Handbook of Magnetic Materials, Volume 8

Handbook of Magnetic Materials, Volume 8 Elsevier, 1995 Edited by: K.H.J. Buschow ISBN: 978-0-444-81974-1

by kmno4

PREFACE TO VOLUME 8

The Handbook series Magnetic Materials is a continuation of the Handbook series Ferromagnetic Materials. The original aim of Peter Wohlfarth when he started the latter series was to combine new developments in magnetism with the achievements of earlier compilations of monographs, producing a worthy successor to Bozorth's classical and monumental book Ferromagnetism. This is the main reason that Ferromagnetic Materials was initially chosen as title for the Handbook series, although the latter aimed at giving a more complete cross-section of magnetism than Bozorth's book. Magnetism has seen an enormous expansion into a variety of different areas of research in the last few years, comprising the magnetism of several classes of novel materials that share with truly ferromagnetic materials only the presence of magnetic moments. For this reason the Editor and Publisher of this Handbook series have carefully reconsidered the title of the Handbook series and changed it into Magnetic Materials. It is with much pleasure that I can introduce to you now Volume 8 of this Handbook series. Artificial multilayered structures are prominent examples of such classes of novel materials. Progress in the field of molecular beam epitaxy has made it possible to tailor-make layered metallic materials having sharp interfaces, crystalline coherence and superlattice periods of the order of lnm. These materials have opened a new field of magnetism that permits detailed studies of the propagation of magnetic order as a function of separation and crystallographic orientation, as well as studies of the interplay of strain and magnetic properties. A detailed account of achievements on rare earth based artificial multilayered structures is presented in the first chapter of this volume. Magnetostriction refers to any dimensional changes of a magnetic material caused by changes in its magnetic state. Magnetostriction can originate from changes in magnitude or direction of the applied field or from changes in temperature. The former type of magnetostriction is particular pronounced in rare earth compounds of the type RFe2, as has been described in detail in Chapter 7 of Volume 1. The second type of magnetostriction is largest near the Curie temperature in ferromagnetic materials. This volume magnetostriction gives rise to the technically important Invar alloys, and the associated moment-volume instabilities in transition metal alloys have extensively been discussed in Chapter 3 of Volume 5. The large body of experimental results that have become available for the many intermetallic compounds in which

vi

PREFACETO VOLUME 8

rare earths are combined with 3d transition metals is described in the third chapter of the present volume. The ferrites form a large class of magnetic materials and some of these materials are of considerable technical importance. The properties of hard ferrites as well as soft ferrites have been described in several chapters in Volumes 2 and 3. Since the appearance of these chapters substantial progress has been made in the understanding of the physical and chemical properties of these materials which made it necessary to update the results described in the preceding chapters. New results obtained on ferrites are described in Chapter 3, where the emphasis is on spinel ferrites. Of substantial technical importance is, furthermore, the group of so-called soft magnetic materials. A detailed description of several important classes of soft magnetic material has been presented already in Chapter 6 of Volume 1 and Chapter 2 of Volume 2. Supplementary results, dealing mainly with laminated amorphous alloys and electrical steels and the problem of the loss producing effect of the rotational magnetisation are highlighted in Chapter 4. A survey of the magnetic properties of various types of rare earth intermetallics was given already in Volume 1 of the Handbook series. Since then proliferation of scientific results, obtained with novel techniques, and made for a large part on single crystals, have led to a more complete understanding of the basic magnetic interactions in these materials. This requires a major updating of the experimental results presented in Volume 1. However, the experimental and theoretical material that has accumulated over the years is so extensive that it is hardly possible to condense it in a single chapter. In the preceding volume, Vol. 7, supplementary information was presented already for intermetallics in which rare earths are combined with 3d transition metals. In the present volume the updating process has been continued by means of a chapter on rare earth copper compounds of the type RCu2. Volume 8 of the Handbook on the Properties of Magnetic Materials, as the preceding volumes, has a dual purpose. As a textbook it is intended to be of assistance to those who wish to be introduced to a given topic in the field of magnetism without the need to read the vast amount of literature published. As a work of reference it is intended for scientists active in magnetism research. To this dual purpose, Volume 8 of the Handbook is composed of topical review articles written by leading authorities. In each of these articles an extensive description is given in graphical as well as in tabular form, much emphasis being placed on the discussion of the experimental material in the framework of physics, chemistry and material science. The task to provide the readership with novel trends and achievements in magnetism would have been extremely difficult without the professionalism of the NorthHolland Physics Division of Elsevier Science B.V., and I wish to thank Joep Verheggen and Wim Spaans for their great help and expertise. K. H.J. Buschow Van der Waals-Zeeman Laboratory University of Amsterdam

CONTENTS Preface to Volume 8 .

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vii

Contents of Volumes 1-7

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ix

List of Contributors

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xi

1. Magnetism in Artificial Metallic Superlattices of Rare Earth Metals J.J. RHYNE and R.W. ERWlN . . . . . . . . . . . . . .

1

2. Thermal Expansion Anomalies and Spontaneous Magnetostriction in Rare-Earth Intermetallics with Cobalt and Iron A.V. A N D R E E V . . . . . . . . . . . . . . . . . . .

59

3. Progress in Spinel Ferrite Research V. A.M. BRABERS . . . . . . . . . . . . . . . . . .

189

4. Anisotropy in Iron-Based Soft Magnetic Materials M. SOINSKI and A.J. MOSES . . . . . . . . . . . . . .

325

5. Magnetic Properties of Rare Earth-Cu2 Compounds Nguyen Hoang L U O N G and J.J.M. FRANSE . . . . . . . . .

415

Author Index

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493

Subject Index

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521

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527

Contents .

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Materials Index

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CONTENTS OF VOLUMES 1-7 Volume 1 1. 2. 3. 4. 5. 6. 7.

Iron, Cobalt and Nickel, by E.P. Wohlfarth . . . . . . Dilute Transition Metal Alloys: Spin Glasses, by J.A. Mydosh Rare Earth Metals and Alloys, by S. Legvold . . . . . . Rare Earth Compounds, by K . H . J . Buschow . . . . . . Actinide Elements and Compounds, by W. Trzebiatowski . . Amorphous Ferromagnets, by F.E. Luborsky . . . . . . Magnetostrictive Rare Earth-Fe 2 Compounds, by A.E. Clark

. . . . . . . . . . and G.J. Nieuwenhuys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

71 183 297 415 451 531

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

Ferromagnetic Insulators: Garnets, by M.A. Gilleo . . . . . . . . . . . . Soft Magnetic Metallic Materials, by G.Y. Chin and J.H. Wernick . . . . . . . Ferrites for Non-Microwave Applications, by P.L Slick . . . . . . . . . . Microwave Ferrites, by J. Nicolas . . . . . . . . . . . . . . . . . . Crystalline Films for Bubbles, by A.H. Eschenfelder . . . . . . . . . . . Amorphous Films for Bubbles, by A.H. Eschenfelder . . . . . . . . . . . Recording Materials, by G. Bate . . . . . . . . . . . . . . . . . . Ferromagnetic Liquids, by S. W. Charles and J. Popplewell . . . . . . . . .

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55 189 243 297 345 381

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509

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Volume 3 1. Magnetism and Magnetic Materials: Historical Developments and Present Role in Industry and Technology, by U. Enz . . . . . . . . . . . . . . . . . . . . . . 2. Permanent Magnets; Theory, by H. Zijlstra . . . . . . . . . . . . . . . . 3. The Structure and Properties of Alnico Permanent Magnet Alloys, by R.A. McCurrie 4. Oxide Spinels, by S. Krupidka and P Novcik . . . . . . . . . . . . . . . . 5. Fundamental Properties of Hexagonal Ferrites with Magnetoplumbite Structure, by H. Kojima

6. 7. 8. 9.

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Properties of Ferroxplana-Type Hexagonal Ferrites, by M. Sugimoto Hard Ferrites and Plastoferrites, by H. Stiiblein . . . . . . . . . . . . . . . . . . . . Sulphospinels, by R.P. van Stapele Transport Properties of Ferromagnets, by LA. Campbell and A. Fert

ix

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1

37 107 189 305 393 441 603 747

x

CONTENTS OF VOLUMES i - 7

Volume 4 1. 2. 3. 4. 5.

Permanent Magnet Materials Based on 3d-rich Ternary Compounds, by K.H.J. Buschow Rare Earth-Cobalt Permanent Magnets, by K.J. Strnat . . . . . . . . . . . . . Ferromagnetic Transition Metal Intermetallic Compounds, by J. G. Booth . . . . . . Intermetallic Compounds of Actinides, by V Sechovsk~ and L. Havela . . . . . . . Magneto-optical Properties of Alloys and Intermetallic Compounds, by K.H.J. Buschow

1

131 211 309 493

Volume 5 1. Quadrupolar Interactions and Magneto-elastic Effects in Rare-earth Intermetallic Compounds, by P. Morin and D. Schmitt . . . . . . . . . . . . . . . . . 2. Magneto-optical Spectroscopy of f-electron Systems , by w. Reim and J. Schoenes . . 3. INVAR: Moment-volume Instabilities in Transition Metals and Alloys, by E. E Wasserman 4. Strongly Enhanced Itinerant Intermetallics and Alloys, by P.E. Brommer a n d J . J . M . Franse 5. First-order Magnetic Processes, by G. Asti . . . . . . . . . . . . . . . . . 6. Magnetic Superconductors, by 0. Fischer . . . . . . . . . . . . . . . . .

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133 237 323 397 465

Volume 6 1. Magnetic Properties of Ternary Rare-earth Transition-metal Compounds, by H.-S. Li and J.M.D. Coey

2. 3. 4. 5. 6.

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Magnetic Properties of Ternary Intermetallic Rare-earth Compounds, by A. Szytula Compounds of Transition Elements with Nonmetals, by O. Beckman and L. Lundgren Magnetic Amorphous Alloys, by P. Hansen . . . . . . . . . . . . . . . . Magnetism and Quasicrystals, by R. C. O'Handley, R.A. Dunlap and M.E. McHenry Magnetism of Hydrides, by G. VCiesinger and G. Hilscher . . . . . . . . . . .

1

85 181 289 453 511

Volume 7 1. Magnetism in Ultrathin Transition Metal Films, by U. Gradmann . . . . . . . . . 2. Energy Band Theory of Metallic Magnetism in the Elements, by V.L. Moruzzi and P.M. Marcus

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3. Density Functional Theory of the Ground State Magnetic Properties of Rare Earths and Actinides, by M. S. S. Brooks and B. Johansson . . . . . . . . . . . . . . . 4. Diluted Magnetic Semiconductors, by J. Kossut and W. Dobrowolski . . . . . . . . 5. Magnetic Properties of Binary Rare-earth 3d-transition-metal Intermetallic Compounds, by J.J.M. Franse and R.J. Radwahski

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1

97 139 231 307

6. Neutron Scattering on Heavy Fermion and Valence Fluctuation 4f-systems, by M. Loewenhaupt and K.H. Fischer

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503

chapter 1 MAGNETISM IN ARTIFICIAL METALLIC SUPERLATTICES OF RARE EARTH METALS

J.J. RHYNE Research Reactor Center and Department of Physics University of Missouri-Columbia Columbia, Missouri 65211 U.S.A.

and

R.W. ERWlN Materials Science and Engineering Laboratory National Institute of Standards and Technology Gaithersburg, Maryland 20899 U.S.A.

Handbook of Magnetic Materials, Vol. 8 Edited by K. H.J. Buschow ©1995 Elsevier Science B.V. All rights reserved

CONTENTS 1. Introduction to superlattices and rare ealths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Neutron scattering and artificial metallic superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Magnetic scattering, structure, and coherenee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Superlattices with e-axis growth directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Superlattices with a basal plane growth direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Interlayer magnetic coupling in superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Effect of applied magnetic fields on interlayer coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Incommensurate magnetic periodicity - interplanar turn angles, spin slips, and their temperature dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Magnetoelasticity in lare earth superlattices and films and the suppression of ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Theory of magnetoelasticity in superlattices and films . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Magnetoelastie energies in Dy superlattices and films . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Coherent magnetic m o m e n t s and disorder at interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Residual m o m e n t effects in superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. S u m m a r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 6 11 11 26 30 32 35 41 43 44 48 50 54 55 55

1. Introduction to superlattices and rare earths Intense interest has been generated over the past several years in the growth and properties of layered magnetic materials, both from a fundamental point of view and for applications. Layered structures have been prepared by a variety of techniques such as sputtering, electro-deposition, and evaporation, and include semi-conducting, metallic, and insulating materials. These systems can consist of crystalline layers of one element or compound interleaved with layers of a different element or compound, or alternatively may be built of amorphous layers or amorphous layers alternated with crystalline layers. Depending on the materials and growth techniques, these multilayers may be produced (a) with no uniform crystallographic alignment or coherence from layer to layer, (b) with alignment of one specific crystallographic axis direction along the growth (stacking) direction, or (c) with true three-dimensional atomic order (epitaxy) in which there is multilayer atomic registry both along the growth axis and also within the growth planes. For the purposes of this review, the term artificial metallic superlattice will be reserved for this latter category of true threedimensionally coherent layered structures, while the term multilayer will be used for layered structures in which coherence is present in less than three dimensions. Refinements in computer-controlled molecular beam epitaxy (MBE) techniques for the growth of single crystal artificial superlattices of two or more distinct compositions have unearthed vast possibilities for the production of tailor-made artificial superlattices with controlled film thicknesses down to atomic dimensions and with highly reproducible stacking sequences. This has provided previously unavailable opportunities to examine problems of interaction ranges, tunneling distances, and other coherent phenomena which are dependent on the superlattice periodicity. The lanthanide elements and their alloys with non-magnetic but chemically and electronically similar elements such as yttrium, scandium, and lutecium (collectively known as the rare earths), have long provided a fertile area for the study of longrange indirect exchange interactions, crystal field anisotropy, and magnetostrictive effects. The elements have weak exchange compared to the 3d transition elements as illustrated by their low ordering temperatures (e.g., Tb, TN = 230 K) and have anomalously large crystal field and magnetoelastic interactions that arise from the strong spin orbit coupling and highly non-spherical 4f charge distribution. Below their magnetic ordering temperatures, many of the lanthanide elements exhibit one or more forms of periodic incommensurate spin structures such as helices or longitudinal spin density waves, etc., and transitions among them as the temperature is varied. The occurrence of these periodic magnetic orderings below their Nrel

4

J.J. RHYNEand R.W. ERWIN

temperatures in the heavy lanthanides (except for the S state ion Gd) is a consequence of electronic effects, in particular the occurrence of nearly two-dimensional parallel sections (nesting) on the hole Fermi surface that are spanned by a Q-vector whose magnitude determines the initial stable periodicity of the magnetic ordering. At lower temperatures, the free energy associated with the magnetostrictive and anisotropy interactions becomes significant and strongly perturbs the basic periodic magnetic orderings leading to phase transitions to a ferromagnetic state in Tb, Dy, Ho, and Er. These transitions are largely driven by a lowering of the magnetoelastic energy arising from a coupling of the local moments to the hexagonally symmetric lattice strains. It has been suggested by Larsen, Jensen, and Mackintosh (1987) that dipole-dipole interactions of extremely long range may be also responsible for perturbing the e-axis modulated moment states found in Ho and probably Er. The development (Nigh 1963) of techniques for producing high-quality large single crystals of the heavy rare earth elements and alloys provided an opportunity to study the anisotropy in the magnetic properties of the rare earths that led to much of our current insight into the fundamental orderings and interactions in the rare earth metals. For details and references on these works, one is referred to the many review works available, including, but certainly not limited to, Elliott (1972), Gschneidner and Eyring (1979), Coqblin (1977), and the extensive treatise by Jensen and Mackintosh (1991). A major breakthrough in rare earth materials occurred in 1984 with the development (Durbin, Cunningham, and Flynn 1982, Kwo et al. 1985a, 1985b) of MBE growth procedures for single crystal rare earth metal superlattices. These exotic materials have made possible prototypical tests for verifying many of the theoretical concepts of magnetic exchange, anisotropy, and magnetostrictive effects in the rare earths that could not previously be examined in as controlled a way using conventional bulk materials. Superlattices consisting of magnetically concentrated layers (e.g., Dy) interleaved in a controlled fashion with magnetically 'dead' layers (e.g., Y) offer a near-ideal opportunity to investigate these basic interactions. It should be noted that such a system is unique and can never be simulated by bulk dilute alloys because of the attendant reduction in the average exchange interaction with the decreased density of magnetic ions and the probability of some nearest neighbors even in very dilute samples. Y and Lu have similar physical and electronic properties to the magnetic heavy rare earths and good epitaxial growth is achieved because of the relatively small mismatch between the basal plane lattice parameters (e.g., 1.6% for Dy and Y). The key to the growth (see fig. 1) of rare earth superlattices lies in the use of a [110] Nb buffer layer evaporated onto a [1120] sapphire substrate beneath the rare earth metals. This buffer layer prevents a chemical reaction between the sapphire and the rare earths during growth. A strain-relieving Y overlayer is placed between the Nb and the rare earth bilayers. Y and Nb have a nearly perfect 3:4 atomic registration sequence that allows good epitaxial growth in spite of the 33% lattice parameter mismatch. Lattice parameter mismatch is a moderately serious constraint for MBE-produced materials, since this mismatch must generally be taken up by lattice dislocations. As shown in the figure for a [DyIY] superlattice, the thick Y layer is followed by a constant bilayer repeat sequence of I atomic planes of Dy and

MAGNETISM IN ARTIFICIAL METALLIC SUPERLA'ITICES

5

RARE EARTH SUPERLATTICE STRUCTURE bilayer / / ."

Y 100021 Nb [110]

,-

~

/ :- d I ( A / p l a n e )

Y ~ 2oJ e (rlldlans/plane)

2~1

221 B

lCb:'.,277.oo,

soo i - i 5 0 0 ,~

." repeat N times

sapphire

II1~o1

substrate Fig. 1. Schematic drawing of a rare earth artificial metallic superlattice structure [NAINB] consisting of N bilayers each with NA atomic planes of element A and NB atomic planes of a dissimilar element B. The expanded view of a bilayer, of thickness L, lists the physical parameters appropriate to A (and B) layers. (See text.)

m atomic planes of Y. This is designated [DytlYm]N where N is the total number of bilayers. Rare earth superlattices have been produced by similar procedures in the [GdlY] (Kwo et al. 1986, Majkrzak et al. 1986), [DylY] (Durbin, Culmingham, and Flynn 1982, Rhyne et al. 1987, 1989 and references therein, Salamon et al. 1986, Erwin et al. 1987, Borchers et al. 1987, and Majkrzak et al. 1988), [ErlY] (Erwin et al. 1988a, 1988b, Borchers et al. 1989a, 1989b, 1991a, 1991b, Rhyne et al. 1991), [HolY] (Majkrzak et al. 1988), and [DyIGd] (Majkrzak et al. 1986, 1988) systems. This chapter concentrates primarily on experimental and theoretical studies of [Dy[Y] and [ErIY] superlattices and the Sc and Lu counterparts of Y. A review of rare earth superlattice systems with emphasis on [Gd[YI, [HolY], and [GdlDy] materials is found in Majkrzak et al. (1991). The [DyIY], [Er[Y], and counterpart Lu and Sc superlattices were grown at the University of Illinois. In the procedures developed there (Dnrbin et al. 1982), the chamber pressure during evaporation was maintained in the 10 - 9 Tort range to minimize oxidation of the rare earths which were evaporated at a rate of only about 0.5-1.0 A per second with a substrate temperature of approximately 300°C. These conditions were chosen to minimize interdiffusion between the lanthanide layers. Analysis of the neutron or X-ray diffraction data (Botchers 1989b) for the intensity of the bilayer harmonics (see next section) confirms that this interdiffusion is quite minimal and that the composition reaches 85% of the pure element (Y or Er, Dy) within about two planes on either side of the interface. (See fig. 2.) The intensity analysis of the same [Erz3.5[Y19]10o superlattice also confmns that the d-spacing variation is quite abrupt as shown in fig. 2, reflecting the relatively large (2.5%) lattice mismatch between Er and Y.

6

J.J. RHYNE and R.W. ERWIN

© O

t.O

.-4 v

0.6 0 ~3

fl) o

0.2

0 C3 #3

i

-0.2 0.0

I

r

I

20.0

i

I

40.0

I

I

60.0

1

80.0

ALomic P l a n e I n d e x 2.92

o
corresponds to a weighted average d-spacing of the A and B layers of the bilayer. Other peaks (bilayer harmonics) in the neighborhood of the principal ones fall off in intensity as controlled by the relative nuclear scattering amplitudes bA,s, layer thicknesses, and the lattice parameter mismatch. The last quantity is additionally responsible for intensity asymmetry, which can occur for the harmonics on opposite sides of the fundamental peak. In general, the number of harmonics present is an indication of the crystal perfection of the superlattice and the sharpness of the interface boundary. In the extreme limit of a sine wave composition modulation, only one harmonic exists. The addition of incommensurate magnetic order, such as found in Dy and Er, produces additional principal scattering peaks of solely magnetic origin at Q's dictated by the magnitude crf the magnetic propagation vector. The principal satellite peaks occur along the c* direction on opposite sides of the nuclear (000l) peaks (for the helical structures) and are designated Q+ and Q - . They are again individually part

MAGNETISMIN ARTIFICIALMETALLICSUPERLATFICES

9

of an infinite repeating set of magnetic harmonics with amplitudes controlled by factors analogous to the nuclear case - the magnetic scattering amplitudes PA,B (PB = 0 for Y) and the magnetic propagation vectors ~A,B in both A and B layers. Note that even though Y has no 4f magnetic electrons and is only paramagnetic, a spin density wave may be formed in the conduction band with characteristic propagation vector ~B as discussed later. In Er supeflattices, the principal moment component is parallel to the c-axis and thus gives rise to satellites along c* about (h, k, h + k, l) {h, k # 0} class nuclear reflections, but not about the (000l) peaks. (See eq. (3).) As discussed later, at low temperature the transverse moment components order into a helix that then gives rise to weak [000l] ± satellites as well. The neutron scattering intensity for the c* direction (z) for a superlattice can be written as the sum of the coherent nuclear and magnetic scattering terms as:

I(Q) -- Snuc(q) + Smag(+Q) + Smag(-q),

(1)

where the nuclear structure factor has the form:

N_ l bm eiQ.z,, ' 2

s°oo(q) =

(2)

m----O

and the magnetic structure factor can be written:

Smag(+Q) =

N-lprn' ~ l'eiQ'z"~:t=i¢"~m=O

2,

(3)

bm is the nuclear scattering amplitude of atoms in the m atomic plane, PmJ. is the coherent magnetic scattering amplitude proportional to the moment component perpendicular to the scattering vector, and Cm is the cumulative layer magnetic phase shift of the propagating magnetization wave. In the case of an idealized [nAlns]u superlattice with perfectly sharp interfaces, the nuclear scattering intensity is:

Snuc(Q)-

sin2(NqL/2) sin2(nAqdA/2) b2A+ sin2(nBqdB/2) bZB+ sin2(QL/2) { sin2(QdA/2) sin2(QdB/2) + 2bAbB cos (QL/2)

sin (nAQdA/2) sin (nBqdB/2) ~ sin (qdB/2)

}'

(4)

where N is the total number of bilayers each of thickness L = n A d A + nBdB, and d is the atomic d-spacing of the respective A and B layers The same functional form as eq. (4) also describes the X-ray diffraction profile. For a superlattice in which only the A layer contains magnetic atoms (e.g., PB = 0) and for which the total magnetic phase shift across a bilayer is given by # = nA~CAdA + nBXBdB, the corresponding magnetic scattering is:

10

J.J. RHYNE and R.W. ERWlN

sin2[N(QL q- ~ ) / 2 ] s i n 2 [ n A ( Q 4. ~¢A)dA/2] "1 p 2 Smag(q-Q) =

s i n 2 [ ( Q L 4- ~ ) / 2 ]

sin2[(Q 4- tCA)dA/2]

~

2

(5)

Details of the calculation of these structure factors are given by Erwin et al. (1987). Each of the above structure factors consists of a product of terms - the first produces the infinite set of bilayer peaks and the second is a much broader envelope function (i.e., generally nadA , loo 50

o/ 1.80

2.00

2.20

2.40

2.60

Oz (A"1) Fig. 4. Neutron diffraction scans along (0001) in a [DY16[Y2o]89 superlattice for temperatures below TN. Note the temperature independence of the (0002) peak at Q z = 2.215 .~- 1 . The small peak to the right is a bilayer harmonic. The fundamental and two bilayer harmonics are shown for both Q - (~ 2.02 A - l ) and for Q+ (~, 2.42 ,~-]) magnetic satellites. Tc denotes the helical ordering temperature TN = 167 K.

MAGNETISMIN ARTIFICIALMETALLICSUPERLATTICES

11

3. Magnetic scattering, structure, and coherence One of the intriguing concepts in the study of artificial metallic superlattices is the magnetic exchange coupling of spins in one atomic layer to those in adjacent magnetic layers separated by non-magnetic intervening layers. For the rare earth systems, different superlattices have been examined to determine the dependence of the magnetic coherence on (a) the direction of the spin alignment (e.g., basal plane or c-axis), (b) the type of spin structure (ferromagnetic or periodic antiferromagnetic), (c) the direction of the superlattice growth (basal plane or c-axis), and (d) different non-magnetic interlayers (e.g., Y, Lu, or Sc). The first two case studies involve varying the magnetic rare earth element and uniformly result in coherent multilayer coupling. The Dy superlattices discussed in subsection 3.1.1 are examples of basal plane helical spin structures with spins perpendicular to the propagation vector along the e* axis, while [ErlY] superlattices discussed in subsection 3.1.2 are an example of an Ising-like incommensurate antiferromagnetic system with spins principally parallel to the c-direction. [GdiY] superlattices (subsection 3.1.3) are examples of a system with pure ferromagnetic coupling of spins within the layers for which either longrange ferromagnetic or antiferromagnetic interlayer coupling is found, depending on the Y thickness as shown by Majkrzak et al. 1988. The definitive case of rotating the growth axis relative to the spin propagation direction does in fact destroy the longrange magnetic coherence in [DylY] and [Gdly ] superlattices as will be reviewed in subsection 3.2. Changing the non-magnetic interlayer has the effect of enhancing or suppressing transitions between spin configurations as given in subsections 3.1.1.1 and 3.1.1.2, and of suppressing long-range order in the case of Sc (subsection 3.1.1.3).

3.1. Superlattices with c-axis growth directions As discussed above, rare earth superlattices are typically grown with layers stacked up along the c-axis crystallographic direction. Within this category are three distinct classes of magnetic structures. These are (a) magnetic layers in which the spins lie perpendicular to the growth direction and are ferromagnetically aligned in each atomic plane with the alignment direction precessing in a helical fashion from one plane to the next, (b) layers in which the order parameter is aligned along the caxis and is modulated from one atomic plane to the next, and (c) layers for which spins in all planes are ferromagnetically aligned and parallel, but are coupled either ferromagnetically or antiferromagnetically to spins in adjacent magnetic layers. Other more complex structures have been envisioned that are combinations of these three basic types.

3.1.1. Basal plane helical spin configurations 3.1.1.1. Superlattices with Y interlayers. Figure 4 shows the results of a Qz = (O00l) scan around the (0002) principal Bragg peak for a [Dy16]Y20189 superlattice for several temperatures near and below TN = 167 K. Note that there is one temperatureindependent central peak at Q = 27r/(d) ofstructural origin with one harmonic visible and displaced by AQ = 27r/L 27r/(102 A) = 0.0308 A -1. Higher order harmonics

12

J.J. RHYNE and R.W. ERWlN

are not visible on this scale. The magnetic scattering arising from the incommensurate spin structure produces the temperature-dependent Q- and Q+ satellite peaks on either side of the nuclear structure peaks. As discussed later, the ferromagnetic transition that occurs at 85 K in bulk Dy is suppressed in the superlattices. This would have been manifested by additional intensity appearing at low temperatures on the nuclear peaks. The presence of the fully resolved bilayer harmonics on either side of the Q+ and Q - satellites as shown in fig. 4 is dramatic evidence that the magnetic structure is coherent over many supedattice periods and is n o t interrupted at the layer boundaries or by the intervening Y layers. Also important is the fact that the chirality of the helix is maintained across the bilayer boundary. If the magnetic order were confined within single Dy layers, the Q width of the central magnetic satellite peak would encompass the bilayer harmonic positions rendering them unresolved. This is the case found for a superlattice of [DY14]Y34174 (see fig. 5 and later discussion) in which the coherence range has dropped to less than one bilayer due to the increased number of intervening layers of Y. From the scattering data, the coherence ranges of the atomic (nuclear) and magnetic order can be calculated from the intrinsic Q-width of the respective nuclear and magnetic peaks after deconvolution of the instrumental resolution width. The nuclear peaks typically give a coherence range of 500-700 ~. The resulting magnetic coherence distance, after correcting for the nuclear coherence, ranges from 580 ~ in a sample of 46 ~ of Dy separated by 26.5 ,~ of Y, down to 245 ,~ in a 46 ~ Dy, 56 ~ ' Y sample, and finally to 80 ~k in a 40 ~ Dy, 95 ~k Y specimen. All values but the last correspond to the propagation of magnetic phase coherence across multiple bilayer cells. o

[DYl4 ]Y34 ]74 ~(XI-

.'tZ- .,n

%

1 1.80

1.90

2.00

2.10

2.20

2.50

2.40

2.50

Qz Fig. 5. Broadened magnetic scattering peaks from a [Dy141Y34174 superlattice showing the loss of magnetic coherence at larger Y layer thicknesses.

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATTICES

13

r(~,) 2 0 0 I00

600

i

i

50

50

i

~

#

r-' Dependence of Coher

25 12I

500 A

-r 4 0 0 I-Z W

-J 3 0 0 bJ U Z W

~ 200 "Io I00

,,~.. > 1 4 0 A

,~f" 0

0.00

I

0.01

single Dy layer I

I

0.02 0.03 r-I=(y THICKNESS)-I(~, -I )

0.04

Fig. 6. The magnetic coherence length for a series of [DyIY] superlattiees as a function of the reciprocal of the thickness of the Y layer. The number of Dy planes was held approximately constant at 15 4- 1 planes (~ 43 ,~). The plot illustrates the 1/r falloff of the coherence, and shows that the layers would become non-interacting for Y thickness greater than approximately 140 ,~.

The decrease in the correlation length with increasing Y thickness has been examined in a series of [DyJY] superlattices with essentially fixed Dy layer thicknesses of 15 4- 1 atomic planes (~ 43 ~) and varying Y layer thicknesses as shown in fig. 6. The coherence range dependence quite accurately reflects a 1/rv falloff where rv is the thickness of the intervening planes (~, 2.87 ,~ per plane). As discussed later, the exchange coupling in the c-direction is of finite range, and thus this behavior suggests the presence of a decorrelation mechanism (viz., Dy basal plane anisotropy, thermal disorder) that competes with the exchange and leads to the loss of coherence for increasing Y thickness. The coherence length extrapolates to the non-interacting single layer limit of one Dy layer thickness (43 ~ ) at approximately 140 ~ of Y. Superlattices consisting of alternating layers of Ho and Y have also been fabricated and studied by neutron diffraction by Bohr et al. (1989) and by magnetic X-ray scattering by Gibbs et al. (see Majkrzak et al. 1991). The results are similar to those in Dy[Y superlattices and exhibit long-range coherence of the Ho moment across multiple bilayers. At low temperatures additional fifth and possibly seventh harmonics of the magnetic structure were observed that, as in bulk holmium, signal a spin-slip structure or equivalent bunching of the moments induced by the large 6-fold basal plane crystal field anisotropy. The presence of higher harmonics was also confirmed in more recent results on HolY superlattices by Jehan et al. (1993).

14

J.J. RHYNEand R.W. ERWIN

They also deduced from the width of the higher harmonics relative to the first order peaks that there is no spatial coherence between the spin-slip blocks across multiple layers, even though long-range coherence of the helical chirality and average turn angle exists from layer to layer similar to that in DyIY superlattices. The light rare earths (La-Eu) form a series of magnetic elements (except for La) where the magnetic exchange is considerably weaker than in the heavy series and becomes comparable in magnitude to the crystal field anisotropy energies. In addition departures from hexagonal close packed crystal symmetry are found based on an ABAC stacking of hexagonal layers instead of the ABAB sequence for conventional hexagonal close packed structures. This can be described in terms of two distinct atomic sites, one of hexagonal symmetry and one of cubic symmetry. In bulk Nd the ordering is of a multi-q incommensurate structure (see Jensen and Macintosh 1991, and Moon and Nicklow 1991, and references therein) in which the hexagonal site moment orders at 19.9 K with spins along the b crystal direction in an incommensurate modulation. At 8.2 K the cubic site moments order with a distinct incommensurate wave vector. When fabricated as one component of an [Nd[Y] supeflattice, the effects of epitaxial strain produce ordering characteristics in Nd (Everitt et al. 1994) that are quite distinct from those of the bulk element. The Nrel temperature is enhanced by almost 30% in the superlattice and the hexagonal sites order as ferromagnetic sheets in the basal plane with a small sinusoidal component superposed. Even though grown along the c crystal axis, magnetic order is not propagated between layers of Nd in contrast to the heavy rare earth Y superlattices. Below about 6 K order of the cubic site moment was also observed for thin Y interlayer superlattices.

3.1.1.2. Superlattices with Lu interlayers. Superlattices grown with Lu as the nonmagnetic spacer layer [Beach et al. 1993] show similar long-range coupling and loss of coherence as the spacer layer thickness is increased. In dramatic contrast to the [DyIY] superlattices, the ferromagnetic transition is enhanced by approximately a factor of two in [DylLu] superlattices as shown in fig. 7. (Also see plot of turn angles in fig. 25 of section 6.) This effect on the magnetic structure is discussed in section 7. A second difference observed in the [DylLu] superlattices compared to the [DyIY] superlattices is that the long-range coupling through the Lu has a different character that apparently depends on the fact that the Dy layers become ferromagnetic. When the Lu spacer layers are thin the coupling between layers is ferromagnetic, whereas for Lu layer thicknesses greater than approximately 10 atomic planes, antiparallel alignment of the Dy layers is always observed. Also the coherence of the helimagnetic state is reduced to a single layer for a Lu layer thickness of 80 ,~. However, the coherence of the low temperature antiparallel state remains at about two bilayers for this sample. These effects can be explained by a model for the interlayer coupling that includes a competition between indirect exchange and dipolar energies. As shown in section 3.2, the dipolar coupling of 300 ~ domains can approach several kilogauss. This is the approximate size of the field required to align the Dy layers in the antiparallel state, a value which was shown to be nearly independent of Lu layer thickness, as would be the case

MAGNETISM IN ARTIFICIALMETALLICSUPERLATIqCES I

10

I

I

I

t

150 K 170 K

lLu

8-

I

15

Q O o ___

o

4

1.8

2.0

2.2

2.4-

2.6

ez Fig. 7. Diffractionscans for a [DY21lLul0] superlattice show that below 160 K the helical magnetism of the Dy layers is transformed into ferromagnetismwith antiparallel alignment of the layers. The peak widths demonstrate that the helimagneticstate is of shorter range than the ferromagneticstate, and there is some remanence of the helix in the ferromagnetic state. The centroids of the magnetic scattering at 170 K and Q ,,~ 2.05 ,~-1 and Q ,,~ 2.4 ~-1 determine the turn angle in the Dy layers. for dipolar coupling. The domain size has been confirmed from X-ray diffraction peak widths of basal plane reflections in the ferromagnetic state. This domain size is limited by epitaxial strain energies as the lattice obtains a local orthorhombic distortion to minimize the magnetoelastic energy. If the domain size becomes too large, the energy minimizing distortion would force a large number of energetically costly atomic dislocations. The modification of the magnetic structures in the superlattices compared to the bulk magnetic structures provides mfique insights into the physics that controls these magnetic structures. The ferromagnetic transitions are altered as well as the temperature dependence of the turn angles that describe the magnetic structures. These magnetic structures are summarized for Dy-based materials in section 6. The classical theory of the ferromagnetic transition in Dy will be modified to account for epitaxial strain in section 7. Theories of the temperature dependence of the turn angle have concentrated on superzone-gap effects. The data for [DylY ] and [DylLu ] superlattices indicate a direct relation of the turn angle to the strains in the system.

16

J.J. RHYNE and R.W. ERWIN

While strain effects might be included in the band theories of the temperature dependence of the turn angle, it is interesting that the superlattice data can be described by a simple phenomenological theory. It is also significant that strain produces strong effects on the turn angle at the initial ordering temperature. 3.1.1.3. Superlattices with Sc interlayers. Scandium, like Y and Lu, is a non-magnetic hcp metal with electronic and physical properties similar to the heavy rare earths, and Sc forms solid solution alloys with them. However, in marked contrast to Y (e.g., Gotaas et al. 1988), dilute alloys of the heavy metals with Sc do not show longrange magnetic order (Child and Koehler 1968) below concentrations of 20-30% of the magnetic rare earth. The lower concentration alloys instead exhibit spin glass behavior. Superlattices of Dy layers alternated with Sc layers have been prepared and studied by magnetization and neutron diffraction by Tsui et al. (1993). The 8% basal plane lattice mismatch between Dy and Sc presented a major obstacle to production of highquality superlattices that was solved by reducing the deposition temperature and rate. A superlattice of [DY25 AIScao /~]66 was produced with crystal coherence of about 500 ~ along the c growth axis. Neutron diffraction scans along c* are shown in fig. 8 8000

I

o Sc (0002)

l ~ 400C

.0

2.3

2.6

Qc~(~-I) Fig. 8. Neutron diffraction scans along the [0002] direction from a [Dyz5 i]Sc4o A]66 superlattice: (a) Nuclear intensity at 160 K showing five structural superlattice sidebands and a (0002) reflection from the Sc buffer layer, (b) a zero field scan at 10 K showing the short ranged ferromagnetic order along the growth direction as indicated by the thick line underneath the structural superlattice peaks which remains unchanged, and (c) zero-field-cooled scan at 10 K with a 60 kOe field applied along the a-axis showing the magnetic superlattice intensities on top of the structural peaks. This latter result indicates a coherent ferromagnetic order with vanishing short-range order. Scans (b) and (c) are displaced by 2000 and 4000 counts for clarity. Lines through the data points are Gaussian fits, and arrows indicate superlattice reflections.

MAGNETISM IN ARTIFICIALMETALLICSUPERLA'ITICES

17

at 160 K (a), which is above the magnetic ordering temperature, and at 10 K (b). The observation of superlattice sidebands in (a) attests to the quality of the superlattice; however, there is no significant change in or additional peaks on cooling to 10 K (b) which would signal long-range magnetic ordering. Rather, as shown by the heavy line in (b), the magnetic scattering appears as a broad diffuse background about the nuclear peak positions, characteristic of short-range ferromagnetic correlations. The width of the peak corresponds to a coherence range of 24 ± 3 ~, which is close to the Dy layer thickness of 25 ~ indicating that the Dy layers individually order ferromagnetically but with no interlayer coupling. The ferromagnetic peak intensity decreases at elevated temperatures and becomes broader. The onset of irreversibility in the bulk magnetization data indicates a Te of 147 K considerably above the To of bulk Dy, reflecting strain effects as discussed in section 7. Figure 8c shows the diffraction results of application of a 60 kOe field in the basal plane to align the spins in different Dy layers. The structural peaks show markedly enhanced intensity and the broad diffuse peak is essentially removed, both corresponding to the long-range coherence induced by application of the field. The lack of long-range order in the c-axis Dy-Sc superlattices, as well as in the dilute alloys of Dy and Sc, suggests that the features of the generalized susceptibility x(q, ~) in Sc may be very different from those in Y or Lu.

3.1.2. c-axis modulated spin systems Superlattices consisting of alternate layers of Er and Y represent a more complex magnetic ordering than the [DyIY] helical and [GdIY] antiferromagnetic or ferromagnetic configurations. The c-axis is the favored moment direction in Er due to a change in sign of the 4f electron quadrupole moment compared to Dy. Thus the spin alignment is parallel to the stacking direction of the superlattice and also parallel to the c-axis magnetic propagation vector. In bulk elemental form (Cable et al. 1965, Habenschuss et al. 1974), Er initially orders at 84 K into a c-axis modulated moment structure in which the parallel (c) component of the magnetization has a sine-wave amplitude modulation with a period of approximately seven atomic layers. The wave vector of the modulated structure progresses through a series of lattice-commensurate lock-in states with decreasing temperature as derived by Gibbs et al. (1986) and discussed in section 6. The transverse moment components in Er are disordered down to 56 K and were originally considered (Cable et al. 1965) to exhibit helical ordering at lower temperatures. More recent neutron scattering data, supplemented by detailed calculations of exchange and two-ion couplings (Jensen and Cowley 1993), show that the intermediate temperature state (18 K < T < 56 K) is described by a wobbling cycloidal ordering in which there is a b-axis moment perpendicular to an a--c plane cycloid that oscillates with a different period from the basic structure. Below 18 K the c-axis order becomes ferromagnetic, resulting in a conical moment state with apex angle about 280 and with a bunching of moments around the a-axis directioxts. The scattering geometry for the Er-Y superlattice studies (Erwin et al. 1988a, 1988b, Borchers 1989a, 1989b, 1991a) is somewhat more complex, in that predominately basal plane reflections, for example [10i0], must be used to detect the c-axis

18

1 I RHYNE and R.W. ERWIN

moment components, since the neutron scattering is sensitive only to moment components perpendicular to Q. Basal plane ordering was detected as previously by re-orienting the sample with [0002] parallel to the scattering vector. Figure 9 shows temperature-dependent scans along c* at (1010) and along c* at (0002) (fig. 10) from a [Er32[Y21] superlattice demonstrating that well-defined nuclear peaks and harmonics as well as magnetic satellite peaks and harmonics are observed. The magnetic peak widths again confirm that the magnetic order is long range. The reduced intensity of the basal plane nuclear satellites results from the near equality of Er and Y scattering lengths and possible interface defects resulting from the nearly 3% lattice mismatch. The TN for the superlattice, marked by the onset

/

!

""

70K

"

65K

250 "~

200

C

--

150

.~

ioo

c

__c

,

5O

,20K

,6K

-0.5

-0.5 -0.1

0.1

0.5

0.5

K z ( A -I )

Fig. 9. Diffraction scans along the c* direction through (10i0) for [Er32]Y21] at several different temperatures showing the development of a linear spin density wave with principal moment component along the c-axis. At lower temperatures the ordering becomes more complex as indicated by the appearance of higher-order harmonics [e.g., (1010)4"3 harmonic of the (1010) reflection.

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATI'ICES

19

70

60

5O C

~- 4 0 ~ 3O 55 K

20

I0

'

/ - - - - " - ' - '

/

20K

61< 1,8

2.0

2.2

2.4

2.6

Fig. 10. Diffraction scan through (0002) for [Er321Y21]. Below about 30 K, the basal plane component of the moment exhibits periodic order as indicated by the satellites of (0002) 4-.

of magnetic satellites in the (10i0) scans, is 78 4- 1 K which is 7% lower than for bulk Er and similar to the reduction seen in Dy superlattices. The initial ordering is nearly sinusoidal for the c-axis moment components with the transverse components disordered. Below 35 K, additional magnetic scattering appears which can be indexed as higher harmonics [e.g., (1010) +3 and (1012) -3,-5] of the fundamental magnetic satellites (101 l) ±. These reflect the presence of a more complex intermediate spin state as seen in bulk Er, but at a correspondingly lower temperature than in the bulk. Below about 30 K, the basal plane moment components order into a periodic state; however, no conical ferromagnetic ordering is observed down to 5 K in contrast to bulk Er. The magnetic coherence range derived from the magnetic peak width, after

20

J.J. RHYNE and R.W. ERWIN

,-... 600 i Z

~ 400 W 0 Z W W -r

o o 200 Z

[Dy, I Y,]

0;

I

[] I

I

I

I

10 20 Y THICKNESS (plones)

i

30

Fig. 11. Coherence length for [Er321Y21 ] and [ErlaIY26 ] superlattices derived from both basal plane and c-axis moment component satellites. The basal plane values from the [DyIY ] superlattices are shown for comparison, along with 1/,rv and 1/r 2 fits to the [DyIY] data.

deconvolution of the instrumental resolution, is shown in fig. 11 as a function of the Y thickness. It is noted that the coherence range derived from the separate widths of basal plane and c-axis moment components is different. The figure shows the coherence range for [DyIY ] superlattices for comparison. The basal plane coherence length is comparable to that found for [DyIY ] superlattices, while the c-axis values are relatively larger. It is noted that the Er layer thicknesses were not held constant as was the case for the [DyIY ] superlattices in fig. 6.

3.1.3. Ferromagnetism and antiferromagnetism in Gd-Y superlattices 3.1.3.1. Polarized neutron scattering. Superlattices consisting of alternate layers of Gd and Y have been studied extensively with neutron scattering by Majkrzak et al. (1986, 1988, 1991). Gadolinium is an S-state ion and does not possess the Fermi surface nesting features that lead to an incommensurate periodic magnetic structure and is thus ferromagnetic in bulk form. A superlattice of alternating Gd and Y layers would then be expected to show a collinear alignment of Gd spins on opposite sides of the Y layer, assuming long-range order to be developed. In contrast to superlattices of the other heavy lanthanides with yttrium in which the magnetic and nuclear scattering are well separated in Q due to the incommensurate structure, the ferromagnetic coupling expected for some Gd-Y multilayers virtually necessitates the use of polarized beam techniques such as used in reference (Majkrzak et al. 1986) and also in the studies of Gd-Dy superlattices described in reference (Majkrzak et al. 1988). In the polarization analysis, four structure factors can be

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATTICES

21

derived, corresponding to the combinations of neutron spin-flip and non-spin-flip scattering as follows (Majkrzak et al. 1986): (A) non-spin-flip scattering: N

F+±(Q) = - y'~(-bj 4- pj cos ¢)e iqu~,

(6)

j=l

and (B) the spin-flip scattering: N

F±a:(Q) = - y~(pj sin ¢)eiQ~'j,

(7)

j=l

where ¢ is the angle between the spin direction and the neutron polarization which is parallel to the applied guide field. The variable uj is the position of the jth atomic plane along the c* stacking direction. The non-spin-flip scattering thus contains information about the alignment of the spins along the applied field (polarization) axis, while the spin-flip scattering contains information about only spin components perpendicular to the field direction. Performing the lattice sums for the multilayer results in expressions for the spin-flip and non-spin-flip cross sections analogous to eqs (4) and (5) that then contain terms separating out the magnetic only scattering amplitudes. Using the same reflection geometry described above for the Dy-Y multilayers, Majkrzak et al. 1986 obtained the intensity distribution in fig. 12 (plotted on a logarithmic scale) for a [Gd10[Y101225 superlattice. Non-spin-flip magnetic harmonic peaks were observed at spacings AQ = 27r/L (L = bilayer thickness) corresponding to long-range ferromagnetic alignment of spins in the Gd layers and are designated by the even numbers in the figure. However, additional peaks arising from spinflip scattering were observed at half this Q separation and thus correspond to an antiferromagnetic alignment of spins on opposite sides of the Y interface. The spin alignment is then an antiphase domain structure as shown in fig. 13 where the canting angle (90 o - e ) is induced by the 150 Oe field applied as shown at 75 K. In contrast to these results for repeated bilayers of 10 planes each of Gd and Y, other Y thicknesses, in particular 6 and 20 planes, showed no evidence for the antiferromagnetic coupling of alternate Gd layers. Rather they exhibited a ferromagnetic coupling of Gd spins on opposite sides of the Y layer at all temperatures below Te. Further confirmation of the unique ferromagnetic-antiferromagneticexchange coupling oscillation in the [GdlY] superlattices was provided by studies of a [GdsIY~IGdsIY10] quadlayer as discussed by Majkrzak et al. (1991). Neutron diffraction studies indeed showed reflections confirming the parallel coupling of spins between Gd layers separated by 5 atomic planes of Y and antiferromagnetic coupling of spins between Gd layers separated by 10 atomic planes of Y. Thus the exchange oscillations found in discrete single-period superlattices were confirmed in a superlattice of quadlayers for which the magnetic unit cell was found to be twice the length of the chemical unit cell.

22

J.J. RHYNE and R.W. ERWIN

r

r

!

(0002)

40 I- HSLR(III)- 801-S-20 / k,=2.67 ~-' il T= 150K H= 150 Oe ] 104

-s t£)

103

t6 ~3

g 8 $-I H 4I I H

g -

J 5

!

102 i

1.600

,.900

2.200

2.500

0(~.-') Fig. 12. Polarized neutron scattering data from a [Gda0JY101225 superlattice. The odd numbered peaks represent spin flip scattering and correspond to antiferromagnetic coupling of the Gd spins across the Y layers. The even numbered peaks are non-spin-flip scattering and represent the ferromagnetic coupling of the spins within a Gcl layer (after Majkrzak et al. 1986).

Yafet et al. (1988) have calculated the exchange coupling across Y layers of various thicknesses using an indirect exchange model that is based on the generalized conduction electron susceptibility calculated by Liu, Gupta, and Sinha (1971). This model correctly predicts the crossover from ferromagnetic to antiferromagnetic spin configurations with Y thickness as shown in the inset to fig. 13. 3.1.3.2. Bulk magnetization. The bulk magnetic response of the Gd-Y superlattices has been studied by SQUID magnetometry. As shown in fig. 14, the magnetization of a [GdsJYs]80 superlattice (Kwo et al. 1985b) with the field applied parallel to the film plane shows a square loop behavior, while that with the field applied perpendicular to the film plane shows a near linear field dependence. This is the behavior expected for a thin film with relatively small anisotropy. The loop feature at low field is not explained and was not observed in other superlattices. The large slope of the Hparallel curves above technical saturation Hs reflects the induced magnetization of disordered spins in the interface region as discussed in section 8. The spontaneous magnetization or(0) determined from the post-saturation magnetization extrapolated back to H = 0 is significantly lower than the 250 emu/g value expected for ferromagnetic Gd +++, again the result of the interface disorder. The spontaneous moment is observed to

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATIqCES

(+)

-,

~

. . . . . . . . .

0 ...........

(-~

~i

S .

.

.

23

.

.

.

~ .

.

~ .

.

.

.

.

J

. . . . . . . . . 4. . . . . . . . . . .8. . .

12

t6

Ny Gd

XSL

Y +~

Q'II C-AXIS

Fig. 13. Schematic representation of the antiphase domain configuration of Gd moments in the [Gd101Y101225 superlattice at 75 IC The angle c is approximately 80*, and its departure from 90 o is a consequence of the 150 Oe applied field. The inset at the top illustrates the oscillatory ferromagnetic and antiferromagnetic spin couplings across the Y layer depending on the thickness of Y. The solid line is the result of an indirect exchange calculation of Yafet et al. (1988) (after Majkrzak et al. 1986).

depend linearly on the inverse of the number of Gd planes in each bilayer of the superlattice as shown in fig. 15a, which includes data for a Gd fihn. In addition to the increase in the spontaneous moment toward the bulk value for thicker Gd layers, there is a concomitant reduction in the induced moment A~r = ~r(15) - cr(0), where tr(15) is the measured moment at 15 kOe. The quantity A~r is also proportional to the inverse number of Gd planes per bilayer as shown in fig. 15b and is essentially 0 for a pure Gd fihn. The oscillation in the interlayer exchange coupling with increasing number of Y intervening planes in [GdIY ] superlattices was observed definitively in the neutron diffraction studies described above. As shown by Kwo et al. (1986), the property is also manifested in the bulk magnetization, particularly the remanent magnetization err and the field Hs required for moment saturation defined as the field at which the

24

J.J. RHYNE and R.W. ERWIN

I

I

I

l-

I

T

ZOO

r

.

r -----

i

---t

HII

I00

.[ o

I

i

,1

I 6

12

q.) v

b -100

-200 I

a

I

I

t

H

J

I

t

I

(kOe)

Fig. 14. Magnetization data for a [GdsIY518o superlattice for field applied parallel to the film plane HII and perpendicular to the plane Ha. (after Kwo et al. 1985b).

300 i 0

l

----_____

t

I

--.-_____

200

E Q; b

I00

I

I

0

0.1

(O)

I

0.2

t/NGd Fig. 15. (a) Linear dependence of the spontaneous moment 0-(0) and (b) the excess moment A~r = o'(15) - ~(0) on the inverse of the number of Gd planes in a bilayer of superlattices with 5, 10, and 20 Gd planes. Data for a Gd film are also shown. The Curie temperatures of the superlattlces deviate very little from the bulk value of 293 K except for the 5 plane sample that shows a lower Tc possibly from alloying effects (after Kwo et al. 1985b).

MAGNETISM IN ARTIFICIAL METALLIC SUPERLA'ITICES

25

Y THICKNESS (,~)

0

20

40

60

8O t

.od=4

,

.od--lo+-t

"I,O-

s

!÷i I I

I I

/5 ÷

i i

I I

%'

e

1

/ \ /~ ,

5

i

0

I

I

i

:,,

p

t

t0 20 N y ( ATOM IC LAYERS)

30

Fig. 16. The oscillations in [top] the normalized remanent magnetization ~rr/O'(0), and [bottom] the field for technical saturation Hs plotted as a function of the number of intervening Y layers for [GdIY ] superlattices with both 4 and 10 atomic planes of Gd. The out-of-phase oscillations reflect the oscillations in the interlayer exchange coupling as illustrated in the inset of fig. 13 (after Kwo et al. 1985b).

rapid initial rise in the moment for O'parallelis complete. The definitions of Hs and o"r can be visualized from the HII curve of fig. 14. In fig. 16 the remanent magnetization ~r/~r(0) (normalized by the spontaneous magnetization) and the saturation field Hs are plotted for a series of [GdlY] superlattices as a function of the thickness of the intervening Y layer, Ny. Remarkably, these two quantities show out-of-phase oscillations with Ny and have identical values for superlattices with both 4 and 10 atomic planes of Gd. A comparison of these curves with the coupling oscillations shown in the inset of fig. 13, for which the solid line gives the result of the calculation by Yafet et al. (1988) and the solid points are experimental results from the neutron diffraction, shows a consistent behavior with the number of Y atomic planes.

3.1.4. Superlattices with two magnetic layers The helical ordering of magnetic layers and the long-range interlayer phase coherence in D y - Y superlattices are well confirmed as reviewed in previous sections. Likewise, in Gd-Y superlattices, the occurrence of ferromagnetic layers and an oscillation of

26

J.J. RHYNEand R.W. ERWIN

the interlayer coupling is fully established. Superlattices made up of alternate layers of Gd and Dy would then be expected to show a highly complex ordering as found by Majkrzak et al. (1988, 1991) from neutron diffraction studies. The bulk magnetization results show that as the temperature is lowered, the Gd layers first become ferromagnetically aligned (To ~ 250 K for [GdsIDys], reduced from the 293 K of bulk Gd by interface alloying). At about 200 K the Dy layers begin to order non-colinearly. As shown from the neutron diffraction (Majkrzak 1988), the turn angle of the Dy is significantly perturbed from the bulk values, showing a tendency toward ferromagnetic alignment for planes adjacent to the Gd layers with a gradually increasing magnitude toward the center of the Dy layers. For only 5 Dy planes, the distorted ordering is essentially a fan structure or incomplete helix. Gd-Dy superlattices with thicker Dy layers show a somewhat distinct ordering described as a 'superspiral.' Neutron data of Majkrzak et al. (1988) confirm that the Gd planes of a [GdsIDYlo] superlattice initially order ferromagnetically, and then below 200 K the onset of Dy ordering induces a change in the direction of adjacent Gd layers. Initially parallel spins in adjacent Gd planes are found to rotate by about 900 (angle specific to this superlattice) as a result of the influence of the intrinsic Dy helical ordering. The turn angle of the Dy helix is in turn perturbed by coupling to the Gd spins, and is again found to vary from near 0 (ferromagnetic) for spins near the Gd-Dy interface to essentially the bulk Dy value in the center of the Dy layer. Superlattices of alternate layers of Dy and Er present the opportunity to study the interacting order of two systems that intrinsically both show incommensurate magnetic structures of different types. A series of c-axis [DylEr ] superlattices has been studied by magnetization and by X-ray (Dumesnil 1994a) and neutron diffraction (Lee 1994 and Dumesnil 1994b). In these results distinct ordering temperatures were found for the two dissimilar magnetic layers. The initial ordering temperature is close to TN of bulk Dy and involves helical ordering of the Dy spins with no involvement of the Er 4f spins. Spiral coherence is still propagated across multiple bilayers via the ostensibly paramagnetic Er layers in a manner similar to Y superlattices. At temperatures below 100 K the Dy helical order is gradually supplanted by a ferromagnetic intralayer moment structure. The coupling between layers is antiferromagnetic, except for very thin Er layer thicknesses (~< about 15 ~) in which the interlayer coupling is ferromagnetic. At temperatures below the TN of bulk Er, incommensurate ordering of the c-axis component of the Er spin structure was observed, and at 10 K evidence was found of basal plane moment components also ordering in an incommensurate structure.

3.2. Superlattices with a basal plane growth direction 3.2.1. Helical spin configurations As discussed in the previous subsection, the occurrence of long-range coherent magnetism in e-axis rare earth superlattices is independent of the primary orientation of the spins and independent of whether the spin order is ferromagnetic or incommensurate antiferromagnetic. All of these systems have the common feature that the superlattice growth axis is along the c* propagation direction of the spin system.

MAGNETISM IN ARTIFICIALMETALLICSUPERLATTICES

27

In order to test the dependence of the ordering on the superlattice growth direction relative to the c* magnetic propagation vector, superlattices were prepared by Flynn et al. (1989) with the growth axis along a basal plane direction. This required new preparation procedures due to the unavailability of conventional substrate materials with an appropriate lattice matching to the a-c plane of the hexagonal rare earths. Highly polished single crystal slabs of Y were used as a substrate material with the superlattice of alternate Dy and Y layers grown on top, following a thick Y buffer layer. In the neutron scans along the a* (growth axis) direction, the near coincidence of the Y lattice parameters and that of the superlattice prevented the direct observation of the (0002) structural peak. Instead, as illustrated in fig. 17 for a [Dy261Yg]s2 superlattice, the bilayer harmonics of the (0002) (enhanced by applying a magnetic field which produced ferromagnetic ordering) were scanned and their width confinned the quality and chemical order coherence of the superlattice as shown by the (Q, O, 2.221) nuclear harmonics of near resolution width. Below 170 K, helical magnetic ordering was found in the basal plane of the Dy layers; however, as shown in I

60-

160

09

I

I

I

[DY291Y1°]

I

(qb 0 2.221)

120

CO

Qu 0 1.99!

80 O C)

40

=

I

-0.12

27t/~

!

=

0.078

A -t !

r

-0.04

0.[)4

Q (A -I)

0.12

Fig. 17. Magnetic and structural peaks of the b-axis superlattice [Dy26[Yg]82scanned along the a* direction. The narrow resolution-limitedpeaks are the bilayer harmonics of the (0002), and broad peak is the helimagnetic satellite. The width of this peak corresponds to a coherence range nearly identical with the Dy layer thickness showing the loss of long-range magnetic coupling.

28

J.J. RHYNE and R.W. ERWIN

the figure the (0, 0, 1.995) Q - magnetic satellite scanned along a* is quite broad and there are no resolved bilayer harmonics of the magnetic ordering. Indeed the intrinsic width of this peak, after deconvolution of the resolution broadening, is ~ = 80 ~. This is approximately the 7 3 / ~ thickness of the Dy layers and is compelling evidence that the coherence between the helices in adjacent Dy layers is destroyed in these basal plane superlattices. This result thus established that the multiple layer coherence indeed depends on the magnetic propagation vector (c') being parallel to the growth axis. For the b-axis growth superlattices, the helical order which develops within each Dy layer also is observed to have finite range. Transverse scans show that the width of the magnetic satellite peaks along the (11"~0) direction is essentially resolution limited (~ 1> 500 ~), indicating that the ferromagnetic sheets in the basal planes of each layer are uniformly ordered and probably limited in extent only by the sample size (along a) and by the layer thickness (along b). However, the coherence range measured along the c* in-plane direction is significantly smaller and decreases with temperature to about 250 ~ below 100 K. Thus the helical stacking of the a--b ferromagnetic sheets within each layer is within a finite and temperature-dependent domain size. Similar results for the range of magnetic coherence were obtained for a [DyTIY25167 basal plane superlattice. I I I l - 6 0 0 - ~) Cd(43A)/Y(52A) ,

-

2oo

I

•%1~1200

I

I

b) (~d(OOA)/V(ZGA)

4OO

Q× (~-') Fig. 18. (a) Neutron diffraction scans at 80 K (circles) and at 315 K (triangles) along the b* direction for a [Gd43 AIYs2 :,]85 superlattice with a b-axis growth direction. At 80 K peaks develop with twice the bilayer periodicity indicating that the Gd layer moments are anti-aligned. (b) Scans along the b* direction for a [Gd6o :,1Y26 :,]so superlattice at 90 K (circles) and at 315 K (triangles). The extra intensity evident at the superlattice satellite positions at 90 K indicates that the Gd layer moments are ferromagnetically aligned.

MAGNETISM IN ARTIFICIALMETALLICSUPERLATTICES

29

3.2.2. Dipole-dipole coupling in Gd-Y superlattices The coupling in Gd-Y superlattices with a c-axis growth direction has been observed to oscillate with the thickness of the Y layer with a period of approximately 25 A as reviewed in section 3.1.3. The behavior is consistent with a simple RKKY indirect exchange model of the interlayer coupling. This coupling should decay quite rapidly along the basal plane directions as was observed for b-axis Dy-Y superlattices (see section 3.2.1). Neutron diffraction studies of two Gd-Y superlattices grown along the b crystallographic direction are shown in fig. 18. Scans along the a* direction above Tc for both [Gd43.~IY52/~]85 and [Gd_60.~lY26/~]80 show the presence of structural harmonic peaks separated from a (1010) dominant central peak arising from the Y substrate by AQ = 21r/L where L is the bilayer thickness. Below Tc, the superlattice with the thicker (52 .~) Y interlayer shows new peaks with a periodicity of 2x AQ reflecting an antiparallel alignment of Gd moments across the Y interlayer. In contrast, 80 K data (below Tc) on the superlattice with the thinner 26 ,~ Y interlayer show extra intensity developing on the structural harmonics, indicating a parallel alignment of Gd moments between layers. At elevated temperatures (> 120 K) the structure reverts to antiferromagnetic alignment. Studies of the external magnetic fields required to flip the antiferromagnetic spins into parallel alignment yield a coupling strength of 80 and 20 G for the thick and thin Y interlayer superlattices, respectively, which is significantly smaller than the critical fields measured in c-axis Gd-Y systems (~ 1 kG) (Majkrzak et al. 1986, Kwo et al. 1986). This sharp contrast in flipping fields for b-axis and c-axis samples again reflects the anisotropic nature of the exchange interaction discussed in section 4, which suggests that the range of the coupling along basal plane directions is smaller by a factor of at least 10 compared to the c-axis direction. The fact that b-axis Gd-Y superlattices do retain an antiferromagnetic interlayer coupling while b-axis Dy-Y superlattices show no interlayer coupling requires additional explanation. The basal plane helical order of Dy with propagation direction along c* leads to a complex spin configuration with spins at various angles to the layer interface. Such a structure has potential spin frustration and boundary effects that may adversely affect spin ordering compared to the relatively simple ferromagnetic or antiferromagnetic spin configurations found in the Gd-Y superlattices where the spins are all parallel to the layer interface. The Gd-Y case has a clear analog with antiparallel dipole coupling between two ferromagnetic sheets. The critical coupling field for such a dipole configuration is: He = 4M W In

tGa +

ty

'

(8)

where M is the magnitude of the Gd moment, W is the lateral width of the Gd layers, and tC,d and ~y are the Gd and Y layer thicknesses. The fields required to flip the antiparallel layer spins into ferromagnetic alignment as calculated from this expression agree quite well with the experimentally determined fields given above assuming intralayer domains of approximately 4000 .~ dimensions. The transition from parallel to antiparallel interlayer coupling with Y interlayer thickness in Gd-Y

30

J.J. RHYNE and R.W. ERWIN

b-axis superlattices can then be rationalized as arising from a relatively short-range ferromagnetic RKKY exchange contribution and a longer ranged antiferromagnetic dipole coupling favoring antiferromagnetism. In simplest form, both these interactions should exhibit an r -3 fall off; however, the RKKY mechanism may be subject to more complex electronic and mean free path damping effects which can further limit the range of the exchange coupling, but would have no effect on the dipole coupling. 4.

Interlayer magnetic coupling in superlattices

The above results illustrate that for periodic moment systems, long-range interlayer magnetic coupling is present in superlattices for which the stacking direction is parallel to the c-axis propagation direction of the periodic magnetic system and that such coupling is destroyed for stacking sequences along basal plane directions. The interlayer exchange coupling is not simply ferromagnetic or antiferromagnetic (except for Gd and Lu superlattices) but is of a more complex form as can be demonstrated from the effective phase shift of the magnetic ordering across the Y or Lu layers. This phase shift can be calculated from the asymmetry in the intensities 6.00-

5.50 Y layer [DY16 1Y2o ]sg 5.00

4.50

LAYER TOTAL PHASE SHIFT

tl.

o ¢n 4.00

3.50

3.00

2.50

2.00

-

~

Y layer [DYt6 I Y9 1100

T

40

-w

80 T(K)

,

,

120

160

Fig. 19. The total phase shift of the spin density wave across Y and Dy layers as a function of temperature for [DYl6]Y20]89 and for [Dy161Yg]loo. Note that the Y phase shift is not a multiple of lr and is independent of temperature. The variation in the Dy shift reflects the T-dependence of the turn angle. (See text.)

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATFICES

31

of the magnetic satellites Q - and Q+ using eq. (5). This has been done for the [DyIY] superlattices, and as shown in fig. 19, the phase shift is not a multiple of ~r (as for purely ferromagnetic or antiferromagnetic interactions), but is completely prescribed by the number of interleaving Y atomic planes. The Y layeroPhase shift corresponds to an effective turn angle of 51-520 per Y layer (~ = 0.31 A-I). The theory by Yafet et al. (1988), invoking only the RKKY interaction between Dy layers, successfully predicts chiral coherency and the correct order of magnitude for the interlayer interactions (as measured, for example, by the magnitude of the applied field required to break down helical order). However, in this calculation the Y turn angle is not a constant independent of Y thickness. The experimental results suggest that the magnetic structures are determined by the separate conduction electron susceptibilities of each layer material. The complexity of the ordering in [Dy[Y] and [ErIY] superlattices and its dependence on propagation direction has led to the suggestion (Rhyne et al. 1987) that the mechanism behind the long-range spin coupling is the stabilization of a spin density wave (SDW) in the Y and 4f lanthanide conduction bands via RKKY coupling to the 4f local moments in the lanthanide. In linear response theory, the real space exchange coupling J(R) can be expressed (see Elliott 1972) in terms of a Fourier transform of a q-dependent exchange j(q): qmax

J(R) = ~ j(q)e -iq'n,

(9)

q=O

where j(q) is in turn proportional to a conduction electron generalized susceptibility x(q) through an exchange matrix element Jsf(q) in the following form:

j(q)-= Ij~f(q)12x(q)/2, x(q) has

(10)

been calculated for Dy and Y from the band structure (Gupta and Freeman 1976, Liu, Gupta, and Sinha 1971). This function exhibits strong positive maxima along the c* direction at q = qmax ~ 0 (q = reduced wave vector), where qmax is only minimally different for Dy and Y and is in general prescribed by 'nesting' features of parallel sheets of the Fermi surface (Keeton and Loucks 1968). In Dy and other lanthanide elements this calculated wave vector qmax is also very close to the measured magnitude of the helical wave vector at the initial ordering temperature. In the a* and b* directions the peaked behavior at q 5~ 0 is not observed and j(q) falls off in a manner that reflects the core size and can be approximated by a Gaussian of width 0.63 ]k-1. Flynn et al. (1989) derived the three-dimensional representation for j(q) shown in fig. 20a. by using the Gaussian form in the basal plane direction and the expression for x(q) calculated by Liu, Gupta, and Loucks (1971) for the c* direction. The real space exchange J(R) calculated from eq. (9) is also shown in fig. 20b, along with the overall envelope function for the exchange. The salient feature of this calculation is that the range of the real space exchange coupling is

32

J.J. RHYNE and R.W. ERWIN

j (F)

Fig. 20. (a) Schematic representation of j(q) as discussed in the text for which the width along a* reflects mainly the exchange matrix element while that along c* reflects x(q). (b) The envelope function for the Fourier transformreal space exchange, showing the highly anisotropic spatial range. The oscillatorycurve is the actual transformfunction. highly anisotropic and extends beyond 130 ,~ with significant amplitude in the cdirection, but falls rapidly to negligible values within about 12 ~ along the basal plane direction. This result clearly explains the existence of long-range coupling through Y layers exceeding 100 ,~ in thickness (e.g., fig. 6) in superlattices grown along the c-direction and also accounts for the lack of such coupling through Y layers as thin as 26 ,~ in [Dy[Y] superlattices with growth axis along b (e.g., fig. 17).

5. Effect of applied magnetic fields on interlayer coherence The response of the superlattice systems to an external applied field has been studied by both magnetization (SQUID magnetometry) and by neutron diffraction. The results of magnetization studies on [DyIY] superlattices can be found in Borchers et al. (1987), and on [ErIY] in Erwin et al. (1988b), and Borchers et al. (1989a, 1989b, 1991a, 1991b). In elemental lanthanide periodic moment structures (e.g., Dy, Ho, Er) the effect of applying an external field in the plane containing the spins is to effect a transition to a ferromagnetic state. This transition can either be of first order in which a discontinuous jump from a near zero net moment helical state to a saturated ferromagnetic state is measured by a magnetometer at a critical applied field, or the transition may proceed through a series of intermediate or 'fan-type' moment states occurring before final ferromagnetic alignment. The details of the magnetization process in bulk crystals reflect a free energy balance between the Zeeman energy and the exchange, crystal field anisotropy, and magnetostriction contributions (see Elliott 1972, and Jensen and Macintosh 1991).

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATI'ICES

33

In superlattices, magnetometry results are complicated by the large addenda correction to the data for the paramagnetism of the substrate and buffer layers. Although highly precise qualitative moment values are difficult to obtain, critical field and saturation effects are conveniently measured. Neutron diffraction data taken as a function of applied field provide straightforward insight into the magnetization process, including the critical fields, intermediate moment states, magnitude of the ferromagnetic and helical moments, and details of the transfer of the moment components from a helical configuration to ferromagnetism. This transfer is signaled by a loss of scattered intensity in the satellite reflections at Q- and Q+ and a concomitant increase in the intensity on the fundamental structural reflection and harmonic peaks (e.g., (0002)). In addition, the results continuously monitor the effect of the applied field on the long-range interlayer spin coherence. As an example, fig. 21 and fig. 22 show the comparative effects of a field applied in the basal plane of a [Dy14]Y14164 superlattice at 10 K (0.059Tc) and at 130 K (0.77Tc), respectively. At 10 K , the intensity of the Q - satellites was observed to decrease above about 3 kOe and to essentially collapse in fields of 10 kOe and !

B

~2

7

[Dy14Y14164 1OK ~-8 (0002)

E

24

20

c

.25 kOe Ferro

= O

% 16 1 0 kOe_ m r-

$

12

c

-

H:O

8-

1.8

1.9

2.0

Oz (A -~)

2.1

2.1

2.2

2.3

2.4

Oz (A -1)

Fig. 21. (Left) Applied field effect at 10 K on the Q - magnetic satellite reflection in [DYI41Y]4164 showing the continuous transition in intensity from the (0002)- incommensurate structure reflections into added ferromagnet intensity on the (0002) structural peak. (Right) The original helical phase is not restored after application of the field until the sample is annealed.

34

J.J. R H Y N E

5

Magnetic , Satellites ?

,~

and R.W. ERWIN

[DY14Y14164 130 K

20i [DY14Y14164 130 K 16~- (0002) .~25 kOe

Ferro

H=O, " Ae. This trend correlates with the increasing effective valence z through the series. The ws dependence on the effective R valence is not monotonic, a maximum occurs at z = 4 ( C e ) , Ws = 1 6 x 10 -3 at 5 K. However, the relatively small Ws = 5.4 x 10 -3 in UCo5.3 originates from the low Co magnetic moment. This means that the magnetoelastic-coupling coefficient ncoco increases monotonically with increasing z and reaches very high value 27 x 10-3/.tB 2, the latter being the maximum value known.

4.1.2. Compounds with magnetic R The RCo5 compounds are ferromagnets for the light rare-earth metals and ferrimagnets for the heavy rare earth. The easy-magnetization direction of the R sublattice is primarily determined by the second-order Stevens factor aj. So, in the cases of Pr, Nd, Tb, Dy and Ho, the competition between the magnetic anisotropies of the R and Co sublattices leads to spin reorientations. Both types of the magnetic phase transitions - magnetic ordering and spin reorientation - influence the thermal expansion. Results obtained on single crystals are available for most of the RCo5 compounds. These compounds have a rather wide homogeneity range. The results presented concem the R-rich boundary of this range which is close to the exact 1 : 5 stoichiometry

86

A.V. ANDREEV

for most R. The considerable off-stoichiometry of this boundary is found only in HoCo5.5. The temperature range of stability of the CaCus-type compounds, i.e. the range between the melting point and eutectoid decomposition point, is narrow for ErCos.9 and nearly absent for TmCo 6 (Buschow 1972) (see also section 4.1.1.4). Since it is necessary for the thermal-expansion measurement to expose the sample at high temperature during several hours, it has been impossible to perform proper measurements of these most unstable compounds. Irreversibility of the a, c(T) dependence has been observed due to a change of composition (Andreev and Zadvorkin 1991). For the more stable compounds, only data on LaCo5 are lacking. After presentation of the results of the thermal-expansion studies of the concrete compounds, we will discuss first the volume effects connected with the exchange magnetostriction, and then the anisotropic magnetostriction which manifests itself at spontaneous spin-reorientation phase transitions and in peculiarities of the temperature behaviour of the linear strains.

4.1.2.1. PrCo5 and Pr(Col_~N ix )5. PrCo5 is uniaxial ferromagnet above 107 K. At this temperature the cone-axis type spin reorientation occurs due to a competition between the uniaxial Co and the basal-plane Pr contributions to the magnetocrystalline anisotropy (the first anisotropy constant K1 passes through zero). However, the large value of K2 leads to the situation that even at the lowest temperatures K1 does not exceed -2K2, which is necessary for the plane-cone transition. At 4.2 K the tilt angle 0 between the e axis and the easy-magnetization direction is equal to 230 (Tatsumoto et al. 1971). If the Co sublattice anisotropy is weakened by Ni substitution, the cone-axis reorientation shifts toward higher temperatures, and the plane-cone transition appears (Andreev et al. 1985c). Structural and magnetic properties of Pr(COl_xNi~)5 are presented in table 4.3 and 4.4, respectively. The thermal expansion of PrCo5 and the Pr(Col_~Ni~:)5 solid solutions was studied on single-crystalline samples at 5-1300 K (Andreev et al. 1983, 1985c, Andreev and Zadvorkin 1991). The temperature dependences of lattice parameters of PrCo5 are presented in fig. 4.8. One can see that the a parameter is not influenced by the magnetic ordering (910 K). The experimental and extrapolated a(T) curves coincide above about 450 K. In this range the thermal expansion is similar to that of YCo5 (fig. 4.2). However, at low temperatures some negative basal-plane deformation appears. Along the c axis considerable strain appears just below To and increases with decreasing temperature, as in YCos. But in the c direction some additional positive strain occurs in the same temperature range where the negative strain appears in the a direction, leading to the Invar effect below 400 K. The negative basalplane contribution to Ws predominates, so that Ws changes nonmonotonically with temperature. At 5 K Ws in PrCo5 is considerably lower than in YCo5 (table 4.4). These peculiarities of the thermal expansion can be seen in Pr(Col_~Niz)5. Above about 0.5To the a, c(T) curves are similar to those of the corresponding Y(Col_zNiz)5 compounds. However, with decreasing temperature the negative Aa contribution and additional positive contribution to Ac appear. PrNi5 does not undergo magnetic phase transitions and its a, c(T) dependences have been used for testing the correctness of the extrapolation, as in the case of YNis. But PrNi5 exhibits

THERMAL EXPANSIONANOMALIES

87

V (nm3) 0.090

0.088 c (pro) 400 0.086 a (pm)

398

/ff~

508

396

504 ~J /o 500 0

I

I

I

I

1

200

400

600

800

1000

T (K)

Fig. 4.8. Temperature dependences of the lattice parameters a and c and the unit-cell volume V for PrCo5. The dashed curves represent the phonon contribution to the thermal expansion. (After Andreev and Zadvorkin 1991.) small anomalies of nonmagnetic origin below about 30 K. The latter contributions are negative in the basal plane and positive along the e axis (table 4.4). They were attributed by Andreev et al. (1985c) to crystal-field effects, as was done before with respect to the anomalies in the susceptibility and the specific heat behaviour (Craig et al. 1972). The compound PrNi5 has been extensively studied because of its nonmagnetic singlet ground state that originates from crystal-field interactions (Barthem et al. 1988, Franse and Radwanski 1993, and references therein). PrCoNi4 is a ferromagnet below 40 K, but it exhibits a similar behaviour due to the very low magnetic moment, which cannot provide a noticeable magnetostriction. Since the ferromagnetic Pr(Co]_zNi=)5 compounds have a multiaxial (of the cone or plane type) magnetic anisotropy at low temperatures, an orthorhombic distortion can occur in the case that the 7-magnetostriction constants A7 take large values. No visible distortion was found. This means that the value of the A7 contribution is lower than 10 -4 . The spin reorientation does not reveal itself in the thermal expansion of PrCo5 because the tilt angle 0 is low and its temperature dependence is not sharp. In the solid solutions where 0 increases with Ni concentration, the spin reorientation

88

A.V. ANDREEV

influences the linear thermal expansion. This will be discussed in section 4.1.2.8 dealing with anisotropic magnetostriction.

4.1.2.2. NdCos. Structural and magnetic properties of ferromagnetic NdCo5 are listed in tables 4.3 and 4.4. Its thermal expansion behaviour is similar to PrCos, as follows from single-crystal results (Andreev et al. 1982a, Andreev and Zadvorkin 1991). There is the Invar effect along the c axis and negative strain in the basal plane at low temperatures, as well as a nonmonotonic ws(T) dependence and the absence of the orthorhombic distortion in the multiaxial anisotropy range below 295 K. A complete spin reorientation from the basal plane to the c axis occurs in NdCo5 via an intermediate cone range (235-295 K). This behaviour originates from the competition between the basal-plane contribution of Nd and the uniaxial contribution of Co to the magnetic anisotropy (Tatsumoto et al. 1971). The spin reorientation additionally influences the linear thermal expansion due to a large anisotropic magnetostriction and does not manifest itself in the V(T) dependence due to the different signs of the linear strains. This will be discussed in section 4.1.2.8. 4.1.2.3. SmCos. In ferromagnetic SmCo 5 both the Sm and Co sublattices have uniaxial magnetic anisotropy. Therefore, the compound does not undergo a spin reorientation. Its structural and magnetic properties are presented in tables 4.3 and 4.4. A thermal-expansion anomaly connected with the magnetic ordering in SmCo5 has been observed by Buschow et al. (1974) on polycrystalline samples studied in the range 273-973 K. However, the temperature range considered is not large enough exceeding Tc for making a proper extrapolation to the ferromagnetic range. The thermal expansion behaviour of SmCo5 was studied on single-crystalline samples in the more extended range 5-1300 K by Andreev and Zadvorkin (1991). The compound differs from PrCo5 by the absence of an Invar effect along the c axis and positive strain in the basal plane. The latter strain, like negative strain in PrCos, appears much below To. Despite the fact that the absolute values of the linear strains are smaller than in PrCo5 and NdCos, the volume effect in SmCo5 is larger and practically equal to that in YCos, because the linear strains have the same sign. 4.1.2.4. GdCos. The thermal expansion of ferrimagnetic GdCo5 is very similar to that of YCos. It is characterized by the absence of basal-plane strain, and the )~c and Ws values are close to those of YCo5 (Andreev and Zadvorkin 1991). Structural and magnetic properties of GdCo5 are presented in tables 4.3 and 4.4. 4.1.2.5. TbCos+x. From this compound the equilibrium composition of the CaCustype phase starts to deviate from the exact 1 : 5 ratio, as has been mentioned above. The magnetic data are available for TbCos.1 and TbCos.2 single crystals. The compounds are ferrimagnets with low magnetic moments at 4.2 K (0.55#B for = 0.1, Yermolenko 1983, and 0.25#a/f.u. for z = 0.2, Ballou et al. 1989). There is a compensation point at about 90 K and the anisotropies of the Tb and Co sublattices are competitive. Two spin-reorientation phase transitions of second order, at which

THERMAL EXPANSION ANOMALIES

89

a (pm)

V (rim3) .

.

.

.

.

0.088

c (pm) 401 -

°'°86

399

397

395

494

-

0

-""

'

I

'

I

200

400

600

800

1000

T (K)

Fig. 4.9. Temperature dependences of the lattice parameters a and c and the unit-cell volume V for TbCosA. The dashed curves represent the phonon contribution to the thermal expansion. (After Andreev and Zadvorkin 1991.) the easy-magnetization directions change from the basal plane to the c axis via an easy cone, occur in a narrow temperature range around 400 K. These results were obtained by Okamoto et al. (1973) on single-crystalline samples. Structural and magnetic properties of TbCos.1 are presented in tables 4.3 and 4.4. The thermal expansion of TbCos.1 w a s studied on single crystals at 5-1300 K (Andreev et al. 1983, Andreev 1990, Andreev and Zadvorkin 1991). The temperature dependences of the lattice parameters and the unit-cell volume are shown in fig. 4.9. Well-pronounced anomalies in the linear thermal expansion can be seen, which reflect the spin reorientation and will be discussed separately in section 4.1.2.8. Due to the opposite signs of these linear effects they do not show up in the V(T) dependence. The basal-plane strain above the spin reorientation range is negligible. If one shifts the a(T) curve downwards, so as to exclude the influence of the spin reorientation, Aa becomes negligible at low temperatures as well, like in YCo5 and GdCos. No orthorhombic distortion was found in the compound in multiaxial anisotropy range below 410 K. This means that the gamma-magnetostriction is lower than 10 -4.

