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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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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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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37 107 189 305 393 441 603 747

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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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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<

i

i

i

i

2.88

v

0

2.84

\

O'3

I q~ 2.80

2.76

! 0.0

I 20.0

,

I 40.0

,

I 60.0

,

I BOO

At,o m i e P l a n e I n d e x

Fig. 2. The composition profile (top) calculated from the X-ray diffraction intensities for an [Er23.5] Y19]100 superlatticeshowing that interdiffusion is limited to approximately5 total planes. At the bottom is shown the d-spacing variation for the same superlatticeillustrating the ratherabrupttransitionresulting from the relatively large Er-Y lattice mismatch. 2. Neutron scattering and artificial metallic superlattices Neutron scattering has played a seminal role in elucidating the spin structures that occur in the lanthanide metals and in detailing the perturbations of these structures at temperatures below the N6el point and those induced by the application of external fields. Much of this work was done by W.C. Koehler (who once described the multiplicity of spin states as a 'panoply of exotic spin configurations') and his associates at Oak Ridge more than 30 years ago. (See Elliott 1972.) More recently, highly precise synchrotron radiation studies, conducted at very close temperature intervals, have s h o w n additional details of the magnetic ordering in Ho (Gibbs, Moncton, and

MAGNETISM IN ARTIFICIALMETALLICSUPERLATrlCES

7

D'Amico 1985) and Er (Gibbs et al. 1986) including the occurrence of lock-in transitions in which the periodic moment wave-vector passes through a series of lattice-commensurate phases as the temperature is lowered, according to a model of spin-slip states. (See above references and later discussion.) The resulting steplike changes contrast to the more monotonic linear variation reported in the early neutron measurements taken with much coarser temperature intervals and have been confirmed also by neutron scattering studies. For artificial metallic superlattices, neutron diffraction provides atomic structural information complementary to that obtained from X-ray diffraction. In addition, the magnetic interaction of the neutron with ordered spins in the superlattice provides the opportunity to study the magnetic spin configurations and the range of coherence of the magnetism. In particular, it can be used as a probe of the propagation of magnetic phase information from a magnetically ordered layer (e.g., Dy) across an intervening non-magnetic layer to additional ordered layers. Neutron diffraction studies on supeflattices are typically performed using a triple axis spectrometer with the sample in reflection geometry. An analyzer crystal set for elastic scattering is used to suppress background scattering. Both conventional unpolarized neutron beams and beams polarized by reflection from a magnetic crystal have been used for studying superlattice properties. The polarization technique offers the advantage of extracting only the magnetic component of the scattering (see later discussion), however, with a significant reduction in beam intensity. Astute choices of polarization techniques and also selection of optimal resolution conditions have been the keys to successful diffraction experiments on superlattices. The trade-off between intensity and enhanced resolution is intensified by the available sample volumes of these superlattices that typically range from 0.01 to 0.05 mm 3. Even considering the high crystal perfection, this represents a major challenge for neutron scattering experiments with presently available fluxes. Most of the superlattice studies reviewed here were performed at the National Bureau of Standards Reactor at the National Institute of Standards and Technology. For a typical geometry in which the growth axis (perpendicular to the layer planes) is oriented along the neutron scattering vector, the basic unit cell of an A-B superlattice is one bilayer (e.g., one Y + one Dy layer). If the growth axis is along the c crystal direction [0001] then, considered apart from the effects of intralayer atomic ordering, the bilayer sequence would give rise to a set of peaks along the c* [000l] direction of reciprocal space corresponding to diffraction from the bilayer cell at evenly spaced values of AQ = 2~r/L where L is the bilayer thickness. The Bragg diffraction conditions can be satisfied in two ways: (a) the above diffraction from the bilayer unit cells that can be directly observed at low angle experiments (e.g., [Mo]Si] artificial superlattices, Fernandez et al. 1987) and (b) conventional diffraction from the atomic order within the bilayers seen at large angles. In the small angle case N - 2 subsidiary maxima, due to the finite size of the superlattice, are present between the principal bilayer diffraction peaks, where N is the total number of bilayers. These subsidiary maxima are only observable if the bilayer repeat sequence is nearly perfect and the spectrometer resolution is at least comparable to 2~r/NL. In the large angle regime, illustrated by the X-ray diffraction pattern for

8

' J.J. RHYNE and R.W. ERWIN

10 7

~o~

,- i0 ~ ~0 C.

10 ~

10~

2.0

2.1

2.2

(i-')

2.3

2.4

2.5

Fig. 3. Room temperature X-ray (0001 scan) diffractionpattern for an [Er23.s[Y19]10o superlattice showing the presence of harmonicsup to fifth order. The arrow marks the position of the central (0002) Bragg peak. (Note K and Q are used interchangeablyto denote the wave vector in inverseangstroms.) an [Er23.51Y19]lOO superlattice in fig. 3, the atomic order within a bilayer modulates the intensity of the bilayer Bragg peaks and has the effect of singling out sets of predominant (000/) peaks from the infinite set of bilayer reflections. If the lattice parameter mismatch is reasonably small, these groups of peaks occur at multiples of Q = 2~r/(d) where (d> 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 << NL) which, for the nuclear case, has its maximum on one of the bilayer Bragg peaks only if there is no lattice mismatch. The product of these two terms gives the intensity modulation of the diffracted intensity peaks as shown in fig. 3 for the atomic structure and in fig. 4 for the magnetic structure.

/ [l~/ls IY=o]. T - Dependence Tc = 167K

I /~ 250

, 160 K

i

~200 t--

.~ 1so n-

=>, 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, "<10 kOe 81t.O C

4b

1

0i

.8

1.19

2:0 Qz (A'I)

2:1 Qz (/~-1)

Fig. 22. (Left) Field dependence of the Q- magnetic satellite on the same superlattice as in fig. 15 but at the elevated temperature of 130 K. The satellite and magnetic bilayer harmonics are observed to broaden in low fields reflecting a loss of coherence. (Right) Only at fields higher than 10 kOe is the intra-layer helimagnetic structure gradually removed by the field and the resulting ferromagnetic component appears as intensity on the (0002). higher. This corresponds to the destruction of the helix by the applied field and its conversion to ferromagnetic order. This is the same as in bulk Dy, although without the low temperature discontinuous transition. The intensity removed from the Q± helix peaks reappears as ferromagnetic intensity added to the (0002) peak and its harmonics. In contrast, at 130 K the Q - satellites are observed to first broaden increasingly with field. This corresponds to a decrease in the magnetic coherence length and at the higher fields the coherence is lost to a degree that merges the three satellite peaks into one broad peak. This initial loss in coherence occurs before an appreciable ferromagnetic moment component is developed as shown in fig. 22 where additional (000l) intensity is evident only above 10 kOe. Erwin (1987) suggested that this loss of coherence may arise from a Zeeman coupling of the applied field to the uncompensated moment, which is the vector sum of the moments in a Dy layer. It will be in general non-zero because of an incomplete helical period at the layer boundary. As shown in fig. 23, the net uncompensated ferromagnetic moment can be quite large and varies with the Dy layer thickness and also with temperature because of the variation of both the turu angle and the order parameter. At temperatures near TN it is suggested that the coupling of the external field to the helically distorted (superspiral) uncompensated moment in each layer acts like a random field and breaks the weak interlayer exchange coupling of the Dy spins. This results in independent layer helical magnetism as is reflected in a single broad magnetic satellite at the centroid of the three original zero field satellites. At low temperatures the decreased thermal spin disorder and strongly increased anisotropy

MAGNETISM IN ARTIFICIALMETALLICSUPERLATFICES

35

16

[DYI61Yg]Io0

14 12[ o

0

o

0 30 60 T(K) 90 120 150 180

Fig. 23. Uncompensatednet ferromagnetic moment resulting from the incomplete helices in the Dy layers. The coupling of the applied field to this net moment at temperaturesapproachingTN is suggested to destroy the long-range interlayer coupling. and magnetostriction interactions result in a lower ferromagnetic transition field and apparently preclude the loss of coherence before the helical order is eliminated. This explanation is also consistent with the behavior of a [DYl61Y9]too sample that has relatively less Y and a larger zero field coherence length. Due to the thinner Y layer this sample is expected to have a stronger interlayer exchange coupling, and relatively little broadening of the Q - satellite was found at 130 K (or at lower T) on application of a field. This suggests that the applied field Zeeman coupling to the net moment is insufficient to break the relatively stronger c-axis exchange coupling in this sample. The onset transition fields (< 1 kOe at 10 K) for the helical to ferromagnetic conversion were lower than in the thicker Y sample, reflecting again the larger magnetostriction and anisotropy energy densities.

6. Incommensurate magnetic periodicity - interplanar turn angles, spin slips, and their temperature dependence The fit of eq. (5) to the magnetic diffraction data for the superlattices yields values of the propagation vector t~A for the magnetic layer and a total phase shift across the bilayer, #i. From these, the layer-to-layer turn angle ~A in the Dy (or Er) layers can be determined: 180 co ~A - - ~A. (11) ~" 2

36

J.J. RHYNE and R.W. ERWlN

Similarly, an effective turn angle can be determined for the spin density wave in the non-magnetic Y or Lu that represents the turn angle to which the 4f spins would lock if present. These turn angles can be temperature dependent with the initial (high T) value prescribed by the nesting wave vector coupling parallel sheets of the Fermi surface. For example, in bulk Dy at Trq, w (Wilkinson et al. 1961) has the value 43.20 , which decreases to 26.50 just above the first-order transition to ferromagnetism at Te. The variation of w with temperature has been suggested to arise from the development of superzone gaps in the Fermi surface coupled to the order parameter (Elliott and Wedgwood 1963), or from the effects of the magnetostrictive interactions (Evenson and Liu 1969). In the superlattices (or bulk crystals) the temperature dependence of the turn angle is reflected in a shift of the Q-position of the magnetic satellites. Figure 24 shows this effect for [Dy16[Y20189 for which the satellite position (marked with the vertical "

,h,

DY161Y2o] 89 120 / Q'Mag. Satellites Ill 108/Tc= 167K

~t

96

84 r-

= 72

zILl p.~ 48

36 24 12 0 1.76 1.86 1.96 2.06 2.16 Q, (A") Fig. 24. Expanded detail of the temperature dependence of the (0002)- satellites in a [Dy16]Y20189 superlattiee illustrating the temperature dependence of the turn angle (w). As shown by the vertical line, the magnetic peaks move to lower Qz (larger w) as the temperature is increased.

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATrlCES

I

60

I

oo

o

I

37

I

o

yttrium ~'40 co k_ •

-El

20 DylLu

0

I

0

I

80 T(K)

160

Fig. 25. The zero field magnetic structures in Dy superlattices described by the temperature dependence of the interplanar turn angle w. The low temperature ferromagnetic transitions and high temperature turn angles, compared to bulk Dy, are dramatically shifted in opposite directions for Y and Lu superlattice interlayers due to epitaxial constraints. (See text.)

lines) moves to lower Q (larger w) as the temperature in increased. The turn angles derived from the scattering data are shown in fig. 25 for several Dy superlattices, and the turn angles for the c-axis modulated spin structure of the Er superlattices are shown in fig. 26. It is noted that w in the Y remains constant at about 5152 degrees/Y layer (x -- 0.31 /~-I), corresponding closely to the peak q position (Gupta and Freeman 1976, Liu, Gupta, and Sinha 1971, Child et al. 1965) in x(q). The turn angle between the Dy planes varies with temperature as in bulk Dy (Wilkinson et al. 1961) but, for Y superlattices, without the first order drop to w = 0 at 85 K which signals the onset of the ferromagnetic state. The suppression of Tc in Y superlattices and its enhancement in Lu superlattices compared to bulk values is a consequence of epitaxial clamping of the magnetostrictive modes as discussed later. In the superlattice [DY161Yg]10o the turn angle is observed to lock in to 300 below about 50 K, presumably driven by the increase in the sixfold basal plane single ion anisotropy energy. For Er superlattices, using an analysis including the effects of strain modulation, distinct turn angles were obtained for the c-axis and the basal plane moment components which also order at different temperatures. However, measurements at low scattering angles, for which strain modulation effects are negligible, produced identical values for the turn angles of both components as in bulk Er. As shown in fig. 26, the c-axis turn angle is about 500 (~-, 27r/7) per layer corresponding to the high temperature initial state of bulk Er and shows relatively little temperature dependence compared to bulk Er which is indicated by the solid

38

J.J. RHYNE and R.W. ERWIN

60

C-axis basal [Er23[Y19]

A

[En3~%]

A

•

[]

layers

~o Q)

t~ t

50

| ..... s~/1~ 417 --

12/2~

bulk Er C-axis-basal

1/2

10/21

40

• ,

0

I Q

I

I I Y layers

I

40 T(K)

I

55

8045

Fig. 26. The temperature dependence of the turn angles in the Er and Y layers for three [Er[Y] superlattices compared to bulk Er. The superlattice Er layer to is 'clamped' near the high-temperature lock-in value of bulk Er (27r/7). The basal plane to appearing at low temperature has a somewhat smaller value than the c-axis to. The to in the Y layers is near the 50* value prescribed by the electronic structure. The fractions give (two times) the commensurate lock-in state values as developed by Gibbs et al. (1985, 1986) here referenced to the magnetic unit cell.

line. The plateaus in the bulk turn angle (indexed by the fractions) correspond to stable commensurate moment configurations (a 'spin-slip' structure) as derived by Gibbs et al. (1986) from an analysis of the magnetic X-ray scattering using synchrotron radiation. The denominator of the fraction gives the number of atomic planes required for a complete period of the spin structure and thus depends on the number of slips in the structure. The effect of the slip structure is to produce a small net moment to the configuration (e.g., a complete period in the 2/7 slip structure in Er consists of 3 spins 'up' followed by 4 spins 'down' for a net moment of 1 per 7 atomic planes). This is observed in the e-axis magnetization data on the superlattices (Borchers et al. 1991a, 1991b) as shown in fig. 27 for [Er23.5[Y1911oo. The nearplateau below 18 kOe corresponds to a moment approximately 1/7 of the saturation moment as consistent with the 2/7 spin-slip configuration. The abrupt transition fields of 18 kOe and above are those required to convert the c-axis periodic moment structure to a ferromagnetic spin alignment. Moment data taken at constant applied field show inflections corresponding to the development of spin slips as illustrated in fig. 28 for [Er13.5]Y25h00. The solid curve is for field cooled, the dashed line is for zero field cooled magnetization, and the lines marked B and D represent the

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATIqCES

250 0

q-

,

~

39

,

1OK __--l--t~-~lp~_%~ 2000

e ~ - o

_

20 30

K K

tl S, 60K tq

1000

X

(1(}

,

1. . . . . . . . .

__

0 0

~ ........

10 (1

L _

~

.

.

.

.

20 0

t

_

:~O 0

,10 0

lld,er'nal Fiohl (kOu) Fig. 27. G-axis magnetization data corrected for demagnetization effects taken with a SQUID magnetometer for the superlattice [Er23.slY1911o0 at several temperatures. The near-plateau below 18 kOe reflects the net moment of approximately 1/7 saturation produced by the 2/7 spin-slip structure. The transition fields reflect the energy required to uncouple the periodic moment structure and produce ferromagnetic spin alignment along the c-axis.

D B

20.0

¢ E

15.0

o-t" 1"-, X,

0 10.0 tq

5.0

I

0.0 0.0

,I

,

,

,

20.0

400

60.0

0o.o

loo.o

Temperakure (K) Fig. 28. Field cooled (solid line) and zero-field-cooled (dashed line) isofleld magnetization data taken on the same [Er23.5 [Y19]10o superlattice. The vertical lines marked B and D mark the temperatures where the 2/7 and 3/11 spin-slip states, respectively, develop.

40

J.J. RHYNE and R.W. E R W l N

I

I

!

I

(a) 120 eTr/7 state

80

51.4 °

413

600

~.a~

I

I

zero [

I

I

I (b)

T = 40K

°

480

H 11 too2 [Er13~26]

x, = 2.45 A CoUimation

"o 360

k remanence

-

40v 50h 30~

14°h

~ 240

120

-~.6

"0.2

q -

0.2

7.0

016

(A-b

Fig. 29. Neutron diffraction scans along (117.~) for [Er131Y26 ] at 10 K with a c-axis applied field showing the shift in ~v to the 21r/7 commensurate state. (b) At 40 K the zero field w = 2 / 7 and there is no shift at 8 kOe. Higher fields induce fan states that are also observed as inflections in the SQUID magnetization data. The zero field remanence curve indicates that the helical state is reformed with slightly reduced coherence after removal of the applied field. The quantities 40h, 40v, 50h, and 30v indicate the horizontal and vertical instrumental collimation in minutes of arc before and after the monochromator and analyzer, respectively.

MAGNETISM IN ARTIFICIALMETALLICSUPERLATTICES

41

temperatures at which the 2/7 and 3/11 commensurate spin-slip states are assumed to develop. The neutron diffraction results on the [Er13.slY2511oo superlattice clearly demonstrate the lock in effect to the 2/7 stable commensurate spin-slip state under the application of an applied field. At low temperatures the turn angle in zero field was found to be 50.3 °. As shown in the left portion of fig. 29(a) on application of a c-axis field (H = 7 kOe internal), the turn angle shifts to the commensurate value w = 27r/7 = 51.40 with no loss in the long-range magnetic coherence. The relatively poor resolution of the principal and superlattice peaks is a consequence of the outof-scattering plane (112.~) scan. At higher fields no further shift in q is observed, indicating that the 27r/7 state is the stable configuration. At 40 K (fig. 29b, left), the zero field state already has the stable w = 27r/7 value and is thus not shifted by application of an 8 kOe field. However, larger fields strongly shift the q-centroid and produce broadening, indicating a loss of coherence. This shift corresponds to the development of linear fan states with an average w ~ 35 °, which is significantly lower than any turn angle in bulk Er. The fan states also appear as higher field pseudo-plateaus in the SQUID magnetization data (Borchers 1991a) for temperatures of 20 K and higher. A similar remanent fan state shifted to longer wavelength is seen in bulk Ho (Koehler et al. 1967), presumably due to the dependence of the exchange interaction on lattice parameter as discussed in section 7.

