Frontiers in Magnetic Materials

A.V. Narlikar Frontiers in Magnetic Materials

A.V. Narlikar (Ed.)

Frontiers in Magnetic Materials With 491 Figures and 22 Tables

Prof. Dr. Anant V. Narlikar UGC-DAE Consortium for Scientific Research University Campus Khandwa Road Indore-452017, MP India

ISBN-10 3-540-24512-X Springer Berlin Heidelberg New York ISBN-13 978-3-540-24512-4 Springer Berlin Heidelberg New York Library of Congress Control Number: 2005923313 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2005


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CONTRIBUTORS

AKIMITSU, J. Department of Physics and Mathematics Aoyama-Gakuin University 5-10-1 Fuchinobe Sagamihara Kanagawa 229-8558 JAPAN

ANDO, Yoichi Central Research Institute of Electric Power Industry Komae Tokyo 201-8511 JAPAN

ASAKA, T. High Voltage Electron Microscopy Station (HVEMS) National Institute for Materials Science (NIMS) Tsukuba 305-0044 JAPAN

AWANA, V.P.S. National Physical Laboratory, Dr K.S. Krishnan Road, New Delhi-110012, INDIA

BAJPAI, A. UGC-DAE Consortium for Scientific Research University Campus Khandwa Road Indore 452017 INDIA Formerly : Inter University Consortium for DAE Facilities Present address: Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, INDIA

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BANERJEE, A. UGC-DAE Consortium for Scientific Research University Campus Khandwa Road Indore 452017, INDIA Formerly : Inter University Consortium for DAE Facilities

BENNEMANN, K. H. Institut für Theoretische Physik Freie Universität Berlin Arnimallee 14 D-14195 Berlin GERMANY

BERTSCHAT, H. H. Bereich Strukturforschung Hahn-Meitner-Institut Berlin GmbH, D-14109 Berlin GERMANY

BLUMBERG, G. Bell Laboratories Lucent Technologies Murray Hill NJ 07974 USA

BOECHAT, B. Instituto de Física Universidade Federal Fluminense Av. Litorânea s/n, Campus da Praia Vermelha Niteroi, 24.210-340, RJ BRAZIL

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BOOTHROYD, A.T. Clarendon Laboratory Department of Physics University of Oxford Parks Road Oxford OX1 3PU, UK.

BREWER, William D. Fachbereich Physik Freie Universität Berlin 14195 Berlin GERMANY

BÜNEMANN, Jörg Fachbereich Physik and Material Sciences Center Philipps-Universität Marburg D-35032 Marburg GERMANY

CHOUTEAU, G. Grenoble High Magnetic Field Laboratory, MPI-FKF and CNRS, B.P. 166, 38042 Grenoble cedex 9, FRANCE

CHOVAN, J. Department of Physics University of Crete and Research Centre of Crete Heraklion GREECE

CONTINENTINO, M.A. Instituto de Física Universidade Federal Fluminense Av. Litorânea, s/n, Campus da Praia Vermelha Niteroi, 24.210-340, RJ BRAZIL

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de BRION, S. Grenoble High Magnetic Field Laboratory, MPI-FKF and CNRS, B.P. 166, 38042 Grenoble cedex 9, FRANCE

DOWBEN, P.A. Department of Physics and Astronomy, and The Center for Materials Research and Analysis (CMRA) 255 Behlen Laboratory of Physics University of Nebraska P.O. Box 880111 Lincoln Nebraska 68588-0111 USA

EISAKI, H. Nanoelectronic Research Institute AIST Tsukuba Ibaraki 305-8568 JAPAN

FERNANDES, J.C. Instituto de Física Universidade Federal Fluminense Av. Litorânea s/n, Campus da Praia Vermelha Niteroi, 24.210-340, RJ BRAZIL

GEBHAARD, Florian Fachbereich Physik and Material Sciences Center Philipps-Universität Marburg D-35032 Marburg GERMANY

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GOZAR, A. Bell Laboratories, Lucent Technologies Murray Hill, NJ 07974 USA and University of Illinois at Urbana-Champaign Urbana, IL 61801 USA

GUIMARÃES, R.B. Instituto de Física Universidade Federal Fluminense Av. Litorânea s/n, Campus da Praia Vermelha Niteroi, 24.210-340, RJ BRAZIL

JENKINS, S.J. Department of Chemistry University of Cambridge Lensfield Road, Cambridge CB2 1EW U.K.

JENSEN, P. J. Institut für Theoretische Physik Freie Universität Berlin Arnimallee 14 D-14195 Berlin GERMANY

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JUNGWIRTH, Tomas School of Physics and Astronomy University of Nottingham Nottingham NG7 2RD U.K. and Institute of Physics ASCR Cukrovarnicka 10 162 53 Praha 6 CZECH REPUBLIC

KAGEYAMA, Hiroshi Department of Chemistry Graduate School of Science Kyoto University Kyoto 606-8502 JAPAN

KARPPINEN, M. Materials and Structures Laboratory Tokyo Institute of Technology Yokohama 226-8503 JAPAN

KIMOTO, K. High Voltage Electron Microscopy Station (HVEMS) National Institute for Materials Science (NIMS) Tsukuba 305-0044 JAPAN and Advanced Materials Laboratory (AML) National Institute for Materials Science (NIMS) Tsukuba 305-0044 JAPAN

KOMIYA, Seiki Central Research Institute of Electric Power Industry Komae Tokyo 201-8511 JAPAN

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MAKAROVA, Tatiana Umeå University 90187 Umeå SWEDEN

MATSUDA, M. Advanced Science Research Center Japan Atomic Energy Research Institute Tokai Ibaraki 319-1195 JAPAN

MATSUI, Y. High Voltage Electron Microscopy Station (HVEMS) National Institute for Materials Science (NIMS) Tsukuba 305-0044 JAPAN and Advanced Materials Laboratory (AML) National Institute for Materials Science (NIMS) Tsukuba 305-0044 JAPAN

MOSHNYAGA, V. I Physikalisches Institut Universität Göttingen Friedrich-Hund-Platz 1 D-37077 Göttingen GERMANY

MOTOYAMA, N. Department of Physics and Mathematics Aoyama-Gakuin University 5-10-1 Fuchinobe Sagamihara Kanagawa 229-8558 JAPAN

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NAGAI, T. High Voltage Electron Microscopy Station (HVEMS) National Institute for Materials Science (NIMS) Tsukuba 305-0044, JAPAN

NAIR, Sunil UGC-DAE Consortium for Scientific Research University Campus Khandwa Road Indore 452017 INDIA Formerly : Inter University Consortium for DAE Facilities Present address: Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, INDIA

NAKAJIMA, T. Materials Design and Characterization Laboratory, Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, JAPAN

NÚÑEZ-REGUEIRO, M.D. Laboratoire de Physique des Solides, Bâtiment 510, Université Paris-Sud, 91405 Orsay, FRANCE

OHM, Torsten Institut für Physik Universitat Dortmund D-44221 Dortmund GERMANY

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PAPANICOLAOU, N. Department of Physics University of Crete and Research Centre of Crete Heraklion GREECE

PRABHAKARAN, D. Clarendon Laboratory, Department of Physics University of Oxford Parks Road Oxford OX1 3PU U.K.

SAGUIA, A. Instituto de Física Universidade Federal Fluminense Av. Litorânea s/n, Campus da Praia Vermelha Niteroi, 24.210-340, RJ BRAZIL

SINOVA, Jairo Physics Department Texas A&M University College Station Texas 77843-4242 USA

TERASAKI, Ichiro Department of Applied Physics, Waseda University, Tokyo 169-8555, JAPAN

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UEDA, Y. Materials Design and Characterization Laboratory, Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, JAPAN

UEHARA, M. Yokohama National University Faculty of Engineering Division of Intelligent Systems Engineering 79-1 Tokiwadai Hodogaya-ku Yokohama Kanagawa 240-8501 JAPAN

WEBER, Werner Institut für Physik Universitat Dortmund D-44221 Dortmund GERMANY

WEISER, Stefan Institut für Physik Universitat Dortmund D-44221 Dortmund GERMANY

YAMAUCHI, H. Materials and Structures Laboratory Tokyo Institute of Technology Yokohama 226-8503 JAPAN

