Electronic, Magnetic, and Thermoelectric Properties of Spinel Ferrite Systems: A Monte Carlo Study, Mean-Field Theory, High-Temperature Series ... Calculations (SpringerBriefs in Materials) 3031406125, 9783031406126

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SpringerBriefs in Materials Rachid Masrour

Electronic, Magnetic, and Thermoelectric Properties of Spinel Ferrite Systems A Monte Carlo Study, Mean-Field Theory, High-Temperature Series Expansions, and Ab-Initio Calculations

SpringerBriefs in Materials Series Editors Sujata K. Bhatia, University of Delaware, Newark, DE, USA Alain Diebold, Schenectady, NY, USA Juejun Hu, Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA Kannan M. Krishnan, University of Washington, Seattle, WA, USA Dario Narducci, Department of Materials Science, University of Milano Bicocca, Milano, Italy Suprakas Sinha Ray , Centre for Nanostructures Materials, Council for Scientific and Industrial Research, Brummeria, Pretoria, South Africa Gerhard Wilde, Altenberge, Nordrhein-Westfalen, Germany

The SpringerBriefs Series in Materials presents highly relevant, concise monographs on a wide range of topics covering fundamental advances and new applications in the field. Areas of interest include topical information on innovative, structural and functional materials and composites as well as fundamental principles, physical properties, materials theory and design. SpringerBriefs present succinct summaries of cutting-edge research and practical applications across a wide spectrum of fields. Featuring compact volumes of 50 to 125 pages, the series covers a range of content from professional to academic. Typical topics might include • A timely report of state-of-the art analytical techniques • A bridge between new research results, as published in journal articles, and a contextual literature review • A snapshot of a hot or emerging topic • An in-depth case study or clinical example • A presentation of core concepts that students must understand in order to make independent contributions Briefs are characterized by fast, global electronic dissemination, standard publishing contracts, standardized manuscript preparation and formatting guidelines, and expedited production schedules. Indexed in Scopus (2022).

Rachid Masrour

Electronic, Magnetic, and Thermoelectric Properties of Spinel Ferrite Systems A Monte Carlo Study, Mean-Field Theory, High-Temperature Series Expansions, and Ab-Initio Calculations

Rachid Masrour Laboratory of Solid Physics Faculty of Sciences Dhar El Mahraz Sidi Mohamed Ben Abdellah University Fez, Morocco

ISSN 2192-1091 ISSN 2192-1105 (electronic) SpringerBriefs in Materials ISBN 978-3-031-40612-6 ISBN 978-3-031-40613-3 (eBook) https://doi.org/10.1007/978-3-031-40613-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 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 publisher, 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 publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

General Introduction

AB2 O4 spinel oxides comprising two transition metal elements (i.e., A and B) and oxygens exhibit interesting and varying structural, electrical, and magnetic properties. Among them, NiCo2 O4 has an inverse spinel structure in which half of Co ions occupy tetrahedral sites (A-sites), whereas the other half of Co ions and all Ni ions occupy octahedral sites (B-sites) [1, 2]. In recent years, Ferrimagnetic Nanoparticles (FMNPs) have witnessed extensive nanoscience growth due to their unique magnetic properties of nanoparticles (Fe3 O4 , Fe2 O3 ) and synthesis methods [3]. Fe2 O3 exhibits solid magnetic characteristics due to its high magnetization saturation (Ms ), 92 emu/g of material at ambient temperature, and high curie temperature (Tc ) of 577 °C [4]. Antiferromagnetic (AFM) spintronics is an emerging field aiming to manipulate and control spins for future data storage applications [5–7]. Such interest is owing to the promising features hosted by AFM materials, including non-stray fields, high exchange interaction, manipulation of the spin wave at the terahertz frequencies, and efficiency in transport mechanisms compared to Ferromagnetic (FM) counterparts [8–10]. The ferrimagnetic systems are well adapted to study magnetic properties of a certain type of magnetic materials which are solicited for the aforementioned technological applications as well as academic researches [11, 12]. Theoretically, several studies on mixed-spin systems have been carried out to investigate their magnetic properties by using different numerical techniques of statistical physics, including renormalization group technique [13, 14], mean-field approximation [15, 16], effective-field theory [17, 18], Monte Carlo simulations [19, 20], or exact recursion relations on various structures such as square [22], honeycomb [23], Bethe–Heitler production of dileptons with high invariant mass [24], cubic lattices, and hexagonal core–shell structure [26, 27]. The novel dynamic behaviors in a ferrimagnetic Gdx (FeCo)1−x nanosphere model with different Gd compositions ranging from x = 0 to 0.44 at finite 0–1200 K temperatures using stochastic atomistic numerical calculations [28]. Stanciu et al. experimentally observed the temperature dependence of dynamic modes in amorphous GdFeCo using an all-optical pump– probe technique; the Ferromagnetic Resonance (FMR) frequency rapidly increased when the temperature approached the angular momentum compensation point. Also,

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General Introduction

Kim et al. experimentally determined that domain wall mobility in a GdFeCo ferrimagnet is enhanced at the angular momentum compensation temperature. Zhu et al. proposed a robust means of determining the angular momentum compensation point in ferrimagnets based on the Curie–Weiss theory. Yamamoto et al. [32, 33] employed Density Matrix Renormalization Group Technique (DMRG) and Quantum Monte Carlo (QMC) method to calculate the thermodynamic properties of the Heisenberg ferrimagnetic mixed-spin chain. By using the DMRG and Spin-Wave Theory (SWT), Langari studied the phase diagram of XXZ anisotropic ferrimagetic spin-(1/2, 1) chain under the presence of a transverse magnetic field. Chen [35, 36] investigated the excited states and thermodynamic properties of the Heisenberg ferrimagnetic spin chain by using Dyson–Maleev Mean-Field (DMMF) theory and Bond Operator (BO) method. The Double Perovskite Oxides (DPO) with general formula A2 BB' O6 (where A is alkaline or rare earth metal and the cation B(B' ) having 3(4/5)d states are taken from two different transition metals) have attained much interest due to their unusual and marvelous physical properties such as colossal magnetoresistance [37, 38], multiferrocity [39], thermo-electricity [40], structure stability [41], magnetodielectricity [42], giant anisotropic magneto-caloric effect, etc. [43]. The density functional theory, Monte Carlo simulations, Green function, and high-temperature series expansions were applied for a series of spinel systems [44–47].

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Contents

1

Exchange Interaction Types in Magnetic Materials . . . . . . . . . . . . . . . 1.1 Magnetic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Magnetic Dipole Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Direct Exchange Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Indirect Exchange Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Super-Exchange Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Ruderman–Kittel–Kasuya–Yosida Exchange Interaction . . . . . . . 1.7 Double Exchange Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Spin Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 Potts Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.3 Continuous Spin Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.4 N-Component Vector Models . . . . . . . . . . . . . . . . . . . . . . . . 1.8.5 The Spherical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.6 Model of D’Edwards Anderson . . . . . . . . . . . . . . . . . . . . . . 1.8.7 Model of Sherrington and Kirkpatrick . . . . . . . . . . . . . . . . 1.9 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.1 Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 1.9.2 Periodic Boundary Conditions Screw . . . . . . . . . . . . . . . . . 1.9.3 Anti-periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . 1.9.4 Free Edge Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 1.10 Calculation of the Values of the Exchange Integrals by Mean Filed Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 High-Temperature Series Expansions . . . . . . . . . . . . . . . . . . . . . . . 1.12 Critical Exponents and Scaling Laws . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 2 3 4 5 5 6 6 8 8 9 9 9 10 10 10 12 12 12 12 13 14 17

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Computational Methods: Ab Initio Calculations and Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Born–Oppenheimer Approximation . . . . . . . . . . . . . . . . . . 2.1.2 The Hartree and Hartree–Fock Approximations . . . . . . . . 2.1.3 Thomas–Fermi Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Theorems of Hohenberg and Kohn (H.K) . . . . . . . . . . . . . . 2.1.5 Kohn–Sham Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Functional Exchange and Correlation . . . . . . . . . . . . . . . . . 2.1.7 Pseudo-Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Simulation Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( )[ 3+ 2+ ] 2− Fe M O 4 (M = Co, Cu, Magnetic Properties of Fe3+ 2 Ni and Fe) Inverse Ferrite Spinels: A Monte Carlo Study . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 20 21 22 24 27 28 31 32 39 44 45 49 49 51 52 53 58 64

Thermoelectric and Spin–Lattice Coupling in a MnCr2 S4 Ferrimagnetic Spinel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Ab Initio Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Monte Carlo Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 68 68 69 74 74

Magnetic Properties of LiMn1.5 Ni0.5 O4 Spinel: Ab Initio Calculations and Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Ab Initio Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 78 78 79 83 83

Magnetic Properties of Inverse Spinel Fe3 O4 Nano-Layer: A Monte Carlo Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Ising Model and Monte Carlo Simulations . . . . . . . . . . . . . . . . . . .