4.1.2.6. DyCos+=. The magnetic properties of DyCos.2 are similar to those of TbCos.1. It is also a ferrimagnet with a compensation point in the M(T) dependence and two second-order spin-reorientation phase transitions. Structural and magnetic properties of the compound are presented in tables 4.3 and 4.4. Since they slightly depend on concentration in the homogeneity range, we have listed the data that have been

90

A.V. ANDREEV

obtained on the same single crystals on which the thermal expansion has been studied. This data correspondence is especially important with regard to the spin-reorientation temperatures. The plane-cone transition temperature determined from magnetization measurements shows a large error due to its field dependence. Therefore, the data are taken from neutron-diffraction results obtained in zero field by Chuev et al. (1981a). The thermal expansion of DyCos.2 has been studied on single-crystalline samples at 5-1300 K (Andreev et al. 1983, Andreev 1990, Andreev and Zadvorkin 1991). The temperature dependences of the lattice parameters and the unit-cell volume are shown in fig. 4.10. They differ considerably from the compounds discussed above, for magnetic as well as for nonmagnetic R. First of all, the thermal-expansion coefficient along the c axis ~c is much higher above To, while aa within the basal plane is slightly lower. This leads to the higher value of the volume coefficient c~v. i

i

i

i

i

(

V (nm a)

c (pm)

0.086

401

0.084

III I I I I I I I I I I I I I /I .. f

397

393

0.082

a (pm) 499

889 /

a

495

iiI i~Iy 385

200

t

i

~

~

400

600

800

1000

491 T (K)

Fig. 4.10. Temperature dependences of the lattice parameters a and c and the unit-cell volume V for DyCos.2. The dashed curves represent the phonon contribution to the thermal expansion. (After Andreev and Zadvorkin 1091.)

THERMALEXPANSIONANOMALIES

91

Whereas C~v at 1200 K is practically the same for the compounds with R = Y, Pr-Tb, (4.1 + 0.1) x 10 -5 K -I, it is 1.5 times larger for DyCos.2. The increase of av can be understood, since it might be connected with the decrease of the structure stability. But the volume strain ws exceeds that of other RCo5 by a factor 4. This huge value appears therefore not only to be due to a softness of the lattice. It might even be larger if the linear strains did not have different signs. The difference between the extrapolated and experimental a(T) curves appears just below To, unlike in PrCo5 and NdCos, where ha is also negative. No orthorhombic distortion is found for DyCos.2 in the multiaxial anisotropy range below 370 K (the cone or basal-plane anisotropy types). Therefore, the 7-magnetostriction is lower than 10 -4. The spin-reorientation at 300-370 K influences the linear thermal expansion (on the scale of fig. 4.10, the effect in the basal plane is almost invisible)and does not manifest itself in the V(T) dependence due to the different signs of the linear strains. This will be discussed in section 4.1.2.8.

4.1.2.7. HoCos+x. This ferrimagnet has magnetic properties similar to those of TbCosa and DyCos.2. However, the spin reorientation is rather sensitive to the composition. Decroep et al. (1982, 1983) observed on a single crystal with the composition HoCo5.6 that there is a cone-axis reorientation at 180 K. The cone is stable down to 4.2 K (0 = 72 °) and the cone-plane transition does not occur. In a compound with the composition HoCo5.5 both transitions were observed. Structural and magnetic properties of HoCo5.5 are presented in tables 4.3 and 4.4. As for DyCos.2, we took the plane-cone transition temperature from neutron-diffraction results in zero field (Chuev et al. 1981b), because of its strong field dependence in magnetization measurements. The thermal expansion of HoCo5.5 was studied on single-crystalline samples at 5-1300 K (Andreev et al. 1983, Andreev 1990, Andreev and Zadvorkin 1991). The temperature dependences of the lattice parameters and the unit-cell volume are similar to those for DyCos.z. A negative basal-plane strain appears just below To, and a huge positive strain exists along the e axis. The spin reorientation influences the linear thermal expansion relatively weakly and is not seen in the V(T) dependence due to different signs of the linear strains. This will be discussed in the next section. The values of C~v, Ac and Ws are even somewhat larger than in DyCos.2. The value ws = 32x 10 -3 at 5 K seems to be the largest volume effect observed thus far in 4f-3d intermetallics. As in other RCo5 compounds with multiaxial magnetic anisotropy (of the cone or plane types) at low temperatures, no orthorhombic distortion has been found in HoCo5.5 below 170 K, which indicates negligible 7-magnetostriction.

4.1.2.8. Anisotropic magnetostriction. In all RCo5 compounds which undergo a spin reorientation, except PrCos, the reorientation affects the thermal expansion. Upon rotation of the magnetic moment from the basal plane towards the e axis, the e parameter increases additionally, while the basal-plane thermal expansion becomes smaller. As a result, the V(T) dependence has no anomaly at T1 (the plane-cone transition) and at T2 (cone-axis). In figs 4.11 and 4.12, the thermal expansion of NdCo5 and

92

A.V, A N D R E E V

90, o

"0

®

o o

60

i

0

o

0

o 0 o

30

o t

0 a (pm)

,

I

J

J

o

c (pro)

503.4

397,8 503,0 397.6

0 o°

502.6

0

~

0 0

o

o

. o . o o 9 o o 0 9 -° 200

o

&e

_L

;

397.4

I

240 i

fo

o

I

280 i

i

,

,

X (K) i

4

.I::

O(

-4 0

0.4

0.8

cos2G

Fig. 4.11. Spin reorientation in NdCo 5. Upper graph: Temperature dependences of the angle 0 between

the easy-magnetization direction and the c axis and of the lattice parameters a and c. The dashed curves are the extrapolation of the a, c(T) curves to the coae-anisotropy range. Lower graph: Dependences of the additional strains Aa/e o and Ac/e o on cos 2 0. (After Andreev et al. 1982a.)

ill the spin-reorientation range is presented together with the 0(T) dependences (Andreev et al. 1982a, 1983, Zadvorkin 1987). The observed anomalies are especially well-pronounced in TbCos.1, where the Tt-T2 interval is the most narrow among all RCos. The transition temperatures Tx and T2 correspond to the points on the a, c(T) curves in which these curves start to deviate from the extrapolations from the basal-plane and uniaxial ranges to the cone-anisotropy range. The values of T1 and Tz determined in this way are in good agreement with the magnetometric data. WbCo5.1

THERMAL EXPANSION ANOMALIES

93

90 ¸ o

~" 60

0 Q

30 0 I

I

I

~

I

...........~o O0 0 0 0 0

0

c (pm) 398.5

a (pm) 496.6

o

o o

o

0 °°

i

398.1

/

l

I

360

I

I

400 i

~"

o

~.0 /

0

496.2

oo

JL.[

°8

398.3

440 T (K)

D

i

f

i

i

4

v C

0

-4

i

0

0.4

f

0.8

COS2E)

Fig. 4.12. Spin reorientation in TbCosa. Upper graph: Temperature dependences of the angle 0 between the easy-magnetization direction and the e axis and of the lattice parameters a and c. The dashed curves are the extrapolation of the a, c(T) curves to the cone-anisotropy range. Lower graph: Dependences of the additional strains An/a0 and Ae/e0 on cos2 0. (After Andreev et al. 1983.) In the uniaxial magnetic anisotropy range above T2, the basal-plane and uniaxial strains are described by the eqs (4.1) and (4.2). In the basal-plane anisotropy range below T1, analogous formulae can be deduced from (2.4) provided the 7 - m a g n e t o striction can be neglected, which is possible in RCos: 1 )~a(T) = ,k~'°(T) - ~ ~ ? 2 ( T ) ,

(4.5)

1 ~ ),~'2(T).

(4.6)

~e(T) = ~ ' ° ( T ) -

94

A.V. A N D R E E V T A B L E 4.5

Magnetostriction constants for RCo 5 compounds with magnetic R. A?'2(Tm) and A~'2(Tm) are the anisotropic magnetostriction constants in the basal plane and along the c axis at the middle of spinreorientation interval; A~'2(0) and A2' a 2(0) with index 'extr' are these constants extrapolated to 0 K from the spin-reorientation range; A~'2(0) and A~'2(0) with index 'calc' are these constants at 0 K calculated on the basis of extrapolated constants for TbCo5.1; A~'°(0) and A2' (0) are the exchange magnetostrietion constants in the basal plane and along the c axis. R

Pr Nd Sm Gd Tb Dy Ho Er Tm

;~?,2(Tm) ;~,2(Tm) ~,2(0 )

~ 2(0) ~2'

~,2(o)

,2 (0) ~2'

~,°(o)

~ o(o) ;~2'

extr 10-3

calc 10-3

calc 10-3

10-3

10-3

-1.2 -1.0 1.4 0 - 1.92) -1.8 -0.7 0.7 1.6

2.4 2.0 -2.8 0 3.82) 3.7 1.4 - 1.3 -3.3

-3.2 -3.6 -0.2 0 0.4 -4.3 -2.7 -

8.7 8.5 6.5 6.1 6.6 34.6 37.7 -

10-3

10-3

extr 10-3

-0.4 . . -0.3 ,-, - 0 . 4 ,-~ - 0 . 3 . .

0.8

-1.11) -1.1

2.21) 2.2

- 1.9 -1.8 -0.6

3.8 3.7 1.2

. .

. .

. .

0.6 0.8 0.5 . .

. .

. .

1) The data obtained on Pr(Col_xNix)5 with x = 0.2 and 0.4. 2) For TbCos. 1 the extrapolated data are taken.

Therefore, the step changes inth~ a, e(T) dependences during the complete spin reorientation just correspond t o a d - and A~,2, respectively. In general, it is not so simple to make the proper extrapolations necessary to determine these values. But if the T1-T2 interval is narrow, like in TbCo5A, or the thermal-expansion coefficients below and above the spin-reorientation range are the same, like in NdCos, the linear extrapolations presented in figs 4.11 and 4.12 seem to be realistic. Pourarian et al. (1981) studied the thermal expansion of NdCo5 in the spin-reorientation range on aligned polycrystalline samples using a strain-gauge technique. They found a change of 9.6 x 10 -4 along the c axis between the plane-cone and cone-axis transitions. This is in satisfactory agreement with x~'2 8 x 10 -4 in table 4.5. In figs 4.11 and 4.12 a comparison is also made between the observed additional strains Aa/a, Ac/c and cos 2 0. The linearity of the Aa/a(cos 2 O) a n d Ac/c(cos 2 O) dependences shows that the second-order magnetostriction constants Xtx'2 "'1 and x~'2 "'2 are actually sufficient for the description of these strains, according to eq. (2.4) and its consequences leading to eqs (4.1), (4.2), (4.5) and (4.6). It should be noted that the temperature dependences of )~,2 and )~2'2 within the spin-reorientation range are neglected here because of narrowness of the T1-T2 interval. This presents no difficulties in the cases of NdC05 and TbC05.1, since the results obtained from the 'up' and 'down' extrapolations coincide. The values of "u x c~'2 and '~2c~,2 determined in such a way are attributed to the middle of the spin-reorientation ranges and are presented in table 4.5. In the cases of DyCos.2 and HoCo5.5 the results of the 'up' and 'down' extrapolations differ considerably (20-30%). The numbers presented are average values between two extrapolations and also refer to the middle of the T1-T2 interval.

THERMAL EXPANSION ANOMALIES

95

T h e "'1 Ira'2 and %a,2 "'2 constants obtained for the different RCo5 are difficult to compare because they correspond to different temperatures. In order to compare them, they should be extrapolated to zero temperature. Assuming only a single-ion interaction of the 4f-electron shell with the crystal electric field and neglecting the Co contribution to the anisotropic magnetostriction, Andreev et al. (1983) have determined "'1 ~,a,2 and ,~,2 for zero temperature using formula (3.1). The temperature dependences of/~R needed for the extrapolation are obtained as differences between #m of RCo5 and YCos. The results are presented in table 4.5. In fig. 4.13 the thermal expansion of Pr(Col_=Ni=)5 in the spin-reorientation range is shown together with the O(T) dependences (Andreev et al. 1985b). One can see that anomalies similar to those in NdCo5 appear in the a, c(T) curves, the cone angle O increasing with increasing Ni content. The anomalies can be observed even in the z = 0.2 compound, in which the reorientation is still incomplete. For both the a: = 0.2 and a: = 0.4 compound, it is possible to determine "'l)a'2and )~2°42 and extrapolate them to T = 0. The same values have been found for both compounds, 90

i

o o

60

o

)O~O~qDo~qD~ 0 ~ 30 DOoo •go @O O () i P : : ©~>-c "o

@x=O ¢ 0.2 o

o o

?o-o@@

502

@@ a

0.4

, a (pro)

•• 501

+j°

qo I)

qB

qD

500

499

498 c (pm) 399.6

oo o

°

°

497

~

399.2

398.8 1O0

200

T (K)

Fig. 4.13. Temperature dependences of the angle O between the easy-magnetization direction and the c axis, and temperature dependences of the lattice parameters a and c in Pr(Col_xNi=)5 in the spin-reorientation range. (After Andreev et al. 1985c.)

96

A.V. A N D R E E V

A~'2(0) = -1.1 x 10 -3 and A~'2(0) = 2.2 x 10 -3. In order to obtain the #pr(T) dependence, we subtract the #m(T) of corresponding Y(Col_xNix)5 from #m(T) of Pr(COl_~Niz)5. In PrCozNi3 (x = 0.6) the uniaxial Co sublattice anisotropy becomes so weak that the basal-plane Pr sublattice anisotropy dominates in the whole magnetically ordered range, and there is no spin reorientation. Since the obtained values of A~'2 and A~'2 do not depend on the Co content (this is in agreement with a negligible Co sublattice contribution to the anisotropic magnetostriction), they are presented in table 4.5 as being representative for PrCos. In the R series the second-order anisotropic magnetostriction constants at 0 K can be determined on the basis of expression (3.2). Using this formula, A~'2(0) and A~'2(0) have been calculated for all RCo5 in which R is magnetic. The B2 coefficient has been derived from the data obtained for TbCos.1 because in this compound the spin reorientation is very pronounced and the/~Tb(T) dependence is well determined owing to the large Tb moment. Results of the calculation are presented in table 4.5. There is rather good agreement between the anisotropic magnetostriction constants determined from the anomalies of the thermal expansion in RCo5 with R = Pr, Nd, Dy and Ho and those calculated on the basis of the TbCos. 1 results. This means that the calculated values presented in table 4.5 for RCo5 which do not undergo a spontaneous spin reorientation can be considered as realistic. Similar self-consistent results have been obtained by Deryagin et al. (1984b, 1985b) from acoustic measurements on NdCos, T'oCo5.1 and DyCos.2 single crystals, also neglecting the Co contribution.

4.1.2.9. Exchange magnetostriction. The A~'° and A~'° exchange-magnetostriction constants at 0 K for the studied RCo5 compounds with magnetic R have been de~2 termined by substitution of An, Ac, Aal'2 and A2' into the eqs (4.1), (4.2), (4.5) and (4.6). The values are presented in table 4.5. These constants strongly depend on the type of rare-earth metal. The basal-plane strain x ~'° is practically always negative and much smaller in absolute value than the uniaxial strain A2o~,0. Both strains have minimum values in the middle of the R series. In all compounds studied, the basal-plane anisotropic and exchange magnetostriction constants are comparable in absolute value, but along the c axis the exchange constants ~c~,0 "'2 are always much larger than the anisotropic constants. As has been already mentioned, the spin reorientation does not manifest itself in the V(T) dependences. However, not only the direction of the magnetic moment changes during the spin reorientation. Neutron-diffraction studies have shown that the magnitudes of #R and /-tCo also change. #a decreases and #Co increases upon rotation from the basal plane to thec axis. This can be observed as a step up in the #re(T) dependence because of the larger values of A/~R than 5A#co (Pirogov et al. 1982, Kelarev et al. 1983, Yermolenko 1983, Tsushima and Ohokoshi 1983). This can lead to an additional volume effect at the spin reorientation. The absence of such an effect in RCo5 is apparently the result of a cancellation of the contributions to this effect from the Co-Co and R-Co exchange interactions due to different signs of A#R and A/leo. In several other compounds (Sm2Co7, Tm2Co7, Tm2Fe17, NdzFe14B) this

THERMAL EXPANSION ANOMALIES

I

97

i

32

• PrCo 5 O NdCo 5 ® SmOo 5 ~GdCo 5 @ TbCos. I

24

~) DyC°5.2

@

@ H°C°5 •5 @ O

(D @

16 l) 0 0

@

@

eO

%

eo 0 rn

~L

I

21

@

@

@ 4D

t)

t)

@

@

@

I @

I @

@

ID

ID

I

(D

(D

ID

O v

8rr

• -2

I

0

,

0.4

~

• i 0

Oi

0.8

TFFc

Fig. 4.14. Temperature dependences of the spontaneous volume magnetostriction ;Vs and the intersublattice magnetoelastic-coupling coefficient nRc o for RCo 5 with magnetic R. (After Andreev and Zadvorkin 1991.)

additional volume effect has been observed. This phenomenon will be discussed in the corresponding section. Figure 4.14 shows the temperature dependences of volume effect ~os. They have been analyzed using formula (2.6) and neglecting the weakest contribution due to the R - R exchange interaction. The Co-Co exchange-interaction contribution can be obtained from the results obtained on YCos. The ngco intersublattice magnetoelasticcoupling coefficients have been calculated under the following assumptions: 1. The Co moment in RCo5 is the same as in YCos, and the R moment is equal to the difference between/~m of RCo5 and YCos. 2. The ncoco coefficient in RCo5 is the'same as in YCos, provided the difference in compressibility is negligible.

98

A.V. ANDREEV

3. The compressibility is proportional to the volume thermal expansion coefficient av in the paramagnetic range, so that the difference in compressibility can be taken into account by means of the relative difference between the av values in RCo5 and YCos. 4. ncoco is temperature-independent, as found for YCos. The results obtained are presented in fig. 4.14. For SmCos, nsmco is not determined due to the low Sm moment and the consequently large error in its determination as difference between #m of SmCo5 and YCo~. The n~co coefficients are weakly temperature-dependent. The dependence of naco on the atomic number throughout the rare-earth series looks rather strange. At first sight, naco should correlate with To, i.e. it should be maximum in GdCo5 and decrease on both sides of Gd due to the weakening R-Co exchange interaction. But one can see in fig. 4.14 that nRco is negative for the light R and positive for the heavy R while nRco ~ 0 for those R, which are expected to have a maximum R-Co interaction (Gd, Tb). A similar situation has been observed for the R2Co 7 compounds (see below). The difference in the signs of nRco for light and heavy R correlates with the different type of magnetic arrangement. The absolute values of nRco are several times smaller than ncoco. Nevertheless a considerable contribution to the volume effects results from the R-Co exchange interaction, due to the large moment of R. This is a main difference between the Co-rich and Fe-rich rare-earth intermetallics (see also section 5). In Fe-rich compounds practically only the Fe-Fe exchange interaction contributes to the thermal-expansion anomalies.

4.2. R2Co7 R2Co7 intermetallics exist in ahnost all R-Co binary systems and are interesting representatives of magnetic 4f-3d systems. They display various types of magnetic anisotropy (Deryagin et al. 1979, Kudrevatykh et al. 1983, Tarasov 1987), undergo spontaneous and field-induced spin-reorientation phase transitions (Andreev et al. 1988b, Tarasov 1987, Bartashevich 1992), and can reversibly absorb large amounts of hydrogen. The latter leads to drastical changes in the magnetic properties (Bartashevich et al. 1983, Andreev et al. 1986c, Bartashevich 1987). However, the R2Co 7 compounds have been studied less extensively than the RCo5 compounds due to the lower practical importance of the former compounds. A further reason is the much more complicated metallurgy of R2Co7. Owing to the complex peritectic reactions of the formation of the R2Co 7 compounds, the preparation of single-phase samples and, moreover, single crystals is rather difficult. Nevertheless, the data on magnetic properties obtained on single crystals are available for about half of the R2Co7 series. The basic element of the crystal structure of R2Co7 can be obtained by stacking hexagonal structural blocks RCo5 and cubic blocks RCo2 along the common hexagonal (trigonal for RCo2) axis according to the scheme: 2RCo5 + 2RCo2 = 2R2Co7 (Lemaire 1966a, b). This is possible since both the RCo5 and RCoz lattices contain the same hexagonal Co net with coordination number 4 (so-called Kagome net). The structural element, obtained in this way (2R2Co7), has the dimensions a ~ 0.5 ran, c ~. 1.2 nm. The double-layer stacking of these elements with

THERMALEXPANSIONANOMALIES

99

RCo; RCo,

RCo,

2RCo

RCo,

OR(~f~) ~ ( 4 ~ a) O~(za) ~(~e) ® M(6h) ¢~(q2k)

RCo: RCo: Fig. 4.15. Unit cell of the crystalstructureof the hexagonalmodificationof R2Co7 (the Ce2Ni7 type). c ~ 2.4 nrn, when the sequence of the cubic blocks corresponds to the hexagonal MgZn2-type Laves phase, leads to the hexagonal modification of the R2Co7 phases (Ce2Ni7 type, space group P63/mmc). The R2Co7 compounds formed with the light R element crystallize in this structure type. It is shown in fig. 4.15. A three-layer stacking (c ~ 3.6 nm), with the sequence of the cubic blocks as in the cubic MgCu2-type Laves phase, leads to the rhombohedral modification of the R2Co7 phases (Gd2Co 7 type, space group R3m). This structure is preferred by R2Co 7 with heavy R. In the cases of R = Y, Gd and Tb, both modifications have been observed. In both structures the R atoms occupy two nonequivalent sites: R1 with a hexagonal local environment (as in RCos) and R2 with a quasicubic local environment (as in RCo2, with some distortion). The Co atoms occupy five nonequivalent sites in R2Co7. However, a common Co sublattice with uniaxial magnetic anisotropy can be considered as a good approximation, as follows from magnetic property studies. The total magnetocrystalline anisotropy is the result of the competition between the R1, R2 and Co sublattice contributions. The thermal expansion of R2Co7 bears a substantial resemblance with that of RCos. However, in some peculiarities, such as the orthorhombic distortion, they differ considerably. As in the case of RCos, we will discuss the thermal expansion of R2C07 starting from Y2Co7 with nonmagnetic Y in order to determine the Co sublattice contribution to the magnetoelastic properties.

100

AN. ANDREEV

TABLE 4.6 Structural data for R2Co 7 compounds, a (300 K), a (5 K), e (300 K) and c (300 K) are the lattice parameters of the hexagonal unit cell of hexagonal (Nd, Sm), rhombohedral (Y, Cd, Tm) or both (Tb) structure modifications; at 5 K, in the case of orthorhombic distortion, the third lattice parameter b is also presented; c~a, c~e are the linear thermal expansion coefficients in the paramagnetic range (1000 K) in the basal plane and along the c axis, respectively; c~v is the volume thermal expansion coefficient at 1000 K. Compound a (300 K) c (300 K) a (5 K) pm pm pm

b (5 K) c (5 K) pm pm

C~a c~e c~v Ref. 10 -5 K -1 10 -5 K -1 10 - 5 K -1

Y2Co7 Nd2Co7 Sm2Co 7 Gd2Co 7 Tb2Co 7 (r) Tb2Co 7 (h) Tm2Co 7

874.0 862.8 865.9 855.1

1.58 1.74 1.66 1.88 1.64 1.64 1.80

500.32 506.00 504.90 502.79 498.65 500.68 496.18

3622.4 2444.6 2432.7 3638.4 3623.2 2418.8 3607.4

498.95 503.40 503.32 500.95 496.62 498.42 495.42

References: [1] Zadvorkin (1987). [2] Andreev (1990). [3] Andreev et al. (1988a).

3620.0 2442.8 2427.2 3633.9 3618.0 2414.5 3603.4

1.12 0.89 1.27 1.40 1.10 1.11 0.94

4.3 4.4 4.6 5.2 4.4 4.4 4.5

[1-3] [1-3] [4] [1-3] [1-3] [1-3] [1, 2,5]

[4] Bartashevich et al. (1993a). [5] Andreev et al. (1992b).

TABLE 4.7 Magnetic and magnetoelastic data for R2Co 7 compounds. #m is the magnetic moment per formula unit at 4.2 K; type of magnetocrystalline anisotropy is given for 4.2 K; Te is the Curie temperature; T 1 and T2 are the temperatures of the plane-cone and cone-axis transitions, respectively; -~a, he are the linear spontaneous magnetostrictive strains in the basal plane and along the e axis, respectively. For compounds with orthorhombic distortion, ,~a means isotropic expansion in the basal plane, which corresponds to a change in the average R-R distance; ws is the volume spontaneous magnetostriction; e is the orthorhombic distortion ( b / V ~ - a)/a due to a gamma-magnetostriction. Compound Um ,haiso- Tc #B/f.u. tropy K

7'1 K

7"2 K

Aa (5 K) ,~e (5 K) ws (5 K) e (5 K) Ref. 10 -3 10 -3 10 -3 10 -3

Y2Co7 Nd2Co7 Sm2Co 7 Gd2Co 7 Tb2Co 7 (r) Tb2Co 7 (h) Tm2Co 7

230 405 -

290 68 450 450 45

1.2 0.9 2.1 2.7 2.5 2.5 3.2

9.6 14.6 10.1 4.2 7.5 7.5 4.0

axis plane canted axis canted plane canted

639 613 670 775 720 720 640

References: [1] Zadvorkin (1987). [2] Andreev (1990). [3] Andreev et al. (1988a). [4] Tarasov (1987). [5] Bartashevich et al. (1992).

[6] [7] [8] [9] [10]

3.4 3.2 3.6 5.6 3.0 3.1 2.7

5.8 5.0 7.8 11.0 8.0 8.1 6.6

0 2.38 0 0 3.08 2.96 -3.50

[1--4] [1-3, 5, 6] [4, 7] [1-3, 8] [1-3, 9] [1-3, 9] [1, 2, 4, 10]

Andreev et al. (1988b). Bartashevich et al. (1993a). Andreev et al. (1985g). Andreev et al. (1982b). Andreev et al. (1992b).

4.2.1. Data on particular compounds 4.2.1.1. Y 2 C 0 7 . Magnetic properties of YzCo7 have been studied on single crystals of the rhombohedral structure by Deryagin et al. (1979), Kudrevatykh et al. (1983),

THERMAL EXPANSION ANOMALIES

101

Tarasov (1987), Ballou (1987). This compound is a ferromagnet (structural and magnetic properties are listed in tables 4.6 and 4.7) with uniaxial magnetic anisotropy in the whole ordered range. The thermal expansion of both rhombohedral and hexagonal single crystals has been studied by Andreev et al. (1988a). In contrast with YCos, the compound Y2Co7 exhibits magnetostrictive strain not only along the e axis, but also within the basal plane. However, ,Xc at 5 K is 3 times larger than )~a (table 4.7). Both linear strains follow the square of the magnetic moment. As in the case of RCo5 with nonmagnetic R (and Gd, which does not contribute to the magnetic anisotropy and anisotropic magnetostriction), this shows that there is negligible anisotropic magnetostriction of the Co sublattice in comparison with exchange magnetostriction. Consequently, the data presented for Aa and he at 5 K in table 4.7 are equal to the exchange magnetostriction constants "'l x ~'° and Az~,0, respectively. The magnetoelastic as well as the magnetic properties of the hexagonal Y2Co7 modification are the same as those of the rhombohedral modification. This confirms the negligible anisotropic magnetostriction of the Co sublattice in YzCo7, because structural modification strongly influences the anisotropic magnetostriction, if the latter is large (Tb2Co7, Andreev et al. 1982b, 1988a). The volume effect in Y2Co7 0~s is slightly lower than in YCo5. However, this is due to the considerably smaller Co moment #co = 1.37#B, instead of 1.65/*B in YCos. The magnetoelastic-coupling coefficient ncoco is 3.1 x 10-3/*B 2, which is larger than in YCos, although it is difficult to directly compare ncoco of the compounds with different crystal structure and Co content. The decrease of/*co and Te in R2Co7 in comparison with RCo5 is generally connected with additional filling of the Co 3d band by electrons from R due to larger R content in R2Co 7. In a way it has the same effect as the increase of the effective valence z of R atoms within the RCo5 series. The appearance of the basal-plane strain and the increase of ncoco in Y2Co7 in comparison with YCo5 are in agreement with the observed trend in the magnetoelastic properties of RCo5 with nonmagnetic R having different z (see section 4.1.1.5).

4.2.1.2. Nd2Co7. Nd2Co7 has the hexagonal structural modification (Buschow 1980). Studies of magnetic properties, carried out on single crystals (Andreev et al. 1988b, Bartashevich et al. 1992), show that this highly anisotropic ferromagnet exhibits a plane-cone spin reorientation at 230 K and a cone-axis transition at 290 K. There is a strong anisotropy within the basal plane at low temperatures, and the a axis [100] is the easy-magnetization direction, which corresponds to a positive third anisotropy constant K3. The thermal expansion of Nd2Co7 (Andreev et al. 1988a, b) is similar to that of NdCos. Both types of magnetic phase transition influence the a, c(T) dependences (fig. 4.16). At the scale of this figure, the anomaly at spin reorientation in a(T) curve, being twice smaller than in c(T), is invisible. The volume expansion does not sense the spin reorientation because of different signs of the linear effects. NdzCo7 differs from NdCo5 by a large positive orthorhombic distortion e = (b/v~- a)/a,

102

A.V. ANDREEV

a (pm) 511

' "' V (rim3)

E~MD a

'oooo--O °°°°oooo a oO0°

507

c(pm)

~0 2454

oY

bX"~.x~/;/ ocO.~oO-/ ~530~

0.552 503

oO02 7

o

_ooO o

° ou

o

~)20~X~O~:~00 000~ /, oO° / /

.~.//

0.544

oO~It

/

/

/ ~2446

o°

o

o

VooO~OCP 000/

2438

ooO°//

°oo°// 0.536 / I

0

200

I

400

I

I

600

800

T (K)

Fig. 4.16. Temperature dependences of the lattice parameters a,b (in the orthorhombically distorted range), c and the unit-cell volume V for Nd2Co 7. The dashed curves represent the phonon contribution to the thermal expansion. Inset: correlation between the orthorhombic axes and the easy-magnetization direction. (After Andreev et al 1988a.)

which appears in the multiaxial-anisotropy range and increases sharply with decreasing temperature. (In the case of distortion, the third parameter b is needed to describe the lattice. For a better presentation, we use in fig. 4.16 and further figures the values b/v/3 for compounds with distortion, which is equal to the a parameter when distortion disappears). The 7-magnetostdction which manifests itself in this distortion, as well as the anisotropic a-magnetostriction, responsible for anomalies at spin reorientation, will be discussed in sections 4.2.2 and 4.2.3, respectively. Structural and magnetic properties of NdeCo7 are presented in tables 4.6 and 4.7.

4.2.1.3. Sm2Co7. The compound Sm2Co 7 crystallizes in the hexagonal structural modification (Buschow 1980). Magnetic properties of the Sm2Co7 single crystals were studied by Tarasov (1987) and Bartashevich et al. (1993a). This compound is a highly anisotropic ferromagnet with the easy magnetization direction along the c axis at room temperature. No significant anisotropy was observed within the basal

THERMAL EXPANSION ANOMALIES

103

plane. At low temperatures a substantial component of the spontaneous magnetization appears in the basal plane. It means that the easy magnetization direction deviates from the c axis. The angle/9 of the cone of easy axes reaches 15 o at 4.2 K. The cone-axis spin-reorientation temperature T2 has been found to be 68 K from ac susceptibility measurements as well as from the usual analysis of the magnetization curves (Sucksmith and Thompson 1954). The structural and magnetic characteristics of Sm2Co7 are collected in tables 4.6 and 4.7. A positive spontaneous magnetostriction has been found by Buschow et al. (1974) in S m 2 C o 7 o n polycrystalline samples at 273-973 K. Thermal expansion of the Sm2Co7 single crystals has been studied by Bartashevich et al. (1993a). In fig. 4.17 the temperature dependences of the lattice parameters are presented. In the uniaxial anisotropy range (68 K < T < To) they are similar to those of Y2Co7. Both the basal-plane and uniaxial strains A, and Ac are positive, appear just below To, and 512

2440

508

2420

504

L12428

~

504.0

? 503.6

2426 • •

503.2

•

I

•m '

0

2400

i ~

40

50

2424 120

T (K)

500

~

0

,

,

,

I

60O

300

900

T (K) Fig. 4.17. Temperature dependences of the lattice parameters a and c for Sm2Co 7. The full curves represent phonon contributions to the thermal expansion. Inset: the low-temperature part of the a(T) and c(T) curves. Here, the curves show the extrapolation from the uniaxial-anisotropy range (above 68 K) to the cone-anisotropy range. (After Bartashevich et al. 1993a.)

104

A.V. ANDREEV lO

BEll

.'¢ "% cos

8

nmmmimmmmmn

6

\

4

~D~DDD 2

\

vVvvvvvvvvv

0

i_ t0 -

300

600

,

900

T (K) Fig. 4.18. Temperaturedependencesof the linear (,~a and ,~c)and volume (Ws)spontaneousmagnetostrietion for Sm2Co7. The dash curve represents the ws(T) dependence for Y2Co7. (After Bartashevich et al 1993a.) increase monotonically with decreasing temperature. However, the spin reorientation is accompanied by additional linear and volume effects, all of which are negative with respect to the uniaxial state and by one order of magnitude lower than the total value of the corresponding strain. The temperature dependences of Aa, Ac and Ws are presented in fig. 4.18. Extrapolation of the a, c(T) dependences from the uniaxial range to zero temperature, as shown in the inset in fig. 4.17, gives Aa(0) = 2.5 x 10 -3 and ,~e(0) = 3.8 x 10 -3. This corresponds to Ws(0) = 8.7x 10 -3. The observed values of ,Xa, )~c and Ws at the lowest temperature (5 K) are equal to 2.1 x 10 -3, 3.6 x 10 -3 and 7.8 x 10 -3, respectively. The occurrence of a spin reorientation in Sm2Co7 and Tm2Co7 (Tarasov 1987, Andreev et al. 1992b) is not expected on the basis of the competition in the anisotropy between the Sm and Co sublattices. Indeed, the uniaxial magnetic anisotropy of Sm2Co7 is much stronger than that of Y2Co7 at room temperature. This means that the anisotropies of the Sm and Co sublattices do not compete. The magnetic anisotropy of the SmCo 5 blocks is uniaxial. The cubic Laves phase compound SmCox has 4 equivalent easy-magnetization axes of the (111) type. In Sm2Co7 one of these axes in the cubic block coincides with the c axis and the others deviate by the angle about 700 from the e axis. In the presented scheme there is no reason for a spin reorientation in Sm2Co7, because the magnetic moment in the SmCo2 blocks

THERMALEXPANSIONANOMALIES

105

should choose one of the (111) axes along the e axis under the influence of the uniaxial SmCo5 environment. Here it should be noted that the RCo2 blocks are not exactly cubic in R2Co7. In the hexagonal description the compound SmCo 2 has the lattice parameter ahex = aeub/V/2 = 512.5 pm (at 300 K) but it is only 504.9 pm in 5m2Co7 . This means that the SmCo2 blocks are compressed perpendicular to the c axis with a deformation of -1.5 x 10 -2. This deformation is almost temperature independent. We can neglect the possible deformation along the e axis, because the relative difference between the observed c parameter of Sm2Co7 and the value calculated as the sum of the heights of the RCo5 and RCo2 blocks is only 0.3 x 10 -2. SmCo2 has a huge spontaneous magnetostriction described by a ,)tll I constant which reaches - 5 x 10 -3 at the lowest temperature (Korolyov et al. 1990a). It is important that the spontaneous magnetostriction is negative. Therefore, due to a reverse magnetostrictive effect the compression should favour the magnetic moment in the SmCo 2 blocks to lie along the (111) axes with an angle of 700 from the c axis of Sm2Co7. The possibility of the magnetoelastic origin of the observed spin reorientation in Sm2Co7 and Tln2Co7, when the R atoms on the quasicubic sites are considered to be responsible for this phenomenon, has been discussed by Bartashevich et al. (1993a), taking into account a large negative magnetoelastic contribution to the first anisotropy constant from the compression. The presence of basal-plane anisotropy in SmCo3 at 4.2 K has been found by Ioshie and Nakamura (1990). It can be explained by the same mechanism as in S m 2 C o 7. In the cone range no orthorhombic distortion of the hexagonal lattice was observed. There are two reasons for the absence of noticeable distortion in Sm2Co7. First, the values of the A%4 magnetostriction constants, which contribute a dominant part to the 7-magnetostriction of R2Co7 are calculated to be smaller by factor 4 for R = Sm than for R = Nd, Tb, Tin (Andreev et al. 1988a). Second, the distortion e strongly depends on the angle 0o Even at the lowest temperature, e in Sm2Co7 cannot reach its maximum possible value due to a small deviation of the easy-magnetization direction from the e axis. The 7-magnetostriction of R2Co7 will be discussed in section 4.2.2.

4.2.1.4. Gd2Co7. This compound can crystallize in both structural modifications. However, its magnetic properties have been studied only on rhombohedral single crystals (Deryagin et al. 1979, Tarasov 1987, Andreev et al. 1985g, Ballou et al. 1992). Gd2Co7 is a ferrimagnet with a compensation point at 440 K. The magnetic anisotropy is uniaxial in whole ordered range. The structural and magnetic properties of G-d2Co7 are presented in tables 4.6 and 4.7. Buschow et al. (1974) have studied thermal expansion of Gd2Co7 on polycrystalline samples at 273-973 K using a strain-gauge technique and have observed a positive spontaneous magnetostriction, in agreement with a negative pressure derivative of the Curie temperature dTe/dP (Brouha and Buschow 1973). The thermal expansion on single crystals has been studied by Andreev et al. (1985g, 1988a). Temperature dependences of the lattice parameters and the unit-cell volume are qualitatively similar to those of Y2Co7 . Basal-plane and uniaxial strains are positive. They appear just below Te and increase monotonically with decreasing temperature. The

106

A.V. ANDREEV

compensation of magnetic moments of the Gd and Co sublattices does not manifest itself in the thermal expansion. 4.2.1.5. TbzCo 7. Tb2C07, like Gd2Co7, crystallizes in both modifications. Magnetic properties of rhombohedral and hexagonal single crystals have been studied by Deryagin et al. (1979), Kudrevatykh et al. (1983), Tarasov (1987) and Andreev et al. (1982b). The values of/~m and Tc of this ferrimagnet do not depend on the modification (table 4.7). The same holds for the compensation point (420 K), although the magnetic anisotropy differs considerably. In the hexagonal crystal, basal-plane anisotropy is found below T1 = 405 K and uniaxial anisotropy above T2 = 450 K, with a cone of easy directions between these temperatures. A strong anisotropy exists within the basal plane at low temperatures, and the b axis [210] is the easiest-magnetization direction (K3 < 0). The rhombohedral modification also displays uniaxial anisotropy above 450 K. But below this temperature it has a special anisotropy type which can be called 'cone' only conditionally. There are three easy axes that deviate from the basal plane at low temperature by 140 (their projections on the basal plane coincide with [210] axes). With increasing temperature, they slowly move towards to the basal plane, pass it at 330 K, and rotate further to the c axis reaching it at the same temperature as in the hexagonal crystal. The difference in the anisotropy reflects the presence of the symmetry plane perpendicular to the c axis in the hexagonal structure and its absence in the rhombohedral one (Andreev et al. 1982b). The structural and magnetic properties of both modifications of Tb2Co7 are presented in tables 4.6 and 4.7. The thermal expansion of hexagonal TbeCo7 and the rhombohedral solid solutions (Tbl_~:Y~)2Co 7 (0 x< z ~< 1) have been studied on single crystals by Kudrevatykh et al. (1983) and Andreev et al. (1988a). Among the already mentioned compounds, the behaviour of both hexagonal and rhombohedral Tb2Co7 is similar to that of Nd2Co 7. Only the sign of the orthorhombic distortion is opposite. The volume effect is the same for both modifications, as are the other properties originating from the exchange interactions (am and To). The linear strains and the orthorhombic distortion are also nearly the same at 5 K for hexagonal and rhombohedral crystals, but a slight difference exists in their temperature dependences. The 7-magnetostriction and the anisotropic a-magnetostriction of Tb2Co7 and its solid solutions (Tbl_~Y~)2Co7 will be discussed in sections 4.2.2 and 4.2.3, respectively. 4.2.1.6. Tm2Co7. The compound Tm2Co7 has the rhombohedral crystal structure (Gd2Co7 type). Magnetic properties have been studied by Tarasov (1987) on the samples cut from the large grains of an ingot. The misorientation of subgrains over samples was about 10 °, so that they could not be considered as real single crystals but more as strongly aligned polycrystals. TmzCo7 is a ferrimagnet with compensation point at 80 K (structural and magnetic properties are presented in tables 4.6 and 4.7). The compound has uniaxial magnetic anisotropy above 45 K. A spin reorientation leading to a cone of easy directions has been observed below this temperature. The cone angle 0 at 4.2 K is estimated as 454-150 from the projections of the spontaneous magnetization on the basal plane and the e axis. A considerable error is due to the

THERMAL EXPANSIONANOMALIES

107

large misorientation of the subgrains. For the same reason, the easy direction within the basal plane, which corresponds to a projection of spontaneous magnetic moment, is not found from the magnetization measurements. It is determined as [210] from the signs of the orthorhombic distortion and the 7-magnetostriction coefficients (see section 4.2.2). The thermal expansion has been studied on the same samples mentioned above (Zadvorkin 1987, Andreev et al. 1992b). In addition to the positive magnetostrictive strains, as found in all studied compounds R2Co7, there is a large negative orthorhombic distortion in the cone range. The spin reorientation, like in Sm2Co7, is accompanied by additional negative linear strains and volume effect. The additional strains decrease the corresponding normally found strains by 15% of their total values and will be discussed in section 4.2.4. The spin reorientations in Tm2C07 and Sm2Co 7 are not expected to originate from the competition in anisotropy between the Tm (Sm) and Co sublattices. Like the SmCo 2 blocks in Sm2Co7, the TmCo 2 blocks in Tm2Co 7 are compressed with a deformation of - 1 . 4 x 10 -2 perpendicular to the c axis. This occurs because in the hexagonal description the TmCo2 Laves phase has a lattice parameter abex = 503 pm (at 300 K) but it is only 496 pm in Tm2Co7. TmCo2 has the same negative )qll magnetostriction constant as SmCo2 ( - 5 × 10 -3 at 5 K, Andreev and Zadvorkin 1985). For Sm2Co7, the spin reorientation has been discussed by Bartashevich et al. (1993a) taking into account a large negative magnetoelastic contribution to the first anisotropy constant due to the compression. Therefore, the explanation of multiaxial anisotropy at low temperatures in Sm2Co7 (Bartashevich et al. 1993a) can be also applied to TmECO7. A similar situation can be expected in Er2Co7, where the signs of the Stevens coefficients ~ and/3, which determine the signs of anisotropy and magnetostriction constants of the R ions, are the same. The properties of this compound have not yet been studied on a single-crystal sample. However, singlecrystal measurements on the related ErCo3 compound, having an even larger part of the R2 ions in the quasicubic blocks (Bartashevich et al. 1992), did not reveal any spin reorientation. Therefore, we can suppose that in Er2Co7 the anisotropy is uniaxial within the whole magnetically ordered range. In this case, the absence of the spin reorientation in Er2Co7 can be associated with the low value of the Stevens coefficient/3, which determines the anisotropy of the R 2 ions, and with the relatively low value of ~111 - " --2.5 X 10 -3 in ErCo2 (Andreev et al. 1985e).

4.2.2. 7-magnetostriction First of all, several words should be said about signs of the orthorhombic distortion e, the 7-magnetostriction A"~ and the third anisotropy constant K3, which determines the easy-magnetization direction within the basal plane. The values of e, presented in table 4.7, are determined from X-ray diffraction data as e=

~

a

a.

(4.7)

It is positive, if the basal plane is stretched along the b axis, and negative, if stretched along the a axis. A positive value of K3 corresponds to an easy [100] axis

108

A.V. ANDREEV

(a), and a negative value to an easy [210] axis (b). As other type of magnetostriction, )cr is positive when the lattice is stretched along the easy axis, and negative when it is compressed in this direction. All possible combinations of signs of e, )(r and K3, can be presented as follows: K3

+

+

-

e

+

-

+

~'r

_

+

+

m

(4.8)

Therefore, the sign of )Cr depends not only on the sign of e, but also on the sign of /t23. For this reason, the distortion, both positive in NdzC07 and Tb2C07, corresponds to negative )(r in Nd2C07 (the lattice is stretched perpendicular to the easy axis) and positive in Tb2C07 (stretching parallel to the easy axis). For negative e in TmeC07 there are two possibilities, and we need to know either the sign of Ks to determine sign of .vr, or the sign of )(r to determine easy direction within the basal plane. The temperature dependences of A'r for Nd2Co 7 and ( T b l _ x Y x ) 2 C o 7 are presented in fig. 4.19. The fact that the magnitude of ,Vr is larger than 2 x 10 -3 can be taken as evidence that it originates from the single-ion mechanism. Two features of ,Vr should be noted. First, the Nd 3+ and Tb 3+ ions have the same sign of the c~ Stevens factor and hence the same sign of the second-order )(r,2 coefficient, while )(r < 0 in NdeCo7 and )(r > 0 in TbzCo 7. Second, ,Vr decreases with increasing temperature much faster than expected on the basis of the A-~,2 behaviour, neglecting the Cosublattice contribution. The huge 7-magnetostriction in R2C07 should be attributed to the quasicubic blocks RC02. This is because in RC05, where R content differs from that in R2C07 only by 33%, but where RC02 blocks are absent, no orthorhombic distortion is found. Since the Co sublattice plays an important role in the formation of anisotropic magnetostriction of the RCo2 compounds, and the structure of R2Co 7 contains RC02 blocks, one could suppose that the Co sublattice gives a large contribution to Xr. However, it cannot lead to both a negative ~'Y in Nd2C07 and a positive A'Y in Tb2C07 in the same way. At least in one of these compounds, a large Co contribution should make the A'r(T) decrease slower due to much slower decrease of IZco(T) than #u(T). The 7-magnetostriction in R2C07 has been satisfactorily described by Andreev et al. (1988a) taking into account the fourth-order constants ,,~13''4and ,~27'4. The formulae for the description of the magnetostriction up to sixth-order have been published by Clark (1980). For the 7-magnetostriction they can be reduced to the following:

)(r = ~'r'2sin2 0 + ~7'4 ( 6 - sin2 8) sin2 0 - 2~'~'4 sin4

(4.9a)

)(r = )(r'Z sinZ O + "~71'4( 6 - sinZ O) sine O + 2)~'4 sin4

(4.9b)

THERMAL EXPANSION ANOMALIES

109

a~

'o ¢
ac (Andreev et al. 1985d, in agreement with results of Fujii et al. 1987) for this temperature. However, at 1000 K e (pm)

o

1222

oJ°

1218 V (nm3)

o oOooo 0.944 / / / / / /

/

/ /

0.936 /

/

/

/ /

~'~

/

876

/

/

872

/ / /

0.928 ./

0

/ I

|

I

200

400

600

T (K)

Fig. 5.13. Temperature dependences of the lattice parameters a and e and the unit-cell volume V for Nd2Fe14B. The dashed curves represent the phonon contribution to the thermal expansion. The solid lines show the extrapolation from the uniaxial-anisotropy range (above 135 K) to the cone-anisotropy range. (After Andreev et al. 1985d.)

164

A.V. ANDREEV

c~c reaches the same value as c~a (Fujii et al. 1987). For the volume expansion a v the following values are found: 3.4 x 10 -5 K -1 at 800 K (Andreev et al. 1985d), 4 . 2 x 10 -5 K -1 at 1000 K (Fujii et al. 1987) and 4 . 8 x 10 -5 K -1 at 1000 K (Buschow and Grrssinger 1987). This disagreement between the a v values is the main reason for a similar disagreement in the spontaneous magnetostriction, since the use of different TD values (450 K in paper of Buschow and Grfssinger 1987) for the extrapolation does not lead to considerable differences in ws in the case of such huge ws. The results of Buschow and Grrssinger (1987), which have studied in the same way nearly all R2Fe14B, are presented in table 5.5. In the ferromagnetic range, the temperature dependences of both lattice parameters of Nd2Fe14B mimic the a, c(T) curves of Y2Fe14B, but only above the spin reorientation. Below 140 K additional negative magnetostrictive strains appear in the basal plane and along the c axis. They reach value of about - 0 . 3 x 10 -3 at 5 K, which corresponds to Aws ~, - 1 x 10 -3, or about 5% of the total Ws at this temperature (Andreev et al. 1985d). A small basal-plane anomaly, corresponding to negative additional magnetostriction, has been also observed by Fujii et al. (1987). However, no anomaly is found along the c axis. The anisotropic o~-magnetostriction in NdzFe14B does not exceed 5 x 10 -5 at all temperatures, as found on single crystals by Andreev et al. (1986d). Therefore, the observed anomalies at the spin reorientation are attributed to a change in the exchange a-magnetostriction. The ws values and their temperature dependence are found in all studies to be very similar to those in Y2FeI4B (if one neglects the additional effect and extrapolates Ws to low temperatures from the uniaxial range). This means that w~ in NdzFe14B is completely (or almost completely, according to Buschow and Gr6ssinger 1987) determined by the Fe sublattice. For this reason, the observed Aws effect at the spin reorientation apparently reflects a change in the magnitude of/ZFe, as discussed in the case of Tm2Fel7 (section 5.1.1.6). The change in magnetic moment, A#Fd#Fe, corresponding to Aws/Ws, is estimated to be 2-3% (Andreev et al. 1985d) or even smaller~ if one takes the larger ws value (Buschow and GrOssinger 1987). Fruchart et al. (1987) have studied 57Fe M6ssbauer spectroscopy and found a difference of about 2% (just on the level of the experimental error for such complicated spectra with six Fe sublattices) in the average hyperfine field between samples of Y2Fel4B, saturated in the easy and hard magnetization directions. This shows a possible anisotropy in magnitude of/~F~ in R2Fe14B and is, in principle, in good agreement with the above derived A#Fd#F~ values. But the sign of the obtained effect is the opposite, #F, should be larger when it is directed in the basal plane (Hhf = 30.8 T) than along the c axis (Hhf = 30.2 T). Moreover, Fruchart et al. (1987) have found no indication of this effect in the case of Nd2Fe14B. It should be noted that the volume effect Aw~ at the spin reorientation, found by Andreev et al. (1985d), only exceeds the experimental error by a factor of 3. According to Fujii et al. (1987), Aws has to be several times smaller than reported by Andreev et al. (1985d). In such case, a possible change in #F~ becomes lower than 1%, and it is nearly impossible to detect it experimentally. At least this change has not been observed from magnetization curves of single crystals. However, the effect seems to exist, being found in two independent studies. Another possible reason of this effect can be a noncollinearity

THERMAL EXPANSION ANOMALIES

165

between the Nd and Fe sublattices in the canted range, which is proposed on the base of neutron diffraction (Givord et al. 1985b), 57Fe-M6ssbauer spectroscopy (Onoreda et al. 1987) and 145Nd-M6ssbauer spectroscopy (Nowik et al. 1990). The appearan~ of any noncollinearity can influence the exchange-magnetostriction constants without a change in the Fe magnetic moment. As in the already discussed R2Fe14B compounds, ~s disappears not just at To, but at somewhat higher temperatures (about 40 K higher than Te according to Andreev et al. 1985d, and about 100 K higher according to Buschow 1986b; the latter discrepancy being due to the difference in thermal expansion coefficient in the paramagnetic range).

Sm2Fel4B. The magnetic properties of Sm2Fel4B have been studied on single crystals by Andreev et al. (1985d, h), Hiroyoshi et al. (1985). This ferromagnet is the only compound of the R2Fel4B family where the basal-plane contribution to the magnetic anisotropy from the R sublattice is strong enough to dominate in whole ordered range. Sm2FeInB has also large anisotropy within the basal plane, and the [100] axis is the easy-magnetization direction. Structural and magnetic properties of Sm2Fe14B are listed in tables 5.3 and 5.4. The thermal expansion of Sm2FeI4B has been studied on single crystalline samples by Andreev et al. (1985d) using X-ray dilatometry in the temperature range 5-800 K. Buschow (1986b) and Buschow and Gr6ssinger (1987) have studied isotropic polycrystalline samples by standard dilatometry in the range 4.2-1000 K (their data are shown in table 5.5). Despite the presence of magnetic Sm instead of nonmagnetic Y and a different type of magnetic anisotropy, the thermal expansion of Sm2Fe14B, linear as well as volume, is found to be remarkably the same as in Y2FelnB. This

5.2.1.4.

TABLE 5.5 Malume thermal-expansion coefficient and volume spontaneous magnetostriction for R2Fe14B compounds (After Buschow and Grfssinger 1987). txv is the volume thermal expansion coefficient at 1000 K; ws is the volume spontaneous magnetostriction; W~d is the Fe-sublattice contribution to the volume spontaneous magnetostriction; a~f is the contribution to the volume spontaneous magnetostriction, connected with the rare-earth sublattice. Compound

av 10-5 K-1

~os (4.2 K) 10-3

~O~d (4.2 K) 10-3

~%f (4.2 K) 10-3

La2Fel4B Ce2Fel4B PrEFe14B Nd2Fel4B Sm2Fe14B Gd2Fel4 B Tb2Fel4 B Dy2Fe14B Ho2Fe14B Er2Fel4B Lu2Fel4B

4.95 6.06 4.86 4.77 4.65 4.68 4.68 4.68 4.65 4.77 4.68

25.2 32.1 27.9 27.9 27.3 33.6 33.6 35.1 35.4 31.2 27.0

25.2 32.1 25.3 25.5 25.8 26.1 26.1 26.4 26.4 26.5 27.1

0 0 2.5 2.4 1.5 7.5 7.5 8.7 9.0 4.6 0

166

A.V. ANDREEV

indicates again the dominant role of the Fe sublattice in the spontaneous magnetostriction of R2Fe14B and a negligible anisotropic t~-magnetostriction compared to the exchange striction. Also the 7-magnetostriction, which could manifest itself in SmzFe14B due to basal-plane arrangement of the magnetic moments, is found to be lower than 1 x 10 -4 from the absence of an orthorhombic distortion (Andreev et al. 1985d).

5.2.1.5. Gd2Fe14B. Single-crystal data on the magnetic properties of GdzFe14B have been obtained by Bog6 et al. (1985), Hirosawa et al. (1986), Nakagawa et al. (1987). Single crystals of this compound and its hydride GdEFe14BH3. 4 have been studied by Bartashevich and Andreev (1989,1990). Bartashevich et al. (1990) have investigated single crystals of C,-d2FeI4B and of the C,-d2(Fel_xCox)14B solid solutions up to z = 0.3. These are collinear ferrimagnets with uniaxial magnetic anisotropy in the whole ordered range. The value of K1 is about 25% lower than in Y2Fe14B and mimics its nonmonotonic temperature dependence. Structural and magnetic properties of Gd2Fe14B are presented in tables 5.3 and 5.4. Thermal-expansion data on Gd2Fel4B are available from the single-crystal studies (Bartashevich and Andreev 1989, 1990, Bartashevich et al. 1990) and isotropicpolycrystalline studies (Buschow 1986b, Buschow and Grrssinger 1987). The latter results are presented in table 5.5. The thermal expansion, linear as well as volume, is found to be similar to that of Y2Fe14B. About of 10% (Bartashevich et al. 1990) or 20% (Buschow and Gr6ssinger 1987) of the total Ws is attributed to the R-Fe interaction contribution. 5.2.1.6. Tm2Fel4B. The magnetic properties of Tm2Fel4B have been studied on single crystals by Hirosawa and Sagawa (1985), Yamada et al. (1985), Hirosawa et al. (1986). This ferrimagnet has at low temperatures basal-plane magnetic anisotropy with the [100] axis as easy-magnetization direction. Spin reorientation to the uniaxial anisotropy, which occurs at 310 K, is a first-order phase transition, as found from magnetization measurements (Hirosawa and Sagawa 1985) and confirmed by specific-heat measurements (Fujii et al. 1987). However, in the latter study the spin reorientation has been observed at 325 K. An anomalously large anisotropy in the saturation magnetization is observed around the spin reorientation, ~m along the c axis just above Ts is by 6% (about of 1.2#B) larger than #m in the basal plane just below Ts. The magnetic structure of Tm2Fel4B is found by neutron diffraction to be noncollinear below Ts (Yamada et al. 1985), which is in principle confirmed by the 169Tm MSssbauer study (Gubbens et al. 1987). Structural and magnetic properties of the compound are presented in tables 5.3 and 5.4. The thermal expansion has been measured on aligned polycrystalline samples by Fujii et al. (1987) at 4.2-1100 K. It is found to be generally similar to that in Y2Fel4B and Nd2Fel4B (fig. 5.14). The values of Aa is also much larger than he, ;as is the same as in the two latter compounds and varies with temperature in proportion to the square of #Fe- Additional positive spontaneous magnetostrictive strains are observed at the spin reorientation both in the basal plane and along the c axis. The corresponding volume effect Aws can be estimated as 0.6 x 10 -3 or about 3.5% of

THERMALEXPANSIONANOMALIES

....~ I

I

I

t

]

Y2Felt'B /

167

i

~

~. ,R

.t2

1__.

0

20o

40o

600

8oo

~000 ~200

T (K)

Fig. 5.14. Temperaturedependences of the magnetostrictivestrains along the c axis (ell) and in the basal plane (c.L) for R2Fel4B with R = Y, Nd and Tin. Inset: temperature dependence of volume magnetostrictionfor Y2Fel4B. (After Fujii et al. 1987.) O~s(Ts) which should have some correlation with the observed jump in #m at the spin reorientation. This jump can be attributed either to one of the three effects: 1) an increase of/~F~, 2) a decrease of #Tin, 3) a change of the angle between the sublattices (including 'noncollinear-collinear structure' transition), or to their combination. It is difficult to explain the jump by means of effect 3), because an increase of total magnetic moment of Tm2Fe14B would correspond to a more noncollinear magnetic structure at high temperatures than at low temperatures. This seems to be unlikely, since the noncollinearity can be taken to originate from the Tm sublattice. The 57Fe MOssbauer study (Price et al. 1986) has shown that the Hhf are lower in the uniaxial state on all Fe sites except 8jl. The average Hhr decreases from 28.6 T at 294 K to 26.4 T at 318 K, which reflects a decrease in #F~ upon rotation from the basal plane to the c axis (if we attribute the volume effect only to the Fe sublattice). This correlates with the sign of the observed Aws. However, a value of A].tFe/#Fe of about 8% corresponds to AWs/Wsof about 16%, which is much larger than the observed effect. From these results, it appears that there is an additional difficulty in explaining the jump in the #re(T) dependence. Using the conversion factor 14.8T/#B between the hyperfine field and the Fe magnetic moment (Buschow 1986a), not an

168

AN. ANDREEV

increase of about 1.2#s, but a decrease in/tin of about 2.1#B can be evaluated from these data. In such case, a decrease of about 1.6#a in #Tin is needed for explaining /zm(T). From the results of the 169Tm MOssbauer study (Gubbens et al. 1987) some decrease of #Tin could be evaluated, but this is not larger than 0.5#a.

5.2.1.7. Other R2Fel4B. The thermal expansion of R2Fel4B with R = La, Ce, Er has been studied by Buschow (1986b) above room temperature on polycrystalline samples. Buschow and Gr6ssinger (1987) have investigated the thermal expansion in R2Fe14B with nearly all R (except R = Y, Tm, Yb) on polycrystals by standard dilatometry at 4.2-1000 K. The results on some of these compounds, considered in sections 5.2.1.2-5.2.1.5, have been already discussed in the corresponding sections. Here we will mention the results on the thermal-expansion and spontaneousmagnetostriction properties of compounds, which are reported only in the original papers. The data are presented in table 5.5. For all studied compounds the high-temperature volume-expansion coefficient (well above ire, at ,,, 1000 K) is found to be nearly the same, C~v ~ 4.8 x 10 -5 K -1. The only exception is CeEFeI4B , where ~v is higher (see table 5.5). The value TD = 450 K is taken for the extrapolation of the paramagnetic curves to the ferromagnetic (ferrimagnetic, in the case of heavy R) range. The values of Ws in compounds with trivalent nonmagnetic R (La, Lu) fall in the range (25-27) x 10 -3. In Ce2FelaB with tetravalent Ce, ~os is noticeably higher. The difference between compounds with magnetic and nonmagnetic R is attributed to the R-sublattice contribution (the Fe-sublattice contribution ws,d, shown in table 5.5, is determined for compounds with magnetic R by interpolation of the data of LazFe14B and LuzFe14B). The ~0s,fcontribution originating from the R sublattice is found to be always positive. For light R it is less than 10% of the total ~o~. In the case of heavy R, w~,f is considerably larger and reaches 25% of ~0s in HozFelaB. Buschow and Gr6ssinger (1987) have concluded that ws,f is roughly (we should add, very roughly) proportional to the total R moment and do not find a satisfactory explanation for this behaviour. No thermal-expansion anomaly is reported to accompany the spin reorientation in Er2FelaB, which occurs at 325 K. A very large magnetovolume effect is found in all compounds studied at the Curie temperature, (5-12) x 10 -3, which is equal in some cases to 30% ofws (4.2 K), while ws disappears far above ire. A strong decrease of the ws(T) dependence occurs in an interval of about 50 K above ire. Then, ~s slowly vanishes within a 100-150 K temperature range. 5.2.1.8. R2Fe14C. Intrinsicproperties of the R2FelaC carbides are not studied so extensively as their isostructural boron counterparts. The R2Fe14C compounds do not crystallize from the melt and must be formed by a solid-solid transformation in an appropriate temperature range (Gueramian et al. 1987, Buschow et al. 1988, De Mooij and Buschow 1988, van Mens et al. 1988). For this reason, no singlecrystal results are reported. The R2Fel4C compounds exist for all R for which the corresponding borides R2Fe14B exist, except R = Yb and Th. The compounds

THERMALEXPANSIONANOMALIES

169

RzFe14B and R2Fel4C have almost the same lattice parameter a, but e and, consequently, the unit-cell volume is smaller by about 1.5% in carbides. The values of Tc are lower by about 50 K, and/~Fe is also smaller by several percents (exact data on magnetic moment are not available because of the absence of single crystals). However, the magnetic properties of R2Fel4C closely resemble those of R2FelnB. The thermal expansion of R2Fe14C with R = Nd, Gd, Tb, Dy, Ho~ Er and Lu has been studied by Buschow (1988b) from room temperature up to 1000 K by standard dilatometry, so that the results obtained pertain only to the spontaneous volume magnetostriction. They have been compared with the magnetostriction of R2FelaB only qualitatively because of the absence of low-temperature data. It is shown that R2Fel4C exhibit similar but less-pronounced anomalies in the thermal expansion compared to those in R2Fel4B. The lower spontaneous magnetostriction is connected with lower /zFe in the carbides than in borides. In the carbides the spontaneous volume magnetostriction extends to temperatures considerably above the Curie temperature, as in the borides. In a later study of Lu2Fel4C and Lu2Fe14B the measurements are extended to 10 K (Kou et al. 1991a, b) and the conclusion about lower ~s in the carbides is confirmed..

5.2.2. Discussion All studies of the thermal expansion in R2Fe14B show a huge spontaneous volume magnetostriction, which is substantially higher than that previously found in other magnetically ordered materials. The quantitative differences between the various results reported appear first of all to be due to different values of volume thermal-expansion coefficient in paramagnetic range and, consequently, due to different extrapolation of the paramagnetic V(T) dependences to the ordered range. Indeed, for NdEFe14B etv = 3.4 x 10 -5 K -1 (Andreev et al. 1985d) leads to ~Os = 20 x 10 -3, av = 4.1 x 10 -5 K -1 (Fujii et al. 1987) gives ~0s = 24 x 10 -3 and a v = 4.8 x 10 -5 K -1 (Buschow and Grrssinger 1987) gives 0~s = 28 x 10 -3. The same origin leads to a disagreement in the results about the volume effect at Te. All studies find it to be very large, but it varies from 15% to 30% of a~s (0 K). The reason of this disagreement in data on c~v is not known. A further contradiction concerns the contribution to ~Os from the R sublattice through R-Fe intersublattice interaction. This contribution varies from negligible (Fujii et al. 1987, Andreev et al. 1985d) or about 10% of total a;s (Bartashevich et al. 1990) to rather substantial values (up to about 25%, Buschow and Grrssinger 1987). However, a large scatter in this contribution through the R series exists in the latter report and the observed behaviour has no satisfactory explanation. We should note that a possible reason of the scatter in the data can be crystallographic texture present in the samples. The samples used for the measurement are cut from the arc-melted ingots which always have large crystallographic texture due to the solidification conditions. In the case of anisotropic thermal expansion, as present in R2Fe14B below Te, this texture is very important, because the usual relation AV/V = 3AI/l used for the determination of volume effects from linear effects becomes incorrect. (This reason cannot explain the above mentioned difference in av

170

A.V ANDREEV

at high temperature, because the thermal expansion of R2Fel4B in the paramagnetic range is found to be isotropic.) Now we will discuss a possible correlation between the two anomalies of R2Fel4B which have been found soon after the discovery of these compounds as permanentmagnet materials: the huge spontaneous magnetostriction leading to a negative thermal expansion in a wide temperature range and the nonmonotonic behaviour of the Fe-sublattice anisotropy. Such a correlation means that the anomalous thermal expansion is the reason for the nonmonotonic K1 behaviour, as is proposed for Y2Fel4B by Bolzoni et al. (1987b):

KI(T)/KI(O)- [M(T)/M(O)]3 [ 1 - o~(c/a)2],

(5.2)

where M is the magnetization. The second factor in this formula (Carr 1958) should take into account the temperature dependence of the c/a ratio. The best fit to the KI(T) data has been obtained with the coefficient a = -0.52 using the thermalexpansion data of Givord et al. (1985a). The description has been developed further by Cadogan and Li (1992). First of all, if one accepts this model, it is seen from eq. (5.2) that not large spontaneous magnetostriction, but a large temperature variation of the c/a ratio can influence the magnetic anisotropy. This can occur not only due to spontaneous magnetostriction but also due to anisotropy of the 'normal', phonon, thermal expansion. On the other hand, an isotropic distribution of the magnetostrictive strains over main axes does not change c/a, so that it does not influence the magnetic anisotropy even when Ws is large. Therefore, it is unreasonable to attribute the anomaly in KI(T) only to the large Ws without any data on the anisotropy in linear thermal expansion. This has been done, for example, by Kou et al. (1991b), who have found no KI(T) anomaly in Lu2FelaC, in contrast to Lu2Fel4B, and associated this with less-pronounced Invar-like behaviour of the carbide. As already is mentioned in section 5.2.1.1, the thermal-expansion results of Givord et al. (1985a) used for fitting of KI(T) by eq. (5.2) are in disagreement with the other data available. In all other studies on single crystals and aligned polycrystalline samples, the c(T) dependence, as well as c/a(T), is found to be monotonic. When substituted into (5.2), it does not lead to an anomalous KI(T) dependence at any o~ value. Several other facts can be presented as evidence for no connection between the two anomalies: 1. The thermal expansion of ThzFe14B is very similar to that of Y2Fel4B with an even more pronounced Invar effect (see section 5.2.1.2). However, the KI(T) dependence in this compound has no anomaly. 2. Figure 5.15 shows the temperature dependences of the c/a ratio in the Y2(Fel_~Cox)14B solid solutions. One can see that these curves are similar for all compounds studied, but the anomaly in KI(T) rapidly disappears with increasing Co content. The compounds with x = 0.1 and 0.2 have completely the same c/a(T) dependences below ,~ 500 K. However, KI(T) is nonmonotonic for x = 0.1 and normal for x = 0.2 (Bartashevich et al. 1990). 3. Bartashevich and Andreev (1989, 1990) have studied the thermal expansion as well as the magnetic anisotropy on single crystals of the hydrides R2Fe14BH3.4

T H E R M A L EXPANSION A N O M A L I E S

171

c/a 1' 2 3 ~ 1.380

1.375

1.370

I

o

T

400

I

I

8oo

I

T

K)

Fig. 5.15. Temperature dependences of the c/a ratio for the Y2(Fel_xCox)14B solid solutions. 1: a: = 0; 2: x = 0.1 3: x = 0.2; 4: x = 0.3. The dashed curves are extrapolations from the paramagnetie range. T h e arrows indicate the Curie temperature. (After Andreev and Bartashevich 1990a.)

with R = Y and Gd. They have found that the monotonic increase of c/a with increasing temperature for the pure R2Fe14B compounds changes into a decrease in the hydrides. However, the KI(T) dependences remain nonmonotonic. Moreover, K1 at low temperature is negative in GdzFe14BH3.4 and the plane-cone-axis spin reorientation occurs at 280-315 K. Such KI(T) curve cannot be fitted using formula (5.2). 4. The difference in lattice parameters between La2Fel4B and Lu2FeI4B is 1.4% (a parameter) and 4.1% (e). The e/a ratio differs by 2.7% (data are taken from Herbst 1991). However, such large variations do not change the character of the Kx(T) curve. For both compounds, KI(T) is nonmonotonic and nearly the same. Thermal expansion influences the interatomic distances much more gently, all changes in a, e and e/n do not exceed 0.5% below To. Point-charge-model calculation shows (Bartashevich et al. 1990) that the change in Kx in Y2Fea4B owing to the thermal expansion would be less than 1% in the temperature interval 0-0.5To where nonmonotonic behaviour of Kx is observed. Extrapolation of the high-temperature part of the KI(T) curve to 0 K, using the third power of the magnetic moment, gives a much larger difference (about 40%) with the actual K1 (4.2 K) value, which cannot be explained by a possible influence of the thermal expansion. Therefore, it has to be concluded that there is no correlation between these two well-known anomalies observed in RzFea4B.

5.3. Other compounds 5.3.1. RFe12_xM~: The RFe12_~M¢ compounds have the tetragonal crystal structure of the ThMn12 type. The structure is shown in fig. 5.16. It belongs to the I4/mmm space group. Unit cell

172

A.V. ANDREEV

2(o)~

8(f)0

8(i)0 8(j)@

Fig. 5.16. Unit cell of the tetragonalcrystalstructureof the RFel2_xMx compounds (ThMn12type). contains 2 formula units. The T h i n l 2 structure is slightly related with the hexagonal Th2Nil7 type. As in the case of R2Fe17, the crystal structure of RFea2-rMr can be derived from the CaCu5 type when replacing half of R atoms by the Fe 'dumbbells'. The replacement can be described by the scheme 2RFes-R+2Fe = REel2. For more detailed comparison of the crystal structure of 1 : 5, 2 : 17 and 1 : 12 compounds, see Li and Coey (1991). The R atoms in RFe12_=M~ occupy a single 2a thorium site, while Fe and the third element M are distributed over the 3 manganese sites, 8f, 8i and 8j. The Fe atoms are found to occupy the 8f and 8j positions fully while the 8i site is populated by a mixture of Fe and M atoms. Such distribution is found for all M except Si. The Si atoms occupy the 8f and 8j sites. The 8i position in the ThMnl2 type of structure corresponds to the 'dumbbells' having the shortest Fe-Fe distance dFeFe in the Th2Nil7 type. However, the shortest dFeFe in RFelx_=Mr is not between atoms on the 8i sites but between those on the 8f sites (in YFellTi, 297 pm and 249 pm, respectively, Moze et al. 1988). The RFe12_rMx compounds exist for R elements from Ce to Lu (except Pm, Eu, Yb) as well as for R = Y. The family also contains three actinide representatives, UFel0Si2, UFel0MO2 (Suski et al. 1989) and ThFel0V2 (Buschow and de Mooij 1989). The magnetic properties of RFel2-=Mr have several general features. The compounds with nonmagnetic and light R elements are ferromagnets and those with heavy R elements are ferrimagnets. The average Fe moment is found to be 1.35--1.9#B, depending on the specific M element (the minimum value of/IFe corresponds to the Mo compounds). The relatively low/iFe compared to other Fe-rich intermetallics is attributed to a filling of the 3d band by electrons from M atoms. The Curie temperatures vary from 400 K to 650 K (260-500 K for the Mo series) with the highest value for R = Gd in each series. The magnetic anisotropy of the Fe sublattice is uniaxial. Its magnitude is close to that of R2Fel4B, but monotonically decreases

THERMAL EXPANSION ANOMALIES

173

with increasing temperature. The R-sublattice anisotropy is uniaxial for R = Sm, Tm and multiaxial for R = Pr, Nd, Tb, Dy in agreement with the different sign of the second-order Stevens factor a for these groups of R ions. The compounds with Er and Ho deviate from this trend in all M series. In the Ho compounds the uniaxial anisotropy is enhanced and in the Er compounds one has negative contribution to the total K1. This leads to a spin reorientation in the Er compounds as in the compounds where the R ion has a negative a value. These exceptions are explained by a strong influence of higher-order crystal-field terms (Li and Coey 1991, and references therein). The thermal expansion of RFet0Si2 (R = Nd, Sm, To, Ho, Er, Y), YFel0MO2 and YFelo.sTil.2 has been measured by Buschow (1988b) by means of standard dilatometry on polycrystalline samples from room temperature to 1000 K. The data are compared with the thermal expansion of the R2Fe14C carbides measured in the same way. Spontaneous magnetostriction has not been calculated, but it is clearly shown that strains in RFeI2_xM~ are considerably smaller than in R2Fel4B and R2Fe14C. A negative thermal expansion is observed only for Ho and Er compounds in limited temperature range just below Te. The average linear thermal-expansion coefficient in the 300-550 K temperature range is found to be rather low, 1.3 x 10 -6 K -1, in YFel0.sV1.5. The compound is considered as good Invar-type alloy. Andreev et al. (1989) have studied thermal expansion by X-ray dilatometry on polycrystalline samples at 5-800 K for three representatives of the RFel2_~M~ series (YFenTi, YFel0.sVL5 and ErFel0.sV1.5). The results are in general agreement with data of Buschow (1988b), showing that O~s is considerably lower than in compounds with the Nd2Fe14B structure. The thermal expansion in the paramagnetic range (at 800 K) is found to be nearly isotropic (t~a = 1.25 x 10 -5 K -1, t~e = 1.1 X 10 -5 K -1) which is similar to R2Fel7 and Nd2Fe14B. The linear magnetostrictive strains Aa and Ae in YFen Ti at 5 K are equal to 4.5 x 10 -3 and 1.0 x 10 -3, respectively. Thus, in contrast to the R2Fel7 and R2Fe14B compounds, the magnetostriction strain along the shortest Fe-Fe distance in YFenTi (between the Fe atoms on the 8f sites) is considerably smaller than that in the perpendicular direction. The volume effect is equal to (10 4- 1) x 10 -3 for all studied compounds. The value of Ws depends on the temperature as #2e. Despite the fact that a~s is only slightly smaller than in RxFe17, there is no Invar-like behaviour in the studied RFe12_~Mx compounds due to the nearly twice larger Curie temperature. The value nFeFe = 3.2 X 10-3#B 2 has been found using #Fe = 1.73#B at 4.2 K in the single-crystal data for YFenTi by Kudrevatykh et al. (1990). The thermal expansion of YFet0.5Vl.5 and ErFel0.sV1.5 is very similar to that of YFenTi. The spin reorientation of the cone-axis type in the Er compound, which occurs at 65 K, is not accompanied by a noticeable anomaly in the a(T)and e(T) dependences. This shows that anisotropic magnetostriction is lower than 2 x 10 -4. The volume effect at Tc is smaller and vanishes faster with increasing temperature above Te than in R2Fel4B and R2Fe14C , in agreement with the conclusion of Buschow (1988b). For the RFe12-rMr compounds, thermal-expansion data obtained on single crystals are available only for one compound. Andreev et al. (1990b) have studied the magnetic properties and the thermal expansion of DyFenTi. This compound is a

174

A.V. ANDREEV

strongly anisotropic ferrimagnet with Tc = 550 K and #,,~ = 1 0 . 0 # B at 4.2 K. The magnetic anisotropy at high temperatures is uniaxial and its temperature dependence above room temperature is close to that measured on single crystals of YFellTi and LuFellTi (Andreev et al. 1988c, Kudrevatylda et al. 1990). Below 220 K, the multiaxial anisotropy of the Dy sublattice dominates. At 4.2 K the magnetic moment is oriented at an angle 0 = 800 with the [001] axis, and its projection on the basal plane coincides with the [110] axis. A very large anisotropy exists within the basal plane, even larger than the uniaxial anisotropy. The easy-magnetization direction approaches the [001] axis in two steps. At 120 K a sharp decrease of 0 to 450 occurs by a first-order transition (Andreev et al. 1990b have suggested the transition to be of second order because of the absence of an observable hysteresis, but actually it should be of first order). The cone angle gradually decreases at 160-220 K and a usual second-order transition of the 'cone-axis' type is observed at 220 K. The magnetic properties of DyFellTi have been also studied on a single crystal by Hu et al. (1990). According to these results, 0(4.2 K) = 900 (i.e. the anisotropy is not of the cone type but of the basal-plane type), and the easy-magnetization direction is the [100] axis, the first-order transition occurring at 58 K and the 'cone-axis' reorientation at 200 K. The temperature dependences of the lattice parameters and the unit-cell volume for DyFellTi are shown in fig. 5.17. In the paramagnetic range (at 800 K) O~a - 1.2 x 10 -5 K -1, ae = 1.30 x 10 -5 K -1. The magnetic ordering is accompanied by positive strains which reach the values Aa = 3.2 x 10 - 3 and A~ = 1.2 x 10 - 3 at 5 K. This corresponds to the volume effect 7.6 x 10 -3. The low-temperature spin reorientation does not influence the thermal expansion. The second spin reorientation (at 220 K) is accompanied by additional negative magnetostrictive strains along the e axis and within the basal plane. Figure 5.18 presents the temperature dependence of Ws. The ws(T) curve is fitted in the uniaxial-anisotropy range with nFeFe X #2F~(T) using nF~F~ = 2.8 x 10-3/tB 2, which is slightly lower than found by Andreev et al. (1989), and using #F~(T) taken from YFellTi single-crystal data (Kudrevatykh et al. 1990). Extrapolation to 0 K gives ws = 9x 10 -3 and an additional effect at the second spin-reorientation of Aws = - 1 . 4 x 10 -3 or 15% of extrapolated ws. The extrapolated value of Ws is close to those found in YFellTi, YFel0.sV1.5 and E r F e l 0 . 5 g a . 5. According to the 57Fe-M6ssbauer-effect measurements of Hu (1990), Hu et al. (1989), a discontinuity of about 5% is observed in average hyperflne field (and, consequently, in #F~) at the second spin-reorientation. The expected value of Aw~ due to this hyperfine field anisotropy and concomitant A#ve is about 10%, which roughly corresponds to the observed value. The sign of Aws is also in agreement with A#F~. The absence of a noticeable anomaly at the low-temperature spin reorientation shows that #F, does not change. Andreev et al. (1990) have attributed this reorientation to a transition from a noncollinear to a collinear structure within the Dy sublattice. In this case, due to negligible contribution from the Dy-Fe exchange interaction and anisotropic magnetostriction (this is confirmed by absence of an orthorhombic distortion in the multiaxial range), no anomaly is expected in the thermal expansion. 57Fe-M6ssbauer-effect results show some change in #Fe, but this is rather questionable, at least it is considerably smaller than at second spin reorientation.

THERMAL EXPANSION ANOMALIES

175

a(pm) 854 852 850

C(pm)

S

848 VtnmJ~- - " 0.350[ _

ao,.

/

482

Tc

J

480

/ ff

,"

~

478

~

I 2"," Vo/z

0.346 ~

/ i /

f

0.342 i

r

l

400

I

800

T(K) Fig. 5,17. Temperature dependences of the lattice parameters tz and e and the unit-cell volume V for DyFellTi. The dashed curves represent the phonon contribution to the thermal expansion. The solid curves show the extrapolation from the uniaxial-anisotropy range (above 220 K) to the eone-anisotropy range. (After Andreev et al. 1990b.)

The thermal expansion of the actinide representative of RFe12_xM~, UFel0Si2, has been studied by Andreev et al. (1991b). In the few known U compounds of high 3d content (UCo5.3 and RI_~U~Co5), the U atoms do not carry a magnetic moment. The values of #d and Te and the magnetic anisotropy are considerably smaller compared to RCo5 with nonmagnetic R (La, Y) due to a filling of 3d band by additional electrons from U (Deryagin and Andreev 1976, Deryagin et al. 1978, Andreev 1979). One would expect that in UFel0Si2 the situation is similarly relative to the compounds YFel0Si2 and LuFel0Si2. However, the magnetic properties of UFel0Si2 differ considerably from such a prediction. One finds a much higher Te in UFel0Si2 compared with YFel0Si2 and even with GdFel0Si2, pointing to a considerable U-Fe exchange interaction. This means that U atoms have a noticeable spin moment S. On the other hand, the large contribution to the magnetic anisotropy and the field-induced phase transition in the hard direction point to a large orbital moment L of U. The situation seems to be qualitatively similar to that in UFe2 (Andreev 1979, Andreev et al. 1979, Popov et al. 1980, Brooks et al. 1987, Wulf et al. 1989). In UFe2 the spin and orbital moments of U cancel each other almost completely, #ti < 0.1/-tB. The Fe moment in UFe2 is reduced to 0.6#B in comparison

176

A.V. ANDREEV

,(lo 3)

i

i

i

30O

i

i

600

T(K) Fig. 5.18. Temperaturedependencesof the spontaneousvolume magnetostriction;as for DyFellTi. The solid curve is a fit of ws(T) with n FeFeX/~Fe' 2 using/ZFe(T) from YFellTi. (After Andreevet al. 1990b.) with #Fe = 1-5/-tB in YFe 2 due to an additional filling of the 3d band by electrons transferred from U. In UFel0Si2, #Fe is reduced to 1.6#B (1.8#B in YFel0Si2). The cancellation of the L and S moments of U is not complete, /1o = 0.5/~B (Andreev et al. 1993). The thermal expansion of UFel0Si2 is shown in fig. 5.19. At 5 K Ws is equal to 10.2 x 10 -3 like in other RFe12_=M=. However, its distribution over the main axes is opposite. The linear strains Aa and Ao are equal to 2.1 x 10 -3 and 6.1 x 10 -3, respectively. Another peculiarity of the thermal expansion of UFel0Si2 is the sharp decrease of o~s with increasing temperature. Since # o is rather small compared to 10#Fe, the temperature dependence of #m is nearly completely determined by the Fe sublattice. As seen in fig. 5.19, ws(T) does not scale with #Zm(T). This can be considered as evidence for a large contribution to Ws from the U-Fe exchange interaction, in contrast with other RFe12_=M~ compounds, where the analogous R-Fe contribution does not exceed 10% of ws. An attempt to estimate the U-Fe-interaction contribution is shown in fig. 5.20. The second curve in this figure represents the Fe-Fe contribution calculated using nFeF~ = 2.8 X 10-3#B 2 from above discussed results on DyFellTi, for the YFellTi single crystal and #Fe (4.2 K) = 1.59/~B in UFel0Si2 found from the 57Fe-Mrssbauer-effect measurements of the UI_~Y=Fel0Si2 solid solutions (Andreev et al. 1992c). The U-Fe-interaction contribution is estimated to be unexpectedly high, about 30% of ws, which is much larger than in the case of rare-earth metals. This generally correlates with the higher Curie temperature of UFel0Si2. The reason of unusual U behaviour in UFel0Si2 is still unknown.