PREFACE The field of Magnetism, perhaps the oldest branch of physics, today provides the world’s second largest industry, with magnetic materials claiming a market next only to semiconductor industry. In the current scenario the demand for magnetic materials is fast rising and Magnetism has taken central stage in science and technology. Even more excitingly, several exotic magnetic phenomena and novel materials have recently been discovered, many of which seem to hold the vital keys to unlocking new technology as well as challenging physics. This edited book, Frontiers in Magnetic Materials, focuses on the current achievements and state of art advancements in the frontal magnetic materials. The phenomenal success of magnetic materials may be ascribed to four relevant happenings in the last 10-15 years; (1) A dramatic refinement and development of new methods and probes for magnetic materials characterization, (2) overall improvements in general expertise and techniques in sample synthesis, (3) discovery of high Tc superconductivity phenomenon in rare-earth based cuprate and related systems, and (4) the global realization of importance of nano-science and –technology. During mid-80s, the traditional characterization techniques like magnetic susceptibility, magnetometry, magnetic resonance, neutron scattering, heat capacity etc., got a tremendous impetus through the use of microprocessors and commercial SQUIDS. As a consequence, the measurement capabilities for bulk magnetic properties got vastly refined. Interestingly, the same materials, which had already been exhaustively studied 40-50 years ago, now began yielding exciting new findings and deeper insights, which had hitherto not been possible. Likewise, the atomic level sensing capability of the microscopic nuclear probes, like Mössbauer and perturbed angular correlation spectroscopies also got enhanced. The sample processing too became more reliable to successfully synthesize reproduceable good quality single crystals and epitaxial films of complex materials in varied structures like, ultra-thin monolayers, multilayers, sandwiches, etc., with intriguing magnetic properties. The remarkable discovery of high temperature superconductivity in La-based cuprate in 1986 triggered a renewed interest in RE-based perovskite manganites and related magnetic systems that had already been exhaustively studied many years earlier. But now with improved sample processing and characterization, the studies led to the exciting discovery of the CMR and charge/orbital ordering effects and the start of a gigantic activity in fundamental research and technology. Some of these along with a few double perovskites and other magnetic systems have interestingly been found half-metallic ferromagnets possessing a pronounced spin polarization at very low temperatures. This makes them strong candidates for the prospective ‘spintronic’ and tunneling magneto-resistance (TMR) devices. Similarly, several other fascinating magnetic systems, such as spin ladders and chain materials, presenting interesting situations of qunatum magnetism, have really been the off-shoots of high Tc cuprates. This edited book takes stock of the above developments and many more. For instance, it presents detailed coverages of magnetism of two dimensional borates, and a host of highly

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anisotropic oxy-borates, field induced transitions from incommensurate magnetism to commensurate magnetism in layered antiferromagnets, curious differences in orbital and spin states of two apparently similar nickel oxides, detailed ac susceptibility studies probing, spin glass, superparamagnetic and long range ordered systems, site-dependent order/disorder effects in perovskite manganite, and growth and properties of bi-layer manganite single crystals. The success in sample processing and characterization during the last decade has led to the synthesis of unusual magnetic systems showing dual properties. The examples of these are ferromagnetic semiconductors, magneto-superconductors and magneto-thermoelectrics. Furthermore unconventional magnetism has been discovered in various forms of carbon, such as graphite, fullerenes, nanotubes etc.,where processing the materials differently seem to change them from ferromagnetic to superconducting ! Mounting importance of nano-level studies has led to the ingenious ways of producing nanoparticle samples and the advent of powerful new techniques for element specific studies, layer-by-layer and even leading to atomic resolution, suitable for investigating magnetism of nanoclusters, bi- and trilayer systems and even single atoms at surfaces and interfaces. The book carries an interesting overview of these modern methods of characterising magnetic materials. The success at the nano-level has opened-up new frontiers, namely magnetism of nanoclusters, surface and interface magnetism, low dimensional magnetism, interacting nanostructure magnetism etc., all holding an intense technological prospects, are covered in this edited book. The potential of high resolution TEM and the Lorentz TEM for observational studies of magnetic materials has been convincingly demonstrated. Finally, turning from these exotic new materials to the perennial problem of magnetism of nickel, this edited book presents an interesting approach that applies the original Gutzwiller variational scheme to multiband problem. Frontiers in Magnetic Materials comprises 24 chapters on the mentioned topical areas, written by noted international experts in the field and should be of central interest to researchers and specialists in physics and materials science faculties of academic institutes as well as relevant industries. At the same time, the book is also likely to be of relevance for final year undergraduates reading physics and materials science. I would like to thank all the contributors for their splendid and timely cooperation. I am grateful to William Brewer for his help and suggestions in planning the book and to Alok Banerjee for useful comments. Thanks are further extended to the IUC, Indore and UGCDAE Consortium for Scienific Research, Indore for providing the necessary infrastructure facilities.

November, 2004

Anant V. Narlikar

CONTENTS CONTRIBUTORS

V

PREFACE

XV

MODERN METHODS FOR INVESTIGATING MAGNETISM William D. Brewer Introduction Topics Treatesd Modern Methods Conclusions References

1 2 2 36 37

PROBING MAGNETIC PHASES IN DIFFERENT SYSTEMS USING LINEAR AND NON LINEAR SUSCEPTIBILITY A. Banerjee, A. Bajpai, and Sunil Nair Introduction Probing Spin Glass and Superparamagnetic Systems Probing Long Range Ordering Investigating Weakened Charge Ordered State and Electronic Phase Separation References

43 46 53 63 68

TRANSMISSION ELECTRON MICROSCOPY STUDY ON MANGANESE OXIDES T. Asaka, T. Nagai, K. Kimoto, and Y. Matsui Introduction Experiments Results and Discussion Conclusion References

71 74 75 93 94

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SINGLE CRYSTALS OF BILAYER MANGANITES D. Prabhakaran and A.T. Boothroyd Introduction Experimental Details Results and Discussion Conclusions References

97 100 101 112 113

GUTZWILLER-CORRELATED WAVE FUNCTIONS: APPLICATION TO FERROMAGNETIC NICKEL Jörg Bünemann, Florian Gebhard, Torsten Ohm, StefanWeiser, and Werner Weber Introduction Gutzwiller Variational Theory Results for Ferromagnetic Nickel Diagrammatic Approach Exact Results for Infinite Coordination Number Variational Ground-State Energy Landau-Gutzwiller Quasi-particles Outlook References

117 120 124 135 138 141 145 148 149

CHEMISTRY OF HALF-METALLIC AND RELATED CATION ORDERED DOUBLE PEROVSKITES M. Karppinen and H. Yamauchi Introduction Cation Ordering in the Perovskite Structure B-Site Ordered Double Perovskites A-Site Ordered Double Perovskites References

153 155 157 171 179

DILUTE MAGNETIC SEMICONDUCTORS Jairo Sinova and Tomas Jungwirth Introduction The Basic Picture of DMS Theoretical Models of DMS Transport Properties of DMS Systems Summary References

185 187 189 191 205 205

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UNCONVENTIONAL MAGNETISM IN CARBON BASED MATERIALS Tatiana Makarova Introduction Magnetic Properties of Diamond and Graphite Graphite with Structural Defects Experimental Data on High Temperature Magnetism in Graphitic Carbons Nanotubes Fullerenes The Role of Impurities in the Magnetism of Carbon Structures Possible Applications of Magnetic Carbon References

209 209 212 221 228 229 236 241 242

ORBITAL AND SPIN ORDER IN THE TRIANGULAR S=1/2 LAYERED COMPOUND (Li,Na)NiO2 S. de Brion, M.D. Núñez-Regueiro, and G. Chouteau Introduction Orbital Properties of NaNiO2 Orbital Properties of LiNiO2 Magnetic Properties of NaNiO2 Magnetic Properties of LiNiO2 Solid Solution of Li1-xNaxNiO2 Conclusion References

247 250 252 255 256 265 268 271

STRUCTURES AND ELECTROMAGNETIC PROPERTIES OF THE A-SITE ORDERED PEROVSKITE MANGANITE Y. Ueda and T. Nakajima Introduction Sample Preparation The A-Site Ordered Manganites RBaMn2O6 The A-Site Disordered Manganites R0.5Ba0.5MnO3 Summary References

273 275 276 285 292 293

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THE LIMITS TO SPIN-POLARIZATION IN FINITE-TEMPERATURE HALF-METALLIC FERROMAGNETS P.A. Dowben and S.J. Jenkins Introduction The Problem with Magnons The Problem with Spin-Polaronic Non-quasiparticle States The Problem of Magnon-Phonon Coupling The Problem of Interfaces The Problem of Interface Composition Experimental Proof Half-Metallic Character ? Summary References

295 296 299 300 305 309 314 317 319

LAYERED COBALT OXIDES AS A THERMOELECRIC MATERIAL Ichiro Terasaki Introduction Brief Summary of Thermoelectrics Layered Cobalt Oxides Origin of the Large Thermopower Unconventional Electronic States Summary and Future Prospects References

327 328 332 338 339 343 344

COMMENSURATE AND INCOMMENSURATE MAGNETISM IN LAYERED ANTIFERROMAGNETS J. Chovan and N. Papanicolaou Introduction Symmetry Constraints in Ba2CuGe2O7 The KSEA Limit Commensurate Phase Nonlinear σ Model Dynamics of the Commensurate Phase Incommensurate Phases In-Plane Magnetic Field Magnetic Properties of K2V3O8 Conclusion References

347 349 352 352 355 357 361 371 374 381 383

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MAGNETISM IN HIGHLY ANISOTROPIC BORATES: EXPERIMENT AND THEORY M.A. Continentino, J.C. Fernandes, R.B. Guimarães, B. Boechat, and A. Saguia Introduction The Anhydrous Borates Magnetic Properties Charge Ordering and Structural Transition Random Magnetic Chains References

