. The parameter 0.18 is calculated on the basis of the electronegativity of Li, which is 1.0 [1]. The electronegativity values of the elements are listed in Table 5.1 [3]. The variation characteristics of electronegativity may be found in Table 5.1: electronegativity gradually increases from left to right and decreases from up to down in the periodic table. In 1970, Phillips [2] reviewed the reported ionicity investigations and introduced the studies of his group. On the basis of spectroscopic methods, they proposed a definition of ionicity fi =

Eh2

C2 , + C2

(5.3)

where Eg2 = Eh2 + C 2 and E g represent the energy gap in the molecular orbital theory. For a purely Group IV crystal, such as diamond, Si, or Ge, E h = E g and C = 0; for an ideal ionic crystal, C = E g and E h = 0. However, in reality, no truly ideal ionic crystal exists. Phillips [2] listed the ionicity data, as shown in Table 5.2, where f IP is the Phillips ionicity obtained by spectroscopic methods; f i and f i1 are the ionicities reported in 1932 and 1939 by Pauling [2], and the remaining parameters are introduced in Sect. 5.2.

5.2 Study of the Ionicity of Spinel Ferrites

45

Table 5.2 Ionicities of II–VI compounds. Here, f IP and f i (f i1 ) are the Phillips and Pauling ionicities (from Phillips [2]). f IT is the ionicity as calculated by Ji et al. [4], where R is the probability ratio, between the question and Sr compounds, of the second electron in the cation jumping to the anions through the potential barrier with height V C and width r, where V C is the second ionization energy of the question cation and r is the effective radius of the anions with coordination number N Material

f IT

f IP [2]

f i [2] Pauling (1932)

f i1 [2] Pauling (1939)

N

VC

r

R

BeO

0.568

0.602

0.63

0.81

4

MgO

0.641

0.841

0.73

0.88

6

18.21

0.138

0.159

15.04

0.14

CaO

0.831

0.913

0.79

0.97

0.330

6

11.87

0.14

0.778

SrO

0.926

0.926

0.79

ZnO

0.572

0.616

0.59

0.93

6

11.03

0.14

1.000

0.80

4

17.96

0.138

CdO

0.589

0.785

0.168

0.55

0.85

6

16.91

0.14

0.210

BeS

0.543

MgS

0.606

0.312

0.22

0.61

4

18.21

0.181

0.105

0.786

0.34

0.67,

6

15.04

0.184

CaS

0.257

0.804

0.902

0.43

0.81

6

11.87

0.184

0.735

SrS

0.914

0.914

0.43

0.91

6

11.03

0.184

1.000

ZnS

0.546

0.623

0.18

0.59

4

17.96

0.181

0.112

CdS

0.562

0.685

0.18

0.59

4

16.91

0.181

0.151

BeSe

0.538

0.299

0.18

0.59

4

18.21

0.195

0.092

MgSe

0.599

0.790

0.29

0.65,

6

15.04

0.198

0.237

CaSe

0.801

0.900

0.39

0.90

6

11.87

0.198

0.723

SrSe

0.917

0.917

0.39

0.80

6

11.03

0.198

1.000

ZnSe

0.541

0.676

0.15

0.57

4

17.96

0.195

0.099

CdSe

0.556

0.699

0.15

0.58

4

16.91

0.195

0.134

BeTe

0.530

0.169

0.09

0.55

4

18.21

0.218

0.073

MgTe

0.584

0.554

0.18

0.59

6

15.04

0.221

0.208

CaTe

0.783

0.894

0.26

0.88

6

11.87

0.221

0.702

SrTe

0.903

0.903

0.26

0.75

6

11.03

0.221

1.000

ZnTe

0.532

0.546

0.06

0.53

4

17.96

0.218

0.079

CdTe

0.545

0.675

0.04

0.52

4

16.91

0.218

0.112

5.2 Study of the Ionicity of Spinel Ferrites Although ionicity has been studied for several decades, few studies have been performed on the ionicity of three-element compounds, and to the best of our knowledge, no systematic ionicity study has been conducted for spinel ferrites. To explore the magnetic properties and cation distribution of spinel ferrites, our group systematically investigated the ionicity of spinel ferrites using a quantum-mechanical potential barrier model [4] by referring to the data reported by Phillips [2].

46

5 O 2p Itinerant Electron Model for Magnetic Oxides

5.2.1 Quantum-Mechanical Potential Barrier Model Used to Estimate Cation Distributions According to quantum mechanics [5], the transmission coefficient of electrons through a square potential barrier, as shown in Fig. 5.1, may be approximately expressed as 16k02 k 2 −2ka T≈ ka >> 1. 2 e k02 + k 2

(5.4)

where k0 =



2mE/2 , k =



2m(V0 − E)/2 .

(5.5)

Here,  is the Planck’s constant, and m and E are the mass and kinetic energy of the electron, respectively. In addition, V 0 and a are the height and width of the potential barrier, respectively. From Eqs. (5.4) and (5.5), we derive T≈

16E(V0 − E) −2a√2m(V0 −E)/2 e , ka >> 1. V02

(5.6)

Based on both the above equation and that the binding energy (BE) of an electron in a cation increases with the ionization energy, as shown in Fig. 2.2, in 2007, our group proposed a quantum-mechanical potential barrier model to estimate the probability ratio between two cations losing their electrons using the ionization energies [6]. We supposed that a square potential barrier was present between a cation–anion pair. The height of the potential barrier is proportional to the ionization energy of the cation and the width is related to the distance between the nearest cations and anions. Therefore, the content ratio, R, of cations with different valence, as shown Fig. 5.1 Diagrmmatic sketch of a square potential barrier

5.2 Study of the Ionicity of Spinel Ferrites

47

Fig. 5.2 An anion and its two adjacent cations with electrons

in Fig. 5.2, is related to the probabilities of their last ionized electrons transmitting through the potential barrier. Using Eq. (5.6), we derive the approximate expression [6]: R=

  VD PC 1/2 1/2 , = exp 10.24 rD VD − cv rC VC PD VC

(5.7)

where the units of energies and lengths are nm and eV, respectively. PC (PD ) is the probability of the last ionized electron of C (D) cations transmitting to the anions through a potential barrier with height V C (V D ) and width r C (r D ). V C (V D ) is the ionization energy of the last ionized electrons of the C (D) cation and r C (r D ) is the distance from the C (D) cation to the anion. cv is a correction parameter for the two potential barriers that deviated from the square shape with different extents, and cv = 1.0 when V C = V D and r C = r D .

5.2.2 Study of the Ionicity of Group II–VI Compounds Using the Quantum-Mechanical Potential Barrier Model According to the ionicity data of the II–VI compounds, as shown in Table 5.2, f IP is calculated by Phillips, f i and f i1 are given by Pauling, and the Sr compounds have the maximum ionicity values. By fitting the ionicity values of the Sr compounds, the ionicity values of other compounds may be estimated using Eq. (5.7). For II–VI compounds, such as SrO, if all Sr atoms lose two electrons, then f i = 1.0; if each Sr atom loses one electron, then f i = 0.5. If some Sr atoms lose two electrons, the ionicity may be expressed as fIT = 0.5 + 0.5 × cf × R.

(5.8)

48

5 O 2p Itinerant Electron Model for Magnetic Oxides

R is calculated by Eq. (5.7), where PC (PD ) is the probability of the second electron of the cation transmitting to the anions through the potential barrier with height V C (V D ) and width r C (r D ). Here, V C (V D ) is the second ionization energy of the cations in the question compound (the Sr compound), r C (r D ) is the effective radius of the anions in the compounds, and cv is assumed to be 1.0. For the oxides shown in Table 5.2, the ionicity value of SrO is the maximum given by Phillips [2], f IP = 0.926, and the value of R is set as 1.0. Then the value cf = 0.852, which is obtained by fitting the ionicity (f IP = 0.926) of SrO. Similarly, the values of cf were obtained as 0.828, 0.834, and 0.806 for SrS, SrSe, and SrTe, respectively, by fitting the ionicities (see the f IP values in Table 5.2) of these compound. In Table 5.2, the values of iconicity, f IT , were calculated using Eqs. (5.7) and (5.8); the parameters used in the calculation process are shown in Table 5.2. N represents the ion coordination number in the compounds; V C (or V D ) represents the second ionization energy of the cations in the question (or Sr) compound; and r = r C = r D is the effective radius of the anions with coordination number N. The calculated f IT values are close to the ionicities given by Phillips, indicating that the estimation method of the compound ionicity using Eqs. (5.7) and (5.8) is reasonable. Next, we estimate the ionicity of spinel ferrites using a similar method.

5.2.3 Study of Ionicity of Spinel Ferrite Fe3 O4 As mentioned in Sect. 4.1, we assume that Fe3 O4 possesses an inverse spinel structure (see Fig. 4.1). Eight molecules (56 ions) are present per crystal cell with the space group Fd3m and crystal lattice constants a = b = c = 8.39 Å and α = β = γ = 90°. The arrangement of O anions belongs to the close-packed, fcc lattice structures. All (A) sites are occupied by Fe3+ cations, while eight Fe2+ and eight Fe3+ cations occupy the [B] sites [7, 8]. To estimate the ionicity, we calculated the density of states (DOSs) for the valence electrons of Fe3 O4 [4] using the plane-wave pseudopotential DFT method [9–14]. The obtained DOS of Fe 3d and O 2p are shown in Fig. 5.3; the DOSs of Fe 4s and O 2s are shown in the inset of Fig. 5.3. The figure shows that near the Fermi energy level, only the DOSs of Fe 3d and O 2p orbitals are present. Thus, the elementary factor to affect magnetic order of the material is the itinerant electrons in the material hopping between Fe 3d and O 2p levels. By performing the integral for the DOSs below the Fermi energy levels in Fig. 5.3, the average number, n, of 3d electrons in Fe cation at the (A) and [B] sites of Fe3 O4 was obtained as 5.975 and 6.089, respectively; the average number of 2p electrons of O ions was obtained as 4.894. Obviously, the calculated average number of O 2p electrons is lower than six in O2− anion, in contrast to the traditional view; the calculated average number of 3d electrons in, 6.051[= (5.975 + 6.089 × 2)/3, there are one and two Fe at the (A) and [B] sites], is more than the value 5.33[= (6 + 5 × 2)/3, one Fe2+ and two Fe3+ ] as in the traditional view. Consequently, the valence absolute values of cations and anions calculated by the DOSs are obviously smaller

5.2 Study of the Ionicity of Spinel Ferrites

49

Fig. 5.3 DOSs of Fe 3d and O 2p electrons in Fe3 O4 with (A)[B]2 O4 spinel structure (from Ji et al. [4]). The insert shows the DOSs of Fe 4s and O 2s

than those in the traditional view, suggesting that Fe3+ ions are difficult to form, due to the second electronic affinity energy of oxygen (8.08 eV) being far lower than the third ionization energy of Fe (30.65 eV). In the traditional view, Fe3 O4 has an (A)[B]2 O4 inverse spinel structure; all cation magnetic moments couple ferromagnetically with each other at low temperatures in (A) and [B] sites, while cation magnetic moments in (A) sites couple antiferromagnetically to those in [B] sites. According to the above calculated 3d electron numbers for Fe, the calculated magnetic moment per Fe3 O4 molecule, 3.8 μB [(10.000−6.089)×2−(10.000−5.975)], is smaller than the experimental value of 4.2 μB [7, 8]. Therefore, the calculated Fe 3d numbers need correction to more closely match the experimental result. Let mA (μB ) and mB (μB ) represent the magnetic moment of one Fe cation at the (A) and [B] sites, respectively. If the observed average molecular magnetic, 4.2 μB , is to be satisfied, then 2mB −mA = 4.2.

(5.9)

Letting the 3d electron average number ratio (nA /nB ) in the Fe cations at (A) and [B] sites to be equal to the calculated value, nA /nB = 5.975/6.089.

(5.10)

Since nA and nB are greater than the energy level 5, as mentioned in Sect. 4.1.2, according to the atomic physics and the traditional ferrite theory [7, 8], mA = 10 − nA , mB = 10 − nB , and we obtain

(5.11)

50

5 O 2p Itinerant Electron Model for Magnetic Oxides

nA = 5.584, nB = 5.692; mA = 4.416, mB = 4.308,

(5.12)

indicating that the numbers of 3d electrons in the Fe cations at the (A) and [B] sites, 5.584 and 5.692, are lower than the calculated values, 5.975 and 6.089, and are higher than the values in the traditional view of 5 [one Fe3+ per (A) site] and 5.5 {one Fe3+ and one Fe2+ per pair of the [B] sites}, respectively. Supposing that other 3d and all 4s electrons of Fe cations are obtained by oxygen anions, the average valences of Fe cations at the (A) and [B] sites are +2.416 (= 8 − nA , 8 origins from the total number of 3d and 4s electrons in one Fe atom) and +2.308 (= 8 − nB ), respectively. These valence values are very close to the results reported by Jeng et al. [15]. Therefore, the ratios of the trivalence and divalence Fe 2+ cations at the (A) and [B] sites can be easily obtained, Fe3+ A /FeA = 0.416/0.584 and 3+ 2+ 3+ FeB /FeB = 0.308/0.692. Thus, the number of Fe cations per molecule is only 1.032 (= 0.416 + 0.308 × 2), rather than 2.00 in the traditional view; the average valence of oxygen anions V alO is −1.758 [= (2.416 + 2.308 × 2)/4]. Since ionicity f i = 1.0 when the valence of oxygen anion is −2.0, we define the ionicity as fi = |ValO |/2.00,

(5.13)

and obtain the ionicity of Fe3 O4 as 0.879 [= 1.758/2.000].

5.2.4 Estimation of the Ionicity of Spinel Ferrites M3 O4 Using the Quantum- Mechanical Potential Barrier Model Third ionization energies, V (M 3+ ), of Cr, Mn, Fe, Co, Ni, Cu, and Zn are 30.96, 33.67, 30.65, 33.50, 35.17, 36.83, and 39.72 eV, respectively, wherein the V (Fe3+ ) value is the minimum. Therefore, Fe3 O4 should have the maximum ionicity in M 3 O4 (M = Cr, Mn, Fe, Co, Ni, Cu, and Zn, if all of them may be formed). It is similar to estimate the ionicity of MO in Sect. 5.2.2, f IT of the spinel ferrites with (A)[B]2 O4 structure may be estimated by fitting the ionicity of Fe3 O4 (0.879). If the eight electrons of the three Fe atoms are obtained by the O anions, f i = 1.00. If only two electrons per Fe atom are obtained by the O anions, f i = 0.75. Thus, the ionicity of spinel ferrites can be estimated using fIT = 0.75 + 0.25 × cf × R.

(5.14)

For Fe3 O4 , f IT = 0.879, let R = 1.0, then cf = 0.516. Let cf be a constant for the spinel ferrites M 3 O4 (M = Cr, Mn, Fe, Co, Ni, Cu, and Zn), whose ionicities can be calculated using Eqs. (5.7) and (5.14). Here, for Eq. (5.7), cv is set to 1.0 and V C (V D ) is the third ionization energy of the cation M (Fe). r C = r D = 0.144 nm is the

5.2 Study of the Ionicity of Spinel Ferrites

51

Table 5.3 Ionicities of spinel ferrites, f IT [4], where R is the probability ratio, between the question ferrite and Fe3 O4 , of the third electron in the cation jumping to the O anion through the potential barrier with height V C and width r, where V C is the third ionization energy of the question cation and r is the effective radius of the O anion with coordination number 12 M 3 O4

f IT

Total valences of cations

VC

r

R

Cr3 O4

0.8726

6.9805

30.96

0.144

0.9501

Mn3 O4

0.8293

6.6343

33.67

0.144

0.6146

Fe3 O4

0.8790

7.0320

30.65

0.144

1.0000

Co3 O4

0.8314

6.6515

33.50

0.144

0.6313

Ni3 O4

0.8129

6.5029

35.17

0.144

0.4873

Cu3 O4

0.7990

6.3916

36.83

0.144

0.3795

Zn3 O4

0.7822

6.2573

39.72

0.144

0.2493

Note Assume that all of these spinel ferrites may be formed, although it has been found no report about Ni3 O4 , Cu3 O4 , and Zn3 O4

Table 5.4 Estimated ionicities and total valences of spinel ferrites MFe2 O4 (M = Cr, Mn, Co, Ni, Cu, Zn, Cr)

Ferrite

Ionicity, f IT

Total valence of cations

MnFe2 O4

0.8624

6.8994

CoFe2 O4

0.8631

6.9052

NiFe2 O4

0.8570

6.8556

CuFe2 O4

0.8523

6.8185

ZnFe2 O4

0.8467

6.7738

CrFe2 O4

0.8769

7.0148

effective radius of an oxygen ion with a coordination number of 12. The calculated ionicities and the parameters used in these calculations are listed in Table 5.3. In Table 5.3, we considered that the ionicity of M cation equals the ionicity of M 3 O4 . We estimated the ionicities of MFe2 O4 (M = Cr, Mn, Co, Ni, and Cu) compounds by averaging the ionicities of one M and two Fe cations. The calculated ionicities and related total valence of cations are shown in Table 5.4.

5.3 Experimental Studies of O 2p Holes in Oxides As mentioned in Sect. 2.1, the second electronic affinity energy of oxygen equals 8.08 eV, which is higher than the first ionization energy but lower than the second ionization energy of most metal atoms, as shown in Appendix A. Therefore, an ideal state where all oxygen anions are O2− ions is difficult to form even in a monoxide. In general, some O1− ions must be present in an oxide. The third ionization energy of a cation is far higher than its second ionization energy; the fourth ionization energy of a cation is far higher than its third ionization energy. Thus, as the anion/cation ratio

52

5 O 2p Itinerant Electron Model for Magnetic Oxides

increases, the content ratio of O1− ions in an oxide increases and the absolute value of its average valence decreases. This has been confirmed by many experimental and theoretical investigations [16–23]. Earlier studies found that the O 2p holes exist in oxides. An O2− ion has a stable outer electron shell with eight electrons; if a hole is present in the outer shell of an oxygen anion, then it is an O1− ion.

5.3.1 O 2p Hole Studies Using Electron Energy Loss Spectroscopy Using the inelastic scattering of incidence electrons in a sample, the losing energy of the incidence electrons can be measured by E = E0 − Emin .

(5.15)

Then, electron energy loss spectrum (EELS) is obtained. EELS is used to analyze the chemical composition, thickness, and electronic structure of materials. In Eq. (5.15), E 0 is the energy of incidence electrons, E min is the remaining energy of incidence electrons subjected to inelastic scattering in the sample. E is closely related to the electronic structure of the sample [24]. In 1988, Nücker et al. [16] reported their investigation for the superconductor material YBa2 Cu3 O7-y . They employed high-energy transition EELS with an incidence electron energy of 170 keV and the resolution of the energy and wavenumber of 0.4 eV and 0.2 Å−1 , respectively. The samples had a thickness of 1000 Å and were cut from the bulk material. Due to the transition spectroscopy used, the data of the samples came from the entire sample and not only from the surface. The variation of the oxygen vacancy ratio (y) in the samples was obtained by annealing in super-high vacuum. They found that for these superconductors, the DOS at the Fermi energy has predominantly oxygen 2p character. Therefore, the charge carriers in the superconducting compounds are O 2p holes. Ju et al. [17] analyzed EELS of La1-x Srx MnO3 (0 ≤ x ≤ 0.7) films, which were epitaxially grown on (100) LaAlO3 substrates via a polymeric sol–gel technique and finally annealed at 700 °C for 1 h in air. XRD indicated that the films had an ABO3 perovskite structure. The resistivity curves as a function of temperature are shown in Fig. 5.4a, which are very similar to those of La1-x Srx MnO3 single-crystal films reported by Urushibara et al. [25]. Figure 5.4b shows the region near the threshold of the O-K edge in the EELS of La1-x Srx MnO3 films and the LaAlO3 substrate. They concluded that the prepeak around 529 eV corresponds to the empty states in O 2p. However, this prepeak was not observed for the LaAlO3 substrate. Therefore, La1-x Srx MnO3 materials are charge-transfer-type oxides with carriers having a dominant O 2p hole character. The experimental phenomenon in Fig. 5.4b may be explained using ionization energy as follows: (i) Since LaAlO3 has no O1− peak in Fig. 5.4b, as an approxima-

5.3 Experimental Studies of O 2p Holes in Oxides Fig. 5.4 Curves of the resistivity versus the test temperature a and EELS b of La1-x Srx MnO3 reported by Ju et al. [17]

53

54

5 O 2p Itinerant Electron Model for Magnetic Oxides

tion, we may assume that all oxygen ions are O2− anions. (ii) Since the third ionization energy of Sr (43.60 eV) and the fourth ionization energy of Mn (51.20 eV) are distinctly higher than the third ionization energies of Al (28.45 eV) and Mn (33.67 eV), La1-x Srx MnO3 includes no Sr3+ and Mn4+ cation but includes Sr2+ and Mn3+ cations and O1− anions. In addition, the content ratio of O1− anions increases with increasing Sr (x). This conclusion, that no Mn4+ cation is present in La1-x Srx MnO3 materials, is important evidence for the following discussion of magnetic ordering and electrical transport property.

5.3.2 Several Other Experimental Investigations for O 2p Holes Mizoroki et al. [18] prepared the polycrystalline perovskite manganites La1-x Srx MnO3 (x = 0.1, 0.2, 0.3, 04, and 0.5) and performed magnetic Compton scattering measurements on the samples. They concluded that doped holes predominantly enter into the O 2p states in the lightly doped region. Grenier et al. [19] observed the superstructures associated with the oxygen 2p states in La7/8 Sr1/8 MnO3 manganites using X-ray diffraction at the O-K edge. They found a 2p charge ordering described by alternating hole-poor and hole-rich MnO planes. Ibrahim et al. [20] synthesized colossal magnetoresistance (CMR) manganites Pr1-x Srx MnO3 (x = 0.0 and 0.3) and analyzed the X-ray absorption and Auger spectra of the samples. They demonstrated the existence of an oxygen 2p hole state and showed its importance in the electronic processes. They concluded that the DOS of the oxygen 2p holes increases with Sr doping. Papavassiliou et al. [21] prepared the polycrystalline manganites La1−x Cax MnO3+δ and analyzed the nuclear magnetic resonance (NMR) and X-ray absorption spectra. They found the spin-polarized hole arrangement of the oxygen 2p orbitals in the samples. After reviewing the above experimental results, Alexandrov et al. [22, 23] indicated that the conventional DE interaction model contradicts these experiments. These experiments have shown that the current carriers are oxygen p holes rather than d electrons in ferromagnetic manganites.

5.4 Study of Negative Monovalent Oxygen Ions Using X-Ray Photoelectron Spectra In the conventional analysis method for XPS, all oxygen ions in an oxide were assumed to be negative bivalent. However, in the last three decades, the theoretical calculation and XPS investigations indicated that O1− ions coexist with O2− ions in an oxide.

5.4 Study of Negative Monovalent Oxygen Ions …

55

Fig. 5.5 A binding energy scale for the O 1s peaks proposed by Dupin et al. [28]

Cohen and Krakauer [26, 27] investigated the effective charges of O, Ti, and Ba ions in BaTiO3 using the full-potential linearized-augmented-plane-wave method. They concluded that the average valences of Ba, Ti, and O are +2, +2.89, and − 1.63, respectively. The calculated valence of Ba ions is the same as the value in the traditional view, but the absolute values of average valences of Ti and O are obviously lower than the conventional values of 4 and 2, respectively. This result can be understood using the ionization energy of the free atoms. (i) The valence (+2.00) of Ba cation is the same as the value in the conventional view because the second ionization energy (10.00 eV) of Ba is close to the second electron affinity energy (8.08 eV) of O. (ii) The valence (+2.89) of Ti cation is distinctly lower than the value (+4.00) in the conventional view since the third and fourth ionization energies (27.49 and 43.27 eV) of Ti are far higher than the second electron affinity energy (8.08 eV) of O. Thus, it is difficult for O to obtain the third electron of Ti, and it is nearly impossible for O to obtain the fourth electron of Ti. In 2000, Dupin et al. [28] analyzed the XPS of O 1s for several oxides and proposed that the spectrum peak located from 527.7 to 530.6 eV corresponds to O2− ions; the spectrum peak located from 531.1 to 532.0 eV belongs to O1− ions and the spectrum peak of the weakly adsorbed oxygen may be located from 531.1 to 533.5.0 eV, as shown in Fig. 5.5. Based on these results, several XPS investigations were performed, as described in the next section.

5.4.1 Study of Ionicity of BaTiO3 and Several Monoxides Using O 1s XPS Applying the analysis method proposed by Dupin et al. [28], our group examined the XPS [29] of BaTiO3 and monoxides CaO, MnO, CoO, ZnO, NiO, and CuO. XRD analyses indicated that these samples possess a single-phase crystal structure, crystal lattice constants, and crystallite sizes are shown in Table 5.5 [29]. For BaTiO3 , Fig. 5.6 shows the O 1s spectrum (points) and the fitted results (curves) using Gaussian– Lorentzian functions. Here, the O 1s spectrum is fitted using three peaks with BEs of approximately 528.5, 530.7, and 532.6 eV. Referring to the interpretation proposed by Dupin et al. [28], the lower BE peak is assigned to O2− ions, the middle BE peak

56

5 O 2p Itinerant Electron Model for Magnetic Oxides

Table 5.5 Molar mass, purity, crystallite size, crystal system, and crystal lattice constants for the samples. [29] Molecular formula

Molar mass (g/mol)

Purity (%)

Crystallite Size (nm)

Crystal system

Crystal lattice constants (nm, °)

BaTiO3

233.19

99.5

>100

Tetragonal

a = 0.3995, b = 0.3995, c = 0.4033

CaO

56.08

98.0

>100

Cubic

a = 0.4810

MnO

70.94

99.5

>100

Cubic

a = 0.4446

CoO

74.93

99.0

47.3

Cubic

a = 0.4261

ZnO

81.39

99.0

>100

Hexagonal

a = 0.3250, b = 0.3250, c = 0.5207

NiO

74.69

99.0

13.6

Cubic

a = 0.4181

CuO

79.55

99.0

27.7

Monoclinic

a = 0.4682, b = 0.3428, c = 0.5130 α = γ = 90.00, β = 99.41

Fig. 5.6 O 1s photoelectron spectrum (points) with fitting results (curves) for BaTiO3 [29]

is assigned to O1− ions, and the higher BE peak is assigned to OChem , which is the chemically adsorbed oxygen on the surface. The fitting data for BaTiO3 are shown in Table 5.6. From the relative peak areas, S 1 and S 2 , the ratio of O1 to O2 of O1− and O2− main components can be deduced. Note that O1 + O2 = 1, O1 /O2 = S1 /S2 ;

(5.16)

5.4 Study of Negative Monovalent Oxygen Ions … Table 5.6 Fitting results [29] of O 1s photoelectron spectra for BaTiO3 and several monoxide samples. According to the view proposed by Dupin et al. [28], the lower BE (binding energy) peak is assigned to O2− ions, the middle BE peak to O1− ions, and the higher BE peak to OChem , chemically adsorbed oxygen on the surface

57

Composition

Peak Position (eV)

FWHMa (eV)

Peak area (%)

BaTiO3

528.48

1.55

50.9

530.68

2.10

41.8

532.62

1.67

530.65

1.90

87.1

532.15

1.55

12.0

533.60

1.05

529.25

1.45

66.3

530.80

1.52

27.4

532.04

1.62

529.25

1.56

62.4

530.95

1.79

26.9

532.60

1.99

10.7

529.60

1.21

64.7

531.20

1.87

31.4

532.95

1.27

528.70

1.07

67.2

530.50

1.50

29.5

532.13

1.30

529.85

1.95

51.8

531.70

1.90

40.5

533.35

1.79

CaO

MnO

CoO

ZnO

NiO

CuO

a FWHM

7.20

0.90

6.30

4.00

3.30

7.70

= full width at half maxima

then O2 =

1 , O1 = 1 − O2 . 1 + S1 /S2

(5.17)

Furthermore, the average valence of oxygen ions ValO = −2O2 − O1 .

(5.18)

Table 5.6 shows that S 1 /S 2 = 0.418/0.509 for BaTiO3 . Using Eq. (5.17) and (5.18), we can calculate that O1 = 0.45, O2 = 0.55, and V alO = −1.55. The V alO value is close to the value (−1.63) calculated by Cohen [26, 27]. Figure 5.7 shows the O 1s spectra with fitted results for the monoxides CaO, MnO, CoO, ZnO, NiO, and CuO. The fitted data are listed in Table 5.6. According to the above-described analyses for the O 1s spectrum of BaTiO3 , the V alO values of these monoxides were calculated. According to the definition of ionicity in Eq. (5.13), f i = |V alO |/2.00, the f i values were calculated. The data of V alO , f i , and the second

58

5 O 2p Itinerant Electron Model for Magnetic Oxides

Fig. 5.7 O 1s photoelectron spectra (points) with fitting results (curves) for monoxides CaO, MnO, CoO, ZnO, NiO, and CuO

ionization energy, V (M 2+ ), of the cations are listed in Table 5.7. In addition, the f i values of SrO, CaO, MgO, and BeO reported by Phillips [2] are listed in Table 5.7. Using the data in Table 5.7, ionicity f i may be dependent on the second ionization energy V (M 2+ ), as shown in Fig. 5.8. Figure 5.8 shows that values of f i nearly linearly decrease with increasing V (M 2+ ) and that the two lines, for XPS data [29] and the data reported by Phillips [2], are close to each other. The reason for the V (M 2+ )

5.4 Study of Negative Monovalent Oxygen Ions …

59

Table 5.7 The average valence of oxygen, V alO , and the ionicity, f i , measured using XPS reported by Wu et al. [29] and as reported by Phillips [2] for monoxides MO. V (M 2+ ) is second ionization energy of the M cations Reported by Wu et al. [29]

Reported by Phillips [2]

Monoxides MO

V (M 2+ ) (eV)

V alO

fi

Monoxides MO

V (M 2+) (eV)

fi

CaO

11.87

−1.879

0.940

SrO

11.03

0.926

MnO

15.64

−1.707

0.854

CaO

11.87

0.913

CoO

17.06

−1.699

0.849

MgO

15.04

0.841

BeO

18.21

0.785

ZnO

17.96

−1.673

0.837

NiO

18.17

−1.695

0.847

CuO

20.29

−1.560

0.780

Fig. 5.8 The dependence of the ionicities f i , measured using XPS [29] and reported by Phillips [2] on the second ionization energies V (M 2+ ) of the cations

dependence of f i can be explained as follows. The second electron affinity energy of oxygen is 8.08 eV, which is lower than the second ionization energies of metal cations, between 11.87 and 20.29 eV for these oxides. These data of the free atoms must affect how ions gain and lose electrons in oxides. Therefore, it is more difficult for a cation with higher V (M 2+ ) to lose its second electron, and its oxide has lower ionicity. Guo et al. [30] and Raddy et al. [31] reported the ionicities, f i , of several dioxides MO2 (M = Th, Zr, Hf, Sn, Pb, Ti, Si, and Ge), where f i as the function of the fourth ionization energy V (M 4+ ) of the cations is shown in Fig. 5.9. The figure shows that f i is less than 0.75 when V (M 4+ ) > 44 eV, indicating that no quadrivalent cation is present when V (M 4+ ) > 44 eV. The relations of the ionicity with the ionization energy of the cations in Figs. 5.8 and 5.9 can be explained as follows: since it is difficult for cations to lose their electrons with high ionization energy, the valence absolute value of each oxide is less than its value in the conventional view. Therefore, the absolute value of the average valence for oxygen ions |V alO | in oxide is less than 2.0.

60

5 O 2p Itinerant Electron Model for Magnetic Oxides

Fig. 5.9 The ionicity of the dioxides MO2 (M = Th, Zr, Hf, Sn, Pb, Ti, Si and Ge) reported by Guo et al. [30] (❚) and Raddy et al. [31] (●) as a function of the fourth cation ionization energy, V (M 4+ )

5.4.2 Effect of Argon Ion Etching on the O 1s Photoelectron Spectra of SrTiO3 The effect of argon ion etching on the O1s photoelectron spectra of two SrTiO3 samples was analyzed by our group [32, 33]. We found that, as the argon ion etching time increases, the peak intensity ratio of O1− /O2− decreases, while the peak intensity ratio of Ti2+ /Ti3+ increases, which is similar to the results reported by Steinberger et al. [34]. Two commercial SrTiO3 samples, a single-crystal thin piece (SSTO), and a polycrystalline bulk (PSTO) were obtained, and sample surfaces were cleaned using different procedures. By analyzing XRD data, the samples were confirmed to have a single cubic ABO3 perovskite phase; the crystal lattice constants of the SSTO and PSTO were 0.3869 and 0.3906 nm, respectively; the average crystallite size for the PSTO sample was greater than 100 nm. XPS of the samples were measured using an X-ray photoelectron spectroscopy PHI5000 Versa Probe with monochromatic Al Kα radiation (1486.6 eV). For each sample, seven sets of photoelectron spectra were measured after the sample was etched with an argon ion beam for 0, 30, 60, 90, 120, 150, and 180 s. The XPS data were analyzed using a method similar to that in Sect. 5.4.1. Figures 5.10a and 5.11a show the O 1s spectra without etching, which were fitted using three peaks, P1, P2, and P3, with BEs E 1 < E 2 < E 3 . The chemical shift from P1 to P2, E = E 2 −E 1 , has the values 1.88 < E < 2.08 eV, while the chemical shift, E = E 3 −E 1 , has the values 2.88 < E < 3.22 eV. As mentioned in Sect. 5.4.1, peaks P1, P2, and P3 correspond to O2− , O1− , and OChem on the surface. O 1s spectra with etching for 30, 60, 90, 120, 150, and 180 s, Figs. 5.10b–g and 5.11b–g, were fitted using only two peaks. The lower BE peak corresponds to O2− ions, and the higher BE peak corresponds to O1− ions. No chemically adsorbed oxygen was present in the spectra since the etching and measurements were performed in vacuum. Using the relative peak areas of O1− and O2− , the average

5.4 Study of Negative Monovalent Oxygen Ions …

61

Fig. 5.10 O 1s photoelectron spectra (points) with fitting results (curves) for SSTO with different etching time 0, 30, 60, 90, 120, 150, and 180 s [32, 33]

62

5 O 2p Itinerant Electron Model for Magnetic Oxides

Fig. 5.11 O 1s photoelectron spectra (points) with fitting results (curves) for PSTO with different etching time 0, 30, 60, 90, 120, 150 and 180 s [32, 33]

5.4 Study of Negative Monovalent Oxygen Ions …

63

valences, V alO , of the oxygen anions for all samples were calculated by the method mentioned in Sect. 5.4.1, and they were listed in Tables 5.8 and 5.9. In both tables, E, E, FWHM, and S represent the peak position, chemical shift (from P1 to P2 or P3), full width at half maximum, and the area of the peak, respectively. Figure 5.12 shows the Ti 2p3/2 spectra for the SSTO sample with etching time t = 0, 30, 60, 90, 120, 150, and 180 s. As seen in the figure, the intensity of the lower BE peak increases with increasing t. Referring to the calculated results reported by Cohen [26, 27], no Ti4+ cations are present, and the higher BE peak is assigned to Ti3+ cations, while the lower BE peak is assigned to Ti2+ cations. Using the analysis method similar to that for the O 1s spectra, the average valence, V alT , of the Ti cations was estimated. Using the peak area ratio, S 2 /S 3 , of Ti2+ / Ti3+ , the content ratio of Ti2+ to Ti3+ cations, T 2 and T 3 , may be calculated. Since T 2 + T 3 = 1, T 2 /T 3 = S 2 /S 3 , T3 =

1 , T2 = 1 − T3 . 1 + S2 /S3

(5.19)

The average valence of Ti cations can be obtained by ValT = 3T3 + 2T2 .

(5.20)

Table 5.8 For SSTO, the fitting results [32, 33] of O 1s photoelectron spectra with different etching time t. E, E, FWHM, and S represent the peak position, chemical shift (from P1 to P2 or P3), full width at half maximum, and the area of the peak, respectively. V alO represents the average valence of oxygen ions t(s)

E(eV)

E(eV)

0

529.189

–

531.265

2.076

FWHM(eV)

S(%)

V alO

1.300

75.103

−1.762

1.856

23.487

532.410

3.221

1.130

1.41

529.187

–

1.490

90.735

530.620

1.433

1.500

9.265

529.250

–

1.530

91.956

530.851

1.601

1.481

8.044

90

529.314

–

1.555

94.094

531.000

1.686

1.326

120

529.245

–

1.600

94.18

531.140

1.895

1.310

5.82

529.191

–

1.630

94.197

531.180

1.989

1.290

5.803

529.255

–

1.610

94.197

531.215

1.96

1.310

5.803

30 60

150 180

−1.907 −1.920 −1.941

5.906 −1.942 −1.942 −1.942

64

5 O 2p Itinerant Electron Model for Magnetic Oxides

Table 5.9 For PSTO, fitting results [32, 33] of O 1s photoelectron spectra with different etching time t. E, E, FWHM, and S represent the peak position, chemical shift (from P1 to P2 or P3), full width at half maximum, and the area of the peak, respectively. V alO represents the average valence of oxygen ions t(s)

E(eV)

E(eV)

0

529.270 531.150

FWHM(eV)

S(%)

V alO

–

1.550

58.136

−1.617

1.88

1.600

36.054

532.150

2.88

1.560

5.811

529.510

–

1.690

81.548

531.253

1.743

1.767

18.452

60

529.590

–

1.660

84.71

531.309

1.719

1.776

15.29

90

529.597

–

1.670

85.32

531.257

1.66

1.750

14.68

529.549

–

1.640

85.258

531.270

1.721

1.750

14.742

150

529.590

–

1.700

85.889

531.290

1.7

1.750

14.111

180

529.563

–

1.680

85.843

531.270

1.707

1.770

14.157

30

120

−1.815 −1.847 −1.853 −1.853 −1.859 −1.858

Figure 5.13 shows the Ti 2p3/2 spectra for PSTO etched for 0, 30, 60, 90, 120, 150, and 180 s. The calculated data using Eqs. (5.19) and (5.20) for SSTO and PSTO are listed in Tables 5.10 and 5.11, where E, FWHM, and S represent the peak position, full width at half maximum, and the area of the peak, respectively. V alT represents the average valence of Ti ions. For the Sr 3d 5/2 and Sr 3d 3/2 spectra, there is no indication of peaks arising from different charge states. Referring to the calculated result for Ba in BaTiO3 reported by Cohen [26, 27], the average valence, V alS , of Sr is, therefore, assumed to be 2.000. Summarizing the above analyses for O 1s, Ti 2p3/2 , and Sr 3d 3/2 photoelectron spectra of SSTO and PSTO, the average valences, V alS , V alT , and V alO , of Sr, Ti and O ions, respectively, as well as the total valence of cations, V +al = V alS + V alT , are listed in Table 5.12. The curves of the absolute values of V alO and V +al versus etching time t are shown in Fig. 5.14. Table 5.12 and Fig. 5.14 show the interesting phenomena: (i) absolute values of V alO increase and V +al decrease as the etching time t increases; (ii) the change rates of V alO and V +al gradually decrease with increasing t when t ≤ 90 s; (iii) V alO and V +al have no significant change when t ≥ 90 s. These observations are similar to those reported by Steinberger et al. [34] and can be explained as follows. Steinberger et al. [34] reported the dependence of Zn and O contents on the argon ion etching time t in ZnO. They concluded that the ratio of Zn increased with increased t from 45% (0 s) to 55% (400 s), while the ratio of O decreased with

5.4 Study of Negative Monovalent Oxygen Ions …

65

Fig. 5.12 Ti 2p3/2 photoelectron spectra (points) with fitting results (curves) for SSTO with different etching time 0, 30, 60, 90, 120, 150 and 180 s [32, 33]

66

5 O 2p Itinerant Electron Model for Magnetic Oxides

Fig. 5.13 Ti 2p3/2 photoelectron spectra (points) with fitting results (curves) for PSTO with different etching time 0, 30, 60, 90, 120, 150 and 180 s [32, 33]

5.4 Study of Negative Monovalent Oxygen Ions …

67

Table 5.10 For SSTO, fitting results [32, 33] of Ti 2p3/2 photoelectron spectra with different etching time t. Here, E, FWHM, and S represent the peak position, full width at half maximum, and the area of the peak, respectively. V alT represents the average valence of Ti ions t(s)

E(eV)

FWHM(eV)

0

456.461

0.775

2.097

458.059

1.170

97.903

456.034

1.557

11.561

457.804

1.330

88.439

456.095

1.950

18.033

457.785

1.360

81.967

90

456.120

1.810

21.731

457.778

1.340

78.269

120

455.990

1.860

22.332

457.750

1.440

77.668

455.981

1.830

22.911

457.681

1.370

77.089

455.985

1.800

23.001

457.705

1.420

76.999

30 60

150 180

S(%)

V alT 2.979 2.884 2.820 2.783 2.777 2.771 2.770

Table 5.11 For PSTO, fitting results [32, 33] of Ti 2p3/2 photoelectron spectra with different etching time t. Here, E, FWHM, and S represent the peak position, full width at half maximum, and the area of the peak, respectively. V alT represents the average valence of Ti ions t(s)

E(eV)

FWHM(eV)

0

456.260

0.590

1.55

457.885

1.330

98.45

456.090

1.250

7.079

457.860

1.480

92.921

60

456.210

1.400

10.733

457.915

1.470

89.267

90

456.220

1.450

13.476

457.887

1.440

86.524

456.289

1.450

14.604

457.889

1.460

85.396

456.310

1.470

15.402

457.856

1.490

84.598

456.260

1.460

15.243

457.963

1.520

84.757

30

120 150 180

S(%)

V alT 2.985 2.929 2.893 2.865 2.854 2.846 2.848

68

5 O 2p Itinerant Electron Model for Magnetic Oxides

Table 5.12 Average valences, V alS and V alT , of the Sr and Ti cations in the SSTO and PSTO samples with different etching times Δt. V +al = V alS + V alT , represents the total valence of cations, V alO represents the average valence of oxygen ions [32, 33] Δt (s)

SSTO

PSTO

V alO

V alS

V alT

V +al

V alO

V alS

V alT

V +al

0

−1.762

2.000

2.979

4.979

−1.617

2.000

2.985

4.985

30

−1.907

2.000

2.884

4.884

−1.815

2.000

2.929

4.929

60

−1.920

2.000

2.820

4.820

−1.847

2.000

2.893

4.893

90

−1.941

2.000

2.783

4.783

−1.853

2.000

2.865

4.865

120

−1.942

2.000

2.777

4.777

−1.853

2.000

2.854

4.854

150

−1.942

2.000

2.771

4.771

−1.859

2.000

2.846

4.846

180

−1.942

2.000

2.770

4.770

−1.858

2.000

2.848

4.848

Fig. 5.14 Curves of the absolute values of V alO and V +al versus the etching time t of SSTO and PSTO samples

increased t from 55% (0 s) to 45% (400 s). Thus, this oxide varied from Zn0.82 O with 18% Zn vacancies (0 s) to ZnO0.82 with 18% O vacancies (400 s). This is because there is a tendency that during etching, the lost ions with small atomic numbers are more than those with larger atomic numbers.

5.4 Study of Negative Monovalent Oxygen Ions …

69

In the SSTO and PSTO samples, the ratio of O to metal ions decreased and the calculated oxygen vacancy ratio increased in a parallel manner with increased etching time t. Here, the sample composition was assumed to change from the standard ABO3 to ABO3-δ with the increase of t, where δ represents the oxygen vacancy content. Since the sum [(3−δ)|V alO |] of the anion valence absolute value must equal the sum of the cation valences (V +al = V alS + V alT ), for ABO3-δ oxides, (3−δ)|V alO |= V +al ; then δ=3 −

Val+ . |ValO |

(5.21)

Based on Table 5.12 and Eq. (5.21), δ and the percentage of oxygen vacancies, V V = (δ/3) × 100%, were calculated; the results are shown in Table 5.13. The results show that the δ values of the SSTO and PSTO are 0.17 and −0.08 when Δt = 0, suggesting that the average oxygen contents per molecule were 2.83 and 3.08, respectively. This indicates that SSTO has O vacancies and that PSTO has cation vacancies at Δt = 0. These phenomena, that O ratio is lower or higher than the standard ratio, can also be observed in literature [34–36]. Moreover, Table 5.13 shows that the maximum oxygen vacancy ratio in SSTO sample is 18% when t ≥ 90 s, which is the same as that in ZnO0.82 , as reported by Steinberger et al. [34]. The different Δt values for similar O vacancy ratios are likely due to the different intensities of the argon ion beams and different hardness of the samples. According to the above discussion, the relative intensity of the P2 peaks with chemical shift E ≈ 2 eV in O 1s spectra decreases with Δt (Figs. 5.10 and 5.11), while the O vacancy ratio δ increases with Δt (Table 5.13). These results indicated that the P2 peaks should be regarded as they arise from O1− . The assumption [37, 38] that the P2 peaks arise from oxygen vacancies is incorrect. Table 5.13 Oxygen vacancy content, δ, per formula ABO3-δ , for samples SSTO and PSTO with different etching times Δt, and the content percentage of vacancies V V = (δ/ 3) × 100% Δt (s)

SSTO δ

PSTO formula

V V (%)

δ

formula

V V (%)

0

0.174

SrTiO2.83

5.8

−0.082

SrTiO3.08

–

30

0.439

SrTiO2.56

14.6

0.285

SrTiO2.71

9.5

60

0.490

SrTiO2.51

16.3

0.351

SrTiO2.65

11.7

90

0.536

SrTiO2.46

17.9

0.375

SrTiO2.62

12.5

120

0.540

SrTiO2.46

18.0

0.380

SrTiO2.62

12.7

150

0.543

SrTiO2.46

18.1

0.393

SrTiO2.61

13.1

180

0.544

SrTiO2.46

18.1

0.392

SrTiO2.61

13.1

70

5 O 2p Itinerant Electron Model for Magnetic Oxides

5.5 O 2p Itinerant Electron Model for Magnetic Oxides (IEO Model) As mentioned in the previous sections, the coexistence of O1− and O2− ions in an oxide has been proven through theory and experiments. Therefore, Alexandrov et al. [22, 23] indicated that the DE interaction model contradicts the experimental results for the valence electron states, which have proved that the charge current carriers in ferromagnetic manganites are O 2p holes rather than 3d electrons. They, therefore, studied the electrical transport property of manganites using the O 2p hole carrier model. However, the magnetic ordering of oxides has not been studied using their model. Similar to the O 2p hole carrier model, our group proposed an O 2p itinerant electron model for magnetic oxides (IEO model) [39, 40], which includes the following features: (i)

O2− (2s2 2p6 ) and O1− (2s2 2p5 ) anions simultaneously coexist in an oxide. The outer orbit of an O1− anion has an O 2p hole. In a given sublattice, an O 2p electron with a constant spin direction can hop from an O2− anion to the O 2p hole of an adjacent O1− anion with a metal cation acting as an intermediary. This hopping process can, of course, also be understood as a 2p hole hopping in the opposite direction. The reasons why the itinerant electrons originated from O 2p electron rather than the 3d electrons may be explained using the data of the electron affinity energy (E af , representing the ability to gain an electron) of oxygen and the ionization energies [V (M N+ ), representing the ability to bind an electron] of the cations. As mentioned in Sect. 2.1 and Appendix A, for 3d transition metals from Ti to Zn, the V (M 2+ ) and V (M 3+ ) values are between 13.58 and 39.72 eV; however, the second E af value of oxygen is only 8.08 eV. These data suggest that the ability of O2− (2s2 2p6 ) anions to bind the 2p electrons is weaker than that of monovalent and divalent cations to bind 3d electrons in 3d transition metals. Therefore, the itinerant electrons in spinel ferrites and perovskite manganites originated from the O 2p electrons of the oxygen anions, rather than from the 3d electrons of the divalent and trivalent cations [22, 23].