#~e(T/Te)

THERMAL EXPANSION ANOMALIES

177

a(pm) 840

c(pm) 474

836

472

/ /

832 /

/

/

/,

I

t

/

C,3s(IO-3', ,~//

12

I 470

I 100

100

0

400

Tc 800

T(K) Fig. 5.19. Temperature dependences of the lattice parameters a, c, the spontaneous volume magnetostriction ~as and the square of molecular magnetic moment # 2 for UFeloSi2. The dashed curves represent the phonon contribution to the thermal expansion. (After Andreev et al. 1991b.)

5.3.2. Compounds with NaZn13 type of structure Rare-earth intermetallic compounds with cubic crystal structure of the NaZn13 type exist only for R -- La. The compound LaCo13 has very a high Te (1280 K, Buschow 1980) making it difficult to observe thermal-expansion anomalies at magnetic orderhag. The anisotropic magnetostriction in this compound is found to be 1 x 10 -4 at room temperature (Mushnikov et al. 1992). Although no intermetallic compounds are known to exist in the binary La-Fe system, the NaZnl3-structure compounds based on Fe and La can be stabilized by third element, either Si or A1. Compounds with Si, La(Fe~Sil_r)13, are found to exist in the range 0.81 ~< z ~< 0.88, A1 compounds, t a ( F e r m l l _ r ) 1 3 , exist in the range 0.46 ~< z ~< 0.92 (Palstra et al. 1983, 1985). In this limited concentration range the compound of A1 system may be found ha three magnetic structures: (i) Micromagnetism at low Fe content. (ii) At higher Fe content a soft ferromagnetic state exists where the Fe moments follow closely the Slater-Pauling curve. (iii) An antiferromagnetic state at the highest Fe content, with sharp metamagnetic transitions to the fully saturated ferromagnetic state accompanied by a large hysteresis of the order of 5 T (Palstra et al. 1985, 1986).

178

A.V. ANDREEV

12

I

I

I

~

I

S i2

cb .r-v

2.8"1 0-3 ~Fe2

~'X~

4

,

0

I

0.2

,

I

,

0.4

I

0.6

,

I

n

0.8

T/T c Fig. 5.20. Temperature dependences of the spontaneous volume magnetostriction Ws for UFel0Si 2. The dashed curve is a fit of ~s(T) with nF~v~ X ~ (see text).

A large spontaneous magnetostriction is found to appear in both the ferro and antiferromagnetic compounds below about 280 K, independent of the ordering temperature of the compound (which is within the 180-250 K interval for compounds studied in this way, Palstra et al. 1985, 1986). At 4.2 K, ~Vs adopts values reaching from 7 x 10 -3 for x = 0.65 to 19 x 10 -3 for z = 0.86. The forced volume magnetostriction about 10 x 10 -3 accompanies the metamagnetic transition in compound with x = 0.89. The total volume effect for the ferromagnetic state in this compound (spontaneous magnetostriction a;s in antiferromagnetic state and forced magnetostriction w) exceeds 24 x 10 -3, one of the largest values known. The authors discuss the results on the base of Shiga's theory (Shiga 1981) taking into account both localmoment and band-part contributions. Yermolenko et al. (1988) have performed thermal-expansion studies by X-ray diffraction of the antiferromagnetic compound La(Fe0.ssAI0.12)13 and of the Co solid solutions of this compound with small amounts (up to 1%) of Co instead of Fe. The Fe-Co substitution leads to a very sharp decrease of the metamagnetic-transition field and compounds with more than 0.5% of Co are ferromagnets. The values of ~s are found to be 13 x 10 -3 and 21 x 10 -3 at 5 K for the antiferromagnetic and ferromagnetic samples, respectively. This is in good agreement with data of Palstra et al. (1985, 1986). Buschow (1985) has studied the thermal expansion of variety of La(Fe, X, Co)13 compounds (X = Si, A1) La(Fe, X, Co)13 (X = Si, A1) above room temperature and has shown that they

THERMAL EXPANSION ANOMALIES

179

are good Invar alloys whose thermal-expansion properties can be widely regulated mainly by variation of the Co content.

5.3.3. Y-Fe binary compounds Andreev et al. (1990a) have studied the thermal expansion of all intermetallic compounds existing in the Y-Fe binary system. The study has been performed on single crystals at 5-900 K together with magnetization measurements. The results on YFe2 and Y2Fel7 have been discussed already in sections 3.1 and 5.1.1.1, respectively. The other studied compounds are the cubic Y6Fe23 (Th6Mn23 type of structure, Fm3m space group) and the rhombohedral YFe3 (PuNi3, R3m). YFe3 is a ferromagnet with/~,n = 4.9#B (/~F~ = 1.65/~B) and Tc = 545 K. The magnetic anisotropy is of the basal-plane type. The anisotropy within the basal plane is negligible. In the paramagnetic range (900 K), Ota = 1.55 × 10 - 5 K - 1 and c~ = 1.35 x 10 -5 K -1 which corresponds to C~v = 4.5 x 10 -5 K -1. Spontaneous magnetostriction is found only within the basal plane. The experimental e(T) curve coincides with that extrapolated from the paramagnetic range and Us reaches the value 3.3 x 10 -3 at 5 IC In accordance with basal-plane anisotropy, an orthorhombic distortion can appear below To. Absence of this distortion shows that corresponding 7-magnetostriction is lower than 1 x 10 -4. Y6Fe23 is a ferromagnet with #m = 40/IB (#F~ = 1.7#B) and T~ = 483 K. The easy-magnetization direction is the [100] axis. The value of C~v at 900 K is equal to 4.6 x 10 -5 K -1. Compared to YFe2 and YFe3, a sharp increase of 0:s is observed in Y6Fe23, ws = 10.2x 10 -3 at 5 K. No tetragonal distortion (which is possible according to the symmetry of easy axis) was found. Therefore, anisotropic-magnetostriction constant ),100 is lower than 5 x 10 -5. 6. Acknowledgements I thank very much my colleagues of Permanent Magnets Laboratory, especially Dr. Sergey M. Zadvorkin and Dr. Mikhail I. Bartashevich, for the long-time collaboration. The manuscript has been written during my stay in Megagauss Laboratory of the Institute for Solid State Physics, University of Tokyo. I am grateful to the friendly team of Megagauss Laboratory, especially to Prof. Tsuneaki Goto, who has provided excellent conditions for my work. I thank the Ministry of Education, Science and Culture of Japan for financial support. References Abbundi, R. and A.E. Clark, 1978, J. Appl. Phys. 49, 1969. Abbundi, R., A.E. Clark and N.C. Coon, 1979, J. Appl. Phys. 50, 1671. Akulov, N.S., 1939, Ferromagnetism (ONTI

Publishing Co., Moscow, in Russian). Andreev, A.V., 1979, Crystal Structure and Magnetic Properties of Some Uranium Intermetallics, PhD Thesis (Ural State University, Sverdlovsk, USSR, in Russian).

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Andreev, A.V., 1990, Magnetic and Magnetoelastic Properties of Rare-Earth and Actinide Intermetallics,Dissertation of Doctor of Sciences (Ural State University, Sverdlovsk, USSR, in Russian). Andreev, A.V., 1991, Physica B 172, 377. Andreev, A.V. and M.I. Bartashevich, 1990a, J. Less-Common Metals 162, 33. Andreev, A.V. and M.I. Bartashevich, 1990b, J. Less-Common Metals 167, 107. Andreev, A.V. and M.I. Bartashevich, 1990c, Soy. Phys. Solid State 32, 669. Andreev, A.V. and S.M. Zadvorkin, 1985, unpublished result. Andreev, A.V. and S.M. Zadvorkin, 1990, Phys. Met. Metallogr. 69(4), 85. Andreev, A.V.. and S.M. Zadvorkin, 1991, Physica B 172, 517. Andreev, A.V., A.V Deryagin, R.Z. Levitin, A.S. Markosyan and M. Zeleny, 1979, Phys. Status Solidi A: 52, K13. Andreev, A.V., A.V. Deryagin and S.M. Zadvorkin, 1982a, Phys. Status Solidi A: 70, Kl13. Andreev, A.V., E.N. Tarasov, A.V. Deryagin and S.M. Zadvorkin, 1982b, Phys. Status Solidi A: 71, K245. Andreev, A.V., A.V. Deryagin and S.M. Zadvorkin, 1983, Soy. Phys. JETP 58, 566. Andreev, A.V., A.V. Deryagin, S.M. Zadvorkin, R.Z. Levitin, R. Lemaire, J. Laforest, A.S. Markosyan and V..V. Snegirev, 1984a, Soy. Phys. JETP 60, 1280. Andreev, A.V., M.I. Bartashevich and A.V. Deryagin, 1984b, Soy. Phys. JETP 60, 356. Andreev A.V., A.V. Deryagin and S.M. Zadvorkin; 1985a, Phys. Met. Metallogr. 59(2), 116. Andreev A.V., A.V. Deryagin, S.M. Zadvorkin and G.M. Kvashnin, 1985b, Soy. Phys. Solid State 27, 1905. Andreev A.V., A.V. Deryagin and S.M. Zadvorkin, 1985c, Phys. Met. Metallogr. 60(4), 96. Andreev A.V., A.V. Deryagin, S.M. Zadvorkin and S.V. Terent'ev, 1985d, Soy. Phys. Solid State 27, 987. Andreev A.V., A.V. Deryagin, S.M. Zadvorkin, V.N. Moskalev and E.V. Sinitsyn, 1985e, Phys. Met. Metallogr. 59(3), 57. Andreev A.V..; A.V. Deryagin, S.M. Zadvorkin, N.V. Kudrevatykh, R.Z. Levitin, V~N. Moskalev, Yu.E Popov and R.Yu. Yumaguzhin, 1985f, in: Fizika Magnitnykh Mate-

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THERMAL EXPANSION ANOMALIES Dissertation of Doctor of Sciences (Institute of Metal Physics, Sverdlovsk, USSR, in Russian). Yermolenko, A.S., A.V. lvanov, V..Ye. Komarov, LM. Magat, G.M. Makarova and Ya.S. Shur, 1977, Phys. Met. Metallogr. 43(2), 195. Yermolenko, A.S., Ye.V. Shcherbakova, A.V. Andreev and N.V. Baranov, 1988, Phys. Met. Metallogr. 65(4), 117.

187

Zadvorkin, S.M., 1987, Spontaneous Magnetostriction of Some Rare-Earth Compounds with Metals of Fe Group, PhD Thesis (Ural State University, Sverdlovsk, USSR, in Russian). Zhong, X.P., R.J. Radwanski, ER. de Boer, T.H. Jacobs and K.H.J. Buschow, 1990a, J. Magn. Magn. Mater. 86, 333. Zhong, X.P., R.J. Radwanski, ER. de Boer, R. Verhoef, T.H. Jaeobs and K.H.J. Buschow, 1990b, J. Magn. Magn. Mater. 83, 143.

chapter 3 PROGRESS IN SPINEL FERRITE RESEARCH

V.A.M. BRABERS Department of Physics Eindhoven University of Technology P.O. Box 513 NL-5600 MB Eindhoven The Netherlands

Handbook of Magnetic Materials, Vol. 8 Edited by K. H.J. Buschow ©1995 Elsevier Science B.V. All rights reserved 189

CONTENTS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Spinel ferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. The chemical composition of spinel ferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. The crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. The cation distribution between the octahedral and tetrahedral sublattice . . . . . . . . . . . . 1.4. Structural phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Magnetization and magnetic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Magnetic interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Magnetic structure of the simple ferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Magnetic structure of the Zn-substituted ferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Disorder and spin-glass behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191 191 191 193 195 204 209 209 210 217 221

3. The magnetic permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Domain wall movement and rotational contribution to the permeability . . . . . . . . . . . . . 3.2. The Globus model for polycrystalline ferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Magneto-acoustic emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Non-magnetic grain boundary model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Magnetic losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Microstructure and additives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Magneto-optical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Optical and magneto-optical properties of magnetite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Magneto-optical properties of mixed ferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

226 226 231 237 240 248 251 253 253 256 259

5. Magnetite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Structure and the physical properties in relation to the Verwey transition . . . . . . . . . . . . 5.2. Magneto-elecaic properties . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. The electrical conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Galvano-magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. M6ssbauer spectra and the electrical conduction process . . . . . . . . . . . . . . . . . . . . . . . . . 5.7. Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8. Magnetic after-effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Nanoscalc structures in spinel ferrites; bulk materials, particles and films . . . . . . . . . . . . . . . . 6.1. Grain boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Ferrite films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Multilayers and supcrlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References of Tables 2 and 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Ferrite Conferences proceedings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265 265 272 274 283 285 286 290 295 295 297 301 305 309 311 324

190

Introduction Nothwithstanding that spinel ferrite materials are already for more than forty years widely used in electrotechnical equipment, much research and development in this field is still going on, from basic as well as from the applicational point of view. Spinel ferrite materials have been covered by several authors in earlier volumes of this series: P.I. Schlick, vol. 2, chapter 3, Ferrites for Non-Microwave Applications; J. Nicolas, vol. 2, chapter 4, Microwave Ferfites and S. Krupi~ka and P. Nowik, vol. 3, chapter 4, Oxide Spinels. Particulate spinel ferrites have been covered in the contributions of G. Bate, vol. 2 chapter 7, Recording Materials and S.W. Charles and J. Popplewell, vol. 2, chapter 8, Ferromagnetic Liquids. The present chapter is a supplement to these contributions. Thousands of papers on ferrite spinels have been published during the last 25 years; a limited selection of topics has been made, in which important progress has been realised, i.e. the magnetic and crystallographic structure, the magneto-optical properties, the magnetic permeability of polycrystalline ferrites, nanoscale structures and the basic properties of magnetite. More general information on ferrites can be found in: 'Ferrites', by J. Smit and H.P.J. Wijn, 1959, Philips Technical Library and 'Physik der Ferriten' by S. Krupi~ka, 1973, Vieweg and Son, Braunschweig. Properties of commercial available spinel ferrite materials can be found in 'Soft Ferrites' by E.C. Shelling, 1988, Second ed., Butterworths publ. co., and additional numerical data on spinel ferrites in 'LandoltBrrnstein, Numerical Data and Functional Relationships in Science and Technology, New Series', 1970, 1980, 1981 volumes III/4b, III/12b and III/27d.

1. Spinel ferrites

1.1. The chemical composition of spinel ferrites A large number of oxides with a metal-oxygen ratio of 0.75 as composition is known to crystallize into the spinel structure, which is called after the mineral spinel MgAI204. Among these oxides, magnetite Fe304 is an important compound, from which the spinel ferrites can be derived by partial substitution of the iron ions by other cations. Between most of the spinel oxides solid solutions can be formed, which means that a great variety of spinel oxides is possible. If we restrict ourselves to the t 3+ tt spinel ferrites, the general chemical formula of these materials is Me Fez_~M%O4, in 191

192

V..A.M. BRABERS

which Me' represents a divalent cation, or a combination of cations with an average valency of two, and Me" a trivalent cation or a combination of cations with an average valency of three. The composition parameter can range between zero and two, but it is obvious that if z is close to two, these oxides cannot be considered anymore as ferrites. The so-called simple spinel ferrites are now those ferrites in which z = 0 and the bivalent metal ion is Mg or Cd or one of the bivalent transition metal elements Mn, Fe, Co, Ni, Cu and Zn. The solid solutions between the simple ferrites are called mixed ferrites. A complete substitution of the bivalent Me t ion by a combination of ions with an average valency of two is known to exist for the combinations Li+-Fe 3+ and Cu+-Fe 3+ in Li0.5Fe2.504 and Cu0.5Fe2.504. However, the Cu+-Fe 3+ valencies are only stable at high temperature (1200-1350°C) at which the spinel phase is formed (Mexmain 1971). By quenching to room temperature, a meta-stable spinel structure can be retained in which an electron transfer between the cations occurs resulting in a mixture of mono- and bivalent copper and bi- and trivalent iron ions (Singa 1969). A special case of the Met-substitution is the 7-phase of Fe203, maghemite, which is a spinel compound with a large number of vacancies on 3+ 3+ the regular cation sites. The chemical formula can be written as l"]l/3Fe2/3Fe 2 04, which shows that one bivalent Me' is replaced by 1/3 vacancy [] and 2/3 Fe3+-ion. (H~igg 1935, Verwey 1935). Similar meta-stable cation defective spinel structures have been observed in sub-micron particles of some simple and mixed ferrites, which can be considered as solid solutions between these ferrites and the 7-Fe203 phase. (Gillot et al. 1985, Rousset et al. 1987, Bendaoud et al. 1987). The complete substitution of the trivalent iron by another trivalent ion in magnetite and the formation of solid solutions is possible for Me" = AI, Cr, Ga, Rh and V. Blasse (1964) reported a systematic investigation of the solid solutions of some mixed ferrites substituted with trivalent aluminium, chromium, vanadium and rhodium, Lensen (1959) reported similar results for gallium and vanadium. Besides the Fe 3+ substitution by trivalent cations, complete mixed series of fenites exists with the substitution of a 1:1 mixture of two- and four valent metal ions, in which the four-valent cations are Ti 4+ and Ge 4+ in combination with Fe 2+, Ni 2+, Co 2+ or Mn 2+. The substitution of Fe 3+ by a combination of one pentavalent ion and two bivalent ions has been reported to be possible for antimony in combination with Ni 2+, Co 2+ or Mg 2+ (Blasse 1964). Partial solid solutions of the simple ferrites are also possible with compounds which either do not exist, or which do not have the spinel structure. Gorter (1955) showed that, although Na0.sFe2.504 does not exist, solid solutions with Lio.sFe2.504 are possible up to a replacement of 40% of lithium by sodium. A corresponding solubility range 0 ~< z ~< 0.2 for sodium in the Znl_2~Na,Fe2+~O4 system was found by Obradors et al. (1985). Fe2SnO4 and Feln204 are not known as compounds but spinel solid solutions Fe3_~Sn~O4 with 0 ~< z ~< 0.5 (Basile et al. 1974) and the spinels Me'FelnO4 with Me t = Ni, Mn, Co, Mg and Fe are reported to exist (Gerardin et al. 1988). Calcium ferrite CaFe204 is orthorhombic (Decker and Kaspar 1957) but up to 20% of the Fe 2+ ions in the spinel structure of magnetite can be replaced by calcium (Gerardin et al. 1989).

P R O G R E S S IN S P I N E L F E R R I T E R E S E A R C H

193

There are numerous papers on the various types of substitutions in and the formation of solid solutions between spinel ferrites; more detailed references can be found hi Landolt B6rnstein (1970, 1980 and 1991). From the foregoing it is clear that a great variety in the chemical composition of spinel ferrites is possible, which is shown by the thousands of papers which appeared during the last fifty years dealing with these materials. The diversity in composition results also into a large range of physical properties which permits the tuning of the properties for specific applications and makes spinel ferrites of particular interest. 1.2. The crystal structure The spinel crystal structure was first reported for magnetite (Fe304) and spinel (MgAl204) by Bragg (1915) and Nishikawa (1915). The ideal spinel structure consists of a cubic closed packing of the oxygen ions with in between a large number of holes, which are partialy filled with the metal ions. The structure is cubic and belongs to the space group O7h-F3dm. The coordinates of the equivalent positions of this space group which are relevant for the spinel structure are given in table 1. The first column gives the number of the equivalent positions in the set, which is denoted by a letter in the second column. The third gives the point symmetry of the position of each set and the next column presents the coordinates of all the equivalent positions in fractions of the lattice parameter (Henry et al. 1965). Because the spinel structure is Face-Centered-Cubic, the coordinates of the additional positions can be found by the translations (0, 1/2, 1/2), (1/2, 0, 1/2) and (1/2, 1/2, 0). The unit cell contains 8 'molecules' MgAlzO4; 32 O2-ions occupying the positions e, 16 TABLE 1 The sets of the equivalent s y m m e t r y points of the space g r o u p Fd3m~O 7, which are relevant for the spinel structure. The origin is taken at a tetrahedral site and the value of/~ is approximately 3/8.

48

0 2 - -sites

32

f

e

mm

3m

~,,0,o; o,~,o;

-if,0.0; o,~,o;

1 1 1 7+~z, 7.~; 1 1 ~,~+~,~;1

1 1 1 ~-~,. ~, ~; 1 ~ , ~ , - . , ~;

0, 0, . ;

0,0,7;

14,14, 1~ Z"~;4

1'1'~

~, ~__,~'_;

1 ~1 - ~',~~ , 1~ - " ;

i

~,£, ~,;

i

A-sites

16

d

3m

i+

~1 + . , ~1- . , ~ +1~ ; 1

B-sites

1 1

1

1

5 55 8' 8 ~8'

577 8' 8' 8'

757 8' 8' 8'

775

!!!.

! 3 _ 3.

3!L

8' 8 ' 8 '

8' 8 ' 8 '

8' 8' 8'

3 3 1 8'8'8"

3 3 3 4' 4' 4'

16

c

3m

8

b

43m

1 1 1 2' 2' 2'

8

a

43 m

o,0,0;

1

8'8'8"

1

-/z"

194

V.A.M. BRABERS

ml3+-iOns the positions d which are in the center of an oxygen octahedron and the 8 MgZ+-ions in the positions a which are in the center of an oxygen tetrahedron. The positions f and b are interstitial tetrahedral sites and the positions c interstitial octahedral sites. In real spinels the closed cubic packing of the oxygen ions is deformed by the cations. The tetrahedral sites are usually too small to contain the metal ions and the oxygen ions will be displaced from their ideal position into a (111) direction away from the central tetrahedral ion. A quantitative measure of this displacement is the oxygen parameter #, which determines the exact positions e of the oxygen lattice sites as indicated in table 1. If there is no enlargement of the tetrahedrons, the oxygen parameter # equals 0.375. Figure 1 gives a schematic picture of the unit cell of the spinel structure. The cubic unit cell can be devided into 8 octants as shown in fig. la. All octants have the same occupation for the oxygen ions but there are two types of octants, shaded and non shaded, with different positions for the metal ions as is shown in fig. lb. In fact the oxygen ions form a f.c.c, lattice with an edge 1/2a. The tetrahedral sites form two interpenetrating f.c.c, lattices with an edge a, shifted relative to each other over a distance 1/2 a ~ in the direction of one body diagonal of the cube; the octahedral sites form four f.c.c, lattices with the edge a, which are shifted relative to each other over a distance 1/2 a V/2 in the directions of the face diagonals of the cube. The oxygen parameter is also indicated in fig. lb. The symmetry of the space group O7-Fd3m is only applicable for the spinel structure in case that each sublattice is occupied by one type of cation, which means for instance that for MgAI204, all Mg 2+ ions are in tetrahedral sites and all AI3+-ions in octahedral sites. This type of cation distribution is called the normal distribution. It has been suggested by Grimes (1972) that due to a possible displacement of the B-site ions along a (111) direction, the B-site symmetry in certain spinels is not anymore centro-symmetric and the space group has to be F43m. Based on the observation of the reflections with h ÷ k = 4n ÷2 in electron, X-ray and neutron diffraction experiments, which are forbidden for Fd3m symmetry, some polemics is going on in literature about the real space group of spinels. Hwang et al. (1973) found in electron diffraction experiments (200) and (420) reflections for MgAl204, which is consistent with the F43m symmetry. However, Smith (1978) pointed out that these reflections might originate from double diffraction effects. Steinsvoll et al. (1981) improved the accuracy of their previous neutron scattering experiments and reconfirmed the Fd3m synunetry for MgAI204. Fleet (1986) arrived at the same conclusion for magnetite using X-ray diffraction. Schmocker et al. (1976) reported a temperature dependent partial inversion of the site occupancy for MgAl204, which means that tetrahedral Mg 2+ ions are partially interchanged with octahedral A13+ ions, and a lowering of the Fd3m symmetry for MgAl204 was suggested, if the inversion is substantial. In fact, the crystal symmetry of partial or complete inverse spinels and of spinel solid solutions is only in first approximation Fd3m, so far no difference is made between the various metal ions. However, not only the crystal syImnetry is affected by the cation distribution, also a number of physical properties is influenced by the presence of certain cations in either the octahedral or tetrahedral sites in the spinel structure. This means that the cation distribution, characterizing the structure of ferrites, is another important parameter for changing the physical properties.

PROGRESS IN SPINEL FERRITE RESEARCH

-f

i

.. .........~,l ..........

L--5--->7 .1

195

........

I

'

....

E

3---q 1

T-r--( [

i

!

........

t

i [

I

i : i

i

L/"

,,

i :t

i i

-'/~............... !i......... 7~..--"

/

L......... " i E !

...........

i

i

I I

i l ! I

J ! ~

i

~...... h--*"

j/~

Fig. 1 (a and b). Schematic presentation of the spinel structure. The large spheres are the oxygen ions, the small spheres the tetrahedral ions and the small dark spheres are the octahedral ions. The dashed oxygen is outside the two depicted octants.

1.3. The cation distribution between the octahedral and tetrahedral sublattice The distribution o f the metal ions a m o n g the tetrahedral and octahedral sites in spinel ferrites has d r a w n m u c h attention in literature since the p i o n e e r i n g w o r k o f Barth and P o s n j a k (1932), in w h i c h they p r o v e d with X-ray diffraction studies, that in a n u m b e r o f spinels M e I M e ~ n O 4 au inverse distribution o f catious occurs; in that case the d i v a l e n t ions are located on the octahedral sites and the trivalent ions

196

V.A.M. B R A B E R S

are uniformly distributed over the remaining octahedral and tetrahedral sites. Nrel (1950) and Pauthenet (1950) showed further, using magnetization measurements on CuFe204 and MgFe204 specimens quenched from various temperatures, that the cation distribution in these ferrites is temperature dependent and can be described by an equilibrium K -

(1 - 6 ) ( 2 - ~5) 62

- exp

{-E/kT}

(1)

in which the inversion parameter 6 represents the octahedral Cu or Mg concentration and the value of the energy E was found to be about 0.14 eV. The actual cation distribution of a certain spinel oxide is determined by the total energy of the crystal which depends on a number of factors: the ion size, the Coulomb energy of the charged ions in the lattice and the short-range Born repulsion energy, crystal field effects, the ordering of the cations and covalency and polarization effects. Simple geometric arguments concerning the ionic radii suggest that the smaller cations would prefer the smaller tetrahedral sites. However, very often the opposite effect is found; f.i. in ZnFe204 and MnFe204 the larger Mn 2+ and Zn2+-ions (r A = 0.66 and 0.60 A, respectively) prefer the A-site positions, whereas in MgFe204 the smaller Mg z+ (r A = 0.57 ~) the octahedral sites. This indicates that the other factors are more dominant in determining the cation distribution. In fact, the largest contribution to the crystal energy originates from the Coulomb interactions between the charged ions: _e 2

U~ =

a

.M

(2)

with e the elementary charge, a the lattice parameter and M the Madelung constant. Verwey et al. (1948) determined the Madelung constants as function of the oxygen parameter #, with the valencies of the octahedral and tetrahedral cations as additional parameters. They found that normal 2-3 spinels have the highest Madeling constant for oxygen parameters above # ~ 0.379, which means that these spinels have the highest electrostatic stability in this cation arrangement. After the paper by Verwey et al. (1948), several authors recalculated the Madelung constants for 2-3 and other valencies. (Fisher et al. 1975, Thompson and Grimes 1977, Hermans et al. 1974 and references cited). There were only slight differences between the results which can be attributed to the different calculation methods used by the various authors. (Evjen 1932, Ewald 1921, Hund 1935, Bertaut 1952). For instance, the results obtained by Thompson and Grimes (1977) can be presented by the formula M =139.8 + 1186.A - 6483A 2 - (10.82+ 412.2A- 1903A2) qA+

(3) + 2.609q 2,

PROGRESS IN SPINEL FERRITE RESEARCH

197

in which A = # -- 0.375 and qa is the mean electric charge of the tetrahedral ions. This result shows that in case of an inverse cation arrangement, qA -= 3 and qB = 2.5, the Madelung constant decreases slightly with increasing oxygen parameter, while Verwey et al. reported an increase. The general trend, that for 2-3 spinels the normal distribution is favoured by the Madelung energy at higher #-values is still supported by these data. However, the actual cation distributions and oxygen parameters found in some simple spinels does not agree with this result. In table 2 the lattice parameter, cation distribution, oxygen parameter and the X-ray density are listed of the possible 'stoichiometric' spinel ferrites, i.e. ferrites of which the cation ratios are integer numbers. The cation distribution is indicated by the octahedral metal ions between the brackets and the tetrahedral metal ions before the brackets. Considering only the simple ferrites, it can be seen that Zn and Cd ferrite have a normal distribution in agreement with the high oxygen parameter (0.387 and 0.390) but Ni ferrite is an inverse spinel and the distributions in Mn, Co, Cu and Mg ferrite TABLE 2a Crystallographic parameters of 'stoiehiometric' spinel ferrites. Ferrite

Lattice parameter ~

Cation distribution

Oxygen parameter

Density g/cm 3

Ref. *

Fe304

8.398

Fe 3+ [Fe 2+ Fe 3+ ]

0.379

5.193

[1, 2]

7-Fe203

8.3396(a)

Fe 3+ [Fe~+['-]o.33 ]

-

4.887

[3]

8.3221

(x3 = c) FeAI204

8.151

,~ Ve2+ [AI23+]

0.387

4.26

[4--61

Fe2AIO 4

8.263

,,~ Fel2+_aFe]+ 3"t- 34- ] [Fe 24e Fel_eAI

0.385

4.77

[6, 7]

CdFe204

8.70

Cd [Fe~ l

0.390

5.81

[8, 9]

CoFe20 4

8.381

CoeFel_6[COl_rFel+6]

0.3852

5.294

[10, 11]

0.07 < 6 < 0.24 Co2FeO 4

8.22

F o3+ vO.18('n24~0.82 2434- 34[C00,18Co Fe0.82]

0.382

5.69

[12, 13]

Cuo.5 Fe2.504

8.413

Cu0.22 Feo.78 [C u0.28Fe 1.72]

0.388

5.25

[14, 15]

8.222 (a)

C uo.08Feo.92 [C uo.92Fe Los ]

-

5.39

[16]

C 8.382

C u0.33 Feo.67 [Cu0.67 Fe 1.33]

0.378

5.40

[16]

FeCr20 4

8.378

Fe 2+ [Cr~ + ]

0.388

5.07

I17, 18]

Fe2CrO 4

8.396

F,~3+ r~o2+ ~0.3 --~0.7 24- Feo.7 34- Cr 34- ] [Feo.3

0.385

5.11

[17-19]

CuFe204

8.715 (c)

*See the list on p. 309.

198

V.A.M. BRABERS TABLE 2b Crystallographic parameters of 'stoichiometric' spinel ferrites.

Ferrite

Lattice parameter ,~

Cation distribution

Oxygen parameter

Density g/cm 3

Ref.*

FeGa204

8.363

Fe0.05Ga0.95 [Ga 1.05Fe0.95]

0.3822

5.89

[20]

Fe2GaO 4

8.3815

Ga0.63 Fe0.37[Ga0.37Fel.63 ]

0.3800

5.54

[20]

Fe2GeO 4

8.411

Ge[Fe2]

0.375

5.54

[21]

Lio.sFez504

(o) 8.3143 (d) 8.2923

Fe[Li0.sFe2.5] Fe[Li0.sFe2.5]

0.382

4.79 4.82

[22] [22, 23]

MgFe204

8,372

MgeFel_6[Mgl_rFel+r]

0.3856

4.53

[10, 24]

8.398

0.14 < 6 < 0.26

MnFe204

Mn2FeO 4

4.49

8.525

Mnl_rFer[MneFe2_e ]

8.515

0.07 < 6 < 0.23

0.390

4.94

[10, 25]

8.90 (c)

Mn[MnFe]

-

4.89

[25]

4.96

8.37 (a) MoFe204

8.509

Fe[FeMo]

0.383

5.86

[261

NiFe204

8.337

Fe[FeNi]

0.386

5.38

[10, 27]

Fe2TiO4

8.538

Fe[FeTi]

0.390

4.77

[1, 28]

Fe2VO4

8.418

F~2+ ~03+ ~0.4 *~0.6

0.378

5.05

[29-31]

[~2+ ~ 3 + V3+ 1 " t'0.6 ~'0.4 J FeV204

8.456

'~ Fe2+ [V3+ ]

0.386

4.87

[31, 32]

ZnFe204

8.443

Zn[Fe]

0.387

5.32

[33]

*See the list on p. 309.

are intermediate, ill spite of the large values of the oxygen parameter. This implicates that the other factors contributing to the total energy of the crystal are competing with the Madelung energy. McClure (1957) and Dunitz and Orgel (1957) estimated the crystal field stabilization from optical spectra for the transition metal ions in octahedral and tetrahedral symmetry and introduced the concept of the octahedral site preference energy for the respective ions. Miller (1959) included further a shortrange term which is opposed to the Coulomb term. For Zn, Cd and Mn ferrites the normal structure was found and for, Ni, Co, Cu, Fe and Mg ferrite the inverse structure, which is in a first approximation in agreement with the experimtal data. However, the intermediate distributions found experimentally for Mn, Co, Cu and Mg ferrites are not accounted for. Another approach to the problem of the cation distribution in spinels was made by O'Neill and Navrotsky (1983). They presented a refined set of cation ionic radii, derived from an optimised fit to the experimental spinel lattice parameters, on which they based their electrostatic lattice energy calculations. Investigating the cation ex-

PROGRESS IN SPINELFERRITERESEARCH

199

change reaction between the octahedral and tetrahedral lattice (AI_ x B~ [A~B2- z ] 04) (4)

A + [BI = B + [A],

and supposing a linear dependence of the lattice and oxygen parameter upon x, a quadratic z-dependence of the electrostatic lattice energy was found:

a U ~ = c~Ex + 5x 2.

(5)

Kriessman and Harrison (1956) reported an empirical quadratic z-dependence of the enthalpy of the inversion reaction in mixed MgMn ferrites, which was claimed to be an experimental support for the O'Neill-Navrotsky model. O'Neill and Navrotsky made further a rough estimate for the short range force term in the crystal energy, which turned out to be also quadratic in z. Their final conclusion was that the energy of the inversion reaction was quadratic dependent on the inversion x, resulting in the equilibrium equation X2

(1 - z ) ( 2 -

x)

1 exp RT

+

exp { A S / R } .

(6)

RT

Possible volume effects due to the cation migration are small and not included in this equation. The excess entropy AS is usually small compared with the influence of the change in energy and is often ignored (Driessens 1968 and Reznickiy 1977). Although O'Neill and Navrotsky considered only the electrostatic contribution in detail and disgarded crystal field and other effects. In spite of this and the nonrealistic conclusion that all simple ferrites are normal spinels, eq. (6) might be useful to describe the T-dependence of the cation distribution in certain ferrites. A nice theoretical study of the cation distribution in spinels was presented by Connack et al. (1988), which allowed to predict correctly the structure of a large number of 2-3 spinels. Their approach of the problem is based on the interatomic potentials which are derived from the relevant binary oxides. Electrostatic and short range energy as well as crystal field terms were considered. The analytical form for the short-range potentials used is of the Born-Mayer or Buckingham type: V(r) = A. exp ( - r / p ) - Cr -6.

(7)

The parameters A, p, and C for the cation-oxygen interactions were obtained from the relevant binary oxides. Because the cation-oxygen bond length depends on the coordination, the pre-exponential part A of the short range potential was also modified by a factor exp {At~p), in which Ar is the difference between the cation radii in octahedral and tetrahedral sites. In fact, this modification is an attempt to include covalency effects. In this model the displacement polarisability is automatically taken into account but to include also the ionic polarisability the shell model by Dick and Overhauser (1958) was used. The cation distribution was studied by calculating the energy difference AE between the normal and inverse structure, without the entropy term. The results for a nmnber of ferrites are given in table 3. Columns 1, 2 and 3

V.A.M. BRABERS

200

TABLE 3 Calculated energy differences between the normal and inverse structure. Compound*

Lattice parameter (,~)

Observed u value

AE (eV)

AE I (eV)

FeCr204 N

8.376

-

0.939

FeAI204 N

8.119

0.390

1.561

NiFe204 I

8.3532

0.381

0.127

-0.737

-0.892

-1.629

0.3795

MgFe204 1

8.389

0.382

0.175

-0.673

0.0

-0.673

0.3788

ZnFe204 N

8.443

0.3888

0.629

0.203

0.0

0.203

0.3898

CoFe204 I

8.390

-

0.301

-0.497

-0.322

-0.819

0.3791

FeFe204 1

8.3963

0.379

0.430

-0.338

-0.176

-0.514

0.3788

MnFe204 M

8.511

0.3846

0.836

0.208

0.208

0.3842

0.201 1.265

O.S.E. (eV) ** 1.403 -0.176

0.0

AEt~ot (eV)

Calc. u value

1.664

0.3893

1.089

0.3924

*The actual cation distribution is indicated as: N: normal, I: Inverse and M: intermediate. **O.S.E.: octahedral crystal field stabilization energy, which takes the preference energy of the cations into account. list the ferrites and the experimental lattice and o x y g e n parameters. Column 4, A E the crystal energy without correction for the coordination, column 5 A E ~, the crystal energy with this correction, column 6, the crystal field stabilization energy O.S.E., columal 7 the total energy difference ( A E t + O.S.E.) and column 8 the o x y g e n parameter obtained from the calculations. All the cation distributions are correctly predicted, although, the obtained values of AE~ot need some discussion. NiFe204 is kalown to be a very stable inverse spinel (see table 2) which is in agreement with the high negative value AEt~ot = - 1 . 6 2 9 eV. MnFe204 and ZnFe204 have both energy values o f about 0.2 eV; Vologin et al. (1976) found experimentally from a kinetic study of the cation redistribution rate in neutron irradiated ferrites that this energy is about + 1 . 4 eV, in agreement with a very stable normal distribution. MnFezO4, however, is an intermediate ferrite with an energy difference between N and I-state in the range of + ( 0 . 3 - 0 . 6 ) eV determined from the electrical properties (Brabers 1970), and for Fe304 and M g F e 2 0 4 values of - 0 . 2 4 eV and - 0 . 1 5 eV are reported, respectively. (Wu and Mason 1981, and Brabers and Klerk 1977a). The inversion energies calculated by C o r m a c k et al. are not in all cases comparable with the experimental data, although a closer agreement is found compared with previous studies. The lattice and o x y g e n parameters used in the calculations are slightly different f r o m the most reliable values listed in table 2, which could influence the results in the w r o n g direction. Aa~other important item in the discussion of the cation distribution is whether short range cation order within the octahedral or tetrahedral sublattice occurs. It was already pointed out by Anderson (1956) that the substitution of two-valent ions on the octahedral sites is not at random but that 'perfect' shortrange cation order can occur in inverse spinels. In fact, this short range order gives an enthalpy contribution which is only partially included in the lattice energy

PROGRESS IN SPINEL FERRITE RESEARCH

201

calculation by Cormack et al. For instance, entropy effects and magnetic interactiolts are not included. The influence of short-range order is clearly demonstrated by the Verwey transition in magnetite (Fe304) , which is caused by the long range ordering of Fe 2+ and Fe 3+ ions at 125 K. The low value of the transition temperature and the low transition energy are strong indications that the long range order is not a transition from a complete at random distribution of the ions to an ordered structure, but that substantial short range order has to exist above the transition (Verwey et al. 1948, Anderson 1956). For the experimental determination of the cation distribution, several techniques are applicable: X-ray and neutron diffraction, nuclear magnetic resonance and M6ssbauer spectroscopy, which are in principal direct methods to determine the distribution. However, the possibilities of X-ray diffraction techniques are rather limited because of the similar values for the X-ray scattering of the 3d-metal ions, but by using a wavelength close to the absorption edge of one of the metals in the ferrite, anomalous scattering takes place. A suitable choice of the wavelength results in significantly different 'effective' scattering factors for elements whose scattering would otherwise be substantially identical. Skolnick et al. (1958) and Rieck et al. (1966) applied this anomalous X-ray dispersion for the distribution study of NiFe204 and manganese ferrites, respectively. Neutron diffraction (ND) has the advantage that the scattering factors for the 3d-metal ions are quite distinct, and in addition to the crystallographic structure, information can be obtained about the magnetic structure from the magnetic scattering. The remaining problems with this technique concern the high absorption of thermal neutrons by Cd and the inelastic scattering by hydrogen, which can be present in wet-prepared ferrites. Hastings and Corliss (1956) were the first to find with ND that MnFe204 is only for 20% inverse, and not complete as reported earlier (Gorter 1954). Mn-NMR-spectroscopy has been applied to the study of the distribution in wet-prepared MnFe204 by Yasuoka et al. (1967), who found that the inversion in this type of materials is much higher, up to 50%. 57Fe-M6ssbauer spectroscopy is a very powerful tool in the determination of the ionic structure of ferrites, especially because of the large differences between the hyperfine parameters of Fe 2+ and Fe 3+ . However, there are some critical points in the determination of the cation distribution from Mrssbauer spectroscopy. The hyperfine interactions in octahedral and tetrahedral sites are only slightly different and particularly in mixed ferrites a distribution of hyperfine fields occurs. It is therefore difficult to discern the subspectra or to make assignments to the respective lattice sites; the use of external magnetic fields to separate the subspectra is indispensable. Co~tsequently, cation distributions determined from Mrssbauer spectra in zero field have to be considered as unreliable. A good review of the application of Mrssbauer spectroscopy to spinel ferrites has been given by Vandenberghe and De Grave (1989). In addition to the direct techniques, indirect methods to derive the cation distributions from physical properties like saturation magnetization, electrical properties and lattice parameter have also been applied. Saturation magnetization data at 4 K have been frequently used to calculate the cation distribution in ferrites, but the results are only correct as far as the magnetic structure or spin configuration is exactly

202

V.A.M. BRABERS

known. For example, high-field magnetization measurements on the Fe3_~,TizO4 system showed that for Ti-concentrations below x = 0.5, the magnetic structure is of the collinear N6el type, (magnetization in A and B antiparallel) but for higher titanium concentration spin canting occurs. This implies that the ionic structure determined from saturation magnetization is only reliable for the lower Ti-concentrations. [Li et al. 1992]. The combined analysis of the thermoelectric power and electrical conductivity to determine the cation distribution has been proposed by Mason (1987) for spinel ferrites containing Fe 2+ and Fe 3+ ions. The Seebeck coefficient Q and the electrical conductivity cr depend on the octahedral [Fe2+ ] and [Fe3+] concentration according the equations: -k[ [Fe3+] ] Q=-TIn2 +A [Fe2+ ]

(8)

and [Fe2+][Fe3+] o" = g N

[Fee+] + [Fe3+]

e2a2 - kT

(--EH) Uo exp

;

(9)

--£-T--

k is the Boltzmann constant, e the electron charge, A a kinetic term which is zero for small polaron hopping, g a geometrical factor involving the coordination number, N the density of the octahedral sites (cm-3), a the jump distance, u0 the frequency of the phonon involved in the hopping process and E H the hopping energy. From the experimental data for Q and o', the octahedral Fe 2+ and Fe 3+ concentration can be calculated. The restriction of this analysis is that small polaron hopping of electrons on the octahedral Fe-sites must be the only conduction process. Mason et al. derived from the electrical properties the temperature dependence of the cation distribution for a number of ferrous ferrite systems and analyzed the results with the O'Neill-Navrotsky model. The ~ and /3 values of this model are given in table 4 for a number of ferrites. For magnetite (Fe304), an unusual feature has been observed by Wu and Mason (1981) that the random value of x = 2/3 is reached at high temperature, and perhaps will become slightly smaller at still higher temperatures. O'Neill et al. (1983) suggested that the excess nonconfigurational entropy AS in eq. (6) has to be taken into account. The temperature dependence of a obtained for the Ti ferrite system may also be suggestive of a negative nonconfigurational entropy term (Trestmann et al. 1983). A temperature dependent/3-factor is only found for the Ti-system. For 0 K, the calculated cation distributions in the Ti ferrite system are in agreement with the Nrel-Chevallier model, i.e. with increasing Ti 4+ concentration the octahedral sites are first occupied by Ti 4+ and Fe 2+ ions. Li et al. (1992) used high field magnetization measurements at 4 K and proved that this is not the case but that the additional Fe 2+-ions introduced by the presence of Ti4+ are already partially located on the tetrahedral sites for low Ti-concentrations. Brabers (1992a)analyzed

PROGRESS IN SPINEL FERRITE RESEARCH

203

TABLE 4 The parameters ~ and/3 of the O'Neill-Navrotsky model, describing the temperature dependence of the cation distribution. Ferrite

Cations involved

~ kJ/mol

/3 kJ/mol

References

MgFe204

[Mg2+ ]Fe3+

17.4

- 16.4

Trestman et al. (1984)

Fe3_xTixO4

[Fe2+ ]Fe3+

31.06 (1300 *C) 22.15 (1000 oC) 13.84 (630'C)

-24.59 -19.08 -14.03

Trestman et al. (1983)

Fe304

[Fe2+]Fe3+

24.3

-23.01

O'Neill et al. (1983)

CoFe204

[Co2+]Fe3+

35.0

-25.9

Erickson et al. (1985)

Fe3_xAlxO4

Fe2+ [AI3+ ]

59.4

-30.3

Mason (1985)

CuFe204

[Cu2+]Fe3+

13.8

-16.7

O'Neill et al. (1983)

1 plus a nonfigurational entropy A S ° = -3.27 J/mol.K included.

0.'~ E ev

0

0.5

1.0

x Fig. 2. Tentative energy level scheme for the electrons on Fe2+-sites in the mixed series Fea_zTixO4 (Brabers 1992a). the electrical properties o f the titanium ferrite s y s t e m in terms o f h o p p i n g c o n d u c tion in d i s o r d e r e d systems, including a r a n d o m i z a t i o n o f the site energies due to e l e c t r o n - e l e c t r o n and e l e c t r o n - t i t a n i u m interactions, w h i c h could account for the observed tetrahedral Fe 2+ concentrations. T h e obtained tentative energy level s c h e m e for the electrons on the F e 2+ sites in F e 3 _ x T i x O 4 is shown in fig. 2. The dashed line represents the F e r m i level. The lattice and o x y g e n p a r a m e t e r are other properties from w h i c h the cation distribution could be derived, using ionic radii and g e o m e t r i c a l factors only. Unfortunately, this a p p r o a c h is v e r y unreliable because a n u m b e r o f effects like c o v a l e n c y and p o l a r i z a t i o n are not included. However, if the relation b e t w e e n lattice p a r a m e t e r and cation distribution is k n o w n f r o m other experiments, the kinetics o f the cation

204

V.A.M. BRABERS

redistribution can be studied by measuring the lattice parameter as function of time. For instance in MgFe204, a non equilibrium distribution obtained by quenching from high temperature is re-equilibrated at a constant temperature between 300 and 500 ° C. The observed mechanical creep is analyzed in terms of the kinetics of the cation redistribution (Brabers and Klerk 1977a). A finite time (a few hours) to achieve equilibrium is found between 350 and 450°C. Similar kinetic studies have been reported for Mn ferfites, using the relation between the cation distribution and the electrical conductivity (Sim~ova and Sirn~a 1974) or the Curie temperature (Brabers 1992b) and for MgFe204, relating the magnetization With the cation distribution (Waiters and Wirtz 1971). The cooling rate dependence of the cation distribution and the redistribution kinetics in CoFe204 have been studied by De Guire et al. (1989) by means of Mrssbauer spectroscopy in high magnetic fields. The important result of these kinetic studies is that the exchange of cations between the sublattices is achieved at relatively low temperatures (200-450 ° C), which might have implications for the preparation of well characterized ferrites and the occurrence of aging effects.

1.4. Structural phase transitions The normal spinel structure belongs to the space group O7-Fd3m, but the discussion on the cation distribution in the preceding section shows already that changes in the structure can occur due to a redistribution of the cations between the octahedral (B) and tetrahedral (A) sublattices. Usually, this disordering process is not considered as a phase transition and is certainly not a first order transition (Haas 1965). However, due to the large unit cell of the spinel structure which consists of two interpenetrating A-site f.c.c, lattices, four B-site f.c.c, lattices and eight oxygen f.c.c, lattices, all with the same edge a, superstructures in spinel compounds occur by ordering of the cations TABLE 5 Superstructures in spinel compounds. Ordering type

Space group of the ordered phase

Example

Predicted type of change from 07

1:1 A-sites

F43m; T2

Lio.5Feo.5[Cr2]O 4

1st or 2nd order

1:2 A-sites

I 41/amd; D~

1"-11/31n2/3S 4

1st order

1:1 A-sites 1:3 B-sites }

P21;T4

. • 4+ LIo.sZno.5 [Llo.sMnl. 5 ]04

1st or 2nd order

1:1 B-sites

P4322; D47 P4122; D43

Mn[MnTi]O4 Zn[ZnTi]O4

1st order 1st order

1:1 B-sites

Imma; D~

(Fe[Fe2+ Fe3+ ]04)? Zn[LiSb]O4

ist order

1:3 B-sites

P4332; 0 6 P4132; 07 P43212; D~

Fe[Lio.sFezs]O 4

1st order

Fe[l"ll/3Fe5/3]O 4

1st order

1:5 B-sites

PROGRESS IN SPINEL FERRITE RESEARCH

205

o Fe 2+ • Fe 3+

Fig. 3. The ordering of Fe2+ and Fe3+ ions in magnetite below the Verwey-transitionand the resulting charge density wave. Nearest-neighbourFe2+ ions are linked and form a wave with polarization along the b-axis, which is clearly shown in the projection onto the a-b plane (Zuo et al. 1990). among the two A- and four B-f.c.c. lattices. Based on symmetry considerations, Haas (1965) predicted the nature of these phase transitions which occurred when the ordering takes place. In table 5 the different types of ordered phases found in a number of spinel compounds are given. For the spinel ferrites, the ordering in fe304, 7-Fe203, Li0.sFe2.504 and Li0.sFe0.sCr204 are of particular interest. The transition in Fe304 concerns an electronic ordering of Fe 2+ aud Fe 3+ ions on the B-sites with an ordering temperature near 125 K. Verwey et al. (1941) suggested a simple charge order model which led to the orthorhombic symmetry D2~ for the low temperature phase. Later structural refinements of this low temperature phase, however, indicated a lower monoclinic symmetry Cs4 - Co. (Yoshida et al. 1977, Iizumi et al. 1982 and Zuo et al. 1990). Data of the magnetoelectric effect are even indicative of a still lower triclinic symmetry, because of the absence of an ac mirror plane (Miyamoto and Shindo 1993). Based on a convergent-beam electron diffraction study Zuo et al. (1990) proposed a realistic charge ordering model, different from the Verwey-order scheme, in which charge-density waves occur as shown in fig. 3. The structural transition modifies also a number of physical properties like the electrical conduction and is the cause of the observed metal-insulator, the so-called Verwey transition. The Verwey trausition temperature Tv is very sensitive for impurities and non-stoichiometry and decreases to about 80 K for impurity levels of a few percent as is shown in fig. 4 (Kakol et al. 1992). The disjunction in the decrease of Tv near

206

V.A.M. BRABERS

•

120

Fe 3-x O4

o Fe 3-x ZnxO4 .

~___~>110

•

Fe 3-x Tix 04

100 O

•

•

%%%

...

•

"%,%

90 %%%% I

I I q l T P

0

0.01

r

r

i

0.02

%~" , [

0.03

Fig. 4. The Verwey transition temperature of non-stoichiometric and Ti- and Zn-doped magnetite as function of the non-stoichiometry and impurity content (Kakol et al. 1992).

z = 0.012 coincides with a change in the nature of the Verwey-transition from first order for low x-values to second order above z = 0.012. Experimental support for this first to second order transition as function of the composition has been reported by Shepherd et al. (1991) in their study of the heat capacity and entropy data of nonstoichiometric magnetites Fe3_~O4. The latent heat of the transition is suddenly lost for a cation deficiency in excess of 0.012. In addition, neutron scattering experiments on non-stoichiometric magnetite with x = 0.018 showed no long range order below the second order transition temperature ( ~ 95 K). The long range Verwey ordering disappears for compositions x > 0.012 due to the deficiency of charge per unit cell, i.e. the ratio Fe2+/Fe 3+ is to small (Aragon et al. 1993). An explanation of this change in the order of the transition has been given by Aragon and Honig (1988) based on a mean field model and the Str/issler-Kittel formalism (1965) for phase transitions. The 7-phase of Fe203 has a cation deficient spinel structure, in which the cation vacancies can give rise to a 1:5 order on the B-sites resulting in a tetragonal superstructure, with a tripled cell dimension in the c-direction. The cell dimensions are a = 8.3396 .~ and c = 24.966 ,~ (Greaves 1983). By raising the temperature, a transition from an ordered to a disordered state has not been observed because 7-FezO3 transforms into c~-Fe203 above 400°C. However, due to the low temperature preparation of the 7-phase by oxidation of Fe304 or dehydration of ferric hydroxide, poorly crystallized particles can be formed in which a disordered structure can be realized at room temperature (Boudeulle et al. 1983, Cormack et al. 1985). In Li0.sFe2.504 the Li- and Fe-ions order on the B-sites in the ratio 1:3 and the transition to a disordered structure occurs near 750°C (Braun 1952). As was demonstrated by Anderson (1956), short range order on the B-sites can be achieved

PROGRESS IN SPINELFERRITERESEARCH

207

while maintaining a finite entropy. The octahedral sublattice can be considered to be formed by tetrahedrons of four octahedral ions. If the octahedral sites are occupied by cations with different valencies, the charge of the tetrahedrons of octahedral ions must be more or less constant over the crystal because of the electro-neutrality. This means that the Coulomb energy induces predominantly the short range order. The long range order is governed by the ordering of the tetrahedral sets of octahedral ions, which is in fact a second-neighbour ordering (De Bergevin and Brunel 1966, Vandenberghe et al. 1980). Because of this second-neighbour effect it is clear that the physical properties of Li ferrite will be less affected as compared with ordering effects in case of alloys, where nearest neighbour interactions play an important role. A clear evidence for the occurence of short range order in disordered Li0.sFe2.504 is given the M0ssbauer spectra of the ordered and disordered compound reported by Dormann et al. (1980, 1983). The magnetic hyperfine fields, which are mainly determined by the nearest neighbour configuration, change only slightly (,,~ 0.5%) and the isomer shifts and quadrupole splittings are identical for the ordered and disordered state. The 1:1 ordering of Li and Fe on A-sites in Li0.sFe0.sCr204 has been reported by Gorter (1954). Another cause of structural phase transitions in oxidic spinels is the ordering of Jahn-Teller deformations, orginating from transition metal ions with an orbitally degenerate electronic ground state. These Jahn-Teller cations lower their energy by a spontaneous deformation of their octahedral or tetrahedral environment. An effective coupling between the electronic states on different cations through the interaction between the degenerate states and lattice vibrations can, due to the ordering of the electronic states, led to a cooperative macroscopic deformation above a critical concentration of Jahn-Teller cations. Jahn-Teller type transitions in spinel ferrites are encountered in spinel ferrites containing Cu 2+, Mn3+, and Fe2+ ions. Copperferrite 'CuFe204' is dimorphic at room temperature, depending on the octahedral Cu 2+concentration established by the thermal history of the material; quenched from 900 ° C, the structure is cubic and slowly cooled to room temperature, the structure is tetragonal, space group D419. It has to be noted that stoichiometric Cul.0Fe2.004 cannot be made by conventional high-temperature techniques but that stabilization of the spinel phase requires some copper deficiency: Cu0.96Fe2.040 4 (Brabers and Klerk 1977b; Tang et al. 1989). The transition from the tetragonal to the cubic structure for slowly cooled copperferrite occurs near 4150 C, but lower transition temperatures and even multiple tetragonal-cubic-tetragonal transitions have been reported for samples quenched from various temperatures below 9000 C. Brabers and Klerk (1977b) related these multiple transitions to the interference of a cation redistribution process, going on at a temperature T~x between 250--400 of which the exact value depends on the cation vacancy concentration established at the quenching temperature. Tang et al. (1989) explained the multiple transitions by a reoxidation process of oxygen deficient copperferrite, CuFe204_7, which takes place in the same temperature range. The octahedral Mn3+-jahi~-Teller ions are responsible for the tetragonal structure of Hausmamfite Mn304 (Jarosch 1987) and the tetragonal structure of the ferrites in the manganese ferrite system Mn, Fe3_zO4 with z > 1.95 (Brabers 1971). Due to the distortive effect, Jahn-Teller active cations influence significantly the free mixing

208

V.A.M. BRABERS

160

(a)

cubic

14C

..~tet[agonal >1

12( tetragonal ............ -cf f < l ~- 100

)

I

orthorhombic I

T

I

~ 160

14C

12C

100 ..,

magnetic ordered phase

f

80 2.0

:"

t

I

i

1.8

I

I

1.6 X

Fig.

5.

Crystallographic and magnetic phase diagram of the Fea_zCr:cO4system below 160 K (Kose and lida 1984).

energy3of the spinel lattice, which in the case of Mn ferrites gives rise to clustering of Mn +-ions and in extreme form to the miscibility gap in the Mn-rich part of this system (Brabers 1971; Holba et al. 1973)• Tetrahedral Fee+-ions are known to introduce Jahn-Teller phase transitions in the fe3_xTi~O4 and Fe3_xCrzO4ferrite systems. A small tetragonal distortion occurs near 140 K for Fe2.05Fi0.9504 due the Jahn-Teller effect of the A-site FeZ+-ions (Ishikawa et al. 1971). In the Fe3_~Cr~O4 system transitions to a tetragonal structure with c/a > 1 as well as with c/a < 1 are reported, depending on the composition of the spinel (Levinstein et al. 1972, Kose and lida 1984)• In a certain temperature and composition range as shown in fig. 5 the structure is orthorhombic due to interference of the magnetic ordering with the structural Jahn-Teller deformation (Kose and Iida 1984)•

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209

2. Magnetization and magnetic structure

2.1. Magnetic interactions The magnetic ordering in the simple spinel ferrites is one of the first examples of the application of N6el's two sublattice model of ferrimagnetism (N6el 1948). In this model it is assumed that the magnetic interaction between the tetrahedral and octahedral metal ions (AB interaction) is strongly negative and the interactions between the ions of the same sublattice (AA and BB interactions) weak. These interactions favour an antiparallel arrangement of the A- and B-lattice, the so-called collinear N6el configuration and consequently the resultant magnetization is the difference between the A- and B-lattice magnetizations. Using the molecular field concept for ferrimagnetic spinels, N6el explained the convex curvature of the reciprocal susceptibility vs temperature and predicted anomalous magnetization vs temperature curves, i.e. the occurrence of a compensation temperature and a second type of Ms(T) with a maximum magnetization at a finite temperature T ¢ 0. The former was proved by Gorter and Schulkes (1953) to appear in the Li0.sFe2.5_~Cr~vO 4 system and the second by Maxwell and Pickart (1953), in nickel aluminiumferrites The magnetic ordering in oxide spinels is mainly due to superexchange interactions via the oxygen anions. As pointed out by Anderson (1950, 1959, 1963) the theory of the superexchange interaction forecasts a maximum antiparallel interaction for a 1800 AOB configuration in case of certain 3d-transition metal ions, including Fe a+, while for the 90 ° BOB configuration the interaction is weak or small (see for more details also Krupi~ka, S. and E Novak, 1982, Oxide Spinels, in: Ferromagnetic Materials, Vol. 3, ed. Wohlfarth, p. 216). Although the 1800 configuration does not occur in the spinel structure, the 125 ° AOB angle still results in a significant strong interaction for Fe 3+ ions, in contrast with the 900 BOB interaction. The AOA interactions are very weak, because of the smaller angle and the larger distance and do not affect the magnetic order as far as there are sufficient magnetic ions on the B-sites. So in ferrites with a substantial concentration of Fe 3+ in both lattices, the prerequisite to obtain a collinear N6el arrangement, i.e. the relative strengths of the magnetic interactions seems to be fulfilled. Experimentally it has been confirmed that the magnetic moment of a number of simple ferrites, at least semi-quantitatively, can be fitted within the N6el model (Smit and Wijn 1959). However, very soon after the introduction of N6els model it was demonstrated that many other magnetic structures exist in spinel compounds, if substituted with various magnetic or non-magnetic ions. Particularly this is the case when the negative B-B interaction is compatible with the negative AB interaction. If the exchange integral [JAB[ is smaller than 3/2 [JBBI '-q'B/SA the B-lattice will then be split into two sublattices with magnetizations making an angle between each other and a resultant moment antiparallel to the A-site moment. This triangular uniform canting configuration was introduced by Yafet and Kittel (1952). Lyons and coworkers (1962) pointed out that the Yafet-Kittel triangular configuration is not stable in the cubic spinel symmetry but can be stabilized by a sufficient tetragonal distortion and that otherwise a spiral configuration represents the ground state.

210

V.A.M. BRABERS

A remark has to be made about the origin of large B - B interactions in certain spinels, which finds its roots in the direct exchange which is very prominent for Cr 3+ ions. Essentially the B - B exchange consists of an antiferromagnetic direct exchange and a superexchange interaction, which includes an antiferromagnetic and a ferromagnetic component. The superexchange, prevalently ferromagnetic for the 3d 3 electron configuration, is enhanced by covalency effects (Huang and Orbach 1968), whereas the direct contribution is strongly dependent oli cation-cation overlap. Consequently for the Cr 3+ ions with strong d-orbital overlap in oxides, the antiferromagnetic direct exchange usually exceeds the superexchange, while Cr3+ in sulfide and selenide spinels and Mn 4+ (also 3d 3) in oxidic spinels, exhibit ferromagnetic behaviour due to the larger covalency. A third exchange mechanism which can be found in oxidic spinels concerns the so called double exchange (Zener 1951), a ferromagnetic interaction, which is due to the hopping of electrons between metal ions in different valence states in one sublattice. In ferrites this double exchange will be masked by the strong negative A - B interaction, although the small positive JBB value for Fe304 derived from susceptibility measurements (Ndel 1948) might be indicative of such a contribution. Clear evidence for the occurrence of the double exchange • 3+ 2+ was found by Lotgering and Vall Diepen (1977) in the Znl.5_~Ti0.5+xFel_2~Fe2~ 04 system, which consists of spinels with only Fe z+ and Fe 3+ on B-sites• From the composition dependence of the paramagnetic Curie temperature the exchange integrals for the octahedral pairs Fe34-Fe 3+, Fe2+-Fe 2+ and Fe3+-Fe 2+ were determined to be J u = -1.370 K, J22 = -3.333 K and J12 -- +1.580 K, respectively, showing a clear positive value of J12 for the Fe3+-Fe 2+ interaction. The advantage of the ZnTi ferrite system studied by Lotgering and van Diepen is that the A-sublattice is only occupied by diamagnetic ions, so that A B and A A superexchange interactions do not interfere. Also in other spinel compounds with one sublattice exclusively occupied by diamagnetic ions, peculiar magnetic properties have been observed. This type of compounds is either antiferromagnetic or ferromagnetic, according to the sign of the interactions in the magnetic sublattice. It has been found that the magnitude of these interactions depends on the kind of the diamagnetic ion present in the other sublattice. Especially, relatively strong A A interactions are observed in Mn 2+ [Ga]204 and Co 2+ [Co3+]204 (octahedral Co 3+ is diamagnetic in the low-spin state), although the A - O - A superexchange is expected to be very weak. Both observations denote the presence of long range superexchange interactions of the type A - O - B - O - A , the strength of which depends mainly on the covalency of the B-ion. For compounds with only B - B interactions the situation is more complex, because of the anomalous property of the octahedral lattice which achieves 'perfect' short range order while maintaining a finite entropy. Nearest neighbour interactions alone can never lead to long range magnetic order, which means that other much weaker forces like magnetic anisotropy and dipole interactions are responsible for the long range order (Anderson 1956). 2.2. Magnetic structure of the simple ferrites Most of the technically applied spinel ferrites are based on a limited number of chemical systems. Depending on the specific properties desired, like high permeability,

PROGRESS IN SPINEL FERRITE RESEARCH

211

high induction, temperature and time stability, low losses and operating frequency, a choice has to be made for a certain composition. However, the final specification of the physical properties of a ferrite material is not only determined by the basic composition but also by the presence of small concentrations of impurities and the characteristics of the microstructure, i.e. grain size, porosity, grain boundaries, crystallographic texture etc. It is obvious that before developing the technology for the manufacturing of a specific ferrite material, including the microstructure, an appropriate choice of the composition has to be made on the basis of the intrinsic magnetic properties: saturation magnetic moment, magnetic anisotropy and magnetostriction. In table 6, the Curie temperature, the saturation magnetization at 300 K, the magnetic moment at 0 K, the magnetic anisotropy K1 and the magnetostriction coefficients at room temperature are listed for the same 'stoichiometric' ferrite spinels, of which the crystal structure parameters are given in table 2. From the high saturation magnetization at room temperature it is evident that ferrites based on Co, Li, Mg, Mn and Ni ferrite are the most prospective for applications, so much the more that the saturation magnetization can be enhanced by decreasing the A-site moment by substitution with non magnetic Zn-ions. Though, cobalt ferrite has the disadvantage of a high magnetic anisotropy, which makes ferrites with high Co-concentration not suitable for high permeability applications. Most of the technical ferrites can be found in the Mn-Zn, NiZn, Mg-Mn and Li-Zn-Ti ferrite systems. M6ssbauer spectroscopy has been proved to be a powerful tool to disclose the magnetic structure of ferrites. Particularly the presence of the Am = 0 lilies in the spectrum recorded with a magnetic field parallel to the 7-ray beam is indicative of spin canted magnetic structure. Neutron diffraction and high field magnetization experiments give supplementary information. Most of the important simple ferrites (Ni, Co, Cu, Mg and Li) are collinear Nrel type ferrimagnets with probably one exception MnFe204. Leung et al. (1973) showed that NiFe204 has a collinear magnetic structure, which is agreeing with the complete inverse structure, in spite of the ferromagnetic interaction between octahedral Fe 3+ and Ni 2+, suggested by Morel (1967). Lithium ferrite, Li0.sFe2.504, in the ordered as well as in the disordered crystallographic structure appeared to be a collinear N6el ferrimagnet (Dormann et al. 1983). Moreover, no substantial differences were found in the M6ssbauer hyperfine parameters for the two crystallographic structures, which is due to the short range order persisting in the disordered phase. Among the six B-site nearest neighbours of an octahedral ion, there are always two lithium ions, one in each tetrahedral group of B-sites which is in line with Anderson's description (1956) of the short range order in the spinel structure. The two crystallographic structures in which copperferrite can appear, remain also collinear magnetic ordered down to 4.2 K in a field of 6 T (Hannoyer et al. 1985, Janicki et al. 1982). Although Narayanasany and Hagstr6m (1983) reported a small canting in lower fields (4 T). Some doubt may be put forward about the nature of their specimens because of the high preparation temperature (1200°C) at which 'stoichiometric' copperferrite in fact does not exist (Brabers and Klerk 1977b).

212

V.A,M,

BRABERS

O t~

I

¢q

t~

.+.

+

I

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+ t¢3

4-

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r~

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+

+

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+

+

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+

+

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tr~ G'~

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V

P R O G R E S S IN SPINEL FERRITE R E S E A R C H

213

x~ I

i

I

i

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x

-I-

+

-l-

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+

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o~ O 0q

d.

F.©

oq

214

V.A.M. BRABERS

The cation distribution in M g F e 2 0 4 , which can be changed by the thermal treatment of the sample, may affect the magnetic structure. For a slowly cooled specimen with 94% of the Mg-ions in B-sites a collinear structure down to 4.2 K is observed (De Grave et al. 1979). Specimens quenched from above 1000°C showed broadened M6ssbauer spectra and some intensity around the Am = 0 lines at 77 K and in fields up to 4.5 T (Sawatzky et al. 1969). Neutron diffraction experiments (Wieser et al. 1970) indicated that the spin canting in the quenched specimens, if present, is not caused by a uniform long range Yafet Kittel order, but concerns a local canting of the Fe-spins due to the cation disorder. The precise determination of the spinstructure of CoFezO 4 is hampered by the high magnetic anisotropy and the possible variation of the inversion degree between 80 and 95%. Originally, the magnetic structure was reported to be collinear, supported by the neutron diffraction experiments of Prince (1956), but evidence for non-aligned spins was found at 4 K in magnetic fields up to 6 T, with canting angles 0A ,~ 0B ~ 250 (Petitt and Forester 1971 and Hauet et al. 1987). However, Persoons et al. (1993) argued that the non-alignment of the spins in the M6ssbauer spectra is not due to a canted structure but to the non-saturation of the sample caused by the high magnetic anisotropy; the nearly equal A- and B-site angles are also in favour of a non-alignment of the collinear structure in an external field. In addition, highfield magnetization measurements on a single crystal, fig. 6, show already saturation below 1 T for the easy direction (100), with no substantial high field susceptibility,

4

:~2

0

• [lOO] • [111] • [110]

6r

, 12

1'8

H (Tesla) Fig. 6. Magnetization curves of single-crystaUine CoFe204 at 4.2 K for the crystallographic main directions (Guillot et al. 1988).

PROGRESS IN SPINEL FERRITE RESEARCH

215

which denotes a collinear spin structure (Guillot et al. 1988). A similar conclusion was drawn by Teillet et al. (1993) from neutron diffraction experiments and highfield magnetization data on polycrystalline cobalt ferrite; for specimens with higher concentrations of Co in A-sites, i.e. lower inversion degree, it was not excluded that the long range collinear order might be interrupted by local canted spins. The magnetic structure of MnFe204 is from the simple ferrites the most polemical. No evidence has been found for spin canting of the Fe-ions from M6ssbauer spectroscopy, (Sawatzky et al. 1969), but the experimental magnetic moment at 4.2 K (4.5-4.8#B/formula unit) is too low to account for a collinear structure (5.0~B/formula unit). The partial inverse structure together with an anomalous valence state of the cations on the B-sites (Fe z+ and Mn 3+) have been suggested to induce the lower moment but M6ssbauer studies clearly show that the Fe 2+ and Mn3+-valencies are not the ground state (Sawatzky 1969 and Lotgering et al. 1973). Sawatzky et al. (1969) suggested that only the octahedral Mn 2+ spins are canted, which idea was further refined by Sin~a et al. (1975). The observed small highfield susceptibility and magnetic moment as function of the inversion degree was quantitatively explained by a non-collinear magnetic structure, in which the B-site Fe 3+ spins are antiferromagnetic coupled with the A-site lattice and the B-site Mn 2+ spins make an angle of 530 with the B-site Fe 3+ spins, which is in compliance with the empirical relation between moment and inversion degree y: m0(PB) = 5 - 2y. The inversion degree y of ceramic Mn-ferrite is usually in the range of 0-25%, but for small particles, prepared by precipitation from aqueous solutions, an inversion degree above 50% can be obtained (Yasuoka et al. 1967, Sakurai and Shinjo 1967, Corradi et al. 1977), which results in a lower magnetization and higher Curie temperature. Based on the two sublattice molecular-field model, Dionne (1988) calculated the thennomagnetization curves for MnFe204 as function of the inversion degree shown in fig. 7, using effective averaged values for the exchange interactions and averaged magnetic sublattice moments, derived from the site occupancy and the 53°C canting of the B-site Mn spins. Figure 7 clearly shows that the room temperature magnetization can be enhanced by lower inversion. For small MnFe204

6

2 I

I

200

I

I

400 T (K)

y=0 .2.4.6.8 1.C I

I

II

|

600

~1~

800

Fig. 7. Thermomagnetizationcurves for Mnl_uFe~[Mn~Fe2_~]O4 with the inversion y as parameter (Dionne 1988).

216

V.A.M. BRABERS

particles with high inversion this implies that the saturation magnetization can be increased by annealing at temperatures between 300 and 400°C, where a cation rearrangement takes place, resulting in a lower inversion. Experimentally this effect has been confirmed by Corradi et al. (1977) and similar behaviour has also been found for MnZn ferrite particles (Michalk et al. 1985, Petrera et al. 1982). Zinc ferrite, ZnFe204, as well as CdFe204, are special simple ferrites because of the normal cation distribution, so that there are only magnetic ions on the Bsites. At 10 K ZnFe204 magnetically orders with a doubling of the unit cell because of the non-collinear antiferromagnetic ordering of the B-site Fe 3+ depicted in fig. 8 (Kfnlg et al. 1971, Boucher et al. 1971). Srivastava et al. (1984) pointed out that the anisotropic exchange interaction arising from the hybridization of the orbitals on the oxygen ions is responsible for the two dimensional character of the antiferromagnetic order. The early neutron diffraction study by Hastings and Corliss (1956) suggested the antiferromagnetic nature of the magnetic structure, but the observed magnetic lines were broad, typical for short range order. Krnig et al. (1971) found that the effect of stoichiometry upon the magnetic ordering is large and that long range order can only be achieved in highly stoichiometric ZnFe204. More recent, Vologin (1987) showed by neutron diffraction experiments that Li and Co impurities up to 2% frustrate already the long range order, leading to a short range antiferromagnetic order. Although ZnFe204 has a strong preference for the normal cation distribution (Vologin et al. 1976), small concentrations in the order of 1 percent of Fe3+-ions 1

-1

~-o III IV~

II

~

5

~

3

1

-1

1

3

5

-1

5

3

1

-1

5

3

5

-1

1

3

1

-1

5

3

1

w

A w

Fig. 8. Magnetic order of het B-site Fe3+ ions in stoiehiometricZnFe204. The magnetic unit cell is tetragonal with a doubling of the c-axis. An essential feature of the model is the occurrence of parallel c-componentsof groups of four neighbouring iron spins. Open circles have +components and full circles -componentsalong the c-axis. Roman numbers correspond to the four reference spins, arab numbers refer to the position in units 1/8a (a is the chemical unit cell parameter) and the translations 001 and 1/2 1/2 0 are antitranslations. A relatively low Fe3+ moment, 4.2~B, is found which might be due to covalencyeffects (Krnig et al. 1971).

PROGRESS IN SPINELFERRITERESEARCH

217

in A-sites are supposed to be present in order to explain the smaller than expected Curie constant derived from the paramagnetic susceptibility (Lotgering 1966, Ligenza 1976, Ligenza et al. 1983). The tetrahedral Fe3+-ions create ferrimagnetic clusters with octahedral Fe 3+ neighbours due to the strong A - B interactions. Currently it is not clear if the A-site Fe 3+ presence is an intrinsic property of stoichiometric ZnFe204 or that Zn-deficiency due to the evaporation of zinc during the ceramic processing is the reason. Studies on zinc ferrite particles, synthesized by coprecipitation revealed a larger magnetization than found for bulk material (Sato et al. 1990b). Maximum magnetization values were found for particles with sizes of about 6 nm, which was caused by a long range ferrimagnetic ordering below 30 K. Similar as for Mn ferrite particles, it was proved by neutron diffraction that the diverging magnetization arises from a considerable Fe 3+ occupancy of the A-sites, which increases with decreasing particle size, i.e. 0.11 for 96 nm and 0.14 for 29 nm particles (Kamiyama et al. 1992).

2.3. Magnetic structure of the Zn-substituted ferrites The enhancement of the spontaneous magnetization by the dilution of the tetrahedral magnetic ion concentration with nonmagnetic Zn-ions is used in a large number of technical soft ferrites and is a direct consequence of the antiferromagnetic coupling between the tetrahedral and octahedral sublattices. In fig. 9 the magnetic moment at 0 K is plotted for the relevant mixed zinc ferrite series as function of the zinc concentration. In general, the magnetic collinear N6el structure is well able to account for the initial increase of the magnetic moment at low zinc concentrations but does not predict the subsequent decrease at higher concentrations.

6 e-,

~

M

e

=Mn Co

zI.

Ni Fe 2 Li

0 MeFe204

0.2

0.4

0.6 5

0.8

1.0 ZnFezO4

Fig. 9. The magnetic moment per formula unit of the mixed Mel_6Zn6Fe204 ferrites extrapolated to OK.

218

V.A.M. BRABERS

For the Nil_xZn~Fe204 system, Daniels and Rosencwaig (1970) and Leung et al. (1973) introduced uniform spin canting according to the Yafet-Kittel configuration to explain the decrease in magnetic moment above z = 0.5. In fact these authors approximated the uniform YK spin canting by an average value of the canting angle of the Fe3+-ions, discarding the variation in local canting due to the chemical disorder. Suggestions of paramagnetic isolated Fe 3+ centers (Gilleo 1960) and superparamagnetic clusters (Ishikawa 1962) to explain the magnetization at higher Zn-concentrations were not confirmed by the M6ssbauer experiments at 77 K and at room temperature reported by Daniels and Rosencwaig. The MOssbauer experiments at 4.2 K by Leung et al. (1973) revealed no significant canting below the composition z = 0.5, although neutron diffraction data (Satya Murthy et al. 1969) proved canting to exist also below that value. As a matter of fact, neutron diffraction probes the total spin canting whereas MOssbauer spectroscopy detects only the canting of the iron spins, which means that Ni2+-canting might be substantial in the range of lower Zn-concentrations. Moreover, Leung et al. suggested that in the composition range 0.5 ~< z ~< 0.8 not only the B-site moments are canted but also the A-site moments, which implies that the resultant magnetization in an external field is neither parallel to the A- or B-site spin moments. Taking into account the differences in the canting angles for the Fe 3+ and Ni 2+ B-site spins and different values for the two B magnetic sublattices, Leung et al. presented a schematic diagram of the spin --ME --ND configuration at 4.2 K for 0.5 ~< z ~< 0.8 as shown in fig. 10, 0 yK(B) and 0 yK(B) are the canting angles determined with MOssbauer and neutron diffraction experiments, respectively, and (S)B = 1/2(B1 + B2) with Bi = SFe(Bi) + SNi(BI) for i = 1, 2.

J

SFe(B I ) ~ N i ( B 1)

Fig. 10. Schematic presentation of the spin configuration for Ni1_xZnxFe20 a with 0.5 ~ x N< 0.8 at 4.2 K (Leung et al. 1973).