385 386 388 394 396 410

METAL-INSULATOR TRANSITION AND MAGNETORESISTANCE IN MANGANITE THIN FILMS: LATTICE STRAIN AND DISORDER EFFECTS V. Moshnyaga Introduction Basic Properties of Perovskite Manganits Metalorganic Aerosol Deposition (MAD) Technique Magnetotransport in Single La0.7(Ca,Sr)0.3MnO3 Films Nanocomposite Manganite Based Films Photoinduced Electronic Inhomogeneity in Manganite Films Summary and Outlook References

415 417 423 425 436 444 451 454

MAGNETISM OF INTERACTING TWO-DIMENSIONAL NANOSTRUCTURES P. J. Jensen and K. H. Bennemann Introduction The Model System Two Dimensional Disordered Particle Ensembles Dipole Coupled Magnetic Particles Located on Hollow Microspheres Growth of Thin Magnetic Films Conclusion and Outlook Appendix A: Dipole Energy and Blocking Temperature Appendix B: Cluster Monte Carlo Method References

459 463 465 473 480 487 490 492 494

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SURFACE AND INTERFACE MAGNETISM ON THE ATOMIC SCALE H. H. Bertschat Introduction Motivation Historical Survey Three Types of Principal Experiments Combined Hyperfine Interactions and Perturbed Angular Correlation Spectroscopy (PAC) Sample Preparation and Measurements at the Online Mass Separator ISOLDE/CERN Adatoms on Ferromagnetic Surfaces Coordination-Number Dependence of Magnetic Hyperfine Fields at Cd Impurities on Ni Surfaces Symmetry Independence of Impurity-Induced Magnetic Units Mixed Coordination Numbers Conclusions and Outlook References

503 503 505 507 508 511 513 516 521 523 525 527

MAGNETO–SUPERCONDUCTIVITY OF RUTHENO-CUPRATES V.P.S. Awana Introduction Experimental Details Results and Discussion Summary and Concluding Remarks References

531 534 535 567 570

SUPERCONDUCTIVITY AND MAGNETISM IN LADDER AND CHAIN COMPOUNDS-PHYSICS OF (Sr,Ca)14Cu24O41 M. Uehara, N. Motoyama, M. Matsuda, H. Eisaki, and J. Akimitsu Introduction Magnetism in the Edge-Sharing CuO2 Chains Physical Properties of Two-Leg Ladders in (Sr,Ca)14Cu24O41 Under Ambient Pressure Physical Properties of Two-Leg Ladders in (Sr,Ca)14Cu24O41 Under High Pressure Summary of Spin Gap and Superconductivity of Two-Leg Ladder References

573 579 587 595 603 606

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AN EXPERIMENTAL REALIZATION OF THE SHASTRY-SUTHERLAND MODEL Hiroshi Kageyama Introduction Geometrically Frustrated Spin Models Crystal Structure Single Crystal Growth Spin-Gap Formation Exact Dimer Singlet Ground State Three-Dimensional-Shastry-Sutherland Lattice Quantum Phase Transition at Zero Field Extremely Localized Triplet Excitations Multi-triplet Excitations Quantized Magnetization Plateaus Dzyaloshinsky–Moriya Interaction Search for New Shastry-Sutherland Compounds Conclusions References

611 613 616 617 619 622 624 628 630 633 636 641 644 646 648

COLLECTIVE SPIN AND CHARGE EXCITATIONS IN (Sr,La)14-xCaxCu24O41 QUANTUM SPIN LADDERS A. Gozar and G. Blumberg (Sr,La)14-xCaxCu24O41: The Structure and General Properties Magnetic Properties of Sr14Cu24O41 Effects of Temperature and Ca(La) Substitution on the Phononic and Magnetic Excitations in Sr14Cu24O41 Density-Wave Correlations in Doped Two-Leg Ladders Summary References

653 655 666 674 692 693

ELECTRONIC PROPERTIES OF α’ – NaV2O5 A. Gozar and G. Blumberg General Properties of α’-NaV2O5 and Motivation for a Spectroscopic Study Magnetic Raman Continuum in the High Temperature Phase (T > 34 K) Collective Excitations in the Low Temperature Phase of NaV2O5 (T < 34 K) Summary References

697 702 717 732 733

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COLLECTIVE MAGNETIC EXCITATIONS IN SrCu2(BO3)2 A. Gozar and G. Blumberg Introduction: Why SrCu2(BO3)2 Low Temperature Phononic Spectra in SrCu2(BO3)2 Magnetic Properties of SrCu2(BO3)2 Summary References

735 738 741 753 754

MAGNETIC AND CHARGE CORRELATIONS IN La2-x-yNdySrxCuO4 : RAMAN SCATTERING STUDY A.Gozar, Seiki Komiya, Yoichi Ando, and G. Blumberg The Phase Diagram and Structural Properties of the High Temperature Superconductor La2–xSrxCuO4 Magnetic and Electronic Properties of Macroscopically Orthorhombic La2–xSrxCuO4 at Light Doping (0 < x < 0.03) Spin and Lattice Dynamics at Commensurate x=1/8 Sr Doping in La2-x-yNdySrxCuO4 Summary References

776 786 787

SUBJECT INDEX

791

755 758

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PROBING MAGNETIC PHASES IN DIFFERENT SYSTEMS USING LINEAR AND NON LINEAR SUSCEPTIBILITY A. Banerjee, A. Bajpai∗, and Sunil Nair∗ UGC-DAE Consortium for Scientific Research†, University Campus, Khandwa Road, Indore 452017, India 1. INTRODUCTION Study of the magnetic phases in various systems is an important activity of condensed matter physics and materials science research. Variety of phases found in magnetic systems are probed by many different measurement techniques depending on the nature of the magnetism as well as on the associated magnetic transitions. A system is explored from the response it gives to the excitations induced by the probes. Depending on the technique used, the magnetic system is explored in different length and time scales. When the response is collected directly from atoms or their electrons then it is broadly called a microscopic probe like neutron scattering, Mössbauer effect, Magnetic Circular Dichroism etc. On the contrary, in conventional dc-magnetization or ac-susceptibility measurements the response is collected from the magnetic induction or force exerted by the bulk of the sample that defines the length scale of the measurement as macroscopic. In between, there is the mesoscopic length scale when the primary response originates from the collections of atoms whose dimensions are much smaller than the bulk of the sample like magnetic domains or magnetic clusters. The measurement techniques such as small-angle neutron scattering, neutron de-polarization and magnetic force microscopy belongs to this category because for them the response originates from the magnetic clusters or magnetic domains present in the sample. However, these length scales of the probes are neither directly related to the intrinsic length scales of magnetic order ∗ †

Present address: Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India. Formerly: Inter University Consortium for DAE Facilities.

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of the system nor with the penetration depth of the probe. For example, neutron scattering, Mössbauer effect or ac-susceptibility measurements contain signatures of both magnetic longrange and short-range orders. Likewise, magnetic circular dichroism and magneto-optic Kerr effect are efficient tools to probe the surface magnetism having penetration depths restricted to only a few tens of nanometers. Nevertheless, the former is a microscopic probe where the response depends on the absorption of x-rays by individual atoms whereas response for the latter depends on the total magnetization of the surface produced by a large collection of atoms. Similarly, the time scales involved in various measurement techniques varies from about 10-12 second for neutron spin-echo measurement to 1 second or more for dcmagnetization measurement. The importance of the time scale of the measurement arises mainly to probe the magnetic relaxation or the spin dynamics. The measurement techniques like Mössbauer effect, muon depolarization, nuclear magnetic resonance (NMR) and electron spin resonance (ESR) probe the local spin dynamics in time scale < 10-5 seconds. Whereas, global dynamics or long wavelength relaxation processes of metastable magnetic systems are probed by techniques like dc-magnetization or ac-susceptibility measurements. Thus different measurement techniques probe the magnetic systems with different approaches and very often one needs to use a combination of them to develop reasonable understanding about any system. However, it is also possible to do in-depth study of magnetic systems by exhaustively utilizing the potential of a particular technique. In this article, an attempt is made to exemplify how ac-susceptibility (ac-χ) can be used to unravel the magnetism of interesting metastable magnetic systems and the associated short-range order as well as to effectively probe the critical behaviour of systems with long-range magnetic order. 1.1. LINEAR AND NON LINEAR AC-χ AS A PROBE FOR MAGNETIC SYSTEMS A magnetic specimen, when placed in an alternating magnetic field ( H = H0sin2πft) results in a time varying magnetization (M), which is measured by using a simple set-up consisting of a primary and two oppositely wound secondary coils. In general M can be written in terms of H as (1) M = M0 + χ1H + χ2H2 + χ3H3 +…. Where M0 is the magnetization created by the internal field and χ1, χ2, χ3 are the first-, second- and third-order susceptibilities respectively, which can be directly measured using a mutual inductance bridge in ac-susceptibility measurement [1]. The magnetization M has an inversion symmetry with respect to the sign of H in the absence of any superimposed dcmagnetic field or a symmetry-breaking internal field as a consequence of which the even order susceptibilities χ2 , χ4 ….remain absent. The small amplitude of the ac field (H) allows one to accurately measure the true initial susceptibility ( χ ) which otherwise has to be estimated from the extrapolation of high field data. The added advantage of measuring χ as a function of the probe frequency (f ~ 1/τprobe) of the applied field is that it provides useful information of the various time constants intrinsic to the magnetic system (τsystem) . In fact, the use of linear and non linear ac-χ as a probe for characterizing a magnetic materials initially gained popularity, due to the easy manifestation of metastability through the frequency dependence of χ1 in a variety of canonical spin glasses (SG) and an assembly of small magnetic particle termed as superparamagnets (SPM).