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6.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Electronic and Magnetic Structures of Fe3 O4 Ferrimagnetic: Ab Initio Calculations, Mean-Field Theory, and Series Expansion Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.2 Electronic Structure Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 7.3 Theories and Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

8

Magnetic Properties of Mixed Ni–Cu Spinel Ferrites Calculated Using Mean-Field Approach . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 High-Temperature Series Expansions . . . . . . . . . . . . . . . . . . . . . . . 8.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 103 105 107 109 111 111

Effect of Cobalt on NiCr2 O4 : Calculation of Critical Temperature and Exchange Interactions . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Green’s Functions Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Mean Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 113 114 114 115 116 119 119

10 Studying the Effect of Zn Substitution on NiFe2 O4 Spinel Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 121 121 123 125 125

9

General Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

About the Author

Prof. Rachid Masrour from Morocco, actually, is Research Professor in the Faculty of Sciences Dhar El Mahraz at Sidi Mohamed Ben Abdellah University, Fez, Morocco. He completed his Ph.D. in March 2006 at same University. His research interests lie in the area of Condensed Matter Physics, Material Sciences, Material for energy, Magnetism,…. He has 322 articles published in Web of Science and 10 book chapters published in international journal and 2 books (https://www.bookpi.org/bookstore/pro duct/new-frontiers-in-physical-science-research/) with h-index 32. He participated in more 100 Moroccan and international conferences. He’s a Referee of a several articles and also Editorial Board Members in different international journals. He has a lot of collaboration with laboratories from abroad. He has been honored with International Association of Advanced Materials Young Scientist Medal in recognition for their contribution to “Magnetism, Electromagnetism and Spintronics” and delivered a lecture in the Advanced Materials World Congress October 11–14, 2022. One of the world’s most cited top scientists in material physics (top 2 %, Stanford University Ranking, US, 2020). He won the prize for best oral presentation Award for MCGPD-2021, by Indian Association for Crystal Growth & Indian Science; Technology Association International Organization for Crystal Growth, July 5–8, 2021. The prizes are awarded during the closing ceremony of the congress; and Outstanding Scientist Award (VDGOOD PROFESSIONAL ASSOCIATION, 15-02-2020, India). It was among the most critical in 2020 by Elsevier. Has been xiii

xiv

About the Author

admitted as a fellow of IAAM (International Association for Advanced Materials, Stockholm, Sweden) in recognition of their contribution to the Li-ion battery as well as the right to use the “FIAAM” designation letters in 2023.

Chapter 1

Exchange Interaction Types in Magnetic Materials

1.1 Magnetic Interactions The exchange interactions can play a pivotal role in the phenomenon of long-range magnetic order. The exchange effect is subtle and somewhat mysterious, because it seems surprising to have to think of identical exchange operators and particles when we are only dealing with a bar magnet and a pile of iron filings. But this is, as often with the subject of magnetism, a demonstration of how quantum mechanics underlies many everyday phenomena. Exchange interactions are nothing but electrostatic interactions, due to the fact that charges of the same sign cost energy when they are close and save it when they are far away.

1.2 Magnetic Dipole Interaction The magnetic dipole interaction is the first interaction whose importance can be magnetic dipoles μ)) predicted by Refs. [1, 2]. The energy of 1 and μ2 separated (→two ( →− )(− → − → → − → μ0 − 3 − μ μ μ r r − by the distance r is given by E = 4πr μ 1 2 1 1 2 2 . 3 r2 This energy depends on the distance between the spins and their degree of mutual alignment. We can easily estimate the order of magnitude of this interaction by considering two magnetic moments μ1 = μ2 = 1 μB separated by a distance equal to r = 2A˚: E=

μ0 μ2B = 2.110−24 J 2πr 3

which is equivalent to about 1 K of temperature (E = kB T where kB is the Boltzmann constant and T is the absolute temperature). As many materials orders at much higher temperatures (around the 1000 K), the magnetic dipole interaction is too weak to explain the ordering of most magnetic materials [1]. Nevertheless, it can play an © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. Masrour, Electronic, Magnetic, and Thermoelectric Properties of Spinel Ferrite Systems, SpringerBriefs in Materials, https://doi.org/10.1007/978-3-031-40613-3_1

1

2

1 Exchange Interaction Types in Magnetic Materials

important role in the properties of materials which are ordered at temperatures of the order of the millikelvin.

1.3 Direct Exchange Interaction Assume a fairly simple model with only two electrons which exhibit position vectors r1 and r2 . Moreover, we consider that the total wave function is composed of the product of the single electron states ψa (r1 ) and ψb (r2 ). The belonging electrons are indistinguishable; therefore, the squared wave function must be invariant for the exchange of the two electrons. Since electrons are fermions, the Pauli exclusion principle must be respected, which leads to an antisymmetric wave function. By taking into account the spin of the electrons, two possibilities are given: a symmetric spatial part ψ in combination with an antisymmetric spin part χ or an antisymmetric spatial part in combination with a symmetric spin part. The first situation represents a singlet state with Stotal = 0, the second represents a =) 1. The( corresponding triplet state [ with (→)Stotal(− ) (→)] total wave functions are given by → u S = √12 ψa − r 1 ψb → r2 + ψa − r 2 ψb − r1 χ S ) (→) (− )] 1 [ (→) (− uT = √ ψa − r 1 ψb → r2 − ψa − r 2 ψb → r 1 χT 2 The energies of the singlet and triplet states amount to { ES = { ET =

ψ∗S Hψ S d V1 d V2 ψ∗T HψT d V1 d V2

taking into account the normalized spin parts of the singular and triplet wave − → (− → − →)2 − →− → functions, i.e.: S 2 = S1 + S2 = S12 + S22 + 2 S1 . S2 − →− → we obtain S1 S2 = 21 Stotal (Stotal + 1) − 21 S1 (S1 + 1) − 21 S2 (S2 + 1) = =

{ −3 4 +1 1

3 1 Stotal (Stotal + 1) − 2 4 f or Stotal = 0 singlet state f or Stotal = 1 tri plet state

The effective Hamiltonian can be expressed as

1.4 Indirect Exchange Interaction

H=

3

1 → − →− (E S + 3E T ) − (E S − E T ) S1 S2 4

The first term is constant and often included in other energy contributions. The second term is spin dependent and is important for ferromagnetic properties. The exchange constant or the exchange integral J can be defined by J= { J=

E S − ET 2

(→) ∗ (− ) (→) (− ) r 1 ψb → r2 Hψa − r 1 ψb → r2 d V1 d V2 ψa∗ −

Then, the spin-dependent term in the effective Hamiltonian can be written as − →− → follows: H = −2J S1 S2 . If the exchange integral J is positive, then ES > ET , i.e., the triplet state with Stotal = 1 is energetically favored. If the exchange integral J is negative, then ES < ET, i.e., the singlet state with Stotal = 0 is energetically favored. We see that this situation, which only considers two electrons, is relatively simple. But, in magnetic systems, the atoms have a lot of electrons. The most important part of this interaction applies mainly between neighboring atoms, this consideration leads, within the framework of the Heisenberg model, to a term in the Hamiltonian of ∑ − →− → H =− Ji j Si S j ij

with Jij being the exchange constant between spin i and spin j, a good approximation ⎧ ⎨ 1 f or neighboring spins of Jij is often given by: J for neighboring spins Ji j = ⎩ 0 other wise In general, J is positive for the electrons of the same atom while it is often negative if the two electrons belong to different atoms.

1.4 Indirect Exchange Interaction Magnetic interactions can also be indirect, the different classes of indirect exchange depend essentially on the type of magnetic material.

4

1 Exchange Interaction Types in Magnetic Materials

1.5 Super-Exchange Interaction Ionic solids experience this type of exchange interaction. A non-magnetic ion placed between two magnetic ions mediates the exchange contact between the nonneighboring magnetic ions. There cannot be a direct exchange due to the huge gap between the magnetic ions: MnO is an example of an antiferromagnetic ionic solid (Fig. 1.1) where this type of interaction occurs. Due to Hund’s rule, each Mn2+ ion has five electrons in its d-shell, all of which have parallel spins. The electrons in the p orbitals of the O2 ions are fully occupied and have antiparallel spin alignments. The relative alignment of the spins of neighboring Mn atoms can occur in two ways. A ferromagnetic arrangement results from a parallel alignment, while an antiparallel alignment results from an antiferromagnetic arrangement, due to the energy advantage of the latter configuration, the electrons involved may delocalize as their kinetic energy will be reduced. Electrons with their ground state shown in (a) in the antiferromagnetic case can be transferred through the excited states shown in (b) and (c), causing delocalization. The Pauli exclusion principle prohibits the arrangements shown in (e) and (f) for ferromagnetic alignment with the corresponding ground state shown in (d) and (f). Therefore, there is no delocalization, and, as shown in Fig. 1.1, the antiferromagnetic coupling between two Mn atoms is energetically preferred. The oxygen atom’s electrons must be in the same orbital, which means that the atom must connect the two Mn atoms.