(ii)

Since an itinerant electron has a constant spin direction in a given sublattice, the two O 2p electrons in the outer orbit of an O2− anion, which have opposite spin directions, become itinerant electrons in two different sublattices. That is, in a magnetic oxide below Curie temperature, the itinerant electrons must have opposite spin directions in two sublattices, such as (A) or [B] sites of spinel ferrites. Subjected to the constraints of Hund’s rule [7] and that an itinerant electron has a constant spin direction in a given sublattice, the magnetic moments of the cations with 3d electron numbers nd ≤ 4 (such as Mn3+ or divalent/trivalent Ti or Cr cations) are antiparallel to those of the cations with nd ≥ 5 (such as

(iii)

5.5 O 2p Itinerant Electron Model for Magnetic Oxides (IEO Model)

(iv)

71

Mn2+ or divalent/trivalent Fe, Co, or Ni cations), regardless of whether they are located at the (A) or [B] sublattice. In the hopping process of an itinerant electron, if it passes through the highest energy level of the intermediary cation (for example, Mn3+ (3d 4 ) in ABO3 perovskite manganites or Fe3+ (3d 5 ) in (A)[B]2 O4 spinel ferrites), it consumes only a small amount of the system energy. Otherwise, it spends more energy. This property may be the reason that the average molecular magnetic moments μobs and Curie temperature T C rapidly decrease and present the canting magnetic structures, since the ratio of Fe is less than 2.0 in (A)[B]2 O4 spinel ferrites [41] and since a portion of Mn cations is substituted by other cations in ABO3 perovskite manganites [40]. In particular, the doping antiferromagnetic composition in spinel ferrites or ABO3 perovskite manganites decreases their T C . These features are discussed in detail in Sects. 8.1, 8.2, and 9.3–9.6.

To discuss the transition of an itinerant electron, we should become familiar with how the electrons arrange at discrete energy levels. The spin- and angle-resolved photoemission spectra of metals were reviewed [42], including the investigation for Fe by Kisker et al. [43], as shown in Fig. 5.15. As is well known, five energy levels are present in the 3d subshell of a 3d transition metal atom. From Fig. 5.15, we find that the valence electrons of Fe are in a nondegenerate state and that the down-spin electrons are distributed near the Fermi level. Figure 5.15 may be explained as follows. The 3d electrons with up-spin in the cations (when nd ≤ 5) of 3d transition metals, called the majority spin, arrange from the low- to high-energy level for one electron per energy level; when nd > 5, extra 3d electrons arrange from the high- to low-energy level for one electron per energy level, called the minority spin. Similarly, for 4f electrons of rare earth metals with seven energy levels in the 4f subshell, when the number of 4f electrons nf ≤ 7, they arrange from the low- to Fig. 5.15 Spin- and angle-resolved photoemission spectra for Fe metal (from Kisker et al. [43]), measured at 0.3 T C , T C represent Curie temperature. The arrows represent the spin directions of the electrons

72

5 O 2p Itinerant Electron Model for Magnetic Oxides

high-energy level for one electron per energy level; when nf > 7, extra 4f electrons arrange from the high- to low-energy level for one electron per energy level. According to the IEO model, all cation magnetic moments align parallel to each other in a given sublattice {(A) or [B] sublattice} of spinel ferrites MFe2 O4 (M = Fe, Co, Ni, and Cu) since all cations have nd ≥ 5. If an Mn3+ , Cr2+ , Cr3+ , Ti2+ , or Ti3+ cation with nd ≤ 4 is doped in a spinel ferrite, the magnetic moment must be antiparallel to that of the Fe cations in the (A) or [B] sublattice. As an example, a transition process of an itinerant electron is discussed below. Figure 5.16a shows an ion chain, O↓↑ –Mn2+ –O↓↑ –Mn3+ –O↓ , in which the right 1− O anion absents an up-spin 2p electron in the outer orbit (a p hole is present) represented by “.” An arrow drawn on a 3d energy level represents an electron with a specific spin direction. The sign “e− ↑” represents an itinerant electron with an up-spin. An itinerant electron can transit from the middle oxygen anion to the 2p hole of the right oxygen anion via the Mn3+ cation, as shown in Fig. 5.16b, forming the state O↓↑ –Mn2+ –O↓ –Mn3+ –O↓↑ , as depicted in Fig. 5.16c. Alternatively, an O 2p electron can hop from the left O ion to the middle O ion via Mn2+ , as illustrated in Fig. 5.16d, forming the O↓ –Mn2+ –O↓↑ –Mn3+ –O↓↑ state, as shown in Fig. 5.16e. However, if the direction of the Mn2+ cation magnetic moment is parallel to that of the Mn3+ cation magnetic moment, an up-spin itinerant electron cannot transit from the left oxygen anion to the middle oxygen anion via Mn2+ , as depicted in Fig. 5.16f, since it is constrained by Hund’s rules: no two electrons with the same spin direction can occupy the same energy level. In addition, compared to an itinerant electron with an up-spin (e− ↑) that transits via Mn3+ , as shown in Fig. 5.16b, more energy will be consumed when it transits via Mn2+ , as shown in Fig. 5.16d. This will be further discussed in Chap. 9. It is similar to the spin-polarized O 2p holes [21]: an O1− anion has its magnetic moment direction. At the three 2p energy levels of an O1− anion in Fig. 5.16, the major spin direction is down, indicating that the magnetic moment direction of the O1− anion is down, similar to that of the Mn2+ cation. Figure 5.17a shows an ion chain, O↓↑ –Fe3+ –O↓↑ –Fe3+ –O↑∇ , in which the right 1− O anion loses a down-spin 2p electron in the outer orbit (a p hole is present) represented by “∇.” An electron with a down-spin, “e− ↓,” may hop from the middle O ion to the 2p hole of the right O ion via Fe3+ , with a certain probability, as depicted in Fig. 5.17b, yielding the state illustrated in Fig. 5.17c. Similarly, an electron with a down-spin, “e− ↓,” may hop from the left O ion to the 2p hole of the middle O ion via Fe3+ , with a certain probability, as depicted in Fig. 5.17d, yielding the state illustrated in Fig. 5.17e. In Fig. 5.17, the major spin direction of O1− anion is up because the itinerant electron spin is down. Therefore, the magnetic moment direction of the O1− anion is up. If the left Fe3+ is replaced by Mn3+ (3d 4 ) [or Cr2+ (3d 4 )], the magnetic moment direction of Mn3+ (or Cr2+ ) must be antiparallel to that of the right Fe3+ to perform the transition of the itinerant electron “e− ↓,” as shown in Fig. 5.17f. According to Fig. 5.17, the unsolved issues of the magnetic moment (μobs ) mentioned in Sect. 5.4.1 and the Curie temperature (T C ) for spinel ferrites MFe2 O4 may be explained. (1) When M = Fe, Co, Ni, and Cu, the cation magnetic moments

5.5 O 2p Itinerant Electron Model for Magnetic Oxides (IEO Model)

73

Fig. 5.16 a–e An itinerant electron with up-spin can transit in the given sublattice only when the magnetic moment of Mn2+ is antiparallel to that of the Mn3+ . f Transitions are prevented when the magnetic moment of Mn2+ is parallel to that of the Mn3+ . An arrow drawn on a 3d energy level represents an electron with a specific spin direction. The symbol  represents a 2p hole, which in the illustrated case represents the absence of an up-spin electron

at each sublattice, (A) or [B] sublattice, align parallel to each other because of the 3d electron number of all divalent and trivalent Fe, Co, Ni, and Cu cations, nd ≥ 5; but the magnetic moment of (A) sublattice is antiparallel to those of the [B] sublattice since the spin direction of itinerant electrons in the (A) sublattice is antiparallel to that in the [B] sublattice. Consequently, the total magnetic moment per molecule is slightly higher than that of M 2+ cation. (2) When M = Cr, due to the 3d electron numbers of Cr2+ and Cr3+ , nd = 4 and 3, respectively, the magnetic moments of Cr2+

74

5 O 2p Itinerant Electron Model for Magnetic Oxides

Fig. 5.17 a–e An itinerant electron with down-spin can transit along an ionic chain O↓↑ –Fe3+ – O↓↑ –Fe3+ –O↑∇ . f Transitions may perform when the magnetic moment of Mn3+ is antiparallel to that of the Fe3+ . An arrow drawn on a 3d energy level represents an electron with a specific spin direction. The symbol ∇ represents a 2p hole, which in the illustrated case represents the absence of a down-spin electron

and Cr3+ are antiparallel to that of the Fe cation, either at the (A) or [B] sublattice, resulting in a distinctly lower total magnetic moment per molecule than that of the Cr2+ cation. (3) When M = Mn, a few of Mn3+ (nd = 4) cations are present with magnetic moments antiparallel to those of Mn2+ (nd = 5) and Fe (nd ≥ 5) cations, either at the (A) or [B] sublattice, resulting in the total magnetic moment per molecule being slightly lower than that for the Mn2+ cation. (4) According to the IEO model

5.5 O 2p Itinerant Electron Model for Magnetic Oxides (IEO Model)

75

(iv), the antiferromagnetic composition from Mn3+ cations decreases T C for spinel ferrites. This is the reason why the T C (570 K) of MnFe2 O4 is far lower than that (T C = 860, 793, 863, and 766 K) of MFe2 O4 (M = Fe, Co, Ni, and Cu) [7, 8]. The antiferromagnetic composition of the Cr cation decreases T C of spinel ferrites Cu1-x Crx Fe2 O4 from 652 K (x = 0.1) to 420 K (x = 0.5) [44]. These explanations are further discussed in Chaps 6–9.

5.6 Relationship Between the IEO Model and the Conventional Models The views based on the IEO model are related to the conventional SE and DE interaction models. First, the IEO model accepted the view that the spin direction of an itinerant electron is constant during the transition process in the DE model, as shown in Fig. 3.8. Second, the view that the two electrons with opposite spin directions in the outer orbit of an O2− anion become the itinerant electrons of the two sublattices is similar to the antiferromagnetic ordering of MnO explained by the SE model [8], as shown in Fig. 3.7. The main characteristics of the IEO model that differ from the SE and DE models are as follows. In the SE and DE models, all oxygen ions are assumed to be O2− anions, and the itinerant electrons originate from 3d electrons of the cations in perovskite manganites. In the IEO model, the experimental result that a fraction of O1− anions exist in an oxide in addition to O2− anions is accepted, and the itinerant electrons are thought to originate from the 2p electrons of oxygen anions in a magnetic oxide. A major difference between the IEO model and the conventional models is the opposing views about the magnetic coupling between Mn3+ and Mn3+ magnetic moments. (1) For LaMnO3 , the conventional view assumes that the antiferromagnetic ordering originates from the superexchange interaction between Mn3+ and Mn3+ magnetic moments; in the IEO model, this is the antiferromagnetic coupling between Mn3+ and Mn2+ magnetic moments. (2) For La0.85 Sr0.15 MnO3 , the conventional view assumes that the ferromagnetic ordering originates from the DE interaction between Mn3+ and Mn4+ magnetic moments; in the IEO model, the ferromagnetic coupling between Mn3+ and Mn3+ magnetic moments is proposed. This will be further discussed in Chap. 9. Based on the above discussion, the IEO model can be compared to the traditional SE and DE interaction models and the explanation of the magnetic ordering in oxides by different models can be addressed, as shown in Table 5.14. This suggests that the traditional SE and DE interaction models may be improved using the IEO model.

(continued)

There is only one magnetic sublattice, the SE interaction between Mn3+ cations magnetic moments of Mn3+ (3d 4 ) cations are antiparallel to those of Mn2+ (3d 5 ) cations For the different preparation conditions, Mn cations have different average valences. The different Mn2+ /Mn3+ ratios result in different sample magnetic moments

↓↑↓↑↓↑↓↑↓

LaMnO3

↑↑↑↑↑↑↑↑↑

↓↓↓↓↓↓↓↓↓

The cation magnetic moments at (A) sites SE interaction between divalent cations are antiparallel to those at [B] sites because the spin directions of itinerant electrons are antiparallel in the two sublattices The two magnetic sublattices have the same crystal structure, and the distribution of various cations is the same or nearly the same. The total magnetic moment of the sample is always very small

MO(M = Mn, Fe, Co, Ni)

Explanation using SE and DE models

Illustration of magnetic ordering Explanation using IEO model

Materials

Table 5.14 Comparison of the explanations of magnetic ordering in typical oxides according to the IEO, SE, and DE models

76 5 O 2p Itinerant Electron Model for Magnetic Oxides

MnFe2 O4

MFe2 O4 (M = Ti, Cr)

MFe2 O4 (M = Fe, Co, Ni, Cu)

Materials

Table 5.14 (continued)

↓↓↑↓↓↑↓↓↑

↑↑↓↑↑

↓↓↑↓↓↑↓↓↑

↑↑↓↑↑

↓↓↓↓↓↓↓↓↓

↑↑↑↑↑

Explanation using SE and DE models

(continued)

The magnetic moments of Mn3+ cations (nd There is no satisfactory explanation to be found = 4) are antiparallel to those of Mn2+ and Fe cations (nd ≥ 5) in each sublattice

The magnetic moments (nd ≤4) of Ti and Cr There is no satisfactory explanation to be cations are antiparallel to those (nd ≥ 5) of found Fe cations in each sublattice

3d electron number, nd ≥ 5, of all cations. (1) SE interaction, the cation magnetic All cation magnetic moments at (A) or [B] moments are parallel when sites are parallel at low temperatures, and cation-O-cation bond angle, θ, is close the cation magnetic moments at (A) sites to 90°,while the cation magnetic are antiparallel to those at [B] sites. moment is antiparallel when θ > Because the spin directions of itinerant 120°[8] electrons are antiparallel to each other in the (2) The cation magnetic moments are parallel in [B] sublattice due to DE two sublattices interaction. The cation magnetic The two magnetic sublattices have different moments in the (A) sublattice are crystal structures, resulting in a larger total antiparallel to those in [B] sublattice magnetic moment due to SE interaction [45, 46]

Illustration of magnetic ordering Explanation using IEO model

5.6 Relationship Between the IEO Model and the Conventional Models 77

La0.8 Sr0.2 Mn0.9 Cr0.1 O3

La0.8 Sr0.2 Mn0.9 Fe0.1 O3

↑↑↑↑↓↑↑↑↑

DE interaction between Mn3+ and Mn4+ cations The calculated magnetic moment of the sample is distinctly lower than the experimental magnetic moment

La0.6 Sr0.4 MnO3

Explanation using SE and DE models

↑↑↑↑↑↑↑↑↑

La0.85 Sr0.15 MnO3

SE interaction between Fe3+ and Mn3+

DE interaction between Cr3+ and Mn3+

The magnetic moments of Fe3+ (3d 5 ) are approximately antiparallel to those of Mn3+ (3d 4 ) The magnetic moments of Cr3+ (3d 3 ) exhibit canted ferromagnetic coupling with those of Mn3+ (3d 4 )

The magnetic moments of Mn3+ (3d 4 ) show DE interaction between Mn3+ and Mn4+ canted ferromagnetic coupling with each cations other

There is only one magnetic sublattice, all of Mn3+ (3d 4 ) cation magnetic moments are parallel to each other. No Mn4+ cations The calculated magnetic moment of the sample is very close to the experimental magnetic moment

Illustration of magnetic ordering Explanation using IEO model

Materials

Table 5.14 (continued)

78 5 O 2p Itinerant Electron Model for Magnetic Oxides

References

79

References 1. Huang K, Han RQ (1988) Solid state physics (in Chinese). Higher Education Press, Beijing 2. Phillips JC (1970) Rev Mod Phys 42:317 3. Ren H (2015) Introduction to inorganic chemistry of energetic materials (in Chinese). Beijing Institute of Technology Press, Beijing 4. Ji DH, Tang GD, Li ZZ, Hou X, Han QJ, Qi WH, Bian RR, Liu SRJ (2013) Magn Magn Mater 326:197 5. Merzbacher E (1970) Quantum mechanics (Second Edition). Wiley, New York 6. Tang GD, Hou DL, Chen W, Zhao X, Qi WH (2007) Appl Phys Lett 90:144101 7. Chen CW (1977) Magnetism and Metallurgy of soft magnetic materials. North-Holland Publishing Company, Amsterdam 8. Chikazumi S (1997) Physics of Ferromagnetism 2ed. Oxford University Press, London 9. Payne MC, Teter MP, Allen DC, Arias TA, Joannopoulos JD (1992) Rev Mod Phys 64:1045 10. Milman V, Winkler B, White JA, Packard CJ, Payne MC, Akhmatskaya EV, Nobes RH (2000) Int J Quantum Chem 77:895 11. Perdew JP, Wang Y (1992) Phys Rev B 45:13244 12. Anisimov VI, Zaanen J, Andersen OK (1991) Phys Rev B 44:943 13. Antonov VN, Harmon BN, Yaresko AN (2003) Phys Rev B 67:024414 14. Zhu L, Yao KL, Liu ZL (2006) Phys Rev B 74:035409 15. Jeng HT, Guo GY, Huang DJ (2004) Phys Rev Lett 93:156403 16. Nücker N, Fink J, Fuggle JC, Durham PJ (1988) Phys Rev B 37:5158 17. Ju HL, Sohn HC, Krishnan KM (1997) Phys Rev Lett 79:3230 18. Mizoroki T, Itou M, Taguchi Y, Iwazumi T, Sakurai Y (2011) Appl Phys Lett 98:052107 19. Grenier S, Thomas KJ, Hill JP, Staub U, Bodenthin Y, García-Fernández M, Scagnoli V, Kiryukhin V, Cheong SW, Kim BG, Tonnerre JM (2007) Phys Rev Lett 99:206403 20. Ibrahim K, Qian HJ, Wu X, Abbas MI, Wang JO, Hong CH, Su R, Zhong J, Dong YH, Wu ZY, Wei L, Xian DC, Li YX, Lapeyre GJ, Mannella N, Fadley CS, Baba Y (2004) Phys. Rev. B 70:224433 21. Papavassiliou G, Pissas M, Belesi M, Fardis M, Karayanni M, Ansermet JP, Carlier D, Dimitropoulos C, Dolinsek J (2004) Europhys Lett 68:453 22. Alexandrov AS, Bratkovsky AM (1999) Phys Rev Lett 82:141 23. Alexandrov AS, Bratkovsky AM, Kabanov VV (2006) Phys Rev Lett 96:117003 24. Egerton RF (1986) Electron energy loss spectroscopy in the electron microscope. Plenum Press, New York 25. Urushibara A, Moritomo Y, Arima T, Asamitsu A, Kido G, Tokura Y (1995) Phys Rev B 51:14103 26. Cohen RE (1992) Nature 358:136 27. Cohen RE, Krakauer H (1990) Phys. Rev. B 42:6416 28. Dupin JC, Gonbeau D, Vinatier P, Levasseur A (2000) Phys Chem Chem Phys 2:1319 29. Wu LQ, Li YC, Li SQ, Li ZZ, Tang GD, Qi WH, Xue LC, Ge XS, Ding LL (2015) AIP Adv 5:097210 30. Guo YY, Kuo CK, Nicholson PS (1999) Solid State Ionics 123:225 31. Reddy RR, Gopal KR, Nazeer Ahammed Y, Narasimhulu K, Siva Sankar Reddy L, Krishna Reddy CV (2005) Solid State Ionics 176:401 32. Wu LQ, Li SQ, Li YC, Li ZZ, Tang GD, Qi WH, Xue LC, Ding LL, Ge XS (2016) Appl Phys Lett 108:021905 33. Wu LQ (2016) Study of oxygen ionic valence and its influence on the magnetic property of the perovskite manganites La1-x Srx MnO3 (Master’s thesis). Hebei Normal University, Shijiazhuang 34. Steinberger R, Duchoslav J, Arndt M, Stifter D (2014) Corros Sci 82:154 35. Guo YQ, Zhu MG, Li W, Roy S, Ali N, Wappling R (2004) J Magn Magn Mater 279:246 36. Mahendiran R, Tiwary SK, Raychaudhuri AK, Ramakrishnan TV (1996) Phys Rev B 53:3348

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37. Bogle KA, Bachhav MN, Deo MS, Valanoor N, Ogale SB (2009) Appl Phys Lett 95:203502 38. Zhang P, Gao CX, Lv FZ, Wei YP, Dong CH, Jia CL, Liu QF, Xue DS (2014) Appl Phys Lett 105:152904 39. Xu J, Ma L, Li ZZ, Lang LL, Qi WH, Tang GD, Wu LQ, Xue LC, Wu GH (2015) Phys Status Solidi B 252:2820 40. Tang GD, Li ZZ, Ma L, Qi WH, Wu LQ, Ge XS, Wu GH, Hu FX (2018) Phys Rep 758:1 41. Liu SR, Ji DH, Xu J, Li ZZ, Tang GD, Bian RR, Qi WH, Shang ZF, Zhang XY (2013) J Alloy Compd 581:616 42. Johnson PD (1997) Rep Prog Phys 60:1217 43. Kisker E, Schroder K, Campagna M, Gudat W (1984) Phys Rev Lett 52:2285 44. Ravinder D, Sathi Reddy K, Mahesh P, Bhaskar Rao T, Venudhar YC (2004) J Alloy Compd 370:L17 45. McQueeney RJ, Yethiraj M, Chang S, Montfrooij W, Perring TG, Honig JM, Metcalf P (2007) Phys Rev Lett 99:246401 46. Moyer JA, Vaz CAF, Arena DA, Kumah D, Negusse E, Henrich VE (2011) Phys Rev B 84:054447

Chapter 6

Magnetic Ordering of Typical Spinel Ferrites

In Sect. 4.1, we discussed the crystal structure of (A)[B]2 O4 spinel ferrites. Moreover, we indicated the difficulties related to explaining magnetic ordering for these materials using conventional magnetic ordering models, such as for the ratio of Cr or Mn cations at the (A) sites to those at the [B] sites in spinel ferrites. As mentioned in Chap. 5, to solve these difficulties, our group [1, 2] proposed an O 2p itinerant electron model for magnetic oxides (the IEO model) on the basis of the experimental results that O1− ions and O2− ions may coexist in an oxide and confirmed the results via EELS, XPS, and NMR. In this chapter, we introduce the investigations for the typical spinel ferrites containing Cr and Mn cations using the IEO model, wherein the cation magnetic moments are parallel or antiparallel to each other (no canting angle) at low temperatures (we discuss only the low-temperature state unless stated otherwise).

6.1 Method Fitting Magnetic Moments of Typical Spinel Ferrites As mentioned in Sect. 4.1, for (A)[B]2 O4 spinel ferrites MFe2 O4 , when M = Fe, Co, Ni, and Cu, the observed values of average molecular magnetic moments, μobs , is slightly higher than their calculated values, μcal , by the conventional model. When M = Mn, μobs is slightly lower than μcal ; when M = Cr, μobs is only half of μcal . Thus, the reasons for the occurrence of such differences have become essential for improving our understanding of the magnetic structures. These difficulties have been solved by the IEO model [1, 2]: when M = Fe, Co, Ni, and Cu, the 3d electron number of all cations nd ≥ 5 and the magnetic moments of all cations in a given sublattice align parallel to each other; when M = Mn, nd = 4 for Mn3+ (3d 4 ), nd = 5 for Mn2+ (3d 5 ), the magnetic moments of Mn3+ cations align antiparallel to those of Fe cations, and

© Science Press 2021 G.-D. Tang, New Itinerant Electron Models of Magnetic Materials, https://doi.org/10.1007/978-981-16-1271-8_6

81

82

6 Magnetic Ordering of Typical Spinel Ferrites

the magnetic moments of Mn2+ cations align parallel to those of Fe cations; when M = Cr, nd = 4 and 3, respectively, for Cr2+ (3d 4 ) and Cr3+ (3d 3 ) cations and the magnetic moments of all Cr cations align antiparallel to those of Fe cations. Using the IEO model and the quantum-mechanical potential barrier model [3] to fit the observed sample magnetic moments, our group reported several series of investigations for the cation distributions of spinel ferrites containing Cr or Mn ions [1, 2, 4–12]. In this section, as an example [5], by fitting the magnetic moment of spinel ferrites Ni0.7 Fe2.3 O4 doped Cr and Co, we introduce the application of the IEO model. Ferrite samples, CoNi, CrNi, and CrFe, with nominal compositions Cox Ni0.7−x Fe2.3 O4 , Crx Ni0.7−x Fe2.3 O4 , and Crx Ni0.7 Fe2.3−x O4 were prepared via the chemical co-precipitation method. The samples had a single-phase cubic spinel structure. The observed average molecular magnetic moments (μobs ) increased with increasing Co ratio, but μobs decreased with increasing Cr ratio. Based on the IEO model [1, 2] and a quantum-mechanical potential barrier model [3], the cation distributions of the three series of samples were estimated by fitting the dependencies on the doping ratio x of μobs , which was measured at 10 K. The obtained Cr cation distributions at (A) and [B] sites are very close to those obtained elsewhere using neutron diffraction [13].

6.1.1 X-ray Diffraction Analysis By analyzing the XRD data [14], the crystal structure and the lattice parameters of all the CoNi, CrNi, and CrFe samples were obtained. All samples had a single-phase cubic spinel structure; the crystal lattice constant, a, the bond lengths from O anions to the cations at (A) and [B] sites, d AO and d BO , respectively, and the bond length from the cations at (A) sites to those at [B] sites, d AB , are listed in Table 6.1. For all three series of samples, from Table 6.1, we can conclude that the observed average √ values of d AO and d BO are 1.05 and 0.98 times the ideal values, √ 3a/8 and a/4, respectively. However, the value of d AB is equal to its ideal value, 11a/8, that is, dAO

√ √ 3 11 1 a dBO = 0.98 × a dAB = a. = 1.05 × 8 4 8

(6.1)

Figure 6.1 shows the dependence of the crystal lattice constant a on the doping ratio x for the three series of samples: the a of the CrFe samples decreases as x increases, while the a of CrNi and CoNi samples increases with increasing x. The variation in the lattice constant is effected by the cation radii [15] and the cohesive energies of the samples. To study the intrinsic magnetic property of a material, the crystallite sizes need to be larger than or close to 100 nm to lower the surface effects. The volume-averaged crystallite sizes of all samples were estimated using the Scherrer equation [16]:

6.1 Method Fitting Magnetic Moments of Typical Spinel Ferrites

83

Table 6.1 Rietveld fitting results for the XRD patterns of the samples, where a is the crystal lattice constant; d AO and d BO are the distances from the O anion to the cations at the (A) and [B] sites, respectively, and d AB is the distance from the cations at the (A) sites to those at the [B] sites [5] x

a (Å)

d AO (Å)

d BO (Å)

d AB (Å)

1.894

2.038

3.461

Cox Ni0.7−x Fe2.3 O4 0.00

8.348

0.10

8.355

1.896

2.040

3.464

0.20

8.359

1.897

2.041

3.465

0.30

8.365

1.898

2.042

3.468

1.894

2.038

3.461

Crx Ni0.7−x Fe2.3 O4 0.00

8.349

0.15

8.357

1.896

2.040

3.465

0.25

8.359

1.897

2.041

3.466

0.30

8.362

1.897

2.042

3.467

Crx Ni0.7 Fe2.3−x O4 0.00

8.352

1.895

2.039

3.462

0.05

8.349

1.894

2.038

3.461

0.10

8.347

1.894

2.038

3.461

0.15

8.347

1.894

2.038

3.461

0.20

8.345

1.894

2.037

3.460

0.25

8.343

1.893

2.037

3.459

0.30

8.341

1.893

2.036

3.458

Fig. 6.1 Dependence of the crystal lattice constant a on the doping ratio x for the three series of samples [5]

84

6 Magnetic Ordering of Typical Spinel Ferrites

D=

Kλ , B cos θ

(6.2)

where K represents the Scherrer constant, B represents the full width at the halfmaximum height of the XRD peak, and θ represents the diffraction angle. This equation is suitable only when crystallite sizes are smaller than 100 nm. The volume-averaged crystallite sizes of all CoNi, CrNi, and CrFe series of samples were larger than 100 nm; thus, the surface effects were very weak in all the samples.

6.1.2 Magnetic Property Measurements The magnetic hysteresis loops of the CoNi, CrNi, and CrFe series of samples at 10 K are shown in Figs. 6.2, 6.3, and 6.4, respectively. The specific saturation magnetization of all samples at 10 and 300 K, σ S−10 K and σ S−300 K , respectively, and the magnetic moments, μobs , calculated using σ S−10 K are listed in Table 6.2. The dependencies of μobs on the doping ratio x for three series of samples are shown (points) in Fig. 6.5, wherein the curves represent the fitted results, and the fitting method is provided in the following subsection. Figure 6.5 shows that the μobs values of the CoNi samples increase with increasing x. However, the μobs values of the CrNi and CrFe samples decrease with increasing x. Fig. 6.2 Magnetic hysteresis loops for Cox Ni0.7−x Fe2.3 O4 samples measured at 10 K [5]

6.1 Method Fitting Magnetic Moments of Typical Spinel Ferrites

85

Fig. 6.3 Magnetic hysteresis loops for Crx Ni0.7−x Fe2.3 O4 samples measured at 10 K [5]

Fig. 6.4 Magnetic hysteresis loops for Crx Ni0.7 Fe2.3−x O4 samples measured at 10 K [5]

6.1.3 Primary Factors that Affect Cation Distributions Based on the IEO model and the quantum-mechanical potential barrier model [1–4], the observed dependences of the magnetic moments μobs on the doping ratio x were fitted for all samples; the cation distributions of the samples were obtained from the fitting process. In the fitting process, the following factors were considered. Factor 1: Ionization energy and the distance between cation and anion Since the electron binding energy of the cation in the materials increases as the ionization energy of the free atom increases (see Fig. 2.2), the average valence of cations may be investigated using the ionization energies by the quantum-mechanical

86

6 Magnetic Ordering of Typical Spinel Ferrites

Table 6.2 Specific saturation magnetization for the three series of samples measured at 10 K (σ S−10 K ) and 300 K (σ S−300 K ). μobs is the observed magnetic moment per molecule for the three series of samples at 10 K [5] x

σ s −10 K (Am2 /kg)

σ s−300 K (Am2 /kg)

μobs (μB /formula)

Cox Ni0.7−x Fe2.3 O4 0.00

65.59

60.74

2.743

0.10

70.86

65.95

2.964

0.20

72.68

68.59

3.040

0.30

76.51

72.10

3.200

Crx Ni0.7−x Fe2.3 O4 0.00

67.40

62.79

2.818

0.15

66.25

60.02

2.758

0.25

64.98

58.00

2.698

0.30

62.99

57.11

2.611

Crx Ni0.7 Fe2.3−x O4 0.00

66.28

61.64

2.772

0.05

63.59

59.79

2.657

0.10

60.33

56.96

2.519

0.15

56.87

53.67

2.372

0.20

55.99

51.09

2.334

0.25

55.07

49.19

2.293

0.30

52.66

46.82

2.191

Fig. 6.5 Observed values μobs (points) and fitted values μcal (curves) of the sample magnetic moments as a function of doping ratio x for the three series of samples [5]

6.1 Method Fitting Magnetic Moments of Typical Spinel Ferrites

87

potential barrier model mentioned in Sect. 5.2.1. The ratio (R) of different cations is related to the probability of their last ionized electrons transmitting through the potential barrier and is given as R=

   VD PC 1/2 1/2 , = exp 10.24 r D VD − cv rC VC PD VC

(6.3)

where the units of length and energy are nanometers (nm) and electron volts (eV), respectively; PC (PD ) is the probability of the last ionized electron of C (D) cations transmitting to the anions through a potential barrier of height V C (V D ) and width r C (r D ). In addition, V C (V D ) is the ionization energy of the last ionized electron of cations C (D), and r C (r D ) is the distance from cations C (D) to the anion. In this section, as an approximation, parameter cv is assumed to be 1.0. Factor 2: Pauli repulsion energy of the electron cloud between adjacent cations and anions This is considered by the effective ionic radius [15]: smaller ions should be present at the sites with smaller available space in the lattice. For the spinel ferrites, the volume of an (A) site is smaller than that of a [B] site (see Sect. 4.1.1). Factor 3: Tendency toward charge density balance In the thermal treatment process of a spinel ferrite sample, a tendency toward charge density balance (see Sect. 4.1.1) forces some divalent cations (with large effective ionic radius) to enter the (A) sites (with small available space) from the [B] sites (with large available space), jumping an equivalent potential barrier V BA , because cations at the (A) and [B] sites have four and six adjacent oxygen ions, respectively. V BA is affected by the ionization energy, ionic radius, and the thermal treatment temperature. We suppose that VBA (Ni2+ )=

VBA (M 2+ )V (Ni3+ )r (Ni2+ ) , V (M 3+ )r (M 2+ )

(6.4)

VBA (Fe2+ )=

VBA (M 2+ )V (Fe3+ )r (Fe2+ ) , V (M 3+ )r (M 2+ )

(6.5)

where r(X 2+ ) (X = M, Ni, and Fe, M = Cr and Co) is the effective radii of the X 2+ cation with coordination number 6 and V (X 3+ ) is the third ionization energies of the X cation, as shown in Appendix A. Factor 4: Ionicity As mentioned in Sects. 5.1–5.4, negative divalent and monovalent oxygen anions coexist in an oxide. Thus, the average valence of the (A)[B]2 O4 spinel ferrites and

88

6 Magnetic Ordering of Typical Spinel Ferrites

their total number (N 3 ) of trivalent cations are less than the values (8 and 2, respectively) in the traditional view. In the following discussion, the ionicity values of various cations in (A)[B]2 O4 spinel ferrites are given by the method in Sect. 5.2.4. Factor 5: IEO model According to the IEO model in Sect. 5.5, the magnetic moment directions of Cr3+ (3d 3 ) and Cr2+ (3d 4 ) are antiparallel to those of the divalent and trivalent Fe, Co, and Ni cations (3d electron numbers are more than or equal to 5) either at the (A) sites or [B] sites.

6.1.4 Fitting the Magnetic Moments of the Samples The chemical formulas for the CoNi, CrNi, and CrFe spinel ferrites with the (A)[B]2 O4 structure may be described as M x1 Nix2 Fe3−x1−x2 O4 ; then, the cation distributions can be written as 3+ 3+ 2+ 2+ 2+ Ni y2 Fe3+ (M y1 y3 M y4 Ni y5 Fe y6 ) 2+ 2+ 3+ 3+ 3+ [Mx1−y1−y4−z1 Ni2+ x2−y2−y5−z2 Fe3−x1−x2−y3−y6−z3 Mz1 Niz2 Fez3 ]O4 .

(6.6)

From Eq. (6.6), we have y1 + y2 + y3 + y4 + y5 + y6 = 1,

(6.7)

y1 + y2 + y3 + z 1 + z 2 + z 3 = N3 .

(6.8)

Here, N 3 is the average number of all trivalent cations per molecule. 8 N3 = [ f M x1 + f Ni x2 + f Fe (3.0 − x1 − x2 )] − 6.0, 3

(6.9)

where f Fe , f M (f Cr , f Co ), and f Ni are the ionicities of the Fe, Cr (Co), and Ni ions. Obviously, if the ionicities of all cations are equal to 1.0; then, N 3 = 2.0 from Eq. (6.9), the total valence of the cations is equal to 8.0, and all oxygen ions are negative divalent, i.e., the traditional valence value. Here, the ionicity of Cr, Co, Fe, and Ni were calculated using Table 5.3: f Cr = 0.8726, f Co = 0.8314, f Fe = 0.8790, and f Ni = 0.8129. From Eq. (6.6), we have RA1

x1 y1 = , 3 − x1 − x2 y3

x2 x1 y2 y4 = , RA4 = , 3 − x1 − x2 y3 3 − x1 − x2 y3 x2 y5 y6 RA5 = , RA6 = (6.10) 3 − x1 − x2 y3 y3 RA2

6.1 Method Fitting Magnetic Moments of Typical Spinel Ferrites

RB1

x1 − y1 − y4 z1 = , 3 − x1 − x2 − y3 − y6 z3

RB2

89

x2 − y2 − y5 z2 = , 3 − x1 − x2 − y3 − y6 z3

(6.11)

where RA1 , RA2 , RA4 , RA5 , and RA6 are the probability ratios of M 3+ (Cr3+ or Co3+ ), Ni3+ , M 2+ (Cr2+ or Co2+ ), Ni2+ , and Fe2+ , respectively, taken with respect to the Fe3+ ions at the (A) sites, and RB1 and RB2 are the probability ratios of the M 3+ (Cr3+ or Co3+ ) and Ni3+ , respectively, with respect to the Fe3+ ions at the [B] sites. From Eqs. (6.7) and (6.10), we get y3 =

3 − x1 − x2 . (RA1 + RA4 )x1 + (RA2 + RA5 )x2 + (1 + RA6 )(3 − x1 − x2 )

(6.12)

Equations (6.8) and (6.11) yield

z3 =

  N3 − 1 + RA1 3−xx11−x2 + RA2 3−xx12−x2 y3 −y1 −y4 −y2 −y5 1 + RB1 3−xx11−x + RB2 3−xx12−x 2 −y3 −y6 2 −y3 −y6

.

(6.13)

Referring to the quantum-mechanical potential barrier model, Eq. (6.3), the ratios RA1 , RA2 , RA4 , RA5 , RA6 , RB1 , and RB2 can be given as RA1 =

   V (Fe3+ ) P(M 3+ ) exp 10.24dAO V (Fe3+ )1/2 − V (M 3+ )1/2 , (6.14) = 3+ 3+ V (M ) P(Fe ) P(Ni3+ ) P(Fe3+ )    V (Fe3+ ) exp 10.24dAO V (Fe3+ )1/2 − V (Ni3+ )1/2 , = 3+ V (Ni )

RA2 =

RA4 = =

(6.15)

P(M 2+ ) P(Fe3+ )



 V (Fe3+ ) 3+ )1/2 − d V (M 2+ )1/2 − d V (M 2+ )1/2 , V (Fe exp 10.24 d AO AO AB BA V (M 2+ )

(6.16) RA5 = =

P(Ni2+ ) P(Fe3+ )



 3+ )1/2 − d V (Ni2+ )1/2 − d V (Ni2+ )1/2 , exp 10.24 d V (Fe AO AO AB BA V (Ni2+ ) V (Fe3+ )

(6.17)

90

6 Magnetic Ordering of Typical Spinel Ferrites

RA6 = =

P(Fe2+ ) P(Fe3+ ) V (Fe3+ ) V (Fe2+ )



 exp 10.24 dAO V (Fe3+ )1/2 − dAO V (Fe2+ )1/2 − dAB VBA (Fe2+ )1/2 ,

(6.18) RB1 =

   V (Fe3+ ) P(M 3+ ) 3+ 1/2 3+ 1/2 exp 10.24d = V (Fe , (6.19) ) − V (M ) BO 3+ V (M 3+ ) P(Fe )

RB2 =

   V (Fe3+ ) P(Ni3+ ) = exp 10.24dBO V (Fe3+ )1/2 − V (Ni3+ )1/2 , (6.20) 3+ 3+ P(Fe ) V (Ni )

where M = Cr or Co and V (X N+ ) (X = M, Ni or Fe, N = 2 or 3) represents the Nth ionization energies of X cation, whose values may be found in Appendix A. Similarly, V BA (X 2+ ) represents the heights of the equivalent potential barriers, all with width d AB , that must be jumped by X 2+ ions as they move from the [B] sites to the (A) sites during thermal treatment processes of the samples. The values of d AO , d BO , and d AB were measured and are listed in Table 6.1. As mentioned above in Factor 5, the magnetic moment direction of CrN+ (N = 2 or 3) is antiparallel to those of other cations in the same sublattice. Therefore, the magnetic moments of M 2+ , M 3+ , Ni2+ , Ni3+ , Fe2+ , and Fe3+ ions are m2 , m3 , 2, 3, 4, and 5 μB , respectively, where for M = Cr, m2 = −4 μB and m3 = −3 μB , and for M = Co, m2 = 3 μB and m3 = 4 μB . Thus, we can calculate the average magnetic moment per molecule, μcal , in M x1 Nix2 Fe3−x1−x2 O4 using Eq. (6.6), ⎫ ⎪ μcal = μBT − μAT ⎪ ⎪ ⎪ ⎪ μAT = m 3 y1 + 3y2 + 5y3 + m 2 y4 + 2y5 + 4y6 , ⎪ ⎪ ⎬ μB1 = m 2 (x1 − y1 − y4 − z 1 ) + m 3 z 1 , , ⎪ μB2 = 2(x2 − y2 − y5 − z 2 ) + 3z 2 = 2(x2 − y2 − y5 ) + z 2 , ⎪ ⎪ ⎪ μB3 = 4(3 − x1 − x2 − y3 − y6 − z 3 ) + 5z 3 = 4(3 − x1 − x2 − y3 − y6 ) + z 3 , ⎪ ⎪ ⎪ ⎭ μ BT = μ B1 + μ B2 + μ B3 , (6.21) where μAT and μBT are the magnetic moments of the (A) and [B] sublattices, respectively, and μB1 , μB2 , and μB3 are the magnetic moments of Cr (Co), Ni, and Fe ions at [B] sites, respectively. Note that Eq. (6.10) contains five equations and Eq. (6.11) contains two equations; in total, there are 20 independent equations, including Eqs. (6.4), (6.5), (6.7)–(6.11), and (6.14)–(6.21), and 21 parameters, including y1 –y6 , z1 –z3 , N 3 , RA1 , RA2 , RA4 , RA5 , RA6 , RB1 , and RB2 , V BA (M 2+ ), V BA (Ni2+ ), V BA (Fe2+ ), and μcal for each value of the doping ratio x. Therefore, for a given sample, we need to obtain a value for at least one independent parameter to fit the observed magnetic moment μobs .

6.1 Method Fitting Magnetic Moments of Typical Spinel Ferrites

91

Fig. 6.6 Dependence of the equivalent potential barrier V BA (M 2+ ) on the doping ratio x for the three series of samples, where M are Cr, Cr, and Co [5]

Here, we assume that V BA (X 2+ ) linearly varies with the doping ratio. Then, we have only two independent fitting parameters for each series of samples, as shown in Fig. 6.6. Using the two values of V BA (M 2+ ) (M = Co, Cr, Cr) for each series as the fitting parameters, the dependence of the magnetic moments (μobs ) at 10 K on the doping ratio x for the three series of samples were fitted. The obtained data are shown in Tables 6.3, 6.4, and 6.5 for CoNi, CrNi, and CrFe series samples, respectively. For the three series of samples, the dependences on doping ratio x of fitted values μcal (curve) and observed values μobs (points) of the magnetic moments are shown in Fig. 6.5. The figure shows that the μcal curves are very close to the μobs values, suggesting that the assumption of the magnetic moments of Cr cations being antiparallel to those of Fe, Co, and Ni cations in a given sublattice is reasonable. Figures 6.7, 6.8, and 6.9 show the calculated cation distributions of CoNi, CrNi, and CrFe series of samples, respectively.

6.1.5 Discussion on Cation Distributions The cation distributions of CoNi, CrNi, and CrFe series of samples shown in Tables 6.3, 6.4, and 6.5 and in Figs. 6.7, 6.8, and 6.9, respectively, are discussed as follows: (1)

The Cr cation ratios at the (A) sites of the samples calculated by us are close to those obtained using neutron diffraction.

Figure 6.10 shows the calculated ratios of Cr cations (sum of Cr2+ and Cr3+ ) at the (A) sites in CrNi and CrFe as well as the experimental results obtained by Ghatage et al. [13] using neutron diffraction from NiCrx Fe2−x O4 . The figure shows that the calculated results are close to the neutron diffraction experimental results, indicating that the Cr cation distributions in Tables 6.4, and 6.5 are reasonable.

92

6 Magnetic Ordering of Typical Spinel Ferrites

Table 6.3 Cation distribution of Cox Ni0.7−x Fe2.3 O4 , obtained by fitting the dependence on x of the magnetic moments in the samples. V BA (Co2+ ), V BA (Ni2+ ), and V BA (Fe2+ ) are the heights of the equivalent potential barriers that must be jumped by the Co2+ , Ni2+ , and Fe2+ ions when moving from the [B] sites to the (A) sites during the thermal treatment of the samples; N 3 is the total number of trivalent cations per molecule. μcal and μobs are the fitted and observed values of the sample magnetic moments at 10 K [5] x

0.0

0.1

0.2

0.3

N3

0.9086

0.9137

0.9186

0.9233

V BA (Co2+ ) (eV)

1.3720

1.2950

1.2180

1.1410

V BA (Ni2+ ) (eV)

1.3341

1.2592

1.1843

1.1094

(Fe2+ )

1.3143

1.2405

1.1667

1.0930

Co3+

0.0000

0.0126

0.0237

0.0334

Ni3+

0.0669

0.0544

0.0428

0.0321

Fe3+

0.5428

0.5150

0.4863

0.4566

Co2+

0.0000

0.0109

0.0233

0.0372

Ni2+

0.0552

0.0504

0.0447

0.0380

Fe2+

0.3351

0.3567

0.3792

0.4026

Co2+

0.0000

0.0682

0.1349

0.2000

Ni2+

0.5377

0.4574

0.3782

0.3000

Fe2+

1.1633

1.1427

1.1212

1.0987

Co3+

0.0000

0.0083

0.0180

0.0294

Ni3+

0.0401

0.0378

0.0344

0.0298

Fe3+

0.2588

0.2856

0.3134

0.3420

μcal (μB /formula)

2.778

2.916

3.057

3.201

μobs (μB /formula)

2.743

2.964

3.040

3.200

V BA

(eV)

(A) sites

[B] sites

(2)

The Cr cation distributions at the (A) and [B] sites are very close to those of Fe cations.

Using the cation distribution data in Tables 6.4 and 6.5, the percentage of Cr3+ , Cr2+ , Fe3+ , and Fe2+ ions occupied at the (A) and [B] sites in the total Cr and Fe ions in a given sample can be obtained. The results are listed in Table 6.6, which indicate that the percentages of Cr cations are close to those of Fe cations, except that the magnetic moment directions of Cr2+ and Cr3+ cations are antiparallel to those of Fe cations, since the third ionization energies (30.96 and 30.65 eV) of Cr and Fe are very close each other.