45

Consequently, a frequency dependence in ac-χ (χ1) together with a bifurcation of field-cooled (FC) and zero-field-cooled (ZFC) magnetization (i.e. history dependent magnetization) were considered sufficient to assign a spin-glass phase in many systems. However, a cursory glance at literature over the last 3 decades show that many long-range order systems like ferromagnets (FM) antiferromagnets (AFM) show typical time and history dependence in bulk susceptibility (both ac and dc) similar to what one observes in a conventional spin glass (SG) or superpramagnets (SPM) [1]. However, it is to be noted that the physical origin and the underlying physics behind this qualitatively similar time and history effect in first order susceptibility may have completely different physical origins. For instance, the origin of metastability in a spin-glass like system is the ‘frustration effect’ arising from random competing interactions leading to a co-operative freezing of the spin system on a broad timescale. Superparamagnets consisting of small magnetic particles or clusters give rise to metastability purely because of dynamic phenomenon called blocking arising out of competition between thermal and anisotropy energies of the individual magnetic entities [1,2]. However, in conventional ferro/antiferromagnets the time and history effects can arise from domain wall dynamics/pinning and issues related with anisotropy [3]. It is thus clear that the mere observation of frequency dependence in ac-χ, particularly in the first-order susceptibility or history dependence in dc-magnetization is not conclusive enough for exact determination of the magnetic ground state. This is especially true for those samples which are neither tailor made small particle systems nor from the family of canonical spin glasses. It is here that the measurement and analysis of the higher order susceptibility becomes crucial. As it will become clear in the later part of this article, at times the higher order susceptibility can provide much needed information to differentiate among a variety of magnetic orders such as FM, AFM, Ferrimagnet, SG, SPM etc., hidden in the magnetic specimen. For instance, non-linear susceptibility was introduced as a direct probe for the divergence of Edwards-Anderson order parameter, signifying the onset of spin-glass transition [2]. Though in the past, there have been efforts to discern a spin glass phase from a superparamagnetic phase in conventional small particle systems using higher order susceptibility [4], there have been little effort for in-depth study in a vast variety of bulk magnetic materials including transition metal oxides, magnetic ceramics, metals etc., exhibiting signatures of metastability in bulk magnetization studies, the origin of which remains ambiguous. The endeavor of this article is to bring out the subtle features exhibited by higher order magnetic susceptibility in a variety of systems and to emphasize the role which it can play in the determination of magnetic phase. We have broadly divided our study in three parts. Section 2 deals with studies on SPM and SG phases and Section 3 is devoted to the conventional long range magnetic orders like FM and AFM. We have elaborated more on our experimental results in a series of a diluted antiferromagnets and in a series of amorphous ferromagnets. This is to lay emphasis on the subtle experimental observations and the robustness of the underlying theoretical predictions related to higher order susceptibility, even in such unconventional magnetic specimens. Apart from this, it is experimentally shown that all the critical exponents required to characterize a FM phase transition can be determined using linear and non-linear low-field susceptibility alone in conjunction with the relevant scaling equations. Finally, a brief account of a study on a weakened charge-ordered system and electronic phase separation using linear as well as non-linear ac-susceptibility is given in

46

the Section 4 of this article to elucidate the relevance of this technique in investigating systems of current interest. 2. PROBING SPIN GLASS AND SUPERPARAMAGNETIC SYSTEMS Both SG and SPM systems exhibit qualitatively similar features in the first order susceptibility, i.e., a sharp peak in χ1 which is frequency dependent[5]. Added to this, the third order susceptibility χ3 is theoretically expected to show a negative peak as a function of temperature in both the systems. In a situation like this, when both χ1 and χ3 show exactly similar temperature dependence around TG or TB for a SG and SPM phase respectively, a detailed study of their corresponding field and frequency dependence of χ1 and χ3 can provide the missing link as guided by theories and earlier experiments [2,6-12]. A pioneering work addressing this issue has been taken up for Au96Fe4 system known to be a spin glass and Cu97Co3, which is a conventional SPM. This work explores the subtle differences in the nature of χ3 to differentiate SPM from the SG phase, the details of which can be found in Ref.[4] and also in references therein. Here we elaborate on this issue using some experimental results obtained on a series of polycrystalline AFM sample, which, on dilution has shown typical signatures of metastability in bulk magnetization. The samples in this particular series of compounds are neither from a family of a canonical spin glass nor a tailor made SPM and hence our purpose is to illustrate that how χ3 can be effectively used to identify the underlying magnetic phase in this series of compounds, LixNi1-xO . 2.1. METASTABLE MAGNETISM IN Li-Ni-O Pure NiO is a well known antiferromagnetic insulator and Li was substituted to tune the band gap accordingly to change the conductivity properties. This substitution not only generated a series of compounds with interesting physical properties but also generated a technologically important material for use as an insertion electrode in Li batteries. The systematic study of the structure and the magnetic phases of LixNi1-xO over the whole composition range was initiated many decades ago by Goodenough et.al. [13]. Since then a variety of magnetic orders have been proposed for different composition ranges of this series [14-18] and various aspects related to the origin of the magnetic ordering of this system is being debated even in the recent times [19,20]. In this section a systematic study of the magnetic phases around two important compositions i.e. x = 0.35 and 0.5 is presented to demonstrate how the long standing controversies related to the magnetic phases are resolved through the measurement of low-field linear (χ1) and non-linear (χn) ac-susceptibilities [21-23]. The details about sample preparation and structural characterization can be found in reference [21]. Since high magnetic fields can smear the transition and mask the intrinsic signatures of the magnetic system, only low field dc-magnetization and ac-susceptibility measurements were performed to characterize this magnetic system using home made set-ups [24,25]. Figure 1 shows the real part of first-order ac-susceptibility (χ1r ) for x = 0.35 sample of the LixNi1-xO series as measured at different frequencies of the exciting ac-field depicting the probe time effect on the magnetic transition. Inset of the same figure shows the bifurcation of zero-field-cooled (ZFC) and field-cooled (FC) dc-susceptibilities below the transition for this sample indicating the dependence on the field history of the measurement.

47

0.50

0.25

ZFC 0.00

12

120

160

200

T (K)

-2

χ1 (10 emu/mol)

FC

16

x = 0.35

χDC (emu/mol)

20

r

8 13 33 333 1.3

4

205

210

Hz Hz Hz kHz

215

220

T (K) Fig 1. Temperature dependence of χ1 for the LixNi1-xO sample with x = 0.35 at the Field 1 Oe and in different frequencies. Inset shows FC and ZFC dc-susceptibility for x = 0.35 measured at 20 Oe r

FC

2

r

15

0

100

125

150

T (K)

10 13 H z 73 H z 13 3 H z 43 3 H z 1 kHz

1

-2

χ (10 emu/mol)

ZFC

x = 0 .5

χDC (emu/mol)

4

20

5

80

90

100

110

120

130

T (K ) Fig 2. Frequency dependence of χ1r for the sample LixNi1-xO with x = 0.5 at 1 Oe. Inset shows FC and ZFC dc-susceptibility for x = 0.5 measured at 20 Oe

48

210

21 5

2 20

225

x = 0.35

-2

χ (10 emu Oe /mol)

T (K ) 2 05 0 .0

3

r

-3

-0 .1

-0 .2

Fig 3. The negative peak in the real part of the χ3r as a function of temperature for the sample with x = 0.35. x = 0.5 -1

-2

r

χ3 (10

-4

-2

emu Oe /mol)

0

-3 90

100

110

120

130

T (K)

Fig 4. The negative peak in the real part of the χ3r as a function of temperature with x = 0.5 The frequency dependence of ac-χ for x = 0.5 sample is shown in Figure 2 and the inset of this figure shows the history dependence of dc-susceptibility for the same sample. Thus samples of both the compositions have shown time and history dependent magnetization which are generic features of magnetically disordered systems showing metastable magnetism like SG, cluster-glasses, SPM and even inhomogeneous ferromagnets [2,5]. We show that in such cases, the third order susceptibility χ3 can be used as a probe to discern the cause of metastability, guided by theories and earlier experiments [2, 6-12]. The third-order ac-susceptibility (χ3) for the x = 0.35 and 0.5 samples are shown in figures 3 and 4. Both of them show qualitatively similar pronounced negative peaks at temperatures where χ1 has shown peak for the respective samples (figures 1-2). Since both systems like SG or SPM are expected to show negative peak in χ3 around the transition, it becomes non-trivial to identify the exact nature of magnetic transitions in these two samples only on the basis of qualitative features of χ3.