7. Magnetoelasticity in rare earth superlattices and films and the suppression of ferromagnetism The Hamiltonian used to describe heavy rare earths is of the general form H = Hex + Hme + Hel + Her,

(12)

where the terms are exchange, magnetoelastic, elastic, and crystal field components. Magnetoelastic energies refer to the general dependence of magnetic energies upon the spacing between atoms, so that the magnetoelastic term contains the explicit dependence of the exchange and crystal field energies on atomic spacing. It has usually been found sufficient to consider only the first derivatives of the exchange and crystal field energies in the magnetoelastic term; however, the second-order derivatives must be included in the elastic energy term. These second-order derivatives of the exchange and crystal field energies could possibly be linked to the anomalous behavior of the measured elastic constants in the bulk rare earths. The most important effect of the crystal field term in the heavy rare earths is to determine the easy directions for the magnetic moments. At low temperatures the crystal field anisotropy energies can contribute to the stabilization of a ferromagnetic phase. Although there is a continuing interest in looking for band structure effects in superlattices, the experimental data suggest that it is the elastic constraints on rare earth superlattices and fihns grown by MBE that strongly perturb their magnetic structure compared to the bulk materials, as shown by studies of [Dy]Y], [ErIY ], [DylLu], and [DylSc] superlattices and Dy and Er thin films by neutron scattering

42

J.J. RHYNE and R.W. ERWIN

and SQUID magnetometry. The strains induced by epitaxy could in fact perturb the band structure. These studies have provided new insight into the interplay of magnetoelastic and exchange interactions in rare earths. The most striking result of the epitaxial clamping to the substrate and intervening non-magnetic layers is the suppression or enhancement of the ferromagnetic transitions found in bulk Dy and Er. The helical and linear spin density wave orderings are also significantly perturbed. For example, in the [Dy]Y] superlattices the temperature dependence of the interlayer turn angle is weakened as the elastic clamping is increased, while in [DylLu ] and [DylSc] superlattices the initial turn angle near the Nrel temperature is much smaller than in bulk Dy. This temperature dependence is completely suppressed in the Er superlattices where the turn angle is typically fixed at the commensurate lock-in value 21r/7. In the thin films, however, the turn angles are closer to the bulk values. There is also a 50% reduction of the basal plane ordering temperatures compared to bulk Er. These dramatic effects are also evident in quantitative calculations of the magnetoelastic energies and equilibrium spin configurations. In the bulk elements the first-order ferromagnetic phase transition is driven by a reduction in the magnetoelastic energy associated with the ferromagnetic state compared to the helical (or c-axis modulated) configuration (Cooper 1967, 1968, Evenson and Liu 1969). In Dy for example, the discontinuity in the magnetoelastic energy at the ferromagnetic transition is 3.6 K/atom as calculated from the bulk elastic constants and measured anomalous strains (Rosen and Klimer 1970 and Erwin et al. 1989). The total magnetoelastic energy reaches nearly - 6 K/atom just below the ferromagnetic transition. It is presumed that this energy is just sufficient to overcome the part of the exchange energy that is independent of atomic spacing. Elliot (1972) has also calculated the driving energy for ferromagnetism using the measured critical fields. That formalism assumes that the driving energy becomes negligible, compared to the exchange energy near the Nrel transition. While this is certainly true for the single-ion part of the driving energy, the two-ion exchange part will have approximately the same temperature dependence as the exchange, so that it is not clear that it can be neglected. Proceeding in this way, Elliot obtains good agreement between the calculated single-ion driving energy and the value extracted from the critical field measurements of about 1 K/atom at the ferromagnetic transition. The magnetoelastic energy can arise from two principal strain modes: (1) an anisotropic 7-mode strain of 0.5% at Te (Rhyne and Legvold 1965, Callen 1965, 1968) that corresponds to an orthorhombic symmetry distortion of the basal plane and that is effectively 'damped' in the helical state and (2) an c~-mode c-axis expansion of 0.25% and an isotropic basal plane dilation (both partially clamped in the helical state). In bulk Er the ferromagnetic alignment is along the c-axis so that there is no "r-mode distortion. The magnetoelastic driving energy at Te (20 K) is found to be 2.3 K/atom (Rosen and Klimer 1970). There is also a decrease of the magnetoelastic energy associated with the ordering of the basal plane spin components in Er below 55 K. In the superlattices there is further clamping by the presence of the non-magnetostrictive epitaxial Y layers attached on either side of the magnetic layers. In Dy supeflattices, this clamping strongly inhibits the 7-mode strain and is relatively weaker

MAGNETISM IN ARTIFICIAL METALLIC SUPERLA'ITICES

43

for the a-modes. The a-mode energy (calculated to be ~ 0.6 K/atom at 85 K by Erwin et al. 1987) is insufficient alone to drive the transition, and the clamping of the 7-mode strain forces the Dy in the superlattice to remain helical down to low T. In Er superlattices and films, the situation is more complicated, but again the absence of the conical ferromagnetic state and the weaker temperature dependencies of the turn angles compared to bulk Er are also ascribed to magnetostrictive effects as will be discussed in the next section. The effects of the perturbed magnetoelastic energies in the superlattice can be seen from the magnetization data of fig. 27 in which an external field of 18 kOe is required to induce ferromagnetism at 10 K, a regime in which bulk Er is intrinsically ferromagnetic.

7.1. Theory of magnetoelasticity in superlattices and films This section focuses on the role of magnetoelastic energetics in determining the magnetic structure of rare earth superlattices and films. While the ideas presented here are applicable to all magnetic thin film materials, the giant magnetostriction of the rare earths emphasizes the magnetoelastic effects. Dy and Er superlattices and films, which have received the most attention with regards to magnetoelastic effects, will be discussed here. The magnetoelastic free energy in Cartesian coordinates can be expressed as 1

fine = "~ E C#u~'u#£uu -- E K#~.##. ,u,u #

(13)

The magnetoelastic coefficients, Ku, appear in terms linear in the strains as they arise from derivatives of the exchange and single-ion energies with respect to the strains. The equilibrium strains satisfy the equation

E cu~-=Ku

(14)

and the equilibrium magnetoelastic free energy is

-

1

fme = - - - ~ E C , u u g # # ' u v #,u

= -- 21 E Ku'(uu"

(15)

#

When a magnetic material is grown with a non-magnetic material in a fihn or multilayer additional elastic terms are added to the free energy, 1

re1 = "~r,~ E 5uu(?uu - ~'°u)(2"~ - ?'0u),

(16)

#u

where r~ represents the ratio of non-magnetic to magnetic material and a labels the growth direction. The ~'o~ are the strains in the free non-magnetic material measured with respect to the non-magnetic strains in the free magnetic material. In the case

44

J.J. RHYNE and R.W. ERWIN

that the strains in the growth plane are coherent (guu ~ euu for/.t # c~) the additional elastic energy is 1 iel ---- ~ 7"or E

(C,av

- -

-

-

G).

(17)

This adds terms to free that are both quadratic and linear in the strains so that the effective elastic coefficients for #, v # c~ are

(18) and the effective magnetoelastic coefficients for # # c~ are K . = K . + r~ ~

(19)

ro~(~.v - ~ . o 6 ~ / ~ ) .

7.2. Magnetoelastic energies in Dy superlattices and films A first attempt to treat magnetoelastic effects in multilayers and fihns assumes that the basic exchange interactions and anisotropy energies can be transferred directly from the bulk magnetic materials. This assumption is likely to be incorrect at the interfaces, yet it provides reasonable explanations for many of the experimental results. The magnetoelastic coefficients of the bulk materials are to be applied in eqs (13)(14). Care must be taken to correct for the magnetic structure dependence of the coefficients, since the structures are typically different in the layered materials compared to the bulk. For example, using eq. (13) and the symmetry arguments for the spin correlation functions (Callen and Callen 1965), the magnetoelastic coefficients in the basal plane helimagnetic structure of Dy at T = 0 K are i

Kx,y(T = 0) = +0.686~,0 - 0.06(3 + cos w)/4

Kz(T = 0) = 1.4 cos w

(103 K/atom),

(103 K/atom),

(20) (21)

where 6,o,0 selects the large constant term 0.68 only if the structure is ferromagnetic = 0). Consider the case that the layered rare earth material is grown along the hcp eaxis. If the elastic constants are equal in the magnetic and non-magnetic material then eq. (13) for the equilibrium anomalous strains simplifies to ~'xx + gyy = ( g b + 2r'zCb~'0x,y)/((1 "[- rz)Cb),

(22)

gxx - "~yy= (Kx - Ky)/((1 + rz)(C11 -- e12)),

(23)

Fzz = (Kz -- Cl3('~xx + ~yy))/c33,

(24)

where Cb = Cll + c12 - 2c23/C33 and Kb = Kx + Ky - 2KzC13/c33.

MAGNETISM IN ARTIFICIALMETALLICSUPERLATFICES

45

Assuming for the moment that these expressions for the equilibrium strains are correct, then the magnetoelastic driving energy for the ferromagnetic transition, A = fm~(W = 0 °) - fme(W) can be calculated. If the low temperature helimagnetic state has w = 30 °, then Az = -1.25 - (3.95 - 91.0 rze'0x,y)/(1 + rz)

(K/atom).

(25)

This magnetoelastic driving energy is plotted in fig. 30 for rz ranging from zero to infinity as a function of ~'0x,y. It is believed that the exchange energy barrier to ferromagnetism is equal to the discontinuity in magnetoelastic energy found at Tc (there is about 5 K of temperature hysteresis in bulk Dy at Tc), which is 3.6 K/atom at 85 K as shown in fig. 31. This number may become slightly larger at 0 K. Figure 30 suggests qualitatively that ferromagnetism is obtained for ~"~ 1.5% (as applies for Dy on yttrium) in the weak clamping limit, rz < 1, which should apply in sufficiently thick films. Note also that the driving energy for weak clamping becomes stronger than the bulk driving energy when ~" x< -4.4%. For example, the lattice mismatch of Dy on Lu is only - 2 . 5 % at room temperature. The data for a 4000 ,~ Dy fihn shown in fig. 32 are quite similar to those obtained for the bulk material (Wilkinson et al. 1961). The helimagnetic structure collapses in a first-order phase transition to a ferromagnetic state just below Tc = 80 K (compared to 85 K for the bulk). However, the temperature hysteresis for this transition is about 15 K (compared to 5 K for bulk Dy), and on cooling the transition may require as long as one hour for completion. The temperature dependence of the turn angle is identical to that of the bulk except in a 10 K wide region near the Curie temperature, where the angle is slightly larger than in the bulk. Although all of the magnetic intensity collapses onto the nuclear (0002) Bragg peak below 80 K, the width of this magnetic scattering is not resolution limited. The peak shape is no longer Gaussian (as is the resolution function), and the width is estimated to correspond to a magnetic correlation range of 200 ~. No uniform -y-mode magnetostrictive distortion was observed. In the ferromagnetic state when a saturating field of 25 kOe is applied along the easy a-axis direction, the (0002) Bragg peak sharpens to a resolution limited width, consistent with the formation of a single domain. The resulting anomalous b-axis strain from just above Tc in zero field to below Tc with the field applied was found to be only about 1/3 of the discontinuity at Tc in bulk Dy (Rhyne and Legvold 1965). This is a peculiar result since eq. (21) then suggests that the effective clamping parameter, rz, is 2, and the magnetoelastic driving energy should be considerably reduced as in fig. 31. Furthermore it was found that the Nb buffer layer undergoes no anomalous strains at all. This indicates that inelastic processes are involved, probably at the interface between the rare earths and the Nb. It can be concluded that the results for the Dy film are not fully explained, and will require a more careful investigation of all of the relevant strains. The diffraction from a superlattice consisting of bilayers of 16 atomic planes of Dy alternated with 9 atomic planes of Y is compared to the film in fig., 32b. Although the initial ordering temperature, TN, is within 7 K of the

46

J.J. RHYNE and R.W. ERWIN

2O

0 _x¢

15

o

10

i -0

i r r l l l . , , t

. . . .

Dy (a)

Er (b)

5

/

bY_ 0

(2 o

o ~

r=O

-5

0 -10 -

0

1

-1

0

1

sfrain ¢ (10 -2) Fig. 30. Critical fields for Dy (a) and for Er (b) c-axis superlattices calculated as a function of the lattice mismatch e, based on a perfect epitaxy model. The experimental critical field values are shown by the triangles for comparison. The parameter r represents the ratio of non-magnetic to magnetic material. The model predicts that the growth of Er on a substrate with a smaller basal plane lattice parameter than Er would produce a ferromagnetic transition temperature higher than in the bulk.

o

I

I

I J

I

I f

o co - 4 -6 ©

( b ) I~r ,

0

100 W

(K)

i

,

50

7 - - - - - -

100

Fig. 31. Magnetoelastie energy as a function of temperature for (a) bulk Dy and (b) bulk Er.

bulk and film values (185 K), there is evidently no collapse to the ferromagnetic state down to 4 K. Note, however, that additional weak magnetic peaks develop near the (0002) nuclear peak at low temperature. These peaks are separated by 2~r/L (where L is the bilayer thickness) from the helimagnetic peaks, so that they represent the development of local ferromagnetic order within the helimagnetic framework. As in the case of the Dy film, this low temperature ferromagnetic-like state is extremely time dependent and temperature hysteretic. In fact this feature was not found in the

MAGNETISM IN ARTIFICIAL METALLIC SUPERLAqTICES

100

I

I

I

I

I

I

(o) [Dy,olVg]

Dy film

47

(b)

80 .~

60

4K

qO

-E 4.0

~

I

72K

20-

lsoK,AI,, A

0 2.0

2.2

2.4

L,- T

2.0

2,2

2.4

Qz Fig. 32. Diffraction scans along (000~) for a Dy film (a) and for a [DylY] superlattice (b). The film has a ferromagnetic transition where the magnetic intensity collapses onto the nuclear Bragg peak, whereas the multilayer does not. The helimagnetic intensity is shaded.

lO- ~ ) 5

~E

o

[Erz3IYI9 ]

-

(d) ~

5

0

"~

0

__i

[DY16~Yg] ''''''' 20 40 20 0 Infernal Field (kOe) ,

,

i

I

r

,

t

I

Fig. 33. Magnetization plotted versus internal field at 10 K for two [DyIY] superlattices (a, b), and for two [Er]Y] superlattices (c, d). The critical field for ferromagnetism is determined qualitatively as the point where the magnetization first increases sharply toward saturation.

other D y - Y superlattices where the ratio of Dy to Y was smaller. The energy difference between the exchange barrier and the magnetoelastic driving energy for growth on Y (e ,~ 1.5%) is calculated to range from 1.2 K/atom at rz ~ 1 to 2.0 K/atom at rz = 2 to 3.8 K/atom at rz = ~ . These differences correspond to critical fields of 2.0, 3.3, and 6.6 kOe on 10#B. Magnetization measurements for two of the DylY superlattices are shown in fig. 33. These data were obtained from the diffraction intensity of the (0002) Bragg peak as a function of the magnetic field applied

48

J.J. RHYNE and R.W. ERWIN

along the easy a-axis direction. In agreement with the above calculations, the low temperature critical fields are below 5 kOe. The calculation of critical fields is complicated by the contribution of the sixfold basal plane anisotropy. Also the effective clamping parameters and non-magnetic strains, ~'0, may depend sensitively on the growth conditions. There are not yet data for the anomalous growth plane strains for Dy multilayers where rz is expected to be of order 1 or greater. From clamping considerations the anomalous strain along the c-axis should be ezz = Kz/c33. The measured c-axis anomalous strain in Dy superlattices (Erwin et al. 1987) is approximately 44- 1 x 10 -3 at low temperatures, while the calculated value is 6 x 10 -3 for w = 30 °. This is not a serious discrepancy considering the assumptions concerning the multilayer structure that are required to extract the anomalous strains from the neutron diffraction data and the neglect of interface effects on the magnetoelastic coupling coefficients. The additional forced magnetostriction in a saturating magnetic field was measured to be 1.0 x 10 -3 (Erwin et al. 1987) for [DY151Y14], and the calculated value is also 1.0 x 10 -3. The case of Dy-Lu superlattices is significantly different since there is an extreme enhancement of the Curie temperature from 85 K to the range 140-160 K. Calculations attempting to model these magnetic structures are difficult because the temperature dependence of the different terms in the Hamiltonian must be included. A reasonable model for bulk Dy can be constructed based on the wave-vector dependence of the exchange energy extracted from inelastic neutron scattering measurements (Nicklow 1971, Nicklow et al. 1971). The temperature dependence of this exchange function can be interpreted as arising from its strain derivatives. Using a simple mean-field theory for the temperature dependence of the magnetic moment, one can solve for the temperature dependence of the magnetic structure and the anomalous strains using appropriate values for the magnetoelastic constants. Unfortunately, when the derived exchange and magnetoelastic constants are applied to the superlattice model, the extreme enhancement of the Curie temperature is not obtained. Typically what is found is that the Dy turn angle is considerably reduced but the Curie temperature may only rise 10-20 K. The reason for this is that the added terms in the Hamiltonian of the superlattice due to epitaxial strain give a direct perturbation of the exchange energy, just as the corresponding terms in bulk Dy can account for the temperature dependence of the turn angle. The bare exchange and its strain derivatives have the same dependence on temperature through the renormalization of the magnetic moment. A strong enhancement of the Curie temperature cannot be obtained from this theory, since it has been assumed that a large fraction of the driving energy arises from the single-ion gamma-mode strains which renormalize so strongly with temperature that their contribution is negligible compared to the exchange at high temperatures. The failure of this theory is interesting since it suggests that some other physics related to the superlattice structure might be responsible for the enhancement of Te [Beach et al. 1993]. 8. Coherent magnetic moments and disorder at interfaces The coherent 4f atomic magnetic moment in magnetic layers of the Dy and Er superlattices can be calculated from the observed magnetic scattering intensity relative

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATTICES

49

12

10

Function ( J = 15/2)

••i•ouin

8

zk

2

00

[DY'6 IY2°]8' I Oy Coherent q~ [DylslY,,]~ tL~yerMoment

3b

60

90

m\ m \

120

180

T(K) Fig. 34. Temperaturedependence of the coherent Dy layer moment in [DYlrlY20189 and [DYlsIY14164 superlattices compared to a Brillouin function. Also shown is the total integrated magnetic intensity. (See text.) to the nuclear intensities. The resulting temperature dependence of the moment is shown in fig. 34 for Dy superlattices compared to a Brillouin curve (J = ~ for Dy) and is given in fig. 35 for Er superlattices compared to the experimental bulk c-axis and basal plane moments. For the Er superlattices the data analysis was based on a simple parameterization of the intermediate temperature spin state. Note that the low-T value of the c-axis moment is 8#B as in bulk Er (total free ion moment = 9/zB), and with a rapid fall off at increasing temperatures. The 5 K basal plane moment (helical) is about 3.5#B, which is reduced from the bulk value of 4#B. For Dy, the calculated atomic moments for both superlattices fall below the Brillouin curve for the 10#B free ion moment of Dy. Figure 34 also shows the total overall magnetic intensity obtained from a difference pattern between scans taken at temperature (T) minus that for T > TN. This difference intensity, which should include all the magnetic scattering (coherent and incoherent), does accurately follow the Brillouin curve. These results imply some degree of disordered moment, presumably in the interface layers, which does not contribute to the coherent moment. Additional evidence for this is found in a temperature-dependent weak broad diffuse scattering that underlies the [0002] nuclear peak. This is shown by the enlarged plot in fig. 36 for a [DY15[Y14164 superlattice at 10 K. The shaded area is a representation of the broad incoherent scattering underlying the (0002) diffraction peak at Qz = 2.24 ,~-1. The broad distribution reflects ferromagnetic moment correlations of a range of order

50

J.J. RHYNE and R.W. ERWIN

l g

C-axis

•

[Er23~{t9] zx

•

[grl3lYz6]

3

basal

[Er321Y21] 0

%

% % ~

_

a

bulk

•

% %

% %

% %

~ 2 3 1 1 9 ]

{D bulk g r C-axis - -

o

A

-

2

0

%

\ k

o .1"4 _

{D

•

\\ \

•

G}

\ \

II 0

¢

\\

t

I

I

I

40

0

30

T(K) Fig. 35. The c-axis and basal plane moment components for [Er32]Y21], [Er23[Y19], and [Eq31Y26 ] superlattices (TN = 78.0, 78.5, and 72.5 4- 1 K, respectively) compared to bulk Er (TN = 84 K) given by the continuous lines. The ordering temperature for the basal plane components is about half the value for bulk Er. The cartoons at the top of the figure represent the ordering progression of the c-axis

moment componentas the temperature is lowered. 10 ~ . The reduction of the coherent moment in the interface layers has also been found in [Gd-Y] superlattices by Majkrzak et al. (1986) and has been calculated directly from magnetic diffraction data on [GdIY] using polarized synchrotron radiation data by Vettier et al. (1986). The best fit model to the diffraction data as shown in fig. 37 assumes the full Gd moment (5.6#B for T = 150 K) at the center of the Gd layer with a reduction in the planes near the interface. Interface disorder effects are presumably responsible for the decreased coherent moments in the Er superlattices also.