6.1 Method Fitting Magnetic Moments of Typical Spinel Ferrites

93

Table 6.4 Cation distribution of Crx Ni0.7−x Fe2.3 O4 , obtained by fitting the dependence on x of the magnetic moments in the samples. V BA (Cr2+ ), V BA (Ni2+ ), and V BA (Fe2+ ) are the heights of the equivalent potential barriers that must be jumped by the Cr2+ , Ni2+ , and Fe2+ ions when moving from the [B] sites to the (A) sites during the thermal treatment of the samples; N 3 is the total number of trivalent cations per molecule. μcal and μobs are the fitted and observed values of the sample magnetic moments at 10 K [5] x

0.00

0.15

0.25

0.30

N3

0.9086

0.9326

0.9483

0.9563

V BA (Cr2+ ) (eV)

1.3080

1.2850

1.2697

1.2620

V BA (Ni2+ ) (eV)

1.2816

1.2590

1.2440

1.2365

(Fe2+ )

1.2625

1.2403

1.2255

1.2181

Cr3+

0.0000

0.0307

0.0496

0.0587

Ni3+

0.0648

0.0486

0.0385

0.0337

Fe3+

0.5257

0.5021

0.4871

0.4798

Cr2+

0.0000

0.0260

0.0431

0.0516

Ni2+

0.0580

0.0450

0.0366

0.0324

Fe2+

0.3516

0.3476

0.3451

0.3439

Cr2+

0.0000

0.0754

0.1261

0.1515

Ni2+

0.5346

0.4207

0.3446

0.3064

Fe2+

1.1472

1.1528

1.1562

1.1578

Cr3+

0.0000

0.0179

0.0312

0.0382

Ni3+

0.0426

0.0357

0.0303

0.0274

Fe3+

0.2755

0.2976

0.3116

0.3185

μcal (μB /formula)

2.819

2.752

2.682

2.640

μobs (μB /formula)

2.818

2.758

2.698

2.611

V BA

(eV)

(A) site

[B] sites

(3)

An explanation for why the curve of the magnetic moment versus doping ratio for CoNi series samples has opposite variation tendency to those for CrNi and CrFe series samples.

Figures 6.7, 6.8, and 6.9 show that the ratio of Fe at the [B] sites is greater than those of Cr, Co, or Ni cations at the (A) and [B] sites. This results in magnetic moment directions of the samples being the same as those of Fe cations at the [B] sites for each sample in CoNi, CrNi, and CrFe series samples. For CrNi or CrFe samples, the ratio of Cr2+ (−4 µB ) at the [B] sites increases with increasing x, while the ratio of Ni2+ (2 µB ) or Fe2+ (4 µB ) at [B] sites decreases with increasing x; both of which result in the decrease of the sample magnetic moments, since the magnetic moment direction of Cr2+ cations is antiparallel to those of Ni2+ and Fe2+ cations. For CoNi samples,

94

6 Magnetic Ordering of Typical Spinel Ferrites

Table 6.5 Cation distribution of Crx Ni0.7 Fe2.3−x O4 , obtained by fitting the dependence on x of the magnetic moments in the samples. V BA (Cr2+ ), V BA (Ni2+ ), and V BA (Fe2+ ) are the heights of the equivalent potential barriers that must be jumped by the Cr2+ , Ni2+ , and Fe2+ ions when moving from the [B] sites to the (A) sites during the thermal treatment of the samples; N 3 is the total number of trivalent cations per molecule. μcal and μobs are the fitted and observed values of the sample magnetic moments at 10 K [5] x

0.00

0.05

0.10

0.15

0.20

0.25

0.30

N3

0.9087

0.9078

0.9069

0.9060

0.9052

0.9044

0.9035

V BA (Cr2+ ) (eV)

1.4204

1.4250

1.4296

1.4342

1.4388

1.4434

1.4480

V BA (Ni2+ ) (eV)

1.3917

1.3962

1.4007

1.4052

1.4097

1.4142

1.4187

(Fe2+ )

1.3710

1.3755

1.3799

1.3843

1.3888

1.3932

1.3977

Cr3+

0.0000

0.0115

0.0230

0.0345

0.0461

0.0578

0.0695

Ni3+

0.0692

0.0693

0.0695

0.0696

0.0698

0.0700

0.0701

Fe3+

0.5611

0.5501

0.5390

0.5279

0.5167

0.5055

0.4942

Cr2+

0.0000

0.0079

0.0157

0.0234

0.0310

0.0386

0.0461

Ni2+

0.0523

0.0521

0.0518

0.0516

0.0514

0.0512

0.0510

Fe2+

0.3175

0.3092

0.3011

0.2929

0.2849

0.2769

0.2690

Cr2+

0.0000

0.0258

0.0517

0.0777

0.1037

0.1297

0.1559

Ni2+

0.5411

0.5413

0.5415

0.5417

0.5419

0.5421

0.5423

Fe2+

1.1805

1.1559

1.1313

1.1066

1.0818

1.0570

1.0322

Cr3+

0.0000

0.0048

0.0096

0.0144

0.0191

0.0238

0.0285

Ni3+

0.0374

0.0373

0.0371

0.0370

0.0369

0.0367

0.0366

Fe3+

0.2410

0.2348

0.2287

0.2226

0.2166

0.2106

0.2046

μcal (μB /formula)

2.734

2.641

2.547

2.454

2.360

2.265

2.171

μobs (μB /formula)

2.772

2.657

2.519

2.372

2.334

2.293

2.191

V BA

(eV)

(A) site

[B] sites

the ratio of Co2+ (3 µB ) at [B] sites increases with increasing x, and the ratio of Ni2+ (2 µB ) decreases with increasing x, which increases the sample magnetic moment, since the magnetic moment direction of Co2+ is parallel to that of Ni2+ .

6.2 Cation Distribution Characteristics in Typical Spinel Ferrites So far, the cation distributions of several series of typical spinel ferrites have been investigated by the IEO model and the method in Sect. 6.1. Moreover, the dependencies of the observed (μobs ) and fitted (μcal ) average molecular moments at 10 K on Cr or Mn ratio are shown in Fig. 6.11 for Mnx Ni1−x Fe2 O4 [2], Crx Fe3−x O4 [4], Crx Co1−x Fe2 O4 [6], Crx Ni1−x Fe2 O4 [7], and Cux1 Crx2 Fe3−x1−x2 O4 (0.0 ≤ x 1 ≤ 0.284,

6.2 Cation Distribution Characteristics in Typical Spinel Ferrites

95

Fig. 6.7 For per molecule of Cox Ni0.7−x Fe2.3 O4 samples, Co (a), Ni (b), and Fe (c) cation contents and the total content (d) of Co, Ni, and Fe at the (A) and [B] sites as a function of the doping ratio x. Where the contents of Co2+ and Co3+ cations at the (A) and [B] sites are represented by Co2A , Co2B , Co3A , Co3B , the relative contents of Ni and Fe cations are represented by Ni2A , Ni2B , Ni3A , Ni3B , Fe2A , Fe2B , Fe3A , Fe3B . The content sum of divalent and trivalent Co, Ni, and Fe at the (A) and [B] sites is represented by CoA , CoB , NiA , NiB , FeA , FeB

1.04 ≥ x 2 ≥ 0.656) [8]. The figure indicates that the fitted results (curves) are very close to the observed values (points) for each series samples, regardless of whether the samples contain Cr or Mn. Notably, only two fitting parameters are needed for a series of samples, suggesting that the IEO model as well as the fitting method is reasonable. Table 6.7 lists the fitted result data and the related parameters for MFe2 O4 (M = Cr, Mn, Fe, Co, or Ni). Here, r 2 (M 2+ ) represents the effective radius of the divalent M cations with a coordination number of 6 [15]; V (M 2+ ) and V (M 3+ ) represent the second and third ionization energies, respectively; μm2 and μm3 represent the magnetic moments of the M 2+ and M 3+ cations, respectively, where the negative signs in the magnetic moments values represent the fact that their magnetic moments are antiparallel to those of other cations. Furthermore, d AO , d BO , and d AB represent the lengths of the A–O, B–O, and A–B bonds, respectively, in samples at room temperature obtained by XRD; μobs and μcal represent the observed (at 10 K) and fitted average molecular magnetic moments of the samples, respectively; and f i

96

6 Magnetic Ordering of Typical Spinel Ferrites

Fig. 6.8 For per molecule of Crx Ni0.7−x Fe2.3 O4 samples, Cr (a), Ni (b), and Fe (c) cation contents and the total content (d) of Cr, Ni, and Fe at the (A) and [B] sites as a function of the doping ratio x. Where the contents of Cr2+ and Cr3+ cations at the (A) and [B] sites are represented by Cr 2A , Cr 2B , Cr 3A , Cr 3B , the relative contents of Ni and Fe cations are represented by Ni2A , Ni2B , Ni3A , Ni3B , Fe2A , Fe2B , Fe3A , Fe3B . The content sum of divalent and trivalent Cr, Ni, and Fe at the (A) and [B] sites is represented by Cr A , Cr B , NiA , NiB , FeA , FeB

and N 3 represent the ionicity of the M cations in the samples and the number of trivalent cations, respectively. T TH represents the final thermal treatment temperature employed during the sample preparation. Table 6.7 shows four interesting characteristics: (1)

(2)

As long as the magnetic moment directions of all cations obey the limits of the IEO model, the distribution patterns of cations are similar for these spinel ferrites: all (A) sites are not occupied by the divalent cations as in the normal spinel structure or by the trivalent cations in the inverse spinel structure. The percentages of M cations (including M 2+ and M 3+ ) occupying the [B] sites are in the range of 64–82%, indicating that the distributions of Cr and Mn cations are similar to those of Fe, Co, and Ni, except that the directions of the magnetic moments of Cr2+ , Cr3+ , and Mn3+ cations obey the limits of the IEO model.

6.2 Cation Distribution Characteristics in Typical Spinel Ferrites

97

Fig. 6.9 For per molecule of Crx Ni0.7 Fe2.3−x O4 samples, Cr (a), Ni (b), and Fe (c) cation contents and the total content (d) of Cr, Ni, and Fe at the (A) and [B] sites as a function of the doping ratio x. Where the contents of Cr2+ and Cr3+ cations at the (A) and [B] sites are represented by Cr 2A , Cr 2B , Cr 3A , Cr 3B , the relative contents of Ni and Fe cations are represented by Ni2A , Ni2B , Ni3A , Ni3B , Fe2A , Fe2B , Fe3A , Fe3B . The content sum of divalent and trivalent Cr, Ni, and Fe at the (A) and [B] sites is represented by Cr A , Cr B , NiA , NiB , FeA , FeB

Fig. 6.10 The Cr contents at (A) sites in the Crx Ni0.7−x Fe2.3 O4 ( ) and Crx Ni0.7 Fe2.3−x O4 ( ) samples estimated by our group and results for the NiCrx Fe2−x O4 ( ) samples from neutron diffraction reported by Ghatage et al. [13] as functions of the Cr doping ratio x

98

6 Magnetic Ordering of Typical Spinel Ferrites

Table 6.6 For the all samples in the CrNi and CrFe series of samples, the content ratios of Cr3+ , Cr2+ , and Fe3+ , Fe2+ , at the (A) sites taken with respect to total contents of Cr and Fe ions in a given sample, and the content ratios of Cr3+ , Cr2+ , and Fe3+ , Fe2+ , at the [B] sites taken with respect to total contents of Cr and Fe ions in a given sample [5] Cr3+ (%)

Fe3+ (%)

Cr2+ (%)

Fe2+ (%)

(A) sites

21.4 ± 1.8

22.8 ± 1.9

16.4 ± 1.0

14.0 ± 1.0

[B] sites

11.1 ± 1.6

12.0 ± 1.8

51.0 ± 1.0

50.8 ± 0.9

Fig. 6.11 For spinel ferrites M x N 1−x Fe2 O4 under 10 K temperature, M cation content x dependences of the observed value μobs (points) and the fitting value μcal (curves) of average molecular magnetic moments [2, 4, 6–8]

(3)

(4)

The estimated Cr cation ratio at the (A) sites taken with respect to the total Cr content for the Crx M 1−x Fe2 O4 samples (M = Fe, Co, and Ni) is close to the observed results obtained by neutron diffraction in Crx NiFe2−x O4 (0.0 ≤ x ≤ 1.0), as reported by Ghatage et al. [13] and shown in Fig. 6.10. As mentioned in Sect. 6.1, V BA (M 2+ ) is the height of the equivalent potential barrier that must be jumped by M 2+ ions as they move from [B] to (A) sites during the thermal treatment of samples. In Table 6.7, for all samples, V BA (M 2+ ) ranged between 0.81 and 1.37 eV, which is reasonable, considering the potential barrier that could be transited by M 2+ ions moving from [B] to (A) sites during the thermal treatment.

6.2 Cation Distribution Characteristics in Typical Spinel Ferrites

99

Table 6.7 Cation distributions of the ferrites MFe2 O4 (M = Cr, Mn, Fe, Co, or Ni) obtained by fitting the magnetic moments per molecule of material measured at 10 K. Here, r 2 (M 2+ ) is the effective ionic radius of M 2+ cation; V (M 2+ ) and V (M 3+ ) are the second and third cation ionization energies; μm2 and μm3 are the magnetic moments of the M 2+ and M 3+ cations; d AO , d BO , and d AB are the lengths of the A–O, B–O, and A–B bonds, respectively; μobs and μcal are the observed and fitted magnetic moments per molecule at 10 K; f i is the ionicity of the M cation; N 3 is the average number of trivalent cations per molecule; and V BA (M 2+ ) is the height of the equivalent potential barrier that must be jumped by the M 2+ ions passing from the [B] to the (A) sites when the samples were treated at high temperatures (The magnetic moment of the Fe3 O4 sample was measured at 116 K, at which the Verwey transition occurs) M r2

Cr (M 2+ )

Fe

Co

Ni

0.80

0.83

0.78

0.745

0.69

V (M 2+ ) (eV)

15.50

15.64

16.18

17.06

18.17

V (M 3+ ) (eV)

30.96

33.67

30.65

33.50

35.17

μm2 (µB )

−4

5

4

3

2

μm3 (µB )

−3

−4

5

4

3

d AO (Å)

1.938

2.014

1.883

1.936

1.892

d BO (Å)

2.031

2.037

2.062

2.029

2.036

d AB (Å)

3.480

3.505

3.481

3.476

3.457

μobs (µB /formula)

2.044

4.505

3.927

3.344

2.105

μcal (µB /formula)

1.998

4.477

4.201

3.266

2.104

Ionicity of M ion f i

0.8726

0.8293

0.8790

0.8314

0.8129

N3

1.015

0.899

1.032

0.905

0.856

(eV)

0.8617

1.129

0.815

1.2477

1.773

V BA (Fe2+ ) (eV)

0.8760

1.098

0.815

1.2390

1.784

Fe3+

0.271

0.403

0.277

0.440

0.614

Fe2+

0.368

0.319

0.390

0.318

0.206

M 3+

0.127

0.096

0.139

0.122

0.123

M 2+

0.234

0.182

0.195

0.120

0.058

Fe2+

0.932

0.961

0.923

1.017

1.087

Fe3+

0.429

0.317

0.411

0.225

0.094

M 2+

0.451

0.639

0.461

0.639

0.794

M 3+

0.188

0.083

0.205

0.118

0.025

0.639

0.722

0.666

0.757

0.819

[6]

[2]

[4]

[6]

[7]

V BA

(Å) [15]

Mn

(M 2+ )

(A) site

[B] sites

M 2+ +

M 3+

References

100

6 Magnetic Ordering of Typical Spinel Ferrites

References 1. Tang GD, Li ZZ, Ma L, Qi WH, Wu LQ, Ge XS, Wu GH, Hu FX (2018) Physics Reports 758:1 2. Xu J, Ma L, Li ZZ, Lang LL, Qi WH, Tang GD, Wu LQ, Xue LC, Wu GH (2015) Phys. Status Solidi B 252:2820 3. Tang GD, Hou DL, Chen W, Zhao X, Qi WH (2007) Appl. Phys. Lett. 90:144101 4. Tang GD, Han QJ, Xu J, Ji DH, Qi WH, Li ZZ, Shang ZF, Zhang XY (2014) Physica B 438:91 5. Xue LC, Lang LL, Xu J, Li ZZ, Qi WH, Tang GD, Wu LQ (2015) AIP Advances 5:097167 6. Shang ZF, Qi WH, Ji DH, Xu J, Tang GD, Zhang XY, Li ZZ, Lang LL (2014) Chinese Physics B 23:107503 7. Lang LL, Xu J, Qi WH, Li ZZ, Tang GD, Shang ZF, Zhang XY, Wu LQ, Xue LC (2014) J. Appl. Phys. 116:123901 8. Zhang XY, Xu J, Li ZZ, Qi WH, Tang GD, Shang ZF, Ji DH, Lang LL (2014) Physica B 446:92 9. Lang LL, Xu J, Li ZZ, Qi WH, Tang GD, Shang ZF, Zhang XY, Wu LQ, Xue LC (2015) Phys. B 462:47 10. Xu J, Ji DH, Li ZZ, Qi WH, Tang GD, Zhang XY, Shang ZF, Lang LL (2015) Physica Status Solidi B 252:411 11. Xu J, Qi WH, Ji DH, Li ZZ, Tang GD, Zhang XY, Shang ZF, Lang LL (2015) Acta Phys. Sin. 64:017501 12. Du YN, Xu J, Li ZZ, Tang GD, Qian JJ, Chen MY, Qi WH (2018) RSC Advances 8:302 13. Ghatage AK, Patil SA, Paranjpe SK (1996) Solid. State. Commun. 98:885 14. Rietveld HM (1969) J. Appl. Cryst. 2:65 15. Shannon RD (1976) Acta Crystallogr. A 32:751 16. Zhang LD, Mu JM (1994) Nano-materials (in Cninese). Liaoning Science and Technology Publishing House, Shenyang

Chapter 7

Experimental Evidences of the IEO Model Obtained from Spinel Ferrites

In Chap. 5, we introduce experimental bases of the IEO model, including ionicity studies, O 2p hole investigations by EELS and NMR, and the average valence examinations of oxygen ions by XPS. In this chapter, we introduce experimental evidences of the IEO model obtained from spinel ferrite investigations, including additional antiferromagnetic phase presented in Ti-doped spinel ferrites, the magnetic moment increase due to Cu(3d10 4s1 ) substituted for Cr(3d5 4s1 ), and the unusual infrared spectra of CrFe2 O4 .

7.1 Additional Antiferromagnetic Phase in Ti-Doped Spinel Ferrites Since free Ti atom has the valence electron state 3d 2 4s2 , many researchers have investigated spinel ferrites in which Ti cations were assumed to be Ti4+ , whose ionic magnetic moment is zero due to the 3d 0 state. However, disputes remained about the Ti cation distribution among (A) and [B] sites in (A)[B]2 O4 spinel ferrites. Jin et al. [1] prepared Fe3−x Tix O4 (0 ≤ x ≤ 0.09) films on (001)-oriented MgO substrates and concluded that all Ti4+ cations occupied (A) sites. Kale et al. [2] prepared Ni1+x Tix Fe2−2x O4 (0 ≤ x ≤ 0.7) polycrystalline samples and found that both the lattice constants and saturation magnetizations of the samples decreased with increasing x. They concluded that the percentage of Ti4+ cations occupying (A) sites increased from 5% (when x = 0.1) to 71% (when x = 0.7). Dwivedi et al. [3] prepared CoTi2x Fe2−2x O4 (x = 0.0, 0.05, or 0.1) and concluded that tetravalent Ti cations enter [B] sites. Srinivasa Rao et al. [4] prepared Ti-substituted Co ferrite with the general formula CoFe2−x Tix O4 (0 ≤ x ≤ 0.3) and found that the magnetization of the samples decreased with increasing x. In addition, they considered that all Ti cations, being tetravalent, entered at [B] sites. Srivastava et al. [5] prepared Ni0.7+x Zn0.3 Tix Fe2−2x O4 (0 ≤ x ≤ 0.08) ferrite samples and concluded © Science Press 2021 G.-D. Tang, New Itinerant Electron Models of Magnetic Materials, https://doi.org/10.1007/978-981-16-1271-8_7

101

102

7 Experimental Evidences of the IEO Model Obtained …

that all Ti4+ cations entered the lattice at [B] sites, resulting in a canted spin structure based on Mössbauer spectra analysis. Chand et al. [6] prepared spinel ferrites Ni1+x Tix Fe2−2x O4 (0 ≤ x ≤ 0.1) and studied the samples by electron paramagnetic resonance, Mössbauer spectroscopy, and magnetization measurements at various temperatures. They found that the samples have canted spin structure and that the magnetic moment decreases with increasing Ti concentration. They concluded that all Ti4+ occupy [B] sites. Kobayashi et al. [7] prepared Zn0.6−x Ni0.4+x Tix Fe2−x O4 (x = 0.0, 0.2, or 0.3) films and analyzed the photoemission and X-ray absorption spectra of the samples. They claimed that most of the Ti ions in the samples were Ti4+ cations. To explain why there are such large discrepancies among the reported Ti cation distributions, our group [8–13] investigated the magnetic properties of several series of Ti-doped spinel ferrite powder samples and found that an additional antiferromagnetic phase exists. Using the method in Sect. 6.1 to fit the sample magnetic moments at 10 K, we obtained the cation distributions. The results indicated that most Ti ions are Ti2+ cations in addition to a few Ti3+ cations, but there is no Ti4+ cation. The distributions of Ti ions at (A) and [B] sites are similar to those of Ni ions; the only difference is that the magnetic moment direction obeys the IEO model; thus, Ti cations occupied either at (A) or [B] sites couple antiferromagnetically with Fe cations, supporting the suitability of the IEO model. In addition, this proves that the average valence of spinel ferrites is distinctly lower than the value in the traditional view. We introduce these studies as follows.

7.1.1 X-ray Diffraction Spectra of the Samples Figure 7.1 shows the XRD patterns of Ni0.68−0.8x Tix Fe2.32−0.2x O4 (x = 0.0, 0.078, 0.156, 0.234, 0.312), Ni0.68+0.26x Tix Fe2.32−1.26x O4 (x = 0.00, 0.08, 0.16, 0.24), and Ni1−x Tix Fe2 O4 (x = 0.0, 0.1, 0.2, 0.3, 0.4). As in Sect. 6.1, the XRD data of the samples were fitted by the Rietveld powder diffraction profile fitting technique using the X’Pert HighScore Plus software. The Rietveld fitting results, including the crystal lattice constant a, the distances from O anions to the cations at (A) and [B] sites, d AO and d BO , and the distance from the cations at (A) sites to those at [B] sites, d AB , are listed in√ Table 7.1. For the √ cubic spinel structure, the ideal values of d AO , d BO , and d AB are 3a/8,a/4 and 11a/8, respectively. However, the observed values of d AO and d BO in Table 7.1 are 1.05 and 0.98 times the ideal values, respectively, while the value of d AB is equal to its ideal value. Figure 7.2 shows the dependencies of crystal lattice constant a on the Ti content x. It may be seen that the a value increases with increasing x for the three series of samples. The crystalline sizes were estimated using the Scherrer equation (Eq. 6.2). It was found that the volume-averaged diameters of the crystallites in all samples were >100 nm; thus, surface effects are expected to be very weak in all samples.

7.1 Additional Antiferromagnetic Phase in Ti-Doped Spinel Ferrites Fig. 7.1 XRD patterns for the three series of samples at the room temperature [8–11]

103

104

7 Experimental Evidences of the IEO Model Obtained …

Table 7.1 Rietveld fitted results for the XRD patterns of the samples obtained using the X’Pert HighScore Plus software, where a is the crystal lattice constant; d AO and d BO are the distances from the O anion to the cations at the (A) and [B] sites, respectively; d AB is the distance from the cations at the (A) to those at the [B] sites [8–11] Samples

x

a (Å)

d AO (Å)

d BO (Å)

d AB (Å)

Ni0.68 Fe2.32 O4

0.000

8.350

1.895

2.039

3.462

Ni0.618 Ti0.078 Fe2.304 O4

0.078

8.362

1.897

2.042

3.467

Ni0.555 Ti0.156 Fe2.289 O4

0.156

8.373

1.900

2.044

3.471

Ni0.493 Ti0.234 Fe2.273 O4

0.234

8.387

1.903

2.048

3.477

Ni0.430 Ti0.312 Fe2.258 O4

0.312

8.402

1.906

2.051

3.483

Ni0.68 Fe2.32 O4

0.000

8.350

1.894

2.038

3.461

Ni0.70 Ti0.08 Fe2.22 O4

0.080

8.353

1.895

2.039

3.463

Ni0.72 Ti0.16 Fe2.12 O4

0.160

8.364

1.898

2.042

3.467

Ni0.74 Ti0.24 Fe2.02 O4

0.240

8.371

1.899

2.044

3.470

NiFe2 O4

0.000

8.348

1.894

2.038

3.461

Ni0.9 Ti0.1 Fe2 O4

0.100

8.354

1.896

2.040

3.463

Ni0.8 Ti0.2 Fe2 O4

0.200

8.368

1.899

2.043

3.469

Ni0.7 Ti0.3 Fe2 O4

0.300

8.388

1.903

2.048

3.477

Ni0.6 Ti0.4 Fe2 O4

0.400

8.403

1.907

2.052

3.484

Fig. 7.2 Dependences of crystal lattice constant a on the Ti content x of the three series of samples [8–11]

7.1.2 X-ray Energy Dispersive Spectra of the Samples When a thin electron beam emits to the surface of a sample, if the energy of the electron beam is higher than the critical ionization excitation energy of the inner electrons in an element, the inner electrons are ionized, while the outer electrons transit to the inner shell and radiate the energy by X-ray. This X-ray has different wavelengths for different elements. Therefore, the incident electrons with different

7.1 Additional Antiferromagnetic Phase in Ti-Doped Spinel Ferrites

105

energies may excite the X-ray with different wavelengths. By measuring the dependence of accepted X-ray intensity on the incident electron energy, various element ratios can be obtained. and The actual compositions of Ni0.68−0.8x Tix Fe2.32−0.2x O4 Ni0.68+0.26x Tix Fe2.32−1.26x O4 samples were analyzed using the X-ray energy dispersive spectra (EDS), as shown in Figs. 7.3 and 7.4. It may be seen from Figs. 7.3b and 7.4b that the compositions obtained from EDS (points) are very close to the nominal chemical compositions (lines).

Fig. 7.3 For the series samples Ni0.68−0.8x Tix Fe2.32−0.2x O4 , a X-ray energy dispersive spectra (EDS), b dependences on Ti doping level x of the element contents (Fe, Ni, and Ti), obtained from EDS (points) and the nominal compositions (lines) [8–11]

(a)

(b)

Fig. 7.4 For the series samples Ni0.68+0.26x Tix Fe2.32−1.26x O4 , a X-ray energy dispersive spectra (EDS), b dependences on Ti doping level x of the element contents (Fe, Ni, and Ti), obtained from EDS (points) and the nominal compositions (lines) [8–11]

106

7 Experimental Evidences of the IEO Model Obtained …

7.1.3 Magnetic Measurements and Analysis of the Results Figure 7.5 shows the magnetic hysteresis loops of the three series of samples measured at 10 and 300 K. It may be seen that the specific saturation magnetizations, σ s , of the three series of samples, decrease gradually with increasing Ti doping level x at both 10 and 300 K, in a manner similar to that reported by Kale et al. [2]. Interestingly, when x < 0.3, the specific saturation magnetizations at 300 K, σ S−300 K , is lower than that at 10 K, σ S−10 K ; when x ≥ 0.3, however, σ S−300 K > σ S−10 K . The values of σ S−300 K , σ S−10 K , and the magnetic moments per molecule, μobs , calculated using σ S−10 K are listed in Table 7.2. Figure 7.6 shows the dependence of the specific magnetization and its differential, σ and dσ /dT, of three series of samples on the test temperature, T, measured from 300 to 10 K, under an applied magnetic field of 50 mT. It can be seen that the variation trend of the σ – T curves for the sample with x < 0.15 is very close to that for the sample with x = 0.00. However, when x > 0.15, there was a transition temperature, T N , at which dσ /dT = 0, and a transition temperature, T L , at which dσ /dT reached a maximum value. When T < T N , σ decreased distinctly with decreasing T. In addition, the values of T N and T L for the samples increased with increasing x, as listed in Table 7.2. Figure 7.7a and b show σ and dσ /dT as functions of T for the sample with x = 0.312 under applied magnetic fields of 50 and 10 mT. It can be seen that the values of T N and T L decrease with decreasing applied magnetic field. The values of T N and T L measured at 10 mT for temperatures of 120 and 169 K are lower than those measured at 50 mT for 156 and 263 K. The characteristics of the curves of σ versus T suggested that the additional antiferromagnetic structures arise within (A) and [B] sublattices of the traditional spinel phase when the Ti doping level x > 0.15. This antiferromagnetic structure is similar to those of the typical antiferromagnetic materials, MnO, FeO, and NiO [14–16]. This, in turn, requires that Ti cations appear as Ti3+ or Ti2+ cations with 3d1 or 3d2 electron states and magnetic moments of 1 or 2 μB , rather than as Ti4+ cations with zero magnetic moments. According to the above discussions, we assume that the magnetic moments of Ti cations are nearly antiparallel to those of Fe and Ni cations at (A) and [B] sublattices at a temperature of 10 K, and the canting angle of the magnetic moments between Ti cations and Fe (Ni) cations increases with increasing temperature. This canting angle rapidly increases when the measurement temperature approaches T L , decreasing the effect of the canted antiferromagnetic Ti cation magnetic moments on the total magnetic moment of the samples, and concomitantly, increasing the total magnetic moment of the samples. The magnetic moments of the Ti cations then translate to the paramagnetic state when T > T N . If the moment for Ti cations are parallel to those of Fe and Ni cations, the temperature dependencies of the sample magnetic moments when x ≥ 0.15 would be similar to that when x = 0.00, rather than decrease with decreasing temperature when T < T N.

7.1 Additional Antiferromagnetic Phase in Ti-Doped Spinel Ferrites

107

Fig. 7.5 Magnetic hysteresis loops for the three samples measured at 10 K and 300 K

The preference for Ti3+ or Ti2+ is related to the ionization energies of the cations. For Ti, Fe, and Ni atoms, second ionization energies are 13.58, 16.18, and 18.17 eV, and third ionization energies are 27.49, 30.65, and 35.17 eV, respectively. However, the fourth ionization energy of Ti is 43.27 eV. Therefore, there should be no Ti4+

108

7 Experimental Evidences of the IEO Model Obtained …

Table 7.2 The transition temperatures T L , T N and the specific saturation magnetization of the samples at 300 and 10 K, σ s−300 K and σ s−10 K . μobs is the magnetic moment corresponding to σ s−10 K Sample composition

x

T N (K)

T L (K)

μobs (μB )

σ s (Am2 /kg) σ s−300 K

σ s−10 K 62.22

Ni0.68 Fe2.32 O4

0.000

–

–

2.601

58.16

Ni0.618 Ti0.078 Fe2.304 O4

0.078

–

–

2.437

54.82

58.49

Ni0.555 Ti0.156 Fe2.289 O4

0.156

133

104

2.160

49.29

52.03

Ni0.493 Ti0.234 Fe2.273 O4

0.234

181

138

1.942

45.60

46.94

Ni0.430 Ti0.312 Fe2.258 O4

0.312

263

156

1.627

41.15

39.45

Ni0.68 Fe2.32 O4

0.00

–

–

2.579

57.68

61.69

Ni0.70 Ti0.08 Fe2.22 O4

0.08

–

–

2.221

49.67

53.27

Ni0.72 Ti0.16 Fe2.12 O4

0.16

151

101

1.915

43.27

46.03

Ni0.74 Ti0.24 Fe2.02 O4

0.24

230

160

1.450

34.64

34.96

NiFe2 O4

0.00

–

–

2.339

52.35

55.73

Ni0.9 Ti0.1 Fe2 O4

0.10

–

–

1.986

44.17

47.55

Ni0.8 Ti0.2 Fe2 O4

0.20

219

143

1.653

38.17

39.75

Ni0.7 Ti0.3 Fe2 O4

0.30

278

186

1.298

32.76

31.36

Ni0.6 Ti0.4 Fe2 O4

0.40

–

216

0.918

28.24

22.29

cations when there are Fe2+ and Ni2+ cations in a spinel ferrite. Thus, we assume that Ti cations in a spinel ferrite are either Ti3+ or Ti2+ .

7.1.4 Cation Distributions of the Three Series of Ti-Doped Samples By referring to the analysis method in Sect. 6.1, the cation distributions of the three series of samples were estimated by fitting the dependence of the sample magnetic moments at 10 K on the Ti doping level x [9, 10]. According to the IEO model, the magnetic moments of Ti3+ (3d1 ) and Ti2+ (3d2 ) are opposite to Fe3+ (3d5 ), Fe2+ (3d6 ), Ni3+ (3d7 ), and Ni2+ (3d8 ) in the same sublattices of (A) or [B] in the samples at 10 K, while the other factors to affect the cation distributions were taken account, including the ionization energy and the distance between cation and anion, the Pauli repulsion energy of the electron cloud between neighboring cations and anions, and the tendency toward charge density balance and the ionicity. We noticed that the second and third ionization energy of Ti, V (Ti2+ ) = 13.58 eV, V (Ti3+ ) = 27.49 eV, are distinctly lower than those of Fe and Ni, V (Fe2+ ) = 16.18 eV, V (Fe3+ ) = 30.65 eV, V (Ni2+ ) = 18.17 eV, and V (Ni3+ ) = 35.17 eV. In addition, the effective radius of Ti2+ cation (0.086 nm, coordination number 6) is distinctly larger than that of Fe2+ (0.078 nm) and Ni2+ (0.069 nm) [17]. In the process fitting

7.1 Additional Antiferromagnetic Phase in Ti-Doped Spinel Ferrites

109

Fig. 7.6 Dependences on temperature T of the specific magnetizations σ and dσ/ dT for the samples under a 50 mT applied magnetic field

110

7 Experimental Evidences of the IEO Model Obtained …

Fig. 7.7 Dependences on temperature T of a the specific magnetizations σ and b dσ/ dT for the sample Ni0.43 Ti0.312 Fe2.258 O4 under a 50 and 10 mT applied magnetic field

the sample magnetic moments, the potential barrier shape-correcting constant cv in Eq. (6.3) related to potential barriers of Ti2+ , Ti3+ ions should be different from those of Fe and Ni cations. Here, cv = 1.0 for Fe and Ni cations, while cv = 1.19 for Ti cations determined in the fitting process [9–11]. Tables 7.3, 7.4, and 7.5 list the fitting results and relative parameters used in the fitting process of the three series of samples [10, 11]. The fitted magnetic moments (curves) and the measured magnetic moments (points) at 10 K as a function of the Ti doping level (x) are shown in Fig. 7.8. It may be seen that the fitted results are very close to the observed values.

7.1 Additional Antiferromagnetic Phase in Ti-Doped Spinel Ferrites

111

Table 7.3 Fitted results for the observed magnetic moments of the samples Ni1−x Tix Fe2 O4 : the cation distributions and the fitted magnetic moments μcal of the samples. V BA (Fe2+ ), V BA (Ti2+ ), and V BA (Ni2+ ) are the heights of the equivalent potential barriers which must be jumped by the Fe2+ , Ti2+ , and Ni2+ ions from the [B] to the (A) sites during thermal treatment. N 3 is the total content of trivalent cations per molecule x

0.00

0.10

0.20

0.30

0.40

N3

0.8558

0.8981

0.9403

0.9827

1.0251

V BA (Ti2+ ) (eV)

1.3025

1.4560

1.6095

1.7630

1.9165

(Ni2+ )

(eV)

1.3982

1.5630

1.7278

1.8926

2.0574

V BA (Fe2+ ) (eV)

1.4075

1.5734

1.7393

1.9051

2.0710

Ti3+

0.0000

0.0082

0.0174

0.0277

0.0386

Ni3+

0.1072

0.1048

0.0996

0.0922

0.0828

Fe3+

0.5291

0.5754

0.6166

0.6530

0.6850

Ti2+

0.0000

0.0084

0.0144

0.0185

0.0210

Ni2+

0.0803

0.0616

0.0465

0.0345

0.0251

Fe2+

0.2835

0.2417

0.2054

0.1742

0.1476

Ti3+

0.0000

0.0030

0.0060

0.0093

0.0131

Ni3+

0.0455

0.0395

0.0350

0.0314

0.0284

Fe3+

0.1740

0.1672

0.1657

0.1691

0.1772

Ti2+

0.0000

0.0804

0.1621

0.2446

0.3273

Ni2+

0.7671

0.6941

0.6189

0.5419

0.4638

Fe2+

1.0134

1.0157

1.0124

1.0037

0.9902

μcal (μB /molecule)

2.3330

1.9860

1.6400

1.2970

0.9590

V BA

(A) sites

[B] sites

By comparing Tables 7.3, 7.4, and 7.5, it may be seen that the three series of samples have similar variation tendencies for cation distribution. As an example, the characteristics of Ni0.68−0.8x Tix Fe2.32−0.2x O4 samples are discussed as follows: (1)

The Ti2+ cations at [B] sites constitute about 80% of the total Ti content (x) for all samples.

Figure 7.9a–c shows the dependencies of the divalent and trivalent Fe, Ti, and Ni cation ratios at the (A) and [B] sites of the samples on the Ti doping level x, respectively. Figure 7.9d shows the dependencies of Fe, Ti, and Ni cation (sum of divalent and trivalent cation) ratios at the (A) and [B] sites on the Ti doping level x. It may be calculated that the percentage of the Ti2+ cations at [B] sites is about 80%, and that of the Ti3+ cations at both (A) and [B] sites are few. This result is close to the view that all Ti cations occupied [B] sites [3−5] but is different from the view that assumed all Ti cations are Ti4+ cations.

112

7 Experimental Evidences of the IEO Model Obtained …

Table 7.4 Fitted results for the observed magnetic moments of the samples Ni0.68−0.8x Tix Fe2.32−0.2x O4 : the cation distributions and the fitted magnetic moments μcal of the samples. V BA (Fe2+ ), V BA (Ti2+ ), and V BA (Ni2+ ) are the heights of the equivalent potential barriers which must be jumped by the Fe2+ , Ti2+ , and Ni2+ ions from the [B] to the (A) sites during thermal treatment. N 3 is the total content of trivalent cations per molecule x

0.0

0.078

0.156

0.234

0.312

N3

0.9121

0.9423

0.9726

1.0030

1.0331

V BA (Ti2+ ) (eV)

1.3523

1.4380

1.5237

1.6093

1.6950

(Ni2+ )

(eV)

1.4517

1.5437

1.6357

1.7276

1.8196

V BA (Fe2+ ) (eV)

1.4614

1.5539

1.6465

1.7391

1.8317

Ti3+

0.0000

0.0059

0.0123

0.0192

0.0265

Ni3+

0.0700

0.0667

0.0626

0.0578

0.0523

Fe3+

0.5898

0.6151

0.6387

0.6606

0.6807

Ti2+

0.0000

0.0063

0.0115

0.0159

0.0194

Ni2+

0.0484

0.0403

0.0332

0.0269

0.0215

Fe2+

0.2918

0.2657

0.2417

0.2196

0.1995

Ti2+

0.0000

0.0632

0.1268

0.1905

0.2544

Ni2+

0.5288

0.4803

0.4314

0.3823

0.3331

Fe2+

1.2188

1.2019

1.1828

1.1618

1.1389

Ti3+

0.0000

0.0026

0.0054

0.0084

0.0117

Ni3+

0.0327

0.0303

0.028

0.0258

0.0234

Fe3+

0.2196

0.2217

0.2256

0.2312

0.2385

μcal (μB /molecule)

2.7064

2.4377

2.1681

1.8980

1.6278

V BA

(A) sites

[B] sites

(2)

Reason for the decrease of sample magnetic moment with increasing x

It may be seen from Fig. 7.9d that the Fe ion ratio at [B] sites, FeB , is far more than the ratios of Ni and Ti ions, NiB and TiB . Thus, the magnetic moment direction of samples is the same as that of Fe ions at [B] sites. The reason for the decrease of magnetic moment with increasing x is that Ti cations (coupled antiferromagnetically with Fe) replaced Ni cations (coupled ferromagnetically with Fe): TiB increases with increasing x, and NiB decreases with increasing x. In addition, it may be seen from Fig. 7.9a that, with increasing x, Fe3+ (5 μB ) ratio increases and Fe2+ (4 μB ) ratio decreases in the (A) sublattice; thus, magnetic moment decreases with increasing x since the magnetic moments of the (A) sublattice is opposite to that of the [B] sublattice.

7.1 Additional Antiferromagnetic Phase in Ti-Doped Spinel Ferrites

113

Table 7.5 Fitted results for the observed magnetic moments of the samples Ni0.68+0.26x Tix Fe2.32−1.26x O4 : the cation distributions and the fitted magnetic moments μcal of the samples. V BA (Fe2+ ), V BA (Ti2+ ), and V BA (Ni2+ ) are the heights of the equivalent potential barriers which must be jumped by the Fe2+ , Ti2+ , and Ni2+ ions from the [B] to the (A) sites during thermal treatment. N 3 is the total content of trivalent cations per molecule x

0.0

0.08

0.16

0.24

N3

0.9121

0.9283

0.9444

0.9604

V BA (Ti2+ ) (eV)

1.4493

1.5810

1.7127

1.8443

(Ni2+ )

(eV)

1.5559

1.6972

1.8386

1.9799

V BA (Fe2+ ) (eV)

1.5662

1.7085

1.8507

1.9930

Ti3+

0.0000

0.0066

0.0141

0.0226

Ni3+

0.0735

0.0820

0.0908

0.0997

Fe3+

0.6192

0.6418

0.6586

0.6701

Ti2+

0.0000

0.0057

0.0101

0.0136

Ni2+

0.0438

0.0401

0.0366

0.0335

Fe2+

0.2635

0.2239

0.1898

0.1606

Ti2+

0.0000

0.0656

0.1317

0.1979

Ni2+

0.5342

0.5512

0.5675

0.5827

Fe2+

1.2464

1.1853

1.1200

1.0513

Ti3+

0.0000

0.0022

0.0041

0.0059

Ni3+

0.0285

0.0274

0.0268

0.0265

Fe3+

0.1909

0.1683

0.1500

0.1356

μcal (μB /molecule)

2.6359

2.2211

1.8146

1.4175

V BA

(A) sites

[B] sites

Fig. 7.8 Fitted μcal (curves) and measured values μobs (points) of the average molecular magnetic moments at 10 K vs the Ti doping level x of the three series of samples

114

7 Experimental Evidences of the IEO Model Obtained …

Fig. 7.9 Curves of Fe (a), Ti (b), Ni (c) content average values versus Ti-doped level x per molecule of the samples Ni0.68−0.8x Tix Fe2.32−0.2x O4 . Where the contents of divalent and trivalent Ti ions at the (A) and [B] sites were represented by Ti2A , Ti2B , Ti3A , Ti3B , the corresponding contents of Ni and Fe were represented by Ni2A , Ni2B , Ni3A , Ni3B , Fe2A , Fe2B , Fe3A , Fe3B . d Curves of the sum of divalent and trivalent Ti, Ni, and Fe at the (A) and [B] sites

(3)

Trivalent cation ratio per molecule of the sample

In the traditional view, the total valence of the cations in an (A)[B]2 O4 molecule is 8.0 (one divalent cation and two trivalent cations). It may be seen from Tables 7.3, 7.4, and 7.5 that the total number of the trivalent cations is close to 1.0 rather than 2.0, and other cations are divalent cations. The average valence of oxygen is close to −1.75 calculated according to the valence balance.

7.1 Additional Antiferromagnetic Phase in Ti-Doped Spinel Ferrites

115

7.1.5 Magnetic Ordering of Spinel Ferrites Tix M1−x Fe2 O4 (M = Co, Mn) Magnetic structure and cation distribution of Ti-doped spinel ferrites Tix M 1−x Fe2 O4 (M = Co, Mn, Ti-Co, Ti–Mn series samples) were studied by our group [12, 13]. XRD analyses indicated that the two series of samples possess the cubic crystal ¯ The crystal lattice constant a of structure with a single phase and space group Fd3m. Ti–Co series samples increases with increasing Ti doping level x, a of Ti–Mn series samples decreases with increasing x. The crystallite sizes of all samples are larger than 100 nm. Therefore, the effect of the crystallite surface on the sample magnetic property may be neglected. The magnetic moments of the two series of samples decrease with increasing x. The XPS analyses of the two samples (Ti0.2 Co0.8 Fe2 O4 , Ti0.4 Co0.6 Fe2 O4 ) indicated that the average valences of oxygen ions are −1.69 and − 1.72, respectively, suggesting that the total valences of the cations are also less than the values in the traditional view. Using the methods in Sect. 6.1 and 7.1.4, the magnetic moments of the two series of samples measured at 10 K were fitted, as shown in Fig. 7.10. It may be seen that the fitted values (curves) are very close to the observed values (points). In the fitting process, the magnetic moments of Ti2+ , Ti3+ , and Mn3+ are assumed to exhibit antiferromagnetic coupling with Fe cations at the given sublattice. The cation distribution states obtained in the fitting process are similar to those in Sect. 7.1.4. This further indicates that most Ti cations occupy [B] sites with divalent state and that the assumption of cation magnetic moment directions in the IEO model is reasonable. Fig. 7.10 Dependences of observed (points) and fitted (curves) average molecular magnetic moments of the samples Tix M 1−x Fe2 O4 (M = Co, Mn) at 10 K on Ti-doped level x [12, 13]

116

7 Experimental Evidences of the IEO Model Obtained …

7.2 Amplification of Spinel Ferrite Magnetic Moment Due to Cu Substituting for Cr Our group synthesized spinel ferrites with nominal composition Cux Cr1−x Fe2 O4 (0.0 ≤ x ≤ 0.4) using a chemical co-precipitation method [18, 19]. The saturation magnetization of the samples increased when Cu2+ or Cu3+ (with a magnetic moment of 1 or 2 μB of) substituted for Cr2+ or Cr3+ (with 4 or 3 μB ), which cannot be explained if the magnetic moments of Cr2+ and Cr3+ cations are assumed to be parallel to those of Fe and Cu cations. However, according to IEO model, the magnetic moments of Cr2+ and Cr3+ cations are antiparallel to Fe and Cu cation moments in spinel ferrites at the (A) or [B] sublattice. The dependence on the Cu doping level of the sample magnetic moments at 10 K was fitted successfully, supporting the accuracy of the IEO model. The experimental results and the analyses are provided in this section.

7.2.1 X-ray Energy Dispersive Spectrum Analysis Spinel ferrites with nominal composition Cux Cr1−x Fe2 O4 (0.0 ≤ x ≤ 0.4) were synthesized using a chemical co-precipitation method. The final thermal treatment was performed in a tube furnace with argon flow at 1673 K for 4 h and then cooled to room temperature in the furnace. Figure 7.11 shows X-ray EDS. Figure 7.12 shows the dependencies of the observed ratios, x 1 , x 2 , and x 3 , of Cu, Cr, and Fe in the end samples on the Cu nominal composition x, where x1 = 0.71x, x2 = 1.04 − 0.96x, x3 = 3.00 − x1 − x2 (x ≤ 0.4).

(7.1)

The observed Cu ratio, x 1 , is significantly less than the nominal composition x, while Cr and Fe ratios, x 2 and x 3 , are also different from their nominal composition. Fig. 7.11 X-ray Energy dispersive spectra of the samples with nominal composition Cux Cr1−x Fe2 O4

7.2 Amplification of Spinel Ferrite Magnetic Moment Due …

117

Fig. 7.12 Cu, Cr, and Fe contents of the samples with nominal composition Cux Cr1−x Fe2 O4 (0.0 ≤ x ≤0.4) measured by EDS versus Cu nominal content x

The reason for why the Cu ratio x 1 is less than its nominal ratio x is that some Cu cations were lost during the thermal treatment process at 1673 K. Therefore, the actual samples can be represented by Cu×1 Crx2 Fe3.0−x1−x2 O4 (0.0 ≤ x 1 ≤ 0.284, 1.04 ≥ x 2 ≥ 0.656).