49

13 H z 133 H z

30

-2

I (10 emu Oe /mol)

However, a detailed analysis of χ3 can unambiguously indicate the true nature of these transitions. It is well known theoretically, that for SG systems, χ3 is expected to have a negative divergence at the glass transition temperature (TG) as the amplitude of the measuring ac-field (H) and the frequency of the ac-field (f) tend to zero [6]. Further, it is also shown that the divergence of χ3 as the spin-glass transition is approached from the high temperature is given by χ3 = τεγ3 , where ε is the reduced temperature (T-TG)/TG , γ3 is the critical exponent and τ is the related critical amplitude. This behavior have been observed experimentally for many SGs; for instance, see reference [8]. Here it should be borne in mind that the f- and Hdependent peak in χ1 is non-critical for a SG transition. Contrary to this, since the SPM is understood as a progressive blocking of moments of the magnetic particles or clusters in the experimental timescales it neither shows criticality in χ1 nor in χ3 with H, f or T. The linear as well as higher order susceptibility has normal field and temperature dependence for SPM above the Blocking temperature (TB), that is the temperature at which the peak in susceptibility occurs and Wohlfarth’s model can be used to calculate the temperature dependence of χ1 and χ3 [4,5, 8]. The total magnetization (M) of a system of non-interacting SPM particles can be given as M = n<µ>L(<µ>H/kBT), where n is the number of particles per unit volume, <µ> is the average magnetic moment of a single entity ( magnetic particle), kB is the Boltzman constant and L(x) is the Langevin function. The linear and non-linear susceptibilities above TB can thus be calculated to be χ1 = n<µ>/3kBT and χ3 = −(n<µ>/45)(<µ>/3kBT)3 . Thus χ1 shows a 1/T dependence and χ3 shows a –1/T3 dependence. Figure 5 and 6 show the results of the H and f dependence of the peak value of χ3 for samples with x = 0.35. The measured |χ3max| for sample with x = 0.35 clearly indicates a diverging behaviour of χ3 in the limit H→ 0, f → 0 at T = TG. It is to be noted that χ1 does not show such diverging behavior with H as is shown in ref. [21] The diverging behaviour of χ3 is further substantiated in the figures 7 and 8 where the log-log plots of |χ3max| against H or f follows straight lines.

-4

20

Iχ3

max

10

0

0

5

10

15

20

H

Fig 5. |χ3(r)max| Vs. applied ac field for the sample x = 0.35 measured at frequency 13 and 133 Hz.

1 Oe 2.3 Oe

30

-2

I (10 emu Oe /mol)

50

-4

20

I χ3

max

10

0.0

0.3

0.6

0.9

1.2

f (kHz) Fig 6. |χ3(r)max| Vs. frequency for the sample x = 0.35 measured at ac field of 1 and2.3 Oe.

ln ( |χ3max| )

3 2 1

x =0.35 f =13 Hz

0 0

1

2

3

ln(H) Fig 7. A log-log plot of |Ȥ3max| against ac field for the sample x = 0.35

51

ln (| χ3ma x| )

4

x = 0.35 h = 1 Oe 3

2 2

3

4

5

6

7

8

ln (f) Fig 8. A log-log plot of |Ȥ3max| against frequency for the sample x = 0.35

-7

2.5

-6

-5

-4 2.0

2.0

1.5

x = 0.35

- χ3 (r)

1.5

1.0

333 Hz slope = -1.18 (0.04)

0.5

1.0

0.5

433 H z 1.3 kH z

0.0 -5.5

-5.0

-4.5

-4.0

(T - T G )/ T G Fig 9. Log-Log plot of χ3 against reduced temperature ε = (T-TG)/TG for x = 0.35 sample at frequency 433 Hz and 1.3 kHz. Inset shows the same at 333 Hz

52

12 10 8

-2

I (10 Oe emu/mole)

Finally, the criticality of χ3 as a function of temperature is illustrated in the figure 9, where a log-log plot for χ3 against the reduced temperature (T-TG)/TG is shown. From this plot, it is found that the critical exponent associated with χ3 is close to that of the randombond Ising spin-glass (γ3 = 1.18 ± 0.04). This analysis of the χ3 data shows that the low temperature magnetic phase of the x = 0.35 sample can be unambiguously concluded to be a spin-glass like. In a complete contrast to this, when similar measurements were repeated for the x = 0.5 sample no such criticality in χ3 could be found as is shown in figures 10 and 11. Moreover, the variation of |χ3max| with H or f is much smaller than what is found for the sample with x = 0.35 in figures 5 and 6. Thus χ3 does not become critical in the limit H→ 0, f → 0 for x = 0.5 sample. More significantly, it is shown in figure 12 that χ3 has a T−3 dependence as predicted by the Wohlfarth’s model for superparamagnetic particles. It needs to be mentioned here that χ1 follows a 1/T dependence as described earlier.

Iχ3

m ax

-4

6

4 2 0

5

10

15

H (Oe)

0 .7 5

-2

|χ3 | (10 emu Oe /mol)

Fig 10. |χ3(r)max| Vs. ac field for the sample x = 0.5.

max

-3

0 .6 0

0 .4 5 0 .0

0 .3

0 .6

0 .9

1 .2

f (k H z )

Fig 11. |χ3(r)max| Vs. frequency for the sample x = 0.5 measured at ac field of 2.26 Oe.

emu Oe /mol)

53

-2

12 Oe

-0.1

-0.2

χ3 (10

-3

0.0

0.0

-0.1

0.05

0.06 -3

0.07 -5

0.08

-3

T (10 K )

χ3 (10

-3

-2

Oe emu/mol)

0.1

-0.2

8 Oe

-0.3 0.055

0.060

T

-3

( 10

0.065 -5

K

-3

0.070

)

Fig 12. T dependence of of Ȥ3 above the blocking temperature for the sample x = 0.5 measured at 8 Oe. Inset shows the same for 12 Oe field. -3

Hence, it can be concluded from the preceding analysis that the magnetic transition observed for the sample with x = 0.5 arises from the thermal blocking of the superparamagnetic particles contrary to the spin-glass like freezing observed in the sample with x = 0.35 [21-23].

3. PROBING LONG RANGE ORDERING A universality class based on the dimensionality of the magnetic lattice and the orderparameter is assigned to a system depending on the values of the critical exponents which characterize the phase transition [26,27]. Hence, to develop a proper understanding about the long-range ordered state it is essential that the critical exponents associated with the phase transition are precisely determined. Conventionally, exhaustive measurements of both lowfield ac-susceptibility and high field dc-magnetization are needed to obtain the relevant critical exponents required for the complete characterization of the paramagnetic to ferromagnetic phase transition [27,28]. Considering the fact that ideal second-order paramagnetic to ferromagnetic phase transition is defined in the zero external field, a large applied field is likely to smear the true critical behaviours apart from introducing fieldinduced effects in many systems thus making low field susceptibility measurements advantageous. Moreover, the Kouvel-Fisher (K-F) analysis of the ac-susceptibility data is a well established method to independently determine both the transition temperature (TC) as well as the susceptibility exponent γ [29]. Thus, it is not surprising that the conventional second-order paramagnetic to ferromagnetic phase transition has been extensively studied

54

using linear susceptibility measurements to characterize the nature of the transition for a variety of systems [27,28]. In spite of the fact that non-linear effects become more pronounced in the vicinity of a phase transition the behaviour of the higher-order susceptibilities across conventional phase transitions has remained relatively less explored. This is primarily because of the fact that the non-linear susceptibilities are usually a couple of orders smaller in magnitude than their linear counterparts, thus making it more difficult to measure. Apart from this, the theoretical understanding about the non-linear susceptibilities is far from being satisfactory as far as these magnetic systems are concerned, though treatments of the critical regimes based on the simple mean field models [30] as well as more complicated Sherrington-Kirkpatric [31], Bethe approximation models [32] exist. However, there are a few experimental studies of the third ordered susceptibility (χ3) on ferromagnetic systems [30,33,34] which though predominantly qualitative in nature have included attempts on characterizing these transitions by the determination of the critical exponents. Recently, a study has been made to understand the hysteresis effect in ferromagnet through the measurement of non-linear susceptibilities [35]. Measurements of the non linear susceptibility in antiferromagnetically ordered systems are even rarer as compared to that reported in ferromagnets and these investigations have relied on indirect measurement techniques like the magneto-optical Faraday Rotation to characterize the magnetic phase transition [36,37]. Besides these conventional long range order spin transitions, χ3 has also been used in studying quadrapolar interactions in some rare earth intermetallic compounds [38] and to probe the presence [39] or absence [40] of these multispin correlation functions in some heavy Fermion systems. The ferromagnetic order is accompanied with internal field which acts like a symmetry-breaking field and gives rise to the second-order susceptibility (χ2 ). This aspect has been exploited to investigate the coexistence of ferromagnetic order and spin-glass phase in the system (NiFe)25Au75 [41]. Though underutilized till date, χ2 is a promising tool for discerning the presence and dynamics of mixed phase systems, like transition metal oxides, where such a mixed phase can arise due to electronic phase separation. In this section, a calculation based on the molecular-field approach [30] is presented to show the true nature of criticality of χ3 across a paramagnet to ferromagnetic phase transition. This is substantiated by the detailed measurement of linear as well as nonlinear susceptibilities on a series of amorphous ferromagnetic samples [42]. 3.1 THEORY When an external field H is applied to a collection of ‘n’ particles, the net magnetic moment is given by ª µΗ º (2) m = nµtanh « » ¬ kT ¼ where µ is the Bohr’s Magneton and k is the Boltzmann constant. The total magnetic field which the dipoles experience is given by H = H + λM . Where λ is the molecular-field constant and the magnetization (M) can now be written as