PROGRESS IN SPINEL FERRITE RESEARCH

219

For a high degree of magnetic dilution x ~ 0.75, Bhargava and Zeman (1980) found that superparamagnetic clusters are indeed effective at temperatures above 200 K, but that at lower temperatures a localized-canting model of the magnetic spins represents the magnetic structure. Vologin et al. (1979; 1980) concluded for the same composition from neutron diffraction experiments that below T = 5 K the magnetic structure might be helicoidal. Clark and Morrish (1973) introduced a spin reversal of those octahedral Fe 3+ ions, which are surrounded by only Zn ions as tetrahedral nearest neighbours, which allowed a refinement of the cation distributions. No experimental evidence was found for A-site spin canting in magnetic fields as low as 5 T, which implies that the two B magnetic sublattices are equal and consequently the distribution of the Ni 2+ and Fe 3+ ions on the B-sites must be at random. The MnZn ferrite system (Mnl_~Zn~Fe204) is from a M6ssbauer spectroscopy point of view an interesting system because the concentration of A-site Fe 3+-ions is low and ranges from about 20% for a: = 0 to about zero for z > 0.5. For large zinc concentration the spectra are only due to B-site contributions. Morrish and Clark (1975) showed that below x = 0.5 the magnetic structure at 4.2 K is collinear, as far as the iron spin configuration is concerned and that for x > 0.5 an increasing canting of the B-site Fe spins goes on with increasing Znconcentration. The canting angles are field dependent and are reduced in larger external fields; however, for z = 0.8 the structure seems to be quite stable because no decrease is observed in external fields between 5 and 9 T. The canting is also temperature dependent and disappears at 30 and 20 K in an external field of 5 T for x = 0.8 and 0.7, respectively (Morrish and Schurer 1977). Roughly, the magnetic structure of MnZn ferrites with large Zn content can be respresented with a four sublattice non-collinear model, one A-site lattice and three B-site lattices B1, B2 and B3, the latter being caused by the local reversal of the spins of the B-site Fe 3+ ions, which are only surrounded by Zn ions in the six nearest neighbour A-sites (Morrish and Clark 1975). The experimental determination of the magnetic structures in the CoZn ferrite (Col_~Zn~Fe204) system is obscured by the high anisotropic properties of the Co 2+ ions, similar as in the case of pure cobalt ferrite (CoFezO4). Non-alignment of the magnetic spins at 4.2 K in fields up to 5 T was reported by Petitt and Forester (1971), from which they concluded that spin canting occurs for all compositions in this system. However, for low Zn-concentrations it is more likely that non-saturation due to the high magnetic anisotropy causes the non-alignment and that the magnetic structure is coUinear at concentrations below • = 0.4. For z >/ 0.5 the non zero Am = 0 lines in the Mrssbauer spectra can be assigned to a canted structure because the anisotropy is strongly lowered with increasing z. At still higher Zn-concentrations localized spin canting with a distribution of canting angles certainly appears and evidence for paramagnetic clusters in C o 0 . 6 Z n 0 . n F e 2 0 4 has been found in the temperature range just below the Curie temperature of 322 K (Petitt and Forester 1971). 2+ Fe 2O 4) was studied by The spin structure in the zinc ferroferrite system (Zn~Fel_~: Dickof et al. (1980), who applied the local canting model, developed by Rosencwaig (1970) for substituted ytrium ions garnet, to this spinel system. A consistent picture of the composition dependence of the magnetization was obtained if the possibility

220

V.A.M. BRABERS

of multiple spin reversals due to next nearest neighbour interactions was taken into account. The critical composition for spin canting is near z = 0.4, for which an average angle of 200 at 4.2 K was found in an external field of 1 T and no appreciable canting in a field of 5 T. The magnetic structures in the LiZn ferrite system (Li0.5(l_~)Fe0.5(l_~)Zn~Fe204) are well accessible by Mrssbauer spectroscopy, because the only magnetic ions are Fe 3+ ions. Rosenberg et al. (1982) found evidence for a canted B-site structure for zinc concentrations above x = 0.6 and in fields up to 7 T. At higher Znconcentrations, reversed spins were supposed to be present in order to explain the low magnetization. Similar evidence for the B-site canting in the LiZn ferrite system was obtained by Dormann (1980), who found also evidence of A-site spin canting for the Ti substituted Li ferrites. From neutron diffraction experiments on the LiZn ferrite system, Vologin and Mal'tsev (1984, 1987) derived a non-collinear structure for the composition range 0.5 < z < 0.8; for zinc concentratiolts above x = 0.8 no long-range magnetic order is observed but the magnetic behaviour could be described by ferrimagnetic and antiferromagnetic clusters. In a subsequent paper by Vologin and Prokopov (1988) concerning neutron diffraction on a single crystal with composition z = 0.85, it was shown that the cluster idea is only a rough approximation, but that there exists also a long range modulation of the ferrimagnetic Order exceeding the dimensions of the clusters. Several models have been put forward to describe the magnetic structures occurring in the LiZn ferrite system. Uniform Yafet-Kittel canting using effective molecular field :coefficients depending on the average composition was proposed by Dionne (1974), but this model failed for large Zn concentration. Partial spin reversal (Rosenberg et al. 1982) is a special case of the local canting model applied by Patton and Liu (1983) and Camley and Patton (1986) to analyze the composition dependence of the saturation moment and high field susceptibility. Later, Zhang et al. (1989) refined the analysis with this local canting model by introducing empirical values for the exchange integral ratio JAB/JBB in terms of the molecular field coefficients derived from the experimental magnetization data versus temperature. By self-consistent iterative computation a good description of the experimentally determined composition dependence of the magnetic moment at 0 K according to Gorter (1954) was obtained. However, Patton and Liu (1983), reported magnetic moments derived from high field magnetization measurements, which are 10-20% higher than Gorter's data in the composition range 0.2 < x < 0.7, which implies that the model calculations by Zhang et al. have to be reconciled with these more realistic data. In all mixed Zn ferrite systems, Zn substitution above a concentration of x ~ 0.5 has been shown to decrease the resultant magnetic moment due to spin canting on the B-sites. Dormann (1980) found that Ti substitution in Li ferrite induces A-site spin canting and diminishes the B-site canting due to the presence of Ti 4+ on B-sites. This could lead to a small increase in net magnetization of the double substituted TiZn lithium ferrites but the effect of the zero contribution o f the Ti 4+ ions to the B-sublattice magnetization is usually larger than the increase of the magnetization

PROGRESS IN SPINELFERRITERESEARCH

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caused by the decreasing B-site canting. The net result is a decrease in magnetization, except for very high Zn concentration (Camley and Patton 1986 and Dionne 1974). Some LiZnTi ferrites (Lio.5(l_r+y)ZnxTiyFe2.5-0.5(r+3y)O4) show interesting microwave properties like a narrow ferrimagnetic resonance line width and a rectangular hysteresis loop (Baba et al. 1972, Nicolas 1980, Konwicki et al. 1977), which can be used in microwave applications. For the Ti-rich part of the system (y > 0.9) Piotrowski et al. (1987) and Ligenza et al. (1988) found with low angle neutron diffraction experimental evidence for the existence of superparamagnetic clusters, which are caused by a segregation of Fe 3+ into clusters below 800°C. 2.4. Disorder and spin-glass behaviour Up to this point, we have considered the magnetization of ferrites mainly from the point of view of the magnetic long range ordering. However, the chemically disordered spinel ferrites are typical candidates to realize several types of disordered magnetic phases. The general increased interest in disordered phenomena in the seventies had also an impact in magnetism. The spin-glass phase was introduced to describe the properties of magnetically diluted intermetallic compounds, which became known as canonical spin-glasses, f.i. the Cul_rMnx and AUl_~Fex intermetallics with x ~ 0.01. In these intermetallics, the long range, spatially oscillating RKKY interactions are the cause of the so-called spin frustration, resulting in a freezing of the spins in random orientation at low temperature, which can be probed by a cusp in the ac susceptibility at the freezing temperature Tt. The particular situation for magnetic spinels is that the magnetic interactions are only determined by the nearest-neighbour, involving generally antiferromagnetic exchange integrals with [JAB[ >/ [JBB[ > [JAA[. Thus, the two magnetic sublattices A and B are antiparallel, and the A A and B B interactions are frustrated. Moreover, on dilution the magnetic lattices with non-magnetic or weakly interacting ions, additional frustration of certain local moments will occur. In fact, increasing dilution causes a competition between percolation and frustration, which are basically different phenomena disturbing the magnetic long range order. In Systems with non-competing interactions i.c. one ferromagnetic sublattice or an antiferromagnet one with IJABI >> JAn, JBB, there is a critical value of dilution, the percolation threshold, at which there does not exist an infinite cluster of atoms connected by magnetic interactions and magnetic long range order is not possible. Several authors carried out Monte-Carlo calculations to determine the percolation limits for the magnetic ordering on dilution in spinels. Hubsch et al. (1978) calculated the percolation thresholds including AB, AA and B B bonds, arriving at a value of 0.33 for both A-site and B-site percolation. These data deviate strongly from the results of Scholl and Binder (1980), who found xc = 0.429 4- 0.003 for A-sites and x¢ = 0.390 4- 0.003 for B-sites by extrapolation of the Monte-Carlo simulation to infinite cluster sizes. It is clear that finite-size effects restrict the accuracy and the value of the percolation threshold if only finite clusters are considered. In view of this, the deviating value x¢ = 0.401 4- 0.012 for B-site percolation obtained for a finite cluster of 11664 sites (Fiorani et al. 1979) can be explained, whereas with a

222

V.A.M. BRABERS

series extrapolation method an estimate of zc(B) = 0.428 4-0.004, obtained by Sykes et al. (1976) is in good agreement with the result of Scholl and Binder. However, in systems with competing interactions, the magnetic long range order may break down at a lower concentration of the nonmagnetic ions than the percolation threshold due to the frustration effects of the competing interactions. The mixed spinel system z MgzTiO4-(1-x) MgFezO4 has become a classical example to show this behaviour. Scholl and Binder (1980) calculated numerically the percolation threshold for the spinel structure, considering a magnetic A or B ion to be part of an ordered cluster only if it is linked to it by at least one nearest JAB-bond. For the MgTi ferrite system, taking into account a statistical distribution of the Mg 2+-ions among A and B-sites and Ti 4+ only in B-site positions, a percolation threshold of z ,~ 0.68 is found which is far above the experimental concentration z ~ 0.4 at which the magnetic long range order breaks down (De Grave et al. 1977). This early breakdown of the magnetic order is indicative of the frustration effects due to the competing next nearest A A and B B interactions, which are shown to have finite non zero values for pure MgFe204. JAA, JAB and JBB are --5, --23 and - 9 K, respectively (De Grave et al. 1979). Scholl and Binder (1980) constructed a qualitative magnetic phase diagram for the MgTi ferrite system, in which the magnetic long range order persists up to a critical concentration z ~ 0.4 and spin-glass behaviour above that composition up to ~ 0.75 in the low temperature range (10-20 K). Based on M6ssbauer and magnetic susceptibility data, Brandt et al. (1985) arrived at a different phase diagram presented in fig. 11. The ferrimagnetic order in the system Mgl+xFe2_2,Ti, Oa becomes re-entrant near the percolation threshold I for A B bonds (x "~ 0.68) and in addition a transition to a spin-glass like state (SGL) occurs at low temperature, where the transverse components of the magnetic moments freeze. This second transition is equivalent with the semi-spin-glass transition introduced by Villain (1979). At higher dilution only a transition to the spin-glass state

loo ! collinear / paraferrimagnetic magnetic 50

pin 0

0.5 Dilution x

t

~

I

II

1.0

Fig. 11. Magneticphase diagramfor the Mgl+xFe2_zxTixO4system,deducedfrom M6ssbauerspectra and magetizationdata (Brandtet al. 1985).

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appears, of which the spin-glass freezing temperature extrapolates with increasing dilution to the percolation threshold II including AB, AA and BB bonds: z _ 0.83 (Hubsch et al. 1978). An important contribution to the understanding of the magnetic structure in disordered spinels with non-magnetic substitutions has been given by Villain (1979). V'dlain's first result concerns spinels with only magnetic ions oll B-sites, which, due to the high ground state degeneracy, remain in principle paramagnetic down to 0 K, the so-called 'cooperative paramagnets'. However, the degeneracy of the ground state is reduced by the introduction of non-magnetic impurities and a spin-glass state might be created with increasing non-magnetic substitution due to frustration effects. The second important outcome of Villain's paper is that an intermediate concentration of non-magnetic impurities in magnetic spinels give rise to a new state, in which a longitudinal component of long range magnetic order coexists with a transversal component exhibiting spin-glass behaviour. This magnetic state is based on the concept of canted local states occurring only in the vicinity of the impurities. These canted local states are characterized by XY components of the local moments perpendicular to the direction Z, the average direction of spins, i.e. the direction of the ferrimagnetic spontaneous magnetization in case of the MgTi ferrite system. The interaction between these XY components leads to the possibility of freezing in random orientations below some transition temperature TF, which is designated as the semi-spin-glass structure. This semi-spin-glass state is phenomenologically equivalent to a localised canted state at low temperatures, with the only difference that at higher temperatures a collinear structure appears. For diluted spinels Ao:AI_~[ByBI_~]04 ' ' with A ~ and B' non-magnetic atoms, a schematic magnetic phase diagram at T = 0 as shown in fig. 12 has been proposed by Coey (1987), which is partly based on the earlier results by Vilain (1979) and Poole and Farach (1982). A large number of papers have been published concerning the magnetic structure of specific magnetic diluted spinel systems, which papers are

sping,ass

para

' ~ .................. af

Fig. 12. Schematicmagneticphase diagram for magneticdilutedspinels AxA~_x [ByB~_~] 0 4 at 0 K. Aj and B ~are nonmagneticions; af = antiferromagnetic,fi = ferrimagneticand r.c. = random canting (Coey 1987).

224

V..A. M. BRABERS

1.0

netic ~t~-antiferromag - - perturbed I antiferromagnetic . J ~

Y 0.5 •

ferrimagnetic

LCS

spi: ,,

\

\gla •paramagnetic /

\

~erromagnetic 0.5

1.0

x Fig.

13.

Experimental magnetic phase diagram for iron spinels FexM]_x[FeuM~v]O4 at 0 K (Dormann et al. 1990).

of transient interest, but which might still contribute to the construction of universal magnetic phase diagrams. Dormann and Nogues (1990) reviewed the magnetic structures of substituted ferrites, and used published data to derive a tentative experimental • . 3+ • 3+ t 3+ It phase diagram presented in fig. 13 for the Fe spmels: Fe z MI_~(Fev M1_v)204. A spin glass or spin glass like phase is usually observed between the percolation limit lines II and I for A B + A A .4- B B interactions and only A B interactions, respectively. If only one sublattice is nearly completely occupied with Fe 3+ ions (x ~ 1, V ~ 0 or z ,~ 0 and V ~ 1), antiferromagnetic or perturbed antiferromagnetic phases are found. The line E separates the ferrimagnetic region from the localised canted state (LCS). In a certain range around this line semi-spin-glass behaviour might be expected• In ferrites with anisotropic magnetic ions like Co 2+, Mn3+ and Fez+, however, the uniaxial anisotropy of these ions in random positions could lead to a ferromagnetic wandering axis (FWA), which is phenomenologically equivalent to a local canting state. The net effect of these anisotropic ions will be an extension of the spin-glass region and a shift of the LCS region. The LCS region will be observed in a concentration range where normally ferrimagnetic order exists in case of isotropic ions and the spin-glass-like phases replaces the isotropic LCS range. A recent experimental study of such a system concerns the Co 2+ effect and the A-site dilution in the mixed Zn~COl_~FeCrO4 ferrites (Nogues et al. 1991), which show spin-glass-like behaviour below the A-site percolation limit but ferrimagnetic behaviour in high magnetic fields. However, Chakravarthy et al. (1991), using neutron diffraction techniques reported the existence of finite magnetic clusters in this system, which give rise to perturbed ordered states, and which are more complex than the semi-spin-glass structure or an mfiaxial ferromagnetic wandering axis. A

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225

Lil. 125Ti1.25Fe0.62504 4oo 6"

0.6-

200

-=

0.4 "V 0 . 2 ~

~

TF

TN

\ 0

I

)

50

100

T (K) Fig. 14. Temperature dependence of the mean hyperfine fields deduced from M6ssbauer spectra in zero field and of the mean canting angles (sin2 0A) and (sin 2 0B) in an external field of 1 T for Lil.125Til.25Feo.62504 (Dormann et al. 1987).

clear description of this particular magnetic phase is not yet presented but might emerge if the structure and dynamics of the finite clusters will be characterized by small-angle neutron scattering. In addition to the Mg-Ti ferrite system, Ti/Li-, AI, and Zn/Mg-substitutions in lithium ferrite (Li0.sFez.504) have to be mentioned as particular systems, in which semi-spin-glass structures are-encountered (Nogues et al. 1990). As an example, the results of the M6ssbauer experiments on Lil.125Til.25Fe0.62504 are shown in fig. 14. The temperature dependence of the mean hyperfine field in zero field and the mean canting angles (sin2 0a) and (sin 2 0B) measured in a field of 1 T indicate two anomalies: one at the Nrel temperature TN and a second one atthe spin freezing temperature Tf. Between TN and Tf evidence is found for the relaxation of the transverse spin component, which follows from the observation that a collinear structure is probed by the M6ssbauer spectra due to the smaller relaxation time 7compared with the measuring time 0"m ~ 10-8 s) and a non-collinear structure is probed by high field static magnetization measurements (Dormann et al. 1987).

226

V.A.M. B R A B E R S

3. The magnetic permeability 3.1. Domain wall movement and rotational contribution to the permeability

Spinel ferrites are widely used as soft magnetic materials which implies that the magnetic permeability as function of frequency, magnetic field and temperature as well as the losses are important factors in designing components based on these ferrites. The study of the magnetization processes in ferrites and the related losses has been an extended topic for over nearly fifty years. Basically, two processes are distinguished: rotation of the magnetization in domains and domain wall displacement. In principle, the domain wall displacement can give a much larger contribution to the permeability than the rotational process and accounts for the high permeability observed in certain non-microwave ferrites. However, in microwave ferrites the rotational permeability is the dominant mechanism. The pioneering work by Snoek (1948) showed the importance of the rotational process. Based on the prediction of the occurrence of the natural spin resonance with a resonance frequency Wr = "/HA (Landau and Litshitz 1935) which is caused by the anisotropy field H A, Snoek (1948) derived a relation between the static initial susceptibility Xi of a polycrystalline ferrite and the resonance frequency of the natural spin resonance (NSR): frXi =

4/3 7Ms

(10)

in which Ms is the saturation magnetization and 7 = ge/2mc. For a number of polycrystalline ferrites a reasonable agreement was found between the observed dispersion frequency in the susceptibility spectrum and the frequency calculated with Snoek's formula (10), which seems to justify the conclusion that in these ferrites the magnetization process is only of rotational origin (Smit and Wijn 1959). For a series of NixZnl_xFe204 ferrites see fig. 15. However, Rado et al. (1950) observed two resonance dispersions in the spectrum for certain Mg ferrites at about 30 and 1500 Mc/s, respectively. The frequency dependence of the permeability of small particles of the same ferrite embedded in wax showed only the microwave dispersion (see fig. 16). The microwave dispersion frequency agrees very well with the magnetic resonance frequency determined by the anisotropy field caused by the crystal anisotropy K1; the lower frequency dispersion is supposed to be related to the resonance of the domain walls. However, in certain cases the experimentally found lower frequency dispersions might also originate from dilnensional resonance of the samples due to the high magnetic and dielectric permeability as pointed out by Brockmann et al. (1950). For typical Mn-Zn ferrites (pi ranging between 2000 and 5000), the dimensional resonance can occur in cores with cross section dimensions between 0.5 and 10 cm in the frequency range 10-2-10 MHz (Snelling 1988). Other substituted Mg ferrites show only one broad resonance ranging from radio to microwave frequencies, which is now supposed to be an overlap of the two types of resonance (Rado 1953). In fact these observations suggest that both magnetization processes can contribute to the magnetic susceptibility.

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227

10000

1000

1000

~ ~wWh~

t00

~

10 ~t"

P" 100

10

1

0.1

0-60. .

10

1

100

1000

4000

f (MHz) Fig. 15. Frequency dependence of ~z' and #", the real (o) and imaginary (o) parts of the initial permeability for a series polycrystalline NizZnl_ x Fe204 with x = 0.36; 0.50; 0.64; 0.80 and 1.0 for the curves labelled with A . . . . . . E, respectively. (Smit and Wijn 1959.) 24 Lt' -1 0.8 16

(b)

0.6

12 ~t

0.4

8 0.2 4

o °

0 I

I

r

10

I

I

I

t 00

f (MHz)

p

u

1000

I

-0.

i

10000

_J

I

I

10

I PI f I I i 1 O0

1000

10000

f (MHz)

Fig. 16. Frequency dependence of #' and /~" for polycrystalline MgFe ferrite at room temperature (a) and for a suspension of fine particles of this ferrite in wax (b) (Rado et al. 1950).

228

v..A.M. BRABERS

The initial susceptibility Xirot due to the rotation of the magnetization M in a polycrystalline material is given by: rot _ Xi

(11) K1 '

where the coefficient c depends on the sign of the anisotropy constant (c = 1/3 for K1 > 0 and c = 1/2 for K1 < 0, Chikazumi 1966). In principle this contribution does not depend on the grain size of the ferrite because it originates from the rotation of the magnetization in the domains or grains and gives the lower limit of the susceptibility, which depends only on the intrinsic properties of the ferrite: the magnetic anisotropy and saturation magnetization. But, as we will see later on, the effective rotational susceptibility of certain polycrystaUine samples might be still dependent on the microstructure; i.e. the grain size, in the case that grain boundaries with a finite thickness and a much smaller susceptibility than the bulk of the grains are present. The contribution of the domain wall motion to the initial susceptibility, however, is strongly dependent on the grain size as was experimentally shown by the linear dependence of X on the grain size D, first by Globus (1963a) for low- X nickel ferrite and later for high-x MnZn ferrite by R6ss et al. (1964). R6ss et al. supposed that with increasing grain size the internal stress decreased, which could explain qualitatively the increasing susceptibility. Globus (1963b) suggested a simple model, in which the grains are considered as spheres which contain

,H Q

L x LI

1"

f

D

i

t Fig. 17. Domainwall bulging, under the influenceof an external field H. The magneticpoles appearing are indicated.

PROGRESS IN SPINEL FERRITE RESEARCH

229

in the demagnetized state only one diametrical 1800 domain wall pinned at the periphery of the grain and which can bulge out when a small external field is applied. In equilibrium, the gain in field energy is in balance with the increased wall energy, created by the increase in wall area and the demagnetizing energy due to the magnetic poles formed on the bulged wall (see fig. 17). Smit and Wijn (1959) suggested that the demagnetizing energy could be substantially larger than the increase in wall energy and calculated for the initial susceptibility M~4D

Xi '~ 20 ~

K12.d

,

(12)

where d represents the distance between the domain walls and D z the surface of the wall. In deriving eq. (12), it was supposed that the crystal anisotropy was in the same order of magnitude as the demagnetizing energy. However, Hoekstra et al. (1978) showed that for low anisotropy MnZn-ferrites, i.e. M 2 >> K1, this is not correct. The ratio of the demagnetizing energy AEd and anisotropy energy AEk equals:

AEd/AEk ~--~ ( 3 71")(.~I/M:).

(13)

For small values of K1/M2s, the aIfisotropy energy is dominant! In the special case of the MnZn ferrites studied by Hoekstra et al., an estimate of the wall energy showed that this energy is also negligible and that the susceptibility for spherical bulging can be calculated from the balance between AEk and the magnetostatic energy: Xi '~ 1 / I O M 2 / K I •

(14)

Hoekstra et al. (1978) discussed several cases of domain wall bulging as indicated in fig. 18. In fig. 18a, the wall is only pinned along the two vertical edges and this pinning causes cylindrical bulging. No magnetic poles are formed on the wall and consequently no demagnetizing energy is involved; the susceptibility calculated for this type of bulging results in:

Xi-'- ~1 M2"(AK1) -112 "L

(15)

with L the distance between the domain walls and A the exchange constant. The temperature dependence of Xi of some metals like Fe and Ni could be well described with the formula (Kersten 1956): xi(T) ,-~ L M ( T ) / [ K I ( T ) ] 112

(16)

which is obtained by substituting the approximated temperature dependence of A ( T ) = A(O)[M(T)/M(O)] 2 in eq. (15). In polycrystalline ferrites it is very likely that the domain walls are pinned by the grain boundaries and move only within the grains.

230

V.A.M. BRABERS

In a certain range of grain diameters, in which there is only one wall in every grain, L equals the grain diameter D and the linear dependence (16) between #i and grain size is obtained. However, the cylindrical wall pinning (fig. 18a) seems to be rather unrealistic for polycrystalline ferrites with more or less spherical grains; spherical bulging, including demagnetizing effects should be considered, which does not lead to a linear dependence between #i and the grain size (compare eqs (12) and (16)). Further the experimental ~i values obtained by Hoekstra et al. for their MnZn fen-ites were in between the theoretical values calculated with their formulas for cylindrical and spherical bulging (eq. (14) and (16)) which suggested that a finite pinning as indicated in fig. 18c occurs with a mixed spherical-cylindrical bulging as result. Kersten (1956) argued that for small K 1 / M 2 value the demagnetization energy ED could be small, due to the capability of the magnetization close to the wall to rotate away from the preferred orientation, more into the direction parallel to the bulged wall, also suggested by NEd (1946). Globus (1963b, 1977a, b) used this idea that ED could be small also for the simple spherical bulging and proposed a model in which the experimentally found linear grain size dependence of/_t i was explained by spherical bulging, ignoring the demagnetizing energy. Although this model is basically a rough approximation, a number of experimental features and relations were established which could be described within the model, and which are worthwhile to discuss in more detail. For polycrystalline ferrites there are two effects which have to be considered when the grain size becomes very small. The first item concerns the wall thickness 6w; for low anisotropy values (K1 "- 103 erg/cm 3 = 102 J/m 3) the wall thickness ~Sw= ~ / ~ / K 1 becomes of the order of 1 #.

/ MI

!iiii~'

/

M

/

(a)

(b)

/ M

i~i~N ii!IN

M

/

(c) Fig. 18. Models for domain-wall bulging. (a) The wall is pinned along two edges, resulting in cylindrical bulging. (b) The wall is pinned along its circumference causing spherical bulging. (c) The wall is pinned along its circumference with a finite pinning, resulting in a mixed spherical-cylindrical bulging (Hoekstra et al. 1978).

PROGRESS IN SPINEL FERRITERESEARCH

231

The second item concerns the critical size of the spherical grains below which the grains are monodomains (Smit and Wijn 1959): Dcr

1.47 =

(17)

,

where 7 = 2 Av/-~x represents the wall energy. Both these effects put a lower limit to the grain size for the application of magnetization models based on domain wall movements in polycrystalline ferrites. In low anisotropy ferrites the domain walls are rather thick and can reach values up to 1/z, which means that domain walls if present in the 1 # grains cannot move. In this case the magnetization process may be caused by a contraction of the domain wall, pinned in the microstructure, which mechanism was suggested by Kornetzki (1962) and R6ss (1965), and is in principle a spin rotation process. In higher anisotropy ferrites (K1 " 106 erg/cm3), domain walls are only possible in a spherical configuration as in fig. 18b in grains with D in the range larger than 0.5 # eq. (17). A very interesting technique to study the magnetization in magnetic materials is the neutron depolarization technique. In contrast with other techniques, where only the surface domain structure can be studied, the neutron depolarization technique probes the domain size in the bulk of the material (see for a review of the ND-technique Rekveldt 1989). With this technique it was shown by Rekveldt (1977) that in a MgMn ferrite (Mgo.68Mn0.52Fea.sO4) with small grain size, 0.8 /z, the magnetic domain size is of the order of a few microns, much larger than the grain size, indicating a cooperative effect of the grain size magnetizations. Similar experiments on NiZn ferrite (Ni0.49Zno.50Fe2.0104) with a grain size of 2 # indicated a domain size of the order of the grain size. For both ferrites it is clear that the moving domain wall mechanisms cannot be the origin of the magnetization process.

3.2. The Globus model for polycrystalline ferrites In the model proposed by Globus (1963b) the demagnetized state of a polycrystalline ferrite is given by the diametrical 180 o domain walls pinned at the periphery of spherical grains with diameter D = 2r (fig. 19). Applying a weak magnetic field H H=0

®H

grain Fig. 19. Model of a domain wall pinned to the grain boundaryin a spherical grain (Globus 1963b).

232

V.A.M. BRABERS

spherical bulging occurs; the wall surface is 7r(r2 + X2) and the volume of the black spherical segment is ~r/6(3r 2 + ~e2). The equilibrium state in a field H is now determined by the minimization between the magnetostatic energy of the reversed magnetization in this segment and the wall energy, resulting in the relation (18) in which 7 represents the wall energy w 3Tr 2 Xi -- ( # i - 1) w - - - M s D / 7 . 4

(18)

Relation (18) is a lineair relation between Xw and the grain size, which was experimentally found for NiFe204 (Globus 1962), NiZn ferfites (Globus and Duplex 1966) and MnZn ferrites (R6ss and Hanke 1970, and Perduyn and Peloschek 1968). However, instead of the experimental Xw value, the value (xw)c has to be introduced in formula (18); (x.W)c is the initial susceptibility corrected for the porosity (xW)~. = Xidx/d, with dx tl~e X-ray density and d the density of the polycrystalline speomens. The experimental linear grain size dependence of the corrected susceptibility found for NiZn ferrites and YFe-garnets (Globus and Duplex 1966, 1968) indicated, provided no pores are present inside the crystallites, that the intergranular pores do not affect the susceptibility. Apparently, the contribution of the intergranular porosity to the total auisotropy by means of demagnetizing fields is for these materials negligible or constant. Relation (18) was also derived by Guyot et al. (1988) from the equation of motion of a 180 o domain wall (D6ring 1948): m5 +/3;2 + az = 2Ms.H.

(19)

This equation describes the motion of a unit surface of a domain wall in an infinite medium; m represents the domain wall mass, /3 the damping coefficient and c~ the restoring force per unit surface and 2Ms.H is the magnetic pressure due to the external magnetic field. Replacing eq. (19) by its integral over the whole surface of the individual domain wall, the dynamic response to a small ac field H = H0 exp (jwt) can be calculated with the additional assumptions that: -

-

-

all the domain parameters are frequency independent the mass of the domain wall is small, so that the term m~ can be neglected and the damping is dominant the response of the domain wall to the field is linear and consequently the bulging remains spherical.

If we take z0 as the displacement of the center of the wall, z0 = Z0 exp (j~t), the domain wall energy 7 (per surface unit) acts as a surface tension which gives a total restoring force ~ 4rrTz 0. The total damping force equals jw/3zo~r2/2. The sum of both forces is opposite to the magnetic thrust 2.Ms.H.~rr 2 and the solution of this equation is a relaxational type equation Ms.Hr 2 exp (jwt) z0 = ~ 27 1 + jwflr2/87

(20)

PROGRESS IN SPINEL FERRITE RESEARCH

233

From (20) the quasi-static wall contribution (18) can be derived as well as the wall relaxation frequency: fo =

47

(21)

~-flr2

Gieraltowski and Globus (1977) found experimentally for NiFe~O4 and YFe-garnet samples indeed a relaxation frequency proportional to 1/D z and a proportionality between the maximum in the frequency dependence of the magnetic losses # " and the mean grain size Din. The practical interest of these findings is that the dispersion of the susceptibility can be shifted to higher frequencies and the magnetic losses can be suppressed by low grain sizes. Another remarkable result is that for the domain wall bulging susceptibility of a ferrite with a fixed grain size Xw.fO z ~ constant (Globus 1977a, b), with f0 the wall resonance frequency while for the spin rotation susceptibility Xs'fr = constant (formula (10)). The bulging of the domain wall in a small magnetic field is a reversible process, which is supposed to contribute not or olfly to a minor extent to the losses and gives a constant value of the permeability up to a critical field Her above which a sudden increase of the permeability is observed. A typical example of the field dependence of the permeability and the occurrence of a critical magnetic field is shown in fig. 20 4000

3000

j / /395K ~

H~C

2000

1000

77 K 0

I

I

I

,

200

400

600

800

1000

H (mOe) Fig. 20. The permeabilityof polyerystallineNio.sZno.sFe204(grain size D = 3.0 #m and porosity 0.03) as function of the measuring field H at various temperatures(Globus and Duplex 1971).

234

V.A.M. BRABERS created ring omain wall

grain

Fig. 21. The irreversible domain wall displacementfrom its diametrical positionin a spherical grain. (Globus and Duplex 1971). Guyot and Globus (1973, 1977) attributed the appearance of this critical H value to the depinning of the domain wall, extending the wall motion with irreversible displacements as indicated in fig. 21. For Hcr, the wall detaches and moves away from its diametrical position in the Globus model, which leads to an increase of the susceptibility. The value of H~r for a certain composition must be proportional to D -1, since the magnetic thrust acting on the wall is proportional to D e and the total pinning force is proportional to D. In the remanent state the domain wall with radius r0 is fiat and located at a distance a from the diametrical position. With a simplified calculation, Guyot and Globus determined the energy dissipation in the quasi-static state, i.e. at low frequencies, which is composed of two contributions. For the continuous pinning and depinning of the wall at the grain boundaries, the peripheric pinning force F = 27rrf act as a friction force giving rise to an energy loss. The second contribution is proportional to the wall energy in the peripheric ring of the domain wall which is created and annihilated when the wall is moving. For the hysteresis loss E eq. (22) is obtained: ED - -

8mr

n7 ~

f + --

3 mr

(22)

with m r the relative remanent magnetization, D the diameter of the grain, f the pinning force per length unit, 7 the wall energy and n a proportionality constant (~, 1). Equation (22) relates the hysteresis losses to the relative remanent magnetization mr and by plotting E,D/8m, versus mr, the wall energy 7 can be determined from the slope of this curve. In fig. 22 the temperature dependence of the wall energy for NiFe204 and Ni0.47Zn0.53Fe20 4 is presented and compared with the calculated value 7 = 2x/-)I-RT1 (dashed lines) in which K1 is the magnetocrystalline anisotropy and A the exchange constant, determined from the emperical relation

kTc (1 A(T) = --~

-

T/Tc) 112

with a, the lattice parameter and Tc the Curie temperature.

PROGRESS IN SPINEL FERRITE RESEARCH

235

2.0 NiFe204

1.0

O3 I

I

I

I

I

I

I

1.0 ~r

0.~

\

•,,

Nio.47Zno.53Fe204

i

0

I ' ~t~l~-~ "¢-r~+-- ~ I

500 T (K)

Fig. 22. Temperature dependence of the wall energy. The dashed curves are calculated (Guyot and Globus 1977).

A final outcome of the Globus model is the possibility of the construction of normalized hysteresis loops by plotting M/Ms versus HD/Hett and universal hysteresis loops, Plotting M / M s . K 1 / K versus HD/Hefr, with M the magnetization, Ms the saturation magnetization, K the total magnetic anisotropy of the polycrystalline sample and Heft the effective magnetic field, a universal hysteresis loop can be obtained, not dependent on the material parameters (Globus 1977). Important parameters in this analysis are the total anisotropy and the effective anisotropy field caused by the total anisotropy. The total anisotropy of a polycrystalline ferrite consists of three components, the magnetocrystaline anisotropy (K1), the magneto-elastic energy (~.As) and a dipolar term proportional to M 2. Since the intergranular porosity did not affect the permeability, the dipolar contribution is not caused by the porosity (Globus and Duplex 1966, 1968). For single domain polycrystalline NiZn and MnZn ferrites it was proved that the dipolar contribution is negligible, which means that the total anisotropy K can be presented by K = K 1 + AK = K1 + tr.As, with As the saturation magnetostriction and tr an elastic stress (Globus and Duplex 1968). This elastic stress was related by Pascard and Globus (1981) to the exchange stresses 'produced by the volume magnetostriction' A V / V , which is the magnetic volume expansion due to the spontaneous magnetization. In fig. 23, the specific polycrystalline anisotropy AK(T) for NiFe204 is plotted versus the 'exchange stress' trn(T) = trn "- (AV(T)/v)/x, with the compressibility and A V ( T ) / V the volume magnetostriction, deduced from a comparison of the thermal expansion of NiFezO4 and a nonmagnetic spinel with the same Debye temperature (fig. 24).

236

V.A.M. BRABERS

= 855 K -1 ~. -21

o9

"~'-3 .
> D, the measured permeability depends linearly upon the grain size and for very large grains #e is equal to #i. The initial permeability values obtained on single crystals of ahnost the same compostion were indeed in the same range as the fitted values. A similar curved grain size dependence of #i was also reported by Gieraltowski (1989) for cobalt doped NiZn ferrites with average grain sizes between 1 and 4 #m which was explained with a model of wall displacements in the grains with a large distribution of the grain sizes (Postupolski and Wisniewska 1978). However, Co-doping in NiZn ferrites stabilizes the domain wall, decreases the wall permeability and favours the rotational magnetization (de Lau 1975). Moreover, neutron depolarization experiments on undoped NiZn ferrite with D ~ 2 #m indicated that the domain size was identical to the grain size, which makes the wall mechanism very unlikely in these materials (Rekveldt 1977). In fig. 28c it is shown that Gieraltowski's data can be well analysed with the non magnetic grain boundary model, with #i -- 100 and 6 - 12 run (Johnson and Visser 1990). The non-magnetic grain boundary model seems to be complementary to the Globus domain wall size model and can be applied particularly to those ferrites in which the rotational permeability is dominant and the Globus model is not appropriate. Other implements of this model are in agreement with experimental findings; the resonance frequency, at which #" = #', increases with decreasing grain size, according to the Snoek relation (10) for rotational permeability and the temperature dependence of the polycrystalline permeability can be described without assuming an additional anisotropy originating from stresses like in the Globus model (see fig. 29). However, a serious drawback of the grain boundary model is the fact that the temperature factor 1/#2.d#/dT as well as the loss factor ].ttt/(pt) 2, both important parameters

242

V.A.M. BRABERS

2{.){10

. . . . . . . . . . . . . . . . . . . . tL

-e

-

•

1500 1000 500

o -

fo

20(}0

?5

2o

b 1500 -I

t 00O

t

U~ +

500 /'* ~+~

0

~"

L_

~ Si containing . Pure •

J ___.L ....

2

4

7

L--+

6

,-----. ~ - - ~ - -

8

I0

5

5(1 25

0

I

2

3

4

Grain size (I-tin) Fig. 2& (a) Grain size dependeneeofthe rotationalpermeabilityofpolycrystallineMn0.68Zno.z4Fe~.osO4. The solid curve fits eq. (25) with #i = 2500 and 6 = 1.5 nm (Visser and Johnson 1991). (b) Similar plot as (a) for Mno.6oZno.35Fez0504, including samples with grain sizes down to 0.3 ~zm, obtained by sintering wet chemical prepared particles (Johnson et al. 1992 and Noordermeer et al. 1988 and 1991). The solid curve represents the fit to eq. (25) with ~zl = 3000 and 6 = 1.5 nm. (e) Grain size dependence of the permeability for Co doped NiZn ferrites (Gieraltowski 1989). Solid curve is the fit to eq. (25) with #i = 100 and 6 = 12 nm (Johnson and Visser 1990).

for applications, are independent of the grain size, which is not in agreement with experiments (Visser and Johnson 1991). Both effects are observed to be strongly affected by the microstructure (Snelling 1988). Another aspect of the permeability which remains unexplained in the grain boundary model is the occasionally occurrence of a peak in the frequency dependence of the permeability just below the resonance frequency, indicating a resonance character of the dispersion. In terms of Globus' domain wall size model, Gieraltowski (1989) related the occurrence and the magnitude of these peaks to the standard deviation of the grain size distribution.

PROGRESS IN SPINEL FERRITE RESEARCH

i0000

Singlecrystal

243

.~

5000 / _- 150.0 r"j .... ~,--,,-o'-~-'~\g~ = 2500

0 Poly crystals 3000 :~ ]

j J

.oool 1

_ ^ aD=9gm

_ 0

0

0

~ i

• D = 2gm'

-50

0

5'0

100

T (°C)

Fig. 29. Temperature dependence of the initial permeability in single and polycrystalline Mn0.68Zn0.24Feg_.0804. The large symbols are the calculated values, using the NMGB model with 6 = 1.5 nm and the #i values of the single crystal permeability (Visser and Johnson 1991).

High-permeability ferrites with low magnetic anisotropy are the potential candidates for which the non-magnetic grain boundary model can be applied, i.e. the manganese zinc ferrites. Knowles discussed already in 1977 the permeability mechanisms in these ferrites and emphasized the importance of the rotational permeability especially at low anisotropy. Supposing a rotational contribution as well as wall susceptibility based on the Globus model, he calculated the total susceptibility of a K1 randomly oriented assembly of spherical grains with diameter D. In fig. 30, the bro2 ken line presents the calculated normalized permeability (l~-l)/M;/2K1 of a MnZn ferrite as function of the normalized grain diameter Dx/A/K 1 cx D/6, intercepting the vertical axis at a value of 2/3, the latter value being determined by the anisotropy constant K1. The solid line presents the experimental permeability; the extrapolation of this line will intercept the vertical axis at a considerably lower value than 2/3, which means that the effective anisotropy is higher than K1. Kerr observations on these ferrites, subjected to a small ac field indicated also that the magnetization is pinned at the grain boundaries as if subjected to a large local anisotropy. This large grain boundary anisotropy gives also evidence for the non-magnetic grain boundary model, since the permeability in the grain boundary region will be substantially lowered by the enhanced local anisotropy. Further, the ratio between grain size and wall thickness D/6 must be at least two, if 1800 walls are to exist, which implies that D/x/-A/K > 14. The calculations of Knowles are only of interest for those materials

244

V.A.M. BRABERS

e~

2

"-"

1 0

'

20

'

40

D/~ A/K 1 Fig. 30. Normalized susceptbility as function of the normalized grain diameter (l~mwles 1977).

where the grain diameter is in the order of the wall thickness and for those having a small anisotropy, permitting a substantial rotational susceptibility like in the MnZn ferrites. An essential feature of the NMGB model is that the presence of intra-granular domain walls is not a prerequisite like in the case of the domain wall models. Moreover, as mentioned before, there exists a critical grain diameter below which domain walls cannot exist in the grains due to the competition between the magnetostatic energy and wall energy. Certainly, below this critical diameter, the NMGB model can describe the permeability behaviour of polycrystalline MnZn ferrites very well as was shown in a paper by van der Zaag et al. (1993). In their study the domain size of a series of mangenese zinc ferrites with constant composition Mn0.60Zn0.35Fe2.0504 and variable grain size was determined with the neutron depolarization tectmique (ND) and related to the initial permeability. The ND experiments were performed on demagnetized samples; the results are given in fig. 31, where the domain size A is plotted versus the grain size. Below diameters of 4 #, a 1:1 relation is obtained and from 4 to 8 # m a 1:2 relation, which indicates that 4 # m is the critical diameter Dc for domain wall formation in this ferrite. However, a theoretical estimate of De, using eq. (26), a modification of eq. (17), in which the magnetic permeability of the surrounding medium of the grains was taken into consideration, gives Dc values always lower than the grain size.

De ~ #o M [

(26)

For instance, inserting the experimentally determined parameters in (26) with #¢~f = 1200, the macroscopic effective permeability of the polycrystalline ferrite with a grain diameter 4 # yielded a critical diameter of 0.8 #m. (The other parameters of this ferrite are #o.Ms = 0.52 T, Tc = 430 K, K = 32 J / m 3 and a ~ 3 ,A,) Also the calculated domain wall thickness (~ 1 pro) does not exclude the domain formation in 4 # m grains.

PROGRESS IN SPINEL FERRITE RESEARCH

245

J f

f f

f

f

2

I

0

I

2

I

I

4

I

I

6

I

I

I

8

I

10

~

I

12

I

b

14

I

I

16

D (btm) Fig. 31. Magnetic domain size as function of the grain size of polycrystallineMn0.6oZn0.35Fe2.0504, determined with neutron depolarizationexperiments (Van der Zaag et al. 1993). The solid line corresponds to AID = 1, which reflects the monodomainstate and the dashed line AID = 0.63 is the ratio calculated for the two-domainstate of the grains. The large discrepancy between the experimentally and theoretically obtained critical diameters is still puzzling. In fact, the introduction of the macroscopic permeability in eq. (26) is a rough approximation. Nevertheless, the experimental neutron depolarisation data indicate clearly that the critical diameter for this ferrite is near 4 #m. The initial permeability, measured at the induction level below 0.1 m T is plotted in fig. 32 versus the grain diameter. The experimental data fit well to eq. (25) with 5 = 1.39 nm and #i = 2240 (solid curve). No discontinuity was observed near D = 4/am, the critical diameter, which is illustrative that the intra-granular domain wall does not affect the permeability significantly at this low induction level and the permeability must be predominantly of rotational origin in this regime. The basic question concerning the NMGB-model is now whether the non-magnetic grain boundary has a real physical foundation and might be inherent in the ceramic nature of ferrites. It is obvious that the grain boundary chemistry will play an important role: chemical inhomogeneities, preferentially located near the grain boundary might be a major cause of the low permeability layer 8. Especially, to suppress eddy current losses in commercial MnZn ferrites, additives like CaO and SiO2 are used, which are known to segregate at the grain boundaries (Franken and Stacey 1980). Tsunekawa et al. (1979) reported even a 3-5 nm segregation of a non-crystalline phase at the grain boundaries in MnZn ferrites for impurity levels of 0.3% CaO and 0.04% SIO2, which affected also the lattice parameter of the spinel phase over a distance of about 20 nm. The permeability of these high-impurity level ferrites is much lower (1000-3500) than for more pure ferrites (18500), where no evidence

246

V..A.M. BRABERS

2000 i

1500 :£

1000

500

I

0

I

2

I

I

4

I

I

6

I

i

I

8

1'0

'

' 12

'

1'4

'

16

D (/am) Fig. 32. The initial permeabilityat 180C as function of the grain size for Mno.6oZno.35Fe2.osO4. The solid line is a fit to the NMGB model with ~i = 2240 and 6 = 1.39 nm (Van der Zaag et al. 1993). of a glassy phase at the grain boundary could be found. Sundahl et al. (1981) investigated the effect of the grain boundary chemistry of polycrystalline MnZn ferrites (Mno.50Zno.42Fe2.0804) with identical microstructure (7 # grains) in relation to the observed permeability. The Zn depletion in the grain boundaries detected with Auger spectroscopy is supposed to account for the degrading of the permeability due to microstresses at the grain boundaries. No significant correlation was found between permeability and the segregation of Si and Ca at the grain boundaries, but a large permeability was observed if the boundary composition is oxygen deficient compared with the stoichiometric spinel composition M304. Consequently, the lower permeability for ferrites with oxidized grain boundaries can be explained by the low permeability of the grain boundary region in terms of the NMGB model. The bulk composition of the ferrite is chosen to have a low magnetic anisotropy, but oxydation of the grain boundary will change the anisotropy substantially. The balance between the anisotropy contributions o f the different ionic species is lifted due to the changing concentration of the cations with various valencies. Particularly, the oxidation of the highly anisotropic Fe2+-ions may change the anisotropy. An indirect indication that grain boundary oxydation in ferrites exists can be found in the dielectric behaviour. Due to the lower Fe z+ concentration the electrical resistivity of the grain boundary is larger than the grain resistivity and a Maxwell-Wagner dispersion of the dielectric constant can be observed (Koops 1951). Zn depletion in the grain boundaries, as well as Mn-depletion was also reported by vail der Zaag et al. (1993) over a range of about 10 rim. The interfaces between the grains in their samples, studied by TEM, showed only the spinel structure and no second phases. It has to be mentioned that the studied ferrites have a low impurity level (< 100 ppm).

PROGRESS IN SPINEL FERRITE RESEARCH

247

To justify the non-magnetic boundary thickness of 1.4 nm in their analysis of the permeability data, they suggested that also spin canting occurs in the surface as a result of the discontinuity of the structure at the interface of the grains, and because of this, large anisotropy fields will be produced at the interface. Similar effects were suggested to create magnetic inactive layers in small particles of various types of ferrites, with a thickness ranging between 0.3 and 1.5 nm (Sato et al. 1987). A combination of Zn-depletion and spin canting at the surface is proposed by van der Zaag et al. (1993) as explanation for a low permeability grain boundary layer in their samples. An interesting analysis of the literature permeability data of coarse grained MnZn ferrites has been reported by Visser et al. (1992). When measured at medium inductance, the domain walls in larger grains will contribute strongly to the reversible permeability, the rotational contribution in the grains is only a minor effect. The Globus model and the NMGB model can now be combined by neglecting the rotational contribution which denotes that #i of the grains is proportional to D (Globus model) and introducing the effect of the non-magnetic grain boundary eq. (25) can be rewritten as: D -]-Ze

D -

--

-#i

6

+

(27)

with D/i.ti only dependent on temperature. The experimental permeability data #e of Perduyn and Peloschek (1968) given in fig. 33a are replotted in fig. 33b as D/#e versus temperature; the curves are more or less independent of the grain diameter. For this particular ferrite, Te ,~ 1450 at which #i should be infinite and D/#~ has to be equal to the non magnetic grain boundary thickness 6, which is about 0.6 nm as follows from fig. 33b. If 6 is supposed to be temperature independent, experimental evidence of which is given in another paper

(b) 4.0

40.000 ~,

30.000 :f 20.000 10.000

f

0 -50

6

3.0

: : 1::k ° 2 .0

D D' ~

21~m ~30gm

1.0

5b

T (°C)

lbo

150

0 -50

0

50 100 T (°C)

150

Fig, 33. (a) The permeabilityof coarse grained, high purity MnZn ferrites according to Perduyn and Peloschek (1968). (b) The data of (a) replotted as D/,e versus temperature(Visser et al. 1992).

248

V.A.M. BRABERS

• Perduijn and Peloschek o ROsset al [2 40.000 Beer and Schwartz /J

~

2. o ""~-"'"'~.........................~.

¢q

•

"~20.000'

/

~"

,4 .i.o"

0

(b)

?

/./°

? 30.000

10.000

(a)

~,0" o°"

u

o//°°° o°.°

:::k ~.-'--~-¥~.......

~t~.o

0" " 0 r

I

~

I

I

i

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i

5O Grain size D (gm)

I

I

100

0

t

I

I

[

I

I

50 Grain size D (gm)

I

I

100

Fig. 34. (a) The permeability of coarse grained MnZn ferrites as function of the grain size. (b) The grain boundary thickness plotted versus grain size (Visser et al. 1992).

by Visser and Johnson (1991), D/#i can be determined as function of temperature (dotted curve in fig. 33b). The room temperature value of D/#i ~ 0.59 is now used to analyse the other literature data of #e, given in fig. 34a, by plotting 6 = D/Ize - D/#i versus the grain size as shown in fig. 34b. Two sets of data are found, those with (5 ~ 1.3 nm and those with t5 ~ 0.6 nm for ferrites specially prepared from very pure raw materials with carefully processing to obtain a high perlneability (Perduyn and Peloschek 1968). This analysis suggests that impurities as well as the ceramic processing are important factors to realize a particular microstructure which affects the permeability and losses of high quality ferrites. Whether the Globus wall size model or the NMGB model is applicable to describe the permeability depends on the combination of several items; the intrinsic properties K1 and Ms, the mictrostructure, i.e. grain size and non-magnetic grain boundary and the induction level at which the permeability is determined. Both models seem to have mutually excluding regions of validity but may also be combined in the intermediate region as suggested by the above-mentioned analysis of Visser et al. (1992). In the design of ferrites, these models can serve as guide lines to produce materials with improved propertie s .

3.5. Magnetic losses In addition to the permeability, the magnetic loss is also an important factor in the quality of a magnetic ferrite material. The total magnetic loss consists of three contributions: hysteresis loss, eddy current loss and residual loss, which for # >> 1 can be presented by a total loss factor tan 6t/# = #~/(#,)2 (Snelling 1988): tan 6t

tall 6h

tan 6e

-~-- - - - T - + - T -

tan ~r

+ --if-

PROGRESS IN SPINELFERRITERESEARCH

4v B -

-

-

7rl.tod2f +

-

37r#0#3

=aB+b.f

249

tan ~r +

16p

-

(28)

#

+e,

where v is the Rayleigh hysteresis coefficient, (#(H) = ~i + / / H ) , 1~ is the peak value of the flux density perpendicular to a cross-section of a cylinder of the ferrite material with radius d, f is the frequency and p the bulk resistivity, a and c are hysteresis and residual loss coefficients, which are material properties, b is the eddy current loss coefficient also depending on the shape. The total power loss is related to the loss factor tan 6/# by eq. (29) c.g.s, units, or (30) S.I. units. p = tan...._~6._1 yB2.10_ 7 # 4 P =

B2 tan6 7r.f# /to

watt/cm 3,

watt/m 3.-

(29)

(3o)

The eddy current loss in ferrites is usually small compared to the total loss and is caused by the eddy currents when the skin depth is large compared with the dimensions of the sample (Chikazumi 1966). To obtain high permeability MnZn ferrites, i.e. to obtain a low anisotropy K1, anisotropy-compensation by substitution of Fe2+-Ti 4+ in spinel ferrites is often used. The introduction of Fe 2+ enhances the electrical conductivity in the grains; to lower the bulk conductivity, other additives like Ca and Si are used, which segregate at the grain boundaries, producing highresistive grain boundaries and lowering the eddy current loss, but lowers also the permeability according the NMGB model. The major contribution to the total loss is the hysteresis loss, which is govenled by many factors. Hysteresis losses are due to irreversible wall displacements, which increase with increasing amplitude of the ac field and which are influenced by the wall pinning. To decrease the hysteresis losses, elimination of the wall permeability by small grains is one option, and another possibility is the domain wall pinning by CoZ+-substitution which gives rise to a local induced anisotropy by ionic short range order. The losses in MnZn ferrites, used in power applications up to 500 kHz, have been improved by simultaneous substitution with Co 2+ and Ti 4+ and taking special precautions in the processing of the ferrites (Stijntjes and Roelofsma 1986). hi fig. 35 the temperature dependence of K1, /-ti and the power loss is shown for the ferrite Mno.65Zn0.25Fe2.104, which shows that the minimum loss coincides with the secondary maximum in the permeability curve, caused by the zero crossing of K1. For low hysteresis losses, the magnetic anisotropy, mechanical stresses, including magnetostriction effects and the porosity should be low, while the magnetization must be high. The level of the power loss at the compensation point (K1 = 0) in fig. 35 is due to microstresses in the ferrite, which are caused by large calcium iolts, as impurities incorporated in the bulk of the grains. An improvement of the losses

250

V.A.M. BRABERS

+

i

J 4

~3 2

J

200

F = 100 kHz A

B = 100

mT

~-~ 100 ~d

~)

P

I

100

r

20O

1

P

3O0

T (°C) Fig. 35. The temperature dependence of the anisotropy constant K1, the initial permeability and tbe power loss of polyerystalline Mn0.65Zno.25Fe2.1004 (Stijntjes and Roelofsma 1986).

was obtained by an optimized thermal processing of the ferrite, by which the concentration of Ca in the bulk of the grains decreased and segregates more at the grain boundary. (Stijntjes and Roelofsma 1986). Further improvement was obtained by a simultaneous CoZ+-Ti 4+ substitution, as call be seen from the frequency dependence of the power loss in fig. 36. Especially, the development of ferrites with smaller power losses at higher frequencies are important for the size reduction of the power supplies operating at higher frequencies. The hysteresis loss instigates a loss tangent proportional to the magnetic induction and the eddy current loss gives rise to a loss tangent proportional to the frequency. The third term in eq. (28) is the residual loss tan 6r/#, which is the remaining loss usually defined as the loss measured when the frequency and induction is lowered to small values. At frequencies below the ferromagnetic resonance, the residual loss can partially originate from magnetic after-effects due to thermally activated domain wall motions, in addition to the gyromaguetic losses, which are dominant at higher frequencies.

PROGRESS IN SPINEL FERRITE RESEARCH

251

104 5

at 85 °C

//

2 g-"

1/111 /111

10 3 5

x,_.,

200 mT//~ /

////

2 o 102 o 5

2 I

10

- -

2

T

I

I

)

5

10 2

2

5

103

Frequency (kHz) Fig. 36. Power loss versus frequency for polycrystalline Mno.65Zno.asFezlO4 cooled after sintering with a cooling rate of 150*C/hr in a controlled oxygen atmosphere (dashed lines). An improvement of the losses is obtained by Co-Ti substitution: Mno.715Zno.2osCoo.oorTio.03Fe2.1osO4 (solid lines) (Stijntjes and Roelofsma 1986).

3.6.Microstructureandadditives Many efforts in soft ferrite research are focussed on the improvement of the temperature and time stability of the permeability, increasing the operational frequency range and on the decrease of the magnetic losses. See, for instance, the proceedings of the series of International Conferences on Ferrites, Kyoto (1970), Paris (1976), Kyoto (1980), San Fransisco (1985), Bombay (1989) and Tokyo (1992). From the foregoing it may be clear that, although the intrinsic properties like crystal anisotropy, saturation magnetization and magnetostriction are well known to choose a basic composition for the ferrite material, the tayloring of the temperature, time, field and frequency dependence of the permeability and the lowering of the magnetic losses are largely determined by the processing of the polycrystalline ferrites. The microstructure, i.e. grain size and grain size distribution, grain boundary composition, porosity, distribution of the pores inside the grains or at the grain boundaries, chemical homogeneity of the grains and the mechanical stress caused by the processing of the material are decisive parameters to achieve special properties. Since the development in the early sixties of low loss Mn-Zn ferrites by modification of

252

V.A,M. BRABERS

the grain boundaries with small additives of CaO and SiO2, the role of additives in the processing of polycrystalline ferrites has become an important item to vary the microstructure. Unfortunately, some prerequisites for the microstructure to change the value of a property are sometimes opposing each other. For instance a small grain size is favourable for low eddy current loss, but larger grains may in certain cases decrease the hysteresis loss. Both effects have to be taken into consideration for the optimization of the total loss. Depending on the objectives, several types of additives can be distinguished. The first type concerns additives designed to segregate at the grain boundaries, like Si and Ca, in order to increase" the grain boundary21_resistivity in MnZn ferrites and reducing the eddy current losses. However, Ca ions enter also the spinel lattice under reducing atmosphere and introduce microstresses in the grains, which enhance the hysteresis losses; if reoxidation can take place during the cooling cycle of the ceramic processing, Ca segregates again from the grain into the boundary and the hysteresis losses are lower. (Stijntjes and Roelofsma 1986; Nomura et al. 1982, Otsuki 1992, Mochizuki 1992). In addition to the formation of the high resistance grain boundary, the addition of CaO and SiO2 promotes the sintering of the polycrystalline ferrite by the formation of a 'glass' phase at the grain boundary, so that sintering can be carried out at lower temperatures, preventing exaggerated grain growth. A second class of additives concerns those which promotes or prevents grain growth. Yah and Johnson (1978) investigated the effect of a number of additives on the grain growth in MnZn ferrites at 1300°C. The grain growth was promoted by addition of dopants of CaO, SrO, V205, Nb2Os, SbaOs, PbO and CuaO, by means of the formation of liquid phases with the ferrite. TiO2 and SiO2 also sustain the growth of exaggerated grains without a liquid phase; the effect on the grain growth might be explained by the formation of cation vacancies and diffusion of Ti and Si into the grains. These additives are used for high permeability materials of which the crystallite size must be large. For MnZn ferrites, B203 is a special agent for the crystal growth since a small amount of 50 ppm shows already a remarkable effect; however in combination with MoO3 and Na20 this effect can be cancelled. Ta205 and ZrO2 are agents which inhibit the particle growth and are particular of interest to lower the eddy current losses in high frequency low loss MnZn ferrites (Ishino et al. 1992 and 7.nidarsi~ et al. 1992). However, above 12500 Ta205 dissolves into the spinel, leading to an increasing grain growth (Znidarsi~ et al. 1992), which involves that an additive can give different effects upon the grain growth depending on the sinter temperature. Similar behaviour is found for BaO. BaO and Fe203 form lowmelting eutectics below 1370°C and it has been reported that BaO promotes grain growth in MnFe204 at 1375°C substantially (Paulus and Haineliu 1968). However, for BaO doped NiZn(Co) ferrites, sintered at 1240°C, segregation of Ba z+ in the grain boundaries was observed, preventing the grain growth during sintering (Drofenik, Beseni~ar and Kolar 1984). An extra benifit of the use of additives segregating on the grain boundaries is the improvement of the mechanical strength as is reported for ZrO 2 doped NiZn ferrite (Beseni~ar et al. 1988) and for Na20, CaO and ZrO 2 doped MnZn ferdtes (Hirota et al. 1986). A third class of additives are those which substitute for the main components in the technically important MnZn and NiZn ferdtes, and which effect in principle not

PROGRESS IN SPINEL FERRITE RESEARCH

253

3000 300

g

P-i 2000 200

losses "'---c.....~~._~

1000 0

.tJ2

.()4

.68

.16

O

.32

.64

100

Ge 0 2 CONTENT (tool %) Fig. 37. Initial permeability and power losses (at B = 0.2 T and 15.75 KHz) versus Ge content in Mno.565Zno.374Fe2.o604(Jain et al. 1980). the microstructure but control mainly the saturation magnetization, Curie temperature and the thermal characteristics of the permeability. These additives comprise MgO, NiO, CuO, CoO, A1203, TiO2, GeO2, etc. An illustrative example involves the addition of GeO2 to reduce the temperature coefficient of the permeability of Mn0.51Zno.44Fe2.0504 (Gallagher et al. 1983). The permeability of the unsubstituted ferrite shows a temperature variation of 0.2% in the temperature range of - 5 0 ° + 50°C, whereas with a substitution up to Ge0.05 for Zn the variation is below 0.01%. The presence of Fe z+, stabilized by the Ge 4+ ion during the sinter process, increases the permeability and controls the secondary peak position. Both effects reduce the temperature coefficient. Improvements of the permeability and losses by GeO2 substitution are shown in fig. 37. The improved properties are associated with the increased density. At higher Ge-concentrations, the properties degraded, due to discontinuous grain growth caused by a liquid phase segregation at the grain boundaries (Jain et al. 1980). There have been numerous papers and patents appeared during the last thirty years dealing with the effect of processing and additives upon the properties and microstructure of technical applicable ferrites. For the interested reader, more information about this topic can be found in the proceedings of the series of ferrite conferences quoted before.

4. Magneto-optical properties 4.1. Introduction

During the eighties, the interest in the magneto-optical properties of spinel ferrites has increased because of the possible application of these materials ill high density

254

V.A.M. BRABERS

magneto-optical recording media. In general, magneto-optical effects result from the interaction of electromagnetic waves with a magnetic field or with the magnetization in a material. The material properties can be described by a magnetic permeability and a dielectric tensor; at optical frequencies, however, the magnetic permeability can be taken to be unity, which means that the m.o. effects are well described by the complex dielectric tensor, only. There are now several types of magneto-optic phenomena which can be separated into transmission or reflection effects. If the magnetization is directed along the light propagation direction in the transmission configuration, the Faraday effect is observed which is the rotation of the plane of polarization of a linearly polarized light beam on passing through the specimen. The rotation is proportional to the thickness of the sample. The Faraday rotation can also be described as a magnetic circular birefringence effect: (31)

2c (n+ - n - ) = ~ (n + - n - ) ,

where n + and n - are the refractive indices for right and left hand circularly polarized fight and A the wavelength. If the material absorbs light the refractive indices are complex, which implicates that the rotation is also a complex quantity of which the real part corresponds to the Faraday rotation and the imaginary part to the ellipticity giving rise to a circular magnetic dichroism. This dichroism, which originates from a difference in attenuation for the left and right hand circularly polarized waves, can convert an initially linearly polarized wave into an elliptically polarized wave on passing through a specimen. The Cotton-Mouton effect is observed in case that the magnetization is perpendicular to the light propagation direction in the transmission configuration, which means that the normal modes of the propagated waves are linearly polarized with the direction of polarization perpendicular or parallel to the magnetization. The difference between these two waves corresponds with the linear magnetic birefringence, the Cotton-Mouton effect, and for absorbing material a linear magnetic dichroism occurs which is related to the difference in attenuation of the two perpendicularly polarized waves. The magneto-optical effects on reflection, the Kerr effects, are classified into three groups, depending on the different configuratimts schematically shown in fig. 38. In the Polar-Kerr effect the magnetization is normal to the reflecting surface (fig. 38a), in the longitudinal or meridional effect the magnetization is parallel to the surface in the plane of incidence (fig. 38b) and in the transversal or equatorial (a)

Polar

(b)

Longitudinal or M-eridional

(c)

Equatorial or lransverse

Fig. 38. Configurationsfor the three Kerr effects.

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255

50 (a) 40

0.85

g- 30 20 10

J I

0

2

I

4

I

6

8

I

10

12

Photon energy (eV) Fig. 39. (a) The optical reflectivity of magnetite at 300 K after Schlegel et al. (1979).

Kerr effect, the magnetization is parallel to the surface, perpendicular to the plane of incidence (fig. 38c). Although magneto-optic effects can arise from a wide variety of submicroscopic effects on an atomic scale, it is very convenient to describe the magneto-optical properties of a material in a macroscopic phenomenological way in terms of the dielectric tensor (Landau and Lifshitz 1960, Wettling 1976). For the Polar-Kerr effect of a cubic crystal with the magnetization along the Z-direction this tensor reduces to =

exx --¢xy 0

exy £xx 0

0 0 e~

(32)

where the diagonal components do not depend on the magnetization and the offdiagonal components are linearly dependent on the magnetization. For moderately absorbing crystals, exx > exy, the Polar-Kerr rotation Ok and ellipticity ~?k can be related to the complex dielectric coefficients exx and ~xy by -- ~xy

Ok + it/k =

ev,W~.(exx - 1)

(33)

and the Faraday rotation 0F 7rl exy OF + irlF -- /~ ~

(34)

with I the thickness of the sample and rlF the Faraday ellipticity or magnetic circular dichroism (Wettling 1976). For a more detailed description of magneto-optical effects the reader is referred to the review of Reim and Schoenes (ch. 4, p. 58 of vol. 5).

256

V.A.M. BRABERS

"7,

0

v

4

E~ 2

0

I

2

I

I

i

4 6 8 Photon energy (eV)

I

10

12

Fig. 39. (b) The optical absorption coefficient at 300 K after Schlegel et al. (1979).

4.2. Optical and magneto-optical properties of magnetite The analysis of the magneto-optical and optical spectra of spinel ferrites in terms of assignments to specific phenomena or electronic transitions has been a controversial item. Different assignments for the specific characteristics were made by several authors which is distinctly demonstrated for pure magnetite Fe304. The reflectivity and the absorption of magnetite has been reported by Schlegel et al. (1979) for a large energy interval between 1 meV and 12 eV (fig. 39) and the complex Polar-Kerr effect by Sim~a et al. (1980a) and Zhang et al. (1981) (fig. 40). From these data, Zhang et al. (1981, 1983) derived the off-diagonal elements Cxy t and ~xy, " shown ill fig. 41. Schlegel et al. (1979) explained the optical spectra in the energy interval of 0.5 eV-5 eV as mainly being caused by a number of Fe 3d'V~ 3d '~-1 4s orbital promotion processes, taking also into account the final state effects of the atom-like 3d '~-1 configurations; charge transfer transitions from the oxygen 2p bands to the metal 4s states were suggested to start above 5 eV. In subsequent papers on the optical properties and the Polar-Kerr effect of magnetite, MgFe204 and Li0.5Fe2.504, Zhang, Schoenes and Wachter (1981, 1983) gave the same interpretation for the magnetooptical properties, which they supported by the argument that the energy distributions of the photoelectrons observed in photo-electron spectroscopy (Alvarado et al. 1976) are quite similar as in the optical spectrum. However, in several band structure calculations for Fe304 (Yanase and Siratori 1984, Feil 1987, Zhang and Satpathy 1991 and P6nicaud et al. 1992), the Fe 4s band is more than at least 2 eV situated above the highest occupied 3d states, from which Feil (1987) concluded that the peak absorption at 0.6 eV cannot be assigned to the first 3d-4s orbital promotion process (Degiorgi et al. 1987). Moreover, inverse photoemission experiments by Sancrotti

PROGRESS IN SPINEL FERRITE RESEARCH

2

257

rl K

l 0 O

-] iI ,-M

~. -3 I !

U

I I J t

1

2 3 4 Photon energy(eV)

Fig. 40. The complex Polar-Kerr effect of magnetite at 300 K (Zhang et al. 1981).

et al. (1991) on Fe304 indicate that the energy levels in the range up to 4 eV above the Fermi-level are unoccupied Fe-3d states and that Fe 4(sp) states are 5 eV well above the Fermi level. Muret (1974) and Sim~a (1979) assigned the optical and magneto-optical spectra below 1 eV to the small polaron conduction model. Below the Verwey transition, no transition is expected, in contradiction with the observed PKR spectra at low temperature (Sin~a and Hamernikova 1981, Matsumoto et al. 1978); the 0.6 eV peak cannot be attributed to this model. The features in the off-diagonal element spectra around 2 eV (see fig. 41) were attributed by Sim~a et al. (1980a) to crystal field transitions of tetrahedral Fe 3+ ions and at higher energies, assignments to charge transfer transitions between cations in different crystallographic sites or between oxygen 2p and cation 3d levels were proposed. However, 3d crystal-field transitions of Fe 3+ ions in tetrahedral surroundings are slightly parity allowed, from which Feil (1987) concluded that the oscillator strength of these transitions are too low to account for the observed spectra. He proposed quite a different alternative explanation for the magneto-optical properties based on the concept of intervalence charge transfer transitions. Intervalence charge transfer transitions (IVCT) are optical transitions, where an electron is optically transferred from one site to another adjacent site. Strens and Wood (1979) found two peaks in the diffuse reflectance spectrum of magnetite, one at 0.85 eV and a second one near 2.4 eV. Sherman (1987) applied the self-consistent field

258

V.A.M. BRABERS

.06 I

1.85

4.0

.04 ~-

.02

0

0

© -.02

-.04

-.06

! I ! ! -0.5 I

l,

241',./T

3.35

2.55

!

I I I I ! I

t

I

I

1

2

3

4

~aco(eV) Fig. 41. The complex off-diagonaldielectric tensor elements e~y and dx~ of magnetite at 300 K. The indicated transitionenergies are assignedto orbital promotionprocesses by Zhang et al. (1981, 1983). X a scattered wave method (SCF-Xa-SW) for the molecular orbital calculations of mixed valence (Fe2Ol0) 15- clusters and related the two peaks observed in Fe304 to optical IVCT transitions. The first at 0.85 eV to the optical transition between Fe2+(B) ~ Fe3+(B) and the second to the Fe2+(tZg) --* Fe3+(eg) or to the pair excitation 6Alg + 6Alg --~ 4Tlg + 4Tlg , also found in Fe(III) oxides (Sherman and Waite 1985). For the thermally induced electron transition between Fe2+(B) and Fe3+(B) which causes also the electrical conductivity, a small negative energy was derived (--, -0.01 eV), which would imply that the electrical transport in magnetite in the high temperature phase is not a hopping type conduction (Sherman 1987). /t In compliance with Sherman, Feil (1987) attributed the negative peak in the exy spectrum near 0.6 eV to the IVCT transition between Fe z+ --, Fe 3+ ions on B-sites, and the positive peak near 2 eV to the Fe2+(t2g) --+ Fe3+(eg) 1VCT transition, which assignments are supported by the disappearance of these peaks if Fe 2+ is substituted by Mn 2+ (Sirn~a et al. 1980b,c). There is further no linear dependence of the peak

PROGRESS IN SPINEL FERRITE RESEARCH

259

intensity with the Fe 2+ concentration, which points to a 2 ion-mechanism. The negative peak around 3 eV is assigned to another IVCT transition from tetrahedral to octahedral sites and above 4 eV the Ozp ~ Fe3d charge transfer transitions dominate the spectrum. Similar assigmnents of the magneto-optical spectra around 3 eV and above 4 eV has been made for iron garnets (Wittekoek et al. 1975) but later it was also suggested that in these garnets, Ozp ~ Fe3d transitions are causing the peculiarities in the 2-4 eV range of the magneto-optical spectra (Moskvin et al. 1991). However, this is not in agreement with the estimates of Zaanen et al. (1985) for this transfer energy. Since spin allowed crystal field transitions for d 5 ions are not possible and transitions for 3d 6 ions are expected to have a low oscillator strength, the overall assignments of the magneto-optical spectrum of magnetite by Fell seems to be a good starting point for the interpretation of the magneto-optical spectra of spinel ferrites.

4.3. Magneto-optical properties of mixed ferrites Polar-Kerr effect studies on manganese containing ferrites are reported by Nakagawa ~'n2+ sub et al. (1980), Sim~a et al. (1980b,c) and Feil (1987). With increasing M,l stitution in Fe304, the features in the M.O. spectra at low energy related with the octahedral Fe z+ ions disappear (Sin~a et al. 1980b and c), but some additional contributions to the M.O. activity appear as can be clearly seen in fig. 42 for the absorptive " for some Mn ferrites (Fell 1987). For the MnZn part of the off-diagonal element exy ferrite a dip is seen in the t2g ~ eg IVCT transition around 1.5 eV, which is probably due to an enhanced Fe 2+ crystal field transition. This enhancement might originate from the cationic disorder resulting in the distortion of the octahedral symmetry, removing the inversion centre. The influence of the lattice distortion upon the magneto-optical activity is distinctly demonstrated by the effect of the substution of 1-2% AI3+ in Fe304, by which the intensity of the Polar-Kerr spectra changes 10-30% in certain energy intervals (Fontijn et al. 1993). The radius of the octahedral AI 3+ ion (0.53 ]k) is much smaller than the Fe 3+ ions (0.645 ~), which might be the reason for this large effect. In MnFe204 the main peak around 3 eV is shifted a few tenth of an electronvolt to lower energy with an additional shoulder near 3 eV and a small peak at 4 eV. Since MnFe204 has a partial normal structure (4-80% of Mn in tetrahedral sites), these results can be interpreted as IVCT transitions like in Fe304, in which tetrahedral Mn 2+ as well as tetrahedral Fe 3+ ions are involved. Similar changes in the M.O. spectra compared with Fe304 have been reported for MgFe204 and Li0.sFe2.504 (Zhang et al. 1983, Vigfiovsky et al. 1979, 1981), which can, according to Feil, also be interpreted as additional enhanced IVCT transitions because of the distortion of the lattice site symmetry by the cationic disorder. It has to be mentioned that the spectra reported by Vi~ilovsky et al. showed some discrepancies compared with the spectra given by Zhang et al. which can be attributed to the various surface treatments of the samples. Martens et al. (1984a, 1986a) showed that polishing or etching of the crystal surface introduces a surface layer, which indeed affects the Polar-Kerr effect. Nakagawa et al. (1980) measured besides the MnxFe3_~O4system also the Polar-Kerr rotation of nickel and zinc substituted magnetite and

260

V.A.M. BRABERS

0.04

0.02

-0.02

-0.04 0

I

2

3

4

5

Photon energy(eV) " of the off-diagonaldielectric tensor element at 300 K of Fe304 (full Fig. 42. The absorptive part ~xy curve), Mno.aZn0.3Fe2.304(dotted curve) and MnFe204 (dashed curve) after Feil (1987). tried to substract the contributions of the specific ions, assuming that the spectrum is composed by a simple superposition of the individual ions. A similar analysis has been proposed by Vigfiovsky et al. (1979) for Li0.sFe2.504 and MgFe204. However, in view of the foregoing discussion this approach, i.e. a simply superposition seems to be questionable. In the Polar-Kerr rotation spectrum (Kahn et al. 1969) and the infrared Faraday rotation spectrum of NiFe204 (Zanmarchi and Bongers 1969) some evidence is reported for the magneto-optical activity of Ni 2+ ions near 1 eV, which might correspond to a 3 A 2 --~ 3Tx crystal field transition and other transitions near 2, 3 and 5 eV (Nakagawa et al. 1980). The Polar as well as the equatorial-Kerr effect spectra of NiFe204 were also measured by Krinchik et al. (1977, 1979) and Khrebtov et al. (1978) who related the M.O. activity near 3 eV to the 3A2g(F ) --~ 3Tlg(P ) electrical di.peole transition of the octahedral Ni 2+ ions, also found in the absorption spectra of Ni + containing crystals. Furthermore interesting results were obtained by Krinchik et al. (1979) in their systematic study of the optical spectra and the equatorial Kerr effect of the mixed

PROGRESS IN SPINEL FERRITE RESEARCH

261

o

~xy

0.01

-C °

-? -0.0l [~

I

e"xy

-0.02

2

I 3

I 4

5

K {eV) Fig. 43. The complex off-diagonal dielectric tensor elements exy ' and ~xy " of Fe [CoCr]O4 at room temperature (Krinehik et al. 1979).

series CoCrxFe2_~O4, NiCrzFe2_~O 4 and CoAI~Fe2_~O4. With increasing substitution of Fe 3+ by Cr 3+, the optical absorption decreases in the 2-5 eV range and the magneto-optical activity due to Fe3+ above 2 eV disappears almost completely when the octahedral Fe 3+ is completely replaced by Cr 3+ as is shown in the spectral dependence of the off-diagonal components ~xy , and exy , for Fe [Co Cr] O4 (fig. 43). This underlines that the transitions attributed to Fe 3+ are due to pair transitions between Fe-ions in different sublattices, as was already suggested by Wittekoek et al. (1975) and Scott et al. (1974, 1975a, b, 1977) because of the quadratic dependence of the intensity of these transitions on the ion concentration (Wood and Remeika 1967). Kfinchik et al. (1979) discussed the two most probable mechanisms for these pair excitations i) two-exciton and ii) charge transfers between magnetically active ions on different sites, which are in fact the before-mentioned IVCT transitions. Based on the available experimental data Krinchik favoured the charge transfer transitions as the origin of the magneto-optical activity of the Fe-ions, in compliance with the IVCT assignments made by Feil. However, the disappearance of the spin allowed

262

V.A.M. BRABERS

0.50

(a)

0.25

o

-0.25

-0.50 ~ 0

1

2

3

L 4

--__ 5

Photon energy (eV) Fig. 44. (a) The Polar-Kerr rotation Ok and ellipticity for CoFe204 (Martens et al. 1985).

crystal field transition for Ni 2+ in the 3 eV region observed in the NiCr~Fe2_zO4 series remains obscure (Krinchik et al. 1979). Special characteristics are reported for the magneto-optical properties of Co conmining ferrites, which are usually related to the relatively strong optical absorptions at 1.98 eV (4A2 ---+4TI(P)) and 0.86 eV (4A 2 ---+4TI(F)) for Co 2+ in tetrahedral sites • and the much weaker absorptions for Co 2+ in octahedral sites at 2.48 and 1.05 eV, respectively (Papalardo et al. 1961). The strong effects of tetrahedral Co 2+ ions upon the M.O. properties were already demonstrated in the seventies by Ahrenkiel et al. (1975), who reported reflectance circular-dichroism data for the CoRh~Fe2_~O4 seties, in which the Co 2+ ions are shifted to the tetrahedral (Td) sites with increasing Rh-content. The same assignments to the Co 2+ (Td) crystal field transitions were also made for the transitions found in the equatorial Kerr spectra by Khrebtov et al. (1978) for CoAl~Fez_xO4 spinels, for CoCr~Fe2_~O4 by Krinchik et al. (1977, 1979) and in the Polar-Kerr spectra by Abe and Gomi (1982) for CoMxFe2_~O4,

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263

0.50 (b)

0.25 rl k oO°~

0

o

-0.2

Ok

-0.50 0

I 1

I I I 2 3 4 Photon energy (eft)

I 5

Fig. 44. (b) Col.5Fel.504 at 300 K (Martens et al. 1985).

with M = Mn, Cr and AI. Peeters and Martens (1982), in their study of the PolarKerr rotation of the CoAI~Fez_~O4 and Co~Fe3_~O4 series and the Faraday rotation of CoAI~Fe2_~O4 films discussed the spectra in terms of the same CoZ+(Td) crystal field transitions, but found an additional broad paramagnetic intense transition around 2 eV, which was assigned to an IVCT transition between octahedral Co 2+ and Fe 3+ ions. In two subsequent papers by Martens et al. (1984b, 1986b) further evidence for these assignments were given. Moreover, the magneto-optical spectra of cobalt ferrites with excess cobalt (Col+~Fe2_xO4) showed increasing intensity of the two tetrahedral Co z+ crystal field transitions (fig. 44a and b) because of the larger Co (Td) concentration and a decrease of the octahedral Co2+-Fe 3+ charge transfer transition (Martens et al. 1985). In the optical spectra of these ferrites, additional crystal field transitions are probed due to octahedral Co 3+ ions at 0.8, 1.6 and 2.6 eV as well as a Co2+-Co 3+ octahedral charge transfer transition near 0.8 eV. For the application of spinel ferrites as magneto-optical storage medium, the use of cobalt substituted ferrites seems to be promising because of the enhanced magnetooptical activity due to the cobalt ions. However, the 'intrinsic' magneto-optical

264

V.A.M. BRABERS

(a)

~2-

o

/

ii/

v

x =

~ g

--

I"]

..... 0.56 ........... 0.69

V I

I

400

o

I

I

800

I

I

1200

~

I

1600

I

2000

Wavelength (nm) Fig. 45.

(a) Faraday rotation spectra at room temperature of 200 nm polycrystalline films of the Col_xZnzFe204 series (Suzuki et al. 1988).

~l~

(b)

"~2

:di!l

0

=

,-.,

0

0

1 ~-2

~

x=

"-'= 050

"

~ 1

~1 J

400

....... o.81

kJ

-4

J

800

0

...........

1.49

J

1200

1600

2000

Wavelength (nm)

Fig. 45. (b) CoAlxFe2_xO4 series (Suzuki et al. 1988).

PROGRESS IN SPINELFERRITERESEARCH

265

properties are not the only decisive parameters. For optical writing with a laser beam, the Curie temperature of the pure cobalt ferrite is rather high, which can be lowered by the substitution with non-magnetic ions. But the substitution will in turn lower the M.O. activity. Further, if reflection magneto-optical properties are used, surface effects (Martens et al. 1984a, 1986a) as well as mechanical stresses are affecting the performance. The perpendicular magnetic anisotropy in films, necessary for magneto-optical storage, can be caused by a combination of the crystal anisotropy and stress induced anisotropy. Martens and Peeters (1986c) showed for (100) oriented CoFezO4 films that the additional stress-induced anisotropy is a necessity to overcome the large shape anisotropy brought on by the high magnetization. Fujii et al. (1992) found for Co ferrite films with (100) oriented columnar structures perpendicular to the surface, tensional stresses in the order of 2.108 N/m 2, which induce a stress-induced perpendicular anisotropy of about 1.18.105 joule/m s. This implies that besides the intrinsic properties of the basic material, the technology of the deposition of the films is another important factor to produce usable magneto-optic memories. Most of the magneto-optical studies reported so far are concerned with the Kerr effects, since the optical absorption of spinel ferrites increases rapidly above 1 eV and even for conducting ferrites, in which mixed Fe 2+ and Fe 3+ valencies are present, the absorption is also substantially below 1 eV. Yamazaki et al. (1987) reported the Faraday rotation spectra measured on 200 nm CoZn ferrite fihns and Suzuki et al. (1988) for zinc and aluminium substituted cobalt ferrites, shown in fig. 45a and b. Osterero et al. (1991, 1992) reported the Faraday rotation at 1.08 eV on 30/1 single crystal platelets of CoFe204 and AI and Cd substituted CoFeaO4; a strong anisotropy of the Faraday rotation was found between (111) and (100) orientations, originating from the non-saturation due to the high magnetic anistropy. Similar anisotropy effects were also found for the Kerr effect of Co ferrite films grown in the (100) and (111) orientation (Peeters et al. 1980). 5. Magnetite 5.1. Structure and the physical properties in relation to the Verwey transition Although magnetite is one of the best known magnetic compounds, the investigations of the physical properties is still an intriguing field of research. Especially, the properties in relation to the electronic ordering of the Fe z+ and Fe s+ ions on octahedral sites and the phase transition itself have been subject of a number of recent papers. The room temperature dc-conductivity of FeaO4 is about 2 x 104 ~ - l m - t and the qualitative features of the temperature dependence of the conductivity shown in fig. 46 are very common for all reports. A broad maximum occurs around room temperature and a minimum at about 100 K below the magnetic Curie temperature Tc of 860 K. The maximum conductivity observed is 3 x 104 Q - l m - 1 . Stoichiometry and hydrostatic pressure have only a small effect on the conductivity data above the Verwey transition Tv, the maximum is shifted to lower temperature by compression and to higher temperature by oxidation (Kakudate et al. 1979). Usually, the

266

V.A.M. BRABERS

,

[ 105[ -

~ xlO 3 ~ [- /

~

If,n,

o

10-5

m

0

p

~

~

~

i

n

8

~

~

16 1oo /

~.