7.2.2 X-ray Diffraction Analysis Figure 7.13 shows the XRD patterns of all samples. The XRD spectra indicate that ¯ The the samples had a single-phase cubic spinel structure with space group Fd3m. XRD data were fitted using the X’Pert HighScore Plus software with the Rietveld Fig. 7.13 X-ray diffraction patterns for samples Cux1 Crx2 Fe3−x1−x2 O4

118

7 Experimental Evidences of the IEO Model Obtained …

Table 7.6 Rietveld fitted results for the XRD diffraction patterns of the samples obtained using the X’Pert HighScore Plus software, where a is the lattice constant; d AO and d BO are the distances from the O anion to the cations at the (A) and [B] sites, respectively; d AB is the distance from the cations at the (A) to those at the [B] sites Sample composition

x1

x2

a (Å)

d AO (Å)

d BO (Å)

d AB (Å)

Cr1.040 Fe1.960 O4

0.000

1.040

8.393

1.890

2.057

3.480

Cu0.071 Cr0.944 Fe1.985 O4

0.071

0.944

8.390

1.889

2.056

3.478

Cu0.142 Cr0.848 Fe2.010 O4

0.142

0.848

8.388

1.889

2.056

3.477

Cu0.213 Cr0.752 Fe2.035 O4

0.213

0.752

8.391

1.889

2.057

3.479

Cu0.284 Cr0.656 Fe2.060 O4

0.284

0.656

8.395

1.890

2.058

3.480

powder diffraction profile fitting technique. The fitting results, including the crystal lattice constant a, the distances, d AO and d BO , from O anions to the cations at (A) and [B] sites, and the distance, d AB , from the cations at (A) sites to those at [B] sites, are listed in Table 7.6. It can be seen that a (8.392 ± 0.005 Å), d AO (1.890 ± 0.001 Å), d BO (2.057 ± 0.001 Å), and d AB (3.478 ± 0.001 Å) are approximately constant for various Cu doping levels. For the √ √ cubic spinel structure, the ideal values of d AO , d BO , and d AB are 3a/8, a/4, and 11a/8, respectively. However, the observed values of d AO and d BO in Table 7.6 are 1.040 and 0.980 times the ideal values, respectively, while the value of d AB is equal to its ideal value. By using the X’Pert HighScore Plus software to analyze the XRD patterns, we found that the volume average diameters of crystallites in all samples were above 100 nm so that surface effects are expected to be very weak in all samples.

7.2.3 Magnetic Measurement and Magnetic Moment Fitting Results Figure 7.14a and b shows the magnetic hysteresis loops of the samples measured at 10 and 300 K, from which the saturation magnetizations, σ s−10 K and σ s−300 K , of the samples at 10 and 300 K were obtained. The average molecular magnetic moments of the samples at 10 K, μobs , were calculated using σ s−10 K . Figure 7.15 shows the dependence of μobs (points) on the Cu ratio, x 1 , in which the curve represents the fitted magnetic moments, μcal , using the method in Sect. 6.1. In the fitting process, the magnetic moment directions of Cr2+ and Cr3+ cations are assumed to be antiparallel with those of Fe and Cu cations at the (A) or [B] sublattice according to the IEO model. The data of σ s−10 K , σ s−300 K , μobs , and μcal are listed in Table 7.7. Figure 7.15 shows that the magnetic moment gradually increases with increasing Cu doping level x 1 ; the fitted magnetic moments are close to the observed values. Various cation distributions of the fitting results are listed in Table 7.8.

7.2 Amplification of Spinel Ferrite Magnetic Moment Due … Fig. 7.14 Magnetic hysteresis loops for the samples Cux1 Crx2 Fe3−x1−x2 O4 measured at 10 K (a) and 300 K (b)

Fig. 7.15 Dependence of average molecular magnetic moments of the samples at 10 K on the Cu content x 1 , in which the points and curve represent observed and fitted values, μobs and μcal , respectively

119

x1

0.000

0.071

0.142

0.213

0.284

Sample

Cr1.040 Fe1.960 O4

Cu0.071 Cr0.944 Fe1.985 O4

Cu0.142 Cr0.848 Fe2.010 O4

Cu0.213 Cr0.752 Fe2.035 O4

Cu0.284 Cr0.656 Fe2.060 O4

σ s−300 K

46.49

44.87

37.03

30.92

25.51

(Am2 ·

kg−1 )

74.25

73.57

57.31

51.21

40.87

σ s−10 K

Specific saturation magnetizations (Am2 ·

kg−1 )

3.088

3.045

2.360

2.098

1.666

μobs (μB /molecule)

3.169

2.865

2.499

2.092

1.664

μcal (μB /molecule)

The magnetic moment measured at 10 K

Table 7.7 Specific saturation magnetizations of Cux1 Crx2 Fe3−x1−x2 O4 measured at 300 and 10 K, expressed as σs−300 K and σs−10 K , μobs and μcal are the observed and fitted magnetic moment at 10 K

120 7 Experimental Evidences of the IEO Model Obtained …

7.2 Amplification of Spinel Ferrite Magnetic Moment Due …

121

Table 7.8 Cation distributions and relative parameters of the samples Cux1 Crx2 Fe3.0−x1−x2 O4 . Here, μAT , μBT , and μcal are the fitted magnetic moments per molecule of the (A) sublattice, the [B] sublattice and the samples, respectively; V BA (Cu2+ ), V BA (Cr2+ ), and V BA (Fe2+ ) are, respectively, the height of the equivalent potential barriers which must be jumped by the Cu2+ , Cr2+ , and Fe2+ ions from a [B] site to an (A) site during thermal treatment; N 3 is the number of trivalent cations per molecule x

0.0

0.1

0.2

0.3

0.4

x1

0.000

0.071

0.142

0.213

0.284

x2

1.040

0.944

0.848

0.752

0.656

N3

1.0143

1.0007

0.9872

0.9737

0.9602

V BA (Cu2+ ) (eV)

1.1745

0.9270

0.6795

0.4320

0.1845

V BA (Cr2+ ) (eV)

0.9833

0.7761

0.5689

0.3617

0.1545

V BA (Fe2+ ) (eV)

1.0008

0.7899

0.5790

0.3681

0.1572

μAT (μB /molecule)

1.5140

1.5798

1.6472

1.7320

1.8669

μBT (μB /molecule)

3.1784

3.6713

4.1462

4.5967

5.0357

μcal (μB /molecule)

1.6644

2.0915

2.4990

2.8647

3.1688

Fe3+

0.3141

0.2923

0.2675

0.1633

0.1037

Fe2+

0.3164

0.3490

0.4004

0.5125

0.6118

Cr3+

0.1563

0.1357

0.1070

0.0579

0.0287

Cr2+

0.2133

0.2191

0.2162

0.2438

0.2256

Cu3+

0.0000

0.0018

0.0035

0.0056

0.0048

Cu2+

0.0000

0.0022

0.0054

0.0170

0.0256

Fe2+

0.9598

0.9389

0.9255

0.7889

0.7282

Fe3+

0.3698

0.3998

0.4466

0.5454

0.6263

Cr2+

0.4964

0.4509

0.3807

0.2834

0.2022

Cr3+

0.1741

0.1743

0.1661

0.1749

0.1536

Cu2+

0.0000

0.0331

0.0739

0.1845

0.2533

Cu3+

0.0000

0.0029

0.0072

0.0230

0.0363

(A) sites

[B] sites

Figure 7.16a–c shows the cation distribution curves for Fe, Cu, and Cr at the (A) and [B] sites versus the Cu ratio, x 1 . Figure 7.16d shows the total ratio of Cu, Cr, and Fe at (A) and [B] sites versus x 1 . In Fig. 7.16, Fe3A , Fe2A , Fe3B , and Fe2B represent the ratios of the trivalent and divalent Fe cations at (A) and [B] sites; Cu3A , Cu2A , Cu3B , and Cu2B represent the corresponding ratios of the trivalent and divalent Cu cations; Cr 3A , Cr 2A , Cr 3B , and Cr 2B represent the corresponding ratios of the trivalent and divalent Cr cations. A2 , B2 , A3 , and B3 represent the sum of divalent and trivalent cation ratios at (A) and [B] sites. Figure 7.16 shows that most Cu ions occupying [B] sites are Cu2+ . For the sample without Cu, most Fe and Cr cations occupying [B] sites are divalent. In the doped Cu samples, Cu2+ gradually substituted for Cr2+ at [B] sites.

122

7 Experimental Evidences of the IEO Model Obtained …

Fig. 7.16 Dependences on the Cu content x 1 of Fe (a), Cu (b), Cr (c) cation contents at the (A) and [B] sites per molecule of sample Cux1 Crx2 Fe3−x1−x2 O4 . Where the contents of divalent and trivalent Cu ions at the (A) and [B] sites were represented by Cu2A , Cu2B , Cu3A , Cu3B , the corresponding contents of Cr and Fe were represented by Cr 2A , Cr 2B , Cr 3A , Cr 3B , Fe2A , Fe2B , Fe3A , Fe3B . d Dependences on x 1 of the sum of divalent and trivalent cation contents of Cu, Cr, and Fe at the (A) and [B] sites, represented by A2 , B2 , A3 , B3

Similar to that in Sect. 6.1, only two independent fitting parameters were used in the fitting process for this series of samples: two V BA (Cu2+ ) values with x 1 = 0.071 and x 1 = 0.213, indicating that the fitting method in Sect. 6.1 is reasonable.

7.3 Unusual Infrared Spectra of Cr Ferrite According to the IEO model, the magnetic moment of Cr cation is antiparallel to Fe cations regardless at the (A) or [B] sublattice in Cr-doped spinel ferrites. This had been proved by using infrared spectra [20].

7.3 Unusual Infrared Spectra of Cr Ferrite

123

7.3.1 Infrared Spectra of Spinel Ferrites MFe2 O4 (M = Fe, Co, Ni, Cu, Cr) Infrared spectra of (A)[B]2 O4 spinel ferrite samples MFe2 O4 (M = Fe, Co, Ni, Cr) and Cu0.85 Fe2.15 O4 were measured at room temperature, and the results are shown in Fig. 7.17. Table 7.9 lists the preparation conditions and the cation distributions of the samples [19, 21–23] estimated by the IEO model. In Table 7.9, T TH represents the final thermal treatment temperature, r 2 (M 2+ ) is the effective ionic radii of M 2+ cations with coordination number 6; V (M 2+ ) and V (M 3+ ) are the second and third ionization energies of M cations, respectively; μm2 and μm3 are magnetic moments of M 2+ and M 3+ cations, respectively; d AO , d BO , and d AB are the lengths of A–O, B–O, and A–B bonds, respectively; μobs and μcal are the observed (at 10 K) and estimated magnetic moment per molecule of the sample, respectively; N 3 is the number of trivalent cations per molecule; V BA (M 2+ ) is the heights of the equivalent potential barriers, which must be jumped by the M 2+ ions from [B] to (A) sites when the samples are treated at high temperature. In Fig. 7.17, the absorption peak positions, the centers of the two absorption bands for each sample, were signed by the narrows. The higher frequency band (peak position wave number ν 1 ) corresponds to the vibrations of the tetrahedral sites (O–A–O bonds), and the lower frequency band (peak position wave number ν 2 ), to the vibrations of the octahedral sites (O–B–O bonds), all of which are similar to those reported in Refs. [24–27]. For the MFe2 O4 samples, the wave number ν 2 shifted from 378 cm−1 for M = Fe to 400 cm−1 for M = Cu. However, the ν 2 shifted from 400 cm−1 for M = Cu to 479 cm−1 for M = Cr. This phenomenon has not yet been explained by the traditional view. Fig. 7.17 Infrared spectra of spinel ferrites MFe2 O4 (M = Fe, Co, Ni, Cr) and Cu0.85 Fe2.15 O4 measured at room temperature. Where ν is the wave number of the incident light [20]

124

7 Experimental Evidences of the IEO Model Obtained …

Table 7.9 Cation distributions of the spinel ferrites samples MFe2 O4 (M = Fe, Co, Ni, Cr) and Cu0.85 Fe2.15 O4 estimated by fitted sample magnetic moments [19, 21–23]. Where T TH represents the final thermal treatment temperature, r 2 (M 2+ ) is the effective ionic radii of M 2+ cations with coordination number 6; V (M 2+ ) and V (M 3+ ) are the second and third ionization energy, respectively; μm2 and μm3 are magnetic moments of M 2+ and M 3+ cations, respectively; d AO , d BO , and d AB are the lengths of A–O, B–O, and A–B bonds, respectively; μobs and μcal are the observed (at 10 K) and fitted magnetic moment per molecule of sample, respectively; N 3 is the number of trivalent cations per molecule; V BA (M 2+ ) is the heights of the equivalent potential barriers which must be jumped by the M 2+ ions from the [B] to the (A) sites when the samples were treated at high temperature MFe2 O4

Fe3 O4

CoFe2 O4

NiFe2 O4

Cu0.85 Fe2.15 O4

CrFe2 O4

T TH (°C)

1450

1400

1400

1200

1400

r 2 (M 2+ ) (Å) [17]

0.78

0.745

0.69

0.73

0.80

V (M 2+ )

(eV)

16.18

17.06

18.17

20.29

15.50

V (M 3+ ) (eV)

30.65

33.50

35.17

36.83

30.96

μm2 (μB )

4

3

2

1

−4

μm3 (μB )

5

4

3

2

−3

d AO (Å)

1.883

1.936

1.890

1.886

1.938

d BO (Å)

2.062

2.029

2.034

2.053

2.031

d AB (Å)

3.481

3.476

3.454

3.473

3.480

μobs (μB /formula)

3.927a

3.3437

2.3426

2.109

2.0442

μcal (μB /molecule)

4.201

3.2661

2.3603

2.105

1.9982

N3

1.032

0.9051

0.8557

0.8506

1.0149

(eV)

0.815

1.2477

1.3714

0.7323

0.8607

V BA (Fe2+ ) (eV)

0.815

1.2390

1.3608

0.6240

0.8760

Fe3+

0.2772

0.4396

0.5160

0.3412

0.2708

Fe2+

0.3895

0.3181

0.2975

0.5643

0.3680

M 3+

0.1386

0.1221

0.1047

0.0402

0.1268

M 2+

0.1948

0.1201

0.0818

0.0544

0.2343

Fe2+

0.9225

1.0174

1.0003

0.8417

0.9320

Fe3+

0.4108

0.2249

0.1862

0.4028

0.4291

M 2+

0.4612

0.6393

0.7647

0.6890

0.4507

M 3+

0.2054

0.1184

0.0488

0.0665

0.1881

References

[21]

[22]

[23]

[19]

[22]

V BA (M

2+ )

(A) sites

[B] sites

a Measured

at 116 K, below which Verwey phase transition happens

7.3 Unusual Infrared Spectra of Cr Ferrite

125

Fig. 7.18 Dependence of infrared absorption peak position ν 2 of samples MFe2 O4 (M = Fe, Co, Ni, Cr) and Cu0.85 Fe2.15 O4 on the magnetic moment μm2 of M 2+ cation [20], including the data reported by Ati et al. ( ) [24], Gabal et al. ( ) [25], Wahba et al. ( ) [26]

7.3.2 Dependency of the Peak Position ν 2 on the Magnetic Moment (μM2 ) of Divalent M Cations in MFe2 O4 (M = Fe, Co, Ni, Cu, Cr) It may be seen from Table 7.9 that, for MFe2 O4 (M = Fe, Co, Ni, Cu, Cr), the ratios of M 2+ cations at the [B] sites are between 0.45 and 0.76, the ratios of Fe2+ cations at [B] sites are between 1.02 and 0.84. The sum of these M 2+ and Fe2+ cations is about 3/4 of the total number of cations at [B] sites. Due to the uniform distributions of M 2+ and Fe2+ cations, the most magnetic interactions between cations at the [B] sublattice happen between M 2+ and Fe2+ . When the magnetic moment of M 2+ varies, the magnetic interactions between M 2+ and Fe2+ must vary. This results in variation of their thermal vibration energy because the magnetic moments of M 2+ ions are, μm2 = 4, 3, 2, and 1 μB for M = Fe, Co, Ni, and Cu. Figure 7.17 indicates that the ν 2 value increases with decreasing μm2 . According to the IEO model, μm2 = −4 μB for M = Cr because Cr cations are antiferromagnetically coupled with Fe cations. Then, an interesting result was found: the ν 2 value increases almost linearly with decreasing μm2 from M = Fe to M = Cr, as shown in Fig. 7.18, which shows the data included in Fig. 7.17 and those reported by Ati et al. [24], Gabal et al. [25], and Wahba et al. [26].

7.3.3 Infrared Spectra of Co1−x Crx Fe2 O4 and CoCrx Fe2−x O4 Figures 7.19a and b show infrared spectra of Co1−x Crx Fe2 O4 (0.0 ≤ x ≤ 1.0) and CoCrx Fe2−x O4 (0.0 ≤ x ≤ 1.0) measured at room temperature [28]; the results are similar to those in Sect. 7.3.2. It may be seen from Figs. 7.19a and b that, when Crsubstituted for Co, ν 2 moved from 388 to 479 cm−1 ; when Cr substitutes for Fe, ν 2 moved from 394 to 491 cm−1 ; the ν 2 values moved for 91 and 97 cm−1 , respectively.

126

7 Experimental Evidences of the IEO Model Obtained …

Fig. 7.19 Infrared spectra of spinel ferrites Co1−x Crx Fe2 O4 (a) and CoCrx Fe2−x O4 (b) measured at room temperature [28]. Where ν is the wave number of the incident light

These infrared spectra results prove the suitability of the IEO model: in a given magnetic sublattice, all cation magnetic moments couple ferromagnetically with each other when their 3d electron number nd ≥ 5; the magnetic moments of the cations with nd ≤ 4 coupled antiferromagnetically with those when nd ≥ 5.

References 1. Jin C, Mi WB, Li P, Bai HL (2011) J Appl Phys 110:083905 2. Kale CM, Bardapurkar PP, Shukla SJ, Jadhav KM (2013) J Magn Magn Mater 331:220 3. Dwivedi GD, Joshi AG, Kevin H, Shahi P, Kumar A, Ghosh AK, Yang DD, Chatterjee S (2012) Solid State Commun 152:360 4. Srinivasa Rao K, Mahesh Kumara A, Chaitanya Varma M, Choudary GSVRK, Rao KH (2009) J Alloy Compd 488:L6

References 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

127

Srivastava RC, Khan DC, Das AR (1990) Phys Rev B 41:12514 Chand P, Srivastava RC, Upadhyay A (2008) J. Alloys Comp 460:108 Kobayashi M, Ooki Y, Takizawa M, Song GS, Fujimori A (2008) Appl Phys Lett 92:082502 Xu J, Ji DH, Li ZZ, Qi WH, Tang GD, Shang ZF, Zhang XY (2015) J Alloy Compd 619:228 Xu J, Ji DH, Li ZZ, Qi WH, Tang GD, Zhang XY, Shang ZF, Lang LL (2015) Phys Status Solidi B 252:411 Xu J, Qi WH, Ji DH, Li ZZ, Tang GD, Zhang XY, Shang ZF, Lang LL (2015) Acta Phys Sin 64:017501 Xu J (2015) Magnetic moment directions of Ti, Mn cations in spinel ferrite and O2p electron itinerant mechanism (Doctoral dissertation). Hebei Normal University, Shijiazhuang Du YN, Xu J, Li ZZ, Tang GD, Qian JJ, Chen MY, Qi WH (2018) RSC Adv 8:302 Du YN (2018) Study of magnetic structure and cation distributions of spinel ferrites Tix M 1−x Fe2 O4 (M = Co, Mn) and Mn1+x Fe2−x O4 (Master’s thesis). Hebei Normal University, Shijiazhuang Dai DS, Qian KM (1987) Ferromagnetism. Science Press, Beijing (in Chinese) Chikazumi S (1997) Physics of Ferromagnetism (2e). Oxford University Press, London Chen CW (1977) Magnetism and metallurgy of soft magnetic materials. North-Holland Publishing Company, Amsterdam Shannon RD (1976) Acta Crystallogr A 32:751 Zhang XY, Xu J, Li ZZ, Qi WH, Tang GD, Shang ZF, Ji DH, Lang LL (2014) Phys B 446:92 Zhang XY (2014) Study on magnetic structure and cation distributions in Cu–Cr ferrites Cu–Cr (Master’s thesis). Hebei Normal University, Shijiazhuang Tang GD, Shang ZF, Zhang XY, Xu J, Li ZZ, Zhen CM, Qi WH, Lang LL (2015) Phys B 463:26 Tang GD, Han QJ, Xu J, Ji DH, Qi WH, Li ZZ, Shang ZF, Zhang XY (2014) Phys B 438:91 Shang ZF, Qi WH, Ji DH, Xu J, Tang GD, Zhang XY, Li ZZ, Lang LL (2014) Chin Phys B 23:107503 Lang LL, Xu J, Qi WH, Li ZZ, Tang GD, Shang ZF, Zhang XY, Wu LQ, Xue LC (2014) J Appl Phys 116:123901 Ati AA, Othaman Z, Samavati A (2013) J Mol Struct 1052:177 Gabal MA, Angari YMA, Kadi MW (2011) Polyhedron 30:1185 Wahba AM, Mohamed MB (2014) Ceram Int 40:6127 Pervaiz E, Gul IH (2012) J Magn Magn Mater 324:3695 Shang ZF (2014) Study on magnetic properties and cation distributions in Co–Cr ferrites (Master’s thesis). Hebei Normal University, Shijiazhuang

Chapter 8

Spinel Ferrites with Canted Magnetic Coupling

In the spinel ferrites discussed in Chaps. 6 and 7, all cation magnetic moments are coupling co-linearly (parallel or antiparallel) at 10 K. In this chapter, the spinel ferrites with canted magnetic coupling is discussed, including the systems in which the Fe ratio per molecule is less than 2.0 and those doped by nonmagnetic ions.

8.1 Spinel Ferrites with Fe Ratio Being Less Than 2.0 Per Molecule Many experimental results indicated that, when the Fe ratio is less than 2.0 for an (A)[B]2 O4 spinel ferrite, canted magnetic coupling occurs, while the canting angle increases with decreasing Fe ratio [1–5]. This phenomenon may be explained by using the IEO model: It can be seen from Table 6.7 that the states of Fe ions are Fe3+ (3d 5 ) or Fe2+ (3d 6 ) in a spinel ferrite. When an itinerant electron with down-spin transits via Fe3+ (3d 5 ), it occupies the highest energy level of this Fe ion, as shown in Fig. 5.17, consuming little system energy. When an itinerant electron with downspin transits via Co2+ , Ni2+ or Cu2+ , the occupied energy level in these cations lowers successively, increasing the consumed system energy and decreasing the magnetic ordering energy (absolute value), E M . When the Fe ratio is less than 2.0, the E M value is insufficient to maintain the co-linear magnetic coupling of adjacent magnetic ions, resulting in the canted magnetic coupling to decrease the magnetic repulsion energy between adjacent magnetic ions. As an example, the magnetic property of Co1+x Fe2-x O4 (0.0 ≤ x ≤ 2.0) was investigated by our group [1, 2]. X-ray powder diffraction analyses indicated that the samples exhibited a single-phase cubic spinel structure with space group Fd3m and that the crystal lattice constant decreased linearly with increasing x, as shown in Fig. 8.1. Figure 8.2 shows the magnetic hysteresis loops of the samples measured at 10 K. Figure 8.3 shows the dependence of the specific saturation magnetization σ s on the © Science Press 2021 G.-D. Tang, New Itinerant Electron Models of Magnetic Materials, https://doi.org/10.1007/978-981-16-1271-8_8

129

130 Fig. 8.1 Dependence of the crystal lattice constant a for Co1+x Fe2-x O4 (0.0≤x≤2.0) on the x [1]

Fig. 8.2 Magnetic hysteresis loops of Co1+x Fe2-x O4 (0.0 ≤ x ≤ 2.0) at 10 K [1]

Fig. 8.3 Dependence of the specific saturation magnetization σ s on the Co increased level x of Co1+x Fe2-x O4 at 10 K [1]. The insert is the result of Co1+x Fe2-x O4 at 77 K reported by Takahashi et al. [5]

8 Spinel Ferrites with Canted Magnetic Coupling

8.1 Spinel Ferrites with Fe Ratio Being Less Than 2.0 Per Molecule

131

Co increased level x. It can be seen that σ s decreases with increasing x, when x ≤ 1.4. However, σ s increases first from 6.84 with x = 1.4 to 8.83 Am2 /kg with x = 1.6, then decreases again with increasing x. This dependence of σ s on x is very similar to that measured at 77 K by Takahashi et al. [5], as shown in the inset of Fig. 8.3. If the cation magnetic moments are assumed to be parallel to each other at the (A) and [B] sites of Co1+x Fe2−x O4 (0.0 ≤ x ≤ 1.4), respectively, the calculated average molecular magnetic moments (μcal ) are larger than the observed values (μobs ). Therefore, we assumed that the canting angle between the sample magnetic moment and the cation magnetic moments increases with increasing the Co2+ ratio and fitted the curve of μobs versus x of the samples Co1+x Fe2-x O4 (0.0 ≤ x ≤ 1.4), by applying the fitting method in Sect. 6.1. The various cation distributions were obtained in the fitting process. The only difference of the fitting method from that in Sect. 6.1, is that the magnetic moment attenuation factors, [1 − c1 (C2A − C2A0 )1.2 ] for (A) sublattice

(8.1)

[1 − c1 (C2B − C2B0 )1.2 ] for [B] sublattice

(8.2)

and

were used in Eq. (6.21). In Eqs. (8.1) and (8.2), C 2A and C 2B represent Co2+ ratios at the (A) and [B] sublattices, respectively, corresponding to Co increased level x. C 2A0 and C 2B0 represent the values of C 2A and C 2B when x = 0.0. It may be seen that the canting angle between the sample magnetic moment and the cation magnetic moments is zero, attenuation factors are 1.0, when x = 0.0. We found that c1 in Eqs. (8.1) and (8.2) equaled 0.420 in the fitting process. The fitting results are given in Table 8.1, including the cation distribution, the equivalent potential barrier height V BA (Fe2+ ) and V BA (Co2+ ), the fitted average molecular magnetic moment (μcal ), and the magnetic moments of (A) and [B] sublattices, μAT and μBT . It may be seen from Table 8.1 that the ratio of trivalent cations is between 0.905 and 0.727, as were obviously less than the value 2.0 in the traditional view. The various cation ratios as a function of x, are shown in Fig. 8.4, where the ratios of Co2+ and Co3+ cations at (A) and [B] sites are represented by Co2A , Co2B , Co3A , and Co3B , the corresponding ratios of Fe cations are represented by Fe2A , Fe2B , Fe3A , and Fe3B . It may be seen from Fig. 8.4 that the trivalent cations dominate (A) sites compared to divalent cations, while most cations at [B] sites are divalent. This result is close to that of the inverse spinel structure in the traditional view. Figure 8.5a shows the magnetic moments as a function of x. Where μobs (points) and μcal (curve) represent the observed and fitted sample magnetic moments per molecule, respectively; μAT and μBT represent fitted magnetic moments of (A) and [B] sublattices per molecule, respectively. It can be seen that μcal is very close to μobs . Figure 8.5b shows the dependence on x of the canting angle between the sample magnetic moment and the cation magnetic moments at (A) sites (φ A ) and [B] sites (φ B ).

132

8 Spinel Ferrites with Canted Magnetic Coupling

Table 8.1 Fitted results for the dependence on the Co increased ratio, x, of the cation distributions and the magnetic moments of the samples Co1+x Fe2-x O4 (0.0 ≤ x ≤ 1.4). Here, μAT , μBT and μcal are the fitted magnetic moments per molecule of the (A) sublattice, the [B] sublattice and the samples, respectively; V BA (Fe2+ ) and V BA (Co2+ ) are respectively, the height of the equibalent potential barriers;N 3 is the number of trivalent cations per molecule x

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

V BA (Fe2+ )(eV)

1.0950

1.1260

1.1570

1.1880

1.2190

1.2500

1.2810

1.3120

V BA (Co2+ )(eV)

1.1020

1.1330

1.1640

1.1960

1.2270

1.2580

1.2890

1.3200

μAT (μB /molecule)

4.3028

4.2328

4.1534

4.0676

3.9760

3.8786

3.7753

3.6657

μBT (μB /molecule)

7.6023

7.1501

6.6355

6.1050

5.5719

5.0432

4.5238

4.0170

μcal (μB /molecule)

3.2995

2.9173

2.4821

2.0374

1.5959

1.1647

0.7486

0.3512

N3

0.9051

0.8796

0.8543

0.8289

0.8035

0.7781

0.7527

0.7274

Fe3+

0.4276

0.4037

0.3763

0.3454

0.3105

0.2715

0.2280

0.1795

Co3+

0.1224

0.1543

0.1893

0.2272

0.2686

0.3136

0.3627

0.4162

Fe2+

0.3253

0.2924

0.2599

0.2276

0.1955

0.1635

0.1314

0.0991

Co2+

0.1247

0.1495

0.1746

0.1998

0.2254

0.2514

0.2779

0.3052

Fe2+

0.9784

0.8795

0.7804

0.6813

0.5823

0.4834

0.3849

0.2869

Co2+

0.6665

0.7988

0.9309

1.0624

1.1933

1.3236

1.4531

1.5814

Fe3+

0.2687

0.2244

0.1833

0.1456

0.1116

0.0816

0.0557

0.0345

Co3+

0.0864

0.0972

0.1054

0.1107

0.1128

0.1114

0.1063

0.0972

(A) site

[B] sites

According to Table 8.1, we found that the above ratio of Co cations (including Co2+ and Co3+ ) occupied at (A) sites to those at [B] sites in CoFe2 O4 and Co2 FeO4 [1, 2] is in the range of those reported by Chandramohan et al. [6], Ferreira et al. [7], and Murray et al. [8] using the Mössbauer spectrum, and the data are listed in Table 8.2. The ratio of CoFe2 O4 estimated by us is 0.247/0.753, while those obtained by the Mössbauer spectrum are between 0.23/0.77 and 0.38/0.62. The ratio of Co2 FeO4 estimated by us is 0.56/1.44, while others have reported values between 0.44/1.56 and 0.70/1.30. These data indicated that our estimation methods here and in Sect. 6.1 are reasonable.

8.2 Spinel Ferrites Containing Nonmagnetic Cations Spinel ferrites containing nonmagnetic cations, such as Zn, Mg, and Al, were investigated by many scholars. However, the reported sample magnetic moments and cation distributions have distinct differences [9–18].

8.2 Spinel Ferrites Containing Nonmagnetic Cations

133

Fig. 8.4 The cation ratios at the (A) and [B] sites in Co1+x Fe2-x O4 as the function of the Co increased ratio x. Where Co2A , Co2B , Co3A , Co3B , Fe2A , Fe2B , Fe3A , Fe3B represent the ratios of divalent and trivalent Co and Fe at the (A) and [B] sites per molecule, respectively

8.2.1 Disputation of Nonmagnetic Cation Distribution The magnetic moments of Zn2+ (3d 10 4s0 ), Mg2+ (3 s0 ), and Al3+ (3 s0 ) are zero. No reports have yet addressed the magnetic moment of Al2+ (3 s1 ) in an oxide. All Zn cations in ZnFe2 O4 were assumed to be located at (A) sites in the traditional view [9]. Siddique et al. [10] concluded based on Mössbauer measurements that all Zn cations in Cu1−x Znx Fe2 O4 occupied at (A) sites. Gul et al. [11] studied the crystal structure, magnetic, and electrical properties of Co1−x Znx Fe2 O4 , they also considered that all Zn cations occupied at (A) sites. Oliver et al. [12] concluded that the ratio of Zn cations at the (A) sites to those at [B] sites in ZnFe2 O4 , RAB = 0.55/0.45 according to extended X-ray absorption fine structure measurements. Mathur et al. [13] studied Znx Mn1−x Fe2 O4 and obtained RAB = 0.1/0.0, 0.2/0.1, 0.35/0.15, 0.45/0.25, and 0.5/0.4, respectively, when x = 0.1, 0.3, 0.5, 0.7, and 0.9. Sakurai et al. [14] grew single-crystal Mn0.80 Zn0.18 Fe2.02 O4 and reported the

134

8 Spinel Ferrites with Canted Magnetic Coupling

Fig. 8.5 a Dependence on the Co increased ratio, x, of the magnetic moments for the samples Co1+x Fe2-x O4 (0.0≤x≤1.4) at 10 K. Here, the curves of μAT , μBT and μcal are the fitted magnetic moments per molecule of the (A) sublattice, the [B] sublattice and the samples, respectively; μobs (points) is the observed magnetic moments per molecule of samples. b Dependence on the Co increased ratio, x, of the canted angle between the sample magnetic moment and the cation magnetic moments at the (A) sites (φ A ) and the [B] sites (φ B )

Table 8.2 Under different preparation conditions, the ratio of Co cations (including Co2+ and Co3+ ) occupied at the (A)/[B] sites in CoFe2 O4 and Co2 FeO4 estimated by our group [1, 2] and reported by other authors [6–8] by Mössbauer spectrum Samples

Annealed

Cooled

Ratio

References

CoFe2 O4

for 3 h at 1173 K

with furnace

0.247/0.753

Our group [1, 2]

CoFe2 O4

for 6 h at 1170 K

with furnace

0.23/0.77

Ferreira et al. [7]

CoFe2 O4

for 6 h at 1273 K

rapidly quenched

0.31/0.69

Chandramohan et al. [6]

Co1.04 Fe1.96 O4

for 48 h at 1320 K

rapidly quenched

0.38/0.62

Murray et al. [8]

FeCo2 O4

for 3 h at 1173 K

with furnace

0.56/1.44

Our group [1, 2]

FeCo2 O4

for 17 h at 1170 K

3 K/min

0.45/1.55

Ferreira et al. [7]

FeCo2 O4

for 17 h at 1170 K

rapidly quenched

0.50/1.50

Ferreira et al. [7]

FeCo2 O4

for 6 h at 1170 K

rapidly quenched

0.44/1.56

Ferreira et al. [7]

FeCo2 O4

for 72 h at 1193 K

rapidly quenched

0.70/1.30

Murray et al. [8]

2+ 3+ 2+ 2+ 3+ cation distribution as (Mn2+ 0.71 Zn0.10 Fe0.19 )[Mn0.09 Zn0.08 Fe1.83 ]O4 by using XANES measurements and XMCD spectroscopy. In these references, the Zn ratio at (A)/[B] sites are shown in Table 8.3. There are distinct differences for the Mg cation distributions in spinel ferrites, as shown in Table 8.4. Singh et al. [15] prepared Mgx Mn1−x Fe2 O4 (0.0 ≤ x ≤ 0.8) by the conventional and hot-pressed techniques. They concluded that, when x = 0.2, all Mg ions entered (A) sites; when x = 0.4, 0.6, and 0.8, the ratio of Mg occupying (A)/[B] sites are 0.3/0.1, 0.4/0.2, and 0.4/0.4, respectively. Antic et al. [16] synthesized MgFe2 O4 nanopowder and reported the Mg distribution as RAB ≈ 0.31/0.69 by Rietveld fitting. Khot et al. [17] synthesized Mg1−x Mnx Fe2 O4 (x =

8.2 Spinel Ferrites Containing Nonmagnetic Cations Table 8.3 Content ratio, RAB , of Zn at the (A)/[B] sites of spinel ferrites in several references

Table 8.4 Content ratio, RAB , of Mg at the (A)/[B] sites of spinel ferrites in several references

135

Materials

RAB

References

ZnFe2 O4

1.0/0.0

Dai and Qian [9]

Cu1-x Znx Fe2 O4

x/0.0

Siddique et al. [10]

Co1−x Znx Fe2 O4

x/0.0

Gul et al. [11]

ZnFe2 O4

0.55/0.45

Oliver et al. [12]

Zn0.1 Mn0.9 Fe2 O4

0.1/0.0

Mathur et al. [13]

Zn0.3 Mn0.7 Fe2 O4

0.2/0.1

Mathur et al. [13]

Zn0.5 Mn0.5 Fe2 O4

0.35/0.15

Mathur et al. [13]

Zn0.7 Mn0.3 Fe2 O4

0.45/0.25

Mathur et al. [13]

Zn0.9 Mn0.1 Fe2 O4

0.5/0.4

Mathur et al. [13]

Mn0.80 Zn0.18 Fe2.02 O4

0.10/0.08

Sakurai et al. [14]

Materials

RAB

References

Mg0.2 Mn0.8 Fe2 O4

0.2/0.0

Singh [15]

Mg0.2 Mn0.8 Fe2 O4

0.0/0.2

Khot et al. [17]

Mg0.4 Mn0.6 Fe2 O4

0.0/0.4

Khot et al. [17]

Mg0.6 Mn0.4 Fe2 O4

0.0/0.6

Khot et al. [17]

Mg0.4 Mn0.6 Fe2 O4

0.3/0.1

Singh [15]

Mg0.6 Mn0.4 Fe2 O4

0.4/0.2

Singh [15]

Mg0.8 Mn0.2 Fe2 O4

0.4/0.4

Singh [15]

MgFe2 O4

0.31/0.69

Antic et al. [16]

Mg0.8 Mn0.2 Fe2 O4

0.06/0.74

Khot et al. [17]

MgFe2 O4

0.12/0.88

Khot et al. [17]

Ni0.5 Mg0.5 Fe1.2 Cr0.8 O4

0.10/0.40

Hashim et al. [18]

Ni0.5 Mg0.5 Fe1.0 Cr1.0 O4

0.14/0.36

Hashim et al. [18]

0, 0.2, 0.4, 0.6, 0.8, 1.0). They showed that, when Mg ratio (1−x) ≤ 0.6, all Mg ions entered [B] sites; when (1-x) = 0.8 and 1.0, RAB = 0.06/0.74 and 0.12/0.88, respectively. Hashim et al. [18] prepared Ni0.5 Mg0.5 Fe2−x Crx O4 (0.0 ≤ x ≤ 1.0). They gave the Mg ion distributions as RAB = 0.1/0.4 for x ≤ 0.8; RAB = 0.14/0.36 for x = 1.0. There are also distinct differences for Al cation distributions in spinel ferrites, as shown in Table 8.5. Patange et al. [19] prepared NiAlx Fe2−x O4 and reported that the ratio of Al occupying (A) sites is less than or equal to 0.06 for 0.2 ≤ x ≤ 1.0, indicating that almost all Al cations occupied [B] sites. Pandit et al. [20] prepared CoAlx Fe2-x O4 (x = 0.0, 0.2, 0.4, 0.6, 0.8). They found the Al ion distributions as 0.04/0.16 ≤ RAB ≤ 0.26/0.54 when 0.2 ≤ x ≤ 0.8. Mane et al. [21] synthesized CoAlx Crx Fe2−2x O4 (0.0 ≤ x ≤ 0.5). They gave the Al ion distributions, RAB = 1/4 for all samples.

136

8 Spinel Ferrites with Canted Magnetic Coupling

Table 8.5 Content ratio, RAB , of Al at the (A)/[B] sites of spinel ferrites in several references

Materials

RAB

References

NiAl0.2 Fe1.8 O4

0.02/0.18

Patange et al. [19]

NiAl0.4 Fe1.6 O4

0.05/0.35

Patange et al. [19]

NiAl0.6 Fe1.4 O4

0.06/0.54

Patange et al. [19]

NiAl0.8 Fe1.2 O4

0.04/0.76

Patange et al. [19]

NiAl1.0 FeO4

0.02/0.98

Patange et al. [19]

CoAl0.2 Fe1.8 O4

0.04/0.16

Pandit et al. [20]

CoAl0.4 Fe1.6 O4

0.10/0.30

Pandit et al. [20]

CoAl0.6 Fe1.4 O4

0.19/0.41

Pandit et al. [20]

CoAl0.8 Fe1.2 O4

0.26/0.54

Pandit et al. [20]

CoAl0.5 Cr0.5 Fe1.0 O4

0.10/0.40

Mane et al. [21]

In summary, distinct differences were found for the ratios of Zn, Mg, and Al cations occupying (A) sites to those occupying [B] sites in spinel ferrites reported in the literature.

8.2.2 Fitting Sample Magnetic Moments Our group [22, 23] synthesized the spinel ferrites powder samples M x Mn1-x Fe2 O4 (M = Zn, Mg, Al). The crystal structures of the samples were analyzed using XRD patterns. The magnetic hysteresis loops were measured using a vibrating sample magnetometer attached to a Physical Properties Measurement System. The distributions of various cations in the samples were given by fitting sample magnetic moments using the IEO model and the method in Sect. 6.1. XRD analyses indicated that the three series of samples have a single-phase cubic spinel structure, the crystal lattice constant a of Mg- and Al-doped spinel ferrites decreased more rapidly with increasing x than that of the Zn-doped spinel ferrites, as shown in Fig. 8.6. The magnetic hysteresis loops of the Zn-, Mg-, and Al-doped spinel ferrites measured at 10 K are shown in Fig. 8.7. The dependencies of the magnetic moments, μobs , at 10 K on the doping level, x, are shown by points in Fig. 8.8. For the Zn-doped ferrites, μobs increases with increasing x when x ≤ 0.4; μobs decreases with increasing x when x ≥ 0.4. For all Mg- and Al-doped samples, μobs decreases with increasing x. In Fig. 8.8, the curves show the fitted results of the sample magnetic moments, μcal , and the magnetic moments of (A) and [B] sublattices, μAT , and μBT , for the three series of ferrites. The fitting method is the same as that in Sect. 6.1, except Eqs. (6.4) and (6.5) are substituted by VBA (Fe2+ )=

VBA (Mn2+ )V (Fe3+ )r (Fe2+ ) . V (Mn3+ )r (Mn2+ )

(8.3)

8.2 Spinel Ferrites Containing Nonmagnetic Cations

137

Fig. 8.6 Dependences of the crystal lattice constant a on x in M x Mn1-x Fe2 O4 (M = Zn, Mg, Al)

It is used to fit the relationship between equivalent potential barriers V BA (Fe2+ ) and V BA (Mn2+ ). The equivalent potential barriers V BA (M 2+ ) with M 2+ = Zn2+ , Mg2+ , and Al2+ were obtained in the fitting process. The results are shown in Fig. 8.9. In addition, the attenuation factors corresponding to the canting angle similar to Eqs. (8.1) and (8.2), [1 − c1 (y1 + y4 )1.2 ] for (A) sublattice

(8.4)

[1 − c1 (x1 − y1 − y4 )1.2 ] for [B] sublattice

(8.5)

and

are introduced in Eq. (6.21), where y1 + y4 and x 1 − y1 − y4 represent the content sum of divalent and trivalent nonmagnetic cations at (A) and [B] sublattices, respectively. In the fitting process, we obtained c1 = 2.0, 0.5, and 0.5 in Eqs. (8.4) and (8.5) for Zn-, Mg-, and Al-doped samples, respectively. The fitted results of the canting angles between the sample magnetic moment and the cation magnetic moments at the (A) sublattice (φ A ) and [B] sublattice (φ B ), similar to those in Fig. 8.5b, are shown in Fig. 8.10. The fitted results of the cation distributions for the three series of samples are shown in Figs. 8.11, 8.12, and 8.13.

8.2.3 Discussion on Cation Distributions On the basis of Figs. 8.11, 8.12, and 8.13, the cation distributions of the three series of spinel ferrites are discussed as follows:

138 Fig. 8.7 Magnetic hysteresis loops of the M x Mn1-x Fe2 O4 (M = Zn, Mg, Al) samples at 10 K

8 Spinel Ferrites with Canted Magnetic Coupling

8.2 Spinel Ferrites Containing Nonmagnetic Cations

139

Fig. 8.8 Dependences on the doped level x of the observed (points) and fitted (curves) magnetic moments, μobs and μcal , of M x Mn1-x Fe2 O4 (M = Zn, Mg, Al) samples at 10 K. Where μAT and μBT represent the fitted magnetic moments of the (A) and [B] sublattices, respectively

(1)

Dependence of trivalent cation ratio on the third ionization energy

The third ionization energies, V (M 3+ ), of Al, Fe, Mn, Zn, and Mg are 28.54, 30.65, 33.67, 39.72, and 80.14 eV (see Appendix A). Figures 8.11, 8.12, and 8.13 show that their trivalent cation ratios decrease with increasing V (M 3+ ). It can be seen from

140

8 Spinel Ferrites with Canted Magnetic Coupling

Fig. 8.9 Fitted equivalent potential barriers V BA (M 2+ ), M 2+ = Zn2+ , Mg2+ , Al2+ (a) and V BA (Mn2+ ) (b) as the functions of x in M x Mn1-x Fe2 O4 (M = Zn, Mg, Al) samples

Fig. 8.12 that only Mg2+ exist at the (A) or [B] sites in the Mg-doped samples. Figure 8.11 shows that very few Zn3+ cations occupy either the (A) or [B] sites in the Zn-doped samples. For example, for ZnFe2 O4 , only 1.0% and 3.4% Zn3+ cations exist in the (A) and [B] sites, respectively. Such a small Zn3+ ratio can be neglected. These calculated results are very close to those in the conventional view in which no Mg3+ and Zn3+ but Mg2+ and Zn2+ are thought to exist in spinel ferrites [9], indicating that the cation distributions in Figs. 8.11, 8.12, and 8.13 are reasonable because it is difficult for oxygen ions to obtain electrons from cations with high ionization energies. (2)

Magnetic moment directions of the samples are the same as those of the [B] sublattice.

For all three series of samples, the percentage of Fe (Fe2+ and Fe3+ ) cations occupying [B] sites lies between 62 and 74% of the total Fe, and the ratio of Mn2+ cations occupying [B] sites lies between 55 and 65% of the total Mn, indicating that the magnetic moment directions of the samples are the same as those in the [B] sublattice.

8.2 Spinel Ferrites Containing Nonmagnetic Cations Fig. 8.10 Fitted canting angles between the sample magnetic moment and the cation magnetic moments at the (A) sublattice (φ A ) and the [B] sublattice (φ B ) as the functions of x in M x Mn1-x Fe2 O4 (M = Zn, Mg, Al) samples

141

142

8 Spinel Ferrites with Canted Magnetic Coupling

Fig. 8.11 In each molecule of Znx Mn1-x Fe2 O4 , Zn (a), Mn (b) and Fe (c) cation ratios in average and the total content (d) of Zn, Mn and Fe at the (A) and [B] sites as functions of the Zn-doping level x. Where divalent and trivalent Zn cation ratios at the (A) and [B] sites are represented by Zn2A , Zn2B , Zn3A , Zn3B ; corresponding ratios of Mn and Fe are represented by Mn2A , Mn2B , Mn3A , Mn3B , Fe2A , Fe2B , Fe3A , Fe3B ; the total content of divalent and trivalent Zn, Mn and Fe are represented by ZnA , ZnB , MnA , MnB , FeA , FeB

This may be seen clearly in Fig. 8.8: the fitted magnetic moments of [B] sublattices are larger than those of (A) sublattices for the three series of samples, except that the magnetic moments of the two sublattices are approximately equal for ZnFe2 O4 . (3)

Effects of Mg2+ (Al) cation distribution on the sample magnetic moments

For Mgx Mn1-x Fe2 O4 (0.0 ≤ x ≤ 1.0) and Alx Mn1-x Fe2 O4 (0.0 ≤ x ≤ 0.5), it can be seen from Figs. 8.12d and 8.13d that the ratios of Mg2+ and Al (including Al2+ and Al3+ ) cation increase approximately linearly at both (A) and [B] sites, with the increased ratio at [B] sites being greater than that at (A) sites for every doping level. This is the underlying reason why the magnetic moments of [B] sublattices decrease more rapidly than those of (A) sublattices, while the total magnetic moments decrease approximately linearly with increasing x [see Figs. 8.8b and c] since the magnetic moments of Mg2+ (3 s0 ) and Al3+ (3 s0 ) are zero, and no studies have been reported about the magnetic moments of Al3+ (3 s1 ) cations in an oxide.