µ

(H + ȜM) kT where N is the particle density (N = n/V). M = Nµtanh

(3)

55

Now, for temperature above the ferromagnetic transition temperature (TC) i.e. in the paramagnetic region M has inversion symmetry with respect to the applied field H, hence the magnetization can be written as M = χ1H + χ3H3 + χ5H5 +…. Substituting this in equation (3),

χ 1 H + χ 3 H3 = Nµtanh

µ kT

(H + Ȝ[ Ȥ1H + Ȥ3H3+……] )

(4)

(5)

1 2 x and using the expression for TC from the mean-field 3 theory i.e. kTC = Nµ2λ; and comparing the coefficients of H, H3 etc.

Using the expansion tanh(x) = x -

χ1 =

1 Nµ 2 T kTc § · ¨ −1¸ © Tc ¹

χ3 = −

1 Nµ 4 T 4 3 3 3k T Tc § T · ¨ − 1¸ © Tc ¹

(6)

(7)

It is clear from equations (6) and (7) that as the TC is approached from the high temperature + side, that is in the limit of T → TC , χ1 has a positive divergence, whereas χ3 is expected to show a negative divergence. Now, for temperature below the ferromagnetic transition temperature (TC) i.e. for T < TC, because of the presence of spontaneous magnetization, M does not have the inversion symmetry with respect to the applied field H, hence the magnetization can be written as M = M0 + χ1H + χ2H2 + χ3H3 +….

(8)

where M0 is the spontaneous magnetization. Substituting equation (8) in the equation (3), M = Nµtanh{

µ kT

[ȜMo+( ȜȤ1+1)H+ ȜȤ2H2+ ȜȤ3H3]}

(9)

Expanding tanh(x) and comparing the coefficients of different powers of H, we get

ª µλΜ 0 1 µ 3 λ3 Μ 0 3 º − M0 = Nµ « » 3 k 3T 3 ¼» ¬« kT

χ1 =

Nµ 2 kTc

ª § Tc · 2 § M 0 · 2 º ¸¸ » «1 − ¨ ¸ ¨¨ «¬ © T ¹ © Nµ ¹ »¼ ª T § § Tc · 2 § M 0 · 2 · º « − ¨1 − ¨ ¸ ¨¨ ¸¸ ¸» «¬ Tc ¨© © T ¹ © Nµ ¹ ¸¹»¼

(10)

(11)

56

1 kTcM 0 χ 13 2 3 2 N 3µ 4 ª § Tc · § M 0 · º «1 − ¨ ¸ ¨¨ ¸¸ » «¬ © T ¹ © Nµ ¹ »¼

χ2 = -

χ3 =

χ 14

§ ¨¨ ª §T · §M0· º © ¸¸ » «1 − ¨ ¸ ¨¨ «¬ © Tc ¹ © Nµ ¹ »¼ 2

2

4

ª « λ · « λT c M 0 2 ¸ «2 N 2 µ 2 ¸¹ « T Nµ 2 « «¬

(

)

(

)ª

(12)

χ1

§ Tc · «1 − ¨ ¸ «¬ © T ¹

2

§ M0· ¨¨ ¸¸ © Nµ ¹

2

º » »¼

º » 1T» − » 3 Tc » » »¼

(13)

Equation (10) is solved for the spontaneous magnetization and yield for T < TC 2

2

§ M0· §T · § T · ¨¨ ¸¸ = 3 ¨ ¸ ¨1 − ¸ © Tc ¹ © Tc ¹ © Nµ ¹

(14)

Substituting this value of (M0/Nµ)2 in the value of χ1 and simplifying,

χ1

=

º ª T 3 − 2» Nµ 2 «¬ Tc ¼ 2kTc ª T º «¬1 − Tc »¼

(15)

Hence it is clear that χ1 diverges in the positive side as TC is approached from the low − temperature side, that is in the limit of T → TC . Now substituting equations (14) and (15) in the equation (13) and simplifying, 4

§ Nµ 2 · ¨ ¸ λT 8 ¨ 2kTc ¸¹ χ3 = © 3 § T · 4 ( Nµ )2 Tc ¨1 − ¸ © Tc ¹

(16)

−

where it is obvious that χ3 also has a positive divergence in the limit T → TC . Hence it is clear from the above that for a paramagnetic to ferromagnetic transition χ3 + shows divergence in the negative side in the limit of T → TC whereas it shows a divergence − in the positive direction in the limit of T → TC . Thus at TC, χ3 would be expected to cross the temperature axis. 3.2 EXPERIMENTS To experimentally verify the true critical behavior χ3 as suggested by these calculations, an extensive study of linear and non-linear ac-susceptibility measurements were performed on a well characterized ferromagnetic system so that the advantages of the present study become

57

obvious. The samples chosen for the present study is from a series of amorphous ferromagnetic metallic glass Fe5Co50Ni17-XCrXB16Si12 with x = 5, 10 and 15 referred as A2, A3 and A4 respectively. This series was extensively studied in the past, details of which can be found in the references [43-45]. Figures 13(a), 13(b) and 13(c) show the Kouvel-Fisher plots i.e. 1/χ1 d/dT(χ1-1) versus T for the A2, A3 and A4 samples respectively. The inverse of the slope directly gives the value of the susceptibility exponent γ and the intercept of the straight line on the temperature axis gives the value of TC. The values of TC and γ found form this measurement are in reasonable agreement with the values reported earlier as is shown in the Table 1. Table 1. Values of the susceptibility exponent γ and the transition temperature TC as determined from the Kouvel-Fisher analysis of the first-order susceptibility. The values determined by the earlier workers are given for comparison.

A

B

A2

A3

A4

TC (K)

267.44 267 A

222.76 222.2 A

174.37 174 B

γ

1.16±0.0008 1.19 A

1.388±0.01 1.38 A

1.41±0.01 1.73 B

Values determined in the reference [43] using ac-susceptibility measurements. Values determined in the reference [43] using high field dcmagnetization.

It is clear from the mean field calculation given earlier that χ3 is expected to diverge on both sides of phase transition and change sign at TC. However, this critical behavior is not easy to observe primarily because the expansion of M in terms of higher powers of H, which was used for the calculation is strictly valid for small values of H. Hence, to observe true critical behavior, measurements have to be done at very low fields, and higher fields tend to smear the transition . It is difficult to predict a priori the fields at which this critical behavior will be seen, as it depends on the extent of non-linearity in the system and thus varies from sample to sample. It is also to be noted that at higher measuring fields, the contributions from domains can become dominant and mask the true critical behavior of the spin system. Careful low-field measurements have shown the predicted critical behaviour in χ3 for this series of amorphous ferromagnets. Figures 14(a), 14(b) and 14(c) show the critical behaviour for the A2, A3 and A4 samples respectively. It can be clearly seen from the figure 14 that χ3 changes sign across TC and the exact TC can be directly determined from the crossover of χ3 on the temperature axis. These experimental TC values are given in Table 2. It is to be noted that TC determined from the crossover of χ3 matches well with those determined from the Kouvel-Fischer analysis (Table 1). This matching of TC from two different measurements not only substantiates the fact that the measured χ3 is a genuine response of the spin system but also the crossover of χ3 can be used as a direct method to determine TC .