. . . . .

,

~

24

T (K-1)

Fig. 46. Temperaturedependenceof the electrical conductivityof Fe304 (Miles et al. 1957). appearance of the room temperature maximum is attributed to correlation or polaronic effects of the charge carriers, though Parker et al. (1976) and Tinsley (1980) argued that the entire temperature dependence of the conductivity above Tv may be explained by magnetic scattering and the appearance of the spontaneous spin-order at To. Similar magnetic order effects upon the conductivity have also been observed in Ni-substituted magnetite (Whall et al. 1984). Near 125 K the electronic ordering of the ferrous and ferric ions causes a decrease in the conductivity of about two orders of magnitude, which is known as the Verweytransition. Resembling charge ordering transitions have also been found in other mixed valence oxide compounds such as TiaOT, YFe204 and ErFe204. The RFe204 compounds, in which R stands for a rare-earth metal ion, do not crystallise into the spinel structure but have a hexagonal layered structure composed of a R203 layer and a Fe202.5 layer stacked alternately along the c-axis (Kato et al. 1975). In certain stoichiometric RFe204 compounds of which the ionic radius of the rare-earth ion is close to 0.90 .A, i.e. Y, Er and Ho, two Verwey-type transitions in the electrical properties occur (fig. 47), the first near the antiferromagnetic ordering temperature near 240 K and the second about 25 K lower (Sakai et al. 1985, 1986, Iida et al. 1990). In YFe204 these first order transitions are accompanied by the distortion of the hexagonal lattice to a monoclinic and subsequently to a triclinic lattice (Nakagawa et al. 1979) but in ErFe204 only a monoclinic structure has been found (Iida et al. 1990). The electrical conduction mechanism in these rare-earth compound seems to be quite different compared with Fe304. The electrical conductivity shows a large aifisotropy, along the c-axis about 2 orders of magnitude lower than in the ab plane, and an in-plane room temperature value of 10 -3 ff2-1m-1 (Tanaka et al. 1982). The temperature dependence of the resistivity above the transitions is thermally

PROGRESS IN SPINELFERRITERESEARCH

267

activated with an energy of 0.23 eV and below the transitions, the resitivity can be approximated with the formula p = poexp

[(TolT) '/3]

(35)

which suggests a two-dimensional variable range hopping mechanism for the electrical transport (Sakai et al. 1986, Hurd 1984). The relatively high activation energy for the layered compounds compared with the metallic-like behaviour of magnetite, may be related to the larger nearest neighbour distance between the iron ions, which is 3.12 and 2.97 ,~ for YFe204 and Fe304, respectively. Further, a notable difference between magnetite and the RFe204 compounds is the lower transition temperature for magnetite and the different pressure effect upon the Verwey transition temperature. For magnetite, Tv shifts to lower temperature by applying a hydrostatic pressure with - 0 . 2 7 K/kb (Kakudate et al. 1979) and the two transitions in YFe204 with -10.5 K/kb and - 2 . 6 K/kb, respectively, and disappear entirely at pressures above 15 kb (Siratori et al. 1990). The compressibility of both compounds are comparable (2-3 x 10 -4 (kb)-l), which implies that the Verwey transition in YFe204 is more sensitive to atomic distances than in Fe304. The Verwey transition in YFe204 is clearly a two-step transition because of the three different crystal structures observed and is very sensitive to stoichiometry as follows from the complete disappearance of the transition for 1% oxygen deficiency (Tanaka et al. 1982). A bifurcated heat capacity anomaly found by Evans and Westrum (1972) for polycrystalline Fe304, later also confirmed for a powdered high purity single crystal (Rigo et al. 1983, 1989), suggested that the characteristic intrinsic transition in magnetite is also a two-step Verwey transition. Heat capacity on high-quality single crystals, however, revealed only one narrow specific heat

109

1011

~-

10 7

10 5 I

140

I

l

180

I

I

220

I

I

260

I

300

T (K) Fig. 47. Temperaturedependenceof the electrical resistivityof stoiehiometrieYFe204, measured upon cooling and heating (Sakai et al. 1986).

268

V.A.M. BRABERS

!A 14o 135

25 6.0 ,"4

~

-

20

'

.~ 5.0 r~

15 u

4.o lo I

3.0

100

!

I

110

120 130 T (K)

I

)

140

Fig. 48. Temperature dependence of the specific heat of magnetite near the Verwey transition according to Rigo et al. (1983) and the initial susceptibility of a stoichiometrie single crystal of Fe304 (curve I) and of a powdered single crystal (curve II) after Walz et al. (1991).

anomaly near 124 K, designating the single stage character of the Verwey transition in magnetite (Gmelin et al. 1983, Shepherd et al. 1985). In fig. 48 the specific heat data of the ground single crystal by Rigo et al. (1983) are shown; a major peak occurs at 124 K and a smaller one at 113 K. Walz et al. (1991) performed initial susceptibility measurements on the same powdered material and on an identical bulk single crystal, also shown in fig. 48. For the single crystal, a steep stepwise transition in the xi-T curve occurs, while for the ground single crystal an extended transition zone is observed starting at 113 K, and ending into a first-order transition at 124 K. Moreover, magnetic after-effect measurements indicated that lattice defects like cation vacancies were introduced by the powdering of the specimens, which could explain the bifurcation of the specific heat curve (Walz et al. 1991). The Verwey-transition temperature depends strongly on the stoichiometry, the impurity content and mechanical stresses. As already shown in fig. 4, Tv decreases with the cation vacancy concentration and the Ti and Zn impurity concentration in a similar way. Miyahara (1972) reported the effect of a large number of impurities upon the shifting of Tv, but no regularity could be established with the charge of the

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269

substituent or the site of the impurity. Partially, this might be caused by the simultaneous effect of the oxygen non-stoichiometry and impurity concentration, since both have an equivalent effect on the transition temperature. A systematic study of stoichiometric magnetite single crystals doped with up to z = 0.03 Ni, Co or Mg showed a Tv-shift, which is about half the effect of Ti or AI substitution and which is three times as large as for Ga-substitution (Brabers and Van Der Vleuten 1984). The transition temperatures were determined from resistivity versus temperature curves; for concentrations below x = 0.01, sharp transitions within 1 K and for larger concentrations, except for Ga, more gradual transitions were observed, in agreement with the change of the transition from first to second order with increasing impurity concentration as proposed by Aragon and Honig (1988). The small ilffluence of the Ga-impurities can be explained by the position of most of the Ga ions in tetrahedral sites, which obviously does not act on the octahedral ordering. Even for a Ga-concentration of x = 0.1 only a shift of 15 K was observed with a sharp transition. The observed Tv shift for the Ti-substitution was in agreement with Kakol's data (1992); Ti-ions enter on substitution the octahedral site with a simultaneously appearance of additional Fe2+-ions, which might explain the twice larger effect for Ti compared with the two-valent Ni, Co and Mg. However, the large effect of the AI 3+ -ions, comparable with the Ti 4+ -Fe 2+ substitution, is puzzling. Apparently, the rather small A13+-ions (r = 0.53 fk) modify the ordering in a stronger wa~. than the two-valent ions, of which the ionic radii are in between those of the Fe + and Fe3+-ions (table 7, Shannon and Prewitt 1969). An increase of Tv has also been reported for the isotopic substitution of 160 by 1SO; with 43% of 180 an increase of 6 K is observed from 119 to 125 K (Terukov et al. 1979). However, the exact stoichiometry of the samples is not reported, which makes it doubtful if this effect is entirely due to the isotopic substitution, since for stoichiometric non lSO-substituted magnetite transition temperatures near 125 K are also reported (Walz et al. 1990, 1991). Tamura (1990) reported the pressure dependence of the Verwey-transition for stoichiometric and non-stoichiometric magnetite; the pressure coefficient for Tv of non-stoichiometric magnetites F e 3 _ ~ O 4 , is somewhat lower than for stoichiometric magnetite, - 0 . 2 7 K/kb, and ranges between -0.20 and -0.25 K/kb, without a clear correlation with 6, the non-stoichiometry parameter. Honig and coworkers (Honig and Spalek 1989, Aragon and Honig 1988, Honig 1988) applied a mean field thermodynamic approach, originally advanced by Str~issler and Kittel (1965), to the Verwey transition in no nstoichiometric Fe3_ ~04. Parametrization of the internal energy in terms of the experimental dependence of the transition TABLE 7 Ionic radii in octahedral sites. Ion

Radius (A)

Ion

Radius (A)

Mg 2+

0.72

Fe 2+

0.77

Ni 2+

0.70

Fe3+

0.645

Co 2+

0.735

Ti4+

0.605

270

V.A.M. BRABERS

temperature on the non-stoichiometry gave results, consistent with the existence of two regimes on either side of the critical value 6c = 0.0117. For 6 > re, the transition is second order, and for 6 < 6c first order, which was experimentally confirmed by heat capacity measurements on magnetite samples Fe3_604 with varying 6. For 6 > 36~ ~ 0.036 the transition disappears altogether (Shepherd et al. 1991). A single long range order parameter was derived to describe the density of states, from which the Fermi potential could be calculated; the associated electrical conductivity and thermopower were calculated and agreed reasonably well with experiment (Honig and Aragon 1988). An equal approach was applied to zinc and titanium doped magnetites in the succeeding papers (Wang et al. 1990, Kakol 1990, Honig et al. 1992, Kakol et al. 1992, Kozlowski et al. 1993). The electrical properties of the complete mixed series Fe3_ xTi~ 04 and Fe3_xZn~ 04, i.e. the thermopower and conductivity in the temperature range of 100 to 300 K, have also been rationalized in terms of a similar order-disorder theory (Hijmans and De Boer, 1955), supposing that the electrical transport mechanism in these oxides is of a small polaron type (Kim and Honig 1994). A serious drawback of the mean field treatment and the order- disorder formalism implemented by Honigs group is that use is made of simple nearest neighbour models. As was already pointed out by Anderson (1956) nearest neighbour Coulomb interactions in magnetite led to transition temperatures higher than 104 K. The Verwey transition is exceptional in the sense that the driving mechanism is the Coulomb interaction and still occurs around 120 K. That the Verwey transition in magnetite is not primarily driven by magnetic interactions is supported by the small value of the magnetic anisotropy energy accompaning the magneto-electric effect at 77 K (Kita et al. 1979), the small change of the saturation magnetization (< 0.1%) at Tv (Umemura and Iida 1976 and Matsui et al. 1977) and no anomalies of the magnetic anisotropy constants on passing the critical non-stoichiometry parameter 6c for Fe3_,O4 (Kakol and Honig 1989). The octahedral sites in the spinel structure are arranged in tetrahedrons, with each site belonging to two tetrahedrons. In this arrangement of octahedral sites it is possible to keep strong short-range order, imposed by the so-called Anderson condition: the charge of the individual tetrahedron must be constant, i.e. for stoichiometric magnetite two Fe 2+ ions and two Fe3+ions, which means an effective charge of 10+ per tetrahedron. In a magnetite crystal it is possible to keep this short range order without long-range order and the number of ionic distributions which fulfil Anderson's condition is very large. The energy differences between the various states with short-range order will be very small. If the interaction which led to the Verwey transition is restricted to nearest neighbour pairs, the energy is identical for all configurations which fulfil Anderson's condition and no transition will occur. This means that long range order is not achieved by nearest neighbour interactions but next nearest neighbour Coulomb interactions are the driving mechanism. Anderson's condition is not restricted only to FesO4 but is in fact a property of the spinel lattice. This is demonstrated by the nearly identical Mrssbauer hyperfine parameters for the crystallographic ordered and disordered phase of Lio.sFe2.504, which depend

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271

primarely on the magnetic nearest neighbour interactions and consequently the same short range order must be present in both phases (Dormann et al. 1983). Matin et al. (1990) treated the Verwey transition in a percolative way. The B-site tetrahedrons in the spinelstructure are arranged in such a way that the edges of the tetrahedrons form hexagons. If any of these Fe-hexagons are occupied by three electrons in alternating positions (Fe 2~'-sites), the Fe 2+ and Fe3+-sites in these hexagons can be interchanged with preserving Anderson's condition. Thus, in hexagons with alternating electrons, short range order conserving transitions can go on by the simultaneous exchange of three electrons to neighbouring sites, by which long range order can be built up at lower temperature. The introduction of an impurity or vacancy in the hexagon blocks the transition. With percolation calculations, in which the hexagons are taken as 'sites' of the percolation problem Matin et al. arrived at a critical impurity concentration of x = 0.039 for the possibility of long range electronic rearrangements, which means the complete disappearance of the Verwey transition, in good agreement with the experimental value obtained for cation deftcient Fe3_~O4 from heat capacity experiments for the disappearance of the second order phase transition (Shepherd 1991). Using the Bethe lattice model to calculate the percolation limit of the nonstoichiometry for long range order, Aragon et al. (1993) found a critical concentration of x = 0.012, consistent with the first composition range for the first order type transition, and a critical concentration of 0.036 for the disappearance of the second order type transition. The low temperature structure of magnetite is in first approximation monoclinic Cc with a unit cell which corresponds tot the cubic spinel structure with av/2 x av/2 x a with additional small deformations (Yoshida et al. 1979 and Iizumi et al. 1982). The lattice parameters of the monoclinic phase at 10 K are a = 11.868 ~; b = 11.851 ,~; e = 16.752 ~ and fl = 90.20 °, with 32 formula units per unit cell (Iizumi et al. 1982). Equivalent data are reported for temperatures between 80 K and Tv (Yoshida et al. 1977, 1979). The original charge ordering scheme proposed by Verwey et al. (1947), i.e. the ordering of the Fe e+ ions in rows along the (110) direction, resulting in an orthorhombic structure turned, out to be not correct (Shirane et al. 1975). A detailed charge ordering scheme based on NMR-data required a doubling of the unit cell in the (100) and (010) directions as discussed by Mizoguchi (1978), but which is not consistent with the neutron diffraction data (Iizumi et al. 1982). Later a twinning structure in the crystals was proposed to explain this discrepancy (Mizoguchi 1985a, b). From a combined study of the magnetic, magneto-electric and diffraction data, Iida (1980) inferred a nearly monoclinic phase, spacegroup Pc, with a small triclinic term, in line with the observations of the magneto-electric effect observed by Siratori et al. (1979) and Rado et al. (1975), which implies the space group 1. However, the Cc space group has been found from convergent beam electron diffraction by Tanawaka (1987), who observed reflections due to the c-glide symmetry. Zuo et al. (1990) presented the charge ordering scheme presented in fig. 3, based on simulated dynamical electron diffraction patterns, which are consistent with Cc symmetry and electron diffraction experiments.

272

V.A.M. BRABERS

Near 10 K, an additional phase transition in magnetite has been suggested by anomalies observed in magnetization (Matsui et al. 1977), specific heat (Todo et al. 1977, Rigo et al. 1983) and the magneto-electric effect (Kato et al. 1981 and Miyamoto et al. 1983). However, in other reports, Fe304 did not show an anomaly in specific heat near 10 K (Rigo et al. 1983, Gmelin et al. 1983, Koenitzer et al. 1989 and Shepherd et al. 1991), nor in the NMR frequencies (Yanai et al. 1981). The anomalous effects around 10 K may originate from ferroelectric domains near 10 K (Iida et al. 1980) as well as from changes in magnetic domain structure as suggested by Rigo et al. (1983) and probed with NMR by Mizoguchi (1985b), which means that there is no additional phase transition at 10 K. At room temperature magnetite can undergo a phase transition at pressures above 25 GPa to a monoclinic structure with 'tentative' lattice parameters a = 4.22 ~; b = 5.43 A; c = 5.60 ,~ and fl = 106 °, with two formula units per unit cell and a volume change of -7.2% compared with the cubic structure, which is consistent with a density predicted for a structure with all cations in sixfold coordination (Mao et al. 1974). However, in the study of the phase diagram of iron-oxygen at high pressure, and between 300 and 900 K, the large volume change and the exact lattice parameters are questioned (Huang and Bassett 1986). The structure of the spinel phase of magnetite under pressure up to 4.5 GPa has been studied by Nakagiri et al. (1986) and Finger et al. (1986). The isothermal bulk modulus KT and its pressure derivative is reported to be 181 (2) GPa and 5.5 (15), respectively. The oxygen parameter # was found to be 0.3810 at atmospheric pressure with a pressure dependence of - 4 . 6 x 10-4 GPa (Nakagiri et al. 1986), while Samara et al. (1969) derived a small positive value of d#/dp = 5.8 x 10 -4, due to the deviating analysis of their data. The small value of the pressure dependence of # is indicative of the equal compressibility of the tetra- and octahedrons. The lattice parameter decreases from 8.3949 ]k to a = 8.3332 .~ at 4.44 GPa (Finger et al. 1986). Stable non-stoichiometric magnetite is usually to be known with an excess of oxygen fe304+ 7 because of the stable framework of the oxygen lattice. Several successful and unsuccessful attempts have been made to prepare oxygen deficient magnetites (Moukassi et al. 1982 and ref. cited), but the reports are contradictory. McCannon et al. (1986) succeeded to prepare iron rich magnetite (0.5 #m) particles (so-called hyper-magnetite) with composition Fe32604 and a slightly increased lattice parameter (8.401 /~) for the freshly prepared specimens compared with the stoichiometric spinel phase (8.397 ~). This hypermagnetite turned out to be very unstable and oxidizes readily in air at room temperature; X-ray data of aged samples revealed a phase intermediate between 7-Fe203 and Fe304, which could explain the contradicting results about the existence of iron-rich magnetites.

5.2. Magneto-electricproperties As already referred to before, magnetite shows interesting magneto-electric properties at low temperature. Rado and Ferrari (1975) reported a static electric polarization in the (110) direction, induced by a static external magnetic field at 4.2 K and which

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273

depends nonlinearly upon the external magnetic field components in the (110) plane. The orthorhombic notation is used since the deviations from this structure are small and in first approximation it is a pseudo-orthorhombic symmetry. From the analysis of these data Rado and Ferrari concluded that there is no symmetry in the low temperature structure of magnetite at all, i.e. point group 1 and that the macroscopic anisotropy energy depends on the electric field. In the following paper, Rado and Ferrari (1977) investigated the linear and bilinear magneto-electric effects. Two types of experiments were performed: I. with a biasing magnetic field H0 in the bc-plane at some angle ~ from the c-axis, a perturbing 1-kHz electric field ea was applied ill the a-direction and the 1-kHz component of the induced magnetization in the c-direction was measured and with the application of a perturbing 1-kHz magnetic field h c in the c-direction, the 1-kHz part of the electric polarization in the a-direction was measured II. the same experiments were also performed with a biasing magnetic field in the b-direction, in which case the crystal is supposed to contain two sets of magnetic domains as distinct from the first case in which the crystal is single domain. In the second case the 1-kHz, as well as the 2-kHz response was measured. It was found that the electric field dependence of the macroscopic magnetic anisotropy energy produces a linear as well as two kinds of bilinear magneto-electric effects, which appear in the free energy density as function of the perturbing electric and magnetic fields as the terms - o~ij ei h i , - f l i j k ci hj hk/2 and - 7ij k ei ej h k/2. The dependencies on H0 of the measured magneto-electric susceptibilities aij and/3ijk could be well explained on the basis of the electric field dependence of the macroscopic magnetic anisotropy energy, but not the susceptibility -71jk, for which experimentally relaxation effects were found, not included in the proposed electric-field-dependent anisotropy mechanism. After the pioneering work of Rado and Ferrari, the magneto-electric effects in magnetite have been studied by several authors in relation to the structure. Magnetoelectric measurements at 4.2 K (Kato et al. 1981), at 4.2-60 K (Miyamoto et al. 1988) and at 77 K (Siratori et al. 1979) disclosed that the real magnetic crystal symmetry is triclinic, point group 1 but that the deviations from the monoclinic structure are very tiny, especially the breaking of the mirror symmetry parallel to the (110) plane is very small (Siratori et al. 1979). A complication in the study of the low temperature behaviour of magnetite is the conversion into a multiple twinned structure on cooling through the Verwey transition. By squeezing the crystal along the (111) direction, which is achieved by a tight fixing of the crystal in a sample holder which shrinks on cooling and the application on cooling of an external magnetic field in the direction +400 tilted from the [100] to the [110] axis, a single magnetic domain crystal might be obtained. By this procedure the magnetically easy c-axis, the intermediate b-axis and the hard a-axis are aligned along the [001], [110] and [1 ]'0] axis of the originally cubic phase, respectively (Kato et al. 1981).

274

V.A.M. BRABERS

97"8 I

I

0

I

I

40

i

t

I

80

I

120

i

I

160

T (K) Fig. 49. Temperature dependence of the saturation magnetization of a single crystal of magnetite, squeezed along the [100] axis and field-cooled with a magnetic field of 10 kOe (Matsui et al. 1977b). Near the Verwey transition a small decrease is observed but below 10 K an upturn of the magnetization is observed, which is also found in powdered crystals (Rigo et al. 1983).

The magnetic anisotropy energy accompanied by the electric polarization in magnetite was estimated to be in the order of 102 J/m 3 (Kita et al. 1979). Ferro-electric behaviour, i.e. ferro-electric hysteresis, was reported between 4 and 70 K, with spontaneous polarization of 4.8 and 1.5 # C/cm 2 for the a- and c-axes, respectively (Kato et al. 1982, 1983). Further anomalous behaviour in the temperature dependence of the spontaneous and magneto-electric electrical polarization was reported near 40 and 6 K by Inase and Miyamoto (1987). The anomalous behaviour around 40 K could be attributed to the switching of the a- to b-axis by an external magnetic field (Inase et al. 1985, Shindo et al. 1992). The anomalous behaviour below 10 K, also reported in earlier papers (Miyamoto et al. 1979, 1983 and 1986a, b) is not accompanied by a change in crystal structure. These effects have been suggested to be caused by a change in the magnetic structure (Miyamoto 1990), indications of which have also been found in the temperature dependence of the magnetization shown in fig. 49 (Matsui et al. 1977b). 5.3. The electrical conductivity A basic item in the understanding of the electrical conduction in magnetite is the question of the itinerant vs localized character of the charge carriers. The existence

PROGRESS IN SPINELFERRITERESEARCH

275

of the Verwey transition marks the importance of correlation effects and the Coulomb interactions between the ions play a dominant role in the phase transition. In the charge ordering process, long range as well as short range, the electrons are considered to be localized on the Fez+ ions, and the electrical conduction process will be quite distinct from that of itinerant electrons because of the thermally activated mobility. Even if there is a finite transfer integral t, the energy band of the electrons will split, supposed that the correlation energy U is large compared to t. Ihle and Lorenz (1980) estimated that for the formation of this Coulomb gap U/t must be larger than 3, with U = U1 + 2Uz, in which U1 and U2 the effective Coulomb integrals between nearest neighbour and next nearest neighbour sites_respectively. Goodenough (1971) estimated a critical cation-cation separation of ,-~ 2.95 A for Fea+-Fe 3+, below which localized electrons become itinerant and will show metallic conduction. The Fe-Fe octahedral distance in magnetite, 2.97 ,~, is close to this value, which suggests that U/t in the Ihle-Lorenz approach is near the critical value. Moreover, the pressure coefficient of Tv is negative and rather large, between - 2 . 0 and - 4 . 7 K/GPa (Kakudate et al. 1979, Samara 1968, Schloessin et al. 1984, Tamura 1990) and the electrical conductivity in the temperature range of Tv up to 900 K increases with pressure (Kakudate et al. 1979, Schloessin et al. 1984), which is in agreement with the expected increase of the overlap integral t with pressure. From the experimental pressure dependence of the Verwey-temperature, a value of 0.08 eV was deduced for the overlap integral (Lorenz and Ihle 1982). In a number of papers, Lorenz and Ihle (1977, 1979, 1980) and Ihle (1984) discussed the effect of electron correlations as well as the influence of electron-phonon interactions on the short and long range electron ordering in magnetite. A microscopic theory for the electrical conductivity, based on the small-polaron (SP) correlation model in the narrow-band limit has been proposed by Ihle and Lorenz (1985, 1986), taking into account the polaronic short-range order due to the intersite small polaron interaction. The conductivity above and below Tv results in this model from the superposition of small polaron band and small polaron hopping condition: the SP-band conduction is the dominant mechanism concerning the dc-conductivity below room temperature, while the observed maximum in the optical conductivity near 0.2 eV is attributed (Schlegel et al. 1978, 1980) to hopping conduction. An analysis of the dc-conductivity and of the optical conductivity determined from reflectivity data, displays within the framework of this theory a good agreement between the theoretical fit and the experimental data as is shown in fig. 50 and 51 (Degiorgi et al. 1987). Because of Anderson's condition, it is not unrealistic to introduce a bipolaron concept to explain the Verwey ordering and the electrical conduction in magnetite (Chakraverty 1980). Especially if charge ordering under the influence of the Coulomb interaction as discussed by Wigner (1938), and a narrow spin polarized d-band for the conduction electrons (Camphausen et al. 1972, Yanase et al. 1984) is considered, Wigner crystallization can take place, if the ratio of the interatomic Coulomb interaction to the band width is larger than 3 (Cullen and Callen, 1973). The ground state of magnetite can be considered to be one where Wigner-localization has taken place. As can be seen in fig. 46, the conductivity above Tv is not metallic and

276

V.A.M. B R A B E R S

'U

6

0 v

.S ""i ' S • °.s

'°~........

4

. fit o 0 (to)

..../

// //d-s

Osc.

0 U

.....

c~ 2 0

/"

ptl. d-s Osc.

0

............... "J'.~ ~,.j = ~

0

0.1

!

I

I

I

I

0.2

0.3

0.4

0.5

0.6

Photon energy (eV) Fig. 50. The real part of the optical conductivity ~r0(w) at 300 K for magnetite between 0.08 and 0.65 eV. A good agreement is obtained between the experimental and fitted data, which are composed of a small-polaron band contribution %(w), a small-polaron hopping contribution ~,h(W) with a m a x i m u m near 0.2 eV and a d - s oscillator contribution with a m a x i m u m near 0.6 eV. The assignment of the 0.6 eV contribution to a d - s transition is questionable and might be as well ascribed to an IVCT transition as discussed before in relation to the magneto-optical properties. The contribution of all the transitions are phenomenologically described as oscillators (Degiorgi et al. 1987).

an increasing conductivity is observed up to room temperature. Mott (1979, 1980) suggests that in this temperature range the magnetite-phase is a 'Wigner-glass' of bipolarons, some of which dissociate with rising temperature. Nearest-neighbour bipolaron hopping is proposed as most likely mechanism just above Tv (Mott 1980). Some indications for the existence of bipolarons in magnetite have been obtained from diffuse neutron scattering experiments (Yamada 1980, Yamada et al. 1980). In muon spin relaxation experiments an anomaly in the temperature dependence of the muon hyperfine frequency is observed around 250 K depending on the external magnetic field, which provides evidence that this anomaly results from cross relaxation between the muon Larmor precession and the electron-correlation process. These data support Mott's suggestion that above Tv magnetite is in the Wigner glass state (Boekema et al. 1986). Degiorgi et al. (1990) considered the sharp lines they observed in the far infrared spectra (0-0.08 eV) overlapping the phonon spectra in the low temperature phase of magnetite (Kuipers and Brabers, 1977) as the fingerprint of the bipolaronic ordered ground state. Coey et al. (1992), observed with scanning tunnelling microscopy, using an atomically sharp iron tip, certain features in the microscopic images on (100) surfaces, which could be explained as static arrays of pairs of Fe z+ ions with short range order and a charge fluctuation time greater that 103 s. This is interpreted as a Wigner glass state on the (100) surface.

PROGRESS IN SPINEL FERRITE RESEARCH

277

or-4 v .~ ' - '.~ ' ~ .

f

..-.""" . . . . " "

.

U f

o

0 i ¢J

• f /s

,;

I

100

.+,"

....."

"~.~

..,."y '

~"~..

........ o h I

!

200

I

300

I

I

400

T (K) Fig. 51. Temperature dependence of the experimental and theoretical dc-conductivity of magnetite above Tv composed of a small-polaron band and hopping contribution a b and ~rh, respectively (Degiorgi et al. 1987).

Self-consistent-field-Xa molecular orbital calculations have been used by Sherman (1987) to study intervalence charge transfer (IVCT) in ( F e 2 + F e 3 + O l o ) 15- clusters, with a Fe-Fe distance of 2.937/~. For the thermally induced IVCT transition between the two ions, which is in fact the electrical conduction process, a small negative activation energy is calculated (,-- -0.016 eV) which implies that the electron is not trapped at a given site of the dimer. The lack of an activation energy barrier means that small polarons are not the charge carriers. However, the actual Fe-Fe distance in Fe304 is slightly higher (2.97 A), which denotes that the additional electrons of the Fe 2+ ions are close to a delocalized or an itinerant state. The non-metallic conductivity behaviour above the Verwey transitions, i.e. the positive temperature coefficient of the conductivity indicates that, at least in this temperature range, a small polaron hopping model is still not excluded. Several reports on band structure calculations have been published for magnetite (Yanase et al. 1984, De Groot et al. 1986, Feil 1987, Zhang et al. 1991 and P6nicaud et al. 1992). In all reports the majority-spin electrons are semiconducting with a sizeable energy gap, whereas minority spin electrons are present at the Fermi energy which is taken as a sign that magnetite should be considered from an itinerant electron point of view (Yanase et al. 1984, P6nicaud et al. 1992). Though band structure calculations for 'insulating' Zn, Co and Mn ferrites gave also the 'metallic phase' with energy bands crossing the Fermi level (P6nicaud et al. 1992), which makes it questionable that a calculated final density of states near the Fermi level might be taken as evidence for an itinerant character of the charge carriers. The critical parameter for itinerant or localized character of the 3d electrons is U/t, where U is the electron-electron Coulomb interaction and t the 3d bandwidth (Hubbard 1963, 1964).

278

V.A.M. BRABERS

However, with photo-electron spectroscopy, valence band emission was observed to start from the Fermi level, which was taken by Siratori et al. (1986) as proof that F e 3 0 4 is a 'metallic' conductor. Subsequent photo-emission studies by Lad and Henrich (1989), inverse photo-emission studies by Sancrotti et al. (1991) and O-ls X-ray absorption experiments by de Groot et al. (1989) showed that strong charge transfer exists from the O 2p ligands to the Fe 3d states, and a strong covalent admixture of Fe (3d) and O (2p)-derived states exists above as well as below the Fermi level. Based on either pure band-like or duster type approaches, these states can be predicted to appear close to the Fermi energy, but it is not yet clear in how far these states might be considered to form broad bands, small polaron bands or localized states. The character of the charge carriers is not yet definitely settled. The electrical conductivity of magnetite above Tv is only slightly dependent on stoichiometry and impurity concentration, but can easily change a factor ten below Tv by an impurity concentration of z = 8 x 10 -3 in Fe3_~Ti~O4 (Kuipers et al. 1979) or by an iron deficiency in the same order (Kuipers et al. 1976, Aragon et al. 1986). Above Tv the conductivity decreases slightly, whereas below Tv the conductivity increases with impurity concentration (fig. 52a), which led Kuipers and

(a)

il

\

ca

v

o

8.1o-'

":':),,×2-o " , ~ 10-4 ' " ~ 10 .3 ""--4.104

.A

I

4

r

f

i

8

I

1

I

I

12 103/T (K-1)

I

I

I

16

Fig. 52. (a) Logarithm of the electrical conductivity of Fe3_xTixO4 vs reciprocal temperature.

PROGRESS IN SPINEL FERRITE RESEARCH

279

(b)

4C

x= 104~ ~ = 0

-4~ 1.4

~o -80 o

~

"~ -120

-160

-200 I

60

t

100

/]1 Ill

x= 10.3

/

x=3.10 3 1

140

I

I

180

i

I

220

T CK) Fig. 52 (b) Absolute thermoelectric power of Fe3_zTizO 4 vs temperature (Kuipers and Brabers 1979).

Brabers (1979) in their study of the thermoelectric power of Ti-doped magnetite to a two level model for the conduction below Tv. The conduction takes place on two energy levels, separated by a gap of about 0.12 eV and the experimental results are compatible with mixed p- and n-type conduction over the two levels in this model (Kuipers and Brabers, 1976, 1979, Brabers, 1980). Supposing a constant ratio of the mobilities of the p- and n-type charge carders as function of temperature in the low-T-phase, the step in the conductivity at Tv could be calculated from which it followed that the nature of this step is a change in the number of charge carders and not a sudden change in the mobility. From this analysis, it is also concluded that the mobility of the electrons is thermally activated above, as well as below Tv, indicating hopping conduction (Kuipers and Brabers 1979). Hurd et al. (1982) and McKinnon et al. (1981) emphasized in their analysis of the electrical conductivity ~ below Tv that in the range Tv/2 < T < Tv, log cr is better described with a linear function of T than by an Arrhenius law which refers to a mechanism based on incoherent tunnelling of electrons between neighbouring sites. Pai and Honig (1983), however, reported that an Arrhenius law is obeyed, but that aging at room temperature or

280

V.A.M. BRABERS

magnetic annealing of the specimens could change this behaviour. In fact all these measurements referred to were performed on magnetite single crystals cooled down through the Verwey transition, without taking precautions to prevent twinning in the low temperature phase. As was shown by Matsui et al. (1977a) on mechanically squeezed and in magnetic field cooled specimens to obtain monoclinic single domains, the conductivity is anisotropic with ~c ~ 2~% and can be approximated with an Arrhenius law between (1/2)Tv and T; below (1/2)Tv larger deviations occur, with Cb > 0"a > ~re. Lenge et al. (1984) analyzed the low temperature conductivity with multiple-range hopping for which the temperature dependence can be expressed as (Mott 1974) ~r=o'oexp

-

(36)

~

For stoichiometric Fe304, two breaks in the In ~r versus T -1/4 were observed at 53 and 24 K (fig. 53), while for non-stoichiometric magnetite only one break occurs

200 100 5 ee~.[ ,

T (K) 50 ,

30

20

,

,

,

\ O

D

-15

3 TI] 0.25

2

0.35 T-l/4 (K-1/4)

0.45

Fig. 53. Logarithm of the electrical conductivity of a stoichiometric Fe304 crystal plotted versus T-l/4. No precautions are taken for twinning below Tv (Lenge et al. 1984).

PROGRESS IN SPINEL FERRITE RESEARCH

281

near 29 K, comparable with the data of Drabble et al. (1971). By combination with magnetic after-effect measurements, which will be discussed later, Lenge et al. concluded that above 50 K the electrical conduction in stoichiometric F e 3 0 4 is due to thermally activated hopping processes and below that temperature to thermally assisted tunnelling of electrons and to an electron diffusion process between specific Fe-ion pairs (activation energy 0.055 eV; r0 = 4 x 10 -9 sec), which is only observed for stoichiometric magnetite and causes the additional break at 53 K in the In ~r vs T -1/4 curve. Just above Tv, according to Lenge et al. the conductivity follows the equation cr = a - b / T

(37)

which is the Taylor series development of an Arrhenius law tr ,-~ exp [ - A w / k T ] , in which the hopping energy is small compared with the thermal energy: tr = a(1 - A w / k T ) .

(38)

This implies that from 125 to 300 K, electron hopping should occur with a small activation energy in the order of 0.01 eV. The ac-conductivity of magnetite trae shows dielectric relaxation behaviour in two temperature ranges (Akishige et al. 1985, Kobayashi et al. 1986, 1988). In the temperature range of 30-50 K, a frequency dependence of trae c~ Aw~ is observed, which is characteristic for hopping of bound charge carriers among randomly distributed centers like in the impurity conduction model proposed by Pollak and Geballe (1961) and below 20 K a Maxwell-Wagner relaxation takes place which is related to spacecharge layers due to the twinning or ferro-electric domains in the specimens. The 30-50 K relaxation is suggested to be caused by bound charge carrier hopping around cation vacancies and from the shift of the maxima in the U-T curves an activation energy of about 0.07 eV is estimated (fig. 54). With increasing nonstoichiometry 6, the 30-50 K anomaly becomes less well defined and impurity conduction dominates the electrical properties (Kobayashi et al. 1988). From dc-measurements on untwinned specimens in the temperature range of 30-450 K, a T -1/4 dependence of the conductivity was observed, which indicates a variable range hopping mechanism for this impurity conduction (Kobayashi et al. 1986). I n n n"xed ferro-ferdtes, M~ 2+ Fel_~Fe 2+ 3+ 2 O4, where Fe 2+ and Fe 3+ ions are present on octahedral sites, and which in fact are partial substitutions of the Fe 2+ ions in magnetite by M 2+ ions, the electrical conductivity is always thermally activated over a large temperature range of several hundred degrees (~ 100-500 K) and can be attributed to the electronic conduction on B-sites. However, it is remarkable that the observed activation energy for ferro-ferrites with Mn or Zn as substituents entering the tetrahedral sites, the activation energy is decreasing with increasing substitution from about 0.06 to 0.03 eV (Funatogawa et al. 1959, Lotgering 1964, Brabers and Scheerder 1988) whereas for Mg, Co and Ni entering octahedral sites, the activation energy increases from 0.06 to 0.1 eV (Miyata 1961, Yamada 1973, Brabers 1993). This different behaviour for A- or B-site substitution reflects the

282

V.A.M. BRABERS

a)

]

1°"I 5

2/--',/

4

,/ d 10 lo~_ 1/

_i ! /

5:106 I-Iz

/ dc 0

'2;

'4; T (IO

16;

I

8;'

0

' "'2;

'

zlO

'

60

'

810

T (K)

Fig. 54. The dielectric constant eI (a) and the ac-conductivity (b) of nonstoichiometric Fe3_604 with 6 = 0.0013 (Kobayashi et al. 1988). The sample is single crystalline above Tv, with no precautions taken for twinning below Tv. The arrows in (a) indicate the positions of the two dielectric relaxations.

B-site disorder effect upon the conduction. With increasing A-site substitution, the available number of charge carriers (Fe2+) decreases, whereas the B-site lattice remains uniformly occupied with Fe. For lower Fe z+ concentrations (z > 0.5) the intersite Coulomb interactions between Fe 2+ ions will decrease, which might explain the lower activation energy. For B-site substitution, however, the intersite Coulomb interactions are not decreasing since the B-site two-valent Fe 2+ ions are replaced by other two-valent ions. These interactions will even increase due to the different ionic size of the substituent (see table 7, p. 269) or to a more statistically distribution of the two-valent ions, which might be conserved from the high temperature at which the preparation was performed due to the finite diffusion rates of the ionic species at lower temperature. This more statistically distribution is a disturbance of the short range order and a violation of Anderson's criterion for the short range order. Due to the larger disorder on B-sites, localization of the charge carriers will be promoted, which will particularly for high substitutions result in small polaron hopping. An experimental indication of the effect of local disorder can be found in the dispersion of the magnetic perlneability of Mn0.8Fe2.204. Two electronic magnetic relaxations are found with activation energies of 0.03 and 0.07 eV, respectively; the first is assigned to the charge transport in non-disturbed regions in the crystals, whereas the second, of which the magnitude depends on the B-site Mn 2+ concentration, is attributed to electron hopping between iron ions around B-site Mn2+ ions (Brabers and Scheerder 1988). The effect of the B-site disorder in nickel ferrous ferrites NixFe3_~O4 has been discussed in detail by Whall et al. (1986a, b). For high nickel concentrations (z > 0.4), nearest neighbour and variable range hopping can

PROGRESSIN SPINELFERRITERESEARCH

283

describe the experimental conductivity and thermo-electric results quite well, for lower concentrations the formation of a Coulomb gap seems to take place, evidence of which is also found in the zinc and manganese ferrous ferrite systems (Philips et al. 1994).

5.5. Galvano-magnetic properties The galvano-magnetic properties of magnetite has been subject of several reports. The Hall resistance of magnetic materials is usually decomposed into two terms, the ordinary R0 and the extraordinary Hall effect RI: PH = R0" B q- RI" M.

(39)

The extraordinary Hall coefficient is negative above the Verwey temperature and decreases in absolute value with increasing temperature up to 650 K where it changes sign to positive (Todo et al. 1992). Below Tv, the Hall resistivity depends in a more complicated way upon the magnetic induction, which is apparently due to the microstructure of the twinned crystals (Kostopoulis et al. 1970). The ordinary Hall effect is generally separated from the extraordinary effect by plotting the Hall voltage against the external magnetic field and considering the high field part of the curve only. In all the experiments, at least at room temperature (Lavine 1959, Kostopoulos et al. 1970, Siemons 1970, Feng et al. 1975), the absolute value of the negative Hall voltage increases with the external field. Siemons (1970) attributed this increase at high fields to an increase of the extraordinary term introduced by an increase in the high field magnetization and arrived at a positive sign for the ordinary Hall coefficient, inconsistent with the negative sign in other reports (Lavine 1959, Feng et al. 1975). However, a major problem with Hall effect measurements on magnetic samples with finite dimensions is the inhomogeneous effective field in the specimens due to the demagnetization effect and the non-homogeneous current distributions caused by the internal fields (Shiozaki et al. 1981). This problem was discussed in detail by Siratori et al. (1988) who suggested also the existence of higher order terms in the Hall voltage, proportional to M 3, which makes the analysis of the Hall effect of Fe304 even more problematic. The Hall effect measurements ~fig. 55) reported by Feng et al. (1975) were performed on polycrystalline 2500 A films, which approximates an ideal two-dimensional configuration and restrains the above mentioned problems. These data in combination with resistivity data yield a thermal activated Hall mobility in the range of 1-10 -1 cm-2/V.s with an activation energy of 0.039 eV. As to how far these data are representative for bulk magnetite, one has still to remind that the samples are polycrystalline and fixed to glass substrates. Particularly magnetoresistance may be effected by grain boundaries. The transverse magnetoresistance has been reported to be positive and negative depending on the magnitude of the applied field, crystal orientation, temperature and the origin of the sample. Negative values were reported by Domenicalli (1950), Samokhvalov et al. (1960), Zalesskii (1961), Balberg (1970), Feng et al. (1975), whereas positive values are given by Kostopoulos (1972) and Shiozaki et al. (1981).

284

V.A.M. BRABERS T (K) 150

250200

125

110100

4. Tv • , ~ / /

/

"O~ il/I~l

'e~ 10"9"~o ~ 9

• ~.....~''°n/~

o....e ''''~"

,.....,.......

..~....~.....

>~

t

°.°...o'

t:l

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.

J

.O..~...q-~"'" [] 10-10 lC I

4

I

I

6

I

I

8 1000/r (K-1)

J

I

10

Fig. 55. Temperature dependence of the ordinary (n) and extraordinary Hall coefficient (o) determined on polycrystalline 2500 ,~ thick magnetite films (Feng et al. 1975). The magnitude of the longitudinal magnetoresistance is comparable with the transversal magneto-resistance (Domenicalli 1950 and Samokhvalov et al. 1960). Besides the differences in sign, there exists a large scattering in the absolute value of the magnetoresistance. In particular below the Verwey transition temperature, complex behaviour is observed due to magnetic and crystallographic domains, which cannot be analyzed in a simple way (Kostopoulos et al. 1977, 1980, 1981). Kostopoulos and Alexopoulos (1976) analyzed for a synthetic single crystal the room temperature transverse magnetoresistance, which they separated into a negative isotropic term in the order of 10 -4 and a positive anisotropic term in the order of 10 -3, which saturates with the magnetization. A similar separation was performed by Shiozaki et al. (1981) but they found for the isotropic component a ten times higher value. Samokhvalov et al. (1960) reported negative magnetoresistance at room temperature in the order of 3% in a field of 1.5 T and the polycrystalline films of Feng et al. (1975) showed a similar large magnetoresistance. This large variation in MR-data seems puzzling but might be explained by the microstructure of the specimens as was shown by experiments on manganese substituted magnetite. The magnetoresistance for polycrystalline samples is about one order of magnitude larger than for single crystals, which is caused by the grain boundaries (Brabers and Willems 1968). The low magnetoresistance reported by Kostopoulos et al. (1976) was determined on a synthetic crystal, which is supposed to be a single phase crystal. Samokhvalov's data were obtained on a natural crystal and Shiozaki's data on a crystal grown by chemical transport and Feng's data on polycrystalline films. Since the equilibrium

PROGRESS IN SPINELFERRITERESEARCH

285

stability range of magnetite, i.e. the range of the iron-oxygen ratio at temperatures below 1000 K is quite small, a-Fe203 segregations in the form of very thin plates might be present in this type of crystals, and the crystals resemble a polycrystalline structure. The observed large magnetoresistance might originate from the 'polycrystalline' nature of the specimens and particular from the grain boundaries.

5.6. MOssbauer spectra and the electrical conduction process Besides the numerous papers on MOssbauer experiments concerning the magnetic and crystallographic structure of magnetite (see for a review Vandenberghe et al. 1989 and De Grave et al. 1993), several authors related the B-site line broadening in the MOssbauer spectra of magnetite to the electron exchange process (Kiindig and Hargrove 1969, Sawatzky et al. 1969). The typical probing time in MSssbauer experiments is in the order of 10 -8 s, which means that if electron hopping between Fe-ions occurs with a frequency 108 s -1 distinct Fe 2+ and Fe3+ -B-lines will not be observed and strong line broadening will be observed. Further increase of the hopping frequency, i.e. temperature, sharpening of the broad absorption region to a mixed Fe2+-Fe 3+ line will occur. Sawatzky et al. (1969) related the differential line broadening, AW = F B - P A , to the thermal activated hopping ( A W ) - 1 ,~ e x p { - E / k T } , and found two distinct temperature regions, above T = 250 K with E = 0.065 eV and below 250 K with a nearly constant AW. Later, it was recognized in several reports that the B-site component of the spectra may consist of two distinct subspectra because of two different orientations of the electric field gradient principal axis with respect to the hyperfine field direction (Boekema et al. 1977, Van Diepen 1976, H~iggstrrm et al. 1978. and Evans et al. 1987), which causes line broadenhag. The recent results of De Grave et al. (1993), however, showed that after taking into account the two B-site subspectra, a considerable differential line broadening persists, which is plotted in fig. 56. For cobalt doped magnetite samples according Sawatsky's analysis. The activation energy above 250 K is in the order of 0.04 eV whereas below 250 K a smaller energy of about 0.01 eV is obtained, which suggests according to De Grave et al., a band like process at lower temperatures and local hopping at higher temperatures. The temperature dependence of the A and B-site magnetic hyperfine fields deduced from the M6ssbauer spectra were analyzed by De Grave et al. (1993). Unlike the A-site hyperfine fields, the temperature dependence of the B-site fields could not be calculated adequately with the Heisenberg-type exchange model, but instead a non-localized electron model led to an excellent agreement with the experimental data (fig. 57). This model was originally developed by Kubo and Ohata (1972) to explain the magnetization of Ca 2+ substituted LaMnO3 perowskite. The spins of the Mn 4+ ions are localized and the additional electrons of the Mn 3+ ions are delocalized within an energy band with a finite bandwidth. Applying this model to the temperature dependence of the B-site hyperfine field of magnetite, De Grave et al. calculated a band width of 0.82 eV at room temperature which conflicts with the earlier supposed hopping conduction at this temperature. If fact, such a large band width is quite unrealistic if the electrical properties of magnetite are taken into consideration.

286

V.A.M. B R A B E R S

5

-

~ •

4_

"XX~ _

..

0.01

•

0.02

• 0.04

A

2

I

1

I

3

I

[

I

5

[

I

7

9

1000/T (K1) Fig. 56. Logarithm of the inverse differential M6ssbauer line broadening A W = F B - _r'a vs the inverse temperature for cobalt doped magnetite F e 3 _ x C o x O 4 after De Grave et al. (1993).

5.7. Magnetic properties The room temperature intrinsic magnetic properties of magnetite like magnetization, magnetostriction and magnetic anisotropy are well established (see table 6) and are not very sensitive for the oxygen stoichiometry. However, certain anisotropic impurities, like Co 2+, might affect the anisotropy (Bickford et al. 1957) and magnetostriction (Leyman et al. 1985) substantially. The temperature dependence of the anisotropy constant K1 below room temperature does not decrease monotonically but shows a minimum near 250 K and becomes even positive near 140 K (fig. 58). For the cubic phase, the magnetocrystalline anisotropy is given by a series development of the squares of the direction cosines c~i with respect to the crystal axes: E A = K 1 (O~lC~ 2 22 "t- 0~20~ 2 32 -t- 0~30~1) 2 2 "t- K20~1o~20~3 2 2 2 ••• ,

(40)

where K1 and K2 are the first and second-order anisotropy constants. For the low temperature monoclinic phase this energy is given by Chikazumi (1976) as: 2 b2 + Kaaa 4 + Kbba~ + KabaaC% 2 2 _ Kuai01, 2 EA = Kac~a + Kba

(41)

PROGRESS IN SPINEL FERRITE RESEARCH

287

500

400

©

300

200

100

I 0

[

I

300

I

[

I

I

600

900

T (K) Fig. 57. Temperature dependence of the B-site hyperfine field for magnetite, deduced from M6ssbauer spectra (De Grave et al. 1993). The solid curve is calculated based on the delocalized-electron model by Kubo and Ohata (1972).

and a b are the direction cosines with respect to the monoclinic a- and b-axes and c~i01 with respect to the monoclinic [101] axis, which is due to the small distortions close to the high temperature cubic [111] axis. Without the last term eq. (40) is the expression for the orthorhombic symmetry, earlier used by Calhoum (1954) and the last uniaxial term takes account of the nearly rhombohedral deformation. The data of K1 below Tv were approximated by Abe et al. (1976), taking an average value of the monoclinic constants: w h e r e ot a

1

K 1 = ~'~ ( - - 2 0 K a a -

20Kbb -- 6K.b)

(42)

and an anomalous positive contribution AK1 was obtained above Tv compared with the dashed line, which was estimated from the high temperature data by Smith (1956) and the calculated values below Tv. Chikazumi (1976) and Abe et al. (1976) suggested this positive contribution to be related with some 'hypothetic' anisotropic ions like Fe 1+ or Fe 2+ with a doublet as ground state, which could explain the 2 + lOllS In positive contribution in a similar way as the anisotropy contribution ot ~~t, " " Co-doped magnetite (Co-dope about 1%) as explained by Slonczewski (1958, 1961).

288

V.A.M. BRABERS

100 i

~

200 i

i

T (K) 400 500

300 i

i

t

i

i

i

Tv

;

600 i

i

700 ~

i

800 i

i

L

Tel

..5/

Q

oe°/

o

2~ -2

~ J

~

[] Sm!th (! 956) • Bickfordet al. (1957) o Abe et al. (1976)

-3 Fig. 58. Temperaturedependenceof the magneticanisotropyconstantK 1 of magnetite. However, the nature of this small concentration of anisotropic ions is suspicious, as well as the exact position of the dashed line which is based on Smith's data. The high temperature data by Abe et al. (1976), Bickford et al. (1957) and the recent data by Kakol et al. (1989) suggest that the negative K1 values by Smith (1956) are too large. Siratori and Kino (1980) argued that the short range order and atomic displacements, preceding the Verwey transition are the cause of the extra part of the cubic anisotropy above Tv, which should also affect the elastic and magnetostrictive properties. Indeed a strong softening in C44 (Siratori and Kino 1980) and an anomalous decrease of the cubic magnetostriction constant AlU below 250 K is observed (Brabers et al. 1980). The entity of the short range order, according to Siratori and Kino, is specified by a (100) axis and two (110) axes perpendicular to it. That the short range order preceding the Verwey transition is indeed acting on the anisotropy and magnetostriction is supported by the comparison of the behaviour of AI3+ and Ga 3+ substition in magnetite. For a concentration of 0.4 AI3+ per formula unit, the anomalous temperature behaviour of Kt and "~111 disappears already while for the same Ga concentration it is still present, which is a consequence of the larger disturbance of the ordering by AI than by Ga, also observed in the shift of the Verwey transition (Brabers and Hendriks 1982, Merceron et al. 1986, Brabers, Merceron and Porte 1988). The magnetic anisotropy as function of the nonstoichiolnetry of magnetite, Fe30_004 has been studied by Kakol and Honig (1989). With increasing nonstoichiometry the absolute value of the negative K1 decreases and varies for 0 < 6
50 nm) extending over many layers. Magnetite layers are ferrimagnetic with a Curie temperature close to the bulk value but due to the crystallographic stacking fault incoherence, the magnetic coherence is confined to one chemical layer thickness. An increase of the Nrel temperature of NiO, due to the exchange coupling is observed, in agreement with mean field estimates. Further, the linear behaviour of the magnetization with applied field for thin superlattices (< 8.0 nm) show a return to 5'-shape hysteresis loops above the Nrel temperature of NiO similar to bulk Fe304, but with a reduced magnetization (Berry et al. 1993). This result suggests that the linear behaviour is not a paramagnetic ordering, but is rather a frustration effect both by strain-induced anisotropy and an exchange coupling persisting to high temperature, even above the TN of NiO. It is clear that the new effects found in these superlattices will certainly provoke new developments in ferrite research in the near future. References of Tables 2 and 6 [1] Brabers, V.A.M., T.E. Whall and P.S.A. Knapen, 1984, J. Cryst. Growth 69, 101. [2] Abrahams, S.C. and B.A. Calhoun, 1953, Acta Crystallogr. 6, 105. [3] Greaves, C., 1983, J. Solid State Chem. 49, 325. [4] Chassagneux, E and A. Rousset, 1976, J. Solid State Chem. 16, 161. [5] Slack, G.A., 1964, Phys. Rev. A134, 1268. [6] Mason, "17.O., 1985, J. Am. Ceram. Soc. 68, C74. [7] Petric, A., K.T. Jacobs and C.B. Alcock, 1981, J. Am. Ceram. Soc. 64, 632. [8] Firsov, Y.E and G.E Popov, 1981, Zh. Fiz. Khim. 55, 603. [9] Otero Arean, C., E. Garcia Diaz, J.M.

[10]

Satay Murthy, N.S., M.G. Natera, R.J.

[11]

Begum and S.I. Youssef, 1970, in: Proc. 1st Int. Conf. Ferrites, Tokyo, p. 60. Sawatzky, G.A., E van der Woude and

Rubio Gonzalez and M.A. Villa Garcia,

[18]

1988, J. Solid State Chem. 77, 275.

[12] [13] [14] [15] [16] [17]

A.H. Morrish, 1969, Phys. Rev. 187, 747. Kawano, S. and N. Yamamoto, 1974, Jpn. J. Appl. Phys. 13, 1891. Roiter, B.D. and A.E. Paladino, 1962, J. Am. Ceram. Soc. 45, 127. Mexmain, J., 1971, Ann. Chim. (Paris) 6, 297. (2ervinka, L. and Z. Sim]a, 1970, Czech. J. Phys. 20, 470. Mexmain, J., 1969, Pron. Chim. (Paris) 4, 429. Levinstein, H.J., M. Robbins and C. Capio, 1972, Mater. Res. Bull. 7, 27. Derbyshire, W.D. and H.J. Yearian, 1958, Phys. Rev. 112, 1603.

310 [19]

[20] [21]

[22]

[23]

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[31] [32]

[33] [34] [35] [36] [37]

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V.A.M. BRABERS Robbins, M., G.K. Wertheim, R.C. Sherwood and D.N.E. Buchanan, 1971, J. Phys. Chem. Solids 32, 717. Oles, A., 1966, Acta Phys. Pol. 30, 125. Durif-Verambon, A., E.E Bertaut and R. Pauthenet, 1956, Ann. Chim. 13, Serie 1, 34. Tomas, A., P. Laruelle, J.L. Dormann and M. Noguess, 1983, Acta Crystallogr. C39, 1615. Dubrinova, I.N., R.G. Zacharov, A.A. Shchepetkin and V.E Balikierev, 1981, lzv. Akad. Nauk SSSR, Neorg. Mater. 17, 699. Brabers, V.A.M. and J. Klerk, 1977, J. Phys. (Paris) 38, C1-207. Brabers, V.A.M., 1969, Phys. Status Solidi 33, 563. Abe, M., M. Kawachi and S. Nomura, 1972, J. Phys. Soc. Jpn 33, 1296. Gorter, E.W., 1954, Philips Res. Rep. 9, 295. Blasse, G., 1964, Philips Res. Rep. 19, Suppl. 3. Riedel, E. and A. Hiimeyra, 1983, Z. Naturforsch. 38b, 1630. Rogers, D.B., R.Y. Arnott, A. Wold and J.B. Goodenough, 1963, J. Phys. Chem. Solids 24, 347. Riedel, E., J. K~hler and N. Pfeil, 1989, Z. Naturforsch. 44b, 1427. Reuter, B., E. Riedel, E Hug, D. Arndt, U. Geisler and J. Behnke, 1969, Z. Anorg. Allg. Chem. 369, 306. K6nig, U. and G. Chol, 1968, J. Appl. Crystallogr. 1, 124. Smit, J. and H. Wijn, 1959, Ferrites (Philips Technical Library, Eindhoven). Samara, G.A. and A.A. Giardini, 1969, Phys. Rev. 186, 577. Flanders, P.J.E, 1971, J. Appl. Phys. 42, 1653. Pauthenet, R., 1983, in: High Field Magnetism, ed. M. Date (North-Holland, Amsterdam) p. 77. Bickford, L.R., J.M. Brownlow and J.M. Penoyer, 1957, Proc. IEE 104B, Suppl. 5, 238. Brabers, V.A.M. and J.H. Hendriks, 1982, J. Magn. Magn. Mater. 26, 300. Brown, W.E and C.E. Johnson, 1962, J. Appl. Phys. 33, 2752.

[41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65]

Bate, G., 1980, in: Ferromagnetic Materials, Vol. 2, ed. E.E Wohlfarth (North-Holland, New York) p. 381. Kaneko, M., 1980, IEEE Trans. Magn. 16, 1319. Babkin, E.V., K.E Koval and V.G. Pynko, 1984, Thin Solid Films 117, 217. Zhou, ZJ., Y. Mei, L. Chen and H.L. Luo, 1986, IEEE Trans. Magn. 22, 1341. Pickart, S.J. and A.C. Tumock, 1959, J. Phys. Chem. Solids 10, 242. Mereeron, Th., M. Porte, V.A.M. Brabers and R. Krishman, 1986, J. Magn. Magn. Mater. 54-57, 909. Tellefsen, M., L. Carreiro, R. Kershaw, K. Dwight and A. World, 1984, J. Phys. Chem. Solids 88, 754. Pauthenet, R., 1950, C. R. Acad. Sci. Paris 230, 1842. Takahashi, M. and M. Fine, 1972, J. Appl. Phys. 43, 4205. Pines, B. and N. Gumen, 1963, Fiz. Tverd. Tela 5, 3486. Bozorth, R.M., E.E Tilden and A.J. Williams, 1955, Phys. Rev. 99, 1788. Perthel, R. and W. Keilig, 1963, Monatber. Deut. Akad. Wissenschaften (Berlin) 5, 109. Blasse, G., 1963, Philips Res. Rep. 18, 383. Miyahara, S. and Y. Kino, 1965, Jpn. J. Appl. Phys. 4, 310. Zinovik, M.A. and A.-G. Davidovich, 1981, Zh. Neorg. Khim. 26, 1586. Onyszkiewicz, I., N.T. Malafaev, A.A. Murakhovskii and J.J. Pietrzak, 1982, Phys. Status Solidi A: 73, I(243. Nagata, H., T. Miyadai and S. Miyahara, 1972, IEEE Trans. Magn. MAG-8,451. Rechenberger, H. and K. Stierstadt, 1964, Z. Angew. Phys. 17, 242. Pauthenet R. and L. Bochirol, 1951, J. Phys. Radium 12, 249. Arai, K.I. and N. Tsuya, 1974, Phys. Status Solidi B: 66, 547. Kose, J. and S. lida, 1984, J. Appl. Phys. 55, 2321. Belov, K.E, A.N. Goryaga, V.N. Sheremetev and O.A. Naumova, 1985, J.E.T.P. Lett. 24, 118. Lensen, M., 1959, Ann. Chim. (Paris) 2, 81. Brabers, V.A.M., T. Merceron and M. Porte, 1988, J. Phys. (Paris) 49, C8-929. Varret, E and P. lmbert, 1974, J. Phys. Chem. Solids 35, 215.

PROGRESS IN SPINEL FERRITE RESEARCH [66] Folen, V.E, 1960, J. Appl. Phys. 31, 62. [67] Arai, K.I. and N. Tsuya, 1972, J. Phys. Soc. Jpn 33, 1581. [68] Gorter, E.W., 1950, Nature 165, 798. [69] Kriessman, C.J. and S.E. Harrison, 1956, Phys. Rev. 103, 857. [70] Belson, H.S. and C.J. Kriessman, 1959, J. Appl. Phys. 30, 170. [71] Arai, ICI. and N. Tsuya, 1973, J. Phys. Chem. Solids 34, 431. [72] Brabers, V.A.M., Th. Merceron, M. Porte and R. Krishnan, 1980, J. Magn. Magn. Mater. 15-18, 545. [73] Hastings, J.H. and L.M. Corliss, 1956, Phys. Rev. 104, 328. [74] Guilland, G., 1951, J. Phys. Radium 12, 239. [75] Brabers, V.A.M., 1992, Phys. Rev. Lett. 68, 3133. [76] Penoyer, R.E and M.W. Sharer, 1959, J. Appl. Phys. 30, 315 S.

311

[77] Brabers, V.A.M., J. Klerk and Z. Simga, 1977, Physica B86-88, 1461. [78] Boucher,B., R. Buhl and M. Perrin, 1969, J. AppI. Phys. 40, 1126. [79] Buhl, R., 1969, J. Phys. Chem. Solids 30, 805. [80] Abe, M., M. Kawachi and S. Nomura, 1971, J. Phys. Soc. Jpn 31,940. [81] Krishnan, R. and M. Rivoire, 1971, Phys. Status Solidi A: 7, I 2 A/m, with a maximum error of 3% (Soinski 1987 0. This accuracy is comparable to the precision of commercial devices produced for measuring coercive force greater than 40 A/m. The sample thickness has an obvious influence on the value of coercive force shown in table 7 but halving the length of the standard Epstein steel strips brings about an increase of coercive force of 6%. The influence of width deviation for samples of standard Epstein length, however, can be ignored.

ANISOTROPY IN IRON-BASED SOFT MAGNETIC MATERIALS

349

4.5. Anisometric methods As has already been mentioned in section 4.2, the Epstein test cannot be applied to measuring anisotropic properties of short electrical sheets. This limitation is a problem for producers of small electrical devices which use electrical strip shorter than 280 mm. Epstein testing is also troublesome when the homogeneity needs to be checked along the length of the strip (Wiglasz et al. 1984, Soinski 1984b, Tobisch et al. 1985, Pastor et al. 1986). Figure 13 illustrates the principle of anisotropy measurement. A sample is put into an induction anisometer in a measurement position and turned by an angle relative to the external magnetic field. The 25 m m diameter disc sample is placed in a strong d.c. magnetic field of 14.5 x 104 A / m or a 50 I-Iz a.c. field of 8 x 104 A/m. The measuring coil indicated by C in fig. 13a has an axis which is vertical to the direction of the applied magnetic field H0. Such a coil position allows changes of the vertical component of the magnetic polarisation vector J.L to be directly measured a)

lz El.

----:

/,¢ I

l+,l

/

t

,"," i ,," rolling diroctlon

b)

Fig. 13. An induction anisometer with a sample, a) Sample position after a rotation ~p relative to the external magnetic field, b) an example of a magnetic vector distribution in the sample plane after rotation ¢; J, J.L, JII the magnetic polarisation vector and its vertical and horizontal components, respectively; Ho, Ha, H are field vectors, magnetizing, demagnetizing and total, respectively; EL being poles of the electromagnet; C' being the measuring coil with X axis perpendicularto magnetic field H0; S is a sample; X, Z are measuring coil axis and sample rotation axis, respectively; Y is a direction of magnetizing field; a, ~p are angles between rolling direction and the direction of the magnetic polarisation vector J, and between rolling direction and the direction of the magnetizing field direction H0, respectively; ~b is a deviation angle of the total magnetic field vector H from the magnetic polarisation vector d.

350

M. SOINSKIand A.J. MOSES

when the sample is rotated. This component is proportional to the torque directly measured in more complicated types of anisometers. In the case of electrical steel sheets or amorphous ribbons the vector of the total magnetic polarisation J lies in the material plane due to the thickness. For this reason it is sufficient to know only the dependence of the anisotropy energy on a given angle, c~' - see eq. (1) and angle v (eq. (2)), between the direction of the magnetic polarisation vector and the reference direction. Application of an external magnetic field brings about a change of the position of the magnetic polarisation vector from the minimum energy position. A given value of c~ or v, say c~' is established between the direction of magnetic polarisation J and the reference direction, e.g., the rolling or the casting direction. Equations (1) and (2) take the following form for a unit volume E = Kf(c~'),

J / m 3,

(8)

where K is the anisotropy constant for the magnetic material, f ( W ) is a function determining the position of the magnetic polarisation vector J relative to the rolling or casting directions. When the demagnetization is negligible the sample is under the influence of the external magnetic field H0, causing an energy Em given by Era = ( - H o ) J cos(~ - a'),

J / m 3.

(9)

As can be seen from fig. 13b, if the vertical component J± = 0, then the direction of the vector J is the same as the direction of the magnetizing field Ho. When the direction of the sample magnetic polarisation vector d differs from the direction of the magnetic field H0, the sample is under the influence of the torque (J± x H0). The total free energy in the sample is equal to the sum of E and Era. By minimisation of eq. (8) and eq. (9) we obtain dEk/d~' = -dEm/dcx' and this leads to an expression J.LHo - - K df(t~')

d(a') '

Nm,

(10)

where J± = J sin(~ - c~'). When Ho is known it is easy to use Fourier analysis of eq. (10) to calculate the anisotropy constant measuring directly the value of the vertical component J± (Soinski 19870. The value of the torque as well as the value of the vertical component of magnetic polarisation J± affecting the sample is strongly influenced by the anisotropic properties of the material. Generally the angle ~ in fig. 13 (b) is known but the angle c~ not. When the resulting magnetic field intensity H is sufficiently high (see fig. 13b) o~ approaches ~ and it is possible to assume that o~ ~ ~. Knowing the dependence of the component J± on the angle ~, i.e. J±(~), it is possible to determine the values of the magnetocrystalline anisotropy constant KI and the uniaxial anisotropy constant Kj, in the case of polycrystalline and amorphous materials, respectively.

ANISOTROPY IN IRON-BASED SOFT MAGNETIC MATERIALS

351

Applying a homogeneous magnetic field H0 to the sample leads to demagnetization of the material (see section 4.3). The influence of demagnetization for disc samples with various thicknesses is discussed in (Soinski 1985b). It should be noted that the position of the magnetic polarisation vector J inside a sample may be changed due to the demagnetization of the sample (the vectors J and demagnetizing field Hd in fig. 13b are parallel to each other). When the magnetic field H0 applied is too low, the calculation of the results becomes more difficult. The influence of the demagnetization decreases with an increase of the magnetizing field and with a decrease of the sample thickness. However, using the summation method mentioned earlier it was stated in (Soinski 1985b) that the deviation angle ~b (fig. 13b) between vectors H and H0 in an anisometer powered by a magnetic field H0 >/8 x 104 A/m amounts to less than 0.1 ° and is negligible even for samples 0.7 mm thick. The error of the anisometric test proposed here, amounts, however to 5%, as was shown on the basis of the metrological analysis in (Soinski 1984b). On the other hand, statistical analysis, presented in (Soinski 1982), confmns very good consistency of results obtained in an induction anisometer and in the standard Epstein frame with samples cut at various angles to the rolling direction. Measurements of the anisotropy of non-oriented sheets or amorphous ribbons made using the anisometric and the Epstein method are shown in section 5. The nonoriented materials have smaller amplitudes J.L = f(~) than those of grain oriented steel sheets (figs 38 and 30, respectively). However, in silicon iron sheets possessing weak Goss texture the direction along which the magnetic properties are poorest ranges between 500 and 700 to the rolling direction, while in highly oriented samples it is 55 °, as can be seen from figs 38 and 30a, respectively. The rolling direction in all samples with Goss texture remains the easiest for magnetisation. As has already been mentioned, plastic and thermomagnetic treatment may bring about preferred magnetization directions in the material. Because of this, most silicon sheets contain some Goss orientation despite various technological attempts to remove it. The degree of texture determined by X-ray diffraction, amounts up to 10% relative to the orientation in single crystal as can be seen from table 2 (Soinski 1987e, Lyadkovsky et al. 1986). Publications (Soinski 1982, 1983) show four basic types of mixed orientation met in isotropic silicon-free sheets (cf. data on fig. 35 in section 5.4). The best and the worst magnetization for texture type (1) are ~ = 0 ° and 90 °, respectively. For texture type (2), ~ = 30 o and 900 when intermediate magnetization is present at ~ = 0 °. For type (3), the best magnetization direction is along ~ = 0 ° and the worst between ~o = 800 and 90 °. Finally type (4) occurs when the best and worst magnetization is at ~ = 0 ° to 10° and ~ = 90 °, respectively. In iron-based amorphous materials (fig. 43), the easiest magnetization occurs along the casting direction, while magnetization is most difficult along the transverse direction. The anisotropic properties of such materials are significantly lower than those of grain oriented sheets (Chambron et al. 1982). The 6.5% silicon iron alloys of microcrystaUine tapes possess relatively high magnetocrystalline constants but the effect of anisotropy is diminished by grain misorientation (Tanaka 1992).

M. SOINSKI and A.J. MOSES

352

TABLE 2 Degree of sheet orientation and directional changes of coercive force. Sheet srade Parameter

Anisotropic sheets

Isotropic silicon sheets

M4

M5

M6

M7

M9

M43

M45

95

87

85

65

45

10

10

Degree of texture (%) (anisometric method)

95

85

79

59

47

15

12

11

Coercive force Hex in A/m X=0* X=55' X=90*

7.0 39 51

8.5 36 41

9.5 35 43

10.5 32 36

12 29 35

40 50 56

52 63 66

75 77 84

84

82

79

53

35

Degree of texture (%) (X-ray diffraction

M47

method)

K1/KM (%) changes of anisotropy constant K1 of sheet in relation to constant g M for a single crystal of 3.85% Si-Fe (anisometric method)

3.7

1.7

1.2

n: Si chemical composition (% wt)

3.4

3.0

2.8

3.2

3.1

1.50

1.55

1.60

Changes (K1/KM) n (%)

0.55

0.55

0.52

0.13

0.04

0.007

0.002

0.001

It is worth noting that anisotropy energy can also be determined from the magnetization curve by calculating the difference between energy needed to magnetise a sample to saturation state along X and Y axes, thus,

E = { ffo'r"[H(X)- H(Y)]dJ} V,

J.

(11)

The energy stored in a silicon-iron crystal when it is saturated in the X = [100] direction is proportional to the area between the ordinate axis and the magnetization curve for this direction. Calling this energy Eloo, the directional cosines in eq. (1) are t~l = 1, 0~2 = e~3 = 0 and the isotropic energy contribution Ko = Eloo. Finding the energy E n o along the crystal direction [110] and the energy/3111 in the crystal direction [111] in a similar way the following values for K1 and K2 are obtained (Brailsford 1954): K 1 = 4 ( E l 1 0 - E 1 0 o ) and = 27(El11-E10o)-36(El10-E10o). This confirmes the minor importance of the second anisotropy constant K2, as can be seen in fig. 14. This method of calculating anisotropy energy is not frequently used in practice because of difficulty in establishing full saturation of the sample (see fig. 14).

Kz

ANISOTROPY IN IRON-BASED SOFT MAGNETIC MATERIALS

353

[1oo]

2.1 2.0

1.5

io] i.O H

0

20 000

*0 dO0

gO ~00

~/m

Fig. 14. Magnetisation curves for three principal crystallographic directions in an iron single crystal (after Bozorth 1951).

The use of wound samples is, unfortunately, not recommended. The reason for this is that because of the narrow width of the strip it is difficult to prepare a suitable test sample apart from with the strip parallel to the rolling direction (IEC Publication 1978). Also punched rings (IEC Publication 1982a) are not reliable to give any directional data as the magnetic flux has to flow along the full circumference of the sample hence only average of magnetic properties can be revealed.

4.6. Non-magnetic anisotropicproperties The specific resistivity p(T) of electrical sheet samples of metals and alloys at a given temperature T can be calculated from

s(T)

p(T) : R(T) I(T)'

g~m,

(12)

where R(T) is the electrical resistivity, and l(T) and s(T) are the sample length and cross-sectional area, respectively. The influence of sample length on the precision of the resistivity calculation in the sheets changes at different temperatures from T -- 4.2 to 293 K. It is less important than cross-sectional area changes (Soinski 1984a). This is the reason why a system of correctly measured voltage allows the change in resistivity along a direction relative

354

M. SOINSKI and A.J. MOSES

7

mI-I

I-,I Iw

~l-tlmrlm

liquid He~ or Nz current and voltaeje circuit

--,~ ~-I[i ii

~ ~"

,

~;

tube to )our In liquid He2 or N= samples cut under "'different angles heater.

Fig. 15. 1"he detail of a unit for measuring directional resistivity changes in electrical steel sheets.

to a reference direction of thin samples to be determined (Soinski 1985a). It has been also assumed that l(T) and s(T) in eq. (12) are considered as constant. An experimental system shown in fig. 15 can measure the resistance of several dozen samples 80 mm x 10 mm x 0.3-0.5 mm in size. The samples are narrow and short enough to obtain a homogeneous temperature. A drive current of 1 Amp energises the sample and the sample voltage is measured. Ni-Cu thermocouples are placed in the middle of the sample packet. The resistivity error p(T) calculated from the results is less than 4% (Soinski 1985a), whereas more complicated bridge circuits (IEC Publication 1978) lead to an error of 3.5%. The main causes of these errors are deviations in calculations of the geometric dimensions rather than the precision of the resistance measurement. When the resistivity measurements are taken at high temperatures a compensation method has to be applied (Soinski 1985c). To measure the directional variation of the mechanical reverse bending, rectangular samples 150 mm x 30 mm x 0.3-0.5 mm in size are used (see fig. 16a). The test specimen is initially bent through 900 before starting the test which consists of bending the specimen through 1800 to each side alternatively and counting the number of bends made before the first cracks in the base metal are visible (British Standard BS 1973). Static tensile testing of samples according to ASTM standards (ASTM Standard 1985) can be carried out in the set up shown in fig. 16b. The tensile testing machine consists of two essential parts, the strain device and the mechanism for applying for the load to the specimen. The specimen is a strip cut at an angle to the rolling direction.

ANISOTROPY IN IRON-BASED SOFT MAGNETIC MATERIALS

355

a) J

r

~

0,I ram, clearance each.:,lde

Jockey weight. Fulcrum

b)

r~

J

Scale Specimen

t Tensile machine Fig. 16. Tests for measurement of mechanical anisotropy, a) Reverse bending test; b) standard sample for static tensile testing.

356

M. SOINSKI and A.J. MOSES

Reverse bending and static tensile tests, allow anisotropic properties of electrical sheets to be determined. For this purpose a number of specimens must be cut at various angles to the rolling direction. If the width of the material is not sufficiently great for cutting transverse specimens the non-magnetic test strips are taken in the direction of rolling. Again this must be carefully specified (IEC Publication 1991).

4. 7. Polar figures and angular dependencies As has already been suggested, changes in directional properties of magnetically soft materials can be measured by means of anisometric systems or other methods, such as the Epstein square or single sheet tester. To evaluate the sample anisotropy it is common to use the concept of degree of sample texture, which is a ratio of oriented volume to the total volume of the sample. By calculating the anisotropy constants or the degree of sample texture only the quantitative characterisation of the material can be assessed. However the directional changes in magnetic parameters are often presented in the form of polar figures. These are more useful for the users than some texture coefficient (table 2) or anisotropy constants (fig. 27) which are more of academic interest. Polar figures are drawn to represent the qualitative anisotropic properties of magnetic materials. They show the change of a particular property with magnetization direction in a graphical way. Figure 17 shows a typical polar fig. of induction, magnetic field intensity and core loss in different types of magnetic sheets. a)

L

b)

,

rolling

~ .

\~ 19o°" /r

0,4 0,8 lJ 1.4 2,0 "-T

c)

ll~'~~-.;I.

I~,o~

0 400 I100lzoolSO0000 ~m

rolling I bdirection 0°

150

~ o

°

75 o ,

o lo~3~

4,o5~e.o W/kg

Fig. 17. Polar figures showing relationship between the magnetization direction and: a) induction, b) magnetic field intensity, c) core loss. Curves 1, 2 and 3 are for grain oriented silicon steel (grade M4), non-oriented silicon steel (grade M45) and for non-oriented silicon-free, respectively.

ANISOTROPY IN IRON-BASED SOFT MAGNETIC MATERIALS

357

The variation shown graphically in fig. 17, are important in the design of various electrical devices but are labour consuming to obtain. The polar figures of another magnetic parameter, namely apparent core loss, are also sometimes used, for instance, when designing measurement or shunt transformers (see fig. 18). These figures are rather easy to obtain from only a few samples. Polar figures are often given by the producers of anisotropic sheets in the form shown in fig. 19. These figures show also the great importance of anisotropy for highly oriented or conventionally oriented electrical steel sheets. On the other hand, as was mentioned in section 4.2, in the case of the isotropic sheets Standards allow measurements to be made on mixed samples cut parallel and perpendicular to the rolling direction. This can give rise to misleading assessment of true material properties. Further research, for example (Napoli et al. 1983), has led to a few more suggestions for polar figure formats of greater use in developing theoretical models of the influence of anisotropy on the properties of magnetic cores. It is very easy to establish from fig. 20 the commonly used maximum value of induction anisotropy ^u0-60 ~ 2 5 0 0 " This is the value of the difference between the magnetic flux density produced along the rolling direction and that produced at 600 to the rolling direction at the same field strength of 2500 A/m. The results of the study given in this chapter suggest that the properties along the rolling or casing directions should be determined using the standard Epstein frame, single sheet tester or in a wound/ring sample test. The magnetically difficult directions and the degree of orientation should be determined by means of anisometric methods. On the basis of relationships, to be described in section 5, the directional changes of parameters such as magnetic induction, magnetic field strength, apparent

a) lv 2001 150 100 50

/

20 15 10

I

d/

Y

5.0

kg

~V

¢-/

v" 'V io /~ 'v0

20O 150 100 50-

J • v'lO d

J,

~vSS.r

2O 15

¢Vo

5~0

,o

A']

S

2.0

2.0 1.5 1.0

., ¢1

i I I

O.5 0.2'

I~ VA,

,

B

L

O.2

O.6

1.0

1.4 1.7 T

o.2

0.2

0.6

1.0

I

B

1.4 1.7 T

Fig. 18. Angular dependencies of apparent core loss for three directions of magnetic flux: a) grainoriented (anisotropic) silicon steel sheet (grade M4), b) non-oriented (isotropic) silicon sheet (grade 45).

358

M. SOINSKI and A.J. MOSES

a}

b)

Le

"R~g"k P

Hr.

.

Jooo.~l,.,, l$OOAIm ~ 000 Aim

/ --90

-60

-30

/ A/ /~' ~ "

3,o

k

,///,/

30

-,.o,

)/josT

\

0

'

14T

A/m

500

//o,

B, 1.5T

~

90

60

-90

-60

-30

0

30

60

Fig. 19. Polar figures of grain-oriented (anisotropic) silicon steel sheet (grade M4) drawn relative to the magnetization direction ~: a) induction, b) core loss, at 50 Hz.

a)

T~8

b) A/m~H

L'

H:2500

1,9 1,8 - 0 1,7 B2e°° E

~,8 ~

e

/-

/~_

.

,~ ..'.

. k

tein t e s t

I_-,o . o

1,5

- - - ~

1,4

27.5 °

c) v~g, v .:t5 300 150 ' loo I/ 20

/~o Epstein ,~1 O01 t,5 - t e s t I

55 e

~~'

~

0

50 °

10 °

P

I

30 °

50 °

70 °

90 °

B=1.5 i

,...o:

! ~ I

~

30 °

90*

726 °

d) wy~g

10 tein ~'5,0 /~"lis" teLst " T 2,0 10*

)°° I

4 Joo

B:1,5

"=

> 70"

!~

90 ° 0

~pstein test 100

30 °

50"

700

90*

Fig. 20. Dependence of parameters of non-oriented (isotropic) silicon sheets M45 on magnetizing angle ~#: a) induction, b) magnetic field strength, c) apparent core loss at 50 Hz, d) core loss at 50 Hz.

ANISOTROPY IN IRON-BASED SOFT MAGNETIC MATERIALS

359

core loss, core loss, coercive force, magnetostriction, sample demagnetization, resistivity and some mechanical properties are easy to estimate in this way. Figure 20 shows four such basic angular dependencies of silicon iron sheet with weak Goss texture. Angular variation of properties for other types of electrical material are described in the next section and are summarised in section 7. The way in which polar figures such as fig. 20 are obtained is by plotting sample properties for angle ~o = 0', i.e. along the rolling direction (see for instance curve /3000 in fig. 20a which is taken from the Epstein test). This figure shows the way of constructing the polar plot of magnetic induction. The characteristic points of any angle changes for ~o = 27.5 °, 550,72.5 ° and 90 °, i.e. anisotropy of induction in this case are determined by the data established via an anisometric method. The corelation between the anisometric method and Epstein test is used here, as shown in table 5. This means that the results from the anisometer are confirmed when samples were cut at different angles (cf. table 4). The whole curve B = f(~o) is obtainable by a graphic or computer approximation of the data for the five angles ~o and it represents any required property relative to the changes of magnetizing direction.

5. Main properties of electrical sheets, amorphous ribbons and microcrystalline tapes 5.1. Anisotropic and isotropic silicon iron electrical sheets It is customary to call sheets which have more than 0.3% weight of silicon, silicon iron or silicon steel sheets. There are high silicon sheets and low silicon sheets with 1.5% silicon being a border line. The amount of silicon closely determines the value of core loss, which in turn, is the basis for the classification of electrical steel sheets. Core loss and the penetration depth of the magnetic field under a.c. magnetisation is essentially dependent on the sheet resistivity. This, in turn, is dependent on the amount of silicon or other alloy additives. The chemical composition also determines other physical properties of a sheet (Wohlfarth 1980), while the crystal structure determines the anisotropic sheet properties. Alloy and metal electrical resistivities can be represented at a given temperature T as the sum of the following components:

p(T) = pi(T) + pr(T) "1- pro(T, H) + pg,

am,

(13)

where: pi - ideal resistivity, Pr - residual resistivity, P m - magnetic resistivity, pgdimension-based resistivity. The value of pi is determined by the material resistivity of an ideally periodical crystallographic lattice and results from thermal vibration of the lattice. Residual resistivity Pr, is brought about by additives, deformations and impurities found in the lattice. In the case of electrical steel sheets magnetic resistivity Pm does not play a major role. Geometric dimensions of samples used for resistivity measurements are large enough to make pg negligible (Soinski 1984a, 1985a).