Fig. 1.1 Crystalline and magnetic structures of MnO

1.7 Double Exchange Interaction

5

Fig. 1.2 Graphical representation of the function F vs.x

1.6 Ruderman–Kittel–Kasuya–Yosida Exchange Interaction The RKKY (Ruderman–Kittel–Kasuya–Yosida) exchange interaction is a mechanism for coupling magnetic moments of localized internal electron spins of d or f electron shells in a metal by means of interaction by the electrons of conduction. The RKKY interaction was originally proposed by Ruderman and Kittel [3] to explain the unusually broad nuclear spin resonance lines that had been observed in natural metallic silver. The explanation uses second-order perturbation theory to describe an indirect exchange coupling whereby the nuclear spin of an atom interacts with a conduction electron through the hyperfine interaction, and this conduction electron then interacts with another nuclear spin, thus creating a correlation energy between the two nuclear spins. (Alternatively, instead of the nuclear spins coupling to the conduction spins through the hyperfine interaction, another scenario is that the inner electron spins couple to the conduction spins through the exchange interaction.) The derived exchange interaction takes the following form: J R K K Y (r ) ∝ F(2k F r ). where kF is the Fermi energy, r is the distance separating the two interacting local magnetic moments, F is a function which takes the following form: F(x) = sinx−xcosx , due to the long range and anisotropy of this exchange coupling, complex x4 spin configurations frequently appear and it also exhibits an oscillatory characteristic (Fig. 1.2). Therefore, the type of coupling depends on the distance between the magnetic moments.

1.7 Double Exchange Interaction The double exchange mechanism is a type of magnetic exchange that can occur between ions in different oxidation states. First proposed by Zener [4], this theory predicts that the ease with which an electron can be exchanged between two species has important implications for determining whether materials are ferromagnetic,

6

1 Exchange Interaction Types in Magnetic Materials

Fig. 1.3 Example of the double exchange interaction

antiferromagnetic, or exhibit spiral magnetism [5]. For example, consider the 180degree interaction of Mn-O-Mn in which the “eg ” orbitals of Mn interact directly with the “2p” orbitals of O, and one of the Mn ions has more electrons than the other (Fig. 1.3). In the ground state, the electrons of each Mn ion are aligned according to Hund’s rule: If O gives up its upper spin electron to Mn4+ , its vacant orbital can then be filled by an electron from Mn3+ . At the end of the process, an electron has moved between neighboring metal ions, retaining its spin. The double exchange predicts that this movement of electrons from one atom to another will be easier if the electrons do not have to change spin direction to conform to Hund’s rules when on the atom accepting. The ability to jump (to relocate) reduces kinetic energy. Thus, the overall energy saving can lead to a ferromagnetic alignment of neighboring ions. This model is superficially similar to the super-exchange interaction; however, in the super-exchange interaction, a ferromagnetic or antiferromagnetic alignment occurs between two atoms having the same valence (number of electrons), whereas in the double exchange, the interaction occurs only when one atom has an extra electron compared to the other [4, 6].

1.8 Spin Models 1.8.1 Ising Model The most influential model of a system capable of phase transition is the Ising model. It was invented by W. Lenz in 1920 as a simple model of a ferromagnetic system, but it can be interpreted as a model for other physical systems. It was first solved by E. Ising in 1925, who treated the one-dimensional case D = 1. In 1944, Onsager solved the model for D = 2 in the absence of an external magnetic field and showed that the critical exponents of the model were very different from those predicted

1.8 Spin Models

7

Fig. 1.4 2D Ising model

by Landau’s theory (Fig. 1.4). An exact solution for the D = 2 model with a nonzero external field appeared only recently. Despite decades of intensive effort, we still do not have an exact solution for this D = 3 model, even in the absence of a magnetic field. The simulation of such a system can be developed by the Monte Carlo method, thus making it possible to calculate the values of physical quantities such as magnetization, energy, specific heat, and susceptibility at a given temperature. This system, which occupied some of the most brilliant physicists for almost three quarters of a century, is easily described, it is generally assumed to be a cubic lattice and that each point of the lattice is associated with a number S equal to 1 or −1. The study of magnetic phenomena in the solid, the atoms of the Ising model are arranged on a plane. A magnetic moment μB (Bohr magneton) exists for each of these atoms and is randomly oriented in spin up (+μB ) and spin down (−μB ). For the simplest case, we consider that only the nearest neighbors have an interaction energy between them. ∑ ∑ For such a system, the Hamiltonian writes H = 21 i j Ji j Si S j − H i Si , where H is an external field, the indices i and j designate the lattice sites, and Jij is the coupling constant between the spin and its neighbors defined so that ⎧ J ⎪ ⎪ ⎨ i f i and j ar e neighboring sites Ji j = ⎪ 0 ⎪ ⎩ other wise The [partition function for this )]simple model is Z I sing = ( ∑ ∑ 1 ∑ , where the sum is extended to i Si + 2 i j Ji j Si S j (Si ) ex p −β −H all possible ± 1 configurations of the network sites. The physical image is that of an array of magnets which must be parallel or antiparallel to a uniform magnetic field. The (−) signs in Hamiltonian system are standard. They simply dictate the choice of sign for the interaction parameter J and the external field H. If one sets J < 0, neighboring spins try to align parallel to each other and parallel to H, the system in this case is ferromagnetic. If J > 0, neighboring spins try to align antiparallel to each other.

8

1 Exchange Interaction Types in Magnetic Materials

1.8.2 Potts Model An extension of the Ising model is the Potts model, this and the Ising model are comparable [7, 8]. The difference is that unlike the Ising model where only two sound states are possible, for the Potts model the spin Si on each site i can take one of the q possible values, distributed uniformly around the circle, according to angles: θn =

2πn q

with q = 0, 1, 2, q − 1, and the Hamiltonian of the system is expressed as Hc = Jc

∑ i, j

( ) cos θsi − θs j

the sum being calculated over the pairs of nearest neighbors < i, j > on all the sites of the network, and Jc is the coupling constant which determines the strength of the interaction. This model is now known as the Potts vector model or the clock model. Potts provided the two-dimensional localization of the phase transition, for q = 3 and 4. In the limit q → ∞, this becomes the XY model.

1.8.3 Continuous Spin Model XY Model Heisenberg Model: While the Ising model describes magnets with extreme uniaxial anisotropy, the Heisenberg model is intended to describe fully isotropic spin systems. Assuming the external field H is along one of the spatial directions. In the Heisenberg model, the spin variables Si are three-dimensional vectors. The spins can be represented either by the elements Sx and Sy and Sz , which respect the constraint S 2 = Sx2 + S y2 + Sz2 , either by two variable angles θ and φ in spherical coordinates [7], the Hamiltonian for the Heisenberg model is given by H H eis = ∑ − →− → ∑− →− → −J i, j Si S j − i H Si . The XY model is a simplification of the Heisenberg model where the spins are vectors with two components, which can be oriented in any direction in the plane (x, y) [7], and which satisfy the constraint S2 = Sx 2 + Sy 2 . This model can be described either by an angular variable which indicates the spin direction [8]. In this description, the Hamiltonian of the model is written in the same form as in for the Heisenberg model, i.e., as a function of the angular variables. In this description, the Hamiltonian of the model is written as H X Y = −J

∑ i, j

( ) ∑[ ] cos θi − θ j − h x cos(θi ) + h y sin(θi ) i

1.8 Spin Models

9

1.8.4 N-Component Vector Models A theoretical generalization of the Heisenberg model is obtained by extending the number of components of the spin vector to an arbitrary value n > 3. The spin variables will then be vectors with n components. Classical models of this type are often called n-component (or vector) models. The Heisenberg model is a three-component vector model and the XY model is a two-component vector model. These models are used to describe at the phenomenological level magnets with complex structure or structural phase transitions [9, 10].

1.8.5 The Spherical Model In 1952, Berlin and Kac introduced a simplified version of the Ising model which can be solved exactly for an arbitrary spatial dimension D. ∑ The constraint N Si2 = N , Si = ± 1 or Si2 = 1 in the Ising model is replaced by a weaker: i=1 where N is the number of vertices in the network. The spherical condition means that the square Si 2 of each variable Si can take any value provided that the previous constraint is respected.

1.8.5.1

Blume–Emery–Griffiths Model

This model was introduced by Blume, Emery, and Griffiths (BEG) in 1971 to describe phase separation in mixtures He3 − He4 : H B E G = −J

∑ i, j

∑ ∑ Si S j [1 + aSi S j] + ∆ Si2 − H Si i, j

i

In the equation J is the usual Ising coupling, the parameter a > 0 gives the constant K = aJ of the bi-quadratic exchange and the parameter ∆ represents the interaction of the spins with the crystal field at each point i, H is the external magnetic field [11].