9. Residual m o m e n t effects in superlattices The effects of anisotropy, magnetostrictive, and epitaxial strains are also evident in the remanent magnetic state examined in a series of [DyIY] superlattices. Ill diffraction studies fields were applied along a basal plane direction that induced a partial or complete collapse of the helical state in favor of ferromagnetic order. Between each isothermal field application, the field was reduced to zero and the

MAGNETISM IN ARTIFICIAL METALLIC SUPERLAI"FICES

51

12 IDylslY14164 10K 10

:>,

8 6

42 jiJ 1.80

1.95

2.10

2.25

2,40

2.55

2.70

ez(~, 1) Fig. 36. Enlarged plot of the (000/) scattering in the [DY151Y14164superlattice showing the broad magnetic peak underlying the pattern at the (0002) position. This peak, reproduced by the shaded area, reflects short-range ferromagnetic correlation of some of the moments, presumably those in the interface layers. This explains the reduced coherent layer moment (fig. 34).

.:L n-"

Gd21Y21 T=ISOK

6.0

g.. 5°t '1 ~

4.(?

~ s°l ~ 2.0 ~

1.0

W

"

20

i

r

2O O LAYER NUMBER

Fig. 37. Models for the spatial magnetic moment modulation in a [Gd211Y21] superlattice. The dashed curve follows the chemical modulation, and the solid curve, which gives the best fit to the data, represents the full 6.0/~B moment (T = 150 K) at the center of the Gd layer with a smooth decrease in the interface planes (after Vettier et al. 1986).

52

J.J. RHYNE and R.W. ERWIN

after 18 kOe

[DYlsIY9 ]100 I0 K Residual Intensity Nuc,. Peaks 1

2'°1

after IOkOe /'~

after

[ /

/

~

t

).,Je~Ul

H=O

l

/1~/

"l

~

36

J~ l~"

H=O

/'~

30

" II

24

I ~

after 18 kOe after 3kOe

after i IOkOe

.,~

~

LU 18"

~ 120

12

lay,. Iv. ],oo 0 rL~/"°'~ ~ ..,~t,~:

IOK

0 tgo 2.oo 2.1o

2.10 2.20 2.30 2.40 Oz(~," )

Residual Intensity Q- Mog Satellites

o z (~-')

Fig. 38. (Left) Excess ferromagnetic intensity remains on the (0002) nuclear peaks following application of the field, which indicates the strong metastability of the induced ferromagnetic state at 10 K for the superlattice with only 9 Y planes (cf. fig. 39 for the 20 Y plane sample). (Right) Residual intensity for H = 0 at the Q - satellite positions in the [DY161Yg]10o superlattice after applying each of the fields shown.

warmed to 80 K

r

I

after / ' ~ 60kOe ~" I~

[DYI61¥20]S9 Residual Intensity NucLPeake

after

;5 kOe I/" fl

H-O

Y

~- 90

warmed1/q80 K

~1

]

~/~

| ~ 1 /~T

1

[

II

~__~ ~

after ~ / / ' /

IX

after

240

H-O

/

~'

~-,80

1,84 1.94 2.04 Oz(~'-I)

2) ~

~o 0 2.12 2.22 2.32 Oz (~")

otter 60 kOe

.

. 'joy

[DYL61Y20]SS iOK Residual Intensity Q" Satellites

•4

Fig. 39. (Left) Residual intensity for H = 0 at the (0002) peak position in the [DY16]Y20]s9 superlattice that shows that all the ferromagnetic intensity is lost following removal of the applied field. (Right) Except for 3 kOe, the Q - satellites are reformed with significantly broadened peaks, indicating a partial loss of coherence in the remanent state. The coherence is recovered on warming, for example to 80 K.

MAGNETISM IN ARTIFICIAL METALLIC SUPERLA'ITICES i

after

after

~ 40 koe

- after

~ /I

=

I zskoe

,~ | ,~

t

/

~

53

/

. oI ri , zt -

| = ~ 1,90

,

Residual Intensity 0 - Satellites

2.00 2.10 Qz

Fig. 40. Residual intensity at 130 K on the Q - satellites in the [DYlfIYg]89 superlattice following magnetization at the fields shown. The initial state is essentially recovered at high temperatures (e.g., 130 K = 0.79Tc) in all superlattices studied.

residual magnetic intensities at both the Q+ satellite position and at the (0002) ferromagnetic peak were measured. At 10 K in a [Dy 161Y9 ] 100 sample, the remanent state after magnetization showed little tendency to restore the initial zero field helical state (except for magnetizing fields less than about 3 kOe) as confirmed by the weak intensity reappearing at the magnetic sublattice positions in fig. 38 (right), and by the large ferromagnetic intensity remaining on the (0002) peaks in fig. 38 (left). After the field was removed the spins remain metastably locked into the ferromagnetic configuration until the sample temperature was warmed above 100 K in zero field. In a thicker Y sample [Dyl6lY20]89 with lower anisotropy energy density, essentially all the ferromagnetic intensity was lost from the (0002) peak when H was reduced to 0 (see fig. 39 (left)). However, as illustrated in fig. 39 (right), the intrinsic helimagnetic state was only partially reformed, and in particular the remanent state Q peaks are significantly broadened suggestive of a partial loss of interlayer coherence following the magnetization. The total wave vector integrated magnetic intensity in the Q+ remanent peaks is close to that found for the initial H = 0 state. This result is consistent with the disappearance of the ferromagnetic intensity on the (0002) positions in the remanent state. The original coherence was restored by warming the sample as shown by the scan at 80 K. At 130 K where the anisotropy and magnetostrictive energies have become negligible, the original zero field patterns were essentially recovered in all samples for all fields applied up to 40 kOe as illustrated in fig. 40 for the remanent state in a [DY161Y9189 superlattice at 130 K following magnetizations in the fields given. Er superlattices also show the effect of inhibited low temperature reformation of the intrinsic periodic moment state after magnetization. For [Er13.slY251100 in fig. 29b (left) the zero field remanent state following application of a 27 kOe field at 40 K

54

J.J. RHYNE and R.W. ERWIN

(~, 0.5TN) recovers the zero-field-cooled turn angle but the range of coherence is appreciably reduced. 10. Summary Artificial metallic superlattices of the heavy lanthanide elements with yttrium have been shown to exhibit incommensurate (except Gd) periodic magnetic order that is coherent over several bilayer periods. The features of the ordering, including the phase and chirality coherence observed in [Dy[Y] systems, suggest that the order is propagated via a spin density wave in the Y conduction bands stabilized by the 4f spins. Superlattices with the S-state ion Gd behave similarly, except that the interlayer coupling is only collinear ferromagnetic or antiferromagnetic. The existence of the long-range propagating order is found to be independent of the direction of the net moment component (viz., Dy vs. Er) as long as the stacking direction of the superlattice is parallel to the propagation vector of the magnetic structure (e*). In the [Dy[Y] superlattices, the magnetic coherence length is found to be proportional to the reciprocal thickness of the intervening Y layer. However, [DylY] superlattices produced with the growth axis in a basal plane direction are observed to have no long-range interlayer coupling consistent with the large anisotropy in the conduction electron generalized susceptibility, which leads to a large anisotropy in the real space exchange coupling range. Gd-Y and Dy-Lu superlattices show planar ferromagnetic and antiferromagnetic coupling between layers in both c-axis and b-axis growth direction superlattices, where for the latter case the coupling is aided by dipole-dipole energies. The transition from periodic moment states to ferromagnetism observed in bulk samples of Dy and Er is completely suppressed in the Y-based superlattices due to clamping of the magnetoelastic strains. An enhancement in Tc is observed in Lu-based superlattices also arising from stain considerations. In the Dy-Y superlattices, the application of an external field effects a gradual transition to an aligned ferromagnetic state. This transition is preceded at temperatures approaching TN (in samples with relatively thick Y layers) by a loss in interlayer coupling before the helical ordering is broken. The extensive array of interlayer coupling and phase transition phenomena discovered in the superlattices and the disparate nature of many of these phenomena compared to the bulk metals, have provided added insight into some of the fundamental magnetic interactions in the rare earth metals. It is also clear that many facets remain unexplained and are motivation for future theoretical and experimental investigations. For example, the effect of forming a superlattice structure on the electronic band structure and on the detailed forms of x(q) and the exchange interaction, plus the dependence of magnetic coherence range on the non-magnetic interlayer thickness are ripe for further study. A more detailed analysis of the effect of interface strains and dislocation densities on the elastic, magnetoelastic, and exchange energies and their interplay may be expected to lead to a better description of the modification of ferromagnetic transition temperatures and associated transition fields.

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATYICES

55

In the experimental arena, the formation and properties of superlattices with light rare earths are unexplored. These systems can provide information on the effect of epitaxy between hcp metals and the double hexagonal close packed (dhcp) and Sm crystal structures, and the resulting perturbation of the complex multi-Q states in Nd and the exchange-driven ordering of the single ground state in Pr. The growth of superlattices with non-constant layer repeat sequences and of superlattices with multiple magnetic metals (e.g., the Dy-Gd system of Majkrzak 1988) promise to produce new unique complex ordering phenomena. These and other superlattice investigations not yet envisioned can be expected to further enhance our basic understanding of the fascinating realm of rare earth magnetism.

11. Acknowledgements The authors particularly want to thank their long-term collaborators in the study of magnetic superlattices, Myron Salamon, Julie Borchers, and C. Pete Flynn and their students who worked on these problems, Jack Cunningham, Roy Du, Frank Tsui, Shantanu Sinha, Bob Beech, Tony Matheny, and B. Everitt. In addition, the authors have particularly benefited from discussions about magnetic superlattices with Chuck Majkrzak, J. Kwo, Denis McWhan, and Christian Vettier among others, and with Alan Mackintosh, Jens Jensen, and AI Overhauser concerning spin density waves.

References Beach, R.S., J.A. Borchers, R.W. Erwin, C.E Flynn, A. Matheny, J.J. Rhyne, and M.B. Salamon, 1992, J. Magn. Magn. Mater. 104, 1915. Beach, R.S., J.A. Borchers, A. Matheny, R.W. Erwin, M.B. Salamon, B. Everitt, K. Pettit, J.J. Rhyne, and C.P. Flynn, 1993, Phys. Rev. Lett. 70, 3502. Bohr, J., Doon Gibbs, J.D. Axe, D.E. Moncton, K.L. D'Amico, C.E Majkrzak, J. Kwo, M. Hong, C.L. Chien, and J. Jensen, 1989, Physica B 159, 93. Borchers, J., Shantanu Sinha, M.B. Salamon, R. Du, C.P. Flynn, J.J. Rhyne, and R.W. Erwin, 1987, J. Appl. Phys. 61, 4043. Borchers, J.A., G. Nieuwenhuys, M.B. Salamon, C.P. Flynn, R. Du, R.W. Erwin, and J.J. Rhyne, 1989a, J. Phys. (Paris), Colloque C8, 49, 1685. Borchers, J.A., 1989b, PhD Thesis, University of Illinois. Borchers, J.A., M.B. Salamon, R.W. Erwin, J.J. Rhyne, R.R. Du, and C.P. Flynn, 1991a, Phys. Rev. B 43 3123.

Borchers, J.A., M.B. Salamon, R.W. Erwin, J.J. Rhyne, G.J. Nieuwenhuys, R.R. Du, C.E Flynn, and R.S. Beach, 1991b, Phys. Rev. B 44 11814. Cable, J.W., E.O. Wollan, W.C. Koehler, and M.K. Wilkinson, 1965, Phys. Rev. 140 A1896. Callen, E., and H.B. Callen, 1965, Phys. Rev. 139, A455. Callen, E., 1968, J. Appl. Phys. 39, 519. Child, ILR., W.C. Koehler, E.O. Wollan, and J.W. Cable, 1965, Phys. Rev. 138, A1655. Child, H.R., and W.C. Koehler, 1968, Phys. Rev. B 174, 562. Cooper, B.R., 1967, Phys. Rev. Lett. 19, 900. Cooper, B.R., 1968, Phys. Rev. 169, 281. Coqblin, B., 1977, The Electronic Structure of Rare Earth Metals and Alloys: The Magnetic Heavy Rare Earths (Academic Press, London). Durbin, S.M., J.E. Cunningham, and C.P. Flynn, 1982, J. Phys. F 12, L75. Dumesnil, K, C. Dufour, M. Vergnat, G. Marchal, Ph. Mangin, M. Hennion, W.T. Lee,

56

J.J. RHYNE and R.W. ERWIN

H. Kaiser, and J.J. Rhyne, 1994a, Phys. Rev. B (to be published in 1994). Dumesnil, K., C. Dufour, Ph. Mangin, G. Marchal, M. Hennion, W.T. Lee, H. Kaiser, and J.J. Rhyne, 1994b, J. Magn. Magn. Mater. (to be published in 1994). Elliott, R.J., and F.A. Wedgwood, 1963, Proc. Phys. Soc. (London) 81, 846. Elliott, R.J., ed., 1972, Magnetic Properties of Rare Earth Metals (Plenum, London). Erwin, R.W., J.J. Rhyne, M.B. Salamon, J. Borchers, Shantanu Sinha, R. Du, J.E. Cunningham, and C.P. Flynn, 1987, Phys. Rev. B 35, 6808. Erwin, R.W., J.J. Rhyne, J. Borchers, M.B. Salamon, R. Du, and C.P. Flynn, 1988a, J. Appl. Phys. 63, 3461. Erwin, R.W., J.J. Rhyne, J. Borchers, M.B. Salamon, R. Du, and C.E Flynn, 1988b, J. Phys. (Paris), Colloque C8, 49, 1631. Erwin, R., J. Rhyne, J. Borchers, R. Du, M. Salamon, and C.P. Flynn, 1989, Physica B 161, 260. Evenson, W.E. and S.H. Liu, 1969, Phys. Rev. 178, 783. Everitt, B.A., J.A. Borchers, M.B. Salamon, J.J. Rhyne, R.W. Erwin, B.J. Park, and C.E Flynn, 1994, J. Magn. Magn. Mater. (to be published in 1994). Fernandez, E, C.M. Falco, P. Dhez, A. Khandar-Shahabad, L. Nevot, B. Pardo, J. Corno, and B. Vidal, 1987, Appl. Phys. Let. 51,880. Flynn, C.P., E Tsni, M.B. Salamon, R.W. Erwin, and J.J. Rhyne, 1989, J. Phys.: Condens. Matter 1, 5997. Gibbs, Doon, D.E. Moncton, and K.L. D'Amico, 1985, J. Appl. Phys. 57, 3619. Gibbs, Doon, Jacob Bohr, J.D. Axe, D.E. Moncton, and K.L. D'Amico, 1986, Phys. Rev. B 34, 8182. Gotaas, J.A., J.J: Rhyne, L.E. Wenger, and J.A. Mydosh, 1988, J. Appl. Phys. 63, 3577. Gschneidner, ICA., Jr., and L. Eyring, eds, 1979, Handbook on the Physics and Chemistry of Rare Earths, Vol. l-2 (North-Holland, Amsterdam). Gupta, R.E and A.J. Freeman, 1976, Phys. Rev. B 13, 4376. Habenschuss, M., C. Stassis, S.K. Sinha, H.W. Deckman, and EH. Spedding, 1974, Phys. Rev. B 10, 1020. Jehan, D.A., D.E McMorrow, R.A. Cowley, R.C.C. Ward, M.R. Wells, and N. Hagmann, 1993, Phys. Rev. B 48, 5594.

Jensen, J. and A. Mackintosh, 1991, Rare Earth Magnetism, Structure and Excitations (Oxford University Press, Oxford). Jensen, J. and R.A. Cowley, 1993, Europhys. Lett. 21, 7O5. Keeton, S.C. and T.L Loucks, 1968, Phys. Rev. 168, 672. Koehler, W.C., J.W. Cable, H.R. Child, M.K. Wilkinson, and E.O. Wollan, 1967, Phys. Rev. 158, 450. Kwo, J., D.B. McWhan, M. Hong, E.M. Gyorgy, L.C. Feldman, and J.E. Cunningham, 1985a, in: Layered Structures, Epitaxy, and Interfaces, eds J.H. Gibson and L.R. Dawson, Proe. Materials Research Society Symposia 37 (Mater. Res. Soe., Pittsburgh, PA) p. 509. Kwo, J., E.M. Gyorgy, D.B. McWhan, M. Hong, EJ. DiSalvo, C. Vettier, and J.E. Bower, 1985b, Phys. Rev. Lett. 55, 1402. Kwo, J., E.M. Gyorgy, EJ. DiSalvo, M. Hong, Y. Yafet, and D.B. McWhan, 1986, J. Magn. Magn. Mater. 74-77, 771. Larsen, C.C., J. Jensen, and A.R. Mackintosh, 1987, Phys. Rev. Lett. 59, 712. Lee, W. T., I-L Kaier, J.J. Rhyne, K. Dumesnil, C. Dufour, Ph. Mangin, G. Marchal, R.W. Erwin and J.A. Borchers, 1994, J. Appl. Phys. (to be published in 1994). Liu, S.H., R.E Gupta, and S.K. Sinha, 1971, Phys. Rev. B 4, 1100. Majkrzak, C.E, J.W. Cable, J. Kwo, M. Hong, D.B. McWhan, Y. Yafet, J.V. Waszczak, and C. Vettier, 1986, Phys. Rev. Lett. 56, 2700. Majkrzak, C.E, D. Gibbs, E Boni, Alan I. Goldman, J. Kwo, M. Hong, T.C. Hsieh, R.M. Fleming, D.B. McWhan, Y. Yafet, J.W. Cable, J. Bohr, H. Grimm, and C.L. Chien, 1988, J. Appl. Phys. 63, 3447. Majkrzak, C.E, J. Kwo, M. Hong, Y. Yafet, Doon Gibbs, C.L Chien, and J. Bohr, 1991, Adv. Phys. 40, 99. Moon, R.M., and R.M. Nicklow, 1991, J. Magn. Magn. Mater. 100, 139. Nicklow, R.M., 1971, J. Appl. Phys. 42, 1672. Nicklow, R.M., N. Wakabaysashi, M.K. Wilkinson, and R.E. Reed, 1971, Phys. Rev. Lett. 26, 140. Nigh, H.E., 1963, J. Appl. Phys. 34, 3323. Rhyne, J.J. and S. Legvold, 1965, Phys. Rev. 138, A507. Rhyne, J.J., R.W. Erwin, Shantanu Sinha, J. Borchers, M.B. Salamon, R. Du, and C.P. Flynn, 1987, J. Appl. Phys. 61, 4043.