8.2 Spinel Ferrites Containing Nonmagnetic Cations

143

Fig. 8.12 In each molecule of Mgx Mn1-x Fe2 O4 , Mg (a), Mn (b) and Fe (c) cation ratios in average and the total content (d) of Mg, Mn and Fe at the (A) and [B] sites as functions of the Mg-doping level x. Where divalent and trivalent Mg cation ratios at the (A) and [B] sites are represented by Mg2A , Mg2B , Mg3A , Mg3B ; corresponding ratios of Mn and Fe are represented by Mn2A , Mn2B , Mn3A , Mn3B , Fe2A , Fe2B , Fe3A , Fe3B ; the total content of divalent and trivalent Mg, Mn and Fe are represented by MgA , MgB , MnA , MnB , FeA , FeB

(4)

Effects of Zn2+ cation distribution on the sample magnetic moments

For the cation distributions of the Zn-doped ferrites, as shown in Fig. 8.11, when x < 0.4, the Zn2+– ratio at (A) sites increases rapidly and Fe2+ , Fe3+ , and Mn2+ ratios at (A) sites decrease gradually with increasing x, resulting in the magnetic moment of the (A) sublattice decreasing rapidly and the total magnetic moment (μobs ) of the samples increasing. When x > 0.4, the Zn2+ ratio at [B] sites increases rapidly and the Mn2+ ratio at [B] sites decreases gradually with increasing x, resulting in the magnetic moment of the [B] sublattice and the total magnetic moment of the samples decreasing rapidly. When x = 1.0, μobs is close to zero for ZnFe2 O4 , the Zn cation ratio occupied at the (A)/[B] sites, RAB = 0.47/0.53, which is close to RAB = 0.55/0.45 (see Table 8.3), based on the X-ray absorption fine structure measurements reported by Oliver et al. [12]. The reasons for why Zn cations have the distributions in Fig. 8.11a may result from the crystal lattice energy: (a) It can be seen from the discussion in Sects. 4.1

144

8 Spinel Ferrites with Canted Magnetic Coupling

Fig. 8.13 In each molecule Alx Mn1-x Fe2 O4 , Al (a), Mn (b) and Fe (c) cation ratios in average and the total content (d) of Al, Mn and Fe at the (A) and [B] sites as functions of the Al-doping level x. Where divalent and trivalent Al cation ratios at the (A) and [B] sites are represented by Al 2A , Al 2B , Al 3A , Al 3B ; corresponding ratios of Mn and Fe are represented by Mn2A , Mn2B , Mn3A , Mn3B , Fe2A , Fe2B , Fe3A , Fe3B ; the total content of divalent and trivalent Al, Mn and Fe are represented by Al A , Al B , MnA , MnB , FeA , FeB

and 6.1 that the observed distance from an √ O anion to an adjacent cation at the (A) site, d AO , is greater than its ideal value, 3a/8, while the observed distance from an O anion to an adjacent cation at the [B] site, d BO , is smaller than its ideal value, a/4. This may be caused due to the repulsive energy between A-O ions being higher than that between B-O ions, with the ideal bond lengths. The actual bond lengths are different from the ideal value, which can lower the total lattice energy. This repulsive energy may include the Pauli repulsive energy of the electron cloud between AO (B-O) ions and the magnetic repulsive energy between A-A (B-B) cations with ferromagnetic order. (b) The effective radius of Zn2+ (0.074 nm) is larger than those of Fe3+ (0.0645 nm) and Mn3+ (0.0645 nm) [24]. When x < 0.4, the ratio of Zn2+ at (A) sites increases rapidly since doping Zn2+ cations with zero magnetic moments and the lager radius can decrease the magnetic repulsive energy in the (A) sublattice. (c) It can be seen from Fig. 8.11 that the ratios of Fe3+ and Mn3+ cations at (A) sites decrease with increasing Zn2+ ratio. Zn2+ cations (with the larger radius) substituting for Fe3+ and Mn3+ cations (with the smaller radius) result in the Pauli repulsive energy that increases at the (A) sublattice. When x > 0.4, the (A) sublattice can, therefore,

8.2 Spinel Ferrites Containing Nonmagnetic Cations

145

no longer have more Zn2+ cations. This rapidly increases the Zn2+ ratio at the [B] sites. Mg and Al cation distributions have no variation tendency similar to that in Fig. 8.11a because the effective radii of Mg2+ (0.072 nm) and Al [Al2+ (0.060 nm), Al3+ (0.0535 nm)] are smaller than that of Zn2+ (0.074 nm) [24]; thus, the effect of Mg2+ or Al cation entering an (A) site on the Pauli repulsive energy and the magnetic repulsive energy is lower than that of Zn2+ cation, resulting in the monotonic variation in the sample magnetic moments and cation distributions. (5)

Dependence of the average canting angles (φ) between the sample magnetic moments and cation magnetic moments on the doped cation radii

For the same doping level, it can be seen from Fig. 8.10 that the φ values of Zn-doped samples are larger than those of Mg- and Al-doped samples. This may be due to the effective radius of Zn2+ (0.074 nm) being larger than the radii of Mg2+ and Al2+ . This may also be the reason why the magnetic moment of ZnFe2 O4 is close to zero, while the magnetic moment of MgFe2 O4 is 1.68 µB /molecule. (6)

Equivalent potential barrier V BA

As mentioned in Sect. 6.1, a tendency toward electrical charge density balance forces a fraction of divalent cations, jumping an equivalent potential barrier, V BA , to enter (A) sites from [B] sites. It can be seen from Fig. 8.9 that the values of V BA (Zn2+ ) are between 0.10 and 0.29 eV, while V BA (Fe2+ ), V BA (Mn2+ ), V BA (Mg2+ ), and V BA (Al2+ ) are between 0.31 and 1.40 eV for the samples studied here. These may be one of the reasons for why the Zn2+ cation ratio at (A) sites in Znx Mn1-x Fe2 O4 (0.0 ≤ x ≤ 0.4) increases more rapidly with increasing x than the ratios of Mg2+ and Al2+ at (A) sites in Mg- and Al-doped samples.

References 1. Liu SR, Ji DH, Xu J, Li ZZ, Tang GD, Bian RR, Qi WH, Shang ZF, Zhang XY (2013) J Alloy Compd 581:616 2. Liu SR (2013) Study on structure, magnetic properties and cation distributions in Co1+x Fe2-x O4 and Znx Co1-x Fe2 O4 ,(Master’s thesis). Hebei Normal University, Shijiazhuang 3. Bian RR (2013) Study on cation distributions and magnetic properties of ferrites NiCrx Fe2-x O4 and Ni1-x Cox Fe2 O4 (Master’s thesis). Hebei Normal University, Shijiazhuang 4. Shang ZF (2014) Study on magnetic properties and cation distributions in Co-Cr ferrites (Master’s thesis). Hebei Normal University, Shijiazhuang 5. Takahashi M, Fine ME (1972) J Appl Phys 43:4205 6. Chandramohan P, Srinivasan MP, Velmurugan S, Narasimhan SV (2011) J Solid State Chem 184:89 7. Ferreira TAS, Waerenborgh JC, Mendonca MHRM, Nunes MR, Costa FM (2003) Solid State Sci 5:383 8. Murray PJ, Linneit JW (1976) J Phys Chem Solids 37:1041 9. Dai DS, Qian KM (1987) Ferromagnetism. Science Press, Beijing in Chinese 10. Siddique M, Khan RTA, Shafi M (2008) J Radioanal Nucl Chem 277:531

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8 Spinel Ferrites with Canted Magnetic Coupling

11. Gul IH, Abbasi AZ, Amin F, Anis-ur-Rehman M, Maqsood A (2007) J Magn Magn Mater 311:494 12. Oliver SA, Harris VG, Hamdeh HH, Ho JC (2000) Appl Phys Lett 76:2761 13. Mathur P, Thakur A, Singh M (2008) Phys Scr 77: 14. Sakurai S, Sasaki S, Okube M, Ohara H, Toyoda T (2008) Phys B 403:3589 15. Singh M (2006) J Magn Magn Mater 299:397 16. Antic B, Jovic N, Pavlovic MB, Kremenovic A, Manojlovic D, Vucinic-Vasic M, Nikolic AS (2010) J Appl Phys 107:043525 17. Khot VM, Salunkhe AB, Phadatare MR, Thorat ND, Pawar SH (2013) J Phys D Appl Phys 46:055303 18. Hashim M, Alimuddin, Kumar S, Shirsath S E, Kotnala RK, Chung H, Kumar R (2012) Powder Technol 229:37 19. Patange SM, Shirsath SE, Jangam GS, Lohar HS, Jadhav SS, Jadhav KM (2011) J Appl Phys 109:053909 20. Pandit R, Sharma KK, Kaur P, Kotnala RK, Shah J, Kumar R (2014) J Phys Chem Solids 75:558 21. Mane DR, Devatwal UN, Jadhav KM (2000) Meter Lett 44:91 22. Ding LL, Xue LC, Li ZZ, Li SQ, Tang GD, Qi WH, Wu LQ, Ge XS (2016) AIP Adv 6:105012 23. Ding LL (2017) Study of magnetic ordering and cation distributions in spinel ferrites M x Mn1-x Fe2 O4 (M = Zn, Mg, Al) (Master’s thesis). Hebei Normal University, Shijiazhuang 24. Shannon RD (1976) Acta Crystallogr A 32:751

Chapter 9

Magnetic Ordering and Electrical Transport of Perovskite Manganites

In Sect. 4.2, the crystal structure of ABO3 perovskite manganites and the difficulties to explain their magnetic structure based on the traditional view were introduced. Especially, it was unclear why the magnetic moment of Re1–x Aex MnO3 (Re = La, Pr, Nd, etc., Ae = Ca, Sr, Ba, Pb, etc.) varies with the ratio of divalent cation Ae based on the classical theories. Our group explained this issue using the IEO model, proposed that the electrical transport of perovskite manganites below the Curie temperature is spin-dependent, and discussed in detail the canted magnetic structure of those with divalent cation ratio 0.15 ≤ x ≤ 0.40.

9.1 Ferromagnetic and Antiferromagnetic Coupling in Typical Perovskite Manganites La1–x Srx MnO3 (0.00 ≤ x ≤ 0.40) materials are typical perovskite manganites, which have rich magnetic and electrical transport properties. Our group explained the magnetic property using the IEO model, and fitted the curves of the sample magnetic moment versus the Sr ratio x.

9.1.1 Crystal Structure and Magnetic Measurement Results of La1–x Srx MnO3 Polycrystalline Powder Samples Our group [1, 2] synthesized La1–x Srx MnO3 polycrystalline powder samples using the sol–gel method. The XRD analyses indicated that all samples had a single ABO3 perovskite phase with the space group R3c. The volume-averaged diameters of the crystallites in each sample were estimated based on the Scherrer equation. In every case, these diameters were found to be greater than ~100 nm. Thus, it was that the surface effects would be very weak. © Science Press 2021 G.-D. Tang, New Itinerant Electron Models of Magnetic Materials, https://doi.org/10.1007/978-981-16-1271-8_9

147

148

9 Magnetic Ordering and Electrical Transport …

Rietveld fittings of the XRD data of the samples were performed. The fitting results for the crystal lattice constants, a and c, the crystal cell volume, v, the B–O (Mn–O) bond length, d BO , and the B–O–B (Mn–O–Mn) bond angle, θ, are shown in Fig. 9.1. As the doping level, x, increases, several distinctive features are observed: (i) there is a characteristic doping level, x 1 = 0.15, such that for x < 0.15, the magnitude of the Mn–O bond length, d BO , changes only slightly, while it undergoes a more significant reduction with an increase in x when x > 0.15; (ii) there is another characteristic doping level, x 2 = 0.20, for the lattice constant a and the crystal cell volume v such that both a and v change only slightly when x < 0.20, while both of them decrease with an increase in x when x > 0.20; and (iii) the Mn–O–Mn bond angle, θ, increases gradually with an increase in x. We would like to note that these dependencies of the crystal structure parameters on the doping level, x, are related to each sample’s magnetic property, as discussed below. The magnetic hysteresis of La1–x Srx MnO3 samples at 10 K are shown in Fig. 9.2. The curves of specific magnetizations σ under 50 mT applied field versus test temperature T are shown in Fig. 9.3. The curves of the average molecular magnetic moment (at 10 K), μobs , and the Curie temperature, T C , versus the Sr ratio, x, are shown in Figs. 9.4 and 9.5. The data reported by Urushibara et al. [3] and Jonker and Van Santen [4] are also shown in Fig. 9.4. It can be seen μobs increases rapidly with 353

(a)

5.52 5.50

351

3

5.48

5.44

0.0

0.1

350 349

c 6

5.46

348 0.2

x

0.3

1.964

dBO (Å)

(b)

352

v (Å )

a, c (Å)

a

0.4

(c)

1.960

0.0

0.1

0.2

x

0.3

0.4

(d)

170 168

1.956 166 1.952 164

1.948 1.944

0.0

0.1

0.2

x

0.3

0.4

162

0.0

0.1

0.2

0.3

0.4

x

Fig. 9.1 Dependences on Sr doped level x of the crystal lattice constants, a and c (a), the crystal cell volume, v (b), Mn–O bond length, d BO (c) and Mn–O–Mn bond angle, θ (d) of La1−x Srx MnO3 polycrystal powder samples [1, 2]

9.1 Ferromagnetic and Antiferromagnetic Coupling … 120

x =0.15

80

x =0.05

40

σ (Am2/kg)

Fig. 9.2 Magnetic hysteresis of La1−x Srx MnO3 (0.05 ≤ x ≤ 0.40) polycrystalline powder samples at 10 K [1, 2]

149

x=0.20 x=0.25 x=0.30

0 -40

x=0.35 x=0.40

-80

x=0.10

-120

-2

-1

0

1

2

μ 0 H (T) 50

Fig. 9.3 Curves of specific magnetizations σ of La1−x Srx MnO3 under 50 mT applied field versus test temperature T [1, 2]

45

x=0.35

40

x=0.40

σ (Am2/kg)

35 30

x=0.15

25

x=0.20

x=0.10

20

x=0.05

15

x=0.25 x=0.30

10 5 0 -5

0

50

100

150

200

250

300

350

400

T (K) 5 La 1- x Srx MnO3

4 3

μ ( μB )

Fig. 9.4 Dependences of the average molecular magnetic moment on Sr ratio x of La1−x Srx MnO3 samples. ( ) and ( ) are observed values reported by Urushibara et al. [3] and Jonker et al. [4], ( ) is observed value by our group [1, 2], curve is the fitted result by our group [1, 2]

2 1

μ obs -Urushibara μ obs -Jonker

0

μ obs -Wu

Fit -- Wu

0.0

0.1

0.2

x

0.3

0.4

150 400 350

Urushibara Wu

300

TC (K)

Fig. 9.5 Curie temperature T C of La1−x Srx MnO3 as the function of the Sr ratio x. ( ) points are the results of single-crystal samples reported by Urushibara et al. [3], ( ) points are the observed values of polycrystalline by our group [1, 2]

9 Magnetic Ordering and Electrical Transport …

250 200 150 100 50 0.0

0.1

0.2

0.3

0.4

Sr doping level x

an increase in x when x ≤ 0.15 but decreases slowly for x ≥ 0.15. The maximum observed value, μobs = 4.19 μB , occurs when x = 0.15. Notably, observed trend in μobs by our group is very similar to those reported by Urushibara et al. [3] and Jonker and Van Santen [4]. In Fig. 9.4, the points represent the observed results while the curve represents the fitted result obtained from the analysis presented in the following section.

9.1.2 Study of Valence and Ionicity of La1–x Srx MnO3 According to the method in Sect. 5.4, the valence of La1–x Srx MnO3 powder samples was studied by XPS with a monochromatic Al Kα radiation (1486.6 eV) source. The C1s binding energy (284.8 eV) of carbon as a contaminant was used for calibration to compensate for the charging effects. The computer program XPSPEAK (Version 4.1) was used to fit the narrow-scan spectra after the Shirley-type background subtraction [5]. The narrow-scan spectra of the O1 s peaks were fitted using symmetric Gaussian– Lorentzian product functions. Figure 9.6a–e is the O 1s photoelectron spectra (points) and the fitted results (curves) of La1–x Srx MnO3 (x = 0.05, 0.10, 0.15, 0.20, and 0.25) powder samples. According to the analysis method in Sect. 5.4, each spectrum was fitted by the three peaks with different E b (binding energy, BE). The lower BE peak was assigned to O2− ions, the middle BE peak to O1− ions, and the higher BE peak to OChem , chemically adsorbed oxygen on the surface [6]. The fitted results are listed in Table 9.1, where E, FWHM, S, V alO , and f O represent the peak position, full width at half maximum, area percentage of the peak, average valence of oxygen, and ionicity, respectively. Here, V alO of O anions was calculated by using Eqs. (5.17) and (5.18). Then, f O of O anions was calculated using f O = |V alO |/2.00. Figure 9.6f shows the unitary O 1s photoelectron spectra, where the intensity ratio of O1− /O2− peaks has the minimum value at x = 0.15, where |V alO | = 1.71 and f i = 0.855 have the maximum value.

9.1 Ferromagnetic and Antiferromagnetic Coupling …

151

(b) La0.9 Sr0.1 MnO3

(a) La 0.95 Sr 0.05 MnO3

O1s

2O

OChem

2O

OChem

I (a.u.)

I (a.u.)

O1s

O1-

1O

538

536

534

532

530

528

526

538

536

534

532

530

O1s

2O

OChem

I (a.u.)

I (a.u.)

O 1s

2O

OChem

O1-

O1-

536

526

(d) La0.8 Sr0.2 MnO3

(c) La0.85Sr0.15MnO3

538

528

534

532

530

528

538

526

(e) La0.75 Sr0.25MnO3

536

534

532

(f)

O1s

530

528

526

0.25

O 1s

0.05

I (a.u.)

OChem

538

I (a.u.)

0.10

2O

0.20 0.15

O1-

536

534

532

530

Eb(e V)

528

526

538

536

534

532

530

528

526

Eb(e V)

Fig. 9.6 O 1s photoelectron spectra (points) and the fitted results (curves) of La1−x Srx MnO3 powder samples

To confirm that the valence of Mn cations in La1−x Srx MnO3 materials is lower than 3.0, the O 1s photoelectron spectra of MnO, Mn3 O4 , CaMnO3 , and SrMnO3 were analyzed. Their O 1s photoelectron spectra are shown in Fig. 9.7, from which the average valences of oxygen were obtained as –1.73, –1.78, –1.54, and –1.66. Then, the average valences of Mn cations were calculated as 1.73, 2.37, 2.62, and 2.98, as listed in Table 9.2. In the calculation process, the valences of Ca and Sr were assumed to be 2.0 due to their second ionization energies (11.87 and 11.03 eV),

152

9 Magnetic Ordering and Electrical Transport …

Table 9.1 Fitted results for O 1s photoelectron spectra of La1−x Srx MnO3 powder samples. E, FWHM, S, V alO , and f O represent the peak position, full width at half maximum, area percentage of the peak, average valence, and ionicity of oxygen, respectively Sample

E (eV)

FWHM (eV)

S(%)

V alO

fO

La0.95 Sr0.05 MnO3

529.43

1.54

47.37

−1.65

0.825

531.04

1.41

25.15

532.14

1.85

27.48

529.23

1.59

49.33

−1.68

0.840

530.81

1.50

23.36

532.06

2.11

27.31

529.31

1.40

55.05

−1.71

0.855

530.92

1.56

22.20

532.09

2.07

22.75

529.19

1.40

51.28

−1.68

0.840

530.99

1.56

24.39

532.36

2.07

24.33

528.98

1.96

51.13

−1.67

0.835

530.72

1.88

25.75

532.16

2.30

23.12

La0.9 Sr0.1 MnO3

La0.85 Sr0.15 MnO3

La0.8 Sr0.2 MnO3

La0.75 Sr0.25 MnO3

which are lower than the second ionization energy (15.64 eV) of Mn. Meanwhile, Ca and Sr cannot form the trivalent cations since their third ionization energies (50.91 and 43.6 eV) are far higher than the third ionization energy (33.67 eV) of Mn. It is seen from Table 9.2 that the valence of Mn cation increases with the oxygento-cation ratio, while the valences of Mn in CaMnO3 and SrMnO3 are lower than 3.00, in which has no Mn4+ . Therefore, the valences of Mn in La1−x Srx MnO3 should not be higher than 3.00.

9.1.3 Fitting of the Curve of the Magnetic Moment Versus Sr Ratio for La1-x Srx MnO3 The above investigation indicated that there are 30–35% of O1− anions but no Mn4+ cation in La1–x Srx MnO3 . There is an O 2p hole in an O1− anion. Therefore, the DE interaction between Mn3+ and Mn4+ cations cannot be used to explain the magnetic and electrical transport properties. The IEO model was used to explain the magnetic structure of La1–x Srx MnO3 by our group as follows. Firstly, the transition mechanism of the itinerant electrons in La1–x Srx MnO3 is that a 2p electron of O2− ion transits to the 2p hole of the adjacent O1− ion that mediates the cation. Secondly, the two 2p electrons with opposite spin directions of an O2− ion become the itinerant electrons of the A sublattice (containing La and

9.1 Ferromagnetic and Antiferromagnetic Coupling …

153

(a) MnO

(b) Mn3O4

O2-

I (a.u.)

I (a.u.)

O2-

OChem OChem

O1-

538

536

534

532

O1-

530

528

526

538

O

536

534

532

530

2-

O

I (a.u.)

I (a.u.) 538

534

528

526

528

526

(d) SrMnO3

(c) CaMnO3

O Chem

536

O Chem

O 1-

532

2-

O 1-

530

528

526

538

536

534

E b (e V)

532

530

Eb(e V)

Fig. 9.7 O 1s photoelectron spectra (points) and the fitted results (curves) of MnO, Mn3 O4 , CaMnO3 , and SrMnO3 powder samples

Table 9.2 Fitted results for O 1s photoelectron spectra of MnO, Mn3 O4 , CaMnO3 , and SrMnO3 , including the average valences of O, Sr (Ca), and Mn cations, V alO , V alS(alC) , and V alM

Sample

Average valence V alO

V alS(alC)

V alM

MnO

−1.73

–

1.73

Mn3 O4

−1.78

–

2.37

CaMnO3

−1.54

2.00

2.62

SrMnO3

−1.66

2.00

2.98

Sr) and B sublattice (containing Mn), respectively. Thirdly, the magnetic ordering of La1–x Srx MnO3 originates only in the B sublattice, resulting in magnetic coupling that depends on the 3d electron number (nd ) of Mn cations (including local and itinerant 3d electrons). Because nd = 4 for Mn3+ , nd = 5 for Mn2+ , this results in Mn3+ coupling ferromagnetically with Mn3+ , while Mn2+ couples antiferromagnetically with Mn3+ according to the IEO model and the detailed explanations in Sect. 5.5. According to the observed results reported by Urushibara et al. [3] and our group [1, 2], the maximum magnetic moment value of La1–x Srx MnO3 , 4.2 μB , was reported at x = 0.15 (see Sect. 9.1.1), which is slightly higher than the magnetic moment (4

154

9 Magnetic Ordering and Electrical Transport …

μB ) of a Mn3+ cation. It was assumed that all Mn cations are Mn3+ cation at x = 0.15 and that the ionicity f M0.15 = 1.00. Because LaMnO3 has antiferromagnetic structure [3, 4, 7, 8] and zero magnetic moment [4], and the ratio of Mn2+ /Mn3+ magnetic moments is 5/4, the ratio of Mn2+ /Mn3+ contents can be calculated as 4/5. Then, the ionicity of Mn in LaMnO3 f M0.00

  5 4 /3 = 0.8519. = 2× +3× 9 9

(9.1)

In Eq. (9.1), the denominator 3 is attributed to the fact that the ionicity of Mn is 1.0 when the maximum valence value of Mn in La1−x Srx MnO3 is 3.0. To fit the observed trend in the sample magnetic moments as a function of x when x ≤ 0.15 in Fig. 9.4, we assumed that the ionicity of the Mn cations varied as f Mx = sin(θ1 + cx), (0.00 ≤ x ≤ 0.15). By applying the conditions, f M0.00 = 0.8519 and f M0.15 = 1.00, we could then easily calculate the fitting parameters and obtain a fit: f Mx = sin(1.0196 + 3.6747x), (0.00 ≤ x ≤ 0.15).

(9.2)

In Eq. (9.2), the angle is expressed in radian. Using Eq. (9.2), f Mx = 0.933, 0.981, and 1.00 when x = 0.05, 0.10, and 0.15, which are higher than the f O values of oxygen in Table 9.1. This may be caused by two factors: first, there may be La2+ cations in addition to La3+ cations. Second, the surface effects of the powder samples may be playing a role. Lee et al. [9] found that the ratios of O1− /O2− peak intensities of La0.7 Sr0.3 MnO3 films are about 0.89, 0.63, and 0.52 when the incident photon energies are 100, 200, and 300 eV, respectively. Let M 2 and M 3 represent the ratios of Mn2+ and Mn3+ cations, using f M0.15 = 1.00, we have 2M2 + 3M3 = f Mx , and M2 + M3 = 1. 3

(9.3)

Then, M2 = 3 − 3 f Mx ,

M3 = 1 − M2 (0.00 ≤ x ≤ 0.15).

(9.4)

Based on these expressions and the magnetic moments of Mn2+ and Mn3+ , 5 μB and 4 μB , the magnetic moments of the samples could then be obtained as follows: μcal = 4M3 − 5M2 , (0.00 ≤ x ≤ 0.15).

(9.5)

9.1 Ferromagnetic and Antiferromagnetic Coupling …

155

To explain the fact that the magnetic moments decrease with an increase in x when x > 0.15, we assumed that the samples have a canted ferromagnetic structure and that the angle between the sample magnetic moment and the magnetic moments of Mn3+ cations is greater than zero when x > 0.15. We also assumed that the magnetic moments vary linearly with x, as shown below: μcal = 4[1 − 0.72(x − 0.15)],

(0.15 ≤ x ≤ 0.40),

(9.6)

where the parameter 0.72 was obtained by fitting the model to the curve in Fig. 9.4. As was mentioned earlier, Fig. 9.4 shows the measured and calculated dependencies of the magnetic moments of La1−x Srx MnO3 on the Sr-doping level, x, where the solid curve represents the fitted result obtained using Eqs. (9.5) and (9.6). The different points represent the magnetic moments observed by our group ( ) [1, 2], those reported by Urushibara et al. ( ) [3], and those observed by Jonker and Van Santen ( ) [4]. One can see that our fitted curve agrees well with the average value of the observed results. Finally, the average angle φ between the sample magnetic moment and the magnetic moments of Mn3+ cations could also be obtained from Eq. (9.6) and is given by φ=

180 arccos[1 − 0.72(x − 0.15)], (0.15 ≤ x ≤ 0.40), π

(9.7)

where the angle φ is expressed in degrees. A simple calculation shows that, at the highest level of doping considered, that is, at x = 0.40, the canting angle is significantly larger than zero, i.e., φ = 34.9°.

9.2 Spin-Dependent and Spin-Independent Electrical Transport of Perovskite Manganites There have been many investigations on the electronic transport and magnetic properties of perovskite oxides [10–23] because of their potential applications in both electronic and magnetic information storage. Several models have been proposed for the electronic transport properties of these materials. One of the models is that there is polaron hopping below the Curie temperature (T C ) on the basis of the doubleexchange interaction model [7, 8]. In this view, the current carriers of perovskite manganites originate from the d electrons of manganese cations. Another view was proposed by Alexandrov et al. [24, 25], who argued that the current carriers of perovskite manganites originate from the p holes of oxygen anions rather than the d electrons of manganese cations, based on the experimental results of EELS [26,

156

9 Magnetic Ordering and Electrical Transport …

27] and X-ray absorption spectroscopy [28]. Alexandrov et al. [24, 25], and therefore established a current-carrier density collapse (CCDC) model for the electrical transport mechanism, and many researchers [29–32] continued to use this model. However, this model falls short of explaining why current-carrier density collapses near the Curie temperature. To better understand the relationship between magnetic and electrical transport properties of perovskite manganites, our group [10, 11] proposed a model with two channels of electrical transport (TCET) based on the IEO model as follows.

9.2.1 A Model with Two Channels of Electrical Transport for ABO3 Perovskite Manganites Figure 9.8a and b shows the equivalent circuit of the TCET model for single-crystal and polycrystalline perovskite manganites, respectively. As mentioned in Sect. 4.2, there are two sublattices in ABO3 perovskite manganites: the A–O sublattice and the B–O sublattice. According to the IEO model, the two O 2p electrons in the outer orbit of an O2− anion, which have opposite spin directions, become itinerant electrons of the two sublattices. In La1−x Srx MnO3 , the movement of O 2p electrons along the first channel, i.e., the O–A–O–A–O ionic chain where La and Sr occupy A sites, is a spin-independent process because there is no magnetic ordering. The movement of O 2p itinerant electrons along the second channel, i.e., the O–Mn–O–Mn–O ionic chain where Mn occupies B sites, is a spin-dependent process below the Curie temperature because there is magnetic ordering, but it is a spin-independent process above the Curie temperature. The resistance of the spin-dependent channel is represented by two resistors in series, R1 and R2 (with an equivalent resistivity of ρ 1 and ρ 2 ), where ρ 1 includes the residual resistivity and the resistivity resulting from the scattering due to the

Fig. 9.8 Equivalent circuit of the TCET model for a single-crystalline and b polycrystalline perovskite manganites. R1 includes the residual resistance and the resistance resulting from scattering by the crystal lattice. R2 originates from the spin orientations of the itinerant electrons deviated from the orientation of their ground states because of thermal fluctuation. R3 is spin-independent resistance. R4 originates from the scattering by the crystallite interfaces in the polycrystalline sample [10, 11]

9.2 Spin-Dependent and Spin-Independent Electrical …

157

crystal lattice vibration, and ρ 2 originates from the spin orientations of the itinerant electrons to deviate from the orientation of their ground states because of thermal fluctuation. The resistance of the spin-independent channel is labeled R3 (with an equivalent resistivity of ρ 3 ). Therefore, according to the equivalent circuit in Fig. 9.8a, the resistance, R, and resistivity, ρ, of a single-crystal sample can be calculated as follows: (R1 + R2 )R3 (ρ1 + ρ2 )ρ3 , ρ= . R1 + R2 + R3 ρ1 + ρ2 + ρ3

R=

(9.8)

In Fig. 9.8b, R4 represents the resistance originated from scattering by the crystallite interfaces in a polycrystalline sample, which is discussed in Sect. 9.2.4.

9.2.2 Fitting the Curves of Resistivity Versus Test Temperature of Single-Crystal La1−x Srx MnO3 The temperature dependence of the electrical resistivity of the single-crystal La1−x Srx MnO3 samples was fitted by using the TCET model by our group [10, 11]. The fitted results are shown in Fig. 9.9. It can be seen that the fitted values (curves) are very close to the observed values (points originated in the curves reported by Urushibara et al. [3]). When 0.175 ≤ x ≤ 0.40, the resistivities in Eq. (9.8) can be expressed as follows:     E2 E3 , ρ3 = a3 exp , ρ1 = ρ0 + a1 (T1 + T ) , ρ2 = a2 exp − kB T kB T 3

(9.9)

La1-xSrxMnO3

cm)

10

3

x=0.00

x=0.10

10 1

x=0.05 x=0.15

10 -1

x=0.175

10

x=0.20 x=0.30

-3

x=0.40

0

100

200

300

400

500

T (K)

Fig. 9.9 Fitted (curves) the temperature dependence of electrical resistivity for the single-crystal La1−x Srx MnO3 samples [10, 11], where observed results (points) originated the curve report by Urushibara et al. [3]

158

9 Magnetic Ordering and Electrical Transport …

where k B is the Boltzmann constant. The parameters in Eq. (9.9) were determined by fitting the experimental curves of ρ versus T, and the results are listed in Table 9.3. For example, the parameters for the sample with x = 0.20 were determined as follows. First, it can be seen that the curve of ρ versus (T 1 + T )3 can be approximated as a straight line at lower temperatures by adjusting the parameter T 1 , as shown in Fig. 9.10a. We then determined the residual resistivity, ρ 0 , and the parameter a1 from the intercept and slope, respectively, of the straight line in Fig. 9.10a. Second, an approximately straight line of ln ρ versus 1/k B T can be seen at higher temperatures, as shown in Fig. 9.10b. We obtained the approximate values of ln a3 and E 3 from the intercept and slope, respectively, of the straight line in Fig. 9.10b. Third, we determined the parameters a2 and E 2 by fitting the curves of ρ versus T. In this fitting process, the parameters ρ 0 , a1 , and T 1 remained constant, but a3 and E 3 required slight adjustments. The curves of observed ρ, and fitted ρ 1 , ρ 2 , and ρ 3 versus T are shown in Fig. 9.10c. Finally, the dependence of the observed (data points) and fitted (curve) ρ of La0.8 Sr0.2 MnO3 on the test temperature, T, is shown in Fig. 9.10d. It can be seen that the fitted curve is very close to the observed results. For the antiferromagnetic semiconductor samples with x = 0.00 and 0.05, the curves of ρ versus T measured   above the Néel temperature can only be fitted by the equation ρ = a3 exp kEB 3T . The values of the fitted parameters a3 and E 3 are listed in Table 9.3. For the samples with x = 0.10 and 0.15, the curves of ρ versus T  E1 can be fitted using Eqs. (9.8) and (9.9), but the expression ρ1 = a11 exp kB T must substitute for ρ 1 in Eq. (9.9). The values of the fitted parameters a11 , E 1 , a2 , E 2 , a3 , and E 3 are listed in Table 9.3.

9.2.3 Fitting the Curves of Resistivity Versus Test Temperature of La0.60 Sr0.40 Fex Mn1−x O3 Polycrystalline Samples Our group prepared polycrystalline La0.60 Sr0.40 Fex Mn1−x O3 samples (0.00 ≤ x ≤ 0.30) [10, 11]. The XRD data analyses indicated that all samples had only a single ABO3 perovskite phase with space group R3c and that the crystallite sizes of all samples are larger than or close to 100 nm. Using the above TCET model, we fitted the resistivity curves of the samples versus test temperature, as shown in Fig. 9.11. It can be seen that the fitted results are very close to the observed values, indicating that the TCET model is reasonable. The fitted parameters are listed in Table 9.4. The resistance originated from crystallite interface scattering is discussed in the following subsection.

5.5 × 10−11

10−11

2.3 × 10−4

8.5 × 10−5

10−5

9.5 ×

0.175

0.20

0.30

0.40

7.5 ×

1.5 × 10−10

–

0.15

4.2 ×

2.4 ×

–

–

–

10−10

–

0.10

10−4

–

0.05

–

–

0.00

a1 ( cm/K3 )

ρ 0 ( cm)

Sr ratio x

35

30

20

10

–

–

–

–

T 1 (K)

–

–

–

–

0.0014

0.009

–

–

a11 ( cm)

–

–

–

–

0.08

0.15

–

–

E 1 (eV)

E 2 (eV)

7×

109

9 × 109

4.5 × 1010

0.880

0.860

0.730

0.650

0.515

1 × 1011 8×

0.115

1010

–

–

–

4 × 105

–

a2 ( cm)

0.0041

0.0084

0.0095

0.0112

0.0115

0.0050

0.0220

0.2440

a3 ( cm)

5 × 10−5

1 × 10−4

0.0354

0.0515

0.0750

0.1500

0.1700

0.1750

E 3 (eV)

Table 9.3 Parameters of La1−x Srx MnO3 obtained by fitting the experimental curves of resistivity (ρ) versus test temperature (T ) reported by Urushibara et al. [3]. The ρ 0 is the residual resistivity in ρ 1 ; the a1 and T 1 are the parameters (in the ρ 1 ) which result from scattering by the crystal lattice; the a11 and E 1 are the parameters in the ρ 1 of the samples with x = 0.10 and 0.15; the a2 and E 2 are the parameters (in the ρ 2 ) which originate from the spin orientations of the itinerant electrons deviated from the orientation of their ground states because of thermal fluctuation. The a3 and E 3 are the parameters (in the ρ 3 ) which originate from the spin-independent transition

9.2 Spin-Dependent and Spin-Independent Electrical … 159

160

9 Magnetic Ordering and Electrical Transport …

0.6

-3.4

T1+T)3

=2.28456E-4 +1.49857E-10

(a) La0.8Sr0.2MnO3

-3.8

2 4 6 3 3 (T1+T ) (10 K )

24

6

(d)

(c)

cm)

3

La 0.8 Sr 0.2 MnO 3

10 1

(m

cm)

1/kBT)

-3.6

0

(m

ln = -4.658+0.0354 La0.8Sr0.2MnO3

0.3

100

(b)

ln

cm)

0.9

(m

1.2

28 32 -1 (1/kBT) (eV )

36

La0.8Sr0.2MnO3

10

1

1 2

0.1 0

100 200 300 400 500 T (K)

0.1

0

100 200 300 400 500 T(K)

Fig. 9.10 Fitting process of the temperature (T ) dependence of the resistivity (ρ) for singlecrystalline material La0.8 Sr0.2 MnO3 . The data points in a, b, d represent values observed by Urushibara et al. [3]. a Fitting results of the curve of ρ versus (T 1 + T )3 at lower temperatures. b Fitting results of the curve of lnρ versus 1/k B T at higher temperatures. c Curves of observed ρ (squares) [3], fitted ρ 1 (triangles), fitted ρ 2 (inverted triangles), and fitted ρ 3 (circles) versus T. (d) Dependence of observed (data points [3]) and fitted (curve) ρ on T

9.2.4 Discussion on Factors Affecting Electrical Transport Property (1)

Spin-dependent and spin-independent electrical transport in perovskite manganites

Figure 9.10c shows that at lower temperatures, the spin-independent resistivity, ρ 3 , of La0.8 Sr0.2 MnO3 prepared by Urushibara et al. was so high that the electrical transport occurred along the spin-dependent channel, and the value of ρ was close to ρ 1 , which increased with the test temperature increase owing to the thermal vibration of ions. When T was close to T C , the transition probability of the itinerant electrons decreased and ρ 2 increased because the spin orientations of the itinerant electrons deviate from the direction of their ground states. When T was higher than T C , the spin orientations of the itinerant electrons changed into disorder, and the electrical transports are spin independent along both O–A–O–A–O and O–B–O–B–O ionic chains.

9.2 Spin-Dependent and Spin-Independent Electrical … 2

10

1

cm)

La 0.60 Sr0.40 Fex Mn 1- x O3

x =0.30 x =0.25 x =0.20

10

161

x =0.15 0

10

10

x =0.10

-1

x =0.05 x =0.00

-2

10

0

100

200

300

400

T (K) Fig. 9.11 Temperature dependence of observed (points) and fitted (curves) electrical resistivity of the polycrystalline samples La0.60 Sr0.40 Fex Mn1−x O3 [10, 11]

Table 9.4 Parameters of polycrystalline La0.6 Sr0.4 Fex Mn1−x O3 obtained by fitting the experimental curves of ρ versus T from our work [10, 11]. The ρ 0 is the residual resistivity in ρ 1 ; the a1 and T 1 are the parameters (in the ρ 1 ) which result from scattering by the crystal lattice; the a2 and E 2 are the parameters (in the ρ 2 ) which originate from the spin orientations of the itinerant electrons deviate from the orientation of their ground states because of thermal fluctuation. The a3 and E 3 are the parameters (in the ρ 3 ) which originate from the spin-independent transition. All these parameters include the effect of the crystallite interfaces Fe ratio x

ρ 0 ( cm) a1 ( cm/K3 )

0.00

0.0072

2.794 × 10−10

T 1 (K) a2 ( cm) E 2 (eV) a3 ( cm) E 3 (eV) 150

10−9

107

0.05

0.060

0.0044

0.0650 0.0865

0.05

0.0172

1.050 ×

0.40

0.090

0.0020

0.10

0.0737

1.116 × 10−8

63

25

0.110

0.0019

0.0995

0.15

1.1000

1.159 × 10−6

20

20,000

0.120

0.0018

0.1070

0.20

–

–

–

–

–

0.0030

0.1140

0.25

–

–

–

–

–

0.0056

0.1200

0.30

–

–

–

–

–

0.0114

0.1245

The resistivity of La0.6 Sr0.4 MnO3 , prepared by Urushibara et al., was 84 μ cm at 2 K [3], which is far higher than the resistivities of magnetic metals Fe (8.6 μ cm), Ni (6.14 μ cm), and Co (5.57 μ cm) at 0 °C. Therefore, the spin-dependent transport of itinerant electrons in perovskite manganites below T C is distinctly different from the spin-independent transport of free electrons in magnetic metals. In Sect. 12.5, the spin-independent transport of free electrons and the spindependent transport of itinerant electrons below Curie temperature in magnetic Ni (metal) and NiCu (alloys) are discussed. The spin-dependent transport of itinerant electrons in NiCu alloys is similar to that in perovskite manganites.

162

9 Magnetic Ordering and Electrical Transport …

Fig. 9.12 Schematic diagrams of the transit of itinerant electrons in a spin-dependent process. a An itinerant electron with up-spin moves along the O2− –Mn3+ –O1− –Mn3+ –O2− ion chain. b An itinerant electron moves along the O2− –Fe3+ –O1− –Mn3+ –O2− ion chain. Here, the majority spin orientation of 3d electrons in Mn3+ cations is up; the majority spin orientation of 3d electrons in Fe3+ cations and O 2p electrons in O1− anions is down. The arrows “↑” and “↓” represent the electron with up-spin and down-spin, respectively. The symbol “” represents a 2p hole, which in the illustrated case represents the absence of an up-spin electron

(2)

Effect of proportion of the antiferromagnetic phase on the resistivity of La0.6 Sr0.4 Fex Mn1−x O3 (0.00 ≤ x ≤ 0.30)

According to Sect. 9.1, all Mn and Fe ions are trivalent in La0.6 Sr0.4 Fex Mn1−x O3 . On the basis of the IEO model, an itinerant electron with up-spin in a sample with no Fe doping (La0.6 Sr0.4 MnO3 ) moves along the O2− –Mn3+ –O1− –Mn3+ –O2− ion chain, as shown in Fig. 9.12a. The itinerant electron always occupies the highest energy level of Mn3+ and the lowest energy level O2− (the magnetic moment of O2− is antiferromagnetic coupling with that of Mn3+ ). In this transition process, the itinerant electron consumes a small fraction of the energy of the system. When an itinerant electron with up-spin moves along the O2− –Fe3+ –O1− –Mn3+ –O2− ion chain, it occupies the highest energy level of Fe3+ , as shown in Fig. 9.12b because Fe3+ (3d5 ) couples antiferromagnetically with Mn3+ (3d4 ). In this transition process, the itinerant electron must consume more energy, lowering transition probabilities for the itinerant electrons and increasing resistivity. Therefore, the resistivity increases with increasing antiferromagnetic composition. The difference between the energy consumed by the itinerant electrons in polycrystalline La0.6 Sr0.4 MnO3 and La0.6 Sr0.4 Fe0.1 Mn0.9 O3 was estimated to be 8.4 meV, based on their Curie temperatures (364 and 267 K, respectively; see the data of Sr0.40 series samples in Tables 9.5 and 9.6 of Sect. 9.3). (3)

Resistivity (ρ 4 ) originating from scattering at the crystallite interfaces of polycrystalline samples

It is interesting to compare the resistivity of the polycrystalline La0.6 Sr0.4 MnO3 sample (ρ P ) prepared by our group [10, 11] with that of the single-crystal La0.6 Sr0.4 MnO3 (ρ S ) reported by Urushibara et al. [3], as shown in Fig. 9.13a. We found that ρ P is 97 times (at 50 K) and 9 times (at 360 K) the value of ρ S , suggesting that the resistivity (ρ 4 ) originating from scattering at the crystallite interfaces of

9.2 Spin-Dependent and Spin-Independent Electrical …

10

-3

10

80 40

-4

100

200

300

24

cm)

10

-2

120

La0.6Sr0.4MnO3

(m

cm)

(a)

163

(b) La0.6Sr0.4MnO3

20 16

287 K

12 8

0 400

100

200

T (K)

10

3

10

2

10

1

10

LaMnO3

400

-2

3.94

ae (Å)

cm)

(c)

300

T (K)

10 La0.6Sr0.4Mn0.7Fe0.3O3

(d) La1-xSrxMnO3

3.92 3.90

-3

3.88 La0.6Sr0.4Mn1-xFexO3

0

10 100

200

T (K)

300

0.0

0.1

0.2

0.3

0.4

Doping level x

Fig. 9.13 a Temperature dependence of the resistivities, ρ P [10] and ρ S [3], of polycrystalline and single-crystalline La0.6 Sr0.4 MnO3 , and their ratio, ρ P /ρ S . b Temperature dependence of the observed resistivity, ρ P , and the estimated resistivity resulting from crystallite interface scattering, ρ 4 , of polycrystalline La0.6 Sr0.4 MnO3 . c Temperature dependence of ρ P of La0.6 Sr0.4 Fe0.3 Mn0.7 O3 and ρ S of LaMnO3 , and their ratio, ρ P /ρ S . d Dependence of the equivalent cubic lattice constant, ae , of the single-crystalline La1−x Srx MnO3 samples [3] and the polycrystalline La0.6 Sr0.4 Fex Mn1−x O3 samples [10] on the doping level, x. The ae was calculated using the crystal cell volume

the polycrystalline sample was far higher than those originating from lattice thermal vibrations (ρ 1 ) and spin-dependent scattering (ρ 2 ). Therefore, for the polycrystalline manganites, the TCET model can be represented by the equivalent circuit shown in Fig. 9.8b. Figure 9.13b shows a comparison of ρ 4 with ρ P , where ρ 4 was estimated using ρ 4 = ρ P − ρ S . According to the report by Urushibara et al. [3], the temperature, T MI , with maximum resistivity is very close to the value of T C for single-crystal La1−x Srx MnO3 (0.175 ≤ x ≤ 0.40). However, the values of T MI were distinctly lower than T C for the polycrystalline La0.6 Sr0.4 Fex Mn1−x O3 (0.00 ≤ x ≤ 0.15) samples [10, 11]. For the La0.6 Sr0.4 MnO3 sample, the temperature at which ρ 4 reached a maximum, 287 K (see Fig. 9.13b), is distinctly lower than its T C (364.5 K) due to the scattering at the crystallite interfaces of the polycrystalline sample. (4)

Effect of crystal lattice constants on the resistivities of samples

For the antiferromagnetic semiconductor, polycrystalline La0.6 Sr0.4 Fe0.3 Mn0.7 O3 , its resistivity was far lower than that of single-crystal LaMnO3 , with a ρ P /ρ S ratio

164

9 Magnetic Ordering and Electrical Transport …

of 0.51% at 270 K and 0.58% at 300 K, as shown in Fig. 9.13c. A comparison of Fig. 9.13a and c shows an interesting phenomenon: the change in the amplitude of the resistivity from that of the polycrystalline ferromagnetic conductor, La0.6 Sr0.4 MnO3 , to that of the antiferromagnetic semiconductor, La0.6 Sr0.4 Fe0.3 Mn0.7 O3 , is far lower than the change in the amplitude of the resistivity from that of the single-crystal ferromagnetic conductor, La0.6 Sr0.4 MnO3 , to that of the single-crystal antiferromagnetic semiconductor, LaMnO3 . The primary causes of this phenomenon may be the distinct difference in the crystal lattice constants of the two systems, and the proportion of the antiferromagnetic phase. The equivalent cubic lattice constants (ae ) of single-crystal La1−x Srx MnO3 [3] and polycrystalline La0.6 Sr0.4 Fex Mn1−x O3 [10] are shown in Fig. 9.13d, which were calculated using the average volume per molecule of each sample. The samples of La0.6 Sr0.4 Fex Mn1−x O3 (0.0 ≤ x ≤ 0.3) and La1−x Srx MnO3 (0.175 ≤ x ≤ 0.40) exhibited rhombohedral structures, while La1−x Srx MnO3 (0.00 ≤ x ≤ 0.15) was reported to have an orthorhombic structure [3]. The value of ae of the polycrystalline La0.6 Sr0.4 MnO3 is very close to that of the single-crystal La0.6 Sr0.4 MnO3 . However, the value of ae of polycrystalline La0.6 Sr0.4 Fex Mn1−x O3 [10] increased for 0.006 Å, from 3.872 Å (x = 0.00) to 3.878 Å (x = 0.30), while the value of ae of single-crystal La1−x Srx MnO3 increased for 0.068 Å, from 3.874 Å (x = 0.40) to 3.942 Å (x = 0.00). According to Sect. 9.1, there are only Mn3+ cations in La1−x Srx MnO3 (0.15 ≤ x ≤ 0.40). Therefore, in La1−x Srx MnO3 (0.15 ≤ x ≤ 0.40), the increase in resistivity from x = 0.40 to x = 0.15 occurred because of the increase in the crystal lattice constant and the decrease in the canting angle between Mn3+ ionic magnetic moments and the sample magnetic moments, while the increase in resistivity from x = 0.15 to x = 0.00 occurred because of the increases in both the crystal lattice constant and the proportion of the antiferromagnetic phase. (5)

Discussion for curves of resistivity versus test temperature of single-crystal La0.9 Sr0.1 MnO3 and La0.85 Sr0.15 MnO3

An increase (in increments of 0.01 Å, see the following example in Sect. 9.3.3) in ae may rapidly reduce the transition probability of the itinerant electrons whenever spin-dependent or spin-independent transitions take place, increasing the resistivity. This may be the reason for why the single-crystal samples of La0.9 Sr0.1 MnO3 and La0.85 Sr0.15 MnO3 exhibited high resistivity at low temperatures, as shown in Fig. 9.9. The ae values of these samples are distinctly larger than that of the single-crystal samples of La0.6 Sr0.4 MnO3 , as shown in Fig. 9.13d. For La0.85 Sr0.15 MnO3 , ρ > 500  cm when T < 10 K because of the large crystal lattice constant, ae , which results in a lower spin-dependent transition probability of the itinerant electrons. The value of ρ decreased with increasing T below 202 K because of the thermal energy of the itinerant electrons, which increased their spin-dependent transition probability. In the temperature range of 202–234 K, ρ increased with increasing T because the spin orientation of the itinerant electrons deviated from their ground-state direction, decreasing their spin-dependent transition probability.