58

2.5

1.2

-1

1/χ (d(χ )/dT)

1.5

Sample A2 γ = 1.16+0.008 Tc= 267.44 K

-1

1/χ(d(χ )/dT)

2.0

1.0

(a)

0.5 0.0 266

0.8

0.4

(b) 0.0

267

268

269

Sample A3 γ = 1.388+0.01 Tc= 222.76 K

270

222

223

3

-1

1/χ(d(χ )/dT)

223

224

224

225

T (K)

T (K) Sample A4 γ= 1.410+0.01 TC= 174.37 K

2

1

(c) 0

174 175 176 177 178 179

T (K)

Fig. 13. Kouvel-Fisher plots of the first order susceptibility for the samples A2, A3 and A4. The value of the γ and TC as determined from the plots are given in the respective graphs. It can be seen in Figure 14(a) that the A2 sample shows a double transition accompanied by two crossovers. This sample has gives an uncharacteristically low value for susceptibility exponent γ, a fact observed by earlier workers as well. Considering the fact that this sample has the largest percentage of Ni and is close to the critical concentration, this double transition could be arising from the clusters with a distribution of TCs, which is common for this class of materials. However, it is interesting to note that no direct evidence of such a distribution is evident from the linear susceptibility measurements. This only goes on to show that low-field nonlinear susceptibility is more sensitive tool than the linear susceptibility, as far as studies of spin systems near the transition is concerned. As mentioned earlier, the field in which the measurement is performed is an important consideration in the study of criticality in χ3. This can be clearly seen in figure 15, which shows the field dependence of χ3 for the A3 sample at measuring frequency of 133.33 Hz. It is clear that the critical feature is sharpest at the lowest measuring field and increasing the measuring field tend to wipes off this critical behavior indicating that larger applied fields destabilizes the system from its equilibrium ground state.

59

Sample A3

0.25

2

χ3(emu/gm Oe )

Sample A2

2

χ3 (emu/gm Oe )

0.50

0.25

(a) 0.00

0.00

(b)

-0.25

-0.50

264

265

266

267

268

269

220

222

T (K)

0.4

226

Sample A4

2

χ3(emu/gm Oe )

224

T (K)

0.2

0.0

(c) -0.2 153

163

173

184

194

T (K)

Fig. 14. The critical feature of χ3 as observed in the third-order ac-susceptibility for A2, A3 and A4 samples. The measurements were done at frequency 133.33 Hz and exciting field of 200 mOe.

Accurate determination of the transition temperature (TC) is an important consideration in the study of critical behavior across the ferromagnetic phase transition. The most popular techniques used for this purpose are Kouvel-Fischer plot or Arrot’s plot [46]. However, in both these techniques the TC is determined from the fitted data. Apart from this, the Arrot’s plot is made from the high field measurements which is associated with the complications related to high field measurements used to probe the critical regime as mentioned earlier [47]. An attempt is made here to show how the crossover in χ3 on the temperature axis can be used to directly as well as accurately determine the TC in the limit of H → 0. Figure 16 shows the variation of the crossover temperature of the χ3 with the applied field. The critical exponent associated with the third-order susceptibility, γ3, is determined from the slope of the straight line in the double-logarithmic plot of (-3/4)χ3H2 versus ε, where ε = [(T-TC)/TC] and the value of TC is taken from crossover point χ3 on the temperature axis as mentioned earlier. Figures 17(a), 17(b) and 17(c) show these plots for A2, A3 and A4 + samples respectively as T → TC . The exponents γ3 determined from these plots for A2, A3 and A4 samples turns out to be 4.57, 4.80 and 5.04 respectively which match well with the 3D Heisenberg value of 4.88. A deviation from the straight-line can be seen as T approaches TC.

60

0.50 0.15 Oe 0.19 Oe 0.23 Oe

2

χ3(emu/gm Oe )

0.25 0.00 -0.25 -0.50 -0.75 220

222

224

226

T (K)

Fig. 15. The field dependence of χ3 for the A3 sample measured at frequency of 133.33 Hz. It is clear that the critical behaviour is suppressed with applied field.

Temperature of Crossover

222.65 222.60 222.55 222.50 222.45 222.40 0.0

Intercept on T axis : 222.43 K

0.1

0.2

0.3

Applied ac field ( Oe)

Fig. 16. The temperature of the crossover TC as a function of measuring ac-field for the A3 sample. A linear behaviour is observed and the interpolation to zero field provides the unique way to determine the true transition temperature.

These curvatures in the plots close to TC arise mainly due to the dominating presence of higher-order terms close to the transition temperature apart from the inhomogeneities present in the samples. Thus from the measurements of low-field linear and non-linear ac-susceptibilities the exponents γ and γ3 for this series of samples are determined. The other critical exponents α, β, δ and ∆ associated with specific heat, spontaneous magnetization, field dependence of magnetization at TC and gap exponent respectively needed for the complete characterization of the phase transition can be calculated using the following scaling relations

61

-3

6x10

-3

(-3/4)χ3H (emu/gm)

-3

(b)

2

2x10

2

(-3/4) χ3H (emu/gm)

-2

10

(a)

4x10

Sample A2 linear fit γ3 = 4.57+0.08

-4

8x10 -4 6x10

Sample A3 linear fit γ3= 4.805+0.12

-3

10

-4

4x10

5

6

7

8

9

2

10

3

4

5

6 7 8 9 10 -3

-3

ε (x10 )

ε (x 10 ) -2

8x10

-2

(c) -2

4x10

2

(-3/4)χ3H (emu/gm)

6x10

-2

2x10

Sample A4 linear fit γ3= 5.04+0.09

-2

10

80

90

100 -3

ε (x10 )

Fig. 17. Double-logarithmic plots of χ3 versus reduced temperature (ε) for the A2, A3 and A4 samples. The slopes of the straight lines give the value of the exponent γ3 for the respective samples. γ3+ = γ + 2∆, ∆ = γ + β, δ = 1 + γ/β, and α + 2β + γ = 2 . All the exponents required for a complete characterization of this series is calculated and given in Table 2. It can be clearly seen that the values determined in the present study mach reasonably well with those given in reference [43]. The only exponent in which a large difference is seen is α which is calculated by using Rushbrooke equality α + 2β + γ = 2. The ambiguity in the value of this exponent is probably because the values determined in the reference [43] from the combination of both low ac-field and high dc-field measurements, as is the general practice. Whereas in the present case they are all determined from the low-field measurements alone.

62

Table 2. Values of all the critical exponents as determined from the linear and non-linear acsusceptibility measurements and calculated using scaling relations. TC (χ3) are the value of transition temperatures determined from the crossover of χ3 in the temperature axis. The values quoted from the previous work for comparison.

A

A2

A3

A4

γ

1.16±0.0008 1.19 A

1.388±0.01 1.38 A

1.41±0.01

γ3

4.57 ± 0.08

4.805 ± 0.12

5.04 ± 0.09

∆

1.705

1.708

1.815

β

0.545 0.35A

0.32 0.41A

0.405 0.52A

α

-0.25 0.2A

0.03 -0.2A

-0.22 -0.7A

δ

3.12 4.42A

5.33 4.49A

4.48 4.32A

TC (χ3)

267.05

222.55

173.45

Values determined in the reference [43] .

Hence, the following advantages of the low-field linear and non-linear susceptibilities as an effective probe to study the paramagnetic to ferromagnetic phase transition is unambiguously demonstrated : (i) Determination all the relevant critical exponents needed to characterize the magnetic phase transition only from the low-field linear and non linear susceptibility measurements. (ii) Directly ascertain the TC and accurately determine the zero-field TC, which otherwise has to be estimated from extrapolation of high field data. (iii)

Detecting subtle features not seen in the linear susceptibility measurements.

63

4.

INVESTIGATING WEAKENED CHARGE ORDERED STATE AND ELECTRONIC PHASE SEPARATION

A study on the so called ‘glassy’ behaviour in weakened charge-ordered state and the phenomenon of electronic phase separation is presented in this section to show how low-field linear as well as non-linear susceptibility measurements culminate as valuable probe for interesting magnetic systems. Hole doped manganites systems have attracted enormous attention in the recent time because they have displayed many interesting phenomenon like colossal-magnetoresistance, charge-ordering, electronic phase separation etc [48-50]. Manganites with hole concentration close to 50% have an important role and are being extensively studied since the Coulombic repulsive forces are known to stabilize the formation of charge and orbital ordered antiferromagnetic ground state in these systems. A variety of relaxation effects possibly arising out of phase competition are observed around this compositions, but whether these ‘glassy’ phases mimic the canonical spin-glasses or whether electronic phase separation leads to a new form of ‘glassiness’ remains to be understood. In order to understand these, a half doped charge-ordered system, Pr05Ca0.5MnO3 which is known to be a narrow bandwidth system with robust charge-ordering is selected for this study. Mn site substitution in the half doped charge ordered manganites has emerged as a popular means to comprehend the charge ordering and the concomitant antiferromagnetism. It is now well known that Mn site substitution affects the magnetism by tailoring the strength of the magnetic exchange and also induces various interesting phases in this class of materials [51-55]. Though many magnetically metastable phases are observed at low temperatures in the Mn site substituted weakened charge ordered state, the exact nature of the ground state remains to be properly understood. It is important that the Mn site substitution should not lead to the significant change in the structure or add any magnetic interaction to the parent compound. Hence, non-magnetic Al is chosen for Mn site substitution since its ionic radius match very well with that of Mn4+ present in the system. It has been shown from the detailed analysis of the structure that Al substitution in Pr0.5Ca0.5MnO3 up to 10% of Al does not have any significant effect on the structure [56]. Further, it is shown that in Pr0.5Ca0.5Mn1-XAlXO3 the strength of charge ordering progressively decrease with the increase in Al. In an attempt to understand the evolution of magnetic phases in weakened charge ordered system with quenched non-magnetic impurities, a detailed study is made on the Pr0.5Ca0.5Mn0.975Al0.025O3 compound using linear as well as nonlinear susceptibilities and dc-magnetization. Since large fields can mask the intrinsic signatures of inhomogeneously magnetized systems, the lowfield measurements are made to discern the magnetic ground state as shown in the figure 18. The inset of Figure 18 shows the weakening of charge ordering on Al substitution in Pr0.5Ca0.5MnO3 which otherwise shows a sharp peak around 240K arising from the charge ordering. On 2.5% Al substitution this peak gets broadened and shifts to lower temperature indicating the weakening of charge ordering. It is known that the parent Pr0.5Ca0.5MnO3 has an antiferromagetic order below about 175K. Though the signature of this antiferromagnetic transition is not seen in the inset of figure 18, it is detected from the magnetocaloric effect determined from the dc-magnetization measurement [56]. Figure 18 shows that the dilution of magnetic lattice by 2.5% Al substitution causes the long-range antiferromagnetic transition temperature (TN) to drop to about 50K. Interestingly, an additional feature is observed at lower temperature as evident by a cusp in the zero field cooled (ZFC) dc-magnetization at