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATTICES Rhyne, J.J., R.W. Erwin, J. Borchers, M.B. Salamon, R. Du, and C.P. Flynn, 1989, Physica B 159, 111. Rhyne, J.J., R.W. Erwin, J.A. Borchers, M.B. Salamon, E Tsui, R. Du, and C.E Flynn, 1991, in: Science and Technology of Nanostructured Magnetic Materials, eds C.G. Hadjipanayis and G.A. Prinz (Plenum, New York) p. 117. Rosen, M. and H. Klimer, 1970, Phys. Rev. B 1, 3748. Salamon, M.B., Shantanu Sinha, J.J. Rhyne, J.E. Cunningham, R.W. Erwin, J. Borchers, and

57

C.P. Flynn, 1986, Phys. Rev. Lett. 56, 259. Tsui, E, C.E Flynn, R.S. Beach, J.A. Borchers, R.W. Erwin, and J.J. Rhyne, 1993, J. Appl. Phys. 73, 6904. Vettier, C., D.B. McWhan, E.M. Gyorgy, J. Kwo, B.M. Buntschuh, and B.W. Batterman, 1986, Phys. Rev. Lett. 56, 757. Wilkinson, M.K., W.C. Koehler, E.O. Wollan, and J.W. Cable, 1961, J. Appl. Phys. 32, 48S. Yafet, Y., J. Kwo, M. Hong, C.F. Majkrzak, and T. O'Brien, 1988, J. Appl. Phys. 63, 3453.

chapter 2 THERMAL EXPANSION ANOMALIES AND SPONTANEOUS MAGNETOSTRICTION IN RARE-EARTH INTERMETALLICS WITH COBALT AND IRON

A.V. ANDREEV Permanent Magnets Laboratory Institute of Physics and Applied Mathematics Ural State University 620083 Ekatednburg, Russia

Handbook of Magnetic Materials, Vol. 8 Edited by K. H.J. Buschow ©1995 Elsevier Science B.V. All rights reserved 59

CONTENTS 1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. L i m i t a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61

2. G e n e r a l r e m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

3. S p o n t a n e o u s m a g n e t o s t r i c t i o n in R F e 2 a n d R C o 2 L a v e s p h a s e s . . . . . . . . . . . . . . . . . . . . . . . . 3.1. R F e 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. R C o 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. S p o n t a n e o u s m a g n e t o s t r i c t i o n in R - C o i n t e r m e t a l l i c s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. R C o 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. R 2 C o 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. R C o 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. R a C o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. C o n c l u s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66 67 70 72 72 98 116 127 136

5. S p o n t a n e o u s m a g n e t o s t r i c t i o n in R - F e i n t e r m e t a l l i c s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. R 2 F e l 7 a n d t h e i r c a r b i d e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. R 2 F e 1 4 B a n d R 2 F e 1 4 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. O t h e r c o m p o u n d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~. . . . . . . . . . . . . . . .

137 138 154 171

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

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

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

60

179 179

To unforgettable Japan and its perfect people

1. Introduction

Rare-earth (R) intermetallic compounds with 3d metals (T) form a wide class of magnetic materials. They combine a localized magnetism of the R sublattice with an itinerant (in a good approximation) magnetism of the 3d sublattice. This makes such intermetaUics very interesting subjects from the scientific point of view. On the other hand, large magnetic anisotropy and magnetostriction, originating from the R sublattice, in combination with high magnetic moment and Curie temperature, originating from the 3d sublattice (first of all, Fe and Co), have led to the discovery of several excellent materials for application as permanent magnets and magnetostrictors. For these reasons, the R-T intermetallics have been extensively studied and large information has been collected on the magnetic moments, magnetocrystalline anisotropy, temperatures of the magnetic ordering and spin-reorientation phase transitions. In this Handbook the intrinsic as well as hysteresis properties of the R-T compounds have been reviewed by Buschow (1980, 1988a), Strnat (1988), Li and Coey (1991), Franse and Radwanski (1993). Magnetoelastic properties have not been studied so extensively, although, the information collected on this topic is also large enough to be reviewed. Clark (1980) has considered the anisotropic magnetostriction of cubic Laves phases RFez. Morin and Schmitt (1990) have discussed magnetoelastic behaviour of rare-earth compounds mainly with nonmagnetic partners, having low ordering temperature. The present chapter deals with thermal expansion anomalies originating from the spontaneous magnetostriction in the R-T intermetallics. Therefore, it has some relation with the review of Wasserman (1990) on the Invar problem in volume 5, but different magnetic materials are under consideration here.

1.1. Limitations Wasserman (1990) has discussed in great details the classical Invar alloys based on 3d metals. They have only a single type of electrons responsible for the magnetic properties and have a highly symmetrical structure. Therefore, these alloys can be considered in a good approximation as an isotropic media. They have been extensively studied for a long time, and Wasserman (1990) has mentioned more than 20 different theoretical models proposed for the understanding the Invar effect. In a brief presentation of several R-T intermetallics, which display Invar-like behaviour in some temperature range (owing to a particular combination of the phonon 61

62

A.V. ANDREEV

thermal-expansion contribution, spontaneous magnetostriction and ordering temperature), Wasserman (1990) has stated that very large spontaneous magnetostriction in the R2Fe14B compounds 'is far from being understood'. The same can be said about the spontaneous magnetostriction of many other R-T compounds. Indeed, the R-T intermetallics are much more complicated materials than the 3d-metal alloys, since they have two different types of 'magnetic' electrons and form complex anisotropic lattices. Moreover, the experimental results of magnetovolume effects of most R-T compounds have become available only relatively recently. For these reasons, the author has limited himself to presentation of the experimental data and giving the phenomenological description of the spontaneous magnetostriction. The analysis consists of the separation of anisotropic and isotropic (in respect to the direction of the magnetization) parts of the magnetostriction with further separation of the contributions to these parts from the R and T sublattices (where it is possible). If in the case of anisotropic magnetostriction the dominant role of the single-ion mechanism (interaction between the 4f orbital moment of R and the crystalline electric field) is unquestionable (so-called giant magnetostriction), the microscopic mechanism of the magnetovolume effect and its distribution over the main axes is less easily given. The author leaves this problem to theoreticians and hopes that the data presented in this chapter will help them to develop proper models. The second limitation concerns the Laves phases RFe2 and RCo2. Magnetoelastic properties of these compounds have been studied much more extensively than those of all other intermetallics. They should be reviewed separately (supplementing the excellent chapter in volume 1 on the anisotropic magnetostriction of RFe2 written by Clark, 1980). In the present chapter, only a brief summary on the spontaneous magnetostriction of RT2 is given, which is needed for a comparison with other compounds discussed. On the other hand, several representatives of actinide compounds, related to the rare-earth compounds (ThC05, UC05, Th2Fe14B, UFeloSi2), will be included in this chapter. 2. General remarks Magnetostriction means a change of dimensions and shape of a sample due to a change of its magnetic state. Magnetostriction reflects dependences o f magnetic and exchange interactions on interatomic distances and manifests itself in different ways. A well-known type of this phenomenon is the magnetostriction induced by an external magnetic field. Another type is the spontaneous magnetostriction. It appears when the magnetic state changes by a temperature variation (the magnetic ordering or change of the type of magnetic arrangement, including spin-reorientation transitions). Figure 2.1 presents the temperature dependences of the lattice parameters a and e and the unit-cell volume V of the tetragonal intermetallic compound SmaFel4B (Andreev et al. 1985d). The compound orders ferromagnetically at 620 K. One can see that thermal expansion drastically changes in the vicinity of this temperature. At

THERMAL EXPANSION ANOMALIES

63

c (pm) 1220 c

1212

a (pro)

f

J

f

fJ

88O a

V (nm3) v

0.94

/

/

/

/ /

J

0.93

/ /

/ 0.92

/

/

/

/

/

876

/

/

872

/

J I

I

I

200

400

600

T (K)

Fig. 2.1. Temperature dependences of the lattice parameters a and c and the unit-cell volume V for Sm2Fel4B. The dash curves represent the phonon contribution to the thermal expansion. (After Andreev et al. 1985d.)

higher temperatures, a normal phonon thermal expansion is observed both along the tetragonal e axis and in the basal plane (along the a axis). Of course, the unit-cell volume also increases with increasing temperature. The dash lines show the extrapolation of the a, c, V ( T) dependences from the paramagnetic to the ferromagnetic range. These curves correspond to a hypothetic nonmagnetic sample with parameters a0, c0, V0. The extrapolated curves are obtained using the Debye theory and the Griineisen relation. It should be mentioned that the extrapolation process includes several assumptions. The correctness of such extrapolations is a point of discussion even in classical Invar alloys with cubic structure (Wasserman 1990). The difference in extrapolation is the main source of the disagreement existing in the literature regarding spontaneous magnetostriction, especially in the case of compounds with high (room temperature or larger) ordering temperature. Below Tc all the experimental curves deviate from the extrapolations. The magnetic ordering reduces the thermal expansion coefficient C~ealong the c axis, but it is

64

A.V. ANDREEV

positive at all temperatures. In the basal plane a magnetic contribution to the thermal expansion is so large that it leads to a negative aa in a wide temperature range. A similar Invar-like anomaly is also observed in the volume thermal expansion. The relative differences between observed and extrapolated curves

ha = (a - ao)/ao; hc = ( c - co)/~o (2.1) are spontaneous linear magnetostrictive strains in the basal plane and along the e axis, respectively. They are temperature dependent and (usually) reach their maximum values at 0 K. The corresponding relative difference ws -

(2.2)

(V - Vo)/Vo

defines the spontaneous volume magnetostriction. In the case of uniaxial lattices (i.e. tetragonal, hexagonal and rhombohedral): Ws = 2ha + he. In compounds with a cubic structure: Ws = 3ha. In classical Invar alloys based on 3d metal ws can reach at 0 K the value 1.9 x 10 -2 (Wasserman 1990 and references therein). In R - T intermetallics ws at 0 K reaches similar or even larger (3 x 10 -2) values. In the given example of Sm2Fel4B it varies in literature from 1.92 x 10 -2 (Andreev et al. 1985d) to 2.73 x 10 -2 (Buschow and Grrssinger 1987) which is mainly due to different extrapolation, see section 5.2. In this compound the spontaneous magnetostriction does not change the symmetry of the crystal structure (at least within the experimental error). In general, not only the interatomic distances change due to the spontaneous magnetostriction but also a lattice distortion can be observed. Linear magnetostriction strain for a crystal with cubic symmetry up to second-order terms is given by the phenomenological expression (Akulov 1939): h --" h0 + 1.Sh10O(ax23x + (~YflY 2 2 + O~zflz 2 2 -- 1/3)

(2.3)

+ 3h111(~/~x~y~y + ~;~y~z~z + ~xZx~zZz). Here ai are the cosines of the angles between the magnetization direction and the coordinate axes, fli are the cosines of the strain measurement direction, h0, hi00 and h i l l are the magnetostriction constants. Figure 2.2 shows the magnetostrictive strain corresponding to each constant. The quantity h0 describes the isotropic magnetostriction which does not depend on the magnetization direction nor on the strain direction. This type of magnetostriction does not reduce the symmetry of crystal. The constants hlO0 and h l l I describe the anisotropic magnetostriction, which reduces the lattice symmetry to tetragonal (hl0O) or rhombohedral (hill). In the case of uniaxial crystals, the linear magnetostriction up to second-order terms can be described by the following expression (Clark et al. 1965): ,~ = ~a,OEa2 R2~ ..t- ~a,0(42 "'1 V-'x -[" t-'y] ~ "'2 t-'z

+ h i~,2(~x2 + / ~2) ( ~ -2

-~,2~2, 1/3)+~2 ~,z~z2 - 1/3)

+ h%2[O.5(3x2 _ 3~)(~2 _ %2) + 2O,xCxay3y] + 2A*'2(Otx/3x+ Olyfly)O~zflz.

(2.4)

THERMAL EXPANSION ANOMALIES

2kIa

65

~.2a

I I

i I

I

I

l/

k

~.o

~q oo

XIl I

Fig, 2.2. Magnetostriction modes for the uniaxial and the cubic symmetries.

The z axis coincides with the c axis in the crystal. The magnetostriction constants with index a (a-magnetostriction) do not reduce the lattice symmetry and describe the changes in interatomic distance within the basal plane (A~) and along the c axis x~'° and ,~,0 (A 2~) , fig" 2.2. The terms with zero-order c~-magnetostriction constants "'1 do not depend on the magnetization direction and represent the isotropic (with respect to magnetization direction) part of magnetostriction, similar to )~0 in cubic crystals. However, this type of magnetostriction cannot be called 'isotropic'. As seen in expression (2.4), there are no terms independent of the strain measurement direction owing to the anisotropy of lattice. In the present review the isotropic (with respect to magnetization direction) magnetostriction is called 'exchange magnetostriction' following Belov et al. (1983a). However, the term 'exchange magnetostriction' is completely correct only in the localized model, where 'isotropic magnetostriction' is due to the interatomic-distance dependence of the exchange interaction. In the band model this type of magnetostriction originates from the band splitting and the increase in kinetic energy due to the transfer of electrons from the minority to the majority band. The system reduces this increase by expanding. I use the expression 'exchange magnetostriction' regardless to the model, only in order to avoid the expression 'anisotropy of isotropic magnetostriction' when considering magnetostriction in noncubic compounds. The second-order a-magnetostriction has anisotropic character and reflects the change in interatomic distances within the basal plane (A~''2) and along the c axis

66

A.V. ANDREEV

(

)

•

o

•

,

2

-

•

2 ], when the magnetization rotates in the crystal under the action of a magnetic field or during a spontaneous spin reorientation. The quantity A"r (7-magnetostriction) describes an orthorhombic distortion in the basal plane when the magnetization direction does not coincide with the e axis. The quantity ,V describes a deviation of the angle between the basal plane and the c axis from 90 °. The phenomenological expressions for magnetostrictions up to sixth-order terms are given by Clark (1980). The magnetovolume effect Ws and the atomic magnetic moment # for a singlesublattice ferromagnet are connected by following phenomenological expression: Ws = k C p 2 = n # 2,

(2.5)

where k is the compressibility, C is the magnetoelastic-coupling constant. The magnetoelastic-coupling coefficient n = k C is used when the factors k and C cannot be found separately. In the case of R - T intermetallics, Ws is described by three terms (Lee and Pourarian 1976): Ws --= nRR/-tl + nRdlgRlgd + ndd/gd2,

(2.6)

where #R and #d are the magnetic moments of R and 3d atom, respectively, nRR and ndd are the magnetoelastic-coupling coefficients in the R and T sublattices; n~a is the intersublattice magnetoelastic-coupling coefficient. Magnetic anisotropy constants used in this chapter correspond to generally accepted phenomenological expressions for the magnetic anisotropy energy ER. In a cubic crystal Ea

22 22 22 222 = K l ( ~ x ~ y + ~x~z + ~y~z) + K 2 ~ x ~ y ~ z + 22 22 222 + ~ ( ~ x ~ y + ~x~z + ~y~z) ,

(2.7)

where K1 and K2 are first and second anisotropy constants, ai are the cosines of the angles between the magnetization direction and the coordinate axes. In an uniaxial crystal Ea = K1 sin 2/9 + K2 sin 4 O + K3 sin 6/9 + K 4 sin 6 0 cos(n~),

(2.8)

where n = 4 corresponds to the tetragonal structure and n = 6 to the hexagonal structure. 0 is the angle between the magnetization direction and the [001] axis (c axis). ~ is the angle between the projection of magnetization on the basal plane and the [100] axis (a axis).

3. Spontaneous magnetostriction in RFe2 and RCo 2 Laves phases Laves phases having the A n 2 stoichiometry are the largest class of binary intermetallics. In the R - T alloy systems this type of compounds exists with T = Mn,

THERMALEXPANSIONANOMALIES

67

) (

J

( Q

R (8a)

• T(16d)

Fig. 3.1. Unit cell of the cubic crystalstructureof the RT2 Laves phases (MgCu2 type). Fe, Co and Ni. Here a brief summary is given on the magnetostriction in RFe2 and RCo2. RFe2 and RCo 2 crystallize in a relatively simple cubic structure of the MgCu 2 type (Fd3m space group). The unit cell of this structure contains 8 formula units and is shown in fig. 3.1. The R atoms occupy single position (Sa), the R sublattice has diamond structure. The 3d atoms occupy single position (16d) and form 4 tetrahedrons around [3/4, 3/4, 3/4] and equivalent sites.

3.1. RFe2 These compounds are ferromagnets (with nonmagnetic or light R element) or ferrimagnets (with heavy R) and have the largest Curie temperatures among all binary R-Fe alloys. The magnetic moment of the Fe atom/~Ve at 0 K is about 1.4 #B in the cases of trivalent R. Magnetic and magnetoelastic properties of RFe2 are presented in table 3.1. The highest known room-temperature anisotropic magnetostriction is found in TbFe2 (Clark and Belson 1972) and attracted the attention to RFe2 as magnetostrictive materials. The )1111 constant reaches the value 2.4 x 10 -3. SmFe2 exhibits a Alll value of similar magnitude, but negative. These two compounds are the basis of the multicomponent systems, including the famous 'Terfenol' Tb0.3DY0.7Fe2. The Fe sublattice contribution to the total anisotropic magnetostriction of RFe2 is negligible compared to the R sublattice contribution. It follows from the fact that the magnetostriction in the compounds with nonmagnetic R = Y or Lu does not exceed 8 x 10 -5 (Clark 1980; single-crystal data of Deryagin et al. 1985a and Andreev et al. 1990a). This is by two orders lower than in the cases of R ions having

68

A.V. ANDREEV

TABLE 3.1 Magnetic and magnetoelastic properties of RFe 2 and RCo 2. a is the lattice parameter at room temperature; EMD is the easy-magnetization direction at 4.2 K; tzm is the magnetic moment per formula unit at 4.2 K; Te is the Curie temperature; A100 and Am are the anisotropic magnetostriction constants at 4.2 K; Ws is the spontaneous volume magnetostriction at 4.2 K. Compound

a pm

EMD

#m #B

Te K

)q00 10 -3

)qll 10 -3

CeFe 2 SmFe 2 GdFe 2 ThFe 2 DyFe 2 HoFe 2 ErFe 2 TmFe 2 LuFe 2 YFe 2 LIFe2 SmCo 2 GdCo 2 TbCo 2 DyCo 2 HoCo 2 ErCo 2 TmCo 2

730.2 741.5 740.5 734.1 733.8 731.3 728.4 724.5 722.3 736.9 705.7 726.4 726.0 721.1 719.2 717.4 715.7 713.6

[100] [110] 2 [100] [111] [100] [100] [111] [111] [100] 4 [111] [111] [111] [100] [111] [100] [110] 6 [111] [111]

2.3 3.1 3.8 5.8 6.8 6.9 6.0 3.7 2.8 2.8 1.2 1.5 5.0 6.3 7.0 7.5 6.9 3.8

230 660 790 697 624 586 577 560 581 552 170 208 398 235 137 78 31 5

0.061 0.03 <0.051 -0.07 -0.75 -0.091 0.23 -1.2 -1.25 -1.9 -2.2 -1.05 0.755

-4.1 4.5 3.03 0.783 -1.85 -3.6 <0.051 3.0 -5.0 0.15 4.5 5.05 0.55 -2.5 -4.1

Ws 10 -3 5.4 6.3 8.8 4.1 2.2 1.6 6-7.2 6.8 5.5 5.5 4.7 -

Ref. [1, 2] [3-5] [1, 6] [3, 6, 7] [3, 8-10] [3, 8-11] [8, 9, 12] [3, 13] [2, 9, 14] [9, 15] [4, 16, 17] [4, 5] [1, 18, 19] [18, 19] [18, 19] [18, 19] [12, 18, 19] [18]

1 Data on polycrystals are attributed to )q0o or )~m according to the easy-magnetization direction. 2 Spin reorientation to the [111] axis at 160-200 K. 3 The values extrapolated to 0 K from the high-temperature range (180-300 K). 4 The easy-magnetization direction does not coincide with a main axis, the [100] axis is the nearest of them. 5 The values obtained from the extrapolation of quasibinary system data. 6 Spin reorientation to the [100] axis above 18 K. References: [1] Franse and Radwanski (1993). [2] Deryagin et al. (1985a). [3] Clark (1980). [4] Andreev (1990). [5] Korolyov et al. (1990a). [6] Barbara et al. (1977). [7] Clark et al. (1978). [8] Deryagin et al. (1981). [9] Zadvorkin (1987). [10] Kudrevatykh (1994).