1.8.6 Model of D’Edwards Anderson The first spin glass model was developed by Edwards and Anderson [15]. They were the first to introduce an order parameter to describe this unusual phase of magnetic order. q = q i = lim qi (t)qi (t)= lim>Si (0)Si (t)< with >...< denotes the thermal average. q is a homogeneous parameter that does not depend on the site i and which describes the limitations of temporal fluctuations of a given spin. From this point of view, it is very different from the usual order parameters which are related to broken symmetry and long-range order. q is directly related to the magnetic

10

1 Exchange Interaction Types in Magnetic Materials

susceptibility χ and specific heat. These two quantities must present a peak at the freezing temperature, all the more rounded as the field is high. The above equation is the spin autocorrelation function Si . It becomes non-zero below Tg and represents a phase transition of a new type with local character. This model uses mean-field theory with an order parameter. The randomness of exchange interactions is considered a fundamental element. It uses a Gaussian ( ) integrals which can be written in the form: P Ji j = distribution (P(Ji j ) exchange ) ] [ 2 J −J √1 exp − ( i j 0 ) . This Gaussian is centered in J and length 2∆. 2π∆

0

2∆

Whereas in conventional second-order magnetic phase transitions, one observes the appearance at Tc of a global order inherent in the discontinuity of a spatial order parameter linked to a broken symmetry, we do not observe such an order in spin glasses, but an order of a new type. It is to describe this new order that Edwards and Anderson proposed a temporal order parameter.

1.8.7 Model of Sherrington and Kirkpatrick The model of Sherrington and Kirkpatrick is the infinite range version of the model of Edwards and Anderson for an assembly of N Ising spins. The Hamiltonian is Hi j = Ji j S-i S- j , with S = ±1. The disorder is also in this case introduced through the interactions which are described by random variables Ji j with a law of probability P(Ji j ) integrals of ] 2 ( ) [ J −J exp − ( i j 0 ) . exchanges which are written in the form: P J = √ 1 ij

2π J

2J 2

This Gaussian is centered in J0 and length 2J . Independent of the distance between the spins. Spins all interact regardless of the distance between them.

1.9 Boundary Conditions During a simulation, we can only consider a system of finite size, whereas we are generally interested in the properties of an infinite system. To make a meaningful extrapolation to the thermodynamic boundary, the problem of boundary conditions must be solved. It is crucial to consider both the structure of the system and the way the problem was formulated. This led to many approaches.

1.9.1 Periodic Boundary Conditions The problem of surface effects can be overcome by implementing periodic boundary conditions (introduced by Born and von Karman in 1912), such as given in Fig. 1.5.

1.9 Boundary Conditions

11

The cubic box is replicated throughout space to form an infinite lattice. During the simulation, when a molecule moves in the original box, its periodic image in each of the other boxes moves in exactly the same way. Thus, when a molecule leaves the central box, one of its images enters it from the opposite side. There are no walls at the boundary of the central box, and no surface molecules. Periodic boundary conditions effectively eliminate surface effects, but retain finite lattice size effects L since the maximum length of the correlation is limited to L/2 [12].

Fig. 1.5 Boundary conditions a–b: periodic, c: screw, and d: free

12

1 Exchange Interaction Types in Magnetic Materials

1.9.2 Periodic Boundary Conditions Screw For this type of condition, the lattice spins are represented as entries in a onedimensional vector that surrounds the system [12]. The last spin of a row and the first of the next row are considered near neighbors.

1.9.3 Anti-periodic Boundary Conditions The anti-periodic conditions give rise to an interface that will be added to the edge of the system [13], so we can apply the anti-periodic conditions to the studied interface while the periodic conditions are applied in the other directions.

1.9.4 Free Edge Boundary Conditions We speak of free conditions if no connection exists between the end of a line and any other line in the network. Therefore, the spins at the end of the lattice have no near neighbors. This type of boundary conditions is used to study the impacts of surfaces and irregularities, because large changes related to surface tensions can occur near surfaces where the behavior of the system is not uniform. It is also used for simulating the behavior of particles or nanoparticles. Systems with free boundaries generally exhibit different characteristics than an infinite system with periodic boundary conditions [13].

1.10 Calculation of the Values of the Exchange Integrals by Mean Filed Theory Starting with the well-known Heisenberg model, the Hamiltonian of the system is ∑ given by H = −2 Ji j S-i S- j , where Ji j is the exchange integral between the spins i, j

situated at sites I and j. S-i is the operator of the spin localized at the site i. In this work, we consider the nearest neighbor (nn), next nearest neighbor (nnn), and the third next nearest neighbor (T nnn), J1 , J2 , and J3 respectively. H = −J1

∑ i, j

S-i S- j − J2

∑ i,k

S-i S-k − J3

∑

S-i S-l

i,l

The sums over i j, ik, and il include all (nn), (nnn), and (Tnnn) pair interactions, respectively. In the case of spinels containing the magnetic moment only in the

1.11 High-Temperature Series Expansions

13

octahedral sublattice, the mean-field approximation of this expression leads to simple relations between the paramagnetic Curie temperature θ p and the critical temperature TC , the angle of helices ϕ, respectively, and the considered three exchange integrals J1 , J2 , and J3 . Following the method of Holland and Brown [14], the expressions of TC and θ p that can describe the system spinel are Tc (K ) = θ p (K ) =

5 [2J1 − 4J2 − 4J3 ](K ) 2K B

5 [6J1 + 12J2 + 12J3 ](K ) 2K B

where k B is the Boltzmann constant. Concerning the antiferromagnetic side, we have used the formula of Néel temperature TN given by [14] TN (K ) =

2 S(S + 1)λ(ϕ) 3

For helimagnetic B-spinel, the angle of helic ϕ [14] is given by cos(φ) = −

1 J1 + 2J2 4 J2 + 2J3

1.11 High-Temperature Series Expansions In this section, we shall derive the high-temperature series expansions (HTSE) for the both the zero-field magnetic susceptibility χ and two-spin correlation functions ξ to order six in β. The relationship between the magnetic susceptibility per spin and the correlation functions may be expressed as follows: χ (T ) =

β ∑ /< - - > Si S j N ij

< > where β = 1/k B T and N is the number of magnetic ions. Si S j = T r Si S j e−β H /T r e−β H is the correlation function between spins at sites i and j. In the ferromagnetic case we get k = 0. The high-temperature series expansion of ξ 2 gives the function: ξ 2 (T ) =

n ∑ 6 ∑ m=−n n=1

b(m, n)y m τ n

14

1 Exchange Interaction Types in Magnetic Materials

/ where y = J2 J1 and τ = 2S(S + 1)J1 /k B T . In spin glasses, the magnetic susceptibillty near to TSG is expected not in the linear parameter par χ0 of the dc susceptibility χ , but in the nonlinear susceptibility χs = χ − χ0 . This is due to the fact that the order q in the spin glass state is ] ∑ [parameter >Si 2 ] ∑ [< si s j , and Anderson [15], leading to an associated susceptibility χs = N1T 3 av ij [< ] >2 possibly diverges where the correlation length of the correlation function Si S j at T = TSG . The simplest assumption that one can make concerning the nature of the singularity of the magnetic susceptibility χ (T ) is that the neighborhood of the critical point above two functions exhibits an asymptotic behavior: χ (T ) ∝ (T − TC )−γ ξ 2 (T ) ∝ (Tc − T )−2ν

1.12 Critical Exponents and Scaling Laws Critical phenomena are those phenomena that occur in the immediate vicinity of a second-order transition point. Approaching the critical point, the ordered phase has a stability comparable to that of the disordered phase. In the vicinity of a critical point (second-order transition), it is possible to describe the behavior of certain physical quantities of the system by power laws, not necessarily integral, of the reduced temperature. These exponents are called critical exponents. The fact that these exponents are not necessarily integers is particular because it implies that the thermodynamic quantities are not analytical functions of their variables. Many theoretical and experimental works endeavored to determine the asymptotic laws which govern the approach of the point of transition. The behavior of each of the singular physical quantities is characterized by a critical exponent. The existence of relations between the various critical exponents is easily interpreted, if we assume that in the neighborhood of the transition point, the free energy and the correlation functions obey homogeneity properties [16–20]. The values of the critical exponents are relatively universal. That is to say, they are independent of many characteristic details of the system. This universality shows that they do not depend on the microscopic structure of material, but that they are just related to the phenomena of second-order phase transitions. In materials having a layered structure such that the interactions between spins of the same layer are strong while the interactions between spins belonging to different