360

M. SOINSKI and A J . MOSES

Electrical conductivity 7 of metals and alloys can be calculated from: "r =

-x,

(14)

where: n~ - number of electrons per unit volume (n~ does not depend on temperature), e - electron charge, I - average free path, m - electron mass, u~ - average electron velocity in an electric field. The average electrons velocity ue depends on the temperature, on the direction of the applied electric field and on the crystalline structure. The presence of a strong texture changes the isotropic character of resistivity bringing about its anisotropy. At low temperatures, however, we should expect a 30-50% increase of the magnetic anisotropy constant K1 in electrical sheets (Bozorth 1951) and this fact should be taken into consideration when designing any cryomagnetic cores (Soinsld 1984a). The anisotropy of resistivity in electrical sheets is brought about additionally by the chemical non-homogeneity of the alloy, non-homogeneous distribution of alloy components and additives, a changing number of grain boundaries for different crystallographic directions and a change of the position of grain boundaries with time (Soinski 1984a, 1985a, Coulomb 1972, McGuire et al. 1975, Miedema et al. 1975, Smith 1951). The problem of anisotropy of resistivity seems to have a significant influence on the development of electrical cryogenic devices (Soinsld 1984a, Inui et al. 1977). Figure 21 shows the temperature dependence of electrical resistivity measured in samples cut at different directions to the rolling direction. It can be seen in fig. 21a that the resistivity along the 900 direction lies between the 0 ° and 55 o values and all three curves are identical for non-oriented steel sheet grade M43. This enables a quantitative analysis of different types of electrical sheets to be made (Soinski 1985a). The grade M9 oriented steel sheets and grade M43 non-oriented silicon steel sheet possess significantly less degree of texture than the grain oriented steel sheet grade M4 (see table 2). The resistivity values p0/> pgO t> p55 in fig. 21 can be explained by a change of the free path along those three directions (Soinski 1985a). It should be emphasized, however, that the additives and impurities found in electrical sheets have a tendency to align along to rolling direction, which additionally leads to an increase of p0 relative to pg0 and p55. As can be seen, the anisotropy of resistivity is determined by the degree of sheet orientation. In spite of this, the data shown in fig. 21 allows us, by graphical extrapolation to 0 K, to estimate the values of residual resistivity. This gives Pr as approximately 0.4 #ff2m for anisotropic sheets with around 3% silicon content and 0.25 #ff2m for isotropic material with around 1.5% silicon content. In the case of isotropic silicon-free steel sheets to be decribed later, Pr is approximately 0.03 #g2m. The dependence of the anisotropy of resistivity on the anisotropy of induction, core loss and apparent core loss, is presented in fig. 22. The relationship shown in this figure was confirmed by statistical computation (Soinski 1985a). The anisotropy of mechanical properties of electrical sheets can be generally calculated from: AM x - ° = M x - M °

(15)

ANISOTROPY IN IRON-BASED SOFT MAGNETIC MATERIALS

°m5I, 0.4 _

0.2

0

361

., ~

........~ ~

50

100

I 200 250 300 "K

150

b) ~ m l d e'

M 43

0.5

i

I

0.4 0.3 0.20

50

100

150 200 250

T 300 *K

Fig. 21. The influence of temperature on electrical resistivity of electrical steel sheets measured along directions at various angles to the rolling direction: a) grain oriented steel sheet (grade M4); b) nonoriented silicon steel sheet (grade M43); p0, p55, pgo _ resistivities at the angles of 0', 55* and 90* to the rolling direction.

3O I~ 25 20

"

...'

15

/

/

I

...'•"

1.o

/I

¢" 5.0

.~."

(

/

/

x --,--

M 4

I

~ / = - -

.~

~ _ , ~ .~.'~, ~

Me

.9 M43

...... i'66"'~//i~'"'~i~i~"~'6- i~/b"k.~-~ Fig. 22. Dependence of the anisotropy of resistivity Ap on the anisotropy of induction AB, on anisotropy of core loss AP (at 50 Hz) and on the anisotropy of apparent core loss AV (at 50 Hz).

362

M. SOINSKI and A.J. MOSES

AZ,A'~% • ~:.~ ~2o A 90 A

......... /o O~

418

Zl/

114

&.,.o.'"

• .......

/%0

12 /

•

I0

Az=28%

~A55

.1% i

,

M47~45,43

4T

59

M9

M7

i

r985 9s

2 0

K%

elect~cal M8 M5 M4 sheet

Fig. 23. Percentage changes of relative elongation AA and relative reduction Az, with the degree of sheet texture K; K % - degree of sample texture obtained by X-ray diffraction method as the ratio of oriented volume to the total volume of the sample, in percent.

or from AM =

(AMX-°/Mx)

%100,

(16)

where M x, M ° are values of a given mechanical parameter obtained by a reverse bending test and static tensile test when investigated along directions X = 55 o or X = 90 °, relative to the rolling direction. Reverse bend testing carried out for samples cut along various directions provides data from which the anisotropy of the tested sheets can be determined. The following average numbers of bends lr were obtained in a group of grain oriented silicon iron grades M4, 5, 6, 7 and 9. Along the rolling direction, l ° = 11, lr55 = 10 for ~ = 550 and lr9° = 8 for ~ = 900 . The average data for grades M43, 45 and 47 of non-oriented steel sheets with about 1.5-1.6% Si wt. were as follows, l ° = 21 for ~ = 00; l~5 = 18 for ~ = 550 and l ~ = 14 for ~ = 90 °. In general it can be seen that the number of bends counted for high silicon anisotropic sheet is lower than for non-oriented silicon steel sheets. The minimum number of bends was found for ~ = 900 in both types of electrical sheets. The relationship: l° > lr55 > Ir9° is different to the directional relationship of induction and core loss but the same as that of coercive force and magnetostriction. Figure 23 shows the influence of sheet orientation on elongation A and reduction Z obtained by static tensile test. It can be seem from fig. 23, that elongation AA and reduction Az, both calculated according to eq. (16), diminish with a decrease of sheet orientation (Soinski 1990). Data for grain-oriented sheets and silicon non-oriented sheets calculated for samples 0.3 and 0.5 mm thick, respectively, are also shown in fig. 23. The data are in general agreement with (Rawlings 1970), where elongations

ANISOTROPY

IN IRON-BASED

SOFT MAGNETIC

MATERIALS

363

14%, 8% and 26% were stated for crystallographic directions [001], [111] and [110], respectively, in 3% silicon iron sheet. Values of anisotropy (eq. 16) of conventional grades yield strength Ro.2 obtained for different types of materials are A55 " 20% for M4 grade grain oriented steel R0.2 - sheets and A55 e~ 15% for non-oriented M45 grade silicon steel sheets (Soinski R0.2 - 1990), (the ratio calculated from eq. (16) describes here the anisotropy between 550 and the rolling direction in the yield point stress from static tensile tests). These are relative data concerning stresses at the beginning of inelastic behaviour of the specimens, i.e. at the lyield point which results in a specific elongation of A1/l = 0.2. Specimens were cut at ~o = 55 ° and ~ = 0* to enable percentage anisotropy values to be evaluated. The changes of anisotropy of tensile strength ARm calculated as above amount to maximum values of A55 "~ 25% for M4 steel sheet and 10% for Rm M45, respectively. However, for other sheets with orientations between electrical steels with grades from M4 to M45, the values are nearly proportional to the degree of sample texture, defined earlier in fig. 23. More data on anisotropy of mechanical properties of electrical steel sheets are available, for example in references (Soinski 1990, Rawlings 1970). In order to establish the true magnetic properties of ferromagnetics, in open magnetic circuits, it is necessary to know the demagnetization factor N. The value of this coefficient is calculated from eq. (6) and additionally from magnetic permeability given in (Soinski 1984c, 1985'o). The magnetic field distribution in samples cut at different angles and magnetized in an open magnetic circuit is shown in reference (Soinski 1984c). As can be seen from fig. 24 the sample ends indicated with j = 1 and j = 28 where the demagnetization coefficient has a higher value are demagnetized most strongly. This is so because magnetic poles are formed there. The middle part of the sample, with j = 14 and 15, is the hardest to demagnetize. Changes of the anisotropy of demagnetization coefficient along the directions 550 and 900 can be calculated from the following equation (Soinski 1984c, 1983"o):

A, x(s) = { tNo0(s)- Nx0(s)l/ o0(s)}

%100,

(17)

where: j refers to numbered 10 mm lengths of the sample, NX0 is the demagnetization coefficient along directions X = 550 and 90 °, NOo is the demagnetization coefficient along the rolling direction. Figure 25 shows changes relative to the rolling direction (eq. (17)) for grain oriented electrical steel samples cut at 550 and 900 . This data given in this simple way should be taken together with the data of characteristics along the rolling direction presented in fig. 24. From the data obtained in fig. 24a and fig. 25 and from eq. (17) it is easy to calculate the value of the demagnetization coefficient of grain oriented sheet along the 550 and 900 directions and at any part of the sample. In the case of non-oriented silicon and silicon-free sheets the curves are the same as in fig. 25, but the values of anisotropy from eq. (17) are significantly lower, as shown in table 3. The table shows

364

M. SOINSKI and A.J. MOSES

a)

b)

N~

10-4 I00

10-4 5O

50

anisotropic

4O

isotropic

40

20

~ = ~ NN~N~

10 8 6

280

~.

20

g~-direction

,,., Ho = (5; 1)×102 A/m " ~ ' ~

Ho = (5; 1)× 1{)2 A/m Ho = 2 5 0 0 ~ _ ~ ~ _ _ ~

4 H

o

2

1

.....

H°-' '°°°~'~'m;>x~ J----

1

3 4 5 6 7 8 9 10 11 12 I3 14 28 27 26 25 24 23 22 21 20 19 18 17 16 15 2

=

5

~

Ho = 10000 A/m/

2 3 4 5 6 7 8 9 10 11 12 13 14 28 27 26 25 24 23 22 21 20 19 18 17 16 15

Fig. 24. Axial distribution of demagnetization coefficient Nl0to (eq. (6)) for samples cut along the rolling direction and magnetized in a homogenous field H0: a) for a strip cut from grain oriented (anisotropic) steel sheets (grade M4), 0.3 mm thick; b) for a strip cut from non-oriented (isotropic) silicon steel sheet (grade M43) and for a strip cut from non-oriented (isotropic) silicon-free sheet, both 0.5 mm thick; j is a number of each 10-ram-long segments along the sample.

a)

b) % J ~AAD55 16 M

Ho=5000

12

Ho =2500 M

/

/

m

~

m

% J k AAD9 o 10

~

//~Ho=IOOOA/m o'=(

--J

i

H° = 5000Mrn

2 -2

/

/

J

-4

'

~ H o - - ' (5;' 1)x'102'~ m

Ho = 1000 Mm -6

-8 -10

i

-12 28 -16

i 28

i

i

i i 5 24

i

i

i

i i I0 19

i

i

i

r

i

5 24

i

i

I

i

i

10 19

i

i

i

i

14 15

i 14 15

Fig. 25. Axial distribution of anisotropy of demagnetization coefficient (eq. (17)) for a sample cut from the grain oriented steel sheet grade M4: a) at 55* to the rolling direction, b) 900 to the rolling direction.

ANISOTROPY IN IRON-BASED SOFT MAGNETIC MATERIALS

365

TABLE 3 Ratios indicatingchanges of the anisotropyof demagnetizationcoefficient(as defined in eq. (17)) of grain oriented electrical silicon steel sheets in relation to non-oriented silicon and silicon free electrical steels for the angles X = 550 and X = 90°.

External magnetizing field H0 in A/m 10000 5000 2 500 1000 500 and 100

Angle of cut to rolling direction AAD55(j) for anisotropic sheet M4 divided by AAD55(j)for isotropic silicon sheet M43 8 9 10 13 9

Angle of cut to rolling direction AAD90(j) for anisotropic sheet M4 divided by AAD90(j)for isotropic silicon or silicon-freesheets 7 8 8 9 4

how many times the value of the anisotropy of demagnetization coefficient obtained from eq. (17) is higher in grain oriented M4 steel sheets than in non-oriented silicon steel sheet grade M43, cut at 55 °, and non-oriented silicon steel sheet grade M43, or non-oriented silicon-free steel sheet cut at 900 to rolling direction. These data depend on magnetizing field Ho as shown in table 3. However the data given for an angle 900 includes non-oriented silicon-free steel sheets of mixed type of texture, whereas electrical steel sheets M4 and M43 have strong and weak Goss textures, respectively. In general, demagnetization anisotropy in isotropic sheets is an order of magnitude smaller than in anisotropic sheets due to their relatively smaller degree of orientation (see table 2). The anisotropy of demagnetization discussed so far, is related to sheets in open magnetic circuits. This has an advantage of allowing more precise determination of the directional changes of some magnetic parameters, such as magnetostrictive elongation when short strips are often used. References (Soinski 1984c, 1985b) discuss research on the influence of shortening or lengthening the strips on the demagnetization coefficient distribution. Magnetostriction shows changes in value not only due to texture but also due to the influence of the external magnetic field and sample demagnetization. In polycrystalline materials magnetized in relatively weak magnetic fields, only so-called linear magnetostricti0n is observed. It consists of dimensional changes whereas the total volume remains unchanged. In strong magnetic fields the magnetostriction phenomenon brings about volume changes of the sample with its shape unchanged. A strain gauge is often glued to a sample to measure magnetostrictive elongation in the central region of a short sheet strip and readings are taken of the relative changes of resistance brought about by changes of the sample geometry. Measurable magnetostrictive elongation is a function of many factors such as state of magnetization, temperature, internal stresses, shape and shape dependent sample demagnetization. The problem of magnetostrictive elongation in Si-Fe single crystals

366

M. S O I N S K I

and A.J. MOSES

and in commercial electrical sheets cut at various angles to the rolling direction is discussed in a number of publications (for example in (Shilling et al. 1974, Zaykowa et al. 1964, Zaykowa et al. 1966, Simmons et al. 1971, Taguchi et al. 1974, Hall 1959, Stanbury 1987, Imamura et al. 1983, Soinski 1989)). The anisotropy of magnetostrictive elongation depends greatly on the relative degree of grain orientation as can be seen from data for three electrical steel sheets with different degree of Goss texture shown in fig. 26. The samples were studied, in a homogeneous external magnetic field H0 ~< 20 kA/m in an open magnetic circuit. The sample intrinsic magnetic field H according to eq. (7) is indicated on the figure. a) ~.xl06 J

12

10 8 6 4 2 0 -2 -4 -6

.......".......................................................... /........................... / .......... : P.551+ I~1 . -..."':'20"" - . . 80 100 ~'"'", , I I"'" ~ I ~ 8 25 "'64.

,.. ~.90o 160

I

a

84

200

I

'

( x l 0 0 ) A / m Ho

I

114

164

-8

"-

(x I00) A/m H

~.55°

b) ~.x 1()6 J

10 8 6 4 2 0 -2 -4 -6

....

.~.~.

..'/"

•................

IX~sl+ 1~- . . . . . SS~'T.........................

~¢C- 40 ""6"0. . . . . . I"00"" -. """~1"

,

I

I

]

I

'

~.9oo

140

200

"'t.

(xl00)A/m Ho

I

(xl00)A/m H

-8

• ~55o

c) ~xl06

•

6

17~551+ 13.ol

.-:....L..-:...:....-:.. ............

4

...........

. .....

2 0 -2 -4 -6

.... I , '

8o I

"'"''1"

-11

08

27

"...........''~.....L.. loo ..... I I

46

65

14o ' "'1-.

~.9o~ .............

104 " .

"

200 I

(xlO0) A/m Ho

142

(×lOO)Mm H

F i g . 26. Directional changes o f magnetostrictive elongation in an external m a g n e t i z i n g field H o : a ) for anisotropic sheet grade M 4 , 0 . 3 m m thick; b) for anisotropic sheet grade M 9 , 0 . 3 m m thick; c) for isotropic silicon sheet grade M 4 3 , 0 . 5 m m thick.

ANISOTROPY IN IRON-BASED SOFT MAGNETIC MATERIALS

367

&H e 90

I

80 7O /

60

Y"

I

50

40

~J

30

2oi

/

I v f

,.I

I

/

(K,: K H )*

y-

\

/

\ 1

2

3

'4

5

6 7 8 9 10

20

30

40

K~]K~ 5O 60 7O 8O9OIO0 %

Fig. 27. Dependence of the anisotropy of coercive force on the ratio of the anisotropy constants (eq. (1)) in various samples.

The maximum magnetostrictive elongation is observed when magnetised at 900 to the rolling direction and not at 550 , as might be expected. The difference between the 0 ° and the 900 elongation is strongly influenced by the degree of sheet orientation (Zaykowa et al. 1964, Hall 1959, Soinski 1989, Narita et al. 1982, Kan et al. 1982). It is found that magnetostrictive elongation falls drastically when the sheet thickness diminishes to below 0.1 mm. This can be explained by annihilation of subdomain structures (Yamaguchi 1984, Mei et al. 1986). There is also a visible relationship between the directional changes of magnetostrictive elongation and the directional changes of coercive force presented in table 2. Coercive force is an important parameter because it has an influence on hysteresis loss which can constitute around 20% of the total loss in anisotropic sheets and around 40% in isotropic sheets. Coercive force is usually measured by the methods described (Soinski 1987e) and its anisotropy is defined in a similar way, to that shown in eq. (15) and eq. (16). The directional changes of coercive force of some samples are given in table 2. From this table it can be seen that the minimum coercive force occurs when the material is magnetized along the rolling direction; the maximum, however, is along the 900 direction and not at 550 , as might be expected. This distribution is analogous to the distribution of magnetostriction. The data in table 2 indicates also that the anisotropy of coercive force increases with the increase of sheet orientation. It is worth noting that typical values of Hc are as follows, 70 A/m for commercial iron, 30 A/m for heat-treated iron, 12 A/m for chemical pure iron, 8 A/m for electrolytic pure iron and 5.6 A/m for a (110) [001] 3.2% Si-Fe single crystal.

368

M. SOINSKI and AJ. MOSES

a)

B°

TJ 1.8

I AB0-55 2500

1.6

/

d)

~

AHt,5

M43 1.7T B A

~ l l i n g / - - / .i: • directton

_ B5~

Bgo B° B55

~

1.6

AHLs

1.4

~

~

M

4

1.4

1.2

r

1.0

I

I

1000 2500 5000

b)

T 1.8

,w t

~

direction

H

1.2 1000 2500

A/m

5000

10000 Aim

M45 Bo B90 B55

0-55 [ /AHLS~...--" ~ ~

1,6 )[1.4

~

10000

~

/

tt

)

e)

T t BA 1.8 1.6

~

B° B90 B55

o-s5

1.2 1.0

c)

1000 2500 5000

t

H ~

10000

A/m

T ¸LB 1.8

t

I

i

1000 2500

5000

0 B9o B° B55

1.6 1.4

1.4 /

~

H

10000 A/m

M 47 T 'tB A 1.8

B° Bgo B55

~

- AHI 5

"~

1.6

1.2

1.0

I

I

I

1000 2500 5000

k 10000

tt

~

Aim

1.4 !

I

I

1000 2500

I 5000

H I ~10000 Aim

Fig. 28. Static magnetizing characteristics of: a), b), c) - grain oriented electrical steel sheets grades M4, M5, and M9; d), e), f) - non-oriented silicon steel sheets grades M43, M45, and M47.

Only a very small concentration of non-metallic additives and impurities must be present in order to produce sheet material with a low coercive force. This is especially important if the sizes of the impurities are close to that of the domain wall thickness. Figure 27 shows changes of the anisotropy from eq. (16) relative to the quotient of the anisotropy constants K1/KM(from eq. (1) in a sheet K1 and in a single crystal Kra). The results given in fig. 27 represent a quantitative influence of Goss texture on the magnitude of the coercive force. The changes are commonly represented in the form of the equation He = ( K i ? (Druzinin 1974). To represent the changes in fig. 27 a formula (K1/KM)"can be also easily used (cf. data in table 2). As has already been mentioned, cold-rolled electrical steels show strong anisotropy in the plane of the sheet (compare figs 17-19). For this reason maximum induction and minimum core loss can be obtained along the rolling direction very easily. Magnetization in other directions is much more difficult.

A N I S O T R O P Y IN I R O N - B A S E D S O F T M A G N E T I C M A T E R I A L S

369

A quantitative analysis of induction anisotropy in an external magnetic field of 2500 A/m can be calculated from the following equations: Y-X

Y

X

AB25oo = B25oo - B25oo, AA(B) =

Y-X Y {AB2.soo/B~oo)

(18)

T,

(19)

%100,

Y where: B2~00 is the induction in the Y direction (for isotropic silicon-free sheets the direction may differ from 00 as in fig. 36), BX00 is the induction in the X direction (where X = 550 or X = 90°). The values of anisotropy from eq. (18) are generally used in classifying directional properties of sheets, as shown in fig. 28. Typical values of anisotropy of induction AB°-0~5 are not higher than: about 0.550.60 T in grain oriented and high permeability sheets and not higher than about 0.13-0.17 T in non-oriented silicon sheets. It should be added, however, that the values of anisotropy of induction ^u0-55 ,,~ ~2500 - - 0.12 T correspond to the presence of about 25% Cross texture in the material (Matheisel 1978). This dependence, however, shows a marked non-linear characteristic with changes of the degree of Goss texture. When designing magnetic circuits it is useful to know the value of the anisotropy of magnetic field AH1.5 of the material because the designers prefer to know the given value of induction rather than the given value of magnetic field. The results of measuring AHx.5 anisotropy are also shown in fig. 28. In recent years the notion of anisotropy of core loss was introduced into norms and catalogues. In isotropic sheets its upper value is usually not higher than 10-14% and it is calculated in a similar way to the ratio expressed in eq. (4) but it is commonly written in catalogues as the formula:

AA(P1.0) = 4-( 1 -

2P°o/[P°o+ P19.°] )

%100,

(20)

where: P°o, P19° - core losses in directions 0 ° and 900 at induction B = 1.0 T. Figure 29 shows core losses measured in grain oriented and non-oriented silicon steel sheets cut at 00, 550 and 900 indicating the very high values of core loss anisotropy in grain oriented electrical steel sheets. The data in the figure enables easy estimation of anisotropy from eq. (20) and calculation of absolute core loss differences between the hard and easy directions of magnetization. The results given in fig. 29 prove that both anisotropic and isotropic sheets show significant anisotropy of core loss (see also (Soinski 1984b, 1982, 1987e, 1983)). This is also confirmed by the anisotropy of apparent core loss shown in fig. 18. The changes of the vertical component of magnetic polarisation J± = f(~), recorded by an induction anisometer, are shown in fig. 30. The curves show amplitude dependence on the degree of sheet orientation. This dependence on anisotropy of magnetic properties (see table 4) can be used in establishing the angular variation shown for example in fig. 20.

M. SOINSKI and A.J. MOSES

370

a)

d~

pO P"

p

~

pS.S

/

p°

6 5 3

M4

2 ~

p

/d'

O

4

pg0

~

I,,P,.,

M 43

3

1

2

0 O,2 O,3 1

1,4 17

. . . . . . .

B

B T

0 0,2 0,6 e) Wfig'

1

1,4 1,7 T

pgO

bp

7 4

pgO

P'~#

6

3

2

P~~~j~B

1

C)

012 0,6 W/k~p

/

M 45

5

'~'~

4 3

1

1,4 1,7 plo

0 0,2 0~6 1

T f)

W/kg 6

pS5

p

1,4 1.7 T peOpss

M 47 2

p ~ ~

4

j r

1

B

0 0,2 "0.6

O

pO

1

1,4 1,7

T

3

0 0,2 0.6

I

1

i

r

r

i

r

1,4 1,7

T

Fig. 29. Dependence of core loss on induction in electrical silicon steel sheets: a), b), c) - grain oriented electrical steel sheets grades M4, M5, and M9; d), e), f) - non-oriented silicon steel sheets grades M43, M45, and M47.

As can be seen from figs 28-30, the zero points of the curves J.L = f ( ~ ) identify the directions of the extremes of sheet magnetic properties. The results in table 4 point to a quantitative connection between the vertical component of magnetic polarisation J± and anisotropy of induction, core loss and apparent core loss. This may serve as an argument for practical application of the anisometric method in establishing angular dependences of various properties of electrical sheets, shown in table 4.

ANISOTROPY IN IRON-BASED SOFT MAGNETIC MATERIALS

:J f:,

bj

60

096

~,

o,o:,

/11

6Q

40 20,0,t 24

t _

"~-'~

.,

4(]

1253° 1

/

20

71-,=U -

100

371

-0,~

,,o;_-

'

10 I-O,O'12

~t0* -101

a0*

, ¢2 60 , , , ~

Y~ ~.-

~

d) 10dF___~

30°

V,.3Zf/60'a~

'3zl~Z S2z'

Fig. 30. Examples of crystal orientation and the dependence of the vertical component of magnetic polarisation J.L, on the sample rotation angle ~: a) for an iron single crystal, b) for grain oriented steel sheet grade M4, c) for non-oriented silicon steel sheet grade M45, d) linearized curve ,71 = .COP)and broken line shows curves drawn for alternating magnetic field.

In table 4 the values of anisotropy AB, AH, AV, and AP (as in eq. (18)) are calculated on the basis of data obtained using the 25-cm Epstein frame. The orientations of the strips were established by means of an induction anisometer and are shown in table 2. The dependence of the sheet orientation and their anisotropies AB, AH, AV, and AP obtained are illustrated in more detail in (Soinski 1984b). Data in this table etl, 0/2, 'tg[~blz[', 'tg[~b2z[' and ' S l z ' or 'S2z' are taken from the characteristic points of the induction anisometer measurements (see fig. 30). These data give a comparison between both measurement methods and are used in

372

M. SOINSKI and A.J. MOSES

TABLE 4 Dependences of the J ± = f(~o) parameters on induction anisotropy AB (at H = 2500 A/m), magnetic field anisotropy AH, apparent core loss anisotropy AV and core loss anisotropy AP (at B =1.5 27). Parameter

Anisotropic sheets

Isotropic silicon sheets

M4

M5

M6

M7

M9

M43

M45

M47

87.5 29.0 38.5 49.5 29.45 5.81

78.5 27.5 34.5 44.5 25.35 5.64

72.5 24.5 32.0 41.0 24.35 4.91

54.5 17.0 24.0 30.0 18.43 3.24

43.0 12.0 19.5 23.5 14.73 2.27

14.5 3.0 6.5 5.5 5.53 0.53

12.5 2.5 5.5 4.5 4.73 0.33

8.0 0.5 3.5 2.5 4.01 0.03

AB~.~o55 T

0.645

0.521

0.473

0.414

0,311

0.078

0.078

0.057

A B ~ o9° T

0.466

0.370

0.346

0.301

0,248

0.068

0.074

0.029 0.043

ctl deg 1 2 deg

tgl~lzl tgl¢2zI 51z cm 2 ,..q2z cm 2

A B ~ 5 ~ "5 T

0.270

0.210

0.200

0.210

0,200

0.054

0.056

AB°~o72'5 T

0.570

0.440

0.420

0.360

0,270

0.084

0.086

0.063

AH155-0 A/m

14180

9180

8880

7750

7200

800

800

550

AHI.~- ° A/m

4880

3980

4180

4650

4600

650

800

550

AV15.5-° VA/kg

306.5

253.6

236.9

214.8

176.8

17.3

22.2

12.4

AVI.~- ° VA/kg

113.6

101.5

107.5

-

-

12.6

18.6

9.9

AP15.~- ° W]kg

2.47

2.16

2.11

1.72

1.42

0.72

0.67

0.44

API.~-° W/kg

2.67

2.48

2.27

2.01

1.67

0.78

0.71

0.47

further numerical and statistical evaluation. For this reason all numerical data are given in this table with greater precision than the measuring system allowed. The numerical calculations give however definite functions p(z) representing analytically the relationship between anisotropies AB, AH, AV, and AP, as indicated in table 4 and independent random variables x (parameters t~l,~2,tg[~blzl,tgl~b2zI, S l z and S2z). The dependence f(x) on the set X for electrical steel sheet of grades M4, M5, M6, M7, M9, M43, M45 and M47 (all with different degrees of Goss texture) are polynomial expressions: n

p(x) = ~_, alJl xn-j,

(21)

j=0

where: n = 1, 2, 3 - degree of the polynomials (linear, parabolic and 3rd degree functions), a[j] - investigated numerical coefficients. The values of the a[j] coefficients in the polynomials which approximate to experimental induction, apparent core loss and core loss anisotropy, are given in tables 5 and 6, respectively. It is clear that the approximation error diminishes with increase of the degree of the polynomial (see approximation errors for polynomials of different degree in both tables, columns 6 and 7). However, tables 5 and 6 show maximum approximation errors, i.e. the discrepancies between Epstein measuremerits (data AB, AV and AP from table 4) and prediction of these anisotropies from

ANISOTROPY IN IRON-BASED SOFT MAGNETIC MATERIALS

373

TABLE 5 Numerical values of the coefficients of polynomials of the 3rd degree, according to eq. (21)* approximating the anisotropy of induction. Numerical values of the coefficients of polynomials Anisotmpy of of the 3rd degree according to eq. (21) induction ~Epstein

Maximum approximation error for polynomials of respective degree

~st)

Parameter of the anisotmpy CUrve

sx = .f(,p) (fig. 30) as an independent vat. z

~[o]

~[1]

~[2]

~[3]

]-9.28010-7 1.31410 - 4 2.04010 - 3 2.79510 - 2

n=2

n=l

12%

16%

11%

18%

a2

17%

23%

tg(~blz)

20%

24%

tg(~b2z)

approximation error enf = 8% A B 2 ~9°

I-3.8771o-6 - 1 , 7 3 0 1 o - 5 1.7381o - 2 2.31110 - 2

approximation error enf = 10% AB~5~o 27"5 -4.07310-6 1.0191o - 4 7.85910 - 3 7.90710 - 3

approximation error enf = 15% AB~5~o72'5 4.73810-6

- 5 . 3 9 6 1 0 - 4 2.51110 - 2 - 3 . 2 6 9 1 0 - 2

approximation error enf = 19% *Approximating anisotropy of induction AB250o at 2500 A/m for different magnetization directions is given with regard to the parameters of the vertical component of magnetic polarisation curve d.t = f ( ¢ ) and initial data are provided in table 4.

the texture (data AB or AV and AP in tables 5 or 6, respectively). This points to strong non-linearity of the approximation for electrical steel sheet with different degrees of texture. Equations (21) should be used when a high degree of texture exists and this allows the establishment of close angular dependence as shown in fig. 20. The values of magnetic field anisotropy ^r_r55-0 ^rr90-o f(Slz~ 1.5 "-- f(Slz) and ~'~'~ 1.5 S2z) as in table 4 permit the formulation of other equation not shown here (Soinski 1987a). Errors between the points given in table 4 and the approximate curves derived from eq. (21) amount to not more than 20% which corresponds to the precision of graphic approximation. Table 6 also indicated changes of apparent core loss anisotropy AV and core loss anisotropy AP relative to changes of the J.L = f(~) curve parameters. In this case maximum errors between given points (table 4) and approximation curves (from eq. (21)) in table 6 are greater than those for induction anisotropy in table 5, amounting to up to 30% for polynomials of the 3rd degree. The errors can be reduced

374

M. SOINSKI and A.J, MOSES

TABLE 6 Numerical values of the coefficients of polynomials of the 2nd or 1st degree, according to ex (21)* approximating anisotropy of apparent core loss AV and core loss AP. Anisotropy of Numerical values of the coefficients of polynomials AV and AP )f the 2rd or 1st degree according to eq. (21) (Epstein test)

Maximum approx- Parameter imation error for of the polynomials of anisotropy respective degree Curve

J± = f(~) (Og. 30) as an independen variable x in eq. (21)

~[01

~[11

AVI.~-° VA/kg 3.78310 - 2 1.315

~[21

~[31

-4.378

0.0

approximation error enf = 31% (n = 2) 1.1671o + 1 0.0 AVI.~-° VA/kg 1.03310 - 1 1.350 approximation error enf = 25% (n = 2) AP~-°W/kg 3.19710 - 2 1.94610 - 2 0.0 0.0 approximation error enf = 37% (n = 1) APl.~-°W/kg 9.04210 - 2 3.8551o - 1 0.0 0.0

rim3

nml

20%

43%

n=3 15%

n=l 35%

n=3 30%

n=2 37%

n=3 13%

n---2 14%

c~2

c~2

approximation error enf = 16% (n = 1) *Approximating anisotropy of apparent core loss AV and core loss AP at 1.5 T for different magnetization directions is given with regard to the parameters of the vertical component of magnetic polarisation curve J.L = f(¢) and initial data are provided in table 4.

if the m e a s u r e m e n t s are t a k e n w i t h an a n i s o m e t e r u n d e r d.c. and a.c. m a g n e t i s a t i o n (see fig. 30d) b e c a u s e hysteresis and e d d y current losses affect the m e a s u r e d data. I n this case in o r d e r to relate the data resulting f r o m the a.c. and d.c. m a g n e t i s a t i o n it is o n l y necessary to c o m p a r e percentage values. Data discussed in this section refer to the m o s t i m p o r t a n t m a g n e t i c and n o n m a g netic a n i s o t r o p y o f electrical steel sheets w i t h Cross texture, but with different degrees o f orientation. T h e results d e s c r i b i n g the a n i s o t r o p i c b e h a v i o u r o f g r a i n o r i e n t e d and n o n - o r i e n t e d silicon steel sheets have the a d v a n t a g e o f b e i n g o f c o m m o n use. This is of great significance for a q u i c k e s t i m a t i o n o f the sheet p r o p e r t i e s and their role in the construction and a s s e m b l y o f m a g n e t i c cores. T h e i n f o r m a t i o n g i v e n b y eq. (21) gives n e w possibilities o f e x p l o i t i n g the k n o w l e d g e o f a n i s o t r o p i c p r o p e r t i e s o f the m a t e r i a l in d e s i g n i n g m a g n e t i c cores. This in turn leads to an i m p r o v e m e n t in efficiency o f electrical devices.

ANISOTROPYIN IRON-BASEDSOFT MAGNETICMATERIALS

375

5.2. High induction (permeability) steel sheets Cold-rolled high permeability electrical sheets have the Cross texture and display a very strong directional anisotropy resulting from orientation of the Fe-Si crystal and the effect of a stress coating. The use of these steel sheets in building large transformers is desirable for energy saving, lowering size and weight and improving working efficiency. These sheets have better magnetic properties than other commercial materials as they have higher magnetic induction and lower core loss due to the good crystal orientation (Honma 1983, Sokolov et al. 1989, Moses 1990). A better texture of the high permeability sheet is obtained by an increase of steel purity and from the application of special inhibitors. This leads to an increase of induction Bl0oO (at H = 1000 A/m) by about 0.1 T relation to conventional oriented silicon iron. The strong anisotropy affects the directional mechanical, electrical and magnetic properties, most importantly, induction and core loss. In the high permeability sheets the average deviation between the [100] direction in ~ystals and the rolling direction is decreased from 7 ° to 3 °, relative to conventional electrical sheets (Groyecki 1981). The [111] direction, 54.70 to the rolling direction, displays cleary the worst magnetic properties, while the [110] direction, 900 to the rolling direction, also shows poor values. At lower flux densities, the difference between the core losses of conventional grain oriented and high permeability electrical steel decreases due to the effect of the grain size. The core loss at the lower flux density is closely related to the 1800 domain wall spacing and material with smaller grains has lower eddy current loss and consequently lower total loss (Honma 1983). This means that high permeability steel sheets are more suitable for transformers operating at high flux density. The improvement of the magnetic properties of high permeability sheets is also due to optimization of stresses brought about by selecting desirable electrical insulating coatings. Modern coatings induce a large tensile stress in the steel and reduce losses further. Tensile stresses appearing at the sheet surface essentially influence the domain structure in two ways. Firstly, it decreases 1800 domain widths and secondly, it reduces the number of 900 magnetic domain walls (Honma 1983). Material with superior (110) [001] orientation contains fewer 900 domain walls whose motion causes positive magnetostriction during the magnetization process. This is why higher permeability materials exhibit lower magnetostriction. This in turn reduces transformer noise. The stresses, are caused by the difference in thermal expansion between the insulating layer and the sheet. Research in (Pftitzner 1981b) shows that the core loss resulting from the insulating coating decreases along the rolling direction but increases when magnetised perpendicular to the rolling direction. High permeability sheet, as it has been noted, possesses high permeability over the whole range of magnetic field intensities compared with conventional grain oriented silicon iron sheets and they also have lower core loss, especially at high induction. They possess higher anisotropy as can be seen from fig. 31. The data are given at peak inductions of 1.0, 1.3, 1.5 and 1.7 T indicated in fig. 31 as 10, 13, 15 and 17, respectively. It can be seen that the high permeability material exhibits a higher increase in core loss with

376

M. SOINSKI and A.J. MOSES Conventional grain oriented electrical steel sheets

r

i

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-_

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i I

-_

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,

t

s_~.r

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\ " ~ / / / ~ J ~

I~.=~//

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Fig. 31. Comparison of core loss of 0.3-ram thick conventional (grade M5) and high permeability grain oriented silicon steel sheets, as a function of angle of magnetisation from the rolling direction, at 50 Hz (taken from Honma 1983).

increasing angle from the rolling direction than that which occurs in conventional steel. This fact needs to be carefully taken into account in practical use of this material. The angular dependence of properties of high permeability steel sheets obtained in a similar way to that shown in fig. 20 is shown in fig. 32. It describes quantitatively the anisotropic properties of magnetic induction and core losses, respectively and gives an indication of the very high orientation met in these materials. Beside the application of stress coatings to high permeability sheets domain size can be controlled by surface etching, mechanical scribing, spark ablating or laser scribing. It is very benefitial to refine the domain wall spacing in this way in order to reduce eddy curent loss of any highly oriented material whose grain size is large (Honma 1983, Sokolov et al. 1989). The major breakthrough in applying this domain refinement process commercially was when instead of applying scribed lines mechanically, a high power laser beam was used (Moses 1990). Small localised stressed areas caused by lines of such spots running transverse to the rolling direction scribed 5 nun apart produce improvements. The damaged surface needs recoating but the important feature to be noted is that the effect for core loss reduction of laser irradiation disappears by annealing at a temperature higher than 500°C. So no further stress relief anneal can be carried out. Heat proof domain refinement processes have however now been developed. These domain-controlled steels do not possess any other significantly different type of texture or magnetic anisotropy than conventional grain oriented steels although

ANISOTROPY IN IRON-BASED SOFT MAGNETIC MATERIALS

8)Ti Bzsoo

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377

test

1.a

17 11 1t

'

1-2 o

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'5

'

*

t

go

4 85 67.8 Angle from rolllng dlrectlon

tJ) Pt0 2

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/ 0

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s'5 61J

g',

Angle from rolling direction Fig. 39- Angular dependence in 0.27-ram thick, high permeability electrical steel sheets of: a) induction, at a magnetic field strength of 2500 A/m; b) core loss, at a peak induction of 1.0 T, at 50 Hz.

it is harder to magnetise at 90 ° to the rolling direction than what is expected from crystall orientation alone. 5.3. Thin gauge anisotropic steel sheets less than 0.20 mm thick Improvements in core loss of electrical steel sheets has been made not only by a higher degree of texture and refining of domain wall spacing but also by developing thinner gauge steel or a higher silicon content. Thinner grades of electrical steels will almost certainly become more common in the future because they have lower core loss due to reduced eddy currents (Moses 1990), The disadvantages of their normally poorer orientation can be overcome, because of their greater susceptibility to domain refinement than thicker grades. The sheets with silicon content up to 3.5%

378

M. SOINSKI and A J . MOSES

or higher (see section 3) usually range in thickness between 0.05 and 0.20 mm. The improvement does not carry on by thinning materials indefinitely, because of hysteresis loss due to surface inclusions, which increases as the material becomes thinner. The grain size also tends to increase, hence requiring even better domain refinement methods or stress coatings to sustain the downward trend of losses. However, the results of many domain studies have shown that the optimum orientation is not the perfect (110) [001] orientation, but a texture with a mean angle of tilt of around 2.5 °. This texture has not been achieved in practice even in thin gauge highly oriented electrical steel sheets. The materials available commercially are magnetically acceptable but their width only ranges from 2.5 to 120 mm (Bak et al. 1990, Yamaguchi et al. 1987). They are usually produced in small quantities which possess a Goss texture (Benford 1984, Yamaguchi et al. 1985). The anisotropic properties of thin gauge steel ribbons are rarely discussed in techrtical papers. Thin anisotropic electrical sheets usually possess a lower degree of grain orientation than M4 grade grain oriented steel (see also fig. 30). It is best if magnetic wound cores are manufactured from steels having as high as possible degree of texture but usually stacked core material should possess a lower degree of grain orientation if low losses can be achieved (see data in table 1 for two types of thin gauge electrical steel sheets). These electrical sheets, as any oriented silicon steel, are sensitive to directional changes of magnetic parameters, more noteable however in sheets having greater grain orientation (fig. 33). Thinness of the sheets is an additional factor strongly affecting the directional properties, particularly coercive force and core loss. Magnetic properties of narrow commercially available anisotropic sheets cannot be investigated in the standard Epstein frame due to the small sample width (IEC Publication 1978). Instead a yoke frame (Soinski et al. 1987c, Brix et al. 1984) aJ

b}

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Fig. 33. Changes in the 50 Hz a.c. magnetization loops in thin gauge steel sheets, 0.20-ram thick with: a) 75% Cross texture, b) 90% Goss texture.

ANISOTROPY IN IRON-BASED SOFT MAGNETIC MATERIALS

379

TABLE 7 Anisotropy of coercive force AHev - x (calculatedaccording to eq. (18) for ABY'x and as a percentageAA (He) accordingto eq. (19) as for AA(B)) obtained from a.c. magnetization loops at 50 Hz at a peak flux density of 1.5 T and d.c. values (eoercivemeterwith samples in an open magnetic circuit at H = 1 x 10 4 A/m). 8-era Epstein IGoss orientation (%) 75% 90% 90% 90%

Sheet thickness (ram) 0.20 0.20 0.10 0.05

AH55-0 (A/m) 10 25 32 54

AAHe (%) 14 52 82 123

Square Coercivemeter AHe55-° (A/m) 10 29 24 46

AAH¢ (%) 24 132 92 144

AHe9°-° (A/m) 16 30 40 56

AAHe (%) 38 136 154 175

or non-standard permeameter must be used. Results using these methods do not differ by more than 5% from the values obtained by the standard test frame (Bak et al. 1990). The geometry of thin individual strips has a significant influence on the measured values of coercive force due to demagnetization in magnetically open circuits. Because of this, the geometrical dimensions, especially width and length of the strips, should be selected so that even the normal 5% error in sample preparation cannot influence the value of coercive force (IEC Publication 1978). The measurements of coercive force is strongly sensitive, however, to the changes of strip thickness and may increase by up to 50% as the it drops from 0.20 to 0.05 mm (Druzinin 1974). Figure 33 shows directional changes of a.c. hysteresis loops of thin gauge electrical steel with two values of sheets texture. Hysteresis loops calculated at peak inductions of 1.0 T and 1.5 T for the thinner strips have shapes similar to those shown in fig. 33, although the coercive force is different as shown in table 7. This table also gives detail of the anisotropy of coercive force of the same sheets. From this table it can be seen that the sample with 75% Goss orientation has a relatively small anisotropy of coercive force. The greatest anisotropy of coercive force in all anisotropic sheets under investigation occurs between 00 and 900 directions, as for other types of electric sheets described earlier in table 2. Directional changes of induction and core loss are shown in fig. 34. The magnetization characteristics for other thicknesses are similar to those shown in fig. 34. The curves for both types of sheets when magnetised at 550 to the rolling direction are almost the same. The difference in induction and core loss anisotropy is due to different values in the rolling direction measured in sheets with 90% and 75% Goss texture. Another publication (Bak et al. 1990) shows numerical values of induction anisotropy measured in sheet samples under investigation. It is valuable to note that sheets with 90% Goss orientation have two or three times greater anisotropy than sheets with 75% Goss orientation because the dependence is strongly nonlinear. It should also be mentioned, that correlation between the magnetic properties of anisotropic thin sheets and their thickness is extensively covered in (Benford 1984).

380

M. SOINSKI and A.J. MOSES

B 1.8 1.8 1&

1.2 oj

_ _ 0 / ~

sheet

1.o

90%

orientation/~ ~

o.8

/

///~y

0.4

Z/

sheet

~

75%

orientation

o,2

H

3 45 7

20 40 100 200

1000 3000

Fig. 34a. Directional characteristics of 0.20-ram thick, steel sheets with 75% and 90% Cross texture: d.c. magnetization.

W~kg' 3.6 S.4 S.2 3.0 2,8 2,8 2,4

2a

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/

l:

I

J /

"

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/

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Fig. 52. Comparison between rotational power loss, rotational hysteresis loss, alternating power loss, and alternating hysteresis loss in a sample of grain-oriented silicon iron.

402

a)

M. SOINSKI and AJ. MOSES

~ L D O . ~ O F ROTA° B(T) 2 TING FLUX - - LOCUSOF ROTA° TINGFIELD f.

--LOCUS OF ROTATING F[UX - - LOCUSOF ROTATIN(; FIELD

GDOH(A/m)

B(r)

• GO0 H(A/m)

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-6o0

-i

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,_J -2,

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--"~II~ FLUX TIN(; FIELD . .LOCUS OF ROTA- B(T)

/

H(A/rn)

6OO | 2

2 ]T 3000 H(A/m)

-600

LOCUS OF ROTA- - TING FLUX - - LOCUS OF ROT~ B(T) 2 'i" TIN(; FIELO / 3000 .H(A/m)

/ 2.Non-Orlanteci 3.POWERCORE~t~

.cA/.~

/

2,NlxI-Drlanted

\\ I /

l

,.

\ .CA/.~

I

~

/

\

3000 2

e(r)

s(r

30O

Fig. 53. Loci of rotating field needed to produce clockwise and antielockwise rotational flux density in non-oriented electrical steel, grain-oriented electrical steel and amorphous (POWERCORE strip); (a) 1.0 T, (b) 1.2 T.

grains according to whether the magnetisation vector is moving towards or away from the rolling direction.

6.3. Importance of anisotropy in transformers The efficiency of transformers can be as high as 95-99.5% because the magnetic circuit guarantees relatively full magnetic coupling of the electric circuit. A reduction in the amount of losses and progress in transformer building lead to reduction of weight and dimensions in the magnetic cores, as well as to better environmental protection. Core losses in laminated and wound transformer cores consume at least 2% of all

ANISOTROPY IN IRON-BASED SOFT MAGNETIC MATERIALS

403

electricity which is generated. Consequently the producers of electrical steels and the manufacturers of transformer cores are attempting to develop new materials and to use them more efficiently. Smallers cores are usually of the strip wound design in which the flux will travel along the strip direction most of the time in all parts of the magnetic circuit. Here the main priority for efficiency is to have low losses and high permeability along the strip longitudinal direction. Normally grain oriented silicon iron is used because of its good properties when magnetised along the rolling direction. The material is of course highly anisotropic but the anisotropy plays only a minor role in core performance. J

f

"----i

4,.~..,,~...~>

//t'

~,~

/

-""

~,

.

~

.

.

.

----

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l/I .../._!../...

/I/

Ill LOCUS IN |'-i : AHORPHOU5CORE! ~

(a) ~

|

Bpeak

-

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i 1

Tesla

-""

I

I

l

I

I I I I

I

I

I

I

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/11 !11 Ill

Ill

Ill

I

(b) --

I

I

LOCUS

IN

""-

~

"--

i -St-Fe. CORE

---

i Bpeak - 1,2T

=

! t

Tesla

Fig. 54. Locus of localised fundamental component of flux density in model three-phase, three-limb cores assembled from (a) amorphous ribbons and (b) grain-oriented silicon iron. Limb flux density of 1.2 T, 50 Hz in each ease.

404

M. SOINSKI and A.J. MOSES

The stacked laminated core is used in large single phase transformers and also in 3 phase transformers from a low power rating upwards. Figure 54 (Moses 1992b) shows the flux distribution in two such stacked cores: one made from grain oriented 3% silicon iron and the other from amorphous POWERCORE TM strip. These cores have very simple corner joints where yoke and limbs laminations are interleaved, the magnetizing coils (not shown) are placed on the 3 vertical limbs. The flux density is represented as loci at various points in the core. The straight line loci show where flux varies cyclically with time but not in direction whereas the eUiptical loci shows where the flux undergoes spacial rotation during the cycle. There are obvious differences, rotation occurs in one core but not the other, the alternating flux density mainly remains in the rolling direction even at the corners of the grain oriented core and the flux density uniformity is greater in the amorphous core. Such results and similar studies highlight several aspects of the effects of anisotropy in stacked cores. The flux uniformity is vital to give the desired low building factor (i.e. only a small increase in loss of the built up core compared with the loss under ideal conditions in a single strip tester or Epstein frame). It is more uniform in cores with lower anisotropy as can easily be seen in the example given in fig. 54. Poor building factors in cores made from low loss, high permeability, highly oriented steel have been atributed to this increase in non uniformity of the flux. The anisotropy tends to reduce the amount of rotational magnetisation which occurs and forces flux transfer at joints to occur by more complex methods which can increase losses. Although at first sight rotational magnetization might appear harmful its presence can cause a net saving in energy loss by avoiding more lossy processes such as normal or interlaminar flux transfer between layers of laminations in the transformer joint regions. It is widely thought that the 3 phase stacked core should be made of highly oriented material. However the only reason for using highly anisotropic material is that it does have low losses along its rolling direction. Isotropic material will give a lower building factor even if the losses along its longitudinal direction are higher than conventional steel. A novel way of reducing the loss of 3 phase stacked cores build from anisotropic grain oriented silicon iron is to assemble the core with laminations cut at a small angles to the rolling direction. As in the Epstein frame mentioned earlier, the stacking method is again important. Figure 55 shows two methods of stacking laminations into a single phase core and fig. 56 shows the resulting power loss (Davies et al. 1984). It can be seen that a significant improvement can be obtained by cutting laminations at 60 to the rolling direction. This method of loss saving depends on the complex flux transfer which occurs at mitred and overlapped corner joints. The limb losses are believed to increase slightly but lower corner losses occur. The effectiveness of the method depends on the core geometry and on the anisotropy of the material. A way of predicting the effectiveness of the method has not yet been reported. The largest market for amorphous ribbon is at present in single phase transformers in the U.S.A. and in Japan. More efforts have been made in Europe for 3-phase distribution transformers with four wound, cut amorphous cores. Again, the main

ANISOTROPYIN IRON-BASEDSOFT MAGNETICMATERIALS

405

(a) adjacent laminations ~

~

.

.~

Co) Fig. 55. Representationof the two stacking methods: (a) A-type and (b) C-type; Rolling direction (RD). priority for efficiency is to have low losses along the longitudinal direction, but permeability is lower than in the best electrical steels. This is the reason why amorphous cores are most attractive when loss cost capitalization during the transformer life time is taken into account.

6.4. Importance of anisotropy in motors The efficiency of industrial motors varies from around 50% to 90% partly depending on the anisotropy of the magnetic core material, power rating and rotational speed. Energy efficient motors are becoming more important in the need to protect the environment by cutting down the huge amount of energy wasted in rotating electrical machines. Some 20% of motor losses are due to the magnetic hysteresis and eddy current occurring in stator laminations of induction machines. Large generator stator cores are often segmented and are assembled from either grain oriented or non-oriented steel. To achieve a more isotropic magnetic circuit, the segments are positioned carefully relative to the core geometry. Joining the segments requires, according to fig. 7, reversal of laminations every second segment to make the magnetic core more homogeneous. Moreover when grain oriented steel is used it is sometimes assembled with the rolling direction parallel to the teeth and sometimes perpendicular to them according to the designers preference. In order for the stator losses to be minimised, the flux must flow along the easy direction of magnetisation as much as possible in segmented cores. It is usually stated that multipole cores with narrow core backs should have the rolling directions of the segments in a circumferential direction whereas, segments in 2 pole machines should be opposite. However the predominant factor is rotational flux in the core back and this should be seriously considered when making this decision. It is difficult to conclusively establish which assembly is most beneficial. Figure 57 shows the loci of the 50 Hz component of flux density in a grain oriented segment.

406

M. SOINSKI and A.J. MOSES W/kg 1.3

,/ I'1

0"9

0-7

0"5

B !

l

I

t

I

,

I-3

1.4

1.5

1.6

1.7

T

Fig. 56. Variationof power loss with flux density in the large core: (a) a-type stacking laminations cut at 6* to RD (cf. fig. 7), (b) c-type stacking laminations cut at 6* to RD, (c) laminations cut parallel to RD, (d) laminations cut parallel to RD tested in single strip tester (from Gumaidh et al. 1992, Davies et al. 1984). The widespread occurrence of rotational flux is apparent (Moses et al. 1981, Moses et al. 1989). The dotted line indicates three position of the butt joint between segments in adjacent layers. Series of experiments have shown that the flux and resulting loss distribution depends on the texture of the material and also on the position of the butt joint. This effect of anisotropy in a machine is a 3D problem, it has not been analysed by any FEA study because of the basic difficulty of modelling anisotropy. Smaller stator cores are assembled from unsegmented laminations when there cannot be any economic advantage in using grain oriented material. However when non-oriented steel is used there are still differences in flux distribution and local loss caused by a small degree of texture particularly because of saturation close to the teeth roots (Huttenloher et al. 1984, Moses 1988). Figure 58 shows the variation in localized loss caused by applying radial uniform stress to a lamination (Moses et al. 1989). The loss increases by different amounts in different parts of the core depending on the magnitude and direction of the localised stress relative to the localised flux density. It is generally noted that the losses become lower as the material becomes more isotropic (partly because of lower rotational magnetisation and higher rotational permeability) but this has not been proven to be a general rule.

ANISOTROPY IN IRON-BASED SOFT MAGNETIC MATERIALS

11

407

I'" l I

@

~

,/

8

10

,

II

~.~t=

~ s

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16

/

/ ,~

testpo~tnumbers-,~ 7

flux ~nsity

mtlingdirection

~

0L_L._I 1 2 Testa

....

~ornple1 s~e

2

Fig. 57. Loci of fundamental (50 Hz) component of flux density in the plane of the sheet for 28M4 grade samples.

7. Concluding remarks The production of soft magnetic materials in a form of electric sheets on an industrial scale still includes cold-rolling. Thin amorphous ribbons or tapes, on the other hand, are usually produced by rapid quenching of the liquid alloy changing its liquid state into an amorphous or microcrystalline state. Furthermore, 6.5% silicon microcrystalline tapes have been manufactured by chemical vapour diffusion. Electrical sheets are characterized by anisotropy of magnetic parameters resulting from the crystalline structure of the material as well as from the type and degree of orientation. Oriented transformer sheets display the so-called Cross texture and the anisotropy resulting from this texture is smaller than in the case of an Fe--Si single crystal. Isotropic silicon sheets with weak Goss texture and isotropic silicon-free sheets with mixed texture possess grain orientation which is an order of magnitude smaller. Uniaxial induced anisotropy is also present in amorphous materials. This is two orders of magnitude smaller in the iron-based amorphous ribbons than the magnetocrystalline anisotropy in highly oriented electrical sheets. Microcrystalline tapes

408

M. SOINSKI and A.J. MOSES

W/kg

5.

region 2 ,a

/

/ .p /

region 2 region 3 /s /

/

~ d

/

~

• d'

IS

•

,

,•e'

region 3

J

region

/

4-

region region 5

3.

region'5

__,behind .... behind

slots teeth

I

compressive stPess Fig. 58. Variation of iocalised loss with radial compressive stress in regions of the core assembled from annealed laminations at 1.0 T, 50 Hz.

have magnetocrystalline anisotropy significantly smaller than that of grain oriented electrical steel sheets. This is due to high silicon content and grains misorientation. The values of induction, apparent power and core loss are the most important industrial parameters. Measurements of directional properties form a basis for classification of any magnetic materials. Determination of the anisotropic properties of the materials under discussion in this chapter depends largely on improved measurement methods. This concerns mainly the lamination stacking technique applied in the Epstein frame and a method for quantitative determination of sample demagnetisation. The anisotropic properties of materials with strong Goss texture, of the thin and less anisotropic materials isotropic silicon sheets, as well as non crystalline materials

ANISOTROPYIN IRON-BASEDSOFT MAGNETIC MATERIALS

409

can be calculated from the anisometric methods. The method determines angular dependence of properties while avoiding the necessity of taking many measurements in the Epstein frame. This can be very useful for users of electrical sheets. Further development in applied research and experimental techniques in the field of soft ferromagnetic materials should result in the discovery of new magnetic materials having still new applications particularly in the electronics industry. Conventional grain oriented silicon iron is continually being improved particularly due to cleaner and more homogeneous steels and better surface coatings. More attention will be focussed on anisotropic properties of all of these materials as users become more skilled at taking full advantage of such properties or avoiding their effects as appropriate. Quicker and better ways of anisotropy measurement will be called for at the same time. It has already been mentioned that texture affects magnetic characteristics of ma'terials yielding them more or less anisotropic. Anisotropy also plays a part in directional changes of coercive force or magnetostriction, especially when measured in magnetic open circuits because demagnetisation is also anisotropy sensitive. A knowledge of the anisotropy of demagnetisation is however exceptionally useful in magnetic metrology. It seems also that directional changes in coercive force, magnetostriction and demagnetization are correlated to the degree of material orientation and the data presented in this chapter confirm the influence of crystalline structure on the directional changes of properties considered in this work. A harmfuU influence of anisotropy of mechanical properties is expected in motors and generators of high power rating and the designers should be well aware of them. The phenomenon of anisotropy of resistivity in electrical sheets is crucial in the design of cryogenic devices. Thin electric sheets are produced nowadays to meet demands of industry which exploits them in building magnetic cores to operate at high frequencies or under impulse changes of magnetic field. The study of the anisotropic properties presented in this chapter constitutes a good supplement to the characteristics of thin gauge steel sheets rarely available in the technical literature. It is worth noting, that existing thin steel sheets have lower grain orientation than M3 or M4 grade grain oriented sheets. Amorphous ribbons have lowest core losses but their saturation magnetization is smaller than of any other magnetic material discussed in this chapter. Wider use of cut cores wound with amorphous material should lead to considerable saving of electrical energy in transformers. However, at present use of amorphous ribbon is limited due to such factors as the need for thermomagnetic treatment of cores and a tendency for brittle ribbon to fracture. Tapes with a microcrystalline structure seem to be a material for the future especially in high frequency applications. These tapes display magnetostriction close to zero and combine the advantages of electrical sheets, such as relatively high induction, and of amorphous material, such as relatively low core loss. The demand for energy-saving electrical machines calls for radical changes in design, to take into account anisotropic properties of soft magnetic materials to a greater extent than at present.

410

M. SOINSKI and A.J. MOSES

TABLE 12 Anisotropic properties of high induction (permeability) and grain oriented sheets, non-oriented silicon and silicon free sheets, iron based amorphous ribbons and iron based microcrystalline tapes. Position (1) (table 1) Designation

(2)

HI-BM4

(6)

(3)

(4)

(8)

(5)

(7)

(9)

(11)

(12)

thin M5 gauge

M6

thin M7 gauge

M9

M43 M45

M47

(13)

(14)

(10)

Fe78 Fe93.5 0% Si B13 Sir. 5 Si9 (13) mixed text. (14) amorphous (10) mierocryst.

strong Goss texture

weak Goss texture

Degree of orient.

97% 95% 90% 87% 85% 75% 65% 45%

10% 10%

9%

7%

0.1%

Rank

1

9

10

11

12

13

14

11 9

12 11

13 12

14 14

10 13

1 : AB 2 : AP

1 1

2 2 2

3

4

6 7

Anisotropy under examination 3 4 8 5 7 9 3 4 10 5 6 8

5

6

7

8

Here: the rows of table designate data for anisotropy of magnetic flux density (induction) (AB) and anisotropy of core loss (AP), according to eq. (18) and eq. (5), respectively. < 7% - below 7%, >0.1% - above 0.1%

Isotropic silicon-free sheets generally display a mixed texture and grain orientation of a degree smaller than that of grain oriented steel with the Goss texture. This means that silicon-free sheets have a lower value of anisotropy of magnetic flux density and core loss than other materials. Frequently, the directions of easy and hard magnetisation do not lie along the rolling or tranverse directions. The classification of different sub-types of grain orientation also covers the cases when the direction of best and worst magnetic properties corresponds to about 30 and 80-900 to the rolling, respectively. The value of uniaxial anisotropy constants and the anisotropy of microcrystalline tapes are small. Their semicrystalline structure can be attributed to small randomly oriented grains and there is a lack of induced anisotropy. The trend in the anisotropy of magnetic properties of ferromagnetic alloys investigated here can be seen from a close analysis of the data given in table 12. The properties in this table are ordered according to the degree of crystalline orientation due to the necessity for statistical evaluation (see also table 1). The correlation coefficients calculated according to (Kendall 1970), for the above pairs treated as a random variables are: 0.91 for line depicted as 'rank' and for line 1, i.e. evaluating correlation between the degree of orientation and the induction anisotropy AB, and 0.9 evaluating in a similar way correlation between the degree of material orientation and the anisotropy of core loss (AP). The consistency coefficient (Kendall 1970) of the whole test shown in table 12 (all ranks from 1 to 14 in all lines together) is also about 0.9 and this value confirms a strong relationship between crystalographic structure and orientation versus different types of anisotropy considered as a statistical variable.

ANISOTROPY IN IRON-BASED SOFT MAGNETIC MATERIALS

411

Thus when we have one reference, e.g., degree of orientation, with a high degree of probability - other magnetic, mechanical or electrical values of iron-based soft magnetic materials under consideration can be determined. It should be emphasized that the rank arrangement for all magnetic parameters (tables 1 and 12) given by the consistency coefficient is also affected by the material thickness or domain structure. These were not normalized in any way when the correlation was calculated. Considering these factors the coefficient of the test consistency would be higher but this would reduce the clarity of results. References Anderson, J.C., 1968, Magnetism and Magnetic Materials (Chapman and Hall, London) p. 113. Arai, IC and N. Tsuya, 1980, IEEE Trans. Magn. MAG-16, 126. Arai, K., N. Tsuya and K. Ohomori, 1981, IEEE Trans. Magn. MAG-17, 3154. Arai, K.I., IC Ishiyama and H. Mogi, 1989, IEEE Trans. Magn. MAG-25, 3949. Armeo Catalogue, 1976. ASTM Standard, 1985, E 8-85: Tensile Testing of Metallic Materials. Bailey, D.J. and L.A. Lowdermilk, 1985, Rapidly Quenched Metals (Elsevier, Amsterdam) p. 1625. Bak, Z., P. Bragiel and M. Soinski, 1990, IEEE Trans. Magn. MAG-26, 3076. Benford, J.G., 1984, IEEE Trans. Magn. MAG-20, 1545. Beromi, Y., 1992, Rotational Power Loss Measurement System under Controlled Magnetisation, PhD Thesis, University of Wales, U.IC Boll, R. and H.R. Hilizinger, 1983, IEEE Trans. Magn. MAG-19, 1946. Boon, C. R. and .I.E. Thompson, 1969 Proe. IEE 3, 605. Boyd, E.L and J.D. Borst, 1984, IEEE Trans. Pow. Appar. Syst. PAS-103, 3365. Bozorth, R.M., 1951, Ferromagnetism (D. van Nostrand, Toronto) pp. 61 and 446. Brailsford, E, 1954, Magnetic Materials (Wiley, New York) p. 33. Brissonneau, E, 1980, J. Magn. Magn. Mater. 19, 52. Brissonneau, E, 1984, J. Magn. Magn. Mater. 41, 38 Brix, W., K.A. Hempel and EJ. Schulte, 1984, IEEE Trans. Magn. MAG-20, 1708

Brugel, L, E Brissonneau, A. Kedous and J.C. Perrier, 1984, J. Magn. Magn. Mater. 41, 230. British Standard BS 601, 1973, Part 5: Mechanical Measurements. Bucholtz, E, K.E Koo, A. Dandridge and G.H. Sigel, 1986, J. Magn. Magn. Mater. 54-57, 1607. Chambron, W. and A. Chamberod, 1981, J. Phys. (Paris) 42 (10), C5 511. Chambron, W., E Lancon and A. Chamberod, 1982, J. Phys. (Paris) Lett. 43, L 55. Chang, C.E, R.L Bye, V. Laxmanan and S.IC Das, 1984, IEEE Trans. Magn. MAG-20, 553.

Coulomb, E, 1972, Les textures dans les mrtaux de rrseau cubique (Dunod, Paris) p. 168. Davies, D. and A.J. Moses, 1984, IEEE Trans. Magn. MAG-20, 559. Druzinin, V.V., 1974, Magnetic Properties of Electrical Steel Sheets (Energija, Moscow) p. 52. Eadie, G.C., 1984, J. Magn. Magn. Mater. 41, 1. Electrical World, 1985, High-tech Core Alloy Slashes No-load Loss (McGraw-Hill, New

York). Enokizono, M., N. Teshima and K. Narita, 1982, IEEE Trans. Magn. MAG-18, 1007. Enokizoao, M., T. Todaka and K. Yuki, 1992, Int. J. Appl. Electromagn. 3, 389. Ferro, A. and G.P Soardo, 1980, J. Magn. Magn. Mater. 19, 6. Fish, G.E., V.R.V. Ramanan, R. Hasegawa and A.C. Diebold, 1983, IEEE Trans. Magn. MAG-19, 1937. Frankel, D. and R. Nuscheler, 1980, Arch. Elektrotech. 62, 327. Groyecki, J., 1981, Wiad, Elektrotech. 6, 145.

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Gumaidh, A. M., W.L. Mahadi, Y. AlinejadBeromi, A.J. Moses and T. Meydan, 1992, Int. Workshop Proc. (PTB Braunsehweig). Hall, R.C., 1959, J. Appl. Phys. 30, 816. Heek, C., 1975, Magnetische Werkstoffe und ihre technisehe Anwendungen (Springer-Verlag, Heidelberg) p. 90. Honma, K , 1983, CIGRE Study Committee 12. Huttenloher, D., H.W. Lorenzen and R. Nuseheler, 1984, IEEE Trans. Magn. MAG-20, 1968. IEC Publication 404-2, 1978, Magnetic Materials, Part2: Methods of Measurement of Magnetic, Electrical and Physical Properties of Magnetic Sheet and Strip (Geneva). IEC Publication 404-1,1979, Magnetic Materials, Part 1: Classification (Geneva). IEC Publication 404-4, 1982, Magnetic Materials, Part 4: Methods o f Measurement of the d.c. Magnetic Properties of Solid Steels (Geneva). IEC Publication 404-7, 1982, Magnetic Materials, Part 7: Method of Measurement of the Coereivity of Magnetic Materials in an Open Magnetic Circuit (Geneva). IEC Publication 404-8-2, 1985, Magnetic Materials, Part 8: Specificationfor Cold-rolled Magnetic Alloyed Steel Strip Delivered in the Semi-processed State (Geneva). l i C Publication 404-8-4, 1986a, Magnetic Matrials, Part 8: Specification for Cold-rolled Non-oriented Magnetic Steel Sheet and Strip (Geneva). IEC Publication 404-8-5, 1986b, Magnetic Materials, Part 8: Specification for Grain Oriented Magnetic Steel Sheet and Strip

(Geneva). IEC Publications 404-8-8, 1991, Section 8.8: Specification for Thin Magnetic Steel Sheets for Use at Medium Frequencies (Geneva). IEC Publication 404-3, 1992, Magnetic Materials, Part 3: Methods of Measurement Properties of Magnetic Sheet and Strip by Means of a Single Sheet Tester (Geneva). lmamura, M., T. Sasaki and H. Nishimura, 1983, IEEE Trans. Magn. MAG-19, 20. Inui, Y., Y. Tsutsumi, T. Miyashita, S. Masuda, T. Sakamoto, Y. Kako and K. Okuyama, 1977, IEEE Trans. Pow. Appar. Syst. PAS-96, 1831. Jacobs, I.S., 1979, J. Appl. Phys. 50, 7294. Johnson, L.A., E.E Cornell, D.J. Bailey and S.M. Hegyi, 1982, IEEE Trans. Pow. Appar. Syst. PAS-101, 2109.

Kan, T., Y. Ito and H. Shimanaka, 1982, J. Magn. Magn. Mater. 26, 127. Kendall, M.S., 1970, Rank Correlation Methods (Griffin, London). Kneller, E.F., 1962, Ferromagnetismus (Springer, Berlin). Kopcewicz, M., H.-G. Wagner and U. Gonser, 1985, J. Magn. Magn. Mater. 51, 225. Kraus, L, L Tomas, E. Kratochvilova, B. Spfingmann and E. Muller, 1987, Phys. Status Solidi A: 100, 289. Kuo, Y.C., L.S. Zhang and R.W. Gao, 1983, J. Magn. Magn. Mater. 31-34, 1563. Luborsky, F.E. and J.D. Livingston, 1982, IEEE Trans. Magn. MAG-Ig, 908. Luborsky, EE., J. Reeve, ~ A . Davies and I-LH. Liebermann, 1982, IEEE Trans. Magn. MAG-I8, 1385. Luborsky, EE., ed., 1983, Amorphous Metalic Alloys (Butterworths, London). Lyadkovsky, G., P.K. Rastogi and M. Bala, 1986, J. Met. No 1, 18. Matheisel, Z., 1978, Cold-rolledElectrical Sheets (WNT, Warsaw). Matsubara, I, K. Maeda, K. Itisatake and T. Miyazaki, 1987, Pine. Conf. Soft Magnetic Materials 8, 51. Matsuoka, I-L and M. Okamoto, 1985, Rapidly Quenched Metals (Elsevier, Amsterdam) p. 1633. McGuire, T.R. and R.I. Potter, 1975, IEEE Trans. Magn. MAG-11, 1018. Mei Yu, H.L Luo and C. Politis, 1986, IEEE Trans. Magn. MAG-22, 448. Metglas Catalogues 198%1992, Magnetic Alloys. Technically Superior, Allied Metals Products. Miedema, A.R. and W.E Dorleijn, 1975, Philips Tech. Rundsch. No. 3. Moorjani, K. and J.M.D. Coey, 1984, Magnetic Glasses (Elsevier, Amsterdam) pp. 110, 392 and 409. Moses, A.J. and P.S. Phillips, 1977, Proc. liE, 124(4), 413. Moses, A.J. and G.S. Radley, 1981, liEE Trans. Magn. MAG-17, 1311. Moses, A.J. and S. Hamadeh, 1983, IEEE Trans. Magn. MAG-19, 2705. Moses, A.J., 1988, Physica Scripta I"24, 49. Moses, A.J. and I-L Rahmatizadeh, 1989, IEEE Trans. Magn. MAG-25, 4003. Moses, A.J., 1990, Proc. l i E 137, Pt. A, No 5, 233. Moses, A.J., 1992a, J. Mater. Eng. Perform. 1, 235.

ANISOTROPY IN IRON-BASED SOFT MAGNETIC MATERIALS Moses, A.J., 1992b, Int. 1. Appl. Electromagn. 3, 317. Nagel, W.D. and W.J. Ros, 1987, Prec. Conf. American Power, Chicago, 460. Nakata, T., N. Takahashi and Y. Kawase, 1985, Prec. Coal Soft Magnetic Materials 7, 1. Napoli, A. and R. Paggi R., 1983, IEEE Trans. Magn. MAG-19, 1557. Nafita, K., M. Enokizono, N. Teshima and Y. Mori, 1980, J. Magn. Magn. Mater. 19, 143. Narita, K. and Y. Yamishiro, 1980, IEEE Trans. Magn. MAG-16, 728. Narita, K., N. Teshima, Y. Mori and M. Enokizono, 1981, IEEE Trans. Magn. MAG-17, 2857. Narita, K. and T. Yamaguchi, 1982, IEEE Trans. Magn. MAG-11, 1661. Nathasingh, D.M. and H.H. Liebermann H.H., 1987, IEEE Trans. Pew. Delivery PWRD-2, 843. Nippon Kokan, K. IC Publication, 1991, Super-E Core (6.5% Si Steel Plate). Novak, L , L Potocky, E. Kisdi-Koszo and A. Lovas, 1981, Acta Phys. Slov. 31,101. O'Handley, R.C., 1987, J. Appl. Phys. 62(10), R15. Okada, K., 1992, Technical Leaflet (Materials and Processing Research Centre, Nippon Konan K.IC). Overshott, ICJ., 1979, Electron. Power. 347. Pastor, G. and A. Ferreiro, 1986, J. Magn. Magn. Mater. 62, 101. PftRzner, H., 1981a, Elektrotech. Maschinenbau 98, 109. PfiRzner, H., 1981b, J. Appi. Phys. 52, 3708. PfiRzner, H., 1982, IEEE Trans. Magn. MAG-18, 961. Potocky, L, J. Kovac, L Novak, A. Lovas, L Pogany and E. Kisdi-Koszo, 1987, Prec. Conf. Soft Magnetic Materials 8, 166. Rawlings, R.D., 1970, Mechanical Anisotropy. A Study of 3% Silicon Iron Sheet with a Marked (110) [001] Texture (JARN Metallurgical Materials, London). Salzmann, E, W. Grimm and A. Hubert, 1983, J. Magn. Magn. Mater. 31-34, 1599. Sate, Y., T. Sate and Y. Okazaki, 1987, Prec. Conf. Rapidly Quenched Metals, Montreal, 6, EPE7. Sate, J. and T. Yamada, 1992, IEEE Trans. Magn. MAG-28, 2775. Sievert, J., 1992, Int. J. Appl. Electromagn. 3, 321.

413

Sharp, M.R.G. and K.J. Overshott, 1973, Prec. lEE 120, 1451. Shilling, J.W. and G.L Houze, 1974, IEEE Trans. Magn. MAG-10, 195. Shishido, I-I., T. Kan, Y. Ito and H. Shimanaka, 1982, IEEE Trans. Magn. MAG-18, 1412. Simmons, G.H. and J.E. Thompson, 1971, Prec. IEE 118, 1302. Slama, J. and M. Prejsa, 1980, J. Magn. Magn. Mater. 19, 42. Smith, 1., 1951, Physica 16(6), 612. Smith, C.H., Nathasingh D. and H.H. Liebermann, 1984, IEEE Trans. Magn. MAG-20, 1320. Soinski, M., 1982, Materialpriifuag 24(9), 321. Soinski, M., 1983, Materialpriifung 25(1-2), 31. Soinski, M., 1984a, Cryogenics No 3, 133. Soinski, M., 1984b, IEEE Trans. Magu. MAG-20, 172. Soinski, M., 1984c, Arch. Elektrotech. 67, 403. Soinski, M., 1985a, J. Magn. Magn. Mater. 53, 54. Soinski, M., 1985b, Materialpriifung 27 (12), 390. Soinsld, M., 1985c, Steel Res. 56, 525. Soinsld, M., 1987a, Modelling Simulation and Control A 14(3), 1. Soinski, M.B. Szymanski and W. Wilczynski, 1987b, Przegl. Elektrotech. No 2, 59. Soinski, M. and L. Kieltyka, 1987c, Prec. Conf. Soft Magnetic Materials 8, 101. Soinski, M., 1987d, Przegl. Elektrotech., No. 11-12, 333. Soinski, M., 1987e, Wiad. Elektrotech., No 13-14, 337. Soinski, M., 1987f, IEEE Trans. Magn. MAG-23, 3878. Soinski, M., 1989, IEEE Trans. Magn. MAG-25, 3166. Soinski, M., 1990, J. Mater. Shaping Technol. 8, 187. Sokolov, B.IC, Y.N. Dragoshanskiy and V.V. Gubernatorov, 1989, Phys. Met. Metall. 68(3), 150. Stanbury, H.J., 1987, Prec. Conf. Soft Magnetic Materials 8, 190. Szymanski, B., M. Soinski and I. Skorvanek, 1987, Prec. Syrup. Magnetic Properties of Amorphous Metals: Reprinted in Magnetic Properties of Amorphous Metals (Elsevier, Amsterdam, 1987) p. 154. Taguchi, S., T. Yamamoto and A. Sakakura, 1974, IEEE Trans. Magn. MAG-10, 123. Takahashi, M. and Ch.e. Kim, 1977, Jpn. J. Appl. Phys. 16, 2061.

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chapter 5 MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS

Nguyen Hoang LUONG Van der Waals-Zeeman Laboratorium University of Amsterdam The Netherlands (On leave from the Cryogenic Laboratory Faculty of Physics, University of Hanoi Vietnam)

and

J.J.M. FRANSE Van der Waals-Zeeman Laboratorium University of Amsterdam The Netherlands

Handbook of Magnetic Materials, Vol. 8 Edited by K. H.J. Buschow ©1995 Elsevier Science B.V. All rights reserved 415

CONTENTS 1. I n t r o d u c t i o n

.................................................................

417

2. B r i e f s u r v e y

.................................................................

418

2.1. R a r e - e a r t h a n d r a r e - e a r t h - - C u c o m p o u n d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. R C u 2 c o m p o u n d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

......

418 420

3. T h e o r e t i c a l a s p e c t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Crystal-field i n t e r a c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. E x c h a n g e i n t e r a c t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. D e s c r i p t i o n o f m a g n e t i c p r o p e r t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

421 421 424 425

4. M a g n e t i c p r o p e r t i e s o f R C u 2 c o m p o u n d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. C e C u 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

432 432

4.2. P r C u 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. N d C u 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. S m C u 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

436 441 447

4.5. G d C u 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. T b C u 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

451 457

4.7. D y C u 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8. H o C u 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9. E r C u 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.TmCu 2 .................................................................

463 468 473 480

5. C o m p a r i s o n o f i s o s t r u c t u r a l c o m p o u n d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

485

6. A c k n o w l e d g e m e n t s

488

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References .....................................................................

416

488

1. Introduction

Among the rare-earth intermetallic compounds, the RT2 (R = rare earth, T = transition metal) with the cubic Laves phase structure have been widely studied in the past decades. Initially, less attention has been paid to the RCu2 compounds after the early magnetic investigations performed by Sherwood et al. (1964). However, during the last decade substantial progress has been achieved in the study of the magnetic properties of the RCu 2 compounds. The RCu 2 compounds, except for LaCu2, crystallize in the orthorhombic CeCu2 structure, i.e. belong to the systems with low symmetry. The compounds show a large variety of magnetic behaviour. Most of the RCu 2 are antiferromagnetic. T b C u 2 has the highest N6el temperature with a value for TN near 50 K. The magnetic properties of the RCu 2 are largely affected by the crystal field (CF) effects. The magnetic ordering temperatures in the RCu 2 compounds are low and crystal-field and exchange interactions can be studied by varying temperature and/or magnetic field. Changes in the magnetic structure occur in most of the compounds below the N6el temperature and most of the compounds exhibit metamagnetic transitions below the magnetic ordering temperature. The metamagnetic transition in Z b C u 2 has been discussed by Gignoux and Schmitt (1991) as a typical example of a single-step transition in a metallic system. Unlike the RnTm compounds where T is a magnetic transition metal, 4f magnetism can be investigated in the RCu 2 without disturbing effects of the d magnetism. An extensive description of the magnetic properties of 4f-3d intermetallic compounds has been presented in the reviews by Buschow (1977), Kirchmayr and Poldy (1979) and Buschow (1980). The magnetic properties of binary rare-earth 3d-transition metals intermetallic compounds with the transition metals Mn, Fe, Co, and Ni have been reviewed by Franse and Radwanski (1993). It turns out to be of considerable interest to carry out in a systematic way investigations of the physical properties within isostructural series generated by the various rare-earth elements. As far as the binary rare-earth intermetallic compounds with non-magnetic partners are concerned, to our knowledge, only a review is available for the magnetic properties of the RA12 compounds (Purwins and Leson 1990). In the present work, the magnetic properties of the RCu 2 are reviewed. Transport properties of these compounds are not particularly considered in this review. For these properties we refer the reader to the review of Gratz and Zuckermann (1982). Thermal conductivity and electrical resistivity of some heavy RCu 2 compounds have been reported by Bartkowski et al. (1992). LaCu2, which crystallizes in the hexagonal A1B2 structure, is not included in this work. EuCu 2 and Y b C u 2 a r e both intermediate-valence compounds and are not considered in detail. We mention that EuCu 2 is known to be antiferromagnetic at 417

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N.H. LUONGand J.J.M. FRANSE

4.2 K (Sherwood et al. 1964, Wickman et al. 1966). M6ssbauer studies have been performed on this compound by Nowik et al. (1973), Abd-Elmeguid and Kaindl (1978) and Abd-Elmeguid et al. (1981). YbCu2 is paramagnetic (Sherwood et al. 1964). The anomalous properties of Yb-Cu (as well as Ce-Cu) compounds have been reviewed by Bauer (1991). CeCu2 is a Kondo compound, and hence, is discussed not in detail. By lack of information we have also to omit the radioactive compound PmCu2. YCu2 and, occasionally, LuCu2, are treated as reference materials in connection with the other magnetically ordered compounds. Summarising, we will deal with a systematic discussion of the RCuz compounds where R = Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er and Tm. A brief survey on all existing R-Cu compounds is given in section 2. In section 3 we present a short theoretical consideration of crystal-field and exchange interactions as well as of some important magnetic properties. Experimental results on RCu2 compounds are reviewed in section 4. A comparison of different RCu 2 compounds is given in section 5. 2. Brief survey 2.1. Rare-earth and rare-earth-Cu compounds In this chapter, the term lanthanides refers to the elements from La to Lu. Lanthanides to the left of Gd in the periodic table are referred to as the light rare earths, and the remaining ones as the heavy rare earths. The rare earth elements are of fundamental interest because most of them behave as well-defined ions in solids. Exceptions are known to be the elements Ce, Eu and Yb. The number of binary R - T (T = 3d transition metals Mn, Fe, Co, Ni) intermetallic compounds, available at present, amounts to nearly 200 (Franse and Radwanski 1993). As far as the R-C~ compounds are concerned, the nature of the compound formation is summarized in fig. 2.1. The light lanthanides RCu phases as well as YbCu crystallize in the orthorhombic FeB-type of structure. The remaining RCu compounds are formed within the cubic CsCl-type of structure. SmCu and EuCu can adopt both structures. ScCu and YCu have the CsCl-type of structure as well. C e C i l 2 crystallizes in an orthorhombic structure which is the prototype for the RCu2 series. I - ~ C u 2 and S c C u 2 a r e exceptions, forming the hexagonal A1BE-type and the tetragonal MoSi2-type of structure, respectively. The light rare earths RCu5 compounds have the hexagonal CaCus-type of structure, whereas the RCu5 compounds with the cubic AuBs-type of structure are formed for the heavy rare earths. An exception is YbCus, that crystallizes in the CaCus-type of structure. The RCu 6 compounds crystallize in the orthorhombic CeCu6-type of structure and have been reported to exist for the light rare earths from La to Sm. For the heavy rare earths, only Gd and Tb form phases with the RCu6 stoichiometry. The existence of the phases RzCu7, RCu4, R2Cu9 and RCu 7 has been reported for a limited number of rare earth elements (Subramanian and Laughlin 1988).

MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS

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I

I

Sc Y b Ce Pr ~ Pm SmEu Gd Tb Dy ~ Er Tm Yb Lu

Fig. 2.1. Compound formation in the R-Cu systems. After Subramanian and Laughlin (1988).

/

b

F,1 Fig. 2.2. Chemical unit cell of the orthorhombic CeCu2-type of structure. The open and closed circles indicate the rare-earth and copper atoms, respectively. After Hashimoto et al. (1979a).

420

N.H. LUONG and J.J.M. FRANSE TABLE 2.1 Crystallographic data for the RCu2 compounds.

R

Lattice parameters (/~) a b e

Ref.

Melting Reaction point (*C) type

Ref.

Sc Y La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

3.29 4.305 4.346 4.425 4.400 4.387 4.37 4.360 4.45 4.320 4.310 4.300 4.280 4.275 4.266 4.28 4.245

[1] [4] [4] [4] [4] [4] [2] [4] [4] [4] [4] [4] [4] [4] [4] [4] [4]

990 935 834 817 841 840 850 860 597 860 870 890 915 935 960 757 1000

[2, 3] [5] [4] [6] [7] [8] [2] (proposed) [2] [9] [10] [2] [11] [2] [12] [2] [13] [2]

6.800 7.057 7.024 7.059 6.96 6.925 7.25 6.858 6.825 6.792 6.759 6.726 6.697 6.76 6.627

8.388 7.315 3.807 7.475 7.435 7.420 7.40 7.375 7.54 7.330 7.320 7.300 7.290 7.265 7.247 7.40 7.220

congruent congruent congruent congruent congruent congruent congruent peritectic congruent congruent congruent congruent congruent congruent peritectic congruent

[1] Dwight et al. (1967). [2] Subramanian and Laughlin (1988), estimated on the basis of the systematics of crystallographic data in the R--Cu systems. [3] Markiv et al. (1978), quoted by Subramanian and Laughlin (1988). [4] Storm and Benson (1963). [5] Chakrabarti and Laughlin (1981). [6] Rhinehammer et ah (1964). [7] Canneri (1934), quoted by Subramanian and Laughlin (1988). [8] Carnasciali et ah (1983a). [9] Costa et al. (1985). [10] Carnasciali et ah (1983b). [11] Franceschi (1982). [12] Buschow (1970). [13] Iandelli and Palenzona (1971).

2.2. RCu2 compounds The orthorhombic CeCu2-type o f structure has first been determined by Larson and Cromer (1961). The chemistry o f the CeCuz-type structure was the subject o f considerable discussion (e.g., Debray 1973, Michel 1973, Bruzzone et al. 1973). This structure belongs to the space group Imma. The rare-earth atoms and the copper atoms o c c u p y the 4e and 8h sites, respectively, and form a double-layer structure along the c axis as s h o w n in fig. 2.2 (Hashimoto et ah 1979a). Crystallographic data for the RCu2 c o m p o u n d s are collected in table 2.1. As can be seen f r o m this table, the lattice parameters and the melting temperatures o f EuCu2 and YbCu2 are anomalous, indicating the divalent state o f Eu and Yb. For more detailed information we refer the reader to the review paper of Subramanian and Laughlin (1988), in which the R--C-~ phase diagrams have been compiled and evaluated. The Ce--Cu phase diagram is s h o w n in fig. 2.3, as an example.

MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS

421

Weight Percent Cerium

0 10 20 30 40

50

1200~ 1000

60

70

80

r

~

90 r

100

-~. ./~ 817 C

798"( 726°C

:'%111 `°

-2 r'~

•

E

tOO

II

-

2oo -

I

~I I~I

.~

= _.__L__--- . . . . . . . . . . . . . . . . . . . .

0 ~

0

10

20

Cu

/

72

30

40

50

60

70

~

80

61"C

90

AtomicPercentCerium

100

Ce

Fig. 2.3. The Ce-Cu phase diagram. After Subramanian and Laughlin (1988).

3. Theoretical aspects

3.1. Crystal-field interaction The 4f electrons of a rare-earth ion in a solid, being considered as well localized and separated from other charges, experience an electrostatic potential VcF(r) that originates from the surrounding charge distribution. The potential reflects the point symmetry of the site of the rare-earth ion. In case there is no overlap between this charge distribution and the wave functions of the 4f state, VCF satisfies Laplace's equation and can be expanded in terms of the spherical harmonics, Ynm. The value of n in this expansion is limited to 6, as higher multipoles cannot cause electronic transitions between states of the 4f ion. For details see Hutchings (1964), Fulde and Loewenhaupt (1985). It is an experimental fact that for the rare-earth ion, the spin-orbit interaction is much larger than the crystal-field and exchange interactions. A review of experimental data of the spin-orbit interactions based on the observation of intermultiplet transitions has been presented by Osborn et al. (1991). Owing to this fact, it is usually sufficient to consider the lowest multiplet J, given by Hund's rules. This

422

N.H. L U O N G and J.J.M. FRANSE

limitation results in a substantial simplification in computations. Eu +3 and Sm +3 are ions where the analysis requires the involvement of a higher multiplet since the higher multiplet is only 530 K and 1500 K above the ground multiplet, respectively. The crystal-field Hamiltonian that describes part of the electron-electron interactions in the solid due to the electrostatic interaction of the aspherical 4f charge distribution with the aspherical electrostatic field arising from the surrounding, can be written as: oo

n

AT n=0 ra=0

(3.1) i

where f,,m are Tesseral harmonics describing the spatial distribution of the charge associated with the 4f electrons; the summation over i is over all 4f electrons. Anm describes the spatial distribution of the charge surrounding the 4f electrons. Within the Stevens formalism the summation in eq. (3.1) over the 4f electrons leads to matrix elements of the total angular momentum: n rr~ Z fnra(ri) = On(7 ,4f)On (J'), i

(3.2)

where the x, y, z coordinates of a particular electron in the functions fnm(rl) are replaced by the components Jx, Jy, Jz of the multiplet J. The O F are the Stevens equivalent operators (Stevens 1952). On is the appropriate Stevens factor of order n which represents the proportionality between operator functions of x, y, z and operator functions of Jx, Jy, Jz. The parameter On is denoted as aj, flj, 7J for n = 2, 4, 6. The sign of On represents the type of asphericity associated with each Onm term describing the angular distribution of the 4f-electron shell. In particular, the factor ecj describes the ellipsoidal character of the 4f-electron distribution. For ~j > 0, the electron distribution associated with Jz = J is prolate, i.e. elongated along the moment direction whereas for cq < 0 the 4f-electron-charge distribution is oblate, i.e. expanded perpendicular to the moment direction. For ~j = 0 (which is the case of the Gd +3 ion) the charge density has spherical symmetry. (r~f) is the mean value of the n-th power of the 4f radius. Values for the average value (r~f) over the 4f wave function have been computed on the basis of Dirac-Fock studies of the electronic properties of the trivalent rare-earth ions by Freeman and Desclaux (1979) and are presented in table 3.1. Within the ground-state multiplet the crystal-field Hamiltonian (3.1) is written in the conventional form: oo

HCF = E

n

Z

BnmO~n(Y)'

(3.3)

n m=0

where Bnm are called the crystal-field parameters. Usually these parameters are evaluated from the analysis of experimental data. The parameters Bnm can be written as"

B m = On(r4r)A n n

gll

(3.4)

MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS

423

TABLE 3.1 The second-, fourth- and sixth-order multipole moments On(r~f) of the trivalent rare-earth ions. cq, flj and "Vl ate the Stevens factors of the second-, fourth- and sixth-order, respectively. The values for (r2), (r~f) and (r 6 ) have been taken after Freeman and Desclaux (1979). The multipole moments values enter into the relations between the CF parameters B m and the CF coefficients Anm. *-values for Pr have been deduced from interpolation. After Franse and Radwanski (1993).

R

S

I~> (~02)

I~> (%,)

I~> (~g)

Ce Pr Nd Sm Gd Tb Dy Ho Er Tm Yb

5/2 4 9/2 5/2 7/2 6 15/2 8 15/2 6 7/2

1.309 1.208" 1.114 0.9743 0.8671 0.8220 0.7814 0.7446 0.7111 0.6804 0.6522

3.964 3.396* 2.910 2.260 1.820 1.651 1.505 1.379 1.270 1.174 1.089

23.31 18.72" 15.03 10.55 7.831 6.852 6.048 5.379 4.816 4.340 3.932

~,l~> z,l~,'~> ,,/~4~,> (10-3%2) (10-,%,) (10-6:) -74.8 -25.54 -7.161 +40.209 0 -8.303 -5.020 - 1.676 +1.831 +6.976 +21.04

+251.7 -24.95 -8.471 +56.527 0 +2.021 -0.891 -0.459 +0.564 +1.916 -18.857

0 +1141.66 -570.96 0 0 -7.683 +6.260 -6.959 +9.969 -24.33 -581.94

in which expression terms related to the 4f ion, On(r'~f), and the term related to the surrounding charges, AT, are separated. The coefficients A T are known as the crystal-field coefficients. Values for On(r~f) have been collected by Franse and Radwanski (1993) and presented in table 3.1. The computation of the CF coefficients, AT, from microscopic, ab initio, calculations is a difficult problem. A full band-structure calculation of the charge distribution over the unit cell and, consequently, of the full set of CF coefficients, is lacking for almost all compounds. In some cases, the point-charge (PC) model, with electron charges centred at the ion positions in the lattice, can give the correct sign of the leading second-order crystal-field coefficients. However, this model is questioned, especially in metallic systems where the contribution of valence electrons is expected to be significant (Schmitt 1979a, b). This has been confirmed by band-structure calculations by Coehoorn (1991) who concluded that the second-order CF coefficient A ° is mainly determined by the asphericity of the valence-shell electron density of the rare earth under consideration. In band-structure calculations for GdCo 5 Coehoorn and Daalderop (1992) have found that, although the on-site contribution to A ° is the dominant effect, the lattice contribution cannot be neglected. The calculation predicts the correct sign and order of magnitude for A ° but the calculated value exceeds the experimental one by a factor of four. This discrepancy remains to be explained. For more detailed discussion we refer the reader to the reviews of Franse and Radwanski (1991, 1993), Givord and Nozieres (1991), Purwins and Leson (1990). There is only a limited number of compounds for which quantitative data for the CF interactions are available. Discussions are still continuing even for the bestknown systems like the cubic Laves-phase RT2 or the hexagonal RNi5 compounds. The situation becomes more complex for the systems with a lower crystal symmetry.

424

N.H. LUONG and J.J.M. FRANSE

The orthorhombic RCu2 compounds, which we are dealing with in this chapter, belong to these latter cases. For cubic symmetry (in case of the RA12 compounds, for instance) the crystal field is described by only two parameters B4 and B6: n c F - B 4 ( O ° -~- 5 0 4) -+ B 6 ( O ° - 21064),

(3.5)

whereas for the orthorhombic symmetry, nine crystal-field parameters B~ are needed: 00

22

00

22

H ~ = B202 + B202 + B404 + B404 + oo

+ B 606 +

22 B 606

44

66

44 B404+

(3.6)

+ B6 06 + B 606 •

3.2. E x c h a n g e interactions

Since the 4f electrons are strongly localized, direct exchange of the 4f wave functions of different rare-earth ions is precluded. In order to account for the observed ordering temperatures, one has to resort to an indirect exchange between the rare-earth ions via the conduction electrons. This interaction is known as the Ruderman-Kittel-KasuyaYosida (RKKY) interaction (Ruderman and Kittel 1954, Kasuya 1956, Yosida 1957). Within this approach, a spin of a rare-earth ion interacts with the conduction electrons. The effect of this interaction on the conduction electron density is not the same for the spin-up and for the spin-down electrons. Therefore, a spin polarization is created that interacts with the spin of a neighbouring rare-earth ion. The result is an indirect exchange interaction between the rare-earth ions. The RKKY model does not distinguish d- and s-conduction electrons, in contrast to the model of indirect exchange proposed by Campbell (1972) in which the role of the rare-earth 5d electrons is emphasized. Usually, in the RKKY model the simplifying assumption is made that the exchange integral is of the s-f type and constant and that the Fermi surface is spherical (free electrons). Efforts have been made to overcome the shortcomings of the RKKY model (Kirchmayr and Poldy 1979), but in many cases the improvements achieved are minimal. We list in the following some references in which the indirect exchange interaction in rare earths and rare-earth intermetallic compounds is described: Taylor (1971), Taylor and Darby (1972), Coqblin (1977), Kirchmayr and Poldy (1979), Bnschow (1980), Givord and Nozieres (1991), Franse and Radwanski (1993). The exchange interaction is assumed to be of the Heisenberg type: H e x = --

St,

(3.7)

i,j where the summation is over all the magnetic ions in the lattice and where J q is the exchange parameter between local spins at the sites i and j. Limiting ourself to the ground state, we can project the spin onto the total angular momentum and obtain Hex = - E i,j

JRR(gJ -- 1)zJ'J,

(3.8)

MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS

425

where J ~ is an effective exchange parameter between the 4f-spins of the rare-earth ions at sites i and j. In the RKKY model, the ordering temperature is expected to be proportional to the so-called De Gennes factor (g - 1)2j (J + 1). In the molecular-field approximation, the exchange interaction of a single R ion with other R ions is approximated by an effective molecular field B,~. Then one has:

Hm = -gj#BJ.Bm.

(3.9)

The field B,~ is related to the molecular-field coefficient naa by

B,,, = ~,,~R(M),

(3.10)

where "N = 2 ( g j - 1)/gj

(3.11)

and (M) is the thermal average of the magnetic moment. Taking into account the Zeeman magnetostatic energy term caused by an external magnetic field B, the Hamiltonian of the magnetic interactions is given by

Hm = --gJ#BJ" (Bin + B).

(3.12)

Thus, the ground state is described by the total Hamiltonian 6

n

HI~ = ~ , ~

B~O~ - g~u~J.(n,,, + n).

(3.13)

n=0 m=0

Eigenvalues and eigenfunctions are obtained from the diagonalization of this Hamiltonian, provided that the relevant CF and exchange field parameters are known.

3.3. Description of magnetic properties In the following we describe some important magnetic properties which are affected by the crystal-field and exchange interactions and mention some properties which are of help to get information on the energy splitting scheme of the rare-earth ion.

3.3.1. Magnetic moment For describing the magnetic moment of a rare-earth ion one has to calculate the thermal average of the magnetic moment. This quantity is given by: 2J+l

(u) = ( i / z ) ~

~ exp(-E, IkT),

(3.14)

i=1

where #i is the magnetic moment of the ith energy level,

#i = -gJ#B(i~]]i)

(3.15)

426

N.H. LUONG and J.J.M. FRANSE

and where Z is the partition function: 2J+l

Z = E exp(-Ei/kT).

(3.16)

i=1

Ei and Ii) are the eigenvalues and eigenfunctions of the Hamiltonian (3.13). Due to the complicated magnetic structure, the analysis of the magnetization process in case of the RCu2 compounds has been done by decomposing the structure into several sublattices (see, e.g., Iwata et al. 1987, 1988, 1989, Divis et al. 1987, Kimura 1985, 1987a, b). In this way the molecular field Bin(i) acting on the ith sublattice can be written as:

Bin(i) : E nq(Mj), J

(3.17)

where (Mj) is the sum of the thermal average of the magnetic moments per ion in the jth sublattice, nq the molecular field coefficients describing the interaction of ith and jth sublattices. The summation is over all sublattices.

3.3.2. Paramagnetic susceptibility In the paramagnetic region the susceptibility can be described by a Curie-Weiss law: C X - T - 0p'

(3.18)

where C is the Curie constant and Op the paramagnetic Curie temperature. Sometimes, the paramagnetic susceptibility can better be represented by a modified CurieWeiss law in the form: C X = X0 + T - 0p'

(3.19)

where the extra term Xo is a weakly temperature-dependent Pauli susceptibility. Using the eigenvalues Ei and appropriate eigenfunctions Ii) of the crystal-field Hamiltonian, HCF, one can calculate the paramagnetic susceptibility, X~F), along the principal axes (a = a, b, e) by the general Van Vleck formula (see, e.g., Zajac et al. 1987):

- ~

~ ( I(ils~li)12÷ (3.20)

+ 2kT ~j¢, ](JlJ~'li)12 -~j---~i j exp(-Ei/kT),

MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS

427

where J,~ are the components of the operator.J. For a polycrystalline sample the susceptibility arising from crystal-field Hamiltonian, X~, is obtained by the approximation: =

+

+

(3.21)

The crystal-field parameters can be derived from the paramagnetic Curie temperatures 9a, 8b and 8¢ along each principal axis. According to the modified molecular-field model given by Bowden et al. (1971), these paramagnetic Curie temperatures can be expressed in terms of the crystal-field parameters by the following relations (Shohata 1977): Oa = ep +

(2J

1)(2J + 3) (B ° + 10

0b = 0p + ( 2 J - 1)(2J + 3)BO,

5

(3.22)

0c = 0p + (2J - 1)(2J + 3) (B o _ Bo),

10 where B ° and B22 are measured in kelvin. On substituting the measured 0a, 0b and 0e values, the crystal-field parameters B ° and B22 can be derived. The influence of higher-order crystal-field parameters on the paramagnetic Curie temperatures has been studied by Zajac and Maczak (1985). 3.3.3. Specific heat The specific heat is written as the sum of electronic (ce), lattice (phonon) (Cph), magnetic (Cm), and nuclear (e,) contributions: C:

Ce ' + C p h + Cn - ] - c m.

(3.23)

The nuclear part of the specific heat is related to the hyperfine interactions of the 4f shell with the nuclear moment of the 4f ion and is only significant at low temperatures for most of the 4f ions. The largest nuclear contribution is observed in the ordered Ho compounds. This is also the case for the HoCu2 compound (Luong et al. 1985b). The electronic part is written as: ce = 7T

(3.24)

with the coefficient 7 yielding information on the density of states at the Fermi level. The experimental value for the electronic coefficient 7 from specific-heat measurements should be compared with -/-values from band-structure calculations. For the RCu2 compounds, band-structure calculations have been performed for YCu2 (Harima et al. 1990, Harima and Yanase 1992). The resulting value for the specific heat coefficient 7 is 3.47 mJ/K2 mol, which is about two times lower than the experimental value of 6.7 mJ/K2 mol (Luong et al. 1985a).

428

N.H. L U O N G and J.J.M. FRANSE

The phonon part of the specific heat (for a compound with r atoms per formula unit) is approximated by: Cph = 9 r R ( T / O D ) a

00D/T z4ez

(e~ _ 1)2 dz,

(3.25)

where OD is the Debye temperature, R the gas constant. In compounds with a magnetic R element the magnetic contribution, Cm, is associated with the increasing population of excited localized states. These localized states are due to CF and molecular field interactions of the 4f ion which lift the ( 2 J + 1)-fold degeneracy of the ground-state multiplet. Provided that the energy-level scheme is available, it is straightforward to evaluate this contribution by using the expression:

cm(T)

(3.26)

-02F = - - T OT2,

where F is the free energy of the R system. The reverse procedure is rather problematic. Contributions like the electronic and lattice contributions have to be evaluated and subtracted. However, in general it is difficult to separate the magnetic part from the other contributions. Isostructural compounds with non-magnetic R elements are usually employed to estimate the non-magnetic part of the specific heat. In the RCu 2 compounds, LaCu2 possesses a different crystallographic structure and crystallizes in the hexagonal AIB2-type of structure. Therefore, YCu2 is taken as the non-f reference material instead of LaCu 2 and data for YCu 2 are often used in the analysis of the experimental data on RCu 2 compounds. In the paramagnetic state the magnetic part of the specific heat is given by the Schottky contribution (Fulde 1979): CSeh = ~

Z

exp(-Ei/kr)

-

Ei e x p ( - E i / k T )

.

(3.27)

In the simple case of a two-level system, the Schottky specific heat attains a maximum value at an intermediate temperature Tm given by (Gopal 1966):

(g0/al) exp(6/Tm) = [(6/Tm) + 21/[(6/T.) - 21,

(3.28)

where 6 is the energy separation (measured in kelvin) between the ground state and the first excited state; go and gl denote the degeneracies of these two levels. Provided that the temperature Tm and the maximum value of the specific heat, cSch(Tm), are known, one can evaluate 6 according to the expression (Gopal 1966): 6 = [4CSch(Tm)/R + 411/2Tm.

(3.29)

In general, relevant information about the lower part of the energy-level scheme can be obtained from the specific-heat measurements. Experiments in external magnetic fields can give further information as the applied field additionally shifts the energy levels. The shift is not uniform for all levels and is largely dependent on the direction of the applied field. This effect can be detected by the measurements of the specific heat on single-crystalline samples.

MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS

429

3.3.4. Thermal expansion The thermal expansion contains electronic (fie), lattice (phonon) (flph), and magnetic, (tim) contributions:

= Ze + Zph + t - ,

(3.30)

where we neglect a nuclear contribution. The conduction electrons contribute to the thermal expansion through the volume dependence of their entropy. This entropy is related to the density of state at the Fermi surface. The electronic part of thermal expansion is a linear function of temperature, i.e.

fie = aT.

(3.31)

The phonon contribution to the thermal expansion, like the contribution to the specific heat, is approximated by: ~ph = b(T/OD)3

fOOD/T z4ex (e~'_~])2

dx,

(3.32)

where b is a numerical constant. In order to evaluate the magnetic contribution to the thermal expansion, nonmagnetic (electronic and lattice) contributions have to be subtracted from total thermal expansion as in the case of the specific heat. Isostructural compounds with nonmagnetic R elements are used to evaluate this non-magnetic part of the thermal expansion. For RCu 2 compounds, like in case of the specific heat, YCu 2 often serves as the reference material. In the paramagnetic region, the magnetic part of the thermal expansion arises from the thermal excitation of a series of energy levels E0, E b . . . with degeneracies go, g b . . . . This (Schottky) contribution is given by (Barron et al. 1980):

2

(Ei) (E¢/i) },

(3.33)

where 7i = - ( d In Ei/d In V) is the crystal-field Grtineisen parameter of the individual energy level Ei and where ~T is the isothermal compressibility. The brackets denote thermal averages of the form:

(x,) - E glz, e x p ( - E j k T ) / E g' exp(-EjkT). i

(3.34)

i

3.3.5. Graneisen analysis In the study of magnetic systems the specific heat and the thermal expansion are very important. The combined analysis of specific heat and thermal expansion can give valuable information on the system under consideration. Here we briefly describe a procedure that has successfully been applied to several different systems.

430

N.H. LUONG and J,J.M. FRANSE

An arbitrary contribution to the specific heat, el, is related to a corresponding contribution to the thermal expansion, ti, by a so-called Griineisen relation:

.l-'i = Vti/tgei,

(3.35)

where V is the molar volume, t¢ the compressibility and/~i the appropriate Griineisen parameter. For the electronic Griineisen parameter we have (see eqs (3.24) and (3.31)):

-re = V a / x T .

(3.36)

Using eqs (3.25) and (3.32), for the lattice Griineisen parameter we obtain:

l"ph = V b / 9 r Rt¢.

(3.37)

In treating the magnetic contributions to the specific heat and to the thermal expansion, a straightforward approach is to calculate an effective Griineisen parameter, Pen, by the relation: Feff = V t m / t C e m

(3.38)

and to follow its variation with temperature. A pronounced temperature dependence of Fee(T) indicates the presence of several contributions. ErCu2 can serve as an example, in which a change in sign in the parameter Feff is observed upon increasing the temperature (see section 4.9). In this case, at least two different contributions to em and tim can be distinguished. Therefore, we write: e m = err + Cef and

t m = t l r "lt- t e f

(3.39)

in which the Cm and tm are split up into two contributions, the 'long-range' magneticorder contributions % and tmr and the contributions eel and t~f associated with the crystal-field splitting of the energy levels, respectively. We have:

fiFi,

Feff= E

i = lr, cf,

(3.40)

i

with F~ and F~f, in first approximation, temperature-independent Grtineisen parameters and with fi = ci/em. The separated terms in Cm and tm can subsequently be written as: /"lr -- Fef cm'

(3.41)

ccf - /'lr - Fee era,

(3.42)

Clr =

1 - F~f/Fe~

ttr -- ~ - ~

tin,

F~f 1 - F e f d r t r

tef = Fe----ff~ -- ~

(3.43) tim.

(3.44)

MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS

431

Fransse et al. (1985) and Luong et al. (1985b) applied this method in the analysis of the ErCu2 compound (see section 4.9). The method of the combined analysis of specific heat and thermal expansion has been described in more detail by Brommer and Fransse (1990) and successfully applied to a variety of materials (see Brommer and Fransse 1990).

3.3.6. Inelastic neutron scattering Inelastic neutron scattering (INS) gives valuable information on the dynamic properties of the system. INS techniques enable to determine the crystal field scheme of the ground state multiplet of a R a÷ ion as the available neutron energies are of the order of this splitting. The scattering amplitude S(q, w) as function of the momentum transfer q and the energy transfer hw, depends on the matrix element and is given by:

S(q, w) ,~ I(il&LIf)[ z,

(3.45)

where Ja. is the component of the total angular momentum operator perpendicular to q. (i[ and If) denote the initial and final 4f-electron states (= CF levels), respectively. Within the dipole approximation, the intensities for the transitions are determined by the corresponding dipole matrix elements. For a review of the INS spectra in metallic 4f systems see, for instance, Fulde and Loewenhaupt (1985). The INS spectra of a system containing 4f ions depend very much on the relative strength of the CF and exchange interactions. The spectra belonging to only CF transitions contain a few well-defined peaks related to magnetic excitations between CF levels. The number of the transitions as well as the probability for these transitions is determined by matrix elements between the crystal-field eigenstates. In principle, these are governed by the local symmetry of the CF interactions. These transitions are typically single-site excitations. Including the exchange interactions, the transitions on neighbouring ions are coupled and lead to propagating modes. The spectra become more complex and the analysis is not straightforward. For RCu2 compounds, and NdCu 2 is a good example, at least two INS experiments (one above and one below antiferromagnetic ordering temperature) are needed to detect deviations from the CF-only case in the excitation spectrum (Loewenhaupt 1990). For orthorhombic and lower symmetries, the J ground-state multiplet splits into (2J + 1)/2 doublets for half integer values of J and into (2J + 1) singlets for integer d. In the simplest case (Ce 3+, J = 5/2) this gives a level scheme with 3 doublets and 3 possible inelastic transitions. In the most complicated case (Ho 3+, d = 8) a level scheme consists of 17 singlets and up to 136 inelastic transitions are possible (Loewenhaupt 1990). Loewenhaupt (1990) has also pointed out that not all transitions may have non-zero matrix elements but already the clear identification of 10 or 20 inelastic CF transitions is a great challenge for magnetic neutron spectroscopy.

3.3.7. MOssbauer spectroscopy Mrssbauer spectroscopy provides information on at least two important parameters: eft, and the quadrupolar splitting. the hyperfine field, Hhf

432

N.H. LUONG and J.J.M. FRANSE

The hyperfme field is used to evaluate the value for the magnetic moment of the unfilled 4f or 3d shell, usually by taking it directly proportional to the magnetic moment (as long as the additional contributions originating from core polarization and conduction electrons can be neglected). In that case, the temperature dependence of H ~ reflects the temperature dependence of the (4f or 3d) magnetic moment. The quadrupolar splitting gives information about interactions between the quadrupolar moment of the nuclei and the gradient of the electric field. The gradient of the electric field at the rare-earth nuclei results partly from the quadrupolar moment of the 4f shell and partly from that of the surrounding charges. Having an estimate of the 4f-shell contribution, the quadrupolar splitting arising from the lattice-charge arrangement can be evaluated and transformed into the crystal field coefficient A °. Higherorder CF coefficients cannot be determined by the Mrssbauer spectroscopy because of the much lower intensity of the higher-order multipolar interactions. Mrssbauerspectroscopy data for the RCu2 compounds have been presented by Gubbens et al. (1991, 1992).

4. Magnetic properties of RCn2 compounds 4.1. CeCu2 CeCu2 is a Kondo-lattice compound which shows antiferromagnetic order below 3.5 K (Gratz et al. 1985, Onuki et al. 1985). Earlier magnetic measurements have been reported by Sherwood et al. (1964), Olcese (1977) and Hashimoto et al. (1979b). Owing to the interesting magnetic behaviour, the compound has been extensively investigated. A characteristic feature of CeCu2 is the large value for the coefficient of the electronic contribution to the specific heat (between 80 and 105 mJ/K2 mol) (Gratz et al. 1985, Bredl 1987, Bauer et al. 1988, 1989 and Takayanagi et al. 1990). The specific heat of CeCu2 as a function of temperature is shown in fig. 4.1 (Gratz et al. 1985). Althouth magnetic ordering is observed in properties as specific heat and resistivity, a clear antiferromagnetic transition is not found in the low-temperature magnetic susceptibility (Onuki et al. 1990, Trump 1991 and Trump et al. 1991). Figure 4.2 shows the temperature dependence of the magnetic susceptibility in CeCu2 (Onuki et al. 1990). As can be seen from this figure, the susceptibility along the c axis shows a broad maximum near the Nrel temperature, indicating that the magnetic moments are along the e axis, in agreement with the neutron diffraction results of Trump et al. (1991) and Nunez et al. (1992) (see below). The magnetic susceptibility is highly anisotropic over the whole temperature range as revealed from the experiments (Onuki et al. 1985, Takayanagi et al. 1986, Onuki et al. 1990, Trump et al. 1991). However, the sequence of easy-intermediate-hard directions in CeCu2 is not quite clear. From magnetic measurements, Gratz et al. (1985) have determined an effective moment, #etI, of 2.45#B and a paramagnetic Curie temperature, 0p, of 30 K. The magnetization of CeCu2 has been measured on a polycrystalline sample by Hashimoto et al. (1979b) and by Gratz et al. (1985) and on a single-crystalline

MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS

ceCuz

_

0.5t-

_

433

Z

-,=

¢.i

2

0

0

1

2

3

/,

5

6

T (K) Fig. 4.1. Specific heat of CeCu 2 as a function of temperature. Inset: low-temperature data in a plot of e/T versus T2; the full curve represents e/T = "3'+ AT2exp(-AE/kT) with 7 = 82 mJ/K2 mol, A = 1.4J/K4 tool and AE/k = 1.2K. After Gratz et al. (1985).

0.3

..3 0

--

0.2

CeCu2

==

°~%,,o

QU

",o

0.1 c

0

i

i

J

5

10

T (K) Fig. 4.2. Temperature dependence of the magnetic susceptibility in CeCu 2. After Onuki et al. (1990).

434

N.H. LUONG and J.J.M. FRANSE

2.0 CeCuz

a

'6 ~

1.0

0

0

~

8

B(T) Fig. 4.3. Magnetization along each principal axis in CeCu 2 at 1.3 K. After Onuki et al. (1990).

sample by Onuki et al. (1988, 1990) and by Trump et al. (1991). Figure 4.3 shows the magnetization of single-crystalline CeCu2 at 1.3 K (Onuki et al. 1990). At a field of 1.7 T, the magnetization along the a axis exhibits a change of slope, indicating a metamagnetic phase transition. The magnetization is again highly anisotropic. Trump et al. (1991) have reported that a metamagnetic transition occurs along the a axis at a field of 1.1 T at T = 2 K. The metamagnetic behaviour of CeCu2 has received considerable attention (Gratz et al. 1985, Onuki et al. 1988, 1990, Satoh et al. 1990a, Takayanagi et al. 1990, 1991, Trump et al. 1991). Gratz et al. (1985) have reported neutron-diffraction studies on CeCu2. However, the exact nature of the magnetic ordering could not be determined. Later on, the magnetic structure of CeCu2 was studied by Trump et al. (1991) and Nunez et al. (1992). Figure 4.4 shows the magnetic structure of CeCu2 (Trump et al. 1991). In this structure, the magnetic moments of the Ce atoms are parallel and antiparallel to the c axis (Trump et al. 1991 and Nunez et al. 1992). The magnetic moment of the Ce atoms amounts to 0.33#n at 2.5 K (Nunez et al. 1992). Crystal-field effects in CeCu2 have been studied by different methods. From inelastic neutron-scattering experiments, Loewenhaupt et al. (1988) deduced excited levels at 9 meV (105 K) and 23.1 meV (268 K). Morin et al. (1992) have observed one CF transition at 23.4 meV. This value is very close to that obtained by Loewenhaupt et al. (1988). Morin et al. (1992) reported, however, that there does not exist another clear excitation from the ground state. Uwatoko et al. (1990a,b, 1991, 1992) have measured the thermal expansion of single-crystalline CeCu2. Assuming that the phonon contribution to the thermal expansion of CeCu2 equals that of YCu2, the

MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS

435

%c°I i! t.J

Fig. 4.4. Proposed magnetic structure of CeCu 2. After Trump et al. (1991).

magnetic contribution to the thermal-expansion coefficient of C e C u 2 , av(mag), has been estimated as: o t v ( m a g ) = c z v ( C e C u 2 ) - o~v(YCu2).

(4.1)

Figure 4.5 shows the temperature dependence of C~v(mag). In this figure, the calculated magnetic thermal-expansion coefficient of CeCu2 is also shown (see Uwatoko et al. 1991, 1992 and references therein). The level scheme used in these calculations is t/71 = 106 K and E2 = 334 K, which values are in reasonably good agreement with whose obtained by Loewenhaupt et al. (1988) from neutron scattering and by Loewenhaupt et al. (1990) from thermal-expansion measurements. However, the level scheme E1 = 10 K and t/72 = 800 K which has been used by Takayanagi et al. (1986) for describing the temperature dependence and the anisotropy of the magnetic susceptibility of CeCu2, disagrees with the neutron-scattering results. From de Haas-Van Alphen studies, Satoh et al. (1990b) have shown that the cyclotron effective masses of CeCu2 range from 0.5 to 5.3m0, which values are larger than those found in YCu2 (0.1--0.7m0). This underlines the heavy-fermion state of CeCu2. The light masses of Y C u 2 a r e consistent with the small specific-heat coefficient 7= 6.7 nd/K 2 mol in YCu 2 obtained by Luong et al. (1985a, b). The results of de Haas-Van Alphen measurements of Settai et al. (1990, 1992) agree with those obtained by Satoh et al. (1990b).

436

N.H. LUONG and J.J.M. FRANSE

12 lo 8

CeCuz

o ~

4

-2 0

I

I

100

I

I

200

Fig. 4.5. Temperaturedependenceof the magnetic part of the thermal-expansion coefficient,c~v(mag) = av(CeCu2)- txv(YCu2). The circles representthe measured thermal-expansioncoefficient. The solid line is a calculatedresult for the magnetic thermal-expansion coefficient. After Uwatokoet al. (1991). The effect of substituting Y for Ce in CeCu 2 has been studied by Bauer et al. (1989). Long-range magnetic order in CeCu2 rapidly disappears with increasing Y content. The replacement of Cu by Ag in CeCu2 leaves the orthorhombic structure unchanged (Iandelli and Palenzona 1968). Bauer et al. (1990) and Bauer (1991) have substituted Ag for Cu in CeCu2. On both the Cu-rich and Ag-rich sides longrange magnetic order is observed. In the middle of the concentration range some sort of spin-freezing phenomena is supposed to be present (Bauer 1991). The series Ce(Cu~,Gal_x)2 (0.5 x< z x< 1) has been studied by Bauer et al. (1988) and Bauer (1991). The transition temperature shifts from 3.5 K for z = 1 to about 1.9 K for z = 0.95. For higher Ga contents no indications of long-range magnetic order were resolved down to 1.5 K. The magnetic properties of Ce(Znl_~Cu~)2 have been investigated by Morin et al. (1992). When Cu is replaced by Ni in Ce(CUl_~Ni~)2 the orthorhombic phase is found in the range 0 ~< x ~< 0.5 (Olcese 1977).

4.2. PrCu2 Magnetic-susceptibility measurements performed on polycrystalline PrCu2 by Andres et al. (1972) (fig. 4.6) show a nearly temperature-independent Van Vleck paramagnetic behaviour below 4.2 K, confirming that the lowest crystal-field state is a singlet. The specific heat anomaly around 7 K (fig. 4.6) cannot be due to the onset of magnetic order, since there is no susceptibility anomaly around this temperature. This specific-heat anomaly will be discussed in more detail below. Andres et al. (1972) have shown that PrCu z exhibits cooperative nuclear antiferromagnetic order below 54 mK (see fig. 4.7). Using a molecular-field approximation, Andres (1973) has analyzed the nuclear magnetic order in PrCu2. The calculated ordering temperature is between 10 mK and 1 K. The magnetization at 4.2 K and the susceptibility along each principal axis for a single-crystalline sample has been measured by Hashimoto (1979) and Hashimoto

MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS

437

50 ~0

1.5

m: iJ

30

"-6 E

20

"3"

10

0

10

20

30

40

T(K) Fig. 4.6. Molar susceptibility (closed circles) and specific heat (open circles) of PrCu2 above 2 K. After Andres et al. (1972).

.15

.14 T

1.0

w

.13 E

;~ .12

.11 '

~ .I

~' .2

21 .3

i

o--- 0

.4

T (K) Fig. 4.7. Molar susceptibility (closed circles) and specific heat (open circles) of PrCu2 below 0.5 K. After Andres et al. (1972).

et al. (1979b) (figs 4.8 and 4.9). The results show a large anisotropy in the paramagnetic state. The sequence of the easy to hard magnetization direction at low temperatures is the a, b and e axes, which is different from that obtained at high temperatures: the a, c and b axes. The effective magnetic moment,/ten, amounts to 3.4#a and the paramagnetic Curie temperatures along the a, b and c axes are: 0a = 14 K, Ob = --107 K and Oe = - 3 2 K (Hashimoto et al. 1979b). From these values for the paramagnetic Curie temperatures Hashimoto et al. (1979b) have estimated the second-order crystal-field parameters for PrCu2: B ° -- 4.27 K, and B 2 = 2.97 K. By using a point-charge model, Hashimoto et al. (1979b) have also

438

N.H. LUONGand J.J.M. FRANSE

2

PrCuz Q

I

b c

O0

1

5

3 B (T)

Fig. 4.8. Magnetization curves along the principal axes for PrCu 2 at 4.2 IC After Hashimoto et al. (1979b).

10 x104" PrC u2

8

c

"T

~6 E

7o~4

,," ""

0

I

I

I

100

200

300

T (K) Fig. 4.9.

Temperature dependence of the inverse susceptibility along the principal axes of PrCu 2. After Hashimoto et al. (1979b).

calculated the values of B ° and B 2 and arrived at B ° c a l = 4.1 K and B E c a l = 3.48 K, in good agreement with experiment. Wun and Philips (1974) have also measured the specific heat of PrCu2 (fig. 4.10). They have shown that the 7 K anomaly is much sharper than expected for a twolevel Schottky anomaly. The shape of the anomaly cannot be produced by fixed crystal-field levels, but suggests, instead, that it arises from a cooperative transition. These authors attributed this anomaly to Jahn-Teller distortions. The microscopic picture of this class of structural transitions involves phonon-mediated quadrupole interactions between rare-earth ions (Kjems et al. 1978).

MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS

439

12 10

* -

PrCuz MoteculnrFietd /

........ Schoffky

,7

Cuz

--.--Lo

1

oo07

0

I./

O

.-"'"

O O

... .,'"

•/"

5

10

TIK) Fig. 4.10. The zero-field specific heat of PrCu 2 between 0.3 and 10 IC After Wun and Philips (1974).

Thermal-expansion, susceptibility, resistivity and thermal-conductivity measurements performed by Andres et al. (1976), Ott et al. (1977) on single-crystalline PrCu 2 have confirmed that PrCu2 exhibits a cooperative Jahn-Teller effect at TD = 7.5 K. Figure 4.11 shows the temperature dependence of the linear thermal-expansion coefficients, ~i, along the three principal axes in PrCu2 (Ott et al. 1977). Distinct extremal values of oq are observed around 7.5 K giving evidence for an actual distortion of the crystal lattice. The structural distortion below TD has also been measured by neutron diffraction (Kjems et al. 1978). Inelastic neutron scattering in single-crystalline PrCu2 has been performed by Ott et al. (1978) at temperatures below and above the Jahn-Teller transition temperature. It has been shown that the most intense transition occurs at 1.3 meV (15 K). It is assigned as a transition from the ground state to an excited state. The effect of the crystal field on the paramagnetic susceptibility of PrCu2 has been studied by Zajac (1981). In neutron-diffraction measurements, Kawarazaki et al. (1984) and Nicklow et al. (1985) have observed magnetic order in PrCuz below 58 mK. These authors have also reported the magnetic structure of PrCu2 (fig. 4.12). Nicklow et al. (1985) have pointed out that the magnetic structure (electronic + nuclear) is incommensurate with the chemical lattice. Further neutron-diffraction experiments performed by Kawarazaki and Arthur (1988) (part of this work has been published earlier, see Kawarazaki and Arthur 1986) reveal that in the ordered state electronic and nuclear

440

N.H. LUONG and JJ.M. FRANSE

160 140

Pr Cuz

IZO 100

•

o//a

•

e//b

I~ •

o#c

$

80

•

6O

20

-20 -40 -60

I

I

I

2

/,

6 8 T(K}

I

I

I

I

10 12 1/+

Fig. 4.11. Linear thermal-expansion coefficient oti along the principal axes in PrCu2. After Ott et al. (1977).

o~o

\~

•

q

'

o o

\

\

'\

r o Pr • ru

oo

I~

C

---I

Fig. 4.12. Projection of the PrCu2 structure onto the (a, e) plane. The orthorhombic cell is indicated by the dashed lines. The vector ~ gives the direction of propagation and wavelength of the modulated magnetic structure. The indicated diagonal planes are parallel to (103) planes. After Kawarazaki et al. (1984).

MAGNETIC PROPERTIESOF RARE EARTH-Cu2COMPOUNDS

441

PrCu2 *+#~ ~,~ 0:0^.

200

÷4 o'0*°°

÷°

~..~0 ~ ~o ° x~ v

150

E

.. B:OT + B=3 T ° B: 5 T x B=TT

o

~'~x

,,~0 #

"...........,'

Yd lOO

50 NbTi

4

5

6

7

8

9

10

T (K)

Fig. 4.13. Specific heat (per unit volume) of PrCu2 in applied magnetic fields. For comparison the specific heat (per unit volume) of NbTi and Cu is also shown. After Kwasnitzaet al. (1983). moments of the praseodymium atoms are sinusoidally modulated in magnitude and oriented approximately parallel to the crystallographic a axis. PrCu2 may find its way to the applications, for instance, by stabilization of superconductors. Kwasnitza et al. (1983) and Barbisch and Kwasnitza (1984) have measured the specific heat of PrCu2 in magnetic fields up to 7 T (fig. 4.13). The specific heat has a broad maximum around 6 K with values (per unit volume) being about two orders of magnitude larger than those of NbTi (technical superconductor) and copper. An applied magnetic field does not destroy these very high specific heat values. Thus PrCu2, like some other materials (NdSn3, Nd0.9Pr0.1Sn3 and PrB6, see Kwasnitza et al. 1983), is a good candidate for superconductor stabilization. First attempts for application were made (Barbisch and Kwasnitza 1984). These authors have pointed out that PrCu2 powder would be advantageous as a filler in the epoxy used for coil impregnation.

4.3. NdCu2 The antiferromagnetic ordering temperature of NdCu 2 compound is 6 K (Hashimoto et al. 1979b, from ac susceptibility measurements) or 6.5 K (Gratz et al. 1991, from specific heat and resistivity measurements). From specific heat and resistivity measurements (see below), Gratz et al. (1991) have observed a spin reorientation at the temperature, Ts, equal to 4.1 K.

442

N.H. LUONG and J.J.M. FRANSE

The magnetic structure of NdCu2 is complex as revealed from recent neutron diffraction study in the temperature range from 1.4 K up to 8 K in zero external field (Arons et al. 1994). These authors show that two magnetic phases were observed between 1.4 K and TN = 6.5 K. In both phases the magnetic structure can be described by assuming a sinusoidal oscillation of the Nd moments which are oriented along the b direction. In the high-temperature phase between 5.2 K and TN, the magnetic structure is incommensurate with the lattice. In the low-temperature phase (below 4 K) the structure becomes commensurable. Around 4.4 K the neutrondiffraction results suggest a coexistence of the high- and low-temperature phases. The temperature dependence of the inverse susceptibility along each principal axis of NdCu2 is shown in fig. 4.14 (Hashimoto et al. 1979b). The effective magnetic moment,/~eff, amounts to 3.5/~B and the paramagnetic Curie temperatures along the a, b and c axes are: Oa = 17 K, Ob = - 1 6 K and 0e = - 4 K (Hashimoto et al. 1979b). The second-order crystal-field parameters B2° and B22 for NdCu2 were first estimated by Hashimoto et al. (1979b) from measurements of paramagnetic susceptibility in a single-crystalline sample: B ° = 0.8 K, B22 = 1.1 K. Hashimoto et al. (1979b) have also calculated the second-order crystal-field parameters using a point-charge model and obtained: B ° eal = 1.17 K and B 2 eat = 1.01 K. Inelastic neutron scattering investigations on polycrystalline samples have recently been reported by Loewenhaupt (1990) and Gratz et al. (1991). Figure 4.15 shows the inelastic neutron spectra in the paramagnetic state for Ndcu2 (Gratz et al. 1991). These authors observed four inelastic magnetic transitions which were interpreted as crystal-field transitions from the ground-state level to excited crystal-field levels at 34, 58, 84 and 164 K (see inset in fig. 4.15). This is expected for the Nd ion in an orthorhombic symmetry: a splitting of the J = 9/2 ground state into five doublets. Taking into account the values of B ° and B22 obtained by Hashimoto et al. (1979b) 10

"T

xl0/*

8

. ~ N2d C u

_~c

~-1 6 E

~g

"To~L~ 0

0

~

I

100

I

200 T (K)

I

300

Fig. 4.14. Temperature dependence of the inverse susceptibility along the principal axes of NdCu 2. After Hashimoto et al. (1979b).

MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS

140

120

7

100

T=IOK

K

o

Eo=12 m e V

164

x

Eo=17 rneV

*

Eo=50 rrteV

443

meg

84 80

0®

7

®xx

,. 34

60

¢lr co

OLrue' Sehe'ra.e

x

40

_J

20 0 -5

® "

0

I

[

t

5

10

15

ENERGY

TRANSFER

.. .. .. ,

20

(rrte~

Fig. 4.15. Inelastic neutron spectra of NdCu 2 in the paramagnetic state (T = 10 K) obtained with different incident neutron energies E0. The deduced crystal-field transitions are given by the level scheme in the inset. The nine crystal-field parameters Bnm have been derived with the superposition model from these data (full curve). After Gratz et al. (1991). and using a superposition model ( N e w m a n and N g 1989, Divis 1991), Gratz et al. (1991) have derived the following set o f crystal-field parameters which best describe the inelastic neutron-scattering data:

B ° = (1.35 + 0.06) K, B g = (1.56 + 0.08) K, B ° = (2.23 + 0.12) K, B 2 = (1.01 + 0.2)K, B44 = (1.96 + 0.42) K, B6° = (4.89 + 0.43) K, B 2 = (1.35 + 0.45) K,

444

N.H. LUONGand J.J.M. FRANSE

B64 = (4.89 + 0.43) K, B66 = (4.25 + 0.19)K. Magnetization isotherms of single-crystalline NdCu2 measured at 2 K by Svoboda et al. (1992) are shown in fig. 4.16. These results as well as results of Hashimoto et al. (1979b) reveal that metamagnetic transitions are observed only in the b direction, whereas in the a and e directions no transitions are detectable. Bozukov et al. (1992) have performed magnetization measurements on polycrystalline NdCu2 in magnetic fields up to 28 T (see fig. 4.17). They have found a further transition around 12 T. Above 25 T a ferromagnetic spin arrangement seems to exist. At this magnetic-field value, the magnetic moment saturates to the value of 1.9/~B/ion which is about 60% of the gJ value for the trivalent Nd ion (Bozukov et al. 1992). Magnetization measurements of Bozukov et al. (1992) at 1.8 and 4.2 K have shown that the lowest lying field-induced transition disappears at temperatures above 4.2 K. This observation indicates the existence of a spin reorientation in the temperature range 1.8-4.2 K. As mentioned above, measurements without an external magnetic field (specific heat, resistivity) revealed a spin reorientation at the temperature Ts equal to 4.1 K (Gratz et al. 1991). Figure 4.18 shows the temperature dependence of the specific heat of NdCuz and of non-magnetic isostructural LuCu2 (Gratz et al. 1991). From these measurements the N6el temperature and the spin-reorientation temperature have been determined. The electronic specific-heat coefficient 7 is estimated to be around 8-12 mJ/K2 mol, comparable with 7(YCu2) = 6.7 mJ/K2 mol (Luong et al. 1985a) and ?(TmCu2) = 9 mJ/K2 mol (Sima et al. 1988). The magnetic contribution to the specific heat, era, is deduced as: em= c(NdCu2) - c(LuCu2)

(4.2)

2.0 b

Nd Cuz

1.5 i= 1.0 0.5

0.0 0

1

2

3

t~

5

B (T)

Fig. 4.16. Magnetizationisothermsalong the principal axes of NdCu2 at 2 K. After Svoboda et al. (1992).

MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS

445

2.0 1.6

13K 16 K

-6 1.2 l::: .~

/

//

0.8 NdCuz 0.~

~" ""

,_ o., perma.nenf field puLsed field I

0.0

I

S

0

l

I

I

15 B (r)

25

Fig. 4.17. Magnetization curves of NdCu 2 at various temperatures in pulsed fields up to 28 T (solid curves) and in steady fields up to 14 T (dashed curves). After Bozukov et al. (1992).

50

ooO°~

NdCu2 o°°°° "*"t"

L,O

0 O0

O

o

,0°

oo

20 r L

*****

,

e"

e,~

0°1

eeeee

°oo°

30

/

tl°

LuCu2

_."

oO°°°~"/

10

0

10

20

30

,

,

~0

50

60

T (K) Fig. 4.1& Temperature dependence of the specific heat of NdCu 2 and L u C u 2. After Gratz et al. (1991).

and is given in fig. 4.19 as a function o f temperature, cm does not vanish above Try, but shows a broad and pronounced m a x i m u m around 20 K. In this figure, the calculated Schottky a n o m a l y as obtained by using the crystal-field eigenvalues Ei f r o m the neutron experiments is included (Gratz et al. 1991). The curvature of em above

446

N.H. LUONGand J.J.M. FRANSE

10

(experiment)

cm

i 0-~0"0-0-0.0 0

go 06

o

oI

E

/0

1.0

o/°"/ ./~CF ~

(theory)

t~ E t.I

/ ~i'

NdCu2

/ I

I

5

10

I

15 T (K)

I

20

25

Fig. 4.19. Temperaturedependenceof the magnetic contributionto the specific heat, era, of NdCu2, together with the calculated Sehottky specific heat, eCF, which is obtained by using the crystal-field eigenvaluesEi from neutron experiments. After Gratz et al. (1991). TN is clearly related to crystal-field effects, but there are also hints to correlations between the Nd 4f moments, because of the difference between the experimental data and the calculated Schottky peak. Gratz et al. (1991) have calculated the temperature variation of the magnetic entropy Sm. It is shown that Sm does not reach the theoretical value of R In 2 at TN, as one would expect for a complete removal of the two-fold spin degeneracy of the crystal-field ground-state doublet. They attributed this reduction of the entropy to short-range correlation effects in the paramagnetic state. The thermal expansion of NdCu2 has been measured by Gratz et al. (1991). The excess thermal expansion Ac~(T) was deduced by comparing the thermal expansion of magnetic NdCu2 and non-magnetic LuCu2 as: Ao~(T) = c~(NdCu2) - o~(LuCu2) ~-, Otsc(T) + ~cF(T),

(4.3)

where ase is the contribution due to spin correlations either in the ordered state or in the paramagnetic state and ~CF is the crystal-field induced contribution to the thermal expansion. Based on a model for magnetoelastic interactions described by Divis et al. (1990a), the temperature variation of ac~ was calculated and compared with Aa in the paramagnetic state (Gratz et al. 1991) (see fig. 4.20). The temperature variation of the lattice parameters a, b and c in NdCu2 in the temperature range from 4.2 K to 300 K has been reported by Gratz et al. (1993).

MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS

__" Z~a

150 l "

T

447

100

°~CF

~

go -50 0

..~--.~.~. -~' I 50

l 100

~ 150

~ 200

) 250

300

T(K) Fig. 4.20. Temperaturevariationof the spin-dependentcontributionto the thermal expansion(symbols) in NdCu2. The full curve shows the calculatedinfluenceof the crystal field on the thermal expansion. After Gratz et al. (1991). 4.4. SmCu 2

The N6el temperature of SmCu2 compound is TN = 21.7 K as revealed from susceptibility, specific-heat and resistivity measurements performed by Isikawa et al. (1988) on single-crystalline samples. A value for TN of 23 K has been obtained by Gratz et al. (1990) from specific-heat and susceptibility measurements on a polycrystalline sample. Isikawa et al. (1988) have shown that the temperature dependence of the specific heat exhibits, apart from a peak at TN, two distinct anomalies at 9.4 and 16.5 K, while Gratz et al. (1990) did not observe the anomaly around 9 K. The effective magnetic moment, /ze~, amounts to 0.53/ZB; the paramagnetic Curie temperature, 0p, equals - 1 4 K (Gratz et al. 1990). In SmCu2, the Sm ion is in the trivalent state as is concluded from a lattice parameters comparison (Gratz et al. 1990) and from the experimental values of the susceptibility above TN, as obtained by Isikawa et al. (1988). Figure 4.21 shows the magnetization curves along the a and b axes, as obtained by Maezawa et al. (1989). It can be seen from this figure, that a spin-flop transition from the antiferromagnetic state to the ferromagnetic state occurs at 22.5 T. In fields above 23 T, the magnetic moment tends to saturate to the value of 0.3#B/ion, which is about 40% of the gJ value for the Sm 3+ ion. The temperature dependence of the susceptibility of single-crystalline SmCu 2 has been measured by Isikawa et al. (1988) and is shown in fig. 4.22. A sharp peak at TN has been found in the susceptibility curve along the b axis. It is suggested that the b axis is the easy direction and that the antiferromagnetic spin structure is collinear, with the spins aligned parallel to the b axis. Isikawa et al. (1988) did not

448

N.H. LUONG and J.J.M. FRANSE

f

SmCuz

b

/,.2K

EQJ tl

2

/

0

10

30

20

B(T) Fig. 4.21. Magnetization curves along the a and b axes of SmCu 2 at 4.2 K. After Maezawa et al. (1989).

2~L

SmCu2 000000 mOO000

1 lee

"~e~°

°eNeoeuee

° •

Io v--

0 0

I 100

I 200

300

T (K) Fig. 4.22. Temperature dependence of the susceptibility of single-crystalline SmCu 2 along the principal axes. After Isikawa et al. (1988).

MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS

449

observe any other remarkable anomaly than that at TN in the temperature dependence of the susceptibility. These authors have also calculated the anisotropic temperature dependence of the susceptibility of SmCu2 on the basis of the crystal-field and exchange interactions. A good agreement between calculations and experiments is obtained but no explicit data for the crystal-field and exchange interaction parameters are given. Figure 4.23 shows the temperature dependence of the specific heat ep and the magnetic entropy Sm of SmCu2 (Gratz et al. 1990). Tr~ (= 23 K) and Ts (= 17 K) are the N4el and the spin-reorientation temperatures, respectively. The coefficient 7 is estimated to be about 20 mJ/K 2 mol, which value is about twice as high as the 7 values observed for the other RCuz compounds, e.g., 7(YCu2) = 6.7 mJ/K 2 mol (Luong et al. 1985a), 7(NdCu2) _~ 10 mJ/K 2 mol (Gratz et al. 1991) (see sections 4.1 and 4.3) and 7(TmCu2) = 9 mJ/K 2 mol (Sima et al. 1988). However, the value for 7(SmCuz) is four times smaller than that observed for the isostructural Kondo system CeCuz, 7(CeCu2) = 82 mJ/K z mol (Gratz et al. 1985) (see section 4.1). The spin-dependent specific heat, era, is obtained by comparing the specific heat of SmCu2 and that of non-magnetic LuCu2 as (Gratz et al. 1990): em(SmCu2) = e(SmCu2) - e(LuCu2).

(4.4)

Gratz et al. (1990) have obtained a value of 5.5 J/Kmol for the magnetic entropy Sin, which is very close to the value expected for a complete removal of the two-fold spin degeneracy of the crystal-field ground-state doublet (Sin = R In 2 = 5.76 J/K mol). These authors concluded that the ground state is a doublet. The magnetic entropy value estimated by Isakawa et al. (1988) is 88% of the theoretical value. The maximum temperature (45 K) reached in specific-heat measurements performed by Gratz et al. (1990) may be not high enough to observe the Schottky

SmCu2

ee

12

I

0

"6 12

,- ~

0

IIll ~

12

I

I

24 T (K]

I

I

36

Fig. 4.23. The specific heat c and the magnetic entropy Sm of SmCu2 as a function of temperature from 1.5 up to 45 K. After Gratz et al. (1990).

450

N.H. LUONG and J.J.M. FRANSE

anomaly caused by the crystal-field effect. Therefore, these authors have performed thermal-expansion measurements in the temperature range from 4 K up to room temperature. Figure 4.24 presents the coefficient of thermal expansion, a = l - 1 ( A I / A T ) , of SmCu 2 as a function of temperature. Figure 4.24 also shows of isostructural LuCu2. a(T) of SmCu2 shows a pronounced minimum near 40 K. Gratz et al. (1990) have deduced the spin-dependent contribution to the thermal expansion, am(T), for SmCu2 by subtracting a(T) of LuCu2. The results are shown in fig. 4.24. am(T) exhibits a minimum at 45 K, caused by the crystal-field effect. Assuming that the minimum in am(T) represents an effect comparable to the Schottky-type of anomaly in the specific heat (see, for instance, Franse et al. 1985, Sima et al. 1988), Gratz et al. (1990) have used the position of the temperature where the minimum occurs to estimate the splitting energy between the ground-state doublet and the first-excited doublet (see section 3). They have derived a value of about 110 K for this crystal-field splitting. Transport properties (resistivity, thermal conductivity and thermopower) of SmCu2 were measured by Gratz et al. (1990). The curvature of the spin-dependent contribution to the resistivity, Pro, indicates a crystal-field influence on the J = 5/2 ground state, although the effect of the J = 7/2 multiplet on Pm is still an open question. The high-field magnetoresistance in SmCu2 has been measured by Maezawa et al. (1986, 1989). Indications are found that SmCu2 has a complex Fermi surface. Based on the molecular-field theory and using the value of 22.5 T for the flopping field, Maezawa et al. (1989) have estimated the antiferromagnetic-exchangeinteraction parameter for SmCu2 to be 0.11 meV. This value is about half the value calculated from the NEd temperature. Based on the paramagnetic-susceptibility measurements of Gratz et al. (1990), Stewart (1992) has derived a value of 0.061 eV for the exchange interaction between the 4f and the itinerant electrons in SmCu2. 20 15 10

,:

...,,.

~5

~

,.,

. r, 4"

/

:--.

/

j mcu,_ ~,."

5

-10

',-"j

-15

0

_,-~" ~ .,-" ~ -,~ . 100

I

1'0

x.... '

I

~

--,-.-.-/ . '

/

5~/

200 T (K)

Fig. 4.24. Temperature dependence of the thermal expansion, c~, of S m C u 2 and L u C u 2 and the magnetic contribution to the thermal expansion, C~rn. Inset shows details of ~ versus T below 50 IC After Gratz et al. (1990).

MAGNETIC PROPERTIESOF RARE EARTH-Cu2COMPOUNDS

451

To our knowledge, up till now no data on the crystal-field interactions in SmCuz are available. Further experiments are needed to clarify the interesting magnetic properties of this compound. 4.5. G d C u 2

The compound GdCu2 is, by lack of an orbital moment of the 4f electrons, expected to be relatively simple as far as the magnetic anisotropy is concerned. Therefore, in the series RCu 2 this compound is the most convenient object for the study of RKKY exchange interactions. The magnetic properties of GdCu2, among other compounds, were first studied by Sherwood et al. (1964). These authors reported that GdCu2 is antiferromagnetic, with a value for the Nrel temperature equal to 41 K. Poldy and Kirchmayr (1974) have shown that the RKKY model can be successfully applied to GdCu2, i.e. the magnetic arrangement of the Gd a+ ions is due to the indirect exchange interaction via the conduction electrons. Further studies in polycrystalline samples were made on the crystallographic properties (Dwight 1980~ Borombaev et al. 1986a, b), magnetic properties (Gratz and Poldy 1977, Luong and Franse 1981, de Graaf et al. 1982, Borombaev et al. 1986a,b, Borombaev and Markosyan 1987), magnetostriction (Luong and Franse 1981), thermal expansion (Luong and Franse 1981, Luong et al. 1985a,b, Borombaev et al. 1986a,b, Borombaev et al. 1987), specific heat (Luong et al. 1985a, b), transport properties (Gratz 1981, Gratz and Zuckermann 1982) and MSssbauer effect (De Graaf et al. 1982). To our knowledge, only one work on single-crystalline GdCu2 is available (Borombaev et al. 1987). Figure 4.25 shows the results of magnetic measurements in low magnetic field of these authors. In the paramagnetic temperature region the magnetic susceptibility of single-crystalline GdCu2 does not depend on the direction of the external field and obeys the Curie-Weiss law with a paramagnetic Curie temperature, 0p, of (16 -4- 2) K and an effective magnetic moment,/~eff, of (8.14 -40.2)/~B, close to the theoretical value for the Gd 3+ ion (7.94#B). (De Graaf et al. (1982) have obtained the following values for a polycrystalline sample: 0p = 7 K a n d / ~ = 8.7/~B.) The Nrel temperature was derived as the temperature at which the magnetization exhibits a discontinuity and was found to be equal to 42 K, in good agreement with other literature data (Sherwood et al. 1964, Poldy and Kirchmayr 1974, Luong et al. 1985a, b). Interesting magnetization curves were obtained by Borombaev et al. (1987) below the Nrel temperature in high magnetic fields. Figure 4.26 shows the magnetization along the three principle axes of single-crystalline GdCu2 at 4.2 K. In the interval 6-12 T, a metamagnetic transition (in two stages) from antiferromagnetism to ferromagnetism occurs. At Bel = 6.8 T the magnetization sharply increases, reaching approximately 4#B, and then increases more slowly; at Bc2 = 9.5 T again a sharp increase of the magnetization is observed. The saturation magnetic moment is equal to (6.9 + 0.1)/~B, close to the Gd3+ moment (7.0/ZB), indicating that GdCu2 is in the ferromagnetic state in a field B > Be2. Figure 4.27 shows the magnetization curves of GdCu2 at different temperatures in relatively low fields up to 2.8 T. It can be seen from this figure that the magnetization along the b axis increases linearly with

452

N.H. LUONG and J.J.M. FRANSE

Xo(orb. unit) ~TN 200 1.5 150 m. 1.0~

TNI 0

ov -

b,0

I00 ~'a.

0.5 0

50

100

0

200

300

T {K) Fig. 4.25. Temperature dependence of the inverse susceptibility of GdCu 2 along the axes a (o), b (o) and e(o) and of the magnetization along the axis e(+) in a field of 0.15 T. The inset shows the temperature dependence of the initial susceptibility (in a field of 0.001 T). After Borombaev et al. (1987).

GdCu2 6

2

0

~"

0

i

I'"

I

I

10

5

I

I

15

B (T) Fig. 4.26. Magnetization curves along the axes a (solid line), b (dashed line) and c (dotted line) of GdCu 2 at 4.2 K. The inset shows the field dependence of the differential magnetic susceptibility along the a axis. After Borombaev et al. (1987).

field, and that the susceptibility does not depend on the temperature. However, the magnetization along the a and e axes displays a discontinuity near 1 T, in the vicinity of the N6el temperature, the magnetization in this region shows hysteresis. Above 2 T, the susceptibilities Xa and Xe are equal to the susceptibility XB. Borombaev

//JJ

MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS

// 0, / / / 0:////

/

b

o

0"20'~

I

I

0

I

2

I

0

453

2

1

0

I

I

1

2

B (T)

Fig. 4.27. Magnetization curves of GdCu2 along the principal crystallographic axes a (a), c (b) and b (e) at different temperatures: 1) 4.2 K; 2) 15 K; 3) 30 K; 4) 35 K. After Borombaev et al. (1987).

.o~.d'*~

0 1.0

GdCu 2

IE

04v'

~

~* ~ ~ ~ , ~*

~.~,.

___

I'---- 0.5

10

20

30

~0

50

T (K)

Fig. 4.28. The specific heat of GdCu2 at zero magnetic field (o) and in a field of 5 T (+) in a plot of

e/T versus T. The full curve represents the sum of the electronic and lattice contributions to the specific heat of GdCu2. The broken curve represents the specific heat of YCu2. After Luong et al. (1985a). et al. (1987) have explained the transition observed in GdCu 2 by comparing the energies o f the different magnetic phases in the R K K Y model. Their results s h o w that the presence o f a large magnetic anisotropy energy is not a necessary condition for the realization o f metamagnetic transitions. Such transitions are possible in the exchange approximation for ferrimagnetic structures which energies lie between the energies o f the initial antiferromagnetic state and the final ferromagnetic state.

454

N.H. LUONG and J.J.M. FRANSE

Figure 4.28 shows the specific heat of the compound GdCu2 at zero magnetic field and in a field of 5 T in a plot of c/T against T (Luong et al. 1985a). Non-magnetic YCu2 was used in order to separate out the magnetic contribution to the specific heat of GdCu2. A A-type of anomaly is observed around the N6el temperature TN. The value for TN has been determined from the peak in the plot of the specific heat against T and is found to be equal to 40 K, in good agreement with other literature data. Luong et al. (1985a) have also pointed out that their susceptibility experiments above 4.2 K confirm this value of TN. A second anomaly below the ordering temperature is visible in the plot of c/T against T in fig. 4.28 as a broad structure around 10 K. We attribute this anomaly, which is of a Schottky type, to the Zeeman splitting of eight-fold degenerate energy level below the N6el temperature. Finally, a sharp peak is found at 1.5 K that shifts to 1.7 K in a field of 5 T. This peak could indicate a change in magnetic structure. As mentioned before (see section 4.1), the electronic contribution to the specific heat, 7, was derived from the specific heat data of Y C u 2 and found to be equal to 6.7 mJ/K 2 mol. This number is close to the value of 8.2 mJ/K 2 mol reported for Y metal (Stewart 1983). The phonon part of the specific heat of YCu2 follows closely the Debye function, giving 0D = 236 K. Since the 7 value of 6.4 mJ/K 2 mol of Gd metal (Stewart 1983) is close to the above mentioned value for Y metal, Luong et al. (1985a) have taken the 7 value of Y C u 2 for GdCu2 and compared the specific-heat

15 o°

t +4+

~I0 E

4- 0 •

4•

w

4-

4-

E

5

X

g

C 0

g

+

o

4-

%

4-

•

O

I -°°3 %2.~, ° 10 20 30

:

% l z.O T (K)

50

Fig. 4.29. The magnetic contribution to the specific heat for G d = Y l _ x C u 2 as a function of temperature: z = 1 ( . ) , x = 0 . 8 ( + ) , z = 0.6(a), z = 0.4(X), z = 0.2(0). After Luong et al. (1985a).

M A G N E T I C PROPERTIES OF R A R E EARTH-Cu 2 C O M P O U N D S

455

data of YCu 2 and GdCu2 compounds well above the ordering temperature in order to derive the value for the Debye temperature for GdCu2. They obtained a value of 198 K for GdCu2. The electronic and lattice contributions to the specific heat of GdCu2 and YCu2 are represented by the full and broken curves in fig. 4.28, respectively. The magnetic part, Cm, which is obtained by subtracting the electronic and phonon parts from the observed specific heat, is shown in fig. 4.29. This figure also shows this magnetic contribution for the whole series of Gd=YI_=Cu2 compounds which have been studied by Luong et al. (1985a). An analysis of the specific heat data for G d ~ Y l _ x C u 2 has been performed in the same manner as for G d C u 2. We note that G d 0 . 2 Y 0 . 8 C I I 2 , although it has no long-range magnetic order down to 1.2 K, has a small but not negligible magnetic specific heat. From specific-heat measurements Luong et al. (1985a) derived the values of TN and 0D for the Gd=Yl_zCu 2 compounds. The thermal-expansion results for the GdzYl_~Cu 2 compounds obtained by Luong et al. (1985a) are shown in fig. 4.30 in a plot of ot (= I-1AI/AT) against T. The thermal-expansion results for non-magnetic YCu2 (fig. 4.30) are again used to separate out the magnetic contribution to the thermal expansion in the other compounds. The results are shown in fig. 4.31. The (am, T) curves behave similarly to the (Cm,T) curves (see fig. 4.29). The A-type of anomaly around TN as well as the Schottky anomaly below TN both can be observed. Luong et al. (1985a) have discussed the specific-heat and thermal-expansion results in terms of Griineisen parameters. These authors were able to determine the

3O

°o

~20

"7 Y to I o

-¢

:

•

,.k -4-+ o

++...+

el

=e ++;£

.I. .

j:#. ++ ,r ==

.

a~r ~ ' ~ ' - = " J:l~.,m.~ao"

0

-

2O

~

n

• :

.,. ~ ~,,~ t o . ~ "~ " "

+~ - + ÷ 69.~%" "r'l'++'r~ ~ ~LX~

4O

-

6O

80 T (K)

Fig. 4.30. The coefficient of thermal expansion, c~, for G d z Y I _ = C u 2 as a function of temperature: x = l ( o ) , x = 0 . 8 ( + ) , x = 0 . 6 ( a ) , x = 0 . 4 ( x ) , ar = 0.2(o), ~e = 0(--). After L u o n g et al. (1985a).

456

N.H. LUONG and JJ.M. FRANSE

30

@

~

2O

I Y

I 0

-CA: °

++ 4-+

Sg E

~I0

~°+ I

o -t~ ~ x

x+

++

oee

+

°

°

. + + +

Ot 0 a

0 0

+

o++ .:

°

•

•

~.:%N ++H-~%"°k" n

10

2O

30

40 T (K)

50

Fig. 4.31. The magnetic contribution to the coefficient of thermal expansion for GdxYI_=Cu2 as a function of temperature: :e = l(i), a: = 0.8(+), x = 0.6(o), z = 0.4(X), = = 0.2(0). After Luong et al. (1985a). electronic, lattice and magnetic Grtineisen parameters for the Gd~YI_=Cu 2 compounds. Taking the lattice Grtineisen parameter to be equal to 2, the electronic Griineisen parameter is found to be 1.7, quite close to the value Fe = 5/3 which has been obtained for d transition metals (Simizu 1974). The magnetic Grtineisen parameter is at least one order of magnitude larger than the electronic and lattice ones. The magnetic entropy, Sin, in the Gd~YI_~Cu2 compounds was found to be proportional to the number of Gd atoms and the value for S derived from the formula ,5'm -- mR In (2S + 1) is slightly smaller than 7/2 for a free Gd +3 ion (Luong et al.

1985a). Although GdCu2, as mentioned above, is a relatively simple compound, no experimental data about its magnetic structure are available. Because of the large cross section for scattering of neutrons by gadolinium, neutron-scattering studies of the magnetic structure of GdCu2 have not yet been performed. Poldy and Kirchmayr (1974) have calculated the magnetic structure of GdCu 2 in the RKKY model. According to their calculations, GdCuz has a spiral magnetic ordering: all spins in the be plane are parallel to one another, while the directions of the magnetic moments in neighbouring planes differ by 35 °. Studies on single-crystalline GdCu2 by Borombaev et al. (1987) have shown (fig. 4.27) that the curves of the magnetization of GdCu 2 along the a and c axes practically coincide with each other, but differ from

MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS

457

the curve of the magnetization along the b axis. These authors have explained the experimental results by assuming that at the lowest magnetic field (B < 1 T, see above) the spins of the Gd 3+ ions lie in the ac plane (the angle between the spins of neighbouring be layers is 35* (Poldy and Kirchmayr 1974)). Cu in GdCu2 has been partially replaced by Ni (Poldy and Kirchmayr 1974, Gratz and Poldy 1977, Smetana et al. 1985a, Borombaev et al. 1986a, Borombaev and Markosyan 1987), by Co and Fe (Borombaev et al. 1986b, Borombaev and Markosyan 1987) and by A1 (Borombaev et al. 1986a, Borombaev and Markosyan 1987). The aim of these replacements is to observe the effects of changing the conduction-electron concentration, since the exchange via the conduction electrons is sensitive to this parameter. In substituted GdCu2, the electron concentration is decreased with increasing Ni and Co content, whereas it is increased with increasing A1 concentration. It was assumed that Gd, Cu, Ni, Co and AI contribute 3, 1, 0, - 1 and 3 conduction electrons, respectively (Poldy and Kirchmayr 1974, Borombaev et al. 1986b, Borombaev and Markosyan 1987). The limits of structural stability were found to be 35% Ni, 10% Co, 1% Fe and 7% AI (Borombaev et al. 1986a, b, Borombaev and Markosyan 1987). The experimental observations in GdCu2 substituted with Ni, Co and A1 have been discussed in terms of the RRKY model (Poldy and Kirchmayr 1974, Gratz and Poldy 1977, Kirchmayr and Poldy 1979, Borombaev et al. 1986a, b, Borombaev and Markosyan 1987). 4.6. TbCu2

The compound TbCu2 has the highest value of the Nrel temperature in the series RCu 2. Early magnetic measurements of Sherwood et al. (1964) have yielded a value for TN of 54 K for this compound. A value for TN of 53.5 K, i.e. very close to that of Sherwood et al. (1964), has been obtained by Hashimoto et al. (1976) and Hashimoto et al. (1979a) on single-crystalline TbCu2. Somewhat lower values for T N of TbCu 2 have been obtained from magnetic measurements by Poldy and Gratz (1978) and by Smetana et al. (1985a) (TN = 48 K) and from specific-heat measurements by Luong et al. (1985b) (TN = 48.5 K). The early measurements of Sherwood et al. (1964) have shown metamagnetic behaviour of polycrystalline ThCu2. Hashimoto et al. (1976, 1979a) performed magnetization and magnetic-susceptibility studies on single-crystalline samples and observed anisotropies in the magnetic properties. Later, Iwata et al. (1988) have studied the magnetization process of single-crystalline TbCu 2 in the temperature range from 4.2 to 55 K. Their results are shown in fig. 4.32. For a magnetic field applied along the a axis, the magnetization is almost zero below Bc (,'~ 1.9 T) and jumps within one step to a nearly saturated value at higher field. The saturation moment per atom has been determined to be 8.8#B (Hashimoto et al. 1979a), very close to the theoretical value of 9/zB for the free Th3+ ion. Figure 4.33 shows the result of high-magnetic-field experiments on TbCu2 in a pulsed magnetic field performed by Luong and Frame (1981). Due to large magnetostrictive forces the sample was powdered and oriented in succeeding field runs. From these measurements we deduced a value of 1.9 T for the critical field Be.

458

N.H. LUONG and J.J.M. FRANSE

Observed

Calculated

°,,. . . . . . . . . . . .

10 0

2

Observed

10

/*

a

2

4.

a

6f.

::I

:

50.0K i /

..,........ *'I'°''''''''

2

4.

i

30.0K

00

2

4

81 2

c.. 2

4.

B (T)

0

'2

B (T)

o

z~

0

2

55.0K I

4-

=

:

0

50.0,K a

......c

0 L. ,.~:!:;:::': :::~': -''" '~'l 10 0 2 4,

'2

Calculated

o. . . .

4.

55.0K

..*"°'"" "'"""

.5...

::::::::::::::::::::::::::: 0 2 4 B IT)

2

4-

B (T)

Fig. 4.32. Observed and calculated magnetization of TbCu2. After Iwata et al. (1988).

]]I

f

150

100

f

"E

TbCu z

50

O I 0

I

I

10

20

30

B (T)

Fig. 4.33. High-magnetic-fieldexperiments on TbCu2 in a pulsed magnetic field. In succeeding field runs, the sample is powdered and oriented with the easy axis along the field direction. After Luong and Franse (1981).

MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS

•

i

'r',

l!

,~i

"

II

II

ii

II

. . . . . . .

r,'-~

u

!"

. . . . . .

," A A ~

,,'~

C

"

I

Ii

V

I

,6

'

I

'

I

!

I

I

•

,J,

S"

P

J-

I iI

••

iS I

459

,4

:--

!

I

6

I

I

!

i

I

7

,

i

Fig. 4.34. Magnetic structure of TbCu 2 at 4.2 K. After Hashimoto et al. (1979a).

A first neutron-diffraction study on a powdered sample has been carded out by Brunet al. (1971). According to these authors, TbCu2 has a collinear antiferromagnetic structure in which the moments lie along the a axis. The magnetic unit cell is equal to the orthorhombic chemical unit cell along the b and c axes but tripled along the a axis. Further neutron-diffraction studies on a single-crystalline sample by Hashimoto et al. (1979a) and on a powdered sample by Smetana et al. (1983), Smetana and Sima (1985), Sima et al. (1986) and Lebech et al. (1987) have confirmed this magnetic structure. Figure 4.34 shows the magnetic structure of TbCu2 at 4.2 K. A fraction of 2/3 of the Tb magnetic moment in a given e plane is aligned along the +a direction (sublattices A, C) and a fraction of 1/3 along the - a direction (sublattice B). In the adjacent e planes, the magnetic moments of the Tb atoms are all reversed and these antiferromagnetically coupled double layers pile up in the c direction. The temperature dependence of the magnetic structure of TbCu2 has been studied by Brunet al. (1971) and Sima et al. (1986), the results of which are in good agreement with each other. Three distinct regions of magnetic order were identified (Sima et al. 1986, Divis et al. 1987). Between 47 K and the N6el temperature TN (55 K in the studied samples) (region I) the magnetic structure is longitudinally modulated with the propagation vector 1/3a*. At temperatures below 15 K (region III) all the magnetic moments are equal. Finally, for temperatures between 15 and 47 K (region II) there are Tb a+ ions with 'large' magnetic moments and with 'small' magnetic moments. The magnetic structure of TbCu2 in this intermediate temperature region is shown in fig. 4.35 (Divis et al. 1987). Figure 4.36 shows the temperature dependence of the inverse susceptibility along each principal axis of the TbCu2 compound (Hashimoto et al. 1979a). A large magnetic anisotropy in the paramagnetic region has been observed. The susceptibility along the different crystallographic axes obeys the Curie-Weiss law at high temperatures. The effective moment, #eft, amounts to 9.5#a, in good agreement with

460

N.H. LUONG and J.J.M. FRANSE

I°

i

i

iI ,

[

A

.vtl

Fig. 4.35. Magnetic structure of TbCu2 for the intermediate temperature region (15 K < T < 47 K). Numbers 1 to 12 enumerate the positions of moments in the magnetic unit cell and the arrows show their orientation and magnitude. After Divis et al. (1987).

the calculated value for the free T b 3+ ion. The paramagnetic Curie temperatures 0a, Ob and 0c along the a, b and e axes are remarkably anisotropic: 76 K, - 6 K and 36 K, respectively. From these values of the paramagnetic Curie temperatures, Hashimoto et al. (1979a) have estimated the following values for the second-order crystal-field parameters: B ° = 1.23 K, B g = 1.23 K. Hashimoto et al. (1979a) have also calculated the second-order crystal-field parameters on the basis of the point-charge model. Their calculated values are: B ° -- 1.35 K and B 2 = 1.12 K. Experiments and calculations are in reasonable agreement with each other, indicating that the anisotropy of TbCu2 in the paramagnetic state can be explained mainly by the crystal-field effect. Measurements of the specific heat were performed by Luong et al. (1985b) at zero magnetic field and in a field of 5 T. A plot of e/T versus T at zero field is given in fig. 4.37. A A-type of anomaly is observed around the N6el temperature. Apart from this peak at Try, a broad hump was observed around 30 K and a sharp peak occurs at 2.2 K. Kimura et al. (1988) have calculated the specific heat for TbCu2 on the basis of the molecular-field theory including crystal-field interaction. In these calculations the authors have used two different exchange parameters and two crystal-field parameters, B ° and B22, as derived by Hashimoto et al. (1979a) (see above). The results are also shown in fig. 4.37. Based on the calculated temperature dependence of the magnetic contribution to the specific heat, Kimura et al. (1988) concluded that the broad anomaly observed around 30 K in TbCu2 must be attributed to the magnetic heat capacity. Luong and Fransse (1981) and Luong et al. (1985b) have measured the thermal expansion of TbCu2. The coefficient of the thermal expansion of TbCu2 is shown in fig. 4.38 as a function of temperature. Again a A-type of contribution is observed

MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS

10

461

x103 ~ c TbCu2

..