111] Abbundi et al. (1979). 112] Andreev et al. (1985e). 113] Abbundi et al. (1978). 114] Deryagin et al. (1984d). 115] Andreev et al. (1990a). 116] Popov et al. (1980). 117] Andreev (1979). 118] Markosyan (1988). 119] Nakamura (1983).

n o n z e r o orbital m o m e n t L. G d F e 2 w i t h s p h e r i c a l 4 f e l e c t r o n shell o f t h e G d 3+ i o n (L = 0) has also a l o w ( < 5 x 10 - 5 ) m a g n e t o s t r i c t i o n ( B a r b a r a et al. 1977). T h e r a r e - e a r t h c o n t r i b u t i o n to the a n i s o t r o p i c m a g n e t o s t r i c t i o n in R F e 2 is w e l l e x p l a i n e d w i t h i n t h e f r a m e w o r k o f the s i n g l e - i o n m o d e l , w h e r e g i a n t m a g n e t o s t r i c t i o n as w e l l as m a g n e t o c r y s t a l l i n e a n i s o t r o p y o r i g i n a t e f r o m the e l e c t r o s t a t i c i n t e r a c t i o n

THERMAL EXPANSION ANOMALIES

69

between the anisotropic 4f electron shell of the R ion and the crystal field (Clark 1980). An unusual correlation "~111 )) ,'~100 between the magnetostriction constants found in RFe2 is simply explained on the basis of local symmetry of the R sites (Cullen and Clark 1977). In this approximation the single-ion mechanism of magnetostriction gives A10o = 0. Actually, this constant has an order of magnitude not exceeding 10 -4, which is comparable with the magnetostriction in conventional materials, where the single-ion mechanism does not apply due to quenching of the orbital moment of the 3d atoms. In several compounds with an [111] easy-magnetization axis, like TbFe2, AlOOis not determined at low temperatures due to huge magnetocrystaUine anisotropy, but it is estimated to be much smaller than A11x. The only exception from this trend is HoFe2, where IA10o]~ ]Al111, both due to anomalously high Awo and relatively low Al11. The quantity .'~111 monotonically decreases with increasing temperature in RFe2 and exhibits (in the cases where it can be compared with temperature dependences of #~, i.e. if R is a heavy rare-earth metal) very good agreement with single-ion temperature dependence (Callen and Callen 1963):

)~111(T)/,~111(0) _- _T5/2{/-I[IAR(T)I/R(0)] },

(3.1)

where I512(x) is the normalized hyperbolic Bessel function (3x-2+ 1)-(3x-1)ctnh (x); L-l(x) is the function inverse to the Langevin function coth(z) - x -1. On the other hand, A10o(T) does not follow expression (3.1) in DyFe2 and HoFe2, which confirms its not being of single-ion origin. In DyFez, ,'\100(T) is nonmonotonic and even changes sign at room temperature (Clark 1980). In the single-ion model the anisotropic magnetostriction in the R series is described by the expression A = B2otJ(J

-

0.5)(r2),

(3.2)

where B2 is the second-order crystal field coefficient, ~ is the second-order Stevens factor (Stevens 1952), J is the ground state angular momentum for the 3 + ion, and (r E) is the average squared radius of the 4f electron shell. Using experimental data of one compound (TbFe2, Clark 1980) for the determination of B2, it is possible to calculate the )~111 values for all RFe2. The sign of )~111 is found to be in perfect agreement with (3.2), and the magnitude of Alll is also satisfactorily described by this expression. On the other hand, the first anisotropy constant K 1 for cubic lattice is determined not by the second-order Stevens factor but by the fourth-order Stevens factor/3 (assuming IK2[ << IllKII): K1 = B 4 f l J ( J - 0.5)(J - 1)(J - 1.5)
(3.3)

Since o~and ~ change their signs along the R series in different ways, it is possible to prepare quasibinary R'R"Fe2 compounds with high magnetostriction and compensated K1, having high d A / d H characteristics, important for applications. However, due to the different temperature dependences of K1 for different R, the compensation can be achieved only in a narrow temperature range. The basic quasibinary

70

A.V. ANDREEV

compound for such compensation at room temperature is Tb0.3DY0.TFe2. Many multicomponent alloys are developed in order to compensate also higher order anisotropy constants and to increase the temperature interval of low anisotropy (Clark 1980). The principal possibility of a coexistence of low anisotropy and high magnetostriction in a binary compound in the whole temperature range of magnetic ordering is shown for the example UFe2 (Andreev et al. 1979, Popov et al. 1980). The compound has an anisotropy only slightly higher than in YFe2, but )~111 reaches 2.9 x 10 -3 at 4.2 K. Unfortunately, Tc of UFe2 is far below room temperature. The isotropic magnetostriction of RFe2 was determined for several compounds by Zadvorkin (1987). Thermal-expansion studies on single crystals of YFe2 and LuFe2 show a positive volume effect ;as with considerable difference in its values (table 3.1), despite the fact that #F~ is the same in both compounds. The magnetoelastic-coupling coefficient nFeFe = ;as//~2e (see eq. (2.5)) is equal to 1.1 x 10-3/.tB 2 for YFe2 and 2 x 10-a#B 2 for LuFe2 at 5 K. Furthermore, nVeFe is not temperature-independent but decreases linearly with increasing temperature. These results obtained from Xray dilatometry and magnetization measurements of ferromagnets are consistent with those obtained from the high-field magnetostriction and susceptibility measurements of paramagnetic YCo2 and LuCo2, where ncoco is also positive in both compounds. It is larger for R = Lu (3 x 10-3/tB 2 compared to 4 x 10-3#B 2 for R = Y) and decreases linearly with temperature (Belov et al. 1983b, Snegirev 1984). The behaviour of nOd has not been understood. Compounds with R = Dy, Ho and Er exhibit considerably larger COsthan with nonmagnetic R. The difference is attributed to the second term in eq. (2.6) connected with the R-Fe exchange interaction (Zadvorkin 1987), because the volume effect in RNi2 and RA12 does not exceed 10 -4 (Ibarra et al. 1984, 1986). In the latter compounds Ni and Al are nonmagnetic, and only the first t e r m r/RRI22R exists in eq. (2.6). 3.2. RCo2 The RCo2 compounds have much lower Tc values than RFe2. Only in GdCo2 Tc exceeds room temperature (table 3.1). For this reason, these compounds are not interesting for application. But the phenomenon of band metamagnetism found in RCo2 is a point of great attention. The Co sublattice has no intrinsic magnetic moment, YCo 2 and LuCo 2 being paramagnets with an enhanced magnetic susceptibility. The metamagnetic transition to a state with/~co of about I#B was predicted theoretically for fields of 80-350 T (Cyrot and Lavagna 1979, Yamada and Shimizu 1985, Yamada et al. 1984, 1985, Schwartz and Mohn 1984) and has directly been observed by Goto et al. (1989, 1990) at 70 T in both compounds. In the presence of magnetic R, the Co sublattice is magnetized by an effective field H originating from the intersublattice interaction. This field is of order of several hundred Tesla, which is enough to induce #Co values ranging from 0.5#a in TmCo2 to 1.1#a in GdCo2 (Levitin and Markosyan 1988). The main difference in anisotropic magnetostriction between RCo2 and RFe2 is that the Co contribution is not negligible compared to the R contribution. Since YCo2 and LuCo2 are paramagnets, the Co contribution manifests itself directly only

THERMAL EXPANSION ANOMALIES

71

in G d C o 2. Whereas GdAl2, GdNi2 and GdFe2 have magnetostriction not larger than 8 x 10 -5, GdCo2 exhibits at low temperatures a tetragonal distortion corresponding to A10o = - 1 . 2 x 10 -3 (Levitin et al. 1982). Similar or even larger IA1001 values are found directly (in DyCo2 and HoCo2) or from extrapolations of quasibinary data (in the cases of an [111] type easy-magnetization axis, TbCo 2 and ErCo2). As has been already mentioned, the R sublattice does not produce so large a value of )~10o. Additional evidence for the Co origin of high )~1oo is the fact that neither sign nor magnitude follow from the expression (3.2) as expected in the case of an R contribution. In all compounds with #co of about I#B, )~10o is negative. Only in TmCo2 with twice lower #co, A10o is positive and smaller in magnitude. Considering the ~111 constant, a similarity exists between the RFe 2 and RCo2 compounds showing that the R sublattice plays a dominant role in the formation of this type of magnetostriction. However, an additional negative contribution about of I x 10 -3 was certainly found in RCo2 compounds compared to RFe2 for R = Sm, Er, Tm. Z b F e 2 and TbCo2 exhibit the same )~111 value. The data on )qll in compounds with R = Dy and Ho are difficult to compare because of their indirect method of determination (see comments to table 3.1). In the quasibinary system Er(Fel_xCox)2, the temperature dependence of )~111becomes more smooth with increasing Co content than would follow from eq. (3.1), which also points to a considerable Co contribution to '~111 (Andreev et al. 1985e). Large anisotropic magnetostriction of the Co sublattice in RCo2 leads to the assumption of considerable orbital contribution to total Co magnetic moment, in agreement with NMR results (Hirosawa and Nakamura 1982). As regards the isotropic magnetostriction, it is directly connected with band metamagnetism of the Co sublattice. This manifests itself, for example, in the first order magnetic-ordering transition in RCo 2 with R - Dy, Ho, Er. The value of ~s is found to be about 5 x 10 -3 in those compounds where Co has a magnetic moment (Minataka et al. 1976, Lee and Pourarian 1976, Steiner et al. 1978, Gignoux et al. 1979). It is rather difficult to separate the ~s contribution from the Co-Co and the R-Co interactions due to the instability of the Co moment. It is impossible to determine the CO-Co interaction contribution to a;s in RCo 2 compounds with nonmagnetic R, as in other types of R - T compounds. This contribution can be estimated from the above mentioned data on the high-field magnetostriction of YCo2 and LuCo2. Assuming ncoco to be the same in the paramagnetic and ferromagnetic states and using #co = 1.1#B in compounds with R = Gd-Er, O~s varies from 3.5 x 10 -3 to 4.7 x 10 -3 depending on whether ncoco is taken from YCo 2 or Luco2. This estimate shows that the Co-Co interaction contributes considerably to 0;s. However, Ws in Gdco2 and T b c o 2 is certainly larger (table 3.1). This shows the important role of the R-Co interaction contribution. Much information is collected now on magnetic and magnetoelastic properties of binary, quasibinary R'R'Co2 and substituted R(CoI-rMx)2 compounds (M is equal, for example, Al, Ga, Cu) in high and ultrahigh magnetic fields. This exceeds the framework of present chapter and will be considered separately elsewhere.

72

A.V. ANDREEV

4. Spontaneous magnetostriction in R-Co intermetallics In R-Co binary phase diagrams, only the Laves phases RCo 2 have a cubic crystal structure. Other compounds, with the exception of LaCo13, crystallize in lowersymmetry lattices. The R3Co compounds have an orthorhombic structure, R4Co3 and RCo5 are hexagonal. The compounds RCo3, R2Co7, R5Co19 and R2Co17 have hexagonal and rhombohedral structures which are modifications of the Cacu5 type. In the case of noncubic symmetry the error in the determination of spontaneous magnetostrictive strains is much higher than for cubic compounds. Quantitative data with reasonable accuracy usually cannot be obtained on polycrystalline samples. Frequently only an indication of the presence of anomalies in the thermal expansion can be seen. However, for some compounds (RCos, RCo3, R2Co7 and R3Co) data obtained on single crystals are available. The discussion will be restricted mainly to these results. 4.1. RCo5 The relatively simple atomic configuration in the above discussed RCo2 compounds can be found as a structural block R2Co4 in the more complicated crystal structures of RCo3, R5Co19 and R2Co7, where it is combined with another simple structural block, RCos. The magnetic and magnetoelastic properties of these compounds can be discussed as being contributed from the cubic R2Co4 and hexagonal RCo5 blocks. For this reason, the description of the spontaneous magnetostriction in noncubic R-Co intermetallics will start from the RCo5 compounds. The RCo5 phases have an outstanding position among the 4f-3d intermetallic compounds. First, the discovery of a large magnetocrystalline anisotropy in these compounds (Hoffer and Strnat 1966) led to the production of the new generation of powerful permanent magnets based on SmCos, widely used till now. In fact, technical application of rare-earth based magnetic materials began with the RCo5 compounds. Second, a variety of magnetic properties, in combination with rather simple crystal structure, made RCo5 very apt model systems for studying magnetism in rare-earth-3d-metal alloys. These highly anisotropic ferro- (in the case of nonmagnetic or light R) and ferri- (with heavy R) magnets can be prepared relatively easily as single crystals of sufficiently large dimensions (1-3 mm) for magnetometric measurements. For this reason their magnetic properties (the magnitude and orientation of the magnetic moments of the R and Co sublattices, the constants of the magnetocrystalline anisotropy, the temperatures marking the magnetic phase transitions - magnetic ordering and spin reorientations) have been studied in great detail (Tatsumoto et al. 1971, Okamoto et al. 1973, Yermolenko 1974, 1976, 1983, reviews by Buschow 1980, Strnat 1988, Franse and Radwanski 1993). At the same time, the magnetoelastic properties of these alloys have received less attention due to several methodological reasons, connected mainly with the very large magnetocrystalline anisotropy. The study of spontaneous magnetostriction was also difficult because of the high Curie temperatures. The hexagonal crystal structure of the RCo5 compounds presented in fig. 4.1 belongs to the CaCu5 type (the P6/mmm space group). The unit cell contains 1

THERMAL EXPANSIONANOMALIES •

73

a

C

o

C 0 (la) z 0

(20)

• (3g)

Fig. 4.1. Unit cell of the hexagonal crystal structure of RCo5 (CaCus type). formula unit. The structure has one position for the R atoms (la type) and two positions for the Co atoms (2c and 3g types). The lattice parameters a ~ 0.5 nm, cz 0.4m. We will start the discussion of RCo5 with the compounds with nonmagnetic R in order to obtain information on the Co sublattice contribution to the magnetoelastic properties.

4.1.1. Compounds with nonmagnetic R 4.1.1.1. YCo5 and Y(COl_~Nir)5. YCo5 and YNi5 form continuous solid solutions with the CaCu5 structure. According to the neutron diffraction data (Pirogov 1985), the Ni atoms occupy preferably the 2c sites and the Co atoms the 3g sites. However, the site preference is far from perfect and the distribution of the 3d atoms over two positions is close to statistical. Ni does not carry a magnetic moment in RMs. So, the magnetic properties of the Y(Col_xNi~)5 compounds are determined only by Co atoms. All magnetically ordered alloys (z ~< 0.7) are ferromagnets with uniaxial magnetic anisotropy (Buschow and Brouha 1976, Deryagin et al. 1976a, Yermolenko 1983). The structural and magnetic data of Y(CoI_=~.Ni~)5 are listed in tables 4.1 and 4.2, respectively. Magnetic properties here, as well as in tables 4.3 and 4.4 for RCo5 with magnetic R, are presented for the samples used in the X-ray thermal-expansion investigation. The thermal expansion of the Y(Col_~Nix)5 solid solutions was studied by Andreev et al. (1985a) using single-crystalline samples at 5-1300 K. In fig. 4.2 the temperature dependences of the lattice parameters and unit cell volume are presented. The positions of the observed c(T)-curve anomalies (x ~< 0.4) are in good

74

A.V. ANDREEV

TABLE 4.1 Structural data for RCo 5 compounds with nonmagnetic R. a and c are the lattice parameters of the hexagonal unit cell; aa, a e are the linear thermal expansion coefficients in the paramagnetic range (1200 I0 in the basal plane and along the c axis, respectively; a v is the volume thermal expansion coefficient at 1200 K. Compound

a (300 K) c 300 K pm pm

a (5 K) pm

e (5 K) pm

C~a 10 - 5 K -1

ae 10 - 5 K -1

c~v 10 - 5 K -1

Ref.

YCo 5 YCo4Ni YCo3Ni 2 YCo2Ni3 YCoNi 4 YNi5 CeCo 5 Th0.965Co5.07 Th0.950Co5.10 UCos. 3

495.00 493.75 492.15 491.25 490.25 489.07 494.00 499.75 498.10 487.45

493.55 492.25 490.62 489.70 488.96 487.73 492.60 497.60 497.30 477.30

398.35 398.05 397.90 396.70 395.95 395.45 401.20 399.15 401.22 3637.8

1.51 1.50 1.57 1.57 1.55 1.54 1.96 1.49 1.63 1.85

1.25 1.27 1.25 1.23 1.26 1.26 1.58 1.06 1.40 1.05

4.3 4.3 4.4 4.4 4.4 4.3 5.5 4.0 4.7 4.8

[1-3] [1-3] [1-3] [1-3] [1-31 [1-3] [1, 2, [1, 2, [1, 2, [1, 2,

398.62 398.30 398.15 397.50 396.70 396.15 401.05 400.30 401.40 3643.8

References: [1] Zadvorkin (1987). [2] Andreev (1990). [3] Andreev et al. (1985a).

4] 5] 5] 6]

[4] Andreev et al. (1985b). [5] Andreev et al. (1984a). [6] Andreev (1991).

TABLE 4.2 Magnetic and magnetoelastic data for RCo 5 compounds with nonmagnetic R. Te is the Curie temperature; #m and tza are the magnetic moments per formula unit and per 3d atom, respectively, at 4.2 K; )~a, )~e are the linear spontaneous magnetostrictive strains in basal plane and along c axis, respectively; Ws is the volume spontaneous magnetostriction; had is the magnetoelastic-coupling coefficient between 3d atoms at 5 K. Compound

Te K

#m Ud #B/f.u. /2,B

),a (5 K) 10 - 3

.~e (5 K) 10 - 3

Ws (5 K) 10 - 3

nckl Ref. 10--3/ZB2

YCo5 YCo4Ni YCo3Ni 2 YCo2Ni 3 YCoNi 4 YNi5 CeCo 5 Th0.965 Co 5.07 Th0.950Co5.10 UCos. 3

940 780 585 240 637 540 650 365

8.3 6.6 4.9 2.2 0 0 6.5 4.5 7.1 2.37

0 0 0 0 0 0 4.0 -0.2 3.4 2.48

6.8 4.2 2.4 0 0 0 8.3 0.3 5.1 0.41

6.8 4.2 2.4 <0.2 0 0 16.3 0 11.9 5.41

2.5 2.4 2.4 <1 9.6 0 6.7 27.2

References: [1] Zadvorkin (1987). [2] Andreev (1990). [3] Andreev et al. (1985a). [4] Deryagin et al. (1976a). [5] Yermolenko (1983) [6] Andreev et al. (1985b).