9.2 Spin-Dependent and Spin-Independent Electrical …

(6)

165

Activation energy (E 3 ) of spin-independent electron transition

The values of the activation energy, E 3 , of the spin-independent electron transition in the polycrystalline La0.6 Sr0.4 Fex Mn1−x O3 (0.00 ≤ x ≤ 0.30) samples ranged from 65.0 to 124.5 meV (see Table 9.4), which are close to the values (90.5–148.3 meV) of the polaron activation energy, E p , obtained by Liu et al. with the CCDC model for La0.7 Ca0.3 Tix Mn1−x O3 (0.00 ≤ x ≤ 0.07) [29]. The comparable results indicate that the TCET model in our work [10] is similar to the CCDC model [24, 25, 29– 32]. However, the physical mechanism of the TCEP model, which includes spindependent and spin-independent electron transitions, is easier to be understood than that of the CCDC model, which assumes that the current-carrier density changes sharply at the transition temperature from ferromagnetic to paramagnetic phase. Tables 9.3 and 9.4 show that among the ferromagnetic samples, E 3 of polycrystalline La0.6 Sr0.4 MnO3 (65.0 meV) is far higher than that of the single-crystal La0.6 Sr0.4 MnO3 (0.05 meV) owing to scattering at the crystallite interfaces in the polycrystalline sample. However, among the antiferromagnetic samples, E 3 of polycrystalline La0.6 Sr0.4 Fe0.3 Mn0.7 O3 (124.5 meV) is distinctly lower than that of the antiferromagnetic LaMnO3 (175 meV), which may be caused by the difference in crystal lattice constants and different proportion of the antiferromagnetic phase, as discussed in the previous section. The value of the parameter a3 (0.0044  cm, see Table 9.4) of the polycrystalline conductor, La0.6 Sr0.4 MnO3 , is very close to that of the single-crystal conductor, La0.6 Sr0.4 MnO3 (0.0041  cm, see Table 9.3). However, the value of a3 (0.0114  cm) of the polycrystalline semiconductor, La0.6 Sr0.4 Fe0.3 Mn0.7 O3 , is far lower than that of the single-crystal semiconductor, LaMnO3 (0.2440  cm). This may be attributed to the effect of the crystal lattice constant, as discussed in the above section (4). (7)

Regression coefficients of resistivities ρ 1 and ρ 2

As shown in Eq. (9.9) and Tables 9.3 and 9.4, the spin-dependent resistivities, ρ 1 and ρ 2 , are expressed by parameters (or regression coefficients) that play interesting roles: (i)

(ii)

The resistivity ρ 1 in Eq. (9.9) is determined by three parameters: the residual resistivity, ρ 0 ; the parameters a1 and T 1 resulting from scattering by the crystal lattice. All three parameters of the polycrystalline La0.6 Sr0.4 Fex Mn1−x O3 samples—ρ 0 = 0.0072–1.100  cm, a1 = 2.794 × 10−10 –1.159 ×10−6 ( cm)/K3 , T 1 = 150–20 K—are higher than those of the single-crystal La1−x Srx MnO3 samples—ρ 0 = 7.5×10−5 –9.5×10−4  cm, a1 = 4.2 ×10−11 – 2.4 × 10−10 ( cm)/K3 , and T 1 = 35–10 K—because of the effect of crystallite interfaces in the polycrystalline samples. The resistivity ρ 2 in Eq. (9.9) is determined by two parameters, the amplitude a2 and the activation energy E 2 , which originated from spin-dependent scattering. The value of E 2 of the single-crystal La1−x Srx MnO3 samples decreased from 0.880 eV (x = 0.40) to 0.515 eV (x = 0.15) due to the competition between two factors. One factor was the decrease in the canting angle between the sample

166

9 Magnetic Ordering and Electrical Transport …

magnetic moment and Mn3+ cation magnetic moments from 34.9° (x = 0.40) to 0 (x = 0.15) [10]. Another factor was the increase in the crystal cell volume with decreasing x. The value of E 2 of the polycrystalline La0.6 Sr0.4 Fex Mn1−x O3 samples increased from 0.06 eV (x = 0.00) to 0.12 eV (x = 0.15), which may be attributed to the increase in Fe-doping level because the magnetic moments of Fe3+ cations couple antiferromagnetically with those of Mn3+ cations. The a2 values of the polycrystalline La0.6 Sr0.4 Fex Mn1−x O3 samples are far lower than those of the single-crystal La1−x Srx MnO3 samples, possibly because of the intense scattering by the crystallite interfaces to reduce the effect of spin-dependent scattering. In summary, to explain the dependence of the resistivity (ρ) of ABO3 perovskite manganites on the test temperature (T ), our group proposed a model with TCET: a spin-independent channel (with resistivity ρ 3 ) and a spin-dependent channel (with resistivity ρ 1 + ρ 2 ). At low temperatures, ρ 3 is very high, and the electrical transport occurs along the spin-dependent channel. When T is far lower than the Curie temperature (T C ), ρ is very close to ρ 1 , which increases with increasing T owing to the thermal vibrations of the crystal lattice. When T is close to T C , ρ 2 increases rapidly owing to the spin orientations of the itinerant electrons deviated from the orientation of their ground states. When T is higher than T C , the spin orientations of the itinerant electrons become disordered, and electrical transport occurs along both channels with the spin-independent transition. Several determining factors affecting the resistivity have been discussed: (i) The increase in the proportion of the antiferromagnetic phase may increase ρ. (ii) The resistivity ρ 4 originating from crystallite interface scattering in the polycrystalline samples is far higher than ρ 1 and ρ 2 at temperatures below T C . (iii) Increases in the crystal lattice constant, in increments as small as 0.01 Å (see the following example in Sect. 9.3.3), may result in rapid increases in ρ, which may be the reason for the high resistivity of La0.9 Sr0.1 MnO3 and La0.85 Sr0.15 MnO3 at low temperatures.

9.3 Experimental Evidence on the Canting Angle Magnetic Structure in Perovskite Manganites Our group [11, 33] synthesized three series of perovskite manganite polycrystalline samples La0.85 Ba0.15 Fex Mn1−x O3 (0 ≤ x ≤ 0.2), La0.60 Ba0.40 Fex Mn1−x O3 (0 ≤ x ≤ 0.2), and La0.6 Sr0.4 Fex Mn1−x O3 (0.00 ≤ x ≤ 0.30) using the sol–gel method. The three series of samples are hereafter referred to as Ba0.15, Ba0.40, and Sr0.40, respectively. XRD patterns showed that all samples had single-phase rhombohedral ABO3 perovskite structures (space group R3c). The volume-averaged diameter of crystallites in the samples was estimated from the Scherrer equation, which was greater than or close to 100 nm for all the samples. The average molecular magnetic moments of three series of samples at 10 K decrease with increasing Fe doping. Assuming that the magnetic moments of Fe3+ couple antiferromagnetically with those of Mn3+ , the Fe doping dependencies of the sample magnetic moments were fitted successfully according to the IEO model and the method described in Sect. 9.1.

9.3 Experimental Evidence on the Canting Angle Magnetic Structure

167

9.3.1 Analyses for the Crystal Structure of the Samples The structural parameters of the three series of samples were determined by performing Rietveld refinement using the FullProf program, and detailed results of the structural refinement are listed in Table 9.5. Figure 9.14 shows the dependence of the crystal lattice constants (a and c), the crystal cell volume (v), the Mn(Fe)–O bond length (d BO ), and the Mn(Fe)–O–Mn(Fe) bond angle (θ ) on the Fe-doping level (x). The following characteristics were observed: Table 9.5 Crystal structural parameters of Ba0.15, Ba0.40, and Sr0.40 samples at room temperature: where the La, Ba, and Sr cations locate at (0.00, 0.00, 0.25), Mn and Fe cations locate at (0.00, 0.00, 0.00), O anions locate at (u, 0.00, 0.25); a and c are the crystal lattice constants; v is the crystal cell volume; d BO and θ are the Mn(Fe)–O distance and Mn(Fe)–O–Mn(Fe) bond angle, all of which were obtained by fitting the XRD patterns using Rietveld refinement; s is the goodness-of-fit factor, Rp and Rwp are the profile factor and the weighted profile factor, respectively. The ae is equivalent cubic lattice constant, which was calculated using the crystal cell volume Fe a (Å) ratio x

c (Å)

v (Å3 )

ae (Å)

d BO (Å) θ (°)

s

RWP (%) RP (%) u (a)

Ba0.15 0.00 5.5556 13.456 359.68 3.9137 1.9569

165.26 1.27 5.92

4.64

0.4542

0.02 5.5557 13.458 359.75 3.9140 1.9570

165.27 1.36 6.08

4.77

0.4542

0.04 5.5558 13.460 359.80 3.9141 1.9571

165.31 1.32 6.03

4.69

0.4543

0.06 5.5559 13.462 359.86 3.9144 1.9572

165.39 1.35 5.92

4.64

0.4545

0.08 5.5559 13.464 359.92 3.9146 1.9573

165.56 1.35 6.06

4.66

0.4551

0.10 5.5560 13.465 359.96 3.9147 1.9574

165.71 1.29 5.80

4.57

0.4556

0.15 5.5561 13.469 360.07 3.9151 1.9576

166.24 1.36 5.94

4.68

0.4572

0.20 5.5562 13.474 360.22 3.9157 1.9579

166.64 1.30 5.67

4.46

0.4585

0.00 5.5320 13.530 358.57 3.9097 1.9548

175.00 1.69 8.34

4.92

0.4843

0.02 5.5320 13.536 358.75 3.9103 1.9552

175.91 1.36 6.51

4.77

0.4871

0.04 5.5321 13.543 358.94 3.9110 1.9555

176.40 1.45 6.95

4.77

0.4886

0.06 5.5322 13.548 359.09 3.9116 1.9558

176.84 1.39 6.65

4.77

0.4900

0.08 5.5324 13.552 359.22 3.9120 1.9560

177.41 1.47 6.65

4.51

0.4917

0.10 5.5325 13.555 359.33 3.9124 1.9562

178.28 1.38 6.27

4.53

0.4944

0.15 5.5327 13.565 359.62 3.9135 1.9568

179.47 1.46 6.53

4.45

0.4981

0.20 5.5331 13.575 359.93 3.9146 1.9573

179.31 1.43 6.42

4.48

0.5024

0.00 5.4861 13.356 348.14 3.8714 1.9357

172.05 1.34 5.40

4.25

0.4752

0.05 5.4898 13.359 348.69 3.8734 1.9367

171.66 1.32 5.44

4.24

0.4740

0.10 5.4920 13.362 349.04 3.8747 1.9374

171.22 1.30 5.39

4.24

0.4726

0.15 5.4936 13.365 349.30 3.8757 1.9379

170.86 1.30 5.36

4.22

0.4715

0.20 5.4945 13.366 349.44 3.8762 1.9381

170.67 1.37 5.41

4.20

0.4710

0.25 5.4961 13.368 349.69 3.8772 1.9386

170.35 1.27 4.89

3.88

0.4700

0.30 5.4971 13.370 349.88 3.8778 1.9390

170.16 1.26 4.78

3.78

0.4694

Ba0.40

Sr0.40

168

9 Magnetic Ordering and Electrical Transport …

(b)

(a)

a (Ba0.15)

Ba0.15

5.55

360

c / 6 (Ba0.40)

3

c/ 6

5.50

Ba0.40

v (Å )

a, c (Å)

a (Ba0.40) (Ba0.15)

a (Sr0.40)

356 352

5.45

c/ 6

Sr0.40

(Sr0.40) 348

0.0

0.1

0.2

0.0

0.3

0.1

x

(c)

0.3

0.2

0.3

(d)

1.960

Ba0.40

Ba0.15

176

1.955

dBO(Å)

0.2

x

Ba0.40

1.950

172

1.945

168

1.940 1.935

Sr0.40

Sr0.40

Ba0.15

164 0.0

0.1

0.2

x

0.3

0.0

0.1

x

√ Fig. 9.14 For the three series of samples, variations in a the crystal lattice constants a and c/ 6, b crystal cell volume v, c Mn(Fe)–O bond length d BO , and d Mn(Fe)–O–Mn(Fe) bond angle θ as functions of the Fe concentration x, determined by the XRD patterns using Rietveld refinement

(1)

(2)

(3)

(4)

The values of v and d BO of the three series of samples increase with increasing doping level. Comparing the three series samples, v and d BO have similar variation tendencies for every value of x: v(Ba0.15) > v(Ba0.40) > v(Sr0.40), d BO (Ba0.15) > d BO (Ba0.40) > d BO (Sr0.40). The θ values of Ba0.40 and Ba0.15 samples increase with increasing x, while the θ values of Sr0.40 samples decrease with increasing x. Comparing the three series samples: θ (Ba0.40) > θ (Sr0.40) > θ (Sr0.15). The slopes of the curves of v and θ versus x for the Ba0.15 samples are, respectively, smaller than those of the v–x and θ –x curves of the Ba0.40 and Sr0.4 samples. The √values of both a and c increase with increasing x. The values of both a and samples were smaller than a of the Ba0.15 samples but c/ 6 of the Ba0.40 √ larger than c/ 6 of the Ba0.15 samples. For the Ba0.15 samples, both a and c values exhibited only small√variations as x increased, and the a values were distinctly higher than the c/ 6 values for every value of x. For the Ba0.40 √ samples, however, although the a values √ the c/ 6 √ showed little variation, values increased nearly linearly from c/ 6< a (for x < 0.06) to c/ 6> a (for x > 0.10). These characteristics must affect the magnetic and electrical transport properties.

9.3 Experimental Evidence on the Canting Angle Magnetic Structure

169

9.3.2 Magnetic Measurement Results Figure 9.15a–c shows the temperature (T ) dependence of the specific magnetization (σ ) of Ba0.15, Ba0.40, and Sr0.40 samples measured from high to low temperature at an applied field of 0.05 T. Figure 9.16a–c shows the curves of dσ /dT versus T. The Curie temperatures, T C , defined as the temperature where dσ /dT reaches

(a)

La 0.85 Ba 0.15 Fe x Mn 1- x O 3 -1273K-10h

Am2/kg

20

x =0.10 0.08 0.06 0.04 0.02 0.00

0.15

10

0.20

0

0

100

(b)

Am2/kg

200

300

La 0.6 Ba 0.4 Fe x Mn 1- x O 3 -1473K-15h

30 20

x =0.10

0.08 0.06 0.04 0.02 0.00

10 0.15 0.20

0 0

100

(c)

20

300

400

1.8

0.20

1.2

10

200

La 0.6 Sr 0.4 Fe x Mn 1- x O 3 -1273K-10h

30

Am2/kg

Fig. 9.15 Temperature dependence of the specific magnetization σ (T ) of a Ba0.15, b Ba0.40 and c Sr0.40 samples at an applied magnetic field of 0.05 T

0.15

0.6 0.0

x =0.10 0.05 0.00

0.25 0.30

0

100

200

300

0 0

100

200 T (K)

300

400

170

(a)

La 0.85 Ba 0.15 Fe x Mn 1- x O

3

0.0

d /d T(Am2/kg.K)

Fig. 9.16 Temperature dependence of the dσ /dT of a Ba0.15, b Ba0.40, and c Sr0.40 samples at an applied magnetic field of 0.05 T

9 Magnetic Ordering and Electrical Transport …

-0.5 x =0.20 0.15

-1.0

0.10 0.08 0.06 0.04

-1.5 0

d /dT(Am2/kg.K)

(b)

100

x =0.00 0.02

200

300

La 0.6 Ba 0.4 Fe x Mn 1- x O3

0.0

0.15

-0.4

0.00

0.20

-0.03

-0.8 -0.06 0.15

-0.09

-1.2

0

0

100

(c)

d /dT(Am2/kg.K)

100 200

0.10 0.08 0.06 0.04 0.02 x =0.00

200

300

400

La 0.6 Sr 0.4 Fe x Mn 1- x O3 0 0.30

0.00

-1 -0.01

-2

-0.02

0.15

0.25

0.10 0.05

0.20 0

100 200 300

x =0.00

I

0

100

200

300

400

T (K)

its minimum value, are listed in Table 9.6. It can be seen that T C decreases with increasing x. Figure 9.17a–c shows the magnetic hysteresis loops of the samples measured at 10 K. The materials were found to be magnetized to saturation before the applied magnetic field reached 2 T, and the specific saturation magnetization (σ s ) decreased with increasing Fe doping. The average molecular magnetic moment of every sample (μobs ) was calculated using σ s ; values of σ s and μobs are listed in Table 9.6. Figure 9.18a and b shows T C and μobs as functions of x, where the data points

9.3 Experimental Evidence on the Canting Angle Magnetic Structure

171

Table 9.6 Specific saturation magnetization σs; average molecular magnetic moment μobs at 10 K; the Curie temperatures T C ; the temperatures with the maximum resistivity under the applied field 0 T and 2 T, T ρ (0T) and T ρ (2T); the temperature corresponding to the peak of the magnetoresistance, T MR σs (Am2 /Kg)

μobs (μB )

T C (K)

T ρ (0T) (K)

T ρ (2T) (K)

0.00

82.87

3.58

237

–

–

236

0.02

82.16

3.55

224

–

–

223

0.04

78.49

3.40

212

–

–

213

0.06

75.73

3.28

203

–

–

202

0.08

69.84

3.02

192

–

–

193

0.10

68.47

2.96

185

–

–

186

0.15

60.77

2.63

165

–

–

166

0.20

48.84

2.11

143

–

–

149

0.00

76.26

3.29

338

246

255

340

0.02

75.52

3.26

316

230

235

317

0.04

73.96

3.19

289

210

215

290

0.06

73.23

3.16

257

192

196

0.08

68.77

2.97

229

174

178

Fe ratio x

T MR (K)

Ba0.15

Ba0.40

257 230 155a

145

150

131a

65

–

–

–

0.13

53

–

–

–

80.55

3.192

364

292

320

–

0.05

78.53

3.112

321

231

238

–

0.10

78.46

3.110

267

192

196

–

0.15

72.77

2.886

187

133

138

117a

0.20

1.72

0.068

69

–

–

–

0.25

1.64

0.065

51

–

–

–

0.30

0.91

0.036

45

–

–

–

0.10

68.41

2.96

181

0.15

14.68

0.63

0.20

2.96

0.00

Sr0.40

T MR = T MRP , which is distinctly lower than T C , T ρ (0T), and T ρ (2T), other T MR is very close to T C

a This

represent the observed values, and the accompanying curves represent the fitting results (the fitting method is given in Sect. 9.3.4). The data in Fig. 9.18a and b can be summarized by the following characteristics: (i) when x = 0.0, T C (Sr0.4) > T C (Ba0.4) > T C (Ba0.15); however, the trend is opposite for μobs values, μobs (Sr0.4)< μobs (Ba0.4) < μobs (Ba0.15). (ii) μobs of Ba0.15 samples decreased linearly with increasing x up to x = 0.20. However, there was

172

(a) 120

La0.85Ba0.15FexMn1-xO

3

x=0.00

80

Am 2/kg )

40

x=0.20

(

0 -40 -80 -120 -2

(b) 90

-1

0

La 0.60 Ba 0.40 Fe x Mn 1- x O

1

2 x =0.00

3

Am2/kg )

60 30 0.20

(

0 -30 -60 -90

(c) 80

-2

-1

0

1

La 0.6 Sr 0.4 Fe xMn 1- xO 3

2 x=0.00

Am2/kg)

40 0.30

0

(

Fig. 9.17 Magnetic hysteresis loops of the a Ba0.15, b Ba0.40, and c Sr0.40 samples measured at 10 K

9 Magnetic Ordering and Electrical Transport …

-40 -80 -2

-1

0 0 H (T)

1

2

9.3 Experimental Evidence on the Canting Angle Magnetic Structure

(a)

TC /K

Fig. 9.18 a Curie temperature T C as a function of Fe ratio x. b average molecular magnetic moment μobs at 10 K as a function of Fe ratio x; the points represent the experimental values and the curves represent the fitting results

360

173

Sr0.40

270 180

Ba0.15

90

Ba0.40

0.0

0.1

0.2

0.3

(b) 4 3 obs (

B

)

Ba0.15

2 Ba0.4

1

Sr0.4

0 0.0

0.1

0.2

0.3

x

a characteristic doping level, x C = 0.10 and x C = 0.15, for the Ba0.40 and Sr0.40 samples. When x ≤ x C , the decrease in μobs of Ba0.40 and Sr0.40 samples was slower than that observed for the Ba0.15 samples, and the T C values of Ba0.40 and Sr0.40 samples were higher than that of the Ba0.15 samples; when x > x C , μobs decreased rapidly, and the T C values of Ba0.40 and Sr0.40 samples were lower than that of the Ba0.15 samples. In Sects. 9.3.3, 9.3.4, 9.3.5, and 9.3.6, these experimental results are discussed according to the IEO model.

174

9 Magnetic Ordering and Electrical Transport …

9.3.3 Measurement Results of Electrical Transport Property Figures 9.19, 9.20, and 9.21 show the curves of electrical resistivity, ρ 0 and ρ H , versus the test temperature, T, under the applied magnetic fields (μ0 H) of 0.0 T and 2.0 T, respectively, for the Ba0.15, Ba0.40, and Sr0.40 samples. To compare the effect of Fe doping on the resistivity, the Fe doping dependencies of the resistivity for the Ba0.15, Ba0.40, and Sr0.40 samples without the applied magnetic fields are shown in Fig. 9.22. The curves of the magnetoresistance, MR, versus the test temperature T of the three series of samples are shown in Fig. 9.23. The MR values were calculated as M R(%) =

ρ0 − ρH × 100%. ρ0

(9.10)

The data in Figs. 9.19, 9.20, 9.21, 9.22, and 9.23 can be summarized as follows: (i) (ii)

The resistivity of the samples increased with increasing x for all series of samples, as shown in Fig. 9.22. For the samples with the same x, ρ (Ba0.15) > ρ (Ba0.40) > ρ (Sr0.40), which are similar to the relations of the B–O bond lengths (see Fig. 9.14c),

10 1 0.1

10 1 100 10 1 100 10

T C =224K

0T 2T

x=0.04

(c)

x=0.08

(e)

100

T C =203K

10

(f)

T C =192K

T C =185K

x=0.10

(g)

x=0.15 T C =165K

10

150

200

250

1 x=0.20

(h)

0T 2T 300 50 100

1 100

0T 2T

0T 2T

10 1

0T 2T

0T 2T

100

x=0.06

(d)

T C =212K

0T 1 2T 50 100

x=0.02

(b)

0T 2T

(

cm)

100

x=0.00 T C =237K

Ba 0.15

(a)

100 10 1

150

200

250

300

T (K) Fig. 9.19 Test temperature dependences of resistivities of La0.85 Ba0.15 Fex Mn1−x O3 under the applied magnetic fields with 0 and 2 T

9.3 Experimental Evidence on the Canting Angle Magnetic Structure

(iii)

(iv)

175

d BO (Ba0.15) > d BO (Ba0.40) > d BO (Sr0.40). With decreasing d BO , the spindependent transition probability of the itinerant electrons increases; thus, the resistivity decreases. The d BO value of La0.85 Ba0.15 MnO3 (1.9569 Å, see Table 9.5) is only longer for 0.0021 Å than that of La0.6 Ba0.4 MnO3 (1.9548 Å). However, the resistivity of the former (54.6  cm, see Fig. 9.22) is 2482 times that of the later (0.022  cm) at 50 K. This indicates that the discussion for the effect of the crystal lattice constant on the resistivity in Sect. 9.2.4 is reasonable. The ρ 0 –T curves of Ba0.15 samples with x ≤ 0.15 all exhibit a shoulder near T C (Fig. 9.19). However, the shoulder is absent from the ρ H -T curves. The MR peak appears near this shoulder at the transition temperature (T MR ). As shown in Table 9.6, the values of T MR were very close to those of T C . MR decreased with increasing T when T > T MR ; MR decreased with decreasing T when T < T MR , as shown in Fig. 9.23a. In addition, there is a minimum MR at a temperature T L2 , MR increased with decreasing T when T < T L2 and x < 0.06. The ρ 0 –T curves of Ba0.40 samples with x ≤ 0.04 also exhibit a shoulder near T C , as shown in Fig. 9.20. However, the shoulder is absent from the curves of ρ H versus T. The MR peak appears near this shoulder at the transition 0.06

x=0.00

Ba0.40

x=0.02 0.09

(b)

0.04 (a) 0T 2T

0.02 0.2

T C=338K

x=0.04

(c)

0T 2T

0.06 T C=316K

x=0.06

(d)

0.03 0.6

( cm)

0.4 0.1

T C=289K

x=0.08

1.5 (e) 1.0 0.5

0T 2T

0T 0.2 2T x=0.10 12

T C=257K

(f)

8 0T 2T

T C=229K

x=0.15

100 (g) 10

T C=181K

(h)

0T 4 2T 0 x=0.20 100

10 0T 1 2T

0T 2T

1 100

200

300

400 100

200

300

400

T (K) Fig. 9.20 Test temperature dependences of resistivities of La0.6 Ba0.4 Fex Mn1−x O3 under the applied magnetic fields with 0 and 2 T

176

(v)

9 Magnetic Ordering and Electrical Transport …

temperature (T MR ), as shown in Fig. 9.23b. This phenomenon also appears for Sr0.4 series samples, but it is not obvious. For Ba0.40 and Sr0.40 series samples, as mentioned in Sect. 9.3.2, there is a characteristic Fe-doping level, x C = 0.10 and x C = 0.15, respectively, below and above in which there are different variation tendencies of magnetic moments and Curie temperature (see Fig. 9.18). These samples have also different variation tendencies of resistivity and magnetoresistance below and above x C :

When x ≤ x C , there is a resistivity peak at T ρ (0T) and T ρ (2T) in ρ 0 –T curve and ρ H –T curve, respectively, as shown in Figs. 9.20 and 9.21. The T ρ (2T) is slightly higher than T ρ (0T). Both T ρ (0T) and T ρ (2T) are distinctly lower than the Curie temperature T C . For the Ba0.4 samples with x = 0.06 and 0.08, another MR peak appears at T MRP (T MRP = 172 K and 155 K, respectively) in the curve of MR versus T, as shown in Fig. 9.23b and Table 9.6. The values of T MRP are much lower than the values of T C , T ρ (0T), and T ρ (2T). When x = x C , there is a maximum MR peak, 45.5 and 45.2% at T MRP = 131 K and 117 K, respectively, for Ba0.40 and Sr0.40 series sample. When x > x C , there is no peak in both ρ 0 –T and ρ H –T curves, and MR value decreases rapidly with increasing x. 0.024

(a)

Sr0.40

0.018

TC=364K

Resistivity ( cm)

0.012 0.18

x=0.05

(b) x=0.00

x=0.10

(c)

0.12 TC=267K

0.06 100 (e)

0T T =321K 2T C

0T 2T

x=0.15

(d) TC=187K

0T 2T

x=0.20

(f)

0.05 0.04 0.03 0.02 6 4

0T 2 2T 0 x=0.25 100

10

10 0T 2T

1

100

x=0.30

100 (g) 10

200

300

0T 1 2T 400

0T 2T

1 100

200

300

400 T (K)

Fig. 9.21 Test temperature dependences of resistivities of La0.6 Sr0.4 Fex Mn1−x O3 under the applied magnetic fields with 0 and 2 T

9.3 Experimental Evidence on the Canting Angle Magnetic Structure

100

( cm)

La 0.85 Ba 0.15 Fe x Mn 1- x O 3 -0T

(a)

1000

x =0.08

10

x =0.20 x =0.15 x =0.10

x =0.06 x =0.04 x =0.02

1

x =0.00

50

100

150

200

250

300

La 0.6 Ba 0.4 Fe x Mn 1- x O 3 -0T

(b)

100

cm)

10

(

x =0.15

1

x =0.08

0.1

x =0.06 x =0.04

x =0.20

x =0.10

x =0.02 x =0.00

0.01

0

cm)

100

(c)

100

(

Fig. 9.22 Test temperature dependences of resistivities of the three series of samples without the applied magnetic field

177

10

x =0.30 x =0.25 x =0.20

200

300

400

La 0.6 Sr 0.4 Fe x Mn 1- x O 3 -0T

x =0.15

1 x =0.10

0.1

x =0.05 x =0.00

0.01 0

100

200

T (K)

300

400

178

40

La 0.85 Ba 0.15 Fe xMn 1-xO3 TC

MR(%)

30

(a)

TC

TC T C

20

x=0.08 x=0.06 x=0.04

x=0.02 x=0.00

x=0.10 x=0.15 x=0.20

0 100 TMRP=131K

40

MR(%)

TC

TC T TC C

10

200

La0.6Ba0.4FexMn1-xO3 (b) TMRP=155K TMRP=172K T

x=0.10

C

x=0.08

TC

20

300

x=0.06

TC TC TC

TC

x=0.15

x=0.04 x=0.02 x=0.00

x=0.20

0 0

100

200

300

400

(c)

La 0.6 Sr 0.4 Fe xMn 1- xO 3 40

MR (%)

Fig. 9.23 The curves of the magnetoresistance MR versus the test temperature T of samples Ba0.15 (a), Ba0.40 (b), and Sr0.40 samples (c)

9 Magnetic Ordering and Electrical Transport …

T MRP =117K x =0.15 TC

20

x =0.10 TC

x =0.00 TC

x =0.20 x =0.25

0 0

100

TC

x =0.30

200

T (K)

300

x =0.05

400

9.3 Experimental Evidence on the Canting Angle Magnetic Structure

179

9.3.4 Fitting of Sample Magnetic Moment Using the IEO Model Following the investigations in Sect. 9.1, we assumed that all the Mn and Fe cations are trivalent in Ba0.15, Ba0.40, and Sr0.40 samples. According to the IEO model, we assumed that canted antiferromagnetic coupling occurred between the cation magnetic moments of the Fe3+ cations (3d 5 , 5 μB ) and the Mn3+ (3d 4 , 4μB ). Therefore, the average molecular magnetic moment, μcal , per molecule can be fitted by μcal = (4 − 4x − 5x)cosφ.

(9.11)

The fitted curves of μcal versus x are shown in Fig. 9.18b. The fitted curves of canting angles (φ) between the sample magnetic moments and the cation magnetic moments versus x are shown in Fig. 9.24. For Ba0.15 samples, φ decreased linearly with increasing x with x ≤ 0.20, φ=

180 (0.133π − 1.138x), (0.00 ≤ x ≤ 0.20, Ba0.15). π

(9.12)

For Ba0.40 samples, when x ≤ 0.10, φ decreased linearly with increasing x, φ=

180 (0.189π − 2.4x), (0.00 ≤ x ≤ 0.10, Ba0.40). π

(9.13)

100 Ba0.40

Sr0.40

o

()

75 50 25 Ba0.15

0 0.0

0.1

0.2

0.3

x Fig. 9.24 Curves of canted angle φ between the sample magnetic moments and the cation magnetic moments versus Fe doped level x of the Ba0.15, Ba0.40, and Sr0.40 samples

180

9 Magnetic Ordering and Electrical Transport …

When x = 0.15 and 0.20, the φ values were calculated using the observed magnetic moment μobs . For Sr0.40 samples, when x = 0.00 and 0.05, φ values were represented by φ=

180 (0.206π − 3.0x), (x = 0.00, 0.05, Sr0.40). π

(9.14)

When x = 0.10, 0.15, φ = 0°, they are co-linear magnetic structure. When x = 0.20, 0.25, and 0.30, the φ values were calculated using the observed magnetic moment μobs . It is very interesting that the φ values of Ba0.40 and Sr0.40 samples with x = 0.20 reached the maximum, 86.6° and 88.2°, respectively.

9.3.5 Effects of Thermal Excitation, Lattice Scattering, and Spin-Dependent Scattering on the Transition Probability of Itinerant Electrons Figure 9.25 shows the curves of resistivity ρ 0 versus the test temperature T for four samples, including single-crystal La0.85 Sr0.15 MnO3 (SSr0.15) and La0.60 Sr0.40 MnO3 (SSr0.40) [3], and polycrystalline La0.85 Ba0.15 MnO3 (PBa0.15) and La0.60 Ba0.40 MnO3 (PBa0.40) [11, 33]. To conveniently compare the samples, we considered the equivalent cubic lattice constant (ae ) that was calculated using the average crystal cell volume: ae of SSr0.40 (3.8737Å) is 0.0323Å smaller than ae of SSr0.15 (3.9060Å); thus, the transition probability of the itinerant electrons of SSr0.40 was higher than that of SSr0.15. Therefore, ρ(SSr0.40) < ρ(SSr0.15); ae of 10

(a) TL

TLTC=237K

TC=238K

( cm)

0.1

(b)

PBa0.15

10

SSr0.15

0.01

1 TC=338K

0

0

( cm)

1

1E-3

TC=371K

SSr0.40

1E-4 100

200

T (K)

300

400

0.1 0.01

PBa0.40

100

200

300

400

T (K)

Fig. 9.25 Curves of resistivity ρ versus the test temperature T for four samples. a Singlecrystal samples [3] La0.85 Sr0.15 MnO3 (SSr0.15) and La0.60 Sr0.40 MnO3 (SSr0.40); b polycrystalline samples [11, 33] La0.85 Ba0.15 MnO3 (PBa0.15) and La0.60 Ba0.40 MnO3 (PBa0.40). The narrows indicated the Curie temperature T C and the temperature T L corresponding to the resistivity minimum of SSr0.15 and PBa0.15 samples

9.3 Experimental Evidence on the Canting Angle Magnetic Structure

181

PBa0.40 (3.9097 Å) is 0.0040 Å smaller than ae of PBa0.15 (3.9137 Å); thus, the transition probability of the itinerant electrons of PBa0.40 was higher than that of PBa0.15, leading to ρ(PBa0.40) < ρ(PBa0.15). For the SSr0.15 and PBa0.15 samples with higher ae values, a characteristic temperature, T L, was determined. Below T L , ρ 0 decreased rapidly with increasing T. When T L < T < T C , ρ 0 increased rapidly with increasing T. This is attributed to the effects of two elementary factors on the transition of the itinerant electrons: thermal excitation resulting in a decrease in ρ 0 ; spin-dependent scattering resulting in an increase in ρ 0 . Below T L , the spin direction of the itinerant electrons deviated slightly from the direction of their ground state, and the effect of the thermal excitation was stronger than that of the spin-dependent scattering. Above T L , the spin direction of the itinerant electrons deviated rapidly from the direction of their ground state, and the effect of the spin-dependent scattering was stronger than that of the thermal excitation. For the SSr0.40 and PBa0.40 samples with low ae values, the spin-dependent transport at low temperatures could occur under an electric field, and the effect of thermal excitation was not obvious. The spin-dependent scattering strengthened with increasing test temperature, decreasing the transition probability of the itinerant electrons and increasing resistivity. A similar electrical transport property to that of PBa0.15 and SSr0.15 was reported by Zhang et al. [34] (Fig. 9.26). They grew the two NiCo2 O4 films on the MgAl2 O4 substrates under the same oxygen pressure (50 mTorr) but different temperatures (350 and 500 °C, represented by NCO350 and NCO500). They found that the samples have the cubic spinel structure with different crystal lattice constants, a, 8.17 and 8.18 Å for NCO350 and NCO500, respectively. The test temperature T dependencies of resistivity ρ for the two samples are shown in Fig. 9.26. The two ρ−T curves have similar variation tendencies: maximum resistivities were obtained near the Curie 0.60

o 500 C 50 mTorr

(m cm)

0.55 0.50 0.45 0.40

o 350 C 50 mTorr

0.35 0

100

200

300

400

T K Fig. 9.26 Test temperature T dependences of resistivity ρ for the NiCo2 O4 spinel film samples grown on the MgAl2 O4 substrates under the same oxygen pressure and the different temperature [34]

182

9 Magnetic Ordering and Electrical Transport …

temperature T C and minimum resistivities, at T L (T L < T C ). In addition, the resistivity of NCO350 with a = 8.17 Å is distinctly lower than that of NCO500 with a = 8.18 Å.

9.3.6 Effect of Canted Ferromagnetic Coupling on Magnetoresistance According to the data reported by Urushibara et al. [3], the observed magnetic moment of SSr0.15 was 4.2 μB at 10 K. All Mn3+ cations were ferromagnetically coupled without canting angle because the magnetic moment of a Mn3+ cation is 4.0 μB . As listed in Table 9.6, the observed magnetic moments of PBa0.15 and PBa0.40 in our work [11, 33] were 3.59 and 3.30 μB at 10 K, respectively. We, therefore, assume that Mn3+ cations exhibited canted ferromagnetic coupling, which was confirmed by comparing the curves of MR versus T for PBa0.15 and PBa0.40 with that for SSr0.15, as shown in Fig. 9.27. For the SSr0.15 sample, there was a maximum MR of 95% near T C under an applied magnetic field of 15 T; MR decreased whenever T increased or decreased from T C . For the PBa0.15 and PBa0.40 samples, there was also a maximum MR of 34 and 11%, respectively, near T C under an applied magnetic field of 2 T. This is attributed to the spin direction of itinerant electrons deviating from the ground state when the test temperature was close to T C ; this deviation took place to reduce the transition probability of the itinerant electrons and increase the resistivity. An applied magnetic field could reduce the deviation of the spin direction and reduce the resistivity. 100 TC

75

MR(%)

SSr0.15

50 25

PBa0.15

TL2 TC

PBa0.40

TL2 TC

0 0

100

200

300

400

T (K) Fig. 9.27 Curves of magnetoresistance MR versus test temperature T for polycrystalline samples La0.60 Ba0.40 MnO3 (PBa0.40), La0.85 Ba0.15 MnO3 (PBa0.15) [33] and single-crystal sample La0.85 Sr0.15 MnO3 (SSr0.15) [3]. The up arrows indicated the Curie temperature T C . The down arrows indicated the temperature T L2 appeared the MR minimum

9.3 Experimental Evidence on the Canting Angle Magnetic Structure

183

In addition, for the PBa0.15 and PBa0.40 samples, there was a characteristic temperature, T L2 , below which MR increased with decreasing T. This is attributed to the canting angle between the magnetic moments of Mn3+ ions, which reduced the transition probability of the itinerant electrons and increased the resistivity. An applied magnetic field could reduce the canting angle and the resistivity. When the temperature increased, the extent of the electron spin direction deviated from the direction of their ground state increases; the effect of the applied magnetic field weakened, thus reducing MR. This phenomenon of decreasing MR with increasing T below T L2 was not observed for SSr0.15 because there was no canting angle between the magnetic moments of Mn3+ cations in SSr0.15. For Ba0.40 and Sr0.40 samples, as shown in Fig. 9.23b and c, with increasing Fe doping, an MR shoulder appeared below T C ; then, the MR peaks appeared. It is very interesting that at the x values corresponding to the MR peak, 0.10 and 0.15, the canting angles have minimum value, as shown in Fig. 9.24. This is similar to that of the SSr0.15, in which there is no canting angle between cation magnetic moments, and has the maximum MR value in La0.85 Sr0.15 MnO3 [3]. This indicates that the method used for fitting the canting angle in Sect. 9.3.4 is reasonable. In summary, single-phase powder manganites, La0.85 Ba0.15 Fex Mn1−x O3 (0.00 ≤ x ≤ 0.20, Ba0.15 samples), La0.60 Ba0.40 Fex Mn1−x O3 (0.00 ≤ x ≤ 0.20, Ba0.40 samples), and La0.60 Sr0.40 Fex Mn1−x O3 (0.00 ≤ x ≤ 0.20, Sr0.40 samples), with the ABO3 perovskite structure were synthesized using the sol–gel method. By analyzing the XRD patterns, all the samples were found to have the rhombohedral structure with space group R3c. Using the IEO models for perovskite manganites, the magnetic and electrical transport properties and the relationship between them can be explained: (i) (ii)

(iii)

(iv)

In all samples, the Mn and Fe ions were trivalent. In all samples, the magnetic moments of Mn3+ cations exhibited canted ferromagnetic coupling, whereas the magnetic moments of Fe3+ cations exhibited canted antiferromagnetic coupling with Mn3+ cations. The itinerant electrons in the samples were the O 2p electrons. Below the Curie temperature T C , the elementary electrical transport of the samples depended on the spin-dependent transition of the itinerant electrons along the O–Mn(Fe)–O–Mn(Fe)–O ionic chains. Above T C , the electrical transport of the samples was the spin-independent transition of itinerant electrons. For the three samples without Fe, the crystal cell volume decreased successively, v (Ba0.15)> v (Ba0.4) > v (Sr0.4). This has the important effect on the magnetic and electrical properties: The Curie temperature increases successively, T C (Ba0.15) < T C (Ba0.4) < T C (Sr0.4); the canting angle between the sample magnetic moment and cation magnetic moments increases successively, φ(Ba0.15) < φ(Ba0.4) < φ(Sr0.4), and therefore the average molecular magnetic moment decreases successively, μobs (Ba0.15)> μobs (Ba0.4) > μobs (Sr0.4). The B–O bond length d BO value of La0.85 Ba0.15 MnO3 (1.9569 Å) is longer only for 0.0021 Å than that of La0.6 Ba0.4 MnO3 (1.9548 Å). This results in the phenomena: first, the resistivity of Ba0.15 sample (54.6  cm) is 2482 times that of the Ba0.40 sample (0.022  cm) at 50 K; second, at the

184

(v)

9 Magnetic Ordering and Electrical Transport …

low temperature, the resistivity of Ba0.15 sample decreases with increasing temperature, but the resistivity of Ba0.40 sample increases with increasing temperature. There is a characteristic Fe doped level, x C = 0.10 and x C = 0.15, for Ba0.40 samples and Sr0.40 samples, respectively. When x = x C , the canting angle φ reached the minimum, the magnetoresistance appeared the maximum, 45.5 and 45.2% at 2.0T applied magnetic field. When x ≤ x C , with increasing x, φ, μobs , and T C decrease slowly; when x > x C , with increasing x, φ increases rapidly, μobs and T C decrease rapidly.

9.4 Magnetic Coupling Between the Two Sublattices in Perovskite Praseodymium Manganites In ABO3 perovskite lanthanum manganites, the magnetic moment of the A sublattice was not observed due to the valence electron states of La2+ (5d 1 ) and La3+ (5d 0 ). However, in ABO3 perovskite praseodymium manganites, there exists the effect of Pr cation magnetic moment on the sample magnetic moments [35–41] originated from the valence electron states of Pr2+ (4f3 ) and Pr3+ (4f2 ). The relation of cation magnetic moments between the A and B sublattices can be explained using the IEO model. Our group [42] measured the dependencies of the specific magnetization of Pr0.6 Sr0.4 MnO3 polycrystalline powder sample on the test temperature and applied field, as shown in Fig. 9.28. The measurement was performed from 350 to 10 K. A similar experimental result was reported by Maheswar Repaka et al. [40], as shown in the inset of Fig. 9.28. The sample evidently underwent a transition from paramagnetism to ferromagnetism with decreasing temperature. The Curie temperature T CM , defined as the temperature at which dσ /dT reaches its minimum value, was determined to be 304 K. Another transition temperature T CP , defined as the temperature at which dσ /dT reaches its maximum value, can be seen at 55.3 K. A similar transition temperature, T = 65 K, was observed for single-crystal Pr0.6 Sr0.4 MnO3 by Rößler et al. [39]. The transition at T CP can be explained using the IEO model. Below T CP , the magnetic moments of the Mn cations in the B sublattice are assumed to have canted ferromagnetic coupling, and the magnetic moments of the Pr3+ (4f 2 ) and Pr2+ (4f 3 ) in the A sublattice are also assumed to have canted ferromagnetic coupling to each other. However, the total magnetic moment of the A sublattice should be opposite to the moments of the B sublattice because of the following reasons: (i) The spin direction of the itinerant electrons in the A sublattice must be opposite to that of the itinerant electrons in the B sublattice in the IEO model. (ii) In the B sublattice, the number of 3d electrons nd = 4 for Mn3+ , the spin direction of an itinerant electron must be parallel to the spin direction of the local 3d electrons when it hops to a Mn3+ cation. (iii) In the A sublattice, the number of 4f electrons nf in Pr3+ (4f 2 ) and Pr2+ (4f 3 ) cations is 0.08). We assumed that (i) both M 3 /M 2 and C 3 /C 2 were constant when x increased, and (ii) the ratio (C 3 /C 2 )/(M 3 /M 2 ) for La0.95 Sr0.05 Crx Mn1−x O3 , which was equal to that for La0.95 Ca0.05 Crx Mn1−x O3 when x > 0.08, was 5.7757/5.3414 = 1.081. From Eq. (9.27), we obtained M3 /M2 = 0.9133/0.0867 = 10.534,

(9.28)

C3 /C2 = 1.081(M3 /M2 ) = 11.387.