1.12 Critical Exponents and Scaling Laws

15

layers are weak, we find the exponents of a two-dimensional system if the difference between the temperature at which the measurement is made and the transition temperature is large compared to the interactions between layers. But as soon as the temperature difference is of the order of interactions between layers, the critical behavior becomes that of a three-dimensional system [18, 19]. This phenomenon called “Cross-Over” was studied for the first time by Riedel and Wegner [20] who gave a theory based on the properties of homogeneity of free energy. It should be noted that the phase transitions depend enormously on the dimensions of the space, consequently the exponents vary according to the dimension of space and the dimension of the parameter of order n. The first step in the direction of understanding critical phenomena was made by Wilson [21] who gave a perturbation theory allowing to express, for any value of n, the critical exponents in the form of a power series ε = 4 − d. This theory is very interesting and the reasons for its success appear clearly if we appeal to the theory of the renormalization group. This method used by Wilson [22] made it possible to obtain the power expansions of ε and to justify the properties of homogeneity. Now consider a positive real function f (t) with variable t (t being the reduced C ). The behavior of a physical quantity, defined by the function temperature t = T −T TC ] [ Log f (t) =λ Lim Log(t) . f (t), is characterized by a number λ if t →0 We say that λ is a critical exponent and we note f (t) ∼ = t λ with t → 0. If the critical point can be approximated by negative values of t, we can define in an analogous way the critical exponent λ' when tends toward 0− . Table 1.1 gathers the definitions of the main critical exponents. Most of these definitions and notations are due to Fisher [23]. Moreover, the critical exponents, which only depend on the dimension of the parameter of order n and of the physical space d. Moreover, they satisfy a certain number of inequalities, some of which can be rigorously established. Rushbrook [24] α' + 2β + γ ' ≥ 2. Table 1.1 Definition of the main critical exponents and their values predicted by the Ginzburg– Landau theory compared with values found experimentally [17] Properties

Critiques exponents

Susceptibility χ γ γ’

Definitions

Conditions

Theories predictions

Experimental values

t −γ (−t)−γ

T > TC , H = 0 T < TC , H = 0

1 1

1.3–1.4 –

Magnetization M

β δ

(−t)−β H1/δ

T < TC , H = 0 T = TC

0.5 3

0.2–0.4 3–6

Correlation length ξ

ν ν’

t −ν (−t)−ν

T > TC , H = 0 T < TC , H = 0

0.5 –

0.6–0.7 0.6–0.7

Specific heat CH

α α’

t −α (−t)−α

T > TC , H = 0 T < TC , H = 0

Discontinuity Discontinuity

−0.3–0.3 −0.3–0.3

16

1 Exchange Interaction Types in Magnetic Materials

Table 1.2 Hamiltonian and dimensionality of the order parameter

Ising

Expressions of hamiltonian ∑ z z J Si S j

1

XY

∑

y y

2

y y

3

Models

n

i, j

i, j

∑

Heisenberg

i, j

J (Six S xj +Si S j ) J (Six S xj +Si S j + Siz S zj )

∞ ∑

Spherical

p=1

p

∞

p

Si S j

Griffiths [25] α' + β(δ + 1) ≥ 2. Fisher [26] γ ≤ (2 − η)ν. In two dimensions, the Ising model (n = 1) presents a transition with a non-zero order parameter below a temperature T0 . Two points to discuss: • The absence of a long-range order for some two-dimensional systems does not imply a finite value of the susceptibility at any temperature. The Heisenberg model and the XY model even show a divergence of susceptibility below a non-zero temperature. • There are two-dimensional systems which present a long-range order at nonzero temperature such ( as the anisotropic XY ) model whose Hamiltonian has the ∑ y y x x J1 Si S j + J2 Si S j . expression: H = − i, j

Table 1.2 brings together various Hamiltonians of the models that have been studied in detail. The influence of the values of d and “n” has been the subject of several experimental studies. There are a large number of magnetic substances which behave like ideal systems, and whose exponents coincide with those of the theoretical models corresponding to the same values of d and n. Table. 1.3 shows the deduced critical exponent values for various models [27, 28]. Table 1.3 Values of critical exponents for several models, Ising, XY and Heisenberg Models

Ising n=1

Ising n=1

XY n=2

XY n=2

Heisenberg n=3

D

2

3

2

3

3

α

0

0.106

–

−0.01

−0.121

β

0.125

0.326

–

0.345

0.367

δ

15

4.78

15

4.81

4.78

γ

1.75

1.2378 (6)

–

1.316 (9)

1.388 (3)

ν

1

0.6312 (3)

–

0.669 (7)

0.707 (3)

References

17

Many theoretical and experimental results show that the inequalities between the critical exponents obtained from statistical thermodynamics often reduce to equalities. It is possible to interpret this fact if we admit that in the vicinity of a second-order transition point certain thermodynamic functions satisfy the properties of homogeneity. All of these equalities predicted by the homogeneity hypothesis constitute the scaling laws. A function f (x1 , ..., xn ) of n variables x1 , ..., xn is said to be homogeneous of degree p if, regardless of λ, we have f (λx1 , ..., λxn ) = λ p f (x1 , ..., xn ). Assuming that the free energy F is homogeneous in the vicinity of a second-order transition point, one can show the existence of a temperature-independent[ relation] H t = M ship between the variables |t|Mβ et |t|Hβδ which is written as M(H,t) β βδ , |t| , |t| |t| where t is the reduced temperature, M is magnetization, and H is the magnetic field. This relationship is called the state equation (S.E.S.: Scaling Equation of State). Thus, the inequalities of Rushbrook [25] and Griffiths [26] become equalities: α + 2β + γ = 2 . α + β(1 + δ) = 2 Using this formula m = |t|Mβ and h = |t|Hβδ [29–32], we can then write ⎧ ⎨ m = m(h, ±1) . ou ⎩ m = m ± (h) This last relationship implies that m, function of h, follows two curves m + (h) and m − (h) for T > Tc et T < Tc , respectively. The scaling laws make it possible to determine all the critical exponents. The critical exponents that we know how to calculate exactly like those of the three-dimensional spherical model satisfy the scaling laws, the critical exponents associated with the magnetization, magnetic susceptibly, and magnetic field are β = 0.48, γ = 1.11, and δ = 3.1, respectively for Fe2.5 Mn0.5 Al alloy [33] and these are nearly similar to the mean-field theory for β and γ values which have been taken as 0.5 and 1 [33].

References 1. W. Nolting, A. Ramakanth, Quantum Theory of Magnetism (Springer, Berlin Heidelberg, 2009) 2. I.A.S. Getzlaff, Introduction. in Fundamentals of Magnetism (Springer Berlin Heidelberg, 19), pp 1–6 3. M.A. Ruderman, C. Kittel, Indirect exchange coupling of nuclear magnetic moments by conduction electrons. Phys. Rev. 96(1), 99–102 (1954) 4. C. Zener, Interaction between shells in the transition metals. II. ferromagnetic compounds of manganese with perovskite structure. Phys. Rev. 82(3), 403–405 (1951) 5. M. Azhar, M. Mostovoy, Incommensurate spiral order from double-exchange interactions. Phys. Rev. Lett, 118(2) (2017) 6. P.G. de Gennes, Effects of double exchange in magnetic crystals. Phys. Rev. 118(1), 141–154 (1960)

18

1 Exchange Interaction Types in Magnetic Materials

7. B.A. Berg, Markov Chain Monte Carlo Simulations and Their Statistical Analysis (World Scientific, 2004) 8. C. Gaetan, X. Guyon, Mod elisation et statistique spatiales. ematiques et Applications, 2011 edition. (Springer, Berlin, Germany, 2011) 9. M.E.J. Newman, G.T. Barkema, Monte Carlo Methods in Statistical Physics (Clarendon Press, Oxford, England, 1998) 10. A.D. Bruce, Structural phase transitions. II. static critical behaviour. Adv. Phys. 29(1), 111–217 (1980) 11. R.A. Cowley, Structural phase transitions i. landau theory. Adv. Phys. 29(1), 1–110 (1980) 12. S. Krinsky, D. Furman, Exact renormalization group exhibiting tricritical fixed point for a spin-one Ising model in one dimension. Phys. Rev. B 11(7), 2602–2611 (1975) 13. W.S. Kendall, F. Liang, J.-S. Wang, Markov Chain Monte Carlo (Copublished with Singapore University Press, 2005) 14. W.E. Holland, H.A. Broun, Phys-Stat. Sol (a) 10, 249 (1972) 15. S.F. Edwards, P.W. Anderson, J. Phys. F 5, 965 (1975) 16. L. Kadanoff, Physics 2, 263 (1966) 17. N. Boccara, Symétries brisées, ed. Hermann (1976) 18. L.P. Regnault, Thèse d’état (Université Sci. Et Med. Et Inst. Nat. Pol, Grenoble, 1981) 19. K. Hirakawa, H. Ikeda, J. Phys. Soc. Jap. 35, 1328 (1973) 20. E. Riedel, F. Wegner, Z. Physik, 225 (1969) 195; L.P. Regnault, J. Rossat-Mignod, J.Y. Henry, J. Phys. Soc. Jap. 52 (1983) 21. K.G. Wilson, Phys. Rev. Lett. 28(548), 1 (1972) 22. K.G. Wilson, J. Kogut, Phys. Report C12, 75 (1974) 23. M.E. Fisher, Rep. Prog. Phys. 30, 615 (1965) 24. G.S. Rushbrook, J. Chim Phys. 39, 842 (1963) 25. R.B. Griffiths, Phys. Rev. Lett. 14, 623 (1965) 26. M.E. Fisher, Phys. Rev. 180, 594 (1968) 27. G.A.J. Baker, B.G. Nickel, D.I. Meiron, Phys. Rev. B 17, 1365 (1978) 28. M.J. George, J.J. Rehr, Phys. Rev. Lett. 53, 2063 (1984) 29. S. Mandal, J. Panda, T.K. Nath, J. Alloys Compd. 653, 453 (2015) 30. S. Guha, R. Kumar, S. Kumar, L.K. Pradhan, R. Pandey, M. Kar, Phys. B Condens. Matter 579, 411805 (2020) 31. S. Guha, S. Kumar, S. Datta, M.K. Manglam, M. Kar, J. Phys. D Appl. Phys. 52 (2019) 32. E. Tka, K. Cherif, J. Dhahri, E. Dhahri, E.K. Hill, J. Supercond. Nov. Magn. 25, 2109 (2012) 33. S. Guha, S. Datta, S.K. Panda, M. Kar, Mater. Sci. Eng. B 283, 115817 (2022)