1.66 1.32 0.98 0.44 0 0 1.30 1.32 0.45

[7] [8] [9] [10] [11]

Andreev et al. (1984a). Snegirev (1984). Andreev (1979). Andreev (1991). Deryagin and Andreev (1976).

[1-5]

[1-4] [1-4]

[1-41 [1-4] [1-4] [1, 2, 6] [1, 2, 7, 81

[1, 2, 7, 8] [1, 2, 9-11]

THERMAL EXPANSION ANOMALIES

75

TABLE 4.3 Structural data for RCo 5 compounds with magnetic R. a and e are the lattice parameters of the hexagonal unit cell; t~a, C~e are the linear thermal expansion coefficients in the paramagnetie range (1200 K) in the basal plane and along the e axis, respectively; a v is the volume thermal expansion coefficient at 1200 K. Compound

a (300 K) e (300 K) a (5 I4:) pm pm pm

e (5 K) pm

aa c~c 10 - 5 K -1 10 -5 K - I

av Ref. 10 - 5 K -1

PrCo 5 PrCo4Ni PrCoaNi 2 PrCo2Ni3 PrCoNi4 PrNi 5 NdCo 5 SmCo 5 GdCo 5 TbCos. 1 DyCos. 2 HoCos. 5

503.10 500.65 498.35 497.80 497.17 496.50 503.30 498.70 497.90 495.70 492.40 492.00

399.40 399.18 399.22 398.30 397.03 397.07 397.75 397.80 396.70 397.38 398.15 397.15

1.48 1.51 1.47 1.50 1.45 1.40 1.52 1.45 1.49 1.45 1.10 1.20

4.1 4.1 4.1 4.1 4.0 4.0 4.1 4.1 4.1 4.2 6.3 7.2

398.50 399.07 399.75 398.40 397.80 397.65 397.80 398.10 397.20 397.90 398.60 397.90

500.55 498.40 496.65 496.05 495:40 494.50 500.55 497.40 496.32 494.48 491.20 490.62

References: [1] Zadvorkin (1987). [2] Andreev (1990).

1.10 1.05 1.12 1.10 1.12 1.17 1.10 1.18 1.15 1.25 4.10 4.80

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

4] 4] 4] 4] 4] 4]

[3] Andreev et al. (1985c). [4] Andreev and Zadvorkin (1991).

TABLE 4.4 Magnetic and magnetoelastic data for RCo 5 compounds with magnetic R./Zm 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; 7'1 and T2 are the temperatures of the plane-cone and cone-axis transitions, respectively; )~a, ,~e are the linear spontaneous magnetostrictive strains in the basal plane and along the c axis, respectively; ws is the volume spontaneous magnetostriction. Compound

Um #a/f.u.

Anisotmpy

Tc K

T1 K

T2 K

,~a (5 K) he (5 K) Ws (5 K) Ref. 10 -3 10 -3 10 -3

PrCo 5 PrCo4Ni PrCo3Ni 2 PrCo2Ni 3 PrCoNia PrNi5 NdCo 5 SmCo 5 GdCo 5 TbCos. 1 DyCos. 2 HoCos. 5

10.5 8.8 6.9 5.3 0.8 0 10.5 8.7 1.6 0.6 1.2 0.9

canted canted plane plane plane . . plane axis axis plane plane plane

920 750 570 360 40 . 930 1000 1020 980 970 960

125 -

120 150 215 -

-3.8 -1.9 -0.7 -0.4 -0.5 0.7 -3.2 0.8 0 1.0 -3.7 -2.5

References: [1] Zadvorkin (1987). [2] Andreev (1990). [3] Andreev et al. (1985c).

.

. 235 397 300 45

295 410 370 170

9.9 5.8 2.8 2.4 0.5 0.7 7.8 4.6 6.1 5.3 33.4 37.2

[4] Andreev and Zadvorkin (1991). [5] Yermolenko (1983).

2.4 2.0 1.5 1.6 -0.5 -0.7 1.4 6.2 6.1 7.3 26.0 32.2

[1-5] [1-31

[1-31 [1-3] [1-3]

[1-31 [1, 2, 4, 51 [1,2,4,5] [1,2,4,51 [1, 2, 4, 5] [1,2,4,5] [1,2,4,5]

76

A.V. ANDREEV

c (prn)

401

a (pm) 399

500

397

496

395

492

488 0

200

400

600

800

1000

T (K)

Fig. 4.2. Temperature dependences of the lattice parameters a and c for the Y(Col_=Niz)5 solid solutions. The dashed curves represent the phonon contribution to the thermal expansion. (After Andreev et al. 1985a.)

agreement with Te. The magnetoelastic properties of Y(COl_xNi=)5 are strongly anisotropic - there are no noticeable anomalies in a(T). In order to determine the spontaneous magnetostrictive strains, the extrapolations of the a, c(T) curves from the paramagnetic to the ferromagnetic range have been done using the Debye temperatures obtained on the same single crystals by acoustic measurements: TD is found to be 370 K in YCos, 400 K in YNi5 and linearly depends on z for intermediate concentrations. There are two difficulties to make proper extrapolations for YCo5 as well as for the other R-Co intermetallics with high Co content, these difficulties being absent in the case of the above discussed cubic Laves phases that have much lower ordering temperatures. First, the Debye theory has been developed for isotropic materials whereas the compounds studied have rather anisotropic crystal structures. Second,

THERMAL EXPANSION ANOMALIES

77

the extrapolation should be done to 0 K on the basis of the 1100-1300 K data. Therefore, even a small error in the thermal expansion coefficients can lead to a significant error in the extrapolated lattice parameter values. For these reasons a check of the correctness of the extrapolations ought to be made. This test was carded out on the compounds YNi5 and YNinCo (x = 1.0 and 0.8). Both compounds have the same structure, similar composition, Debye temperature and high-temperature thermal expansion as YCos, and do not undergo any phase transition down to 4.2 K. The experimental a, c(T) dependences are in very good agreement with those extrapolated from the 1100-1300 K interval. This confirmed the correctness of the extrapolations, which are considered to be valid also in the case of the magnetically ordered RM5 alloys. For all compounds the experimental a(T) curve coincides with the extrapolated one. This means that the basal-plane strain Aa ~" 0. The axial magnetostrictive strain Ae reaches 6.8 x 10 -3 at 5 K in YCo5 and decreases with increasing Ni content (table 4.2). The volume effect Ws = 2)~a + Ac in these compounds is determined by Ac only. Noticeable anomalies have not been observed in YCo2Ni3 (x = 0.6) despite the fact that T~ of this compound (240 K) is rather high as well as Pro. For the uniaxial anisotropy, formula (2.4) can be transformed as follows (for the basal plane and c axis, respectively): Aa(T) = A~'°(T) + 2 A~,2(T),

(4.1)

2 A[,Z(T), )~¢(T) )~'°(T) + -~

(4.2)

=

and it is difficult to separate the exchange (with c~, 0 index) and anisotmpic (with c~, 2 index) contributions. In the case of the above discussed cubic compounds, the observable anisotropic magnetostriction distorts the lattice, in difference with the exchange one. Therefore, it is easy to separate these contributions. For anisotropic crystal structures the situation is more complicated. The anisotropic magnetostriction with a index does not distort the lattice. From the magnetostrictive strains along main axes, only the combinations of the exchange and anisotropic contributions, similar to those described by formulae (4.1-4.2), can be obtained. However, the information on the ratio between the )~ 0 and )~,2 constants can be derived from the temperature dependences of Aa and Ae. In fig. 4.3, the comparison is made between the temperature dependences of Ws and the squared molecular magnetic moment Pro, the Ws(#Zm)dependence at 5 K being presented for different Ni contents. Since in these compounds Ws ~ Ac, the same dependences exist for Ae. One can see that the spontaneous magnetostriction (linear and volume) has the order of magnitude of 10 -3 and quadratically depends on the magnetic moment when either the temperature or the composition changes. In the case of anisotropic magnetostriction this corresponds to the double-ion mechanism in accordance with Callen and Callen (1965). This mechanism can not provide a contribution to the anisotropic magnetostriction larger than 10 -4. The singleion mechanism of the anisotropic magnetostriction which can provide the observed

78

A.V. ANDREEV 1.0 _

'-"--

......

" 0

........

'

x=O

0.2

~)----

'

0.8

0.6

0.4

E

0.2

0

!

I

I

I

0.2

0.4

0.6

0.8

i

i

20

40

TfTe

t

G" ~5 ,rv

60 I.trn2 (I.tB2)

Fig. 4.3. Comparison between the spontaneousvolume magnetostrictionws and the squared molecular magnetic moment tz2 for Y(Col_xNi=)5 solid solutions. Upper graph: Temperaturedependences of Ws (circles) and/tim (lines), both reduced to the values at 0 K. Lower graph: ws vs ~2 at 0 K. (After Andreev et al. 1985a.) 10 -3 linear strains apparently is not applicable to Y(CoI_=Ni=)5 because it should behave differently from the quadratic Ae(#m) dependence (close to cubic at low temperatures). Therefore, Ac ~, )~,0 and the anisotropic contribution A~,z is negligible (< 10 -4) compared to the exchange contribution. On the another hand, the equation A~'°(T) ~,

2

- ~

A~"2(T)

(4.3)

follows from (4.1) for all temperatures taking into account that Aa(T) ~ 0. Due to different temperature dependences of A~'° and A~'2 this can be valid only in the case of A~'°(T) ~ A~'2(T) ~ 0. Therefore, their possible magnitudes cannot reach 10 -4.

THERMAL EXPANSION ANOMALIES

79

Thus, the magnetoelastic effects in the Co subsystem of the RCo5 compounds are of exchange origin, and the anisotropic magnetostriction constants are relatively small. This conclusion is in agreement with the results of Deryagin et al. (1984a) who studied the magnetoelastic properties of the Ndl_=Y=Co5 solid solutions by means of the acoustic method: The single-ion type magnetoelastic constants, which display themselves at the spontaneous spin-reorientation phase transitions, depend linearly on x and extrapolate to zero for YCos. The magnetoelastic-coupling coefficient ncoco = ~s/# 2 in YCO5 is equal to 2.5 x 10-3/~B 2 and does not depend on temperature.

CeCos. CeCo5 is a highly-anisotropic uniaxial ferromagnet with Tc = 737 K and #m = 6.5#a at 4.2 K (Yermolenko 1983). The lower values of Te and /.tm compared to other RCo5 compounds with trivalent R component as well as the absence of a Ce moment are explained by the fact that the effective valence z of the Ce atoms is equal (or close) to 4, leading to additional electron transfer to the Co 3d band (Lemaire and Schweizer 1967, Wohlleben and Rohler 1984). The thermal expansion of CeCo5 was studied by Andreev et al. (1985b) using a single-crystalline sample. In fig. 4.4, the temperature dependences of a, c and V are 4.1.1.2.

V (nm3).

,

0.085 •

,

_

~

-

xx~o:~...~

,

f

..~- 404

.~:~oc~ ..m~ ~

f

a (pro 498

,

~

t~"

0.083 ~

,

402

f 400

/'~ / ~

c-~o°°°c

~"~

,430°``

398

494

/

/

f 490

I

I

I

I

200

400

600

800

1 00 T(K)

Fig. 4.4. Temperature dependences of the lattice parameters a and c and the unit-cell volume V for CeCo 5. The dashed curves represent the phonon contribution to the thermal expansion. (After Andreev et al. 1985b.)

80

A.V. ANDREEV

presented. One can see that the magnetic ordering is accompanied by the thermal expansion anomalies not only along the c axis like in YCo~, but also within the basal plane. The extrapolation of the a, c(T) curves from the paramagnetic to the ferromagnetic range has been done using TD = 320 K obtained by means of acoustic measurements. The spontaneous magnetostrictive strains are positive and very large. The uniaxial strain )~e is more than twice the basal-plane strain ,Xa, and Ws reaches the value of 1.6% at 5 K. Structural and magnetic data of CeCo5 are collected in tables 4.1 and 4.2, respectively. As in the case of YCO5, the linear deformations are proportional to 1,2. For )~c(T) a small deviation from the linear ,~c(#2m) dependence appears at low temperatures as a result of the pronounced Invar effect along the e axis at T < 300 K (fig. 4.4). An attempt has been made (Andreev et al. 1985b) to explain this effect, as well as the anomalous behaviour of the elastic constants, by a slight variation of the Ce valence with increasing temperature (z ,~ 3.9 at 5 K and increasing to z ~ 4). The effect is rather small in comparison with the total ,~e value. As for YCos, from the proportionality between linear strains and/z 2, the conclusion has been reached that there is a negligible anisotropic magnetostriction compared with the exchange magnetostriction. The values of ndd is almost 4 times larger than that in YCo5 (table 4.2) and does not depend on temperature.

4.1.1.3. ThCo5+~. Although the tetravalent and nonmagnetic state of the Ce atoms in CeCo5 maybe questionable, there is no doubt of the tetravalent state of Th in ThCos. The compound denoted as ThCo5 has actually a wide homogeneity range - from ThCos.0 to ThCos.7 (Van der Goot and Buschow 1975). This notation for off-stoichiometric compounds is not correct from crystallographic point of view, since it does not correspond to a single unit cell. It is known that the variation of composition within the homogeneity range occurs by replacement of 1 Th atom by a Co pair ('dumbbell'). Therefore, an average unit cell can be described as Thl_~Cos+2x. In this notation, the homogeneity range is T h l . 0 C o 5 . 0 - T h o . 9 1 C o 5 . 1 8 . The same remark can be made about other RCo5 with deviations from the exact 1 : 5 ratio composition (TbCos.1, DyCos.2, HoCo5.5, ErCos.9). Both notations are accepted in the literature, and we will use the same as in the original papers. The unique feature of ThCo5 is the band metamagnetism of the Co atoms on the 3g sites. For the 2c-Co sublattice Stoner's criterion of the onset of ferromagnetism is satisfied in zero field for all compositions within the homogeneity range. In the case of the 3g-Co sublattice it is satisfied only for the Co-rich part, where the effective field from the Co atoms on 2c sites and in the 'dumbbells' is large enough to transform the 3g-Co sublattice from the paramagnetic to the ferromagnetic state (Givord et al. 1977, 1979, 1983). On the Th-rich end of the homogeneity range, where the 3g-Co sublattice is paramagnetic, the metamagnetic transition occurs in external fields, the transition field being very sensitive to the composition (Snegirev 1984). Buschow et al. (1975b) have measured the thermal expansion above room temperature on polycrystalline samples and found expansion anomalies in the basal plane and along the c axis which were attributed to the spontaneous magnetostriction.

THERMAL EXPANSION ANOMALIES

81

The room-temperature lattice parameters, as well as Tc = 430 K, correspond to the composition near the exact 1 : 5 stoichiometry. Thermal expansion studies on single crystals of Th0.965Co5.07 and Tho.950Co5.10 have been made by Andreev et al. (1984a) in the temperature interval 5--800 K. In the first compound the 3d-Co sublattice is paramagnetic at low temperatures, in the second one it is ferromagnetic. The structural and magnetic properties of both samples are presented in tables 4.1 and 4.2. One can see that so small a difference in the composition leads to a very large difference in Tc and #m. In Tho.965Co5.07 , according to the neutron-diffraction study (Givord et al. 1979), the magnetic moment of the ferromagnetic Co on the 2c sites is 1.21/JB and that of paramagnetic Co on the 3g sites is 0.96/~B. Such a large magnetic moment of paramagnetic Co is due to the magnetizing influence of the 2c sublattice. Above about 200 K the 3g-Co sublattice also becomes ferromagnetic. It leads to a nonmonotonous temperature dependence of/~m in Tho.965Co5.07. Figure 4.5 shows temperature dependences of a, c and V. The extrapolation of the paramagnetic a, e, V curves into the ferromagnetic range (using To = 300 K) has been made by Andreev et al. (1984a) on the basis of the data obtained within the range 700-800 K. Later (Zadvorkin 1987, Andreev 1990), the lattice parameters have been remeasured up to 1000 K and the extrapolation has been based on the data obtained in the range 800-1000 K. For Th0.965Co5.07 with Tc = 540 K there is no difference in the obtained magnetostrictive strains. In the case of Th0.950Co5.10 with T¢ = 650 K considerably larger strains are obtained. These results show that the 700-800 K interval is not sufficiently extended above the ordering temperature to serve as a basis for a proper extrapolation due to the presence of short-range order effect. The data presented in tables 4.1 and 4.2 for the paramagnetic thermalexpansion coefficients and magnetostrictive strains are taken from results published by Zadvorkin (1987) and Andreev (1990) as being the more reliable ones. The behaviour of Th0.950Co5.10 is similar to that of CeCos, including the Invar effect along the c axis at low temperatures. The value of he exceeds that of ha, and Ws is much larger than in YCo5 (table 4.2). Both linear strains follow the square of the magnetic moment, which shows the domination of the exchange magnetostriction in total magnetostriction, as found in YCo5 and CeCos. The temperature dependences of the lattice parameters in Tho.965Co5.07 are more complicated. Below about 200 K both linear magnetostrictive strains sharply decrease with decreasing temperature to nearly zero at 5 K. Of course, it leads to the disappearance of Ws despite of the fact that in the 200--400 K range Ws of this compound is close to Ws of Th0.950Co5.10. The nonmonotonical behaviour of Wsin Th0.965Co5.07 correlates with the anomalous /.tm(T) dependences and reflects the change of magnetic state of the 3g-Co sublattice. The expression (2.5) is adapted to the two-sublattice case as follows: 2 (4.4) 0:s = 4 nee#2 + 6 neg/.Ze~g + 9 ngg/./g, where nee and ngg are the magnetoelastic-coupling coefficients in the 2c and 3d-Co sublattices, n~g is the intersublattice coefficient, and/~e and #g are the corresponding magnetic moments. Andreev et al. (1984a) have found that the observed difference in spontaneous-magnetostriction behaviour can be described with a negative intersublattice coefficient neg. However, the quantitative data obtained are questionable because they are based on the assumption that nee = ngg.

82

A.V. ANDREEV

V (nm3)

,

,

,

•i c (pro)

0087

4O3

a(pm)502 .z.==z'_--_ • "-: -

-"---- /

/~

~0 O 'J

// //

oo

~OO -

401

•or

399

07 A•."

O•?/~" 0..0//

.00./

/

I 2//

496

-

/ "

OOOO 0000

498

,.od:~ o ° °

ooO/

500"

/

/"~

ooO°'/"

~_~0000000//

OTh0.965Co5.07

10~ ~

• Tho.95oCosA 0

L

0

/

/

/

/

/

I

200

400

600

T (K)

Fig. 4.5. Temperature dependences of the lattice parameters ~ and c and the unit-cell volume V for Th0.965Co5.07 and Th0.950Co5.10. The dashed curves represent the phonon contribution to the thermal expansion. (After Zadvorkin 1987.)