"7. 6

• coO°. °~

E4 X

2

100 i

•"

~"

I

I

l

200

300

T (K}

Fig. 4.36. Temperature dependence of the inverse susceptibility along the principal axes of TbCu2. After Hashimoto et al. (1979a).

1.5 TbCu2 o

1.0 0

==

,//

I--"

"3 0.5

S/

// 0

10

20

30

40

50

60

T (K) Fig. 4.37. The specific heat of TbCu 2 at zero magnetic field (o) and in a field of 5 T (+) in a plot of c/T versus T measured by Luong et al. (1985b). Lines are specific heat (at zero magnetic field) calculated by Kimura et al. (1988).

462

N.H. LUONG and J.J.M. FRANSE

15

TbCu 2

10 "7

o•

Qe•

'o

•o

•

e•

5 0

~'" I

I

I

10

20

30

I

40 r (K)

•

I

I

I

50

60

70

~ g . 4.38. The coemcientofthermalexpansion ~ r T b C u 2 as a ~nction of ~mpera~re. After Luong etal.(1985~.

around TN. Apart from this A-type of anomaly and apart from the lattice contributions, apparently a third contribution to c~ is present above 30 K. Discussions in terms of Grtineisen parameters (see section 3.3 and more details in section 4.9 for ErCuz) lead to the conclusion that for TbCu2 the same sign and approximately the same value of ~¢Fcfas for ErCu2 is observed (Luong et al. 1985b). This observation gives a natural explanation for the third contribution to the thermal expansion. Based on a molecular-field model taking into account crystal-field effects, Iwata et al. (1987, 1988) have performed a quantitative analysis of the magnetization process of single-crystalline "I'bCu2. In the analysis they used two second-order crystal-field parameters determined experimentally by Hashimoto et al. (1979a) (see above). The exchange parameters were chosen to obtain the best fit to the experimental data. The magnetization data are well explained, as shown in fig. 4.32. (We note that the Ngel temperature of their sample is Tr~ = 53.5 K.) Kimura (1985) has analyzed the magnetic structure and the magnetization process using a 12-sublattice model. The crystal-field effect on the paramagnetic susceptibility of TbCuz has been studied by Nowotny and Zajac (1985) using the general Van Vleck formula and by Kimura (1988) with the use of a high-temperature expansion method. Both these studies have shown that anisotropic behaviour of Xi (i = a, b, c) is cancelled in Xp (= (Xa + Xb + Xe)/3) for polycrystalline samples. Nowotny and Zajac (1985) have used only two lowest-order crystal-field parameters as obtained by Hashimoto et al. (1979a) (see above). The higher-order crystal-field parameters have also been taken into consideration but the effect on the paramagnetic susceptibility of RCuz was reported to be negligible (Zajac and Maczak 1985). Magnetic properties of the pseudobinary compounds Tb=YI_=Cu2 have been studied by Hashimoto (1979), Luong et al. (1982), Hien et al. (1983) and Zajac et al.

M A G N E T I C PROPERTIES OF R A R E EARTH-Cu 2 C O M P O U N D S

463

(1988b). When substituting Y for Tb, the antiferromagnetic interactions weaken but the magnetocrystalline anisotropy persists (Hashimoto 1979, Luong et al. 1982). A critical concentration, xc, where the magnetic ordering disappears, was taken by Hien et al. (1983) to be equal to 0.15, following data of Hien et al. (1983) and Hashimoto (1979). The latter author suggests that TN for TbxYI_~Cu2 is proportional to ~2/3 where G is an effective de Gennes factor equal to z(g - 1)2j(j + 1). However, the concentration dependence of TN expressed by the following equation: TN(

)/TN(1)

~

-

(4.5)

where TN(1) is TN(X) for x = 1, is in better agreement with the experimental data obtained by Hien et al. (1983). This concentration dependence of TN is also observed in Dy~YI_~Cu2 compounds (see also section 4.7). Tb0.rY0.4Cu2 has a collinear antiferromagnetic structure, as revealed from neutron-diffraction measurements (Smetana et al. 1983, Sima et al. 1986, Lebech et al. 1987). Hashimoto (1979) has derived values for the crystal-field parameters B ° and B22 in the TbxYl_xCu 2 compounds from values for the paramagnetic Curie temperatures obtained on single-crystalline samples. It was found that the values for B ° and B 2 change very little in the whole composition range. The magnetization process in the ordered Tb~YI_~Cu2 compounds is similar to that of TbCu2. It has been analyzed within the molecular and crystal-field model by Hashimoto (1979) in the paramagnetic state and by Zajac et al. (1988b) in the ordered state. For the crystal-field Hamiltonian only two parameters B ° and B 2 are considered in both studies. Partial replacement of Cu by Ni in TbCu2 leads to a lowering of the electron concentration and an increasing tendency to ferromagnetism (Poldy and Gratz 1978, Hashimoto 1979). This effect has been confirmed by neutron-diffraction studies (Smetana et al. 1983, Sima and Smetana 1984, Smetana and Sima 1985, Lebech et al. 1987). Smetana et al. (1985a) reported the temperature dependence of the ae susceptibility of a Tb(Cu0.77Ni0.23)2 sample. Hashimoto (1979) has derived values for B2° and B22 in Tb(cuxNil_x)2 compounds from values of the paramagnetic Curie temperatures. The magnetization process in single-crystalline Tb(Cu0.7Ni0.3)2 in high magnetic fields (up to 24 T) has been studied by Mamaguzhin et al. (1985) and Divis et al. (1989a, 1990b). In the analysis, the latter authors have taken into account higher-order crystal-field terms. 4.7. DyCu2

Measurements of the temperature dependence of the magnetization have yielded a value for TN of 24 K for the compound DyCu 2 (Sherwood et al. 1964). A value of 31.4 K for TN has been obtained from susceptibility measurements (Hashimoto et al. 1979a). The specific-heat measurements by Luong et al. (1985b) show a sharp A-type of peak at a value for TTq of 26.7 K. A value for the Nrel temperature found with Mbssbauer spectroscopy is (27 4- 0.5) K (Gubbens et al. 1991), in good agreement with that obtained by Luong et al. Lebech et al. (1987) have determined the magnetic structure of DyCu2 at 5 and 15 K by neutron diffraction measurements. The structure is similar to that of TbCu2 as

464

N.H. LUONG and JJ.M. FRANSE

C

!

r" .,

--tJi

.

.

.

.

~ ,~'Ws +~,.'f, !

.

.

t

/

I

b

p,s a°s~/'/' Fig. 4.39. Magnetic structure of DyCu 2. The structure is decomposed into four sublattices A, A', B and B ~. The Dy atoms labeled 4, 5, 9, 12 are on the A sublattice; atoms 3, 6, 10, 11 are on A~; atoms 1 and 8 are on B; atoms 2 and 7 are on B f. After lwata et al. (1989), data from Lebeeh et al. (1987).

10 r 8 t~.2K i f ............. a" :? 6 ', b r

I

I01

8L~5 K

+~

~'

............~

2

2

4

6

8 - 9.5K ,,~ . ....................... Q. •

--

n

2

#

6

~ ................

............. ii"

~.

.

..

2

~- 0

0

2

4

10 8 150K ~ 6

6

I bl

~

0

2

4 B iT}

0

2

6

t,

10 8130"5K

II

6 4.

{ ...,,.•,"

Fig. 4.40.

0

8

':

:~'4"

O t . ~- - ' ~ ,

.............. ...............

O.l~,~-f, 0 2

b

!

, 4 B (T)

Magnetization curves of DyCu2. Solid curves represent the calculated magnetization. After lwata et al. (1989)•

s h o w n in fig. 4.39. H o w e v e r , d u e to the fact that D y has a h i g h n e u t r o n a b s o r p t i o n c r o s s s e c t i o n , the t e m p e r a t u r e d e p e n d e n c e o f the m a g n e t i c s t r u c t u r e has not yet b e e n analyzed.

MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS

10

465

(ii)

0 ~i) c - axis

t~.2K 2 0

0

5

I

I

I

I

10

15

20

25

B (T)

Fig. 4.41. Magnetizationcurves of DyCu2along the c axis in applied field up to 25 T. Curve (i) shows the magnetization process in an initial state. Curve(ii) is the magnetizationcurve obtainedjust after the measurement of (i). After Hashimotoet al. (1990). Measurements of the magnetization at 4.2 K (Hashimoto et al. 1979a) and of the temperature dependence of isothermal magnetization curves (Iwata et al. 1989) have been performed on single-crystalline samples. Figure 4.40 presents the isothermal magnetization curves for DyCu2 (Iwata et al. 1989). The magnetization curve along the a axis has two critical fields, Bel and Be2, and the magnetization increases to about 1/3 of the saturation value at Bel and to nearly its saturation value at Be2. At high temperatures, the two-step process is smeared out. On the other hand, the magnetization along the b and c axes increases monotonically. Magnetization curves have been measured in high magnetic fields up to 30 T by Hashimoto et al. (1990) and by Date (1992). Figure 4.41 shows the isothermal magnetization curves measured along the c axis at 4.2 K (Hashimoto et al. 1990). The magnetization increases abruptly at about 14 T and shows a hysteresis at decreasing magnetic fields (i). The magnetization curve measured just after the measurement of curve (i) is indicated as curve (ii) in this figure. This is quite similar to the curve along the a axis (see fig. 4.40). It turns out that a switching of the magnetic axis is observed. The recovery of the virgin state is obtained either by warming the crystal above about 100 K or by applying a field higher than 5 T along the a axis. Figure 4.42 shows the magnetization curves along the b axis at 4.2 K (Hashimoto et al. 1990). A little jump of the magnetization with a small hysteresis is observed. The critical field decreases by repeating the high-field cycles. The sample was broken in two pieces when a high field was applied along this b axis due to magnetostriction effects. This phenomenon is similar to that observed in the polycrystalline samples of TbCu2 (Luong and Fransse 1981, see fig. 4.33). The results indicate the important role of large magnetocrystalline anisotropy and magnetoelastic energies in DyCu2. Kimura (1987a, b) has made a theoretical study of the magnetization process in RCu2 compounds using the Ising model with four exchange parameters. Necessary conditions for the exchange constants for which DyCu2 exhibits a two-step magnetization process are given (Kimura 1987b).

466

N.H. LUONG and J.J.M. FRANSE

I

2 0~ e-

= .6 0

3

b -axis 4.2 K

0

4

0

I

0

I

I

10

I

I

20

30

B (T) Fig. 4.42. Magnetization curve of DyCu2 along the b axis in applied fields up to 30 T. The number in the figure indicates the order of measurement. The critical field, at which a small jump of the magnetization occurs, decreases in subsequent measurements. After Hashimoto et al. (1990).

10 x103

b

8

DyCu2

"T

a

~6 i=

2

100

200 T (K)

300

Fig. 4.43. Temperature dependence of the inverse susceptibility along the principal axes of DyCu2. After Hashimoto et al. (1979a). The t e m p e r a t u r e d e p e n d e n c e o f the inverse susceptibility along each p r i n c i p a l axis o f D y C u 2 is s h o w n in fig. 4.43 ( H a s h i m o t o et al. 1979a). T h e effective m a g n e t i c m o m e n t , #en, a m o u n t s to 10.3#a. T h e p a r a m a g n e t i c Curie temperatures a l o n g the

MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS

467

a, b and c axes amount to: 0a = 35 K, 0b = --17 K, 0e = 0 K. From these values of the paramagnetic Curie temperatures, Hashimoto et al. (1979a) have estimated experimental results for the two lowest-order crystal-field parameters: B ° = 0.43 K and B 2 = 0.72 K. A point-charge calculation by the same authors results in B ° ~l = 0.89 K and B 2 ~! = 0.71 K, in satisfactory agreement with the experimental values. Nowotny and Zajac (1985) have used the experimental results for the two crystalfield parameters to calculate the paramagnetic susceptibility of DyCu2. They could explain the observed curvature in Xi(T)- 1 CUrveS (i = a, b, C) at low temperatures. The isothermal magnetization curves up to TN have been analyzed using a molecular field model (Iwata et al. 1989). The calculated magnetization curves are presented in fig. 4.40 as the solid lines. A satisfactory agreement between experiments and calculations is obtained. Apart from the experimental results for B2° and B22, a small positive value of 2.47 x 10 - 3 K for B ° was taken into account because its term gives a large contribution to the b axis magnetization at low temperatures. On the other hand, in high field studies Date (1992) has pointed out that a switching of the magnetic axis (see above) implies that this phenomenon cannot be explained by a simple point-charge crystal-field model but should be explained in terms of a Jahn-Teller effect. Specific heat and thermal expansion of DyCu2 have been measured by Luong et al. (1985b). Figure 4.44 shows the specific heat of DyCu2 at zero magnetic field and in a field of 5 T. The A-type of peak at TN disappears in a magnetic field of 5 T. A broad anomaly around 6 K, which is field dependent, is observed. Kimura et al. (1988) have calculated the specific heat of DyCu2 in terms of the molecular-field model including the crystal-field interaction. They have obtained the temperature dependence of the specific heat which is similar in trend with the experimental one. They have used three crystal-field parameters: B °, B22 experimentally obtained by Hashimoto et al. (1979a) and B ° equal to 2.37 x 10 -3 K. According to Kimura et al. (1988), the specific heat hump of DyCu2 near 6 K is associated with the fact that the spacing of the energy between the lowest and the next lowest levels is not large: 10 K. The thermal expansion coefficient of DyCu2 as a function of temperature is shown in fig. 4.45 (Luong et al. 1985b). Apart from the A-type of contribution around TN, an additional peak at about 19.5 K is observed. Up till now no prove has been given whether this peak is due to a change of magnetic structure or not. This should be connected with the difficulty of neutron diffraction studies in Dycu2 as mentioned above. Magnetization measurements for Dy~YI-~Cu2 compounds have been performed by Hien et al. (1983). The magnetization curves obtained for these compounds have a metamagnetic behaviour similar to that for Tb~YI_~Cu2. The Nrel temperatures for Dy,,Yl_rCu 2 with z = 0.2, 0.4, 0.6, 0.8 and 1 are 5, 13, 18, 23 and 26.7 K, respectively. The composition dependence of TN for Dy~YI-~CU2 is similar to that for Tb~YI_~Cu2, i.e. obeys eq. (4.5). For DyrYl_rCu2 the critical concentration, zc, has been evaluated to be equal to 0.15, the same as for the TbrYI-rCu2 compounds (see section 4.6).

468

N.H, LUONG and J.J.M. FRANSE

OyCuz

+/°

1.0 -- ~

E

÷.t14,~

C4 v

0

++%2°°

~- 0.5

...... 0

I

I

I

lO

20

30

T (K) Fig. 4.44. The specific heat of DyCu2 at zero magnetic field (o) and in a field of 5 T (+) in a plot of e/T versus T. After Luong et al. (1985b).

15

DyCu2

I

I,

10 "7x,,

to

~o

5

I

•

o #

~0

•

V

" 5 1P

0

o

° • ~oo

•

•

11

e

°

•

I

I

I

I

I

I

I

10

20

30

40

50

60

70

T (K) Fig.

4.45.

The coefficient of thermal expansion for DyCu 2 as a function of temperature. Luong et al. (1985b).

After

4.8. HoCu2 For HoCu2, a value for TN of 9 K was obtained by Sherwood et al. (1964) from magnetic measurements. From the experimental data on single-crystalline HoCuz, Hashimoto et al. (1979a) and Hashimoto (1979) deduced a value for TN of 9.8 K. Lord and McEven (1980) and Birss et al. (1980) performed neutron-diffraction, resistivity and magnetoresistance measurements on HoCu2. They showed that in

MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS

469

HoCuz, apart from an antiferromagnetic-order transition at 11.4 K, a change of the antiferromagnetic structure occurs at about 8 K. Gratz et al. (1982) carded out measurements of magnetization, resistivity, thermopower and neutron diffraction in HoCu2. They deduced values for TN of 9.7 K and for Ts of 7 K (order-to-order transition). From measurements of the magnetization, Hien et al. (1983) have confLrmed these results. Values for TN of 9.6 K and for Ts of 7 K were derived from specific-heat (Luong et al. 1985b, c) and thermal-expansion (Luong et al. 1985b) measurements. The at-susceptibility measurements by Smetana et al. (1985a) gave values for TN of 9.7 K and for Ts of 6.7 K. Neutron-diffraction studies of the magnetic structure of HoCu2 have been performed on a single-crystalline sample (Hashimoto et al. 1979a) and on powder material (Lord and McEven 1980, Gratz et al. 1982, Smetana et al. 1983, 1985b, Smetana and Sima 1985, Smetana et al. 1986b, Lebech et al. 1987). These studies reveal that the magnetic structure of HoCuz below TN is a collinear commensurable longitudinally modulated one along the a axis (see fig. 4.46). As temperature decreases, a second transition occurs at Ts ~-, 7 K. Below this transition, the magnetic structure has two components. One component is the above mentioned a axis modulated structure and the other component is a c axis incommensurable transversely modulated structure with moments along the b direction. The resulting magnetic structure of HoCu2 below 7 K is then a non-collinear incommensurably modulated structure with magnetic moments lying in the a b plane (Lebech et al. 1987), see fig. 4.47. The magnetic structure of HoCuz has been analyzed by Kimura (1990a) in a molecular-field approximation taking second-order crystal-field effects into account. He has shown that the spin arrangement in the a e plane for 7 K < T < TN can be realized by a competition of three kinds of exchange interactions between neighbouring ions in the same a c plane. He has also shown that the origin of the spin canting at temperatures below Ts is the biquadratic exchange interaction between the nearest Ho ions in the neighbouring a c planes. In later work Kimura (1990b) has investigated the effect of higher-order components of the crystal-field on the spin arrangement of HoCu2. He has shown that one of two possible magnetic structures at 0 K is stabilized by the 6th order component of the crystal field acting on the Ho

--,

JA '

11'

I

'

'

Fig. 4.46. The high-temperature (7 K < T < 10.4 K) magnetic structure of HoCu 2. After Smetana et al. (1986b).

470

N.H. LUONG and J.J.M. FRANSE

2

I

1

I

i

I

I

I I I

I I

I

I

r-"

J

I

I

I

The Mb component is modu[ated o[ong

I

Fig. 4.47. Projection of the magnetic structure of HoCu2 in the ab plane at T < 7 K. After Lebech et al. (1987).

10 xl0 3 b

c

"T

6

E ~u

4

2 0

.,~;" 0

l 100

i 200 T (K)

i 300

Fig. 4.48. Temperature dependence of the inverse susceptibility along the principal axes of HoCu 2. After Hasbimoto et al. (1979a).

ions. Thus the interplay of the biquadratic exchange interaction and the higher-order components of crystal-field has a large effect on the spin arrangement below Ts. Figure 4.48 shows the temperature dependence of the inverse susceptibility along the principal axes for HoCuz in the paramagnetic region (Hashimoto et al. 1979a). The effective magnetic moment, /-teir, amounts to 9.6#n. The paramagnetic Curie temperatures along the a, b and e axes are: 0a = 45 K, t9b = 30 K and 0c = 38 K. From these values for the paramagnetic Curie temperature Hashimoto et al. have estimated the following values for the second-order crystal-field parameters: B ° = 0.14 K, B22 = 0.12 K. On the basis of the point-charge model, Hashimoto et al. (1979a) have calculated the second-order crystal-field parameters and arrived at: B2° ~l = 0.28 K and B 2 c~l = 0,23 K. Using experimentally derived crystal-field parameters of Hashimoto et al., Nowotny and Zajac (1985) have calculated the paramagnetic

MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS

471

200 H°Cu2

o

150 b 100 c

50

0

1

2

3

t~

5

B (T) Fig. 4.49. Magnetization curves along the principal axes of HoCu 2 at 4.2 K. After Hashimoto et al. (1979a).

susceptibility of HoCu 2. They have obtained a total splitting energy of 33 K, which is lowest compared with those of PrCu2, NdCu2, TbCu2, DyCu2 and ErCu 2. The magnetization curves for single-crystalline HoCu 2 at 4.2 K are shown in fig. 4.49 (Hashimoto et al. 1979a). The anisotropy of this compound is not so large compared with T b C u 2 and DyCu z. The specific heat of HoCu2 has been measured by Luong et al. (1985b, c). Figure 4.50 presents the specific heat of HoCuz at zero magnetic field and in a field of 5 T. A A-type of anomaly is observed around TN. Luong et al. (1985b) have obtained the value of 19.9 J/Kmol f.u. for the magnetic entropy. This value points to a multiplicity of at least 11, indicating that the J = 8 (for Ho 3+) multiplet is splitted by crystal-field effects. A sharp peak, which occurs at 7 K and which is strongly field dependent, points to changes in the magnetic structure. Apart from this peak and the A-type of peak around TN, there are two other anomalies at 4.5 K and at temperatures below 1.3 K. The first one is rather broad and is not much influenced by the applied magnetic field. The anomaly below 1.3 K is ascribed to the nuclear contribution of holmium to the specific heat (Luong et al. 1985b). The specific heat in the temperature range from 2 to 20 K and in magnetic fields up to 4 T has also been measured by Bischof et al. (1989). The results at zero magnetic field are in good agreement with the specific-heat data of Luong et al. (1985b, c). Bischof et al. determined the change of specific heat due to the magnetic field, ACCF, by writting ACCF = c c F ( B ) - CCF(0), where CCF is obtained by subtracting contributions of electrons and phonons to the specific heat. Using the values for the CF parameters B2° and B 2 of Hashimoto et al. (1979a) (see above), Bischof et al. have calculated ACCF and compared it with experiments. They observed discrepancies between the calculated and experimental specific-heat curves, which demonstrate the importance of higher-order crystal-field terms. Measurements of the specific heat at higher temperatures (up to 100 K, for instance) would be very useful,

472

N.H. LUONG and J.J.M. FRANSE

2.5

/ f 2.0 0

o

E

HoCu 2

* ~

o

°o ~o

1.5

o~ o

1.0

t3

~o++÷++~+

+ +

°°°°~°°°°°°°o

0.5

0

o o

0 0

o

O O0

0

I

I

I

I

I

I

5

10

15

20

25

30

T (K) Fig. 4.50. The specific heat of HoCu 2 at zero magnetic field (o) and in a field of 5 T (+) in a plot of c/T versus T. After Luong et al. (1985b).

among other experiments, for the determination of the crystal-field scheme of the HoCu2 compound. The coefficient of the thermal expansion of HoCu2 as a function of temperature is shown in fig. 4.51 (Luong et al. 1985b). Two sharp peaks were observed, reflecting 12 HoCu 2 go

~

"7%,"

8

:',,.

'0 o

4.

eo° •

oo

~IL ~

• •

o • o° %oe%eoeeNooeoeoo • •

I

I

I

10

20

30

I

40 T (K)

I

I

I

i

50

60

70

80

Fig. 4.51. The coefficient of thermal expansion for HoCu2 as a function o f temperature. After Luong et al. (1985b).

MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS

473

specific anomalies around Tr~ and Ts. The relative volume change due to magnetic ordering in HoCu2 is -0.25 x 10 -3, which is almost four times smaller than in GdCu2. This fact possibly indicates a reduced polarization of conduction electrons by the holmium moments (Luong et al. 1985b). Neutron-diffraction studies on a powder sample of Ho(Cu0.9Ni0a)2 have been performed by Smetana et al. (1983) and Smetana and Sima (1985). The effect of Ni substitution in the Ho(Cu, Ni)2 system is qualitatively similar to that observed in Tb(Cu, Ni)2.

4°9. ErCu2 The Nrel temperature of ErCu2 has been found to lie in the range 11-13.5 K; 11 K, 13.5 K and 11.8 K, according to the magnetic measurements of Sherwood et al. (1964), Hashimoto et al. (1979a) and Smetana et al. (1985a), respectively and 11.5 K according to the specific-heat measurements of Luong et al. (1985b). In magnetization measurements Hien et al. (1983) have observed, apart from the anomaly that is characteristic for disordering the antiferromagnetic state, a second transition at about Tsl = 6 K. These authors ascribed this transition to the change of the antiferromagnetic structure. This conclusion is supported by ac-susceptibility measurements (Smetana et al. 1985a) and specific-heat measurements (Luong et al. 1985b, c). From specific-heat and thermal-expansion measurements, Luong et al. (1985b) have observed two other anomalies at Ts2 = 4.1 K and Ts3 = 3.3 K. The anomaly around 4 K has also been reported by Smetana et al. (1984, 1985a). These additional peaks at Tse and Ts3 could also point to changes in the magnetic structure. Neutron-diffraction experiments on ErCu 2 have been performed by Smetana et al. (1984), Smetana and Sima (1985) and Lebech et al. (1987). These latter authors 10

x103

°~~/~c

ErCu '7

~6 -i

E aJ

2 J"

¢

,,"

I

I

I

100

200

300

T (K) Fig. 4.52. Temperature dependence of the inverse susceptibility along the principal axes of ErCu 2. After Hashimoto et al. (1979a).

474

N.H. LUONG and J.J.M. FRANSE

have reported that neutron-diffraction powder patterns obtained between 1.5 and 40 K revealed a rather complicated antiferromagnetic structure. In spite of many attempts to interpret the powder patterns, they were not able to describe the magnetic structure of ErCu2 satisfactorily. Single-crystalline data would be helpful. Figure 4.52 shows the temperature dependence of the inverse susceptibility along each principal axis of ErCu2 (Hashimoto et al. 1979a). The effective magnetic moment,/~e~, amounts to 8.9/ZB. The paramagnetic Curie temperatures along the a, b and c axes are: 0a = 18 K, 0b = 53 K and 0c = 36 K. From these data, Hashimoto et al. have estimated the following values for the second-order crystal-field parameters for ErCu2: B ° = -0.35 K, B 2 = -0.36 K. On the basis of the point-charge model, Hashimoto et al. (1979a) have also calculated these second-order parameters and obtained: B ° l = -0.31 K and B2~al = -0.26 K. The magnetization curves for single-crystalline ErCu2 are shown in fig. 4.53. The magnetization along the b axis increases abruptly at about 0.6 T and saturates magnetically at a higher field. From fig. 4.53 it is also clear that ErCu2 has a large magnetocrystalline anisotropy. The high-field magnetization curve of polycrystalline ErCu2 at 4.2 K is shown in fig. 4.54 (Hien et al., unpublished result). A two-step metamagnetic process was observed. In a magnetic field of about 25 T the magnetization is still not saturated. The specific heat and the thermal expansion of ErCu2 have been measured by Luong et al. (1985b, c). Figure 4.55 shows the specific heat of ErCu2 at zero magnetic field and in a field of 5 T. The thermal-expansion coefficient of this compound is presented in fig. 4.56. The anomalies at TN, Tsl, T~2 and Ts3 were observed in the specific-heat and thermal-expansion-coefficient curves. Apart from these anomalies, broad anomalies occurred above TN. Inspecting figs 4.55 and 4.56 Luong et al. (1985b) have assumed that in ErCu2, apart from the 'long-range' magnetic contributions to the specific heat and thermal expansion, crystal-field effects were present. 200

150 Er Cu2

El00

50 -__~.~-~ ~

0

1

2

3

4

5

B (T) Fig. 4.53. Magnetization curves along the principal axes of ErCu2 at 4.2 K. After Hashimoto et al. (1979a).

MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS

150

475

ErCu2

100 E

50

0

0

I

I

I

I

l

5

10

15

20

25

B (T) Fig. 4.54. High-field magnetization curve of ErCu2 at 4.2 IC After Hien et al. (unpublished result).

1.0 -

0

*

O E

ErCu2

#

0

0~

+

o°

÷÷%o

0 00

,,. F-

÷+

00

o°

+

o

+*

÷

4- ÷ 0

4. 0

"l"-I. 0 o

.,I0

0

0

°

0 0 0

o

÷ ÷=-o o

0.5

,I 0

20

10 T

30

t~0

(KI

Fig. 4.55. The specific heat of ErCu2 at zero magnetic field (.) and in a field of 5 T (÷) in plots of e/T versus T. The full curve represents the non-magnetic contribution to the specific heat of ErCu2. The dashed curve represents the specific heat of YCu 2. After Luong et al. (1985b). Luong et al. ( 1 9 8 5 b ) and Franse et al. (1985) have analyzed the e x c e s s contributions to the specific heat and thermal expansion in ErCuz by applying Grtlneisen relations (see section 3.3), s h o w i n g that they consist of a 'long-range' magnetic and a crystal-

476

N.H. LUONGand J.J.M. FRANSE

ErCu 2

10

l

-5 m

-1(3

0

I

I

I

I

I

I

I

10

20

30

40

50

60

70

80

T (K) Fig. 4.56. Thermal-expansioncoefficientof ErCu2. The non-magneticcontribution to the thermal expansion is indicatedby the dashed line. AfterLuonget al. (1985b). field part. The excess specific heat, era, was obtained by subtracting the electronic and phonon parts (non-magnetic) from the observed specific heat. The excess thermal expansion, O~m,is also obtained by subtracting the non-magnetic parts from the observed thermal-expansion coefficient (see fig. 4.57). Cm and am are related by the effective Grtineisen parameter, Fen [see eq. (3.38)]. From the temperature dependence of the Griineisen parameter, ~Fe~, for ErCu2, a change in sign in this parameter was observed. This change in sign indicates that at least two different contributions to am and am have to be distinguished. Values for ~Flr (6×10 -11 me/N) and ~/~ef (-15 x 10 -11 m2fN) have been derived by comparing em and am well below and well above TN, respectively. With these values, Cm and am have been separated into two different contributions (see fig. 4.58). The term Clr is A-type, whereas ccf is Schottky-type. The energy difference between the two lowest doublets was derived to be 76 K. We note that the temperature dependence of cef above and below Tm (at which cef exhibits a maximum, see section 3.3), as reported by Luong et al. (1985b), has not properly been calculated. Franse and Luong (private communication) have calculated ccf as a function of temperature taking into account only two lowest doublets with the experimentally derived energy difference of 76 K between these two doublets. They obtained a similar behaviour of the eel(T) curve compared with the experimental one but the calculated value of ecf is lower (about half the experimental value, at Tin). Using values for the two second-order crystal-field parameters reported by Hashimoto et al. (1979a) (see above), Franse et al. (1985) have calculated the energy levels in the ErCu2 compound. These authors derived a splitting of

MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS

477

10

!s

;, 10

20

30

40

50

T (K)

.I,I.

~s

t,

/.

o

.

i,-. I

i

I

I"

1

20

10 ~t.

t

i

i

I

30

I

40

50

T (K)

44-

4"4" -5 ÷

4"

-10

4" 4"

4" 4" 4÷

4" 4"

4.

4"

~fl.+ + +

Fig. 4.57. The excess specific heat, cm, and the excess thermal expansion, am, of the ErCu2 compound. After Franse et al. (1985).

13 K between the lowest doublets. Apparently, higher-order terms have to be taken into account in order to bring the splitting closer to the experimental value of 76 IC The relative volume dependence of this energy splitting was compared with the volume dependence of the crystal-field parameters as calculated in a point-charge model (Franse et al. 1985). The two results obtained in this way differed in sign. As discussed by Franse et al. (1985), this unexpected volume dependence could originate either from an anisotropic compressibility or from an anisotropic thermal expansion due to preferential orientations in the sample on which the experiments were performed. To clarify this question from the experimental point of view, the thermal expansion in three mutually perpendicular directions of a ErCuz polycrystalline sample has been measured (Duc et al. 1988), see fig. 4.59. Data in this figure clearly prove that the thermal expansion is almost isotropic in the studied sample. These experiments once more show that a crystal-field calculation based on

478

N.H. LUONG and J.J.M. FRANSE

15

0

0

10

oc

(o lr

.°**'m°°

0

.£ u

0

*OOZe :of

0

I I I I

5

0

0

o

o

•

o

0

10

o

20

~

I

I

30

~0

50

I 60

T (K)

t=tr w

~ 0

t

.

.

I

;'0" .

.

t~0 T (K)

o CXrf

-5 0 0 0

-10

O o oo

o o oo

o

O

o

Fig. 4.58. Splitting of cm and C~m of ErCu 2 according to eqs (3.41-3.44) into a long-range magnetic (eir, air) and a crystal-field (c~, c~ce) part with values for n_Fir and ~_r'ef given in the text.

a point-charge model is inadequate to describe the experimental observations for this compound. The ErCu 2 compound has been measured with 166Er M6ssbauer spectroscopy (Gubbens et al. 1991). These authors also reported the results of inelastic neutron scattering, which show a not-yet definitively determined level sequence of doublets above TN at 0, 61, 78, 88, 124, 142, 148 and 160 K. Using a superposition model (Newman 1983, Divis 1991) and by fitting the results of inelastic-neutron-scattering and M6ssbauer spectroscopy, Gubbens et al. (1991) have determined a tentative set

MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS

479

15I ErCu2

10 A

•- !

5

v'

! Q

O

•--

0

~o ~o

-5

o

o°j/

oOJ ,,,/

x x

×

"10

-1_=

0

t

~

10

20

30

40

50

60

TtK)

Fig. 4.59. The linear thermal-expansion coefficient measured in three mutually perpendicular directions of a polycrystalline sample of ErCu2 with a cubic shape (5 x 5 × 5 mm3); the full curve represents the mean value of c~ along the three directions. After Due et al. (1988).

o f all the nine crystal-field parameters. This set is: B ° = - 0 . 2 8 K, B22 = - 0 . 2 2 K, B 0 -- - 0 . 3 0 X 10 - 2 K, B 2 = - 0 . 1 4 x 10 - 2 K, B44 = 0.30 x 10 - 2 K, B ° = - 0 . 2 0 x 10 - 4 K, B62 = - 0 . 4 7 x 10 - 4 K, B64 = - 0 . 9 7 x 10 - 4 K, B 6 = - 2 . 9 6 x 10 - 4 K. The value o f B ° of this set is very close to a value o f - 0 . 2 6 K determined f r o m the quadrupole splitting (Gubbens et al. 1991). One also can see that there is agreement

480

N.H. LUONG and J.J.M. FRANSE

between the values of B ° and B 2 obtained by Hashimoto et al. (1979a) and those obtained by Gubbens et al. (1991). However, the latter authors have pointed out that their set of nine crystal-field parameters is not unique at the present stage of the investigation. Gratz et al. (1993) have measured the anisotropic thermal expansion of polycrystalline Ercu2 sample from 4.2 K to 400 K by means of X-ray powder diffraction. They have described theoretically the obtained data using the above listed set of crystal-field parameters deduced by Gubbens et al. (1991).

4.10. TmCu2 Magnetic measurements on TmCu 2 have been reported by Sherwood et al. (1964) but no value for the ordering temperature has been given. The ae susceptibility (Smetana et al. 1985a), magnetization, susceptibility, resistivity and specific heat (Smetana et al. 1986a) have been measured on polycrystalline TmCu2. These measurements reveal a Nrel temperature value of 6.3 K and magnetic transitions at Tsl = 4.3 K, Ts2 = 3.9 K and Ts3 = 3.3 K which reflect changes of the magnetic structure. The paramagnetic Curie temperatures along the a, b and c axes are reported as: 0a = - 1 5 K, 0b = 49 K and 0c = 18 K (Sima et al. 1989). The effective magnetic moment, #eft, is equal to (7.2 4- 0.2) #a (Smetana et al. 1986a). Neutron-diffraction investigations have been performed by Lebech et al. (1987), Sima et al. (1989) and, later, by Heidelmann et al. (1992). According to the latter authors, TmCuz shows at least three different antiferromagnetic structures. The lowtemperature structure (AFI) (T < 3.1 K) is composed by the first two odd harmonics of a propagation vector (21r/8)a*. In a second temperature region, 3.1 K < T < 4.7 K, the phase is a mixture of two structures (AFII). A third phase, (AFIII), appears in the temperature region 4.7 K < T < TN ~, 8 K. All these structures may be described as sinusoidally modulated with magnetic moments parallel to the b direction, except for one of the components of AFII, where it is necessary to invoke a component perpendicular to the b axis. The result that only one magnetic moment component in the b direction is necessary to describe the observed structure in the case of AFI and AFIII is in agreement with the crystal-field anisotropy studies (Sima et al. 1989). A discrepancy between the Nrel temperature obtained from bulk data mentioned above (TN = 6.3 K) and that obtained from the neutron-diffraction study of Heidelmann et al. (1992) (TN ~ 8 K) is still unexplained. Heidelmann et al. (1992) also pointed out that from the neutrondiffraction data there is no evidence for the existence of the transition at 3.9 K which has been reported from bulk data (see above). The temperature dependencies of the inverse magnetic susceptibility of TmCu2 along each principal crystallographic axis are shown in fig. 4.60 (Sima et al. 1989). From these results the following experimental values for the second-order crystalfield parameters have been derived: B ° = -0.94 K, B 2 = -1.01 K. Figure 4.61 presents the magnetization curves of single-crystalline TmCu2 (in comparison with a polycrystalline sample) at 4.2 K (Svoboda et al. 1989). A large anisotropy is observed. The b axis is the easy magnetization direction. I n this direction, a two-step metamagnetic transition (of possible spin flip type) occurs

MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS

TmCu2

481

a o

3 C

% %

0

100

200

300

T (K) Fig. 4.60. Temperature dependence of the inverse susceptibility along the principal axes of TmCu 2. Symbols are the experimental data. Lines represent the calculated susceptibility according to the set of crystal-field parameters No. 6 (table 4.1). After Sima et al. (1989).

b "R.

6

::3

,oOf,* .

~

o

oe

°

e

po/yory.~tcthne

e

•

•

°

°

°

°

°

°ee e z

2

,see

// •

0

e

C

1

2 B (T)

3

Fig. 4.61. Magnetization curves of TmCu2 along the principal axes (full line) in comparison with the polycrystalline one (dotted line) obtained at 4.2 K. After Svoboda et al. (1989).

with two critical fields Bet = 0.06 T and Be2 = 0.36 T. Above the second step, the magnetization reaches the saturated value of (6.9 4- 0.2)/~B per T m atom which is in good agreement with the value for a free T m 3+ ion. The magnetization along a and e axes is an order of magnitude lower than that along the b axis.

482

N.H. LUONG and J.J.M. FRANSE

Crystal-field effects in TmCu 2 have extensively been studied (Zajac et al. 1987, 1988a, Sima et al. 1988, 1989, Divis et al. 1989b, and Gubbens et al. 1992). Zajac et al. (1987) have calculated the crystal-field splitting and the crystal-field susceptibility along the principal crystallographic axes. In their calculations, they used only the second-order crystal-field parameters B ° = -2.17 K and B 2 = -1.88 K. Their calculations show a total splitting of about 290 K. By comparing the calculations with the experimental results reported by Smetana et al. (1986a), Zajac et al. (1987) have pointed out the need to take higher-order terms into account. Based on the analysis of the specific-heat and thermal-expansion data, Sima et al. (1988, 1989) have studied crystal-field effects in TmCu 2 in more detail. Figure 4.62 shows the experimental and calculated temperature dependencies of the specific heat of TmCu2 in the paramagnetic region. The linear thermal-expansion coefficient of TmCu2 as a function of temperature is shown in fig. 4.63 (Sima et al. 1989). These authors have taken the 7 value equal to 9 mJ/K 2 mol, comparable with the values for the other RCu2 compounds (Luong et al. 1985b). The phonon part of the specific heat of the RCu 2 compounds follows a Debye function rather well (Luong et al. 1985b). From the best fit to the RCu2 data, Sima et al. (1989) obtained 0 D = 194 + 2 K for TmCu2. The values of m and l'ph for TmCu2 are unknown. In case of isostructural YCu2 and (Gd,Y)Cuz, Luong et al. (1985a) have found that x/~ph and 8i) are temperature-independent up to 100 K. Similarly, for TmCu2, Sima et al. (1989) assumed that n_r'ph is also independent of temperature and they have used a ~-r'ph value of 11.9 x 10- 12 m 2 /N, as obtained for YCu2 by Luong et al. (1985a). The problem of analyzing c(T) and ~(T) above TN has been reduced to the determination of parameters OD, Ei and 7i, where 7/[= - d ( l n Ei)/d(ln V)] is the crystal-field Grtineisen parameter of the individual energy

&E{K) 60

• 40

~ "4" % "~

~

~ -

-

////"

~ too

"

~

//

~

/11/

O =/17

//

J,

m

........

0

20

CO T(K)

60

, .......

80

Fig. 4.62. Temperaturedependences of the specific heat of TmCu2 in the paramagneticregion: o, experimental data; - - , calculatedtotal specificheat; - - -, phonon specificheat; •.., electronicspecific heat; - - -, Sehottky specific heat. The arrow indicates the N6el temperature. The inset shows the energies of the CF levels used in calculations. After Sima et al. (1989).

MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS

483

10

T

-10

".......'

-20

i 20

i /40

i 60

t 80

100

T[K~ Fig. 4.63. Temperaturedependences of the linear thermal-expansioncoefficientof TmCu2 in the paramagnetie region: o, experimental data; q , calculated total thermal-expansioncoefficient; - - -, phonon thermal-expansioncoefficient;..., Schottkythermal-expansioncoefficient.The arrow indicates the Nrel temperature. After Sima et al. (1989). level Ei (i = 0, 1, . . . , 1 2 , E0 = 0). To solve this many-parameter (25) problem, Sima et al. (1989) used a Monte Carlo simulation of all parameters in order to find the intervals of appropriate possible values. The final solution for Ei has to be conform to the nine-parameter orthorhombic CF Hamiltonian. This leads to a reduction of the number of parameters to 22 (8D, B ,rr~, 7i). These authors treated the Bnm as adjustable parameters to be determined from the proposed CF level scheme. The number of possible solutions is quite large. About 100 fits were done and the sets of parameters which are in agreement with experimental observations have been selected (see table 4.1). The values of the second-order terms B ° and B 2 are very close to those obtained from susceptibility measurements (see above). From the analysis it appears that the ground state of TmCu2 is an isolated quasi-doublet (El = 5 K, E2 = 68 K). The total splitting is 210 K, i.e. less than that reported by Zajac et al. (1987) (see above). Sima et al. (1989) pointed out that a major contribution to the CF splitting comes from the second-order terms B2° and B 2. However, these terms cannot satisfactorily describe the observed splitting of the quasi-doublet ground state and the inverse susceptibility Xa I (see fig. 4.60) between 20 and 70 K. All nine parameters, Bnm, must be taken into account in order to obtain, at least qualitative, agreement with experiments. The value for E1 agrees with preliminary results of inelastic neutron spectroscopy obtained by Loewenhaupt and Gratz (1989). Gubbens et al. (1992) have studied crystal-field effects in TmCu2 by 169Tm MSssbauer spectroscopy. By combining the results of magnetization (Svoboda et al. 1989, Divis et al. 1989b), specific heat and thermal expansion (Sima et al. 1989)

484

N.H. LUONG and J.J.M. FRANSE TABLE 4.1 Crystal-field parameters, in units of K, for the compound TmCu 2. The sets Nos. 1-6 are the selected by Sima et al. (1989) from an analysis of specific heat and thermal expansion data. Set (i) gives the calculated CF parameters using a modification of the point-charge model (Sima et al. 1989). Set (ii) gives the values obtained by Zajac et al. (1988a) in point-charge model. Set (iii) gives the parameters derived by Gubbens et al. (1992) by combining the results of different experiments.

No

st

B0,

s:,

(10 -3 ) (10 -2 ) 1 2 3 4 5 6 i ii iii

-1.16 -1.30 -1.30 -1.01 -1.01 -0.94 -1.45 -1.13 -0.94

-0.80 -0.87 -1.09 -1.23 -1.23 -1.30 -0.29 0.87 -1.23

-1.81 -0.07 -4.35 0.36 -0.51 0.29 -5.58 0.15 -9.0

-2.90 -1.45 -0.43 -3.26 -3.40 -3.69 -0.36 -0.28 -0.39

B',

s06

(10 -2 )

(10-~

2.17 0.58 -0.07 2.82 3.62 2.90 0.87 0.77 -0.36

0.72 1.45 1.52 -2.32 -0.80 -2.54 3.26 0.43 5.8

(10 -4) -6.52 -9.63 0.72 -7.53 -7.46 -7.82 2.46 0.15 2.47

(10-') 5.79 2.90 -11.66 3.19 2.39 4.06 1.59 0.20 -0.48

(10 -4 ) -7.97 6.59 0.87 4.49 3.19 3.91 -7.17 0.07 6.31

with those of inelastic neutron scattering (Divis, Heidelmann and Loewenhaupt, unpublished results) and 169Tm M6ssbauer spectroscopy, Gubbens et al. (1992) have derived the set of crystal-field parameters for TmCu 2 shown in the last row of table 4.1. However, Gubbens et al. (1992) pointed out that this set of crystal-field parameters still has a preliminary character. By studying the relaxation in TmCu2, these authors have indicated that the proposed crystal-field parameters might need future reconsideration. Javorsky et al. (1992) have measured the specific heat of polycrystalline samples Tm~YI_~Cu2 (z = 0.1, 0.2, 0.3, 0.4, 0.6 and 0.8) at temperatures up to 30 K. The antiferromagnetic ordering has been observed for z = 0.6 and 0.8. For z ~< 0.4, no magnetic ordering has been found and the specific-heat curves turned out to show only a Shottky anomaly due to the crystal-field effect. Using the molecular-field model in a two-level approximation (Fulde 1979), Javorsky et al. (1992) calculated the temperature dependence of the Nrel temperature according to the relation: tanh (E1/2Ttq) = (1/z) tanh (E1/2T~),

(4.6)

where T~ is the NEel temperature for TmCu2 and E1 is the energy separation (measured in kelvin) between the two lowest levels. They have obtained good agreement with experiment. By comparing the experimental data in the paramagnetic region with eq. (3.23), Javorsky et al. derived the best-fit values for El, E2 and E3. They concluded that the value for E1 changes minimally with concentration, (El = (4 4- 2) K). and that the quasidoublet ground state is well separated from the higher excited states for all Tm concentrations, in very good agreement with the preliminary results of inelastic neutron spectroscopy obtained by Heidelmann and Loewenhaupt (private communication), who reported for x = 0.05 the values for El and E2 of 2.1 K and 45 K, respectively.

MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS

485

Specific-heat and magnetization measurements on Tm(Cu~Nil_~)2 have been performed by Divis et al. (1990c). A change from antiferromagnetic (for z --- 0.95 and 0.9) to ferromagnetic ordering (for z = 0.8, 0.75 and 0.7) is observed. For x < 0.7, the CeCu 2 structure becomes unstable. From the analysis of the data, Divis et al. conclude that the crystal-field ground state of the Tm(Cu~Nil_~:)2 compounds (z ~< 0.7) is a quasidoublet.

5. Comparison of isostructural compounds In the preceding section we have discussed the properties of various RCu 2 compounds. It is of interest to make an attempt to see some systematic behaviour, comparing different compounds with the same crystallographic structure. It can be inferred from section 4 that information about the crystal-field interaction ill RCu 2 is not complete. Due to the orthorhombic structure, nine CF parameters are needed to describe the CF Hamiltonian. Up till now, the full set of CF parameters is available only for NdCu2, ErCu2 and TmCu2. Nevertheless, the lowest-order CF parameters Bg and B 2 have been derived for most of the RCu2 compounds. In table 5.1, we collect the second-order coefficients A ° and A2z for the RCu2 compounds. These coefficients are related to the CF parameters B 20 and B 22 by (see section 3): Ao =

B°loej(r2e),

(5.1)

A 2 m B 22/ a J ( r 4 2f ) ,

where the values for the quantity otj(rlf ) are taken from table 3.1. As pointed out in section 4, to our knowledge, CF parameters for SmCu2 are not available. Rather low TABLE 5.1 Crystal-field coefficients, in units of Kao 2, for the RCu 2 compounds. R

A~

A~

ReL

Pr Nd

-168 -112 -188

-117 -154 -218

[1] [1] [2]

Tb

-148

-148

[31

Dy Ho Er

-85.7 -83.5 -190 -153 -134.7 - 134.7

-143.4 -71.6 -194 -120 -186.4 - 176.3

[3] [3] [3] 14] [5] [6]

Tm [1] [2] [3] [4] [5] [6]

Hashimoto (1979). Gratz et al. (1991). Hashimoto et al. (1979a). Gubbens et al. (1991). Sima et al. (1989). Gubbens et al. (1992).

486

N.H. LUONG and J.J.M. FRANSE

values of A ° and A 2 have been obtained by Trump (1991) for the Kondo compound CeCu2, in which anomalous properties are observed. Except for CeCu2, as can be seen from table 5.1, the coefficients A ° and A22 have the same sign and are of the same order of magnitude. From the similarity of the lowest-order CF parameters it seems that the crystal-field model can be used for describing the behaviour of isostructural RCu 2 compounds. At the same time, there is no reason why higherorder CF parameters should be neglected. The necessity to take these higher-order terms into account is indicated in section 4. The importance of the higher-order CF terms is also revealed from the studies on substituted RCu2 compounds. Analyzing the data obtained on a Tb(Cu0.7Ni0.3)2 sample, Divis et al. (1990b) have shown that the step-like appearance in the magnetization curves along the b axis in this compound cannot be explained using second-order terms in CF Hamiltonian only. These authors have shown that in order to account for all features of the magnetization data, the higher-order terms should be included into the Hamiltonian. Divis et al. (1990c) have also used nine CF parameters for describing the specific-heat data on Tm(Cul_xNix)2. Even for simple cubic systems such as RA12, where only two CF parameters are needed, the CF model fails to give a rigorous systematic and quantitative description of the magnetic properties (Purwins and Leson 1990). Little can be said about a systematic analysis of exchange interactions in RCu2. This is probably due to the complicated magnetic structure of these compounds. In some cases, the temperature dependence of the sublattice magnetizations and the magnetization process can be described in the molecular-field and crystal-field model in which, apart from the crystal-field parameters, a limited number of the molecular-field constants is considered. TbCu2 is an example. The magnetic unit cell of this compound consists of twelve magnetic moments but different temperature dependencies of the magnetization are observed in two sublattices only (Sima et al. 1986, Divis et al. 1987). Then the effective molecular fields at these two sublattices can be expressed as (Divis et al. 1987): Bin1 - -

LI(M1) + 2L2(M2),

(5.2)

Bin2 = (51 - L2)(M2) + L2(M1), where M1 and M2 are the sublattice magnetic moments and where LI and L2 are effective molecular-field constants. The observed temperature dependence of the sublattice magnetizations can be well reproduced by a calculation using these two effective molecular-field constants together with two experimentally determined crystal-field parameters (Divis et al. 1987, Iwata et al. 1987). Values for these molecular-field constants are derived from the best fit to the experimental data. The effective molecular-field constants for TbCu2 determined by Divis et al. (1987) and Zajac et al. (1988b) (L1 = 5.77 x 105 A/m/.tB and L2 = 5.42 x 105 A/m#B) are only slightly different from those obtained by Iwata et al. (1987, 1988). Divis et al. (1987) have pointed out that the observed temperature dependence of the sublattice magnetization in TbCu2 can be considered as a result of comparable contributions of the molecular-field and the crystal-field contributions to the total energy in this

MAGNETIC PROPERTIESOF RARE EARTH-Cu2COMPOUNDS

487

compound. Magnetization processes in YbCu 2 compound are satisfactorily explained by Iwata et al. (1988) using the molecular-field model with two crystal-field parameters and five exchange parameters. From the best fit to the magnetization processes in DyCu2, Iwata et al. (1989) have determined the exchange parameters for this compound. Values for these parameters are comparable in magnitude with those of the TbCu2 compound. One of the features of the RCu2 compounds is that the values for the N6el temperature of these compounds are not simply proportional to the de Gelmes factor G = (gj - 1)2j(j + 1) and takes a maximum for TbCu2 (see fig. 5.1). This fact suggests that the RKKY interaction is not sufficient for fully understanding the exchange interactions in the RCu2 compounds. In the discussions of the ground-state spin configurations in the CeCu2-type crystal structure, Kimura (1986) considered four kinds of bilinear exchange interactions. In later work, Kimura et al. (1991), apart from three kinds of bilinear exchange interactions, have introduced the four-spin exchange interaction. These authors show that the four-spin exchange interaction gives rise to new complex states as a possible ground-state spin configuration in the CeCu 2type crystal structure and that, within the 12-sublattice model, 24 types of spin configurations are possible for various values of exchange interactions. Analyzing magnetization process in the TbrYl_=Cu2 compounds in the same molecular-field and crystal-field model as that used by Divis et al. (1987), Zajac et al. (1988b) pointed out that better agreement between calculated and measured magnetization curves can be expected if taking into account higher-order crystal- and molecularfield parameters. In some of the RCu2 compounds, it may be necessary to consider the quadrupole and magnetoelastic interactions for better explanation of the magnetic

16

5O

~0

,-i---

0

12 "7

0

3O

..-i ¢q

a~ 0

{71

2O 0

10

O

0 0

$ O

0

0 0 n ~ I i J i r T r I i ~9 Pm Eu Tb Ho Tm Lu Lo Pr Ce Nd Srn Od Dy Er Yb

Fig. 5.1. Values for the N6el temperature in the R C u 2 series and values for the de Gennes factor (9 - 1)23"(,/+ 1) across the 4f series.

488

N.H. LUONG and J.J.M. FRANSE

behaviour. The role of these interactions in rare-earth intermetallic compounds has been discussed, e.g., by Morin and Schmitt (1978, 1990).

6. Acknowledgements The authors wish to thank Prof. T.D. Hien of the Cryogenic Laboratory of the University of Hanoi for his interest, help and encouragement. The cooperations with Prof. N.P. Thuy, Dr. N.H. Duc and other colleaques of the Cryogenic Laboratory and with Dr. E E Bekker of the University of Amsterdam are kindly acknowledged. The authors are grateful to Dr. Y. Hashimoto, Prof. I. Kimura and Prof. M. Loewenhaupt for providing their work, to Dr. D. Givord for comments on the manuscript, to Dr. N.M. Hang for his comments and assistance in collecting some references and to Dr. R.R. Arons et al. for sending a preprint on the magnetic structure of NdCu2. This work has been supported by the Commission of the European Community (BRITE/EURAM Programme) in the scope of the project 'Basic Interactions in RareEarth Magnets' (BIREM).

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SUBJECT INDEX apparent core loss anisotropy 372, 373 artificial metallic superlattice 3

actinide compounds 62 amorphous ribbons 327 Anderson's condition 270, 275 Anderson's criterion 282 angular dependence of soft magnetic properties 376, 377, 383, 392 anisometric methods 349, 357 anisotropie and isotropic sheets 369 anisotropic energy 66, 328 anisotropic magnetostriction constants of RCo 2 68 anisotropic magnetostriction constants of RFe2 68 anisotropic magnetostriction 91, 125, 137, 153 anisotropic sheets 327 anisotropy constants 328, 329, 356, 360, 368, 410 anisotropy energy, in cubic crystals 66 anisotropy in motors 405 anisotropy in transformers 402 anisotropy of apparent core loss 369, 383 anisotropy of apparent core 382 anisotropy of coercive force 367, 379 anisotropy of core loss 369, 382, 383, 385 anisotropy of demagnetization coefficient 363, 365 anisotropy of demagnetization 365 anisotropy of induction 329, 369, 373, 382, 383, 385 anisotropy of magnetostrictive elongation 366 anisotropy of mechanical properties of Fe alloys 360, 363, 409 anisotropy of resistance 382 anisotropy of resistivity 360, 409 anomalous strain 48 antiferromagnetic 29, 417, 418, 432, 441, 451, 469, 480 antiferromagnetic state 473

band metamagnetism 70, 80 band structure calculations 277 bilayer harmonica 8 buffer layer in superlattices 4 casting direction 386 cation distribution 195, 197, 198 cation exchange reaction 199 Ce2Ni 7 type 99 CeCu 2 type of structure 419, 420 chemical vapour deposition 302 chirality 12 damping 42 classification of electrical sheets 336 coercive force in electric sheets 367, 379, 382, 385, 386 coherence length 34 coherence ranges 12, 28 collinear Ntel configuration 209 collinear Ntel structure 217 core loss 368 core loss anisotropy 372, 373, 381 correlation coefficients 410 Cotton-Mouton effect 254 Coulomb gap 275, 283 critical grain size 231 crystal field anisotropy 14 crystal field 424 crystal-field coefficients 423, 485 crystal-field Hamiltonian 422 crystal-field interaction 421 crystal-field parameters in RCu 5 compounds 422, 427, 437, 442, 460, 467, 470, 474, 479, 480, 482, 484 521

522

SUBJECT INDEX

crystal orientation in Fe alloys 371, 381 crystal structure of CaCu5 type 72 crystal structure of NaZn13 type 177 crystal structure of Nd2Fe14B type 155 crystal structure of ThMn12 type 172 Curie temperature 211-213 Curie temperature of RCo2 68 Curie temperature of RCo 3 129 Curie temperature of RCo 5 74, 75 Curie temperature of RFe2 68 Curie temperature of R2Fe17 142 Curie temperature of R2Fe14B 158 Curie temperature of R2Fe17Cx 142 Curie-Weiss law 426, 451, 459 degree of sample texture in soft magnetic Fe alloys 356, 357 degree of texture in soft magnetic Fe alloys 360, 374 demagnetization anisotropy in soft magnetic Fe alloys 365, 382 demagnetization coefficient in soft magnetic Fe alloys 346, 363, 382, 385 demagnetization factor in soft magnetic Fe alloys 363 demagnetization field in soft magnetic Fe alloys 346 dielectric constant 282 dielectric relaxation 281 dipolar energies 14 dipole coupling 29 directional changes of core loss 394 disaccommodation spectra 291, 294, 296 disaccommodation 293 disordered moment 49 domain observation 345 domain size 244, 245 domain wall bulging 228, 230 domain wall displacement 226 double exchange 210 easy-magnetization direction of RCo 2 68 easy-magnetization direction of RFe2 68 eddy current loss 248 electrical conduction of ferrites 285 electrical conductivity in Fe alloys sheets 360 electrical conductivity of FeaO 4 266 electrical conductivity of magnetite 274 electrical conductivity 202, 278, 280 electrical resistivities in electrical sheets 359 elongation in electrical sheets 362 energy stored in a silicon-iron crystal 352 Epstein frame 337

Epstein test 339 Epstein test, 25-cm Epstein test 341 equatorial Kerr effect 255 exchange, q-dependent exchange 31 exchange coupling 13, 22 exchange interactions 424 exchange magnetostriction 65, 96, 111, 125, 153 extraordinary Hall coefficient 283, 284 fan states 40, 41 Faraday effect 254 Faraday ellipticity 255 Faraday rotation 255 FeaC type 127 ferrimagnets 85 ferrite films 301 ferrite plating 304 ferro-electric hysteresis 274 ferromagnetic phase transition 42 ferromagnetic transitions 14, 42 ferromagnets 85 figure etching 345 fourth-order Stevens factor 69 frustration effects 222 galvano-magnetic properties 283 Gd2Co7-type structure 99 generalized conduction electron susceptibility 22 generalized susceptibility 31 Globus model 231 Cross orientation 379 Goss texture 368, 369, 374, 375, 379 grain boundaries 252, 295 grain misorientation 351 Griineisen analysis 429 Griineisen parameter 430, 476 Griineisen relation 430 high permeability electrical sheets 375 high permeability steel sheets 375 high pressure structure of Fe304 magnetite 272 hydrides of rare earth intermetallics 113, 170 hydrogen absorption in R2Co 7 113 hyper-magnetite Fe3.2604 272 hyperfine fields in ferrites 285 hysteresis loops of electrical steels 379 hysteresis loss in ferrites 234, 248 incoherent scattering 49 incommensurate spin structures 3, 12 indirect exchange 14

SUBJECT INDEX induced anisotropy in soft magneti c materials 389, 410 induction in electric steels 368 induction anisotropy 357, 369, 372, 381 inelastic neutron scattering 431, 439, 434, 442, 478, 484 influence of stacking on properties of electric sheets 341 insignificant Goss texture 383 interdiffusion in superlattices 5 intervalence charge transfer transitions 257 intervalenee charge transfer 277 Invar effect 86 inverse spinels 194, 195 iron-based amorphous ribbons 385 iron-silicon mierocrystalline alloys 327 irreversible domain wall displacement 234 isotropic sheets 327 isotropic silicon-free sheets 381 Jahn-Teller phase transitions in ferrites 208 kinetics of the cation redistribution in ferrites 204 lanthanide contraction in intermatallics 83 lanthanide elements, superlattices of 3 laser ablation deposition 304 lattice mismatch 16 lattice parameter mismatch 4 lattice parameter of ferrites 197, 198, 200 lattice parameter of RCo 2 68 lattice parameters of RCo 3 118 lattice parameters of RCo 5 74, 75 lattice parameters of R2Co 7 100 lattice parameters of RaCo 128 lattice parameters of RCu 2 420 lattice parameter of RFe 2 68 lattice parameters of R2Fel7 141 lattice parameters of R2Fe14B 157 lattice parameters of R2FelTCx 141 laves phases 66 light rare earths, superlattices of 14 light rare-earth metals 85 linear spontaneous magnetostrietive strains of RCos 118, 129 linear spontaneous magnetostrictive strains of RCo 5 74, 75 linear spontaneous magnetostrietive strains of R2Co 7 100 linear spontaneous magnetostrictive strains of R2FeÂ7 142

523

linear spontaneous magnetostrietive strains of R2Fel4B 158 linear spontaneous magnetostrictive strains of R2FelTCz 142 linear thermal expansion coefficients of RCo 3 118 linear thermal expansion coefficients of RCo5 74, 75 linear thermal expansion coefficients of R2Co 7 100 linear thermal expansion coefficients of RaCo 128 linear thermal expansion coefficients of R2Fel7 141 linear thermal expansion coefficients of R2Fel4B 157 linear thermal expansion coefficients of R2FelTCx 141 liquid phase epitaxial growth 301 long-range order in amorphous ribbous 385 loss anisotropy factor (angle) 340 loss anisotropy factor 340 loss factor 241 low temperature structure of magnetite 271 lutecium, superlattices of 3 Madelung constant in ferrites 196 maghemite 192 magnetic after-effects in ferrites 290 magnetic anisotropy constants 66 magnetic anisotropy energy 66 magnetic anisotropy 170, 211-213, 288, 289, 329 magnetic coherence in superlattices 11 magnetic field anisotropy 372, 373 magnetic losses 248 magnetic moment 48, 425 magnetic moment per formula unit of RCo2 68 magnetic moment per formula unit of RCo 3 118, 129 magnetic moment per formula unit of RCo5 74, 75 magnetic moment per formula unit of R2Co 7 100 magnetic moment per formula unit of RFe 2 68 magnetic moment per formula unit of R2Fel7 142 magnetic moment per formula unit of R2Fel4B 158 magnetic moment per formula unit of R2FelTCx 142 magnetic permeability of ferrites 226 magnetic phase diagram of ferrites 223, 224 magnetic phase diagram for Mgl+x Fe2_2xTizO4 222

524

SUBJECT INDEX

magnetic phase shift 9 magnetic relaxations in ferrites 282 magnetic scattering in sperlattiees 9 magnetic structure factor 9 magnetic structure of the simple ferrites 210 magnetic structure of the Zn-substituted ferrites 217 magnetic structure of RCu 2 434, 435, 439, 442, 456, 459, 463, 469, 473 magnetically difficult directions in electric steels 357 magnetization 32, 214, 432, 436, 444, 445, 447, 448, 451, 452, 458, 464, 465, 471, 474, 480, 481 magnetization curves of RCu 2 438 magneto-acoustic emission of ferrites 237 magneto-electric properties of ferrites 272 magneto-electric susceptibilities of ferrites 273 magneto-optical properties of ferrites 253 magneto-optical properties of mixed ferrites 259 magnetocrystalline anisotropy in electric steels 328, 389, 391 magnetoelastic effects in superlattices 44 magnetoelastic energetics in superlattiees 43 magnetoelastie energies in superlattiees 41 magnetoelastie-coupling coefficient 66 magnetoelastie-eoupling coefficient of RCo5 74 magnetometry 22 magnetoresistance 283, 284 magnetostriction 62, 211-213, 288, 289, 291 magnetostriction, c~-magnetostriction 65, 1i0 magnetostriction, -'/-magnetostriction 66, 107 magnetostriction, ~100 anisotropic magnetostriction 64 magnetostriction, Alll anisotropic magnetostrietion 64 magnetostrietion constants 65 magnetostrictive elongation 365, 367 magnetovolume effect 66 maximum energy product 131 measurement of coercive force 347, 379 melting point of RCu 2 420 meridional Kerr effect 254 metamagnetie transitions in RCu 2 444 metamagnetic transition 70, 128, 434, 451,480 method of measuring anisotropic magnetic properties 343 microcrystalline soft magnetic materials 394 microerystalline tapes 391 microstrueture of ferrites 251 mixed orientation in electric sheets 351 mixed texture in steel sheets 381, 383, 384 modelling of anisotropy 395

molecular beam epitaxial growth 305 molecular beam epitaxy 3 multilayers 3, 305, 306 multiple-range hopping 280 M6ssbauer spectra of ferrites 285, 286, 287, 298 M6ssbauer spectroscopy 201, 431, 463, 478, 483 neutron depolarization in ferrites 241, 244 neutron depolarization technique 231 neutron scattering 6 non-magnetic grain boundary model 240 non-magnetic interlayers 11 non-oriented silicon-free electrical steel 385 non-oriented silicon-free electrical steel sheets 381 non-oriented silicon-free steel 381 non-spin-flip scattering 21 non-uniform stacking of steel sheets 341 normal cation distribution in ferrites 194 normal spinels 196 nuclear scattering intensity 9 nuclear structure factor 9 N6el temperature 14, 417, 447, 451, 457, 463, 473, 480, 487 N6el temperatures of RCo3 129 N6el's two sublattice model 209 N6el-Chevallier model 202 optical and magneto-optical properties of magnetite Fe304 256 optical conductivity 276 ordinary and extraordinary Hall effect in ferrites 283 ordinary Hall coefficient in ferrites 284 orthorhombic distortion of RCo 3 118 orthorhombic distortion of R2Co7 100 orthorhombic distortion 106, 113 oxygen parameters in ferrites 194, 197, 198, 200, 272 paramagnetic Curie temperature of RCu 2 427, 432, 437, 442, 447, 451, 460, 466, 470, 474, 480 paramagnetic susceptibility of RCu 2 426 particles of ferrites 216, 217, 297 percolation limit 271 percolation thresholds 221 phase shift 30 plastic and thermomagnetic treatment 351 polar figures 356, 359

SUBJECT INDEX Polar-Kerr effect 254, 255 Polar-Kerr ellipticity rotation 255 polarized neutron scattering 20 power loss in ferrites 251 PuNia-type Structure 116, 179 random stacking, of steel sheets 341 rare earths 3 rare-earth carbides 140 remanent magnetic state 50 remanent magnetization 25 residual loss in ferrites 248 resistivity, of Fe alloys 353 reverse bending of electric sheets 356 RKKY exchange 30 RKKY indirect exchange 29 RKKY model 424, 425, 451 rolling direction of electric sheets 368, 385 rotational component of magnetisation of electric sheets 344 rotational contribution to the permeability in ferrites 226 rotational loss in steel sheets 399 rotational magnetisation in steel sheets 396, 398, 401 saturation field 25 saturation magnetization 211-213, 274, 289 scandium, multilayers of 3 Schottky anomaly 428 Schottky specific heat in RCu2 428 second-order Stevens factor 69 semi-spin glass 222 semi-spin glass structure 223 Sherman transition 277 short range cation order in ferrites 200 silicon and silicon-free electrical steel sheets 327 silicon-free sheets 381, 382 single domain 45 single sheet tester 337, 338 small polaron 275, 277 space group of spinels 194 specific heat of Fe304 268 specific heat 427, 428, 433, 436, 437, 439, 441, 444, 445, 454, 460, 461, 467, 471, 474, 475, 482 spin canting in ferrites 202 spin density wave 31 spin glass 16, 223 spin reorientation 86, 88, 89, 91, 101,106, 121, 149, 152, 162, 166, 173 spin-flip scattering 21

525

spin-glass-like state 222 spin-slip state 41 spin-slip structure 13, 38 spinel structure 193, 195 spontaneous linear magnetostrictive strains 64 spontaneous magnetostriction 62 spontaneous moment 22 spontaneous volume magnetostriction 64 spontaneous volume magnetostriction of RCo2 68 spontaneous volume magnetostriction of RFe2 68 sputtering techniques 302 static tensile testing 354 Stevens factor 422 structure of MgCu2 type 67 structure of Th2Ni17 type 139 superexehange 209 superlattices of ferrites 305, 306 superstructures 204 superzone gaps 36 surface magnetic anisotropy 300 susceptibility 433, 437, 438, 442, 448, 451, 459, 470, 474, 480 tapes, 6.5% silicon microcrystalline tapes 335 tensile strength of steel sheets 363 Th6Mn23 type of structure 179 thermal expansion 429, 434, 436, 439, 440, 446, 447, 450, 455, 460, 462, 467, 468, 472, 474, 476, 477, 479, 482, 483 thermal expansion coefficient 63 thermoelectric power 202, 279 thickness 6w, of domain wall 230 thin gauge anisotropic steel sheets 377 thin gauge steel 378 total magnetic anisotropy 235, 237 turn angles of magnetization 15, 35 type of magnetocrystalline anisotropy of RCo3 118 type of magnetocrystalline anisotropy of RCo5 75 type of magnetocrystalline anisotropy of R2Co7 100 type of magnetocrystalline anisotropy of R2Fel7 142 type of magnetoerystalline anisotropy of R2FeI4B 158 type of magnetocrystalline anisotropy of R2FelTCx 142 uniaxial anisotropy 329, 334, 385, 389 uniform stacking of electric sheets 341

526 variable range hopping 267, 280, 295 vertical component of magnetic polarisation 349, 369, 381, 383 Verwey temperature 205, 206, 268, 275 Verwey transition 201, 265, 270, 288 Verwey-type transitions 266 volume effects on magnetization in R-3d intermetallics 136 volume spontaneous magnetostriction of RCo3 118, 129 volume spontaneous magnetostriction of RCo5 74, 75 volume spontaneous magnetostriction of R2Co7 I00 volume spontaneous magnetostriction of R2Fel7 142 volume spontaneous magnetostriction of R2Fe14B 158, 165 volume spontaneous magnetostriction of R2FelTC~ 142 volume thermal expansion coefficient of RCo3 118 volume thermal expansion coefficient of RCo5 74, 75

SUBJECT INDEX volume thermal expansion coefficient of R2Co7 100 volume thermal expansion coefficient of R2Fe17 141 volume thermal expansion coefficient of R2Fe14B 157, 165 volume thermal expansion coefficient of R2FelTCx 141 volume thermal expansion of R3Co 128 wall energy in ferrites 232, 235 wall thickness in ferrites 230 wet chemical deposition 304 Wigner glass state 276 X-ray techniques 345 Yafet-Kittel configuration 209, 218 yield point 363 yield strength 363 yttrium, multilayers of 3

MATERIALS INDEX Ag substituted in CeCu 2 436 A12Oa, c~-A12Oa 305, 306

DyCo 2 68 DyCo 3 118, 121 DyCo5.2 75, 89 Dy3Co 128, 129, 132 DyCu 2 420, 463, 485, 487 DyFe z 68 Dy2Fel7 141, 142, 149 Dy2Fel4B 165 DyFell Ti 173 Dy-Lu 48 Dy-Sc superlattices 17 DyxYl_xCu 2 467

CaFe204 192 CdFe204 197, 212, 216 CeCo 5 74, 79 CeCu 2 418, 420, 432 Ce(CuxGal_=) 2 436 Ce(Cul_xNix)2 436 CeFe 2 68 Ce2Fe14B 165 Ce2Fe17 146 Ce(Znl_xCux)2 436 Co-coated o,-Fe203 299 Co doped NiZn ferrites 242 Co ferrites 277, 298, 302 Co-modified -y-Fe203 300 CoAlxl?e2_x O 4 261-264 CoCrxFe2_xO 4 261, 262 Co2FeO 4 197, 212 CoFe204 197, 200, 203, 204, 212, 214, 262, 265, 299, 302 Col.5Fel.504 263 Co~Fe3_xO 4 263, 302 Col+x Fe2_xO 4 263 CoMxFe2_xO 4 262 CoMn ferrites 302 CoO 306 Co304 210 CoRh~Fe2_xO 4 262 CoZn ferrite 302 Col_xZnxFe204 219, 264 Cu 441 Cu ferrite 301 CuFe204 196, 197, 203, 207, 211, 212 Cu0.sFeo_.504 192, 197, 212 Cu0.96Fe2.o404 207

Er 17 ErCo 2 68 ErCo 3 118, 123 ErCu 2 420, 430, 431, 473, 485 ErFe 2 68 ErFe204 266 Er(Fel_xCox)2 71 Er2(Feo.6Coo.4)17 149 Er2Fel4B 165 Er2Fe17 141, 142, 149 ErFelo.sV1. 5 173 Er-Y 17 EuCu 2 417, 418, 420 FeAI204 197, 200, 212 Fe2AIO4 197, 212 Fe3_xAIxO4 203 FeCr204 197, 200, 212 Fe2CrO a 197, 212 Fe3_xCrxO 4 208 Fe [Co Cr]O 4 261 FeFe204 200 Fe[l"ll/3Fes/3]O4 204 Fe[Fe2+ Fe3+ ]04 204 527

528

MATERIALS INDEX

FeGa204 198, 212 Fe2GaO4 198, 212 Fe2G-eO4 198 Fe[Lio.5Fe-z5]O,t 204 Fe203, c~-Fe203 197, 205, 206, 212, 285, 295, 298-300, 303-305, 307 FeaO4 191, 193, 197, 200-203, 205, 206, 210, 212, 256, 259, 260, 265-267, 272, 274, 278, 283, 293, 296, 298, 304, 305, 307 Fe~l_6) O4 290 Fe3_604 270, 293 Fe3_xO4 206, 271 Fe304/CoO superlattiees 307 Fe304/MgO superlattiees 307 Fe304/NiO superlattiees 307, 308 Fe304/SiO multilayers 307 Fe3_xSn=O4 192 Fe2TiO4 198, 213 Fea_xTisO4 202, 203, 206, 208, 270, 279 Fe3_xZnxO 4 Fea_xTixO 4 206, 270 Fe2VO4 198, 213 Gd 20 GdCo2 68, 70 GdCoa 119 GdCo5 75, 88, 423 Gd2Co7 100, 105 Gd3Co 128, 129 GdCu2 420, 451 GdCu2 substituted with Ni, Co, Fe, A1 457 GdFe2 68 Gd2Fel4B 157, 158, 165 Gd2Fet7 141,142, 146 GdxYl_xCu2 455, 456 Gd-Y 20 GeFe204 212 HoFe2 68, 69 HoCo2 68 HoCo3 118, 121 HoCos.5 75, 91

Ho(Cuo.gNio.1)2 473 Ho(Cu, Ni)2 473 HoCu2 420, 427, 468, 485 Ho2Fet7 149 Ho2Fe14B 165 LaCol3 177 LaCu2 417, 418, 420, 428, 439 La2Fe14B 165 La(Feo.88Alo.12)l3 178 La(Fe, X, Co)i3, X = Si, AI 178 LaMnO3 285

Lio.sFe~5[Cr2]O4 204 Lio.sFeo.sCr204 205, 207 Lio.sFe2.5_xCrxO4 209 Lio.sFeT.50# 192, 198, 205, 206, 211, 213, 225, 256, 259, 260 Lio.5(l_x)Fe~5(l_=)ZnxFe204 220 Li1.125Til.25F%.625O4 225 Li0.5Zno.5[Lio.sMn~+]O4 204 Li0.5(l_x+~)ZnaTi~Fe2.5_o.5(x+3v)O4 221 Lu 14 LuCo2 70 LuCu2 418, 420, 445, 450 LuFe2 68, 70 Lu2Fel7 141, 142, 152 Lu2Fea,B 165 LuFell Ti 174

Mx ~2+ 2 +~,~3+c~ -~l_x-~2 "-'4

281 maghemite (,7-Fe203) 302 magnetostrietive materials 59 manganese ferrites 201 MCFelnO4 192 3+ u Mei Fe2_=Me~O#, 191 Mel_6Zn, Fe20* 217 MgAI204 191, 193, 194, 301 MgFe20 # 196, 198, 200, 203, 204, 213, 214, 222, 256, 259, 260 Mgl+x Fe/2_2xTi=O4 222 Mg ferrites 226, 227, 301 MgGa204 301 MgMn ferrites 199 Mgo.68Mno.52Fel.80a 231 MgO 301, 305 xMg2TiO4 - (1 - x)MgFe20 * 222 IVlnl_uF% [MBuFe2_u]O4 215 MnFe204 196, 198, 200, 201, 213, 215, 252, 259, 260 Mn=Fe3_xO4 207 Mn ferrites 277 MnGa204 210, 301 Mn[Mn Ti]O4 204 Mn30a 207 Mno.4Zno.3Fex304 260 Mno.5oZno.42Fe2.o804 246 Mno.51Zno.44Fez0504 253 Mno.565Zno.374Fe2.o604 253 Mnct6oZao.35FezosO4 241, 244-246 Mno.65Zno.25FezloO4 249-251 Mno.68Zno.24FezosO4 241-243 Mnl_xZnxFe204 219 Mn-Zn ferrites 226, 251 MnZn ferrites 216,228, 229, 232, 235, 240, 243, 247-249, 252, 296, 302, 304 MoFe204 198, 213

MATERIALS INDEX M6ssbauer spectra of ferrites 201 multilayers, see superlattices

R-Fe intermetallics 137 RNi5 423

NbTi 441 NdCo5 75, 88, 92 Nd2Co7 100, 101, 102, 110 Nd2Co71~ 114 NdCu2 420, 431, 441, 485 Nd2Fel7 141, 142, 146 Nd2Fel4B 157, 158, 162, 165 Ndl_xYxCo5 79 nickel aluminiumferrites 209 NiCrxFe2_xO 4 261, 262 Ni-Co-oxide superlattices 305 Ni-Cu-Zn ferrites 297 NiFe204 198, 200, 201,211, 213, 232-236, 239, 260, 300 N'ixFea_x O4 282 Ni-ferrite 228, 298 NiO 306 NiO-CoO superlattices 306 Nio.sZno.sFe204 233 Nil_xZnxFe204 218 NixZnl_xFe204 226, 227 Nio.47Zno.53Fe204 234, 239 Nio.49Zno.5oFe2.oÂO4 231 NiZn ferrites 232, 235, 252, 301 NiZn(Co) ferrites 252

Scandium 16 ScCu2 418, 420 SmCo2 68 SmCo3 119 SmCo5 75, 88 Sm2Co7 100, 102-104 SmCu2 420, 447 SmFe2 67, 68 Sm2Fel7 146 Sm2Fe14B 62, 157, 158, 165 soft magnetic materials 325 spinel ferrites 189 superlattices of rare earth metals, [DylEr] 26 [DyIC_,d] 5 [DylLu] 14 [DylSc] 16 [DylY] 4, 5, 11, 27

PmCu2 418, 420 Pr(Col_~Ni=)s 86, 95 PrCo5 75, 86 PrCu2 420, 436, 485 Pr2Fe14B 165 RA12 417, 424 RCo2 66, 70 RCo3 116, 126, 136 RCo5 72, 75, 84, 85, 91, 94, 97, 136 R2Co7 98, 100, 111, 136 R2Co7Hx 114 RaCo 127, 134 R--Co intermetallics 72 RCu2 417, 418, 420, 424, 426, 428, 432, 449, 485, 486 RFe2 66, 67 R2Fel7 138 R2Fe14B 154, 168, 169 R2Fel4BH3.4 170 R2Fel7Cx 140, 145, 154, 168 RFel2_x Mx 171 RFe204 266, 267 RFeloSi2 173

[ErlY] 5 [C,dlDY] 25 [GdlY] 5, 29

[no I',rl 5 [NdlY] 14 TbCo2 68 TbCo3 118, 119, 120 TbCos.a 75, 88, 93 Tb2Co7 100, 106, 110 TbaCo 128, 129 Tb(CuxNil_x)2 463 Tb(Cuo.7Nit~3)2 463, 486 TbCu2 417, 420, 457, 465, 485, 486 TbFe2 67, 68 Tb2Fe17 141, 142, 147 Zb2Fe14B 165 Tb2Fe17Cx 147 (Tbl_xYx)2Co 7 106 TbxYl_xCu2 462, 467, 487 Tho.95oCo5.1o 74, 81 Tho.965Co5.o7 74, 81 ThCos+x 80 Th2Co7 115 ZhECo7H5 115 Th2Fel4B 157, 158, 160, 161, 170 Ti407 266 TmCo2 68, 70 TmCoa 118, 123 Zm2Co7 100, 104, 106 TmCu2 420, 444, 480, 485 Tm(Cu~Nil_~)2 485 Tm(CUl_2Nix)2 486

529

530 TmFe2 68 Tm2Fe17 141, 142, 150 Tm2Fel4B 157, 158, 166 Tm2Fe17Cx 150 TmxYl_x Cu2 484 UCos.3 74, 82 LIFe2 68, 70 UFeloSi2 175 Znx Col_xFeCrO4 224 Zn ferrites 277 ZnFe204 196, 198, 200, 213, 216, 217

Z~.F~_+:~204 219 Zn(Ga, A1)204 301 ZaGa204, 301 Zn[LiSb]O 4 204 Znl_2a NaxFe2+~O 4 192 3-12-1- O 4 210 Znl.5-xTio.5+xFel_2xFe2.~ Zn[Zn Zi]O4 204 Yttrium 31 Y(Col_xNix)5 73

MATERIALS INDEX YCo 2 70 YCoa 117, 118 YCo5 73, 74 Y2Co7 100 Y2Co7Hx 114 YaCo 128, 129 YCu 2 418, 420, 428, 444, 454, 455, 475 YFe2 68, 70 YFe3 179 YFe-garnets 232, 233, 238 Y2Fe17 140-142 Y6Fe23 179 Y2Fe14B 156, 157, 158 Y2FelTCx 140 Y2(Fel_~Cox)14B 156, 170 YFe4.951~Ino.o5012 239 YFe204 266, 267 YFell Ti 173 YFelo.sV1.5 173 YIG 238,239 YNi5 73, 74 Y3_x TbxCo 129 Yb substituted in CeCu2 420, 436 YbCu 2 417, 418, 420