Chapter 2

Computational Methods: Ab Initio Calculations and Monte Carlo Simulations

2.1 Density Functional Theory To describe the electronic structure and find the physical properties of a system with several nuclei and electrons, one must solve the fundamental equation of physics established in 1925 [1] by Erwin Schrödinger and which is written as ⎡

∑ N h2 ∑ A h2 ∑ Z e2 | | I − ∇i2 & − ∇ I2 − H u = ⎣− i 2m I 2M i,I |− → →| | ri − R I | ⎤ ∑ ∑ e2 Z I Z J e2 ⎦ |→ − |→ − |+ + →|| u = Eu I i

The susceptibility being the derivative of the magnetization is thus found:

36

2 Computational Methods: Ab Initio Calculations and Monte Carlo …

( ) ∂ = β − 2 ∂B or ) ( χ = β − 2 In the same way we deduce the energy and the specific heat: C=

dE dT

( ) c = kβ2 − 2 What is important here is that the knowledge of the variation of the partition function Z (also called normalization constant) makes it possible to deduce all the information on the macroscopic state of the system [30]. However, the calculation of this function for a system that includes a large number of particles remains one of the major challenges of physics. However, for a system of large number of interacting particles even the enumeration of the states, and therefore of the terms of the partition function becomes impossible, for a system of 10,000 particles, the partition function will contain 210,000 terms. Exact solutions only exist for very small systems, and for simple models (the Ising model and the 2D Potts model), for other systems the use of approximation methods is required such as expansions in series, methods of field theories, and numerical methods. Moreover, if we consider that the system to be studied is placed on a lattice of finite size, the numerical calculation of the partition function becomes much easier. Even if the accuracy of numerical methods depends on the size considered, these methods are a widely used means to understand complex systems and study their behavior and critical properties [31, 32]. Put into practice In a Monte Carlo simulation, we want to calculate the average values of physical quantities such as magnetization M, susceptibility, average energy, and heat capacity. ∑ In general, the average value of a physical quantity A is defined by = Z1 s A(s)e[−βE(s)] , where Z(T) is the partition function of the system studied at temperature T, and E(s) and A(s) are, respectively, the energy of the system and the value of the quantity A in the microscopic state s. In principle, the summation must be made over all the configurations and the microscopic states of the system. But this is numerically impossible when the number of particles in the system is considerable. To circumvent this problem one can try to consider a sampling of the states of the system, two types of sampling prove to be used:

2.2 Monte Carlo Simulations

37

Simple sampling where a large number of microscopic states are randomly generated and the average < A > of the physical quantity is calculated over all the states ∑ C

A(s)ex p[−βE(s)]

s=1 generated, = ∑ . C s=1 ex p[−βE(s)] Obviously, the precision for this type of sampling strongly depends on the number of configurations generated, the greater this number, the more precise the value obtained will be. However, the deeper problem encountered when using simple sampling is that the configurations being randomly generated will correspond to disordered states, so they are not representative of low-temperature states. Sampling by importance where we try to randomly generate dominant configurations that have the most important probabilities of existing [33]. The states in this case have probabilities which respect the Boltzmann probability distribution, which makes it possible to simplify the expression of the average value of the quantity < A > in a simple calculation of average:

=

1 ∑C A(s). s=1 C

Markov chain Creating a random set of states that follow the Boltzmann probability distribution is the most challenging aspect of a Monte Carlo simulation. To begin with, it is impossible to choose states at random and decide to accept or reject them with a probability proportional to e − βEμ. This would not be much better than choosing states at random because we end up rejecting them almost all since the chances of them being accepted would be so slim. Instead, Markov processes serve as the state-generating mechanism employed in virtually all Monte Carlo techniques. A Markov process is a mechanism that creates a new state for a system from its current state μ. When given the starting state I, it generates a new state at random, it never produces the same one. For a true Markov process, all transition probabilities must be dependent only on the two states u and v. The probability w(a → b) from state a to state b must also satisfy the closure relation: ∑ b

w(a → b) = 1

The Markov process is specifically chosen to ultimately create a series of states that arise with the probability provided by the Boltzmann distribution when run long enough from any state in the system. This is done by continuously running a Markov process to create a Markov chain of states (a state a, and using the Markov process to create a new one called v, introduce v into the process to create another called A, and so on) in a Monte Carlo simulation. Thus, we say that we have reached thermal equilibrium when the Markov chain follows the Boltzmann probability.

38

2 Computational Methods: Ab Initio Calculations and Monte Carlo …

Ergodicity To generate states with their Boltzmann probabilities, the state generation process must be able to arrive at any state of the system whatever the initial state considered, since all states having a non-zero probability. Indeed, we are allowed to make certain transition probabilities of the Markov process zero, but there must be at least one path of non-zero transition probabilities between these two states. In practice, the Monte Carlo simulations set almost all of the transition probabilities to zero but with respect to this Ergodicity condition, which speeds up the calculations. Detailed scale Detailed balance is another condition that must be met by the Markov process to ensure that it is the Boltzmann probability distribution after ∑ the balance is larger than any other distribution. At equilibrium we will have b pu w(u → v) = ∑ u pv w(v → u). ∑ By introducing the closure relation, we have pu = v pv w(v → u). However, this condition is not sufficient, to guarantee the arrival at the state of equilibrium it is rather necessary to consider the balance condition in the form [36]: pu w(u → v) = pv w(v → u). This is the condition of detailed equilibrium. It is clear that any set of transition probabilities that satisfy this condition also satisfy before, we can also show that this condition eliminates limit cycles. To see this, let’s first look at the left side of the equation, which is the probability of being in a state u multiplied by the probability of making a transition from that state to another state v. The detailed equilibrium condition tells us that on average the system must go from v to u as often as it goes from u to v. In a limit cycle, in which the probability of occupying some or all of the states changes cyclically, there must be particular states of the Markov chain for which this condition is violated. For the probability of occupying a particular state to increase, for example, there must be more transitions into that state than transitions out of that state, on average. The detailed equilibrium condition prohibits dynamics of this type and therefore limit cycles. Since we want the equilibrium distribution to be the Boltzmann distribution, it is obvious that we must choose the transition probabilities that satisfy the relation: P(μ→ν) = ppμν = e−β( Eν −Eμ ) . P(ν→μ) If the Markov process satisfies this condition, plus the other two discussed in the previous sections, we can be sure that we will arrive at the equilibrium Boltzmann distribution. Acceptance rate As indicated by the previous relation, the choice of the transition probability from a to b depends only on the Boltzmann factor related to the energy of the two states u and v [37].

2.3 Simulation Algorithms

39

pν P(μ → ν) = = e−β( Eν −Eμ ) P(ν → μ) pμ • This is why we use the acceptance ratio to build algorithms to generate the Markov sequence leading to the Boltzmann distribution. • Thus, we will consider only the most probable configurations. Unlikely configurations whose influence on the result is negligible are naturally eliminated by the use of adequate transition probabilities in the Markov chain generation algorithm. • The first configurations of the Markov chain do not respect the Boltzmann distribution, since it would depend on the initial configuration generated randomly, one must therefore run the program for an “equilibrium time” in order to reach equilibrium [35, 37]. The estimation of the magnitude is made by considering only the configurations generated after the equilibrium.

2.3 Simulation Algorithms i. Local algorithms: Metropolis and Glauber An approach often used to achieve detailed equilibrium is to randomly propose a small change in state u, resulting in another state v, such that the reverse process (starting at v then proposing a small change resulting in u) is also likely. More formally, a process in which the condition P(u → v) = P(v → u) holds for all pairs of states (u,v). Thus, taking the example of an Ising model containing N spins, this corresponds to randomly choosing one of the spins on the lattice, so P(u → v) = P(v → u) = 1/ NOT. Detailed balance allows a common scaling factor to be used in the acceptance probabilities for forward and reverse Monte Carlo moves, but being probabilities, they cannot exceed 1. Simulations are then fastest if the greater of the two acceptance probabilities is equal to 1, that is, if P(u → v) or P(v → u) is equal to 1. These conditions (including the detailed equilibrium) are performed by the socalled Metropolis algorithm, in which the probability of acceptance is given by Metr opolis Paccept (u → v) = min[1, Pv /Pu ] = min[1, ex p(−β(E v − E u ))]. Thus, a proposed move that does not increase the total energy is still accepted, but a proposed move that results in higher energy is accepted with a probability that decreases exponentially with increasing energy difference. By way of illustration, let’s describe how a simulation of the Ising model looks like: Initialize all rounds (either random or all the way up). Perform N random trial moves (N = LD): Randomly select a site. Calculate the energy difference ∆E = EB – EA if the test (here the spin flip) induces a change in energy. Generate a random number. Make it uniformly distributed in [0, 1].