4.1.1.4. UCo5.3. The Co-richest compound in the U-Co phase diagram (Elliott 1965) is denoted as U2COll. Deryagin and Andreev (1976) obtained single-phase samples for z = 0.2-0.4 in the UCos+x notation. At z > 0.4, including U2Co11, the alloys contain free Co, and below z = 0.2 the UCo4 phase appears. Dommann et al. (1990) reported the attainment of a single phase for the UCo5 composition and observed free Co in UCo5.3. Magnetic properties of the compound were studied by Deryagin and Andreev (1976). They concluded that UCo5.3 belongs to the series of RCo5 compounds with a nonmagnetic R component. The U does not carry a magnetic moment and transfers its larger (compared to Y, Ce and Th) number (presumably 6)

THERMAL EXPANSION ANOMALIES

83

of electrons into the Co 3d band. This leads to an additional decrease of #Co and Tc (table 4.2). The main structural and magnetic characteristics for UCo5.3 as well as the nonmagnetic state of U have been confirmed in a series of papers (Yermolenko et al. 1977, Dommann et al. 1990, Hillebrecht et al. 1985, Br/indle et al. 1990). This is a highly anisotropic uniaxial Co ferromagnet with relatively low values of #co and Tc. Strictly speaking, UCos.a is not isostructural with RCos. Deryagin et al. (1978) and Andreev (1979) have studied the U solid solutions in RCo5 (R = Y, Nd, Sm and Tb) and found in all cases a solubility limit at x - 0.6. All known compounds with the CaCu5 structure type have c/a ratios within the range 0.73-0.83. The c parameter in RCo5 is determined mainly by the Co lattice and only slightly depends on the character of R. The lanthanide contraction influences the a parameter only, so the c/a ratio in the RCo5 increases towards the heavier R elements and reaches a critical value of about 0.83 for Tm. The stability of the CaCu5 type phases is known to decrease with increasing the R atomic number. Beginning from Tb, the structure is stabilized by partial R-'dumbbell' substitution (see 4.1.1.3) and the possibility to form this structure ends at TmCo6 (Buschow 1972). Since the atomic radius of U is smaller than that of rare earths, a rapid rise of c/a is observed in the RI-rUxCo5 quasibinaries, and the solubility limit at x = 0.6 just corresponds to c/a = 0.83-0.84. The value of c/a extrapolated to x = 1 is equal to 0.865, being far outside of the range of known data for the CaCu5 structure. If we reduce the unit cell of UCo5.3 to the CaCu5 type, the effective c/a ratio is equal to 0.845 and exceeds the indicated limit as well. Andreev (1979) has argued that the structure of UCos.a can be derived from the CaCu5 type by partial U-'dumbbell' substitution and ordering of the associated distortion along the c axis. The results of a neutron-diffraction study (Dommann et al. 1990) are interpreted in terms of a rhombohedral stacking of the ABC type of 3UCo5 blocks without the U-2Co substitution. In any case, UCo5.3 can be considered to have a rhombohedrally distorted lattice closely related to the CaCu5 type and can therefore be meaningfully classified as belonging to the RCo5 family. Its structural and magnetic properties are listed in tables 4.1 and 4.2. A thermal expansion study on a UCo5.3 single crystal has been made by Andreev (1991) in the temperature interval 5-800 K. Below 400 K the experimental a, c(T) curves deviate from the extrapolated ones (TD = 300 K). Figure 4.6 presents the temperature dependences of the linear and volume spontaneous magnetostrictions. The anisotropy of linear strains is opposite to that in YCOs, Ac is nearly negligible and Aa exceeds it by a factor of 5. The temperature dependences )l a and Ws (Ac is too small for an analysis) closely follow that of the square of the magnetic moment. In the case of the linear strain this shows, like in RCo5 with R = Y, Ce, Th, that the contribution of the anisotropic magnetostriction is negligible compared to the exchange contribution. The value of ~;s at 5 K, 5.4 x 10 -3, is considerably lower than in CeCo5 and Thco5 but close to that in YCos. The volume effect in YCO5 is associated with an almost 4 times larger magnetic moment. One can find some analogy between UCOS. 3 and the Y(Cox_xNir)5 system. The compounds with z ~ 0.55-0.60 have similar Tc and/~m, but no spontaneous magnetostriction is observed in the z = 0.6

84

A.V. ANDREEV

6

I 0

I

I

I

'

'

'

I

o 0 0 0

0 0

cO

I

T

0

0

0

0

I

0.)$ o 0

c o

0

-r-t

o

f0 E o

0 0

~a

O

000~00

2

O0

o O

0

%-

0

cu a

0 0

oooo

o

o

T oo,oo

O

oo

? oo,,,~.

0 0

Fig.

4.6.

200 T (K)

400

Temperature dependences of the linear (Aa and Ae) and volume (ws) spontaneous magnetostrictions for UCos. 3. (After Andreev 1991.)

compound. Figure 4.7 shows the temperature dependence of the magnetoelasticcoupling coefficient ncoco = w~/#~o for all RCo5 with nonmagnetic R studied in the manner discussed above. For all compounds ncoco is temperature-independent, at least up to 0.8To. (Due to some short-range order, the spontaneous magnetostriction does not disappear with the spontaneous magnetization at T~, and the determination of ncoco becomes inaccurate when approaching T~. For the UCo5.3, Ws at T~ is equal to 15% of its value at 5 K, and an excess of about 40 K above T~ is required for the volume effect to disappear.) One can see that ncoco sharply increases with increasing effective valence of R, reaching very large value (27 x 10-3#B 2 at 5 K) in the case of U. This conclusion is not changed if we take #Co = 0.54#B for UCo5.3 (Yermolenko et al. 1977). The latter value seems to be overestimated, presumabl2y due to free Co in the sample. Even in this case one finds n¢oco = 19 x 10-3#B , which also is much higher than in YCos. 4.1.1.5. Conclusion. In all studied RCo5 compounds with nonmagnetic R the volume magnetostriction ws(T) depends linearly on/a2(T), at least up to T = 0.8To. In other words, the magnetoelastic-coupling coefficient ncoco does not depend on temperature. The linear magnetostrictions within the basal plane Aa(T) and along the e axis A~(T) are proportional to # 2 ( T ) as well, pointing to a negligible contribution to the total linear effects of the anisotropic magnetostriction in comparison with the exchange magnetostriction. The anisotropic magnetostriction constants A~'2 and A2'2 are lower than 10 -4 .

THERMAL EXPANSION ANOMALIES 40

CU I G3 ZL CO I 0

,

,

,

t

30 0

0

0

000

"o

a

o

o

a

>0

0

0

0

0

I

'

000 0

85

'

'

I

O0 o o R = U

20 C3 C~ I

Th

Ce

7o

o

o

~

o

o

°

0

0

0

~

0

0

o

o

o

°

o

°

o

o

0

0

0

0

0

0

0

0

c" 0

-

¥

,

0 0

~

I

,

200 T

,

,

I

400 (K)

Fig. 4.7. Temperaturedependences of the magnetoelastic-couplingcoefficient nCoCo for RCo5 with nonmagnetic R. (After Andreev 1991.) The distribution of the volume effect over the main crystallographic axes varies from the anisotropic situation Aa << Ac for R = Y through a nearly isotropic situation Aa ~ Ae for R - Ce and Th again to an anisotropic situation for R - U, but with Aa >> 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 ¢<

I

I

I

I

I

I

x=O

qO0

200

300

T,K

Fig. 4.19. Temperature dependences of the "/-magnetostriction A"t for R2Co 7. Top: hexagonal modification of Tb2Co 7. Middle: Nd2Co 7. Bottom: rhombohedral modification of the (Tbl_xYx)2Co 7 solid solutions. Circles show experimental data. The dashed curves show the single-ion-model description with only second-order constant A"v,2. The solid curves represent the single-ion-model description with second- and fourth-order constants ,,V3',2, ,~,4 and Ag'4. (After Andreev et al. 1988a.)

Equation (4.9a), where/9 is the angle between the easy direction and the e axis, corresponds to an easy [100] axis, and eq. (4.9b) to an easy [210] axis. In the case of Nd2Co7 and hexagonal Tb2CoT, where a distortion exists only in the basal-plane anisotropy range, formulae (4.9) take the form ,X'~ =

, V '2 -

~1

A~,4- 2~,4,

(4.10a)

A'Y =

A~, 2 _

1 )l~,4 + 2A7,4.

(4.10b)

According to the single-ion model, A~'4 and A~'4 have different signs for the Nd 3+ and Tb 3+ ions and decrease with increasing temperature much faster than A%2, in agreement with the experimental data of ~7(T) presented in fig. 4.19. The treatment

110

A.V. ANDREEV

of these curves in the framework of the single-ion model made it possible to obtain values for "Vr'2, "'1 x'Y'4, ,~,4 and to estimate the Co contribution to be lower than 2 x 10 -4. The latter value is more than one order of magnitude lower than observed )(Y value. The values obtained from the single-ion model are ,Vy'2 = 0.9 x 10 -3, ,~,4 = 16.7x 10 -3, ,~,4 = 0.5x 10 -3 at 0 K in Nd2CoT, and 1.7x 10 -3, - 1 6 . 7 x 10 -3, - 0 . 5 x 10 -3 in hexagonal Tb2Co7. For the rhombohedral Tb2Co7 the values were found to be larger by 30%. These sets of constants satisfactorily reproduce the experimental values and provide a good description of temperature dependences of ,VY. Using these data, the second- and fourth-order 7-magnetostriction constants have been calculated for all R2Co7 on the basis of single-ion model (Andreev et al. 1988a). Two compounds, Sm2Co7 and Tm2Co7, have been studied later (Bartashevich et al. 1993a, Andreev et al. 1992b). If one substitutes the calculated values (,VY,2 = - 1 . 2 x 10 -3, A~,4 =_. - 4 . 5 x 10 -3, ,,~,4 = - 0 . 1 x 10 -3) and 0 = 150 in eq. (4.9), one finds that A"r is equal to - 0 . 3 x 10 - ~ with a negligible difference between the results obtained with eqs (4.9a) and (4.9b). In the case of Tm2Co 7 (using values calculated for the rhombohedral modification A"r'2 = - 1 . 7 x 10 -3, A~'4 = - 2 3 . 4 x 10 -3, /~,4 = - 0 . 6 x 10 .3 and 0 = 45°), one finds ,VY = - 4 . 7 x 10 .3 according to (4.9a) and - 5 . 3 x 10 -3 (4.9b). Since A"r < 0 and the experiment has shown that e > 0, the easy direction within the basal plane is [210], and the result obtained with eq. (4.9b) should be chosen. In Smzco7 one has )(Y = - 0 . 3 x 10 -3 which should already be seen by means of Xray diffraction measurements as a broadening of the (hhO) or (h00) lines. However, this value is very low, and even a small possible positive Co contribution of about 1 x 10 . 4 will reduce it to the unobservable limit, in agreement with experiment. In Tm2CO7 the observed e = - 3 . 5 x 10 -3 value is by about 50% lower than the predicted A"r value. However, the proposed set of the 7-magnetostriction constants gives only - 3 . 8 x 10 -3 for 0 = 600 (upper limit of the 0 estimate) and this is already close to the observed e. It is interesting that the proposed set gives nonmonotonic A'Y(0) dependences. According to eq. (4.%), A"r changes its sign from negative to positive at 0 = 70 o and becomes equal to +2.8 x 10 .3 for basal-plane anisotropy (0 = 90°). Using eq. (4.9b), one finds that ,Vr ~ 0 at 0 = 830 and also becomes positive for 0 = 90 °. Hence, either a spin reorientation within the basal plane from [210] to [100] or a sign change of the distortion coefficient can be predicted when the cone angle 0 increases to 900 (easy basal plane). In both cases an unusual disappearance of the large distortion with increasing 0 should be seen. The cone of easy directions in TmzCo7 can be opened by the Y substitution (if the larger Y atoms would occupy the R1 sites) or by Ni substitutions, decreasing the uniaxial contribution to the magnetic anisotropy. Finally, the 7-magnetostfiction constants of Sm2Co7 and Tm2Co7 obtained on the basis of the single-ion model and experimental data for R = Nd and Tb are more or less in satisfactory agreement with experimental results.

4.2.3. Anisotropic ~-magnetostriction The spin reorientations in Nd2C07 and Yb2Co 7 influence the thermal expansions in the same manner as in RCos. The orthorhombic distortions, as reflected by

THERMAL EXPANSIONANOMALIES

111

the 7-magnetostriction, disappear in hexagonal Nd2Co 7 and Tb2Co 7 with increasing temperature below the spin reorientation. They disappear in rhombohedral Tb2Co7 for the temperatures below the strong changes of 0, where noticeable anomalies are observed in the a, c(T) dependences. Therefore, the determination of anisotropic a-magnetostriction constants can be made in the same way as in the RCo5 compounds where no orthorhombic distortions are found (see section 4.1.2.8). Like in RCos, the additional strains Aa/a and Ac/c are proportional to cos 2 ~. It means that anisotropic t~-ma~netostriction can be described exlnsively by the secondorder constants hl'zand h~ '~. This is the reason why the anomalies Aa/a = - 0 . 4 x 10 -3 and Ac/c = 0.Sx 10 -~ (in Nd2Co 7 at 250 K and in Tb2Co 7 at 420 K), originating from the anisotropic a-magnetostriction, can be seen at much higher temperatures than the orthorhombic distortion originating from the other types of the anisotropic magnetostriction and involving fourth-order terms. Under the same assumptions made for RCo5 (see section 4.1.2.81, Andreev et al. (19883) have extrapolated the values obtained for h~ '2 and h~ 'z in Nd2Co7 and Th2C07 to zero temperature and have calculated such values for all R2Co7 with a magnetic R component. The temperature dependences of/.t R necessary for the extrapolation have been obtained as differences between pm(T) of R2Co7 and Y2Co7. The calculated values of h~'2(0) and h~'2(0) in R2Co7 have the same sign as in RCos, and are by about 80% larger in magnitude. The increase of anisotropic magnetostriction with increasing R content shows again that it originates from the R sublattice. Since the total R content in R2Co7 is higher than in RCo5 by only 33%, one may conclude that the R2 atoms on the quasicubic sites give a relatively larger contribution to h~ '2 and h~ '2 than the R1 atoms in hexagonal sublattice. However, it should be noted that in addition to the assumptions under which the anisotropic c~-magnetostriction constants have been calculated in RCo5 (see section 4.1.2.8), the use of total values of #R(T) implies the assumption that the temperature dependences of ]/R for both R1 and R2 are equal. Therefore, the values of h~'2(0) and h~'2(0) for R2Co7 published by Andreev et al. (1988a) are much more questionable than those in table 4.5 for RCo5 in which there is only one R sublattice.

4.2.4. Exchange magnetostriction The extrapolated (to 0 K) values of A7'2 and "'2x~'2 together with the experimental values of Aa and he allow the determination of the -lxa'° and h~ '° exchangemagnetostriction constants in Nd2Co7 and Tb2CoT. In order to apply formulae (4.1), (4.2), (4.5) and (4.6) the basal-plane strain h a of the undistorted lattice should be taken. Such strains are presented in table 4.7. In this case,

ha --~ { ~1 [a + (a 2 --[-b2) 1/2]

-

ao)/a0

(4.11)

which means isotropic expansion (or contraction) in the basal plane and corresponds to a change in the average R - R distance. In NdzCo7 one finds A~'°(0) = 0.6 x 10 -3 and A~'°(0) = 3.8 x 10 -3. In Tb2Co7 (independent of structure modification) the corresponding values are 1.6 x 10 -3 and 4.6 x 10 -3. These values of A~'°(0) and A~'°(0)

112

A.V. ANDREEV

can be compared with the data listed for NdCo5 and TbCo5.1 in table 4.5, after taking into account that they have the same deficiencies as those mentioned above for the anisotropic magnetostriction constants, because they were calculated by using A~'2(0) and A~'2(0). The main difference is that uniaxial exchange-magnetostriction constant A~'°(0) is lower than (Tb) or comparable to (Nd) the anisotropic a-magnetostriction constant A~'2(0) in R2Co7, being much larger in RCos. Analyzing the volume effect in R2Co7 in the same manner as in RCos, Andreev et al. (1988a) again have obtained negative nRco values for R2Co7 compounds with light R component (Nd, - 0 . 2 x 10-3I~B2) and positive values for heavy R component (0.4 x 10-3/.tB 2 for Gd and 0.3 x 10-3#B 2 for Tb). The data of Sm2C07 and Tm2C07 have not been analyzed in this way due to low #sin value and due to additional anomalies in thermal expansion at the spin rcorientation, which will be discussed now.

Let us consider again the spin reorientation in Sm2Co7 (fig. 4.18). At the lowest temperatures, it follows from formula (2.4) that )~a "~a

c~0 a,2 ,~,0 "~1' "~ 0.67 Al , Ae = + 0 . 6 7 .~,2,

=)~1~ 0 +0.61A1~,2 , ,~e =A2~,0 +0.61A~ '2,

(4.12a) (4.12b)

where eq. (4.12a) corresponds to the extrapolation from the uniaxial-anisotropy range, and eq. (4.12b) to the strains actually observed in the cone range. If we assume the exchange magnetostriction constants ,~,0 and ~,0 "'2 to be the same for the uniaxial and cone cases, the values of )~,2 and "'2~'2obtained from eq. (4.12) will be equal to 6 x 10 -3 and 3 x 10 -3, respectively. Using the single-ion model in conjunction with the data obtained on Nd2Co7 and Tb2CoT, these constants in Sm2Co7 are calculated as 2.3 x 10 -3 and -4.6 x 10 -3, respectively (Andreev et al. 1988a). The large disagreement points to an incorrectness of the above assumption. Similar results are obtained for Tm2Co7. Thus, the spin reorientation in Sm2Co7 and Tm2Co7 influences the isotropic part of the spontaneous magnetastriction, which is reflected in the volume effect. This is explained by the anisotro]~y of the magnetic moment ~ 0 and "~2' a u anisotropy constants at the values and, consequently, by changes of the )~1' spin reorientation. In fig. 4.18 the dash line represents the temperature dependence of cos for Y2Co7. The difference between the values of COsfor Sm2Co7 and YzCo7 can be attributed to the Sin-Co exchange-interaction contribution, which is about 25% of the total volume effect. (Such a comparison is not valid at high temperature due to a noticeable difference in Te.) Since #Sm is very small compared with other R, the term nRco#Sm#Co in (2.6) becomes sufficiently large only when the magnetoelastic-coupling coefficient nsmco is large. It should be noted that nsmco is certainly positive (from fig. 4.18), in contradiction with the trend observed (Andreev et al. 1988a, Andreev and Zadvorkin 1991) that for the light R nRco is negative. The upturn in COs(T)below 200 K apparently corresponds to a considerable relative increase of #Sm at lower temperatures. A similar peculiarity in the COs(T)behaviour is observed in Tm2Cov and RCo3 with R = Ho, Er, Tm (Andreev and Zadvorkin 1991).