(9.29)

Since M 3 + M 2 = 1 − x, we obtained M2 = (1 − x)/11.534,

M3 = 1 − x − M2 .

(9.30)

198

9 Magnetic Ordering and Electrical Transport …

Since C 3 + C 2 = x, we also obtained C2 = x/12.387, C3 = x − C2 .

(9.31)

Using Eqs. (9.25), (9.30), and (9.31), we calculated the average molecular magnetic moment μcal0 and found that μcal0 decreased from 3.22 μB (x = 0.00) to 3.18 μB (x = 0.30), which is distinctly different from the observed value of μobs . This suggests that as a result of Cr doping, the cation magnetic moments in the samples transitioned from co-linear coupling to canted coupling, where the canting angle (φ) increased with increasing x. Applying Eq. (9.26), we calculated the values of φ, and the results are shown in Fig. 9.40. Interestingly, Fig. 9.40 shows that the dependence of φ on x for the Sr-doped samples (0.00 ≤ x ≤ 0.30) was very similar to that for the Ca-doped samples (0.08 ≤ x ≤ 0.30), while the φ values of 52.3° and 58.1° (x = 0.25 and 0.30) for the Sr-doped samples were close to the φ values of 51.5° and 53.5° (x = 0.25 and 0.30) for the Ca-series samples. These similarities suggest that the above discussions on the canted magnetic structures are reasonable. (4)

Curie temperature of the two series of samples

In the previous sections, we calculated the values of ion ratios, M 2 , M 3 , C 2 , and C 3 . It would be interesting to compare the dependences of T C , M 2 , and C 3 on x, as shown in Figs. 9.38 and 9.41. It can be seen that the curves of T C versus x for both series of samples exhibit similar trends, which can be explained as follows using the IEO model: (i)

An itinerant electron with up-spin moves along the ion chains shown in Fig. 9.42. When it transfers via a Mn3+ cation along the ion chain O2− –Mn3+ – O2− –Mn3+ –O1− , as shown in Fig. 9.42a, it always occupies the highest energy level whenever it reaches a cation or an anion, thus consuming little of the system’s energy in the transition process. When it moves along the ion chain O2− –Mn2+ –O2− –Mn3+ –O1− , which includes Mn2+ and Mn3+ cations, it must

Fig. 9.41 Dependences of ratio of Mn2+ and Cr3+ cations (M 2 and C 3 ) on Cr-doping level (x) for Ca-series and Sr-series samples [46]

9.6 Experimental Evidence for Antiferromagnetic Coupling …

199

Fig. 9.42 Schematic diagrams of the transfer process of itinerant electrons with up-spin along the a O2− –Mn3+ –O1− –Mn3+ –O2− ion chain, b O2− –Mn2+ –O1− –Mn3+ –O2− ion chain, and c O2− – Cr3+ –O1− –Mn3+ –O2− ion chain. Here “↑” and “↓” represent electrons with up-spin and down-spin, respectively, “” represents absence of an O 2p electron with up-spin, that is, an O2p hole [46]

(ii)

(iii)

pass the highest 3d energy level of the Mn2+ cation, as shown in Fig. 9.42b, thus consuming more energy of the system in the transition process because the magnetic moment of Mn2+ is opposite to that of Mn3+ . This increases T C as decreasing the Mn2+ cation ratio. When the itinerant electron moves along the ion chain O2− –Cr3+ –O2− –Mn3+ –O1− , which includes Mn3+ and Cr3+ , as shown in Fig. 9.42c, it must pass the second highest 3d energy level of Cr3+ . In this transition process, the energy consumed by the itinerant electron falls between those shown in Fig. 9.42a and b. As a result, T C decreases with increasing Cr-doping level. In our study, the ratio of Mn2+ and Cr3+ cations, M 2 and C 3 , changed at the same time. M 2 of the Sr-doped samples with all doping levels in the range 0.0 ≤ x ≤ 0.30 was lower than that of the Ca-doped samples (see Fig. 9.41), which is one of the important factors leading to higher T C for the Sr-doped samples compared with those of the Ca-doped samples (see Fig. 9.38). When x < 0.08 (see Fig. 9.41), because C 3 < M 2 and M 2 decreased with increasing x for both series of samples, the effect of M 2 on T C was stronger than that of C 3 . This resulted in increase in T C with decreasing M 2 . M 2 of the Ca-series decreased rapidly with increasing x, resulting in rapid increase in T C , whereas M 2 of the Sr-series decreased slowly, resulting in slow increase in T C with increasing x.

200

9 Magnetic Ordering and Electrical Transport …

(iv)

When x > 0.20 (see Fig. 9.41), because C 3 > M 2 and C 3 increased with increasing x for both series of samples, the effect of C 3 on T C was stronger than that of M 2 . This resulted in decrease in T C with increasing C 3 .

In summary, these unique magnetic properties of both series of samples can be explained using the IEO model: (i) The magnetic moments of Mn2+ cations coupled antiferromagnetically with those of Mn3+ and Cr cations in the Ca-doped samples with x ≤ 0.08 and the Sr-doped sample with x = 0.00. (ii) There was a canted antiferromagnetic structure between the magnetic moments of the Mn2+ cations against those of the Mn3+ and Cr cations in the Ca-doped samples with x ≥ 0.10 and the Sr-doped samples with x ≥ 0.05. (iii) Because the consumed energy increased successively when an itinerant electron transferred via Mn3+ , Cr3+ , and Mn2+ , the IEO model allows us to reasonably explain the dependence of T C on x for both Sr-doped and Ca-doped perovskite manganites, which validated our discussions on cation valence and ratio.

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References 21. 22. 23. 24. 25. 26. 27. 28.

29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52.

201

Singh DJ, Pickett WE (1998) Pseudogaps. Phys Rev B 57:88 Lee JD, Min BI (1997) Phys. Rev. B 55:12454 Hwang HY, Cheong SW, Ong NP, Batlogg B (1996) Phys Rev Lett 77:2041 Alexandrov AS, Bratkovsky AM (1999) Phys Rev Lett 82:141 Alexandrov AS, Bratkovsky AM, Kabanov VV (2006) Phys Rev Lett 96:117003 Nücker N, Fink J, Fuggle JC, Durham PJ, Temmerman WM (1988) Phys Rev B 37:5158 Ju HL, Sohn HC, Krishnan KM (1997) Phys Rev Lett 79:3230 Ibrahim K, Qian HJ, Wu X, Abbas MI, Wang JO, Hong CH, Su R, Zhong J, Dong YH, Wu ZY, Wei L, Xian DC, Li YX, Lapeyre GJ, Mannella N, Fadley CS, Baba Y (2004) Phys. Rev. B 70:224433 Liu XM, Zhu H, Zhang YH (2001) Phys. Rev. B. 65:024412 Wang LM, Wang CY, Tseng CC (2012) Appl Phys Lett 100:232403 Narreto MAB, Alagoz HS, Jeon J, Chow KH, Jung J (2014) J Appl Phys 115:223905 Xu LS, Fan JY, Zhu Y, Shi YG, Zhang L, Pi L, Zhang YH, Shi DN (2015) Chem Phys Lett 634:174 Qian JJ, Li ZZ, Qi WH, Ma L, Tang GD, Du YN, Chen MY (2018) J Alloy Compd 764:239 Zhang KQ, Zhen CM, Wei WG, Guo WZ, Tang GD, Ma L (2017) Denglu Houa and Xiancheng Wu, Insight into metallic behavior in epitaxial halfmetallic NiCo2 O4 films. RSC Adv 7:36026 Rama N, Sankaranarayanan V, Opel M, Gross R, Ramachandra Rao MS (2007) J. Alloys Compd 443:7 Zemni S, Baazaoui M, Dhahri J, Vincent H, Oumezzine M (2009) Mater Lett 63:489 Elleucha F, Trikia M, Bekri M, Dhahri E, Hlil EK (2015) J Alloys Compd 620:249 Ritter C, Radaelli PG (1996) J Solid State Chem 127:276 Rößler S, Harikrishnan S, Naveen Kumar CM, Bhat HL, Elizabeth S, Rößler UK, Steglich F, Wirth S (2009) J Supercond Nov Magn 22:205 Maheswar Repaka DV, Tripathi TS, Aparnadevi M, Mahendiran R (2012) J Appl Phys 112:123915 Boujelben W, Ellouze M, Cheikh-Rouhou A, Pierre J, Cai Q, Yelond WB, Shimizuf K, Dubourdieu C (2002) J Alloys Compd 334:1 Ge XS, Wu LQ, Li SQ, Li ZZ, Tang GD, Qi WH, Zhou HJ, Xue LC, Ding LL (2017) AIP Adv 7:045302 Ge XS, Li ZZ, Qi WH, Ji DH, Tang GD, Ding LL, Qian JJ, Du YN (2017) AIP Adv 7:125002 Ge XS (2017) Study of magnetic ordering in the perovskite manganites Pr0.6 Sr0.4 Mn1−x M x O3 (M = Cr, Fe, Co, Ni) (Master’s thesis). Hebei Normal University, Shijiazhuang Shannon RD (1976) Acta Cryst A 32:751 Li SQ, Wu LQ, Qi WH, Ge XS, Li ZZ, Tang GD, Zhong W (2018) J Magn Magn Mater 460:501 Liu SP, Tang GD, Hao P, Xu LQ, Zhang YG, Qi WH, Zhao X, Hou DL, Chen W (2009) J Appl Phys 105:013905 Liu SP, Xie Y, Xie J, Tang GD (2011) J Appl Phys 110:123714 Liu SP, Xie Y, Tang GD, Li ZZ, Ji DH, Li YF, Hou DL (2012) J Magn Magn Mater 324:1992 Ji DH, Hou X, Tang GD, Li ZZ, Hou DL, Zhu MG (2014) Rare Met 33:452 Hou X, Ji DH, Qi WH, Tang GD, Li ZZ (2015) Chin Phys B 24:057501 Wu LQ, Qi WH, Li YC, Li SQ, Li ZZ, Tang GD, Xue LC, Ge XS, Ding LL (2016) Acta Phys Sin 65:027501

Chapter 10

Antiferromagnetic Ordering in Oxides with Sodium Chloride Structure

Antiferromagnetic oxides, MnO, FeO, CoO, and NiO, with sodium chloride structure are typical antiferromagnetic materials. In this chapter, we explain different characteristics between these materials and antiferromagnetic perovskite manganites, LaMnO3 , using the IEO model.

10.1 Characteristics of Antiferromagnetic Oxides with Sodium Chloride Structure MnO, FeO, CoO, and NiO have sodium chloride crystal structure. This structure can be described as two interpenetrating fcc (face center cubic) lattices displaced by the distance of (1/2) body diagonal of the cube crystal cell; one of the two fcc sublattices is composed by the cations and another by anions. Therefore, the crystal cell has a cubic symmetry with space group Fm3m. There are four molecules per crystal cell, as shown in Fig. 10.1. Table 10.1 shows the Néel temperature (T N ) [1], the crystal lattice constant (a), and the diffraction angles (2θ ) corresponding to (111) lattice planes by X-ray with a wavelength of 1.5406 Å or 1.057 Å, respectively. These structure data came from the ICDD (International Centre for Diffraction Data) cards. Shull et al. [2] confirmed that MnO has the antiferromagnetic structure with Néel temperature T N = 120 K using neutron diffraction with a wavelength of 1.057 Å. They found that the magnetic moments of Mn ions were disordered at 293 K above T N , based on the diffraction pattern shown in Fig. 10.2b, where (111) peak position is very close to that in Table 10.1. The diffraction pattern measured at 80 K, below T N , as shown in Fig. 10.2a, has several peaks more than that in Fig. 10.2b. This result suggests that the neutrons with magnetic moments were scattered by ordering Mn cation magnetic moments. To confirm this, they measured the diffraction patterns at 100–124 K and found that the intensity of the diffraction peaks corresponding to the cation magnetic moments decreases rapidly with increasing test temperature. This indicated that, with increasing temperature, Mn cation magnetic moments switch © Science Press 2021 G.-D. Tang, New Itinerant Electron Models of Magnetic Materials, https://doi.org/10.1007/978-981-16-1271-8_10

203

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10 Antiferromagnetic Ordering in Oxides …

Fig. 10.1 Diagrammatic sketch of sodium chloride structure

Table 10.1 For the typical antiferromagnetic oxides, crystal lattice constant (a) and the diffraction angles (2θ), corresponding to the (111) lattice planes, by the X-ray with wavelength 1.5406 Å or 1.057 Å, respectively, come from ICDD cards. T N is the Néel temperature [1] Oxides

ICDD number

a (Å)

2θ (°) (1.5406 Å)

2θ (°) (1.057 Å)

T N /(K)

MnO

03-065−0641

4.5320

34.243

23.307

120

FeO

00-046−1312

4.2930

36.343

24.710

185

CoO

03-065−2902

4.2603

36.501

24.815

291

NiO

03-065−2901

4.1946

37.093

25.210

515

from order to disorder, that is, they change from antiferromagnetic state to paramagnetic state. In Fig. 10.2a, the crystal lattice constant 8.85 Å came from a large crystal cell containing 32 molecules, to consider the symmetrical characteristic of magnetic ordering. Shull et al. [2] also reported additional results of neutron diffraction analyses for antiferromagnetic materials FeO, CoO, NiO, all of which have the similar magnetic structure with MnO.

10.2 Difference Between Magnetic Structures of Manganese …

205

Fig. 10.2 Neutron diffraction patterns of MnO reported by Shull et al. [2]

10.2 Difference Between Magnetic Structures of Manganese Monoxide and Lanthanum Manganite Importantly, differences exist between magnetic structures of antiferromagnetic materials such as MnO and LaMnO3 . MnO is a typical antiferromagnetic material, which magnetic moment is always very close to zero whenever it was prepared using any condition. However, as mentioned in Sects. 4.2 and 9.1, the average molecular magnetic moment of LaMnO3 may change from 0 to 3 μB under different preparation conditions [3, 4]. This difference in magnetic structures can be explained using the IEO model. As mentioned in Sects. 4.2 and 9.1, since the space of the A site is larger than that of the B site in ABO3 perovskite manganite LaMnO3 , La with large radius and Mn with small radius enter only the A and B site, respectively. The two 2p electrons with opposite spin directions at the outer orbit of an O2− anion transit along the O–La–O–La–O ionic chain of the A sublattice and O–Mn–O–Mn–O ionic chain of the B sublattice, respectively. Due to the La cation has no magnetic moment, the antiferromagnetic structure of LaMnO3 results from the antiferromagnetic coupling between Mn2+ and Mn3+ cations. The different preparation methods can result in different oxygen ratios in LaMnO3±δ , resulting in different valences of Mn cations, Mn2+ /Mn3+ ratio, and average molecular magnetic moments [3, 4]. When a fraction of La cations are substituted by the divalent cations, and the substituted composition is equal to or more than 0.15, all Mn cations are trivalent, and the effect of the preparation condition on the sample magnetic moment can be reduced. Our group [5] examined the effect of the preparation condition on sample magnetic moment

206

10 Antiferromagnetic Ordering in Oxides …

and Curie temperature of La1−x Bax MnO3 and found that the effect was very distinct when x = 0.00 and 0.05, while the effect was weak when x ≥ 0.15. Our group [6, 7] also studied La0.95 Sr0.05 MnO3 and La0.95 T 0.05 Crx Mn1−x O3 (T = Ca or Sr, 0.00 ≤ x ≤ 0.30) and found that there were distinct effects of preparation conditions on the sample magnetic moment and Curie temperature, which are discussed in Sect. 9.6. In antiferromagnetic MnO with sodium chloride structure, the two 2p electrons with opposite spin directions at the outer orbit of an O2− anion transit along the two O–Mn–O-Mn–O ionic chains with opposite Mn cation magnetic moment directions, respectively. The two sublattices have the same crystal structure, resulting in they have the same Mn cation distribution and the same absolute value of the magnetic moments, whenever Mn cation valence change or not. Therefore, the sample magnetic moment of MnO is always very small, even that the valence of Mn cations changed due to the preparation condition was changed. In summary, LaMnO3 has only one magnetic sublattice, which antiferromagnetic structure results from the antiferromagnetic coupling between Mn2+ and Mn3+ cation magnetic moments in this magnetic sublattice. The different preparation conditions change the Mn2+ /Mn3+ ratio, resulting in LaMnO3 having different sample magnetic moments. MnO has two magnetic sublattices with the same magnetic moments but opposite magnetic moment directions, resulting in the sample magnetic moment of MnO being close to zero, even that the valence of Mn cations changed due to the preparation condition was changed.

References 1. 2. 3. 4. 5.

Jiang ST, Li W (2003) Condensed magnetic physics (in Chinese). Science Press, Beijing Shull CG, Strauber WA, Wollan EO (1951) Phys Rev 83:333 Töpfer J, Goodenough JB (1997) J Solid State Chem 130:117 Prado F, Sanchez RD, Caneiro A, Causa MT, Tovar M (1999) J Solid State Chem 146:418 Qian JJ (2018) Study of magnetic ordering and electrical transport properties in the perovskite manganites La1−y M y Mn1−x Fex O3 (M y = Ba0.15 , Ba0.4 , Sr0.40 ). Master’s thesis, Hebei Normal University, Shijiazhuang 6. Wu LQ, Qi WH, Li YC, Li SQ, Li ZZ, Tang GD, Xue LC, Ge XS, Ding LL (2016) Acta Phys Sin 65:027501 7. Li SQ, Wu LQ, Qi WH, Ge XS, Li ZZ, Tang GD, Zhong W (2018) J Magn Magn Mater 460:501

Chapter 11

Itinerant Electron Model for Magnetic Metals

Based on the experimental results for valence electron state and the atomic physics theory for the electron distribution at discrete energy levels in an atom, our group [1, 2] proposed a new itinerant electron model for magnetic metals (the IEM model). According to the IEM model, for magnetic metals Fe, Co, and Ni, we estimated the average numbers of 3d electrons (nd ) per ionic core and the free electrons (nf ) contributed by an atom using the observed average atomic magnetic moment. Then, we found that the resistivities of Fe, Ni, Co, and Cu decrease with increasing nf , providing a new clue for understanding the valence electron structures of metals and alloys.

11.1 Experimental and Theoretical Studies for Atomic Magnetic Moments in Metals The experimental values of average atomic magnetic moments, μobs , for Fe, Co, and Ni are 2.22, 1.72, and 0.62 μB , respectively [3, 4]. In recent decades, some experimental investigations were performed to study the magnetic ordering mechanism of metals. In 1995, Chen et al. [5] grew Fe and Co thin films with 50–70 Å thickness, under ultrahigh vacuum conditions via electron-beam evaporation onto 1-μm-thick semitransparent parylene, (C8 H8 )n , substrates. They obtained high precision L,23 edge X-ray photo-absorption spectra (XAS) and magnetic circular dichroism spectra (XMCD) of Fe and Co thin films by measuring in transmission with in situ grown thin films, using a soft-X-ray sensitive photodiode mounted 0.5 m behind the sample. The two groups of data are particularly noteworthy: (i) For the Fe and Co films, the spin magnetic moments (μS ) are 1.98 μB and 1.55 μB per atom, the orbit magnetic moments (μL ) are 0.086 μB and 0.153 μB , and the ratios of orbit/spin magnetic moments are 4.3% and 9.9%, respectively, which indicates that the contribution of the orbit magnetic moment to the sample magnetic moment is very small. (ii) They © Science Press 2021 G.-D. Tang, New Itinerant Electron Models of Magnetic Materials, https://doi.org/10.1007/978-981-16-1271-8_11

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11 Itinerant Electron Model for Magnetic Metals

adopted the calculated data of 3d electron numbers for Fe and Co, 6.61 and 7.51, respectively [6, 7], which are more than that for Fe and Co free atoms, indicating that a fraction of 4s electrons entered the 3d orbits when the metals were formed from the free atoms. In 2007, Jauch and Reehuis [8] measured the high-accuracy single-crystal structure factors of α-Fe at 295 K using 316.5 keV gamma radiation. By analyzing the experimental results, they concluded that the 3d spin and charge form structure factors favor the occupation of d 7 in the metal instead of d 6 for the free atom. In 2015, Pacchioni et al. [9] investigated the spin, orbital, and total magnetic moments as well as magnetic anisotropy energy for individual Fe atoms and small Fe clusters grown on a Cu (111) crystal. They prepared a series of samples from 0.007 to 0.145 monolayer [one Fe atom is present per Cu (111) crystal cell] in a vacuum chamber. The samples were measured without breaking the vacuum using a scanning tunneling microscope (STM), XAS, and XMCD. They concluded that the orbital magnetic moment μL rapidly decreases as the cluster size increases: (i) For Fe monomers on Cu (111), μL = 0.66 ± 0.04 μB , and the spin magnetic moments μS = 2.6 ± 0.2 μB . The valence electron state of the Fe adatom is close to a 3d 7 4s1 configuration. (ii) For a cluster with five Fe atoms on Cu (111), μL ≈ 0.2 μB and μS = 2.4 μB . The μL /μS ratio is only 8.3%. The μS value is very close to the average atomic magnetic moments in the literature, 2.22 μB [3, 4]. To study the physical mechanism of metal magnetism, many calculations were performed via DFT. Lazarovits et al. [10] reported that a Fe monolayer sandwiched by two semi-infinite Cu (001) substrates has the corresponding spin magnetic moment of 2.54 μB /atom. Stepanyuk et al. [11] predicted that the 3d magnetic nanostructures and superlattices on Cu (111) can be stabilized by surface-state electrons up to 25– 30 K. For metals Ti, V, Cr, Mn, Fe, Co, and Ni, the average atomic magnetic moments are 1.77, 3.15, 4.28, 4.32, 3.17, 1.92, and 0.36 μB , respectively. Santos Dias et al. [12] reported that the spin magnetic moments for Cr, Mn, Fe, and Co adatoms on the Cu (111) surface are 4.07, 4.31, 3.23, and 1.97 μB , respectively. The conclusions of the above experimental and theoretical investigations can be summarized as follows: (i) Differences exist between the data in the above literature [5–12] and the average atomic magnetic moments in bulk metals [3, 4]. (ii) Most studies confirmed that a fraction of 4s electrons enter 3d orbits and transit the 3d electrons during the formation of metals from free atoms. (iii) The effect of the orbital magnetic moments on the bulk metals is far smaller than that of the spin magnetic moments.

11.2 Itinerant Electron Model for Magnetic Metals (IEM Model) X-ray photoelectron spectrum analyses and DFT calculations proved that the valence electrons successively distribute for 6 eV below the Fermi level [13–15]. Figure 5.15 [13] shows the spin-and-angle-resolved photoemission spectra of Fe when the test

11.2 Itinerant Electron Model for Magnetic Metals (IEM Model)

209

temperature is far lower than the Curie temperature, T = 0.3T C , the valence electrons with up-spin distribute for 6 eV below the Fermi level, while the valence electrons with down-spin distribute for 1.5 eV below the Fermi level. The valence electron spectra originated experiments and calculations reported in many studies, including Refs. [13–15], suggested that a metal has only a few free electrons, since only electrons with energies ~0.03 eV below the Fermi level can be thermally excited to above the Fermi level at room temperature. Therefore, the traditional itinerant electron theory needs to be improved, wherein all 3d and 4 s electrons were assumed itinerant electrons itinerating in the crystal lattice [4]. Based on these observations and the investigations mentioned in Sect. 11.1; thus, our group [1, 2] proposed a new itinerant electron model for magnetic metals (the IEM model), which is used to explain the relation between magnetic moments and resistivities of Fe, Ni, Co, and Cu. In this section, we introduce the IEM model as follows. (i)

(ii)

(iii)

Based on gamma radiation diffraction and other observations (see Sect. 11.1) [5–9], in the process of forming a metal solid with a single-crystal or polycrystalline state from free atoms, most of the 4s electrons in 3d transition metals (except for Cu and Zn with a full 3d subshell) enter the 3d orbits to decrease the Pauli repulsive energy between atoms, while the remaining 4s electrons form free electrons. A certain probability exists that the outer orbital 3d electrons transit between the outer orbits of adjacent ionic cores, forming itinerant electrons. The other 3d electrons are local electrons. The resistivity of a metal decreases as the concentration of free electrons increases. The movements of free electrons are subjected by the weak crystal lattice potential field but not by the electron orbits, whose spins have no contribution to the material magnetic moment. The transition of the itinerant electrons is a spin-independent transition above the Curie temperature, but it is a spin-dependent transition below the Curie temperature, and the transition probability decreases with increasing test temperature and rapidly decreases near the Curie temperature.

The valence electron distribution characters of Fe metal can be found from Fig. 5.15 [13]: (i) The valence electrons with up-spin distribute in a range for approximately 6 eV below the Fermi energy level. The distribution probability near the Fermi energy level is very small, and the two spectrum peaks appear at −1.26 and −2.73 eV. (ii) Most valence electrons with down-spin distribute in a range of 1.5 eV below Fermi level and the spectrum peak appears at −0.32 eV, indicating that the valence electron distribution in 3d transition metals is similar to that in the free atom. When the number of 3d electrons nd ≤ 5, there is one electron with up-spin per energy level from low to high energy; when nd > 5, the spins of the excess valence electrons are down, arranged one electron per energy level from high-to low-energy level. The up-spin is named the majority spin, while the down-spin is called the minority spin. According to the observations mentioned in Sect. 11.1, as an approximation, the orbital magnetic moment may be neglected. For a magnetic metal, if nd ≤ 5,

210

11 Itinerant Electron Model for Magnetic Metals

the atomic magnetic moment, μat , equals nd μB ; if nd ≥ 6, only the energy levels containing one electron in the all of five energy levels contribute to μat . Therefore, μat can be written as μat = n d μB , when(n d ≤ 5); μat = (10 − n d )μB , when(6 ≤ n d ≤ 10).

(11.1)

Using this relation, the average 3d electron number nd of Fe, Ni, and Co metals can be calculated using their observed values, μat = μobs . Furthermore, the average free electron number per atom can be obtained as n f = n ds − n d ,

(11.2)

where nds is the total numbers of 3d and 4s electrons in a free atom, as shown in Table 11.1. Using Eqs. (11.1) and (11.2) and the values of μobs and nds , the values of nd and nf can be easily calculated; the results are listed in Table 11.1. The findings are interesting: the electrical resistivities of Fe, Ni, Co, and Cu all decrease with increasing nf , as shown in Fig. 11.1. This indicates that the IEM model can be used to explain the relationship between the magnetic moment and resistivity on the basis of the distinct physical mechanism. This is convenient for experimental researchers to explain the magnetic property of materials. According to the early investigations [4] on the metal energy band theory, the 3d electron numbers for Fe, Ni, and Co are 7.4, 9.4, and 8.3, respectively, which are very close to the values in Table 11.1, indicating that the IEM model can be explained using the metal energy band theory. However, the successive decrease of the observed resistivities of Fe, Ni, and Co cannot be clearly explained via the traditional magnetic ordering theory. We noticed that the average number of 3d electrons in Ni metal is 9.38 per ionic core, implying that 38% of ionic cores have the 3d 10 valence electronic state with a full 3d subshell, which is similar to doping Cu in Ni. Therefore, the T C value (631 K) of Ni metal is far lower than that of Co (1404 K) and Fe (1043 K). This will be discussed further in Chap. 12. Table 11.1 Observed values of the average atomic magnetic moment, μobs , Curie temperature, T C , and electrical resistivity, ρ, for Fe, Ni, Co, and Cu metals, and the total number, nds , of their 3d and 4s electrons, 3d electron number, nd = 10 – μobs , and their free electron number, nf = nds – nd μobs (μB )

T C (K)

ρ(0 °C) (μ·cm)

nds

nd

Fe

2.22

1043

8.6

8

7.78

0.22

Ni

0.62

631

6.14

10

9.38

0.62

Co

1.72

1404

5.57

9

8.28

0.72

Cu

0.00

–

1.55

11

10

1.00

References

[2–4]

[3]

[3]

[4]

[1, 2]

[1, 2]

nf

11.2 Itinerant Electron Model for Magnetic Metals (IEM Model)

211

Fig. 11.1 Dependence of the electrical resistivity (ρ) at 0 °C on the average free electron number (nf ) per atom for Fe, Ni, Co, and Cu [1, 2]

The IEM model is different from the conventional itinerant electron model in two important aspects [4, 5, 16, 17]: (i) The IEM model can be used to explain the relation between the magnetic moment and the electrical resistivity of a metal; (ii) The IEM model can be used to explain why the Curie temperature of Ni metal is far lower than that of Co and Fe, as shown in Sects. 12.4 and 12.5.

References 1. Qi WH, Ma L, Li ZZ, Tang GD, Wu GH (2017) Acta Phys Sin 66:027101 2. Tang GD, Li ZZ, Ma L, Qi WH, Wu LQ, Ge XS, Wu GH, Hu FX (2018) Phys Rep 758:1 3. Ida SK, Ono K, Kozaki H (1979) Data on physics in common use (in Chinese) (Trans by Zhang ZX). Science Press, Beijing 4. Dai DS, Qian KM (1987) Ferromagnetism (in Chinese). Science Press, Beijing 5. Chen CT, Idzerda YU, Lin HJ, Smith NV, Meigs G, Chaban E, Ho GH, Pellegrin E, Sette F (1995) Phys Rev Lett 75:152 6. Wu R, Wang D, Freeman AJ (1993) Phys Rev Lett 71:3581 7. Wu R, Freeman AJ (1994) Phys Rev Lett 73:1994 8. Jauch W, Reehuis M (2007) Phys Rev B 76:235121 9. Pacchioni GE, Gragnaniello L, Donati F, Pivetta M, Autès G, Yazyev OV, Rusponi S, Brune H (2015) Phys Rev B 91:235426 10. Lazarovits B, Szunyogh L, Weinberger P, Újfalussy B (2003) Phys Rev B 68:024433 11. Stepanyuk VS, Niebergall L, Longo RC, Hergert W, Bruno P (2004) Phys Rev B 70:075414 12. Santos Dias MD, Schweflinghaus B, Blügel S, Lounis S (2015) Phys Rev B 91:075405 13. Kisker E, Schroder K, Campagna M, Gudat W (1984) Phys Rev Lett 52:2285 14. Sánchez-Barriga J, Minár J, Braun J, Varykhalov A, Boni V, Di Marco I, Rader O, Bellini V, Manghi F, Ebert H, Katsnelson MI, Lichtenstein AI, Eriksson O, Eberhardt W, Dürr HA, Fink J (2010) Phys Rev B 82:104414

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15. Johnson PD (1997) Rep Prog Phys 60:1217 16. Stearns MB (1973) Phys Rev B 8:4383 17. Stearns MB (1978) Phys Today 31(4):34

Chapter 12

Study on the Origin of Magnetic Ordering Energy for Magnetic Materials

As mentioned in Sects. 3.1 and 3.4, in the traditional view [1], the magnetic ordering energy (the origin of the Weiss molecular field) originated from a quantummechanical effect, which cannot be explained using any classical physical model. In the energy function expression of DFT, a separate component of the phenomenological Coulomb energy is included, but no separate component of the magnetic ordering energy is included, which was present in the exchange–correlation energy. However, to date, no phenomenological expression of the exchange–correlation energy has been found. This causes difficulty in the prediction of new magnetic materials using DFT, although many successful examples for predicting other materials exist. In our opinion, exploring a phenomenological model of magnetic ordering energy is necessary. Therefore, our group proposed a phenomenological Weiss electron pair (WEP) model for magnetic ordering energy. Using the WEP model, our group explained why a rapid increase in crystal lattice constants is present near the Curie temperature. Using the WEP model as well as the IEO and IEM models, our group explained why there are differences in Curie temperatures for several typical magnetic materials and the dependence of electrical resistivity of FeNi alloys on the test temperature.

12.1 Weiss Molecular Field As mentioned in Sect. 3.1, magnetic domains are present in ferromagnetic, ferrimagnetic, and antiferromagnetic materials. Based on the molecular field assumption proposed by Weiss in 1907, the spontaneous magnetization with atomic magnetic moments ordering occurs in a magnetic domain, which was assumed to exist the Weiss molecular field with an intensity H m . The value of H m was estimated as follows [1].

© Science Press 2021 G.-D. Tang, New Itinerant Electron Models of Magnetic Materials, https://doi.org/10.1007/978-981-16-1271-8_12

213

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12 Study on the Origin of Magnetic Ordering Energy …

Table 12.1 Curie temperature, T C , observed average molecular magnetic moment μobs , saturation magnetization, M S, and the molecular field intensity, H m , of La0.8 Ca0.2 MnO3 , La0.75 Ca0.25 MnO3 , La0.70 Sr0.30 MnO3 , and Fe Sample

T c (K)

μobs (μB )

M s (105 A/m)

H m (107 A/m)

H m (105 Oe)

Ca0.20 [2]

198

3.76

6.03

6.24

7.84

Ca0.25 [3]

240

3.13

5.06

9.08

11.41

Sr0.30 [2] Fe [1]

369

3.50

5.61

12.48

15.70

1043

2.22

17.40

55.63

69.91

When the test temperature reached the Curie temperature T C , the spontaneous magnetization disappears at which the thermal energy of the atoms can be assumed to be equal to the magnetic ordering energy corresponding to the H m , kTC = Hm gSμB ,

(12.1)

where Bohr magneton μB = 1.1654 × 10−29 J/(A/m). For metal Fe, Curie temperature T C = 1043 K and the observed average atomic magnetic moment is about gSμB = 2.22μB . It can be calculated as Hm =

kTC 1043 × 1.38 × 10−23 = = 5.5633 × 108 A/m = 6.991 × 106 Oe. gSμ B 2.22 × 1.1654 × 10−29

Similarly, the H m values of the perovskite manganites La0.8 Ca0.2 MnO3 (Ca0.20), La0.75 Ca0.25 MnO3 (Ca0.25), and La0.70 Sr0.30 MnO3 (Sr0.30) were estimated, whose corresponding magnitudes and parameters are listed in Table 12.1. No explanation has been found for such high H m with the phenomenological model, which is a pressing problem to be solved in ferromagnetic physics. Using these data of H m and the corresponding saturation magnetizations M s , we can calculate the molecular field energy density, w, of these materials. Such as, in Fe, M s = 17.40 × 105 A/m, H m = 55.63 × 107 A/m. Then, we get w = μ0 Hm Ms = 4π × 10−7 × 55.63 × 107 × 17.40 × 105 = 1.2164 × 109 J/m3 . (12.2) With the w values, the average molecular field energy per magnetic ion pair, w0 , can be obtained. For a cubic crystal cell of Fe, the crystal lattice constant a = 2.86 Å. Each crystal cell contains two Fe atoms, we have w0 = wa 3 = 1.2164 × 109 × (2.86 × 10−10 )3 = 2.846 × 10−20 J = 0.1778 eV. (12.3) Compared to the average cohesive energy per pair of ions of ~10 eV [4], this value is reasonable. For the oxides Ca0.20, Ca0.25, and Sr0.30 and metal Fe, the calculated

12.1 Weiss Molecular Field

215

values of w and w0 are shown in Table 12.2. In the calculation process, the different crystal structures of these materials were considered. Each crystal cell contains four molecules in orthorhombic Ca0.20 and Ca0.25, w0 = wv/2; each crystal cell includes six molecules in rhombohedral Sr0.30, w0 = wv/3, where v is the cell volume. Table 12.2 depicts that w0 is two times the thermal energy k B T C at Curie temperature T C for all the four materials, indicating that the above calculations are reasonable.

12.2 Thermal Expansion of Perovskite Manganites Near the Curie Temperature The Weiss molecular field energy (magnetic ordering energy), w0 , lowers the system energy and compresses the crystal cell volume. The transition from magnetic order to disorder occurs at the Curie temperature, T C , where a corresponding increase in thermal expansivity takes place. The crystal lattice constants under different temperatures for La0.8 Ca0.2 MnO3 , La0.75 Ca0.25 MnO3 , and La0.70 Sr0.30 MnO3 were measured by Hibble et al. [2] and Radaelli et al. [3]. The obtained dependence on temperature of the Mn–O bond lengths along the different crystal directions is presented in Figs. 12.1, 12.2, and 12.3. The figures show that Mn–O bond lengths rapidly increase near the Curie temperature T C . Only the nonlinear variations in Mn–O bond lengths near T C were assumed to be related to the magnetic ordering. A tangent line along the curves in the low-temperature region may be obtained, as shown in Fig. 12.1. Then, the nonlinear variation amplitude values, d obs , corresponding to the changes in d 1 and d 2 for Ca0.20 sample were found to be 0.00147 Å and 0.00167 Å, respectively. Similarly, the d obs values for the materials Ca0.25 and Sr0.30 were obtained from Figs. 12.2 and 12.3. All the d obs data are shown in Table 12.3. The table shows that the d obs values for Ca0.20, Ca0.25, and Sr0.30 successively increase corresponding to their Curie temperatures, 198 K, 240 K, and 369 K. The other parameters in Table 12.3 are discussed in the next subsection.

12.3 Weiss Electron Pair (WEP) Model for Origin of Magnetic Ordering Energy By comparing the IEM and IEO models, we deduce that the itinerant electrons in magnetic metals and oxides are considered to have similar characteristics; they transit between the outer electron orbits of adjacent ions and their transitions must be constrained by the electron orbits. Therefore, in magnetic metals, the valence electrons were classified as free, itinerant, and local electrons; in magnetic oxides, valence electrons include only itinerant and local electrons.

kT C (eV)

0.0171

0.0207

0.0317

0.0900

Sample

Ca0.20 [2]

Ca0.25 [3]

Sr0.30 [2]

Fe [1]

2

6

4

4

Z

23.394293 K [4]

349.76715 K

231.21523 K

232.63315 K

v (Å3 )

17.40

5.610

5.064

6.030

M s (105 A/m)

55.63

12.48

9.08

6.24

H m (107 A/m)

121.7

8.801

5.778

4.725

w (107 J/m3 )

0.1778

0.0641

0.0418

0.0344

w0 (eV)

1.98

2.02

2.01

2.01

w0 /kT c

Table 12.2 Average energy of Weiss molecular field for La0.8 Ca0.2 MnO3 , La0.75 Ca0.25 MnO3 , La0.70 Sr0.30 MnO3 , and metal Fe. Here, Z is the molecule number per crystal cell, v is the volume of the crystal cell, w is the Weiss molecular field energy density, w0 is the average energy of Weiss molecular field per pair of magnetic ions, and kT C is the thermal energy at T C , M S is saturation magnetization, H m is the molecular field intensity

216 12 Study on the Origin of Magnetic Ordering Energy …

12.3 Weiss Electron-Pair (WEP) Model for Origin …

217

Fig. 12.1 Dependences on the test temperature T of the distances between Mn and O ions, d 1 , d 2 , d 3 , along the three directions of an orthorhombic La0.8 Ca0.2 MnO3 sample [2]

Fig. 12.2 Dependences on the test temperature T of the distances between Mn and O ions, d 1 , d 2 , d 3 , along the three directions of an orthorhombic La0.75 Ca0.25 MnO3 sample [3]

Based on the effective radii, r, given by Shannon [5] (see Appendix B), distinct differences exist in the r values of an element with different valences in compounds. This suggests that the outer electron orbits can be considered as the electron cloud shell including one electron or two electrons with opposite spins. As mentioned in Table 2.1, for the effective radii of several divalent and trivalent cations with a coordination number 6, r 2+ and r 3+ , the difference, r 2+ − r 3+ , ranges between 0.09 and 0.19 Å, indicating that one electron in the outer orbit of an ion moves in a spherical shell with a thickness of approximately 0.1–0.2 Å. Based on the above studies, our group [6, 7] proposed a Weiss electron pair (WEP) model for the origin of the magnetic ordering energy:

218

12 Study on the Origin of Magnetic Ordering Energy …

Fig. 12.3 Dependences on the test temperature T of the distances between Mn and O ions, d 1 and d 3 , along the two directions of a rhombohedral La0.7 Sr0.3 MnO3 sample [2]

Table 12.3 Probability of forming WEPs, D, and the relative parameters of La0.8 Ca0.2 MnO3 , La0.75 Ca0.25 MnO3 , and La0.70 Sr0.30 MnO3 samples. Here, d obs is the nonlinear variation amplitude of the Mn–O bond length near T C , and r e0 and r em are the equilibrium and maximum distances, respectively, between electrons in a WEP, r e = r em − r e0 d obs (Å)

Sample

w0 (eV)

Mn–O bond

Ca0.20

0.0344

d1

0.00147

d2

0.00167

D (%)

r e (Å)

r e0 (Å)

r em (Å)

0.045

0.00889

0.01036

0.00147

0.066

0.0101

0.01177

0.00167

Ca0.25

0.0417

d1

0.00261

0.308

0.01586

0.01848

0.00261

d2

0.00236

0.228

0.01436

0.01673

0.00236

Sr0.30

0.0641

d1

0.00490

3.130

0.02970

0.03460

0.00490

(i)

(ii)

(iii)

Supposing that a moving electron in an outer orbit of an ion has a constant spin direction, the electrons in the outer orbits of the adjacent ions, including the adjacent cations and anions in a compound and the ions (atoms that have lost free electrons) in a metal, may have three states, as shown in Fig. 12.4a, b, and c. If the electrons have the state illustrated in Fig. 12.4a, there are two electrons with opposite spin directions in the outer orbit of each ion. The two electrons located between adjacent ions cannot be exchanged because they have opposite spin directions. The magnetic ordering energy is thus given by the difference between the static magnetic attractive energy and the Pauli repulsive energy of the two electrons. This type of electron pair, which has a particular lifetime and probability of appearing, is called a WEP. When the two electrons between adjacent ions have the state shown in Fig. 12.4b, both the static magnetic and Pauli repulsive energies exist between the two neighboring electrons and they can be easily exchanged since they both have the same spin direction. When the electrons have the state depicted in

12.3 Weiss Electron-Pair (WEP) Model for Origin …

219

Fig. 12.4 Illustrations of a a WEP and b and c itinerant electrons in the outer orbits of adjacent ions

Fig. 12.4c, the middle electron can easily transit the outer orbit of the right ion. Thus, both the electron exchange in Fig. 12.4b and electron transit in Fig. 12.4c may be considered to be simply the transitions of itinerant electrons, wherein the spin directions of the itinerant electrons cannot change. Thus, based on the WEP model, we can successfully explain why an itinerant electron has a constant spin direction, which had not been explained until proposed the WEP model, although the itinerant electron concept has been used for many years. Based on the WEP model, Weiss molecular field energy may be estimated. We assume that D represents the probability of forming a WEP (cf. Fig. 12.4a) and r e represents the distance between the two electrons. Since each electron has an electric charge of −e and a spin of 1μB , in addition to the ionic cohesive energy, the average increase in the system energy is given by u =

(1μ B )2 C − D × . re9 4π μ0 re3

(12.4)

Here, the first component represents the Pauli repulsive energy, and the second one represents the magnetic attractive energy between the two electrons with opposite spin directions. When the two electrons are in the equilibrium state (where r e = r e0 ),  = 0 then the derivative with respect to the distance between electrons, du dre  re =re0

0=−

9C 3(1μ B )2 + D × . 10 4 re0 4π μ0 re0

Therefore, the value of C can be derived as

(12.5)

220

12 Study on the Origin of Magnetic Ordering Energy …

C=

6 3D(1μ B )2 re0 . 36π μ0

(12.6)

Substituting the C value into Eq. (12.4), we have  6  D(1μ B )2 re0 1 . u = − 4π μ0 3re9 re3

(12.7)

Substituting μB = 1.165 × 10−29 J/(A/m), μ0 = 4π × 10−7 H/m, 1 eV = 1.602 × 10−19 J, and the unit of r e is 10−12 m into Eq. (12.7), we obtain  6  re0 1 (1.165 × 10−29 )2 1 D eV − 16π 2 × 10−7 1.602 × 10−19 × 10−36 3re9 re3   6 r 1 u = 53.65 e09 − 3 D. 3re re

u =

(12.8)

When r e = r e0 , we obtain 2 53.65D u 0 = − × 3 3 re0

(12.9)

and 3 re0 =

35.77D . |u 0 |

(12.10)

The maximum distance between the two electrons in a WEP, r em , may be given  ∂ 2 u  = 0, by ∂r 2  e

re =rem

  6 ∂2 90re0 12 u = 53.65 − 5 D, ∂r 2 3re11 re 6 rem =

6 30re0 , rem = 1.165re0 . 12

(12.11)

(12.12)

When r e < r em , the WEP is stable since energy u. When r e > r em due to the increase in temperature, the WEP is broken, causing the molecular field to disappear. This behavior can be described by considering the bond length deviation near to the Curie temperature: dobs = re = rem − re0 .

(12.13)

Here, d obs is the nonlinear thermal expansive length, and its values for La0.8 Ca0.2 MnO3 , La0.75 Ca0.25 MnO3 , and La0.70 Sr0.30 MnO3 are given in Table 12.3.

12.3 Weiss Electron-Pair (WEP) Model for Origin …

221

Furthermore, assuming that |u0 | in Eq. (12.10) is equal to the w0 values in Table 12.3, r e0 , r em , and D can be determined using Eqs. (12.10)–(12.13); they are as listed in Table 12.3. Exploring the data in Table 12.3 for the perovskite manganites La0.8 Ca0.2 MnO3 , La0.75 Ca0.25 MnO3 , and La0.70 Sr0.30 MnO3 , we find that magnetic ordering occurs when the probability D forming WEP (cf. Fig. 12.4a) reaches 0.066%, 0.228%, and 3.13%, respectively; we also find that the r em for the two electrons constituting a WEP is less than 0.035 Å, which is distinctly smaller than the electron moving range (0.09 Å, see Table 2.1) along the ionic radius direction in the outer electron orbit of the ion. Thus, the WEP model is suitable for explaining the magnetic ordering energy of the materials.

12.4 Explanation for the Curie Temperature Difference of Typical Magnetic Materials For typical magnetic materials Co, Fe, Ni, Fe3 O4 , and La0.7 Sr0.3 MnO3 , the Curie temperature T C is 1404, 1043, 631, 860, and 369 K, respectively. Using the WEP model, we can explain why these materials have different Curie temperatures [7, 8]. The magnetic metal Co has a hexagonal close-packed (hcp) crystal structure, wherein each ion (excluding the free electrons) has 12 nearest adjacent ions, in which WEP can form between each ion pair. Therefore, the average number of bonds surrounding an ion capable of forming WEPs, N WEP , is 12. Similarly, for the body-centered cubic (bcc) magnetic metal Fe, N WEP = 8. Thus, the T C of Fe is lower than that of Co. Although each ion in the magnetic metal Ni with an face-centered cubic (fcc) crystal structure has 12 nearest neighbors, its N WEP value is far lower than that of metal Co. According to the IEM model, the average number of 3d electrons, nd , in Ni is 9.38 (cf. Sect. 11.2 and Table 11.1). That is, 38% of Ni ions have full 3d subshells filled by 10 electrons; subsequently, these ions cannot form WEPs with their adjacent ions, similar to those in Cu. Therefore, only 62% of Ni ions can form WEPs with 62% of their nearest adjacent ions, thus obtaining an average value N WEP = 4.61 (= 12 × 0.62 × 0.62). For ferrite Fe3 O4 with the (A)[B]2 O4 spinel structure, a cation has four or six nearest adjacent O anions in the (A) or [B] site, and 8 (A) sites and 16 [B] sites are present per crystal cell. Thus, N WEP = 5.33 [= (4 × 8 + 6 × 16)/24]. A similar calculation can be performed for the manganite La0.7 Sr0.3 MnO3 with ABO3 perovskite structure, where two cations are present per equivalent cubic crystal cell, while WEPs can only be formed at six Mn–O bonds, yielding N WEP = 3. Using these N WEP values and the Curie temperature (T C ) data in Table 4.5, we showed that T C decreases as N WEP decreases (Fig. 12.5). This result indicates that the assumption for the magnetic ordering energy originated from WEPs is reasonable. This view is very important to search new magnetic materials with high T C .