40

2 Computational Methods: Ab Initio Calculations and Monte Carlo …

If ∆E < 0 or if Rend < exp(-∆E): invert the spin. Sampling certain observables. Step 2 corresponds to a unit time step of the Monte Carlo simulation. An alternative to the Metropolis algorithm is Glauber’s dynamics [38]. The trial probability is the same as Metropolis, i.e., P(u → v) = P(u → v) = 1/N. However, the acceptance ex p(−β(E v −E u )) Glauber . which also satisfies the probability is now:Paccept (u → v) = 1+ex p(−β(E v −E u )) detailed equilibrium condition. ii. Cluster algorithms/Wolff’s algorithm Many models encounter phase transitions at some critical temperature. The paradigmatic example for second-order phase transitions is the Ising model defined above. Near the critical temperature, the spins exhibit critical fluctuations. As shown in Fig. 2.3, large domains of aligned spins appear, this phenomenon is associated with: The divergence of the correlation length ζ of the connected spin–spin correlation function. The divergence of the correlation time of the autocorrelation function. Moreover, the correlation time increases with the size of the system as τ ∼ Lz where z is the critical dynamic exponent. For the 2D Ising model simulated with the Metropolis algorithm, z = 2.1665 [39]. This phenomenon of critical slowing reflects the difficulty of modifying the magnetization of a cluster of correlated spins. Consider again the example of a 2D spin system where a spin has four nearest neighbors. If this spin is surrounded by aligned spins, its contribution to the energy is Eu = −4J and after the inversion of this spin, it becomes Ev = 4J. From Tc ≈ 2.269 K, the probability of acceptance is low for the Metropolis algorithm: P(u → v) = ex p(−8βc J ) = 0.0294. Thus, most reversal attempts are rejected. To make matters worse, even if such a spin with aligned neighbors is flipped, the next time it is selected, it will surely flip. Only spin flips at the edge of a cluster have a significant

Fig. 2.3 Snapshots of the 2D ferromagnetic Ising model for two temperature values a: T > TC and b: T ≈ TC

2.3 Simulation Algorithms

41

effect over a longer period, but their fraction becomes extremely small when the critical temperature is approached and the cluster size diverges. One remedy is to develop a non-local algorithm that returns a whole group of spins at once. Such an algorithm was designed for the Ising model by Wolff [40], following the idea of Swendsen and Wang [41] for more general spin systems. An outline of this procedure is shown in Fig. 2.4. The procedure consists of first choosing a random initial site (priming site). Then we add each neighboring spin, provided it is aligned with the Padd probability. If it’s not aligned, it can’t belong to the cluster. This step is repeated iteratively with each neighbor added to the cluster. When no more neighbors can be added to the cluster, all spins in the cluster flip at the same time. The probability of forming a certain spin group in the u state before the Wolff shift is the same as in the v state after the Wolff shift, except for aligned spins that have not been added to the boundary group. By choosing Padd = 1 − exp(2βJ), the acceptance probabilities are simplified. Near the critical point, Wolff’s algorithm significantly reduces the autocorrelation time and the critical dynamic exponent compared to a local algorithm (such as Metropolis or Glauber). iii. Algorithm with continuous time or without rejection As we saw in the previous subsection, with a local algorithm (like, for example, Metropolis) a spin flip of the Ising model at criticality has a high probability of being rejected, and this is even truer in the ferromagnetic phase. A significant part of the computation time will therefore be spent without changing the system. Another method was proposed by Gillespie [42] in the context of chemical reactions, then applied by Bortz, Kalos, and Lebowitz in the context of spin systems [43].

Fig. 2.4 An iteration of the non-local algorithm introduced by Wolff between two spin configurations A and B

42

2 Computational Methods: Ab Initio Calculations and Monte Carlo …

In short, this algorithm lists all possible Monte Carlo moves that can be performed in the system in its current configuration. One of these moves is randomly chosen based on its probability, and the system is forced into that state. The time step of evolution during such a displacement can be rigorously estimated. This time changes with each configuration and cannot be set to unity as in the Metropolis algorithm: it takes a continuous value. This is why this algorithm is sometimes called continuous-time algorithm. On the one hand, this algorithm must maintain a list of all possible moves, which requires a relatively heavy task; on the other hand, the new configuration is always accepted and this saves a lot of time when the probability of rejection would be high. It is also sometimes called the rejection-free algorithm. The efficiency of this algorithm is maximum for T ≤ TC . In detail, an iteration of the continuous-time algorithm looks like this: List all possible moves from the current configuration. Each of these n moves is associated with a probability Pn . { n Calculate the total probability of a hit occurring Q = i=0 Pn . Generate a random number Rend1 uniformly distributed in [0, Q]. This selects the chosen move with the probability Pn /Q. Estimate the time elapsed during the move: ∆t = Q−1 ln(1 − Rend 2 ) where Rend2 is a uniformly distributed random number in [0, 1] [44]. The implementation of this algorithm becomes easier if the probabilities Pn can only take a small number of values. In this case, one can establish lists of all the moves having the same probability Pn . The selection process involves first selecting one of the lists, with the appropriate probability, and then randomly choosing a move from that list. This is the case, for example, in Ising simulations on a square (2D) or cubic (3D) lattice, when the probability Pn is limited to the values 1, exp(−4βJ), exp(−8βJ), or exp(−12βJ) (this last value only appearing in 3D). IVi The worm algorithm We present here another example of a local algorithm, the so-called worm algorithm introduced by Prokofev, Svistunov, and Tupitsyn [45, 46]. The difference with the algorithms presented above is an alternative representation of the system, in terms of graphs instead of spins. The Markov chain is therefore realized in configurations of graphs rather than spins, but still with Metropolis acceptance rates. The principle is based on the high-temperature expansion of the partition function. Suppose we want to sample the magnetic susceptibility of the Ising model. We can access it through the correlation function using the (discrete) fluctuation–dissipation theorem: χ=

β∑ G(i − j ) i, j N

where G(I, j) = − 2 is the connected correlation function between sites i and j. In the high-temperature phase, the mean value of the spin vanishes and G(i − j) = . The first step consists in writing the correlation function G(I, j) of the Ising model in the following form:

2.3 Simulation Algorithms

43

∑ 1∑ Si · S j e−β J k,l Sk ·Sl S Z ∑ || 1 = cosh(βJ ) D N Si .S j (1 + Sk .Sl tanh(βJ )) S k,l Z

G(i, j) =

The configurations that contribute to the sum of equation given the correlation function contain an even number of spins in the product at any given site. Other products involving an odd number of spins in the product have a zero contribution. Each term can be associated with a path determined by the sites that participate in it. A contribution to the sum consists of an (open) path connecting sites i and j and closed loops. The sum over the configurations can be replaced by a sum over such graphs. Importance sampling is no longer performed on spin configurations but on graphs which are generated as follows: One of the two sources, say i, is mobile. At each step, it moves randomly to a neighboring site n. Any closer neighboring site can be chosen with the trial probability P(u → v) = 1/2D, where D is the dimension of the network. If no link is present between the two sites, then a link is created with the probability of acceptance: Paccept (u → v) = min(1, tanh(βJ )). If a link is already present, it is deleted with the probability of acceptance:Paccept (u → v) = min(1, 1/tanh(βJ )). Vi. Histogram Methods—Reweighting The results obtained from standard MC simulations are a set of individual points corresponding to different temperatures, and the interpolation between two successive temperatures is not exact in the transition region, and histogram methods have been proposed to overcome this difficulty [47–49]. Indeed, the configurations generated in a Monte Carlo simulation contain an enormous amount of information. We can thus use “Reweighting” to “extend” the results of the original simulation, carried out at the inverse temperature β0, to any other β sufficiently close to the simulation point without carrying out additional simulations. However, since Metropolis sampling is not a priori restricted to a limited phase space, at least in principle, it is indeed theoretically possible to reweight the Metropolis data obtained for a given temperature T0 = 1/kB β0 to another, T = 1/kB β. The simplest form is based on the fact that the probability of a configuration φ at the inverse temperature β, Pβ (φ), can ' be easily related to the distribution at another temperature β' : Pβ' ∝ e−β Eφ , where C is a proportionality constant (which depends on β and β' , and will remain undetermined). Thus, the expectation value of an operator O(φ) at temperature β' can be written as follows: { 1 β' = pβ' (φ)O(φ)dφ Z { ( ' ) C − β −β = dφ pβ (φ)O(φ)e Z ( ' ) Zβ − β −β E Cβ = Zβ'

44

2 Computational Methods: Ab Initio Calculations and Monte Carlo …

where in the last step, the expectation value is evaluated at the temperature β. In order to get the ratio of partition functions, we can put O = 1, which implies Zβ' Zβ

= C and x z magnetic susceptibility χzz , respectively, for B = 0, and B = 0.1T with x = 0 to 1 and calculated by GFT. The dependence of the thermal magnetization at the field B z = 0.1 T for the different values of dilution x is observed. We see that the typical ferromagnetic behavior is noticed only at T < TN . The curves do not drop at TC as may be expected for a pure ferromagnetic system but have only an inflexion at TN . The values of Néel temperatures found in this work are roughly similar to the values found in Refs. [9, 10].