0.16 Al

0.12

0 % Al

0.016

125 ZFC

-3

2.5 %

R

M (emu/gm)

TN

χ1 (10

FC

0.024

) emu/gm

64

0.08 250

T (K)

0.008

0

50

100 T (K)

150

200

Fig. 18. ZFC and FC dc-magnetization of the Pr0.5Ca0.5Mn0.975Al0.025O3 at a measuring field of 100 Oe. The inset shows the first-order ac-susceptibility at measuring field of 12.5 Oe and frequency of 133 Hz for Pr0.5Ca0.5MnO3 and Pr0.5Ca0.5Mn0.975Al0.025O3 samples. 26K. Moreover, a strong irreversibility or history dependence is observed as indicated by the large bifurcation in the ZFC and field cooled (FC) magnetization measurement. Acsusceptibility measurement has also shown this sharp feature at 26K which was accompanied with signatures of metastable magnetization. In the absence of any observable χ2 arising from the internal fields of the system, this low temperature metastable magnetic phase can be attributed to either spin-glass/cluster-glass or superparamagnetic phases present in the system. However, as mentioned in the section 2, the physical processes which give rise to such cusp in susceptibility and metastable magnetization for these diverse classes of systems is entirely different. As mentioned in the preceding parts of this article, for the spin-glasses or clusterglasses, it is the co-operative freezing of the ‘frustrated’ spins or clusters of spins, arising from the competing ferromagnetic and antiferromagnetic interactions, which give rise to the history and time dependent metastable magnetism in the systems. Whereas, in superparamagnets, it is a purely a dynamic phenomenon arising from the thermal blocking of the magnetization of the individual spin clusters in their easy directions decided by the anisotropies of the respective clusters/entity, which is responsible for the cusp in ZFC dcmagnetizaion or in ac-susceptibility. To identify the origin of the cusp observed at 26 K and to ascertain the nature of the magnetic ground state of the Pr0.5Ca0.5Mn0.975Al0.025O3 sample, a detailed study of the nonlinear ac-susceptibility was undertaken. The χ3 shows a negative peak at the same temperature (TP) where a cusp is seen in χ1 (not shown here). Hence it needs to be verified whether this peak is critical in the limits of H → 0 and T → TP indicating the divergence of the EdwardsAnderson order-parameter which in turn signify the freezing of spins or spin-clusters leading to a spin-glass like transition or it is non-critical as a function of H and T indicating

0.55

0.50

0.45

0.40

R

-6

|χ3 (max)| (10 ) emu/gm Oe

2

65

6

8

10

12

H (Oe)

Fig. 19. The peak value of χ3 plotted as a function of applied ac field H for Pr0.5Ca0.5Mn0.975Al0.025O3 sample, clearly indicate that χ3 is not critical with H. superparamagnet like blocking of clusters of spins. This is vividly demonstrated by plotting the peak value of χ3 as a function of H. This is shown in figure 19, where the value of the real part of χ3 at the negative peak denoted by |χ3R(max)| is plotted as a function of H for Pr0.5Ca0.5Mn0.975Al0.025O3 sample. Here, χ3 is clearly seen to saturate in the limit H → 0, indicating that the low temperature phase occurs due to the thermal blocking of magnetic entities and is not a co-operative freezing phenomenon. To reconfirm this, the temperature dependence of χ3 above TP is analyzed in the similar fashion as it is done in section 2 for LiXNi1-XO series of samples. − Figure 20 shows that the real part of χ3 has a T 3 dependence indicated by the straight line fit above TP similar to what is observed for superparamagnets. This further substantiates the fact that the low temperature phase observed in Pr0.5Ca0.5Mn0.975Al0.025O3 sample is superparamagnetic in nature contrary to the earlier findings where spin-glass like phase was attributed to analogous systems from the linear susceptibility measurements alone. Thus it becomes evident that such low temperature metastable magnetic features in the similar systems need to be properly identified through comparative rigorous measurements. Further, it has been shown that it is possible to estimate the average moment associated with the spin clusters or roughly determine their size from the ratio of χ3/χ1 using the respective equations for superparamagnetic clusters as given in section 2. Finally, an exciting aspect of ac-susceptibility measurements and its merits are presented through a straightforward and simple study to detect the occurrence of electronic phase separation in Pr0.5Ca0.5Mn0.975Al0.025O3 from the thermal hysteresis as shown in figure 21. It is to be noted that ac-susceptibility captures only the dynamic or the reversible part of magnetization unlike the dc-magnetization measurements which measures the total magnetization. Hence no system with static disorder which even exhibit metastable magnetism like spin-glass, cluster-glass or superparamagnets will show the effect of thermal hysteresis in ac susceptibility. The thermal hysteresis in ac-susceptibility can take place only

66

-0.25

-0.30

R

χ3 (10

-7

) emu/gm Oe

2

-0.20

-0.35

4.5

5.0

T

-3

5.5

(10

-5

) K

-3

0 .3 6

0 .3 0 0 .2 7

I

1

χ (10

-5

) emu/gm

χ

1

-3

0 .3 3

R

(10 ) emu/gm

Fig. 20. Temperature variation of χ3 above TP for Pr0.5Ca0.5Mn0.975Al0.025O3 sample. The straight line is a T−3 fit to χ3 measured at 12.5 Oe and frequency 133 Hz.

h e a tin g c o o lin g

*

0 .8 0 0 .6 0 0 .4 0 0 .2 0

h e a tin g c o o lin g

* 15

30

45 60 T (K )

75

Fig. 21. The upper panel shows the real part of first-order ac-susceptibility as measured at a field of 2.5 Oe and a frequency of 733 Hz for the Pr0.5Ca0.5Mn0.975Al0.025O3 sample in heating and cooling cycles. The lower panel shows the imaginary of the first-order ac-susceptibility part as measured during the same run. The asterisk indicate the beginning of the thermal cycle with the arrow indicating the direction.

67

when the system changes dynamically with respect to the temperature cycle. In the present case it happens because of electronic phase separation. The upper panel of figure 21 shows the real part of first-order ac-susceptibility (χ1R) as measured at a field of 2.5 Oe and a frequency of 733 Hz for the Pr0.5Ca0.5Mn0.975Al0.025O3 sample showing large hysteresis behaviour as a function of thermal cycling. It also shows that there is no observable change in the temperature where the cusp in susceptibility is observed (TP) which is identified as the blocking temperature of the superparamagnetic clusters indicating that the cluster volume does not change. However, there is significant change in the temperature of the broad feature observed around 50K. This broad feature is also observed in dc-magnetization measurement (figure 18) and is related to the antiferromagetic order of the weakened charge ordered state. The reduction in the temperature of the peak of this feature in the cooling run indicate that the electronic phase separation reduces the hole concentration from the bulk of the clusters, correspondingly reducing the antiferromagetic transition temperature (TN). This reduction in the hole concentration of the bulk would enhance its concentration at the respective domain walls without any observable change in the size of the clusters. This is evident from the lower panel of the figure 21 which shows that the imaginary part of the firstorder ac-susceptibility (χ1I) is consistently higher in the cooling run. Here, it is to be noted that χ1I arises from the magnetic loss in the system and an enhanced χ1I in the cooling run is presumably arising as a consequence of larger inter-cluster interactions across the hole enriched domain walls. It is important to mention here that there is no observable χ2 in the system, thus ruling out the presence of any ferromagnetic phase in this state. Thus a unique scenario of electronic phase separation is presented by probing a system with linear and non-linear susceptibilities alone, where it is shown that in a weakened charge ordered state the electronic phase separation changes the hole concentration of the bulk of the clusters with respect to the domain walls without invoking any ferromagnetic phase which is shown to be invariably present in all other phase separated manganites studied earlier. A detailed quantitative study on the above mentioned system and the related electronic phase separation using linear as well as non-linear susceptibilities is presented in a recent publication [57].

68

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[40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51]

[52] [53] [54] [55] [56] [57]

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