222

12 Study on the Origin of Magnetic Ordering Energy …

Fig. 12.5 Dependence of the Curie temperature T C on the average number of chemical bonds capable of forming WEPs, N WEP , near a cation [7, 8]

12.5 Explanation for Cu Ratio Dependence of Resistivity and Curie Temperature for NiCu Alloys Based on the IEM model, our group [9] fitted the dependence on test temperature of the resistivity of magnetic NiCu alloys using an equivalent circuit containing free electron (FE) and itinerant electron (IE) channels. The movement of FEs is spinindependent, which moved within a weak potential field of the crystal lattice, were not restrained by the electronic orbit of any ion. The IEs could hop between adjacent ions with a certain probability, which were restrained by the outer electronic orbits of the ions. Below the Curie temperature, T C , the electrical transport of the IEs was spin-dependent. In addition, the dependences of T C of the NiCu alloys on the Cu ratio were explained based on the IEM and the WEP models. Therefore, we provided a scheme to explain both electrical transport and magnetic properties of NiCu alloys using their valence electron structures.

12.5.1 Free and 3d Electron Ratios in NiCu Alloys The average magnetic moment, μobs , per ion of the Ni1−x Cux alloy as function of the Cu ratio (x) is given by Slater–Pauling curves [10], as shown in Fig. 12.6a. This observed result can be fitted using the following equation, μcal = 0.6(1 − x) − 0.57x.

(12.14)

Equation (12.14) can be explained using the IEM model: (i) In 10 valence electrons (3d 8 4s2 ) per Ni atom, an average of 9.4 electrons occupy the 3d orbits and 0.6 electrons form FEs in the metallic Ni. Consequently, the average magnetic moment per ion in metallic Ni is 0.6 μB . (ii) In 11 valence electrons (3d 10 4s1 ) per Cu atom,

12.5 Explanation for Cu Ratio Dependence of Resistivity …

a

223

b

Fig. 12.6 Studies on Ni1−x Cux alloys: a Dependence on the Cu ratio x of the observed (points) [10] and fitted (line) [9] average magnetic moments per atom. b Dependencies on x of the average numbers of 3d electrons per Ni ion (nd ) and FEs per atom of alloy (nf )

10 electrons remain in the 3d orbits of Cu, 0.57 electrons enter the 3d orbitals of the Ni ions in the alloy, and 0.43 electrons contribute to the FEs of the Ni1−x Cux alloys. Then, the dependence on Cu ratio x of the average numbers of 3d electrons per Ni ion (nd ) and FEs per atom (nf ) in the NiCu alloy can be given as n d = 9.40 +

0.57x , (n d ≤ 10.0), 1−x

n f = 10 + x − n d .

(12.15) (12.16)

Figure 12.6b depicts the curves of nd and nf versus x. Since the nd value increases to 10.0 for a full 3d electron shell when x = 0.51, we obtain the following result: nd = 10.0, nf = x, when x > 0.51.

12.5.2 Electrical Transport Model with FE and IE Channels Figure 12.7 shows the temperature dependences of resistivity (ρ) with different Cu ratio (x) for the Ni1−x Cux alloys given by Ref. [11], where both x and T C are signed. These curves are discussed as follows: (i)

For materials with x = 0.00, 0.0961, 0.1954, and 0.2810, the T C is within the test temperature range. We found that the decrease of the ρ value below T C is obviously faster than that above T C . The curves of ρ versus T can be fitted using an equivalent device with two current-carrier channels, as shown in Fig. 12.8. Here, R3 represents the resistance (resistivity ρ 3 ) related to the FEs, which are scattered by a weak, periodic crystal lattice potential field. R1 represents the resistance (resistivity ρ 1 ) related to the transition of IEs between adjacent ions, which are scattered by the crystal lattice. R2 represents the resistance (resistivity

224

12 Study on the Origin of Magnetic Ordering Energy …

Fig. 12.7 Resistivity (ρ) versus test temperature (T ) and Cu content (x) in Ni1−x Cux alloys. The values of x and the Curie temperature, T C , are shown. The data were taken from Ref. [11]

ρ 2 ) related the IEs, whose spin direction rapidly deviates from the ground-state direction when T is close to T C . Thus, the total resistance, R, in Fig. 12.8, can be calculated using the below equation R=

(R1 + R2 )R3 . R1 + R2 + R3

Thus, the resistivity, ρ, of the materials can be given as ρ=

(ρ1 + ρ2 )ρ3 , ρ1 + ρ2 + ρ3

(12.17)

where the resistivity is calculated by ρ1 = 1/σ1 , ρ2 = 1/σ2 , ρ3 = 1/σ3 ,

(12.18)

Fig. 12.8 An equivalent device with two current-carrier channels used to fit the dependencies of ρ on T for a magnetic metal or alloy, where R3 is in the channel of the FEs, and R1 and R2 are in the channel of the IEs

12.5 Explanation for Cu Ratio Dependence of Resistivity …

225

and the conductivity is fitted by  E , σ3 = σ0 − a3 T. σ1 = (a11 − a12 T ) , σ2 = a2 exp kB T 

3

(12.19)

The fitted results (curves) obtained using Eqs. (12.17)–(12.19) for metallic Ni are shown in Fig. 12.9, where the points represent previously derived data [11]. Thus, (a) if T < < T C , ρ 2 is very small; ρ 1 < ρ 3 ; and the electrical transport occurs along both channels. (b) With increasing T, the increase of ρ 1 is obviously faster than that of ρ 3 . (c) If T is close to T C , ρ 2 rapidly increases. (d) If T > T C , both ρ 1 and ρ 2 are much higher than ρ 3 ; the electrical transport occurs along the FE channel, and the contribution of the IEs to the conductivity can be neglected. (ii)

Fitted temperature dependence of ρ for materials with x = 0.00, 0.0961, 0.1954, and 0.2810 are shown in Fig. 12.10. The fitted results (curves) are very close to the observed data points [11]. The fitted parameters (cf. Eq. 12.19) are shown in Table 12.4. The parameters a11 and a12 are related to the effect of lattice vibrations on the spin-dependent transition of IEs. The parameter a2 is related to the effect of the spin direction of IEs because it deviates from the groundstate spin direction when the test temperature increases. The parameters σ 0 and a3 are related to the effect of the periodic crystal lattice potential field on FEs. These effects are related to x.

When x ≤ 0.281, the total resistivity, ρ, increases with increasing x. This may originate from two factors: the effect of the crystal lattice potential field increases with increasing x and no obvious change is observed in the concentration of FEs (Fig. 12.6b). When x > 0.5, ρ decreases with increasing x (see Fig. 12.7). This may result from an increase in the concentration of FEs with increasing x, as shown in Fig. 12.6b. Fig. 12.9 Illustration of the fitted resistivity curves, ρ 1 , ρ 2 , ρ 3 , and ρ, versus the temperature, T, for the magnetic metal Ni. The observed point data were taken from Ref. [11]

226

12 Study on the Origin of Magnetic Ordering Energy …

Fig. 12.10 Fitted resistivity (ρ) curves versus test temperature (T ) for Ni1−x Cux alloys. The point data were taken from Ref. [11]

(iii)

The relations of ρ 1 and ρ 2 to T in Eqs. (12.18) and (12.19) are very similar to those reported for perovskite manganites (ref Sect. 9.2) [12]. in particular, the thermal excitation energy, E, of the spin-dependent resistivity ρ 2 decreases with decreasing T C in the Ni1−x Cux alloys (Table 12.4) or perovskite manganites La1−x Srx MnO3 [12], as depicted in Fig. 12.11. This suggests that the assumption that IEs have similar properties in both magnetic alloys and oxides in the WEP model is reasonable.

(iv)

The view [13] that the magnetic oxides (such as perovskite manganites La1−x Srx MnO3 ) possess metallic electrical conductivity below T C should be improved [12] based on the following reasons. In magnetic metals or alloys, the current carriers include spin-independent FEs and spin-dependent IEs. However, in magnetic oxides such as La1−x Srx MnO3 [12], the current carriers include only IEs, both spin-dependent IEs along the O–Mn–O and spinindependent IEs along the O–La(Sr) –O ion chains, and no FE is involved along any ion chain.

12.5.3 An Explanation of the Curie Temperature Using the WEP Model As mentioned in Sect. 12.4, the Curie temperature, T C , of Co, Fe, and Ni metals decrease with a decrease in the average chemical bond number, N WEP , at which bonds WEPs can form in a certain probability, as shown in Fig. 12.5. In the case of Ni1−x Cux alloys with x = 0.0961, 0.1954, and 0.2810, the N WEP values can be calculated to be 3.49, 2.56, and 1.71, respectively, using the method in Sect. 12.4. In the calculation process, the used average 3d electron number, nd , was given for Ni ions using Eq. (12.15). We, therefore, obtained an approximately linear dependence

a11 (−1/3 cm−1/3 )

0.690

0.470

0.385

0.345

x

0.00

0.0961

0.1954

0.2810



10−13

1.54 × 1.25 ×

10−4

6.00 ×

7.00 ×

10−13

1.82 × 10−13

10−4

2.22 ×

10−13

a2 (−1 cm−1 )

6.00 × 10−4

8.30 ×

10−4

a12 (−1/3 cm−1/3 K−1/3 )

Table 12.4 Fitted parameters of the conductivity σ1 = (a11 − a12 T )3 , σ2 = a2 exp E kB T

0.75

0.85

1.03

1.30

325

415

522

636

T C (K)

0.02246

0.02552

0.03365

0.04975

σ 0 (−1 cm−1 )

4.60 × 10−6

7.00 × 10−6

1.40 × 10−5

2.65 × 10−5

a3 (−1 cm−1 K−1 )

, and σ3 = σ0 − a3 T for Ni1−x Cux alloys. T C is the Curie temperature

E (eV)



12.5 Explanation for Cu Ratio Dependence of Resistivity … 227

228

12 Study on the Origin of Magnetic Ordering Energy …

Fig. 12.11 Thermal excitation energy (E) of the spin-dependent resistivity (ρ 2 ) versus the Curie temperature (T C ) for the Ni1−x Cux alloys [9] and the perovskite manganites (La1−x Srx MnO3 ) [12]

of T C on N WEP for Co, Fe, Ni metals, and Ni1−x Cux alloys, as shown in Fig. 12.12. Therefore, the assumptions for the magnetic ordering energy origination in the WEP model and the classification of electrons as FEs, IEs, and LEs in the IEM model are reasonable. In summary, the WEP model can be used to explain the origin of magnetic ordering energy. WEPs have a certain probability and lifetime. The attractive energy originates from two electrons with opposite spin directions in WEP, while Pauli repulsive energy is present between the two electrons. The two kinds of energies result

Fig. 12.12 Dependence of T C on the average chemical bond number near a magnetic ion, N WEP , at which bonds WEPs can form in a certain probability for Ni1−x Cux alloys and Fe and Co metals [9]

12.5 Explanation for Cu Ratio Dependence of Resistivity …

229

in that there is a balance distance (r e0 ) in the two electrons of WEP. Of course, other factors still affect the system energy. For example, the perovskite manganites La0.8 Ca0.2 MnO3 , La0.75 Ca0.25 MnO3 , and La0.70 Sr0.30 MnO3 have different T C values, 198, 240, and 369 K, respectively; their probability D of WEP formation were calculated in Sect. 12.3, to be 0.066%, 0.228%, and 3.13%, respectively. These differences need to be explored further.

References 1. Dai DS, Qian KM (1987) Ferromagnetism (in Chinese). Science Press, Beijing 2. Hibble SJ, Cooper SP, Hannon AC, Fawcett ID, Greenblatt M (1999) J Phys Condens Matter 11:9921 3. Radaelli PG, Cox DE, Marezio M, Cheong S-W, Shiffer PE, Ramirez AP (1995) Phys Rev Lett 75:4488 4. Gou QQ (1978) Course in solid state physics (in Chinese). People’s Education Press, Beijing 5. Shannon RD (1976) Acta Cryst A 32:751 6. Qi WH, Li ZZ, Ma L, Tang GD, Wu GH, Hu FX (2017) Acta Phys Sin 66:067501 7. Tang GD, Li ZZ, Ma L, Qi WH, Wu LQ, Ge XS, Wu GH, Hu FX (2018) Phys Rep 758:1 8. Qi WH, Li ZZ, Ma L, Tang GD, Wu GH (2018) AIP Adv 8:065105 9. Li ZZ, Qi WH, Ma L, Tang GD, Wu GH, Hu FX (2019) J Magn Magn Mater 482:173 10. Chen CW (1977) Magnetism and metallurgy of soft magnetic materials. North-Holland Publishing Company, pp. 171–417 11. Fang JX, Lu. D (1980) Solid state physics (in Chinese). Shanghai Scientific and Technical Publishers, pp. 310–315 12. Qian JJ, Qi WH, Li ZZ, Ma L, Tang GD, Du YN, Chen MY, Wu GH, Hu FX (2018) RSC Adv 8:4417–4425 13. Urushibara A, Moritomo Y, Arima T, Asamitsu A, Kido G, Tokura Y (1995) Phys Rev B 51:14103

Chapter 13

Prospects and Challenges for Future Work

A series of magnetic ordering rules for magnetic oxides and metals were proposed by our group, which include the IEO model for magnetic oxides, the IEM model for magnetic metals, and the WEP model for the origin of magnetic ordering energy. These new phenomenological models based on experimental results, including electron spectra, are easier to understand than the conventional exchange interaction models. By applying these new models to suitable materials, such as antiferromagnetic MO (M = Mn, Fe, Co, or Ni) oxides, (A)[B]2 O4 spinel ferrites MFe2 O4 (M = Ti, Cr, Mn, Fe, Co, Ni, or Cu), ABO3 perovskite manganites R1−x T x MnO3 (where R represents a trivalent rare earth element, La, Pr, Nd, … and T represents a divalent alkaline-earth element, Sr, Ba, Ca, …), and Fe, Co, and Ni metals, NiCu alloys, different magnetic structures have been explained. These magnetic structures include not only those that could be explained using conventional models but also those that cannot be sufficiently explained using conventional models. The greater success of these new models compared to the conventional models indicates that the traditional models should be improved by the newer models described in this book. However, much effort needs to be made yet to establish a set of new ferromagnetic theories on the basis of the experimental results of electron spectra reported in the past decades. The remaining problems can be stated as follows.

13.1 Other Factors Affecting Magnetic Ordering Energy The three perovskite manganites, La0.8 Ca0.2 MnO3 , La0.75 Ca0.25 MnO3 , and La0.70 Sr0.30 MnO3 , have the same average chemical bond number, N WEP , at which WEPs can form with a certain probability. However, the probabilities of WEP formation were given as 0.066%, 0.228%, and 3.13%, respectively, in Table 12.3. This indicated that there are other factors to affect the magnetic ordering energy in addition to N WEP :

© Science Press 2021 G.-D. Tang, New Itinerant Electron Models of Magnetic Materials, https://doi.org/10.1007/978-981-16-1271-8_13

231

232

13 Prospects and Challenges for Future Work

Fig. 13.1 Illustration of magnetic repulsive energy of the electron spins between adjacent ions, this state may be present in a certain probability

First, the distance between the two ions in Fig. 12.4 has important effect on the probability forming the WEP and the transition probability of the itinerant electrons, and therefore has important effect on the magnetic ordering energy, as discussions in Sects. 9.2.4 and 9.3.5. Second, there is magnetic repulsive energy between the two electrons with the same spin direction when they locate at the state shown in Fig. 12.4b. In addition, there is a certain probability to present the spin state as Fig. 13.1. In this case, since the distance between the electrons with the same spin direction is far shorter than the ionic distance, the corresponding magnetic repulsive energy can be added in Eq. (12.8),  6  re0 1 u = −53.65 3 − 9 (D1 − D2 ) eV, re 3re

(13.1)

where D1 represents the parameter of the magnetic attraction energy between the electrons of WEPs and D2 represents the parameter of the magnetic repulsive energy originated from the state, as shown in Figs. 12.4b and 13.1, which are related to the valence electron state in outer orbits and the electron number in the outer subshell (e.g., 3d electron number). Third, the average molecular magnetic moments and T C decrease rapidly and present the canted magnetic structures for (A)[B]2 O4 spinel ferrites MFe2 O4 (M = Mn, Co, Ni, Cu) when the Fe ratio is less than 2.0, since Fe cations were substituted by any other cations, or for ABO3 perovskite manganites La0.85 Sr0.15 MnO3 , when Mn cations were substituted by any other cations. There are many Fe3+ (3d5 ) cations in the former, and Mn3+ (3d4 ) cations in the later. According to the IEO model, an itinerant electron with up-spin occupies the highest energy level of the 3d subshell in the Mn3+ cation (see Fig. 5.16), and an IE with down-spin occupies the highest energy level of the 3d subshell in the Fe3+ cation (see Fig. 5.17). Otherwise, an IE has to occupy the lower energy level of other cations. This outcome

13.1 Other Factors Affecting Magnetic Ordering Energy

233

indicates that there is a maximum WEP formation probability when an IE with upspin transits along an ion chain O2− −Mn3+ −O1− , or an IE with down-spin transits along an ion chain O2− −Fe3+ −O1− , resulting in that the material has the maximum magnetic ordering energy. Otherwise, the WEP formation probability decreases, and the magnetic ordering energy decreases. In addition, comparing that an IE transits through the highest 3d energy level of the 3d cations, it may consume more energy when it transits by a lower energy levels. Taking into account these factors, Eq. (13.1) may be revised as  6  re0 1 u = −53.65 3 − 9 (D1 − D2 ) + D3 eV. re 3re

(13.2)

For the factors that affect the parameters D1 , D2 , and D3 , it needs to be further explored by means of experimental approaches.

13.2 Magnetic Ordering Energy in DFT Calculations Using DFT to fit physical properties is particularly challenging for magnetic materials [1]. This is due to the inclusion of the magnetic ordering energy in the exchange– correlation energy [2], which has not been phenomenally expressed so far and has to be fitted using various models in DFT calculations. Based on the WEP model, the expression of the magnetic ordering energy may be found and added to the potential energy function of DFT to improve the prediction of the physical properties of magnetic materials.

13.3 Applications of the IEO and IEM Models As mentioned in Chap. 5, the many experimental results indicated that the actual valence states of oxides are lower than their values based on the traditional view. In 1998, the O 2p hole density was taken into account in the Hamiltonian function of the DFT calculation in the book “Physics of high temperature superconductor” by Han [3]. However, the SE and DE interaction models, in which O 2p hole was neglected and all oxygen anions were assumed to be negative divalent, have been used to discuss the magnetic ordering of magnetic oxides. Taking into account that there are O 2p holes (negative monovalent oxygen anions), our group proposed the IEO model. Using the IEO model, we explained the magnetic ordering of spinel ferrites, perovskite manganites, and monoxides with sodium chloride structure. Next, the magnetic ordering of magnetic oxides with other crystal structures, such as garnet or magnetoplumbite, needs to be studied. Specifically, many disputes remain regarding the magnetic ordering of dilute magnetic semiconductor oxides, which may be explained by using the IEO model.

234

13 Prospects and Challenges for Future Work

For nonmagnetic oxides, nitrides, sulfides, selenides, and tellurides, we expect that taking their actual valences into account may lead to ground-breaking discoveries in these fields. It is just the beginning of applications of the IEM model, which should be expanded to various corresponding materials, such as alloys, rare earth metals, and permanent magnetic materials.

References 1. Ströhr J, Siegmann HC (2006) Magnetism: from fundamentals to nanoscale dynamics. Springer, Berlin 2. Xie XD, Lu D (1998) Energy band theory of solid (in Chinese). Fudan University Press, Shanghai 3. Han RS (1998) Physics of high temperature superconductor (in Chinese). Peking University Press, Beijing

Appendix A

Electron Structure and Ionization Energies of Free Atoms

No. Symbol Electron configuration

V 1 (eV) V 2 (eV)

V 3 (eV)

V 4 (eV)

V 5 (eV)

1

H

1s1

2

He

1s2

3

Li

[He] 2s1

5.3917

75.6402

122.4529

4

Be

[He]

2s2

9.3227

18.2112

153.8966 217.7187

5

B

[He] 2s2 2p1

8.298

25.1548

37.9306

259.3752 340.2258

6

C

[He] 2s2 2p2

11.2603 24.3833

47.8878

64.4939

392.087

7

N

[He]

2s2

2p3

14.5341 29.6013

47.4492

77.4735

97.8902

8

O

[He] 2s2 2p4

13.6181 35.1173

54.9355

77.4135

113.8990

9

F

[He] 2s2 2p5

17.4228 34.9708

62.7084

87.1398

114.2428

2s2

21.5645 40.9633

63.45

97.12

126.21

5.1391

47.2864

71.6200

98.91

138.40

10

Ne

[He]

11

Na

[Ne] 3s1

12

Mg

[Ne] 3s2 3s2

2p6

13.5984 24.5874 54.4178

7.6462

15.0353

80.1437

109.2655 141.27

3p1

13

Al

[Ne]

5.9858

18.8286

28.4477

119.992

153.825

14

Si

[Ne] 3s2 3p2

8.1517

16.3459

33.4930

45.1418

166.767

15

P

[Ne] 3s2 3p3

10.4867 19.7594

30.2027

51.4439

65.0251

16

S

[Ne]

3s2

34.79

47.222

72.5945

17

Cl

[Ne] 3s2 3p5

12.9676 23.814

39.61

53.46

67.8

18

Ar

[Ne]

3s2

15.7596 27.6297

40.74

59.81

75.02

19

K

[Ar] 4s1

4.3407

31.63

45.806

60.91

82.66

20

Ca

[Ar] 4s2

6.1132

11.8717

50.9131

67.27

84.50

21

Sc

[Ar] 3d 1 4s2

6.5614

12.7997

24.7567

73.4894

91.65

22

Ti

[Ar] 3d 2 4s2

6.8282

13.5755

27.4917

43.2672

99.30

23

V

[Ar] 3d 3 4s2

6.7463

14.66

29.311

46.709

65.2817

24

Cr

[Ar]

3d 5

4s1

6.7664

16.4857

30.96

49.16

69.46

25

Mn

[Ar] 3d 5 4s2

7.4340

15.6400

33.668

51.2

3p4 3p6

10.36

23.3379

72.4 (continued)

© Science Press 2021 G.-D. Tang, New Itinerant Electron Models of Magnetic Materials, https://doi.org/10.1007/978-981-16-1271-8

235

236

Appendix A: Electron Structure and Ionization Energies of Free Atoms

(continued) No. Symbol Electron configuration

V 1 (eV) V 2 (eV)

V 3 (eV)

V 4 (eV)

V 5 (eV)

26

Fe

[Ar] 3d 6 4s2

7.9024

16.1878

30.652

54.8

75.0

27

Co

[Ar] 3d 7 4s2

7.8810

17.083

33.50

51.3

79.5

28

Ni

[Ar]

3d 8

7.6398

18.1688

35.19

54.9

76.06

29

Cu

[Ar] 3d 10 4s1

7.7264

20.2924

36.84

57.38

79.8

30

Zn

[Ar] 3d 10 4s2

9.3941

17.9644

39.723

59.4

82.6

31

Ga

[Ar]

3d 10

4p1

5.9993

20.5142

30.71

64

32

Ge

[Ar] 3d 10 4s2 4p2

7.8994

15.9346

34.2241

45.7131

93.5

33

As

[Ar] 3d 10 4s2 4p3

9.8152

18.633

28.351

50.13

62.63

9.7524

21.19

68.3

4s2

3d 10

4s2

4s2

4p4

34

Se

[Ar]

30.8204

42.9450

35

Br

[Ar] 3d 10 4s2 4p5

11.8138 21.8

36

47.3

59.7

36

Kr

[Ar] 3d 10 4s2 4p6

13.9996 24.3599

36.950

52.5

64.7

37

Rb

[Kr] 5s1

4.1771

27.285

40

52.6

71.0

38

Sr

[Kr] 5s2

5.6948

11.0301

42.89

57

71.6

39

Y

[Kr] 4d 1 5s2

6.2173

12.24

20.52

60.597

77.0

40

Zr

[Kr] 4d 2 5s2

6.6339

13.13

22.99

34.34

80.348

41

Nb

[Kr] 4d 4 5s1

6.7589

14.32

25.04

38.3

50.55

42

Mo

[Kr]

4d 5

7.0924

16.16

27.13

46.4

54.49

43

Tc

[Kr] 4d 5 5s2

7.28

15.26

29.54

44

Ru

[Kr] 4d 7 5s1

7.3605

16.76

28.47

45

Rh

[Kr]

4d 8

7.4589

18.08

31.06

46

Pd

[Kr] 4d 10

8.3369

19.43

32.93

47

Ag

[Kr] 4d 10 5s1

7.5762

21.49

34.83

48

Cd

[Kr]

4d 10

8.9937

16.9083

37.48

49

In

[Kr] 4d 10 5s2 5p1

5.7864

18.8698

28.03

54

50

Sn

[Kr] 4d 10 5s2 5p2

7.3438

14.6323

30.5026

40.7350

5s1

5s1

4d 10

5s2

5s2

5p3

72.28

51

Sb

[Kr]

8.64

16.5305

25.3

44.2

56

52

Te

[Kr] 4d 10 5s2 5p4

9.0096

18.6

27.96

37.41

58.75

53

I

[Kr] 4d 10 5s2 5p5

10.4513 19.1313

33

4d 10

5s2

5p6

54

Xe

[Kr]

12.1299 21.2098

32.1230

46

55

Cs

[Xe] 6s1

3.8939

23.1575

35

51

56

Ba

[Xe] 6s2

5.2117

10.0039

57

La

[Xe] 5d 1 6s2

5.5769

11.06

19.1773

49.95

61.6

58

Ce

[Xe] 4f 1 5d 1 6s2

5.5387

10.85

20.198

36.758

65.55

59

Pr

[Xe] 4f 3 6s2

5.464

10.55

21.624

38.98

57.53

60

Nd

[Xe]

4f 4

5.525

10.73

22.1

40.4

61

Pm

[Xe] 4f 5 6s2

5.55

10.90

22.3

41.1

62

Sm

[Xe] 4f 6 6s2

5.6437

11.07

23.4

41.4

6s2

(continued)

Appendix A: Electron Structure and Ionization Energies of Free Atoms

237

(continued) No. Symbol Electron configuration

V 1 (eV) V 2 (eV)

V 3 (eV)

V 4 (eV)

63

Eu

[Xe] 4f 7 6s2

5.6704

11.241

24.92

42.7

64

Gd

[Xe] 4f 7 5d 1 6s2

6.150

12.09

20.63

44.0

4f 9

6s2

65

Tb

[Xe]

5.8638

11.52

21.91

39.79

66

Dy

[Xe] 4f 10 6s2

5.9389

11.67

22.8

41.4

67

Ho

[Xe] 4f 11 6s2

6.0216

11.80

22.84

42.5

68

Er

[Xe]

4f 12

6s2

6.1078

11.93

22.74

42.7

69

Tm

[Xe] 4f 13 6s2

6.1843

12.05

23.68

42.7

70

Yb

[Xe] 4f 14 6s2

6.2542

12.1761

25.05

43.56

71

Lu

[Xe] 4f 14 5d 1 6s2

5.4259

13.9

20.9594

45.25

72

Hf

[Xe] 4f 14 5d 2 6s2

6.8251

14.9

23.3

33.33

73

Ta

[Xe] 4f 14 5d 3 6s2

7.5496

16.2

7.864

17.7

7.8335

16.6

8.4382

16.9

4f 14

5d 4

6s2

74

W

[Xe]

75

Re

[Xe] 4f 14 5d 5 6s2 4f 14

5d 6

6s2

76

Os

[Xe]

77

Ir

[Xe] 4f 14 5d 7 6s2

8.967

78

Pt

[Xe] 4f 14 5d 9 6s1

8.9587

18.563

9.2257

20.5

4f 14

5d 10

6s1

79

Au

[Xe]

80

Hg

[Xe] 4f 14 5d 10 6s2

10.4375 18.756

34.2

72

81

Tl

[Xe] 4f 14 5d 10 6s2 6p1

6.1083

20.428

29.83

50.8

6p2

4f 14

5d 10

6s2

V 5 (eV)

66.8

82

Pb

[Xe]

7.4167

15.0332

31.9373

42.32

68.8

83

Bi

[Xe] 4f 14 5d 10 6s2 6p3

7.2856

16.69

25.56

45.3

56.0

84

Po

[Xe] 4f 14 5d 10 6s2 6p4

8.417

85

At

[Xe] 4f 14 5d 10 6s2 6p5

9.3

86

Rn

[Xe] 4f 14 5d 10 6s2 6p6

10.7485

87

Fr

[Rn] 7s1

4.0727

88

Ra

[Rn]

7s2

5.2784

89

Ac

[Rn] 6d 1 7s2

5.17

12.1

90

Th

[Rn] 6d 2 7s2

6.3067

11.5

20.0

28.8

91

Pa

[Rn]

5f 2

92

U

[Rn] 5f 3 6d 1 7s2

6.1941

93

Np

[Rn] 5f 4 6d 1 7s2

6.2657

5f 6

6d 1

7s2

5.89

7s2

6.0262

94

Pu

[Rn]

95

Am

[Rn] 5f 7 7s2

5.9738

96

Cm

[Rn] 5f 7 6d 1 7s2

5.9915

5f 9

7s2

10.14716

97

Bk

[Rn]

98

Cf

[Rn] 5f 10 7s2

6.1979 6.2817

99

Es

[Rn] 5f 11 7s2

6.42 (continued)

238

Appendix A: Electron Structure and Ionization Energies of Free Atoms

(continued) No. Symbol Electron configuration

V 1 (eV) V 2 (eV)

100 Fm

[Rn] 5f 12 7s2

6.5

101 Md

[Rn] 5f 13 7s2

6.58

102 No

[Rn]

5f 14

6.65

103 Lr

[Rn] 5f 14 6d 1 7s2

104 Rf

[Rn] 5f 14 6d 2 7s2

105 Db

[Rn] 5f 14 6d 3 7s2

7s2

4.9

V 3 (eV)

V 4 (eV)

V 5 (eV)

Appendix B

Effective Ion Radii Reported by Shannon1

Symbol

Valence

Coordination number

Effective radius

Note*

Ac

3

VI

1.12

R

Ag

1

II

0.67

IV

1.00

C

V

1.09

C

VI

1.15

C

VII

1.22

VIII

1.28

VI

0.94

2 Al

Am

3

VI

0.75

R

3

IV

0.39

*

V

0.48

VI

0.535

VII

1.21

VIII

1.26

IX

1.31

VI

0.975

VIII

1.09

VI

0.85

VIII

0.95

3

VI

0.58

A

5

IV

0.335

R*

VI

0.46

C*

VI

0.62

A

2

3 4 As

At

7

R*

R R

(continued) 1 Shannon

R D. Acta Cryst. A., 1976, 32: 751.

© Science Press 2021 G.-D. Tang, New Itinerant Electron Models of Magnetic Materials, https://doi.org/10.1007/978-981-16-1271-8

239

240

Appendix B: Effective Ion Radii Reported by Shannon

(continued) Symbol

Valence

Coordination number

Effective radius

Note*

Au

1

VI

1.37

A

3

IVSQ

0.68

B

Ba

Be

Bi

Bk

Br

C

Ca

VI

0.85

5

VI

0.57

3

III

0.01

*

IV

0.11

*

VI

0.27

C

VI

1.35

VII

1.38

VIII

1.42

IX

1.47

X

1.52

XI

1.57

2

2

3

XII

1.61

III

0.16

IV

0.27

A

C

C *

VI

0.45

C

V

0.96

C

VI

1.03

R* R

VIII

1.17

5

VI

0.76

E

3

VI

0.96

R

4

VI

0.83

R

VIII

0.93

R

−1

VI

1.96

P

3

IVSQ

0.59

5

IIIPY

0.31

7

IV

0.25

VI

0.39

4

III

0.08

IV

0.15

P A

2

A

VI

0.16

VI

1.00

VII

1.06

*

VIII

1.12

*

IX

1.18

X

1.23

C (continued)

Appendix B: Effective Ion Radii Reported by Shannon

241

(continued) Symbol Cd

Ce

Ce

Cf

Cl

Cm

Co

Cr

Valence 2

3

4

Coordination number

Effective radius

Note*

XII

1.34

C

IV

0.78

V

0.87

VI

0.95

VII

1.03

C

VIII

1.10

C

XII

1.31

VI

1.01

VII

1.07

E

VIII

1.143

R

IX

1.196

R

X

1.25

XII

1.34

C

VI

0.87

R

VIII

0.97

R

X

1.07

R

XII

1.14

3

VI

0.95

R

4

VI

0.821

R

VIII

0.92

−1

VI

1.81

5

IIIPY

0.12

7

IV

0.08

VI

0.27

A

3

VI

0.97

R

4

VI

0.85

R

VIII

0.95

R

IV

0.58

V

0.67

C

VI

0.745

R*

VIII

0.90

3

VI

0.61

4

IV

0.40

VI

0.53

R

2

VI

0.80

R*

3

VI

0.615

R*

4

IV

0.41

2

P *

(continued)

242

Appendix B: Effective Ion Radii Reported by Shannon

(continued) Symbol

Valence

Coordination number

Effective radius

VI

0.55

R

IV

0.345

R

VI

0.49

ER

VIII

0.57

6

IV

0.26

VI

0.44

1

VI

1.67

VIII

1.74

IX

1.78

X

1.81

XI

1.85

5

Cs

Cu

Cu

Dy

1

2

2

3

Er

Eu

3

2

3

Note*

C

XII

1.88

II

0.46

IV

0.60

E

VI

0.77

E

IV

0.57

V

0.65

VI

0.73

VI

1.07

VII

1.13

VIII

1.19

VI

0.912

VII

0.97

VIII

1.027

R

IX

1.083

R

VI

0.89

R

VII

0.945

VIII

1.004

R R

IX

1.062

VI

1.17

VII

1.20

VIII

1.25

IX

1.30

X

1.35

VI

0.947

VII

1.01

VIII

1.066

*

R

R R (continued)

Appendix B: Effective Ion Radii Reported by Shannon

243

(continued) Symbol F

Fe

Valence −1

Coordination number

Effective radius

Note*

IX

1.12

R

II

1.285

III

1.30

IV

1.31

VI

1.33

7

VI

0.08

2

IV

0.63

IVSQ

0.64

VI

0.78

R*

VIII

0.92

C

IV

0.49

*

V

0.58

VI

0.645

3

A

R*

VIII

0.78

4

VI

0.585

R

6

IV

0.25

R

Fr

1

VI

1.80

A

Ga

3

IV

0.47

*

V

0.55

Gd

Ge

VI

0.62

R*

VI

0.938

R

VII

1.00

VIII

1.053

IX

1.107

RC

2

VI

0.73

A

4

IV

0.39

*

VI

0.53

R*

−0.38

3

R

H

1

I II

−0.18

Hf

4

IV

0.58

R

VI

0.71

R

VII

0.76

Hg

1 2

VIII

0.83

III

0.97

VI

1.19

II

0.69

IV

0.96 (continued)

244

Appendix B: Effective Ion Radii Reported by Shannon

(continued) Symbol

Ho

I

Valence

3

Ir

K

Li

Lu

Mg

1.02

Note*

VIII

1.14

R

VI

0.901

R

VIII

1.015

R

IX

1.072

R

X

1.12

VI

2.20

A

5

IIIPY

0.44

*

VI

0.95

IV

0.42

VI

0.53

IV

0.62

VI

0.8

R*

3

VIII

0.92

RC

3

VI

0.68

E

4

VI

0.625

R EM

5

VI

0.57

1

IV

1.37

VI

1.38

1

La

Effective radius

VI

−1

7 In

Coordination number

3

1

3

2

VII

1.46

VIII

1.51

IX

1.55

X

1.59

XII

1.64

VI

1.032

VII

1.10

VIII

1.16

R

IX

1.216

R

X

1.27

R

XII

1.36

C

IV

0.59

*

VI

0.76

*

VIII

0.92

C

VI

0.861

R

VIII

0.977

R

IX

1.032

R

IV

0.57 (continued)

Appendix B: Effective Ion Radii Reported by Shannon

245

(continued) Symbol

Mn

Valence

2

3 4

Mo

Coordination number

Effective radius

V

0.66

VI

0.72

* C

VIII

0.89

IV

0.66

V

0.75

C

VI

0.83

R*

VII

0.90

C

VIII

0.96

R

V

0.58

VI

0.645

R*

IV

0.39

R

VI

0.53

R*

5

IV

0.33

R

6

IV

0.255

7

IV

0.25

VI

0.46

A

3

VI

0.69

E

4

VI

0.65

RM

5

IV

0.46

R

VI

0.61

R

IV

0.41

R*

V

0.50

6

VI

0.59

Mo

6

VII

0.73

N

−3

IV

1.46

3

VI

0.16

5

III

0.104

VI

0.13

IV

0.99

V

1.00

VI

1.02

VII

1.12

VIII

1.18

IX

1.24

Na

Nb

Note*

1

XII

1.39

3

VI

0.72

4

VI

0.68

R*

A A

C

RE (continued)

246

Appendix B: Effective Ion Radii Reported by Shannon

(continued) Symbol

Valence

Coordination number

Effective radius

VIII

0.79

IV

0.48

VI

0.64

VII

0.69

VIII

0.74

VIII

1.29

IX

1.35

VI

0.983

R

VIII

1.109

R*

IX

1.163

R

XII

1.27

E

IV

0.55

IVSQ

0.49

V

0.63

E

VI

0.69

R*

3

VI

0.6

E

No

2

VI

1.1

E

Np

2

VI

1.10

3

VI

1.01

4

VI

0.87

R

VIII

0.98

R

5

VI

0.75

6

VI

0.72

R

7

VI

0.71

A

−2

II

1.35

III

1.36

−2

IV

1.38

VI

1.40

5

Nd

2 3

Ni

O

Os

2

C C

R

VIII

1.42

4

VI

0.63

RM

5

VI

0.575

E

6

V

0.49

VI

0.545

E

VI

0.525

E

7 P

Note*

8

IV

0.39

3

VI

0.44

A

5

IV

0.17

* (continued)

Appendix B: Effective Ion Radii Reported by Shannon

247

(continued) Symbol

Pa

Valence

Coordination number

Effective radius

V

0.29

VI

0.38

C

3

VI

1.04

E

4

VI

0.90

R

VIII

1.01

VI

0.78

VIII

0.91

IX

0.95

IVPY

0.98

VI

1.19

VII

1.23

C

VIII

1.29

C

IX

1.35

C

X

1.40

C

XI

1.45

C

XII

1.49

IV

0.65

E

V

0.73

E

VI

0.775

R R

5

Pb

2

4

Pd

Pm

Po

Pr

VIII

0.94

1

II

0.59

2

IVSQ

0.64

C

VI

0.86

3

VI

0.76

4

VI

0.615

R

3

VI

0.97

R

VIII

1.093

R

IX

1.144

R

4

VI

0.94

R

4

VIII

1.08

R

6

VI

0.67

A

3

VI

0.99

R

VIII

1.126

R

IX

1.179

R

VI

0.85

R

VIII

0.96

R

IVSQ

0.60

4 Pt

Note*

2

(continued)

248

Appendix B: Effective Ion Radii Reported by Shannon

(continued) Symbol

Valence 4

Pu

Ra Rb

Re

Rh

Ru

S

Sb

Coordination number

Effective radius

VI

0.80

Note* A

VI

0.625

R

5

VI

0.57

ER

3

VI

1.00

R

4

VI

0.86

R

VIII

0.96

5

VI

0.74

E

6

VI

0.71

R

2

VIII

1.48

R

XII

1.70

R

VI

1.52

VII

1.56

VIII

1.61

IX

1.63

X

1.66

XI

1.69

1

E

XII

1.72

XIV

1.83

4

VI

0.63

RM

5

VI

0.58

E

6

VI

0.55

E

7

IV

0.38

VI

0.53

3

VI

0.665

R

4

VI

0.60

RM

5

VI

0.55

3

VI

0.68

4

VI

0.62

RM

5

VI

0.565

ER

7

IV

0.38

8

IV

0.36

−2

VI

1.84

P

4

VI

0.37

A

6

IV

0.12

*

VI

0.29

C

3

IVPY

0.76

V

0.80 (continued)

Appendix B: Effective Ion Radii Reported by Shannon

249

(continued) Symbol

Sc Se

Si Sm

Valence

Coordination number

Effective radius

Note*

VI

0.76

A

5

VI

0.60

*

3

VI

0.745

R*

VIII

0.87

R*

−2

VI

1.98

P

4

VI

0.50

A

6

IV

0.28

*

VI

0.42

C

IV

0.26

*

VI

0.40

R*

VII

1.22

VIII

1.27

IX

1.32

VI

0.958

VII

1.02

E

VIII

1.079

R

IX

1.132

R

XII

1.24

C

IV

0.55

R

V

0.62

C

VI

0.69

R*

VII

0.75

4 2

3

Sn

Sr

Ta

Tb

4

2

VIII

0.81

VI

1.18

VII

1.21

VIII

1.26

IX

1.31

X

1.36

R

C

C

XII

1.44

C

3

VI

0.72

E

4

VI

0.68

E

5

VI

0.64

VII

0.69

VIII

0.74

3

VI

0.923

R

3

VII

0.98

E

VIII

1.04

R (continued)

250

Appendix B: Effective Ion Radii Reported by Shannon

(continued) Symbol

Tc

Te

Valence

Coordination number

Effective radius

Note*

IX

1.095

R

4

VI

0.76

R

VIII

0.88

4

VI

0.645

RM

5

VI

0.60

ER

7

IV

0.37

VI

0.56

A

−2

VI

2.21

P

4

III

0.52

IV

0.66

VI

0.97

IV

0.43

C

VI

0.56

*

VI

0.94

C

VIII

1.05

RC

IX

1.09

*

X

1.13

E

XI

1.18

C

XII

1.21

C

2

VI

0.86

E

3

VI

0.67

R*

4

IV

0.42

C

V

0.51

C

VI

0.605

R*

VIII

0.74

C

VI

1.50

R

VIII

1.59

R

XII

1.70

RE

IV

0.75

VI

0.885

R

VIII

0.98

C

VI

1.03

VII

1.09

VI

0.88

R

VIII

0.994

R

IX

1.052

R

VI

1.025

R

6 Th

Ti

Tl

4

1

3

Tm

2 3

U

3

(continued)

Appendix B: Effective Ion Radii Reported by Shannon

251

(continued) Symbol

Valence

Coordination number

Effective radius

4

VI

0.89

4

VII

0.95

E

VIII

1.00

R*

IX

1.05

XII

1.17

VI

0.76

VII

0.84

II

0.45

IV

0.52

VI

0.73

*

VII

0.81

E

5 6

V

Xe Y

Yb

E E

VIII

0.86

2

VI

0.79

3

VI

0.64

4

V

0.53

VI

0.58

R*

VIII

0.72

E

IV

0.355

R*

V

0.46

*

5

W

Note*

R*

VI

0.54

4

VI

0.66

RM

5

VI

0.62

R

6

IV

0.42

*

V

0.51

VI

0.60

IV

0.40

8 3

2

3

*

VI

0.48

VI

0.90

VII

0.96

VIII

1.019

R*

IX

1.075

R

VI

1.02

VII

1.08

VIII

1.14

VI

0.868

R*

VII

0.925

E

VIII

0.985

R

R*

E

(continued)

252

Appendix B: Effective Ion Radii Reported by Shannon

(continued) Symbol Zn

Zr

Valence

Coordination number

Effective radius

IX

1.042

R

IV

0.60

*

V

0.68

*

VI

0.74

R*

2

VIII

0.90

C

4

IV

0.59

R

V

0.66

C

VI

0.72

R*

VII

0.78

*

VIII

0.84

*

IX

0.89

2

Note*

*Notes Regarding Key R, From r3 versus V plots C, Calculated from bond length—bond strength equations E, Estimated *, Most Reliable M, From Metallic Oxides A, Ahrens (1952) Ionic radius2 P, Pauling’s (1960) Crystal Radius3

2 Ahrens 3 Pauling

L H. Geochim. Cosmochim. Acta, 1952, 2: 155. L. The Nature of the Chemical Bond. Ithaca, New York: Cornell University Press, 1961.

Appendix C

Symbol Notes

Symbol

Quantity

a, b, c,

Crystal lattice constant

A

Mass per mole of material

B

Intensity of magnetic induction

d

Distance between adjacent ions; bond length; Density

DE

Double exchange

DFT

Density functional theory

E

Kinetic energy of the electron

Eb

Binding energy of electron subjected the ion in a crystal

EB

Cohesive energy of a crystal

EF

Fermi energy level

EELS

Electron energy loss spectrum

f

Ionicity

g

Lande factor

h

Planck constant

H

Magnetic field intensity

HC

Coercivity

Hm

Intensity of molecular field

I

X-ray diffraction spectrum intensity; X-ray photoelectron spectrum intensity; Infrared spectrum intensity

J

Exchange integral

IEO model Itinerant electron model for magnetic oxides IEM model Itinerant electron model for magnetic metals k, k B

Boltzmann constant

m

Mass

M

Magnetization per volume (continued)

© Science Press 2021 G.-D. Tang, New Itinerant Electron Models of Magnetic Materials, https://doi.org/10.1007/978-981-16-1271-8

253

254

Appendix C: Symbol Notes

(continued) Symbol

Quantity

MS

Saturation magnetization

n

Number of electrons

N3

Number of trivalent cations

P

Probability

PPMS

Physical property measurement system

r

Effective radius of an ion

re

Distance between the two electrons of a Weiss electron pair

R

Ratio; Resistance

s

Goodness-of-fit factor in Rietveld fitting for X-ray diffraction data

S

Spin quantum number

SE

Super exchange

STM

Scanning tunneling microscope

T

Temperature; Transmission coefficient

TC

Curie temperature

TN

Néel temperature

v

Volume of a crystal cell

V (M N+ ) V BA (M

2+ )

w

Nth ionization energy of M element Height of potential barriers that must be jumped by a divalent cation when it moves from the [B] site to the (A) site during the thermal treatment of the (A)[B]2 O4 spinel ferrite samples Energy density

WEP

Weiss electron pair

XANES

X-ray absorption near-edge structure

XAS

X-ray photo-absorption spectra

XMCD

X-ray magnetic circular dichroism

XRD

X-ray diffraction

XPS

X-ray photoelectron spectra

Z

Number of molecules per crystal cell

α, β, γ

Angles between the basic vectors of a crystal cell

χ

Susceptibility per volume

χm

Susceptibility per mass

χA

Susceptibility per mole

φ

Average canting angle between the sample magnetic moment and the cation magnetic moments in canted magnetic structure

λ

Mean free path; Wavelength

μ

Relative permeability

μ0

Permeability of vacuum

μB

Bohr magneton (continued)

Appendix C: Symbol Notes (continued) Symbol

Quantity

μobs

Observed value of average molecule magnetic moment

μcal

Fitted value of average molecule magnetic moment

ν

Wave number

θ

Bond angle

ρ

Resistivity

σ

Specific magnetization, electrical conductivity

σs

Specific saturation magnetization

σr

Specific remanence magnetization

255