9.3 Results and Discussion

117

110 108

Experiments HTSE GFT

106 104 102 100 98

PM

96

T(K) 94 92 90 88 86

FerriM

84 82 80 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

1,1

x

Fig. 9.1 Magnetic phase diagram of Cx N1-x C2 O4 : A single phase is the transition of PM to FerriM phase in the range 0 ≤ x ≤ 1. The solid squares are the results of magnetic measurements [10]. The solid circles are the results of GFT and the open circles are the results of HTSE 8,350 8,345

ath(Å)

8,340 8,335 8,330 8,325 8,320 8,315 0,0

0,2

0,4

0,6

0,8

1,0

x

Fig. 9.2 The lattice parameter ath as a function of co-substitution content x of Cx N1−x C2 O4

We find significant differences between results in the GFT and HTSE. We attribute this difference to the quantum fluctuations taken into account in the first theory but neglected in second treatment, respectively, which are known to have a much greater influence on the magnetic properties of our system.

118

9 Effect of Cobalt on NiCr2 O4 : Calculation of Critical Temperature … 6,0

x=0 x=0.5 x=0.75 x=1

5,5

4,5 4,0 3,5

2

XZZ10 (emu/mol)

5,0

3,0 2,5 2,0 1,5 1,0 0

20

40

60

80

100

120

140

160

180

200

220

TN(K)

Fig. 9.3 Temperature dependencies of the magnetic susceptibilities XZZ for Bx = 0, and Bz = 0.1 T of Cx N1−x C2 O4 1,2 1,1 1,0 0,9

SZ

S

0,8 0,7 0,6

x=0 x=0.25 x=0.5 x=0.75 x=1

0,5 0,4 0,3 0,2 0,1 0,0 -0,1 0

10

20

30

40

50

60

70

80

90

100

110

120

TN(K) z

Fig. 9.4 Thermal normalized magnetization < SS > for Bx = 0, and Bz = 0.1 T of Cx N1−x C2 O4

References

119

9.4 Conclusion The thermal magnetization and the magnetic susceptibility are calculated by GFT. The analysis of the magnetization evolution shows the existence of exchanges interaction effects on the magnetic order. By applying the GFT to the χ (T ) we have estimated the Néel temperature T N for each dilution x. The obtained MPD is presented in Fig. 9.1. The transition from PM to FerriM is established. In this figure, we have included, for comparison, the theoretical results obtained by HTSE [9] and the experimental results [10]. The obtained values of the critical temperature by GFT show some significant differences.

References 1. K. Tomiyashu, I. Kagomiya, J. Phys. Soc. Jpn. 73, 2539 (2004) 2. G. Lawes, B. Melot, K. Page, C. Ederer, M.A. Hayward, Th. Prffen, R. Seshadri, Phys. Rev. B 74, 024413 (2006) 3. M. Farle, B. Mirwald Schulz, A.N. Anisimov, W. Platow, K. Baberschke. Phys. Rev. B. 55, 3708 (1997) 4. P. Frobrich, J. Jensen, P.J. Kuntz, Eur. Phys. J. B. 13, 477 (2000) 5. S.B. Lazarev, M. Panti, B.S. Toni, Green’s function theory of free electrons in thin films. Physica. A. 246, 53 (1997) 6. O. Schute, F. Klose, W. Felch, Phys. Rev. B. 52, 6480 (1995) 7. P. Frobrich et al., Eur. Phys. J. B. 18, 579 (2000) 8. S.V. Tyablikov, Methods in the quantum theory of magnetism. Ukr. Mat. Zh. 11, 289 (1959) 9. M. Hamedoun, A. Benyoussef, M. Bousmina, J. Magn. Magn. Mater. 322, 3227–3235 (2010) 10. P. Mohanty, A.M. Venter, C.J. Sheppard, A.R.E. Prinsloo, J. Magn. Magn. Mater. 498, 166217 (2020) 11. P. Frobrich, P.J. Jensen, P.J. Kuntz, A. Ecker, Eur. Phys. J. B. 13, 744 (2000) 12. W.E. Holland, H.A. Broun, Phys-Stat. Sol (A) 10, 249 (1972) 13. M.U. Rana, T. Abbas, Mater. Lett. 57, 925 (2002) 14. M. Hamedoun, M. Hachimi, A. Hourmatallah, K. Afif. J. Magn. Magn. Mater. 233, 290 (2001) 15. M.A. Kassem, A. Abu El-Fadl, H. Nakamura., J. Magn. Magn. Mater. 495, 165830 (2020)

Chapter 10

Studying the Effect of Zn Substitution on NiFe2 O4 Spinel Systems

Abstract The effects of Zn2+ on the behaviors of Znx Ni1-x Fe2 O4 (0 ≤ x ≤ 1) system have been studied by Green’s function (GF), mean-field (MF) theories and hightemperature series expansion technique (HTSE). The critical temperature has been obtained by using the GF for several magnetic field and several values of x. The second theory allows us to deduce the values of coupling exchanges. The HTSE is used to calculate the transition temperature. The obtained values by two methods (GF and HTSE) are comparable and are near to those obtained by experiment results.

10.1 Introduction Diluted spinel systems Znx Ni1−x Fe2 O4 are part of this group of Ax A1−x B2 X4 systems with two sublattices (A) and (B). These systems are characterized by magnetic properties dependent on the value of x depending on the composition and types of cations present at different sites. The radii of Zn2+ and Fe3+ are 0.57 and 0.49, respectively [1] and the magnetic moments of Fe3+ and Ni2+ ions are 5 and 2μB , respectively. In previous works [2, 3], GF was used to investigate the systems Znx Ni1−x FeO4 . The magnetization of ferromagnetic system is studied under randomphase approximation by Refs. [4–7]. The magnetization and magnetic susceptibility versus temperature were obtained under effect of x. The MF theory and probability law were used to calculate the values of J1 and J2 . The obtained results by GF are near to those obtained by HTSE [8] and by experiment results [9].

10.2 Theories Heisenberg–Hamiltonian of system is H' = −

∑

( ) ∑( ) y y y H x Six + H y Si + H z Siz Ji j Six S xj + Si S j − i

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. Masrour, Electronic, Magnetic, and Thermoelectric Properties of Spinel Ferrite Systems, SpringerBriefs in Materials, https://doi.org/10.1007/978-3-031-40613-3_10

121

122

10 Studying the Effect of Zn Substitution on NiFe2 O4 Spinel Systems

Here J is the exchange coupling and B(Hx , Hy , Hz ) is the external magnetic field. The thermal GF in the spectral representation is given by [10] G iαmn j (η) (ω) = (

[ α ] α where ωG iαmn j (η) (ω) = Si , H −1 ; S j With the inhomogeneities term is

//

( )m ( )n >> S −j Siα ; S zj

ω

)m (

S −j

)n ω

( )m ( )n [ ( )m ( )n ( α )m ( − )n ] α Sj = Siα S j S −j Siα S j + η S zj Aiαmn j (η) (ω) = Si ; S j GF was defined by the spectral theorem [7, 10, 11]. ( )m ( )n ( ) n z S −j Siα Ciαmn j(η) (ω) = S j We have applied RPA given by Refs. [12, 13] ( )m ( )n β βmn β S −j Siα Sk ; S zj ≈ Siα G i j + S j G iαmn j ( ) The exchange anisotropy ∞Siz Skz is reasonable for RPA-decoupling [12]. We have used this approximation [5, 14]. > 1 [ S(S + 1) − Siz Siz G i±mn Si± Siz + Siz Si± ≈ 2Siz 1 − j 2S

The magnetization is given by Ref. [16] < z > (S − φk )(1 + φk )(1 + φk )2S+1 + (1 + S + φk )φk2S+1 S = (1 + φk )2S+1 − φk2S+1 with ( ) 1 β Ek 1 = coth −1 φk = β E e k −1 2 2 √ The total magnetization and magnetic susceptibility are M(T ) = 2 + 2 and{ z z z (0)> χzz = − , respectively. χx x = −