THERMALEXPANSIONANOMALIES

113

If the values of magnetic moments depend on the orientation, the change of ws can be represented, neglecting the weakest R-R exchange interaction, as follows: AWs -- (2ndd/gd+ nRd/ZR)A/gd+ nRd/ZdAlgR.

(4.13)

In R-Fe compounds with high Fe content (Tmzee17 and Nd2FeI4B, Andreev et al. 1985d, 1985f, 1991a) nRd is negligible and Aws is determined by the first term in (4.13). In R-Co compounds both interactions give considerable contributions to ws. When the R and Co contributions to the magnetocrystalline anisotropy have opposite signs, the corresponding anisotropies of the magnetization, Apd and A/~R, have also opposite signs (Givord et al. 1983), as has been shown experimentally by a neutron-diffraction study of TbCos. 1 at the spin reorientation (Pirogov et al. 1982). Therefore, the first and second terms in (4.13) cancel and Aws is not observed in TbCos.1 (Andreev et al. 1983). Similar cancellations apparently occur in Tb2Co7, OyCo5.3 and HoC05.5. The volume anomaly observed in Sm2Co7 and Tm2Co7 points to the same sign of A#d and A/.ta at the spin reorientation. This is in agreement with positive values of the second-order Stevens factor ctj for Sm3+ and Tm 3+ ions and negative values for Tb 3÷, Dy 3+ and Ho 3÷.

4.2.5. Influence of hydrogen absorption on thermal expansion of R2Co7 Rare-earth-3d-metal intermetallic compounds can absorb large amounts of hydrogen. Interstitial introduction of the H atoms changes the interatomic distances due to a large lattice expansion. It also changes the electron concentration. This may lead to changes in crystal structure, even to amorphization, and to drastic changes in magnetic properties (Wallace 1978, Buschow 1984, Wiesinger and Hilscher 1991, Yamagnchi and Akiba 1993). The R2Co7 compounds are found to be apt subjects for investigation of the influence of hydrogen absorption on magnetism (Bartashevich et al. 1983, Andreev et al. 1985g, Yamaguchi et al. 1985, 1987a, 1987b). Several hydrides have been obtained as single crystals large enough to measure magnetic properties and thermal expansion along the principal axes (Bartashevich 1987). The pressure (p) - temperature (T) - concentration (z) phase diagram of YzCo7H~ features three hydride phases. The a-phase (x x< 0.2) is a disordered hydrogen solid solution in Y2Co7 with slightly larger lattice parameters. The /3-phase has two modifications/3' (1.5 ~< • x< 2.2) and/3" (2.2 ~< x <~ 3.0). The 7-phase corresponds to 6.0 ~< x x< 6.7. Hydrogenation leads to an anisotropic lattice expansion with a volume increase up to 20% (in NdzCo7H8.5, Andreev et al. 1988b). In the/3-phase the lattice symmetry is the same as in the parent compound. The c parameter increases at nearly constant a, whereas in the 7-phase only the a parameter increases and lattice undergoes a large orthorhombic distortion. Such behaviour is satisfactorily explained by the preferential hydrogen occupation of interstitial sites in the quasicubic blocks. The occurrence of the orthorhombic distortion reflects partial occupation of the hexagonal blocks. Owing to the larger atomic radius of Nd and Gd compared to Y, the maximum H content is higher (z = 8.5) in NdzCovH~ and GdzCo7H~v. This corresponds to the "y"-phase. In this phase the orthorhombic distortion is absent because of the complete filling of the H positions in hexagonal blocks.

114

A.V. ANDREEV

With increasing H content both the ordering temperature and the average Co moment decrease in Y2CoTH=. The r - and 7-phases are antiferromagnets with fieldinduced ferromagnetism, through a noncollinear structure in fi-phase and metamagnetic transition in the 7-phase (Bartashevich et al. 1983, Bartashevich and Deryagin 1984, Yamaguchi at al. 1985). Complicated magnetic behaviour with independent ordering of R and Co sublattices and a variety of field-induced transitions were observed in the case of magnetic R components (Bartashevich 1987, Apostolov et al. 1988, Andreev et al. 1985g, 1988b). Onset of antiferromagnetism occurs upon hydrogenation in paramagnetic ThzCo7 (Andreev et al. 1984b). The thermal expansion of the R2CoTH= hydrides was studied only up to 300--420 K because of their decomposition with increasing temperature. Hydrogenation very strongly influences the thermal expansion, up to a sign reversal along the c axis in some temperature range. In the a, c ( T ) curves, anomalies appear with no relation to magnetic phase transitions. They apparently reflect a partial ordering of the H atoms on the different interstitial sites. In fig. 4.20 the thermal expansion of the r-and 7phase Nd2Co7H= hydrides are shown (Andreev et al. 1988b). The 7-phase hydride (x = 6.4) features large orthorhombic distortion - 1 . 0 x 10 -2 at 5 K which magnitude decreases to - 0 . 5 x 10 -2 at room temperature. It has a sign opposite to that of magnetostrictive distortion in the parent compound and is too high to be attributed to 7-magnetostriction. Its nonmagnetic origin is especially clear for Y2Co7H6 where 0.536 ]-

-I ~ ~

0.538

0.534 0.528" o.52z t~

E

0.526 0.525

0.524 0.5?.3

0

I00

200

300

qO0 T,tf

Fig. 4.20. Temperaturedependencesof the lattice parametersa, b (in the case of orthorhombicdistortion) and c for the Nd2Co7Hz hydrides, x = 0: 1 - a, 2 - c. x = 3.1: 3 - a, 4 - c. x = 6.4: 5 - a, 6 - b/V~, 7 - c. x = 8.5:8 - a, 9 - c. (After Bartashevich1987.)

THERMAL EXPANSION ANOMALIES

115

7-magnetostriction cannot be seen in principle due to the prevailing uniaxial magnetic anisotropy, but c ~, 1 x 10 -2. In the/~-phase Nd2CoTH3.1, in addition to the anomalies at the spin reorientation (340-370 K), there is sharp decrease of c with increasing temperature above 240 K without any connection with magnetic properties. From the anomalies at the spin reorientation, the values )~'2(355 K) = - 4 x 10 -4 and A~'2(355 K) = 6.5 x 10 -4 can be determined, pertaining to the midpoint of the spin-reorientation interval. However, it is very difficult to make proper extrapolations from the uniaxial or basal-plane range to the cone range, especially for the c direction, meaning that the presented values have to be considered as estimates. The data are sufficienly accurate, though, to claim that the hydride has an anisotropic a-magnetostriction as large as that of the parent compound. The curves in fig. 4.20 show how important it is in studies of the spontaneous magnetostriction to take into account the probability of other, nonmagnetic, anomalies in the thermal expansion. On the other hand, a very interesting interaction was observed in the Sm(Fe-Co)2 Laves phases between distortions of these completely different origins, the spontaneous magnetostriction and the hydrogen ordering (Gaviko et al. 1990, Korolyov et al. 1990b). Many intermetallic compounds of Th and 3d metals, including ThzCo7, absorb large amounts of H at relatively low pressure and room temperature, forming stable hydrides (Buschow et al. 1975b). The compound Th2Co7, unlike the related ThCo5 compound (which exhibits the coexistence of ferromagnetism and band metamagnetism, see section 4.1.1.3), is an enhanced paramagnet (Buschow 1971) without any field-induced transition up to 100 T (Bartashevich and Goto 1991). The/3-phase hydride Th2Co7Hs, which has the same a parameter as the parent compound but the 18% larger c parameter, is found to be antiferromagnet with TN = 90 K. In applied fields a metamagnetic transition occurs along the c axis at 2 T to a state with a low moment value, #m = 1.4#B/f.u. (0.2#B/Co), which is considered to be a ferromagnetic state (Andreev et al. 1984b). In fig. 4.21 the temperature dependences of the lattice parameters and the unit cell volume of Th2Co7 and its hydride are presented. The curves are matched at the values at 300 K for better comparison because of the large differences in c and V between the alloy and hydride. While Th2Co7 exhibits an anisotropic thermal expansion with aa much larger than ac (2.6 x 10 -5 K -1 against 0.3 x 10-5 K-I), the behaviour in both directions becomes the same in Th2CoTH5 (1.0 x 10 -5 K - l ) . For Th2Co7 the Debye temperature is found to be 320 K from acoustic measurements, and the a, c, V(T) curves can be fitted well using this TD value. This fact was used by Andreev et al. (1988a) to check the correctness of the a, c, V(T) extrapolation from the paramagnetic range into the ordered range of R2Co7, because of the absence of a phase transition in Th2CoT. For Th2CoTH5 these dependences can be described with TD = 600 K (of course, above TN). Some increase of TD would generally be understandable. However, the possibility of gradual ordering of H atoms over the interstices cannot be excluded. In this case, TD cannot be determined correctly from the thermal expansion. Magnetostrictive strains appear below TN, both negative in the basal plane and along the e axis. At 5 K, Aa = Ae = - 7 x 10 -4, and ~s = -2.0 x 10 -3. This

116

A~V. ANDREEV

1. 000 I--

_

_. o.ggq ~

.

~O.g#6 1,000

0.g98 ~-O.9gZ ~. 0.988

I

I

O

100

200

MO

r,K Fig. 4.21. Temperature dependences of the lattice parameters a and c and the unit-cell volume V for Th2Co7 (1) and its hydride Th2Co7H5 (2), matched at the values at 300 K. The dashed curves represent the phonon contribution to the thermal expansion. (After Andreev et al. 1984b.) value is only modest in comparison with other R2Co7 . However, it is associated with a low Co moment. Using the average value Yco = 0.2#B, the ncoco coefficient is calculated to be extremely large, - 5 0 x 10-3~B 2, which is more than twice as large as that of UCo5.3, regarded as a m a x i m u m untill now. Even if #co is different for the Co atoms on the different sites, and even if some of them have magnetic moments larger than 0.2#B, the magnetoelastic C o - C o interaction in Th2Co7H5 can be described as being anomalously strong.

4.3. RCo3 In the RCo3 compounds one finds a stacking of the hexagonal structure blocks RCo5 and the cubic blocks RCo2 along the c axis according to the scheme: RCo5 + 2RCo2 = 3RCo3. The corresponding structural element, 3RCo3, has the dimensions a ~ 0.5 rim, c ~ 0.8 rim. Double-layer stacking of these elements (hexagonal modification) does not occur. All RCo3 compounds have the PuNia-type rhombohedral structure (space group R3m) with c "~ 2.4 nm, corresponding to the mentioned threelayer stacking of the 3RCo3 elements. The structure is shown in fig. 4.22. As in the case of R2Co7 compounds it has two nonequivalent sites for the R atoms. However, the ratio between number of R1 atoms (hexagonal sites) and R 2 atoms (quasicubic sites) is 1 : 2 instead of I : 1 as in RzCo7. The Co atoms occupy three nonequivalent positions. As in the case of RzCo7 a common Co sublattice is usually considered in RCo3 as a good approximation. From this structural viewpoint, one would expect that magnetic and magnetoelastic properties of RCo 3 are in close relation with those

THERMAL EXPANSIONANOMALIES

RC% ~

~

2RCOz~

) C~ crRCo5 .......~J

117

' ~ ~

co\ aco l RCoz

i Rco

Fig. 4.22. Unit cell of the rhombohedralcrystal structure of RCo3 (the PuNi3 type) in comparisonwith the hexagonal RCo5 and cubic RCo2 lattices. of R2C07, but with increasing 'quasicubic' character. In principle, this is actually observed when leaving secondary peculiarities out of consideration.

4.3.1. Data of particular compounds 4.3.1.1. YCo3. From the magnetic viewpoint, YCo3 is on the boundary between the Y-Co compounds with stable Co moment (Y2Co7, YCo5 and Y2Co17) and the band metamagnet YCo 2 (as well as the related LuCo2) , in which Co becomes magnetic only in an external field of about 70 T (Goto et al. 1989, 1990). The compound YCo3 has spontaneous magnetic moment, which is much lower than in compounds with higher Co content. The values of #Co reported in the literature vary from 0.5 to 0.6#n (Lemaire 1966a, b, Burzo and Seitabla 1981, Shcherbakova et al. 1986, Goto et al. 1992). The value 0.6#B is obtained on single crystal along the easy direction (Shcherbakova et al. 1986) and on polycrystals by means of extrapolation of the M(H) dependence on zero field from a range above the anisotropy field (Goto et al. 1992). The latter value will be used below for further considerations. It should be noted that YCo3 undergoes two consecutive metamagnetic transitions at 60 T and 82 T, the final saturation corresponding to 1.23#B/Co, which is already close to the spontaneous moment in Y2Co7 (Goto et al. 1992). The compound exhibits uniaxial magnetic anisotropy in the entire ordered state. Structural and magnetic properties of YCo 3 are listed in tables 4.8 and 4.9. The thermal expansion has been studied on a single crystal at 5-800 K by Andreev and Zadvorkin (1990). No magnetostrictive strain has been observed within the basal plane, like in YCo5 but unlike in Y2Co7. Consequently, the volume effect ~s is equal to the uniaxial strain Ae and reaches only (0.5-4-0.2) x 10 -3 at 5 K. This reflects the

118

A.V. ANDREEV

TABLE 4.8. Structural data for RCo 3 compounds, a (300 K), a (5 K), c (300 K) and e (5 K) are the parameters of the hexagonal unit cell of rhombohedral lattice; at 5 K, in the case of orthorhombic distortion, the third lattice parameter b is also presented; C~a, c~c are the linear thermal expansion coefficients in the paramagnetic range (800 K) in the basal plane and along the e axis, respectively; c~v is the volume thermal expansion coefficient at 800 K. Compound a ( 3 0 0 K ) ¢(300K) a (5K) pm pm pm

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

aa

av 10 -5 K -a

Ref.

10 - 5 K -1 10 -5 K -a

YCo 3 TbCo 3 DyCo3 HoCo 3 ErCo 3 TmCo 3

869.1 864.3 856.2

1.75 1.81 1.69 1.65 1.67 1.66

4.8 4.6 5.0 4.4 4.2 4.6

[1, 2] [1,2] [1,2] [1, 2] [1, 2] [1, 2]

501.84 501.88 499.75 498.85 497.80 496.55

2438.3 2445.2 2428.8 2428.9 2424.8 2423.2

499.86 499.00 498.10 497.85 497.18 495.83

2432.8 2442.8 2426.3 2429.0 2423.9 2416.9

ae 1.31 1.00 1.64 1.10 0.86 1.23

References: [1] Andreev (1990). [2] Andreev and Zadvorkin (1991). TABLE 4.9. Magnetic and magnetoelastic data for RCo 3 compounds. #m is the magnetic moment per formula unit at 4.2 K; type of magnetocrystalline anisotropy is given for 4.2 K; Tc is the Curie temperature; 7"1" is the temperature of a special spin reorientation (see text); 7'2 is the temperature of the cone-axis transition; ha, he are the linear spontaneous magnetostrictive strains in the basal plane and along the c 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 / , ¢ ~ - a ) / e due to a "/-magnetostriction. Compound Um Aniso- Tc #B/f.u. tropy K YCo3 TbCo 3

DyCo 3 HoCo 3 ErCo 3 TmCo 3

0.59 4.75 5.80 6.26 4.84 3.4

axis canted canted plane axis canted

320 483 440 430 430 401

7'1' K

T2 K

290 160 55-62 225 -

References: [1] Andreev (1990). [2] Andreev and Zadvorkin (1991). [3] Shcherbakova et al. (1986). [4] Shcherbakova and Yermolenko (1985).

30

-~a (5 K) he (5 K) ws (5 K) e (5 K) Ref. 10 -3 10 -3 10 -3 10 -3 0 2.80 2.45 1.91 2.62 0.85

0.5 2.34 3.97 3.14 1.96 0.50

0.5 7.9 8.9 7.0 7.2 2.2

0 5.7 2.0 0 0 -3.1

[1-3] [1, 2, 4] [1, 2, 4] [1-3,5,6] [1, 2, 6, 7] [1, 2, 5]

[5] Shcherbakova and Yermolenko (1982). [6] Bartashevich et al. (1992). [7] Georges et al. (1975).

l o w ~zco value. T h e c o r r e s p o n d i n g ncoco c o e f f i c i e n t is e q u a l to ( 1 . 5 4 - 0 . 6 ) x 10-3/.tB 2. In spite o f large e r r o r ( d u e to rather l o w Ws) it is e v i d e n t that ncoco is s m a l l e r t h a n in Y C 0 5 and Y2Co7, w h i c h is in c o n t r a s t to the o b s e r v e d trend o f i n c r e a s i n g ncoco w i t h i n c r e a s i n g filling o f the 3d b a n d b y e l e c t r o n s t r a n s f e r r e d f r o m R. H o w e v e r , the l o w v a l u e ncoco = 1.5 x 10-3/.tB 2 w a s c o n f i r m e d b y an a n a l y s i s o f the w s ( T ) d e p e n d e n c e s in R C o 3 w i t h R = Ho, E r and T m (see b e l o w ) .

THERMAL EXPANSION ANOMALIES

119

Because of small magnetostrictive strains it is difficult to separate the exchange and anisotropic contributions. But from the fact that Aa ~ 0 in the entire ordered range, both -1 x~,o and "'1 x ~'2 are negligible (see section 4.1.1.1). Since the absolute value of A~,2 is usually only about twice that of )~,,z, the anisotropic magnetostriction in YCo3 is considered to be smaller than 10 -4, like in YC05 and Y2C07.

4.3.1.2. SmCo3 and GdCo3. A strong enhancement of the Co magnetic moment has been found in the RCO3 compounds in which the R ions are magnetic. In the collinear ferrimagnetic structure approximation, which is reasonable in the case of GdCo3, and using #c,,l = 7#B, the data reported for #Co vary from 1.27#B (Bartashevich et al. 1993b) to 1.57#B (Burzo and Seitabla 1981). Measurements on a single crystal led to #Co - 1.33#B (Kebe 1983). The value of about 1.3#B, when taken as the maximum possible value for the CO magnetic moment in the RCo3 compounds, is in good agreement with the ultrahigh-field data on YCo3 (Goto et al. 1992). Ioshie and Nakamura (1990) reported basal-plane anisotropy in SmCo3 at 4.2 K derived from nuclear magnetic resonance measurements. The multiaxial anisotropy of SmCo3 correlates with that found at low temperature in Sm2Co7, explained by magnetoelastic mechanism (see sections 4.2.1.3 and 4.2.1.6). The thermal expansion of these compounds was not studied on single crystals. Anomalies below Te were found by Buschow et al. (1974) in SmCo3 and GdCo3, when polycrystalline samples were studied at 273-973 K. The thermal expansion of both compounds is very similar although the first compound is a ferromagnet and the second is a ferrimagnet with rather different temperature dependences of magnetic moment. The anomalies were attributed to the positive spontaneous magnetostriction. 4.3.1.3. Z b C o 3. Magnetic properties of ferrimagnet TbCo3 are reported by Lemaire (1966b). Shcherbakova and Yermolenko (1985), Bartashevich et al. (1993b) have studied the magnetization on single crystals. The compound has a three-axis type anisotropy, like in rhombohedral Tb2Co7 (Deryagin et al. 1979, see section 4.2.5). As in Tb2Co7, the projection of the easy axis in the basal plane coincides with the [210] axis (the b axis in orthorhombic coordinates). Structural and magnetic properties are listed in tables 4.8 and 4.9. The angle between the easy